Metabolic rate constants: some computational
aspects
Stanko Dimitrov1 , Svetoslav Markov2
1
University of Sofia, Faculty of Mathematics and Informatics
2
Institute of Mathematics and Informatics, Bulgarian
Academy of Sciences
November 12, 2015
Abstract. In this work we pose the question how reliable the Michaelis
constant is as an enzyme kinetic parameter in situations when the MichaelisMenten equation is not a good approximation of the true substrate dynamics
as it may be in the case of metabolic processes in living cells. We compare the
Michaelis-Menten substrate-product kinetics with the complete substrateenzyme-product kinetics induced by the reaction scheme originally proposed
by V. Henri. The Henri reaction scheme involves four concentrations and
three rate constants and via the law of mass action is translated to a system
of four ordinary differential equations (denoted HMM-system). We propose a
method for the computation of the three HMM rate constants, which can be
applied in any situation whenever time course measurement data are available. The proposed method has been tested on the case study of acetylcholine
hydrolysis. Our approach provides for the validation of the HMM-system by
taking into account the uncertainties in the measurement data, and focuses
on the use of contemporary computational tools.
Keywords: enzyme kinetics, reaction equations, Michaelis constant, biomasssubstrate-product dynamics, computation of rate constants
1
1
Introduction
Since its discovery around 1900 the Henri enzyme kinetic law has been intensively applied in many fields of life sciences. Two types (or stages) of its
application can be distinguished: a first type of applications in biotechnological (fermentation) processes where the enzymes are often produced by
bacterial cells, and a second type of applications in metabolic processes in
living cells. The first type of applications are often denoted as “in vitro”
whereas the second one—“in vivo”. During “in vitro” processes one is usually interested in the transformation of as much as possible quantity of substrate into product using thereby as little as possible quantity of enzymes.
Thus the ratio enzyme/substrate “in vitro” is usually very small, whereas
“in vivo” this may not be the case. For the “in vitro” case the approximate
Michaelis-Menten model has been proposed together with a simple protocol
for the calculation of the two parameters involved in the model (Vmax and
Km ). The Michaelis-Menten model has proved to be extremely useful and
has become very popular amongst biological scientists. Plausible biochemical
mechanisms (known as Briggs-Haldane interpretations) have been found also
known as (standard) quasi-steady-state approximation (sQSSA) criteria for
the validity of the Michaelis-Menten model. With the necessity of considering “in vivo” processes, two approaches can be possibly followed. One is to
extend the applicability (validity) of the Michaelis-Menten model by finding
more general criteria than the standard enzyme/substrate ratio criterion (the
sQSSA condition), such as the so-called rQSSA and the tQSSA criteria. A
disadvantage of this approach is that these criteria are more complex to check
and involve the computation of new parameters, thereby still producing often
very rough results. Another approach could be to make use of other approximate models together with corresponding validity criteria—such as the first
order decay model. As demonstrated in Section 3 in certain situations the
decay model may produce better approximation than the Michaelis-Menten
model. A third approach would be to make use of the original (exact) Henri
enzyme kinetic law and the induced system of ODEs. The proponents of the
approximate models underline that the Henri law induced system of ODEs
is too complex to use in real applications and cumbersome to handle. In our
opinion this may have been the case some decades ago, however there exist
powerful contemporary mathematical and computational tools to make the
exact Henri model easily applicable. In particular, familiar computer algebra
systems (CAS) such as Mathematica and MATLAB can be used to handle
2
the ODE system induced by the Henri law. In order to apply efficiently the
original Henri law it is necessary to compute the three Henri’s rate constants
from time-course kinetic measurements using computer simulations. However, such time-course measurement data are usually not explicitly available
in the literature, in the best case some graphs are published together with the
computed Michaelis constant Km which is of little use for the computation
of the three Henri rate constants. We remind that the Michaelis-Menten protocol does not make use of time-course measurement data; this probably is
a possible reason for the popularity of the Michaelis-Menten model amongst
experimental biochemists and biologists.
Here we study mathematically and computationally experimental data
paying special attention to the measurement errors involved. We describe and
motivate our modelling approach applying V. Henri’s biochemical reaction
scheme of the simple enzyme-substrate dynamics where two fractions of the
enzyme (free and bound) are involved. Our approach is tested on a set
of simulated time course measurements for the dynamics of acetylcholine
hydrolysis by acetylcholinesterase [34].
Henri’s reaction scheme induces (via the mass action law) a system of
ODEs further briefly denoted as HMM-system. It is well-known that the
Michaelis-Menten equation (MM-equation) for the substrate is an approximation of the HMM-system and this approximation is a good one only under
certain conditions, for recent reviews see [6], [30]. In this work we confirm
that the MM-equation may produce rather rough approximations in certain
situations. We focus on some contemporary computational tools that are
available for dealing directly with the HMM-system, so that there is no need
to make use of approximate models such as the MM-equation.
Section 2 contains a review of the original Henri-Michaelis-Menten reaction scheme (HMM-scheme) and the approximate Michaelis-Menten model
(MM-model). The HMM reaction scheme induces (via the mass action law)
a system of ordinary differential equations (HMM-system) while the approximate MM-model is derived from the HMM-system using well known assumptions [3], [4], [5], [18], [16], [21], [22]). We briefly comment on the criteria for
the “validity” of the MM-model.
Section 3 is devoted to computational experiments that reveal some details on the conditions under which the approximate Michaelis-Menten model
is adequate. We base our analysis on a well defined criterion for the validity
3
of the model as well as general analysis of its dynamics based on the ODE
system. Cases where the biomass solutions of the two models are close, but
the suggested dynamics differ significantly are also presented. The use of
the proposed validation criterion and analysis for the approximate model are
encouraged before any analysis of the process dynamics based solely on the
values of constants such as the Michaelis constant Km .
