Computer Simulations of Mechano-Chemical Networks Choreographing Actin Dynamics in Cell Motility - Computational Modeling of Biological Systems - page 234

Summarizing, actin network dynamics in filopodia and lamellipodia arise from
a complicated mechano-chemical system representing a great challenge for un-
derstanding using both experimental and theoretical approaches. Hence, computer
simulations of this mechano-chemical signaling network, based on microscopic
physics, may shed light on the mechanisms of network growth regulation by various
proteins, complementing and providing guidance to the related experimental efforts.

2.1 Stochastic Simulations of Biological Mechano-Chemical

Networks

Chemical part of the signaling network that regulates actin growth dynamics
consists of various proteins whose numbers evolve in time due to numerous
enzymatic and binary chemical reaction events. Most commonly, chemical reaction
dynamics is analyzed by solving the corresponding system of ordinary first-order
differential equations, with time as the independent variable and concentrations
of interacting species as the time-dependent variables. This approach is known as
chemical kinetics. The continuous concentrations in these equations correspond
to the average numbers of molecules in a unitary volume. However, in reality,
chemical reactions are discrete stochastic processes, where reactants encounter each
other randomly, and may react or not in any given collision. Even unary reactions,
such as radioactive decay are random events at a single atom or molecule level.
However, when the numbers of reacting molecules are large, certainly on the order
of Avogadro's number, the relative stochastic fluctuations of these numbers are
negligible. In such cases, time evolution of averages gives an accurate description
of the system dynamics, and deterministic chemical kinetics can be safely used,
as usually done in case of macroscopic and even mesoscopic chemical reaction
networks.

2.1.1 Reaction-Diffusion Master Equation

A crucial feature of biological signaling networks is that often the average numbers
of molecules of each reacting protein in the relevant spatial region are very low,
on the order of several molecules. In many cases, most of the time, there are no
molecules of certain protein in that spatial region, with only one or two appearing
for short periods of time, which produces average number of molecules lower than
1. In such a case, the fluctuations, which can be roughly estimated as the square
root of the number of molecules, are on the same scale as the average and can even
exceed it by an order of magnitude (in other words, all appearing molecules are
noise). In these cases, chemical kinetics may not provide a physically meaningful
description of the time evolution of a biological signaling network, therefore, the
dynamics of such network has to be treated stochastically.