Lumping the approximate master equation for multistate processes on complex networks

Complex networks play an important role in human society and in nature. Stochastic multistate processes provide a powerful framework to model a variety of emerging phenomena such as the dynamics of an epidemic or the spreading of information on complex networks. In recent years, mean-field type approximations gained widespread attention as a tool to analyze and understand complex network dynamics. They reduce the model\u2019s complexity by assuming that all nodes with a similar local structure behave identically. Among these methods the approximate master equation (AME) provides the most accurate description of complex networks\u2019 dynamics by considering the whole neighborhood of a node. The size of a typical network though renders the numerical solution of multistate AME infeasible. Here, we propose an efficient approach for the numerical solution of the AME that exploits similarities between the differential equations of structurally similar groups of nodes. We cluster a large number of similar equations together and solve only a single lumped equation per cluster. Our method allows the application of the AME to real-world networks, while preserving its accuracy in computing estimates of global network properties, such as the fraction of nodes in a state at a given time

Exact analysis of summary statistics for continuous-time discrete-state Markov processes on networks using graph-automorphism lumping

We propose a unified framework to represent a wide range of continuous-time discrete-state Markov processes on networks, and show how many network dynamics models in the literature can be represented in this unified framework. We show how a particular sub-set of these models, referred to here as single-vertex-transition (SVT) processes, lead to the analysis of quasi-birth-and-death (QBD) processes in the theory of continuous-time Markov chains. We illustrate how to analyse a number of summary statistics for these processes, such as absorption probabilities and first-passage times. We extend the graph-automorphism lumping approach [Kiss, Miller, Simon, Mathematics of Epidemics on Networks, 2017; Simon, Taylor, Kiss, J. Math. Bio. 62(4), 2011], by providing a matrix-oriented representation of this technique, and show how it can be applied to a very wide range of dynamical processes on networks. This approach can be used not only to solve the master equation of the system, but also to analyse the summary statistics of interest. We also show the interplay between the graph-automorphism lumping approach and the QBD structures when dealing with SVT processes. Finally, we illustrate our theoretical results with examples from the areas of opinion dynamics and mathematical epidemiology

Model Reduction for the Chemical Master Equation: an Information-Theoretic Approach

The complexity of mathematical models in biology has rendered model reduction an essential tool in the quantitative biologist's toolkit. For stochastic reaction networks described using the Chemical Master Equation, commonly used methods include time-scale separation, the Linear Mapping Approximation and state-space lumping. Despite the success of these techniques, they appear to be rather disparate and at present no general-purpose approach to model reduction for stochastic reaction networks is known. In this paper we show that most common model reduction approaches for the Chemical Master Equation can be seen as minimising a well-known information-theoretic quantity between the full model and its reduction, the Kullback-Leibler divergence defined on the space of trajectories. This allows us to recast the task of model reduction as a variational problem that can be tackled using standard numerical optimisation approaches. In addition we derive general expressions for the propensities of a reduced system that generalise those found using classical methods. We show that the Kullback-Leibler divergence is a useful metric to assess model discrepancy and to compare different model reduction techniques using three examples from the literature: an autoregulatory feedback loop, the Michaelis-Menten enzyme system and a genetic oscillator

