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Intrinsically Bayesian robust classifier for single-cell gene expression trajectories in gene regulatory networks

  • 1Email author,
  • 1, 2 and
  • 1, 2
BMC Systems Biology201812 (Suppl 3) :23

https://doi.org/10.1186/s12918-018-0549-y

  • Published:

Abstract

Background

Expression-based phenotype classification using either microarray or RNA-Seq measurements suffers from a lack of specificity because pathway timing is not revealed and expressions are averaged across groups of cells. This paper studies expression-based classification under the assumption that single-cell measurements are sampled at a sufficient rate to detect regulatory timing. Thus, observations are expression trajectories. In effect, classification is performed on data generated by an underlying gene regulatory network.

Results

Network regulation is modeled via a Boolean network with perturbation, regulation not fully determined owing to inherent biological randomness. The binary assumption is not critical because the resulting Markov chain characterizes expression trajectories. We assume a partially known Gaussian observation model belonging to an uncertainty class of models. We derive the intrinsically Bayesian robust classifier to discriminate between wild-type and mutated networks based on expression trajectories. The classifier minimizes the expected error across the uncertainty class relative to the prior distribution. We test it using a mammalian cell-cycle model, discriminating between the normal network and one in which gene p27 is mutated, thereby producing a cancerous phenotype. Tests examine all model aspects, including trajectory length, perturbation probability, and the hyperparameters governing the prior distribution over the uncertainty class.

Conclusions

Simulations show the rates at which the expected error is diminished by smaller perturbation probability, longer trajectories, and hyperparameters that tighten the prior distribution relative to the unknown true network. For average-expression measurement, methods have been proposed to obtain prior distributions. These should be extended to the more mathematically difficult, but more informative, expression trajectories.

Keywords

  • Intrinsically Bayesian robust classifier
  • Optimal Bayesian classifier
  • Bayesian trajectory classifier
  • Single-cell expression trajectory
  • Gene regulatory network
  • Boolean network
  • Bayesian partially observed Boolean dynamical system

Background

Phenotype classification is a salient issue for translational genomics, for instance, classification of normal versus cancerous cells, of different stages of tumor development, or different prospective drug response. Both expression microarray and RNA-Seq measurements have received much interest; however, they suffer from a lack of specificity. First, pathway timing is not reflected in the data and, second, expressions are averaged across groups of cells, so that individual cell responses are not detectable. New technologies are being developed for profiling individual cells using RNA-Seq or quantitative PCR [1]. Individual cells can be captured via standard methods, such as flow cytometry, glass capillaries, or laser [2], and be measured at various time points.

Genes have interactions with each other, which can determine how they are behaving over time and define the dynamics of gene regulatory networks (GRNs). One way of showing the dynamics of GRNs over discrete time points is Boolean networks with perturbation (BNp) [3]. A BNp is a Markov chain, in which the state of a gene (0 for off and 1 for on) at the current time is a function of the states of its predictor genes at the previous time plus a small random Boolean noise. Network inference could be done using RNA-Seq time series measurements [4].

Suppose we have single-cell measurements sampled with a sufficient rate to detect regulatory timing. In effect, this would mean that classification would be done on data reflecting an underlying gene regulatory network. In [5] we proposed a classifier to classify the state trajectories of the two classes: wild-type and mutated, each having its own BNp. We derived the Bayes classifier and computed the Bayes error. We analyzed the effects of the length of the trajectories, perturbation probability, and different mutations on the Bayes error.

In [6] and [7] we assumed an observation model on the state dynamics of the BNps, from which the expression values of the genes are obtained. As the parameters of the model were all unknown and the network functions were partially known, we proposed an expectation-maximization (EM)-based algorithm to estimate these parameters and functions, and then plugged them into the Bayes classifier. In [6] we assumed that the expression trajectory data come from single-cell measurements and compared that with a multiple-cell scenario, in which instead of trajectories we have the averaged expressions of all genes over all cells, which translates to an average over all states.

In this paper we extend the single-cell trajectory classification to the Bayesian framework. We propose the intrinsically Bayesian robust (IBR) classifier for the trajectorires. The IBR classifier is a specific type of the obtimal Bayesian classifier (OBC), first introduced in [8, 9] for the classification of static data. In fact, the difference between the OBC and IBR classifiers is that in the OBC the expectation of the class-conditional densities is taken over the posteriors of the parameters to obtain the effective class-conditional densities, whereas in the IBR classifier the expectation is taken over the priors. The IBR/OBC concept has been applied to linear and morphological filtering [10, 11], and IBR Kalman filtering [12]. Regarding the prior distributions, prior construction methods using the pathway knowledge have been studied in the literature, such as [13, 14]. Bayesian methods for finding differentially expressed genes can be used for the aim of classification [15].

Here we apply the IBR classifier to the classification of trajectories. As opposed to [6, 7], where we estimate the parameters, here we assume that the parameters belong to an uncertainty class governed by a prior distribution. We assume that there are two classes: wild-type (S=0) and mutated (S=1). We introduce a Bayesian version of the partially observed Boolean dynamical system (POBDS), proposed in [16], as the observation model. We use a beta prior distribution for the prior probability of the class S=0 and also for the gene perturbation probability. Since the observation model given the states is Gaussian, we employ the normal-gamma distribution as the prior distribution of the mean and precision (inverse of variance) of the Gaussian model.

