Biology Reference
In-Depth Information
numerical procedures for the integration of ODEs and/or NAEs to describe the dynamics of these
models, DBsolve offers explicit solver, implicit solver and bifurcation analyzer. The primary focus of
E-Cell is to develop a framework for constructing simulatable cell models based on the gene sets that
are derived from completed genomes. Contrast to other computer models that are being developed to
reproduce individual cellular processes in detail, E-CELL is designed to “paint a broad-brush picture of
the cell as a whole”. There is another program, named DynaFit (
http://www.biokin.com/dynafit/
), which
is also useful in the analysis of complex reaction mechanism.
In predicting cell behavior, the simulation of a single or a few interconnected pathways can be useful
when the pathway being studied is relatively isolated from other biochemical processes. However, in
reality, even the simplest and best studied pathway, such as glycolysis, can exhibit a complex behavior
due to the high connectivity of metabolites. In fact, the more interconnections exist among different parts
of a system, the harder it gets to predict how the system will react. Moreover, simulations of metabolic
pathways alone cannot account for the longer time-scale effects of processes such as gene regulation,
cell division cycle and signal transduction. When the system reaches a certain size, it will become
unmanageable and non-understandable unless with decomposition of modules (hierarchical models)
and/or presentation of graphs. In this sense, tools mentioned above appear faint. In comparison, Petri nets
capture the basic aspects of concurrent systems of metabolism both conceptually and mathematically. The
major advantages of Petri nets comprise a graphical modeling representation with sound mathematical
background making it possible to analyze and validate the qualitative characteristics and quantitative
behavior of a concurrent system, and to describe the system on different levels of abstraction (hierarchical
models). In addition, the development of computer technology enables Petri net tools to have more
friendly interfaces and possibility of standard data import/export supporting. We are motivated to exploit
the Petri net methodology to model and simulate gene regulated metabolic networks.
The normal discrete system is easy to understand, so we emphasize here on the continuous one
that proves useful to dynamic systems.
We will describe some mathematical formulations that occur
frequently in biology models.
A general differential equation for a single state variable is
dx
/
dt
=
flowin
-
flowout
, while the expressions for the
flowin
and
flowout
can be quite complex as every
bioprocess gives rise to its own system of differential equations involving many dependent variables
(species concentrations) and many free parameters (reaction rate constants). The mass action law
assumes that particles move incessantly. However, cellular metabolites are not like gas molecules. A
metabolic reaction is very complex; interaction delay or saturation effect often exists in a metabolic
system. In these cases, the mass action law becomes violated and should be replaced by equations that
better describe the metabolic interaction while the rest of the algorithm remains the same.
PN model of metabolic reactions
In biochemistry, the most commonly used expression that relates the enzyme catalyzed formation
rate of the product to the substrate concentration is the Michaelis-Menten equation, which is given as
v
=
v
max
S
/
K
m
+
S
. An example of its Petri net model and simulation result is shown as below.
It is clear that such an enzymatic reaction is characterized by these two parameters:
V
max
and
K
m
,
and biochemists are interested in determining these parameters from experiments. Fortunately, there are
several biochemical reaction databases available for public such as BRENDA that provides enzymatic
reaction kinetics. However, only for a subset of the well known pathways those kinetic parameters are
complete, and moreover an enzymatic reaction can be affected by the presence of other compounds,
i. e., the simplest form of the Michaelis-Menten equation does not account for the higher than first order
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