Section 4 considers the computation of the HMM-system rate parameters
using time course data. In particular, we make use of some Mathematica
and MATLAB tools allowing for the estimation of the three rate parameters
of the HMM-system by means of appropriate adjusting simulations of the
system solutions to match available time course measurement data.
2
2.1
Enzyme kinetics basic models
Henri’s reaction scheme
The following reaction scheme of a simple enzyme-substrate dynamics where
two fractions of the enzyme (free and bound) are involved has been proposed
by Victor Henri [7], [13]–[14]:
k
1
k2
S + E −→
←− C −→ P + E.
(1)
k−1
Henri’s reaction scheme (1) describes the reaction mechanism between
an enzyme E with a single active site and a substrate S, forming reversibly
an enzyme-substrate complex C, which then yields irreversibly a product
P . Reaction scheme (1) says that during the transition of the substrate
S into product P the enzyme E bounds the substrate into a complex C
having different properties than the free enzyme and thus being necessarily
considered as a separate substance.
Denoting the concentrations s = [S], e = [E], c = [C], p = [P ] and
applying the Mass Action Law to Henri’s reaction scheme (1) we obtain the
following system of ODEs:
4
ds/dt
de/dt
dc/dt
dp/dt
=
=
=
=
−k1 es + k−1 c,
−k1 es + (k−1 + k2 )c,
k1 es − (k−1 + k2 )c,
k2 c,
(2)
to be further called the HMM-system—in tribute to V. Henri, as well to
L. Michaelis and M. Menten [21]. If the three rate constants k1 , k−1 , k2 are
known, system (2) can be treated as a Cauchy ODE problem with initial
conditions s(0) = s0 > 0, e(0) = e0 > 0, c(0) = 0, p(0) = 0.
In practice the HMM-system rate constants k1 , k−1 , k2 are often not known
and have to be determined for every specific pair enzyme-substrate. The
contemporary approach to this task is to consider the rate constants as parameters in the dynamic HMM-system (2) and to compute them by fitting
the solutions of the system to time course experimentally measured data.
This problem will be considered in Section 4.
2.2
Michaelis-Menten equation
Applying the quasi-steady-state assumption (QSSA) to HMM-system (2) one
derives the following approximate equations for the substrate rate ds/dt and
the complex c [5], [21], [22]:
c=
e0 s
.
Km + s
Vmax s
ds
=−
= −µ(s).
dt
Km + s
(3)
(4)
where the right-hand side function µ(s) = Vmax s/(Km + s) is known as
“specific growth function” in cell growth modelling [19]. The solution sm
of equation (4) is an approximation of the solution s of (2). The QSSA is
reasonably applied e.g. when the ratio [E]/[S] is small so that fermentation
continues for a considerably long time interval during which the concentration
c = [C] of the bound enzyme is nearly constant. The latter can be achieved
e.g. when the condition e0 s0 holds. Let us also remind that under the
validity of QSSA the complex concentration c = [ES] is expressed in terms
of the MM-substrate concentration via (3).
5
In their paper [21] Michaelis and Menten discuss in detail Henri’s reaction
scheme and equation (4) known as Michaelis-Menten equation to be further
denoted as MM-model. In addition Michaelis and Menten proposed a protocol for the practical calculation of the constant Vmax and Km in (4). The
constant Km is known as Michaelis constant [7], [29].
The MM-equation (4) can be written in the form
ds
Vmax s
Vmax
=−
=−
,
dt
Km + s
Km /s + 1
showing that for large values of s the right-hand side is close to the
constant −Vmax ; hence the substrate uptake rate |ds/dt| is nearly equal to
the constant Vmax .
Equation (4) produces a good approximation sm for s under certain conditions (sometimes called validity criteria) [3], [5], [10], [12], [18], [30], [25],
[26]. Thus the condition e0 s0 assures good approximation and is ubiquitous for many fermentation and biotechnological processes, but may not be
present, e.g. in living cells [1], [28]. In the sequel we discuss certain methods and tools for the computation of the Michaelis constant using both the
MM-model (4) and the HMM-system (2). In particular, we are interested
in computational methods for the evaluation of the three rate parameters of
the HMM-system.
2.3
Relations between the rate parameters
In Figure 1 the substrate uptake s is visualized in two different ways. The
two graphics present the approximate solution sm to the MM-model (4) as
well as the original solution s of the HMM-system (2). In this example the
kinetic parameters are chosen so that the validity conditions of the MMmodel do not hold and the difference of the solutions for the substrate of the
two systems is visible. We would also want to compare the two solutions in
case the validity conditions of the approximate model do hold.
In order to correctly compare the solutions s and sm one has to establish
certain consistency relations between the parameters in the MM-equation
and the HMM-system. From the derivation of the MM-equation, see e.g.
[22], we know that the parameters in (4) can be expressed by the parameters
in (2) by means of the relations:
Vmax = k2 e0 ,
Km = (k−1 + k2 )/k1 .
6
(5)
Figure 1: Graphics of the substrate dynamics according to the HMM-system (2) and MM-model (4).
The rate constants of (2) are k1 = 2.62, k−1 = 0.1, k2 = 1.25, initial conditions: s0 = 1, e0 = 1.5 (the
system is dimensionless); the parameters of (4) are Km = (k−1 + k2 )/k1 = 0.51526, Vmax = k2 e0 = 1.875.
On the other side the three parameters in (2) cannot be uniquely determined by the two parameters in (4). The following computational examples
show how the approximate substrate concentration solution sm to the MMequation may look like depending on the initial values of the substrate s0
and the enzyme e0 .
