Algebraic geometry in systems biology - folk.uio.nofolk.uio.no/ranestad/Laubenbacher.pdfAll discrete deterministic models of this type can be translated into the framework of polynomial

Algebraic geometry in systems biology

Reinhard Laubenbacher

Virginia Bioinformatics Instituteand

Mathematics DepartmentVirginia Tech

[email protected]

Reinhard Laubenbacher Algebraic geometry in systems biology

Polynomial dynamical systems

Let x1, . . . , xn be variables with values in a finite field F.

DefinitionA polynomial dynamical system (PDS) over F of dimension n isa time-discrete dynamical system

f = (f1, . . . , fn) : Fn −→ Fn,

with fi ∈ F[x1, . . . , xn].


Example

Let f = (x7, x1x3, x4, x5, x6, x7, x2) : F72 −→ F7

2.

Structure Dynamics


Systems biology

The advent of functional genomics has enabled the molecularbiosciences to come a long way towards characterizing themolecular constituents of life. Yet, the challenge for biologyoverall is to understand how organisms function. By discoveringhow function arises in dynamic interactions, systems biologyaddresses the missing links between molecules and physiology.

Bruggeman, Westerhoff, 2006


The biology

Hannahan and Weinberg (2000) The hallmarks of cancer


Dynamic Model:Collection of variables x1, . . . , xn with states in setsX1, . . . ,Xn;Collection of relationshipsR1(x1, . . . , xn), . . . ,Rn(x1, . . . , xn);Scheme to update the state of each of the variables overtime.

The Xi can be R, Fq, or any other set, such as{active, inactive}.The Ri can be ODEs, logical rules, stochasticrelationships, etc.Update can be in continuous or discrete time.


Discrete models

Assumptions:The Xi are finite sets;The Ri are given as functions fi : X = X1 × · · · × Xn −→ Xi ;Dynamics is generated by iteration of the function

f = (f1, . . . , fn) : X1 × · · · × Xn −→ X1 × · · · × Xn.


Example


Example cont.

Let Xi = {0,1} for all i .

fMcI = xCIT ∨ xCroT ∨ (xMcI ∧ xMcIold)

fMcIold= xCIT ∧ xCroT

xMcI1= xMcI

fMcro = xCIT ∨ (xMcro ∧ xMcroold)

fMcroold = xCIT

xMcro1= xMcro

xMcro2= xMcro1

fCIT = γcI1 ∨ (γcIhigh ∧ (xMcI1∨ (xCIT ∧ (xCIT old

∨ γcI))))


Example cont.

fCIT old= xMcI1

∨ γcIhigh

fCroT = xMcro2∨ (xCroT ∧ xCroT old

)

fCroT old= xMcro2

fγcI = γcI ∨ γcIhigh

fγcI1= γcI

fγcIhigh= γcIhigh .

from F. Hinkelmann and R.L., Boolean models of bistablesystems, Discr. Cont. Dyn. Syst., 2010, in press.


Model types

Boolean networks;Logical models;Bounded Petri nets;Agent-based models.

All discrete deterministic models of this type can be translatedinto the framework of polynomial dynamical systems over afinite field:

F. Hinkelmann, D. Murragarra, A. Jarrah, and R. L., Amathematical framework for agent-based models ofcomplex biological networks, Bull. Math. Biol., 2010, inpress.A. Veliz-Cuba, A. Jarrah, R. L., Polynomial Algebra ofDiscrete Models in Systems Biology, Bioinformatics, 26,1637-1643, 2010.


What the experimentalist needs

Tools to construct models.Tools to analyze models.Tools to use models to generate biological hypotheses anddiscover new biology.

Goal: Provide algorithms and software for all three. Do it insuch a way that the biologist becomes independent of themodeler.


Tools to construct models

Most important tool for ODEs is parameter estimation.Need analogous tool for discrete models.

parameters = unknown coefficients in the polynomials fi .

Choose parameter values by fitting the model to time coursesof experimental data.E. Dimitrova, L.D. Garcia, F. Hinkelmann, A. Jarrah, R.L., B.Stigler, M. Stillman, P. Vera-Licona, Parameter estimation forBoolean models, J. Theor. Comp. Sci. 2010, in press.


Parameter estimation

Input Functions (Optional)

Upload function file: Browse… what is this?

OR (Edit functions below)

f1 = x1

Input Data

Upload data file: Browse… what is this?

OR (Edit data below)

# First time course 1.2 2.3 3.41.1 1.2 1.32.2 2.3 2.40.1 0.2 0.3

Polynome: Discrete System Identification

If this is your first time, please read the tutorial. It is important that you follow the format specified in the tutorial.Make your selections and provide inputs (if any) in the form below and click Generate to run the software.

Input SpecificationType of Data

Number of nodes: 3 what is this?

Type of Model what is this?

Select the type of polynomial dynamical system to be generated: Stochastic Deterministic

Type of Simulation what is this?

Simulate the state space using an update schedule?

Enter update schedule separated by spaces

If none is entered, then the nodes will be simulated in parallel.

Output Options

Select output to view and file format. what is this?

