Polynomial models of finite dynamical systems Reinhard Laubenbacher Virginia Bioinformatics Institute and Mathematics Department Virginia Tech

Goal “This workshop will bring together … with the goal of identifying fundamental scientific questions whose answers could form the basis of a sound mathematical and computational theory for agent based modeling and simulation. “

PathSim Rule-based simulation of host response to infection with viral pathogens. Final version will include system, cellular, and molecular level dynamics. (Inspired by TranSims.) Prototype virus: Epstein-Barr virus (EBV) (ubiquitous human herpes virus that establishes a persistent infection of B lymphocytes). Influenza and respiratory system to be implemented. Useful tool for pathologists/immunologists/epidemiologists

Video available at

PathSim stats 6 tonsillar regions and 3800 germinal centers; Approx mesh points; Approx. 4 million agents.

Reverse-Engineering of Dynamics GOAL: Develop mathematical tools to systematically reverse-engineer desired infection outcomes, e.g., complete viral clearance or induction of a more robust adaptive immune response. APPROACH: Give a mathematical description of PathSim within a framework that admits systematic control theory techniques.

Dynamical Systems over Finite Fields Represent a rule-based deterministic simulation as a finite, time-discrete, parallel-update dynamical system f = (f 1,…,f n ): X n  X n. Assume that X=k is a finite field. Well-known fact: Any such function f over a finite field k can be described by polynomial functions f i in variables x 1,…,x n, with coefficients in k.

Example Boolean networks: X={0,1}; k= Z/2. Then x  y=xy; x  y=x+y+xy;  x=x+1. Any Boolean network can be represented as a polynomial system over the finite field Z/2.

Advantages of Polynomial Viewpoint Computational algebra and algebraic geometry. Specialized symbolic computation software. Well-developed control theory for polynomial systems over finite fields, using tools from algebraic geometry. Similar to the role of a coordinate system in analytic geometry

Reverse-engineering dynamics Aggregate and decompose PathSim to reduce dimension and complexity. Create appropriate time series over a suitable finite field for location nodes in PathSim. (Note: rules are associated to immune cells/virions, not to locations.) Use reverse-engineering algorithm (see poster) to create a “best” polynomial model generating the time series. Analyze its dynamics. Design a controller for this system and appropriate control problems. Requires optimization/metrics. Practice problem: Sim2Virus (exp. det. dynamics)

Structure vs. Dynamics Suppose that k is a finite field and f = (f 1,…,f n ): k n  k n is such that the f i are linear polynomials, with no constant term, so that f is given by a square matrix with entries in k. Theorem: (Hernandez-Toledo) The structure of the state space (number of components, lengths of limit cycles, structure of transients) of f can be completely determined from the characteristic polynomial of f.

f1 := x2+5*x3+2*x4-9*x1 f2 := x1+7*x3+8*x4 f3 := 4*x1-10*x2-4*x3+6*x4 f4 := x1+5*x3-6*x4 Discrete Visual Dynamics (DVD)

Structure vs. dynamics (cont.) Proof: Factor the characteristic polynomial of f into x r p 1 (x) a …p s (x) z. Then x r corresponds to a fixed point system and the other factors correspond to invertible systems. The state space of f is the Cartesian product of the state spaces of the factors.

f1 := x2^2+5*x3+2*x4-9*x1 f2 := x1+7*x3+8*x4 f3 := 4*x1-10*x2-4*x3+6*x4 f4 := x1+5*x3-6*x4

Structure vs. dynamics (cont.) What about the nonlinear case? Find good families of fixed point systems. Find good families of invertible systems. Find good design principles to build up systems from components. The polynomial viewpoint provides a good framework for these tasks. (See the next talk.)

Summary PathSim, multi-scale rule-based simulation of immune response to viral pathogens Goal: control theory using polynomial algebra and combinatorics Fundamental approach: introduce mathematical structure into the object of study (e.g., state sets of agents form a field) to gain access to mathematical design and analysis tools.

Acknowledgements Co-investigators: Karen Duca (VBI), Abdul Jarrah (East Tennessee State U. Math), Bodo Pareigis (University of Munich, Math) Collaborators: Chris Barrett, Madhav Marathe, Henning Mortveit, Christian Reidys (Los Alamos National Laboratory) Edward Green (VT, Math) Pedro Mendes (VBI) Michael Stillman (Cornell U., Math) Bernd Sturmfels (UC Berkeley, Math) David Thorley-Lawson (Tufts U. Medical School) Research Associate: Rohan Luktuke Students: Omar Colon-Reyes Purvi Saraya Jignesh Shah Nicholas Polys Brandilyn Stigler Andrew Ray Maribeth Todd Hussein Vastani Satya Rout John McGee Research Support: National Science Foundation, National Institutes of Health, Los Alamos National Laboratory, and the Commonwealth of Virginia