J. Serb. Chem. Soc. 71 (1) 75–86 (2006)
JSCS – 3401
UDC 577.15+531.9:–519.245
Original scientific paper
About implementing a Monte Carlo simulation algorithm for
enzymatic reactions in crowded media
ADRIANA ISVORAN1,*, EDUALD VILASECA2,3, FERNANO ORTEGA2,4, MARTA
CASCANTE2,4 and FRANCESC MAS2,3
1Department of Chemistry, University of the West Timisoara, Str. Pestalozzi 16, 300115
Timisoara, Romania (e-mail: [email protected]), 2Theoretical Chemistry Research Centre
(CeRQT) of the Scientific Park of Barcelona, (PCB), Barcelona, 3Physical Chemistry
Department, Barcelona University (UB), Barcelona and 4Biochemistry and Molecular Biology
Department, Barcelona University (UB) Barcelona, Spain
(Received 24 January, revised 4 April 2005)
Abstract: In this paper, several aspects of implementing a Monte Carlo simulation
algorithm for studying the Michaelis–Menten mechanism of enzymatic reactions in
crowded media are presented. Using a two dimensional lattice with obstacles, it is
shown how the initial distribution of the reactants and obstacles on the lattice affects
the values of the rate coefficients and the concentration of the reactants. The influence of the number of considered nearest neighbours and of the obstacle concentration on the values of the rate coefficients is also demonstrated. The results strongly
suggest fractal kinetics for enzyme reactions in crowded media.
Keywords: enzymatic reactions, fractal kinetics, crowded media.
INTRODUCTION
The law of mass action considers chemical reactions to be macroscopic, continuous and deterministic even it is known that they involve discrete, random collisions between individual molecules. As smaller chemical systems and intracellular
environments are considered, the validity of the continuous approach is more tenuous. When it is intended to model reaction rates in intracellular media, it must also
be taken into account that this media is crowded and it strongly affects the diffusion
processes.1 Both theoretical and experimental results show that all diffusion-limited reactions are strongly affected by the topology dimension of the media in
which they occur and the law of mass action breaks down in media with low topological dimensions. There are many papers in the literature which consider the effects of macromolecular crowding and a small number of molecules on in vivo
biomolecular reactions. There are two main orientations:
*
Corresponding author.
doi: 10.2298/JSC0601075I
75
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ISVORAN et al.
i) A number of authors consider stochastic modelling of intracellular reactions
to be more realistic because this approach takes into account the discrete character
of the quantity of the components and the random character of the phenomena.2–4
In systems comprising a small number of molecules, the fluctuations may affect
the phenomena and a stochastic approach is more appropriate in those cases, even
though an unrealistic homogeneity is assumed.
ii) Other authors consider that chemical reactions occurring in heterogeneous
media, where the reactants are constrained on the microscopic level, follow
fractal-like kinetics.5–10 This means that the rate coefficients are no longer constant and they depend on time.
In this paper, the fractal-like kinetics approach is considered and some aspects
of implementing a Monte Carlo algorithm for studying Michaelis–Menten enzyme
kinetics in a bidimensional lattice with obstacles, following the Berry algorithm.7
METHOD
Theoretical description of fractal kinetics for enzyme reactions
The fractal-like approach for enzyme kinetics was proposed by Kopelman9
and he considered that the rate coefficients for diffusion-controlled bimolecular reactions are not constants but that they depend on time:
k(t) = k0t–h
(1)
where k0 is a constant and h Î [0,1], called the fractal parameter, is a measure of the
topological dimensionality of the system. In homogenous media, h = 0 and k = k0 =
constant. When applying this equation, for t ® 0 one obtains k ® ¥, hence, it can only
be used for the study of long-term kinetics. Instead of Eq. (1), another equation was
proposed by Schnell and Turner,6 the so-called Zipf–Mandelbrot equation:
k(t) =
k0
(2)
(t + t )
h
where of k0 and h have the same meaning as in Eq. (1) and t is a positive constant.
This equation showed a better fit of simulation data than the Kopelman equation.
