Biology Reference
In-Depth Information
Methods of stochastic modeling
Basic Petri nets are depicted as diagrams that only include the following four modeling elements:
-
Places
, which represent variables of the model - molecular species of the biological system
-
Tokens
, which are contained within places and provide the numerical value associated to the variable
- molecular numbers
-
Transitions
, which represent events affecting the variables. In the Petri net terminology, the occur-
rence of the event associated with a transition is called transition
firing
, which can be correlated with
individual reaction steps
-
Arcs
, which link transitions to places and places to transitions (but not places to places nor transitions
to transitions), define the changes that occur on variables as a result of transitions firings. Incoming
arcs to a place add tokens to the place, whereas outgoing arcs remove tokens - representing how
reactions change the molecular numbers of the reacting molecular species.
The reader interested in a more complete description of the Petri net modeling formalism and a more
detailed explanation of how a set of reactions can be modeled with it is referred to [21].
To provide some hints on the overall modeling process, we describe the encoding of the reactions in
Fig. 1A into the Petri net model shown in Fig. 1B, which has been built using the M obius tool [33]. The
model includes 5 places representing the various biochemical species and 11 transitions, each one having
associated a specific reaction event. The structural description of M obius models, which is the one
graphically rendered by the tool, is to be completed with the details of the kinetics associated to the firing
times of transitions. M obius supports models in which the firing times of reactions are random variables
that follow a user-selectable distribution, which makes it particularly useful to represent stochastic
molecular dynamics. To reproduce a kinetics as the one used in Kar model in [20], we selected for
each transition a firing time distributed according to a negative exponential distribution whose rate is
the propensity function defined as per Gillespie's formulation of stochastic chemical kinetics [23]. The
values of rate constants and the exact form of each propensity function were taken from [20]. The
complete stochastic Petri net model includes the 19 species and 41 reactions of the model in [20], plus
the places and transitions necessary to model the dynamics of cellular mass growth and division [21].
Methods of in silico experiments
Starting from the Petri net defined as described above, we produced a set of modified models, whose
evaluation allows exploring the effects of the mRNA gestation and senescence processes on cell cycle
variability. First of all, we modified the synthesis and degradation rates of Cdh1 mRNA to get a realistic
half-life of 10 minutes together with the realistic average mRNA levels of around 8 molecules per cell [22,
28]. The new rates are
k
smy
=
0.5545 molec min
−
1
for the mRNA synthesis and
k
dmy
=
0.0693 min
−
1
for the mRNA degradation. This led us to modify the Cdh1 mRNA translation rates as well. Since the
average number of mRNA molecules is four times larger this way as in the original model [20] we had
to set the translation rate to one fourth of the value (
k
sy
=
0.40175 molec min
−
1
) to leave unaltered
the average Cdh1 protein numbers. These three rate changes are the only differences compared to the
originals used in [20].
The model was then extended in two different ways. A first modified model considers an
N
step
linear gestation process on the transcription of Cdh1 mRNAs (model “GES”). Another model deploys
an
M
steps linear senescence of Cdh1 mRNAs (model “SEN”). Finally, we also checked the behavior
of that the system with both extensions (model “GES-SEN”). As a basal model we used the system with