Discrete models of biochemical networks

Algebraic Biology 2007RISC Linz, Austria

July 3, 2007

Reinhard LaubenbacherVirginia Bioinformatics Instituteand Mathematics Department

Virginia Tech

Polynomial dynamical systems

Let k be a finite field and f1, … , fn k[x1,…,xn]

f = (f1, … , fn) : kn → kn

is an n-dimensional polynomial dynamical system over k.

Natural generalization of Boolean networks.

Fact: Every function kn → k can be represented by a polynomial, so all finite dynamical systems kn → kn are polynomial dynamical systems.

Example

k = F3 = {0, 1, 2}, n = 3

f1 = x1x22+x3,

f2 = x2+x3,

f3 = x12+x2

2.

Dependency graph(wiring diagram)

http://dvd.vbi.vt.edu

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 functionsand depicted by a wiring diagram similar to the diagram of a digital logic circuit.”

Karp, 2002

Nature 406 2000

Network inference using finite dynamical systems models

Variables x1, … , xn with values in k.(s1, t1), … , (sr, tr) state transition observations with sj, tj kn.

Network inference: Identify a collection of “most likely”

models/dynamical systems f=(f1, … ,fn): kn → kn

such that f(sj)=tj.

Important model information obtained from

f=(f1, … ,fn):• The “wiring diagram” or “dependency

graph” directed graph with the variables as

vertices; there is an edge i → j if xi appears in fj.

• The dynamics directed graph with the elements of kn

as vertices; there is an edge u → v if f(u) = v.

The Hallmarks of Cancer Hanahan & Weinberg (2000)

The model space

Let I be the ideal of the points s1, … , sr, that is,

I = <f k[x1, … xn] | f(si)=0 for all i>.

Let f = (f1, … , fn) be one particular feasible model. Then the space M of all feasible models is

M = f + I = (f1 + I, … , fn + I).

Model selection

Problem: The model space f + I is

WAY TOO BIG

Idea for Solution: Use “biological theory” to reduce it.

“Biological theory”

• Limit the structure of the coordinate functions fi 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.)

Nested canalyzing functions

Nested canalyzing functions

A non-canalyzing Boolean network

f1 = x4f2 = x4+x3f3 = x2+x4f4 = x2+x1+x3

A nested canalyzing Boolean network

g1 = x4g2 = x4*x3g3 = x2*x4g4 = x2*x1*x3

Polynomial form of nested canalyzing Boolean functions

The vector space of Boolean polynomial functions

The variety of nested canalyzing functions

Input and output values as functions of the coefficients

The algebraic geometry

Corollary.

The ideal of relations determining the class of nested canalyzing Boolean functions on n variables defines an affine toric variety Vncf over the algebraic closure of F2. The irreducible components correspond to the functions that are nested canalyzing with respect to a given variable ordering. That is,

Vncf = Uσ Vσncf

Dynamics from structure

Theorem. Let f = (f1, … , fn) : kn → kn be a monomial system.

1. If k = F2, 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)

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 MX?

Open Problems

Study stochastic polynomial dynamical systems.

• Stochastic update order

• Stochastic update functions

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Year-long program 2008-09 at the Statistical and Applied Mathematical Sciences Institute (SAMSI):

Algebraic methods in systems biology and statistics