Rutgers Cytomechanics 2015 - Craelius

CYTOMECHANICS

William Craelius, Ph.D. 14:125:432

Spring 2015

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TABLE OF CONTENTS Preface Chapter 1: Introduction 1 1.1 Background 1.2 How Cells Are Assembled 1.3 Basic Cellular Components 1.4 Tissues 1.5 Tensegrity 1.6 Forces That Hold The Cell Together 1.7 CSK Mechanoreflexes 1.8 Construction of the CSK 1.9 Applications of Cytomechanics Exercises and Review Questions 11 References

Chapter 2: Materials of the Cell and Matrix 14 2.1 Basic Cellular Constituents 2.2 Plasma Membranes 2.3 CSK Components 2.4 Building the CSK 2.5 Cytogel

Exercises and Review Questions 24 References

Chapter 3: Mechanics of Cell Types 25 3.1 A Generic Cell 3.2 Red Blood Cells 3.3 Platelets 3.4 White Blood Cells 3.5 Fibroblasts 3.6 Endothelial Cells 3.7 Myocytes 3.8 Transformed Cells 3.9 Stem Cells

Exercises and Review Questions 32 References

Chapter 4: Material and Structural Mechanics 33 4.1 Background 4.2 Stress Environments of Cells 4.3 Uniaxial Stress and Strain 4.4 Two-Dimensional Stress 4.5 Types of Mechanical Loading 4.6 Measuring Cellular Material Properties 4.7 Material Stiffness and Strength 4.8 Properties of CSK Components 4.9 Flexural Moduli 4.10 Measuring Local Stiffness 4.11 Buckling

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Exercises and Review Questions 48 References Chapter 5: Cytomechanical Tools 50 5.1 Background 5.2 Measuring and Manipulating Cellular Forces 5.3 Measuring Cell-Generated Traction Forces 5.4 Manipulating Cell-ECM Forces 5.5 Atomic Force Microscopy (AFM) 5.6 Magnetic Tweezers 5.7 Micropipette Aspiration 5.8 Cell Moduli from Pipet Aspiration 5.9 Nuclear Probing 5.10 Whole Cell Inflation 5.11 Optical Tweezers 5.12 Hydrostatic Loading 5.13 Shear Flow 5.14 Cell Stretching 5.15 Microelectromechanical Systems (MEMS) 5.16 Particle Tracking Exercises and Review Questions 68 References Chapter 6: Thermodynamics of the Cytoskeleton 70 6.1 The Boltzmann Distribution: Statistical Mechanics 6.2 Diffusion 6.3 Bioelectricity 6.4 Energy Storage 6.5 State Transitions 6.6 Polymerization Exercises and Review Questions 80 References Chapter 7: Kinetic Behavior 82 7.1 Background 7.2 Cell Modeling 7.3 Modeling the Cytoskeleton 7.4 Viscoelastic (VE) Material Exercises and Review Questions 87 References Chapter 8: Micromotors 89 8.1 Introduction 8.2 Muscular Microstructure 8.3 The Pathway to Contraction 8.4 Generation and Regulation of Force 8.5 Skeletal Muscle Energetics Appendix: Review of Biomechanical Terminology 101 Exercises and Review Questions 102 References

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Chapter 9: Cell Propulsion 103 9.1 Introduction 9.2 Cells Swimming Through Fluids 9.3 Cell Substrate Interactions 9.4 Propulsive Motors 9.5 Comparative Motor Analysis Exercises and Review Questions 110 Appendix 111

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PREFACE Cytomechanics studies how cells are influenced by the mechanical stresses and strains that they experience continually throughout their lives. Cytomechanics studies these pico-newton and nanometer quantities, to understand and possibly manipulate the growth, structure and function of cells. This course emphasizes the processes that drive tissue growth, degeneration, and regeneration, with sub-topics including cellular signaling and metabolism, gene mechanics and expression, and the biomechanical properties of cells and their components. Projects in modeling cellular structure and behavior are done with Matlab & Simulink. An important feature of cell mechanics is mechanotransduction: the process whereby cells transform mechanical energy into other forms, including chemical, electrical, and inertial. It is based upon the ability of cells to sense the mechanical forces always present in their environment and react to them on time scales ranging from milliseconds to years in order to achieve biological stability at some level. Mechanotransduction is thus central to cellular structure and function, tissue maintenance, and ultimately, organismal fate. Understanding it requires analysis of extremely small forces and deformations at the nanometer level; this challenge can be met by merging engineering principles with techniques of modern molecular biology such as immunocytochemistry, optical imaging, and microfabrication. Part I (the first 7 chapters) presents the basic structural components, mechanics and energetics of cells, cellular electrophysiology and techniques for cellular visualization and mechanical characterization. Part II (the last five chapters) covers the role of mechanotransduction in complex cell behaviors of signaling, moving, adhesivity, and growth. Quantitative examples and exercises based on traditional biomechanics applied to cells are provided. Software, such as Matlab and Excel will enhance understanding, and specific examples using them are given. The appendices give formulae and data related to cellular mechanical properties and the cytoskeleton, along with a brief tutorial on Simulink. This course packet is a compilation of many textual and research papers, and includes excerpts from same, cited in the text. Referenced or excerpted articles are included online in the supplement. The book attempts to introduce the huge and growing field of cell mechanics, and assumes the reader has background in basic cell biology, biochemistry, and Physics. While reading each chapter, you should find the Appendices useful for definitions and formulae. Companion books in those areas are recommended as reference. For more in-depth texts please read, The Mechanics of the Cell by David Boar, and Biomechanics by Y.C. Fung. This book has liberally adapted ideas and material from those two texts, as well as reviews. Please note that this book will inevitably contain mistakes, so corrections and comments will be appreciated. Permissions for material from research articles are pending.

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Chapter 1: Introduction

CHAPTER 1: INTRODUCTION (With acknowledgements to Ingber 2008; Eyckmans, Boudou et al. 2011)

1.1 Background Biological cells are structures that self-assemble from basic components, adapt their shapes, sizes and strengths to their ongoing needs, and travel to and settle in appropriate locations using their own renewable energy. Many of these abilities result from a particular sensory-reflex system possessed by each cell that responds to mechanical forces within its environment. In fact, in order to properly function and grow, each cell depends upon continual stress and strain within preferred magnitude ranges. When force or strain magnitudes are outside the proper range, either too large or too small, cell function and growth is adversely affected. This inter-relationship between ambient force and cell function underlies a unique design whose principles would be highly useful to engineers. Cytomechanics seeks to uncover those principles using the tools of molecular biology, imaging, biomechanics, and computer modeling. By analyzing forces and deformations on the pico-newton and nanometer levels, cytomechanics seeks to explain and possibly manipulate the growth, structure and function of cells. A prominent example of the application of cytomechanics is accelerating the growth of cells and tissues by exposing them to forces from fluid flow in bioreactors. Specifically, growth of nerves, skin, muscles, bone, and probably all biological tissues can be stimulated by proper application of forces. Other technology is exploiting the highly efficient molecular motors found in bacteria to make nano-scale motors. Although the biological effects of forces are perhaps most evident in the context of physical activity—breathing, heart pumping, blood flow, and physical exercise—such forces also regulate morphogenesis, cell migration, and even cell adhesion to extracellular matrix. Such forces can regulate a wide variety of biological processes, from cell proliferation and differentiation to tissue mass homeostasis and complex inflammatory cascades. The idea that forces can regulate tissue remodeling and development was articulated more than a century ago. In 1892, the surgeon and anatomist Julius Wolff postulated that bone tissue adapts its structure to the mechanical environment based on the observation that trabeculae matched the principal stress lines in bones caused by daily physical loading (Wolff, 1892). Although the alignment of trabeculae could have arisen strictly during prenatal development, he reported this remodeling occurred even after healing of misaligned fractures. In the same era, mechanical forces were proposed to shape tissues and organs during embryonic development (Roux, 1895; Thompson, 1917), but the tools were not available to directly test such ideas experimentally. Nearly a century passed before these concepts began to captivate the scientific community once again. 1.2 How Cells Are Assembled The architecture of biological cells is highly complex, and its elegance can be appreciated in comparison with that of buildings. Building design must meet certain minimum standards: (1) a foundation anchoring it to the correct location; (2) sufficient strength to stand against all expected forces; (3) comfortable internal environment; (4) access portals for incoming and outgoing traffic; and (5) food and waste processing. Using well-

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established formulae from mechanical and civil engineering, the architect can select the type and sizes of structural components, their arrangement, connections, and all the functional components needed to satisfy the building standards. Before construction begins, she knows every line, arc, angle, and load that the building will have. Architecture of buildings is thus laid out as a clearly understood blueprint. Architecture of biological cells, on the other hand, is laid out by a blueprint that is not so clearly understood, in the form of codes on DNA molecules; structurally, cells are squishy, wiggly, and willful. Nevertheless, cellular architecture not only must satisfy all the same standards as buildings, it must be self-renewable. Cells solve the mechanical and civil engineering problems they face in a myriad of elegant ways, using any and every way to live. In fact, cells have much more intelligence than buildings: they can modify their structure to meet changing demands and conditions. Cytomechanics seeks to learn and apply the rules of cellular architecture. While it may never be possible for us to build cells from scratch, we can expect to learn tricks from them that can help us solve many technological problems. 1.3 Basic Cellular Components Basic Plan The basic plan of all animal cells is the same: they have a lipid membrane, skeleton, and internal structures. Unlike most components of buildings, however, which are divided into purely structural or purely functional categories, all cellular components can serve both categories in elegant ways. The membrane is very weak structurally, but nevertheless is a barrier wrap and portal to the outside, as well as a smart skin with sensory and reflex capabilities. Mechanical strength is provided by the cell skeleton, i.e. ‘cytoskeleton,’ (CSK), that supports the membrane and maintains cellular shape. The CSK not only is the backbone and limbs of the cell, it is also a communication network. Within the gel-like cytosol, internal structures include the nucleus, mitochondria, and other organelles Together the 3 components, membrane, CSK, and cytosol provide all the structure and function of the cell. The Lipid Membrane All cells are enveloped by a lipid bilayer that is an aggregation of phospholipids, in a spheroid, separating the cell interior from the exterior environment. Many components other than lipid exist within the bilayer, and perform various functions, as will be seen in Chapter 2. Formation of the bilayer in solution is a spontaneous event, driven by thermal energy, as depicted below in Figure 1.1. Note that the series of time-lapse photographs show the lipid droplet oscillating in shape, due to thermal fluctuations:

Figure 1.1. Time-lapse views of lipid vesicle undergoing thermal fluctuations. See also: http://ftp.aip.org/epaps/journ_chem_phys/E-JCPSA6-119-711337/movie1.gif

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http://ftp.aip.org/epaps/journ_chem_phys/E-JCPSA6-119-711337/movie1.gif




Actin Intermediate Microtubules ……………… .filaments

The Cytoskeleton (CSK) The CSK is a network connecting the outside of the cell directly to points within, including the nucleus. Two views of the CSK shown below (Figure 1.2) illustrate its architectural complexity. On the left is a scanning electron microscope view from outside of a cell. The dense net is composed entirely of the protein, actin. Note the random crisscrossing of the fibers, and the nodes where 2 or more branches connect. Actin filaments, about 10 nM in diameter, run fairly straight between nodes, or connections, and form a thick-tangled 3-dimensional mat at this scale. At a more distant (less magnified) view, the network resembles a gossamer spider web as shown in the cartoon at right, showing a cross section of a segment of the cell border. Note that there is a lipid bilayer, through which the polymer network extends. Proteinaceous channels and integrin molecules traverse the lipid membrane. The extracellular matrix intimately surrounds cells, and is attached at specific sites.

Three types of filaments make up the CSK, schematized below (Figure 1.3). Actin is the thinnest of the 3 filaments, with diameter ≈8 nm. Actin filaments aggregate to form a dense peripheral shell that is the primary component of the CSK of most cells. Intermediate filaments are larger, and connect through the membrane at discrete points. Microtubules are very straight hollow tubes with outer diameter ≈25nM, and can traverse the cell spanning between the cell membrane and the nucleus. Details on the properties of these filaments can be found in Table 2.2.

Figure 1.2. The Cytoskeletal Network

Figure 1.3: Major Cytoskeletal Components

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Fundamental Principles of CSK There are three fundamental principles of the cytoskeleton: (1) the cytoskeleton determines cell function to a large extent, primarily through its architecture and its interaction with the external environment. (2) It self-assembles from various components, without a genetic blueprint. (3) It regulates cell function at long ranges of space and time. The meaning of these principles will be elaborated upon in later chapters. 1.4 Tissues When cells aggregate to form tissues, things become more complicated as shown in the schematic view of 3 skin cells below. Many different types of connections, each with their own biochemical traffic patterns, are made among the various cells and extracellular components.

The schematic above shows the epithelial cells connected together by weld-like connections at the tight junctions, and connecting with the basement membrane by adhesions, mediate by integrin, and other matrix molecules. Stromal cells can interact with epithelial cells through mechanical forces applied to the basement membrane that can regulate nuclear transcription via the Integrin-CSK network. 1.5 Tensegrity (Ingber 2010) The tensegrity model states that cells, tissues, and other biological structures at smaller and larger size scales in the hierarchy of life gain their shape stability and their ability to exhibit integrated mechanical behavior through use of the structural principles of tensegrity architecture. The term, “tensegrity” (contraction of “tensional integrity”) was first created by the architect R. Buckminster Fuller, who first explored use of this form of

Figure 1.4 Epithelial Tissues. BM = basement membrane.

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structural stabilization as early as 1927 in his plan for the Wichita Dymaxion house, which minimized weight by separating compression members from tension members. To create this cylindrical building, Fuller proposed to set a central mast in the earth as a vertical compression strut and to suspend from it multiple circular floors (horizontal wheels) using tension cables. Tensile guy wires that linked the mast to surrounding anchors in the ground provided the balancing tension necessary to stabilize the entire structure. “Fuller called this special discontinuous-compression, continuous-tension system, the Tensegrity” to emphasize how it differs from conventional architectural systems (e.g., brick-on-brick type of construction), which depend on continuous compression for their shape stability. Fuller’s more formal definition in his treatise, Synergetics, is “Tensegrity describes a structural-relationship principle in which structural shape is guaranteed by the finitely closed, comprehensively continuous, tensional behaviors of the system and not by the discontinuous and exclusively local compressional member behaviors”. Note that there is no mention of rigid struts, elastic strings, tensile filaments, internal vs. external members, or specific molecular constituents in this definition. In fact, Fuller describes a balloon with non-compressible gas molecules pushing out against a tensed rubber membrane as analogous to one of his geodesic domes when viewed at the microstructural level (i.e., the balloon is a porous, tensed molecular network on the microscale) and explains that both structures are classic examples of shape stability through tensegrity. Fuller also described hierarchical tensegrity structures in which individual struts or tensile elements are themselves tensegrity structures on a smaller scale; key to this concept is that smaller tensegrity units require external anchors to other tensegrity units to maintain higher order stability. In fact, he argued that nature utilizes this universal system of tensile structuring at all size scales and that it provides a way to mechanically integrate part and whole. In 1948, Fuller’s student, Kenneth Snelson, constructed the first “stick-and-string” tensegrity sculpture, which thrilled Fuller because it visibly communicated the essence of this novel form of shape stability to those who could not “see” it in more complex structures. Snelson’s sculptures contain isolated compression members that are suspended in midair by interconnections with a continuous tensile network. Some of these structures require anchorage to the ground to remain stable (e.g., large cantilevered structures); however, most are entirely self-stabilizing. Similar stick-and string tensegrity models have been used to visualize tensegrity in cells and other biological structures for those who cannot easily visualize them. The appearance of geodesic patterns in biological structures, including viruses, clathrin-coated vesicles, and actin geodomes in the cytoskeleton of mammalian cells, provides additional visual evidence of nature’s use of this form of architecture. From the above discussion, it is apparent that understanding cell structure starts with a look at its basic structural plan. At one level, it appears that the basic architecture of the cell is the same as the geodesic dome, designed by Buckminster Fuller. Geodesic is a highly efficient building, whose structural elements traverse the shortest distance required to hold it up. Many non-cellular structures, including viruses, enzymes, organelles and even small organisms, all exhibit geodesic forms. Stick models of the structure are shown at left in Figure 1.5, and the CSK of a living cell is shown at right.

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1.6 Forces that hold the cell together Tensegrity models can be made from sticks and rubber bands. Their integrity relies on the tension applied to the sticks by the elastic elements; hence the structure has tensegrity. In engineering terms, the sticks represent struts, since they sustain compression and the rubber bands represent ties (or ropes) since they hold tension. While the geodesic dome has a characteristic spherical shape, many other shapes are held together by tensegrity, not the least of which is standing bipeds (see below). Your bones are struts compressed by gravity, while your skeletal muscles act as ropes, applying tension to maintain posture. Another structure is the loaded bow, shown below:

Cellular tensegrity describes a structural plan whereby a network of contractile microfilaments pulls the cell membrane and all its internal constituents centrifugally toward the nucleus at the core. Opposing this centrifugal tensile pull are two types of compressive elements, one of which is outside and the other inside the cell. The external component outside the cell is the extracellular matrix (ECM). Inside the cell there are struts consisting of microtubules and/or bundles of cross-linked microfilaments that resist compression. The third component of the CSK, the intermediate filaments, link microtubules and contractile micro-filaments to each other as well as to the surface membrane and the cell's nucleus. In addition, the intermediate filaments act as guy wires or rods, stiffening the central nucleus and securing it in place. Although the CSK is surrounded by a lipid membrane and lies within a viscous fluid resembling a gel, it is this

Bow and Arrow

Tension

Compression

Muscle Tension mg

Figure 1.5. Living geodesic forms.

Figure 1.6. Biped and bow tensegrity structures.

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hard-wired network of molecular struts and rods that stabilize cell shape. Thus the cell uses tension rods and compression struts to hold itself together. In simplest terms, tensegrity structures maintain shape stability within a tensed network of structural members by incorporating other support elements that resist compression. The stiffness of the stick-and-string tensegrity structures, and hence their ability to resist shape distortion, depends on the level of preexisting tension or “pre-stress” in the structure before application of an external load. The distinguishing microstructural feature accounting for this behavior is that, when placed under load, the discrete structural elements move, changing orientation and spacing relative to one another, until a new equilibrium configuration is attained. For this reason, a local stress can result in global structural rearrangements and “action at a distance.” To visualize tensegrity at work, think of the human body: it stabilizes its shape by interconnecting multiple compression-resistant bones with a continuous series of tensile muscles, tendons, and ligaments, and its stiffness can vary depending on the tone (pre-stress) in its muscles. If I want to fully extend my hand upward to touch the ceiling, I have to tense muscles down to my toes, thus producing global structural rearrangements throughout my body and, eventually, upward extension of my fingers. However, the body is also multimodular and hierarchical: if I accidentally sever my Achilles tendon, I lose form control in my ankle module, but I still maintain structural stability in the rest of my body. Furthermore, every time I breath in, causing the muscles of my neck and chest to pull out on my lattice of ribs, my lung expands, alveoli open, taught bands of elastin in the extracellular matrix (ECM) relax, buckled bundles of cross-linked (stiffened) collagen filaments straighten, basement membranes tighten, and the adherent cells and cytoskeletal filaments feel the pull; however, nothing breaks and the deformation is reversible. Tensegrity provides a structural basis to explain all these phenomena. In the cellular tensegrity model, the stabilizing pre-stress is generated actively by the cell’s contractile apparatus and passively by distension through extracellular adhesions, by osmotic forces acting on the cell’s surface membrane, and, on a smaller scale, by forces exerted by molecular filaments extending through chemical polymerization. The model assumes that the pre-stress is carried by tensile elements in the cytoskeleton, primarily actin microfilaments and intermediate filaments, and that the cell is both a hierarchical and multimodular structure. This pre-stress is balanced by interconnected structural elements that resist being compressed at different size scales, including the cell’s external adhesions to the relatively inflexible ECM and internal cytoskeletal filaments, specifically microtubules that stretch across large regions of the cytoplasm and cross-linked bundles of cytoskeletal filaments that stabilize specialized microdomains of the cell surface (e.g., actin microfilaments in filopodia; microtubules in cilia). In this model, the internal cytoskeleton is surrounded by an elastic sub-membranous cytoskeleton (e.g., actin-ankyrin-spectrin network) and its associated lipid bilayer, which may or may not mechanically couple to the internal, tensed microfilament-microtubule-intermediate filament lattice depending on the type of adhesion complex that forms. The entire cytoskeleton is permeated by the viscous cytosol. Most importantly, this micromechanical model leads to specific predictions relating to the mechanical role of distinct cellular and molecular elements in cell shape control. Contrasting models of cell structure depicts the cell as an elastic cortex that surrounds a viscous cytoplasm with an elastic nucleus in its center. In engineering terms, this is a “continuum” model, and, by definition, it assumes that the load-bearing elements are

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infinitesimally small relative to the size of the cell. It is essentially the balloon model considered by Fuller, but in this case all microstructure is ignored. Because they ignore microstructural features, continuum models cannot provide specific predictions that relate to the functional contribution of distinct cytoskeletal filaments to cell mechanics. Furthermore, although these models can provide empirical fits to measured mechanical properties in cells under specific experimental conditions, they cannot predict how these properties alter under new challenges to the cell. How closely does the geodesic model fit the CSK? To test its validity, suppose you hit one of the struts of the geodesic dome. This would cause the mechanical energy to quickly travel throughout all the struts, reverberating throughout the structure at the speed of sound. Does the cell have a corresponding behavior? The answer is yes, since when the CSK is perturbed at a single site, either by a specific binding event, or experimental poking, the entire structure ‘feels’ it, as the energy is dissipated throughout at the speed of sound. Next compare dome behavior with another cellular characteristic: shape. Geodesic domes would quickly collapse if most of its struts were removed; conversely the dome shape cannot be radically altered by pulling on a few of its struts. Neither of these features of domes is shared by cells. For example, the normally spherical shape of cells in culture will persist even after all microtubules are removed either through drugs or gene knockout. Conversely, cells can flatten when stress is applied to the cell struts by its ECM, as depicted in Figure 1.7 below. If living cells can remain spherical without most of its struts, and then change from spherical to flat when stressed, then their behavior does not closely resemble that of a geodesic dome, and tensegrity must be a more adaptable concept. In other words, the CSK must have built-in redundancy that is provided by the microfilament network, and its structure is highly modifiable. In fact, redundancy of the CSK is to be expected, since its construction can be characterized as fractal, i.e. structural forms are self-similar at different scales; stated another way, the network weight and its volume are independent of each other. Thus the simple geodesic dome model falls short of predicting some cell capabilities. The ability of microfilaments to adapt to stress by either ‘stress-stiffening’ or stress softening will be further discussed in Chapter 10. 1.7 CSK Mechanoreflexes The CSK apparently can sense the forces applied to it, and adjust its size, strength, and orientation in order to resist the forces in an efficient manner. When stresses are highly polarized, such as along the axis of muscles or neurons, filaments of the CSK align themselves according to principal stress directions, as shown in Figure 1.8 below. Besides orienting along stress lines, filaments size themselves according to strength requirements: a conservative architectural practice.

Figure 1.7. Response of a single cell to stress.

