Table 1.

Variant models of the dynamic properties of biochemical reaction networks^*.

Deterministic Models
Time	Concen/Activ	Rate Laws	System of Rate Equations
Discrete	Discrete (0,1)	Discrete	Boolean: Ci(t + 1) = Bi(C1(t), …, CN(t))
Continuous	Continuous	Discrete	Piecewise-linear ODEs: $\frac{d{C}_i}{dt}={\gamma }_i[B_i({\widehat{C}}_1(t),\dots ,{\widehat{C}}_N(t))-C_i(t)]$
Continuous	Continuous	Continuous	Nonlinear ODEs: $\frac{d{C}_i}{dt}=\underset{j=1}{\overset{M}{\Sigma }}{\nu }_{ij}{R}_j(C_1,\dots ,C_N)$
Stochastic Models
Time	Concen/Activ	Rate Laws	System of Rate Equations
Continuous	Continuous	Discrete	Piecewise-linear SDE: $\frac{d{C}_i}{dt}={\gamma }_i[B_i({\widehat{C}}_1(t),\dots ,{\widehat{C}}_N(t))-C_i(t)]+{\sigma }_i\frac{{\xi }_i}{\sqrt{\Delta t}}$
Continuous	Continuous	Continuous	Nonlinear SDEs (Langevin formalism): $\frac{d{C}_i}{dt}=\underset{j=1}{\overset{M}{\Sigma }}{\nu }_{ij}{R}_j(C_1,\dots ,C_N)+{\left[\underset{j=1}{\overset{M}{\Sigma }}\mid {\nu }_{ij}\mid R_j\right]}^{1∕2}\frac{{\xi }_i}{\sqrt{\Delta t}}$
Continuous	Discrete	Continuous	Gillespie SSA: Ci (t + Δt) = Ci(t) + νij where Δt is the time interval until the next reaction and j is the index of the next reaction

^*Explanatory footnotes to Table 1.

Abbreviations: ODE, ordinary differential equation; SDE, stochastic differential equation; SSA, stochastic simulation algorithm; PWL, piecewise linear.

B_i(C₁, C₂, …, C_N) is a Boolean function (i.e., 0 or 1) of Boolean variables.

${\widehat{C}}_i(t)=0$, if 0 ≤ C_i(t) ≤ θ_i; =1, if θ_i < C_i(t) ≤ 1.

R_j(C₁, C₂, …, C_N) is a continuous function (rate law) of continuous variables.

ν_ij is the stoichiometric coefficient of species i in reaction j.

γ_i > 0 are rate constants that govern the rate at which species i approaches its steady-state value, C_i = B_i(…).

ξ_i is a random number chosen from a Gaussian distribution with mean = 0 and variance = 1.

σ_i > 0 are amplitudes for the white-noise terms in the PWL-SDEs.