In the simulation part, we employ a mammalian cell-cycle gene regulatory network [17] consisting of 10 genes as the BNp for the class S=0 and its mutated version as the BNp for the class S=1. We analyze the effects of all the hyperparameters of the model and the length of the observed trajectories on the classification error. The proposed classifier is computationally efficient because we use the sum-product method to reduce the complexity to m×2n, where m is the length of the observed trajectory and n is the number of genes in the network. Being linearly dependent on time points, m, makes the classifier very fast even for longer trajectories.

Methods

In a Boolean network (BN) for n genes, each gene value xi{0,1}, for i=1,,n, at time k+1 is determined by the values of some predictor genes at time k via a Boolean function fi:{0,1}n→{0,1}. In practice, fi is a function of a small number of genes, Ki, called the in-degree of the gene xi in the network. The in-degree of the network is K= maxi=1,,nKi. A gene network can be represented as a graph with vertices representing genes and edges representing regulations. There is a state diagram of 2n states corresponding to the truth table of the BN, representing the dynamics of the network. Given an initial state, a BN will eventually reach a set of states, called an attractor cycle, through which it will cycle endlessly. Each initial state corresponds to a unique attractor cycle, and the set of initial states leading to a specific attractor cycle is known as the basin of attraction (BOA) of the attractor cycle.

State model

We allow stochasticity in our state model by using Boolean networks with perturbation (BNps) instead of deterministic BNs. For BNps, perturbation is introduced with a probability pk by which the state of a gene in the network can be randomly flipped at time k. We assume that there is an independent identically distributed (i.i.d.) random perturbation vector at each time k, denoted by nk{0,1}n, such that the i-th gene flips at time k if the i-th component of nk is equal to 1. Therefore, the dynamical model can be expressed as
$$ \mathbf{X}_{k+1}=\mathbf{f}(\mathbf{X}_{k})\oplus \mathbf{n}_{k+1}, \quad k=0,1,2,\cdots, $$
(1)

where Xk=[x1(k),x2(k),,xn(k)]T is a binary state vector, called a gene activity profile (GAP), at time k, in which xi(k) indicates the expression level of the i-th gene at time k (either 0 or 1); f=[f1,f2,,fn]T:{0,1}n→{0,1}n is the vector of the network functions, in which fi shows the expression level of the i-th gene at time k+1 when the system lies in the state Xk at time k; nk=[n1(k),n2(k),,nn(k)]T is the perturbation vector at time k, in which n1(k),n2(k),,nn(k) are i.i.d. Bernoulli random variables for every k with the parameter pk=P(ni(k)=1) for i=1,,n; and is component-wise modulo 2 addition.

The existence of perturbation makes the corresponding Markov chain of a BNp irreducible. Hence, the network possesses a steady-state distribution π describing its long-run behavior. If pk is sufficiently small, π will reflect the attractor structure within the original network. We can derive the transition probability matrix (TPM) if we know the truth table and the perturbation probability of a BNp. As a result, the steady-state distribution π can be computed as well.

Prior for state parameter

In this paper, we assume that we know the underlying Boolean networks for both the wild-type and mutated classes, and the only uncertain parameter at the state level is the perturbation probability pk. Since 0<pk<1, we can employ the beta prior for pk, for all k=1,2,, as
$$ g(p_{k}) \sim \text{Beta}(a,b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} p_{k}^{a-1} (1-p_{k})^{b-1}, $$
(2)
where a and b are known parameters. Since in reality pk is close to zero, we can choose a and b in such a way that this fact is satisfied. To this end, we can use the mean and variance of pk:
$$ \mathrm{E}\left[p_{k}\right]=\frac{a}{a+b}, \quad \text{Var}\left[p_{k}\right] = \frac{ab}{(a+b)^{2}(a+b+1)}. $$
(3)

Observation model

We define a Bayesian partially-observed Boolean dynamical system (BPOBDS) as the model for the gene expression data. In this model, we assume that the gene expressions come from Gaussian distributions whose parameters are governed by prior distributions whose parameters (the hyperparameters of the observations) are a function of the hidden Boolean states. If yj(k) is the expression value of the j-th gene at time k, then the observation model is
$$ p\left(y_{j}(k)|\theta_{j}(k),\lambda_{j}(k)\right) \sim \mathcal{N}\left(\theta_{j}(k),\lambda_{j}(k)^{-1}\right), $$
(4)

for j=1,2,,n and k=1,2,, where θj(k) and λj(k) denote the mean and precision, respectively, of the Gaussian distribution.

Priors for observation parameters

We employ the well-known normal-gamma prior distribution for θj(k) and λj(k):
$$\begin{array}{@{}rcl@{}} p(\lambda_{j}(k)) &\sim& \text{Gamma}(\alpha_{0},\beta_{0}), \\ p(\theta_{j}(k)| \lambda_{j}(k), x_{j}(k)) &\sim& \mathcal{N} \left(\mu_{j}(k),(\kappa_{0} \lambda_{j}(k))^{-1}\right), \\ \text{where} ~~~ \mu_{j}(k) &=& \mu_{0} + \delta_{0} x_{j}(k), \end{array} $$
(5)

where α0, β0, κ0, μ0, and δ0 are known positive hyperparameters, and xj(k) is the hidden Boolean state of gene j at time k. The intuition behind the prior (5) is that when gene j at time k is on or off, that is, xj(k)=1 or 0, the hyper-mean of the expression for that gene is μj(k)=μ0+δ0 or μj(k)=μ0, respectively, at time k. In (5), μ0 is the baseline expression level and δ0 is the expression coefficient. The hyperparameters α0, β0, and κ0 determine the level of uncertainty, by which we can control the variance of the outputs. We assume the same values of hyperparameters for all genes at all times.

IBR classifier

If one knows the feature-label distribution, then the error of any classifier can be found and an optimal (Bayes) classifier minimizes classifier error. 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. Given a classifier ψ, from the perspective of mean-square error (MSE), the best error estimate minimizes the MSE between its true error (a function of parameter θ) and an error estimate. This Bayesian minimum-mean-square-error (MMSE) estimate is given by the expected true error, \(\widehat {\varepsilon }(\psi)=\mathrm {E}_{\theta }\left [\varepsilon (\psi,\mathbf {\theta })\right ]\), where ε(ψ,θ) is the error of ψ on the feature-label distribution parameterized by θ and the expectation is taken relative to the prior distribution π(θ) [18].

An IBR classifier minimizes the Bayesian MMSE estimate. If ψ(x)=0 if xR0 and ψ(x)=1 if xR1, where x is a multidimensional vector of data, and R0 and R1 partition the sample space, then [8]
$${} \widehat{\varepsilon }\left(\psi \right) =\mathrm{E}_{\pi}[\!c]\int_{R_{1}}f_{\Theta}\left(\mathbf{x}|0\right) d\mathbf{x}+\left(1-\mathrm{E}_{\pi }[\!c]\right)\int_{R_{0}}f_{\Theta}\left(\mathbf{x}|1\right) d\mathbf{x}, $$
where
$$f_{\Theta}\left(\mathbf{x}|y\right) =\int_{\mathbf{\Theta }_{y}}f_{\mathbf{\theta}_{y}}\left(\mathbf{x}|y\right) \pi \left(\mathbf{\theta}_{y}\right) d\mathbf{\theta }_{y} $$
is the effective class-conditional density for class y, Θy being the space for θy, \(f_{\mathbf {\theta }_{y}}\left (\mathbf {x}|y\right)\) is the class-conditional density, and c is the prior probability of the class 0. The IBR classifier is given by [8]
$$ \psi_{\text{IBR}}\left(\mathbf{x}\right) =\left\{ \begin{array}{ll} 0 \quad \text{if ~~ }\mathrm{E}_{\pi }[\!c]f_{\Theta}\left(\mathbf{x}|0\right) \geq \\ \qquad \quad \left(1-\mathrm{E}_{\pi }[\!c]\right)f_{\Theta}\left(\mathbf{x}|1\right) & \\ 1 \quad \text{otherwise} \end{array} \right.. $$
(6)

Trajectory-based IBR classifier

Let Θs=[p2:m,θ1:n(1:m),λ1:n(1:m)] denote the parameters of the class S=s, for s=0,1, where p2:m means the parameters p2,p3,,pm, and similarly for θ1:n(1:m) and λ1:n(1:m). Furthermore, let fs denote the Boolean network function of the class S=s and \(\mathcal {X}=\left [\mathbf {X}_{1},\mathbf {X}_{2},\cdots,\mathbf {X}_{m}\right ]\) denote the Boolean state trajectory at m consecutive times points at which \(\mathcal {Y}\) has been observed. The n×1 Boolean vector Xk=[x1(k),x2(k),,xn(k)]T has the states of the n genes at time k, which are hidden and not observed.

Suppose that we obtain the expressions of n genes at m consecutive time points. Let Yk=[y1(k),,yn(k)]T denote the n×1 expression vector of n genes at time k, and \(\mathcal {Y}=\left [\mathbf {Y}_{1},\cdots,\mathbf {Y}_{m}\right ]\) denote a time trajectory of length m, containing the expression vectors at the m consecutive times. The problem is to optimally classify this observed trajectory \(\mathcal {Y}\) to the class 0 (wild-type) or class 1 (mutated). Let c and 1−c be the prior probabilities of the class S=0 and class S=1, respectively. Since we are uncertain about c, we use a beta prior
$$ g(c) \sim \text{Beta}\left(a_{c},b_{c}\right) = \frac{\Gamma\left(a_{c}+b_{c}\right)}{\Gamma(a_{c})\Gamma(b_{c})} c^{a_{c}-1} (1-c)^{b_{c}-1}, $$
(7)
with mean
$$ \mathrm{E}[\!c]=\frac{a_{c}}{a_{c}+b_{c}}, $$
(8)

where ac and bc are known parameters.

According to (6), the IBR classifier for the trajectories is
$$ \psi_{\text{IBR}}(\mathcal{Y})=\left\{ \begin{array}{ll} 0 \quad \text{if} ~~ \mathrm{E}[\!c]p(\mathcal{Y}|S=0)\geq \\ \hspace{1cm} (1-\mathrm{E}[\!c])p(\mathcal{Y}|S=1) & \\ 1 \quad \text{otherwise} \end{array},\right. $$
(9)

where \(p(\mathcal {Y}|S=s)\) is the effective class-conditional density of the trajectory \(\mathcal {Y}\) in the class S=s for s=0,1.

Effective class-conditional densities of trajectories

The joint distribution of \(\mathcal {Y}\), \(\mathcal {X}\), and Θs given the class S=s can be factorized as
$$ \begin{aligned} p\left(\mathcal{Y},\mathcal{X},\Theta_{s} | S=s\right) &= g(p_{2:m}) P\left(\mathcal{X}|p_{2:m}, S=s\right) \\ & \quad \times p\left(\mathcal{Y}| \theta_{1:n}(1:m),\lambda_{1:n}(1:m)\right) \\ & \quad \times p\left(\theta_{1:n}(1:m), \lambda_{1:n}(1:m)|\mathcal{X}\right). \end{aligned} $$
(10)
In deriving (10), it is assumed that the state parameters {p2:m} are independent of the observation parameters {θ1:n(1:m),λ1:n(1:m)}. Due to the independence assumption in the priors,
$$ \begin{aligned} g(p_{2:m}) &= \prod_{k=1}^{m-1} g(p_{k+1}) \\ &= \prod_{k=1}^{m-1} \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} p_{k+1}^{a-1} (1-p_{k+1})^{b-1}, \end{aligned} $$
(11)
$$ \begin{aligned} &p\left(\theta_{1:n}(1:m), \lambda_{1:n}(1:m)|\mathcal{X}\right) \\ &= \prod_{k=1}^{m} \prod_{j=1}^{n} p(\theta_{j}(k) | \lambda_{j}(k),x_{j}(k)) p(\lambda_{j}(k))\\ &= \prod_{k=1}^{m} \prod_{j=1}^{n} \frac{1}{\sqrt{2\pi (\kappa_{0} \lambda_{j}(k))^{-1}}} \frac{\beta_{0}^{\alpha_{0}}}{\Gamma(\alpha_{0})} \\ & \times \exp \left(-\frac{(\theta_{j}(k)-\mu_{j}(k))^{2}}{2(\kappa_{0} \lambda_{j}(k))^{-1}}\right) \lambda_{j}(k)^{\alpha_{0}-1} \exp(-\beta_{0} \lambda_{j}(k)). \end{aligned} $$
(12)
If we assume that the conditional expressions, given the parameters, of the n genes at m time points are independent, the likelihood of \(\mathcal {Y}\) given the parameters in (10) can be written as
$$ \begin{aligned} &p\left(\mathcal{Y}| \theta_{1:n}(1:m),\lambda_{1:n}(1:m)\right) \\ &= \prod_{k=1}^{m}\prod_{j=1}^{n} p\left(y_{j}(k) | \theta_{j}(k),\lambda_{j}(k)\right)\\ &= \prod_{k=1}^{m}\prod_{j=1}^{n} \frac{1}{\sqrt{2\pi \lambda_{j}(k)^{-1}}} \exp \left(-\frac{(y_{j}(k)-\theta_{j}(k))^{2}} {2 \lambda_{j}(k)^{-1}} \right). \end{aligned} $$
(13)
We should note that the independency assumption in (13) is only for the observations, whereas the genes have interactions at the state level, following the underlying Boolean network, so that they cannot be considered independent. Due to the Markov property in (1), \(p\left (\mathcal {X}|p_{2:m},S=s\right)\) in (10) can be factored as
$$ \begin{aligned} P(\mathcal{X} | p_{2:m}, S&=s) = \\ P(\mathbf{X}_{1}| S&=s) \prod_{k=1}^{m-1} P(\mathbf{X}_{k+1} | \mathbf{X}_{k}, p_{k+1}, S=s), \end{aligned} $$
(14)
where P(Xk+1|Xk,pk+1,S=s) is the probability of transitioning from state Xk at time k to state Xk+1 at time k+1, given the perturbation probability pk+1, in the class S=s, and P(X1|S=s) is the probability of the first state X1 in the class S=s. Let xi denote the n×1 Boolean vector of the state i, for i=1,2,,2n. Given the perturbation probability pk+1, the conditional transition probability matrix (TPM) at time k+1, which is a 2n×2n matrix, in the class S=s can be derived from (1) as
$$ \begin{aligned} \mathbf{A}^{(s)}_{i,j}(k+1) &= P\left(\mathbf{X}_{k+1}=\mathbf{x}^{j} | \mathbf{X}_{k}=\mathbf{x}^{i}, p_{k+1}, S=s\right)\\ &= p_{k+1}^{\mathbf{d}\left(\mathbf{x}^{j},\mathbf{f}_{s}\left(\mathbf{x}^{i}\right)\right)}(1-p_{k+1})^{n-\mathbf{d}\left(\mathbf{x}^{j},\mathbf{f}_{s}\left(\mathbf{x}^{i}\right)\right)}, \end{aligned} $$
(15)

where d(xj,fs(xi)) is the Hamming distance between the two Boolean vectors xj and fs(xi).

For obtaining \(p(\mathcal {Y} | S=s)\), we need to integrate out the joint distribution \(p(\mathcal {Y},\mathcal {X},\Theta _{s} | S=s)\) in (10) with respect to Θs and \(\mathcal {X}\):
$$ {{\begin{aligned} p\left(\mathcal{Y} | S=s\right) &= \sum_{\mathcal{X}} \int_{\Theta_{s}} p\left(\mathcal{Y},\mathcal{X},\Theta_{s} | S=s\right) \\ &= \sum_{\mathcal{X}} \left\lbrace P(\mathbf{X}_{1} | S=s) \prod_{k=1}^{m-1} \int_{p_{k+1}} g(p_{k+1}) p_{k+1}^{\mathbf{d}\left(\mathbf{X}_{k+1},\mathbf{f}_{s}\left(\mathbf{X}_{k}\right)\right)} \right. \\ & \quad\times (1-p_{k+1})^{n-\mathbf{d}\left(\mathbf{X}_{k+1},\mathbf{f}_{s}\left(\mathbf{X}_{k}\right)\right)} d p_{k+1} \\ & \quad\times \prod_{k=1}^{m} \prod_{j=1}^{n} \int_{\theta_{j}(k)} \int_{\lambda_{j}(k)} p\left(y_{j}(k) | \theta_{j}(k), \lambda_{j}(k)\right) \\ & \quad\left. \times p\left(\theta_{j}(k) | \lambda_{j}(k),x_{j}(k)\right) ~ p\left(\lambda_{j}(k)\right) ~ d\theta_{j}(k) d \lambda_{j}(k) \right\rbrace. \end{aligned}}} $$
(16)

Fortunately, as the priors are conjugate, we can analytically solve the integrals in (16). We will use the following lemmas to do so.

Lemma 1

Let P=(0 1) be the domain of pk+1. The following equation holds:
$$ {}\begin{aligned} K_{1} &\triangleq \int_{\mathbf{P}} g(p_{k+1}) p_{k+1}^{\mathbf{d}(\mathbf{X}_{k+1},\mathbf{f}_{s}(\mathbf{X}_{k}))} \\ & \quad \times (1-p_{k+1})^{n-\mathbf{d}(\mathbf{X}_{k+1},\mathbf{f}_{s}(\mathbf{X}_{k}))} d p_{k+1} \\ &= \frac{ \Gamma(\mathbf{d}(\mathbf{X}_{k+1},\mathbf{f}_{s}(\mathbf{X}_{k})) + a) \Gamma(n - \mathbf{d}(\mathbf{X}_{k+1},\mathbf{f}_{s}(\mathbf{X}_{k})) + b)}{\Gamma(a) \Gamma(b) \Gamma(a+b+n)\Gamma(a+b)^{-1}}. \end{aligned} $$
(17)

Proof

See Appendix 1 in Additional file 1. □

We assume that the observations \(\mathcal {Y}\)occur in the steady state. As a result, in (16), P(X1|S=s) is the steady-state probability of the state X1 in the class S=s, for s=0,1. The following lemma gives the steady-state distribution.

Lemma 2

Let \(\pi ^{(s)}_{i}=P\left (\mathbf {X}_{1}=\mathbf {x}^{i} | S=s\right)\) denote the steady-state probability of the i-th state in the class S=s, and \(\pi ^{(s)}=\left [\pi ^{(s)}_{1},\cdots,\pi ^{(s)}_{2^{n}}\right ]\) be the 1×2n vector of the steady-state distribution. Then π(s) can be calculated from
$$ \pi^{(s)} = \pi^{(s)} \mathbf{M}^{(s)}, \quad \sum_{i=1}^{2^{n}} \pi_{i}^{(s)}=1, $$
(18)
where M(s) is the transition probability matrix of the class S=s with the entries
$${} \mathbf{M}^{(s)}_{i,j} = \frac{ \Gamma\left(\mathbf{d}\left(\mathbf{x}^{j},\mathbf{f}_{s}\left(\mathbf{x}^{i}\right)\right) + a\right) \Gamma\left(n - \mathbf{d}\left(\mathbf{x}^{j},\mathbf{f}_{s}\left(\mathbf{x}^{i}\right)\right) + b\right)}{\Gamma(a) \Gamma(b) \Gamma(a+b+n) \Gamma(a+b)^{-1}}. $$
(19)