The fractal kinetics approach to the Michaelis–Menten mechanism of enzyme reactions was applied:
k1
k2
E + S « C ¾¾
®E+P
(3)
k -1
Where E is the enzyme, S is the substrate, C is the enzyme–substrate complex, P is
the product and ki are the rate coefficients. The equations describing this mechanism are
MONTE CARLO SIMULATION FOR ENZYMATIC REACTIONS
77
d[S]
= -k1[E ][S] + k -1[C]
dt
d[E ] = - d[C ] - k [E ][S] + ( k + k )[C]
1
-1
2
dt
dt
d[P ] = k [C]
2
dt
(4)
In classical kinetics based on the law of mass action, the rate coefficients are
constant but fractal kinetics considers the rate coefficients describing second order
reactions as being dependent on time. In the Michaelis–Menten mechanism, only the
k1
first reaction E + S ¾¾
® C is a second order reaction, so only k1 depends on time. If
k1 in Eq. (4) is replaced by k1(t) = k10/(t + t)h, the Equations describing the fractal kinetics of the Michaelis–Menten mechanism in crowded media are obtained.
The Monte Carlo simulation algorithm
Following the Berry algorithm,7 a 100´100 2D lattice with cyclic boundary conditions on which five types of molecules distributed, enzyme (E), substrate (S), complex
(C), product (P) and obstacles (O), was considered. The E, S, C and P types are mobile
on the lattice through diffusion which was modeled by independent random walks of the
individual molecules. The obstacles were considered immobile and non-reactive. At any
moment, one given site of the lattice cannot be occupied by more than one molecule. The
rate coefficients, k1, k–1 and k2 are modeled by reaction probabilities, f, r and g, respectively. For all the calculus in this paper f = 1, r = 0.02 and g = 0.04. At the beginning of
each simulation, the molecules were randomly distributed into the lattice. The rules for
movement and reaction depend on the type of the molecule. One molecule is chosen randomly to move or react according to the following rules:7
i) If the subject molecule is not C:
– randomly choose the nearest neighbour destination site;
– if the destination site is occupied by an obstacle, a complex or a product, the
subject molecule rests in its site;
–if the destination site is free, the subject molecule moves to it;
– if the subject molecule is E (or S) and the destination site is S (or E) then generate a random number between 0 and 1 and compare it with the probability f. If the
random number is lower than the probability, replace the destination molecule by
C, remove the subject molecule and count the number of E–S reactions, g(t);
ii) If the subject molecule is C:
– randomly choose a site among its nearest neighbour empty sites;
– randomly generate a number between 0 and 1.
o It it is lower than r then place E on the subject site and S on a randomly chosen destination site.
o If it is greater than r but lower than r+ g, place E on the subject site and P on a
randomly chosen destination site.
o If it is greater than r + g then C moves to the destination site.
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ISVORAN et al.
A time step, t, is defined as the repetition of this Monte Carlo sequence for
Ntot(t) times, where Ntot(t) is the total number of molecules in the lattice at a time t,
in order to assure that statistically each molecule moves once in the time step. In
this calculation, 600 time steps were used. The values for the reactant concentrations and rate coefficients values were averaged over 200 independent runs.
The number of E–S collisions is related to the rate coefficient, k1, by the equation7
1
g(t) = ò k1( t )[E ]( t )[S ]( t )dt
(5)
ö÷æç dg ( t ) ö÷
k1( t ) = æç 1
E
t
S
t
([
](
)[
](
))
dt ø
è
øè
(6)
0
Thus, one obtaines:
Using Eqs. (4)–(5), the other rate coefficients are obtained:
ö÷æç dg ( t ) + d[S ]( t ) ö÷
k -1 = æç 1
[
](
)
dt
dt ø
C
t
è
øè
P
t
d
[
](
)
ö÷æç
ö
k 2 = æç 1
dt ÷ø
è [C ]( t ) øè
(7)
From the simulation, the concentrations of reactants at any moment t, [E](t),
[S](t), [C](t), [P](t) can be obtained and the data can be differentiated numerically
with respect to time using the first order approximation
dX
dt @
(X (t) - X (t - Dt))
Dt
(8)
Using Eq. (8) in Eqs. (6) and (7), the values of the rate coefficients can be obtained.
For all the calculation, the time was expressed in simulation time units, the reactant concentrations were expressed as the ratios of the number of particles to the
number of lattice sites and the obstacle concentrations were always under the percolation threshold3 ([O]p.t. = 0.407).
