Scalable optimal Bayesian classification of single-cell trajectories under regulatory model uncertainty
Abstract
Background
Single-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.
Results
The 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.
Conclusion
The 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.
Keywords
Optimal Bayesian classification Single-cell trajectory classification Particle filter Probabilistic Boolean networksAbbreviations
- APF-BKF
Auxiliary particle-filter implementations of the Boolean Kalman filter
- BKF
Boolean Kalman filter
- BKS
Boolean Kalman smoother
- BNp
Boolean network with perturbation
- GRN
Gene regulatory network
- IRB
Intrinsically Bayesian robust
- MMSE
Minimum mean square error
- OBC
Optimal Bayesian classifier
- POBDS
Partially observed Boolean dynamical system
- SMC
Sequential Monte-Carlo
- T-LGL
T-cell large granular lymphocyte
Background
A key issue in genomic signal processing is to classify normal versus cancerous cells, different stages of tumor development, or different prospective drug response. Previous gene-expression technologies, such as microarray and RNA-Seq, typically measure the average behavior of tens of thousands of cells [1, 2]. By contrast, the recent advances in next-generation sequencing technologies have allowed in-depth investigation of the transcriptome at a single-cell resolution [3, 4].
Gene regulatory networks (GRNs) govern the functioning of key cellular processes, such as stress response, DNA repair, and other mechanisms involved in complex diseases such as cancer. Often, the relationship among genes can be described by logical rules updated at discrete time intervals with each gene have Boolean states: 0 (OFF) or 1 (ON) [5]. The Partially-Observed Boolean dynamical system (POBDS) model [6, 7, 8] is a rich framework for modeling the behavior of GRNs observed through contemporary gene-expression technologies, as it allows indirect and incomplete observation of gene states. Several tools for the POBDS model have been developed in recent years, such as the optimal filter and smoother based on the Minimum Mean Square Error (MMSE) criterion, termed as the Boolean Kalman filter (BKF) and Boolean Kalman smoother (BKS) [6], respectively.
In [9] and [10], the maximum-likelihood (ML) based classification of single-cell trajectories has been developed. The method uses the ML-adaptive filter proposed in [6] for estimation of the unknown parameters, followed by the Bayes classifier tuned to the ML parameter estimates. The drawback of this method is its inability to use prior knowledge in deriving the classifier. In [11], the intrinsically Bayesian robust (IBR) classifier for the trajectories is developed. This IBR classifier is optimal relative to the prior distribution of unknown parameters.
In this paper, assuming that there are two classes, healthy (c=0) and cancerous (c=1), we derive the optimal Bayesian classifier (OBC) [12, 13] for classification of single-cell trajectories. The difference between the OBC and IBR classifiers is that in the OBC the expectation of the class-conditional densities is taken over the posterior distribution of the unknown parameters [14, 15], whereas in the IBR classifier the expectation is taken over the priors.
Despite the optimality of the developed OBC for single-cell trajectories, its exact computation for large GRNs becomes intractable, due to the large size of the matrices involved. In this paper, we develop a particle-based OBC to scale up the classification of single-cell trajectories. The proposed method contains a bank of Auxiliary Particle-Filter implementations of the Boolean Kalman Filter (APF-BKF) proposed in [16], for both training and test processes.
Our contributions are twofold: 1) Optimal Bayesian Classification: we derive the optimal Bayesian classifiers (OBC) for both single-cell gene expression trajectories and multiple-cell averaged gene expression with uncertain regulatory network prior; and 2) Scalability: the parallel particle filters together with the Monte-Carlo inference have been efficiently used to estimate the likelihood and stationary distributions, making the derived OBC scalable to larger gene regulatory networks.
We apply the APF-BKF-based OBC to classify trajectories of the blood cancer T-cell large granular lymphocyte (T-LGL) leukemia. T-LGL leukemia is a chronic disease characterized by a clonal proliferation of cytotoxic T cells [17]. A Boolean network model of T cell survival signaling in the context of T-LGL leukemia has been constructed by [18] through performing extensive literature search. Then the T-LGL network has been simplified by [17], which constructs the minimum network that preserves the attractor structure of that system. The reduced network contains 18 genes, which has an optimal solution with a transition matrix with 2^{2×18}=68,719,476,736 elements. By contrast, as we will show in our numerical experiments, the proposed APF-BKF-based method captures T-cell dynamics with only 1000 particles.
