discrete models of biochemical networks algebraic biology 2007 risc linz, austria july 3 , 2007
DESCRIPTION
Discrete models of biochemical networks Algebraic Biology 2007 RISC Linz, Austria July 3 , 2007. Reinhard Laubenbacher Virginia Bioinformatics Institute and Mathematics Department Virginia Tech. Polynomial dynamical systems. Let k be a finite field and f 1 , … , f n k [ x 1 ,…, x n ] - PowerPoint PPT PresentationTRANSCRIPT
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Discrete models of biochemical networks
Algebraic Biology 2007RISC Linz, Austria
July 3, 2007
Reinhard LaubenbacherVirginia Bioinformatics Instituteand Mathematics Department
Virginia Tech
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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.
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Example
k = F3 = {0, 1, 2}, n = 3
f1 = x1x22+x3,
f2 = x2+x3,
f3 = x12+x2
2.
Dependency graph(wiring diagram)
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http://dvd.vbi.vt.edu
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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
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Nature 406 2000
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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.
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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.
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The Hallmarks of Cancer Hanahan & Weinberg (2000)
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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).
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Model selection
Problem: The model space f + I is
WAY TOO BIG
Idea for Solution: Use “biological theory” to reduce it.
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“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.)
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Nested canalyzing functions
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Nested canalyzing functions
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A non-canalyzing Boolean network
f1 = x4f2 = x4+x3f3 = x2+x4f4 = x2+x1+x3
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A nested canalyzing Boolean network
g1 = x4g2 = x4*x3g3 = x2*x4g4 = x2*x1*x3
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Polynomial form of nested canalyzing Boolean functions
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The vector space of Boolean polynomial functions
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The variety of nested canalyzing functions
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Input and output values as functions of the coefficients
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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
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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)
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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?
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Open Problems
Study stochastic polynomial dynamical systems.
• Stochastic update order
• Stochastic update functions
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