Partially-observed Boolean Dynamical Systems Research Articles

Kernel-Based Particle Filtering for Scalable Inference in Partially Observed Boolean Dynamical Systems

This paper addresses the inference challenges associated with a class of hidden Markov models with binary state variables, known as partially observed Boolean dynamical systems (POBDS). POBDS have demonstrated remarkable success in modeling the ON and OFF dynamics of genes, microbes, and bacteria in systems biology, as well as in network security to represent the propagation of attacks among interconnected elements. Despite existing optimal and approximate inference solutions for POBDS, scalability remains a significant issue due to the computational cost associated with likelihood evaluations and the exploration of extensive parameter spaces. To overcome these challenges, this paper proposes a kernel-based particle filtering approach for large-scale inference of POBDS. Our method employs a Gaussian process (GP) to efficiently represent the expensive-to-evaluate likelihood function across the parameter space. The likelihood evaluation is approximated using a particle filtering technique, enabling the GP to account for various sources of uncertainty, including limited likelihood evaluations. Leveraging the GP’s predictive behavior, a Bayesian optimization strategy is derived for effectively seeking parameters yielding the highest likelihood, minimizing the overall computational burden while balancing exploration and exploitation. The proposed method’s performance is demonstrated using two biological networks: the mammalian cell-cycle network and the T-cell large granular lymphocyte leukemia network.

Boolean Kalman filter and smoother under model uncertainty

Partially-observed Boolean dynamical systems (POBDS) are a general class of nonlinear state-space models that provide a rich framework for modeling many complex dynamical systems. The model consists of a hidden Boolean state process, observed through an arbitrary noisy mapping to a measurement space. The optimal minimum mean-square error (MMSE) POBDS state estimators are the Boolean Kalman Filter and Smoother. However, in many practical problems, the system parameters are not fully known and must be estimated. In this paper, for POBDS under model uncertainty, we derive an optimal Bayesian estimator for state and parameter estimation. The exact algorithms are derived for the case of discrete and finite parameter space, and for general parameter spaces, an approximate Markov-Chain Monte-Carlo (MCMC) implementation is introduced. We demonstrate the performance of the proposed methodology by means of numerical experiments with POBDS models of gene regulatory networks observed through noisy measurements.

Scalable optimal Bayesian classification of single-cell trajectories under regulatory model uncertainty

BackgroundSingle-cell gene expression measurements offer opportunities in deriving mechanistic understanding of complex diseases, including cancer. However, due to the complex regulatory machinery of the cell, gene regulatory network (GRN) model inference based on such data still manifests significant uncertainty.ResultsThe goal of this paper is to develop optimal classification of single-cell trajectories accounting for potential model uncertainty. Partially-observed Boolean dynamical systems (POBDS) are used for modeling gene regulatory networks observed through noisy gene-expression data. We derive the exact optimal Bayesian classifier (OBC) for binary classification of single-cell trajectories. The application of the OBC becomes impractical for large GRNs, due to computational and memory requirements. To address this, we introduce a particle-based single-cell classification method that is highly scalable for large GRNs with much lower complexity than the optimal solution.ConclusionThe performance of the proposed particle-based method is demonstrated through numerical experiments using a POBDS model of the well-known T-cell large granular lymphocyte (T-LGL) leukemia network with noisy time-series gene-expression data.

Point-Based Methodology to Monitor and Control Gene Regulatory Networks via Noisy Measurements

This paper proposes a methodology to monitor and control gene regulatory networks (GRNs) via noisy measurements in an infinite observation space. Toward this end, we employ the partially observed Boolean dynamical system (POBDS) signal model. The proposed methodology consists of offline and online steps. In the offline step, a family of point-based methods is applied to the POBDS model to gather the necessary control policy prior to the online (execution) step. This is accomplished by developing efficient backup and belief expansion processes to make the computation scale with the log of the number of states, as opposed to the complexity of existing point-based methods, which grows with the number of states. In the online step, simultaneous monitoring and control is achieved by a one-step look-ahead search procedure using the optimal state estimation algorithm for the POBDS model, known as the Boolean Kalman filter (BKF), as well as the information gathered in the offline step. The online one-step look-ahead process confers robustness to changes in system dynamics, possibility of starting the execution process before the completion of the offline step. The use of the BKF for simultaneous monitoring and control during the online stage can be key in assessing possible side effects of intervention. The performance of the proposed methodology is investigated through a comprehensive set of numerical experiments using synthetic gene expression data generated from a melanoma GRN.