Methods
Gene regulatory network model
Gene regulatory networks are modeled as partially-observed Boolean dynamical systems (POBDS). The two components of the POBDS model are a state space model that describes the evolution of the dynamics of the GRN, and an observation model for the measurements. These two components are described below.
GRN state space model
where \(\left \{\mathbf {x}^{1},\ldots,\mathbf {x}^{2^{n}}\right \}\) denotes the set of the corresponding network states in the Boolean vector representation.
Observation model
where \(\mathbf {v}_{k} \sim \mathcal {N}(0,\sigma ^{2}I_{n})\) is an uncorrelated zero-mean Gaussian noise vector, λ= [λ_{1},…,λ_{n}]^{T} is a vector of baseline gene expressions corresponding to the “zero” state for each gene, and D=Diag(δ_{1},…,δ_{n}) is a diagonal matrix containing differential expression values for each gene along the diagonal (these indicate by how much the “one” state of each gene is overexpressed over the “zero” state). Such a Gaussian linear model is an appropriate model for single-cell gene-expression data [19, 20].
Optimal Bayesian classifier (OBC) for single-cell trajectories
Assume there are two POBDSs corresponding to the healthy and cancerous (mutated) classes, each having n genes. The difference between the healthy and mutated classes could be the over-expression or disruption of a value of single or multiple genes in the mutated case. Let \(\mathbb {Y}^{c} = \left \{\mathcal {Y}_{c}^{(1)},\mathcal {Y}_{c}^{(2)},\ldots,\mathcal {Y}_{c}^{(D_{c})}\right \}\) be the set of D_{c} observed trajectories from class c, c=0,1. Let Θ=(θ_{1},…,θ_{M}) be the uncertainty set of M network functions containing the unknown true network functions in (1), indicating M possible Boolean functions as \(\left \{\mathbf {f}^{c}_{\theta _{1}},\ldots,\mathbf {f}^{c}_{\theta _{M}}\right \}\) considering the regulatory model uncertainty for the class c. The prior probability of the model θ for the class c is represented by π(θ∣c), where \({\sum \nolimits }_{i=1}^{M} \pi (\theta _{i}\mid c)=1\), for c=0,1. This uncertainty could arise due to some unknown regulations (i.e. interactions) between some genes in the pathway diagram (more information in “Results and discussion” section). We wish to derive the optimal Bayesian classifier (OBC) under uncertainty using all available data and prior knowledge.
If the feature-label distribution is unknown but belongs to an uncertainty class Θ of feature-label distributions, then we desire a classifier to minimize the expected error over the uncertainty class. This expected error is equivalent to the Bayesian minimum mean-square-error estimate [21] given by \(\hat {\epsilon }(\psi) = E_{\theta |\mathbb {Y}^{c},c} [\epsilon (\psi,\theta)]\), where ε(ψ,θ) is the error of ψ on the feature-label distribution parameterized by θ and the expectation is taken over the posterior distribution of parameters \(\pi (\theta \mid \mathbb {Y}^{c}, c), c=0, 1\).
for c=0,1.
The expectation in (9) is taken with respect to the posterior distribution of θ, i.e. \(\pi (\theta |\mathbb {Y}^{c},c)\), as opposed to IBR classifier [11] that considers the prior distribution π(θ|c). Furthermore, \(\log p_{\theta }(\mathbb {Y}^{c}\mid c)={\sum \nolimits }_{d=1}^{D_{c}} \log p\left (\mathcal {Y}_{c}^{(d)}\mid \theta, c\right)\) is used in (9) due to the independency of the training trajectories.
This vector can be either computed exactly as introduced in [9] or approximated by creating multiple Monte-Carlo trajectories with relatively long horizons.
The complexity of computing the log-likelihood function for a single trajectory of length T is of order O(2^{2n}×T) due to the transition matrix involved in its computation. The whole process of the proposed OBC is presented in Algorithm 1.