RESULTS
In order to implement a simulation algorithm in good agreement with other
simulation data presented in the literature and also to have a reasonable computational time, the effect of the initial distribution of the reactants and obstacles concentrations on the reactant concentrations and rate coefficients values was checked. The time dependences of the number of complex and substrate molecules for
different possibilities of generating the initial distribution of particles are presented
in Figs.1a and b. These dependencies were obtained by randomly distributing the
particles in the lattice in the following manners:
MONTE CARLO SIMULATION FOR ENZYMATIC REACTIONS
79
i) first all the substrate molecules were randomly distributed, then all the enzyme molecules and finally all the obstacles molecules. Hence, the order of distribution was S–E–O;
ii) first all the obstacles molecules were randomly distributed, then all the substrate molecules and finally all the enzyme molecules. Hence, the order of distribution was O–S–E;
iii) first all the obstacles molecules, were randomly distributed, then all the enzyme molecules and finally all the substrate molecules. Hence, the order of distribution was O–E–S;
iv) first the type of particle was randomly chosen and then the number of particle of this type, until all the molecules from all the types were distributed in the lattice. In this case there was not an order for the distribution, it was totally random.
The initial concentrations for the reactants were [S0] = 0.2, [E0] = 0.01 and the
concentration of obstacles was [O] = 0.37, for the cases when they were present.
It can be seen that the order of the distribution of different type of particles
does not affect the results, the lines for the orders S–E–O, O–S–E and O–E–S in
Figs. 1a and b are superposed. There is a small difference between these lines and
that corresponding to the random choice of both type and number of molecules
(solid line in Figs. 1a and b). All these cases clearly differ from that with a lattice
without obstacles, dot dashed lines in Figs. 1a and b. The more realistic situation is
to randomly choose both the type and number of particle when generating the initial distribution in the lattice because it ensures a better mixing in the system and
the increase in the computational time is not significant.
k2
-1 ® E + S and C ¾¾
As the reactions C ¾k¾
¾
® E + P are first order reactions,
the rate coefficients in a low dimensional case do not depend on time, and their values must correspond to the reaction probabilities, k–1 ® r and k2 ® g. The values
of these coefficients in lattices without obstacles but with different initial concentrations of reactants: [S] = 0.2 and [E] = 0.01 for the solid line, [S] = 0.1 and [E] =
0.005 for the dashed line and [S] = 0.05 and [E] = 0.002 for the dotted line are
shown in Fig. 2. The values of the same coefficients in lattices with the same initial
concentrations of reactants ([S] = 0.2, [E] = 0.01) but with different concentrations
of obstacles, [O] = 0.157 for the solid line, [O] = 0.2 for the dashed line and [O] =
0.37 for the dotted line are shown in Fig. 3.
From Fig. 2, it can be seen that in absence of obstacles the values of the rate coefficients k–1 and k2 are not affected by the reactant concentrations and they correspond to the reaction probabilities. However, from Fig. 3, it can be seen that these
values depend on the obstacle concentration and if the obstacle concentration increases, the values of k–1 and k2 are constant but they have lower values than the corresponding reaction probability. The dependence of these rate coefficients on the obstacle concentration is determined by the fact that the probability of a complex molecule having no free neighbour is increase when the concentration of the obstacles is
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ISVORAN et al.
a)
b)
Fig. 1. The time dependence of the number of C molecules (a) and S molecules (b) for different
orders in which the types of particles were initialy distributed in the lattice.
larger. If a complex molecule does not have a free neighbour, it cannot react and, in
this case, the values of k–1 and k2 decrease. Theory predicts that the rate coefficients
k–1 and k2 should be constant and equal to the corresponding reaction probability.
This means that the algorithm must take into account this situation and, in order to
so, an additional condition for C molecules to move or react when they are selected
was introduced. This additional condition implies that if the randomly chosen molecule is a complex and it has no free neighbour, the process is not aborted but another
complex molecule is randomly chosen. If necessary, the process is repeated until a
complex molecule has at least one free neighbour site. In the following simulations
the algorithm with this additional condition was employed.
The time dependencies of the second order reaction rate coefficient k1 are presented in Fig. 4 for the case without obstacles (solid line) and with different concentrations of obstacles: dashed line for [O] = 0.12, dotted line for [O] = 0.25 and
MONTE CARLO SIMULATION FOR ENZYMATIC REACTIONS
81
Fig. 2. The values of the rate coefficients k–1 and k2 in lattices without obstacles. The initial concentrations for the substrate and enzyme were [S0] = 0.2, [E0] = 0.01 for the solid line, [S0] = 0.1,
[E0] = 0.005 for the dashed line and [S0] = 0.05, [E0] = 0.002 for the dotted line.
