Backward-stochastic-differential-equation approach to modeling of gene expression

Abstract

In this article, we introduce a backward method to model stochastic gene expression and protein-level dynamics. The protein amount is regarded as a diffusion process and is described by a backward stochastic differential equation (BSDE). Unlike many other SDE techniques proposed in the literature, the BSDE method is backward in time; that is, instead of initial conditions it requires the specification of end-point ("final") conditions, in addition to the model parametrization. To validate our approach we employ Gillespie's stochastic simulation algorithm (SSA) to generate (forward) benchmark data, according to predefined gene network models. Numerical simulations show that the BSDE method is able to correctly infer the protein-level distributions that preceded a known final condition, obtained originally from the forward SSA. This makes the BSDE method a powerful systems biology tool for time-reversed simulations, allowing, for example, the assessment of the biological conditions (e.g., protein concentrations) that preceded an experimentally measured event of interest (e.g., mitosis, apoptosis, etc.).

Dynamic Bayesian Network Modeling Based on Structure Prediction for Gene Regulatory Network

“…The calculated maximal information coefficients M IC(X; Y ) and its label y j are stored in the parent nodes matrix P3 and sorted by M IC(X; Y ) (lines [14][15][16][17][18]. Fourthly, the gene label y j in the first λ lines of the matrix P1, P2, and P3 are respectively taken as the parent nodes of X i [t] (lines [19][20][21][22]. Finally, the gene label of the parent nodes selected by the three methods are merged and stored in P a i which indicated the candidate parent nodes of X i [t] (line 23).…”

“…With the research on modeling methods of gene regulation network, the methods have evolved from building one-to-one relationships to one-to-multi relationships, from applying simple models to relatively complex models [12]- [14]. At present, the methods of gene regulation network modeling mainly include: Boolean network model [15], [16], correlation model [17], [18], Bayesian network model [19], [20], and differential equation model [21], [22]. Although these models can reconstruct gene regulatory network, they all have some limitations.…”

Gaussian-type density bounds for solutions to multidimensional backward SDEs and application to gene expression

“…BSDEs have numerous applications in stochastic control theory, mathematical finance, and biology (see, for instance, [5,12,25,28]). Several recent papers studied the existence of densities and density estimates for the laws of solutions to one-dimensional BSDEs [1,2,16,17], including non-Markovian BSDEs [17], and fully coupled one-dimensional forward-backward SDEs (FBSDEs) [23].…”

“…Obtaining an upper bound on expression ( Finally, we apply our results to obtain Gaussian-type bounds on the density of the law of the protein level of a gene which is a part of a gene regulatory network. To model stochastic gene expression, we employ the backward SDE approach developed in [25]. Our results apply to a network consisting of more than one gene, and therefore, stochastic gene expression is modeled by a multidimensional BSDE.…”

“…Our results apply to a network consisting of more than one gene, and therefore, stochastic gene expression is modeled by a multidimensional BSDE. For one self-regulating gene, whose expression is also modeled by the BSDE method [25], the problem of existence of a density and Gaussian-type bounds on this density was studied in [17]. Additionally, in [17], a numerical simulation was performed to observe how the density estimates agree with the data produced by the BSDE method.…”