=Paper=
{{Paper
|id=None
|storemode=property
|title=Comparison of Approximate Kinetics for Unireactant Enzymes: Michaelis-Menten against the Equivalent Server
|pdfUrl=https://ceur-ws.org/Vol-827/5_AlessioAngius_article.pdf
|volume=Vol-827
|dblpUrl=https://dblp.org/rec/conf/acsd/AngiusBCHM10
}}
==Comparison of Approximate Kinetics for Unireactant Enzymes: Michaelis-Menten against the Equivalent Server==
https://ceur-ws.org/Vol-827/5_AlessioAngius_article.pdf
Comparison of approximate kinetics for
unireactant enzymes: Michaelis-Menten against
the equivalent server
Alessio Angius1 , Gianfranco Balbo1 , Francesca Cordero1,2, András Horváth1 ,
and Daniele Manini1
1
Department of Computer Science, University of Torino, Torino, Italy
2
Department of Clinical and Biological Sciences, University of Torino, Torino, Italy
Abstract. Mathematical models are widely used to create complex bio-
chemical models. Model reduction in order to limit the complexity of a
system is an important topic in the analysis of the model. A way to lower
the complexity is to identify simple and recurrent sets of reactions and
to substitute them with one or more reactions in such a way that the
important properties are preserved but the analysis is easier.
In this paper we consider the typical recurrent reaction scheme E +
S↽ −−
−⇀
− ES −−→ E + P which describes the mechanism that an enzyme,
E, binds a substrate, S, and the resulting substrate-bound enzyme, ES,
gives rise to the generation of the product, P . If the initial quantities and
the reaction rates are known, the temporal behaviour of all the quantities
involved in the above reactions can be described exactly by a set of dif-
ferential equations. It is often the case however that, as not all necessary
information is available, only approximate analysis can be carried out.
The most well-known approximate approach for the enzyme mechanism
is provided by the kinetics of Michaelis-Menten. We propose, based on
the concept of the flow-equivalent server which is used in Petri nets to
model reduction, an alternative approximate kinetics for the analysis of
enzymatic reactions. We evaluate the goodness of the proposed approx-
imation with respect to both the exact analysis and the approximate
kinetics of Michaelis and Menten. We show that the proposed new ap-
proximate kinetics can be used and gives satisfactory approximation not
only in the standard deterministic setting but also in the case when the
behaviour is modeled by a stochastic process.
1 Introduction
Mathematical models are widely used to describe biological pathways because,
as it is phrased in [1], they “offer great advantages for integrating and evaluating
information, generating prediction and focusing experimental directions”. In the
last few years, high-throughput techniques have increased steadily, leading to
Cordero is the recipient of a research fellowship supported by grants from Regione
Piemonte, University of Torino and MIUR. Horváth is supported by MIUR PRIN
2008.
Recent Advances in Petri Nets and Concurrency, S. Donatelli, J. Kleijn, R.J. Machado, J.M. Fernandes
(eds.), CEUR Workshop Proceedings, ISSN 1613-0073, Jan/2012, pp. 55–69.
56 Petri Nets & Concurrency Angius et al.
the production of a huge volume of data used to derive the complex texture
behind the biological/biochemical mechanisms, and creating in this way the
structure needed for mathematical modelling. Indeed, many models based on
the combination and the integration of various elements in order to investigate
their relationships and behaviour have been devised which become more complex
with the growth of available data. The complexity is reflected in the number of
dynamic state variables and parameters, as well as in the form of the kinetic
rate expressions.
Such complexity leads to difficulties both from the point of view of defining
the model as the parametrisation becomes unfeasible and for what concerns the
analysis of the model. It is often the case hence that in order to have a model
which is feasilble for the analysis simplifications must be performed.
In this paper we focus our attention on the simplified, approximate treatment
of a set of reactions that very often appears as building blocks of complex models.
We consider the reactions
1 k
E + S −→
k2
←− ES −→ E + P (1)
k−1
describing that the enzyme, E, attaches reversibly to the substrate, S, forming
the substrate-bound enzyme ES which gives rise then to the product P releasing
the enzyme. This and similar enzymatic reactions are widely studied in biology.
