A Bayes Regularized Ordinary Differential Equation Model for the Inference of Gene Regulatory Networks

A Bayes Regularized Ordinary Differential Equation Model for the Inference of Gene Regulatory Networks

Nicole Radde (University of Leipzig, Germany) and Lars Kaderali (University of Heidelberg, Germany)
DOI: 10.4018/978-1-60566-685-3.ch006
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Abstract

Differential equation models provide a detailed, quantitative description of transcription regulatory networks. However, due to the large number of model parameters, they are usually applicable to small networks only, with at most a few dozen genes. Moreover, they are not well suited to deal with noisy data. In this chapter, we show how to circumvent these limitations by integrating an ordinary differential equation model into a stochastic framework. The resulting model is then embedded into a Bayesian learning approach. We integrate the-biologically motivated-expectation of sparse connectivity in the network into the inference process using a specifically defined prior distribution on model parameters. The approach is evaluated on simulated data and a dataset of the transcriptional network governing the yeast cell cycle.
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Background

We will now derive the system of differential equations we use to model genetic regulatory networks. The underlying assumption is that these equations describe the true states of the biological system, which is hence a deterministic system. We will discuss this assumption and its consequences in more detail later.

Key Terms in this Chapter

Gene regulatory network (GRN): Here a directed graph G(V,E) with n nodes corresponding to n genes in the network. An edge from node j to node i indicates that gene product j has an influence on the expression rate of gene i. This influence is assumed to be either activating or inhibiting. The dynamics of the system is described by ordinary differential equations.

Bayes regularized differential equation: Specific stochastic modeling approach in which the noise is assumed to stem solely from the measurement process. The state of the system is deterministic and uniquely determined by a differential equation. Observations that are used for network inference are random variables due to measurement noise. This allows for a Bayesian regularization for network inference.

Stochastic modeling approach: In stochastic modeling approaches for GRNs, observed gene expression values are interpreted as random variables, and the network inference problem translates into characterizing their probability distributions from measurements. Contrary to deterministic models, these approaches can capture variability across different cells or experiments. Different stochastic approaches have been introduced, and here we suggest a classification according to whether the system itself is stochastic or noise stems from the measurement process

Quasi-steady state approximation (QSSA): A method to reduce the number of variables of a system that includes processes on different time scales which can be separated into slow and fast. One assumes that the fast processes are always in a steady state, which changes on the slow time scale. For GRNs, the fast time scale corresponds to transcription factor – DNA binding, and the relevant slow time scale is given by the expression rates. Here, the QSSA allows for a functional relation between gene product levels and their effect on the expression rates of regulated genes, as it is implicitly assumed in most network inference approaches

Regulation function ri: Coupling term in the differential equations for GRNs. Function that describes the influence of regulators on the expression rate of gene i. For simplicity, it is often assumed that different regulators act independently, and their influences can be decoupled. An influence of a single regulator j on i is then often described with simple linear functions, or Michaelis-Menten and Hill equations are used, which can be derived from chemical reaction kinetics

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