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Stochastic algebraic models
SAMSI Transition WorkshopJune 18, 2009
Reinhard LaubenbacherVirginia Bioinformatics Institute
and Mathematics DepartmentVirginia Tech
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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
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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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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.
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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.
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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.
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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
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Dimitrova, Hinkelmann, Garcia,Jarrah, L., Stigler, Vera-Licona
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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
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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.
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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
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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
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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.
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General problem
Study function-stochastic polynomial dynamical systems.
Note: Can be viewed as a special family of Markov chains.
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Why polynomial dynamical systems?
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Algebraic models
Most common algebraic model types in systems biology:
• Boolean networks, including cellular automata
• Logical models
• Petri nets
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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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T cell differentiation
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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.
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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?
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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?