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1.8 Construction of the CSK While there are many aspects of cellular architecture remaining to be discussed, lets consider one more question: how is the CSK constructed? Surely there are blueprints, i.e. genes, for each of the protein components, and an overall blueprint laying out their 3-dimensional arrangement, but how do the ropes and rods connect themselves properly, in the right amount and orientation to form the fantastic structures seen above? This is a fundamental problem of biological development. While there is no simple answer, there are two somewhat competing models or theories that provide at least scenarios of how it could happen. One assembly sequence for the CSK could be similar to that of the geodesic dome, with regular sub-structures such as triangles, being welded one by one into a 3 dimensional network. Tensegrity would hold the structure together, and would allow it to adapt and change. The attractive concept that the CSK represents essentially a “fullerene’ structure was originally proposed by Ingber and subsequently supported by many studies. While tensegrity and the geodesic dome definitely apply to cells, it is difficult to imagine how the complex CSK structures seen above assemble and maintain their shape. Geodesic models significantly change shape or collapse when a single strut or rope is cut, a behavior that cells do not share. One way cell stability could occur is by percolation [Forgacs], a process of network formation whereby individual lines grow somewhat randomly from point to point until sufficient connectivity establishes a network. A simple example of percolation is the growth of telephone lines linking the East and West coasts of the U.S. There is no direct line connecting NY with Los Angeles; however, in the development of lines between intervening cities, eventually a continuous pathway was formed, and as lines continued to link cities, more and more pathways were formed. Thus a large number of redundant pathways link the structure end-to-end. Figure 1.9

Figure 1.8. Alignments of CSK filaments.

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LA NY

below shows a hypothetical telephone network in the U.S. many years ago. Note that lines between cities in the Northern sector do link NY with LA. As more cities are connected, it can be seen that more pathways will connect the 2 coastal cities. Since removal of a single line from the network can disrupt the continuously connected pathway from NY to LA, this network has a just critical number of elements for spanning the distance; it is therefore at its, ‘percolation threshold.’

Similarly, Figure 1.10 shows a snapshot of a portion of the Internet web: Do you see any resemblance to the CSK? The percolation model is useful in describing network signaling, as well as elasticity, as will be seen in Chapters 7 and 10. Beginning with the model in a future assignment, you will study the formation of “spanning” networks. Thus there are at least 2 ways to think of the CSK: as a geodesic dome with regular networks of triangular elements or as random networks of lines tied together by percolation. The theories are not mutually exclusive, and both are useful, however neither tells the complete story. 1.9 Applications of Cytomechanics This topic brings us full circle to the first: why study cytomechanics? The simplest answer is that you may find in cells valuable solutions to engineering problems. One example of a structural solution for a strong, lightweight, and flexible material is shown in Figure 1.11 below. Design for this material is taken directly from CSK structure, and is in fact being developed by a company formed by Donald Ingber. Reverse engineering

Figure 1.9. Hypothetical telephone network in the U.S.

Figure 1.10. A portion of the Internet web.

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and exploitation of biological structures can be highly profitable, since cells require no patent royalties. The material below can absorb impact energy well, because shocks are dissipated thoroughly throughout the structure. CSK structure is finding applications in other areas. Bioactive geodesic scaffolds for filtration and catalysis are highly efficient due to their high ratio of surface area-to-volume and low mechanical resistance to flow. One spin-off from this technology is an improved face mask with better protection against pathogens. These filters have adapted many features of cell structure, the CSK, and its bioactive nature. The scaffolds thus have large pores (much larger than the pathogens) for easy air flow, but have a huge surface area and tortuous path to trap particles. Moreover the scaffolds are coated with a synthetic hydrogel "protoplasm" that soaks up pathogens, which are highly hydroscopic (water seeking). The pathogens are killed using solid-state enzymes and other bactericidal agents that are incorporated within the gel. Such ‘CSK- inspired’ materials can have unusual 3-D characteristics, such as the structures shown in Figure 1.12 that have zero mean curvature. The CSK behavior of stress-stiffening has inspired flexible fabrics that get stiffer when stretched, while maintaining their porosity for critical heat exchange. Development of an "artificial gill" for oxygen production is underway, using the high surface area and efficient solid-state catalysis offered by CSK design.

Figure 1.11. Material that mimics CSK structural design.

Figure 1.12. CSK inspired materials.

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Biomimicry of cytomechanical design is advancing many other technologies, including tissue engineering, wound healing, microtubular nanostructures, bioprocess optimization, cryogenics mechanoelectrical signaling, tumor therapies, and genetic regulation. Microtubules serve as perfectly straight templates for fabrication of nanowires. With the preceding cursory view of structural principals of the cell (whose mechanical details are discussed in later chapters, we are now ready to take a closer look. In later chapters we will ask more probing questions: What other roles does the CSK play? How does it interact with the nucleus? How is it formed and maintained? _______________________________________________________

Chapter 1 Exercises and Review Questions

1. Examine Figure 1.1 (and the movie) and explain briefly the events it depicts. 2. What are the general names for structural components that resist compression &

tension? Which cellular components correspond? Compare structural properties of microtubules, spectrin, and actin filaments, and state a different role for each of them.

3. List three examples that could represent tensegrity structures. Sketch them and draw their free body diagrams.

4. State three differences between cellular and building architecture. 5. Demonstrate the percolation threshold of network formation, using

“connect_the_dots.m” on Sakai. What conditions affect formation of a “spanning” network?

6. Examine the Simulink model #1 discussed in class, and make and test your own. Be sure to document the model and label its components.

References Eyckmans, J., T. Boudou, et al. (2011). "A Hitchhiker's Guide to Mechanobiology." Developmental Cell 21(1): 35-47. More than a century ago, it was proposed that mechanical forces could drive

tissue formation. However, only recently with the advent of enabling biophysical and molecular technologies are we beginning to understand how individual cells transduce mechanical force into biochemical signals. In turn, this knowledge of mechanotransduction at the cellular level is beginning to clarify the role of mechanics in patterning processes during embryonic development. In this perspective, we will discuss current mechanotransduction paradigms, along with the technologies that have shaped the field of mechanobiology.

Ingber, D. E. (2008). "Tensegrity-based mechanosensing from macro to micro." Progress in Biophysics & Molecular Biology 97(2-3): 163-179. This article is a Summary of a lecture on cellular rnechanotransduction that was

presented at a symposium on "Cardiac Mechano-Electric Feedback and Arrhythmias" that convened at Oxford, England in April 2007. Although critical mechanosensitive molecules and cellular components, such as integrins, stretch-activated ion channels, and cytoskeletal filaments, have been shown to contribute to the response by which cells convert mechanical signals into a biochemical

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response, little is known about how they function in the structural context of living cells, tissues and organs to produce orchestrated changes in cell behavior in response to stress. Here, studies are reviewed that suggest our bodies use structural hierarchies (systems within systems) composed of interconnected extracellular matrix and cytoskeletal networks that span from the macroscale to the nanoscale to focus stresses on specific mechanotransducer molecules. A key feature of these networks is that they are in a state of isometric tension (i.e., experience a tensile prestress), which ensures that various molecular-scale mechanochemical transduction mechanisms proceed simultaneously and produce a concerted response. These features of living architecture are the same principles that govern tensegrity (tensional integrity) architecture, and mathematical models based on tensegrity are beginning to provide new and useful descriptions of living materials, including mammalian cells. This article reviews how the use of tensegrity at multiple size scales in our bodies guides mechanical force transfer from the macro to the micro, as well as how it facilitates conversion of mechanical signals into changes in ion flux, molecular binding kinetics, signal transduction, gene transcription, cell fate switching and developmental patterning. (c) 2008 Elsevier Ltd. All rights reserved.

Ingber, D. E. (2010). "From Cellular Mechanotransduction to Biologically Inspired Engineering." Annals of Biomedical Engineering 38(3): 1148-1161. This article is based on a lecture I presented as the recipient of the 2009 Pritzker

Distinguished Lecturer Award at the Biomedical Engineering Society annual meeting in October 2009. Here, I review more than thirty years of research from my laboratory, beginning with studies designed to test the theory that cells use tensegrity (tensional integrity) architecture to stabilize their shape and sense mechanical signals, which I believed to be critical for control of cell function and tissue development. Although I was trained as a cell biologist, I found that the tools I had at my disposal were insufficient to experimentally test these theories, and thus I ventured into engineering to find critical solutions. This path has been extremely fruitful as it has led to confirmation of the critical role that physical forces play in developmental control, as well as how cells sense and respond to mechanical signals at the molecular level through a process known as cellular mechanotransduction. Many of the predictions of the cellular tensegrity model relating to cell mechanical behaviors have been shown to be valid, and this vision of cell structure led to discovery of the central role that transmembrane adhesion receptors, such as integrins, and the cytoskeleton play in mechanosensing and mechanochemical conversion. In addition, these fundamental studies have led to significant unexpected technology fallout, including development of micromagnetic actuators for non-invasive control of cellular signaling, microfluidic systems as therapeutic extracorporeal devices for sepsis therapy, and new DNA-based nanobiotechnology approaches that permit construction of artificial tensegrities that mimic properties of living materials for applications in tissue engineering and regenerative medicine.

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Chapter 2: Materials of the Cell and Matrix

CHAPTER 2: MATERIALS OF THE CELL AND MATRIX 2.1 Basic Cellular Constituents All cellular structures are made with same building blocks: atoms of (in descending concentrations) carbon, oxygen, hydrogen, nitrogen and phosphorus, sulphur, and several other atoms in minute quantities. These atomic building blocks are assembled into larger blocks, i.e. amino acids & sugars that are then strung together in even larger blocks, i.e. proteins and carbohydrates, schematically shown in Figure 2.1 below:

Proteins Carbohydrates

Biomembranes

Filaments Amino Acids

Sugars Phospholipids Peptides

Carbon

Oxygen Hydrogen Nitrogen Phosphorous, Sulphur, etc.

As you can see, our study of cell materials is greatly simplified by the fact that all structural components are polymers strung together with 3 types of building blocks. The structural components are mainly proteins, with some carbohydrates. The major categories of cellular constituents are listed in Table 2.1 below:

Table 2.1. Chemical Components of Cells Compound

Fraction in Cell (%)

Relative Size of molecule

Polarity of molecule

Water 70-80 Small Polarized Protein (Polypeptide)

10-20 Large Regionally polarized

Lipid (Fat) 2-20 Medium Non-Polarized Carbohydrate (Sugars)

1-2 Medium to large Regionally polarized

Salts (Electrolytes)

1 Small Polarized

Note from the table that electrical polarity varies among the molecules. The importance of these differences to cell mechanics will be appreciated in our study of Energetics in Chapter 6. Note that lipids are non-polarized, however some amino acids are also non-polarized. Carbohydrates

Size and Complexity

Figure 2.1. Assembly of basic structural components of the cell

Atoms

Monomeric Building Blocks

Polymeric Structural Components

19


can exist as simple sugars, of medium size, or as larger complex chains, such as the backbone of DNA.

Besides polarity and size, other important properties of molecules are their solubility, stability, and shape. As we shall see, proteins are the molecules whose shape, an important determinant of their function, is the most varied. In addition to the structural components, a second category of components is the electrolytes, and monomeric molecules dissolved within the cell soup; these include the major salts, and the building blocks: amino acids, sugars, and cell fuels such as ATP. The major electrolytes consist of (in order of descending concentration in the cell): Cl-, K+, Na++, Ca++, and Mg++. These salts are all surrounded by several water molecules, attached by hydrogen bonds. 2.2 Plasma Membranes To understand the material make-up of cells and matrix, it is entertaining to imagine how they first sprung up in evolution, even though its details will never be known. The following somewhat fanciful history is therefore to be pondered in that light. In the beginning there was primordial soup: a rich sea filled with organic compounds. Over the eons, as the chemicals began to aggregate in the slime, a critical mass of them was captured within a volume surrounded by a barrier: a lipid membrane. Since oil and water do not mix, a lipid bubble is the logical vessel for the aqueous contents of the cell to reside. Thus a thin hydrophobic shell separated the hydrophilic from their surroundings. The snapshots below show a reverse-scan of 10 s of simultaneous data from consisting of both capacitance value and image frame. This ‘X’ defines the trigger moment of the bilayer formation.

Figure 2.1 Capacitance ~solid lines and images ~1–5! of a phosphatidylcholine membrane in development. The white lines in images 1 and 4 correspond to 50 mm. The numbered closed circles represent the capacitance values measured simultaneously with the numbered images. In images 3 and 4, the words ‘‘1st’’ and ‘‘2nd’’ with arrows indicate the first and second regions, respectively. The dashed lines represent the membrane capacitance expected in case of the first thinning continuation without the second thinning. The ‘‘3’’ mark represents the trigger moment of the bilayer formation.

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With time lipid bubbles that were cellular precursors, a prokaryote would have had to develop structures to survive, since lipid bubbles are easily squashed. Hence the CSK was needed, and the rest is added complexity. Now, lets take a look under the hood (Figure 2.2). The first thing we notice is that the biomembrane is a lipid bilayer, arranged as an amphiphile. This means that the molecular structure has both hydrophilic and lipophilic parts, as shown in the schematic of a cross-section below.

The phosphate heads are negatively charged, so the outside layers of the phospho-lipid sandwich quite happily dissolve in water. The core, composed of long hydrocarbon tails, i.e. fatty acids, behaves oppositely, repelling water and all hydrophilic compounds. Thus the bilayer is elegantly arranged to exist with both faces in water, allowing the middle to act as a barrier to movement of water-based materials. Since the membrane separates the cell plasma from the extra-cellular milieu, it is called the plasma membrane (PM). An anecdote about the biomembrane is that the first estimate of its thickness is attributed to Benjamin Franklin. The story goes that he poured a volume of cooking oil into a small pond until it was entirely covered with oil. He then divided the volume of oil poured by the area of the pond, finding the thickness was about 30 A°. The close correspondence of this number with present day measurements of lipid monolayers illustrates the power of simple experiments. It also demonstrates the property of phospholipids to self-assemble into a structure, i.e. a membrane. Due to its lipid structure, the cell membrane is non-polar and hydrophobic. Thus the membrane can sequester charges within the cell, since they cannot penetrate the membrane. It is this feature that is responsible for the bioelectricity of cells, which will be discussed in Chapter 6.

H2O

Hydrophobic core

Polar Phosphate Heads

70 A°

H2O

H2OFigure 2.3. Lipid Bilayer

Figure 2.2. Peeling away the cell membrane

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While the PM can be considered part of the CSK, it is relatively weak. It does have measurable stiffness, however, in shear, compression and bending, as will be seen in Chapter 4. Being essentially a Newtonian fluid, the bilayer by itself has no tensile strength. To illustrate, look at Figure 2.4 below, showing 2 vesicles whose walls are pure lipid bilayers. Starting from the bottom picture, you see the vesicles just touching. The next 2 pictures on top were made as the vesicle at right is pushed out by pressure from the pipette holding it. Note that the vesicles pass into each other. This ghost-like behaviour is due to their liquid nature. 2.3 CSK Components The structural basis of force transmission in cells is the Cytoskeleton (CSK). In the cytoplasm, the CSK is a fundamental structure for mediating force transmission (Wang et al., 1993). The CSK is a highly dynamic cellular scaffolding structure composed of filamentous actin (6 nm in diameter), intermediate filaments (10 nm), and microtubules (23 nm). These three cytoskeletal elements are not single proteins, but consist of many monomers able to span large distances within the cell. Tubulins polymerize to form hollow cylinders known as microtubules and provide a structure for motor proteins such as kinesins and dyneins to travel between different cell compartments. Vimentin, keratin, and lamin monomers form intermediate filaments that connect the nucleus with the endoplasmic reticulum, mitochondria, and Golgi apparatus, providing structural integrity to the cell. Actin monomers assemble into filamentous actin (F-actin) and together with myosin filaments, form the cytoskeletal contractile apparatus. The actomyosin CSK connects multiple parts of the cell membrane as well as the cell membrane to the nucleus (Sims et al., 1992). At the cell membrane, these filaments anchor into clusters of proteins that include focal adhesions (FAs) which link the CSK through transmembrane integrin receptors with the ECM. In the extracellular space, the ECM materializes as a mesh of cross-linked proteins and carbohydrates, and depending on the tissue, can include different constituents including

3. Vesicles pass through each other 2. Pipette pushes vesicle out 1. Vesicles make contact

Figure 2.4. Vesicle ghosts

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collagen, laminin, elastin, and fibronectin fibers interlocked with hyaluronic acid and proteoglycans. From a mechanical standpoint, applying force to this cell-ECM unit leads to structural deformations and rearrangements of the ECM, force transmission through the FA, and (given the highly interconnected nature of the CSK) deformation of nearly every aspect of intracellular structure, including the position of mitochondria, endoplasmic reticulum, and the nucleus (see Figure 3.1). However, ‘‘outside-in’’ force transmission is only half the story, as cells also generate force. Polymerization and depolymerization of microtubules drive pushing and pulling forces, respectively, to control the position of mitotic spindles, chromosomes, and nuclei (Dogterom et al., 2005). The head domain of myosin II pulls on actin filaments to generate traction forces, which are transmitted to focal adhesions and deforms the ECM via an ‘‘inside-out’’ transmission path. Hence, mechanical forces are experienced throughout the cell via the integrated CSK-focal adhesion- ECM architecture. So, underneath the membrane, and penetrating it at many points, is the CSK. The first set of components that is encountered contains different proteins each with separate functions, as depicted in the schematic diagram below (Figure 2.5). The cartoon represents a generic cell; however particular cells may differ in structure. For example, the red blood cell lacks most of the components shown below. The extracellular matrix (ECM) consists of many filamentous proteins, including collagen, fibronectin, vimentin, titin and others. The CSK near the membrane is rich in many other filamentous and small proteins, such as tensin, vinculin, talin, -actinin, etc. The most abundant filamentous protein near the plasma membrane is actin. Note that the CSK connects with ECM via integrin molecules that consist of alpha and beta subunits. These subunits have ligands for specific receptors on matrix proteins. Integrin thus crosses the plasma membrane (PM) to serve as the connector. The standard active ligand of integrin is the 3 amino acid sequence RGD. Circulating cells such as white blood cells and platelets use their integrins as antennae to locate cells and matrix where their function is needed. Figure 2.6 below represents an Integrin of a platelet. With its 2 subunits, integrin resemble both in form and

Figure 2.5. Peri-membranous CSK

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function, a staple. Several other cellular proteins bind with the ECM, as will be detailed in Chapter 11. The most abundant filament in most cells is filamentous actin (F-actin). Theses microfilaments are most prominent around cell perimeters, and serve as ropes tying the network of other proteins together. A notable exception to this cellular dependence on actin for rigidity is the red blood cell, whose CSK is rich in spectrin. While actin (specifically F-actin) is the main contractile generator in cells, it can also sustain some compression. Another type and configuration of actin is globular, or G-actin. Both types exist in cells, as seen below in Figure 2.7, with the F-actin stained green, and the G-actin stained orange (different gray shades if this is black & white rendering). Many other proteins link the main filamentous actin to the membrane, including tensin, talin, and a-actinin, and vinculin, as shown in the cartoon. Although these many proteins, including Integrin, are not abundant, they play crucial roles in mechanical signaling, as we will see in subsequent chapters. Adhesion of cells to the ECM takes place at focal adhesion complexes (FACs), as depicted above. FACs are mechanical linkages that also serve as signal relay stations between the CSK and ECM.

Figure 2.6. Integrin

Figure 2.7. Actin in a fibroblast

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Stress fibers consisting of bundles of actin filaments associated with myosin are usually attached to the FACs. FACs can generate contractile forces during cell crawling, thus serving as mechanical actuators. Recalling from Chapter 1, the CSK has 3 major components: microfilaments, intermediate filaments, and microtubules. Single strand diameter of these proteins ranges from about 8 nM for microfilaments, 15 nM for intermediate filaments, and 25 nM for microtubules. Filaments usually form multi-stranded threads, so large bundles of microfilaments are common. Microtubules (MT) are seen in abundance in the cells below, differently shaded from actin. Note that MTs tend to be straighter and run closer to the nuclei, although they extend to cell perimeters as well. MTs are hollow tubes, whereas the other 2 filaments are like ropes and rods (See Figure 2.8). The CSK is thus structured as a porous network of struts. This plan is, in fact, no different than that of all living materials: even bone, the densest material in the body, is composed of geodesic structures at the meso-scale. The millimeter-size sub–units are arranged in a lattice oriented to maximize strength in the direction of principal stress. Bone, like the CSK, is a strong, shock absorbing material that distributes mechanical energy to the network, but tends to focus large, chronic stresses to stronger and thicker support elements. It should be noted that CSK composition varies from cell to cell, and even region to region within one cell depending on function as well as developmental state. For example, red cells are rich in spectrin, but deficient in actin. Muscle cells, in contrast, have very high actin content. Growing or healing cells may have relatively high actin in their growing portions. Some cells,

MT

Actin

Figure 2.8. Immunostained cells showing actin in orange and microtubules in green.

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such as skin cells of young animals or certain animals, are rich in elastin. Material properties of some CSK components are listed below:

Table 2.2 Properties of Filaments Polymer Typical

Diameter (nM) Persistence Length p (m)

Elastic Modulus E (Gpa)

Mass Density p

(Da/nm) Actin 8 15 2 110 Tubulin 25 6000 2 160 Intermediate Filaments

10-20

Silk 5 Collagen filament

1.5

Collagen fibril 10-300 Elastin 0.002 Cellulose Dry 80 Cellulose Wet 40 Spectrin 0.02 0.002 4500 DNA 0.05 1 1900 2.4 Building the CSK The long-range order of the CSK is generated by simple rules for network assembly and disassembly. In other words, the CSK is a dynamic structure, and static pictures of it just capture one moment of its ever-changing appearance. Examples are in Figures 2.9 below:

Figure 2.9. a. Fluorescence micrograph of a fish keratocyte is shown (with the nucleus in blue). Motile cells such as these form branched actin-filament networks (red) at their leading edge, and these branched networks generate protrusions. Together with coordinated adhesions to a surface (indicated by vinculin, green) and myosin-driven retraction, the protrusions lead to directed movement. Scale bar, 15 μm. b. There are three basic steps involved in the assembly of protrusive, branched actin-filament networks: filament elongation; nucleation and crosslinking of new filaments from filaments close to the membrane; and capping of filaments. Disassembly of the network involves a separate set of proteins that severs the filaments and recycles the subunits. 26


2.5 Cytogel “Cells are Gels.” Gels are relevant to cytomechanics because the internal composition of cells is essentially a gel. We will investigate the mechanical properties of cells by comparing them to a gel surrounded by a smart membrane. In particular, the concept of cell volume regulation will be studied. Gels and Gelatinization Gelatinization refers to the (usually) irreversible loss of the crystalline regions in suspended or dissolved polymers that occurs suddenly when the physico-chemical environment changes. Gel is thus a state of matter between solid and liquid. Such physical changes can be heat, pH, or chemical composition. “Smart” polymer gels actively change their size, structure, or viscoelastic properties in response to external signals. The stimuli-responsive properties, indicating a kind of intelligence, offer the possibility of new gel-based technology. Deformation and the mechanism of polyelectrolyte gel behavior in electric fields are studied experimentally and theoretically, especially, swelling and bending. Gel stiffness can be controlled by electric fields and polarized particles. The cytosol is a gel containing electrolytes, amino acids, carbohydrates, metabolic fuels and products, as well as cellular organelles, such as mitochondria and endoplasmic reticulum, and others depending on cell type. The cytoplasm, since it is a gel, contributes some mechanical stiffness properties to the cell. Ca++ plays a major role in maintaining the ‘cytogel’ since it coordinates polymer-polymer interactions. The cytogel is formed when the negatively charged polymers in the cytosol (proteins and nucleotides) are trapped together, creating osmotic pressure for water influx. A schematic of a gel is shown in Figure 2.11 below. Note that the gel can be modeled as a spring, and that swelling pressure is a function of polymer-polymer interactions, intra-polymer interactions, and osmotic pressure.