According to the MM-model we have ds/dt = Vmax s/(Km + s). The two
constants Vmax and Km can be experimentally determined according to the
MM-protocol. If the MM-model is valid we have Vmax ≈ k2 e0 . From this
expression one can calculate the value kcat = Vmax /e0 . Hence kcat ≈ k2 if the
validity conditions for the MM-model take place.
Assuming that k2 is a universal rate constant for a given specific reaction,
then it should not depend on whether the validity conditions take place or
not (e.g. if k2 is computed in an experiment with e0 s0 , it should be the
same constant in another experiment with e0 > s0 .)
Thus we may assume kcat ≈ k2 . This gives a method for evaluating the
HMM-rate constant k2 . However, it is not possible to obtain any information
about the HMM-rate parameters k1 , k−1 on the basis of the MM-model, resp.
7
the MM-protocol.
3
Computational examples: model comparison
In this section we present the results of several numerical studies aiming at the
comparison of the substrate dynamics of the two models (2), (4). The values
of the dimensionless parameters K = Km /s0 and ε = e0 /s0 play an important
role in our analysis because they can significantly influence the behavior of
the system. The values of the two system characteristic parameters K, ε are
varied in the numerical study in order to present results for the cases when
K ≤ 1 and ε 1, ε > 1, as well as K 1 and ε 1, ε > 1. The values of
the HMM rate constants k1 , k−1 , k2 are kept the same for all cases as follows:
k1 = 0.1, k−1 = 0.01, k2 = 10. The Michaelis constant consistent with these
values is according to (5): Km = (k−1 + k2 )/k1 = 100.1.
3.1
Criterion for good approximation
An important requirement for the validity of the MM-model (4) is the following condition:
e0
1
k2 e0
=
1
(6)
k1 (s0 + Km )2
s0 + Km
1 + (k−1 /k2 ) + (s0 k1 /k2 )
Condition (6) can be derived from the relation between the estimates of
two timescales tc and ts , see [18], or [22], ch. 6.2. The timescale tc corresponds
to the “faster” reaction, involving the enzyme and the complex. Complex
formation is the most significant process during this timescale. The longer
timescale, ts , corresponds to the product synthesis and substrate utilization.
It is natural to assume tc ts which in turn leads to (6).
We see that e0 s0 implies (6) but in case when e0 s0 does not hold,
the MM-model could still be a good approximation if Km is large enough
(Km s0 and also k−1 k2 ). In order to provide a more comprehensive
explanation of this statement we should consider the non-dimensionalized
system proposed in [22], as well as the form of the MM-model under the
considered conditions:
8
τ = k1 e0 t,
u(τ ) = s(t)/s0 , v(τ ) = c(t)/e0 ,
λ = k2 /(k1 s0 ), K = Km /s0 ,
ε = e0 /s0 ,
du/dτ = −u + (u + K − λ)v,
εdv/dτ = u − (u + K)v, u(0) = 1, v(0) = 0.
(7)
(8)
First, the dynamics of v can be summarized as follows: v increases from
t = 0 until it reaches its maximum at v = u/(u + K), after that it decreases
back to 0. An important point is that if K is “big” then the maximum of
v will be smaller. Furthermore, larger K indicates either larger k2 and/or
larger k−1 values in comparison to s0 k1 . The most common ranges of the values of k2 and k1 are considered to be k2 ∈ [103 , 106 ]s−1 , k1 ∈ [107 , 1010 ]m−1 s−1
from many sources (such as [2], [31], [33]). Some sources ([31], [24]) also state
that for “superefficient enzymes” (or enzymes with perfect kinetics) the ratelimiting step is the substrate-enzyme association step. Such enzymes are
said to have reached kinetic perfection or catalytic perfection [32] and that
k2 k−1 holds for them [31]. Additionally, in certain rare cases the concentration of substrate is very small in comparison to the available enzyme [1].
Such cases may include formation of cellular micro-compartments, enzymatic
activity in mammalian muscle tissue and others [1]. We may assume that in
these cases the concentration of substrate can even reach values s0 k2 /k1 .
We will assume the following hold for some of the cases we study: (1) the
substrate concentration is sufficiently low, so that k2 s0 k1 , (2) the formed
enzyme-substrate complex breaks down to product and enzyme faster than it
dissociates to substrate and enzyme, thus k2 k−1 . For the enzymes under
consideration we can summarize that the rate-limiting step of their reactions
is the substrate-enzyme association step and due to their catalytic perfection
a small amount of complex is present during the whole process.
It should be noted that for the case studies in the next section we have
used values of the kinetic parameters different than the ones derived from
real experiments for convenience and simplicity. However, the conditions
we rely on (k2 s0 k1 , k2 k−1 ) are kept valid when examining the case
K 1 and we argue that parameters close to the ranges observed in real
experiments exist, based on the aforementioned statements.
We could also rewrite the Michaelis-Menten model (4) under the assumptions K 1, k−1 k2 in the following form:
9
k1 k2 e0 s
ds
Vmax s
k2 e0 s
=−
=−
= − k−1 +k2
∼ −k1 e0 s.
dt
Km + s
k
+
s
−1 + k2 + k1 s
k
1
The obtained form suggests a much simpler reaction and a negligible role of
the intermediate complex. Thus, the product formation is only limited by
the kinetic constant k1 . The derived equation is also identical to the exact
substrate ODE from (2) if we assume e = e0 , c ∼ 0, which would be natural
given the kinetic parameter assumptions we have made.
