Suppose we have

• \(N\) species: \(X_1, X_2, \ldots, X_N\)
• \(M\) reactions: \(R_1, R_2, \ldots, R_M\)
• In a typical model, \(M = 2N\)

Reaction \(R_i\) takes the form \[ \begin{array}{cccc} R_{i}: \, & u_{i1} X_1+\ldots +u_{ik} X_N &\xrightarrow {\phantom{a}c_{i}\phantom{a}} & v_{i1}X_1+\ldots +v_{ik}X_N \end{array} \] The effect of reaction \(i\) on species \(j\) is to change \(X_j\) by an amount \(v_{ij} - u_{ij}\)

• Straightforward to simulate

1. Initialisation: initial conditions, reactions constants, and random number generators
2. Propensities update: Update each of the \(M\) hazard functions, \(h_i(x)\)
3. Propensities total: Calculate the total hazard \(h_0 = \sum_{i=1}^M h_i(x)\)
4. Reaction time: \(\tau = -ln[U(0,1)]/h_0\) and \(t = t+ \tau\)
5. Reaction selection: A reaction is chosen proportional to it's hazard
6. Reaction execution: Update species
7. Iteration: If the simulation time is exceeded stop, otherwise go back to step 2

Typically there are a large number of iterates

• As the number of reactions (and species) increase, the length of time a single iteration takes also increases
• Clever tricks help
  • Dynamic sorting of the reactions
  • Updating only species and hazards that have changed
  • Avoid simulating two uniform random numbers per iteration
• For models where one of the hazards is large, simulation time can be prohibitively large

Relax some assumptions (e.g. discreteness and stochasticity) in order to make simulation faster and more scalable

• Diffusion approximation / chemical Langevin equation (CLE)
• Hybrid discrete-continuous models
• Linear noise approximation (LNA) / moment closure (2MA)

How do can we assess the appropriateness of these methods?

• Moment closure and the linear noise approximation
• Obtain (differential) equations for the mean and covariance
• Typically assume normality
• Very useful for parameter inference

\[ 2X_1 \longrightarrow X_2 \quad \mbox{and} \quad X_2 \longrightarrow 2 X_1 \] with propensities \(h_1(X_1, X_2) = 0.5 k_1 X_1(X_1-1)\) and \(h_2(X_2) = k_2 X_2\)

Moment equations

The equation for the mean ODE is \[ \frac{dE[X_1]}{dt} = 0.5 k_1 (E[X_1^2] - E[X_1]) -k_2 E[X_1] \] and second moment \[ \frac{dE[X_1^2]}{dt} = k_1 (E[X_1^2X_2] - E[X_1 X_2]) + 0.5 k_1(E[X_1^2] - E[X_1]) k_2 (E[X_1] - 2 E[X_1^2]) \] where \(E[X_1]\) is the mean of \(X_1\) and \(E[X_1^2] - E[X_1]^2\) is the variance

Typically use Gaussian or Log-normal, since it's easy to generalise to the multivariate case

If we consider a slightly more complicated propensity function \[ X \rightarrow \emptyset \] with propensity function \[ h(X) = \frac{k_1 X^2}{k_2 + X^2} \] we get the moment equation \[ \frac{dE(X)}{ dt} =\frac{1}{2k_2}\left[-4k_1 E(X^3) - (2k_2+1)\frac{d E(X^2)}{dt} - 2\frac{d E(X^3)}{dt} - \frac{d E(X^4)}{dt}\right]. \]

• Model building
  • a quick method for estimating the mean/variance of system
• Steady-state analysis
  • set the differentials to zero
• Parameter inference
  • instead of multiple forward simulations (per MCMC iteration), solve a set of ODEs
  • Lotka-Volterra model - \(10^4\) iterations in less than 2 seconds
• Experimental design
  • Each iteration of the algorithm reduces to the parameter inference problem

• Stochastic kinetic model of the heat shock system
  • twenty-three reactions
  • seventeen chemical species
• A single stochastic simulation up to \(t=2000\) takes about \(35\) minutes.
• If we convert the model to moment equations, we get \(139\) equations

• When modelling biological systems, it's unusual to have detailed knowledge about all rate constants
  • Vague parameters over many orders of magnitude
  • Uncertain initial conditions
  • Unobserved species
• How can we assess the appropriateness of the MC/LN approximation?

• Proctor and Grey, 2008
  • 16 chemical species
  • Around a dozen reactions
• The model contains a single event
  • At \(t = 1\), set \(X = 0\)
• If we convert the model to moment equations, we get 139 equations.

• Immigration: \(\emptyset \longrightarrow X + 1\) and \(\emptyset \xrightarrow{k_2 sf} Y + 1\)
• Death: \(X \xrightarrow{k_3 sf} \emptyset\) and \(Y \longrightarrow \emptyset\)
• Interaction: \(X +Y \longrightarrow 2 Y\)

• Accuracy of the mean and variance estimates
  • For any system with a second-order reaction, we can prove that the MC/LN approximations are not exact
• Suitability of the normality assumption
  • Continuous distribution for a discrete state space
• The question is, are the approximations good enough?

