Martina Amongero 
Politecnico di Torino 
Analysing the Covid19 pandemic in Italy with the SIPRO model
We propose a chemical reaction network (CRN) system, called SIPRO, that can be used to analyze the mechanism behind the Covid19 evolution. In particular, we use the deterministic limit of the CRN proposed model, that extends the classical SIR, in order to account for the proportion of unobserved infected people that have not been tested. The model is applied to the epidemic curves of the 21 Italian regions using Bayesian methods for mixedeffects models. In particular we focus on the timedependent effective reproduction number to reflect the impact of containment measures. Our model allows us to describe, estimate and predict the different phases of the pandemic (before the advent of vaccination) with a single model. We compare the performance of the SIPRO model with those of the simple SIR and we show that without loosing predictive power, we can give a reasonable estimate of the proportion of unobserved infections.

Enrico Bibbona 
Politecnico di Torino 
Multiple latent clustering model for the inference of RNA lifecycle kinetic rates from sequencing data
We propose a hierarchical Bayesian model to infer RNA synthesis, processing, and degradation rates from sequencing data, based on an ordinary differential equation system that models the RNA life cycle.
We parametrize the latent kinetic rates, that rule the system, with a novel functional form, and estimate their parameters through 6 Dirichlet process mixture models. We simultaneously perform inference, clustering, and model selection. We apply our method to investigate transcriptional and posttranscriptional responses of murine fibroblasts to the activation of protooncogene Myc. Our approach uncovers simultaneous regulations of the rates, which had not previously been observed in this biological system.

Daniele Cappelletti 
Politecnico di Torino 
Stochastic reaction networks in stochastic environment
Stochastic reaction networks are mathematical models heavily utilized in biochemistry. Usually it is assumed the rates at which biochemical transformations occur only depend on the current chemical configuration. Motivated by biological applications, in this study we considered the more general case of the rates depending on both the current configuration and another stochastic process. We study the positive recurrence of this more general model, and under certain conditions characterize the stationary distribution (when it exists) as a mixture of Poisson distributions, which is uniquely identified as the law of a fixed point of a stochastic recurrence equation. This recursion can be utilized for the statistical computation of moments and other distributional features.

Claudio Del Sole 
Università Bocconi 
Multiscale limits of Chemical Reaction Networks with Fast Absorption and Slow Escape
Motivated by the TogashiKaneko model in dimension two, we define a set of conditions named FASE, Fast Absorption and Slow Escape, under which a properly scaled family of chemical reaction networks allows for a switching limit process that we can fully characterize. FASE networks include a fast subsystem that is quickly absorbed, possibly in different absorbing classes. At a slower time scale, we observe a dynamics within the absorbing class (which remains active in the limit process), while a mechanism can bring the system back into the transient states, restarting the fast cascade, and quickly leading to another absorption. In the limit, such fast sequence of reactions becomes either an instantaneous switch, if the absorbing state is far apart from the latest one visited, or the instantaneous effect of some new reaction that we incorporate in the slower system, if the fast cascade brings the system back close to where it started.

Giulia Della Croce di Dojola 
Politecnico di Torino 
Estimating the ratio between the contagiousness of two viral strains
This work proposes a new method for the estimation of the ratio between the basic reproduction numbers of a new emerging variant and the one currently dominating, based on incidence data from random samples, and given the epidemic curves of total infections and recoveries. Our method is based on a discrete time SIR model with two strains, and it is considered both in the deterministic and in the stochastic version. In the deterministic case we can directly apply the method of maximum likelihood. In the stochastic setting, instead, we need to reconstruct the missing information about the incidence and prevalence of the new variant within a hierarchical Bayesian model. This new methodology is applied to the ISS quick surveys data focusing on the Piedmont Italian region in DecemberJanuary 2022, the period when the omicron variant started to be observed and quickly became prevalent. We show how it is possible, with both approaches, to obtain an estimate of such contagiousness ratio from public data that is consistent with the other studies specifically designed to the aim.

Alberto Lanconelli 
University of Bologna 
Using Gaussian analysis to solve a chemical diffusion master equation
We propose a novel method to solve a chemical diffusion master equation of birth and death type. This is an infinite system of FokkerPlanck 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 FokkerPlanck 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 OrnsteinUhlenbeck 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.