The most common approximate approach to deal with them is provided by the
Michaelis-Menten (MM) kinetics (called also Michaelis-Menten-Henri kinetics)
which, based on quasi-steady-state assumptions, connects the speed of producing
P directly to the concentration of E and P , omitting the explicit modeling of
ES.
Fig. 1. Petri net representation of the reactions given in (1)
System of enzymatic reactions can be described by Petri nets [2] (Figure 1
shows the Petri net corresponding to the reactions given in (1)) and then anal-
ysed by methods developed for this formalism. We propose for the reactions in
(1) an alternative to the approximate Michaelis-Menten kinetics. This new ap-
proximate kinetics is based on a concept widely used in the analysis of Petri nets
and models described by other formalisms like queueing networks and process
algebras. This concept is called the flow equivalent server [3]. The application of
this concept, similarly to the Michaelis-Menten kinetics, leads to a simplified set
Comparison of approximate kinetics Petri Nets & Concurrency – 57
of reactions in which the intermediate complex ES is not modeled explicitly. The
difference is, however, that, since the application of the flow equivalent server
(FES) is based on assumptions that are different and less strict than those used
by the Michaelis-Menten kinetics, the resulting approximation is more robust.
The concept of flow equivalent server has already been used in [4] where a
complex signal transduction model was considered. In that paper we have shown
that this concept can be applied not only to the small set of reactions given in
(1) but also to bigger submodels. This leads to a simplified model which has less
parameters and whose analysis is not as heavy as that of the complete one. For
the model presented in [4] it was shown that the quantitative temporal behaviour
of the simplified model coincides satisfactorily with that of the complete model
and that important qualitative properties are maintained as well. In this paper
our goal is to study in detail the goodness of the FES based approximation for
the reactions in (1) and to compare it to the widely-used approximate kinetics
of Michaelis, Menten and Henri.
The paper is organised as follows. Section 2 provides the necessary back-
ground, Section 3 describes the concept of the flow equivalent server and Section
4 presents the results of the comparison between the approximation approaches.
We conclude with a discussion and an outlook on future works in Section 5.
2 Background
In 1901 Henri [5] proposed a partly reversible reaction scheme to describe the
enzymatic process. According to this scheme the enzyme E and the substrate
S form, through a reversible reaction, the enzyme-substrate complex ES. This
complex can then give rise to the product P through an irreversible reaction
during which the enzyme is freed and can bind again to other molecules of the
substrate. This scheme is summarised in (1) where k1 is the rate of the binding
of E and S, k−1 is the rate of the unbinding of ES into E and S and k2 is the
rate at which ES decays to the product P freeing the enzyme E.
There are two typical approaches to associate a quantitative temporal be-
haviour to the reactions in (1). The first results in a deterministic representation
while the other in a stochastic one. In the following we give a brief idea of both
approaches. For a detailed description see, for example, [6, 7].
The deterministic approach describes the temporal behaviour of a reaction
with a set of ordinary differential equations (ODE). For the reactions in (1) we
have
d[E]
= − k1 [E][S] + (k−1 + k2 )[ES] (2)
dt
d[S]
= − k1 [E][S] + k−1 [ES]
dt
d[ES]
= k1 [E][S] − (k−1 + k2 )[ES]
dt
d[P ]
= k2 [ES]
dt
58 Petri Nets & Concurrency Angius et al.
where [X] is the concentration of molecule X at time t. These equations state
that the rate at which the concentration of a given molecule changes equals the
difference between the rate at which it is formed and the rate at which it is
utilised. The four equations can be solved numerically to yield the concentration
of E, S, ES and P at any time t if both the initial concentration levels ([S]0 , [E]0 ,
[ES]0 , [P ]0 ) and the reaction rates (k1 , k−1 , k2 ) are known. In the determin-
istic approach the concentrations of the molecules are described by continuous
quantities.
In the stochastic approach a continuous time Markov chain (CTMC) is used
to describe the process. Each state of the chain is described by a vector of inte-
gers in which the entries give the quantities of the molecules, which, accordingly,
assume discrete values. These discrete values are resulting either directly from
molecule count or from discretization of continuous values. Reactions are mod-
eled by transitions between the states. For example, from state |x1 , x2 , x3 , x4 |
where x1 , x2 , x3 and x4 are the quantities of the molecules E, S, ES and P ,
respectively, there is a transition to state |x1 − 1, x2 − 1, x3 + 1, x4 | with rate
k1 x1 x2 which corresponds to the binding of one molecule E with one molecule
S to form one molecule of ES. It is easy to see that even for small models the
corresponding CTMC can have a huge state space whose transition rate struc-
ture is non-homogeneous. Exact analytical treatment of these chains is often
unfeasible and in most cases simulation is the only method that can be used for
their analysis.