Show discretized data

Wiring Diagram *.gif Print probabilities in wiring diagram

Show functions for Polynomial dynamical system

State space graph *.gif Print probabilities in state space

Generate

Output will be displayed below. Note: Computation time depends on data size and internet connection.


Tools to analyze models

Computation of model dynamics: steady states and limit cycles.

Letf = (f1, . . . , fn) : kn −→ kn

be a PDS. Let a ∈ kn. Then a is a periodic point of f of period tif f t(a) = a. That is:

f t1(a)− a = 0, . . . , f t

n(a)− a = 0.

For small values of n, the entire phase space of the model canbe computed by exhaustive enumeration of all transitions. Forlarge n use computer algebra.


Dynamics

ADAM – Analysis of Dynamic Algebraic Models h6p://dvd.vbi.vt.edu

F. Hinkelmann, M. Brandon, B. Guang, R. McNeill, G.Blekherman, A. Veliz-Cuba, and R.L., ADAM: Analysis ofdiscrete models of biological systems using computer algebra,2010, under review.


Benchmarks


Example

T cell differentiation

From A. Naldi et al., Diversity and plasticity of Th cell typespredicted from regulatory network modelling, PLoS Comp.Biol., 2010.


Tools to use models for hypothesis generation

Example: Optimal control for drug therapy

From O. Sahin et al., Modeling ERBB receptor-regulated GI/Stransition to find novel targets for de novo trastuzumabresistence, BMC Sys. Biol. 3, 2009


Optimal control for PDS

A controlled PDS (Marchand et al. 1998, On optimal control ofPDS over Z/pZ) is given by

f : Fn × Fm −→ Fn,

with state variables x1, . . . , xn and control variablesu1, . . . ,um.Variety of admissible control inputs:

Qi(x1, . . . , xn; u1, . . . ,um) = 0, i = 1, . . . , r ,

Variety of admissible initial states:

Pj(x1, . . . , xn) = 0, j = 1, . . . , s,

Variety of admissible final states:

Ua(x1, . . . , xn) = 0, a = 1, . . . , t ,


Optimal control cont.

Let X0 be the variety of admissible initializations:

X0 = {x ∈ Fn : Pj(x) = 0, j = 1, . . . , s}

and XM be the variety of final states:

XM = {x ∈ Fn : Ua(x) = 0, a = 1, . . . , t}

XM−1 = {x ∈ Fn : ∃ u,Qi(x; u) = 0, i = 1, . . . , r ⇒ f (x; u) ∈ XM}

XM−2 = {x ∈ Fn : ∃ u,Qi(x; u) = 0, i = 1, . . . , r ⇒ f (x; u) ∈ XM−1}



A valid control sequence of the system is a sequence of controlinputs {u1, . . . ,uM} which drives the system from one of theinitial states to one of the final states:

x1 //

u2

!!BBB

BBBB

B ◦ // . . . //

##

◦

##x0 //

!!

u1==||||||||◦ //

!!

x2 //

==

u3

!!DDDD

DDDD

D. . . // xM−1

uM // xM

◦ // ◦ //// . . .

uM−1;;wwwwwwwww // ◦

;;

X0 X1 X2 . . . XM−1 XM



Problem: Given a controllable system F and an initializationX0. Find a sequence of control inputs which drive the system toan admissible final state XM , in such a way that a given costfunction is minimized.The cost of a trajectory {x0,x1, . . . ,xM} with associated controlsequence Su = {u1, . . . ,uM} is given by a function:

C(Su) =M∑

i=1

c(xi−1,xi ,ui)

where c(xi−1,xi ,ui) is the cost of going from xi−1 to xi by usingthe control input ui .



There exist algorithms using dynamic programming to solvethis problem, but with high combinatorial complexity.

See algorithm by H. Marchand:http://www.irisa.fr/vertecs/Softwares/sigali.html

Computer algebra is used to efficiently compute the Xi .

For large problems heuristic algorithms are needed.


Future work

More analysis tools;New algorithms;Theoretical results;New optimal control algorithms (NIMBioS working group);A better interface for the biologist user;Integration of the different software packages;Stochastic systems.


Theoretical results: example

Study a special class of functions that appears frequently as arule for molecular regulation: nested canalyzing polynomials

Most rules in published models are of this type.Systems constructed from them have the “right” dynamicproperties.The family of nested canalyzing polynomials can beparametrized by binomials.This leads to a closed formula for the number of suchpolynomials.

D. Murragarra and R.L., Regulatory motifs in molecularinteraction networks, 2011, under review.

D. Murragarra and R.L., Nested canalyzing polynomials andgene regulation, 2011.


Contributors and Funding

Contributors: G. Blekherman, M. Brandon, E. Dimitrova, L.Garcia, B. Guang, F. Hinkelmann, A. Jarrah, R. McNeill, D.Murragarra, B. Stigler, M. Stillman, A. Veliz-Cuba, P.Vera-Licona.

Funding: NSF, ARO


Bull. Math. Biol. April 2011 special issue:

Algebraic Methods in Mathematical Biology


Documents

Algebraic geometry in systems biology - folk.uio.nofolk.uio.no/ranestad/Laubenbacher.pdfAll discrete deterministic models of this type can be translated into the framework of polynomial