Biology Reference
In-Depth Information
effects determine the global pattern. Unfortunately, bio-
logical systems tend to exhibit complex network motifs and
a large number of components, making this kind of
modeling extremely challenging.
With the increase in available computational power,
Monte Carlo methods have gained in importance. Here,
many stochastic snapshots of the system are evaluated to
derive dynamic information. While this is robust to error
propagation, the needed computational power is large in
relation to the derived results. Even more computationally
exhaustive is the method of molecular dynamics. Here,
every component molecule and the forces acting upon it are
tracked. This can give exceptional insight into the behavior
of a small number of particles for short periods, but is very
dependent on the assumptions and simplifications used for
molecular shape, behavior and interaction.
As a phenomenological approach, based on the expe-
riences with Turing patterns at a time when computational
power was limited, the so-called cellular-automaton
approach was developed. Similar to PDE, in this approach
space and time are coarsely discretized, but so is the vari-
able in question (e.g., concentration of protein). It thus
becomes a state, in the most fundamental case binary (0s
and 1s). The state at a certain position x at a later time-step
t
is an oversimplification of the complicated auto-catalytic
process on a fast timescale, which is visible as a change of
color in the system and can be triggered by the presence of
minute traces of the activating component, but once this
conversion is locally done, the system needs some time to
recuperate (refractive time) before it again becomes
excitable. This refractive time (e.g., the specific value of n)
combined with the decay strength implemented in rule 3
and the diffusion coefficients determines
the spatial
distance between the spiral wave fronts.
The interaction can be extremely localized (here: acti-
vation within each cell and diffusion to its four neighboring
cells) and still affect a global pattern. We can use this
simple system to illustrate the effect of small differences in
locality of this rule set being reflected in the global pattern.
By changing the size of the diffusion radius or randomizing
the shape of this neighborhood, the shape of the resulting
large-scale spirals is determined (see
Figure 17.1
).
In this example, we minimize the system to one
observable, which hides an interaction network of multiple
agents. Expanding the above rule 3, one realizes that the
topology of this kind of interaction network between the
involved agents can also be a determining factor of a global
pattern. By including a second substance that depends on
the concentration of the first and at the same time influences
it, one arrives at the classic activator
1 is then determined by the state of x and its neighbors at
time t by a fixed set of rules that are loosely related to the
underlying PDEs. The Wolfram automaton is a simple 1D
cellular automaton where time-evolution can be visualized
as the second dimension in a 2D plot. Different rules lead to
different behaviors that can, for example, be mapped to
pigmentation patterns of mollusc shells
[8]
.
As described previously, biological systems tend to
exist in a steady state that is 'waiting' to be perturbed and
thus to switch to a different mode of activity. This resem-
bles a loose description of so-called excitable media, which
are one example that has given prominence to the cellular
automaton. All excitable media share a common 'rule set'
that generates global patterns from extremely localized
interactions. It can be reduced to the following for one
observable substance:
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inhibitor type of
system. As an example, two morphogens that act as acti-
vator and inhibitor of undifferentiated cells of a homoge-
neous tissue can form a spatially distinct pattern
(fingerprint, retinal blood vessel network) of differentiated
cells. These patterns are generated by the network topology
of interacting substances. However, on the one hand its
specific implementation follows stochastic fluctuations,
which is why identical twins have distinct fingerprints and
retinal patterns. On the other hand, global parameters (e.g.,
temperature, foreign substances or overlying gradients) can
also affect the pattern formation and result, for example, in
deformations in embryonal development, or can determine
patterns of revascularization that are essential in wound
healing but disastrous when initiated by tumors.
Similar pattern formation processes are also present at
an intracellular scale. For example, a receptor tyrosine
kinase (RTK) with a transmembrane domain to sense the
presence of its extracellular growth factor ligand can be
activated by ligand-induced di- or oligomerization, which
results in autophosphorylation of the receptor by its
intrinsic tyrosine kinase activity. As a balancing reaction to
this activation, cytosolic protein tyrosine phosphatases
(PTPs) inhibit this RTK activity by dephosphorylation of
the receptors. RTK oligomerization and phosphorylation-
induced increases in activity can be described as a positive
feedback of RTK activity; this activity can also influence
the inhibiting strength of PTPs either by inactivating PTPs
via reactive oxygen species (ROS) or by translocating PTPs
e
1.
Every point in space (called cell) is either in an excit-
able state with an excitation value of 0, or in the excited
state with a value between 1 and n.
2.
Excitation spreads via 'diffusion', i.e., the value of each
cell is distributed among the cells in its neighborhood of
a certain radius.
3.
If a cell has state 0 and 'receives' excitation via diffu-
sion, it becomes excited and its value is set to n;
otherwise, its excitation value decays by 1 (a loss per
cell additional to the redistribution by diffusion).
This phenomenological description of an excitable medium
is the algorithm, which can be translated to a cellular
automaton and renders BZ-like 2D patterns
[9]
. Here, rule 3
