1. Discrete models of biochemical networks Algebraic Biology 2007 RISC Linz, Austria July 3, 2007 Reinhard Laubenbacher Virginia Bioinformatics Institute and Mathematics Department Virginia Tech

2. Polynomial dynamical systems Let k be a finite field and f 1, …, f n  k[x 1,…,x n ] f = (f 1, …, f n ) : k n → k n is an n-dimensional polynomial dynamical system over k. Natural generalization of Boolean networks. Fact: Every function k n → k can be represented by a polynomial, so all finite dynamical systems k n → k n are polynomial dynamical systems.

3. Example k = F 3 = {0, 1, 2}, n = 3 f 1 = x 1 x 2 2 +x 3, f 2 = x 2 +x 3, f 3 = x 1 2 +x 2 2. Dependency graph (wiring diagram)

4. http://dvd.vbi.vt.edu

5. Motivation: Gene regulatory networks “[The] transcriptional control of a gene can be described by a discrete-valued function of several discrete-valued variables.” “A regulatory network, consisting of many interacting genes and transcription factors, can be described as a collection of interrelated discrete functions and depicted by a wiring diagram similar to the diagram of a digital logic circuit.” Karp, 2002

6. Nature 406 2000

9. Network inference using finite dynamical systems models Variables x 1, …, x n with values in k. (s 1, t 1 ), …, (s r, t r ) state transition observations with s j, t j  k n. Network inference: Identify a collection of “most likely” models/dynamical systems f=(f 1, …,f n ): k n → k n such that f(s j )=t j.

10. Important model information obtained from f=(f 1, …,f n ): The “wiring diagram” or “dependency graph” directed graph with the variables as vertices; there is an edge i → j if x i appears in f j. The dynamics directed graph with the elements of k n as vertices; there is an edge u → v if f(u) = v.

11. The Hallmarks of Cancer Hanahan & Weinberg (2000)

12. The model space Let I be the ideal of the points s 1, …, s r, that is, I =. Let f = (f 1, …, f n ) be one particular feasible model. Then the space M of all feasible models is M = f + I = (f 1 + I, …, f n + I).

13. Model selection Problem: The model space f + I is WAY TOO BIG Idea for Solution: Use “biological theory” to reduce it.

14. “Biological theory” Limit the structure of the coordinate functions f i to those which are “biologically meaningful.” (Characterize special classes computationally.) Limit the admissible dynamical properties of models. (Identify and computationally characterize classes for which dynamics can be predicted from structure.)

15. Nested canalyzing functions

17. A non-canalyzing Boolean network f1 = x4 f2 = x4+x3 f3 = x2+x4 f4 = x2+x1+x3

18. A nested canalyzing Boolean network g1 = x4 g2 = x4*x3 g3 = x2*x4 g4 = x2*x1*x3

19. Polynomial form of nested canalyzing Boolean functions

20. The vector space of Boolean polynomial functions

21. The variety of nested canalyzing functions

22. Input and output values as functions of the coefficients

23. The algebraic geometry Corollary. The ideal of relations determining the class of nested canalyzing Boolean functions on n variables defines an affine toric variety V ncf over the algebraic closure of F 2. The irreducible components correspond to the functions that are nested canalyzing with respect to a given variable ordering. That is, V ncf = U σ V σ ncf

24. Dynamics from structure Theorem. Let f = (f 1, …, f n ) : k n → k n be a monomial system. 1.If k = F 2, then f is a fixed point system if and only if every strongly connected component of the dependency graph of f has loop number 1. (Colón-Reyes, L., Pareigis) 2.The case for general k can be reduced to the Boolean + linear case. (Colón-Reyes, Jarrah, L., Sturmfels)

25. Questions What are good classes of functions from a biological and/or mathematical point of view? What extra mathematical structure is needed to make progress? How does the nature of the observed data points affect the structure of f + I and M X ?

26. Open Problems Study stochastic polynomial dynamical systems. Stochastic update order Stochastic update functions

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29. Advertisement 2 Year-long program 2008-09 at the Statistical and Applied Mathematical Sciences Institute (SAMSI): Algebraic methods in systems biology and statistics