3.2
The exponential decay model
We may argue that under the conditions discussed above the second terms
in the non-dimensionalized system (7–8) are almost 0. System (7–8) can be
approximated as follows:
u
; u(0) = 1, v(0) = 0.
(9)
ε
The solution u of this dynamical system is identical to the solution sd of
the substrate dynamics induced by the simple reaction scheme:
du/dτ = −u, dv/dτ =
k
S + E −→ P + E,
(10)
wherein S is the substrate, E is the enzyme and P is the product. Reaction
scheme (10) says, differently to (1), that during the transition of the substrate
S into product P the enzyme E does not bound the substrate into a complex.
Applying the mass action law, assuming e = [E] constant, the kinetic scheme
(10) leads to the following “exponential decay” differential equation for the
substrate concentration sd = [S]:
dsd
= −kesd .
dt
(11)
Solving the Cauchy problem related to equation (11), having set e = e0 ,
we obtain k = k1 . Indeed, the solution of equation (11) is sd (t) = s0 e−e0 kt .
If we then solve the Cauchy problem (9), we would obtain s(t) = s0 e−e0 k1 t
after converting it back to the initial variables.
Our analysis shows that model (11) deserves to be included in the comparison of the two models (2), (4). We shall further show that under the
10
conditions K 1, k−1 k2 the three models (2), (4), and (11) do indeed
provide almost identical results and that the substrate solution of the MMmodel may provide a good approximation even though e0 s0 does not
necessarily hold.
3.3
Computational examples: three cases
Case 1. K ≈ 1 (s0 ≈ Km ) (also valid for the range K ≤ 1 (s0 ≥ Km )).
Computational example 1.1. In this example the MM- HMM- solutions for the substrate are close when ε is “small” as seen in Figure 2.
The values of the initial conditions used in the models are as follows:
s0 = 100, e0 = 0.1, c0 = 0, p0 = 0. The values of the parameters in (4)
are consistent with the initial conditions and rate parameters in (2) and are
calculated using (5) as Vmax = k2 e0 , Km = (k−1 + k2 )/k1 . The parameter k
in the “exponential decay” differential equation (11) is set to be equal to k1
as derived in the previous section. In all the examples that follow, only s0
and e0 will vary, all the other model parameters will have the same values as
in this example or they will be calculated using the corresponding s0 , e0 .
This numerical example demonstrates that when the condition ε 1
holds then the MM-model (4) can be a very good approximation of the
original HMM-system (2). Note that in this example the Michaelis constant
Km used for the computation of the approximate MM-solution is derived from
the HMM-rate parameters k1 , k−1 , k2 . This means that both models describe
the process dynamics using equivalent rate constants and, since we consider
the HMM-system to be true, this implies that the approximate model is also
valid under the assumption ε 1. Our next numerical examples aim to
examine what happens whenever the assumption ε 1 does not hold.
Computational example 1.2. Our second numerical example shows
how the solutions start to deviate when s0 and e0 are close to each other
(Figure 3). We can observe that with ε ∼ 1 the transient phase, related
to the timescale tc , takes much longer and the assumption that tc ts is
violated, thus invalidating the use of the MM-model. The goodness of the
approximation of the complex c can be a better indicator of the applicability
of the MM-model than the approximation of the substrate s, as c undergoes
a boundary layer effect at the beginning of the reaction.
11
Substrate concentration
100
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
80
60
40
20
0
0
100
200
300
400
500
Time
Complex
HMM Enzyme Kinetics
0.05
MM Model
0.04
0.03
0.02
0.01
100
200
300
400
500
Time
Figure 2: The solutions of the substrate (for models (2), (4), (11)) and the complex (for models (2),
(4) ) in the case ε = 10−3 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km =
(k−1 + k2 )/k1 = 100.1, Vmax = k2 e0 = 1; initial conditions: s0 = 100, e0 = 0.1, c0 = 0, p0 = 0. Under
these conditions the MM-model clearly serves as a good approximation of the exact HMM-system.
The values of the initial conditions used are as follows: s0 = 100, e0 =
50, c0 = 0, p0 = 0. As before, we use (5) to calculate Vmax = k2 e0 , Km =
(k−1 + k2 )/k1 .
Computational example 1.3. In this numerical example ε > 1. The
HMM-solution for the substrate is even closer to the exponential decay solution (Figure 4). As observed from the approximations of the complex in
the same figure, with ε > 1, the MM-model is inapplicable due to the much
longer timescale of the complex buildup.
12
Substrate concentration
100
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
80
60
40
20
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Time
Complex
HMM Enzyme Kinetics
25
MM Model
20
15
10
5
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Time
Figure 3: The solutions of the substrate (for models (2), (4), (11)) and the complex (for models (2), (4))
in the case ε = 0.5 ∼ 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km = (k−1 + k2 )/k1 =
100.1, Vmax = k2 e0 = 500; initial conditions: s0 = 100, e0 = 50, c0 = 0, p0 = 0.
The values of the initial conditions used for the given solution are as
follows: s0 = 100, e0 = 400, c0 = 0, p0 = 0. Here, and everywhere in the
sequel Vmax = k2 e0 , Km = (k−1 + k2 )/k1 using (5).
A figure of the specific growth rate function µ(s) = Vmax s/(Km + s) is
included for reference, see Figure 5.
Case 2. K > 1.
Computational example 2.1. It is clear that for values of K larger
than 1 (K ≈ 2 in this example) but not K 1, the relations between
13
Substrate concentration
100
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
80
60
40
20
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Time
Complex
HMM Enzyme Kinetics
200
MM Model
150
100
50
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Time
Figure 4: The solutions of the substrate (for models (4), (2), (11)) and the complex (for models (4), (2))
in the case ε = 4 > 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km = (k−1 + k2 )/k1 =
100.1, Vmax = k2 e0 = 4000; initial conditions: s0 = 100, e0 = 400, c0 = 0, p0 = 0.
the substrate solutions of the three models (2), (4) and (11) are preserved
(Figure 6).