Stochastic spreading on complex networks

Complex interacting systems are ubiquitous in nature and society. Computational modeling of these systems is, therefore, of great relevance for science and engineering. Complex networks are common representations of these systems (e.g., friendship networks or road networks). Dynamical processes (e.g., virus spreading, traffic jams) that evolve on these networks are shaped and constrained by the underlying connectivity. This thesis provides numerical methods to study stochastic spreading processes on complex networks. We consider the processes as inherently probabilistic and analyze their behavior through the lens of probability theory. While powerful theoretical frameworks (like the SIS-epidemic model and continuous-time Markov chains) already exist, their analysis is computationally challenging. A key contribution of the thesis is to ease the computational burden of these methods. Particularly, we provide novel methods for the efficient stochastic simulation of these processes. Based on different simulation studies, we investigate techniques for optimal vaccine distribution and critically address the usage of mathematical models during the Covid-19 pandemic. We also provide model-reduction techniques that translate complicated models into simpler ones that can be solved without resorting to simulations. Lastly, we show how to infer the underlying contact data from node-level observations.Komplexe, interagierende Systeme sind in Natur und Gesellschaft allgegenwärtig. Die computergestützte Modellierung dieser Systeme ist daher von immenser Bedeutung für Wissenschaft und Technik. Netzwerke sind eine gängige Art, diese Systeme zu repräsentieren (z. B. Freundschaftsnetzwerke, Straßennetze). Dynamische Prozesse (z. B. Epidemien, Staus), die sich auf diesen Netzwerken ausbreiten, werden durch die spezifische Konnektivität geformt. In dieser Arbeit werden numerische Methoden zur Untersuchung stochastischer Ausbreitungsprozesse in komplexen Netzwerken entwickelt. Wir betrachten die Prozesse als inhärent probabilistisch und analysieren ihr Verhalten nach wahrscheinlichkeitstheoretischen Fragestellungen. Zwar gibt es bereits theoretische Grundlagen und Paradigmen (wie das SIS-Epidemiemodell und zeitkontinuierliche Markov-Ketten), aber ihre Analyse ist rechnerisch aufwändig. Ein wesentlicher Beitrag dieser Arbeit besteht darin, die Rechenlast dieser Methoden zu verringern. Wir erforschen Methoden zur effizienten Simulation dieser Prozesse. Anhand von Simulationsstudien untersuchen wir außerdem Techniken für optimale Impfstoffverteilung und setzen uns kritisch mit der Verwendung mathematischer Modelle bei der Covid-19-Pandemie auseinander. Des Weiteren führen wir Modellreduktionen ein, mit denen komplizierte Modelle in einfachere umgewandelt werden können. Abschließend zeigen wir, wie man von Beobachtungen einzelner Knoten auf die zugrunde liegende Netzwerkstruktur schließt

Modelling and simulating in systems biology: an approach based on multi-agent systems

Systems Biology is an innovative way of doing biology recently raised in bio-informatics contexts, characterised by the study of biological systems as complex systems with a strong focus on the system level and on the interaction dimension. In other words, the objective is to understand biological systems as a whole, putting on the foreground not only the study of the individual parts as standalone parts, but also of their interaction and of the global properties that emerge at the system level by means of the interaction among the parts. This thesis focuses on the adoption of multi-agent systems (MAS) as a suitable paradigm for Systems Biology, for developing models and simulation of complex biological systems. Multi-agent system have been recently introduced in informatics context as a suitabe paradigm for modelling and engineering complex systems. Roughly speaking, a MAS can be conceived as a set of autonomous and interacting entities, called agents, situated in some kind of nvironment, where they fruitfully interact and coordinate so as to obtain a coherent global system behaviour. The claim of this work is that the general properties of MAS make them an effective approach for modelling and building simulations of complex biological systems, following the methodological principles identified by Systems Biology. In particular, the thesis focuses on cell populations as biological systems. In order to support the claim, the thesis introduces and describes (i) a MAS-based model conceived for modelling the dynamics of systems of cells interacting inside cell environment called niches. (ii) a computational tool, developed for implementing the models and executing the simulations. The tool is meant to work as a kind of virtual laboratory, on top of which kinds of virtual experiments can be performed, characterised by the definition and execution of specific models implemented as MASs, so as to support the validation, falsification and improvement of the models through the observation and analysis of the simulations. A hematopoietic stem cell system is taken as reference case study for formulating a specific model and executing virtual experiments

Approaches to dimensionality reduction and model simplification of dynamics in the chemical context

