Presentation: Random walk models in biology E.A.Codling et al. Journal of The Royal Society Interface R EVIEW March 2008 Random walk models in biology Edward A. Codling 1,*, Michael J. Plank 2 and Simon Benhamou 3 1 Department of Mathematics, University of Essex, Colchester CO4 3SQ, UK 2 Department of Mathematics and Statistics, University of Canterbury, Christchurch 8140, New Zealand 3 Behavioural Ecology Group, CEFE, CNRS, Montpellier 34293, France Presented by Oleg Kolgushev Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Introduction to Random walk theory Fundamentals of Random walks – Simple (isotropic) Random Walks (SRWs)isotropic – Biased Random Walks (BRWs), waiting times, higher dimensions Biased Random Walks – Spatially dependent movement probabilities, Fokker–Planck equation – General diffusive properties and model limitations – Random walk with barriers – Correlated Random Walks (CRWs) and the telegraph equation Correlated Random Walks Random walks as models of biological organism movement – Mean squared displacement of CRWs – Mean dispersal distance of unbiased CRWs – Tortuosity of CRWs Tortuosity – Bias in observed paths – Reinforced random walks – Biological orientation mechanisms Conclusion and future work Contents Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. RWs are traced back to Brownian motion and classical works on probability theory Physicists extended RWs into many important fields: random processes, random noise, spectral analysis, stochastic equations The first simple models of movement using random isotropic walks are uncorrelated and unbiased (SRWs is a basis of most diffusive processes) Correlated random walks (CRWs) involve a “persistence” between successive step orientations (local bias) Paths with global bias in the preferred direction (target) are termed Biased Random Walks (BRWs) CRWs and BRWs produce BCRWs RW theory is applied in 2 main biological contexts: the movement and dispersal of organisms, and chemotaxis models of cell signaling and movement.chemotaxis Introduction Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. A walker moving on an infinite one-dimensional (x) uniform lattice. Motion is random. x 0 = 0, ∆x = δ, ∆t = τ The probability that a walker is at mδ to the right of the origin after n time steps (even) (2.1) Taking the limit δ, τ -> 0 such that δ 2 /τ = 2D gives the Probability Density Function (PDF) of walker location (diffusion equation) (2.2) The mean location and the mean squared displacement (MSD) defined by (2.3) Simple Random Walks Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. A walker moving on an infinite one-dimensional (x) uniform lattice with probabilities moving to right r, left – l, and not moving (1-l-r), x 0 = 0, ∆x = δ, ∆t = τ Taking the limit δ, τ -> 0 and rewriting (2.4) as Taylor series about (x,t) gives partial differential equation (PDE) (2.5) where ϵ=r-l; κ=l+r; Let exists condition (2.6) Under these limits higher order terms in (2.5) tend to zero, giving (2.7) (drift-diffusion equation) Solution of (2.7) with initial condition p(x,0)= δ D (x) (Dirac delta function) is Biased Random Walks Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Substituting (2.8) into (2.3) we can get MSD In contract with SRW the MSD is proportional to t 2 so the movement propagates as wave and more appropriate measure is the dispersal about the origin A similar definitions can be extended into N-dimensional lattice giving standard drift-diffusion equation (where u is the average drift velocity) with solution (2.12) The mean location and MSD are calculated in similar way Biased Random Walks Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Biased Random Walks Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Spatially Dependant Movement Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. General diffusive properties and model limitations Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Random walk with barriers Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Correlated Random Walks and the telegraph equation Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Mean squared displacement of CRWs Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Mean dispersal distance of unbiased CRWs Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Tortuosity of CRWs Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Bias in observed paths Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Reinforced random walks Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Biological orientation mechanisms Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Conclusion Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Introduction to Random walk theory Fundamentals of Random walks – Simple isotropic Random Walk (SRW) References Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30

Presentation: Random walk models in biology E.A.Codling et al. Bias: preference for moving in a particular direction. Isotropic: uniform in all directions. Sinuosity: a measure of the tortuosity of a random walk. Tortuosity: the amount of turning associated with a path. Central Limit Theorem: mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed Correlated random walk (CRW): random walk with persistence Taxis: directional response to a stimulus (cf. kinesis). Examples include chemotaxis, phototaxis, gyrotaxis. Glossary Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas /01/ /01/30