Polynomial dynamical systems over finite fields, with applications to modeling and simulation of bio

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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 StatisticsMarch 6, 2007

• Reinhard Laubenbacher
• Virginia Bioinformatics Institute
• and Mathematics Department
• Virginia Tech

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Polynomial dynamical systems

• Let k be a finite field and f1, , fn ?
  kx1,,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 x1x22x3,
• f2 x2x3,
• f3 x12x22.

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 functions
• and 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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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.

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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 ltf ? kx1, xn f(si)0 for all igt.
• 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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Wiring diagrams

• Problem Given data (si, ti), i1, , r,
• (a collection of state transitions for one node
  in the network), find all minimal (wrt
  inclusion) sets of variables y1, , ym ? x1,
  , xn such that
• (f I) n ky1, , ym ? Ø.
• Each such minimal set corresponds to a minimal
  wiring diagram for the variable under
  consideration.

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The minimal sets algorithm

• For a ? k, let Xa si ti a.
• Let X Xa a ? k.
• Then
• f 0I M f ? kx f(p) a for all p ? Xa.
• 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).

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The algorithm

• Definitions.
• For F ? 1, , n, let
• RF kxi i ? F.
• Let ?X F M n RF ? Ø.
• For p ? Xa, q ? Xb, a ? b ? k, let
• m(p, q) ?pi?qi xi.
• Let MX monomial ideal in kx1, , xn
  generated by all monomials m(p, q) for all a, b ?
  k.
• (Note that ?X is a simplicial complex, and MX is
  the face ideal of the Alexander dual of ?X.)

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The algorithm

• Proposition. (Jarrah, L., Stigler, Stillman) A
  subset F of 1, , n is in ?X if and only if
  the ideal lt xi i ? F gt contains the ideal MX.

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The algorithm

• Corollary. To find all possible minimal wiring
  diagrams, we need to find all minimal subsets of
  variables y1, , ym such that MX is contained in
  lty1, , ymgt. That is, we need to find all
  minimal primes containing MX.

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Scoring method

• Let F F1, , Ft be the output of the
  algorithm.
• For s 1, , n, let Zs sets in F with s
  elements.
• For i 1, , n, let Wi(s) sets in F of size
  s which contain xi.
• S(xi) SWi(s) / sZs
• where the sum extends over all s such that Zs ?
  0.
• T(Fj) ?xi?Fj S(xi).
• Normalization ? probability distribution on F of
  min. var. sets
• This scoring method has a bias toward small sets.

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Model selection

• Problem The model space f I is
• WAY TOO BIG
• 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 x4 f2 x4x3 f3 x2x4 f4 x2x1x3
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A nested canalyzing Boolean network
g1 x4 g2 x4x3 g3 x2x4 g4 x2x1x3
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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 defining the class of
  nested canalyzing Boolean functions on n
  variables forms an affine toric variety over the
  algebraic closure of F2. The irreducible
  components correspond to the functions that are
  nested canalyzing with respect to a given
  variable ordering.
• (joint work with Jarrah, Raposa)

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Dynamics from structure

• Theorem. Let f (f1, , fn) kn ? kn be a
  monomial system.
• 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)
• The case for general k can be reduced 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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