Random Walks and Diffusion. APC 514 December 5, 2002 Cox&Shvartsman

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1 Random Walks and Diffusion APC 514 December 5, 2002 Cox&Shvartsman

2 Cell motility over adhesive substrates: a periodic phenomenon Simple model for linear motion: DIMILLA PA, BARBEE K, LAUFFENBURGER DA MATHEMATICAL-MODEL FOR THE EFFECTS OF ADHESION AND MECHANICS ON CELL-MIGRATION SPEED BIOPHYS J 60 (1): JUL 1991

3 Models and measurements Migration velocity Adhesion strength Ahmed Z, Underwood S, Brown RA. Simple model for linear motion: DIMILLA PA, BARBEE K, LAUFFENBURGER DA MATHEMATICAL-MODEL FOR THE EFFECTS OF ADHESION AND Low concentrations of fibrinogen increase cell migration speed on fibronectin/fibrinogen composite cables. Cell Motil Cytoskeleton May;46(1):6-16. MECHANICS ON CELL-MIGRATION SPEED BIOPHYS J 60 (1): JUL 1991

4 Each step of the process is highly regulated

5 Cell paths are random Neutrophils Epithelial cells 2 different cells 2 different conditions

6 Language for trajectories: random walk theory Probability primer Position jump process Exact solution Binomial distribution Mean and variance Random variables Distribution function Observations of Berg and Brown Exponentially distributed random variable

7 Position Jump Process (1) N steps, net displacement - m: N m N ( N odd/even m odd/even) N! 1 P N( m) = N+ m N m ( )!( )! < >= < < > >= 2 m 0 ( m m ) N Approximate solution: log 1 ± P ( ) 2 N m e π N 2 m 2N use log( n!) ( n+ 1/ 2)log n n+ 1/ 2log π + O( n ) ( ) 2 2 m m m m ± + O( ) 2 2 N N 2N N N 1 & 1/2 1/2 diffusive behavior S. Chandrasekhar, Rev. Mod. Phys. 15, 1, 1-89, 1943

8 N Binomial distribution

9 Position Jump Process (2) step size: l frequency of jumps: λ Continuous space and time: pxtdx (,) = P x ( ) dx e lt N m = λ 2 D λl 2 /2 p ( x, t ) = e 21 π Dt 2l 2πλlt x 2 4Dt 2 p( xt, ) > 0 for all dx t p p = ( D ) with p( x, t = 0) = δ ( x) t x x p p Rewrite: = J flux J D t x x Approximation for large times < x 2 >= 2Dt earlier times: correlations in motion -> cell migration is like that S. Chandrasekhar, Rev. Mod. Phys. 15, 1, 1-89, 1943

10 Model Organism: E.Coli Size: 1 µm Swims by rotating ~ 6 flagella Speeds: up to 10 µm/s

11 Bacteria can be attracted/repelled by chemicals Mechanism? Chemoreceptors in bacteria. attractant Adler, 1969 Science READ! This is sensing, not metabolism Based on genetic approach!!! No molecules yet Macroscopic phenomenon: flux of bacteria = F(gradient of chemicals)

12 Random Motility and Chemotaxis

13 Trajectories In the absence of chemical gradients, a swimming bacterium executes a three-dimensional random walk consisting of runs of swimming in a straight line punctuated by tumbles

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16 From Trajectories to Microscopic Parameters of Cell Migration (Velocity Jump Process)

17 Berg and Brown, 1972 P{ T > t} run or P{ T > t} tumble t run 10t tumble t 1. Runs punctuated by tumbles 2. Both runs and tumbles are exponentially distributed 3. Runs are longer than tumbles 4. Constant velocity MODEL: instantaneous tumbles (neglect tumble time)

18 MODEL: instantaneous tumbles (neglect tumble time) Velocity Jump Process -x λ + - λ v +x 1. Continuous space & Continuous time 2. At every point: right- and left-moving cells 3. Follow a single cell & a population of cells

19 Velocity Jump Process PT { t} = 1 exp( λt) v=± V P{ T t} Inversion method 1 t Simulation λ =1; t=[0]; x=[0]; V=1; STEPS=30 for j=1:5 end; for i=1:steps; T=-log(1-rand(1))/λ; N=length(t); t=[t;t(n)+t]; x=[x;x(n)+v*t]; V=-V; end; plot(x,t); hold on;

20 Velocity Jump Process

21 Velocity Jump Process (1) instantaneous changes in velocity p t p t p + v = λ p + λ p x p v = λ p + λ p x x λ + v - λ +x + pxt (, ) p( xt, ) + p( xt, ) total probability + jxt (, ) vp ( ( xt, ) p( xt, )) flux 2 2 p p 2 p + 2λ = v 2 2 t t x p(0, x) j(0, x) px (,0) = p0( x), = t x v 1 2 t xt () x pxtdx (,) t (1 e λ = ) λ 2λ H.G. Othmer, S.R. Dunbar, W. Alt, J.math.Biol, 26, , 1987

22 Velocity Jump Process (2) ballistic and diffusive regimes xt () 2 = 2 t 1: x ( t) v t t x t 2 1: ( ) vt λ T v D ν 2 2λ 2 λ "persistence time" velocity pxt (, ) jxt (, ) = D only when λ x expression for flux has memory: t 2λt 2 2 λ( t τ) px (, τ ) j( xt, ) = e j( x,0) v e dτ x 0 H.G. Othmer, S.R. Dunbar, W. Alt, J.math.Biol, 26, , 1987

23 Velocity Jump Process (3) fitting data 1. Microscopic parameters can be extracted from data 2. Next step: expressions for macroscopic fluxes BE Farrell et al, Cell Motil Cytoskeleton, 16: , example RB Dickinson, RT Tranquillo, AICHE J, 39 (12): 1995, estimation algorithms

24 Example: Growth Factor-Mediated Cell Motility distributions MF Ware, A Wells, DA Lauffenburger, J.Cell Science, 111, , 1998

25 Cancer Lett 1997 Oct 14;118(2): Locomotory phenotypes of human tumor cell lines and T lymphocytes in a three-dimensional collagen lattice. Niggemann B, Maaser K, Lu H, Kroczek R, Zanker KS, Friedl P. Active cellular locomotion is a feature of such diverse cell types as lymphocytes and cancer cells. The locomotory phenotype of a cell should ultimately reflect the biochemical basis of different migratory strategies. We investigated the locomotory behavior of five epithelial cell lines and one non-epithelial human cell-line as well as human CD4+ T lymphocytes in a three-dimensional collagen type I matrix using time-lapse video microscopy and computer assisted cell-tracking. Migration velocity was up to 70 times lower in tumor cells ( microm/min) as compared to T lymphocytes (7-7.5 microm/min), whereas the percentage of spontaneously active cells was up to twice as high in tumor cells (80-90%) in comparison to T lymphocytes (54%). Persistence, i.e. the degree of roaming, varied appreciably between the different cell types. In conclusion, velocity and persistence may describe distinct migration strategies in different cell types.

26 More Complex Models 1. Higher dimensions: turning operators, anisotropy, etc Dickinson RB.A generalized transport model for biased cell migration in an anisotropic environment. J Math Biol Feb;40(2): Othmer HG, Dunbar SR, Alt W. Models of dispersal in biological systems. J Math Biol. 1988;26(3): Finite tumble time Schnitzer MJ.Theory of continuum random walks and application to chemotaxis. Phys Rev E, 1993 Oct;48(4): Internal state random walks Grunbaum D Advection-diffusion equations for internal statemediated random walks SIAM J APPL MATH 61 (1): JUL

27 From Microscopic Parameters to Macroscopic Balances (Expression for the Chemotactic Flux)

28 Macroscopic Flux (1) n t n t + + ( vn ) = λ n x λ n ( vn ) = λ n x λ n total cell density: n n + n + + flux: j v( n n ) steady state : j eq = n x v x ( λ λ ) + ( λ + λ ) 2 + v nv vn

29 Macroscopic Flux (2) T p µ Tv V c [ λ λ ] persistence time p 2 random motility coefficient + T v( λ λ ) chemotactic velocity p n v jeq = µ + Vcn Tpvn x x in phenomenological models n Three contributions to flux: jeq = µ + αn x 1. random motility 2. chemotaxis (right- and left- moving cells reverse differently) 3. chemokinesis (gradient in cell velocity) To couple to external concentration field, combine with the experimentally determined dependencies of µ and T p

30 Flux in a 1D Gradient (1) Motivated by Berg & Brown 1972 Experiments T runs & tumbles tumble duration is zero use velocity jump process in 1D motion in a gradient λ p p = ( λ + λ ) + / + / T + 1 = p (1 ψ) / 2 T : is the tumbling probability ψ : "directional persistence" probability of reversing after tumbling random motility gradient chemotaxis receptor-mediated mechanism: N # of occuppied receptors B run time B ( ) dn C dt

31 Flux in a 1D Gradient (2) relate to the frequency of tumbles τ = τ dn b dt 0e σ + Nb λ + λ = (1 ψ)cosh( σv ) x + Nb λ λ = p0 (1 ψ)sinh( σv ) x p = 1/ τ = + / + / T time derivative seen by the bacterium p e σ dn N N = ± v dt t x B B B 0 dn B dt 2 v Nb µ = sec h( σv ) p (1 ψ ) x 0 Nb Vc = vtanh( σv ) x

32 Flux in a 1D Gradient (3) Simple Ligand/receptor Equilibrium N B Ntotalc dnb NT Kd = = K + c dc ( K + c) 0 D 2 v c dnb µ = [cosh( σv )] p (1 ψ ) x dc cdnb Vc = vtanh( σ v ) x dc d 2 1 chemotactic coefficient, χ small gradients: 0 v 2 2 B, dn V c c v µ = = σ p (1 ψ ) dc x If the model is correct: macroscopic flux can be estimated from data on binding and microscopic parameters for cell migration

33 Flux in a 1D Gradient (4): Analysis V c µ C x C x 1. Random motility coefficient increases with temporal gradient 2. Random motility coefficient is a decreasing function of spatial gradient: at large gradients all cells swim in one direction 3. Chemotactic velocity has a limiting value: the population can not move faster than the maximal cell speed

34 Chemotactic Wave Paradox Observation aggregation to the source of chemical wave pulse of camp is nearly symmetric Devreotes & Tomchik, Science 212, 443-6, 1981 Simple-model: symmetric chemotactic velocity no net directed motion [camp] c(kx+vt) Worse: cells stay longer in the negative gradient region Prediction: cells move away from the wave source What is the problem? Wave source x Experiment: Cells move only in the wave front and not in the back => chemotactic response can not be determined by the concentration gradient alone χ = χα ( ) chemotactic sensitivity

35 Model: Soll, Wessels, Sylwester, 1993 Translocation phase: Rapid & persistent translocation; suppressed lateral pseudopods formation; elongate shape Peak of the wave: suppression of pseudopod formation and cellular translocation; freeze in cell morphology Decision phase: high frequency of random pseudopod formation; nonpolar cell morphology; no net translocation ~35 sec 10-6 camp ~145 sec ~30 sec ~180 sec Back of the wave: increased frequency of random pseudopod formation; loss of elongate cell morphology; little net translocation 10-8 camp 10-8 camp