The chemical master equation (CME) is frequently used in systems biology to quantify the effects of stochastic fluctuations that arise due to biomolecular species with low copy numbers. The CME is a system of ordinary differential equations that describes the evolution of probability density for each population vector in the state-space of the stochastic reaction dynamics. For many examples of interest, this state-space is infinite, making it difficult to obtain exact solutions of the CME. To deal with this problem, the Finite State Projection (FSP) algorithm was developed by Munsky and Khammash [J. Chem. Phys. 124(4), 044104 (2006)], to provide approximate solutions to the CME by truncating the state-space. The FSP works well for finite time-periods but it cannot be used for estimating the stationary solutions of CMEs, which are often of interest in systems biology. The aim of this paper is to develop a version of FSP which we refer to as the stationary FSP (sFSP) that allows one to obtain accurate approximations of the stationary solutions of a CME by solving a finite linear-algebraic system that yields the stationary distribution of a continuous-time Markov chain over the truncated state-space. We derive bounds for the approximation error incurred by sFSP and we establish that under certain stability conditions, these errors can be made arbitrarily small by appropriately expanding the truncated state-space. We provide several examples to illustrate our sFSP method and demonstrate its efficiency in estimating the stationary distributions. In particular, we show that using a quantized tensor-train implementation of our sFSP method, problems admitting more than 100 × 10 states can be efficiently solved.

Combining single‐cell measurements of ERK activity dynamics with perturbations provides insights into the MAPK network topology. We built circuits consisting of an optogenetic actuator to activate MAPK signaling and an ERK biosensor to measure single‐cell ERK dynamics. This allowed us to conduct RNAi screens to investigate the role of 50 MAPK proteins in ERK dynamics. We found that the MAPK network is robust against most node perturbations. We observed that the ERK‐RAF and the ERK‐RSK2‐SOS negative feedback operate simultaneously to regulate ERK dynamics. Bypassing the RSK2‐mediated feedback, either by direct optogenetic activation of RAS, or by RSK2 perturbation, sensitized ERK dynamics to further perturbations. Similarly, targeting this feedback in a human ErbB2‐dependent oncogenic signaling model increased the efficiency of a MEK inhibitor. The RSK2‐mediated feedback is thus important for the ability of the MAPK network to produce consistent ERK outputs, and its perturbation can enhance the efficiency of MAPK inhibitors.

Stimulation of PC-12 cells with epidermal (EGF) versus nerve (NGF) growth factors (GFs) biases the distribution between transient and sustained single-cell ERK activity states, and between proliferation and differentiation fates within a cell population. We report that fibroblast GF (FGF2) evokes a distinct behavior that consists of a gradually changing population distribution of transient/sustained ERK signaling states in response to increasing inputs in a dose response. Temporally-controlled GF perturbations of MAPK signaling dynamics applied using microfluidics reveals that this wider mix of ERK states emerges through the combination of an intracellular feedback, and competition of FGF2 binding to FGF receptors (FGFR) and heparan-sulfate proteoglycans (HSPGs) coreceptors. We show that the latter experimental modality is instructive for model selection using a Bayesian parameter inference. Our results provide novel insights into how different receptor tyrosine kinase (RTK) systems differentially wire the MAPK network to fine tune fate decisions at the cell population-level.

Measurements of single-cell ERK activity dynamics provide unique insights in the MAPK network topology. We built genetic circuits consisting of optogenetic actuators activating ERK from different nodes within the MAPK network together with an ERK biosensor to measure single-cell ERK dynamics. Evaluating ERK dynamics induced by different temporal optogenetic inputs, in response to a large number of perturbations, shows that the MAPK network is robust to downregulation of most of its nodes. This robustness emerges in part because of the ERK-RSK2-SOS negative feedback. Bypassing this feedback, by direct activation of the RAS/RAF/MEK/ERK submodule, or by RSK2 perturbation, breaks MAPK network robustness. Targeting the RSK2-mediated feedback in a ErbB2-dependent oncogenic signaling model greatly sensitizes ERK to MEK inhibition, allowing efficient ERK activity shutdown within a cell population. Thus, the RSK2-mediated negative feedback is a weak node of the MAPK network whose perturbation enables potent inhibition of ERK.

The development of mechanistic models of biological systems is a central part of Systems Biology. One major challenge in developing these models is the accurate inference of model parameters. In recent years, nested sampling methods have gained increased attention in the Systems Biology community due to the fact that they are parallelizable and provide error estimates with no additional computations. One drawback that severely limits the usability of these methods, however, is that they require the likelihood function to be available, and thus cannot be applied to systems with intractable likelihoods, such as stochastic models. Here we present a likelihood-free nested sampling method for parameter inference which overcomes these drawbacks. This method gives an unbiased estimator of the Bayesian evidence as well as samples from the posterior. We derive a lower bound on the estimators variance which we use to formulate a novel termination criterion for nested sampling. The presented method enables not only the reliable inference of the posterior of parameters for stochastic systems of a size and complexity that is challenging for traditional methods, but it also provides an estimate of the obtained variance. We illustrate our approach by applying it to several realistically sized models with simulated data as well as recently published biological data. We also compare our developed method with the two most popular other likelioodfree approaches: pMCMC and ABC-SMC. The C++ code of the proposed methods, together with test data, is available at the github web page https://github.com/Mijan/LFNS_paper.