Proof

See Appendix 2 in Additional file 1. □

Lemma 3

Let Ω=(− ) and Λ=(0 ) be the domains of θj(k) and λj(k), respectively, for j=1,,n, and k=1,,m. The following equation holds:
$$ \begin{aligned} K_{2} & \triangleq \int_{\mathbf{\Omega}} \int_{\mathbf{\Lambda}} p(y_{j}(k) | \theta_{j}(k), \lambda_{j}(k)) \\ & \quad \times p(\theta_{j}(k) | \lambda_{j}(k),x_{j}(k)) ~ p(\lambda_{j}(k)) ~ d\theta_{j}(k) d \lambda_{j}(k) \\ & =\frac{1}{(2\pi)^{\frac{1}{2}}} \left(\frac{\kappa_{0}}{\kappa_{1}}\right)^{\frac{1}{2}} \frac{\Gamma(\alpha_{1})}{\Gamma(\alpha_{0})} \frac{\beta_{0}^{\alpha_{0}}}{\beta_{1}^{\alpha_{1}}}, \end{aligned} $$
(20)
where
$$ \begin{aligned} \kappa_{1} &= \kappa_{0} + 1, \\ \alpha_{1} &= \alpha_{0}+\frac{1}{2}, \\ \beta_{1} &= \beta_{0} + \frac{\kappa_{0} \left(y_{j}(k)-\mu_{0} - \delta_{0} x_{j}(k)\right)^{2}}{2(\kappa_{0} + 1)}. \end{aligned} $$
(21)

Proof

See Appendix 3 in Additional file 1. □

Summing out \(\mathcal {X}\)

Using (16), (17), (19), and (20), we have
$$ {{\begin{aligned} p(\mathcal{Y}|S=s) &= \sum_{\mathbf{X}_{1}} \cdots \sum_{\mathbf{X}_{m}} \left\lbrace{\vphantom{\left. \quad \times \prod_{k=1}^{m} \prod_{j=1}^{n} \frac{1}{(2\pi)^{\frac{1}{2}}} \left(\frac{\kappa_{0}}{\kappa_{1}}\right)^{\frac{1}{2}} \frac{\Gamma(\alpha_{1})}{\Gamma(\alpha_{0})} \frac{\beta_{0}^{\alpha_{0}}}{\beta_{1}^{\alpha_{1}}} \right\rbrace}} \pi_{\mathbf{X}_{1}}^{(s)} \prod_{k=1}^{m-1} \frac{ \Gamma(\mathbf{d}(\mathbf{X}_{k+1},\mathbf{f}_{s}(\mathbf{X}_{k})) + a) \Gamma(n - \mathbf{d}(\mathbf{X}_{k+1},\mathbf{f}_{s}(\mathbf{X}_{k})) + b)}{\Gamma(a) \Gamma(b) \Gamma(a+b+n) \Gamma(a+b)^{-1}} \right.\\ &\left. \quad \times \prod_{k=1}^{m} \prod_{j=1}^{n} \frac{1}{(2\pi)^{\frac{1}{2}}} \left(\frac{\kappa_{0}}{\kappa_{1}}\right)^{\frac{1}{2}} \frac{\Gamma(\alpha_{1})}{\Gamma(\alpha_{0})} \frac{\beta_{0}^{\alpha_{0}}}{\beta_{1}^{\alpha_{1}}} \right\rbrace. \end{aligned}}} $$
(22)
We use the sum-product algorithm [19] to efficiently compute the summation in (22). Define the 2n×1 vector Φ(k) by
$${} \begin{aligned} \Phi_{i}(k) &= \left(\frac{ \beta_{0}^{\alpha_{0}}}{(2\pi)^{\frac{1}{2}}} \left(\frac{\kappa_{0}}{\kappa_{0}+1}\right)^{\frac{1}{2}} \frac{\Gamma\left(\alpha_{0}+\frac{1}{2}\right)}{\Gamma(\alpha_{0})}\right)^{n} \\ & \quad \times \prod_{j=1}^{n} \left[\beta_{0} + \frac{\kappa_{0}\left(y_{j}(k)-\mu_{0}-\delta_{0}\mathbf{x}^{i}_{j}\right)^{2}}{2(\kappa_{0}+1)} \right]^{-\left(\alpha_{0}+\frac{1}{2}\right)}, \end{aligned} $$
(23)
for i=1,,2n, where \(\mathbf {x}^{i}_{j}\) is the j-th entry of the Boolean state xi. We define an auxiliary 2n×1 vector Π(s)(k) at the time k, for k=1,,m, which is initialized and updated as follows:
$$\begin{array}{@{}rcl@{}} \Pi^{(s)}(1) &=& \left({\pi}^{(s)}\right)^{T} \circ \Phi(1), \\ \Pi^{(s)}(k+1) &=& \left[{\mathbf{M}^{(s)}}^{T} \Pi^{(s)}(k)\right] \circ \Phi(k+1), \end{array} $$
(24)
for k=1,,m−1, where T denotes the transpose, and is the Hadamard product. Once we have calculated Π(s)(m), the summation of (22) is equal to the l1 norm Π(s)(m)1, which is the summation of the all 2n entries of Π(s)(m). Therefore, (22) can be written as
$$ p(\mathcal{Y}|S=s) = \parallel \Pi^{(s)}(m) \parallel_{1}. $$
(25)

Results and discussion

In this section, we consider a mammalian cell-cycle gene regulatory network [17], depicted in Fig. 1, for evaluating our proposed trajectory-based IBR classifier. This GRN consists of n=10 genes, whose interactions are shown in Fig. 1. The Boolean functions of the corresponding Boolean network for this GRN are given in Table 1 [17]. We define class S=0 as the wild-type class, whose network function f0 is in Table 1. The value of the gene CycD is determined by extracellular signals, and we assume it is f1=0. According to [17], one mutated case which leads to cancer is when the gene p27 in the network is shut down and cannot be activated by its regulating genes, that is, f3=0. Therefore, we define class S=1 as the mutated class with the network function f1, which is the same as Table 1 with f3=0. We analyze the effects of the hyperparameters and m on the classification error in Figs. 2, 3, 4, 5, 6, 7, 8 and 9. In the simulations, we have set ac=bc=10 and μ0=10. Therefore, the prior probability c of the class S=0 will have the mean value E[ c]=0.5.
Fig. 1
Fig. 1

