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Compartmentalised biochemical reactions are a ubiquitous building block of biological systems. The interplay between chemical and compartmental dynamics can drive rich and complex dynamical behaviors that are difficult to analyse mathematically -- especially in the presence of stochasticity. We have recently proposed an effective moment equation approach to study the statistical properties of compartmentalised biochemical systems. So far, however, this approach is limited to polynomial rate laws and moreover, it relies on suitable moment closure approximations, which can be difficult to find in practice. In this work we propose a systematic method to derive closed moment dynamics for compartmentalised biochemical systems. We show that for the considered class of systems, the moment equations involve expectations over functions that factorize into two parts, one depending on the molecular content of the compartments and one depending on the compartment number distribution. Our method exploits this structure and approximates each function with suitable polynomial expansions, leading to a closed system of moment equations. We demonstrate the method using three systems inspired by cell populations and organelle networks and study its accuracy across different dynamical regimes.

We develop numerical and analytical approaches to calculate mutual information between complete paths of two molecular components embedded into a larger reaction network. In particular, we focus on a continuous-time Markov chain formalism, frequently used to describe intracellular processes involving lowly abundant molecular species. Previously, we have shown how the path mutual information can be calculated for such systems when two molecular components interact directly with one another with no intermediate molecular components being present. In this work, we generalize this approach to biochemical networks involving an arbitrary number of molecular components. We present an efficient Monte Carlo method as well as an analytical approximation to calculate the path mutual information and show how it can be decomposed into a pair of transfer entropies that capture the causal flow of information between two network components. We apply our methodology to study information transfer in a simple three-node feedforward network, as well as a more complex positive feedback system that switches stochastically between two metastable modes.

The design and implementation of regulation motifs ensuring robust perfect adaptation are challenging problems in synthetic biology. Indeed, the design of high-yield robust metabolic pathways producing, for instance, drug precursors and biofuels, could be easily imagined to rely on such a control strategy in order to optimize production levels and reduce production costs, despite the presence of environmental disturbance and model uncertainty. We propose here a motif that ensures tracking and robust perfect adaptation for the controlled reaction network through integral feedback. Its metabolic load on the host is fully tunable and can be made arbitrarily close to the constitutive limit, the universal minimal metabolic load of all possible controllers. A DNA implementation of the controller network is finally provided. Computer simulations using realistic parameters demonstrate the good agreement between the DNA implementation and the ideal controller dynamics., Comment: 31 pages, 8 figures

Compartmentalization of biochemical processes underlies all biological systems, from the organelle to the tissue scale. Theoretical models to study the interplay between noisy reaction dynamics and compartmentalization are sparse, and typically very challenging to analyze computationally. Recent studies have made progress towards addressing this problem in the context of concrete biological systems but general approaches remain lacking. In this work we propose a mathematical framework based on counting processes that allows us to study compartment populations with arbitrary interactions and internal biochemistry. We provide an efficient description of the population dynamics in terms of differential equations which capture moments of the population and their variability. We demonstrate the relevance of our approach using several case studies inspired by biological systems at different scales., Comment: 9 pages, 3 figures, appendix included

Many systems in biology and beyond employ collaborative, collective communication strategies for improved efficiency and adaptive benefit. One such paradigm of particular interest is the community estimation of a dynamic signal, when, for example, an epithelial tissue of cells must decide whether to react to a given dynamic external concentration of stress signaling molecules. At the level of dynamic cellular communication, however, it remains unknown what effect, if any, arises from communication beyond the mean field level. What are the limits and benefits to communication across a network of neighbor interactions? What is the role of Poissonian vs. super Poissonian dynamics in such a setting? How does the particular topology of connections impact the collective estimation and that of the individual participating cells? In this letter we construct a robust and general framework of signal estimation over continuous time Markov chains in order to address and answer these questions. Our results show that in the case of Possonian estimators, the communication solely enhances convergence speed of the Mean Squared Error (MSE) of the estimators to their steady-state values while leaving these values unchanged. However, in the super-Poissonian regime, MSE of estimators significantly decreases by increasing the number of neighbors. Surprisingly, in this case, the clustering coefficient of an estimator does not enhance its MSE while reducing total MSE of the population.

Cells can utilize chemical communication to exchange information and coordinate their behavior in the presence of noise. Communication can reduce noise to shape a collective response, or amplify noise to generate distinct phenotypic subpopulations. Here we discuss a moment-based approach to study how cell-cell communication affects noise in biochemical networks that arises from both intrinsic and extrinsic sources. We derive a system of approximate differential equations that captures lower-order moments of a population of cells, which communicate by secreting and sensing a diffusing molecule. Since the number of obtained equations grows combinatorially with number of considered cells, we employ a previously proposed model reduction technique, which exploits symmetries in the underlying moment dynamics. Importantly, the number of equations obtained in this way is independent of the number of considered cells such that the method scales to arbitrary population sizes. Based on this approach, we study how cell-cell communication affects population variability in several biochemical networks. Moreover, we analyze the accuracy and computational efficiency of the moment-based approximation by comparing it with moments obtained from stochastic simulations., Comment: 6 pages, 5 Figures

