migration steps

1. protrusion at the frontal edge 2. focal adhesions at the frontal edge 3. contraction at the rear edge 4. focal adhesions at the rear edge

active strains

- decomposition of the deformation gradient - intensity of the strain - cyclic manner - chemotaxis - pulsatile movement

active strains =

protrusions

decomposition of the deformation gradient is measured by

F = Fse * Fsa

intensity of the strain is measured by

Fsa = e0

cyclic manner is measured by

Fsa = e0sin(2π * t/T)

chemotaxis is measured by

Fsa= e0sin(2π * t/T) * i(θ)*i(θ) iθ = cosθ(t)ix+sinθ (t)iy

∂|Fsa|/∂t > 0

protrusion

∂|Fsa|/∂t < 0

contraction

frontal and rear edge adhesion forces are measured by

regularized Heaviside functions

generalized Maxwell model

Fsa = polymerization/depoly. Os/Fse = actin filaments Ofe, Ffe = organelles Ofv, Ffv = cytosol

output of maxwell model

O = Os + Of F = Fs + Ff

numerics generalized maxwell model

2 eqns ρa = Divx(JσF^-T) θ̇ + θ − θinfinity/τinfinity

obstacles

- regularized heaviside function - additional adhesion forces

2 strategies to avoid obstacles

- velocity sensor / "run-and-tumble" - distance sensor / "look-and-run"

velocity sensor equation

Vc = 1/|Ωc|[differential]Ωc vdV

filtering the velocity helps to

avoid oscillations d^2Vcf/dt^2 + 2fωf * dVcf/dt + ωfVcf = Vc

in the "run-and-tumble" model

- if there is no obstacle: cell migrates towards θinfinity - if there is an obstacle: some rendition of regularized heaviside functions

pseudopods

- membrane protrusions - perpendicular to the membrane

splitting of existing pseudopods

zig-zag trajectory

new pseudopods at not active regions

any direction

measuring the active strain within migration with pseudopods

- sum of the strain in each pseudopod - gradient along the pseudopod axis - uniform inside the pseudopod - gradient + uniform

static pseudopods

temporal sensing model

- protrusion - multiple simultaneous pseudopods - minimal distance between pseudopods (5 degrees) - contraction - pseudopod best oriented towards the source - same frame as the source

spatial sensing model

- protrusion; only one pseudopod at the time

durotaxis and polarity

isotropy-anisotropy of the cell

durotaxis - functions to analyze substrates

functions for stiff, soft, and viscous forces stiff: x < x0 soft: x > x0

eqns to analyze equilibrium and constitutive law of the cell

durotaxis - analyzing the solid phase of a cell

σse=1/Jse * FseSseF^Tse anisotropic hyperelastic saint-venant material

durotaxis - anisotropic hyperelastic saint-venant material

Sse = RCloc(R^TEseR)R^T R: found via rotation matrix; ploc = Rp Cloc: found via local elastic tensor Ese: found via green-lagrange tensor

durotaxis - in the local elastic tensor A and B come from their own fcns...

soft substrate: E0 = Ea stiff substrate: Ea = 0.1E0

durotaxis - green-lagrange tensor - fluid phase

diff eqns

durotaxis - adhesion forces

functions i can't make sense of

active strain of cell in durotaxis

- radial protrusion and contraction - lamellipodium in the direction of migration d

durotaxis - soft vs. stiff substrate

there is a difference that can only be seen in images that i can't put here. slides 58-61

durotaxis (discrete approach)

- cellular potts model - boltzmann probability function to validate the trial spin - hamiltonian function - net energy difference

cellular potts model

- 3D regular lattice -> center (x) and label σ(x) - subdomains with same label form discrete objects: medium, cell, substrate - iterative and stochastic reduction of energy - monte carlo-boltzmann dynamics

iterative and stochastic reduction of energy

hamiltonian function (H)

monte carlo-boltzmann dynamics

behavior of biological individuals

each timestep with in cellular potss model

- randomly chosen lattice site (xsource) - allocation of its spin σ(xsource) to one neighbours xtarget randomly selected

durotaxis

does not make sense to me