Inferring the skeleton cell cycle regulatory network of malaria parasite using comparative genomic and variational Bayesian approaches

Abstract

The development of new antimalarial drugs is urgently needed due to elevated drug resistance in the causative agents Plasmodium parasites. An intervention strategy based on the interruption of the parasite cell cycle could be undertaken using a systems-biology aided drug discovery approach. However, little is known about the components or the mechanism of parasite cell cycle control to date. In this proof of concept study, we attempted to infer the skeleton components using comparative genomic analysis and to uncover the genetic regulatory network (GRN) ab initio using a Variational Bayesian expectation maximization (VBEM) approach.

Appendix

Conjugate priors of parameters and latent variables

We choose the conjugate priors for the parameters and latent variables and they are

$$ \begin{aligned} p({\mathbf{b}}_{i})&={\mathcal{N}}({\mathbf{b}}_{i} \vert{\varvec{\upmu}}_{0},{\mathbf{C}}_{0})\\ p({\varvec{\theta}}_{i})&=p({\mathbf{w}}_{i}, \sigma_{i}^{2})= p({\mathbf{w}}_{i}|\sigma_{i}^{2})p(\sigma_{i}^{2})\\ &={\mathcal{N}}({\mathbf{w}}_{i}|{\varvec{\upmu}}_{{\mathbf{w}}_{i}},\sigma_{i}^{2}{\mathbf{I}}_{G})I{\mathcal{G}}\left(\frac{\nu_{0}}{2}, \frac{\gamma_{0}}{2}\right)\\ \end{aligned} $$

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where ${\varvec{\upmu}}_{0}$ and C ₀ are the mean and the covariance of the prior probability density p(b _i ) and they will be discussed in a later section. In general, ${\varvec{\upmu}}_{{{\mathbf{w}}_i}}$ is simply set as a zero vector, and meanwhile ν₀ and γ₀ are set equal to small positive real values. These priors satisfy the conditions for conjugate exponential (CE) models, which will facilitate the derivation of VBE and VEM steps.

Derivation of the VBE and VBM steps

Starting from some arbitrary distribution over the parameters, the VBE step calculates the approximation on the APPs of latent variables. By applying the CE model, q(b _i ) can be shown to have the following expression

$$ q({\mathbf{b}}_{i})={\mathcal{N}}\left({\mathbf{b}}_{i}|{\varvec{\upmu}}_{{\mathbf{b}}_{i}},{\varvec{\Upsigma}}_{{\mathbf{b}}_i}\right) $$

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where

$$ \begin{aligned} {\varvec{\upmu}}_{{\mathbf{b}}_{i}}&=\varvec{\Upsigma}_{{\mathbf{b}}_{i}}(\sigma_{0}^{-2}{\varvec{\upmu}}_{0}+\mathbf {f})\\ \varvec {\Upsigma}_{{\mathbf{b}}_{i}}&=(\sigma_{0}^{-2}I_{G}+D)^{-1}\\ \end{aligned} $$

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with

$$ \begin{aligned} \mathbf {D}&={\mathbf{B}}\otimes\left[({\mathbf{m}}_{{\mathbf{w}}_{i}}{\mathbf{m}}_{{\mathbf{w}}_{i}})^{\rm T} \langle\sigma_{i}^{-2}\rangle_{q({\varvec{\theta}}_{i})}+\mathbf {A}^{-1}\right]\\ \mathbf {f}^{\rm T}&={\mathbf{y}}_{i}^{\rm T}\mathbf {R}\,{\rm diag}\,(\mathbf {m}_{{\mathbf{w}}_{i}})\langle \sigma_{i}^{-2}\rangle_{q({\varvec{\theta}}_{i})}\\ {\mathbf{B}}&={\mathbf{R}}^{\rm T}{\mathbf {R}}\\ {\mathbf {A}}&={\mathbf {I}}_{G}+{\mathbf {K}}\\ {\mathbf {K}}&={\mathbf{B}}\otimes({\varvec {\Upsigma}}_{{\mathbf{b}}_{i}}+{\varvec{\upmu}}_{{\mathbf b}_{i}}{\varvec{\upmu}}_{{\mathbf{b}}_{i}}^{\rm T})\\ \end{aligned} $$

