Stochastic Simulation of Enzyme-Catalyzed Reactions with Disparate Timescales

Abstract

Many physiological characteristics of living cells are regulated by protein interaction networks. Because the total numbers of these protein species can be small, molecular noise can have significant effects on the dynamical properties of a regulatory network. Computing these stochastic effects is made difficult by the large timescale separations typical of protein interactions (e.g., complex formation may occur in fractions of a second, whereas catalytic conversions may take minutes). Exact stochastic simulation may be very inefficient under these circumstances, and methods for speeding up the simulation without sacrificing accuracy have been widely studied. We show that the "total quasi-steady-state approximation" for enzyme-catalyzed reactions provides a useful framework for efficient and accurate stochastic simulations. The method is applied to three examples: a simple enzyme-catalyzed reaction where enzyme and substrate have comparable abundances, a Goldbeter-Koshland switch, where a kinase and phosphatase regulate the phosphorylation state of a common substrate, and coupled Goldbeter-Koshland switches that exhibit bistability. Simulations based on the total quasi-steady-state approximation accurately capture the steady-state probability distributions of all components of these reaction networks. In many respects, the approximation also faithfully reproduces time-dependent aspects of the fluctuations. The method is accurate even under conditions of poor timescale separation.

Reduced linear noise approximation for biochemical reaction networks with time-scale separation: The stochastic tQSSA+

“…This method is termed stochastic QSSA. 20,43 Several studies in the deterministic setting have shown that the validity of the standard QSSA is limited to certain parameter conditions. Thus, the total quasi-steady state approximation (tQSSA) has been introduced as an alternate approximation with increased accuracy.…”

“…This has then been adapted in the stochastic setting where stochastic simulations of the CME are performed using the deterministic quasisteady state expressions of the fast variables obtained using the tQSSA. 20,43,46 There are several studies that investigate the validity of these stochastic quasi-steady state approximations. Particularly, the work by Kim et al demonstrates, via twodimensional systems modeled by the CME and LNA, that the tQSSA provides a better approximation to the original system in comparison to the standard QSSA.…”

Stochastic hybrid modeling of intracellular calcium dynamics

“…8 Mastny et al 54 have carried out in-depth analysis of using QSSA under different conditions through the use of singular perturbation analysis. More recently, Barik et al 13 have extended the QSSA by analyzing the conditions under which the standard QSSA might fail. They have utilized the total QSSA ͑TQSSA͒ and have shown that under certain conditions the method of Rao and Arkin 8 fails.…”

“…Some of the more recent contributions in this area include ͑1͒ speed-up of computation through a binomial tauleaping approach 9 and k-skip method, 10 ͑2͒ time-scale/ reaction partitioning based on the propensity values, 11 a hybrid approach, 12 and quasi-steady-state approximation, 13 ͑3͒ partial-propensity-based approach, 14 and ͑4͒ alternative formulations of CLE. 15 Besides, Cai 16 has developed an approach to perform stochastic simulation of reaction systems with time-delays.…”

“…III C, we present the SHA, which consists of deterministic and stochastic components, explicitly accounts for the presence of slow and fast reactions, and incorporates complex fluxes that cannot be modeled through reactions explicitly. An approach to handle reactions with complex rate expressions is also presented in this section explaining why the existing approaches to deal with complex rates laws such as Michaelis-Menten mechanism 8,13,18 may not be directly applicable. The practical implementation of the SHA to the cytosolic calcium dynamics model 3 is presented in Sec.…”