Adaptive Particle Filtering for Fault Detection in Partially-Observed Boolean Dynamical Systems

We propose a novel methodology for fault detection and diagnosis in partially-observed Boolean dynamical systems (POBDS). These are stochastic, highly nonlinear, and derivativeless systems, rendering difficult the application of classical fault detection and diagnosis methods. The methodology comprises two main approaches. The first addresses the case when the normal mode of operation is known but not the fault modes. It applies an innovations filter (IF) to detect deviations from the nominal normal mode of operation. The second approach is applicable when the set of possible fault models is finite and known, in which case we employ a multiple model adaptive estimation (MMAE) approach based on a likelihood-ratio (LR) statistic. Unknown system parameters are estimated by an adaptive expectation-maximization (EM) algorithm. Particle filtering techniques are used to reduce the computational complexity in the case of systems with large state-spaces. The efficacy of the proposed methodology is demonstrated by numerical experiments with a large gene regulatory network (GRN) with stuck-at faults observed through a single noisy time series of RNA-seq gene expression measurements.

Finite-horizon LQR controller for partially-observed Boolean dynamical systems

This paper proposes an approach for finite-horizon control of partially-observed Boolean dynamical systems (POBDS) with uncertain continuous control input and infinite observation space. To cope with the partial observability of states, the proposed method first maps the POBDS to an unnormalized belief space. The nonlinear dynamics in this continuous belief space are linearized over a nominal trajectory. Then, the optimal feedback controller is derived, based on the well-known linear quadratic regulator (LQR), to push the system to follow the nominal trajectory. This nominal trajectory is computed in a planning stage before starting execution, and updated efficiently during execution, whenever the system is found to deviate from the nominal trajectory. We prove that, under mild regularization conditions, the proposed controller approaches the cost of the nominal trajectory as the linearization error approaches zero. The performance of the proposed controller is demonstrated by numerical experiments with a Melanoma gene regulatory network observed through noisy gene expression measurements.

Control of Gene Regulatory Networks Using Bayesian Inverse Reinforcement Learning.

Control of gene regulatory networks (GRNs) to shift gene expression from undesirable states to desirable ones has received much attention in recent years. Most of the existing methods assume that the cost of intervention at each state and time point, referred to as the immediate cost function, is fully known. In this paper, we employ the Partially-Observed Boolean Dynamical System (POBDS) signal model for a time sequence of noisy expression measurement from a Boolean GRN and develop a Bayesian Inverse Reinforcement Learning (BIRL) approach to address the realistic case in which the only available knowledge regarding the immediate cost function is provided by the sequence of measurements and interventions recorded in an experimental setting by an expert. The Boolean Kalman Smoother (BKS) algorithm is used for optimally mapping the available gene-expression data into a sequence of Boolean states, and then the BIRL method is efficiently combined with the Q-learning algorithm for quantification of the immediate cost function. The performance of the proposed methodology is investigated by applying a state-feedback controller to two GRN models: a melanoma WNT5A Boolean network and a p53-MDM2 negative feedback loop Boolean network, when the cost of the undesirable states, and thus the identity of the undesirable genes, is learned using the proposed methodology.

Gene regulatory network state estimation from arbitrary correlated measurements

BackgroundAdvancements in gene expression technology allow acquiring cheap and abundant data for analyzing cell behavior. However, these technologies produce noisy, and often correlated, measurements on the transcriptional states of genes. The Boolean network model has been shown to be effective in capturing the complex dynamics of gene regulatory networks (GRNs). It is important in many applications, such as anomaly detection and optimal intervention, to be able to track the evolution of the Boolean states of a gene regulatory network using noisy time-series transcriptional measurements, which may be correlated in time.ResultsWe propose efficient estimators for the Boolean states of GRNs using correlated time-series transcriptional measurements, where the nature of the correlation and of the measurements themselves are entirely arbitrary. More specifically, we propose new algorithms based on a hypothesis tree to compute optimal minimum mean square error (MMSE) filtering and smoothing state estimators for a Partially-Observed Boolean Dynamical System (POBDS) with correlated measurements. The algorithms are exact but may be computationally expensive for large state spaces or long time horizons, in which case a process for pruning the hypothesis tree is employed to obtain an approximation of the optimal MMSE estimators, while keeping computation tractable. Performance is assessed through a comprehensive set of numerical experiments based on the p53-MDM2 negative-feedback loop Boolean regulatory network, where the standard Boolean Kalman Filter (BKF) and Boolean Kalman Smoother (BKS) for uncorrelated measurements are compared to the corresponding new estimators for correlated measurements, called BKF-CORR and BKS-CORR, respectively.

