Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. We propose a new modeling framework for gene regulatory networks, that combines the intuitive appeal of a qualitative description of gene states with a high flexibility in incorporating stochasticity in the duration of cellular processes. We apply our methods to the regulatory network of the segment polarity genes, thus gaining novel insights into the development of gene expression patterns. For example, we show that very short synthesis and decay times can perturb the wild-type pattern. On the other hand, separation of time-scales between pre- and post-translational processes and a minimal prepattern ensure convergence to the wild-type expression pattern regardless of fluctuations.

As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network, as well as a high level of synchronisation among the various regulatory processes. Two possible methods of adapting qualitative models to incorporate the continuous-time character of regulatory networks, are discussed and compared. The first method consists of introducing asynchronous updates in the Boolean model. In the second method, the approach introduced by Glass is adopted to obtain a set of piecewise linear differential equations that continuously describe the states of each gene or protein in the network. Both methods are applied to a Boolean model of the segment polarity gene network of Drosophila melanogaster. The dynamics of the model is analysed, and a theoretical characterisation of the model's gene pattern prediction is provided as a function of the timescales of the various processes.

Weakly activated signaling cascades can be modeled as linear systems. The input-tooutput transfer function and the internal gain of a linear system, provide natural measures for the propagation of the input signal down the cascade and for the characterization of the final outcome. The most efficient design of a cascade for generating sharp signals, is obtained by choosing all the off rates equal, and a "universal" finite optimal length.

Abstract:To analyze and gain intuition on the mechanisms of complex systems of large dimensions, one strategy is to simplify the model by identifying a reduced system, in the form of a smaller set of variables and interactions that still capture specific properties of the system. For large models of biological networks, the diagram of interactions is often well represented by a Boolean model with a family of logical rules. The state space of a Boolean model is finite, and its asynchronous dynamics are fully described by a transition graph in the state space. In this context, a method will be developed for identifying the active or operational interactions responsible for a given dynamic behaviour. The first step in this procedure is the decomposition of the asynchronous transition graph into its strongly connected components, to obtain a "reduced" and hierarchically organized graph of transitions. The second step consists of the identification of a partial graph of interactions and a sub-family of logical rules that remain operational in a given region of the state space. This model reduction method and its usefulness are illustrated by an application to a model of programmed cell death. The method identifies two mechanisms used by the cell to respond to death-receptor stimulation and decide between the survival or apoptotic pathways. Key-words:

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A qualitative method to control piecewise affine differential systems is proposed and explored for application to genetic regulatory networks. This study considers systems whose outputs and inputs are of a qualitative form, well suited to experimental devices: the measurements indicate whether the variables are "strongly" or "weakly" expressed, that is, only the region of the state space where trajectories evolve at each instant can be known. The control laws are piecewise constant functions in each region and in time, and are only allowed to take three qualitative values corresponding to no control (u = 1), high synthesis rates (u = umax) or low synthesis rates (u = umin). The problems of controlling the bistable switch to each of its steady states is considered. Exact solutions are given to asymptotically control the system to either of its two stable steady states. Two approximate solutions are suggested to the problem of controlling the system to the unstable steady state: either control to a neighborhood of the state, or in the form of a periodic cycle that passes through the state.

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The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In this paper, we show that, in addition to volume, the geometry of this region has important consequences for the robustness and the fragility of a network. We develop an approximation within which we could algebraically specify the feasible region. We analyze the segment polarity gene network to illustrate our approach. The study of random walks in the parameter space and how they exit the feasible region provide us with a rich perspective on the different modes of failure of this network model. In particular, we found that, between two alternative ways of activating Wingless, one is more robust than the other. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.