The Paradox of Chemical Reaction Networks :Robustness in the face of total uncertainty

By David Angeli:Imperial College, LondonUniversity of Florence, Italy

Definition of CRN

nrnrrnrnrr

nnnn

nnnn

SSSSSS

SSSSSS

SSSSSS

22112211

22221212222121

12121111212111

List of Chemical Reactions:

The Si for i = 1,2,...,n are the chemical species.

The non-negative integers , are the stoichiometry coefficients.

Example of CRNE + S0 ES0 E + S1 ES1 E + S2F + S2 FS2 F + S1 FS1 F + S0

S0

F

E

FS2FS1

ES1ES0

S1 S2

Discrete Modeling FrameworkStochastic:Discrete event systems: PETRI NETS

Reaction rates: mass-action kinetics

Problem : Markov Chain with huge number of states

S0

F

E

FS2FS1

ES1ES0

S1 S2

Continuous Modeling Framework

Deterministic:Continuous concentrations, ODE modelsLarge molecule numbers: variance is neglegible

Isolated vs. Open systems

• Thermodynamically isolated systems:Reaction rates derived from a potential.Every reaction is reversible.Steady-states are thermodynamic equilibria: detailed balance

Passive circuits analog of CRNs.Entropy acts as a Lyapunov function.

• Open systems:Some species are ignored: clamped concentrations.

Partial stoichiometry. Arbitrary kinetic coefficients.No obvious Lyapunov function. Possibility of “complex” behaviour.

Relating Dynamics and Topology• How does structure affect dynamics ?

• How robust is the net to parameter variations ?

• Does the reaction converge or oscillate ?

Qualitative tools: can work regardless of specific parameters values.

• How to define robustness ?

Consistent qualitative behavior regardless of Parameters or kinetics.

MAPK random simulation

More random simulations

What is Persistence• Notion introduced in mathematical ecology: non extinction of species

• For positive systems

it amounts to:

• For systems with bounded solutions equivalently:

)(xfx

itxit

0)(inflim

nRx 0)(

S0

F

E

FS2FS1

ES1ES0

S1 S2

Petri Nets Background

Bipartite graph:PLACES (round nodes)TRANSITIONS (boxes)

Incidence matrix = Stoichiometry matrix = S

P-semiflow: non-negative integer row vector v such that v S = 0

T-semiflow: non-negative integer column vector v with S v = 0

Support of v: set of places i (transitions) such that v_i>0

Necessary conditions for persistence

• Let r(x) denote the vector of reaction rate

• We assume that for x>>0, r(x)>>0

• Under persistence, the average of r(x(t)) is strictly positive and belongs to the kernel of S

• Hence, Persistence implies existence of a T-semiflow whose support coincides with the set of all transitions.

This kind of net is called: CONSISTENT

Petri Net approach to persistence

S0

F

E

FS2FS1

ES1ES0

S1 S2

Assume that x(tn) approachesThe boundary. Let S be the setof i such that xi(tn) 0

Then S is a SIPHON

SIPHON:Input transitionsIncluded in Output transitions

Structurally non-emptiable siphons

A siphon is structurally non-emptiable if it containsthe support of a positive conservation law

S0

F

E

FS2FS1

ES1ES0

S1 S2

P-semiflows:E+ES0+ES1F+FS2+FS1S0+S1+S2+ES0+ES1+FS2+FS1

Minimal Siphons:{ E, ES0, ES1 }{ F, FS2, FS1 }{ S0, S1, S2, ES0, ES1, FS2, FS1 }

All siphons are SNE PERSISTENCE

Network compositions

Full MAPK cascade

22 chemical species7 minimal siphons7 P-semiflows whose supports coincide with the minimal siphons

Hopf’s bifurcations

• Symbolic linearization:

• Characteristic polynomial

• Hurwitz determinant Hn-1 = 0 is a necessary condition for Hopf’s bifurcation (n=6).

)(xfx xx

fx

36

51

6 )...()det( sasasx

fsI

Hurwitz determinant

531

642

531

642

531

5

00

10

00

01

00

aaa

aaa

aaa

aaa

aaa

H

• ai are polynomials of degree i in the kinetic parameters• det(H5) is a polynomial of degree 15 in the kinetic

parameters (12 parameters + 5 concentrations) • Number of monomials is unknown • Letting all kinetic constants = 1 except for k1 k3 k5 k7

yields 68.425 monomials all with a + coefficient

Remarks

• This is much stronger than: det(Hn-1) is positive definite.

• Purely algebraic and graphic criterion for ruling out Hopf’s bifurcations expected.

• Notion of negative loop in the presence of conservation laws.

Conclusions• CRN theory: open problems and challenges• At the cross-road of many fields:

- dynamical systems

- biochemistry

- graph theory

- linear algebra

HAPPY 60 EDUARDO