Random walk models, cell migration

A detailed comparison of the random walk and the Markov chain models using experimental data for migrating endothelial cells has shown [148] that aU five locomotory parameters defined above affect the speed S and persistence time P estimated by the persistent random walk model. The persistent random walk model can provide a measure of the speed of locomotion S, appropriately weighted to account for the frequency and duration of ceU stops (or slow-downs). The persistence time P, however, is a composite measure of all five locomotory parameters mentioned above. Hence, it may not be always possible to extract all the information necessary to describe cell locomotion from the two parameters of the persistent random walk model. The Markov chain approach, however, also has its drawbacks. Perhaps the most serious drawback is that the values of ceU speeds computed according to Equation 35.5 depend on the time interval A t between ceU observations. Accurate estimations of locomotion speeds with Equation 35.5 are possible only when (a) ceU trajectories are not very tortuous and (b) the cell positions are observed at short time intervals At. This time interval must be carefuUy chosen using Richardson plots to minimize errors in the computation of ceU speed [148]. 

Modeling cell migration with persistent random walk models... 

Physical approaches not reqniring the numerical solution of the differential equations have also been developed. For example, an atomistic model considers the cell as a domain filled with a popnlation of particles and diffusion is simulated by the random walk of the particles within the domain (53, 54). The current is computed by counting the number of particles that reach the electrode per unit time. Convection and migration can even be included. Another model, the box method nsed in the early days of electrochemical simulation (55), divides the solntion in thin slabs (boxes wherein the concentration is assumed to be uniform) and calculates the movement of species between slabs nsing Fick s first law of diffusion. Althongh more intuitive, these approaches are in fact eqnivalent to solving the transport eqnation. 

As we mentioned in Section 35.2.2, however, mammalian cells migrating in isotropic environments execute persistent random walks. Dunn [19] and Othmer and coworkers [139] developed the following mathematical model to describe persistent random walks ... 

In accordance with experimental observations, the model assumes that cells migrate by executing persistent random walks [49,121,122] as they go through their division cycle. In a uniform environment, the direction after each turn is randomly selected. However, cell movement can be biased to simulate chemotaxis or haptotaxis. If the cell does not collide with another cell, this persistent random movement continues until the end of the cell s current division cycle upon which the cell stops and divides into two daughter cells. Cell division is asynchronous and the distribution of cell division time tj is a measurable characteristic of each cell phenotype. 

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