February 8, 2005 : HomeworkRead Ch. 4, 5; AppendixBiophysics ReviewCell Types; Mechanical TestingPercolationKinetics

ProkaryotesMost have elevated osmotic pressure, I.e a few tens of atmospheres. Challenger Deep sea dropped to 10,896 m in last 8 million years (Deepest place in ocean) Deficient in CaCO3 at that depthForaminifera (shelled protists) quickly evolved soft, non-calcareous shellsLikely are highly pressurized.

Biomembrane as an isotropic materialBilayer compression resistance, KA = 4 gg= 0.04 J/M2 (g = surface tension)

Homogeneous lipid sheet: BiomembraneStretching membrane thins itexposing hydrophobic core toWater. Rupture occurs at 2-10% area expansion, so say lysis tension ~ 0.016 J/M2. For a 1 mm cell : P= 64,000 J/M3 ~ 0.6 atm. at rupture.10 atm = 106 J/m3

Life @ 1,200 atmoshperesHow thick does the membrane need to be?How thick does the wall need to be?Compressibility properties:

Cell Walls for strengthHow thick does wall need to be to withstand normal pressures inside a bacterium, I.e. 30-60 atm. ?Lets say lysis occurs when wall tension exceeds 5% of KA. We can approximate KA by KVd, and for isotropic wall material, Kv ~ E, so, assume a material E= 3 x 109 J/m3tfailure= 0.05 KA= RP/2=0.05 E d So to not fail, d> RP/E So for R = 0.5 mM, P= 10 atm,

Thick wall sphere

Thick walled sphereEquilibriumPressure insideAverage stress in wall

Pressure from outside

Pressurized both sides

Alternate method (not too thick wall)

Laplace Law for Cylinder under pressureHomework:Find wall stress for cylinder.Calculate stress for a foraminefora notAssuming a thin wall.

Compression of a networkA cartesian lattice of fibers (a) subject to compressive strain (b). The boundary conditions are that the angles between the segments arising from each junction are fixed at 90 and the deformation consists in movement of the junctions toward each other along the three orthogonal axes of the lattice, with no shear deformation. The junctions, however, are allowed to rotate ascompression progresses. This is equivalent to the boundary conditions in Feynman et al.36 in which fiber ends are fixed but direction changes under compression.

Solution to Donnan Problem

A.

B.

C.

D.

Power from electrochemical gradients (I.e batteries)Driving Force determined by NernstDistributed Model

Electrical Model of Cell MembraneMolecular modelCa Wave in Oocyte

dB

Inside

Outside

E Na

gNa

E K

g K

C m

Mechanotransduction What opens channels?

Types of mechanical analysisKinematics - just the connectionsStatics- forces without motionDynamics- forces with motion Rigid versus deformable body FBDsFEL FBL FBR FER

Loading TypesTension- compressionShearReaction TractionFrictionBendingUniaxial/bi-axial

Cytomechanical forces:Gravitational:Muscle contraction:Contact:Buoyant:Hydraulic: (Static or dynamic)Pneumatic

Cell Deformation and StiffnessMost cells are constantly deformed in vivo by both internal and external forces. Experimental deformations can be done by poking, squishing, osmotic swelling, electrical/magnetic fields, drugs, etc. Cells have both area and shear stiffness, mostly due to the cytoskeleton, although lipids contribute some.

Material ParametersModuli: Youngs, area, shear, bending (flexural)Stiff versus compliantStrength versus weaknessBrittle versus ductileIncompressible/CompressibleFailureUltimate tensile strengthHardness: Mohs scale

Comparative Mechanical PropertiesStrain eSteel Wood Bone

CellsCellularpre-stress

Chart2

0.0002

0.007

14

21

210

1200

Material

Modulus (GPa)

Comparative Stiffness

Sheet1

tissue0.2-0.69897000430.0002

rubber70.845098040.007

wood140004.146128035714

bone210004.322219294721

steel2100005.3222192947210

diamond12000006.0791812461200

10.8

299.6

0.0360480641

Sheet1

0

0

0

0

0

0

Sheet2

0

0

0

0

0

0

Material

Modulus (GPa)