1. Integrin acts like a staple holding cells to substrates. Draw a free body diagram showing the

types of forces involved in this interaction. 2. What does polarity have to do with solubility? Give an example.

Gel

Swelling pressure = osmotic pressure- elastic (compressive) pressure

H20

FactorsCa++, pH, heat

Figure 2.10. Gel Model

27


3. Calculate the capacitance of a lipid vesicle, 1 uM in diameter. Show assumptions. 4. Examine Figure 2.1 and explain, using drawings and formulae, what happens. 5. What cellular components serve as ropes or rods? 6. Cite a specific pathway whereby Integrin could be involved in mechanotransduction. 7. Describe what mechanical role the cytogel might serve. Make a Simulink model of it, using

springs. First write the equation of motion, after the gel has been subject to an osmotic perturbation. Estimate the fractional composition of the major components of your model.

References

1. Goldmann, W.H., Mechanical aspects of cell shape regulation and signaling. Cell

Biology International, 2002. 26(4): p. 313-317. 2. http://www.bio.unc.edu/courses/2004spring/biol052-006/ch02final.pdf 3. http://www.science.uwaterloo.ca/~cchieh/cact/applychem/waterbio.html 4. Vuori, K., Integrin signaling: Tyrosine phosphorylation events in focal adhesions.

Journal of Membrane Biology, 1998. 26(3): p. 191.

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http://www.bio.unc.edu/courses/2004spring/biol052-006/ch02final.pdf

http://www.science.uwaterloo.ca/~cchieh/cact/applychem/waterbio.html

Chapter 3: Cell Types

CHAPTER 3: MECHANICS OF CELL TYPES (Adapted from (Bao and Suresh 2003)

3.1 A Generic Cell There are over 200 different cell types in the human body, each with its own specialty, shape, and mechanical properties. It is useful to recognize some basic features common to most, if not all of them, as shown in the generic cell cartoon, Figure 3.1.a below:

Figure 3.1. Generic cell and comparative properties.

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As seen in Figure 3.1.b, the cell membrane consists of a lipid bilayer with integral proteins. Living cells (Figure 3.1.c) are in the micrometer size range, and have stiffnesses, as measured by the elastic modulus, in the sub-MegaPascal range. (Note there are exceptions, i.e., algal cells & neurons, which can be meters in size). Traditional engineering materials, such as ceramics, metals, and plastics, are much larger and stiffer, as shown. Newer materials, i.e. nano-materials, can have much different properties. The intracellular components perform many functions: synthesis, sorting, storage and transport of molecules; storage and expression of genetic information; recognition, transduction and transmission of signals; powering of molecular motors and machines. The organelles also convert (transduce) energy from one form into another, serving as cellular reflexes. These reflexes sense and respond to external environments by continually altering cellular structure. Most living cells can sense and respond to forces. Cells in the human body are highly diverse, ranging from the red blood cell, which is little more than a sack of hemoglobin, to nerve cells 1 meter in length, and a million times smaller in width, with intricate bush-like branches at both ends. Despite this diversity, the mechanical plan is the same for all cells: each is a small glop of gel held together with stiff rods and struts, and surrounded by an oily membrane. A few types of cells are outlined here. 3.2 Red Blood Cells Red Blood Cells (RBCs) are the simplest animal cells: they lack a nucleus and several other organelles, they have a simple CSK, and they assume a very narrow range of sizes and shapes. Despite their simplicity, their mechanical behavior, both in health and disease, represents the foundation of cytomechanics. The RBC travels around the circulation for one purpose only: to exchange gases. It must withstand large shear and deformations as it bumps into obstacles, becomes exposed to variable osmotic pressures, and must squeeze itself, with a 7.0–8.5 µm diameter into tiny (< 3 µm) wide capillaries (See Figure 3.2). Each RBC thus undergoes large elastic strains many times a minute as it speeds through narrow tunnels, a feat that few man-made objects can do. This amazing malleability of RBCs has played a major role in the development of the field of cellular mechanics, since it sparked the curiosity of its founder, Y.C. Fung. Two interrelated structural features of the RBC underlie its mechanical adaptability. The first is its shape under normal conditions. The biconcave disc shape is a brilliant design, since the shallow center has a low bending stiffness, and it can serve as an expandable reservoir to allow swelling without increase in surface area. These properties allow the RBC to squeeze through narrow capillaries, and to swell in hypotonic environments without breaking, as depicted below. Since the plasma membrane of RBCs undergoes lysis at an area expansion of only 3%, the cells would quickly explode when exposed to even slight hypotonicity if the cell did not have this characteristic biconcave shape (See Figure 3.2).

Figure 3.2.a. A red blood cell squeezing through a capillary

RBC in isotonic media

RBC in moderate hypotonicity

RBC in higher hypotonicity

Figure 3.2.b. Shapes changes in red blood cells.

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The key to the structural adaptability of the RBC is in the composition of its CSK, which is the simplest in the animal world. The CSK of RBCs is constructed from a single filament system based on actin, whose polymeric structure is linked with spectrin. Thus RBCs lack the stiffer components, intermediate filaments and microtubules, and contain the flexible polymer spectrin to link a flexible network of actin polymers. RBCs sometimes aggregate in the blood stream to form rouleaux (stacks of coins) as shown below in Figure 3.3:

The elegant mechanical design of RBCs as they travel their tortuous pathways in the vascular system can be appreciated by observing how small defects can affect them. The first recognized molecular disease.is sickle cell, caused by a point mutation that converts normal adult Hemoglobin (HbA) into sickle Hemoglobin (HbS), with a single amino acid difference. The mechanical consequences of this defect begin with polymerization of Hb into straight rods, leading to gelation within the cell. When these abnormally rigid RBCs travel through the vascular system, they initiate a cascade of inflammatory responses that can manifest as hemolytic anemia, vaso-occlusion, and multi-organ damage. The gelation process and its mechanical consequences will be further discussed in Chapter 7.

3.3 Platelets The second simplest cell in the body is the platelet, which, like the RBC, is an anucleate circulating cell. Its role is to protect the vascular system by orchestrating blood clotting at sites of injury. When they encounter an injury, platelets aggregate and transform from discoid into filipodal structures that then spread out to form a physical barrier against blood loss. The characteristic shapes changes, as depicted in Figure 3.4, are accomplished by active work by the actin-myosin network, in concert with microtubules. Similar to contraction by myocytes, the actin-myosin motions within platelets is initiated by Ca++, and is fueled by phosphorylation. Unlike myocytes however, that have an assembled network of actin-myosin filaments ready to work upon demand, the actin/myosin system of platelets is assembled only after an injury. In the absence of injury stimulus, the actin and myosin in circulating platelets exist mostly in the globular form, i.e. as monomers. Actin monomers in the resting cell are sequestered and prevented from polymerization by a Ca++ sensitive protein. Microtubules, in contrast, are elongated, helping to maintain the discoid shape. Upon injury, Ca++ entry causes a cascade of events leads to profilin dissociating the actin-binding protein and commencement of polymerization. When chains elongate to 6 or more actin monomers, myosin can bind to them, setting up the energy-driven movements. In reciprocity to actin polymerization, microtubules tend to de-aggregate, allowing more flexible movements of the filopodia. This remarkable choreography occurs without a nucleus, through the interplay of just 2 filament systems.

Figure 3.3. Red blood cells.

Figure 3.4. Platelet activation

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3.4 White Blood Cells (WBCs) White Blood cells, or leukocytes, are members of a group of circulating cells whose job is responding to immune needs and inflammation. As they speed through the circulatory system, WBCs are incredibly capable of finding the trouble spot infection site, stopping, then attaching themselves to the endothelial cells lining the vessel, and then move through them into the body tissues to attack. WBCs must undergo not only the same passive deformations as RBCs, but must also actively change shape as they perform phagocytosis, i.e. envelope foreign or inflamed objects. Their shapes, unlike RBCs, are generally spherical. How then do WBCs achieve their flexibility and motility? In fact WBCs use an entirely different strategy to accomplish the same feats of deformability and volume expansion without surface area change. The answer is that WBCs can deform and change volume without undue stress on their membranes, because their plasma membranes and CSK are extensively folded. These folds consist of CSK curvatures and membrane microvilli; these serve the same role as the RBC concavities: they are expandable reservoirs. WBCs expansion-shrinkage behavior is much the same as the Hoberman sphere. At low magnification, WBCs appear smooth, but folds and microvilli can be seen at higher magnification as depicted below in Figure 3.5. 3.5 Fibroblasts Fibroblasts serve as a glue holding tissues together and are primary manufacturers of collagen. Shown below in Figure 3.6 are fibroblasts co-cultured with cardiac myocytes, and stained for actin. Myocytes are brilliantly stained (white) showing an abundance of actin, while fibroblasts (left portion) are much darker. Note that the high density of actin sometimes obscures structural details.

Microvilli Outline of WBC

Figure 3.5. WBC microvilli

Figure 3.6. Fibroblasts and cardiac myocytes, stained for actin

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3.6 Endothelial Cells Endothelial cells (EC’s) are a good example of cytomechanical adaptability. The basic role of EC’s is to line the walls of blood vessels. They are directly exposed to flowing blood, as depicted schematically and photographically below.

EC’s change shape and mechanical properties in relation to the forces they experience from the flowing blood. Endothelial cells not only recognize the magnitude of force, but also distinguish between shear and normal forces, and whether they are steady or pulsatile. Abnormally high forces can lead to vascular diseases including thrombosis and atherosclerosis. Cells can 'crawl' like an inchworm by pulling themselves forward using contractile forces.

3.7 Myocytes The three types of muscle cells: skeletal, heart and smooth, each have different mechanical structures and behaviors. Myocytes, or muscle cells, are the primary motor cells of the body, and hence have the highest content of actin. Figure 3.9 below shows a single myocyte that has been stained for actin. Myocytes tend to elongate to form log-shaped structures.

ECs with black nucleus (Fung)

Figure 3.7. Endothelial Cells

Figure 3.6

Figure 3.8. Inside a blood vessel, looking at bulging endothelial cells

Figure 3.9. A cardiac myocyte stained for actin

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3.8 Transformed Cells Cells from almost any tissue can transform into tumor cells that are genetically distinct. Benign tumors grow rapidly, but respond normally to ECM. Malignant cells have mutant actin, disorganized CSK. They lose contact inhibition and invade ECM, and climb over other cells. Cell shape affects malignancy, i.e. imposing a spherical shape on melanoma cells makes them more metastatic. 3.9 Stem Cells (adapted from Lee, Knight et al. 2011) The mechanical properties of stem cells influence their response to their mechanical environment, their ability to migrate and ultimately their differentiation. For example, the amorphous mass of undifferentiated mesenchymal cells, responsible for the development of the skeleton, is susceptible to the influence of mechanical signals mediated through the extracellular matrix. It has long been postulated that adventitious, secondary cartilage develops on cranial membrane bones of the embryo in response to intermittent pressure and tension, associated with movement [Hall, 1972]. The importance of intermittent loading to chondrogenic development has been further supported by studies involving joint immobilization in the developing chick embryo. Indeed following paralysis of skeletal muscles, abnormalities were reported to develop, including the absence of synovial joint cavities, the fusion of long bones by fibrous tissue cartilage or bone, the absence of adventitious and articular cartilages and the distortion of the skeleton. An alternative approach uses theoretical models to predict the effects of mechanical stimuli on lineage-specific stem cell differentiation. These models can predict tissue differentiation during skeletogenesis, fracture healing, bone distraction and the development of pseudoarthroses. As early as 1960, Pauwels proposed that tissue deformation or stretching induces the formation of fibrous connective tissue while compression induces cartilage formation (Figure 3.10.A). More recently researchers have used finite element analysis to estimate the internal mechanical state within structures, to predict the influence of hydrostatic pressure and distortional strain on tissue differentiation. These models suggest a correlation between high levels of compressive hydrostatic stress and chondrogenesis; low hydrostatic stress and osteogenesis; and high distortional strain associated with the formation of fibrous connective tissue or fibro-cartilage (Figure 3.10.B).

Figure 3.10. Models of the relationship between the mechanical environment and differentiation of mesenchymal tissue.

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Further adaptations of the modelling approach enable the establishment of critical values for mechanical parameters in relation to differentiation. For example, a study proposed that local strains lower than 5% induce intramembranous ossification, while hydrostatic pressures greater than 0.15 MPa and local strains smaller than 15% induce endochondral ossification. However, from a biological viewpoint, it is not correct to suggest that in all conditions these critical values represent sharply delineated ‘cut-off’ values that will predict the differentiation of tissues under the influence of distortional strain and hydrostatic stress. Indeed fundamental changes occur within differentiating tissues, which can drastically change the nature of loading, for example, the generation of extracellular matrix. In an attempt to analyze mesenchymal cell differentiation, finite element models were developed, which incorporate the effects of the relative velocity of fluid and solid constituents, fluid pressure and tissue deformation [Prendergast et al., 1997]. The synthesis of extracellular matrix by differentiating MSCs may, or may not, favour the mechanical and perfusion characteristics required for lineage specific differentiation within that tissue, driving the progression of cell phenotype in a step-wise manner. A ‘mechano-regulatory’ pathway (Figure 3.10.C) describes mesenchymal differentiation in a temporal manner, where the emergence of a specific extracellular matrix (Point X—Figure 3.10.C) can favor a divergence in phenotype (red dashed line) from a steady-state condition (solid line). In the presence of significant shear strain and associated motion, fluid velocity and shear forces are also maintained favoring differentiation to fibrous connective tissue. However, an up-regulation or change in collagenous matrix production leads to a higher stiffness and consequent reduction in fluid velocity and shear force leading to osteogenic differentiation. More recent developments on the modelling approach have incorporated other parameters that may interact with the mechanical stimuli to direct differentiation, most notably alterations in the oxygen environment and associated angiogenesis [Checa and Prendergast, 2009]. This model also incorporates cell migration, proliferation and apoptosis and has been further developed to incorporate physiological variation in cellular parameters to predict animal to animal variation in differentiation reported in vivo within a defined bone chamber model subjected to mechanical loading [Khayyeri et al., 2009]. __________________________________________________________________


1. Explain how red blood cells can increase volume without increasing their surface area.

Show formulae. 2. What types of forces would be experienced by ECs in their natural state? Draw a sketch

resembling a free body diagram. 3. Describe two types of transduction that a cell can perform. 4. State 2 structural adaptations of RBCs that relate to its function. 5. Compare the way in which red and white cells adapt to different environments in the

blood stream. 6. Expand your Simulink model of gel (from Chapter 2) to include damping. Show the

various degrees of damping, i.e., under, over, etc.

References

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Bao, G. and S. Suresh (2003). "Cell and molecular mechanics of biological materials." Natural Materials 2(11): 715-725. Lee, D. A., et al. (2011). "Stem Cell Mechanobiology." Journal of Cellular Biochemistry 112(1): 1-9.

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Chapter 4: Material & Structural Mechanics

CHAPTER 4: MATERIAL AND STRUCTURAL MECHANICS

4.1 Background (adapted from Ingber, 2010) Most man-made constructions, such as Stonehenge, gain their stability from continuous compression, as in brick-upon-brick type constructions in which gravitational forces compress one building element down upon another. These are generally stable structures, except when subject to side impacts, when individual building components can fall like dominoes. While the latter phenomenon is the physical basis for a popular ‘app’ (Angry Birds), it is an undesirable structural feature. Tensegrity structures, in contrast, are composed of a network of tensed elements linked to a subset of elements that resist being compressed, and thereby bring the entire system into a state of isometric tension. Buckminster Fuller first coined this term, and a graduate student, Kenneth Snelson, embodied it by constructing sculptures composed of stainless steel beams interconnected by tension cables that hold themselves stable against the force of gravity even though the beams never come in direct contact. 4.2 Stress Environments of Cells Living cells are both affected by and dependent upon mechanical forces in their environment. Cells are specialized for life in their own particular environments, whose physical stress patterns become necessary for normal functioning of the cells. If the forces go outside the normal range, then the cells are likely to malfunction, possibly manifesting as a disease or disability. For example, when people spend more than a few days in zero gravity, many organs, including the bones and heart, decline in mass,. In general, cells need continual stimulation from their usual environmental forces to maintain their size, growth rate, and function. Human cells live in a wide range of mechanical environments. The environments of several cell types are outlined in the table below: Tissue

Mechanical Environment Normal Range Cell Types

Bone, Cartilage Weight bearing forces Continuous: 1X - 4X body weight

Osteocytes, Osteoblasts, Chondrocytes

Arterial Endothelium

Fluid pressure and shear Pulsatile: 60-140 mm Hg;

Endothelial

Tendon Tension Up to 560 +- 9 Kg/cm2

Fibroblast

Skin Compression and shear 0-150 mmHg Epidermal, fibroblast Organ of Corti Fluid shear 100 dB Hair Muscle (Intrafusal)

Tension 50-100 lbs Nerve/specialized muscle

Muscle (Extrafusal)

Tension; active contraction 50-100 lbs Smooth, cardiac, and skeletal myocytes

Mesangium Fluid pressure and shear 120 mm Hg Mesangial While we know some of the forces that are applied to cells and tissues, as listed in the table above, we know much less about how the forces are transmitted throughout the cell. Quantifying cellular force is difficult because they are distributed among the many complex structures within, including the cell membrane, cytoplasm, organelles, and especially the cytoskeleton.

37


4.3. Uniaxial Stress and Strain Perhaps the most fundamental property we would like to know for any material is its strength. To measure this, we can take a representative specimen of the material, with a simple shape such as a bar or rod, and pull it until it breaks. Thus applying constant forces, F, to both sides of the bar shown in Figure 4.1, would translate into the stress, =F/ A, assum ing th were distributed evenly over the cross-sectional area, A. During constant stress, the bar, being deformable, will elongate in the direction of the F vector, along the X direction and would shorten in the y direction. If the initial length in the X direction, were Lo, the deformation could be measured as the strain:

εxo

o

L LL

=−

The resulting stress-strain curve would typically look like one of the following: The curve on the right shows that for low strains, the behavior is linear, but begins to deviate from linearity at higher strain, reaching a maximum stress before curving down, and rupturing. The curve maximum is the ultimate tensile stress (UTS) and the point at which the curve becomes non-linear is

the elastic or proportional limit. The curve on the left is derived from the one on the right, by correcting for changes in area during the test. The elastic region of the stress-strain curve represents the stiffness of the material, defined as Young’s modulus:

Y x

x=

σε ; which is sometimes called the elastic modulus, E. Stiffness can be measured in both

tensile and compressive loads. 4.4. Two-Dimensional Stress The simplest depiction of stress is in a bar or cube, as shown below. Much of the CSK is made from bar or rod-like structures (ropes and struts), and hence experiences these types of stress:

F

F

A

X Y

Figure 4.1

38


Forces on a cube in 1 or 2 dimensions are analyzed as follows. Take the case of unconstrained isotropic object compressed in the y direction:

(Note that for an elastic material the strain occurs almost instantaneously upon application of the stress. Also note that to maintain constant stress, σy , the applied force must be reduced if the face area increases, but this would be a negligible change for all practical

situations). The strain in the y direction is: Ey

y

σε = Because the transverse

direction is unconstrained: 0=xσ and, yx νεε =

Now, Consider the case where the x direction is constrained from movement, i.e. transverse movement is resisted, making:

0=xε To prevent strain in the x direction, a stress σx must be applied:

EE yxx νεεσ == In other words, to prevent transverse strain, you must push on the object with σx. This causes an added stress to the y direction, due to a tendency to strain:

yxy εννεε 2' == and xyy E νσενσ == 2'

Thus the new stress in the y direction is the original unconstrained stress plus the stress caused by transverse constraint:

σy

σy

σy

σy

y

σy

σy

σx σx

Before strain After strain

X

39


xyy E νσεσ +=

Solving for εy we have the biaxial strain equation:

)(1xyy E

νσσε −=

4.5 Types of mechanical loading Measuring the material properties of engineering materials such as metal, ceramics, and polymers is a straightforward and well established procedure: specimens are carefully shaped to fixed geometries, and subjected to forces of various types, while the resulting deformation is measured. The classical method is to take a small piece of the object under test, place it in an Instron machine that applies tension, or compression, and torsion or bending forces as shown below, and measure its deformation until it breaks. This procedure yields parameters such as moduli of elasticity (E) and bending (M), ultimate tensile strength (UTS), and viscosity ( ). To characterize an entire object by these tests on small pieces it must be assumed that the material being isotropic in 3 dimensions, i.e., its material properties are equivalent in all directions and locations. 4.6 Measuring cellular material properties Material properties of cells are much more complicated than most engineering materials. Firstly, most cells and their components are highly anisotropic Secondly, material properties are not constant, and are likely to change during the test procedure. Unlike inanimate objects that possess static properties, cells behave. Thirdly, cells and their components are generally not materials, but are structures, whose properties vary depending on the type of force applied. Despite the complexity and changeability of cell structure, mechanical measurements can yield important information about cell functions. The challenge of cell mechanical measurement has been met with many ingenious testing devices. The basic principle of testing is to apply a small deformation to the specimen while measuring as accurately as possible both the force applied and the deformation as it evolves over time. The most straightforward tests involve directly poking the cell with small flexible probes. A simple tool for applying nano-Newton sized “poking” forces to cells is the fine-tipped

Figure 4.2 Types of forces. (a) tension/compression; (b) bending; (c) torsion; (d) shear.

40


micropipette. These can be readily made by drawing glass pipets under heat. Cell’s stiffness can be roughly estimated by the bending angle of the micropipette’s tip as it pushes against the cell wall with a known force. Devices for applying and measuring forces of < 1 pN and deformations of < 1 nM are now available and continuously being improved. These new techniques can illuminate the processes responsible for operation of cellular machinery, the forces arising from molecular motors and the interactions between cells, proteins and nucleic acids. This topic will be further detailed in the next chapter. A depiction of stress-strain testing is shown. Note the shaded cross-sectional area in the middle of the bar indicates that stress is assumed to be the same in the entire cross section of the bar: Uniaxial stress, as in Figure 4.3, also causes deformations in the other 2 Cartesian coordinates, Y and Z, and can be quantified by Poisson’s ratio:

ε ε υσ

υεε

y y x

y

y

= = −

= −

Figure 4.3. A bar under tension

41


For example, in 2 dimensions, assume the bar shown in Figure 4.3 above was originally a 1 by 1 square, and was stretched to a rectangle 2 by 1. Strain in the x direction is 1, or 100% , while strain in the y direction is 0.3 or 30 %. Poisson’s ratio is then 0.3. Thus the area increased from 1 to 1.4. Since the area, and also volume, of the bar dilate, while the mass does not change, it must be that the material density decreases during deformation. Note that since strain in the non-stressed directions are almost always opposite in sign to that in the stressed direction, the negative sign for the strain ratios, ν, is usually a positive number.

A special case for Poisson’s ratio is ν = 0.5, meaning that the material does not change cross-sectional area (nor volume) when being deformed. This type of material is called ‘incompressible.’ Note that being incompressible does not mean that it is not deformable. While most engineering materials have ν = 0.3 or so, the few biological materials that have been tested, have ν = 0.5 or greater. Bulk muscle, for example, can have ν > 1. This can be seen by contracting your biceps- a very small shortening, i.e. axial compression, can produce a much larger expansion as the muscle bulges upward. This indicates that muscle volume drastically increases during contraction. Note that individual muscle fibers have ν much closer to 0.5, so that the physical arrangement of the bulk muscle is the cause of the large increase. Another special case for Poisson’s ratio is for a cylinder, as shown below. For this case, we have ε ε υσ

υεε

y y x

y

y

= = −

= −

as before, and εy=-γσx

E

and 𝜕𝜕

𝑑= 𝑃

𝐴∗𝐸

so 𝜕𝜕 = 𝑃∗𝑑

14 𝜋𝑑2𝐸

= . (You finish.)

