The concept of robustness of regulatory networks has been closely related to the nature of the interactions among genes, and the capability of pattern maintenance or reproducibility. Defining this robustness property is a challenging task, but mathematical models have often associated it to the volume of the space of admissible parameters. Not only the volume of the space but also its topology and geometry contain information on essential aspects of the network, including feasible pathways, switching between two parallel pathways or distinct/disconnected active regions of parameters. A general method is presented here to characterize the space of admissible parameters, by writing it as a semi-algebraic set, and then theoretically analyzing its topology and geometry, as well as volume. This method provides a more objective and complete measure of the robustness of a developmental module. As an illustration, the segment polarity gene network is analyzed.
For biological networks, the concept of robustness often expresses the idea that the system's regulatory functions should operate correctly under a variety of situations. The network should respond appropriately to various stimulii and recognize meaningful ones (either harmful or favorable), but it should also ignore small (not meaningful) variations in the environment as well as inescapable fluctuations in the abundances of biomolecules involved in the network [1,2,3]. One might even speculate that if the networks malfunctions easily as a result of mutations then it has low chance of being selected by evolution. In that case one might expect a certain degree of mutational robustness [3,4].
While it is difficult to define this robustness property in a precise form, it has been associated to the space of admissible kinetic parameters, its volume [3], and the effect of paramater perturbations on the qualitative behavior of the system [1,2]. Some methods for parameter sensitivity have been developed [5,6], based essentially on derivatives of variables or fluxes with respect to the system’s parameters. The volume of the parameter space can be used as an indication of “how many” parameter combinations are possible, and these are related to the ability of the network to work under a variety of situations. For instance, parameters may range through different orders of magnitude, representing very different environments. However, size is often not a reliable measure for robustness; other quantities, such as shape, play a much more important role, as illustrated in Fig. 1. Analysis of the shape or geometry of the admissible parameter set gives an indication not only of its size, but also how far perturbations around each parameter disrupt the network. A robust biological network will admit small fluctuations in its parameters without changing its qualitative behavior. So, a robust network will be associated to a system whose parameter set has few “narrow pieces” and “sharp corners”. In such sets, reasonable parameter fluctuations may occur without leaving the set, hence maintaining the network’s qualitative behavior (compare Fig. 1 (a) and (b)). We can formalize a measure of robustness that is related to having low rate of exit from the region under random walk [4]. The rate of first exit is intrinsically connected to the geometry of the region and is particularly sensitive to narrow directions and not just the overall volume.
To illustrate the importance of parameter space geometry, and the insight it brings to understanding the network, the model of the segment polarity network developed by von Dassow and collaborators [3] will be analyzed. The segment polarity network is part of a cascade of gene families responsible for generating the segmentation of the fruit fly embryo [7]. Genes in earlier stages are transiently expressed, but the segment polarity genes maintain a stable pattern for about three hours. It has been suggested that the segment polarity genes constitute a robust developmental module, capable of autonomously reproducing the same behavior or generating the same gene expression pattern, in response to transient inputs [3,8,9]. This robustness would be due to the nature of interactions among genes, rather than the kinetic parameters of the reactions. The model [3] describes the interactions among the principal segment polarity genes, is continuous, and involves cell-to-cell communications and around 50 parameters which are essentially unknown. The authors of [3] explored the model by randomly choosing 240,000 parameter sets out of which about 1,192 (or 0.5%) sets were consistent with the generation (at steady state) of the wild type pattern. To explore the robustness of the network as a property of its interactions, Albert and Othmer [9] developed a Boolean model of the segment polarity network, a discrete logical model where each species has only two states (0 or 1; “OFF” or “ON”), but no kinetic parameters need to be defined. This Boolean model is amenable to various methods for systematic robustness analysis [10,11,12]. Ingolia [8] focused on the properties of the (slightly changed) model [3] in individual cells, such as bistability, and extrapolated necessary conditions on parameters to the full intercellular model.
We propose a different approach, that retains the information contained on the kinetic parameters but reduces the model to a logical form with various possible ON levels and species-dependent activation parameters. The admissible set of parameters of the model [3] is analyzed by constructing a cylindrical algebraic decomposition. Among other conclusions, our analysis completely explains the two “missing links” in von Dassow et. al. original model, namely: why the segment polarity pattern can not be recovered without the negative regulation of engrailed by Cubitus repressor protein, and why the autocatalytic wingless activation pathway vastly increases the network robustness. The pre
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