The values of the initial conditions used in the models are as follows:
s0 = 50, e0 = 0.1, c0 = 0, p0 = 0.
Computational example 2.2. In the second numerical example of this
case the substrate solutions deviate when s0 and e0 are close to each other
(Figure 7), just as in example 1.2 and we can draw the same conclusion from
the behavior of the complex, namely the assumption that tc ts has been
14
ΜHsL
0.5
0.4
0.3
0.2
0.1
20
40
60
80
100
s
Figure 5: The specific growth rate function µ(s) = Vmax s/(Km + s) from (4) calculated in the range s
from 0 to s0 = 100. Case - ε = 10−3 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km =
(k−1 + k2 )/k1 = 100.1, Vmax = k2 e0 = 1; initial conditions: s0 = 100, e0 = 0.1, c0 = 0, p0 = 0.
violated, invalidating the use of the MM-model. All this follows from our
choice of s0 and e0 (ε = 1).
The values of the initial conditions used for the given solution are as
follows: s0 = 50, e0 = 50, c0 = 0, p0 = 0.
Computational example 2.3. In this numerical example ε > 1 (Figure 8). The solutions are identical to the solutions of example 1.3. The
approximation of the complex c in Figure 8 can lead us to the conclusion
that the MM-model is inapplicable due to the much longer timescale of the
complex buildup. This again follows from our choice of initial conditions
s0 = 50, e0 = 150, meaning ε = 3 > 1.
The values of the initial conditions used for the given solution are as
follows: s0 = 50, e0 = 150, c0 = 0, p0 = 0.
A figure of the specific growth rate function µ(s) = Vmax s/(Km + s) is
included for reference, see Figure 9.
Case 3. K 1.
Computational example 3.1. In this numerical example ε 1 and
the solutions of the three models (4), (2) and (11) are identical (Figure 10).
The values of the initial conditions used for the given solution are as
follows: s0 = 0.1, e0 = 0.001, c0 = 0, p0 = 0.
Computational example 3.2. Although we have set the initial conditions so that ε 1, the substrate solutions of the three models (4), (2) and
(11) are still very close to each other (Figure 11).
The values of the initial conditions used in the models are as follows:
s0 = 0.1, e0 = 2000, c0 = 0, p0 = 0.
15
Substrate concentration
50
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
40
30
20
10
0
0
100
200
300
400
500
600
Time
Complex
HMM Enzyme Kinetics
0.030
MM Model
0.025
0.020
0.015
0.010
0.005
100
200
300
400
500
600
Time
Figure 6: The solutions of the substrate (for models (4), (2), (11)) and the complex (for models (4), (2)) in
the case ε = 0.002 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km = (k−1 + k2 )/k1 =
100.1, Vmax = k2 e0 = 1; initial conditions: s0 = 50, e0 = 0.1, c0 = 0, p0 = 0.
A figure of the specific growth rate function µ(s) = Vmax s/(Km + s) is
included for reference, cf. Figure 12.
As described above, in case 3: K 1 (also keeping in mind the assumption k−1 k2 ), the MM-model can serve as a very good approximation to
the Henri-Michaelis-Menten law in terms of substrate dynamics, regardless
of the range of initial conditions. As seen from the solutions of the complex
though (Figure 11), the MM-model completely fails to provide a good approximation of the complex dynamics when ε 1. This is expected since
16
Substrate concentration
50
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
40
30
20
10
0
0.0
0.2
0.4
0.6
0.8
1.0
Time
Complex
HMM Enzyme Kinetics
15
MM Model
10
5
0.2
0.4
0.6
0.8
1.0
Time
Figure 7: The solutions of the substrate (for models (4), (2), (11)) and the complex (for models (4),
(2)) in the case ε = 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km = (k−1 + k2 )/k1 =
100.1, Vmax = k2 e0 = 500; initial conditions: s0 = 50, e0 = 50, c0 = 0, p0 = 0.
the approximate model only provides the outer solutions of the system in
the terms of singular perturbation analysis and in this case the inner solution has to be accounted for due to the longer timescale of the complex
buildup process. According to the MM-model, the solution of the enzymesubstrate complex can be derived as c(t) = e0 s(t)/(Km + s(t)), which under
the proposed conditions (K 1, k−1 k2 and ε 1) cannot serve as a
good approximation of the exact complex behavior. If we consider the nondimensionalized system (9) under the same conditions, we can conclude that
17
Substrate concentration
50
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
40
30
20
10
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Time
Complex
HMM Enzyme Kinetics
50
MM Model
40
30
20
10
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Time
Figure 8: The solutions of the substrate (for models (4), (2), (11)) and the complex (for models (4), (2))
in the case ε = 3 > 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km = (k−1 + k2 )/k1 =
100.1, Vmax = k2 e0 = 1500; initial conditions: s0 = 50, e0 = 150, c0 = 0, p0 = 0.
dv/dτ ∼ 0 hence v = const.
The specific growth rate function visualized on Figure 12 is almost linear
in this case (again, regardless of whether ε 1 or ε 1) in contrast to
cases 1 and 2. It is thus natural to expect the approximate model to predict
substrate solutions similar to exponential decay in this case.
Finally, the criterion (6) is obviously violated for computational example
18
ΜHsL
150
100
50
10
20
30
40
50
s
Figure 9: The specific growth rate function µ(s) = Vmax s/(Km + s) from (4) calculated in the range
s from 0 to s0 = 50. Case - ε = 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km =
(k−1 + k2 )/k1 = 100.1, Vmax = k2 e0 = 500; initial conditions: s0 = 50, e0 = 50, c0 = 0, p0 = 0.