Y

0

0.7

X

1

2

For most engineering materials, ν < 0.3

Materials with ν = 0.5 are "Incompressible."Some materials have ν > 1

v =-(.7-1)/1 = 0.3Figure 4.4

42


4.7 Material Stiffness & Strength To begin mechanical analysis of the cell, we need to begin cataloging its mechanical properties. This chapter will present material properties of selected cells and components. Comparing just the elastic properties of cells with other materials, it is seen that cells are much more compliant, and have a pre-stress at zero strain, as shown in Figure 4.5 below:

A more quantitative look at cell (tissue) elasticity is charted in Figure 4.6: Note that mechanical testing of cells is much less standardized compared with solid, engineering materials, as discussed in Chapter 5. Deformation of cells, even those as simple as the human red blood cell, can be properly analyzed only using continuum models. This constitutive behavior includes multiaxial stress–strain relations and changes of mechanical properties with time and/or in response to biochemical or electrical stimulus. However, most living cells are even more complex, so that deformation models must account for internal structure, spatial granularity, heterogeneity and the active features unique to cells. For example, the shape, motility and mechanical properties of tissue cells depend largely on cell cytoskeleton, which is a dynamic system undergoing structural changes during cell spreading and rounding or due to mechanical or chemical signals. Thus the material properties of cells are highly variable and difficult to quantify.

Comparative Mechanical Properties

Strain ε

Steel

Wood BoneSt

ress

σ

Cells

Steel Wood Bone

Cells

Cellular‘pre-stress’

Figure 4.5

0.0002

0.007

14 21

210

1200

0.0001

0.01

1

100

10000

Mod

ulus

(GPa

)

Material

Comparative Stiffness

Figure 4.6

43


The resistance of single cells to elastic deformation, as quantified by an effective elastic modulus, ranges from 102 to 105 Pa (Figure 4.6), orders of magnitude smaller than that of metals, ceramics and polymers. The high degree of elasticity and deformability of cells is illustrated by the red blood cell in Figure 4.7, undergoing strain > 100%: (How is this done? See class notes). Note from the continuum model, strain magnitudes are highly anisotropic, and are greatest at the sites of attachment. Also note, due to the complex shape, it is difficult to calculate Young’s modulus, but the bulk cellular modulus can be estimated assuming a nominal cross-sectional area for the cell. 4.8 Estimating Cell Stiffness (adapted from Lee, Knight et al. 2011) In a recent study, the mechanical properties of MSCs were assessed using micropipette aspiration in their undifferentiated state and during osteogenic and adipogenic differentiation [Yu et al., 2010]. The authors report values for instantaneous and equilibrium Young’s moduli for the undifferentiated cells of approximately 450 and 100 Pa, respectively. These values rise significantly and progressively during osteogenic differentiation over a 21-day period to reach values approximately twice that of the undifferentiated cells. These changes are closely associated with alterations in the cell size and morphology, and with alterations in the cytoskeleton. Indeed a previous study demonstrated a significant reduction in the modulus of MSCs following disruption of the actin cytoskeletal organization [Tan et al., 2008]. Embryonic stem cells have also been studied using micropipette aspiration, with results suggesting a marked stiffening of the nucleus during differentiation, associated with the nuclear skeletal component Lamin A/C, which is only expressed in differentiating cells [Pajerowski et al., 2007]. These findings suggest that both embryonic stem cells and MSCs may be more deformable than their differentiated progeny. This physical plasticity may facilitate the migration of stem cells through solid tissues to the sites of tissue damage.

Figure 4.7

44


Spherical Stress A stressed, thin walled sphere experiences a wall tension, which in the case of a cell, is considered, ‘membrane tension.’ The free-body diagram of a stressed sphere is shown below. To find the membrane tension, T, due to internal pressure, a simple sum of forces will lead to the LaPlace equation P = 2T/R

4.9 Mechanical Properties of CSK components Elastic moduli The deformability of cells is determined largely by the cytoskeleton, whose rigidity is influenced by the mechanical and chemical environments including cell–cell and cell–ECM interactions. Properties of selected polymeric components of the CSK, and other polymers for comparison, are shown below.

Table 3.1 Polymer Typical

Diameter (nM)

Persistence Length, p ( m )

Elastic Modulus, E (Gpa)

Mass Density, p (Da/nm)

Actin 8 15 2 110 Tubulin 25 6000 2 160 Intermediate Filaments

10-20


1.5

Collagen fibril 10-300 Elastin 0.002 Cellulose Dry 80 Cellulose Wet 40 Spectrin 0.02 0.002 4500 DNA 0.05 1 1900 Note that the 2 major CSK components have equal moduli of 2 GPa, but this is 3 orders of magnitude greater than spectrin, the major CSK component of RBCs. Since spectrin is much more dense than either actin or tubulin, this means that its stiffness to weight ratio is very small.

Wall stress in a thick sphere

• To find equilibrium forces:

� Σ Fup = ΣFdown

σ.σ

P

Ri

Ro h

P

Ri

Roh

Figure 4.8

45


Elastin is a major component of skin that imbues it with a high degree of deformability, as illustrated below: A qualitative comparison of a highly compliant polymer like elastin with a stiff one like collagen is shown in Figure 4.10 below. Note that stiffness of both polymers increases drastically at large strains. Also indicated in the figure, is that the stiffness depends on the strain rate, so that collagen can behave more like elastin when stretched at an increased rate. This viscous behavior will be further discussed in Chapter 7.

Low elastic modulus, however, does not imply high deformability, since RBCs lyse at strains > 3% (and skin tissue is not directly comparable to RBCs): Another related issue is that strength, or UTS, is not directly related to stiffness, as seen in the chart below:

Figure 4.11

Figure 4.10

Figure 4.9. Strain > 100%

46


Selected stiffness and strength values are presented in the table below. Note that silk is stronger than steel:

Table 4.2 4.9 Flexural Moduli Since the polymers that make-up the CSK are bar or rod-like structures (ropes and struts), they can and do experiences all types of stresses as shown in Figure 4.1. For rods, like polymers, the most important mechanical mode is bending. Suppose a thin rod of length L is initially upright at zero temperature, but when warmed is bent under the influence of gravity into a curvature with radius R: The rod has bent out of its ‘natural ‘ shape, when R = ∞, under conditions of zero temperature and stress. In this case, we can imagine that the stiffness of the rod determines its curvature, but it cannot be the same stiffness, Y, describing resistance to uniaxial stress. A new measure of stiffness is therefore required, the flexural rigidity. For uniform rods, flexural rigidity is:

κ

πf IY

IR

=

=4

4

Thus the flexural rigidity is based on Young’s modulus multiplied by the geometric parameter for rods, the area moment of inertia, I. To determine the amount of deformation of the rod, we

L bend R

θ

RL

=θ

Figure 4.12

47


now need to know how much force is applied. Since we are dealing with molecules, not big rods, the forces must be considered at the molecular level, and this is where the energetics of the situation must be considered. This is where Boltzmann’s relationship (Chapter 6) is needed. Energy, Earc, expended by (or to) the rod to bend into a given arc, R, is proportional to the flexural stiffness and to the amount of bend. For example, when R = ∞, a straight rod, Earc will be 0. With a large bend, Earc will be large. It can be seen that the ratio, L/R, is roughly proportional to Earc . In fact, elastic theory predicts that:

ELRarc f= κ

2 2

In order to derive a useful description of how bent a given rod would be at a given temperature, Boltzmann’s thermal formula, 3/2KT, the average energy per molecule, is needed. For this, a new parameter is defined, the ‘persistence length’, which represents the length over which the rod will be fairly straight. This length, λp , is defined as the magnitude κf/ KT. To better understand the meaning of λp , consider the case when L = λp and Earc=KT. In other words, the bending energy just equals approximately the average energy of one molecule at temperature T. In this case, we can substitute the conditions:

ER kT

kT pN nmarcf

= = = −κ 2

2241.

Solving for R, we see that:

R

Rradians rees

p

p

=

= =

ξ

ξ2

2 81deg

Thus a rod of the persistence length is curved at 81 degrees when the thermal energy scale reaches kT. If L << p then the rod is relatively straight, otherwise not. Persistence length sets the scale of thermal fluctuations. Filaments with contour length >> p are highly convoluted and can assume many configurations. Persistence lengths for selected polymers are listed in Table 4.2. Formulae related to bending of polymers are listed in Table 4.3:

Please note that the persistence length, λp, is identical to ξp

Table 4.3

48


4.10 Measuring Local Stiffness In general, the mechanical properties of homogeneous materials are determined by applying stresses and measuring the resultant strains (or vice versa). In the regime of linear response, the ratios between stresses and strains—the viscoelastic moduli (or compliances)—are material properties that may show some frequency dependence. They may alternatively be determined from the spectrum of thermal fluctuations. Macroscopic rheological methods are widely used across many disciplines, but the softness of cytoskeletal networks, their limited availability, and the need to resolve the mechanical properties of their local heterogeneities have motivated the development of micro-rheological methods. In the case of ordinary micro-rheology, observing the induced motions of suspended non-magnetic beads yields the deformation field, and thus enables a more-detailed investigation of local heterogeneities and stress propagation. This method (pictured), which is based on a combination of ‘active’ field-driven particles and passive tracer particles, was more recently developed into a purely passive technique, now widely known under the name of two-point micro-rheology. The thermal motion of single particles is determined by either video microscopy or high-speed inter-ferometric detection. Assuming a generalized Stokes–Einstein relation, the frequency-dependent moduli can be extracted. Depending on the bead radius and detection limit, moduli from 10−3 to 102 Pa are measurable.

When microrheological techniques were introduced, discrepancies between the measured network elasticity and macroscopically determined moduli were poorly understood. But it has since been shown that the local compliances depend on the size ratio of the colloidal particle and on some characteristic length scale of the network or of its constituents. Paradoxically, a detailed theory of the micro-rheometry for such heterogeneous networks seems to require a complete understanding of the network mechanics that are to be studied by this method.

Figure 4.13

Figure 4.14

49


Here is an example of testing regional elasticity in a cell:

The picture at left shows the cell with several beads inside that have been colored from blue to red, indicating local stiffness. At right, a close-up of the cell shows records of particle movement over time. Note that the upper particle traversed a relatively tight area, during the same time that the lower particle moved over a much wider area. These random motions indicate how stiffly the beads were trapped within the cytoskeletal network, and hence indicate local stiffness. By measuring the mean squared displacement of a particle <Dr2 > over a period of time, an estimate of stiffness can be estimated as

2r

DGD

=η

where G= Young’s modulus, D= diffusion constant of the cytoplasm, and η= viscosity of the cytoplasm . In the case of active microrheology, magnetic forces are applied and from the resulting displacements (in and out of phase) of magnetic particles in the system, the frequency-dependent susceptibilities are determined. These may be transformed into ‘local moduli’ for a more direct comparison with the macroscopically measured moduli.

50


4.11 Buckling Straight rods subjected to axial loads can buckle, as depicted below. The forces, P1, P2, and P3, required to buckle each loading configuration are shown:


1. Estimate an elastic modulus for the RBC in Figure 4.7 . State assumptions. 2. Human hair has a stiffness of 4 X 1010 dynes/cm2 . When axially strained, it undergoes

brittle breaking at a stress of 106 dynes/cm2 (there is no ductility in hair). What is the radial strain? Assume Poisson’s ratio is close to that of steel, and hair length similar to your own. Show assumptions. (Note, hair is cylindrical).

3. Calculate the energy (in terms of KT) required to bend a microtubule, 20 cm long, into a curve with a 10cm radius? Assume the persistence length to be 2 mm.

4. Now, for the microtubule in #5, calculate the buckling force. 5. Derive Laplace law for a sphere. 6. Model 4:

References

2

2

34L

EIP π=2

2

2 4LEIP π

=2

2

1 LEIP π

=

Figure 4.15

51


Ingber, D. E. (2010). "From Cellular Mechanotransduction to Biologically Inspired Engineering." Annals of Biomedical Engineering 38(3): 1148-1161. This article is based on a lecture I presented as the recipient of the 2009 Pritzker

Distinguished Lecturer Award at the Biomedical Engineering Society annual meeting in October 2009. Here, I review more than thirty years of research from my laboratory, beginning with studies designed to test the theory that cells use tensegrity (tensional integrity) architecture to stabilize their shape and sense mechanical signals, which I believed to be critical for control of cell function and tissue development. Although I was trained as a cell biologist, I found that the tools I had at my disposal were insufficient to experimentally test these theories, and thus I ventured into engineering to find critical solutions. This path has been extremely fruitful as it has led to confirmation of the critical role that physical forces play in developmental control, as well as how cells sense and respond to mechanical signals at the molecular level through a process known as cellular mechanotransduction. Many of the predictions of the cellular tensegrity model relating to cell mechanical behaviors have been shown to be valid, and this vision of cell structure led to discovery of the central role that transmembrane adhesion receptors, such as integrins, and the cytoskeleton play in mechanosensing and mechanochemical conversion. In addition, these fundamental studies have led to significant unexpected technology fallout, including development of micromagnetic actuators for non-invasive control of cellular signaling, microfluidic systems as therapeutic extracorporeal devices for sepsis therapy, and new DNA-based nanobiotechnology approaches that permit construction of artificial tensegrities that mimic properties of living materials for applications in tissue engineering and regenerative medicine.

Lee, D. A., M. M. Knight, et al. (2011). "Stem Cell Mechanobiology." Journal of Cellular Biochemistry 112(1): 1-9. Stem cells are undifferentiated cells that are capable of proliferation, self-maintenance

and differentiation towards specific cell phenotypes. These processes are controlled by a variety of cues including physicochemical factors associated with the specific mechanical environment in which the cells reside. The control of stem cell biology through mechanical factors remains poorly understood and is the focus of the developing field of mechanobiology. This review provides an insight into the current knowledge of the role of mechanical forces in the induction of differentiation of stem cells. While the details associated with individual studies are complex and typically associated with the stem cell type studied and model system adopted, certain key themes emerge. First, the differentiation process affects the mechanical properties of the cells and of specific subcellular components. Secondly, that stem cells are able to detect and respond to alterations in the stiffness of their surrounding microenvironment via induction of lineage-specific differentiation. Finally, the application of external mechanical forces to stem cells, transduced through a variety of mechanisms, can initiate and drive differentiation processes. The coalescence of these three key concepts permit the introduction of a new theory for the maintenance of stem cells and alternatively their differentiation via the concept of a stem cell 'mechano-niche', defined as a specific combination of cell mechanical properties, extracellular matrix stiffness and external mechanical cues conducive to the maintenance of the stem cell population. J. Cell. Biochem. 112: 1-9, 2011. (C) 2010 Wiley-Liss, Inc.

52

Chapter 5: Cytomechanical Tools

CHAPTER 5: CYTOMECHANICAL TOOLS 5.1 Background (with acknowledgement to Bao and Suresh 2003; Eyckmans, Boudou et al. 2011) Understanding how cells generate and sense forces and how the forces are sensed and transduced into biochemical signals is vital to address fundamental questions about cell behavior in both normal and pathological states. Accurate measurement of forces and displacements exerted by cells both in vivo and in vitro is an essential step in this endeavor, and a foundation of mechanobiology. The development of mechanobiology as a field has been enabled by new visualization and measurement technologies. For example, the earliest observations suggesting that mechanical forces drives embryogenesis and bone structure were a natural result of newfound microscopy methods. Mechanobiology received relatively little attention for much of the 20th century as scientists focused on developing molecular biology tools to catalog the genetic basis for life. The recent renaissance in studying mechanics primarily in cell culture has largely been enabled by a suite of tools to measure and manipulate such forces in vitro. For example, in vitro application of strains that would be experienced by bone during physical activity increases proliferation and differentiation of bone cells, and bone matrix deposition, which are all characteristic for mechanically induced anabolic bone growth in vivo. Mechanical stretch that mimics the effect of pulsating blood flow has been shown to trigger many alterations in endothelial and smooth muscle cell signaling, vascular cell proliferation, and expression of inflammatory markers. Methods to apply shear flow on cell cultures have shown that shifts between steady and turbulent shear flow can prevent or promote inflammatory activation, respectively, and may explain the localization of atherosclerotic plaques to specific regions along the vascular tree. Flow rates during development also can drive arterial versus venous phenotype. Bioreactor-type devices have been developed also to model the compression experienced by chondrocytes in the articular joint, tension in muscles, ligaments, and tendons, as well as impact forces associated with trauma. In addition to such externally applied forces, non-muscle cells generate contractile forces on their own. This was first illustrated by Harris and Stopak, showing that cells cultured on soft polymer substrates would wrinkle the substrate surface, implication that forces could be ever present, even in settings without an explicit mechanical stimulus, seemed heretical at the time. However, over the past three decades, it has become clear that (1) most eukaryotic cells can generate intracellular forces that act on the surrounding extracellular matrix (ECM) or neighboring cells, and that (2) this contractile activity is critical for a number of biological processes such as cell migration, mitosis, as well as stem cell differentiation and self-renewal. For physical functions such as mitosis and migration, force is clearly ‘‘essential’’ in the same way that oxygen is essential for life. That is, without force, mitosis and migration cannot proceed. The role of force in genetic responses such as proliferation and differentiation, however, appears to be ‘‘regulatory’’ in the same manner as cytokines might be. Although we have some clues to how forces exert these regulatory functions, clarifying these mechanisms remains a central question for mechanobiology. 5.2 Measuring and Manipulating Cellular Forces Testing mechanical properties of engineering materials such as metal, ceramics, or polymers, is a snap: simply place a specimen in an Instron machine and stretch or compress it in various ways until it breaks. Testing cells is not so easy. Cells are soft, complex, non-linear, and changeable. Unlike inanimate objects, that possess static properties, cells behave. From previous chapters it should now be clear that the cell is not like a hard ball, but like a thin, elastic

53


shell filled with an incompressible fluid, whose fundamental properties can change with circumstance. The challenge of cell mechanical measurement has been met with many ingenious testing devices. Perhaps the simplest tests involve directly poking the cell, as depicted in Figure 5.1 above. A simple probe for applying small nano-Newton level poking forces to cells is a fine-tipped micro-pipet. When the tip bends as it pushes on the cell, the degree of bending indicates at least qualitatively the cell stiffness. So the probes used are more refined than a hammer, but the principle is the same: apply a small deformation to the specimen while measuring as accurately as possible both the force applied and the deformation as it evolves over time. Mechanically probing single cells and even biomolecules is becoming increasingly sophisticated. Devices for applying and measuring forces < 1 pN and deformations < 1 nM are available now. Combining these devices with the latest imaging, molecular biology, and bioinformatics, including modeling and simulation has led cytomechanists to revolutionize our view of the cell. These new techniques can illuminate like never before the processes responsible for operation of cellular machinery, the forces arising from molecular motors and the interactions between cells, proteins and nucleic acids. Several methods have been developed to determine the mechanical characteristics of the cell and to understand how those characteristics change in health and disease. An innovative apparatus was the ‘‘cell poker,’’ which was able to measure the stiffness of living cells by applying an axial strain via a cantilever. This apparatus was also used to demonstrate that neutrophils may be retained in capillaries in the acute inflammatory response due to an increase in stiffness caused by chemoattractants. Another method that has found wide use is atomic force microscopy (AFM). Intrinsic mechanical properties of the cell have been found by applying continuum mechanics models. . Mathematical models have also been applied to endothelial cells to understand how force is transmitted from their apical to basal membranes by combining the AFM with total internal reflection fluorescence microscopy. One of the most widely used methods for studying cellular mechanics is the micropipette aspiration technique. An adaptation of the solution for a punch problem has been used to obtain mechanical properties of bovine aortic endothelial cells, as well as to compare normal and osteoarthritic human chondrocytes. The viscoelastic behavior of cells has also been studied using magnetic bead micro-rheometry. Creep response and relaxation curves were obtained with this technique by applying tangential force pulses on magnetic beads fixed to the integrin receptors of the cell membrane, as shown in section 5.3 below. The aforementioned testing methodologies can measure cellular structural properties and help us understand various biological processes, however, there is currently no technique that can perform stress-controlled indentation testing on single adherent cells. The application of

Figure 5.1. Testing cell mechanical responses

54


controlled and well defined axial stresses to an anchorage-dependent cell has physiological relevance in several cases and may prove vital in understanding how mechanical forces influence cellular behavior. For example, articular chondrocytes are anchorage- dependent cells that attach to their surrounding extracellular matrix and experience many forces in vivo, including compression. 5.3 Measuring Cell-Generated Traction Forces Cells are constantly probing, pushing, and pulling on their microenvironment. A growing body of work is investigating what regulates the cellular generation of contractile forces. To identify when such forces are invoked, several approaches have been developed that allow quantitative measurement of cellular traction forces. In addition, numerous methods are being used to manipulate cell-ECM forces, all of which have been critical in enabling the growing field of mechanobiology. In this section, we will examine the tools used to measure and manipulate cellular forces, and how they have contributed to our understanding of the role of mechanical events in both physiological and pathological settings. Owing to the evolution of biomaterials and polymer chemistry, the concept of soft substrates linked with ECM matrices has enabled the measurement of forces generated by even single cells. The basic concept of using an elastic substrate for force measurement was originally conceived by Harris et al.: when adherent to a thin silicone membrane, non-muscular fibroblasts can cause wrinkles in the substrate (Harris et al., 1981). However, due to the inherent nonlinearity of wrinkling and the complexity of the displacement field generated by a single cell, this technique was not applicable to accurately quantify cell forces. The development of traction force microscopy by Dembo and Wang has been a significant improvement to measure cellular forces. This method uses fluorescent microbeads embedded in a polyacrylamide hydrogel as markers for tracking the deformation of the gel caused by the adherent cell. After obtaining the displacement vector for every bead, the inverse problem is solved to calculate the cell generated force field . With this technique, we can see that in addition to tangential forces, cells also exert forces normal to the 2D planar adherent surface. However, the calculation of forces is computationally intensive, because deformations propagate on these continuous substrates. Furthermore, changing the crosslinking chemistry to manipulate the rigidity of hydrogels may inadvertently affect surface hydration, chemistry, and adhesiveness. To address these limitations in the design of soft substrates, microfabricated arrays of elastomeric cantilever posts have been developed to quantify cell forces by measuring the deflection of the posts under cell tension. The above two force measurement techniques have demonstrated that (1) cells pull harder on stiffer surfaces, affecting cell shape, motility, growth rate, and intracellular signaling and that (2) adherent cells continuously apply tensile forces to substrates that are directed toward the centroid of the cell. Similar observations have recently been reported in cells embedded in a 3D hydrogel, suggesting that mechanotransduction mechanisms might be conserved between 2D and 3D. These methods whereby cells cause strains on their substrate, are critical in identifying a role for cellular forces. For example, it had been observed that cells restricted from spreading against extracellular matrix become growth arrested, regardless of whether that restriction is due to cell crowding upon reaching confluence, decreased ECM ligand coating density on a surface, or micro-patterning to define the area of spreading of a cell. These changes in cell spreading were later found to impact cell contractility. Restricting cell spreading on micro-post arrays revealed that decreased spreading prevented cells from generating traction forces (Tan et al., 2003).

