Least Squares Estimation in Stochastic Biochemical Networks

Abstract

The paper presents results on the asymptotic properties of the least-squares estimates (LSEs) of the reaction constants in mass-action, stochastic, biochemical network models. LSEs are assumed to be based on the longitudinal data from partially observed trajectories of a stochastic dynamical system, modeled as a continuous-time, pure jump Markov process. Under certain regularity conditions on such a process, it is shown that the vector of LSEs is jointly consistent and asymptotically normal, with the asymptotic covariance structure given in terms of a system of ordinary differential equations (ODE). The derived asymptotic properties hold true as the biochemical network size (the total species number) increases, in which case the stochastic dynamical system converges to the deterministic mass-action ODE. An example is provided, based on synthetic as well as RT-PCR data from the retro-transcription network of the LINE1 gene.

Appendix: Density Dependent Markov Processes

Suppose for each n≥1, Z _n ={Z _n (t);t≥0} is a continuous-time Markov process on the d-dimensional lattice $\mathcal{Z}^{d}$ with the jump intensities

$$ q^{(n)}_{z,z+l}=n\beta_l\bigl(n^{-1}z \bigr),\quad z,l\in \mathcal{Z}^d $$

and the transition probabilities

(19)

We assume that the process has a finite number of possible transitions, that is, there exists only a finite number of $l\in \mathcal{Z}^{d}$ such that sup _x β _l (x)>0. We also assume that β _x (l) is a continuous function of x for each l and that the starting point Z _n (0) is nonrandom. Such processes are called density dependent Markov jump processes (DDMJP) since their rates depend on the process density (its state normalized by n). Note that if Y _l ={Y _l (t);t≥0} (indexed by l) is a collection of independent unit Poisson processes then

$$ Z_n(t)=Z_n(0)+\sum _l lY_l \biggl(n\int_0^t \beta_l\bigl(n^{-1}Z_n(s)\bigr)\,ds \biggr). $$

(20)

Note that the above process satisfies (19) and that under our assumption (2), the relation (3) implies that (20) is asymptotically equivalent to the process with the trajectories (1). Indeed, in the current notation the latest process is seen to have the jump intensities given by

$$ \tilde{q}^{(n)}_{z,z+l}=n\beta_l \biggl(n^{-1}z+O \biggl(\frac{1}{n} \biggr) \biggr) . $$

(21)

In order to analyze the asymptotic behavior of (1), it suffices therefore to consider DDMJP given by (20) (see, e.g., Ethier and Kurtz 1986, Chap. 11).

Let us denote by $\hat{Y}_{l}$ the centered Poisson processes, that is $\hat{Y}_{l}(t)=Y_{l}(t)-l$. Also, let $\bar{Z}_{n}(t)=Z_{n}(t)/n$ and F(x)=∑ _l lβ _l (x). Then (20) is equivalent to

$$ \bar{Z}_n(t)=\bar{Z}_n(0)+n^{-1}\sum _l l\hat{Y}_l \biggl(n\int _0^t \beta_l\bigl( \bar{Z}_n(s)\bigr) \biggr)+\int_0^t F \bigl(\bar{Z}_n(s)\bigr)\,ds. $$

(22)

The following result of Kurtz (see Andersson and Britton 2000, Chap. 5) establishes SLLN for $\bar{Z}_{n}(t)$.

Theorem A.1

(SLLN)

Let $\lim_{n\to\infty} \bar{Z}_{n}(0)=z_{0}$ (non random) and suppose that for any compact $K\in{ \mathcal{R}}^{d}$ there exists a constant M _K such that |F(x)−F(y)|≤M _K |x−y|, ∀x,y∈K. Then

$$\lim_{n\to\infty}\sup_{s\le t}\bigl\vert \bar{Z}_n(s)-z(s) \bigr\vert=0 \quad\mbox{\textit{a.s.}}, $$

where z(t) is a unique solution of the integral equation

$$ z(t)=z_0(t)+\int_0^t F\bigl(z(s)\bigr)\,ds. $$

(23)

Thus, the normed jump Markov vector process $\bar{Z}_{n}$, for a large population n, is approximately equal to the deterministic vector function z defined by (23). It turns out that the deviations between the two are of order $\sqrt{n}$. To this end, define first

$$W_l^{(n)}(t) = \sqrt{n} \bigl(n^{-1}Y_l(nt) - t\bigr) = n^{-1/2}\hat{Y}(nt). $$

By Donsker’s theorem (Billingsley 1999) it follows that $W_{l}^{(n)}(t)$ converges weakly to the standard Brownian motion W _l . Define now a scaled and centered process $\bar{Z}_{n}$ as follows:

(24)

Of course, $v_{n}(0) = \sqrt{n} (\bar{Z}_{n}(0) - z(0))$, which by assumption is nonrandom. The second equality above is a direct consequence of the definition of $W_{l}(t), \bar{Z}_{n}$ and z. We can expand the integrand on the far right by Taylor’s theorem, so that

where ∂F=(∂ _j F _i ) is the matrix function of partial derivatives. From Theorem A.1, we know that $\bar{Z}_{n}$ converges to z, and since $W_{l}^{(n)}$ converges to W _l , the standard Brownian motion, one might anticipate that V _n converges to a process V defined by the integral equation

$$ V(t) = v_0 + \sum_l lW_l \biggl(\int_0^t \beta_l\bigl(z(s)\bigr)\,ds \biggr)+ \int_0^t \partial F\bigl(z(s)\bigr)V(s)\,ds. $$

(25)

This is formally stated as the following theorem due to Kurtz (see Andersson and Britton 2000, Chap. 5) where we set G(x)=∑ _l ll ^⊤ β _l (x). Note that β _l (x)∈ℝ and ll ^⊤∈ℝ^d×d, since l is taken as a column vector.

Theorem A.2

(CLT)

Suppose ∂F is continuous and lim _n→∞ v _n (0)=v ₀ (constant). Then V _n ⇒V, the process defined in Eq. (25). This process V is a Gaussian vector process with covariance matrix

$$\operatorname {Cov}\bigl(V(t), V(r)\bigr) = \int_0^{r \wedge t} \varPhi(t,s)G\bigl(z(s)\bigr) \bigl(\varPhi(r, s)\bigr)^\top \,ds. $$

Here, Φ is a matrix function defined as the solution of

$$\varPhi^\prime_2(t,s) = -\varPhi(t, s) \partial F\bigl(z(s)\bigr), \qquad\varPhi(s, s) = I, $$

where $\varPhi^{\prime}_{2}$ denotes the partial derivative with respect to s.