3.2. (in contrast to all other examples where it is evaluated to less than 1):
k2 e0
= 19.9.
k1 (s0 + Km )2
Nevertheless, since K 1, k−1 k2 , the proposed explanations at the
beginning of the section are indeed manifested and the substrate solutions
of all the three considered models are identical. As we also noted, this does
not necessarily mean that the solutions of the other molecular species are
correct, e.g. the MM-model solutions for the complex are incorrect in case
ε 1.
Conclusion. There exist theoretical cases when the decay model produces better approximation to the exact HMM-scheme than the MM-model.
Additionally, in certain cases in which the MM-model can provide good substrate approximations, it may completely fail to give correct estimates of the
complex or enzyme behavior.
3.4
Is the Michaelis constant a reliable kinetic parameter?
Examples 1.2., 1.3., 2.2. and 2.3. demonstrate that the approximate model’s
solutions are far from those of the exact HMM-system despite of the consistency of the parameters used in (4) and (2) even though we derive the Vmax
and Km constants from the solutions of the exact model. This, of course,
follows from the fact that under the initial conditions we have set the approximations embedded in the MM-model do not hold. However, if we free
19
Substrate concentration
0.10
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
0.08
0.06
0.04
0.02
0.00
0
10 000
20 000
30 000
40 000
50 000
Time
Figure 10: The solutions of the substrate (for models (4), (2), (11)) and the complex (for models (4), (2))
in the case ε = 0.01 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km = (k−1 + k2 )/k1 =
100.1, Vmax = k2 e0 = 0.01; initial conditions: s0 = 0.1, e0 = 0.001, c0 = 0, p0 = 0.
the MM-model parameters from the conditions binding them to the exact
scheme (5), we may still be able to find a solution closer to the exact one
(compare Figures 13a and 14 to Figures resp. 4 and 8). On the other hand,
even if such a solution exists, it would correspond to completely different
dynamics as we could easily determine from the solutions of the complex c in
Figure 13b. The duration of the transient phase, as observed in Figure 13b,
is related to the parameter ε (ε > 1 in the considered case) and it indicates
that the MM-model may not be suitable under the given conditions since the
20
Substrate concentration
0.10
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
0.08
0.06
0.04
0.02
0.00
0.000
0.005
0.010
0.015
0.020
0.025
0.030
Time
Complex
HMM Enzyme Kinetics
2.0
MM Model
1.5
1.0
0.5
0.005
0.010
0.015
0.020
0.025
0.030
Time
Figure 11: The solutions of the substrate (for models (4), (2), (11)) and the complex (for models (4),
(2)) in the case ε = 2 ∗ 104 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km =
(k−1 + k2 )/k1 = 100.1, Vmax = k2 e0 = 2 ∗ 104 ; initial conditions: s0 = 0.1, e0 = 2000, c0 = 0, p0 = 0.
approximations embedded in it are in contradiction to the behavior of the
system. As a consequence any parameter estimation would suggest values of
the kinetic parameters much different than the real ones.
Although other sources [26] have also revealed the flaws of the inappropriate application of such approximate models under conditions which do
not hold, some authors still prefer to fit their experimental data with them.
As in Figure 13a the solutions may appear to be close to those of the exact scheme but the dynamics that the approximate model suggests could be
21
ΜHsL
20
15
10
5
0.02
0.04
0.06
0.08
0.10
s
Figure 12: The specific growth rate function µ(s) = Vmax s/(Km + s) from (4) calculated in the range s
from 0 to s0 = 0.1. Case - ε = 2∗104 1. Kinetic parameter values: k1 = 0.1, k−1 = 0.01, k2 = 10, Km =
(k−1 + k2 )/k1 = 100.1, Vmax = k2 e0 = 2 ∗ 104 ; initial conditions: s0 = 0.1, e0 = 2000, c0 = 0, p0 = 0.
completely different than the real one (i.e. the MM-model parameters used
in Figures (13a), (14) are respectively (Km = 200, Vmax = 9000) for Case 1
and (Km = 85, Vmax = 1500) for Case 2; in both cases Km can be evaluated
to Km = 100.1 while Vmax = 4000 in Case 1 and Vmax = 1500 in Case 2).
As explained in the previous section it is always beneficial to verify the
criterion for the validity of the approximate model before applying it. Some
general analysis of the behavior of the ODE system, based on the initial
conditions, could also be valuable.
We can similarly find relatively good fits of the decay model (11) for
Examples 1.1. and 2.1. if we unbind the rate constant k. Figure 15 and
Figure 16 show that in some cases the exponential decay solution may still
approximate the system behavior to a certain degree, although the rate constant k of the decay model may not be strictly connected to the parameters
of the exact HMM-model. Of course, if ε is small enough so that the substrate uptake is almost linear, no solution of (10) would be close to the exact
solutions produced by (2).
Conclusions. The presented computational examples show that the concept of “reliability” of the approximate models (the MM-model and the decay
model) is not well defined. This concept obtains different meanings depending on whether one keeps the parameters of the approximate models bound
to the HMM-system parameters, or not.
22
Substrate concentration
100
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
80
60
40
20
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Time
(a)
(b) The poor approximation of the complex indicates that the MM-model may not
be applicable in the presented case.
