Biology Reference
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and an ensemble average computed. 76 While the topic of SDEs may seem
exotic to many computational biologists, it is more widespread than one
would think. Simulating, for example, a reaction-diffusion model with
stochastic reactions amounts to numerically solving an SDE. 64,65
4.4. Methods for Continuous Deterministic Models
Continuous deterministic models as represented by PDEs can be solved
using any of the discretization schemes from numerical analysis. 77,78 The
most common ones include finite difference (FD) methods, 79 finite
element (FE) methods, 80,81 and finite volume (FV) methods for conser-
vation laws. 82,83 FD methods are based on Taylor series expansions 84 of
the spatial field functions and approximation of the differential operators
by difference operators such that the first few terms in the Taylor expan-
sion are preserved. FE methods express the unknown field function in a
given function space. The basis functions of this space are supported on
polygonal elements that tile the computational domain. Determining
the unknown field function then amounts to solving a linear system of
equations for the weights of the basis functions on all elements. FV
methods make use of physical conservation laws such as conservation of
mass or momentum. The computational domain is subdivided into dis-
joint volumes, for each of which the balance equations are formulated
(change of volume content equals inflow minus outflow) and numeri-
cally solved.
All of these methods have the common property that they require a
computational mesh — regular or irregular — that discretizes the com-
putational domain into simple geometric structures such as lines (FD),
areas (FE), or volumes (FV) with the appropriate connectivity. For
complex-shaped domains as they frequently occur in biological systems
(cf. Fig. 1), it can be a daunting task to find a good connected mesh that
respects the boundary conditions and has sufficient regularity to preserve
the accuracy and efficiency of the numerical method. Mesh-free particle
methods 85-87 relax this constraint by basing the discretization on point
objects that do not require any connectivity information. While particle
formulations are the natural choice for discrete models, their advantages
can be transferred to the continuous domain using continuum particle
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