Kairui Li 
University of St Andrews 
Quantifying Cytoskeletal Dynamics and Remodeling from Liveimaging Microscopy Data
The shape of biological cells emerges from dynamic remodeling of the cell’s internal scaffolding, the cytoskeleton. Hence, correct cytoskeletal regulation is crucial for the control of cell behaviour, such as cell division and migration. A main component of the cytoskeleton is actin. Interlinked actin filaments span the body of the cell and contribute to a cell’s stiffness. The molecular motor myosin can induce constriction of the cell by moving actin filaments against each other. Capturing and quantifying these interactions between myosin and actin in living cells is an ongoing challenge. For example, liveimaging microscopy can be used to study the dynamic changes of actin and myosin density in deforming cells. These imaging data can be quantified using Optical Flow algorithms, which locally assign velocities of cytoskeletal movement to the data. Extended Optical Flow algorithms also quantify actin recruitment and degradation. However, these measurements on cytoskeletal dynamics may be influenced by noise in the image acquisition, by adhoc parameter choices in the algorithm, and by image preprocessing steps. Existing Optical Flow algorithms do not provide tools to estimate uncertainty on inferred velocity fields or remodeling rates that follow from these dependencies. This hinders our progress on understanding actin and myosin dynamics. To address this, we aim to combine Optical Flow algorithms with Gaussian Process regression. Our new method will not be subject to manual parameter optimisation. It will be able to assign velocity values at higher spatial resolution than previous methods and allow for uncertainty quantification of inferred quantities. We test our methods using data on actin and myosin densities in larval epithelial cells of Drosophila pupae. The development of our new method will be a starting point for identifying differences in cytoskeletal movement and remodeling under experimental perturbations. Our method will be applicable to other datasets in which flow fields are present.

Vittoria Martinelli 
Università di Napoli Federico II 
Multicellular PI control for gene regulation in microbial consortia
We describe two multicellular implementations of the classical P and PI feedback controllers for the regulation of gene expression in a target cell population. Specifically, we propose to distribute the proportional and integral actions over two different cellular populations in a microbial consortium comprising a third target population whose output needs to be regulated. By engineering communication among the different cellular populations via appropriate orthogonal quorum sensing molecules, we are able to close the feedback loop across the consortium.
We derive analytical conditions on the biological parameters guaranteeing the regulation of the output of the target population and we validate the robustness and modularity of proposed control schemes via insilico experiments in BSim, a realistic agentbased simulator of bacterial populations.

Anderson Melchor Hernandez 
Politecnico di Torino 
Homogeneous open quantum walks in a Lattice
Homogeneous open quantum walks are a possible generalization of classical Markov chains. The main difference is that the probability transitions are replaced by operators acting on a Hilbert space. Then these walk's can be thought of as Markov processes on a lattice where the evolution of the walker's position depends on local degrees of freedom. In this talk, I will review their mathematical definition and some recent results about their longtime behavior.

Maya Mincheva 
Northern Illinois University 
Reaction Networks with Time Delays
Delay massaction systems provide a model of chemical kinetics in which past states influence the current dynamics. In this work, we provide an algebraic and a graphtheoretic condition for delay stability which is linear stability independent of rate constants and delay parameters. The graphtheoretic condition involves cycles in the directed speciesreaction graph of the network, which encodes how different species in the system interact. Several interesting examples on sequestration networks with delays are presented.
This is a joint work with George Craciun, Casian Pantea and Polly Yu.

Lucia Nasti 
Gran Sasso Science Institute 
Efficient analysis of chemical reaction networks dynamics based on InputOutput monotonicity
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 nonlinear differential equations that rarely can be solved in closedform, 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.

Sarang Nath 
Novo Nordisk Foundation Center for Biosustainability, Technical University of Denmark 
Electrical Representations of (Bio)chemical Reaction Networks
In this research work, we develop and demonstrate a technique to transform reaction networks into modular electrical circuits that embody the same dynamic behaviour. After mathematically proving the equivalence of both representations, we illustrate the potential of the electrical framework to analyse oscillatory or chaotic systems. The approach is then applied to solve for effective rate constants in heterogeneous catalysis, to enumerate flux subcycles in the dihydrofolate reductase (DHFR) reaction pathway, and to simulate a simplified model of E. coli glycolysis. We believe that this methodology can be a valuable tool for (bio)chemists and (bio)chemical engineers to investigate and quantify the dynamics of their specific reaction systems.

Andrés OrtizMuñoz 
Santa Fe Institute 
Combinatorial Semantics for Stochastic Chemical Reaction Networks
We propose a formalism for stochastic chemical reaction networks based on combinatorics. A wellknown method uses probability generating functions to encode the dynamics of SCRNs in the form of a formal partial differential equation. We adopt the probability generating function method and raise it to the level of combinatorics by a process known as categorification. Under this interpretation familiar constructions and theorems in SCRN theory acquire a combinatorial character that offers a new way of thinking about them as well as suggesting new proof methods. This approach is independent of the usual formulation of SCRNs in terms of measure theory and it can be used as a novel foundation for SCRN theory. We reformulate a number of definitions and theorems with this proposed foundation, including general solutions to the chemical master equation, stationary distributions, complexbalance and productofPoisson form, factorial moments, and combinatorial assembly.