Much of the effort in the modern chemical and physical sciences is devoted to the study of complex dynamical phenomena. Such a study is often hampered by the considerable complexity (i.e., the high dimensionality) exhibited by the systems of interest. In this research project, of theoretical and methodological character, we explore some facets of the topics of model reduction and simplification of complex dynamics, both deterministic and stochastic. In particular, in the first part of the work (chs. 2-5), we focus on deterministic systems. In chapter 2, starting from the findings of two previous works [P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234101 (2013) and P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234102 (2013)] we introduce the concept of "canonical format" of the evolution law for mass-action-based chemical kinetics, and show that the study of such a type of formats could lead to the discovery of new interesting features and to a rationalization of already well-known ones. Specifically, we unveil the existence of "attracting subspaces" in an abstract "hyper-spherical" representation of the dynamics of a reacting system. In chapter 3, based on the theory devised in ch. 2, we develop an algorithm (implemented in the companion software DRIMAK, acronym of Dimensional Reduction for Isothermal Mass-Action Kinetics) aimed at detecting the neighborhood of the Slow Manifold, which is a hypersurface, in the concentration space, in the proximity of which the slow evolution takes place. The detection of the Slow Manifold for a reacting system is a potential key-step to elaborate dimensionality reduction strategies. In chapter 4 we extend the theory to open reaction networks, i.e., reaction networks with one or more reactants continuously injected in the reaction environment. Finally, in chapter 5 we further generalize the theory to general phase-space dynamics, possibly damped. The second part of the work (chs. 6-8) is devoted to stochastic systems. In chapter 6 we move the first steps towards the model reduction of stochastic chemical kinetics. Specifically, we show the existence of geometric structures (in the space of the number of molecules of each species) analogous to the Slow Manifold in the macroscopic counterpart. Still in the context of stochastic chemical kinetics, in chapter 7 we make a critical study of two common continuous approximations of the chemical master equation and of the associated Gillespie's stochastic simulation algorithm; namely, we investigate on the physical reliability of the chemical Fokker-Planck and chemical Langevin equations. In particular, we prove that both the approximations suffer from nonphysical probability currents at equilibrium, even for fully reversible and detailed-balanced chemical reaction networks. Finally, in chapter 8 we focus on general overdamped fluctuating systems, which, apart from very simple and low-dimensional cases, are often mathematically intractable. In this context, given the well-known difficulties for the mathematical treatment of such systems, we aim only at achieving a partial, but easily computable, information. Namely, we devise a set of mathematical time-dependent bounds for key-quantities describing the systems of interest

Markov field models of molecular kinetics

Computer simulations such as molecular dynamics (MD) provide a possible means to understand protein dynamics and mechanisms on an atomistic scale. The resulting simulation data can be analyzed with Markov state models (MSMs), yielding a quantitative kinetic model that, e.g., encodes state populations and transition rates. However, the larger an investigated system, the more data is required to estimate a valid kinetic model. In this work, we show that this scaling problem can be escaped when decomposing a system into smaller ones, leveraging weak couplings between local domains. Our approach, termed independent Markov decomposition (IMD), is a first-order approximation neglecting couplings, i.e., it represents a decomposition of the underlying global dynamics into a set of independent local ones. We demonstrate that for truly independent systems, IMD can reduce the sampling by three orders of magnitude. IMD is applied to two biomolecular systems. First, synaptotagmin-1 is analyzed, a rapid calcium switch from the neurotransmitter release machinery. Within its C2A domain, local conformational switches are identified and modeled with independent MSMs, shedding light on the mechanism of its calcium-mediated activation. Second, the catalytic site of the serine protease TMPRSS2 is analyzed with a local drug-binding model. Equilibrium populations of different drug-binding modes are derived for three inhibitors, mirroring experimentally determined drug efficiencies. IMD is subsequently extended to an end-to-end deep learning framework called iVAMPnets, which learns a domain decomposition from simulation data and simultaneously models the kinetics in the local domains. We finally classify IMD and iVAMPnets as Markov field models (MFM), which we define as a class of models that describe dynamics by decomposing systems into local domains. Overall, this thesis introduces a local approach to Markov modeling that enables to quantitatively assess the kinetics of large macromolecular complexes, opening up possibilities to tackle current and future computational molecular biology questions