Biology Reference
In-Depth Information
from noise and delays. The starting point for the analysis is an entirely deterministic model [17] that
captured the dopamine dynamics quite well, but did not account for stochastic perturbations.
Biochemical Systems Theory
Biochemical Systems Theory
(BST) is a firmly established mathematical modeling framework for the
analysis of biological systems. BST is based on ordinary differential equations in which all dynamical
processes are represented with products of power-law functions. Each of these functions consists of a
non-negative
rate constant
, as a multiplicative coefficient, and of all contributing substrates, enzymes,
and modifiers as variables. Each variable is raised to a real-valued
kinetic order
that quantifies the
effect of the variable on a given reaction. A positive kinetic order signifies activation, while a negative
value signifies inhibition and a value of zero corresponds to no contribution at all. BST permits several
variants, among which the format of a Generalized Mass Action (GMA) system is most intuitive. In
this format, each process is separately represented by a power-law term, while the alternative S-system
format first groups all influxes and all effluxes into one term each [18]. The GMA format directly reflects
the stoichiometric connectivity of the system and also indicates in its kinetic orders the strengths of
interactions among the system variables. Its generic format is thus
⎛
⎞
⎛
⎞
P
i
n
Q
i
n
X
f
ijp
X
g
ijq
X
i
=
⎝
a
ip
⎠
−
⎝
b
iq
⎠
,i
=1
,...,n
(1)
j
j
p
=1
j
=1
q
=1
j
=1
where variable
X
i
is affected by
P
i
production and
Q
i
degradation processes;
a
ip
and
b
iq
are rate
constants, while
f
ijp
and
g
ijq
are kinetic orders.
While models within BST consist entirely of ODEs, Mocek et al. showed that processes with a constant
delay can be approximated with arbitrary accuracy within the BST format [19]. This approximation is
accomplished through the introduction of auxiliary variables and equations, which however do not require
additional biological parameters. Wu and Voit further extended Mocek's method to allow for multiple
delays of different types, including discrete, distributed, time dependent, and random delays [15].
Implementation of a GMA model as a Hybrid Functional Petri Net
A Petri net is a mathematical modeling tool for the representation of systems with concurrent processes.
One appealing feature is the graphical depiction of all system components and processes, which facilitates
intuitive, targeted manipulations and simulations. Originally designed for discrete systems, Petri nets
have recently been extended to account for hybrid systems combining both discrete and continuous
events. These Hybrid Functional Petri Nets (HFPN) [13] can be simulated conveniently with the
software package
Cell Illustrator
[20].
As indicated in Fig. 2, it is straightforward to implement a GMA model in the HFPN framework:
each time dependent variable
X
i
is represented in the HFPN as a
continuous place
with the name of the
molecular species, whereas every time independent variable is coded either as a
discrete
or
continuous
place
, depending on its value type. Every production term
n
X
f
ijp
a
ip
j
j
=1