Fig.3. The values of the rate coefficients k–1 and k2 in lattice with obstacles. The initial concentrations of the substrate and enzyme were the same for all situations, [S0] = 0.2, [E0] = 0.01, but
the concentrations of the obstacles were different, as indicated in the Figure.
dash-dotted line for [O] = 0.37. The initial reactant concentrations were the same
for all lines, [S0] = 0.2, [E0] = 0.01. It can be seen that k1 is time dependent also in
absence of obstacles. This fact implies that it is not possible to have good mixing of
the particles in a two-dimensional lattice even in the absence of obstacles. The
non-linear fittings of the lines presented in Fig. 4, made using the Levenber-
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ISVORAN et al.
Fig. 4. The time dependence of k1 for different concentrations of obstacles. The initial concentrations of the reactants were [S0] = 0.2, [E0] = 0.01.
ger–Marquardt method for non-linear curve fitting within the ORIGIN 6.0 package, give the values shown in Table I for the parameters present in the Zipf–Mandelbrot Eq. (2). The values for test differences between the data in plots and data
obtained from the equation using the determined parameters, c2, are also given in
Table I.
TABLE I. The parameters in Zipf–Mandelbrot equation for different concentrations of obstacles
Concentration
k0
t
h
c2
0.37
3.00±0.07
3.7±0.4
0.415±0.005
1.55´10-4
0.25
7.0±0.5
64±5
0.426±0.011
2.12´10-4
0.12
39±0.9
300±30
0.60±0.03
2.37´10-4
0
53±8
410±50
0.60±0.04
2.75´10-4
When the concentration of obstacles was decreased, the behaviour of the system changed and it tended to behaviour in a manner similar to when there were no
obstacles present.
Another consideration that was taken into account was the effect of the number of nearest neighours for each particle. Hence, the case of 8 instead of 4 nearest
neighours, as used by Berry,7 was also analyzed. A comparison between these two
situations is given in Fig 5.
When 8 nearest neighbours were considered (short dotted line), the behaviour
of the system is almost the same as that obtained in the case of no obstacles (solid
line) or low concentrations of obstacles, and it can also be described using the
Zipf–Mandelbrot equation. This fact can be explained by a higher degree of mix-
MONTE CARLO SIMULATION FOR ENZYMATIC REACTIONS
83
Fig. 5. The time dependence of k1 for different numbers of nearest neighbours: 4 neighbours
without obstacles – solid line, 8 neighbours with [O] = 0.37 of obstacles – short dotted line, 4
neighbours with [O] = 0.37 of obstacles – long dashed line.
Fig. 6. The behaviour of k1 for a particular situation in the system. The initial concentrations of
the particles were [S] = 0.1, [E] = 0.02 and [O] = 0.37.
ing of the particles in the case with 8 nearest neighbours, and, hence, the kinetics
tends to those obtained in the absence of obstacles.
After analyzing all these results, the option was to implement an algorithm in
which a random initial distribution of the particles both in terms of their type and
number is considered, with additional conditions for the movement and reactivity
of the C molecules and with 4 nearest neighbours for each molecule. In order to test
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ISVORAN et al.
this final algorithm it was applied to a system having initial concentrations of particles: [S] = 0.1, [E] = 0.02 and [O] = 0.37. The obtained values k1 and k2 were constant and equal to their reaction probabilities (data not shown). The dependence of
k1 on time in this situation is presented in Fig. 6.
Non-linear fitting of the line presented in Fig. 6, made using the Levenberger–Marquardt method for non-linear curve fitting within the ORIGIN 6.0 package, gives the following values for the parameters: k0 = (2.900 ± 0.009), t = (5.9 ±
0.6) and h = (0.395 ± 0.006), with a test difference of c2 = 1.36 ´ 10–4. These values
demonstrate again that the kinetics are fractal-like and can be described by the
Zipf–Mandelbrot equation (2).
DISCUSSION AND CONCLUSIONS
In his paper, Berry proposed an algorithm with which enzyme kinetics on
low-dimensional lattices can be studied and which provided representations of the
spatial distributions of the molecules during the reaction, allowing direct imaging of
their repartition on the lattice.7 When trying to apply the Monte Carlo simulation algorithm as presented in Berry’s paper, some technical problems which were not
mentioned in the paper were encountered. One of them concerns the initial distribution of the particles in the lattice. From our results, a small difference existed between the cases when both the type and number of particles were randomly chosen
and the cases when all the particles of one type were distributed and then from the
other types. As increasing the computational time was not very important, it was
considered better to distribute the particles by randomly choosing both the type and
number of particles, in order ensure a better degree of mixing in the system.