2.1 Michaelis-Menten approximate kinetics
Under some assumptions, the temporal, quantitative dynamics of the mechanism
described by the reactions in (1) can be summarised as follows. Initially we have a
certain amount of substrate, denoted by [S]0 , and enzyme, denoted by [E]0 , and
no complex ES ([ES]0 = 0). Assuming that k2 is significantly smaller than k1
and k−1 , a brief transient period occurs during which the amount of the complex
ES quickly increases up to a “plateau” level where it remains stable for a long
period of time. As the ratio of [S]0 /[E]0 increases, the time needed to reach the
condition d[ES]/dt ≈ 0 decreases and the period during which d[ES]/dt ≈ 0
increases. In this period we have approximately
d[ES]
= k1 [E][S] − [ES](k−1 + k2 ) = 0
dt
from which, considering that the total amount of enzyme is conserved, i.e. [E] +
[ES] = [E]0 , the quantity of ES can be expressed as
[E]0 [S] [E]0 [S]
[ES] = k−1 +k2 = (3)
+ [S] kM + [S]
k1
where the term kM = k−1k+k1
2
is called the Michaelis-Menten constant. Applying
(3), the speed of the production of P can be approximated by
k2 [E]0 [S]
vMM = (4)
[S] + kM
Comparison of approximate kinetics Petri Nets & Concurrency – 59
Accordingly, after the “plateau” level of ES is reached, the kinetic parameters
k1 , k−1 and k2 together with [S] and the initial total quantity of the enzyme,
[E]0 , determine the overall rate of the production of P .
Applying the approximate kinetics of Michaelis and Menten, the differential
equations describing the reactions become
d[E]
=0 (5)
dt
d[S] k2 [E][S]
=−
dt [S] + kM
d[P ] k2 [E][S]
=
dt [S] + kM
3 Approximate kinetics by flow equivalent server
In this section we derive an alternative approximate kinetics for the analysis of
enzymatic reactions, based on the concept of the flow equivalent server. This
technique was originally proposed in the context of the steady-state solution of
queueing networks [3, 8, 9] and can be adapted to our purposes with a proper
interpretation of the assumptions on which it is based. The idea behind this
concept is to consider the reactions given in (1) as a fragment of a large biological
system in which substrate S is produced by an ”up-stream” portion of the system
and product P is used ”down-stream” within the same system. The goal of
the flow equivalent method is to consider the flow of moles that move from
the substrate to the product, in the presence of an enzyme that catalyse this
phenomenon, and to evaluate its intensity in order to define the overall speed of
a ”composite” reaction that captures this situation in an abstract manner.
Figure 2 depicts the Petri net corresponding to the reactions of (1) organ-
ised in order to make explicit the relationship between the substrate S and the
product P , via the enzyme E, enclosing in a dashed box the elements of the
system whose dynamics we want to mimic with the composite transition. This
Fig. 2. Petri net of the reactions in (1) organised for computation of the flow equivalent-
transition (above) and its approximation (below)
60 Petri Nets & Concurrency Angius et al.
picture makes evident the fact that the speed of the composite transition must
depend not only on the speeds of the transitions included in the box but also
on the quantities present in the box, namely, the total amount of enzyme. As-
suming to know the kinetic constants of the reactions inside the box and the
quantity of the enzyme, the speed of the composite transition also depends on
the amount [S] that participates in the reactions and that may change during
the evolution of the whole system. Following this point of view, it is possible
to conceive a characterisation of the speed of the composite transition that is
conditioned on the quantity of S. The flow equivalent approach accounts for this
observation by computing the intensity of the flow of moles that reaches place
P assuming that the total amount of S remains constant. Technically, this is
obtained by short-circuiting the output and input places of the sub-net (intro-
ducing an immediate transition [10] that connects place P with place S) and
by computing the throughput along the short-circuit which will be conditioned
on the initial amount of S and that will thus be computed for all the possible
values of S. In general, this amounts to the construction of a table that looks
like that depicted in Figure 3, where S1 , S2 , ..., Sn represent different values
of the amount of substrate S for which the speeds of the composite reaction
vF ES (S1 ), vF ES (S2 ), ..., vF ES (Sn ) are computed, given that k1 , k−1 , k2 , and
#E are assumed to be the values of the kinetics constant of the reactions in the
box and of the amount of enzyme E.
