Population-Preserving Mass Action

A proposal for a project (3rd year, master)

Mass action systems are used to model the dynamics of chemical and biochemical reactions, the dynamics of populations (prey-predator systems), the spread of ideas and other phenomena that involve the joint action of a large number of agents. A mass action system is defined by a set of reactions, e.g. A+B -> A+C indicating that when two individuals of type A and B meet, the B transforms into C. The analysis of such system is typically done in one of the following two ways: 1) Stochastic simulation where each step a reaction rule is taken and applied at random; 2) Derivation and analysis of a polynomial dynamical system whose variables correspond to the concentrations of the differtent types of agents. In both cases the probability of a reaction like the one described above is proportional to [A][B]. The structure of the corresponding dynamical systems (where are the equlibria, how to tune the parameters in order to steer the system into a desired part of the state space, etc.) pose difficult and fascinating theoretical and computational problems with application to real problems such as drug design.

The proposed project attempts to advance our understanding of mass action systems, taking the following directions:

  • We define a class of pouplation-preserving mass action systems in which every agent is modeled by a probabilistic automaton that can change its state either spontaneously or upon meeting another agent. From these definitions one can construct the appropriate dynamical system. The population preservation feature comes from the fact that agents do not combine, disappear or get created - they only change their state.


  • We develop techniques for the analysis of multi-affine systems, an important subclass of polynomial systems which have interesting properties on rectangles. These proprties allow us to isolate rather easily the equilibrium points.


    • The goal of the project is to connect the two above. That is to try to classify the mass action systems in 2,3 (and hopefully n) dimensions, according to the form of their phase portrait, structure of zeros etc. In particular we sould like to know:

  • Do the polynomial and multi-affine systems derived from our class of mass-action systems have additional constraints on the coefficients that restrict their possible dynamics? In particular if we consider systems that give rise only to bilinear differential equations.


  • Are there good heuristic for choice of variables and dimensions in our algorithms for isolating equilibrium points?


  • What are the principles for controlling mass action systems?


  • How to adapt reachability-based verification techniques to such systems.


    • This is a challenging research direction that can lead to a thesis on topics which are at the intersection of verification, hybrid systems and systems biology. The problem will be attacked using different techniques such as stochastic simulation, reachability computation, but mostly by thinking.

    The work will be conducted at Verimag under the supervision of Oded Maler. Verimag is a leading lab in many domains, away from Parisian provincialism. Atmosphere is cosmopolite, liberal and stimulating. The Alpes are nearby, remuneration is generous, working conditions (space, computers) are good and travels abroad are hard to avoid. The candidate shoud have a solid background in dynamical systems, algorithmics, geometry and programming.

    References:

  • [1 ] Systems Biology

  • [2] The Roles of informatics in Biology

  • [3] Multi affine systems

  • [4] From multi-affine systems to timed automata

  • [5] Introduction to reachability

  • [6] Nonlinear reachability