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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