In practice, this corresponds to the construction and to the (steady state)
solution of the continuous time Markov chain (CTMC) that corresponds to the
sub-model in isolation. Providing the speed of the composite transition in the
tabular form highlighted by Figure 3 is convenient for cases where the domain
of the function is “small”, but may be impractical in many common situations.
Despite the computational complexity of the approach, we must notice that the
equilibrium assumption of the flow equivalent method is used only to obtain
an approximate characterisation of the throughput for different sets of initial
conditions and does not mean that the equivalent speed can only be used for
steady state analysis.
The concept of flow equivalent server described above is used traditionally in
a stochastic setting. However, it can be applied in a deterministic setting as well
using arguments that are summarized by the following points. The complexity
of the approach in the stochastic setting becomes prohibitive when the amount
of the substrate S becomes very large. On the other hand, this is the case in
Given k1 , k−1 , k2 , and #E
S1 vF ES (S1 )
S2 vF ES (S2 )
··· ···
Sn vF ES (Sn )
Fig. 3. Flow Equivalent Server characterisation
Comparison of approximate kinetics Petri Nets & Concurrency – 61
which the stochastic (or at least the average) behaviour of the model is conve-
niently captured by a set of ODE, i.e., by a deterministic model. Moreover, in
the case of our model, the equilibrium solution of the set of differential equa-
tions corresponding to the short-circuited model is simple enough to obtain an
analytic expression for the speed of the composite transition as it is described
in the following.
We assume that the initial condition is [E]0 = M1 , [S]0 = M2 , [ES]0 = 0,
and [P ]0 = 0. We will denote the steady state measures of the compounds by
[E], [S], [ES] and [P ]. In the short-circuited version of the reactions given in (1),
moles transformed in P are immediately moved back to S and consequently its
steady state measure is zero (i.e., [P ] = 0). The steady state measures of the
other compounds can be determined by considering
– the fact that in steady state the rate of change of the quantities of the
different compounds is zero, i.e., we have
d[E](t)
= 0 = −k1 [E][S] + k−1 [ES] + k2 [ES] (6)
dt
d[S](t)
= 0 = −k1 [E][S] + k−1 [ES] + k2 [ES]
dt
d[ES](t)
= 0 = +k1 [E][S] − k−1 [ES] − k2 [ES]
dt
which are three dependent equations;
– and the following equations expressing conservation of mass
[E] + [ES] = M1 , [S] + [ES] = M2 (7)
In (6) and (7) we have three independent equations for three unknowns. There
are two solutions but only one of them guarantees positivity of the unknowns.
The speed of producing P is given by the steady state quantity of ES multiplied
by k2 . This speed is
p
k2 [E] + [S] + kM − ([E] − [S])2 + 2kM ([E] + [S]) + kM 2
vF ES = (8)
2
Accordingly, the set of ordinary differential equations describing the reactions
given in (1) becomes
d[E]
=0 (9)
dt
p
2
k2 [E] + [S] + kM − ([E] − [S])2 + 2kM ([E] + [S]) + kM
d[S]
=−
dt 2
p
k2 2
[E] + [S] + kM − ([E] − [S])2 + 2kM ([E] + [S]) + kM
d[P ]
=
dt 2
which explicitly reflects the assumption of the conservation of E and the obser-
vation that substrate S is transformed into product P .
62 Petri Nets & Concurrency Angius et al.
4 Numerical illustration
In this section, we first compare in Section 4.1 the MM and FES approximate
kinetics from the point of view of the speed they assign to the production of P as
function of the reaction rates (k1 , k−1 , k2 ) and the concentration of the enzyme
and the substrate ([E], [S]). Subsequently, in Sections 4.2 and 4.3 we compare
the quantitative behaviour of the approximations to that of the full model in the
deterministic and in the stochastic setting, respectively.
