Unbiased estimation of second-order parameter sensitivities for stochastic reaction networks

Abstract

Stochastic models for chemical reaction networks are increasingly popular in systems and synthetic biology. These models formulate the reaction dynamics as Continuous-Time Markov Chains (CTMCs) whose propensities are parameterized by a vector $θ$ and parameter sensitivities are introduced as derivatives of their expected outputs with respect to components of the parameter vector. Sensitivities characterise key properties of the output like robustness and are also at the heart of numerically efficient optimisation routines like Newton-type algorithms used in parameter inference and the design of of control mechanisms. Currently the only unbiased estimator for second-order sensitivities is based on the Girsanov transform and it often suffers from high estimator variance. We develop a novel estimator for second-order sensitivities by first rigorously deriving an integral representation of these sensitivities. We call the resulting method the Double Bernoulli Path Algorithm and illustrate its efficiency through numerical examples.

Figures (6)

• Figure 1: Representation of the Double BPA as a balanced binary tree where siblings are coupled. (A) The full tree has $4M^2$ leaves where $M$ is the number of reactions. (B) Bernoulli random variables are symbolised by dots. They are green when they equal 1 (meaning two coupled auxiliary processes get simulated) and red otherwise. After the introduction of these variables, only parts of the tree get generated while still leaving the estimator unbiased.
• Figure 2: Correspondence between the hierarchy of processes defined by the Double BPA and the quantities being updated in the pseudo-code.
• Figure 3: Integration domains over the second-order trajectories. The shape of the domain over which to integrate is dictated by (i) the value of $\min(\sigma_{p+1} \wedge T, T-u)$ (see eq. \ref{['eq:before_switching']} and \ref{['eq:after_switching']}) and (ii) the relative magnitude of $T-\sigma_p - (\sigma_{q+1}^{(p,k)} \wedge (T - \sigma_{p}))$ and $T - (\sigma_{p+1}\wedge T) -\sigma_q^{(p,k)}$.
• Figure 4: Gene expression network. The sensitivity $S_{\theta}^{(i,j)}(x,f,t)$ is computed using $10^4$ DBPA simulations and $5\times10^5$ GT simulations. The parameters of the network are set to $\theta_1 = 0.6$, $\theta_2 = 1.7329$, $\theta_3 = 0.3466$, $\theta_4 = 0.0023$. The initial state is chosen to be $x=(0,0)$ and the output function $f$ to be $f(x) \coloneqq x_2$. In the panels on left-hand side, error bars correspond to two standard deviations. While both methods are unbiased and will lead to the same sensitivity estimate asymptotically, the DBPA can offer large performance improvements over the GT method by having a lower variance-adjusted cost per sample $\mathcal{M}$, as showcased in the panels on the right-hand side.
• Figure 5: Toggle switch network. The sensitivity $S_{\theta}^{(i,j)}(x,f,t)$ is computed using $10^4$ DBPA simulations and $5\times10^5$ GT simulations. The parameters of the network are set to $\theta_1 = 1.0$, $\theta_2 = 1.0$, $\theta_3 = 2.5$, $\theta_4 = 0.5$, $\theta_5 = 0.0023$, $\theta_6 = 1.0$, $\theta_7 = 1.0$, $\theta_8 = 1.0$, $\theta_9 = 0.5$, $\theta_{10} = 0.0023$. The initial state is chosen to be $x=(0,0)$ and the output function $f$ to be $f(x) \coloneqq x_1$. In the panels on left-hand side, error bars correspond to two standard deviations. While both methods are unbiased and will lead to the same sensitivity estimate asymptotically, the DBPA can offer large performance improvements over the GT method by having a lower variance-adjusted cost per sample $\mathcal{M}$, as showcased in the panels on the right-hand side.

Theorems & Definitions (7)

• Theorem 2.1
• Example 4.1: Gene expression network
• Example 4.2: Genetic toggle switch
• Example 4.3: Antithetic integral controller
• Lemma A.1: From the proof of theorem 3.3 in gupta2018estimation
• proof
• proof : Proof of theorem \ref{['thm:main_theorem']}