Classification of State Trajectories in Gene Regulatory Networks

Abstract

Gene-expression-based phenotype classification is used for disease diagnosis and prognosis relating to treatment strategies. The present paper considers classification based on sequential measurements of multiple genes using gene regulatory network (GRN) modeling. There are two networks, original and mutated, and observations consist of trajectories of network states. The problem is to classify an observation trajectory as coming from either the original or mutated network. GRNs are modeled via probabilistic Boolean networks, which incorporate stochasticity at both the gene and network levels. Mutation affects the regulatory logic. Classification is based upon observing a trajectory of states of some given length. We characterize the Bayes classifier and find the Bayes error for a general PBN and the special case of a single Boolean network affected by random perturbations (BNp). The Bayes error is related to network sensitivity, meaning the extent of alteration in the steady-state distribution of the original network owing to mutation. Using standard methods to calculate steady-state distributions is cumbersome and sometimes impossible, so we provide an efficient algorithm and approximations. Extensive simulations are performed to study the effects of various factors, including approximation accuracy. We apply the classification procedure to a p53 BNp and a mammalian cell cycle PBN.

Scalable Optimal Bayesian Classification of Single-Cell Trajectories under Regulatory Model Uncertainty

“…where ||.|| 1 denotes L 1 norm, indicating the summation in (10). M θ,c k is the transition matrix of the Markov state process corresponding to model θ ∈ Θ, with entries given by:…”

“…This vector can be either computed exactly as introduced in [10] or approximated by creating multiple Monte-Carlo trajectories with relatively long horizons.…”

“…This is due to the large transition matrix with 2 2n elements required to compute the loglikelihoods, leading to exponential computational and memory complexity. Thus, the key here is to scale up the OBC for single-cell trajectories by reducing both computational and memory complexity when computing (10). We adopt the Sequential Monte-Carlo (SMC) techniques [24,25,26,27,28,29] for estimating nonlinear state-space models, such as our POBDS here.…”

“…In [10] and [11], the maximum-likelihood (ML) based classification of single-cell trajectories has been developed. The method uses the ML-adaptive filter proposed in [7] for estimation of the unknown parameters, followed by the Bayes classifier tuned to the ML parameter estimates.…”

“…whereL θ c (Y 1:T ) denotes the approximation of L θ c (Y 1:T ). Note that the computational complexity of this algorithm is of order O(N T ) which can be much smaller than O(2 2n T ) that is the complexity of computing the exact log-likelihood function in (10).…”