55


Conversely, up-regulating contractility by activating RhoA rescued proliferation even in un-spread cells, thus demonstrating that the mechanism by which cell shape regulates contractility is mechanotransduction. 5.4 Manipulating Cell-ECM Forces Intracellular forces can be modulated by using traditional molecular methods to directly target the force-generating apparatus. Several pharmacologic agents are available for inhibiting modulators of contractility, including the molecular motor myosin II (blebbistatin), the upstream regulators of myosin phosphorylation Myosin Light Chain Kinase (ML-6, ML-9), and the Rho/ROCK signaling pathway (fasudil, Y27639, C3 botulinum exotoxin), as well as the polymerization processes of actin (latrunculin, cytochalasin D). Similarly, molecular-genetic methods have also been used effectively to target these pathways, including the nonmuscle myosin II isoforms themselves, and have been a mainstay in identifying force in a regulatory pathway. In addition to molecular methods, there are biophysical approaches to manipulate cell-ECM mechanics. One such method is to vary the mechanical stiffness of the substrate to which cells attach. Most commonly, substrate stiffness is manipulated by changing the degree of crosslinking of polymeric hydrogels, the most widely available of which is polyacrylamide. Depending on the chemistry, crosslinking is controlled by the ratio of polymer to crosslinking agents, the duration of exposure to a light source (photopolymerization) or heat, among other factors. Interestingly, whereas the stiffness of most substrates is determined upfront, substrate stiffness of collagen can be altered during the course of the experiment by supplementing ribose to the medium (Girton et al., 1999). The role of ECM stiffness in regulating cell function is now well known. ECM stiffness has been shown to affect migration, proliferation, and differentiation. Whereas increasing stiffness enhances spreading and proliferation of many cell types and facilitates tumor growth, compliant substrates appear to promote branching of neurons or adhesion of and albumin secretion by hepatocytes. Measuring mechanical properties of a cell or any material requires applying some kind of force to it and recording the deformation response. Force can be applied in many modes: tension or compression, uniaxial or biaxial, pure shear, hydrostatic or pneumatic pressure, bending, twisting, and a combination of these. For any of these modes, varying the timing of also helps characterize the material: for example force can be applied suddenly, slowly or rhythmically, with sudden impulse or steps of force, ramps, or sinusoids. The common waveforms used to test mechanical behaviors, both linear and non-linear, are depicted in Figure 5.2 below.

Devices that apply these forces can be conveniently classified into three types (Figure 5.3): Type A, whereby local probes load or stretch a portion of a single cell; type B, mechanical loading of an entire cell; and type C, simultaneous loading of a population of cells.

Tension orCompression

Uniaxial

Biaxial

Shear

Pressure

BendingTwisting

Figure 5.2. Types of force application

56


Several Type A methods are available, as depicted in Figure 5.4 below:

In addition to the modes of cell contact, variations in the type and timing of stimuli are possible, as depicted in Figure 5.5 below:

P

F1

F1 F2

F2 Type A

Type B Type C

Bao & Suresh

Figure 5.3. Classes of Mechanical Testing

Figure 5.4. Type A Cell testing modes

57


Some of the measurement techniques are further described below. 5.5 Atomic Force Microscopy (AFM) In AFM, a sharp tip at the free end of a flexible cantilever (Figure 5.6) generates a local deformation on the cell surface. The resulting deflection of the cantilever tip can be calibrated to estimate the applied force. This is a high-resolution technique that traces the fine structure of nanometer level objects. The tip can be conjugated with an antibody that binds to the cell (‘functionalized’) and then it can be used to pull on it with calibrated force.

(Alenghat, Fabry, Tsai, Goldmann, Ingber) The figure at left (above) is a computer model of the mechanical deformation of a cell by the tip of an AFM cantilever. The model is based on continuum mechanics, and it includes contributions from the elasticity of the cell membrane as well as the interior of the cell. An example of the type of image that AFM can produce is shown in Figure 5.7 below:

Cell surface

Bond to CSK

Pull force

Figure 5.6. Atomic Force Microscopy

Figure 5.7. AFM image of a living cell. Bar= 5 microns

Impulse Step Sinusoid Ramp

TIME

Magnitude

Figure 5.5. Modes (top) and timing protocols (bottom) of force application

58


Single cells can exert large contractile forces that affect their shape. These forces exerted by cells can be measured with AFM, as shown in Figure 5.8. 5.6 Magnetic Tweezers Magnetic beads, a few microns diameter, can be either incorporated into cells, or attached to specific receptors. An illustration of pulling on a cell with a magnet is shown in Figure 5.9.

In this case, the bead has been incorporated into the cell, and is pulled with an external magnetic field. Calibrated stretch of the membrane can be thereby done. A more specific force can be applied to individual molecules using functionalized magnetic beads. These have been coated with antibodies for specific CSK components such as Integrin. For example, when coated with fibronectin, the beads will bind to Integrin. When the beads bind, an external magnetic field can be applied to twist the bead, measuring the response of the CSK portion directly, as shown in Figure 5.10. By using a magnet to pull and twist individual molecules on a cell, this technique represents ‘magnetic tweezers’. These magnetic-twisting studies confirmed that mechanical forces are

B Field Magnetic bead

Membrane

Figure 5.9. Magnetic bead stretching (Wang et al., Science)

Figure 5.8. An osteosarcoma cell attached between the cantilever of an atomic force microscope and a surface can exert contractile forces (red arrow) of more than 100 nN (depicted from left panel to center panel and in a fluorescence micrograph, right panel). Actin filament structures (white), including contractile stress fibres spanning the upper and lower surfaces, are generated in the contracting osteosarcoma cell (right). Scale bar, 10 μm.

Figure 5.10. Pulling on CSK (Wang et al., Science)

59


transmitted only over specific molecular paths, in particular, the Integrin hooks. When the beads were bound to non- CSK receptors, they could not effectively convey force to the inside of the cell. Highly specific molecular "adhesives" were attached to the cells in order to show that tugging on particular receptors at the surface of a living cell triggers nearly instantaneous rearrangements in the nucleus. 5.7 Micropipet Aspiration This technique involves sucking up a small portion (patch) of a cell into a micropipet, while measuring the deformation and pressure force. The geometry and kinematics of the situation shown below have been worked out. Note that the cartoon of Figure 5.11 shows a highly simplified view of what happens to the CSK during aspiration. The pressure clearly deforms it, and by measuring the force required to pull it up the tongue of cell a given length, a measure of stiffness can be obtained. It is also apparent that the deformation is complex, being neither uniform nor isotropic, and applies tension, compression, as well as shear simultaneously. Micropipet aspiration can be either type A or B (Figure 5.3), depending on conditions. If the micropipet is tightly sealed to the cell, there is no movement of the main body of cell into the pipet, and hence only the patch is deformed- this is type A. If there is no seal and no friction between the cell and the pipet, then the whole cell gets squeezed into the pipet, and it is a type B stimulus.

Aspiration

Magnified view of cell patch showing CSK

Drawing of Micropipet Aspiration

Figure 5.11

Negative Pressure

60


Another use of micropipets is to gauge the force produced by a growing cell. At right, you see a photograph of a pipet that has attached to the growth cone, and is being pushed forward by it. 5.8 Cell Moduli from Pipet Aspiration (Lee et al.) The micropipette aspiration technique can measure stiffness of an individual cell with a known aspiration pressure (see Fig. 5.10). The technique has been widely used to determine the mechanical properties of a variety of cell types including stem cells [Hochmuth, 2000]. Micropipettes, with inner diameters typically between 5 and 15 mm, are used to induce deformation of the whole cell, or alternatively discrete regions of the cell, depending on the cell diameter. Moreover, the properties of subcellular components, most notably the nucleus and cytoskeleton may be investigated. With the control of a hydraulic micromanipulator, the micropipette is moved into contact with the cell surface and pressures applied and the cell visualized using bright field or fluorescence microscopy. Two approaches have been used for performing the aspiration technique. For the incremental approach, pressure is applied typically in steps up to 5 cmH20 (0.49 kPa), with the equilibrium cell aspiration length, L, recorded at each pressure. The pipette internal diameter is given by ‘a’ . The apparent Young’s modulus may then be determined using a theoretical model [Theret et al., 1988]. In this model, the cell is assumed to be a homogeneous, elastic half-space material and the Young’s modulus, E, is therefore given as follows, where Φ(η) is defined as the wall function with a typical value of 2.1: Here, Young’s modulus can thus be determined from the slope of the linear regression of the normalized length, L/a, versus the negative suction pressure P. For the alternative approach, in which pressure is applied in a single step, the following equation is fitted to the temporal changes in aspiration length measured experimentally (Eq. 2). This model assumes that the cell behaves as a homogenous linear viscoelastic three-parameter solid half-space.

The relaxation time constant ‘t’ is defined as follows (Eq. 3).

(1)

Figure 5.13. Pipet attached to a growth cone

(2)

(3)

61


The parameter k1, is termed the equilibrium or relaxation modulus (Er or E∞), k1+k2 is the instantaneous modulus (Ei) and µ is the apparent viscosity. Using this viscoelastic model it is

also possible to determine the apparent equilibrium Young’s model given by: It should be recognized that although these models benefit from being simple, they neglect geometrical factors, such as finite cell dimensions, evolution of cell-micropipette contact region and curvature of the micropipette edges. Thus, other models incorporating these geometric factors into a computational form have been developed [Haider and Guilak, 2000], which can also account for the heterogeneity in cellular properties. 5.9 Nuclear Probing Micropipets can even be used to probe into sub-cellular structures, such as the nucleus as shown below. This sequence of pictures shows the pipet drawing a single chromosome into the pipet and gradually pulling it out of the nucleus. This remarkable experiment revealed the surprising, and still unexplained, phenomenon that all the chromosomes and the nucleoli are connected together by strings of DNA. Micropipets have also been used to measure the strength of single chemical bonds that a cell makes with its matrix. The pipet grabs onto the cell like a leech, and then can pull with a calibrated force until the bond breaks (Evans). With a manually 0perated micromanipulation device, the micropipette is moved "like a golf club" to move the beads about 10 micrometers a second, while monitoring the cell with the video microscope. 5.10 Whole Cell Inflation Micropipet aspiration can also be used to test the entire cell, as depicted in Figure 5.14. A direct porthole connection to the entire cell made when sufficient suction is applied to break the membrane and CSK break open. Thus the cell can be inflated with known forces, while its diameter is measured and its surface area estimated. The whole cell swelling technique is powerful because the micropipet can control the internal millieu, including both the cytosol composition and the electrical potential, as depicted in Figure 5.15 below. The total osmolarity of the cell and its external environment is important since slight imbalance can cause large osmotic forces for swelling or shrinking. In fact, using van’Hoff’s formula for osmotic pressure (See Appendix) indicates that for every 1 mOsm difference in salt

Figure 5.14. Pulling out the chromosomes

(4)

62


concentration across the cell membrane, there is a transmembrane pressure gradient of 16.7 mm Hg. Note that the internal and external osmolarities are labelled, ΣC, flows of water are labelled Q, and pressure P. Cell stretch can be applied by applying pressure from the pipet (panel A). Cells can also be stretched by exposing them to hypotonicity, as shown in panel B. 5.11 Optical Tweezers Another type A or B method involves optical tweezers or a laser trap, in which an attractive force is created between a dielectric bead of high refractive index and a laser beam, pulling the bead towards the focal point of the trap. To deform a single cell, 2 microbeads (typically 1 m to several micrometers in diameter) are attached to opposite sides of it and one or both beads are trapped by the lasers, and then pulled apart (Figure 5.16). This technique relies on a high power laser trapping the bead, which involves substantial heating of it. In order not to heat the cell also, the laser beam must be much smaller than the size of the bead. By ‘functionalizing’ the bead, it can bind to specific molecules, allowing the probe to apply force to specific bonds.

Tension

Micropipette

QpQp

Qm

Mesangial Cell

ΣC i

ΣCo

Pp

(K )w

Pi

C =constant Σ i

A. Whole Patched Cell

Micropipette

Stretch

Mesangial CellΣC (t)i

ΣCo

Pi

Qm w(K )

Stretch Solutes

B. Isolated Cell

Solutes

Trap Bead

Actuator Bead

Force

Trap Bead

RNA

Handle

Magnified View

Actuator Bead

Figure 5.16. Optical Tweezers

Figure 5.15

63


Optical tweezers offers several non-contact ways of driving cells. While early optical tweezers simply trapped and moved particles, it is possible to spin particles or cells at high speeds and also to orient them in separate traps and then join them together in lock-and-key assemblies. In a “levitation trap” the laser beam balances the pull of gravity. To “optically” trap a particle in three dimensions it is necessary to exert a “longitudinal” force in the same direction as the laser beam and a “transverse” force at right angles to the beam. The transverse force is created by having the maximum laser intensity at the center of the beam. If the particle is to the left, say, of the center of the beam, it will refract more light from the right to the left, rather than vice versa. The net effect is to transfer momentum to the beam in this direction, so, by Newton’s third law, the particle will experience an equal and opposite force – back towards the center of the beam. In this example the particle is a dielectric sphere. Similarly, if the beam is tightly focused it is possible for the particle to experience a force that pushes back towards the laser beam. We can also consider an energetic argument: when a polarizable particle is placed in an electric field, the net field is reduced. The energy of the system will be a minimum when the particle moves to wherever the field is highest, which is at the focus. Therefore, potential wells are created by local maxima in the fields. Depiction of the apparatus for examining red blood cells being stretched by optical tweezers is shown in Figure 5.17 below:

Figure 5.17. Apparatus for stretching RBCs with optical tweezers

64


5.12 Hydrostatic Loading Groups of cells can be pressurized with devices shown in Figure 5.18. In (a) a piston (platen) pressurizes the fluid above the monolayer of cells. When cells are grown in a 3-dimensional matrix, pressure can be applied directly to them as in (b). Cells can be either confined or without side support. This type of testing is valuable to understand behavior of bone and cartilage cells, as well as cells that may be subjected to high pressures under abnormal conditions.

5.13 Shear Flow A variety of configurations can deliver precise amounts of shear force to groups of cells attached to surfaces. A cone-and-plate viscometer is depicted in Figure 5.19 (a), consisting of a stationary flat plate and a rotating inverted cone with which laminar and turbulent flows can be applied. A parallel-plate flow chamber (b) subjects cells to laminar flow. In both cases the shear stress applied to cells can be readily quantified. Many cellular responses can be measured as a function of shear during and after the force application, including attachment strength, growth, and biochemical responses. These tests are most appropriate for cells for which shear is a way of life, such as blood cells and endothelial cells.

5.14 Cell Stretching When attached to a flexible substrate, cell populations can be stretched by pulling on the substrate. A commercial device, Flexercell, has been extensively used for this purpose. A variety of configurations and actuator types are available as shown below. The procedure involves culturing cells on a thin-sheet polymer substrate, such as silicone, which is coated with ECM molecules for cell adhesion. The substrate is then mechanically deformed while maintaining the cell's viability in vitro. In this manner, the effects of mechanical loading on cell morphology, phenotype and injury can be examined. Furthermore, by systematically altering the mechanical properties of the substrate material through, for example, changing the degree of crosslink in the polymeric gel, the individual and collective interactions of the cells with the substrate can be studied. Such studies have been performed to investigate the propensity for migration of a group of cells towards or away from the region of localized tension or compression in the substrate. Various modes of cell stretching are depicted in Figure 5.20 below.

Figure 5.18. Hydrostatic loading

Figure 5.19. Shear delivery

65


5.15 Microelectromechanical systems (MEMS) (Rajagopalan et Al.) MEMS sensors, because of their small size and fine force/displacement resolution, are ideal for force and displacement sensing at the single-cell level. In addition, the amenability of MEMS sensors to batch fabrication methods allows the study of large cell populations simultaneously, leading to robust statistical studies. Micromechanical sensors are especially suited for these studies because of their small size, which allows for easy interfacing with individual cells, and fine force/displacement resolution that makes them capable of measuring very small forces/displacements. In addition, micromechanical platforms can be batch fabricated cheaply using either integrated circuit (IC) or soft lithography techniques. Studies on mechanobiology range from the tissue level all the way down to individual proteins and DNA, involving a wide range of approaches and instrumentation size scales. The commonly used tools for probing cells and biomolecules, such as AFM, optical and magnetic tweezers etc, have already been the subject of many excellent reviews. Therefore we will survey micromechanical systems developed for cell

mechanics research and the biological insights that have resulted from these studies. Distinct from other reviews on microengineered systems, we highlight two recent developments in this area: (1) micromechanical devices for in vivo and small animal studies and (2) microsystems to manipulate the physiological behavior of cellular organisms through controlled application of forces.

Figure 5.21. MEMS (Bao & Suresh)

Figure 5.20. Cell stretching

66


Substrate topography can be micropatterned so that cells are induced to grow onto tiny elastic platforms that can be moved to apply or measure force. This is a powerful technique, since the platforms can be precisely controlled, and can be functionalized with specific cellular matrix targets. Thus the micropattern can test and perturb cell-matrix interactions in a highly controlled environment. For example, the strength and behavior of specific focal adhesions can be studied. The contractile forces generated by cells during locomotion and mitosis can also been measured with a deformable-substrate method. Micromechanical systems used for cell mechanics studies broadly fall into two major categories. The first category comprises hard silicon-based devices that are fabricated using standard IC manufacturing techniques, whereas the second category comprises systems made of soft polymers and gels. Examples of MEMS cell testing systems are shown above. Two configurations are a cluster of microneedles (Figure 5.21.a and b) or a cantilever beam (Figure 5.21.c). The device in Figure 5.21.d d can apply mechanical force or deformation at several points on a cell in the center. The effect of various MEMS on cellular shape and behavior are shown in Figure 5.22 below.

Figure 4.23 below shows 3D structures with sensors , for growing cells and sensing their forces.

Figure 5.22. Scanning electron micrographs of human mesenchymal stem cells plated on PDMS micropost arrays. The diameters of the posts were the same (1.83 μm) but the lengths (L) were different, as indicated in the figure. Note that the deflection is substantially larger for the 12.9 μm micron length posts (c), which were almost 1000 times softer than the 0.97 μm posts.

67


5.16 Video tracking Cell Movements

Myocytes isolated directly from hearts are log shaped, as shown in Figure 5.24. Their speed and magnitude of contraction can be monitored with a video camera and imaging software that tracks motion of the edges. Records from such an imaging set-up are shown at right.

5.16 Particle Tracking Video tracking of particles injected into cells can be used to estimate regional stiffness within cells. Figure 5.25 shows tracking traces of a particle in a gel. The first pattern (a) shows a relatively wide travel, while the second (b), shows a very tight range of travel. Note that the bar

Figure 5.24. Myocyte contraction

Figure 5.23. A SU-8 force sensing pillar array for cell measurements. (a) A lateral force applied at the tip of the pillar bends the four cantilever beams on which the pillar is suspended. The bending strain is transduced at the base of the cantilever using metal strain gauges. (b) Finite element analysis showing that bending induces alternating regions of compressive and tensile stress in the cantilever beams. (c) A single device viewed from the top. (d) An array of finished devices

68


denotes a distance of 2 micrometers. The difference between a and b is that the gel was polymerized for the b movement, making it more stiff. The mean squared distance traveled for a, is shown in the top curves in c, and for b, shown in the bottom curves. This is an important method, and is discussed more in Chapter 4, section 4.10.

__________________________________________________________________

Chapter 5

Review Questions and Exercises

1. In testing micropipet aspiration of cell patches, what modes of force application are applied? Use a sketch, showing at least 3 modes.

2. Calculate the cell modulus from a pipet aspiration measurement with pipet diameter 1 micrometer, and data attached in Sakai. Show work.

3. Calculate the time constant of mechanical response of a cell to a suddenly applied stress. 4. In a whole cell inflation test, how would the CSK respond to pressurization, based on what

you know of its properties? 5. In a hydrostatic loading test, describe the effects of adding side constraints to the cells.

6. Model 5: Above

Figure 5.25. Particle tracking

69


References Bao, G. and S. Suresh (2003). "Cell and molecular mechanics of biological materials." Nat Mater 2(11): 715-725. Living cells can sense mechanical forces and convert them into biological responses.

Similarly, biological and biochemical signals are known to influence the abilities of cells to sense, generate and bear mechanical forces. Studies into the mechanics of single cells, subcellular components and biological molecules have rapidly evolved during the past decade with significant implications for biotechnology and human health. This progress has been facilitated by new capabilities for measuring forces and displacements with piconewton and nanometre resolutions, respectively, and by improvements in bio-imaging. Details of mechanical, chemical and biological interactions in cells remain elusive. However, the mechanical deformation of proteins and nucleic acids may provide key insights for understanding the changes in cellular structure, response and function under force, and offer new opportunities for the diagnosis and treatment of disease. This review discusses some basic features of the deformation of single cells and biomolecules, and examines opportunities for further research.

Eyckmans, J., T. Boudou, et al. (2011). "A Hitchhiker's Guide to Mechanobiology." Developmental Cell 21(1): 35-47. More than a century ago, it was proposed that mechanical forces could drive tissue

formation. However, only recently with the advent of enabling biophysical and molecular technologies are we beginning to understand how individual cells transduce mechanical force into biochemical signals. In turn, this knowledge of mechanotransduction at the cellular level is beginning to clarify the role of mechanics in patterning processes during embryonic development. In this perspective, we will discuss current mechanotransduction paradigms, along with the technologies that have shaped the field of mechanobiology.

70

Chapter 6: Thermodynamics of the Cytoskeleton

CHAPTER 6: THERMODYNAMICS OF THE CYTOSKELETON

6.1 The Boltzmann Distribution: Statistical Mechanics At the molecular level, mechanical behavior is statistical, and requires special parameters to explain it. The Boltzmann distribution provides a simple explanation for the behavior of molecules exposed to an energy gradient. The energy gradient can be forces applied by mechanical, electrical, or chemical means. It is based on the probabilistic nature of molecules, and has widespread applications. The basic formulation is:

kTF

ePP ∇

−=

2

1 (1)

where P's are the probabilities of states or classes 1 and 2, and DF is free energy. For example Pi can represent the concentration of particles in 2 separate compartments. Lets say the energy in the first compartment is F1 and in the second, it is F2, and DF= F2-F1. Since the particles must either be in compartment 1 or 2, then P1 + P2 =1. So

To illustrate this, take a system whose neutral state (P1=P2= 0.5) is an energy level of 5kT. In other words, the neutral state is when the system can be in either state with equal probability.

One such system would be the particle distribution on earth surface or in the air. The dotted line (P2) would represent the likelihood of finding the particle on the surface (for a mass of 0, P2 = 0), and the dark (red) line would represent finding it above the surface. As expected, as mass of the particle increased, you would find more of it landing on the ground, and less of it in the air. At some mass, represented by 5kT, there is an equal probability of finding the particle on the ground or in the air. Many other compartmental situations can be similarly represented.

P1 x( )1

1 ex 5−+

:= P2 x( )1

1 e x 5−( )−+

:=

0 5 100

0.5

1

P1 x( )

P2 x( )

x

Figure 6.1

71


A mechanical behavior that is well described statistically by the Boltzmann relation is the tendency of polymers to shift from one state to another, depending on energy levels. Examples of these shifts are the opening and closing of ion channels and the folding and unfolding of nucleic acids. The latter behavior can be seen in RNA strands that were pulled by optical tweezers, as shown in Figure 6.2:

The above set-up shows a strand of RNA held at either end by a binding site on the beads. Note that the bead size is much larger than the RNA. When the strand is stretched, it behaves as an ordinary spring at very low and very high levels of tension, but at intermediate ranges, it exhibited unfolding and re-folding behavior, as depicted in Figure 6.3: As the force is slowly raised above 14 pN, the strand will suddenly unfold and then refold again, with the unfolded state becoming more frequent as force increases, until it remains permanently unfolded at a force level above 15 pN. In other words, the probability of unfolding is related to force, as described by the Boltzmann equation. This behavior is quantified in Figure 6.4.