It is easy to check that as the quantity of the substrate tends to infinity the
two approximate kinetics lead to the the same speed of production. In both cases
for the maximum speed of production we have
vmax = lim vMM = lim vF ES = k2 [E] (10)
[S]→∞ [S]→∞
Another situation in which the two approximate kinetics show perfect corre-
spondence is when the quantity of the enzyme is very low. This can be shown
formally by observing that
vMM
lim =1 (11)
[E]→0 vF ES
4.1 Production speeds
A typical way of illustrating the approximate Michaelis-Menten kinetics is to
plot the production speed against the quantity of the substrate. Figure 4 gives
such illustrations comparing the speeds given by the two approximate kinetics.
Reaction rate k2 is either 0.1, 1 or 10 and reaction rates k1 and k−1 are varied
in order to cover different situations for what concerns the ratio k1 /k−1 . Two
different values of [E] are considered. The limit behaviours expressed by (10)
and (11) can be easily verified in the figures. On the left sides of the figure it
can be observed that for small values of [E] the two approximations are almost
identical for all considered values of the reaction rates, thus in agreement with
the trend conveyed by (11). It can also be seen that for larger values of [E] the
two approximations are rather different and the difference is somewhat increasing
as k2 increases, and becomes more significant for higher values of k1 /k−1 . In all
cases the curves become closer to each other when the amount of [S] increases.
4.2 Deterministic setting
In this section we compare the different kinetics in the deterministic setting. Once
the initial quantities and the reaction rates are defined, the systems of differential
equations given in (2),(5) and (9) can be numerically integrated and this provides
the temporal behaviour of the involved quantities, used as references for the
comparisons.
For the first experiments we choose such parameters with which the two ap-
proximate kinetics result in different speeds of production. Based on Figure 4 this
is achieved whenever the quantity of the enzyme is comparable to the quantity
Comparison of approximate kinetics Petri Nets & Concurrency – 63
0.01 1
0.009 0.9
0.008 0.8
0.007 0.7
0.006 0.6
0.005 0.5
0.004 0.4
0.003 FES with 1,1,0.1 0.3 FES with 1,1,0.1
MM with 1,1,0.1 MM with 1,1,0.1
0.002 FES with 10,1,0.1 0.2 FES with 10,1,0.1
MM with 10,1,0.1 MM with 10,1,0.1
0.001 FES with 1,10,0.1 0.1 FES with 1,10,0.1
MM with 1,10,0.1 MM with 1,10,0.1
0 0
0 10 20 30 40 50 0 10 20 30 40 50
0.1 10
0.09 9
0.08 8
0.07 7
0.06 6
0.05 5
0.04 4
0.03 FES with 1,1,1 3 FES with 1,1,1
MM with 1,1,1 MM with 1,1,1
0.02 FES with 10,1,1 2 FES with 10,1,1
MM with 10,1,1 MM with 10,1,1
0.01 FES with 1,10,1 1 FES with 1,10,1
MM with 1,10,1 MM with 1,10,1
0 0
0 10 20 30 40 50 0 10 20 30 40 50
1 100
0.9 90
0.8 80
0.7 70
0.6 60
0.5 50
0.4 40
0.3 FES with 1,1,10 30 FES with 1,1,10
MM with 1,1,10 MM with 1,1,10
0.2 FES with 10,1,10 20 FES with 10,1,10
MM with 10,1,10 MM with 10,1,10
0.1 FES with 1,10,10 10 FES with 1,10,10
MM with 1,10,10 MM with 1,10,10
0 0
0 10 20 30 40 50 0 10 20 30 40 50
Fig. 4. Production speed as function of substrate quantity with [E] = 0.1 for the figures
on the left side and with [E] = 10 on the right side; reaction rates are given in the
legend in order k1 , k−1 and k2
of the substrate. Accordingly, we set [E]0 = [S]0 = 10. For the full model [ES]0
needs to be set too, and we choose [ES]0 = 0. This choice does not help the
approximations. They assume that the total enzyme concentration [E]0 + [ES]0
is immediately distributed between [E] and [ES], thus making possible an im-
mediate (consistent) production of P . On the contrary, in the full model the pro-
duction of [ES] takes time and thus the speed of the production of P must start
from 0, growing to a high value only later. Figures 5 and 6 depict the quantity of
the product and the speed of its production as functions of time for two differ-
ent sets of reaction rates. In both figures the kinetics based on flow equivalence