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We now turn to the VBM step in which we compute $q({\varvec{\theta}}_{i}).$ Again, from the CE model, we obtain

$$ q({\varvec{\theta}}_{i})={\mathcal{N}}({\mathbf{w}}_{i}|{\mathbf{m}}_{{\mathbf {w}}_{i}},{\varvec{\Upsigma}}_{{\mathbf{w}}_{i}})I{\mathcal{G}}\left(\frac{\alpha}{2},\frac{\beta}{2}\right) $$

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where

$$ \begin{aligned} {\mathbf{m}}_{{\mathbf{w}}_i}&=\left({\mathbf {I}}_G+\mathbf {K}\right)^{-1}\left({\mathbf {y}}_i^{\rm T}{\mathbf{RM}}_x\right)^{\rm T}\\ {\varvec{\Upsigma}}_{{\mathbf {w}}_i}&=\sigma_i^2 \left({\mathbf {I}}_G+\mathbf {K}\right)^{-1} \\ \frac{\alpha}{2}=&\frac{N\left({\eta +1}\right)-2}{2} \\ \frac{\beta}{2}=&-\frac{c}{2}\\ \end{aligned} $$

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with

$$ \begin{aligned} {\mathbf{M}}_x&={\rm diag}\left({{\mathbf{m}}_{{\mathbf{b}}_i}}\right) \\ c=&{\mathbf{y}}_i^{\rm T} {\mathbf{RM}}_x {\mathbf{m}}_{{\mathbf{w}}_i}-{\mathbf{y}}_i^{\rm T} {\mathbf{y}}_i-\nu_0 \\ \end{aligned} $$

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Computation of the lower bound ${{\mathcal{F}}}$

The convergence of the VBEM algorithm is tested using a lower bound of ln p(y _i ). In this paper, we use ${{\mathcal{F}}}$ to denote this lower bound and we calculate it using the newest q(b _i ) and q(${\varvec{\theta}}_{i}$) obtained in the iterative process. ${{\mathcal{F}}}$ can be written more succinctly using the definition of the KL divergence. Let’s first review the definition of the KL divergence and then derive an analytical expression for ${{\mathcal{F}}}.$

The KL divergence measures the difference between two probability distributions and it is also termed relative entropy. Thus, using this definition we can write the difference between the real and the approximate distributions in the following way:

$$ \begin{aligned} \left[q\left({\mathbf{b}}_{i}\right)\parallel p\left({\mathbf{b}}_{i},{\mathbf{y}}_i| {\varvec{\theta}}_{i}\right)\right]&=-\int {\rm d}{\mathbf{b}}_i\; q\left({{\mathbf{b}}_i}\right)\ln \frac{p\left({\mathbf{b}}_i,{\mathbf{y}}_i|\varvec {\theta}_i\right)} {q\left({\mathbf{b}}_i\right)}\\ KL\left[q\left({\varvec {\theta}_i}\right)\parallel p\left({\varvec{\theta}}_i\right)\right]&= -\int {{\rm d}{\varvec {\theta}}}_i \ln \frac{p\left({{\varvec{\theta}}_i} \right)} {q\left({\varvec {\theta}_i}\right)}\\ \end{aligned} $$

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And finally, the lower bound ${\mathcal{F}}$ can be written in terms of the previous definitions as:

$$ \begin{aligned} {\mathcal{F}}&=\int {\rm d} {\varvec{\theta}}_i q\left({\varvec{\theta}}_i \right)\left[\int {{\rm d}{\bf b}_i}q\left({\mathbf{b}}_i \right) \ln \frac{p\left({\mathbf{b}}_i,{\mathbf{y}}_i|{\varvec{\theta}}_i \right)}{q\left({\mathbf{b}}_i\right)}+ \ln \frac{p\left({\varvec{\theta}}_i \right)}{q\left({\varvec{\theta}}_i \right)}\right]\\ &=-\int {{\rm d} {\varvec{\theta}}_i}q\left({\varvec{\theta}}_i\right)KL\left[q\left({\bf b}_i \right) \parallel p\left({\mathbf{b}}_i,{\mathbf{y}}_i|{\varvec{\theta}}_i \right)\right]-KL\left[q\left({\varvec{\theta}}_i \right)|{p\left({\varvec{\theta}}_i \right)}\right]\\ \end{aligned} $$

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