BoolFilter: an R package for estimation and identification of partially-observed Boolean dynamical systems

BackgroundGene regulatory networks govern the function of key cellular processes, such as control of the cell cycle, response to stress, DNA repair mechanisms, and more. Boolean networks have been used successfully in modeling gene regulatory networks. In the Boolean network model, the transcriptional state of each gene is represented by 0 (inactive) or 1 (active), and the relationship among genes is represented by logical gates updated at discrete time points. However, the Boolean gene states are never observed directly, but only indirectly and incompletely through noisy measurements based on expression technologies such as cDNA microarrays, RNA-Seq, and cell imaging-based assays. The Partially-Observed Boolean Dynamical System (POBDS) signal model is distinct from other deterministic and stochastic Boolean network models in removing the requirement of a directly observable Boolean state vector and allowing uncertainty in the measurement process, addressing the scenario encountered in practice in transcriptomic analysis.ResultsBoolFilter is an R package that implements the POBDS model and associated algorithms for state and parameter estimation. It allows the user to estimate the Boolean states, network topology, and measurement parameters from time series of transcriptomic data using exact and approximated (particle) filters, as well as simulate the transcriptomic data for a given Boolean network model. Some of its infrastructure, such as the network interface, is the same as in the previously published R package for Boolean Networks BoolNet, which enhances compatibility and user accessibility to the new package.ConclusionsWe introduce the R package BoolFilter for Partially-Observed Boolean Dynamical Systems (POBDS). The BoolFilter package provides a useful toolbox for the bioinformatics community, with state-of-the-art algorithms for simulation of time series transcriptomic data as well as the inverse process of system identification from data obtained with various expression technologies such as cDNA microarrays, RNA-Seq, and cell imaging-based assays.

Particle filters for partially-observed Boolean dynamical systems

Partially-observed Boolean dynamical systems (POBDS) are a general class of nonlinear models with application in estimation and control of Boolean processes based on noisy and incomplete measurements. The optimal minimum mean square error (MMSE) algorithms for POBDS state estimation, namely, the Boolean Kalman filter (BKF) and Boolean Kalman smoother (BKS), are intractable in the case of large systems, due to computational and memory requirements. To address this, we introduce approximate MMSE filtering and smoothing algorithms based on the auxiliary particle filter (APF) method, which are called APF–BKF and APF–BKS, respectively. For joint state and parameter estimation, the APF–BKF is used jointly with maximum-likelihood (ML) methods for simultaneous state and parameter estimation in POBDS models. In the case the unknown parameters are discrete, the proposed ML adaptive filter consists of multiple APF–BKFs running in parallel, in a manner reminiscent of the Multiple Model Adaptive Estimation (MMAE) method in classical linear filtering theory. In the presence of continuous parameters, the proposed ML adaptive filter is based on an efficient particle-based expectation maximization (EM) algorithm for the POBDS model, which is based on a modified Forward Filter Backward Simulation (FFBSi) in combination with the APF–BKS. The performance of the proposed particle-based adaptive filters is assessed through numerical experiments using a POBDS model of the well-known cell cycle gene regulatory network observed through noisy RNA-Seq time series data.

Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems

Partially-observed Boolean dynamical systems (POBDS) are a general class of nonlinear models with application in estimation and control of Boolean processes based on noisy and incomplete measurements. The optimal minimum mean square error (MMSE) algorithms for POBDS state estimation, namely, the Boolean Kalman filter (BKF) and Boolean Kalman smoother (BKS), are intractable in the case of large systems, due to computational and memory requirements. To address this, we propose approximate MMSE filtering and smoothing algorithms based on the auxiliary particle filter (APF) method from sequential Monte-Carlo theory. These algorithms are used jointly with maximum-likelihood (ML) methods for simultaneous state and parameter estimation in POBDS models. In the presence of continuous parameters, ML estimation is performed using the expectation-maximization (EM) algorithm; we develop for this purpose a special smoother which reduces the computational complexity of the EM algorithm. The resulting particle-based adaptive filter is applied to a POBDS model of Boolean gene regulatory networks observed through noisy RNA-Seq time series data, and performance is assessed through a series of numerical experiments using the well-known cell cycle gene regulatory model.