Filtered finite state projection method for the analysis and estimation of stochastic biochemical reaction networks
Authors: D’Ambrosio Elena , Fang Zhou , Gupta Ankit , Kumar Sant , Khammash Mustafa 
Published: October 2022

Time-lapse microscopy has become increasingly prevalent in biological experimentation, as it provides single-cell trajectories that unveil valuable insights into underlying networks and their stochastic dynamics. However, the limited availability of fluorescent reporters typically constrains tracking to only a few network species. Addressing this challenge, the dynamic estimation of hidden state-components becomes crucial, for which stochastic filtering presents a robust mathematical framework. Yet, the complexity of biological networks often renders direct solutions to the filtering equation intractable due to high dimensionality and nonlinear interactions.

In this study, we establish and rigorously prove the well-posedness of the filtering equation for the time-evolution of the conditional distribution of hidden species. Focusing on continuous-time, noise-free observations within a continuous-time discrete state-space Markov chain model, we develop the Filtered Finite State Projection (FFSP) method. This computational approach offers an approximated solution by truncating the hidden species’ state space, accompanied by computable error bounds. We illustrate the effectiveness of FFSP through diverse numerical examples, comparing it with established filtering techniques such as the Kalman filter, Extended Kalman filter, and particle filter. Finally, we show an application of our methodology with real time-lapse microscopy data. This work not only advances the application of stochastic filtering to biological systems but also contributes towards more accurate implementation of biomolecular feedback controllers.

Author Summary

The aim of this paper is to introduce a novel computational approach for numerically solving high-dimensional filtering problems associated with stochastic reaction network models in intracellular processes. This method, termed the Filtered Finite State Projection (FFSP) method, can reliably predict the dynamics of hidden species in reaction systems based on time-course measurements of the stochastic trajectories of certain species. While stochastic filtering is extensively utilised in engineering, its application in biology has been limited, primarily due to the nonlinear nature of biological interactions and the discrete, non-Gaussian nature of state variables. Traditional filtering techniques, such as the Kalman filter, often encounter difficulties under these conditions. We demonstrate that the FFSP method provides an accurate solution to the stochastic filtering problem, complete with a computable error bound. We present several numerical examples to showcase the effectiveness of FFSP and its superior performance compared to other filtering methodologies. Additionally, we apply FFSP to biological data, successfully reconstructing the hidden dynamics of a yeast transcription system from partial measurements obtained through time-lapse microscopy. We believe that FFSP could be a valuable tool for elucidating hidden intracellular dynamics and understanding stochastic cellular behaviours.