We derive an approximate closed-form solution to the chemical master equation describing the Michaelis–Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into an enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme–substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis–Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations, while our approach includes them. We confirm by means of a stochastic simulation of all the elementary reaction steps in the Michaelis–Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis–Menten approximation.
REFERENCES
1.
L.
Michaelis
and M. L.
Menten
, “Die kinetik der invertinwirkung
,” Biochem. Z
49
, 333
–369
(1913
).2.
A.
Cornish-Bowden
, “One hundred years of Michaelis–Menten kinetics
,” Perspect. Sci.
4
, 3
–9
(2015
).3.
S.
Schnell
and P. K.
Maini
, “A century of enzyme kinetics: Reliability of the KM and vmax estimates
,” Comments Theor. Biol.
8
, 169
–187
(2003
).4.
S.
Schnell
and C.
Mendoza
, “Closed form solution for time-dependent enzyme kinetics
,” J. Theor. Biol.
187
(2
), 207
–212
(1997
).5.
A. F.
Bartholomay
, “A stochastic approach to statistical kinetics with application to enzyme kinetics
,” Biochemistry
1
(2
), 223
–230
(1962
).6.
D. T.
Gillespie
, “Stochastic simulation of chemical kinetics
,” Annu. Rev. Phys. Chem.
58
, 35
–55
(2007
).7.
8.
Y.
Ishihama
, T.
Schmidt
, J.
Rappsilber
, M.
Mann
, F. U.
Hartl
, M. J.
Kerner
, and D.
Frishman
, “Protein abundance profiling of the Escherichia coli cytosol
,” BMC Genomics
9
(1
), 102
(2008
).9.
P.
Arányi
and J.
Tóth
, “A full stochastic description of the Michaelis–Menten reaction for small systems
,” Acta Biochim. Biophys.
12
(4
), 375
–388
(1977
).10.
D.
Schnoerr
, G.
Sanguinetti
, and R.
Grima
, “The complex chemical Langevin equation
,” J. Chem. Phys.
141
, 024103
(2014
).11.
É.
Dóka
and G.
Lente
, “Stochastic mapping of the Michaelis–Menten mechanism
,” J. Chem. Phys.
136
(5
), 054111
(2012
).12.
E. L.
Haseltine
and J. B.
Rawlings
, “Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics
,” J. Chem. Phys.
117
(15
), 6959
–6969
(2002
).13.
C. V.
Rao
and A. P.
Arkin
, “Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm
,” J. Chem. Phys.
118
(11
), 4999
–5010
(2003
).14.
M. R.
Roussel
and R.
Zhu
, “Reducing a chemical master equation by invariant manifold methods
,” J. Chem. Phys.
121
(18
), 8716
–8730
(2004
).15.
J.
Goutsias
, “Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems
,” J. Chem. Phys.
122
(18
), 184102
(2005
).16.
P.
Thomas
, A. V.
Straube
, and R.
Grima
, “Communication: Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks
,” J. Chem. Phys.
135
, 181103
(2011
).17.
K. R.
Sanft
, D. T.
Gillespie
, and L. R.
Petzold
, “Legitimacy of the stochastic Michaelis–Menten approximation
,” IET Syst. Biol.
5
(1
), 58
–69
(2011
).18.
P.
Thomas
, A. V.
Straube
, and R.
Grima
, “The slow-scale linear noise approximation: An accurate, reduced stochastic description of biochemical networks under timescale separation conditions
,” BMC Syst. Biol.
6
(1
), 39
(2012
).19.
J. K.
Kim
, K.
Josić
, and M. R.
Bennett
, “The validity of quasi-steady-state approximations in discrete stochastic simulations
,” Biophys. J.
107
(3
), 783
–793
(2014
).20.
J. K.
Kim
, K.
Josić
, and M. R.
Bennett
, “The relationship between stochastic and deterministic quasi-steady state approximations
,” BMC Syst. Biol.
9
(1
), 87
(2015
).21.
H.
Qian
and L. M.
Bishop
, “The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: Linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks
,” Int. J. Mol. Sci.
11
(9
), 3472
–3500
(2010
).22.
X.
Kan
, C. H.
Lee
, and H. G.
Othmer
, “A multi-time-scale analysis of chemical reaction networks: II. Stochastic systems
,” J. Math. Biol.
73
(5
), 1081
–1129
(2016
).23.
D.
Schnoerr
, G.
Sanguinetti
, and R.
Grima
, “Approximation and inference methods for stochastic biochemical kinetics—A tutorial review
,” J. Phys. A: Math. Theor.
50
(9
), 093001
(2017
).24.
N.
Herath
and D.
Del Vecchio
, “Reduced linear noise approximation for biochemical reaction networks with time-scale separation: The stochastic tQSSA+
,” J. Chem. Phys.
148
(9
), 094108
(2018
).25.
H.-W.
Kang
, W. R.
KhudaBukhsh
, H.
Koeppl
, and G. A.
Rempała
, “Quasi-steady-state approximations derived from the stochastic model of enzyme kinetics
,” Bull. Math. Biol.
81
(5
), 1303
–1336
(2019
).26.
