Harissa: Stochastic Simulation and Inference of Gene Regulatory Networks Based on Transcriptional Bursting

Abstract

Gene regulatory networks, as a powerful abstraction for describing complex biological interactions between genes through their expression products within a cell, are often regarded as virtually deterministic dynamical systems. However, this view is now being challenged by the fundamentally stochastic, ‘bursty’ nature of gene expression revealed at the single cell level. We present a Python package called Harissa which is dedicated to simulation and inference of such networks, based upon an underlying stochastic dynamical model driven by the transcriptional bursting phenomenon. As part of this tool, network inference can be interpreted as a calibration procedure for a mechanistic model: once calibrated, the model is able to capture the typical variability of single-cell data without requiring ad hoc external noise, unlike ordinary or even stochastic differential equations frequently used in this context. Therefore, Harissa can be used both as an inference tool, to reconstruct biologically relevant networks from time-course scRNA-seq data, and as a simulation tool, to generate quantitative gene expression profiles in a non-trivial way through gene interactions.

A Appendices

1.1 A.1 Reduced Model

The inference procedure is based on analytical results which are not available for the two-stage ‘mRNA-protein’ model (3). On the other hand, such results exist for a one-stage ‘protein-only’ model that is a valid approximation of the former when proteins are more stable than mRNA (i.e. $d_{0,i}/d_{1,i} \gg 1$). The resulting process $(Z(t))_{t\ge 0} \in \mathbb {R}_+^n$ is also a PDMP, whose master equation can be interpreted in terms of simplified trajectories (Fig. 1B):

$$\begin{aligned} \begin{array}{rll} \displaystyle \frac{\partial }{\partial t}p(z,t) &{} \displaystyle = \sum _{i=1}^n \left[ d_{1,i}\frac{\partial }{\partial z_i}\{z_ip(z,t)\} \right. &{}\\ &{} \qquad \displaystyle + \left. \int _0^{z_i}k_{\text {on},i}(z-he_i) p(z-he_i,t) c_i e^{-c_i h} \textrm{d}h - k_{\text {on},i}(z) p(z,t) \right] . &{} \end{array} \end{aligned}$$

(4)

Given Z(t), mRNA levels X(t) are obtained by sampling independently for every $i\in \{1,\dots ,n\}$ and $t > 0$ from $X_i(t) \sim \textrm{Gamma}(k_{\textrm{on},i}(Z(t))/d_{0,i},b_i$), which is the quasi-steady-state (QSS) distribution of the complete model [5, 6].

1.2 A.2 Inference Algorithm

Now consider mRNA counts measured in m cells, assumed independent, along a time-course experiment following a stimulus. Each cell $k = 1, \dots , m$ is associated with an experimental time point $t_k$. We introduce the following notation:

• $\textbf{x}_k = (x_{ki})\in \{0,1,2,\dots \}^{n}$ : mRNA counts (cell k, gene i);

• $\textbf{z}_k = (z_{ki})\in (0,+\infty )^{n}$ : latent protein levels (cell k, gene i);

• $\alpha = (\alpha _{ij}(t_k))\in \mathbb {R}^{n \times n}$ : effective interaction $i \rightarrow j$ at time $t_k$.

A stimulus is represented as gene $i=0$ and we therefore add parameters $\alpha _{0j}(t_k)$ for $j=1, \dots , n$ and $k=1, \dots , m$. We further set $z_{k0} = 0$ if $t_k \le 0$ (before stimulus) and $z_{k0} = 1$ if $t_k > 0$ (after stimulus). Then, writing $a_i = k_{1,i}/d_{0,i}$, the underlying statistical model of Harissa is defined by

$$\begin{aligned} p(\textbf{z}_k)&= \prod _{i=1}^n {z_{ki}}^{c_i \sigma _{ki} - 1} e^{-c_i z_{ki}} \frac{{c_i}^{c_i\sigma _{ki}}}{{\Gamma }(c_i\sigma _{ki})} \;, \end{aligned}$$

(5)

$$\begin{aligned} p(\textbf{x}_k | \textbf{z}_k)&= \prod _{i=1}^n \frac{1}{x_{ki}!} \frac{{\Gamma }(a_i z_{ki} + x_{ki})}{{\Gamma }(a_i z_{ki})} \frac{{b_i}^{a_i z_{ki}}}{(b_i+1)^{a_i z_{ki} + x_{ki}}} \;, \end{aligned}$$

(6)

with

$$\begin{aligned} \sigma _{ki} = {\left[ 1+\exp (-\{\beta _i + \alpha _{0i}(t_k) z_{k0} + \textstyle \sum _{j=1}^n \alpha _{ji}(t_k) z_{kj}\})\right] }^{-1} . \end{aligned}$$

(7)

Details of this derivation can be found in [4, 5, 8, 14]. Roughly, (5) comes from a ‘Hartree’ approximation of (4), while (6) corresponds to a Poisson distribution with random parameter sampled from the QSS distribution of X(t) given Z(t). Note that $p(\textbf{z}_k)$ is in general only a pseudo-likelihood as $\sigma _{ki}$ depends on $\textbf{z}_k$.

Since the preliminary version of Harissa [5], the global inference procedure has been heavily improved using important identifiability results from [14, 15]. The final algorithm consists of three steps:

• 1. Model calibration: estimate $a_i$ and $b_i$ for each gene individually from (6);

• 2. Bursting mode inference: estimate the frequency mode ($k_{0,i}$ or $k_{1,i}$) for each gene in each cell (can be seen as a binarization step with specific thresholds);

• 3. Network inference: consider $\textbf{z}_k$ as observed from step 2 and maximize (5) with respect to $\alpha$ after adding an appropriate penalization term [14]. Each parameter $\theta _{ij}$ is then set to $\alpha _{ij}(t_k)$ with $t_k$ that maximizes $|\alpha _{ij}(t_k)|$.

1.3 A.3 Repressilator Network

Considering an instance model = NetworkModel(3), the repressilator network simulated in Fig. 3 is defined as follows: