The Structure, Function, and Evolution of Biological Systems Instructor: Van Savage Spring 2011 Quarter 5/5/2011

Global signature of diffusion Random walk x(t+1)=x(t)±1 ->x 2 (t+1)=x 2 (t)+2x(t)+1(1/2 of time) =x 2 (t)-2x(t)+1(1/2 of time) =(1/2)* +(1/2)* = +1= +2 Iterating this gives: =Number of time steps~t

Combined Effects Person trying to walk north (directional) through a busy intersection (nondirectional) Net Flow=Directional Flow+Nondirectional Flow Diffusion Equation (Also known as Kolmogorov forward equation)

Schienbein et al.

Migration of cells Cells move randomly while also guided in direction of inflammation, chemical signal, or electric field. Movement observed in wound healing, embryogenesis, and granulocytes. Can also modify this for chemotaxis.

Langevin equation can describe this White noise defined by the following properties

Langevin movement corresponds to Kolmogorov forward equation/ Fokker-Planck for probability Steady state solution

Match of theory and data

Solution to Langevin and correlation function Correlation Relative Correlation

Match of theory and data for correlation

Similar equations and analyses apply for the distribution of angles Langevin Steady state probability disribution

Distribution for different electric fields

Expected value for cosine angle: theory and data

Expected square of displacement

Predictions and Data

Time evolution of probability distribution with no electric field

Time evolution of probability distribution with electric field turned on

Some discrepancies between theory and data for full case

Sources of discrepancy 1.Not at steady state during shift in electric field 2.Noise may not be white noise 3.Deterministic part may have frequency dependence 4.There is some weak memory to the “noise”