Conference Agenda

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Session Overview
Session
MS152: Stochastic chemical reaction networks
Time:
Tuesday, 09/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F-105
53 seats, 70m^2

Presentations
3:00pm - 5:00pm

Stochastic chemical reaction networks

Chair(s): Michael Felix Adamer (University of Oxford, United Kingdom)

The focus of this minisymposium is on new algebraic and analytic methods for stochastic chemical reaction networks. In contrast to deterministic models, stochastic systems cannot be described by systems of ordinary differential equations and, hence, direct application of algebraic methods is often not possible. We are interested in when the deterministic and the stochastic behaviour of chemical reaction networks diverge and how to analyse this behaviour with a combination of algebra, stochastic analysis and chemical reaction network theory.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Piecewise linear Lyapunov functions for stochastic reaction networks

Daniele Cappelletti
ETHZ

Stochastic reaction networks are mathematical models heavily utilized to describe the time evolution of biological systems, when few active molecules are present. In this case, the system dynamics is stochastic, and the changes of the molecules counts are described by means of a continuous time Markov chain. Despite the large use of these models, simple questions concerning the existence of a stationary distribution are hard to answer to, except for few exceptions, and constitute an active area of research. Often, in order to prove the convergence of a model to a stationary distribution, a suitable Foster-Lyapunov function is sought. I will present a novel and fast convex programming technique to check whether conditions implying the existence of a piecewise linear Lyapunov function hold. Such technique utilizes the geometry of the network to divide the state space in different regions where the calculated Foster-Lyapunov function is linear.

 

Robust stochastic control of reaction networks

Tomislav Plesa
Imperial College

Synthetic biology is a rapidly growing interdisciplinary field of science and engineering that aims to design biochemical systems which behave in a desired manner. With the recent breakthroughs in nucleic-acid-based biochemistry, arbitrary reaction networks can be experimentally implemented using solely DNA molecules, with applications to areas such as medicine, industry and nanotechnology.

In this talk, I will focus on developing mathematical methods for designing reaction networks with predefined stochastic behaviors. In particular, the following fundamental problem in biochemical control theory will be considered: given any well-behaved mass-action kinetics input reaction network, whose structure and dynamics are at-most partially known, the goal is to introduce suitable additional biochemical species and reactions in a systematic manner, such that the resulting enlarged output network has a predefined stationary probability mass function (PMF). For experimental implementability in nucleic-acid-based synthetic biology, it is also required that the output stationary PMF is robust with respect to the initial conditions (ergodicity), and with respect to variations in the rate coefficients of the input network (robust perfect adaptation). I will present a solution to this problem, by embedding a faster controller network, called the stochastic morpher, into any given (slower) input network. Due to the introduced time-scale separation, it will be rigorously shown, using singular perturbation theory, that the controller overrides the firing of the input network, and morphs the input PMF into an output one with a desired form. The morphing will be performed at a lower-resolution level, by mapping the input PMF to the output one taking the form of a linear combination of Poisson PMFs, suitable for designing networks with predefined multi-modality (multi-stability). Higher-resolution morphing will also be presented, with arbitrary output PMFs. The results will be exemplified on relatively simple input networks, whose dynamics will be morphed to display noise-induced multi-modality and predefined PMFs.

 

One-dimensional stochastic reaction networks: Classification and dynamics

Chuang Xu
U Copenhagen

A crucial dynamical property to guarantee the existence of a stationary distribution is positive recurrence. However, it is not easy to provide checkable criteria for stochastic reaction networks, particularly with complex topological or graphical structures.

Motivated by this need, this talk contributes to stochastic dynamics of chemical reaction networks (CRNs) with one-dimensional stoichiometric subspace. I will first present a classification of the state space of the underlying continuous time Markov chain (CTMC) associated with the CRN by identifying all types of states: absorbing (neutral and trapping) as well as escaping states, and open repelling as well as closed attracting non-singleton communicating classes, on each stochastic stoichiometric compatibility class with all large initial states. I will also mention how to use this result to discuss the diversity of long-term dynamics of stochastic CRNs, and point out that limit distributions of CRNs with absolute concentration robustness (ACR) are not necessarily Poisson, which answers a question by Anderson in 2014.

Moreover, I will present checkable necessary and sufficient network conditions for various dynamical properties: Recurrence (positive and null), transience, (non)explosivity, (non)implosivity, as well as existence of moments of passage times of the associated CTMC of one-dimensional stochastic CRNs. As a byproduct, any one-dimensional weakly reversible CRN is positive recurrent, confirming the Positive Recurrence Conjecture proposed by Anderson and Kim in 2018 (in 1-d case). In addition, I will mention how to use these conditions to address a question by Anderson et al. in 2018 on non-explosivity.

Finally, I will emphasize results on one-species CRNs, regarding asymptotics of tails of stationary distributions as well as approximation of an arbitrary discrete distribution by either ergodic stationary distributions or quasi-stationary distributions of one-species mass-action CRNs, and present parameter regions for consistency and inconsistency of stochastic and deterministic one-species CRNs regarding various dynamical properties aforementioned.

 

The geometry and dynamics of spatial networks subject external noise

Michael Adamer
University of Oxford

In this talk I will introduce a way of modelling external noise in chemical reaction networks.
Unlike internal noise, which arises from the finite number of particles present in a chemical reaction system, external noise can have a multitude of origins and also affects systems modelled by ODEs. Due to this fact many stochastic systems subject to external noise retain the geometric features, i.e. the steady state geometry, of a mass-action system. Hence, we propose to treat the system as a mass-action system at steady state with the noise acting as forcing term creating model dynamics. In certain circumstances this approach is mathematically equivalent to the modelling of internal noise and in this talk I am going to explain the assumptions and simplifications behind this.

Further, I am going to show how steady state geometry influences the modelling of external noise systems with particular regard to the multistationarity structure of the system.
To illustrate the dynamical features of external noise system I choose some particular types of noise such as white Gaussian noise and Ornstein-Uhlenbeck noise and show how noise correlations can help to form or destroy spatio-temporal patterns.