provides precise approximation of the production of P . The Michaelis-Menten
64 Petri Nets & Concurrency Angius et al.
kinetics instead fails to follow the full model, but this is not surprising as the
derivation of this kinetics assumes small amount of enzymes. It can also be seen
that high values of k1 /k−1 (Figure 6) lead to worst approximation in case of
Michaelis-Menten kinetics. On the right hand side of the figures one can observe
that for the full model the speed of producing P is 0 at the beginning and then
it increases fast to the speed foreseen by the FES approximation.
10 0.5
full ODE
9 0.45 MM approx.
FES approx.
8 0.4
7 0.35
6 0.3
5 0.25
4 0.2
3 0.15
2 0.1
full ODE
1 MM approx. 0.05
FES approx.
0 0
0 10 20 30 40 50 0 10 20 30 40 50
Fig. 5. Quantity of product (left) and speed of production (right) as function of time
with k1 = 1, k−1 = 10, k2 = 0.1, [E] = 10 and initial quantity of substrate equals 10
12 1
full ODE
0.9 MM approx.
10 FES approx.
0.8
0.7
8
0.6
6 0.5
0.4
4
0.3
0.2
2 full ODE
MM approx. 0.1
FES approx.
0 0
0 10 20 30 40 50 0 10 20 30 40 50
Fig. 6. Quantity of product (left) and speed of production (right) as function of time
with k1 = 10, k−1 = 1, k2 = 0.1, [E] = 10 and initial quantity of substrate equals 10
A second set of experiments is illustrated in Figures 7 and 8. We choose
sets of parameters with which the speed of production of the MM and FES
approximations are similar. In these cases both approximations are close to the
reference behaviour. Still, it can be seen that for high values of k1 /k−1 (Figure
8) the approximation provided by the Michaelis-Menten kinetics is slightly less
precise.
Comparison of approximate kinetics Petri Nets & Concurrency – 65
10 0.25
full ODE
9 MM approx.
FES approx.
8 0.2
7
6 0.15
5
4 0.1
3
2 0.05
full ODE
1 MM approx.
FES approx.
0 0
0 20 40 60 80 100 0 20 40 60 80 100
Fig. 7. Quantity of product (left) and speed of production (right) as function of time
with k1 = 1, k−1 = 10, k2 = 0.5, [E] = 1 and initial quantity of substrate equals 10
12 0.5
full ODE
0.45 MM approx.
10 FES approx.
0.4
0.35
8
0.3
6 0.25
0.2
4
0.15
0.1
2 full ODE
MM approx. 0.05
FES approx.
0 0
0 10 20 30 40 50 0 10 20 30 40 50
Fig. 8. Quantity of product (left) and speed of production (right) as function of time
with k1 = 10, k−1 = 1, k2 = 0.5, [E] = 1 and initial quantity of substrate equals 10
In the following we turn our attention to the cases in which both the approx-
imations are less reliable. In Figure 9 we plotted the case k1 = 0.1, k−1 = 0.1,
k2 = 0.1, [E] = 1, [S]0 = 1 and [ES]0 = 0. As mentioned earlier, with [ES]0 = 0
the initial production speed in the original model is 0 while it is immediately
high in the approximate kinetics. With low values of k1 and k−1 , the time taken
by the system to reach the quasi-steady-state situation assumed by the approx-
imate kinetics is quite long. For this reason there is a longer initial period in
which P is produced by the approximations at a “wrong” speed. Furthermore,
decreasing k1 and k−1 would lead to a longer period in which the approximate
kinetics are not precise (see Figure 9).
Another way of “disturbing” the approximations is to dynamically change
the quantity of the substrate in the system. In the original model, because of
the intermediate step yielding ES, the speed of producing P changes only after
some delay. On the contrary, the approximations react immediately. The harsher
the change in the quantity of the substrate the larger is the difference between
the original model and the approximations. This phenomenon is reflected in the
66 Petri Nets & Concurrency Angius et al.
1 0.035
full ODE
0.9 MM approx.
0.03 FES approx.
0.8
0.7 0.025
0.6
0.02
0.5
0.015
0.4
0.3 0.01
0.2
full ODE 0.005
0.1 MM approx.
FES approx.
