Luca Laurenti

Stochastic analysis of Chemical Reaction Networks using Linear Noise Approximation.

Stochastic evolution of Chemical Reactions Networks (CRNs) over time is usually analyzed through solving the Chemical Master Equation (CME) or performing extensive simulations. Analysing stochasticity is often needed, particularly when some molecules occur in low numbers. Unfortunately, both approaches become infeasible if the system is complex and/or it cannot be ensured that initial populations are small. We develop a probabilistic logic for CRNs that enables stochastic analysis of the evolution of populations of molecular species. We present an approximate model checking algorithm based on the Linear Noise Approximation (LNA) of the CME, whose computational complexity is independent of the population size of each species and polynomial in the number of different species. The algorithm requires the solution of first order polynomial differential equations. We prove that our approach is valid for any CRN close enough to the thermodynamical limit. However, we show on four case studies that it can still provide good approximation even for low molecule counts. Our approach enables rigorous analysis of CRNs that are not analyzable by solving the CME, but are far from the deterministic limit. Moreover, it can be used for a fast approximate stochastic characterization of a CRN.Stochastic evolution of Chemical Reactions Networks (CRNs) over time is usually analysed through solving the Chemical Master Equation (CME) or performing extensive simulations. Analysing stochasticity is often needed, particularly when some molecules occur in low numbers. Unfortunately, both approaches become infeasible if the system is complex and/or it cannot be ensured that initial populations are small. We develop a probabilistic logic for CRNs that enables stochastic analysis of the evolution of populations of molecular species. We present an approximate model checking algorithm based on the Linear Noise Approximation (LNA) of the CME, whose computational complexity is independent of the population size of each species and polynomial in the number of different species. The algorithm requires the solution of first order polynomial differential equations. We prove that our approach is valid for any CRN close enough to the thermodynamical limit. However, we show on four case studies that it can still provide good approximation even for low molecule counts. Our approach enables rigorous analysis of CRNs that are not analyzable by solving the CME, but are far from the deterministic limit. Moreover, it can be used for a fast approximate stochastic characterization of a CRN.

Approximation of Probabilistic Reachability for Chemical Reaction Networks Using the Linear Noise Approximation

We study time-bounded probabilistic reachability for Chemical Reaction Networks (CRNs) using the Linear Noise Approximation (LNA). The LNA approximates the discrete stochastic semantics of a CRN in terms of a continuous space Gaussian process. We consider reachability regions expressed as intersections of finitely many linear inequalities over the species of a CRN. This restriction allows us to derive an abstraction of the original Gaussian process as a time-inhomogeneous discrete-time Markov chain (DTMC), such that the dimensionality of its state space is independent of the number of species of the CRN, ameliorating the state space explosion problem. We formulate an algorithm for approximate computation of time-bounded reachability probabilities on the resulting DTMC and show how to extend it to more complex temporal properties. We implement the algorithm and demonstrate on two case studies that it permits fast and scalable computation of reachability properties with controlled accuracy.

Programming Discrete Distributions with Chemical Reaction Networks

We explore the range of probabilistic behaviours that can be engineered with Chemical Reaction Networks (CRNs). We show that at steady state CRNs are able to “program” any distribution with finite support in \(\mathbb {N}^m\), with \(m \ge 1\). Moreover, any distribution with countable infinite support can be approximated with arbitrarily small error under the \(L^1\) norm. We also give optimized schemes for special distributions, including the uniform distribution. Finally, we formulate a calculus to compute on distributions that is complete for finite support distributions, and can be compiled to a restricted class of CRNs that at steady state realize those distributions.

Stochastic Analysis of Chemical Reaction Networks Using Linear Noise Approximation

Stochastic evolution of Chemical Reactions Networks (CRNs) over time is usually analysed through solving the Chemical Master Equation (CME) or performing extensive simulations. Analysing stochasticity is often needed, particularly when some molecules occur in low numbers. Unfortunately, both approaches become infeasible if the system is complex and/or it cannot be ensured that initial populations are small. We develop a probabilistic logic for CRNs that enables stochastic analysis of the evolution of populations of molecular species. We present an approximate model checking algorithm based on the Linear Noise Approximation (LNA) of the CME, whose computational complexity is independent of the population size of each species and polynomial in the number of different species. The algorithm requires the solution of first order polynomial differential equations. We prove that our approach is valid for any CRN close enough to the thermodynamical limit. However, we show on three case studies that it can still provide good approximation even for low molecule counts. Our approach enables rigorous analysis of CRNs that are not analyzable by solving the CME, but are far from the deterministic limit. Moreover, it can be used for a fast approximate stochastic characterization of a CRN.

Syntax-Guided Optimal Synthesis for Chemical Reaction Networks

We study the problem of optimal syntax-guided synthesis of stochastic Chemical Reaction Networks (CRNs) that plays a fundamental role in design automation of molecular devices and in the construction of predictive biochemical models. We propose a sketching language for CRNs that concisely captures syntactic constraints on the network topology and allows its under-specification. Given a sketch, a correctness specification, and a cost function defined over the CRN syntax, our goal is to find a CRN that simultaneously meets the constraints, satisfies the specification and minimizes the cost function. To ensure computational feasibility of the synthesis process, we employ the Linear Noise Approximation allowing us to encode the synthesis problem as a satisfiability modulo theories problem over a set of parametric Ordinary Differential Equations (ODEs). We design and implement a novel algorithm for the optimal synthesis of CRNs that employs almost complete refutation procedure for SMT over reals and ODEs, and exploits a meta-sketching abstraction controlling the search strategy. Through relevant case studies we demonstrate that our approach significantly improves the capability of existing methods for synthesis of biochemical systems and paves the way towards their automated and provably-correct design.

A Stochastic Hybrid Approximation for Chemical Kinetics Based on the Linear Noise Approximation

The Linear Noise Approximation (LNA) is a continuous approximation of the CME, which improves scalability and is accurate for those reactions satisfying the leap conditions. We formulate a novel stochastic hybrid approximation method for chemical reaction networks based on adaptive partitioning of the species and reactions according to leap conditions into two classes, one solved numerically via the CME and the other using the LNA. The leap criteria are more general than partitioning based on population thresholds, and the method can be combined with any numerical solution of the CME. We then use the hybrid model to derive a fast approximate model checking algorithm for Stochastic Evolution Logic (SEL). Experimental evaluation on several case studies demonstrates that the techniques are able to provide an accurate stochastic characterisation for a large class of systems, especially those presenting dynamical stiffness, resulting in significant improvement of computation time while still maintaining scalability.

Programming discrete distributions with chemical reaction networks

We explore the range of probabilistic behaviours that can be engineered with Chemical Reaction Networks (CRNs). We give methods to “program” CRNs so that their steady state is chosen from some desired target distribution that has finite support in

Experimental Biological Protocols with Formal Semantics

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