## Introduction to geometric control

Course program:

1. Some basic questions in the control formalism, some examples of control systems.

2. Controllability of linear systems. Lie brackets and their relation with controlled motions.

## Systems Biology

Research Topics

Systems Biology aims at providing the mathematical tools necessary to turn the wealth of experimental data that the field of Biology is nowadays producing into models of descriptive nature and predictive power. Our group is currently involved in different research topics. A few samples of our current research interests are listed below.

**Dynamical properties of large-scale biological networks**

Treating a biological network as a dynamical system, different properties can be useful to understand its dynamical behavior: the number of equilibrium points and their stability, the dimension of a "minimal feedback arc set", the tendency of the trajectories of the system to show an ordered behavior, typical of a monotone or quasi-monotone dynamical system. We are investigating these and other similar properties on real large-scale biological networks (gene networks, signaling pathways, metabolic networks) using a minimal amount of kinetic detail for the reactions involved. We have found for example that gene networks tend to be much more monotone than expected.

**Finding putative interactors via gene regulatory network inference**

Several statistical inference methods are been used to identify a graph of putative gene-gene interactions at genome-wide level, using compendia of gene expression profiles. The goal of these reverse engineering studies is to provide genome-wide clues on the potential interactors of a set of genes of interest in an organism, or the mode-of-action of a certain drug.

**Synergistic effect of drugs on metabolic pathways: minimizing side effect**

The aim of this project is to explore the degree of synergism of current drug targets in organism-wide models of metabolic networks, and to optimize the drug "cocktails'' that achieve a certain task in an optimal way. The task could be for instance the suppression of a certain reaction, and the cost functional the minimization of the side effect on the rest of the network.

Flux balance analysis, linear-quadratic programming and combinatorial optimization are the theoretical tools we are using for this purpose.

**Modeling adaptation in sensory transduction**

Adaptation in vertebrate sensory systems corresponds to the ability to shift the dynamical regime of highest input/output sensitivity according to the amplitude of the stimulus, in order to increase the dynamical range of the sensory perception. In control theory, a model for this type of regulation is provided by variants of the so-called integral feedback action. The sensory systems we study (olfactory transduction and phototransduction) adapt but not perfectly, i.e., the step response has a non-zero steady state error. Our work consists in elucidating to what extent they adapt, combining models and electrophysiological experiments.

**Kinetic models of prion replication**

Models based on nucleated polymerization have been introduced in recent years to describe phenomenologically a prion infection, and explain the appearance of the disease by means of a bistability induced by a quadratic term, as in classical epidemic models. Our task has been to include in such models recent experimental data showing a linear relationship between the incubation times and the conformational stability of the amyloid used as inoculum.