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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

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Presentation: Random walk models in biology E.A.Codling et al.

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  1. Journal of The Royal Society Interface • REVIEW • March 2008 • Random walk models in biology • Edward A. Codling1,*, Michael J. Plank2 and Simon Benhamou3 • 1Department of Mathematics, University of Essex, Colchester CO4 3SQ, UK • 2Department of Mathematics and Statistics, University of Canterbury, Christchurch 8140, • New Zealand • 3Behavioural Ecology Group, CEFE, CNRS, Montpellier 34293, France • Presented by Oleg Kolgushev Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  2. Contents 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) • Biased Random Walks (BRWs), waiting times, higher dimensions • 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 • Random walks as models of biological organism movement • Mean squared displacement of CRWs • Mean dispersal distance of unbiased CRWs • Tortuosity of CRWs • Bias in observed paths • Reinforced random walks • Biological orientation mechanisms • Conclusion and future work • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  3. Introduction 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. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  4. SimpleRandomWalks 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. x0 = 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 δ2/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) • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  5. Biased Random Walks 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), x0= 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)=δD(x) (Dirac delta function) is • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  6. Biased Random Walks 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 t2 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 • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  7. Biased Random Walks Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  8. Spatially Dependant Movement Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  9. General diffusive properties and model limitations Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  10. Random walk with barriers Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  11. Correlated Random Walks and the telegraph equation Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  12. Mean squared displacement of CRWs Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  13. Mean dispersal distance of unbiased CRWs Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  14. Tortuosityof CRWs Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  15. Bias in observed paths Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  16. Reinforced random walks Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  17. Biological orientation mechanisms Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  18. Conclusion Presentation: Random walk models in biology E.A.Codling et al. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  19. References 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) • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

  20. Glossary 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. • Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30

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