Stochastic algebraic models

SAMSI Transition WorkshopJune 18, 2009

Reinhard LaubenbacherVirginia Bioinformatics Institute

and Mathematics DepartmentVirginia Tech

Systems biology working group activities

1. Algebraic models of biological networks

2. ODE models of biochemical reaction networks

Foci:

• Structure → dynamics

• Dynamics → structure

• Experimental design

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.

Parameter estimation

Problem: Given experimental time course data and a partially specified model

f = (f1, … , fn) : kn → kn,

with/without information on the function structure, estimate the unspecified functions by fitting them to the data.

Parameter estimation

Variables x1, … , xn with values in k.(E.g., protein concentrations, mRNA concentrations, etc.)

(s1, t1), … , (sr, tr) state transition observations with sj kn, tj k(E.g., consecutive measurements in a time course experiment.)

Network inference: Identify a function

g: kn → ksuch that f(sj)=tj.

The model space

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

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

Let g be one particular feasible function/parameter. Then the space M of all feasible parameters is

M = g + I.

Model selection

In the absence of other information, choose a model which is reduced with respect to the ideal I.

Laubenbacher, Stigler, J. Theor. Biol. 2004

Several other methods.Contributors: E. Dimitrova, L. Garcia, A. Jarrah,

M. Stillman, P. Vera-Licona

Dimitrova, Hinkelmann, Garcia,Jarrah, L., Stigler, Vera-Licona

Model selection

Model selection in original method requires choice of term order

Improvement: Construct a wiring diagram using information from all term orders.

Dimitrova, Jarrah, L., Stigler, A Gröbner-fan based method for biochemical network modeling, ISSAC 2007

Dynamic model

Dimitrova, Jarrah: Construct a probabilistic polynomial dynamical system by sampling the reduced models in all the Groebner cones, together with a probability distribution on the models derived from cone volumes.

Alternative method constructed by B. Stigler.

Probabilistic Boolean networks

For each variable there is a family of Boolean functions, together with a joint probability distribution.

At each update, choose a random function out of this family.

Shmulevich, E. Dougherty, et al.

Dimitrova, Jarrah produce a probabilistic polynomial dynamical system

Update-stochastic Boolean networks

Update variables sequentially, in one of two ways:• At each update, choose at random a permutation,

which specifies an update order.• At each update, choose at random a variable that gets

updated.

Sequential update is more realistic biologically.

See, e.g., Chaves, Albert, Sontag, J. Theor. Biol., 2005

Philosophy: Stochastic sequential update arises through random delays in the completion time of molecular processes.

Consequence: Can approximate update-stochastic systems through systems with random delays, i.e., special function-stochastic systems.

This approach is taken in Polynome.

General problem

Study function-stochastic polynomial dynamical systems.

Note: Can be viewed as a special family of Markov chains.

Why polynomial dynamical systems?

Algebraic models

Most common algebraic model types in systems biology:

• Boolean networks, including cellular automata

• Logical models

• Petri nets

A common modeling framework

1. Boolean networks are equivalent to PDS over the field with two elements.

2. (Jarrah, L., Veliz-Cuba) There are algorithms that translate logical models and Petri nets into PDS.

LM PN

PDS

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Stochastic systems

Instead of f = (f1, … , fn) : kn → kn consider

f = ({f1}, … , {fn}) : kn → kn, together with probability distributions on the sets {fi}. At each update, choose the ith update function from the set {fi} at random.

This is a function-stochastic polynomial dynamical system.

Special case: Delay systems

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

F = ({f1, id}, … , {fn, id}),together with a probability distribution on each set.

Each time id is chosen for an update, a delay occurs in that variable.

What is the effect of delays on network dynamics?

An example

Theorem. (Hinkelmann, Jarrah, L.) Let f be a Boolean linear system with dependency graph D. Let F be the associated delay system. Then F has periodic points if and only if D contains directed cycles (feedback loops).

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Open problems

• Study in more generality the effects of stochastic delays on algebraic model dynamics.

• Can one use stochastic delay systems to efficiently simulate deterministic sequential systems?

• What are good simulation methods for this purpose?