Third ClassTuesday February 2, 2005Previous: Cell Biophysics Protein charge Electroosmosis Swelling & Stretching Equilibrium Gel Model

Diffusion (Correction)Corrections from last time

Types of transportDiffusion:Seconds response, random targeting, gradient driven

Microtubule trackingMilliseconds, point-to-point target, energy drivenActin/myosin scaffoldingSeconds to minutes, gradient targeting, energy driven

CSK Ringingmilliseconds, global or channeled, pre-stress driven

Intelligent GelsCa++ or Mg ++ can do the same and are reversible

Gelation: Phenomena by Non-polar solvent

Electrical conductivity and elastic modulus are zero below threshold, then suddenly In solution, a measure of polymerization is viscosity.

Tissue morphogenesis by gel

Electric gel motor

Chemical versus mechanical signalling

Redundancy=reliabilityTies can break withoutLosing NY-LA cnxn untilp< pcrit

When [branching polymers] > [ ]crit

gel

Ingber, D. E. J Cell Sci 2003;116:1157-1173Sequential images (left to right) from computer simulations of multimodular tensegrities (A,C) or from time-lapse video recording of living cells (B,D)Pre-stressed fabric

AddTrypsin

Rhodamine-phalloidin staining of filamentous actin in HUVECs in the absence (A) and presence (B) of cytochalasin D (0.1 g/ml for 30 min). Note that in B, although some RGD-coated beads were still stained with rhodamine-phalloidin, the integrity of the actin CSK was disrupted, and most actin bundles disappeared. Scale bar = 20 m.

Tensegrity versus PercolationGeodesic dome follows strict rules, fixed tie lengths, fixed tension, and collapses when one is broken.Perc model is random in orientation, tie length, and tension. Properties: Redundancy (reliability), fractal, signal channeling, ie analogy to prey falling on spider web; adaptibility.

Ingber, D. E. J Cell Sci 2003;116:1157-1173Three sequential fluorescent images from a time-lapse recording of the same cell expressing GFP-tubulin showing buckling of a microtubule (arrowhead) as it polymerizes (from left to right) and impinges end-on on the cell cortex at the top of the view [reproduced with permission from the National Academy of Sciences (Wang et al., 2001)]

What determines connections?What happens if the chain density is so low that a connected lattice doesn't exist?Consider the two-dimensional square lattice, on which bonds have been placed randomly

p low p higher

Percolation (gelation)Consider a model: site percolation on a square lattice. This uses an L x L square matrix of 1s and 0s, called the site matrix. A 1 represents an occupied site and a 0 an empty site. The sites are occupied randomly with some site occupation probability, p. A cluster is a set of occupied sites all of which are connected either horizontally or vertically, i.e. an occupied site belongs to a cluster if a member of the cluster is either above, below, left, or right of it. A spanning cluster has an element in both the top and bottom rows of the site matrix.

Bond/site percolation

Probability of connectionhow does the probability of a site matrix having a spanning cluster depend on the site occupation probability p? Site percolation is an example of a critical phenomenon. There is a special value of p, called the critical site occupation probability pc, such that for p < pc spanning clusters never occur. While for p > pc they always occur. The case p=pc is called a critical point.

Critical point: Phase transitionAt critical points a qualitative transition in the behaviour of a system occurs: typically between ordered and disordered states. In gelation, the transition is from no spanning clusters to always spanning clusters. Such qualitative changes are known as phase transitions. Other examples are the freezing of water and demagnetisation of a ferromagnet at the Curie temperature. Strictly, critical points only exist for infinite systems. For finite systems, like we will investigate using the computer, the transition is not sharp but smeared out over a parameter range.

Critical probabilityfor infinite systems, a connecting path appears at the connectivity percolationthreshold pC pC = 0.5 for a square lattice in 2D and pC ~ 0.35 for a triangular lattice in 2D

Direct approachOne approach to finding the critical site occupation probability is to look at every possible LxL matrix. Since there are L2 matrix sites and each can have two values the total number of site matricies is 2L2. Here is how this number grows with L: L

123456

All combinations of matricesL

1234562L2

21651265,53633,554,4326.87 x 1010

Monte Carlo short-cutA powerful method for dealing with the problem is the Monte Carlo method. This involves random sampling of the entire ensemble to obtain a (hopefully) representative sample. Deductions are then made from this sample and assumed to hold for the entire ensemble. We will apply the Monte Carlo method to our problem by generating small samples of random matricies. We use these to estimate the critical site occupation probability and a critical exponent.

Stress and Laplace LawsFFRectangular barCells and balloons

Membrane Tension

Thin walled sphere

Wall stress in a thick sphere

To find equilibrium forces:S Fup = SFdown

Feedback Control of Volume

Anucleated cellsRBCs-

Platelets

Typical Kw = 2 X 10 11 cc./dynes*sec

Water flow

Fibroblasts

Endothelial

Myocytes

TransformationBenign 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.

Polymerization ratedefinitions: ends of filament are not equivalent; n = number of monomers in a single filament; t = time; [M] = concentration of free monomer in solutioncapture rate of monomers by a single filament is proportional to the number of monomers available for capturedn/dt = +kon [M] (capture) (1) kon = capture rate constant, with units of [concentrationtime]-1 release rate does not depend on [M]dn/dt = -koff (release)

Linkers

Kineticsdn/dt = +kon [M]

Transform pairs

Transform backStep inputIf non-zero initial conditions then add :

Cell spreading: polymerizationVp = VL : n = Jp/rn Jp is # of actin per second per area and rn is # actin per volume at interface

Rules for analog simulation1. Write diff-Q with highest order on left2. 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.

Simulink Major Blocks

Block Name

Description

Important Sources

Injects an input to the attached block

Constant

Injects a constant value

Sine Wave

Injects a sine wave

Pulse Generator

Injects a rectangular pulse

Signal Generator

Injects waves of selected forms

Step

Injects one rectangular edge

White Noise

Injects random noise

Important Sinks

Graph

Plots output, axes, and permits printing

Scope

Same as oscilloscope (WYSIWYG)

Kitchen

(Not available in this version)

Important Operators

Discrete

Integrator 1/s

CartilageCell inside is a gel : a state of matter produced by electro-osmosis due to charged polymers : water pressure inside.

Swelling pressure = osmotic pressure- elastic (compressive) pressure

H20

Polymer-polymerIntra-polymerosmosis

FactorsCa++, pH

12

Balance of forces in cytogelNetwork condensed by shielding charge or reducingEnergy - I.e. divalent cations, Acetone, low temp.

Dynamic equilibrium

Mechanical Models

e

s

e

s

Voigt ModelsCRI

Voigt solution

Non-zero I.Cs.

Cell crawling