Biology Reference
In-Depth Information
Continuous models are valid only if the microscopic and macroscopic
scales are well separated, i.e. if Kn
<<
1; for any spatial distribution with
Kn
>>
1, discrete models are the only choice since each particle is impor-
tant and no continuum region exists. Between these two cases lies the
realm of mesoscopic models.
53
Continuous deterministic models are characterized by smoothly
varying (on length scales
L
) field quantities whose temporal and spatial
evolution depends on some derivatives of the same or other field quanti-
ties. The fields can, for example, model concentrations, temperatures, or
velocities. Such models are naturally formulated as unsteady partial dif-
ferential equations (PDEs),
54,55
since derivatives relate to the existence of
integrators, and hence reservoirs, in the system. The most prominent
examples of continuous deterministic models in biological systems
include diffusion models, advection, flow, and waves. Discrete determin-
istic models are characterized by discrete entities interacting over space
and time according to deterministic rules. The interacting entities can,
e.g. model cells in a tissue,
5
individuals in an ecosystem, or atoms in a
molecule.
50
Such models can mostly be interpreted as interacting particle
systems or automata. In biology, discrete deterministic models can be
found in ecology or in structural biology.
>
3.3. Stochastic vs.Deterministic Models
Biological systems frequently include a certain level of randomness, as is
the case for unpredictable environmental influences, fluctuations in mol-
ecule numbers upon cell division, and noise in gene expression levels.
Such phenomena can be accounted for in stochastic models. In such
models, the model output is not entirely predetermined by the present
state of the model and its inputs, but it also depends on random fluctu-
ations. These fluctuations are usually modeled as random numbers of
a given statistical distribution. Continuous stochastic models are charac-
terized by smoothly varying fields whose evolution in space and time
depends on probability densities that are functions of some derivatives of
the fields. In the simplest case, this amounts to a single noise term mod-
eling, e.g. Gaussian or uniform fluctuations in the dynamics. Models of this
kind are mostly formalized as stochastic differential equations (SDEs).
56
