Polynomial dynamical systems over finite fields, with applications to modeling and simulation of biological networks.

Transcription

1 Polynomial dynamical systems over finite fields, with applications to modeling and simulation of biological networks. IMA Workshop on Applications of Algebraic Geometry in Biology, Dynamics, and Statistics March 6, 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 22 +x 3, f 2 = x 2 +x 3, f 3 = x 12 +x 22. Dependency graph (wiring diagram)

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6 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

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10 Motivation (2): a mathematical formalism for agent-based simulation Example 1: Game of life Example 2: Large-scale simulations of population dynamics and epidemiological networks (e.g., the city of Chicago) Need a mathematical formalism.

11 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.

12 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.

13 The Hallmarks of Cancer Hanahan & Weinberg (2000)

14 The model space Let I be the ideal of the points s 1,, s r, that is, I = <f k[x 1, x n ] f(s i )=0 for all 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).

15 Wiring diagrams Problem: Given data (s i,t i ), i=1,, r, (a collection of state transitions for one node in the network), find all minimal (wrt inclusion) sets of variables y 1,, y m {x 1,, x n } such that (f +I) k[y 1,, y m ] Ø. Each such minimal set corresponds to a minimal wiring diagram for the variable under consideration.

16 The minimal sets algorithm For a k, let X a = {s i t i = a}. Let X = {X a a k}. Then f 0 +I = M = {f k[x] f(p) = a for all p X a }. Want to find f M which involves a minimal number of variables, i.e., there is no g M whose support is properly contained in supp(f).

17 The algorithm Definitions. For F {1,, n}, let R F = k[x i i F]. Let Δ X = {F M R F Ø}. For p X a, q X b, a b k, let m(p, q) = pi qi x i. Let M X = monomial ideal in k[x 1,, x n ] generated by all monomials m(p, q) for all a, b k. (Note that Δ X is a simplicial complex, and M X is the face ideal of the Alexander dual of Δ X.)

18 The algorithm Proposition. (Jarrah, L., Stigler, Stillman) A subset F of {1,, n} is in Δ X if and only if the ideal < x i i F > contains the ideal M X.

19 The algorithm Corollary. To find all possible minimal wiring diagrams, we need to find all minimal subsets of variables y 1,, y m such that M X is contained in <y 1,, y m >. That is, we need to find all minimal primes containing M X.

20 Scoring method Let F = {F 1,, F t } be the output of the algorithm. For s = 1,, n, let Z s = # sets in F with s elements. For i = 1,, n, let W i (s) = # sets in F of size s which contain x i. S(x i ):= ΣW i (s) / sz s where the sum extends over all s such that Z s 0. T(F j ):= Π xi Fj S(x i ). Normalization probability distribution on F of min. var. sets This scoring method has a bias toward small sets.

21 Model selection Problem: The model space f + I is WAY TOO BIG Solution: Use biological theory to reduce it.

22 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.)

23 Nested canalyzing functions

24 Nested canalyzing functions

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

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

27 Polynomial form of nested canalyzing Boolean functions

28 The vector space of Boolean polynomial functions

29 The variety of nested canalyzing functions

30 Input and output values as functions of the coefficients

31 The algebraic geometry Corollary. The ideal of relations defining the class of nested canalyzing Boolean functions on n variables forms an affine toric variety 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. (joint work with Jarrah, Raposa)

32 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 the Boolean + linear case. (Colón-Reyes, Jarrah, L., Sturmfels)

33 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?

34 Advertisement 1 Modeling and Simulation of Biological Networks Symposia in Pure and Applied Math, AMS in press articles by Allman-Rhodes, Pachter, Stigler,.

35 Advertisement 2 Special year at SAMSI: Algebraic methods in biology and statistics (subject to final approval)