Comparative Stiffness

Sheet3

Elasticityut tensio sic visYoungs Modulus: Stress over strainShear Modulus: Related to PoissonComparative StrainsComparative Stiffnesses

Poissons EffectswellingIncompressibleMeans no volume change

Elastic Behavioursn < 1 n< 0 E = s/e KA = P/DA/AUnixaxial stress Pressure 1 2

Applying forces (testing types)

Tension or Compression

Uniaxial

Pressure

Biaxial

Shear

Bending

Twisting

Testing methodsQ: What are the relative resolutions?

AFMQ? Why doesnt the AFM needle poke right through?

Micropipet methods

Magnetic tweezersWang et al, Science

Pulling on CSK

Shear and compression

Example: Blood flow forces

Optical TweezersHigh resolutionRefractivity of beadTrapping in the beamLimited force

Optical Tweezer

Necturus erythrocytes loaded with fluo-4 (10 M) and exposed to UV light emitted from a mercury vapor bulb and filtered through a FITC cube (400x). (A) Cells display little fluorescence under isosmotic conditions (n=6). (B) Addition of A23187 (0.5 M) to the extracellular medium increased fluorescence under isosmotic conditions (n=6). (C) Exposure to a hypotonic (0.5x) Ringer solution increased fluorescence compared to basal conditions (n=6). (D) A low Ca2+ hypotonic Ringer solution (5 mM EGTA) did not display the level of fluorescence normally observed following hypotonic swelling (n=6). Light et al. SwellingRBCs

Stimulation Protocols

Impulse Step Sinusoid Ramp

TIME

Magnitude

Figure 4.2 Modes (top) and timing protocols (lower) of force application

Bone Loading Waveforms

Sickle Cell: A gel problemSingle point defect causes Hbs- a polymerizing tendency in deoxygenated stateThe stiff and deformed cells damage vesselsMain approaches:1. Controlling kinetics of polymerization2. Regulating stiffness (rheology) of sickle cells.

Thermal shape variationsStiffFlexible

(a)-(c) Serial images of a 23 mm long relatively stiff fiber. (b) and (c) are, respectively, 21.9 and 41.4 seconds after (a). There is little visible bending (see also Figure 2(a)), consistent with a long persistence length, lp . 12.0 mm. (d)-(f) Serial images of a 20 mm long flexible fiber. (e) and (f) are, respectively, 51.8 and 60.8 seconds after (d). There is marked bending and a short persistence length, lp.0.28 mm (see also Figure 2(b)). The fibers undergo diffusional motion and hence are not adhering to a glass surface, rather are free in solution, a necessary condition for using statistical mechanics to obtain persistence lengths. The width of each frame is 25 mm.

Statistics of fluctuations in 1 dimension

Statistical Mechanics

Dilute Semi-concentrated ConcentratedFloppy Chains

Rods Isotropic Nematic

Harmonic motion (undamped)Gel motion follows simple rulesModel will predict dynamic and Static equilibrium.Natural Frequency

Damped Spring

Viscosity & Elasticity 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 elasticity; it cannot deform as a solid. Cells generally behave as solid-liquid composites. V-E tools can quantify their behaviour, since the models separate viscosity from elasticity in a kind of finite element model.

Maxwell Model: Differential method

Maxwell model: Laplace Method Viscosity: Pascal-sec For a step inputMechanicalImpedance.

Compliance +Slipperinesst=h/E

Gel ModelMake a complete model and label all parametersDescribe the output, relating what happens and why.What is the time constant?State the assumptions and simplifications

Mechanical Terms ReviewStatics and dynamicsKinematics and kineticsVector and scalarsForces, resultantsDeformation

ClassworkAdd damping to your model of cytogelDescribe how you can model thermal fluctuations in cell diameter, and list all the elements. List assumptions.Write the model equation for the above.Complete a simulink model of the above, and do all labelling, including all parameter values.