E.
Levine
and T.
Hwa
, “Stochastic fluctuations in metabolic pathways
,” Proc. Natl. Acad. Sci. U. S. A.
104
(22
), 9224
–9229
(2007
).27.
M. O.
Stéfanini
, A. J.
McKane
, and T. J.
Newman
, “Single enzyme pathways and substrate fluctuations
,” Nonlinearity
18
(4
), 1575
(2005
).28.
B. P.
English
, W.
Min
, A. M.
Van Oijen
, K. T.
Lee
, H. G.
Luo
, B. J.
Sun
, S. C.
Cherayil
, and X. S.
Kou
, “Ever-fluctuating single enzyme molecules: Michaelis–Menten equation revisited
,” Nat. Chem. Biol.
2
(2
), 87
–94
(2006
).29.
R.
Grima
, “Noise-induced breakdown of the Michaelis–Menten equation in steady-state conditions
,” Phys. Rev. Lett.
102
(21
), 218103
(2009
).30.
R.
Grima
, “Investigating the robustness of the classical enzyme kinetic equations in small intracellular compartments
,” BMC Syst. Biol.
3
(1
), 101
(2009
).31.
R.
Grima
and A.
Leier
, “Exact product formation rates for stochastic enzyme kinetics
,” J. Phys. Chem. B
121
(1
), 13
–23
(2017
).32.
D.
Singh
and S.
Chaudhury
, “Single-molecule kinetics of an enzyme in the presence of multiple substrates
,” ChemBioChem
19
(8
), 842
–850
(2018
).33.
R.
Grima
, N. G.
Walter
, and S.
Schnell
, “Single-molecule enzymology à la Michaelis–Menten
,” FEBS J.
281
(2
), 518
–530
(2014
).34.
S.
Bo
and A.
Celani
, “Multiple-scale stochastic processes: Decimation, averaging and beyond
,” Phys. Rep.
670
, 1
–59
(2017
).35.
S.
Smith
and V.
Shahrezaei
, “General transient solution of the one-step master equation in one dimension
,” Phys. Rev. E
91
(6
), 062119
(2015
).36.
G. E.
Briggs
and J. B. S.
Haldane
, “A note on the kinetics of enzyme action
,” Biochem. J.
19
(2
), 338
(1925
).37.
L. A.
Segel
and M.
Slemrod
, “The quasi-steady-state assumption: A case study in perturbation
,” SIAM Rev.
31
(3
), 446
–477
(1989
).38.
J. J.
Tyson
, “Biochemical oscillations
,” in Computational Cell Biology
(Springer
, 2002
), pp. 230
–260
.39.
L. A.
Segel
, “On the validity of the steady state assumption of enzyme kinetics
,” Bull. Math. Biol.
50
(6
), 579
–593
(1988
).40.
N. G.
van Kampen
, Stochastic Processes in Physics and Chemistry
(Elsevier
, 1992
), Vol. 1.41.
C.
Jia
and R.
Grima
, “Dynamical phase diagram of an auto-regulating gene in fast switching conditions
,” J. Chem. Phys.
152
(17
), 174110
(2020
).42.
C.
Jia
, “Model simplification and loss of irreversibility
,” Phys. Rev. E
93
(5
), 052149
(2016
).43.
N. J.
Higham
, Functions of Matrices: Theory and Computation
(SIAM
, 2008
), Vol. 104.44.
K.
Emrah
and P.
Stanica
, “The inverse of banded matrices
,” J. Comput. Appl. Math.
237
(1
), 126
–135
(2013
).45.
J. W.
Brown
, R. V.
Churchill
et al, Complex Variables and Applications
(McGraw-Hill Higher Education
, Boston
, 2009
).46.
G.
Broggi
, L. A.
Lugiato
, and A.
Colombo
, “Transient bimodality in optically bistable systems
,” Phys. Rev. A
32
(5
), 2803
(1985
).47.
M.
Mierzejewski
, J.
Dajka
, J.
Łuczka
, P.
Talkner
, and H.
Peter
, “Dynamical bimodality in equilibrium monostable systems
,” Phys. Rev. E
74
(4
), 041102
(2006
).48.
C. F.
Lam
and D. G.
Priest
, “Enzyme kinetics: Systematic generation of valid King–Altman patterns
,” Biophys. J.
12
(3
), 248
–256
(1972
).49.
P. A.
Sims
, “An “Aufbau” approach to understanding how the King–Altman method of deriving rate equations for enzyme-catalyzed reactions works
,” J. Chem. Educ.
86
(3
), 385
(2009
).50.
51.
J.
Yu
, J.
Xiao
, X.
Ren
, K.
Lao
, and X. S.
Xie
, “Probing gene expression in live cells, one protein molecule at a time
,” Science
311
(5767
), 1600
–1603
(2006
).52.
D. M.
Suter
, N.
Molina
, D.
Gatfield
, K.
Schneider
, U.
Schibler
, and F.
Naef
, “Mammalian genes are transcribed with widely different bursting kinetics
,” Science
332
(6028
), 472
–474
(2011
).© 2020 Author(s).
2020
Author(s)
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