128,960 research outputs found
On the validity of the stochastic quasi-steady-state approximation in open enzyme catalyzed reactions: Timescale separation or singular perturbation?
The quasi-steady-state approximation is widely used to develop simplified
deterministic or stochastic models of enzyme catalyzed reactions. In
deterministic models, the quasi-steady-state approximation can be
mathematically justified from singular perturbation theory. For several closed
enzymatic reactions, the homologous extension of the quasi-steady-state
approximation to the stochastic regime, known as the stochastic
quasi-steady-state approximation, has been shown to be accurate under the
analogous conditions that permit the quasi-steady-state reduction of the
deterministic counterpart. However, it was recently demonstrated that the
extension of the stochastic quasi-steady-state approximation to an open
Michaelis--Menten reaction mechanism is only valid under a condition that is
far more restrictive than the qualifier that ensures the validity of its
corresponding deterministic quasi-steady-state approximation. In this paper, we
suggest a possible explanation for this discrepancy from the lens of geometric
singular perturbation theory. In so doing, we illustrate a misconception in the
application of the quasi-steady-state approximation: timescale separation does
not imply singular perturbation.Comment: 19 pages, 1 Figur
Use and abuse of the quasi-steady-state approximation
The transient kinetic behaviour of an open single enzyme, single substrate reaction is examined. The reaction follows the Van Slyke–Cullen mechanism, a spacial case of the Michaelis–Menten reaction. The analysis is performed both with and without applying the quasi-steady-state approximation. The analysis of the full system shows conditions for biochemical pathway coupling, which yield sustained oscillatory behaviour in the enzyme reaction. The reduced model does not demonstrate this behaviour. The results have important implications in the analysis of open biochemical reactions and the modelling of metabolic systems
The Quasi-Steady-State Approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetic
In this work, we revisit the scaling analysis and commonly accepted
conditions for the validity of the standard, reverse and total
quasi-steady-state approximations through the lens of dimensional
Tikhonov-Fenichel parameters and their respective critical manifolds. By
combining Tikhonov-Fenichel parameters with scaling analysis and energy
methods, we derive improved upper bounds on the approximation error for the
standard, reverse and total quasi-steady-state approximations. Furthermore,
previous analyses suggest that the reverse quasi-steady-state approximation is
only valid when initial enzyme concentrations greatly exceed initial substrate
concentrations. However, our results indicate that this approximation can be
valid when initial enzyme and substrate concentrations are of equal magnitude.
Using energy methods, we find that the condition for the validity of the
reverse quasi-steady-state approximation is far less restrictive than was
previously assumed, and we derive a new "small" parameter that determines the
validity of this approximation. In doing so, we extend the established domain
of validity for the reverse quasi-steady-state approximation. Consequently,
this opens up the possibility of utilizing the reverse quasi-steady-state
approximation to model enzyme catalyzed reactions and estimate kinetic
parameters in enzymatic assays at much lower enzyme to substrate ratios than
was previously thought. Moreover, we show for the first time that the critical
manifold of the reverse quasi-steady-state approximation contains a singular
point where normal hyperbolicity is lost. Associated with this singularity is a
transcritical bifurcation, and the corresponding normal form of this
bifurcation is recovered through scaling analysis.Comment: 50 pages, 10 figures, 1 tabl
Enzyme kinetics far from the standard quasi-steady-state and equilibrium approximations
Analytic approximations of the time-evolution of the single enzyme-substrate reaction are valid for all but a small region of parameter space in the positive initial enzyme-initial substrate concentration plane. We find velocity equations for the substrate decomposition and product formation with the aid of the total quasi-steady-state approximation and an aggregation technique for cases where neither the more normally employed standard nor reverse quasi-steady-state approximations are valid. Applications to determining reaction kinetic parameters are discussed
Rigorous estimates for the quasi-steady state approximation of the Michaelis-Menten reaction mechanism at low enzyme concentrations
There is a vast amount of literature concerning the appropriateness of
various perturbation parameters for the standard quasi-steady state
approximation in the Michaelis-Menten reaction mechanism, and also concerning
the relevance of these parameters for the accuracy of the approximation by the
familiar Michaelis-Menten equation. Typically, the arguments in the literature
are based on (heuristic) timescale estimates, from which one cannot obtain
reliable quantitative estimates for the error of the quasi-steady state
approximation. We take a different approach. By combining phase plane analysis
with differential inequalities, we derive sharp explicit upper and lower
estimates for the duration of the initial transient and substrate depletion
during this transitory phase. In addition, we obtain rigorous bounds on the
accuracy of the standard quasi-steady state approximation in the slow dynamics
regime. Notably, under the assumption that the quasi-steady state approximation
is valid over the entire time course of the reaction, our error estimate is of
order one in the Segel-Slemrod parameter.Comment: 35 pages; 7 figure
The effects of intrinsic noise on the behaviour of bistable cell regulatory systems under quasi-steady state conditions
We analyse the effect of intrinsic fluctuations on the properties of bistable
stochastic systems with time scale separation operating under1 quasi-steady
state conditions. We first formulate a stochastic generalisation of the
quasi-steady state approximation based on the semi-classical approximation of
the partial differential equation for the generating function associated with
the Chemical Master Equation. Such approximation proceeds by optimising an
action functional whose associated set of Euler-Lagrange (Hamilton) equations
provide the most likely fluctuation path. We show that, under appropriate
conditions granting time scale separation, the Hamiltonian can be re-scaled so
that the set of Hamilton equations splits up into slow and fast variables,
whereby the quasi-steady state approximation can be applied. We analyse two
particular examples of systems whose mean-field limit has been shown to exhibit
bi-stability: an enzyme-catalysed system of two mutually-inhibitory proteins
and a gene regulatory circuit with self-activation. Our theory establishes that
the number of molecules of the conserved species are order parameters whose
variation regulates bistable behaviour in the associated systems beyond the
predictions of the mean-field theory. This prediction is fully confirmed by
direct numerical simulations using the stochastic simulation algorithm. This
result allows us to propose strategies whereby, by varying the number of
molecules of the three conserved chemical species, cell properties associated
to bistable behaviour (phenotype, cell-cycle status, etc.) can be controlled.Comment: 33 pages, 9 figures, accepted for publication in the Journal of
Chemical Physic
Model reduction by extended quasi-steady-state approximation
We extend the quasi-steady state approximation (QSSA) as well with respect to the class of differential systems as with respect to the order of approximation. As an application we prove that the trimolecular autocatalator can be approximated by a fast bimolecular reaction system. Finally we describe a class of singularly perturbed systems for which the first order QSSA can easily be obtained
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