0 0
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
Fig. 9. Quantity of product (left) and speed of production (right) as function of time
with k1 = 0.1, k−1 = 0.1, k2 = 0.1, [E] = 1 and initial quantity of substrate equals 1
0.8 0.05
full ODE
0.7 0.045 MM approx.
FES approx.
0.04
0.6
0.035
0.5 0.03
0.4 0.025
0.3 0.02
0.015
0.2
0.01
full ODE
0.1 MM approx. 0.005
FES approx.
0 0
0 5 10 15 20 25 30 0 5 10 15 20 25 30
Fig. 10. Quantity of product (left) and speed of production (right) as function of time
with k1 = 0.1, k−1 = 0.1, k2 = 0.1, [E] = 1, initial quantity of substrate equals 1 and
adding substrate to the system according to (12)
model by adding the following term to the differential equation that describes
the quantity of the substrate:
10(U (t − 5) − U (t − 5.1)) − 10(U (t − 10) − U (t − 10.1)) (12)
where U denotes the unit-step function. The effect of (12) is to add 1 unit of
substrate to the system in the time interval [5, 5.1] and to take away 1 unit
of substrate from it in the time interval [10, 10.1]. The resulting behaviour is
depicted in Figure 10. The approximations change the speed of producing P
right after the change in the quantity of the substrate while the original model
reacts to the changes in a gradual manner. Naturally, if the quantity of the
substrate undergoes several harsh changes then the MM and the FES kinetics
can result in bad approximation of the full model.
4.3 Stochastic setting
In the following we compare the different kinetics in the stochastic setting, by
analysing the corresponding CTMCs. In particular, we determine by means of
Comparison of approximate kinetics Petri Nets & Concurrency – 67
simulation the average and the variance of the quantity of the product as function
of time. The simulations were carried out in Dizzy [11].
The reaction rates for the first set of experiments are k1 = k−1 = k2 = 1.
As in the previous section, this choice allows to test a situation where the speed
of the two approximations are different. For the same reason, we choose the
same initial quantity for the enzyme and the substrate [E]0 = [S]0 = 1. In
the stochastic setting the discretization step, denoted by δ, has to be chosen as
well. This choice has a strong impact because as the granularity with which the
concentrations are modeled is increased, the behaviour of the CTMC tends to
the deterministic behaviour of the corresponding ODE. Figures 11 and 12 depict
the average and the variance of the quantity of the product with δ = 0.01 and
δ = 0.001, respectively. In both figures the approximate kinetics based on flow
equivalence gives good approximation of the original average behaviour while
the Michaelis-Menten approximation results in too fast production of P . On the
right side on the figures one can observe that also the variance is approximated
better by the FES approximation.
1 0.003
full ODE
0.9 MM approx.
0.0025 FES approx.
0.8
0.7
0.002
0.6
0.5 0.0015
0.4
0.001
0.3
0.2
full ODE 0.0005
0.1 MM approx.
FES approx.
0 0
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Fig. 11. The average (left) and the variance (right) of the quantity of the product as
function of time with k1 = 1, k−1 = 1, k2 = 1, [E]0 = [S]0 = 1 and δ = 0.01
1 0.00035
full ODE
0.9 MM approx.
0.0003 FES approx.
0.8
0.7 0.00025
0.6
0.0002
0.5
0.00015
0.4
0.3 0.0001
0.2
full ODE 5e-005
0.1 MM approx.
FES approx.
0 0
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Fig. 12. The average (left) and the variance (right) of the quantity of the product as
function of time with k1 = 1, k−1 = 1, k2 = 1, [E]0 = [S]0 = 1 and δ = 0.001
68 Petri Nets & Concurrency Angius et al.
For the second set of experiments we set k1 = 10 and k−1 = k2 = 1 and
as initial states we choose again [E]0 = [S]0 = 1. In this case too, as it was
shown in Figure 4, the speeds of production of P as predicted by the MM and
FES approximations are quite different. Figures 13 and 14 depict the resulting
behaviour for two different values of δ. As in case of the deterministic setting,
the Michaelis-Menten approximation suffers from the increased k1 /k−1 ratio and
becomes less precise than before. The FES based approach still results in good
approximation for both the average and the variance of the production.
