Stability of Equilibrium

Deficiency 𝜹:

𝜹 = − −

A mass-action system has a stable complex-balanced equilibrium for any rate constants 𝜅𝑗 if and only if it

is weakly reversible and 𝛿 = 0.[4-7] (Fig. 3)

Possible behaviours include bistability and oscillation. (See Fig. 2 for example of bistability.)

Fig. 4. Numerical computation of Turning pattern.[9]

Reaction-Diffusion Equations

For spatially inhomogeneous system with constant diffusion rates

Partial differential equations (concentration)

Models pattern-formation. (Fig. 4)

𝜕𝑡𝑞 = 𝐷 𝛻2𝑞 + 𝑅(𝑞)

reactiondiffusion

Mathematical Models of Biochemical SystemsPolly Yu1, Gheorghe Craciun1,2

Mass-Action Kinetics

For homogeneous dilute solution

Ordinary differential equations (concentration)

Detailed-balanced equilibrium:

At equilibrium, reversible reaction rates are balanced

Complex-balanced equilibrium:

At equilibrium, fluxes at network nodes are balanced

𝑑 Ԧ𝑥

𝑑𝑡= 𝜅𝑗 Ԧ𝑥

𝑦𝑗 Ԧ𝑦𝑗′ − Ԧ𝑦𝑗

stoichiometry

reaction rate

Reaction Networks

Series of elementary chemical reactions

Example: Phosphorylation-dephosphorylation cycle

Goals: Model changes in concentrations 𝑥𝑖 of chemical species X𝑖.Infer qualitative dynamics from network structure.

Stochastic Mass-Action

For homogeneous solution, with low molecular count

Continuous-time Markov process (molecular count)

Chemical master equation (probability distribution)

𝑋 𝑡 = 𝑋 0 + 𝑁 න0

𝑡

𝜆𝑗 𝑠 𝑑𝑠 ( Ԧ𝑦𝑗′ − Ԧ𝑦𝑗)

Poisson process

rate of reaction

1 Department of Mathematics, University of Wisconsin-Madison2 Department of Biomolecular Chemistry, University of Wisconsin-Madison

Domain of applicability

Low conc

High conc

Spatially homogenSpatially inhomogen

Reaction-DiffusionPDE

StochasticMarkov process/ODE

Mass-ActionPolynomial ODE

Power-LawODE

Power-Law Kinetics

For spatially inhomogeneous dilute solution

(Time-dependent) ordinary differential equations (concentration)

𝑑 Ԧ𝑥

𝑑𝑡=𝜅𝑗 Ԧ𝑥, 𝑡 Ԧ𝑥𝑦𝑗 Ԧ𝑦𝑗

′ − Ԧ𝑦𝑗Number

of network nodes

Number of network connected

components

Dimension of stoichiometric

subspace

References

[1] Anderson, Craciun, Kurtz. Product-form stationary distributions for deficiency zero chemical reaction networks. Bull. Math. Biol., 2010.

[2] Cappelletti. (Figure from private communication.)[3] Craciun, Müller, Pantea, Yu. A generalization of Birch’s theorem and vertex-balanced

steady states for generalized mass-action systems. (In preparation.)[4] Feinberg. Existence and uniqueness of steady states for a class of chemical reaction

networks. Arch. Ration. Mech. Anal., 1995.

[5] Feinberg, Horn. Dynamics of open chemical systems and the algebraic structure of the underlying reaction network. Chem. Eng. Sci., 1974.

[6] Horn. Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal., 1972.

[7] Horn, Jackson. General mass action kinetics. Arch. Ration. Mech. Anal., 1972.[8] Kurtz. The relationship between stochastic and deterministic models for chemical

reactions. J. Chem. Phys., 1972.[9] Woolley, Baker, Maini. Turing’s theory of morphogenesis: Where we started, where we are

and where we want to go. The Incomputable, 2017. [10] Yu, Craciun. Mathematical analysis of chemical reaction systems. (Submitted.)

Fig. 3. (a) Trajectories converging to the stable steady state of (b) a weakly reversible deficiency 0 network, under mass-action kinetics for all rate constants.

A

B

(a) (b)

Trajectories towards the steady state

Fig. 2. (a) Trajectories towards the two stable steady states of (b) a bistable mass-action system with given rate constants.

A

BTrajectories of a bistable system

(a) (b)

Fig. 1. Numerical simulation of the network A ⇌ 2A.[2] The stochastic solution with largest volume (𝑉 = 100) best approximates the solution of the ODE.

Solutions to a stochastic mass-action system converge to the solution of a deterministic mass-action system, under appropriate volume scaling.[8] (Fig. 2)

If a mass-action system is complex-balanced, then the stochastic mass-action system has a unique stationary distribution.[1]

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5

[A]

time t

ODE X(t) X(t)/10 X(t)/100

Convergence of stochastic to mass-action

stoichiometry

stoichiometry