Alberto Maria Bersani

Stochastic chemical kinetics and the total quasi-steady-state assumption: application to the stochastic simulation algorithm and chemical master equation

Recently the application of the quasi-steady-state approximation (QSSA) to the stochastic simulation algorithm (SSA) was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions [Rao and Arkin, J. Chem. Phys. 118, 4999 (2003)] and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation (CME) and, in particular, to the finite state projection algorithm [Munsky and Khammash, J. Chem. Phys. 124, 044104 (2006)], in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the (deterministic) total QSSA (tQSSA) and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis-Menten enzyme kinetics, double phosphorylation, the Goldbeter-Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.

Introducing total substrates simplifies theoretical analysis at non-negligible enzyme concentrations: pseudo first-order kinetics and the loss of zero-order ultrasensitivity

The Briggs–Haldane standard quasi-steady state approximation and the resulting rate expressions for enzyme driven biochemical reactions provide crucial theoretical insight compared to the full set of equations describing the reactions, mainly because it reduces the number of variables and equations. When the enzyme is in excess of the substrate, a significant amount of substrate can be bound in intermediate complexes, so-called substrate sequestration. The standard quasi-steady state approximation is known to fail under such conditions, a main reason being that it neglects these intermediate complexes. Introducing total substrates, i.e., the sums of substrates and intermediate complexes, provides a similar reduction of the number of variables to consider but without neglecting the contribution from intermediate complexes. The present theoretical study illustrates the usefulness of such simplifications for the understanding of biochemical reaction schemes. We show how introducing the total substrates allows a simple analytical treatment of the relevance of significant enzyme concentrations for pseudo first-order kinetics and reconciles two proposed criteria for the validity of the pseudo first-order approximation. In addition, we show how the loss of zero-order ultrasensitivity in covalent modification cycles can be analyzed, in particular that approaches such as metabolic control analysis are immediately applicable to scenarios described by the total substrates with enzyme concentrations higher than or comparable to the substrate concentrations. A simple criterion which excludes the possibility of zero-order ultrasensitivity is presented.

A perturbation solution of Michaelis–Menten kinetics in a “total” framework

In this paper we expand the equations governing Michaelis–Menten kinetics in a total quasi-steady state setting, finding the first order uniform expansions. Our results improve previous approximations and work well especially in presence of an enzyme excess.

Is there anything left to say on enzyme kinetic constants and quasi-steady state approximation?

In this paper we re-examine the commonly accepted meaning of the two kinetic constants characterizing any enzymatic reaction, according to Michaelis-Menten kinetics. Expanding in terms of exponentials the solutions of the ODEs governing the reaction, we determine a new constant, which corrects some misinterpretations of current biochemical literature.

Almost-periodicity problem as a fixed-point problem for evolution inclusions

Existence of almost-periodic solutions to quasi-linear evolution inclusions under a Stepanov almost-periodic forcing is nontraditionally examined by means of the Banach-like and the Schauder-Tikhonov-like fixed-point theorems. These multivalued fixed-point principles concern condensing operators in almost-periodic function spaces or their suitable closed subsets. The Bohr-Neugebauer-type theorem jointly with the Bochner transform are employed, besides another, for this purpose. Obstructions related to possible generalizations are discussed.

On Some Almost-Periodicity Problems in Various Metrics

The Bohr-type and the Bochner-type definitions for almost periodic functions are examined in various metrics (Stepanov, Weyl and Besicovitch). The correct definitions of Besicovitch-like multifunctions are given. Weak almost-periodic solutions are proved for differential equations and inclusions. This problem is also discussed as a fixed-point problem in function spaces.

Quasi-Steady State Approximations and Multistability in the Double Phosphorylation-Dephosphorylation Cycle

In this paper we analyze the double phosphorylation-dephosphorylati- on cycle (or double futile cycle), which is one of the most important biochemical mechanisms in intracellular reaction networks, in order to discuss the applicability of the standard quasi steady-state approximation (sQSSA) to complex enzyme reaction networks, like the ones involved in intracellular signal transduction. In particular we focus on what we call “complex depletion paradox”, according to which complexes disappear in the conservation laws, in contrast with the equations of their dynamics. In fact, in common literature the intermediate complexes either are ignored or are supposed to rapidly become negligible in the quasi steady-state phase, differently from what really happens, as shown studying the cycle without any quasi-steady state approximation. Applying the total quasi steady-state approximation (tQSSA) to the double phosphorylation-dephosphorylation cycle, we show how to solve the apparent paradox, without the need of further hypotheses, like, for example, the substrate sequestration.

Deterministic and stochastic models of enzymatic networks-applications to pharmaceutical research

The intracellular transducing device consists of complex networks of enzymatic reactions. Unfortunately, the mathematical models commonly used to describe them are still unsatisfactory and unreliable, even at the level of reproducing simple reaction schemes. The improvement of mathematical models is necessary and can follow different approaches still poorly employed, such as the modeling of spatial structures and phenomena, time delays, stochastic perturbations, only to cite the most relevant ones. In this paper we show some recent results related to the total quasi-steady-state approximation (tQSSA), in a deterministic scenario. Moreover, we show some possible applications of the tQSSA in a stochastic scheme.

Ordering of nested square roots of 2 according to the Gray code

In this paper, we discuss some relations between zeros of Lucas–Lehmer polynomials and the Gray code. We study nested square roots of 2 applying a “binary code” that associates bits 0 and 1 to “plus” and “minus” signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas–Lehmer polynomials, which take the form of nested square roots of 2.

On Stationary States in the Double Phosphorylation‐dephosphorylation Cycle

In this paper we study the double phosphorylation‐dephosphorylation cycle, which is a special case of multiple futile cycle. We study the stationary states, finding some classes of explicit solutions.