Figure 13: The solutions are as in Example 1.3 (ε = 4 > 1; kinetic parameter values: k1 = 0.1, k−1 =
0.01, k2 = 10, Km = (k−1 + k2 )/k1 = 100.1, Vmax = k2 e0 = 4000; initial conditions: s0 = 100, e0 =
400, c0 = 0, p0 = 0), with the only difference in (4) being: Km = 200, Vmax = 9000.
4
4.1
Computation of the HMM-system rate parameters using time course data
General settings
The dynamics of the following biochemical process is studied using the HMMsystem. The evaluation of the models is based on the goodness of fit achieved
with them, keeping in mind the measurement errors contained in the data.
The correctness of the dynamics suggested by the model is assessed by deriving the Michaelis constant for each experiment and comparing it to known
values from the literature corresponding to the same biochemical process.
23
Substrate concentration
50
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
40
30
20
10
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Time
Figure 14: The solutions are as in Case 2, Example 3 (ε = 3 > 1; kinetic parameter values: k1 =
0.1, k−1 = 0.01, k2 = 10, Km = (k−1 + k2 )/k1 = 100.1, Vmax = k2 e0 = 1500; initial conditions: s0 =
50, e0 = 150, c0 = 0, p0 = 0), with the only difference in (4) being: Km = 85
Substrate concentration
100
Henri-Michaelis-Menten Enzyme Kinetics
Michaelis-Menten Model
First Order Reaction Kinetics Model
80
60
40
20
Time
0
0
100
200
300
400
500
600
Figure 15: The solutions are as in Case 1, Example 1 (ε = 10−3 1; kinetic parameter values:
k1 = 0.1, k−1 = 0.01, k2 = 10, Km = (k−1 + k2 )/k1 = 100.1, Vmax = k2 e0 = 1; initial conditions:
s0 = 100, e0 = 0.1, c0 = 0, p0 = 0), with the only difference in (11) being: k = 0.07, rather than k = 0.1
The conclusion is that approximate models should be avoided as much as
possible. In the next section we propose contemporary tools allowing the
computation of the HMM-system parameters directly from time course measurement data.
The process we follow is similar to the parameter fitting procedures and
reverse engineering described in other papers, such as [26], [27], [23].
24
Figure 16: The solutions are as in Case 2, Example 1 (ε = 0.002 1; kinetic parameter values:
k1 = 0.1, k−1 = 0.01, k2 = 10, Km = (k−1 + k2 )/k1 = 100.1, Vmax = k2 e0 = 1; initial conditions:
s0 = 50, e0 = 0.1, c0 = 0, p0 = 0), with the only difference in (11) being: k = 0.08, rather than k = 0.1
4.2
Case study: Fitting the HMM-system against experimental data of acetylcholine hydrolysis
We fit the HMM model to experimental data of acetylcholine hydrolysis by
acetylcholinesterase that has been obtained using the derived rate constants
from the HXA method of the experiments in [34]. The rate constants were
used to compute the solutions of (2) and evaluate them for a number of
time points ti . Finally, a certain amount of noise was added to the data
according to the standard deviations given in [34]. We take into account
the measurement errors in the data and they are displayed as concentration
intervals for each point in the plots.
Our computational problem can be formulated as follows. Given time
course experimental data (together with measurement errors) for the substrate, enzyme and product concentrations find values for the parameters
k−1 , k1 , k2 and initial values e0 , s0 such that the solution of the HMM system
(2) fit well against the experimental data and possibly fit into the measurement intervals. Our procedure for solving this problem passes in two stages:
first we find an initial rough “guess” for the parameter values, then we iteratively improve the parameter set, resp. the solutions, using some optimization
routines of the numerical computing environment MATLAB. More precisely,
25
T
14 000.
e0
4.763 ´ 10-8
s0
0.0025
k-1
21.48
0.0030
0.0025
k1
43 820
k2
10.4
XMIN
0
Out[43]=
YMIN
0
XMAX
14 000.
YMAX
0.003
0.0020
0.0015
0.0010
0.0005
0.0000
0
2000
4000
6000
8000
10 000
12 000
14 000
Figure 17: An example of the usage of dynamical numerical solution of the ODE system and their
graphical representation using the M anipulate and N DSolve functions of CAS Mathematica. An initial
rough fit of the experimental data is obtained in this example.
we have used MATLAB ’s lsqnonlin procedure (optimization algorithm defaults to Trust-Region-Reflective Algorithm), where we aim to minimize the
difference between the experimental data and the computed enzyme kinetics model solutions (using ode23 or ode23s solvers) for a given set of rate
constants and initial conditions.
At the first step, the solution of the ODEs HMM-system (2) with initial
“guess” parameters (Figure 17) is shown on Figure 18. The values of the
parameters are as follows: k1 = 25000m−1 s−1 , k−1 = 0s−1 , k2 = 10s−1 ; s0 =
2.5 ∗ 10−3 m, e0 = 5.4 ∗ 10−8 m.
The results of the iterative optimization procedure we applied using the
initial parameters can be seen on Figure 19. The values of the parameters used for the given solution are as follows: k1 = 16847m−1 s−1 , k−1 =
7s−1 , k2 = 12s−1 ; s0 = 2.5 ∗ 10−3 m, e0 = 5.4 ∗ 10−8 m. Using the parameter set
we have found to solve the HMM-system we get solutions that fit nicely the
experimental data we started with.
Using Km = (k−1 + k2 )/k1 we can derive the Michaelis constant from
the optimal parameters in the HMM-system, Km = 0.00112m. This value is
close to known values of the Michaelis constant from other sources ([9], [11]),
hence we can conclude that the model estimates correctly the behavior of
the underlying biochemical process.