Robert Patterson 
WIAS, Berlin 
Large deviations with vanishing reactant concentrations
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.

Simone Pernice 
Università di Torino 
GreatMod: A general computational framework for modeling Chemical Reaction Networks
We present a new general modeling framework, called GreatMod (freely available https://qbioturin.github.io/epimod) for the analysis of complex systems (e.g., chemical reaction networks, biological and epidemiological systems), characterized by features that make easy its utilization even by researchers without advanced mathematical and computational skills.
The framework’s strengths are based on i) the use of the Petri Net graphical formalism to simplify model design and to provide an intuitive description of system behavior, ii) the implementation of an R package to provide a userfriendly interface, iii) the virtualization of the computational environment to provide a preconfigured setup comprising all the tools of analysis (e.g. model calibration, sensitivity analysis) ensuring portability and reproducibility of results. Furthermore, the framework’s architecture allows the set up of a pipeline of analysis comprising several relevant tools by which it is possible to automatically derive and simulate both the deterministic and stochastic processes underlying the model from its graphical representation.
Finally, we applied GreatMod to analyse the Schlogel model, a quite famous example of a simple reaction network that exhibits bistability, i.e. the solution strongly depends on the initial conditions and parameters, converging to one of the two stable states. Indeed, the framework has been successfully applied to studying different diseases, including COVID19, pertussis, and Multiple Sclerosis.

Tomislav Plesa 
University of Cambridge 
Integral feedback in synthetic biology: Negativeequilibrium catastrophe
A central goal of synthetic biology is the design of molecular controllers that can manipulate the dynamics of intracellular networks in a stable and accurate manner. To address the fact that detailed knowledge about intracellular networks is unavailable, integralfeedback controllers (IFCs) have been put forward for cotrolling molecular abundances. These controllers can maintain accuracy in spite of the uncertainties in the controlled networks. However, this desirable feature is achieved only if stability is also maintained. In this talk, we show that molecular IFCs can suffer from a hazardous instability called negativeequilibrium catastrophe (NEC), whereby all nonnegative equilibria vanish under the action of the controllers, and some of the molecular abundances blow up. We show that unimolecular IFCs do not exist due to a NEC. We then derive a family of bimolecular IFCs that are safeguarded against NECs when uncertain unimolecular neworks, with any number of molecular species, are controlled. However, when IFCs are applied on uncertain bimolecular (and hence most intracellular) neworks, we show that preventing NECs generally becomes an intractable problem as the number of interacting molecular species increases. NECs therefore place a fundamental limit to design and control of molecular networks in synthetic biology.

Francesco Puccioni 
Imperial College of London 
Stochastic modelling of agentbased populations
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 singlecell 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 celltocell 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 celltocell 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, dormancyawakening transition in breast cancer and microbial quiescence.

Francesca Randone 
IMT School for Advanced Studies Lucca 
Refining deterministic approximations of stochastic reaction networks through dynamic boundary projection
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 multistable 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, multiscale 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.

Michiel Renger 
WIAS Berlin 
Time reversal (anti)symmetry in reaction networks
Both on the micro and macroscopic scale, detailed balance is related to a certain timereversal symmetry. In order to analyse nondetailed balanced reaction networks, we propose a certain decomposition into symmetric and antisymmetric dynamics. Consistent with OnsagerMachlup, the symmetric dynamics follow a gradient flow of the free energy. The antisymmetric dynamics are in a sense the exact opposite of a gradient flow.
Joint work with Rob Patterson, Upanshu Sharma, Johannes Zimmer and others.

