Abstract
BackgroundBifurcation analysis has proven to be a powerful method for understanding the qualitative behavior of gene regulatory networks. In addition to the more traditional forward problem of determining the mapping from parameter space to the space of model behavior, the inverse problem of determining model parameters to result in certain desired properties of the bifurcation diagram provides an attractive methodology for addressing important biological problems. These include understanding how the robustness of qualitative behavior arises from system design as well as providing a way to engineer biological networks with qualitative properties.ResultsWe demonstrate that certain inverse bifurcation problems of biological interest may be cast as optimization problems involving minimal distances of reference parameter sets to bifurcation manifolds. This formulation allows for an iterative solution procedure based on performing a sequence of eigen-system computations and one-parameter continuations of solutions, the latter being a standard capability in existing numerical bifurcation software. As applications of the proposed method, we show that the problem of maximizing regions of a given qualitative behavior as well as the reverse engineering of bistable gene switches can be modelled and efficiently solved.
Highlights
Bifurcation analysis has proven to be a powerful method for understanding the qualitative behavior of gene regulatory networks
In the study of such systems, an important goal is to understand how the observed physiological behavior arises out of gene network topology and parameters p. Some of these questions may be studied via examining the bifurcation manifolds Σ of the ordinary differential equations (ODEs) system, which partition the parameter space into regions of different qualitative behavior
In the current context of cell biology, one would like to address problems such as: which parameter configurations lead to an observed qualitative behaviour of the system ("identification")? How can one introduce a certain qualitative behaviour into the system via parameter variations ("design")? We summarize such problems under the name of inverse bifurcation problems, where the task is to map the space of bifurcation diagrams back to the space of parameters
Summary
We demonstrate that certain inverse bifurcation problems of biological interest may be cast as optimization problems involving minimal distances of reference parameter sets to bifurcation manifolds. As applications of the proposed method, we show that the problem of maximizing regions of a given qualitative behavior as well as the reverse engineering of bistable gene switches can be modelled and efficiently solved
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