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A Modified Rao-Blackwellised Particle Filter Based on Updated Process Noise Covariance for Chemical Reaction Networks

Suryasnata Dash, Abhishek Dey

arXiv:2609.31158v1eess.SYcs.SY

TL;DR

Joint estimation of states and multiple unknown parameters in chemical reaction networks requires an appropriate estimation method. The paper combines particle filtering for parameters with EKF state estimation and CLE-based process-noise updates; in a reduced-order gene-expression system, the updated-covariance RBPF balances parameter accuracy, state-estimation error, and innovation whiteness against fixed covariance choices.

  • Problem

    The paper addresses joint estimation of multiple unknown parameters and states in chemical reaction networks, for which extended Kalman filters are typically not feasible for multiple unknown parameters.

    Source: §I · p. 1

  • Method

    The modified RBPF estimates parameters with a particle filter and states with an EKF whose process-noise covariance is updated based on the chemical Langevin equation.

    Source: §I · p. 1

  • Results

    The updated-covariance RBPF balances parameter-estimation accuracy, state-estimation error, and innovation whiteness against fixed process-noise covariance choices.

    Source: §I · p. 1, §VI · p. 6

  • Takeaways & Limitations

    In the reduced-order gene-expression example, updated-covariance RBPF parameter accuracy is comparable to the best-performing fixed-covariance cases.

    Source: §VI · p. 6

Abstract

from arXiv · show

It is necessary to characterize chemical reaction networks (CRNs) accurately to facilitate engineering of both synthetic and naturally occurring biological systems. Estimation of multiple unknown parameters and states in CRNs requires first, an identifiability analysis, and then adoption of a suitable estimation technique. In this paper, we took an example of a reduced order gene expression system, performed parameter sensitivity analysis for multiple unknown parameters, and used a modified Rao-Blackwellised particle filter (RBPF) to estimate parameters and states. The proposed framework estimates parameters using a particle filter and states using an extended Kalman filter (EKF) with process noise covariance updated recursively based on chemical Langevin equation (CLE). We compared the accuracy of parameter estimation, error in state estimation, and whiteness of the innovation sequence for the proposed filter with fixed choices of noise covariance. We found that the RBPF with updated noise covariance finds a balance between these three criteria, demonstrating its suitability for joint state and parameter estimation for stochastic CRNs.

I. INTRODUCTION

Chemical reaction networks model biological systems, but estimating multiple unknown parameters and states requires suitable methods. This paper proposes a modified RBPF for joint estimation in a reduced-order gene-expression system.

I. INTRODUCTION

  • CRNs represent biomolecular systems through chemical reactions and conservation laws, and their mathematical modeling supports biological-system engineering and analysis.The chemical master equation gives a probabilistic representation but faces high-dimensional computational issues; the CLE is an alternative stochastic differential equation representation.Source: §I · p. 1
  • Kalman filters are computationally easier but handle Gaussian noise, whereas particle filters can estimate states and parameters with non-Gaussian noise and multiple parameters.Prior work applied particle filters to cellular responses, heart-rate models, gene regulatory networks, and reduced stochastic reaction networks.Source: §I · p. 1
  • The paper uses a modified RBPF to estimate multiple unknown parameters and states in a chemical reaction network.Parameters are estimated with a particle filter, while states are estimated with a Kalman filter whose process-noise covariance is updated based on the CLE.Source: §I · p. 1
  • The proposed RBPF with updated process-noise covariance balances particle-filter parameter accuracy and EKF state-estimation optimality against fixed covariance choices.The paper also performs parameter sensitivity analysis and compares parameter accuracy, state-estimation error, and innovation whiteness.Source: §I · p. 1

II. RAO-BLACKWELLISED PARTICLE FILTER

The RBPF combines particle filtering for parameters with Kalman filtering for states. Its Gaussian conditional-state posterior is updated using the EKF, while parameter particles are propagated and weighted from observations.

II. RAO-BLACKWELLISED PARTICLE FILTER

  • The RBPF marginalizes the state through a Kalman filter while particle filtering estimates parameters, reducing the estimation problem's dimensionality.The method uses one Kalman filter for the conditional state posterior rather than one filter per particle.Source: §II · pp. 1–2
  • The method assumes the state posterior conditioned on parameters and observations is Gaussian and computes it with a single Kalman filter.The EKF measurement update solves this conditional posterior in closed form.Source: §II · p. 2
  • Parameter-particle weights are updated using the likelihood, then normalized to estimate parameters across the particle set.The state error covariance is denoted Pk.Source: §II · p. 2
  • Resampled parameter particles are propagated from previous posterior parameter values using a Gaussian transition with covariance Σk.Σk is positive definite with small diagonal values, and θk denotes the previous time step's posterior parameter value.Source: §II · p. 2

III. BIOMOLECULAR SYSTEM EXAMPLE

The example is a reduced-order gene-expression system with transcription, translation, and degradation represented as a chemical reaction network. Conservation laws and a quasi-steady-state assumption yield a discrete-time model, with stochastic dynamics represented using the CLE.

