Statistical Model for Biochemical Network Inference

Abstract

We describe a statistical method for predicting most likely reactions in a biochemical reaction network from the longitudinal data on species concentrations. Such data is relatively easily available in biochemical laboratories, for instance, via the popular RT-PCR technology. Under the assumed kinetics of the law of mass action, we also propose the data-based algorithms for estimating the prediction errors and for network dimension reduction. The second algorithm allows in particular for the application of the original algebraic inferential procedure described in [4] without the unnecessary restrictions on the dimension of the network stoichiometric space. Simulated examples of biochemical networks are analyzed, in order to assess the proposed methods’ performance.

Algebraic statistical model for biochemical network dynamics inference

“…In what follows, we assume that the practically computable and consistent (like the least squares) estimates of the linear combinations of κ (cf. (6) below), are available based on the trajectory data (see, e.g., [3] for further discussion). The statistical problem of interest is to identify, based on the estimates of the unknown parameters in (4) and the stoichiometry ν ′ − ν , the true reactions of the network, that is, to identify the values of k for which the rate constants are strictly positive κ k > 0.…”

“…The corresponding likelihood function satisfies $$ℒfalse(\theta false|\mathrm{normal\gamma }false)\propto true\prod _{i=1}^n{p}_i^{u_i}false(\theta false)$$with the log-likelihood $$\ell false(\theta false|\gamma false)=true\sum _{i=1}^n{u}_i\mathrm{normallog}false(p_{i}false(\theta false)false).$$It follows that the likelihood and log-likelihood functions are both maximized in ${\mathbb{R}}^m$ by any ${\widehat{\theta }}_N$ which satisfies $p_{i}false({\widehat{\theta }}_{N}false)=u_i/N$, where u i counts the number of γ points that fall into the chamber S i . The issues of obtaining ${\widehat{\theta }}_N$ in both restricted and unrestricted domains was considered in some detail in [3]. …”

“…However, the SASM method can still be applied when the assumption is violated, although in that case the sparsistency property is not always guaranteed. Also, as discussed in [3], in many practical situations the SASM approach will require a pre-processing step for the network dimension reduction. It is also easy to see that the assumed probability law for the κ coefficients (gamma family) is not particularly relevant for establishing the sparsistency property.…”

“…Whereas multiple such methods exist in the literature (see, e.g., [11]), their statistical properties are often difficult to establish. In order to partially address this issue we study here large sample properties of one such method, the so-called algebraic statistical model for biochemical reaction networks proposed in [2] and further expanded in [3]. …”

“…Regarding (ii), note that since the network trajectories may be observed under varying experimental conditions, like differences in temperatures or catalysts (e.g., enzymes) concentrations, the reaction laws may vary across experiments. In the current paper the assumption (ii) is made tacitly below by considering the family of mass-action rate laws where the kinetic parameter follows a gamma family of distributions as in [3]. Extensions to other rate laws are relatively straightforward, but are not considered here.…”