Bootstrapping least-squares estimates in biochemical reaction networks

Abstract

The paper proposes new computational methods of computing confidence bounds for the least squares estimates (LSEs) of rate constants in mass-action biochemical reaction network and stochastic epidemic models. Such LSEs are obtained by fitting the set of deterministic ordinary differential equations (ODEs), corresponding to the large volume limit of a reaction network, to network’s partially observed trajectory treated as a continuous-time, pure jump Markov process. In the large volume limit the LSEs are asymptotically Gaussian, but their limiting covariance structure is complicated since it is described by a set of nonlinear ODEs which are often ill-conditioned and numerically unstable. The current paper considers two bootstrap Monte-Carlo procedures, based on the diffusion and linear noise approximations for pure jump processes, which allow one to avoid solving the limiting covariance ODEs. The results are illustrated with both in-silico and real data examples from the LINE 1 gene retrotranscription model and compared with those obtained using other methods.

Comprehensive Review of Models and Methods for Inferences in Bio-Chemical Reaction Networks

“…MCMC sampling for mixed-effects SDE models is considered in (Whitaker et al, 2017). In order to overcome ill-conditioned least squares (LS) data fitting and numerical instability, bootstrapped MC procedure based on diffusion and LNA was studied in (Lindera and Rempala, 2015). Particle filter assumes specific type of random processes to identify posteriors while bounding computational complexity for models with large number of parameters is considered in (Mikelson and Khammash, 2016).…”

“…Main research problems considered (Dargatz, 2010) Bayesian inference for biochemical models involving diffusion (Mu, 2010) rate and state estimation in S-system and linear fractional model (LFM) (Palmisano, 2010) software tools for modeling and parameter estimation in BRNs (Mazur, 2012) inference via stochastic sampling and Bayesian learning framework (Srivastava, 2012) stochastic simulations of BRNs combined with likelihood based parameter estimation, confidence intervals, sensitivity analysis (Gupta, 2013) parameter estimation in deterministic and stochastic BRNs, inference with model reduction, mostly MCMC methods (Hasenauer, 2013) Bayesian estimation and uncertainty analysis of population heterogeneity and proliferation dynamics (Linder, 2013) penalized LS algorithm and diffusion and linear noise approximations and algebraic statistical models (Flassig, 2014) model identification for large scale gene regulatory networks (Liu, 2014) approximate Bayesian inference methods and sensitivity analysis (Moritz, 2014) structural identification and parameter estimation for modular and layered type of modes (Paul, 2014) analysis of MCMC based methods (Ruess, 2014) optimum estimation and experiment design assuming ML and Bayesian inference and Fisher information (Schenkendorf, 2014) quantification of parameter uncertainty, optimal experiment design for parameter estimation and model selection (Smadbeck, 2014) moment closure methods, model reduction, stability and spectral analysis of BRNs Langevin equation, moment closure approximations, representations of stochastic RDME (Zechner, 2014) inference from heterogeneous snapshot and time-lapse data (Galagali, 2016) Bayesian and non-Bayesian inference in BRNs, adaptive MCMC methods, network-aware inference, inference for approximated BRNs (Hussain, 2016) sequential probability ratio test, Bayesian model checking, automated and formal verification, parameter discovery (Lakatos, 2017) multivariate moment closure and reachability analysis (Liao, 2017) tensor representation and analysis of BRNs of ABC methods can be found in (Drovandi et al, 2016). The basic idea is to find parameter values which generate the same statistics as the observed data.…”

Algebraic statistical model for biochemical network dynamics inference

“…Throughout the paper all the systems of chemical reactions (1) are meant to represent their limiting ODEs (4). The issue of estimating the ODEs parameters κ from trajectory data has been recently studied in [7, 12] by means of the least squares method. In what follows, we assume that the practically computable and consistent (like the least squares) estimates of the linear combinations of κ (cf.…”

“…Typically, this is not the case in practice, however, it is also easy to see from the proof that the theorem’s hypothesis holds as long as one may substitute for γ their consistent estimates based on the trajectory values. Recall from Section 1 that the estimates satisfying the consistency requirements were recently discussed in [7, 12]. …”

“…Unlike in our previous (in silico) example, the γ coefficient values in (15) are unknown. However, under the assumption (7) we may use the FPKM data from Table II to estimate their values for each trajectory, following the weighted least squares method discussed in [7, 12]. The resulting set of ${\widehat{\gamma }}_2={false({\widehat{\gamma }}_{12},{\widehat{\gamma }}_{22},{\widehat{\gamma }}_{32}false)}^{\mathrm{\top }}$ estimates for the 8 different empirical trajectories considered in the experiment is presented in Table III.…”