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 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.

3. k = F3 = {0, 1, 2}, n = 3 f1 = x1x22+x3, f2 = x2+x3, f3 = x12+x22.
Example
k = F3 = {0, 1, 2}, n = 3
f1 = x1x22+x3,
f2 = x2+x3,
f3 = x12+x22.
Dependency graph
(wiring diagram)

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

7. Nature

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 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.

12. 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.

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

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

15. Problem: Given data (si, ti), i=1, … , r,
Wiring diagrams
Problem: Given data (si, ti), i=1, … , 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) ∩ k[y1, … , ym] ≠ 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 Xa = {si | ti = a}.
Let X = {Xa | a  k}.
Then
f 0+I = M = {f  k[x] | 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).

17. Definitions. For F  {1, … , n}, let RF = k[xi | i  F].
The algorithm
Definitions.
For F  {1, … , n}, let
RF = k[xi | i  F].
Let ΔX = {F | M ∩ RF ≠ Ø}.
For p  Xa, q  Xb, a ≠ b  k, let
m(p, q) = pi≠qi xi.
Let MX = monomial ideal in k[x1, … , xn] generated by all monomials m(p, q) for all a, b  k.
(Note that ΔX is a simplicial complex, and MX is the face ideal of the Alexander dual of ΔX.)

18. The algorithm
Proposition. (Jarrah, L., Stigler, Stillman) A subset F of {1, … , n} is in ΔX if and only if the ideal < xi | i  F > contains the ideal MX.

19. 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 <y1, … , ym>. That is, we need to find all minimal primes containing MX.

20. This scoring method has a bias toward small sets.
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) := ΣWi(s) / sZs
where the sum extends over all s such that Zs ≠ 0.
T(Fj) := ΠxiFj S(xi).
Normalization  probability distribution on F of min. var. sets
This scoring method has a bias toward small sets.

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

22. Limit the admissible dynamical properties of models.
“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.)

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 F2. 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 = (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)

33. What extra mathematical structure is needed to make progress?
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?

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

35. Special year 2008-09 at SAMSI:
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Special year at SAMSI:
“Algebraic methods in biology and statistics”
(subject to final approval)