Living cells encode and transmit information in the temporal dynamics of biochemical components. Gaining a detailed understanding of the input-output relationship in biological systems therefore requires quantitative measures that capture the interdependence between complete time trajectories of biochemical components. Mutual information provides such a measure but its calculation in the context of stochastic reaction networks is associated with mathematical challenges. Here we show how to estimate the mutual information between complete paths of two molecular species that interact with each other through biochemical reactions. We demonstrate our approach using three simple case studies., Comment: 6 pages, 2 figures

In order to function reliably, synthetic molecular circuits require mechanisms that allow them to adapt to environmental disturbances. Least mean squares (LMS) schemes, such as commonly encountered in signal processing and control, provide a powerful means to accomplish that goal. In this paper we show how the traditional LMS algorithm can be implemented at the molecular level using only a few elementary biomolecular reactions. We demonstrate our approach using several simulation studies and discuss its relevance to synthetic biology., Comment: Molecular circuits, synthetic biology, least mean squares estimator, adaptive systems

With the advance of experimental techniques such as time-lapse fluorescence microscopy, the availability of single-cell trajectory data has vastly increased, and so has the demand for computational methods suitable for parameter inference with this type of data. However, most of the currently available methods treat single-cell trajectories independently, ignoring the mother-daughter relationships and the information provided by population structure. This information is however essential if a process of interest happens at cell division, or if it evolves slowly compared to the duration of the cell cycle. In this work, we highlight the importance of tracking cell lineage trees and propose a Bayesian framework for parameter inference on tree-structured data. Our method relies on a combination of Sequential Monte Carlo for likelihood approximation and Markov Chain Monte Carlo for parameter sampling. We demonstrate the capabilities of our inference framework on two simple examples in which the lineage tree information is necessary: one in which the cell phenotype can only switch at cell division and another where the cell type fluctuates randomly over timescales that extend well beyond the lifetime of a single cell.

The dynamics of stochastic reaction networks within cells are inevitably modulated by factors considered extrinsic to the network such as for instance the fluctuations in ribsome copy numbers for a gene regulatory network. While several recent studies demonstrate the importance of accounting for such extrinsic components, the resulting models are typically hard to analyze. In this work we develop a general mathematical framework that allows to uncouple the network from its dynamic environment by incorporating only the environment's effect onto the network into a new model. More technically, we show how such fluctuating extrinsic components (e.g., chemical species) can be marginalized in order to obtain this decoupled model. We derive its corresponding process- and master equations and show how stochastic simulations can be performed. Using several case studies, we demonstrate the significance of the approach. For instance, we exemplarily formulate and solve a marginal master equation describing the protein translation and degradation in a fluctuating environment., Comment: 7 pages, 4 figures, Appendix attached as SI.pdf, under submission

Markovian population models are suitable abstractions to describe well-mixed interacting particle systems in situation where stochastic fluctuations are significant due to the involvement of low copy particles. In molecular biology, measurements on the single-cell level attest to this stochasticity and one is tempted to interpret such measurements across an isogenic cell population as different sample paths of one and the same Markov model. Over recent years evidence built up against this interpretation due to the presence of cell-to-cell variability stemming from factors other than intrinsic fluctuations. To account for this extrinsic variability, Markovian models in random environments need to be considered and a key emerging question is how to perform inference for such models. We model extrinsic variability by a random parametrization of all propensity functions. To detect which of those propensities have significant variability, we lay out a sparse learning procedure captured by a hierarchical Bayesian model whose evidence function is iteratively maximized using a variational Bayesian expectation-maximization algorithm., Comment: 6 pages, 5 figures, under review

Mathematical methods together with measurements of single-cell dynamics provide unprecedented means to reconstruct intracellular processes that are only partly or indirectly accessible experimentally. To obtain reliable reconstructions the pooling of measurements from several cells of a clonal population is mandatory. The population's considerable cell-to-cell variability originating from diverse sources poses novel computational challenges for process reconstruction. We introduce an exact Bayesian inference framework that properly accounts for the population heterogeneity but also retains scalability with respect to the number of pooled cells. The key ingredient is a stochastic process that captures the heterogeneous kinetics of a population. The method allows to infer inaccessible molecular states, kinetic parameters, compute Bayes factors and to dissect intrinsic, extrinsic and technical contributions to the variability in the data. We also show how additional single-cell readouts such as morphological features can be included into the analysis. We then reconstruct the expression dynamics of a gene under an inducible GAL1 promoter in yeast from time-lapse microscopy data. Based on Bayesian model selection the data yields no evidence of a refractory period for this promoter., Comment: 9 pages, 5 figures, peer-reviewed/edited journal version available at Nature Methods: http://www.nature.com/nmeth/journal/vaop/ncurrent/abs/nmeth.2794.html