Mammalian cell-cycle gene regulatory network

Fig. 2
Fig. 2

Classifier error versus m in cell-cycle network

Fig. 3
Fig. 3

Classifier error versus m in cell-cycle network

Fig. 4
Fig. 4

Classifier error versus α0 in cell-cycle network

Fig. 5
Fig. 5

Classifier error versus β0 in cell-cycle network

Fig. 6
Fig. 6

Classifier error versus a in cell-cycle network

Fig. 7
Fig. 7

Classifier error versus b in cell-cycle network

Fig. 8
Fig. 8

Classifier error versus κ0 in cell-cycle network

Fig. 9
Fig. 9

Classifier error versus δ0 in cell-cycle network

Table 1

Definitions of Boolean functions for the wild-type mammalian cell-cycle BN with 10 genes

Order

Gene

Regulating function

x 1

CycD

f1= Extracellular signals

x 2

Rb

\(f_{2}=(\overline {CycD} \wedge \overline {CycE} \wedge \overline { CycA} \wedge \overline {CycB}) \vee (p27 \wedge \overline {CycD} \wedge \overline {CycB})\)

x 3

p27

\(f_{3}=(\overline {CycD} \wedge \overline {CycE} \wedge \overline {CycA} \wedge \overline {CycB}) \vee (p27 \wedge \overline {(CycE \wedge CycA)} \wedge \overline {CycD} \wedge \overline {CycB})\)

x 4

E2F

\(f_{4}=(\overline {Rb} \wedge \overline {CycA} \wedge \overline { CycB}) \vee (p27 \wedge \overline {Rb} \wedge \overline {CycB})\)

x 5

CycE

\(f_{5}=(E2F \wedge \overline {Rb})\)

x 6

CycA

\(f_{6}=(E2F \wedge \overline {Rb} \wedge \overline {Cdc20} \wedge \overline {(Cdh1 \wedge UbcH10)}) \vee (CycA \wedge \overline {Rb} \wedge \overline {Cdc20} \wedge \overline {(Cdh1 \wedge UbcH10)})\)

x 7

Cdc20

f7=CycB

x 8

Cdh1

\(f_{8}=(\overline {CycA} \wedge \overline {CycB}) \vee Cdc20 \vee (p27 \wedge \overline {CycB})\)

x 9

UbcH10

\(f_{9}=\overline {Cdh1} \vee (Cdh1 \wedge UbcH10 \wedge (Cdc20 \vee CycA \vee CycB))\)

x 10

CycB

\(f_{10}=(\overline {Cdc20} \wedge \overline {Cdh1})\)

Figure 2 shows the classification error versus m for a=1,3,5, when the values of the other hyperparameters are b=100, α0=100, β0=104, δ0=40, and κ0=100. We see that the classification error decreases by increasing m and converges to zero. This means that for long enough trajectories we can have perfect classification. The value of the hyperparameter a determines the amount of uncertainty for the gene perturbation probability pk at time k. From a biological perspective, we know that pk should be small. As a result, we have chosen b=100, and a=1,3,5, leading to the mean values of pk, respectively, as \(\mathrm {E}\left [p_{k}\right ]=\frac {a}{a+b}\approx 0.01, 0.03, 0.05\). For a given b, the bigger value of a allows a wider range of pk, which results in higher classification error.

Figure 3 represents the classification error versus m for different values of α0 and β0, when a=1, b=100, δ0=40, and κ=100. As the precision (inverse of variance) has a Gamma(α0,β0) distribution, its mean and variance are equal to \(\mathrm {E}\left [\lambda \right ]=\frac {\alpha _{0}}{\beta _{0}}\) and \(\text {Var}\left [\lambda \right ]=\frac {\alpha _{0}}{\beta _{0}^{2}}=\frac {\mathrm {E}[\lambda ]}{\beta _{0}}\). Figure 3 shows the error curves for the two cases \(\frac {\alpha _{0}}{\beta _{0}}=10^{-3}\) and 2×10−3, each having a different value of β0, leading to a different variance of λ. As such, we can see the effects of both the mean and variance of the precision λ. Whenever \(\frac {\alpha _{0}}{\beta _{0}}\) increases, there is lower variance in the outputs, which results in lower errors. We also notice from Fig. 3 that for a given value of \(\frac {\alpha _{0}}{\beta _{0}}\), increasing β0 decreases the error, the reason being that increased β0 yields lower variance for the precision.

Figure 4 plots the error versus α0 for a=1,3,5, when m=5, b=100, δ0=40, κ0=100, and β0=105. For all values of a, the classification error is a decreasing function of α0. Similarly, Fig. 5 gives the error versus β0 for a=1,3,5, when α0=10 and the other hyperparameters are the same as in Fig. 4. Figure 5 shows an increasing trend of classification error as β0 grows.

