Mapping parameter spaces of biological switches

Abstract

Since the seminal 1961 paper of Monod and Jacob, mathematical models of biomolecular circuits have guided our understanding of cell regulation. Model-based exploration of the functional capabilities of any given circuit requires systematic mapping of multidimensional spaces of model parameters. Despite significant advances in computational dynamical systems approaches, this analysis remains a nontrivial task. Here, we use a nonlinear system of ordinary differential equations to model oocyte selection in Drosophila, a robust symmetry-breaking event that relies on autoregulatory localization of oocyte-specification factors. By applying an algorithmic approach that implements symbolic computation and topological methods, we enumerate all phase portraits of stable steady states in the limit when nonlinear regulatory interactions become discrete switches. Leveraging this initial exact partitioning and further using numerical exploration, we locate parameter regions that are dense in purely asymmetric steady states when the nonlinearities are not infinitely sharp, enabling systematic identification of parameter regions that correspond to robust oocyte selection. This framework can be generalized to map the full parameter spaces in a broad class of models involving biological switches.

Extending combinatorial regulatory network modeling to include activity control and decay modulation

“…As is discussed in [12] and [7] the dynamics captured by the DSGRN computations is highly suggestive of the dynamics exhibited by the ODEs even for moderate levels of exponent h in the Hill functions.…”

“…This approach has been applied to a variety of regulatory networks associated with questions and challenges from systems and synthetic biology including: identification of oscillatory behavior in a simple model of the p53 network [5], identification of minimal models for the switching behavior of the mammalian Rb-E2F system [15], EMT [27], oocyte [7], and design of optimal 3 node hysteretic switches [12]. This variety of applications suggests that DSGRN is a potentially powerful tool for the global analysis of networks.…”

“…. , k q ), and Λ n is given by the formula (7). Observe that the image of Λ n consists of 2 K values…”

“…where Λ n has the form (7) and involves only type 0 edges, that is, it is defined by an interaction function f of order K and depends on the state variables x 1 , . .…”

“…We will refer to Λ n as the classical production rate expression. The function Γ n also has the form (7), but involving only type I edges, that is, it is an interaction function composed of step functions that depend on the state variables xj ∈ {x 1 , . .…”