0.005
1 full ODE
0.0045 MM approx.
FES approx.
0.004
0.8
0.0035
0.003
0.6
0.0025
0.4 0.002
0.0015
0.2 0.001
full ODE
MM approx. 0.0005
FES approx.
0 0
0 1 2 3 4 5 0 1 2 3 4 5
Fig. 13. The average (left) and the variance (right) of the quantity of the product as
function of time with k1 = 10, k−1 = 1, k2 = 1, [E]0 = [S]0 = 1 and δ = 0.01
0.0006
1 full ODE
MM approx.
0.0005 FES approx.
0.8
0.0004
0.6
0.0003
0.4
0.0002
0.2 0.0001
full ODE
MM approx.
FES approx.
0 0
0 1 2 3 4 5 0 1 2 3 4 5
Fig. 14. The average (left) and the variance (right) of the quantity of the product as
function of time with k1 = 10, k−1 = 1, k2 = 1, [E]0 = [S]0 = 1 and δ = 0.001
Comparison of approximate kinetics Petri Nets & Concurrency – 69
5 Conclusion
In this paper we have considered the approximate treatment of the basic enzy-
matic reactions E+S ↽ −−
−⇀
− ES −−→ E+P . In particular, an approximate kinetics,
based on the concept of flow equivalent server, has been proposed for its analy-
sis. This FES approximate kinetics has been compared to both the exact model
and to the most common approximate treatment, namely, the Michaelis-Menten
kinetics. We have shown that the FES kinetics is more robust than the one of
Michaelis-Menten.
The FES approximation for the basic enzymatic reactions is computationally
convenient due to the fact that it has been possible to find an analytic expression
for the speed of the composite reaction in this case. While it is very unlikely for
this to be true in the case of more complex kinetics, the method is very general
and we will study it further within this context to see if it is possible to find other
functional expressions for the speed of the composite reaction. One direction of
research will be computing the flow equivalent characterization of the kinetics
for a number of specific parameter sets and then of constructing the functional
representations via interpolation.
References
1. Fisher, J., Henzinger, T.A.: Executable cell biology. Nature Biotechnology 25
(November 2007) 1239–1249
2. Reddy, V., Mavrovouniotis, M., Liebman, M.: Qualitative analysis of biochemical
reaction systems. Comput. Biol. Med. 26 (1996) 9–24
3. Chandy, K.M., Herzog, U., Woo, L.S.: Parametric analysis of queueing networks.
IBM Journal of R. & D. 19(1) (January 1975) 36–42
4. Napione, L., Manini, D., Cordero, F., Horváth, A., Picco, A., Pierro, M.D., Pavan,
S., Sereno, M., Veglio, A., Bussolino, F., Balbo, G.: On the use of stochastic
Petri nets in the analysis of signal transduction pathways for angiogenesis process.
In: Proc. of The 7th Conference on Computational Methods in Systems Biology
(CMSB 2009). Volume 5688 of Lecture Notes in Bioinformatics., Bologna, Italy
(Sept 2009) 281–295
5. Henri, V.: Recherches sur la loi de l’action de la sucrase. Compt. Rend. 133 (1901)
891–894
6. Voit, E.O.: Computational Analysis of Biochemical Systems. Cambridge University
Press (2000)
7. Gillespie, D.: A rigorous derivation of the master chemical equation. Physica 188
(1992) 404–425
8. Denning, P.J., Buzen, J.P.: The operational analysis of queueing network models.
ACM Comput. Surv. 10(3) (1978) 225–261
9. Stewart, W.J.: Probability, Markov Chains, Queues, and Simulation: The Mathe-
matical Basis of Performance Modeling. Princeton Univ. Press (2009)
10. Ajmone Marsan, M., Balbo, G., Conte, G.: A class of generalized stochastic Petri
nets for the performance analysis of multiprocessor systems. ACM Transactions
on Computer Systems 2(1) (May 1984)
11. Ramsey, S., Orrell, D., Bolouri, H.: Dizzy: Stochastic simulation of large-scale
genetic regulatory networks. J. Bioinformatics Comput. Biol. 3(2) (2005) 415–436