Computer Simulations of Mechano-Chemical Networks Choreographing Actin Dynamics in Cell Motility - Computational Modeling of Biological Systems - page 236

length are well stirred, and may be treated with zero-dimensional kinetics, while
above the Kuramoto length spatial inhomogeneities may manifest themselves, and
diffusion would need to be treated as an explicit additional process. For some
of the actin regulating chemical reactions, the Kuramoto length is on the order
of 100 nm, hence reaction-diffusion treatment is necessary above that length-
scale. In deterministic reaction-diffusion equations, the average concentrations also
depend on spatial coordinates, and chemical kinetics equations are complemented
by diffusion equations. For reaction volumes with 100 nm side, the copy number
of proteins is typically very small, hence randomness of chemical reaction events
plays an important role, and should be taken into account. The corresponding,
spatially resolved stochastic system may be simulated as a collection of connected
Kuramoto volumes, where chemical protein number lattice ( 2 ) is replicated in each
voxel. The resulting reaction-diffusion master equation (RDME) may be written as
[ 19 , 21 , 32 , 33 ],

C D P;

dP
dt

M

D (4)

where M and D are the reaction and diffusion operators, respectively. In general,
exact analytical solution of these equations is usually beyond reach, and even nu-
merical solution is expected to be computationally formidable. As discussed above,
there is direct mapping of these equations into three-dimensional QFT, indicating
both potential challenges in simulating 3D stochastic dynamics, and perhaps hinting
toward using approximate QFT techniques for accelerating simulations.

2.1.2 Detailed Modeling of Filopodia and Lamellipodia

In the models employed in a series of recent works modeling actin-based or-
ganelles [ 19 , 34 - 36 ], the space is discretized into compartments, and the basic
chemical dynamical variables are the copy numbers of all chemical species in
all compartments. Diffusion is realized by reactions of hopping between neigh-
boring compartments, with a compartment size on the order of 100 nm. The rate
of reactions corresponding to diffusional hops is equal to D= l D ,whereD is
diffusion coefficient, and l D is the compartment length. In a recent work, a careful
connection was drawn between the RDME rates and the kinetic rates from a more
microscopic description based on Brownian dynamics of reacting particles [ 32 ]. To
characterize the many-dimensional distribution function, which is the solution of the
master equation, multiple realizations of the random process are run and averages,
variances, joint distributions, and correlation functions are calculated. This approach
is analogous to running multiple Langevin trajectories to obtain characteristics
of probability distribution, which is the solution of the related Fokker-Planck
equation [ 21 ].