Scalable classification of single-cell trajectories
In the previous section, the exact solution for the optimal Bayesian classifier is introduced. However, for large systems with a large number of state variables, the exact computation of Algorithm 1 becomes impractical. This is due to the large transition matrix with 2^{2n} elements required to compute the log-likelihoods, 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 [23, 24, 25, 26, 27] for estimating nonlinear state-space models, such as our POBDS here. These techniques approximate the target distribution using sample points (“particles”) drawn from a proposal distribution, taking advantage of the fact that sampling from the proposal distribution is easier than from the target. This helps alleviate the high computation of the exact filter by using a finite set of Monte-Carlo samples. In this paper, we use the Auxiliary Particle Filter implementation of the Boolean Kalman Filter (APF-BKF) proposed in [16] to deal with large GRNs.
Usually only a few particles have significant weights after a few iterations of the algorithm and most particles have negligible weights. APF-BKF is a look-ahead method that predicts the location of particles with a high probability at time k based on the observations at time step k−1, with the purpose of making the subsequent resampling step more efficient. Without the look-ahead, the basic algorithm blindly propagates all particles, even those in low probability regimes.
for i=1,…,N, where we have used (1) and the fact that the noise is zero-mode (i.e., the Bernoulli noise intensity p is smaller than 0.5).
By simulating the index with the probability v_{k,i}=p_{θ}(Y_{k}∣μ_{k,i},c) w_{k−1,i}, we can sample from P_{θ}(X_{k},ζ_{k} ∣ Y_{1:k},c) and then sample from the transition density given the mixture, P_{θ}(X_{k}∣x_{k−1,i},c).
where Cat (a_{1},…,a_{N}) represents a categorical distribution with the probability mass function \(f(\zeta = i) = a_{i}/{\sum \nolimits }_{j=1}^{N} a_{j}\).
where \(\hat {L}_{c}^{\theta }(\mathbf {Y}_{1:T})\) denotes the approximation of \(L_{c}^{\theta }(\mathbf {Y}_{1:T})\). Note that the computational complexity of this algorithm is of order O(NT) which can be much smaller than O(2^{2n}T) that is the complexity of computing the exact log-likelihood function in (10).
The training process of the proposed method has the computational complexity O(2NMTD_{0}D_{1}), whereas the exact solution has the complexity O(2^{2n+1}MTD_{0}D_{1}). The exponential growth of the complexity with the size of network (i.e., number of genes) for the exact solution precludes its application to large GRNs. However, the number of particles, N, by the proposed method can be chosen relatively small according to the attractor structure of the system (i.e., N<<2^{2n}) [29], allowing the classification of large-scale single-cell trajectories (see “Results and discussion” section). The complexity of the test process for the proposed method is O(2NMT), as opposed to O(2^{2n+1}MT) for the optimal solution.
Optimal Bayesian classifier for multiple-cell scenarios
where “ ∘” is the Hadamard product. Thus, the Boolean nature of the state vector suggests that each element of the multiple-cell measurement is distributed as a mixture of two Gaussian distributions. Replacing (27) and (28) into (26) leads to the OBC for multiple-cell scenarios. Comprehensive comparison results between the OBC in single-cell trajectories and multiple-cells are provided in the next section.
Results and discussion
Node | Regulating function |
---|---|
CTLA4 | TCR ∧¬ Apoptosis |
TCR | ¬ (CTLA4 ∨ Apoptosis) |
CREB | IFNG ∧¬ Apoptosis |
IFNG | ¬ (SMAD ∨ P2 ∨ Apoptosis) |
P2 | (IFNG ∨ P2) ∧¬ Apoptosis |
GPCR | S1P ∧¬ Apoptosis |
SMAD | GPCR ∧¬ Apoptosis |
Fas | ¬ (sFas ∨ Apoptosis) |
sFas | S1P ∧¬ Apoptosis |
Ceramide | Fas ∧¬ (S1P or Apoptosis) |
DISC | (Ceramide ∨ (Fas ∧¬ FLIP)) ∧¬ Apoptosis |
Caspase | ((BID ∧¬ IAP) ∨ DISC) ∧¬ Apoptosis |
FLIP | ¬ (DISC ∨ Apoptosis) |
BID | ¬ (MCL1 ∨ Apoptosis) |
IAP | ¬ (BID ∨ Apoptosis) |
MCL1 | ¬ (DISC ∨ Apoptosis) |
S1P | ¬ (Ceramide ∨ Apoptosis) |
Apoptosis | Caspase ∨ Apoptosis |
As we may not know the true network function, we consider four candidate network functions for each of the healthy and mutated networks as the uncertainty class of possible GRN models. In addition to the true network, we remove the operation ¬ of Apoptosis for the genes sFas and GPCR, which are intermediate nodes. Therefore, this network is very close to the true network. For the third network, we remove ¬ of Apoptosis from two other nodes IAP and P2. In the fourth network of this uncertainty class, we change the operation AND to OR for the gene BID. In this study, we use the observation models described in Eqs. (3) and (25) for the single-cell trajectory and multiple-cell averaging, respectively.
