Analysis of performance and robustness of biological switches: local tools for non-local dynamical phenomena.

Biological switches are frequently encountered in mathematical modeling
of biological systems because binary decisions are at the core of many
cellular processes. A bistable switch presents two stable steady-states,
each of them corresponding to a distinct decision. These two decisions
are assumed to result from the interactions between biochemical effectors
at the molecular level. Because these molecular interactions are
particularly complex, involving many effectors, mathematical models of
biological switches are often high dimensional and nonlinear. Therefore,
an analysis of these systems is challenging. In this dissertation, we try to
identify principles and tools to study the performance and robustness of
biological switches. Our first contribution is to highlight the dynamical
nature of these switches. A biological switch encodes a decision-making
process rather than a static binary code. It captures dynamical phenomena
that are important for the decision-making process, such as decision
latencies and reversibility. Our second contribution is methodological.
While most of the classical analysis tools are based on a linearization
of the system around a stable steady-state, a switch is a non local phenomenon
involving a transition between two stable steady-sates. Rather
than studying the system around stable equilibria, we identify the local
rulers of the decision-making process in both the state and parameter
spaces and propose a local analysis in the vicinity of these particular
points. Our third contribution is to emphasize the added value of an
abstract (that is, mathematical) framework for the analysis of biological
switches. By studying different models, we point out that the same
principles can be used to encode dynamical phenomena in very different
cellular processes. Physiological processes as different as apoptosis, the
cellular choice of death, and action potential, the cellular choice to emit
an electrical spike, share common features when regarded as decision-making
processes.