• If the system contains only zero and first order reactions, the MC/LN moment estimates are exact
• Intuitively, if the MC/LN produce different moment estimates, this suggests an issue with the approximation
  • Note - if the MC/LN approximations agree, this does not imply accuracy
• For a particular set of parameters, \(\gamma_i = (c_1, \ldots, c_L)\), calculate the standardised difference \[ D_{j}^{LM}(\mathbf{\gamma}_i) = \frac{\psi_j^l(\mathbf{\gamma}_i) - \psi_j^m(\mathbf{\gamma}_i)}{\sqrt{\Sigma_{jj}^l(\mathbf{\gamma}_i)}} \] where
  • \(\psi_j(\mathbf{\gamma}_i)\) is the solution to the equation for species \(j\) for parameters \(\gamma_i\) using the LN or MC approximation
  • \(\Sigma_{jj} (\mathbf{\gamma}_i)\) is the variance estimate
• Large values of \(D_j^{LM}\) indicate a potential issues with the approximation

• Simple system with non-linear dynamics \[ X \rightarrow X + 1, \quad X + Y \rightarrow 2Y \quad \text{and} \quad Y \rightarrow \emptyset \]
• Under certain parameter combinations, deterministic model predicts sustained oscillations
  • But the stochastic system can become extinct
• Set-up:
  • Fix \(c_3=0.3\) and \(X(0) = Y(0)=100\)
  • Maximum simulation time \(t=30\)
• Assess the moment closure and the linear noise approximations over the parameter ranges \[ c_1 \sim (10^{-6}, 10^0) \quad \text{and} \quad c_2 \sim (10^{-6}, 10^0) \] on the log scale

• Form a Latin hypercube on the log scale
• At each point, \(\gamma_i\) on the hyper cube calculate the standardised difference \(D_{j}^{LM}(\mathbf{\gamma}_i)\)
  • Values of \(D_{j}^{LM}(\mathbf{\gamma}_i) > 5\) are shown as red triangles

• We can borrow ideas from Gaussian process assessment
• At each point, \(\gamma_i\), on the design space
  • estimate the mean, \(\psi\), and co-variance, \(\Sigma\), using the LN/MC approximations
  • simulate using an exact simulator
• Prediction error for species \(j\),
  \[ e_{i,j} = y_j^*(\mathbf{\gamma}_i) - \psi_j^{l}(\mathbf{\gamma}_i) \]
• Easier to work with the standardised prediction errors \[ e_{i,j}^* = \frac{y_j^*(\mathbf{\gamma}_i) - \psi_j^l(\mathbf{\gamma}_i)}{\sqrt{\Sigma_{jj}^l(\mathbf{\gamma}_i)}} \;. \]
• If the assumptions are correct, then \(e_{i,j}^* \sim N(0, 1)\)

• Construct \(100\alpha\)% confidence interval using the approximate mean \(\psi\) and variance, \(\Sigma\)
• Denote a particular confidence region at design point \(i\), for species \(j\), as \(CI_{i,j} (\alpha)\)
• The proportion of simulated values that land within the confidence region is given by \[ D_j^{CR} = \frac{1}{n_d} \sum_{i=1}^{n_d} \mathbf{1}[y_j^*(\mathbf{\gamma}_i) \in CI_{i,j}(\alpha)], \] where \(\mathbf{1}[\cdot]\) is the indicator function. The value of \(D_{j}^{CR}\) should be approximately equal to \(\alpha\)

• A well known test system that exhibits bi-modal and non-linear characteristics at certain parameter combinations
• The system contains four reactions \[ 2X_1 + X_2 \xrightarrow{\phantom{a}c_{1}\phantom{a}} 3X_1, \quad 3X_1 \xrightarrow{\phantom{a}c_{2}\phantom{a}} 2X_1 + X_2, \quad X_3 \xrightarrow{\phantom{a}c_{3}\phantom{a}} X_1 \\ X_1 \xrightarrow{\phantom{a}c_{4}\phantom{a}} X_3 \] where the distribution of \(X_1\) is bi-modal, the Gaussian assumption would clearly be inappropriate
• Parameters \(\{c_1, c_2\}\) are fixed at \(\{3\times 10^{-7}, 10^{-4}\}\) and the initial conditions are held constant at \[ \{X_1(0), X_2(0), X_3(0)\} = \{250, 10^5, 2\times 10^5\} \;. \]
• Two dimensional hypercube size \(n_d=1,000\) on the \(\log_{10}\) space for the parameters \(\{c_3, c_4\}\)
• At each of the \(n_d\) points on the hypercube, we simulate using the direct method and the LNA to a maximum time of \(t=5\).

• Investigate large standardised errors more carefully

• Moment closure/linear approximations provide a quick, easy to construct approximation to the system of interest
  • Intelligent emulator
• Care needs to be taken to assess the suitability of the approximation
• Assessing the accuracy of other approximate simulators is tricky (arXiv:1409.1096)

References