Sudden extensions of 22 nM (unfolding) when forces above 14 pN are applied

Trap Bead

Actuator

Force

Trap Bead

RNA

Handle

22 nM

Magnified View

Actuator Bead

Figure 6.3 Length-force relationship of a strand of RNA.

Figure 6.2 Stretching of RNA molecule

72


.

This behavior appears to represent a balance between the applied force and the attractive forces within the RNA fold, including base-pairing, hydration, charge shielding, and Van der Wall’s. Thus the fraction of time spent in a state is equivalent to the probability of finding the strand in that state, and is related to force applied, according to the Boltzmann relationship, where Fo is the ‘ground’ energy state, f is applied force, and Δz is displacement:

Within any given time period, T, the probability of being ‘open’ or unfolded can be calculated as the fraction of open time:

kTzfFopen o

eP D−

+=

1

1

Tt

P openopen =

Unfolding upward Folding downward

Figure 6.4.b. Fraction of hairpins folded versus force

Figure 6.4.a. Opening behavior of RNA at selected tensions

73


6.2 Diffusion The diffusion coefficient, D, indicates how fast particles of radius a can move in a solution of

viscosity,η . Units of D are cm2

sec. There are 2 formulae to describe diffusion (in one dimension).

First the flux of particles per unit time, j, (i.e. number of particles p.u. time) passing a given point is directly proportional to the concentration gradient, C:

To describe how concentration changes with time, Fick’s second law is used:

2

22

2

dxd

where

CDdtdC

=D

D−=

A special case of Fick’s 2nd law can be set-up to arrive at a simple formula for estimating diffusion time. This is the case in which an amount of substance is placed in solution at the point x=0, and its concentration spread with time becomes uniform over x. With these initial and boundary conditions, Fick’s 2nd law is solved analytically as:

)4

exp(4

1 2

Dtx

DtC −=

p

This formula is analogous to the Gaussian probability formula:

dxxx

xdxxP )

2exp(

2

1)(2

2

2

−=

Since concentration of particles is directly related to the probability of finding them in a given distance dx, C ≈ P(x)dx. So equating the 2 formulae yields a simple estimate of the time of diffusion for a particular solute, given its coefficient, D, as shown below:

The right hand side of the above should read "X bar squared", and represents the spread of concentration in the x direction. Note that it is analogous to the statistical quantity of variance.

dxdCDj −= Fick’s 1st

Law

Fick’s 2nd Law

Standard estimate of diffusion time. Please note that the bar over the x is not a negative sign.

74


pH pKalog A−( )

HA( )+

δ- = fractional - charge = A−( )A− AH+( )

1

11

alog pH pKa−( )+

δ- =

6.3 Bioelectricity 6.3.a Internal charge and electrochemical equilibrium Cells have trapped negative charges in the form of proteins, amino acids, and nucleic acids. These are charged mostly because of their side groups, in particular carboxyl, that is loses a proton at neutral pH. For example, alanine is shown below:

Note that the net charge of theses amphiphilic molecules can be calculated by summing the charge of each group, using the Henderson-Hasselback equation: AH A H

KA H

AH

KAAH

H

AAH

pK pH

equ

equ

a

⇔ +

=

=

+

= −

− +

− +

−+

−

[ ][ ][ ]

log( ) log[ ][ ]

log[ ]

log[ ][ ]

The above equation gives the proportion of negatively charged molecules within the group, for a given pH. Each side chain has a particular acid-base relationship, with a specific pKa. A corresponding relationship can be derived for the positively charged (i.e. amino) group. Cells thus have a trapped negative charge at neutral pH, in the form of big, impermeable solutes, is the root cause of the electrical potential of all cells. These impermeable solutes are amino acids, nucleic acids, and other compounds with a tendency to become negatively charged. To balance the charge, cations that are mobile and permeant through the

+H3N-CH-COO-

CH3

pKa= 2.7 PKa = 9

75


membrane, enter the cell. The tendency to neutralize charge then competes with the tendency of chemical concentration gradients to nullify, and an electro-chemical equilibrium results, that can be described by 2 simple formulae, the Nernst and Donnan:

inout

in

oution

ionionionionand

ionion

zFRTE

]][[]][[

)][][ln(

−+=−+

=

The Nernst potential is the electrical potential that exactly balances the chemical gradient of mobile ions that balances the trapped negative charge. The Donnan relation is a direct result of the Nernst. You can see this by noting, if the mobile counterions are K+ and Cl- , the Nernst potential for each ion must be the same, since at equilibrium there can be only one potential for the cell. Thus the ions that are freely permeable are those that determine the cell potential. For most cells these are K+ and Cl-. Note that the Donnan equilibrium applies only to K and Cl, and not to other ion pairs, that are not freely permeable. Deriving the Nernst equation comes directly from the Boltzmann relationship, as illustrated below, using the distribution of particles in the atmosphere as an example:

Correction to above formula: note Faraday’s constant.

)ln( 0

iCC

zFRTV =

Deriving the Basic Bioelectric Equation

First there is the distribution of particles subject to gravity:

N h( ) N o ew

hkT.

.

This applies equally well to particles subject to an "E field",since:

q V. w h. Joules( )

and then:

and C C o ez

VRT.

.N V( ) N o e

qVkT.

.

where C = N particles/vol * mass/particle

So for molarquantities: V RT

zln

C oC

.

76


6.3.b Ion channels These are proteins within the cell membrane that open and close to let specific ions in or

out. The illustration below shows that ions will move through the membrane according to their electrochemical (Nernst) gradient.

Figure 4.5

77


Methods to record the flux of ions are shown above. The flux of ions in an electrical gradient, V, is given by Ohm’s Law coupled with the statistical Boltzmann relationship:

FIJ

MassfluxVpNgI open

=

D=

:

6.4 Energy Storage The vast majority of cellular energy is stored in the high energy phosphate bond of the ATP molecule. One ATP molecule is equivalent to 80 pN-nM or 8 N-m/g. Note that the units of torque and energy are equivalent. 6.5 State Transitions The RNA unfolding example represents the general process of state transition. The general way to represent it is: With the 2 compartments or states shown, transition rate K21, represents the flow rate of material from compartment 1 into compartment 2; K12 represents the opposite motion. This is a 2-state, closed system, since no compartment excretes material to the outside or any other compartment. A closed system is the simplest to deal with, since the mathematics is easy (as will be seen in the more advanced treatment at the end). Note that for the RNA example, states 1 and 2 could represent the folded and unfolded conditions, and K21 would represent the rate constant for unfolding, and K12 the rate constant for folding. We can represent the kinetics of the system as follows:

2121212

2121211

XKXKdt

dXand

XKXKdt

dX

−=

+−=

Or better in matrix notation as the state variable representation:

=

•

•

2

1

2

12221

1211

XX

XX

aaaa

(2)

1 2

K21

K12

(1)

78


Or more succinctly as:

XAX =•

(3)

where it is assumed that X and A are matrices. Note that we are analyzing a system with only 2 compartments, but the above state equation (3) applies to systems with any number of compartments. So if there were p components, equation (3) would represent:

ppiii XaXaXaX 12211 ...+++=•

(4) Now back to our 2 state, closed system:

−

−=

1221

1221

kkkk

A

Once you check that this is true, you can ask, how can I analyze the system? It should be apparent that you can solve each differential equation readily using a solver such as Simulink. You can also solve the entire system by coupling the equations. You first must understand what X represents. In our case, it represents probability. One way to look at it is to recognize that the entire two-compartment system can represent many RNA chains, that can either be in state 1 or state 2. The fraction, or state probability, P1, of being folded and the probability of being unfolded, P2, are:

openfoldedopenP

andopenfolded

foldedP

###

###

2

1

+=

+=

and thus P1 + P2 = 1, and X = P.

79


6.6 Polymerization To understand how the CSK and its functional behavior emerges from discrete components, lets look at the kinetics of polymerization. In Figure 6.7, actin filaments were nucleated from beads coated with the nucleation-promoting factor ActA (not shown) and were then crosslinked by fascin inside a unilamellar lipid vesicle

Polymers are assembled from monomers (captured), and disassembled (released) back into monomers, generally as a first order system as depicted below: (capture) dn/dt = +kon [M] (1) (release) dn/dt = -koff (2)

Kopen = 0.9 sec-1

K fold – 8.5 sec-1

Kopen = 7 sec-1

Kfold = 1.5 sec-1

Figure 6.7. Purified proteins were loaded into a vesicle by a microfluidic encapsulation technique that allows the dynamics of filament assembly to be observed immediately after encapsulation. The micrograph (left) shows labeled actin filaments (white) that have polymerized inside the vesicle and have assembled into a fascin-crosslinked network. The diagram (right) is a schematic depiction of the actin filament network present in the inset box of the micrograph

Figure 6.6

80


Definitions: n = number of monomers in a single filament; t = time; [M] = concentration of free monomer in solution kon = capture rate constant, with units of [concentration•time]-1 koff = release rate of monomers; Thus the capture rate of monomers by a single filament is proportional to the number of monomers available for capture. The release rate does not depend on [M]. Note that the front and back end of polymers differ. (Figure 6.8)

__________________________________________________________________

Chapter 6 Review Questions and Exercises

First Week 1. Using the data shown in Figure 6.6 above, and the ground free energy, Fo = 79 kT, graph

the unfolding probability, using Excel or other program. Put data points from empirical experiments for the selected forces on your theoretical curve. (Note data represent numbers obtained from experimental measurements (not calculations); a theoretical curve is generated by formula).

2. Make a Simulink model of the RNA unfolding kinetics. Your model should be

well documented, according to the following guidelines:

• All parameter boxes should be labeled

• Document boxes should be included to describe operations

• Internal parameters, such as initial conditions, should be specified

• Sub-systems should be used so that the entire model can be fit onto 1 page

and each sub-system can be printed separately, with documentation.

• A separate description of the system and all formulae should be made.

• Outputs should be the predicted, as well as measured probabilities

• A reasonable noise level should be placed in the model

Figure 6.8

81


3. A proteoglycan molecule forms a gel. The pKa of its amino group = 10 , pKa of carboxyl = 3 and pKa of the ‘R’ group is 7. What is its net charge at pH = 7 ?

Second Week

1. What is the origin of biopotentials in cells, and how are they calculated? Be specific.

2. Endothelial cells have the usual cell membrane that restrains movement of proteins,

but permits free flow of water and monovalent electrolytes. A Donnan equilibrium of diffusible ions results.

a. Based on this information, complete the table below:

b. What is the transmembrane biopotential? c. What is the concentration of proteins in the cell? ( assume proteins are the only

solute other than electrolyte, and they have a charge of -1). d. At what potential (if any) are all ions at equilibrium?

3. Model 6: Make a Simulink model of the RNA unfolding kinetics. Your model

should be well documented, according to the following guidelines:









References

1. Liphardt, j., Science, 2001. 292.

Ion Outside Concentration, mM Inside Concentration, mM

[Na+] 150 144 [K+] 4 [Cl-] 114 [HCO3-] 28

82


83

Chapter 7: Kinetic Behavior 1

CHAPTER 7: KINETIC BEHAVIOR

7.1 Background (Excerpts from Jamali, Azimi et al. 2010)

One of the important phenomena in cell engineering and developmental biology is the

shape of tissue and the cell’s organization. Depending on the cell type and

environmental conditions, cells can create unique shapes such as flat sheets, self-

enclosed monolayers, cysts, or elongated tubes. The more important question is how

these cells interact and how their local interaction causes a global geometrical

distinctive shape for tissues like the heart or kidney. The geometrical interactions and

coordinated adhesion among neighboring cells and between the cells and local

environment are critical for structure and function of epithelial tissue. Any perturbation

of these orchestrated interactions can cause abnormality in behavior and function of

tissue and often lead to initiation of tumor growth and invasion. Another interesting

subject is embryogenesis, when a stem cell with consecutive rapid divisions and

differentiation can create different tissues, wherein the interactions between cells and

environmental biochemical and biomechanical signals have critical, yet nearly unknown

roles. However, in the last two decades, improved experimental techniques and

developments have allowed for more detailed understanding of cell-cell communication

and the cell’s response to biochemical and biomechanical environmental signals, as

reviewed in Chapter 5. Nevertheless, biological experiments are expensive and depend

on many parameters that are mostly difficult to control and test in isolation.

7.2 Cell Modeling

Mathematical modeling and computational experiments can help explore the behavior

of the individual tissue cells along with investigating their response to environmental

cues. Due to easy isolation in in-silico computational models, incorporating the related

fundamental physical and biological parameters can explain how specific biochemical or

biomechanical parameters may affect the tissue cells and their arrangement. Such a

model can reduce the number of experiments required to obtain meaningful

observations by eliminating unlikely hypothesis while providing a better explanation of

observations. For example, to investigate how individual cells cooperate and contribute

to the overall structure and function of a particular tissue, a proper computational

84


model could define cells as individually deformable shapes, time- and space-dependent

individually regulated cell turnover, and cell-cell and cell-ECM interaction. Many

models have been developed to mimic cell behavior, such as response to external

mechanical and biochemical signals, cell-cell interaction, cell motility, and cell

morphology.

Some models have attempted to mimic collective cell behaviors such as cancer invasion

through the use of continuum and/or discrete approaches where each cell is represented

by a finite element and follows a cellular automata method. Cells can be modeled as

colloidal objects capable of interacting with their environment. In such models, cells are

capable of migrating, growing, dividing, and changing their orientation. In a model

proposed by Galle, Loeffler et al. (2005), the cells move according to the Langevin

dynamics framework and can interact based on a combination of attractive and

repulsive forces. The aim of these models is to replicate the multi-cellular growth

phenomena. By focusing on monolayer culture, Galle et al. have investigated the effect

of key factors on rate and quality of culture growth. They also analyzed the underlying

processes involved in multi-cellular spheroids, intestinal crypts, and other aspects of

developmental biology.

Models can mimic various aspects of cell population, but fail to examine the effects of

cell deformation and morphology on pattern formation and growth processes. Some

models are based on the viscoelasticity of cells, where each cell includes certain elastic

and viscous elements. Such models lend themselves to easy incorporation of the cell-cell

adhesion and repulsion, and various forces acting on individual cells in the cluster. For

example, a 3D deformable cell model with cell adhesion and signaling was developed by

Palsson and colleagues, where each cell is assumed to be ellipsoid shape, with its axis

composed of a combination of springs and viscous elements . This model was used to

investigate the role of cell signaling, cell adhesion, chemotaxis, and coordinated

differentiation in the morphology of a developed organism. Another biomechanical

approach developed by Rejniak and colleagues represents cells as deformable

viscoelastic objects that can be arranged into tissues of various topologies. This

approach joins elastic cell dynamics with a continuous representation of a viscous

85


incompressible cytoplasm. The model covers many aspects of cellular behavior such as

cell growth, division, apoptosis and polarization. With this model it is possible to

investigate the biomechanical properties of cells and cell-cell interaction, the effects of

the microenvironment on a cellular cluster, and how individual cells work together and

contribute to the structure and function of a particular tissue. An application of this

model is tumor growth .

More recently, Coskun et al. developed a mathematical model for amoeboid cell

movement in which a viscoelastic (spring–dashpot) system was used to represent the

cytoskeleton. This model was used to solve an inverse problem of amoeboid cell motility

and to find the variation of spring and dashpots parameters in time. The research shows

that the model and the solution to the inverse problem for simulated data sets are highly

accurate. In general, cell mechanics has been modeled based on non-living structures

using approaches ranging from the soft glassy material model , to the cortical shell–

liquid core model, and tensegrity architecture . Few of the theoretical models that have

been proposed for analyzing the mechanical properties of adherent living cells are

capable of simultaneously incorporating (i) the discrete nature of the cytoskeleton, (ii)

cell–cell and/or cell–extracellular matrix (ECM) interactions, and (iii) the cellular pre-

stress.

The complete cell behavioral model should integrate the tensegrity concept, where each

cell is capable of changing its morphology, and performing the key cellular processes of

growth, division, polarization and death. The modeled cells should interact with each

other and with their environment. Each cell in this model is an individual unit

containing several subcellular elements, such as the elastic plasma membrane,

encompassed by viscoelastic elements that perform the function of the cytoskeleton, and

the viscoelastic elements of the cell nucleus . Additionally, the cell membrane is divided

into segments where each segment (or point) incorporates the cell’s interaction and

communication with its environment, such as adherens junctions.

86


7.3 Modeling the Cytoskeleton.

The mechanical properties of the cytoskeleton, like elasticity and viscosity, are critical to

the validity of the model. Voigt subunits are effective for modeling a viscoelastic system;

the spring constants of the model are linear approximations to the elasticity of the inner

cell. Additionally, all springs are subjected to a damping force resulting from the

viscosity of the cytoplasm, where linear dashpots approximate the viscosity of the

cytoskeleton (Figure 7.1). The cytoskeleton can be divided into uniformly radial

distributed parts, each of which is represented by a Voigt subunit radiating from the

nucleus (Figure 7.1, blue subunits). Each subunit connects two points of the cell and

nuclear membrane, which are aligned in a radial direction from the center of the

nucleus. The nucleo-skeleton is represented as a viscoelastic model involving an

actomyosin system (Figure 7.1, red subunits). The model also contains N Voigt subunits

in the nucleus ( red subunits). The equation below is the sum of all possible forces on

the cell:

Figure 7.1 Cell Model

87


7.4 Viscoelastic (VE) Material

V-E analysis assumes that a complex material can be modeled as a purely viscous

material (a dashpot) combined with a purely elastic material (a spring). It thus

mathematically separates the viscosity of a material from its elasticity. A purely viscous

component is a Newtonian fluid- it has no memory and no elasticity; it cannot deform as

a solid. In cells and tissues, material properties vary from behaving as pure fluids, to

not very pure fluids, to solids, to solid-liquid composites. Despite this diversity, V-E

tools can quantify each of their properties, since the models separate viscosity from

elasticity in a kind of finite element model.

Elastic deformations experienced by structures can be characterized by their particular

stress/strain curves. Examples of such curves for three typical behaviors are shown in

Figure 7.2:

The above curves represent the stiffness of the structure, as measured by their slopes at

any point. The curve on the left represents rubber-type behaviour, since it is relatively

stiff for small strains, relatively compliant (un-stiff) for intermediate strains, and again

is relatively stiff for high strains. Thus there are 3 separate regions of rubber behaviour,

each of which has a molecular explanation. The middle curve is representative of

standard structural materials. The curve on the right is representative of a purely

Newtonian fluid.

An important factor that has been left out of the above analysis of elasticity is time. It

must be recognized that each curve represents a single point in time ( or as close to a

single point in time as possible). That point in time could represent either an

s

e

Figure 7.2 Stress/Strain Curves for Elastic Deformations

88


“equilibrium” or a non-equilibrium situation. The only way to learn the role of time-

dependent changes in stiffness, is to measure it at a sequence of times following each

stress or strain application. In doing this, you would be measuring viscoelastic

properties. In other words, viscoelasticity can be defined as: “stiffness as a function of

time”. Materials that have time dependent stress/strain behaviour are called

viscoelastic.

For example, a hydraulic shock absorber is very stiff, when measured very soon after a

displacement, but is not very stiff, later following the displacement. The key point about

a V-E material, therefore, is its time dependent behaviour, in terms of either stress or

strain.

An example of viscoelastic behavior is shown when cartilage is compressed, as in the

experiment by Grodzinski et al. (Figure 7.3):

Figure 7.3. Cartilage Compression

89


Two distinct phenomena can occur in a VE material: it can creep or it can relax. For

example, play dough, will creep (or flow) from a spherical shape, into a flat puddle,

over the course of several minutes. We could say that a constant force, or stress, of

gravity is acting upon the play dough, and the material creeps slowly. When you pull

suddenly on the material, however, it does not seem to be a liquid, and is rather stiff. So,

again you see that stiffness is ‘relative’ (not in the Einsteinian sense).

Now, getting to specific ways to characterize V-E materials, there are only a few practical

ways you can do so. Testing properties of anything always requires that you ‘perturb’ it

in some way, while measuring its response. There are 2 basic mechanical perturbations:

force (stress) or deformation (strain). There are generally 2 ways you can apply these

perturbations: suddenly, or slowly. The sudden method is referred to as a step input

(impulse is also possible but not practical for most material characterizations); the slow

method can take the form of a ramp, sinusoid, or other wave shape. The complete

perturbation matrix and expected material responses looks like this:

Input Shape Type Stress Strain

Step (sudden) Sigmoidal Creep: Voight

Model

Complete Relaxation:

Maxwell model

Ramp (slow) Ongoing Creep

Ongoing relaxation

The table entries state the type of response to be found for each combination of

perturbation type and shape.

A complex material can be modeled as a purely viscous material combined with a purely

elastic material, thus mathematically separating the viscosity of a material from its

elasticity. A purely viscous component is a Newtonian fluid- it has no memory and no

90


elasticity; it cannot deform as a solid. Cells generally behave as solid-liquid composites.

V-E tools can quantify their behavior, since the models separate viscosity from elasticity

in a kind of finite element model.

The Voigt model is easily quantified, as shown below:

Voigt Model

since s = Ε e + η de/dt

and I = (1/R) V + C dV/dt

then each of the top terms are equivalent to the bottom terms. Then the impedance analysis proceeds:

sCRI

____________________________________________________________

______

Chapter 7

Review Questions and Exercises

1. Calculate the time taken for a microtubule to grow from the centrosome of a

typical cell to the cell boundary. How long would it take to shrink to single

heterodimer length if it undergoes rapid depolymerization? Assume: (1) it grows

from the plus end, (2) [M] = 10 uM/L, (3) rate constants as given, (4) a tubulin

heterodimer is 8 nM long.

MW of monomer = 50,000 Daltons; heterodimer is 2X monomer.

91


e

s

e

s

2. Assume that and endothelial cell length is suddenly doubled. Cytoplasmic

viscosity is 500 dyne-sec-cm-2 and CSK elasticity is 1000 Pa. You may assume

the cell is cuboidal in shape. Describe the stresses within the cell that would

happen immediately after the event, and as time evolves thereafter. What kind of

model could describe the behavior? Draw the model conceptually. What is the

time constant of response? Draw a relevant part of the response, illustrating the

time constant, and be sure to label. What components of the cell does the model

represent? Hint: the initial stress is expressed entirely in the elastic component;

another way to look at it is through the Laplace solution and transform.

3. Recognize the following models and their inputs and outputs: Explain why and

how the outputs differ.

Tubulin

Monomer in

solution

K+ on

(uM-S)-

1

K+ off

S-1

K- on

(uM-S)-1

K- on

S-1

Growing 9.3 44 5.3 23

Rapid disassembly 0 737 0 915

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4. Model 7. Make a Simulink model of the RNA unfolding kinetics. Your model

should be well documented, according to the following guidelines:









References

Jamali, Y., M. Azimi, et al. (2010). "A Sub-Cellular Viscoelastic Model for Cell

Population Mechanics." Plos One 5(8).

