Understanding Biological Phenomena with Control-Theoretic Tools

Abstract

The control community has developed many mathematical tools that are tailored to face impor- tant problems in engineering, but can also be successfully adopted to address problems in systems biology: indeed feedback, which is a fundamental concept at the core of control theory, is ubiquitous in biology, at the point that no living being could survive without the myriad of entangled feedback loops that rule its biological functions. When adopting control-theoretic tools for the study of biological problems, there are two fundamental challenges: deal with the huge complexity of biological systems by means of simplifications that allow us to nicely capture the essence of the system and describe it in our framework; convince biologists that such simplifications are worth adopting because, together with non-trivial mathematical tools, they enable a deeper qualitative and quantitative understanding of biological phenomena. The presentation will focus on the simplification of biological models and the use of the mathematical language to solve problems in biology, which can provide a deep insight when supported by an effective communication between mathematicians and biologists. The talk will start with a discussion on mathematical models and uncertainty in biology, focusing on the concept of robust and structural properties, and on system simplifications that exploit special proper- ties (such as monotonicity, or positivity of the impulse response) to combine into a single aggregate element systems composed of many interconnected units (which can be seen as the nodes of a graph). Then, we will overview the principal mathematical tools that are useful in systems biology, ranging from graph theory to differential equations and frequency methods. Several notions will be surveyed, from sophisticated ones, such as degree theory, to more familiar ones, such as Lyapunov theory. Mainstream tools will also be discussed, some of which are inherited from the theory of parametric robustness. The talk will be concluded by showing how the presented tools have been actually applied to the structural analysis of biological systems, including the structural stability of biochemical networks, the structural steady-state influence, and the classification of biological networks based on the analysis of the cycles in the associated graph structure.