Iacopo Ruolo 
University of Naples Federico II 
Elucidation of TFEB nuclear translocation dynamics in human cells by means of Quantitative Modelling and Microfluidics
The Transcription Factor EB (TFEB) plays a pivotal role in the transcriptional regulation of lysosomal biogenesis and autophagy in response to starvation. TFEB activity is regulated by the kinase mTOR. In nutrientrich conditions, mTOR phosphorylates TFEB by sequestering it in the cytoplasm. During starvation, mTOR is inhibited and unphosphorylated TFEB translocates into the nucleus, where it regulates the expression of its target genes. We performed an experiment monitoring in realtime the nuclear localization of TFEB in individual cells growing in a microfluidics device following alternating pulses of starvation and refeeding. The results revealed the presence of an “overshoot” dynamics with TFEB translocating to the nucleus upon starvation but then partially retranslocating to the cytoplasm. To elucidate the mechanism behind its nuclear shuttling dynamics, we first developed a twocompartment dynamical model (nucleus and cytoplasm) with two different species (dephosphorylated and phosphorylated TFEB) for each of the two compartments. The transport and de/phosphorylation kinetics are described with first order kinetics whose parameters depend on the nutrient status, while the total TFEB is kept constant. This model was not capable to recapitulate the observed overshoot dynamics. Consequently, we investigated two alternative models, hypothesizing that upon starvation, a nuclear export channel is activated either by TFEB itself (negative feedback loop  NFL) or as a direct consequence of mTOR inhibition (incoherent feedforward loop  IFFL). The new models were both able to qualitatively recapitulate the observed dynamics. Finally, we propose an experiment able to distinguish between the two alternative models.

Elena Sabbioni 
Politecnico di Torino 
Bayesian approach for modelling RNA transcription and RNA velocity
Gene expression is regulated through the fundamental processes of transcription, splicing and degradation, that can be modelled as a deterministic chemical reaction network, whose rates need to be estimated from experimental data collected by singlecell RNA sequencing. By this technique, biologists can take only a single snapshot of the cellular states: they obtain the counts of unspliced and spliced mRNA molecules, for each gene and for each cell altogether at the moment of the sequencing, that actually corresponds to different levels of maturity in the evolution of the different cells, and then the cells are destroyed. The aim of this work is to reconsider part of the method introduced in “Generalizing RNA velocity to transient cell states through dynamical modeling (Bergen, V., Lange, M., Peidli, S. et al., Nature Biotechnology, 2020)” to analyze scRNAseq data and to describe the evolution of some cells over different cell types, exploiting the level of expression of their genes and the concept of RNAvelocity. We reformulate it in a way that is mathematically better founded, using Bayesian statistics.

Rachel Topno 
Institut de Génétique Humaine 
Extrinsic and intrinsic transcriptional noise in the control of HIV1 latency
HIV latent reservoirs are cells infected with HIV1 that remain in the body for a long period. These cells express minimally and remain undetected by the host immune system. Stochastically expressing latent cells are hypothesized to be the key drivers of the infection and viral rebounds in HIV patients. From previous studies we understand that this viral rebound is because of the latently infected cells being reactivated. However, the mechanism is not well understood. Latent cells are not completely inactive. They produce small bursts of activity that cannot be detected but can potentially lead to viral rebound. Every now and then a burst of viral particles activates the feedback loop that drives efficient HIV1 transcription and triggers latency exit. This stochasticity is believed to be arising at the level of promoter switching dynamics as a result of two types of transcriptional noise; extrinsic fluctuations (regulated by environmental factors affecting all the gene) and intrinsic fluctuations (the inherent randomness of the reaction system). Studying transcriptional noise can help us understand latency better.
We aim to study the stochastic fluctuation of HIV1 transcription and separate the contribution of extrinsic and intrinsic factors. In this study an HIV1 reporter labeled with the MS2 tag will be integrated in multiple independent copies into a reporter cell line (initially as a homozygous CRISPR integrated in HCT116 diploid cells, then as multiple random viral integrations in Jurkat T cells). We will perform live cell imaging using microfluidic devices that will allow us to image 1001000 cells and generate highthroughput data. The reporter copies sharing the same cellular environment would allow us to follow correlated (extrinsic) as well as uncorrelated (intrinsic to the gene network) fluctuation. The highthroughput live cell imaging data describing the viral transcription will be analyzed using image analysis softwares. We will use a novel autocovariance method to distinguish between intrinsic and extrinsic noise and extract the respective timescales. The statistics of the transcription events will also be extracted using a pipeline developed in the lab. These will then be used to determine the role of transcriptional noise in triggering viral bursting by mathematical modelling of the promoter dynamics. Our study will help determine the respective importance of extrinsic and intrinsic noise for exit from viral latency.

János Tóth 
Budapest University of Technology and Economics 
TBA 
Panqiu Xia 
University of Copenhagen 
Complex balanced distributions for chemical reaction networks and their applications
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.

Romain Yvinec 
PRC, INRAE, CNRS, Université de Tours 
Stochastic BeckerDöring model: large population and large time results for phase transition phenomena
We present results on a stochastic version of a wellknown kinetic nucleation model for phase transition phenomena.
In the BeckerDöring model, aggregates grow or shrink by addition or removal of onebyone particle at a time.
Under certain conditions, very large aggregates emerge and are interpreted as a phase transition.
We study stationary and quasistationary properties of the stochastic BeckerDöring 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