Trajectory based classification results for high-noise scenario (σ=25)
(p=0.05) | (p=0.1) | |||
---|---|---|---|---|
Method | T=3 | T=7 | T=3 | T=7 |
Plug-In | 0.1777 | 0.0922 | 0.2524 | 0.1774 |
IBR | 0.1384 | 0.0723 | 0.1800 | 0.0750 |
OBC | 0 . 1 1 7 3 | 0 . 0 6 7 4 | 0 . 1 6 4 3 | 0 . 0 6 4 6 |
The proposed method does not have any restriction on the noise distribution assumptions due to the generalizability of particle filters. To show this, we also test our method with different noise distributions. While the noise of GRNs is usually Gaussian, sometimes due to up regulation, noise can be Poisson or Negative Binomial (NB). We have simulated Gaussian and NB noise distributions with the same mean and variance while for Poisson noise, the variance is equal to its mean (Details can be found in Additional file 1, the additional experimental results). Additional file 1: Figure S1 shows that our method performs consistently well for all observation noises. The Poisson results are superior to the other models as its variance is smaller.
We also compare the performance of APF with the plain Sequential Importance Resampling (SIR) in high process noise. As Additional file 1: Figure S2 shows, the performance is similar especially when there is enough time points. When the number of time points is low, the SIR-based particle filter performs worse. The superior performance of APF in real-world GRNs is due to the fact that the size of attractors is usually small and the predicted mode values by APF can be a good approximation for the next prediction. Moreover, the initial distribution is assumed to be from the stationary distribution. This makes APF a more desirable approximation solutions due to the lower diversity in the particles.
Conclusions
In this paper, we have developed the optimal Bayesian classifier for binary classification of single-cell trajectories under regulatory model uncertainty. The partially-observed Boolean dynamical system is used for modeling the dynamical behavior of gene regulatory networks. Due to the intractability of the OBC for large GRNs, we have proposed a particle filtering technique for approximating the OBC. This particle-based solution reduces the computational and memory complexity of the optimal solution significantly. The 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 based on noisy time-series gene-expression data.
Notes
Acknowledgment
We thank Texas A&M High Performance Research Computing and Texas Advanced Computing Center for providing computational resources to perform experiments in this work.
Funding
This research has been supported in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Mathematical Multifaceted Integrated Capability Centers (MMICCS) program, under award number DE-SC0019393, NSF Awards CCF-1553281, CCF-1718513 and CCF-1718924, and a grant from the Brookhaven National Laboratory (BNL). Publication costs are funded by DE-SC0019393.
Availability of data and materials
Not applicable.
About this supplement
This article has been published as part of BMC Genomics Volume 20 Supplement 6, 2019: Selected original research articles from the Fifth International Workshop on Computational Network Biology: Modeling, Analysis and Control (CNB-MAC 2018): Genomics. The full contents of the supplement are available online at https://bmcgenomics.biomedcentral.com/articles/supplements/volume-20-supplement-6.
Authors’ contributions
E. H. developed OBC for the single-cell trajectory classification, developed the scalable OBC, performed the experiments, and wrote the manuscript. M. I. wrote part of the code and the first draft, and in conjunction with U. B. N. structured the APF-BKF by integrating their previous partially-observed Boolean dynamical system into this new framework. E. R. D. and X.Q. oversaw the project, proposed the new scalable OBC, and wrote the manuscript. All authors have read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary material
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