Chemical Reaction Networks

We propose a flexible stochastic block model for multi-layer networks, where layer-specific hidden Markov-chain processes drive the changes in the edge clustering. The changes in block membership in a given layer may be influenced by the lagged membership on the rest of the layers. This allows to identify clustering overlap, clustering decoupling, or more complex relationships between layers including settings of unidirectional or bidirectional Granger-block causality. We cope with the overparameterization issue of a saturated specification by assuming a Multi-Laplacian prior distribution. Data augmentation and Gibbs sampling are used to make the inference problem more tractable. Through simulations we show that the standard BVAR models are not able to detect the Granger-block causality under the great majority of scenarios and we present an application to exemplify the use of DSBMM finding new evidence of unidirectional causality from the block structure of the FTA network on the non-observable trade barriers network structure for 159 countries in the period 1995-2017.

We propose a novel method to solve a chemical diffusion master equation of birth and death type. This is an infinite system of Fokker-Planck equations where the different components are coupled by reaction dynamics similar in form to a chemical master equation. This system was proposed in a recent paper for modelling the probabilistic evolution of chemical reaction kinteics associated with spatial diffusion of individual particles. Using some basic tools and ideas from infinite dimensional Gaussian analysis we are able to reformulate the aforementioned infinite system of Fokker-Planck equations as a single evolution equation solved by a generalized stochastic process and written in terms of Malliavin derivatives and differential second quantization operators. Via this alternative representation we link certain finite dimensional projections of the solution of the original problem to the solution of a single partial differential equations of Ornstein-Uhlenbeck type containing as many variables as the dimension of the aforementioned projection space. Our approach resembles and to some extent generalizes the generating function method for solving the chemical master equation.

A Chemical Reaction Network (CRN) is a set of chemical reactions, which can be very complex and difficult to analyze. Indeed, dynamical properties of CRNs can be described by a set of non-linear differential equations that rarely can be solved in closed-form, but that can instead be used to reason on the system dynamics. In this context, one of the possible approaches is to perform numerical simulations, which may require a high computational effort. In particular, in order to investigate some dynamical properties, such as robustness or global sensitivity, many simulations have to be performed by varying the initial concentration of chemical species. In order to reduce the computational effort required when many simulations are needed to assess a property, we exploit a new notion of monotonicity of the output of the system (the concentration of a target chemical species at the steady state) with respect to the input (the initial concentration of another chemical species). To assess such monotonicity behaviour, we propose a new graphical approach that allows us to state sufficient conditions for ensuring that the monotonicity property holds. Our sufficient conditions allow us to efficiently verify the monotonicity property by exploring a graph constructed on the basis of the reactions involved in the network. Once established, our monotonicity property allows us to drastically reduce the number of simulations required to assess some dynamical properties of the CRN.

Large deviation principles for stochastic jump models of chemical reaction networks provide a natural way to connect molecular with continuum and thermodynamic perspectives. However, the first results in this direction required that reaction rates be bounded away from zero even when reactants are missing, which is unphysical. Various authors have contributed to weakening this restriction and this talk will give an overview of a hypothesis on the jumps rates that is necessary under reasonable technical conditions and not merely sufficient for a large deviations principle to hold in the hydrodynamic limit for stochastic chemical reaction networks. In particular arbitrary reaction networks with mass action jump rates will be shown to satisfy a hydrodynamic LDP.
Joint work with Luisa Andreis, Andrea Agazzi and Michiel Renger.

Quantifying cellular growth is crucial to understanding the dynamics of cell populations such as microbes and cancer cells. The standard behaviour of batch cultures is well known and it is usually characterised by a delay before the start of exponential growth, an exponential phase, and a steady phase; however, at the single-cell level, growth varies drastically from cell to cell due to the fluctuations in the cell cycle duration, variability caused by changing environments, and cells interactions.
At the present time, understanding how the cell-to-cell variability affects the evolution of the entire population is still an open challenge; de facto, there are still lacking solid theoretical and simulation methods to forecast the effects of cell heterogeneity on population dynamics.
We propose a novel stochastic model where the cells are represented by agents who divide, die, convert to other species, and rejuvenate in response to an internal continuous state which increases with time. While such models are usually only amenable to simulations, we show that the population structure can be characterized by a functional master equation which can be manipulated to obtain a novel integral renewal equation. Compared to the classic results of renewal theory, as the Bellman Harris branching process, the latter equation takes a step further. In fact, it provides a solid and compact stochastic description of the role played by cell heterogeneity on the population dynamics.
The analytical framework allowed us to fully describe the population size distribution, population growth rate, ancestor and division times distributions. Moreover, we provide an analytical and numerical characterization of the extinction probability and first extinction times distribution for any cell-to-cell heterogeneity range. We also propose a novel way to simulate the evolution of cell populations affected by the variability of the individuals. Such computational tool allowed us to substantiate the analytical and numerical results obtained during this research project.
In conclusion, the following research project proposes a novel methodology to describe the stochastic behaviour of cell structured population with numerical, computational and analytical methods. Our results open a new theoretical path to understanding stochastic mechanisms underlying fluctuations in various biological and medical applications as: extinction of cancer cell populations under treatment, cell population growth in adverse environments, dormancy-awakening transition in breast cancer and microbial quiescence.

To exactly compute the mean dynamics of stochastic reaction networks, the solution of the Chemical Master Equation (CME) is rarely feasible. Often it is necessary to resort to computationally expensive simulations. To reduce the computational costs of this problem, several approximations have been proposed. The most common approximation is obtained considering a smaller set of ODEs, known as deterministic rate equations (DRE), that gives a macroscopic deterministic approximation of the average population dynamics for each species. Unfortunately, it is known that DREs can be inaccurate for systems exhibiting significant intrinsic noise, unstable or multi-stable dynamics. Dynamic Boundary Projection (DBP) is a recently proposed method that couples together a truncated version of the CME, describing the evolution of a subset of states, and a set of DREs, used to shift the observed subset across the state space. It has been shown that it can be successfully applied to refine the estimations of the mean dynamics for systems exhibiting a significant discrepancy between the solution of the DREs and the results of stochastic simulations. I will show how DBP can be successfully applied to SRNs to refine the estimations of their mean dynamics even in the presence of oscillatory orbits, multi-scale populations or multiple stable equilibria. Moreover, I will present ongoing research to reduce the computational costs of the method, by suitably defining a family of rescaled approximating processes.

Stationary distributions is an important characterization of stochastic modeled reactions networks. In a complex system, the existence of stationary distributions is already difficult, providing an explicit formula is even more involved. It is only possible in a very few cases with strict assumptions on kinetics. In this presentation, assuming the stochastic complex balancing, a new criteria for stationary distributions is introduced, which generalizes the classic results. Afterwards, a splitting method is introduced that enlarges the applicability of the new criteria. By making use of the splitting method, it is proved that conditions in this new criteria are necessary for a large class of reaction networks.

We present results on a stochastic version of a well-known kinetic nucleation model for phase transition phenomena. In the Becker-Döring model, aggregates grow or shrink by addition or removal of one-by-one particle at a time. Under certain conditions, very large aggregates emerge and are interpreted as a phase transition. We study stationary and quasi-stationary properties of the stochastic Becker-Döring model in the limit of infinite total number of particles, and compare with results from the deterministic nucleation theory. Our findings are largely inspired from recent results from stochastic chemical reaction network theory. Joint work with Erwan HINGANT