Understanding the biomechanical properties and the effect of biomechanical

force on epithelial cells is key to understanding how epithelial cells form uniquely

shaped structures in two or three-dimensional space. Nevertheless, with the

limitations and challenges posed by biological experiments at this scale, it

becomes advantageous to use mathematical and 'in silico' (computational)

models as an alternate solution. This paper introduces a single-cell-based model

representing the cross section of a typical tissue. Each cell in this model is an

individual unit containing several sub-cellular elements, such as the elastic

plasma membrane, enclosed viscoelastic elements that play the role of

cytoskeleton, and the viscoelastic elements of the cell nucleus. The cell membrane

is divided into segments where each segment (or point) incorporates the cell's

93


interaction and communication with other cells and its environment. The model

is capable of simulating how cells cooperate and contribute to the overall

structure and function of a particular tissue; it mimics many aspects of cellular

behavior such as cell growth, division, apoptosis and polarization. The model

allows for investigation of the biomechanical properties of cells, cell-cell

interactions, effect of environment on cellular clusters, and how individual cells

work together and contribute to the structure and function of a particular tissue.

To evaluate the current approach in modeling different topologies of growing

tissues in distinct biochemical conditions of the surrounding media, we model

several key cellular phenomena, namely monolayer cell culture, effects of

adhesion intensity, growth of epithelial cell through interaction with extra-

cellular matrix (ECM), effects of a gap in the ECM, tensegrity and tissue

morphogenesis and formation of hollow epithelial acini. The proposed

computational model enables one to isolate the effects of biomechanical

properties of individual cells and the communication between cells and their

microenvironment while simultaneously allowing for the formation of clusters or

sheets of cells that act together as one complex tissue.

94

Chapter 8: Micromotors

CHAPTER 8: SKELETAL MUSCLE 8.1 Introduction Cellular motors come in many varieties. The most familiar motor we use every day is our musculature, which consists of skeletal, cardiac, and smooth muscle, all of which move linearly i.e., back-and-forth. Other types of motors operate within most of ours cells that transport things , change cell shape, and move cells around. Most free-living cells, such as bacteria, use rotary engines. Still other motors are used in cell migration and growth. While there is much to study about these different motor mechanisms , our task is simplified by the fact that all the cellular motors use the same fuel: ATP. This chapter will concentrate on the mechanical aspects of muscle cells, and subsequent chapters will cover rotary and other types of motors. The 630 bulk skeletal muscles that move joints in the typical human are composed of > 250 million muscle fibers (cells) in the typical human body. Force generation is accomplished by shortening and/or parallel sliding of two protein filaments: actin and myosin within each fiber. Muscle contraction is driven by the cycling of cross-bridges, whose unitary length change in the nanometer range and its pico-Newton force output are fueled by conversion of chemical energy, stored in the form of adenosine triphosphate (ATP), into a change in myosin protein configuration. The range of force and length changes of a muscle is determined by factors such as muscle cross section, fiber angle, tendon attachment, and lever geometry, but also by the metabolic pathways available for ATP synthesis and by enzymes involved in cross-bridge cycling. In addition, muscle mechanical activity is affected by the extent of actin and myosin filament overlap. Force output can be graded by selective recruitment of motor units and/or by variation of force output from individual units. The most important points of this chapter are:

1. The mechanism of muscle contraction 2. Force output of muscle 3. Neural control of muscle

8.2 Muscular Microstructure Skeletal musculature is made up of anatomically distinct units, which are attached to bones (via tendons) or other muscles (via ligaments) to support specialized movement. In the human, there are 630 individual muscles, which make up ~40% of total weight of the average body. Individual muscles are composed of fascicles—assemblies of muscle fibers that are surrounded by a connective tissue sheath that can span the entire length of a muscle [Fig. 1].

Each fiber (diameter between 10 and 100 m ) is one sin single membrane called the sarcolemma. Glycogen, a chief energy provider for skeletal muscle contraction, is stored in conjunction with this membrane. The sarcolemma protrudes into the muscle fiber at regular intervals forming transverse tubules (T-tubules). T-tubules are conduits for electrical signals to travel from the surface to the core of the cell, where they abut the cisterns of the sarcoplasmic reticulum (SR), the intracellular Ca store of muscle cells.

95


Fibers contain densely packed regular bundles of myofibrils, and are usually multinucleated. Myofibrils are the contractile units of muscle, made up of thick (myosin, diameter 15 nm) and thin (actin, diameter 5 nm) protein myofilaments. These myofilaments are arranged in a hexagonal pattern and interdigitate [Fig. 1]. In humans, each myofibril contains approximately 1500 myosin and 3000 actin filaments. Myofibrils (and, therefore, myofilaments) are organized in highly structured sarcomeres [Figure 8.1.E].

Functionally, sarcomeres consist of two half-sarcomeres

whose force generators have opposite polarity. Actin filaments (each 1 µm long) consist of two intertwined pearl-necklace-like chains composed of globular actin molecules (the pearls, where myosin will bind), tropomyosin (the chain, which covers the binding sites for myosin), and troponin (where Ca will bind to displace tropomyosin and enable actin–myosin bonding) .

Myosin filaments ( 1.6µm long) consist of several hundred myosin molecules that are each around 0.15 µm long and have the appearance of a daffodil (before flowering). Individual molecules are connected by their “stems,” with the bud “head” sticking out to the sides of the assembled filament, as in a tightly wrapped bunch of flowers. Myofibrils also contain mitochondria (the “power station” of the cell), often arranged in parallel with individual sarcomeres.

A broader view of muscle is shown in Figures 8.2 and 8.3. The bulk muscle consists of many fibers, which each consist of many myofibrils. The fibril contains many sarcomeres, oriented in parallel along the length of the fibril.

Figure 8.1. The organization of skeletal muscle, with approximate diameters. Note that the myosin bundle has many ‘heads’ that contact the actin randomly as the 2 components slide past each other.

96


Sarcomeres contain the sliding filaments, actin and myosin, as shown below.

Figure 8.3. A sarcomere, showing actin filaments sliding over myosin, which is connected to the Z-discs with titin springs.

Myosin head

Figure 8.2. Muscle architecture. The sliding filaments, actin and myosin are shown making up a single sarcomere, the fundamental unit of contraction.

Sarcomere

97


The dimensions of typical sarcomeres are shown in Figure 8.4:

8.3 The Pathway to Contraction A. Motor Units In mammalian muscle each individual fiber is innervated by a single motor neuron. Each motor neuron may control a range of homotypic fibers (with the number determining the precision of movement supported). This group of fibers and its controlling neuron is called a motor unit, the functional unit of muscle contraction. The motor unit (below), is comprised of a single alpha motor neuron and all the fibers it enervates. This muscle fiber contracts when the action potential (impulse) of the motor neuron that supplies it reaches a depolarization threshold. The impulse arrives at the neuromuscular junction, which typically extends over an area of about 0.1 cm2. The depolarization generates an electromagnetic field and the potential is measured as a voltage. The depolarization, which spreads along the membrane of the muscle, is a muscle action potential. The motor unit action potential is the spatial and temporal summation of the individual muscle action potentials for all the fibers of a single motor unit, as shown in Figure 8.5.

Figure 8.4. Actin and Myosin

Figure 8.5

98


In this example, there are 4 motor units, each being activated at a different frequency. Their algebraic sum yields a signal that can be registered at the surface of the bulk muscle. Since the motor units are spatially separated, this phenomenon is known as spatial summation.

Motor neurons can connect with between 3-2,000 muscle fibers. This large range relates to the differing functions of muscles. For example, thigh muscles are engaged in forceful, but gross, movements, such as walking and running. Thigh muscles, i.e., quadriceps, therefore, have relatively few motor units that each innervates a large number of muscle fibers. Since there are roughly 300,000 muscle fibers in the quadriceps, and motor units consist of about 1000 fibers, which translates to about 300 motor neurons controlling the muscle. Other large muscles, such as the biceps, are similarly innervated. So when the motor neuron fires, a massive twitch of 300 muscle fibers happens. In contrast, muscles controlling fine movements, such as the eye, have smaller numbers of muscle fibers per motor unit (usually less than 10 fibers per motor unit), which yields much smaller twitches and hence finer control.

During a typical voluntary muscle contraction the nervous system ‘recruits’ motor units in a hierarchical arrangement as motor units with fewer and smaller muscle fibers are activated first, followed by the motor units with larger muscle fibers. The main determinant of muscle force output of each motor unit is their frequency of stimulation. The main determinant of the bulk muscle force is the number of motor units recruited. Thus the biceps theoretically would produce their maximum force (“Maximum voluntary contraction” , MVC) when all 300 or so of their motor units were activating at maximum frequency.

The sequence of action of the power stroke is illustrated in Figure 8.6. In this cartoon, myosin is assumed to be stationary, while pivoting around its bottom joint. Actin is seen to move to the right during the power stroke. B. Excitation–Contraction Coupling Contraction is initiated by an electrical impulse [somewhat paradoxically called an “action potential” (AP)] of neural origin (i.e., originating in the brain or spinal cord) that reaches all

Figure 8.6. The power stroke sequence. The myosin head captures ATP, denoted ‘T’, which is then hydrolyzed to ADP, causing a forward rotation of the myosin, which ‘cocks’ the ratchet. Next the phosphate is released , causing binding to actin, then the ADP is released, causing the ratchet to rotate back, in the power stroke.

Bottom joint of myosin

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muscle fibers within a motor unit via the corresponding motor neuron. Subsequent to neuromuscular transmission, the AP travels along the sarcolemma of each muscle fiber and proceeds into the t-tubules. The t-tubular electrical signal causes release of Ca++ from the adjacent SR. Ca binds to the actin filament, displacing the tropomyosin chain to expose the binding sites for myosin, thereby allowing actin and myosin to interact.

The dependence of force production on Ca is depicted in Figure 8.7.A.

In the presence of adenosine triphosphate (ATP), the key biological energy storage and transfer molecule, actin and myosin filaments will slide past each other, driven by cross-bridge cycling (Figure 6). This causes an increase of force and, if the external load is less than the maximal force developed by the contractile apparatus, shortening of the sarcomere. In either case, the mechanical event is known (again, somewhat paradoxically) as a “contraction”: “isometric” if constant length and “isotonic” if constant force. The AP-induced increase in [Ca ] is counteracted by re-uptake of Ca into the SR. This also is an energy-consuming process that requires ATP. In the absence of maintained electrical stimulation, [Ca ] , therefore, decreases to its resting level, which prevents further formation of new actin-myosin bonds. A single contraction followed by relaxation is called a “twitch.” 8.4 Generation and Regulation of Force A. Temporal Summation of Twitch Tension A key mechanism controlling force in skeletal muscle is AP frequency of motor units. Since AP is an electrical pulse that is much shorter than a mechanical muscle twitch, successive APs can reach the fiber before it has relaxed (i.e., before [Ca ] has returned to its resting value). Re-stimulation of a fiber under these conditions leads to cumulative amplification of [Ca ] . This amplification will cause a summation of force (called “temporal summation”). In the fiber depicted in Fig. 5, this occurs at stimulation frequencies above 5 Hz. Note that temporal summation combines with spatial summation, described in section 2.2 and below to produce smooth muscle forces.

Figure 8.7. (A) Schematic representation of isometric force production (normalized to its maximum value P ) as a function of cytosolic Ca++ concentration (log scale). (B) Relative force versus relative velocity (normalized to maximum.).

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If the interval between APs is less than of the duration of a mechanical twitch ( 10-100 ms, depending on muscle type and excitability ), the individual twitches will fuse, as seen in Figure 5. When twitches completely fuse, a “tetanus” results—a maintained powerful contraction. During a tetanic contraction, the muscle operates on the plateau region of the force-Ca relationship [Figure 8.8]. Upon cessation of electrical stimulation (and provided that ATP is sufficiently abundant), [Ca] is reduced by the reuptake of Ca into the SR and force declines to its resting value. B. Force Production Muscle cross-sectional area provides a good indication of maximum force output. A larger cross-sectional area indicates more fibers or fibers with a larger diameter, both of which translate into larger numbers of parallel myofilaments and, hence, a bigger area for cross-bridge formation. Maximum contractile strength is about 3–4 kg per cm2 of muscle cross section (300–400 kPa. This quantity is called Physiological Cross Section Area (PCSA) and is used to estimate clinical performance of muscles, based on their area. The area of muscles is highly dependent on level of exercise, and hypertrophy and atrophy are constantly occurring in muscles. Typical PCSA for Biceps is 23 N/cm2 with 170 ms minimum response time. For triceps it is 13-23 N/cm2, with 100 ms response time. The table at right summarizes some important properties of muscle. The PCSA for each muscle is influenced by the arrangement of muscle fibers. When fibers are all parallel, fiber direction is parallel to the main axis of force development, so fascicle length and muscle length are equal. In contrast, unipennate fibers run at a “pennation angle”(usually between 10 and 30° ) to the force axis, so fascicle length and muscle length are unequal. (Bipennate muscles have fibers that attach to a

Figure 8.8. Temporal summation of force

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tendon in the middle of the muscle and multipennate muscles are those with multiple tendon attachments and fiber angles.) Whereas unipennate muscles commonly have larger cross sections, thereby generating larger total forces, the net force, contributed by their individual fibers is reduced by the cosine of the pennation angle. The total shortening length of pinnate muscles is reduced by the cosine of the angle. Effective force output and contractile velocity are further affected by the position of tendon attachments to bones (relative to neighboring joints). The mechanical gain of the resulting lever systems influences both force output and fine control of movement. C. Length–Tension Relationship Force output relates to the number of active cross bridges in each half-sarcomere, which is dictated by sarcomere length, which directly relates to the amount of overlap between actin and myosin filaments. The overlap determines the number of cross bridges that may maximally be formed at any time. Thus the highest active tension can be generated when all myosin heads (at either end of the myosin molecule) face actin-binding sites (Fig. 9: b–c). This is the case when sarcomere length in skeletal muscle is around 2.0–2.25 m in hum ans. When muscle fibers are stretched beyond this point, the myosin heads near the M-line will no longer overlap with actin filaments and, therefore, cannot contribute to active force generation (Fig. 9: d). In the extreme of excessive stretch, at around 3.5 m , actin an will have no overlap (Fig. 9: e). At this point, cross bridges cannot form and no shortening or tension may be actively generated. When sarcomere length is less than about 2.0 m , interaction of opposite actin filaments disturbs the hexagonal arrangement required for optimal cross-bridge formation and maximum tension development is reduced [Fig. 9: a–b]. At a sarcomere length of about 1.67 m , m yosin filam en -lines, causing a further decrease in active tension generation.

The length of a sarcomere that optimizes force production, is thus in the midregion near 2.25 m (Fig. 9: c) and presett ing m uscles to th is length m axim izes force output. A comprehensive analysis of sarcomere length changes found that most muscles operate over a surprisingly narrow range at 94.5% of L0. Skeletal muscle is characterized by showing negligible resting (passive) force at sarcomere lengths less than L0. Above L0, passive force rises quasi-exponentially with sarcomere length [the dashed curve in Fig. 9(A)]. This behavior is largely attributable to the elastic characteristics

Figure 8.9. Sarcomere length–tension relationships as measured in a single fiber. A) Piecewise linear nature of the active force–length relationship. The dashed curve denotes the force–extension relation of resting muscle. (B) Sarcomere structural interpretation of panel A.

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of extracellular components, especially the connective tissue that envelops individual fibers, fascicles and the whole muscle (Fig. 1). D. Isotonic Force–Velocity Relationship As stated earlier, the amount of force generated also depends on the velocity of muscle shortening. Faster movements will not permit as much force output as those with a lower velocity. This occurs because the rates of cross-bridge attachment, detachment, and re-attachment are chemically limited by the enzymatic activity of the actin-activated myosin adenosine triphosphatase (ATPase). The number of active cross bridges at any time is, thus, reduced during fast movements of any given muscle. In addition, there are different myosin ATPase isoforms, which differentiate various muscle types with their individual characteristic peak cross-bridge cycling rates . E. Influence of Strain History Notwithstanding the success of the sliding-filament theory of muscle contraction, there are two related phenomena that the above theory is unable to accommodate. Both are functions of strain history. The first phenomenon is reflected in the generation of additional force (in association with reduced metabolic energy expenditure) during an eccentric contraction. If a muscle is stretched during an active contraction, it responds with force in excess of that predicted from its isometric force–length relationship. The metabolic consequences of this behavior were first strikingly demonstrated by use of a pair of back-to-back bicycle ergometers. A diminutive female, pedaling backward (at constant velocity and with leg musculature necessarily contracting eccentrically) could readily exhaust an athletic man pedaling forward (with leg extensor musculature necessarily contracting concentrically). The above phenomenon is attributed to the ability of skeletal muscle to generate forces in excess of in the negative velocity region of the force–velocity relation . At the molecular level, it is presumed to reflect the behavior of negatively strained cross-bridges, which, under sufficient strain, may be broken without concomitant hydrolysis of ATP. This results in diminished heat production , thereby resulting in increased efficiency . The second phenomenon, also strain-history dependent, confers on muscle a degree of “memory.” It is most readily observable on the descending limb of the force–length relation, where a component of force enhancement by an eccentric contraction can be exceptionally long-lived . This behavior has recently been attributed to the behavior of titin (see Figure 2). Because titin contains a segment that is able to unfold reversibly, as the muscle is stretched, it has been implicated as the principle source of energy storage during eccentric contractions. F. Muscle Tone Even in the absence of overt activity, muscles have a certain tautness. This is called “muscle tone” and arises from low-frequency action potentials generated within the nervous system. The maintenance of tone (like that of posture) is also an energy consuming process. At rest, muscle tone accounts for a major portion of baseline energy consumption. 8.5 Skeletal Muscle Energetics A. Efficiency of Skeletal Muscle Efficiency is defined as the mechanical power output divided by the metabolic power input. The peak thermodynamic efficiency of muscle is around 0.4, based on input of the high-energy fuel ATP. This value drops to about half that level when the cost of conversion of dietary substrates into high-energy storage compounds is taken into account. This figure further ignores the

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anabolic costs of muscle fiber synthesis, maintenance, or repair (normal turnover rates are such that the contractile protein of active muscle is totally replaced every 2–4 weeks). Such costs are usually attributed to “resting” or “basal” metabolism. B. Metabolic Pathways The obligatory energy substrate that skeletal muscle utilizes is ATP, which, when broken down into ADP and , releases up to 50 kJ per mole of chemical energy . ATP is needed to fuel cross-bridge cycling, but also to pump Ca back into the SR (where one ATP is needed to transport two Ca ions against their concentration gradient) and to maintain a suitable electrochemical environment for normal AP propagation. ATP concentration in skeletal muscle is about 4 mM/L, which is enough to enable contraction for several seconds or less, since ATP turnover increases linearly with force production . To maintain contraction and force output beyond such a limited period, ATP must be constantly replenished. The immediate source of replenishment is creatine phosphate (CrP), which is present in muscle cells at a concentration of about 20 mM—sufficient to sustain another 10–20 s of contraction. Glucose is the main metabolic source for ATP re-synthesis in muscle; it arises from blood glucose, muscle glycogen, or liver glycogen. The synthesis of ATP from glucose occurs by two main pathways: oxidative phosphorylation and glycolysis. As the name suggests, oxidative phosphorylation requires oxygen (an aerobic process that takes place in the mitochondria), while glycolysis does not (it takes place anaerobically in the cytoplasm). Aerobic ATP synthesis is more efficient than anaerobic synthesis, yielding 36 ATP from one molecule of glucose, compared to only 2 ATP via the glycolytic pathway. Force can be maintained with aerobic ATP synthesis for an extended period of time, provided that there is an adequate supply of nutrients. In contrast, contraction based on ATP arising solely from the glycolytic pathway can maintain maximum force for only around one minute. Besides its lower relative ATP production, another disadvantage of glycolysis is that it produces lactic acid (lactate plus H ), which is not efficiently metabolized by skeletal muscle . Excess production of lactic acid leads to hydrogen ion accumulation within the muscle cell and causes a reduction in intracellular pH. The pathophysiological consequences of this acidification are still under investigation. Recent evidence suggests that lactoacidosis may play a less-prominent role than previously assumed in the impairment of muscle force production upon repetitive activation (e.g., muscle fatigue). Oxidative phosphorylation, therefore, supports the majority of sustained locomotor activities, while anaerobic glycolysis is used for activities that require brief periods of large power output. C. Fiber Types Muscle fibers can be categorized by their dominant metabolic pathway for ATP synthesis. There are three main types (easily identified by their color): slow oxidative fibers, type I (red); fast oxidative/glycolytic, type IIA (pink); and fast glycolytic, type IIB (white). The different fiber colors are related to their content of myoglobin, the oxygen store of muscle cells. Myoglobin, which is red in appearance, is present in particularly high concentration in slow oxidative fibers (which also contain more mitochondria). Most of these fibers tend to be in small motor units and are recruited early during muscle activity. Fast glycolytic cells, in contrast, have very low levels of myoglobin and mitochondria, and large motor unit size. Fast oxidative/glycolytic fibers have mixed properties.

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Fiber types are consistent within a motor unit, but there is usually a mix of types within a given bulk muscle. An exception to this rule is fish, which have distinct and well-defined bands of white or red fibers. To summarize fiber types,

• Fibers are a long cylindrical cell with hundreds of nuclei • 10-100 mm in diameter • 1-30 cm in length • The contractile component is a myofibril • The non-contractile component is endomysium • slow twitch fiber (type I) are red in color because of abundant blood supply and

myoglobin ; they are slower to the peak when contracted and fatigue resistant • fast twitch fiber (type IIB) are pale in color because of less blood supply; they rapidly

peak in force when contracted, but are easy fatigue • intermediate fibers (type IIA) are pink, in between I and II

D. Fatigue Fatigue (characterized by a decrease of force output) is the consequence of prolonged contraction. In the exercising human, fatigue can be subdivided into peripheral and central components . Central fatigue is of neuronal origin, with factors such as decreased motivation, alertness, or neuronal excitability causing a reduction in AP frequency of motor neurons (although the extent and detailed mechanisms of these effects are not very well understood). Peripheral fatigue is caused primarily by biochemical processes within the muscle fiber itself. Depletion of metabolic substrates (e.g., glycogen) has been shown to reduce muscle-force production. In addition, accumulation of metabolic by-products (e.g., glycolytic intermediates and inorganic phosphate), increased osmotic pressure (lacto-acidosis), and elevated muscle temperature all contribute to peripheral fatigue. Secondary to the above biochemical changes, there are intrinsic alterations in either cross bridge cycling and/or SR Ca handling (i.e., release and/or re-uptake) as mechanisms that contribute directly to the reduced force generation in fatigued muscle. E. Muscle Sensory Cells All skeletal muscles have both feed-forward control, through the 1A alpha motor neurons, but feedback control, through muscle spindles, as shown below. Muscle spindles are attached to afferent (sensory) neurons that act essentially as strain gauges, telling the CNS about the state of stretch of muscles. The job of muscle spindles is to report to the central nervous system the state of muscle length. The muscle spindle is composed of ‘intrafusal’ type muscle and its neuronal connections. Ordinary bulk muscle is called ‘extrafusal,’ and this type is much more abundant than the intrafusal type. Each bulk muscle needs only a few muscle spindles as sensors. The cartoon below shows 2 muscle spindles with their innervations. The muscle spindles lie in parallel with bulk muscle, and thus their length changes as the muscle contracts and lengthens. As the spindles are stretched, they send signals, coded as action potentials, to the central nervous system (CNS) , via Ia and II afferents. In addition to the afferent (sensory) nerve fibers attached to the muscle spindle, note that efferent fibers ( efferen ts) with the muscles also. The picture below is a muscle spindle whose nerves are stained , showing them wrap around the muscle.

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Sensory responses from a muscle spindle are depicted in Figure 8.11. Note that as the muscle is stretched, the rate of nerve impulses increases in synchrony. Ia afferents can powerfully excite the ALPHA MOTOR NEURONS of the muscle containing the spindle. This is the basis of the classical STRETCH REFLEX in which extension of the muscle (and thus its spindles) cause a reflex contraction. Golgi tendon organs (not shown) are also involved in reflexes.

The gain of these muscle spindle “stretch receptors” is regulated by the gamma motor neurons, which activate the intrafusal muscle, causing contraction. This can be understood as ‘automatic gain control,’ (AGC) present in many amplifiers. This AGC works by the following mechanism: the muscle spindle length can be changed in 2 ways: (1) it changes its length in sequence with the bulk muscle, and (2) it changes its length when the CNS activates the γ motor neurons, which contract the intrafusal muscle. In case (1), output of the spindle resembles that of Figure 7. Here, the γ motor neurons are off, and the sensory gain is maximal. In case (2) the γ motor neurons are active, and the intrafusal fiber actively contracts during the stretch and hence its output is reduced, as shown below in Figure 8. So for a given stretch of the bulk muscle, the

Figure 8.10. The top image shows a muscle spindle with its innervations. Bottom is a stained micrograph of a muscle spindle.

Figure 8.11. Sensory output from a muscle spindle

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output of the spindle can be reduced. The reason for this control has to do with reflexes, and is beyond the scope of this chapter.

E. Summary In comparison with artificial actuators, muscle surpasses them only in the fuel delivery and renewability. Muscles are linear actuators that can bend around joints. They are adapted for intermittent duty and stiffness (compliance) control. Knowledge of Biological force actuators is crucial to the development of novel actuator technologies. To best mimic the biology, clues derived from skeletal muscle actuator design should drive the design process.

APPENDIX: REVIEW OF BIOMECHANICAL TERMINOLOGY Stress-strain Relations in Muscle (from Madden et Al.[1])

Stress is the typical force per cross-sectional area of the actuator material, in this case, muscle. Peak stress is the maximum force per cross-sectional area that a material is able to maintain position (also known as blocking stress). Thus force developed by actuators scales linearly with cross-sectional area (whose surface normal is parallel to the direction in which actuation is occurring). Strain represents the displacement normalized by the original material length in the direction of actuation. Typical strain is the strain that is often used in working devices, whereas peak or

Figure 8.12. Gain control of a muscle stretch receptor. In each panel (a to e) , the top trace represents afferent neuron pulses, originating from the sensory organ , subjected to constant stretch; the bottom trace represents activation of the control neuronal influence, similar to activation of γ motor neurons. Activation of γ causes slowing the rate of sensory pulses, to a degree proportional to the magnitude of γ activation.

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max strain is the maximum strain reported. The peak strain generally cannot be obtained when operating at peak stress. Strain rate is the average change in strain per unit time during an actuator stroke. The maximum strain rate is usually observed at high frequencies when strains are small. Bandwidth is relevant to some actuators, and is the frequency at which strain drops to half of its low frequency amplitude. Limitations on bandwidth and strain rate can result from a range of factors including speed of delivery of the input energy, rates of diffusion or heating, internal dissipation, inertia related effects including speed of sound, and kinetics associated with the energy transduction method . Work density is the amount of work generated in one actuator cycle normalized by actuator volume. It is very important to note that this does not include the volume occupied by electrolytes, counter electrodes, power supplies, or packaging, unless otherwise stated. These additional contributions to actuator volume do not always scale linearly with work output and therefore need to be considered separately. Also it is emphasized that the work density is not simply the product of peak stress and peak strain. Specific power or power to mass ratio is the power output per unit mass of actuator material. Typically only the mass of the material itself is considered. Peak power density can be found from the maximum product of simultaneously measured stress and strain rate divided by density. Because of the interdependence of load and rate, peak power is in general less than the product of peak stress and peak strain rate normalized by density. Efficiency is the ratio of work generated to input energy expended. Stored electrical energy and even thermal energy can in principle be recovered in order to improve efficiency. Electromechanical coupling in muscle refers to the degree of ratio of electrical activation to mechanical force output. . Elastic modulus is the material stiffness and generally represents the instantaneous value, before any creep is induced. Note that mammalian muscle stiffness at rest is 40 MPa, but this value increased 50X during contraction. Each myosin bundle can produce 5.3 pN of force during its power stroke. There are 100 molecules per actin filament. Each filament has c.s.a. of 1.8 X 10 –15 m2 in the relaxed muscle. Cycle life is the number of useful strokes that the material is known to be able to undergo. Cycle life is often highly strain and stress dependent.

Chapter 8 Review Questions and Exercises

1. How much force does a single myosin-actin bridge attain? How much energy does it take

and how for a single cycle of actin-myosin interaction. ? Calculate the energy efficiency of skeletal muscle.

2. Describe the different roles of IA and II afferents. State what principles and pathways are involved.

3. Workshop: With the “gamma model, “ simulate the operation of a muscle spindle subject to stretch, similar to the traces of Figure 11. Test the effect of gamma motor neuron activity and compare results to Figure 12. Describe the model, and how it works.

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References

1. Madden, J.D.W., N.A. Vandesteeg, P.A. Anquetil, P.G.A. Madden, A. Takshi, R.Z. Pytel,

S.R. Lafontaine, P.A. Wieringa, and I.W. Hunter, Artificial muscle technology: Physical principles and naval prospects. Ieee Journal of Oceanic Engineering, 2004. 29(3): p. 706-728.

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Chapter 9: Cell Propulsion

CHAPTER 9: CELL PROPULSION 9.1 Introduction Cells explore and sense their environment in many ways. Eukaryotic cells extending actin protrusions and form integrin adhesions with their surroundings. These movements are powered by a combination of protrusion generation, adhesion formation and cellular contraction. Directed migration of cells is driven both by chemical signals, i.e. chemotropism, and mechanical signals, i.e., mechanotropism. Movement and migration of cells is governed by the milieu in which cells exist- this is composed of various fluids as well as solid surfaces comprising tissues. Here we will review principles of fluid mechanics, and propulsion mechanisms in various environments. 9.2 Cells swimming through fluids Knowing the forces involved as cells move though fluids or when fluids flow past cells is an important part of Cytomechanics, and requires knowledge of basic fluid mechanics. Here, we outline just the bare minimum of such knowledge, and refer the reader to texts, such as, “Life in Moving Fluids,” by S. Vogel, for more complete treatment. Whether we consider cells migrating through their milieu (i.e. blood cells), or their milieu flowing around them (i.e., Endothelial cells in arteries), we are interested both in the forces involved, and any propulsive or adaptive mechanisms. To estimate these forces we need to know the properties of the surrounding milieu, i.e. blood or plasma, sea water, etc., the shape and size of the cell or cellular component, and relative speeds. Here, we will outline these topics.

Viscosity is the internal resistance to flow, and relates to the thickness, or ‘stickiness’ of the fluid. The concept of viscosity, or ‘lack of slipperiness’, was introduced by Newton. The greater the viscosity, the less easily the moving body slides past the fluid. In other words, viscosity acts as a ‘drag’ on objects moving through it. Viscosity, η, is measured in units of Pa-s, known as Poise. The shear stress in a fluid represents the force needed for one lamina or layer of fluid to slip over another; shear rate is the measure of how fast one lamina slips over another. This model is represented below:

The top plate, area, A, is being dragged over the bottom plate, separated by distance , d,

with fluid of density, ρ, separating them, with force, f, and velocity, Vo. Note that the force (1 & 2) is proportional to the velocity, and inversely proportional to the separation distance.

Drag can be defined as the rate of removal of momentum from a body moving in a fluid.

So, a moving body experiencing a large drag will stop quickly. For example, imagine yourself

d

f A

ρη

η

/

/

2=

−=

crit

o

f

dAvf

(1)

(2)

Figure 9.1

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diving into a pool of molasses- you would glide relatively little, being stopped by the high viscous forces. But if there were a large force pulling you, such that it exceeded the value in equation (2), then only inertial forces would dominate. So for small forces, flow is laminar, dominant force is friction due to viscosity; with larger forces, above the critical force, turbulence occurs, and the dominant force is inertial.

To calculate the drag force on a particle or cellular object, we use the drag equation:

𝑓𝐷𝐷𝐷𝐷 = 12𝐶𝐷 𝜌𝜌𝑉2 (3)

Where CD is the drag coefficient, dependent on Reynold’s number (Re), as shown below:

Now we need to know the relevant Reynold’s numbers, that are dependent on object

shape, size and velocity, and on the fluid properties. Reynolds number quantifies the relative magnitude of inertial and frictional forces in a body moving through a fluid.

Some representative Re numbers, and fluid properties are :

From these data and equation (3) , we can estimate an example: To pull a 5 µm cell

(sphere) at a speed of 1 µΜ/sec, we can estimate f ~ 10-5 pN , based on a CD = 104 , the media being plasma, with density near that of water =1000 .

Fluid

ρm (kg m-3) η (Pa-S) fcrit (N)

Air 1 2 X 10-5 4 X 10-10 Water 1000 0.0009 8 X 10-10 Olive Oil 900 0.08 7 X 10-6 Glycerine 1300 1 0.0008 Corn Syrup 1000 5 0.03

Figure 9.2 Reynolds number for flow around a sphere

Figure 9.3 Reynolds number for flow around a cylinder

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Viscosity is constant for most fluids, but

varies greatly with temperature. Blood is an exceptional fluid, in that its viscosity depends on the nature of the flow, which depends on blood velocity and diameter of the vessel. Typical behavior of blood is shown at right:

In summary, the motion of a body depends on the ratio of viscous and inertial effects: Reynold’s number. Re is small for cells, and large for almost all animals. The cellular world is ruled by friction.

9.2.1 Cell Substrate Interactions

It has been shown that the elasticity of the cellular substrate (characterized by Young’s modulus, E) in contact with the cell influences the direction of cell migration. This process has been termed “durotaxis”. The stiffness of the cellular environment is not only important for cell migration but also appears to influence the metastatic process of cancer cells in vivo. Therefore, a mechanistic understanding of how environmental mechanical signals influence cellular movement and cell dynamics is an important question in cell biophysics. Mechanical properties of the environment not only influence cell migration but also can influence many aspects of the cell life cycle. When cells are plated onto a planar substrate, the morphology and behavior of the cell depends on the stiffness of this substrate. If the substrate is soft, aggregates of adhesion molecules remain small and transitory. In the opposite limit, when the substrate is stiff, a clear network of contractile stress fibers (bundles of F actin) develops, and strong focal adhesions anchoring the cell to the substrate are seen. Substrate stiffness can also influence the long-term fate of cellular development. It has been shown that differentiation of mesenchymal stem cells to more specialized cells is influenced by the stiffness of the cellular substrate. The mechanisms behind this dependence are complex but appear to involve three basic systems: the actin cytoskeleton, cell-surface adhesions and motor-based contraction. Indeed, a simplified mechanically-based model showed that when the time scales of adhesion movement and the cytoskeleton interaction are considered, cells on stiff substrates will form numerous actin filament bundles with many filaments, while cells on soft substrates will have fewer bundles with a smaller number of filaments. This stiffness-dependent organization of the cellular cytoplasm could be an important feature in cellular mechanosensation.

Blood is much more viscous than water because it contains formed elements & plasma proteins, hence it flows more slowly under the same conditions . Water viscosity at room temperature= 0.01 centipoise [1 centipoise (cp)] ; plasma is ~ 1.8 cp; Blood viscosity = 3-10 cp .

Figure 9.4

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It is useful to recognize that cells can move within substrates of pure liquids, semi-solids

and solids. These 3 types of substrates are modeled below. In (A), a pure liquid is represented as a dashpot, with drag force proportional to viscosity times velocity. Applying force, F, leads to a gradual acceleration to a constant velocity. In semi-solid (B), an object can be moved by the force, but will eventually come to rest. In solid (C), the object is trapped, but will oscillate due to the elasticity of the substrate.

Figure 9.5 9.4 Propulsive motors The motor we use every day is our skeletal muscle (Chapter 8, which moves linearly i.e., back-and-forth. Most free-living cells, such as bacteria, use rotary engines. Most cells contain motors that crawl or hop. Still other motors are used in cell migration and growth. While there is much to study about these different motor mechanisms, our task is simplified by the fact that all the cellular motors use the same fuel: ATP. The several known motor types are outlined below: 9.4.1 Rotary Motors The f1-ATPase motor is the best-studied rotary motor, and is found in thermophilic bacteria. The basic plan is shown in the cartoons below. In Figure 9.1, at left, the filament that drives the motion (the ‘tail’ of the cell) is attached to the cell membrane with a kind of bearing composed of proteins. The f1-ATP-ase motor can either make or break ATP, and hence is reversible . It can generate a torque of 40 pN-nM. It can perform work of 80 pn-nM (40 * 2p/3) for each 1/3 revolution. This is equivalent to free energy from ATP hydrolysis.

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The rotations can be visualized by attaching a large actin filament. The key to the high efficiency of this motor is its mechanism of fuel injection.

The ATP synthase (top right) not only is nature’s smallest rotary motor, but also has an important role in producing most of the chemical energy that aerobic and photosynthetic organisms need to stay alive. Operation of this motor was studied by using electromagnets to force it to rotate and generate chemical energy (adenosine triphosphate, ATP). In the process, the direction of its rotation during ATP synthesis was determined. ATP synthase is composed of two linked multi-subunit complexes, called F0 and F1. F0 is embedded in cellular membranes and conducts protons, whereas F1 is a peripheral complex and contains the catalytic sites. Together they couple the flow of protons down an electrochemical gradient to the synthesis of ATP from ADP (adenosine diphosphate) and inorganic phosphate.

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9.4.2 Transport Motors Transport of materials within eukaryotic cells takes place constantly along microtubules (MTs). Materials such as vesicles can be carried by proteins called kinesin or dynein. MTs have 13 protofilaments running parallel to the axis. The MT does not rotate when gliding over kinesin-coatings. The kinesin takes 8 nm steps (or 7 & 9 for waddling), as shown in the figure below. The walking needs high ATPase rate due to large duty cycle. MTs can be fabricated artificially to demonstrate their operation .

The figure below shows cartoons of the carrier molecule walking along. At lower right, real-time snap shots of the motion of a labelled carrier along an MT is shown.

Figure 9.7 The binding-change model for F0F1 ATP synthase. a. Looking up at F1 from the membrane. Each blue or green area represents a pair of α- and β-subunits, in which the catalytic sites are interfacial but mostly on the β-subunit. In step 1, the γ-subunit rotates through 120°, driving conformational changes in the three surrounding catalytic sites that alter their affinities (O, L or T, for open, loose or tight) for substrates and product. In step 2, ATP forms spontaneously from tightly bound ADP and inorganic phosphate (Pi). b. View from the side of F0F1. The α-subunit contains two partial channels. To traverse the membrane, a proton must move through one channel to the center, bind to one of the c subunits, and then be carried to the other partial channel by rotation of the c-ring. The c-subunits are anchored to the ε-subunit (part of the rotor), whereas the a-subunit is anchored through b2 (the stator) to the α3β3 hexamer. Hence, rotation of the c-ring relative to the a-subunit in F0 will drive the rotation of the γ -subunit relative to the α3β3 hexamer in F1. New results show that the γ-subunit rotates in a clockwise direction when the engine generates ATP.

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9.4.3. Walking Motors The actin-myosin (A-M) ratcheting process is shown from the point of view of actin, which is assumed to be stationary, while the myosin ‘walks’ along it (Figure 9.4).

Figure 9.8. The A-M process.

Figure

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9.4.4 Cell Migration Motors

Bacterial pathogens have a cunning way of moving about inside the cells they infect — they harness components of the host’s cytoskeletal machinery, in particular, the protein actin. One such pathogen, Rickettsia conorii, is transmitted to humans through tick bites, and causes Mediterranean spotted fever. This bacterium assembles an elaborate ‘tail’ made of actin filaments (Figure 9.7) to propel itself through the cytoplasm of the infected host cell and to invade neighboring cells. Rickettsia tails consist of parallel, unbranched filaments that closely resemble those present in thread-like cellular filopodia.

Figure 9.8

Figure 9.6. Forward protrusion in concert with contraction of the rear cell body overcomes ECM resistive adhesive tractions to allow forward cell migration.

Figure 9.5. How neurites propel themselves along a substrate

Figure 9.7

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9.3 Comparative Motor Analysis As seen in the Figure, there are three modes of linear cellular motion. Characteristics of each type are shown in the Table.

Motor Step Size Max Force Max Efficiency (%) Processivity Mode

A-M Variable 5 pN 20 None Hops

Kinesin 8 nm 5 pN 50 Good Walks

F1ATPase 120° 40 pN-nm ~100 Good Crawls

Figure 9.8. The difference between walking and hopping propulsion

Figure 9.9

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Chapter 9

Review Exercises and Questions

1. The motor protein, kinesin, can generate a force of 6 pN. How fast could it move a bacterium through a typical cell?.

2. Model the rotary motor of F1 ATPase, showing the behaviors described in class.

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Appendix

Appendix 1 Numerical Modelling with Simulink

Rules for analog simulation •1Write diff-Q with highest order on left •2. Assume you know it, and integrate to get x. (Check integration limits) •3. Perform required operations on lower derivatives. •4. Check and simulate. •5. Scale magnitudes and timing. Dimensional Analysis This is a technique for deriving mathematical descriptions of physical systems, using educated guessing. The key is using standard letters to describe the 4 fundamental physical dimension, and its corresponding SI unit:

Most, if not all, other quantities used in cytomechanics are simple combinations of these dimensions, such as, velocity = LT-1 , acceleration is LT-2 , and so on. As an example of dimensional analysis is finding the period of a simple pendulum, that we know has units of T. Looking at the pendulum, we see that it is a mass, dimension M, at the end of a string, dimension L. The only force acting on the mass is gravity, g, with dimension , LT-2 . So now to derive the formula, we need to write the dimensional relationship as: T f M L LT= −( , , )2 We can see that there is no way that there is no way that the fundamental dimension, M, can be transformed into T, so we can eliminate M, and realize that the period must be independent of mass. So, to get T from L and LT-2 , i.e., g, we can manipulate them as follows:

T fL

LT= −( )2

or, simplifying:

T fLg

= ( )

Note that there will be other dimensionless parameters, in this case, 2π. Other physical systems may not be so easy to decipher, however, it is always useful to

Quantity

Dimension Units

Length 1.1 L (m) Mass M (kg) Time T (s) Charge Q (coul)

120

Appendix

check dimensions and units.

Appendix 2

Fundamental Constants and Relations

Parameter

Symbol Value

Boltzmann constant K 1.38 • 10-23 J/ °K

Thermal energy at room temperature kT ( T = 295°K) 4.1 pN-nM 4.1 • 10-21 J 4.1 • 10-24 erg 2.5 kJ/mole 0.59 kcal/mole 0.025 eV

Energetic Equivalents a•KT Average kinetic energy per molecule = 3/2 KT Persistence Length εp = Kf/KT Thermal Voltage = KT/q Diffusion coefficient =

rKTDπη6

=

Particles in gravity = mgh/KT

Viscosity of water

η 0.001 Pa-s

A.3. Generalized Boltzman Free energy formula:

kT

F

ePP ∇

−=

2

1

where P's are the probabilities of states or classes 1 and 2, and ∆F is free energy. For example Pi can represent the concentration of particles in 2 separate compartments.

121

Appendix

A.4. Mechanical Formulae

A.4.1 Bending energy of a rod: Hence:

Hence,

kT

YIRL

E

fp

f

farc

κε

κ

κ

=

=

= 22

81...22/

==== radiansRL

p

p

εε

θ

2

2

2

:

2

2

=

=⇒

==

==⋅

R

R

kTkTR

E

kTandELwheneSpecialCas

p

p

farc

arcp

ε

ε

κ

ε

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Appendix

A.5 Diffusion Diffusion coefficient indicates how fast particles of radius a can move in

a solution of viscosity, η . Units are cm2

sec .

2

2

2

2

2

2

)4

exp(4

1

)2

exp(2

1)(

xDtDtx

DtC

dxxx

xdxxP

So =→

−=

−=

π

Water flow

V c t( )0

τtQ m t( ) d V c 0( )

Q m K w A c Π P( ).

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Appendix

A.7 Joule-Helmoltz Formula: (Heat produced) = (energy input) X (0.24 cal/joule). A.8 Entropic Spring Constant:

eLengthpersistenclymergthofthePoContourLenL

whereLkTK

p

c

pcsp

==

=

ξ

ξ23

A.9 Properties of Polymers The table below gives Young's modulus, persistence length and mass per unit length of the selected molecules. Note that Tubulin is the protein from which microtubules are made.

Polymer

Typical Diameter (nM)

Persistence Length ξp (µm)

Elastic Modulus E (Gpa)

Mass Density λp (Da/nm)

Actin 8 15 2 110 Tubulin 25 6000 2 160 Intermediate Filaments

10-20


1.5

Collagen fibril 10-300 Elastin 0.002 Cellulose Dry 80 Cellulose Wet 40 Spectrin 0.02 0.002 4500 DNA 0.05 1 1900

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Appendix

A.10. Units and Conversions 1 N=105 dynes

2 1 N=1 Kg-m/sec2

1 dyne=1 g-cm/sec2

1 atm=1.013 * 106 dynes/cm2

1 atm=76 cm Hg

1 bar=106 dynes/cm2

1 pound=4.448 N

1 pound/inch2 (1 psi)=6894 N/m2

1 dyne/cm2=0.1 Pa

1 Pa = 1 N/m2

1 dyne/cm2=0.1 N/m2

1 torr =1 mmHg at 0oC

1 torr =133.32 N/m2

1 cm H2O at 4 o C= 98 N/m2

1 poise (viscosity)=0.1 N-sec/m2

1 joule=107 ergs

1 joule=1 N-m

1 cal=4.186 joules

1 watt = 1joule/sec

1watt-sec= 1 joule

1 hp (horse power) = 745.7 W

1 joule = 6.242 * 1018 ev

1 amp (current)= 1 coulomb/sec

1 volt = joule/coulomb

Boltzmann’s constant(k)= 1.38 * 10-23

joules/oK

1 g(standard gravity)=980.665 c/sec2

gas constant (R)=8.31 joule/mole-oK

log RT/F = 58 mV @ 25 C

1 mole= 6.023 *1023 Avogadro’s

number

1 molar (M)= 1 mol/L

1 molal= 1 mole(solute)/Kg(solvent)

1 Faraday=964800 coulombs/mole

Charge of 1 photon= 1.6 *10-19ev

A.11 List of Symbols Symbol Definition C, Co, Ci Concentration, outside and inside cell (Moles/Liter);

also refers to curvature C= 1/R, where R= radius. D Diffusion Constant ( cm2/sec) Da Dalton- the mass of 1/12 of a C-12 atom E Young’s elastic Modulus (Pa); Also used for energy (J) e Elementary unit of charge

125

Appendix

F Force (Newtons); also Faraday’s constant = 96,500 coul/mole G Gibb’s free energy; also shear modulus G Gravity H Enthalpy

I I Moment of inertia of cross-section

J Mass flux rate Ka, Kv Area and volume expansion moduli κf 3 Flexural bending modulus M Bending moment (N-m) kb Boltzmann’s constant [M] Concentration of monomer M Molecular mass P Pressure (Pa) Q,q Electric charge R Faraday gas constant; principal radii of curvature ree End-to-end length of a polymer S Entropy S Arc length (m) T Temperature (°K) V Volume (m3) Y Young’s Modulus

α Area γ Surface tension ε Strain; also permittivity ν Poisson’s ratio λ Extension τ Time constant ω Angular frequency µ Shear modulus η Viscosity ρ Density (g/m3) Π Osmotic Pressure Γ Reflection coefficient σ Stress (Pa) ξ or λp Persistence length (m)

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Rutgers Cytomechanics 2015 - Craelius