Bifurcation analysis of a modular model of embryonic kidney development

Many biological processes show switching behaviors in response to parameter changes. Although numerous surveys have been conducted on bifurcations in biological systems, they commonly focus on over-represented parts of signaling cascades, known as motifs, ignoring the multi-motif structure of biological systems and the communication links between these building blocks. In this paper, a method is proposed which partitions molecular interactions to modules based on a control theory point of view. The modules are defined so that downstream effect of one module is a regulator for its neighboring modules. Communication links between these modules are then considered as bifurcation parameters to reveal change in steady state status of each module. As a case-study, we generated a molecular interaction map of signaling molecules during the development of mammalian embryonic kidneys. The whole system was divided to modules, where each module is defined as a group of interacting molecules that result in expression of a vital downstream regulator. Bifurcation analysis was then performed on these modules by considering the communication signals as bifurcation parameters. Two-parameter bifurcation analysis was then performed to assess the effects of simultaneous input signals on each module behavior. In the case where a module had more than two inputs, a series of two parameter bifurcation diagrams were calculated each corresponding to different values of the third parameter. We detected multi-stability for RET protein as a key regulator for fate determination. This finding is in agreement with experimental data indicating that ureteric bud cells are bi-potential, able to form tip or trunk of the bud based on their RET activity level. Our findings also indicate that Glial cell-derived neurotrophic factor (GDNF), a known potent regulator of kidney development, exerts its fate-determination function on cell placement through destruction of saddle node bifurcation points in RET steady states and confining RET activity level to high activity in ureteric bud tip. In conclusion, embryonic cells usually show a huge decision making potential; the proposed modular modeling of the system in association with bifurcation analysis provides a quantitative holistic view of organ development.