Figure 6 shows the error as a function of a, when m=5, b=100, δ0=40, κ0=100, α0=100, and β0=105. When a grows, the uncertainty of the perturbation probability grows as well. The error is an increasing function of a. Similarly, Fig. 7 is for error versus b, when a=1, and the others are the same as in Fig. 6. In Fig. 7 the classification error is decreasing as b increases.

Figure 8 illustrates the error versus κ0, for m=5, a=1, b=100, δ0=40, α0=100, and β0=105. From (5), the hyperparameter κ0 controls the variance of the mean parameters θj(k). When κ0 increases, the uncertainty of θj(k) is reduced and its density peaks at μ0 and μ0+δ0 for the unexpressed and expressed states, respectively. Consequently, we expect better error rates for higher κ0. Accordingly, in Fig. 8 the classification error is a decreasing function of κ0.

Figure 9 illustrates the error versus δ0, the expression coefficient, for m=5, a=1, b=100, κ0=100, α0=100, and β0=105. Having larger δ0 means that the mean values of data for each gene in the unexpressed and expressed cases are well separated, which leads to lower classification error. As expected, in Fig. 9 the error is decreasing as δ0 gets larger.

Conclusions

In this paper, we propose a trajectory-based intrinsically Bayesian robust classifier for classification of single-cell gene-expression trajectories. We assume that the expressions of the n genes, whose interactions are known in terms of a Boolean network, are observed in m consecutive time points, for both the wild-type class (S=0) and mutated class (S=1). As the parameters have uncertainty, we assign priors for them. We assume a beta distribution as a prior for both the probability of the class S=0 and the gene perturbation probabilities at each time. We assume a normal-gamma distribution for the mean and precision of the expressions at each time and for each gene, given the underlying states. As such, we derive closed-form solutions for the effective class-conditional densities of the trajectories in each class, by which we define the IBR classifier. The performance of the IBR classifier is evaluated in a cell-cycle gene regulatory network with 10 genes, for which we know the Boolean networks of the two classes. We have analyzed the effects of m and all the hyperparameters on the classification error.

In this paper we have assumed the form of the prior distributions, including the hyperparameter values. Once this is done, the analysis is mathematical: derive the effective class-conditional densities. As is generally the case with Bayesian methods, two issues arise: how are the priors developed and how robust are the methods to the prior assumptions? Both issues have been extensively studied in regard to the OBC, of which the IBR classifier is the special case in which there is no update to a posterior.

In [9] robustness of the OBC relative to various modeling assumptions has been investigated, including false assumptions on the covariance structure and falsely assuming Gaussianity. In [20], similar robustness issues have been considered for the MMSE error estimator, which is required for defining the OBC.

Regarding prior construction, the question is how to transform existing scientific knowledge into a prior distribution. Because the IBR classifier and other IBR filtering methods involve prior distributions on the underlying scientific model, for instance, the stochastic process (power spectra) in Wiener filtering and the feature-label distribution in classification, uncertainty arises from insufficient scientific knowledge, the prior characterizes that uncertainty, and, in effect, constrains the optimization relative to that uncertainty. For IBR/OBC classification in the context of uncertain gene regulatory networks, prior construction methods based on constrained optimization have been developed for both Gaussian [21] and discrete [22] gene regulatory networks under the assumption of static observations. Future work involves extending these prior-construction methods to trajectories, which should be possible, albeit, with more difficult mathematical analysis and substantially more computation.

Another issue to be addressed in future work is extending the trajectory-based IBR classifier to optimal Bayesian classification, which will require analytic representation of the effective class-conditional densities relative to a posterior distribution derived from the prior distribution utilizing sample data.

Abbreviations

BN: 

Boolean network

BNp: 

Boolean network with perturbation

BOA: 

Basin of attraction

BPOBDS: 

Bayesian partially observed Boolean dynamical system

EM: 

Expectation maximization

GRN: 

Gene regulatory network

IBR: 

Intrinsically Bayesian robust

MMSE: 

Minimum mean square error

MSE: 

Mean square error

OBC: 

Optimal Bayesian classifier

TPM: 

Transition probability matrix

Declarations

Acknowledgments

Not applicable.

Funding

Publication cost of this article was funded by the Robert M. Kennedy 26 Chair at Texas A&M University.

Availability of data and materials

Not applicable.

About this supplement

This article has been published as part of BMC Systems Biology Volume 12 Supplement 3, 2018: Selected original research articles from the Fourth International Workshop on Computational Network Biology: Modeling, Analysis, and Control (CNB-MAC 2017): systems biology. The full contents of the supplement are available online at https://bmcsystbiol.biomedcentral.com/articles/supplements/volume-12-supplement-3.

Authors’ contributions

AK mathematically derived the classifier and worked out the example. UB with ED conceived the modeling and edited the original draft. ED structured the classifier, while UB conceived the modeling, and edited the original draft. All authors 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.

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Authors’ Affiliations

(1)
Department of Electrical and Computer Engineering, Texas A&M University, College Station, 77843, TX, USA
(2)
Center for Bioinformatics and Genomic Systems Engineering, College Station, 77843, TX, USA

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