III. BIOMOLECULAR SYSTEM EXAMPLE

  • The gene-expression example models transcription and translation as a two-phase biochemical process represented by chemical reactions.Transcription forms mRNA, while translation uses mRNA to produce protein.Source: §III · p. 2
  • The model includes first-order degradation reactions for both mRNA transcript and protein.The transcript and protein degradation parameters are dT and dX, respectively.Source: §III · p. 2
  • Conservation of total polymerase and ribosome, together with a quasi-steady-state assumption, produces a reduced-order deterministic model in discrete time.A reduced-order CLE model is also used to express the stochastic dynamics in discrete time.Source: §III · p. 2

IV. METHODOLOGY

The methodology uses sensitivity analysis to identify estimable parameters, then applies an RBPF with an EKF whose process-noise covariance is updated using the CLE. It evaluates the approach on SSA-generated gene-expression data with adaptive particle resampling.

Sensitivity Analysis

  • Degradation parameters dX and dT are most sensitive to the measured states, followed by ktl and ktx.The degradation parameters are calculated graphically; these estimates can then be used to estimate ktx and ktl.Source: §IV · pp. 2–3

Data Generation

  • Fifty datasets were generated through stochastic simulation algorithm (SSA), with measurements containing observed states plus MATLAB randint measurement noise in ranges [−1,1] for T and [−3,3] for X.Source: §IV · p. 3

RBPF with updated process noise covariance

  • The RBPF estimates parameters with a particle filter and states with an EKF, updating process-noise covariance from CLE-based state prediction.The CLE uses the stoichiometric matrix V and propensity vector A(ζk,θk) to form the state prediction and noise update.Source: §IV · p. 3
  • Parameter particles are predicted from the previous time step’s posterior parameter values, while the initial covariance Σk decreases before being held constant for ktl and ktx.After the decrease, Σk is held at diag(10^-7,10^-6) for ktl and ktx.Source: §IV · p. 3
  • Stratified adaptive resampling is triggered when effective sample size Neff falls below the threshold N.The stated procedure is intended to reduce unnecessary resampling and preserve particle diversity.Source: §IV · p. 3

V. RESULTS

Across 50 datasets, the updated process-noise covariance produced parameter estimates comparable to the best fixed-covariance cases, but its state-error, convergence-time, and innovation-whiteness results varied by state and covariance. The results also show that parameter convergence was not achieved jointly in every dataset.

V. RESULTS

  • Across 50 datasets, not every run converged for both parameters; some converged for only ktl or ktx.The study used adaptive resampling, and the convergence counts varied with particle size for updated covariance.Source: §V · p. 3, Table II · p. 4
  • The 50 simulations ran for 800 seconds with T and X as measurements and initial ktl and ktx values drawn uniformly from 0.05 to 0.2.The state vector was [T, X] and the parameter vector was [ktl, ktx].Source: §V · p. 3
  • Updated covariance gave a ktx estimated mean better than higher fixed covariances, while ktl estimates had similar means and standard deviations across covariance choices.The text reports these comparisons across 50 datasets; Table III is identified as the parameter-estimation comparison.Source: §V · p. 4, Table III · p. 4
  • The true ktl and ktx values were nearer the estimated bivariate-distribution means for updated covariance and Q = 0.1 × In×n than for the other fixed covariances.Fig. 4 reports marker distances of 0.018 for updated covariance and 0.017 for Q = 0.1 × In×n, versus 0.062 and 0.071 for Q = 1 × In×n and Q = 10 × In×n, respectively.Source: §V · p. 4, Figure 4 · p. 5
  • For T, updated covariance had higher NRMSE and mean ktx convergence time than fixed choices; for X, its NRMSE was below Q = 0.1 × In×n but convergence time remained higher.The convergence-time mean includes only datasets where the respective parameter converged.Source: §V · p. 4
  • Innovation whiteness differed by state: for T, updated covariance was comparable to Q = 0.1 × In×n, while Q = In×n and Q = 10 × In×n were whiter; for X, updated covariance was similar to Q = In×n and Q = 10 × In×n and better than Q = 0.1 × In×n.The whiteness test used 95% confidence limits.Source: §V · p. 5

VI. CONCLUSION

For a reduced-order gene-expression system, sensitivity analysis helped identify parameters for estimation with a modified RBPF. The updated-noise-covariance filter balanced parameter-estimation accuracy with state-filter optimality and innovation whiteness.

VI. CONCLUSION

  • Sensitivity analysis helped obtain identifiable parameters for the reduced-order gene-expression system estimated with the modified RBPF.Source: §VI · p. 5
  • Whiteness testing covers T and X over 100 lags and observed states under different process-noise covariances across 50 datasets.Source: Figure 6 · p. 6, Table IV · p. 6
  • Updated-Q RBPF achieved parameter-estimation accuracy comparable to the best fixed-noise cases while maintaining satisfactory innovation whiteness.The reported conclusion is a balance between state EKF optimality and parameter PF accuracy.Source: Table IV · p. 6, §VI · p. 6
  • Larger process-noise covariance reduced state-estimation NRMSE and improved innovation whiteness, but did not necessarily improve parameter estimates, especially ktx.The paper attributes this trade-off to tighter EKF state correction making competing parameter particles harder for the PF to distinguish.Source: §VI · p. 6
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