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
SSSSSSSSSSSS
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 FrameworkDeterministic:
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 siphonsA 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 xxfx
36
51
6 )...()det( sasasxfsI
Hurwitz determinant
531
642
531
642
531
5
0010
000100
aaaaaa
aaaaaa
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