The MM-model (4) on the other hand provides very different results for
26
−3
x 10
substrate − s
product − p
experimental data − s
experimental data − p
enzyme e
complex c
experimental data − e
errors
3
Concentration − enzyme, complex [mM]
Concentration (substrate, product [mM])
−7
Enzyme Kinetics Model
x 10
2
0.6
0.4
1
0.2
0
0
2000
4000
6000
8000
Reaction time [seconds]
10000
12000
0
14000
Figure 18: The numerical solution of the HMM-system with an initial parameter “guess” set of the
unknown parameters k1 = 25000m−1 s−1 , k−1 = 0s−1 , k2 = 10s−1 ; s0 = 2.5 ∗ 10−3 m, e0 = 5.4 ∗ 10−8 m.
Concentration (substrate, product [mM])
−7
Enzyme Kinetics Model
x 10
substrate − s
product − p
experimental data − s
experimental data − p
enzyme e
complex c
experimental data − e
errors
2.5
2
0.6
1.5
0.5
0.4
Concentration − enzyme, complex [mM]
−3
x 10
1
0.3
0.2
0.5
0.1
0
0
2000
4000
6000
8000
Reaction time [seconds]
10000
12000
14000
Figure 19: The numerical solution of the HMM-system ( henrisystem1) after the fitting of the model.
The parameters and initial values used for the given solution are k1 = 16847m−1 s−1 , k−1 = 7s−1 , k2 =
12s−1 ; s0 = 2.5 ∗ 10−3 m, e0 = 5.4 ∗ 10−8 m.
27
Concentration (substrate, product [mM])
−7
Enzyme Kinetics Model
x 10
substrate − s
product − p
experimental data − s
experimental data − p
enzyme e
complex c
errors
experimental data − e
2.5
2
0.7
0.6
1.5
0.5
0.4
Concentration − enzyme, complex [mM]
−3
x 10
1
0.3
0.2
0.5
0
0
2000
4000
6000
8000
Reaction time [seconds]
10000
12000
14000
Figure 20: The numerical solution of the MM-model after fitting it to the experimental data. Model
parameters obtained from the optimization procedure: Vmax = 7.12 ∗ 10−7 ms−1 ; Km = 2.193 ∗ 10−3 m.
the same experimental data (Figure 20). Although the obtained parameters
(Vmax = 7.12 ∗ 10−7 ms−1 ; Km = 2.193 ∗ 10−3 m) are relatively close to the
real ones, the solutions do not pass through the intervals of the data. As in
Example 2 and Example 3 of the first two Cases, the approximate model’s
solutions deviate from the real ones even though its parameters may be close
to the exact model’s rate constants. This can easily be explained by the fact
that the complex c has more complicated dynamics (see Figure 19 for the
dynamics of c and e) than what the MM-model can estimate, i.e. data points
from the transient phase cannot be accounted for. The obtained parameters
are thus unreliable as they do not correspond to a good approximation of the
complex c and the enzyme e. A rather good fit using the same approximate
model (4) can be achieved by omitting the first observation data point of
the enzyme (Figure 21). The values of the model parameters (Vmax = 6.6 ∗
10−7 ms−1 ; Km = 1.26 ∗ 10−3 m) are even closer to those found using the HMM
model. In general, we cannot expect data points related to the “faster”
reaction (taking place during the tc timescale) to be approximated well with
the MM-model because it is entirely focused on the “slower” reaction and the
solutions we get cannot possibly follow the dynamics during both timescales.
28
Concentration (substrate, product [mM])
−7
Enzyme Kinetics Model
x 10
substrate − s
product − p
experimental data − s
experimental data − p
enzyme e
complex c
errors
experimental data − e
2.5
2
0.6
1.5
0.5
0.4
Concentration − enzyme, complex [mM]
−3
x 10
1
0.3
0.2
0.5
0.1
0
0
2000
4000
6000
8000
Reaction time [seconds]
10000
12000
14000
Figure 21: The numerical solution of the MM-model after fitting it to the experimental data. The first
experimental data point for the enzyme has been omitted in the optimization. Model parameters obtained
from the optimization procedure are now Vmax = 6.6 ∗ 10−7 ms−1 ; Km = 1.26 ∗ 10−3 m.
5
Concluding remarks
In this work we demonstrate that the value of the Michaelis constant may
strongly depend on the way one makes use of the experimental data and
the method of calculation. On the other side the rate constants in the enzyme kinetic Henri’s reaction scheme are well defined and can be efficiently
computed from time course experimental data. Moreover, the modelling of
metabolic pathways makes use of multiple reactions and can be made more
rigorous when based on Henri’s reactions instead of Michael-Menten approximate reactions, being especially imprecise when applied to living cells. Such
arguments confirm the need of refining the general methods for the determination of the Henri’s reaction scheme rate constants. Such general methods
using contemporary software tools are suggested and discussed in the present
work. In addition we wish to point out the need of validation methods in
combination with the availability of experimental data involving measurement errors [17]; a more detailed introduction of the use of validated interval

methods is presented in [8]. In this work we make use of CAS Mathematica
as computational tool, however other suitable software tools can be used,
such as MATLAB and COPASI [15], [20].
29
Ensuring the validity of the Michaelis-Menten model has been identified
as a significant step in its application which has led to the comprehensive
studies of validation criteria [30].
When modelling kinetic reactions we suggest to start with a description
of the process by a biochemical reaction equations (schemes) and then pass
to mathematical differential equations via the mass action principle. Such
an approach is useful in keeping close to the biochemical mechanism of the
modelling process; we recommend the use of this approach not only in fields
like enzyme kinetic and metabolic networks but also in fields like cell growth
and population dynamics [19].
6
Acknowledgements
The authors are very grateful to the anonymous reviewers whose valuable
reports contributed much to the quality of the paper.
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