Another technical problem appeared when the values for the coefficients k–1
and k2 were determined. Their values must be constant and must also correspond to
the imposed probabilities of the reactions. In the simulations, their values were constant but, for higher obstacles densities, they were smaller than the reactions probabilities. This means that for high concentrations of obstacles, the probability that a C
molecules has no free neighour is increased. In order to solve this problem, an additional condition was introduced in the simulation algorithm: if the randomly chosen
C molecule has not free neighbours, another C molecule must be randomly chosen
until a C molecule is chosen which has at least one free neighbour.
After introducing these new conditions into the algorithm, it was applied for different concentrations of obstacles. From the values presented in Table I, it can be
seen that the value of the h parameter decrease a little with increasing concentration
of obstacles but the values of the t parameter decreases considerably under the same
conditions. The physical meaning of the t parameter is that it is a measure of the critical time from which the rate coefficients will be driven by macromolecular crowding
effects.6 As the concentration of obstacles increases, the shorter is this critical time.
Even if h decreases with increasing concentration of particles, the fractality of the ki-
MONTE CARLO SIMULATION FOR ENZYMATIC REACTIONS
85
netics increases because there is a considerable increase in the t parameter, which
means that in the Zipf–Mandelbrot equation the decrease of k is greater.
There are also other papers in the literature3,7 presenting simulations in
two-dimensional lattices with different concentations of obstacles. There the dependencies k1 = k1(t) for different concentrations of obstacles were presented only
in double logarithmic plots, which are easier to be interpreted using the Kopelman
equation. As it was also mentioned in a paper by Schnel and Turner,6 the present
results show that for all the presented situations, the fractal-like kinetics are better
described by the Zipf–Mandelbrot equation than by the Kopelman equation.
The results presented in this paper show that the implemented Monte Carlo simulation algorithm, which is based on the Berry algorithm,3 may be applied to study
enzyme kinetics in two-dimensional media. The Michaelis–Menten mechanism for
enzymatic kinetics was considered and the results demonstrate fractal-like kinetics
in both homogeneous and heterogeneous two-dimensional media. The fractality of
the kinetics depends on the concentrations of the obstacles.
This simulation algorithm could be improved in order to perform simulations
in a three-dimensional lattice and/or to simulate the case of the reversible Michaelis–Menten mechanism.
Acknowledgements: This paper is a result of a short-term research fellowship offered to
Adriana Isvoran by the Federation of European Biochemical Societies to work at the Barcelona University and this financial support is kindly acknowledged. Financial support from Spanish Science
and Education Ministery, Projects BQU2003-09698-CO2-02 and SAF2005-01627, from the Generalitat de Catalunya, are also acknowledged.
IZVOD
O IMPLEMENTACIJI MONTE KARLO ALGORITMA (MONTE CARLO) NA
ENZIMSKE REAKCIJE U ISPUWENIM SREDINAMA
ADRIANA ISVORAN1, EDUALD VILASECA2,3, FERNANDO ORTEGA2,4, MARTA CASCANTE2,4 i
FRANCESC MAS2,3
1Department of Chemistry, University of the West Timisoara, Str. Pestalozzi 16, 300115 Timisoara, Romania, 2Theoretical
Chemistry Research Centre (CeRQT) of the Scientific Park of Barcelona, (PCB), Barcelona, 3Physical Chemistry Department, Barcelona University (UB), Barcelona i 4Biochemistry and Molecular Biology Department, Barcelona University
(UB) Barcelona, Spain
Prezentovani su neki aspekti implementacije algoritma Monte Karlo simulacije za prou~avawe Mihaelis-Menten (Michaelis-Menten) mehanizma enzimskih reakcija u ispuwenim sredinama. Kori{}ewem modela dvo-dimenzionalne re{etke sa
preprekama pokazano je da po~etna raspodela reaktanata i prepreka na re{etki uti~e
na vrednosti koeficijenata brzine i koncetraciju reaktanata. Pokazan je uticaj razmatranog broja najbli`ih susednih ~estica i koncentracije prepreka na vrednost
koeficijenta brzine. Rezultati ~vrsto sugeri{u da za enzimske reakcije u ispuwenim
sredinama va`i "fraktalna" kinetika.
(Primqeno 24. januara, revidirano 8. aprila 2005)
86
ISVORAN et al.
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