Modeling and simulation of genetic regulatory networks

At the interface of biology, computer science and mathematics, modeling, analysis, and simulation methods make it possible to describe and predict the behavior of complex genetic regulatory networks. These methods fall into four main categories depending on the mathematical concepts used, the use of graph theory concepts, the use of differential equations, Boolean formalizations or generalized logic, or a stochastic description of the transitions between molecular states. In view of the genetic and molecular data currently available, qualitative approaches (possibly based on quantitative models) seem particularly suitable. Such approaches and their automation should quickly become essential to characterize normal or pathological dynamic behaviors of biological networks, to predict the effects of disturbances, or even to design a new generation of therapeutic or agronomic tools.

Towards the functional characterization of molecular networks at the genome-scale

At the level of the genomic sequences themselves, serious progress remains to be made as regards the precise characterization of the transcribed regions and, above all, of the cis-regulatory regions, sometimes larger and more complex than the coding regions whose transcription they control. Recent reviews based on a few dozen genes studied in detail suggest that a developmental gene can be controlled by dozens of regulatory factors, each capable of binding at multiple sites, forming regions that work in an integrated fashion. Different combinations of factors can thus lead to different states of expression, depending on the region and the embryonic stage. The individual prediction of promoters or transcriptional regulation sites is made difficult by the small size and/or the low conservation of the DNA regions involved. Most researchers agree on the need to characterize functional combinations of binding sites for proteins, but no tool is yet really operational for this task.

The modeling process

Do we rather wish to account only for the existence of several alternative states of expression, as well as the paths leading to these states? In this case, it will suffice to introduce as many variables as there are genes involved, the molecular details being therefore treated implicitly. 

1. Identification of the interactions between the elements considered. This identification will depend on the level of description. In some cases, it will be enough to describe the regulatory influences between genes, while in other cases, it will be necessary to detail the molecular mechanisms involved in the regulation of genes. 

2. The choice of mathematical functions to represent an interaction or a set of interactions involved, for example, in the regulation of the expression of a gene. 

3. The evaluation of the values ​​or domains of values ​​for the parameters presents in the equations, either according to the experimental data available or with regard to theoretical considerations if necessary. 

These three points do not strictly speaking constitute a series of stages ordered in time. They are generally approached in parallel, and very often in an iterative manner. In fact, over the course of the simulations or the resolution of the equations produced and the comparison of the results obtained with the experimental data, we are often led to re-evaluate one or the other of these points until obtaining dynamic behavior consistent with the available data. This modeling process and its components are, of course, not unique to biology. They can thus be found in any dynamic modeling process, from the “exact” sciences to the human sciences.

Regulatory diagrams to mathematicians' graphs

The first type of formalization is close to the schematic graphic representation to which the biologist is accustomed to resort, at least within the framework of synthetic presentations of the information obtained on the elements and interactions of the biological system studied. From a formal point of view, a graph is defined by the "vertices" which compose it, to which are added "edges" connecting pairs of vertices. These edges can be oriented (we then speak of “arcs”) and signed (minus sign in the case of inhibition, plus sign in the case of an activation). From a biological point of view, the vertices can, for example, represent regulatory genes, and the edges of the interactions between these genes via their products (mRNA or protein). In order to be able to exploit the concepts, analysis tools, and mathematical results of graph theory, it is a question of rigorously defining the rules for the representation of elements and biological interactions in a simplified, standardized language, and most of the unequivocal time. The chosen representation rules assign a vertex per gene and distinguish between positive or negative regulatory interactions (signed arcs).

Stochastic models

The use of differential equations presupposes that the concentrations of the molecules involved vary continuously and deterministically. These considerations can prove problematic when certain molecules are present in small numbers. This may be the case for certain factors transcription, and in any case, for the genes themselves, which are generally present in one or two copies in cells. In addition, different types of fluctuations can affect the time course of many molecular processes, for example, the time necessary for the transcription of a gene. To take into account such effects, several authors have proposed so-called stochastic models, formalizing the evolution of a system by transitions between different states, each transition being affected by a defined probability. A rigorous approach is based on equations called "master equations." Unfortunately, it turns out that most of the time, these equations are impossible to solve analytically. Consequently, we use simulation methods, which allow an approximation of the solutions of the master equations.

Starting from the set of reactions that can occur in the case of the mechanism of inhibition by the substrate, the evolution of the state of the system is the number of molecules of each type, is predicted. The evolution is determined by stochastic variables representing the time interval between two successive reactions, as well as the type of the next reaction. The shape of the concentration curves for the three molecular species resembles that of the differential curves, with the addition of a "noise." The behavior predicted by the differential model can indeed be interpreted as the average of the behaviors predicted by the stochastic model. This is not a problem in this specific case where, given the system's regulatory mechanism, all stochastic behaviors roughly follow deterministic behavior. In other cases, however, divergences may appear between different stochastic simulations of the same system, representing, for example, alternative development paths for the cell. Differential models are unable to explicitly take into account such stochastic effects. Quite cumbersome to implement, stochastic simulations have so far only been applied to a small number of biological molecular networks well characterized experimentally, including the regulation of the expression of the lambda phage and the cascade of phosphorylations involved in the process of bacterial chemotaxis.

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Author: Vicki Lezama