Stochastic analysis of a miRNA–protein toggle switch

Abstract

Within systems biology there is an increasing interest in the stochastic behavior of genetic and biochemical reaction networks. An appropriate stochastic description is provided by the chemical master equation, which represents a continuous time Markov chain (CTMC). In this paper we consider the stochastic properties of a toggle switch, involving a protein compound (E2Fs and Myc) and a miRNA cluster (miR-17-92), known to control the eukaryotic cell cycle and possibly involved in oncogenesis, recently proposed in the literature within a deterministic framework. Due to the inherent stochasticity of biochemical processes and the small number of molecules involved, the stochastic approach should be more correct in describing the real system: we study the agreement between the two approaches by exploring the system parameter space. We address the problem by proposing a simplified version of the model that allows analytical treatment, and by performing numerical simulations for the full model. We observed optimal agreement between the stochastic and the deterministic description of the circuit in a large range of parameters, but some substantial differences arise in at least two cases: (1) when the deterministic system is in the proximity of a transition from a monostable to a bistable configuration, and (2) when bistability (in the deterministic system) is "masked" in the stochastic system by the distribution tails. The approach provides interesting estimates of the optimal number of molecules involved in the toggle switch. Our discussion of the points of strengths, potentiality and weakness of the chemical master equation in systems biology and the differences with respect to deterministic modeling are leveraged in order to provide useful advice for both the bioinformatician and the theoretical scientist.

Some results on the structural properties and the solution of the Chemical Master Equation

“…We consider again Example 3 in Section II (MicroRNA Toggle Switch) [14]. The equations describe an orthogonal one-step process described by a bivariate master equation (M = 2) with transition probabilities and zero otherwise (see [14] and references therein for more details).…”

“…The equations describe an orthogonal one-step process described by a bivariate master equation (M = 2) with transition probabilities and zero otherwise (see [14] and references therein for more details). The chosen parameters areᾱ = 1.68, β = 0.202, δ = 0.2, γ = 0.2, k 1 = 90, k 2 = 0.05, Γ 1 = 10300, Γ 2 = 1006.…”

“…The chosen parameters areᾱ = 1.68, β = 0.202, δ = 0.2, γ = 0.2, k 1 = 90, k 2 = 0.05, Γ 1 = 10300, Γ 2 = 1006. In [14] the stationary distribution for this process was not computed exactly, but the model was reduced to one dimension, by considering a different time scale for the two reactions. In particular, n 2 was considered as a fast variable and the value of its steady state (computed by means of the deterministic concentration equation in Eq.…”

“…CMEs can be usefully exploited also to represent (and simulate, as usual) a biological framework, already modeled by means of ordinary differential equations. In [14], for instance, the following microRNA-protein toggle switch is considered:…”

“…The method to build up the CMEs uses an interesting recursive approach to aggregate the vector of probabilities and the dynamical matrix. The analysis of the dynamical properties is the novelty of the paper, as well as the characterization of the chemical networks of reactions in a graph-theoretical fashion; our results are suitably exploited to find efficiently the exact solution at the equilibrium, in a much faster way with respect to usual Monte Carlo SSA techniques; the method is validated in a well-established biochemical example [14]. The paper is organized as follows: in Section II we recall some preliminary definitions and introduce the general setting, supported by some examples.…”