@article{DibakJdelRazoDeSanchoetal.2018, author = {Dibak, Manuel and J. del Razo, Mauricio and De Sancho, David and Sch{\"u}tte, Christof and No{\´e}, Frank}, title = {MSM/RD: Coupling Markov state models of molecular kinetics with reaction-diffusion simulations}, volume = {148}, journal = {Journal of Chemical Physics}, number = {214107}, doi = {10.1063/1.5020294}, year = {2018}, abstract = {Molecular dynamics (MD) simulations can model the interactions between macromolecules with high spatiotemporal resolution but at a high computational cost. By combining high-throughput MD with Markov state models (MSMs), it is now possible to obtain long-timescale behavior of small to intermediate biomolecules and complexes. To model the interactions of many molecules at large lengthscales, particle-based reaction-diffusion (RD) simulations are more suitable but lack molecular detail. Thus, coupling MSMs and RD simulations (MSM/RD) would be highly desirable, as they could efficiently produce simulations at large time- and lengthscales, while still conserving the characteristic features of the interactions observed at atomic detail. While such a coupling seems straightforward, fundamental questions are still open: Which definition of MSM states is suitable? Which protocol to merge and split RD particles in an association/dissociation reaction will conserve the correct bimolecular kinetics and thermodynamics? In this paper, we make the first step towards MSM/RD by laying out a general theory of coupling and proposing a first implementation for association/dissociation of a protein with a small ligand (A + B <--> C). Applications on a toy model and CO diffusion into the heme cavity of myoglobin are reported.}, language = {en} } @article{KostreSchuetteNoeetal.2021, author = {Kostre, Margarita and Sch{\"u}tte, Christof and No{\´e}, Frank and del Razo Sarmina, Mauricio}, title = {Coupling Particle-Based Reaction-Diffusion Simulations with Reservoirs Mediated by Reaction-Diffusion PDEs}, volume = {19}, journal = {Multiscale Modeling \& Simulation}, number = {4}, publisher = {Society for Industrial and Applied Mathematics}, arxiv = {http://arxiv.org/abs/2006.00003}, doi = {10.1137/20M1352739}, pages = {1659 -- 1683}, year = {2021}, abstract = {Open biochemical systems of interacting molecules are ubiquitous in life-related processes. However, established computational methodologies, like molecular dynamics, are still mostly constrained to closed systems and timescales too small to be relevant for life processes. Alternatively, particle-based reaction-diffusion models are currently the most accurate and computationally feasible approach at these scales. Their efficiency lies in modeling entire molecules as particles that can diffuse and interact with each other. In this work, we develop modeling and numerical schemes for particle-based reaction-diffusion in an open setting, where the reservoirs are mediated by reaction-diffusion PDEs. We derive two important theoretical results. The first one is the mean-field for open systems of diffusing particles; the second one is the mean-field for a particle-based reaction-diffusion system with second-order reactions. We employ these two results to develop a numerical scheme that consistently couples particle-based reaction-diffusion processes with reaction-diffusion PDEs. This allows modeling open biochemical systems in contact with reservoirs that are time-dependent and spatially inhomogeneous, as in many relevant real-world applications.}, language = {en} } @article{DibakdelRazodeSanchoetal.2018, author = {Dibak, Manuel and del Razo, Mauricio J. and de Sancho, David and Sch{\"u}tte, Christof and No{\´e}, Frank}, title = {MSM/RD: Coupling Markov state models of molecular kinetics with reaction-diffusion simulations}, volume = {148}, journal = {Journal of Chemical Physics}, number = {21}, doi = {10.1063/1.5020294}, year = {2018}, abstract = {Molecular dynamics (MD) simulations can model the interactions between macromolecules with high spatiotemporal resolution but at a high computational cost. By combining high-throughput MD with Markov state models (MSMs), it is now possible to obtain long time-scale behavior of small to intermediate biomolecules and complexes. To model the interactions of many molecules at large length scales, particle-based reaction-diffusion (RD) simulations are more suitable but lack molecular detail. Thus, coupling MSMs and RD simulations (MSM/RD) would be highly desirable, as they could efficiently produce simulations at large time and length scales, while still conserving the characteristic features of the interactions observed at atomic detail. While such a coupling seems straightforward, fundamental questions are still open: Which definition of MSM states is suitable? Which protocol to merge and split RD particles in an association/dissociation reaction will conserve the correct bimolecular kinetics and thermodynamics? In this paper, we make the first step toward MSM/RD by laying out a general theory of coupling and proposing a first implementation for association/dissociation of a protein with a small ligand (A + B ⇌ C). Applications on a toy model and CO diffusion into the heme cavity of myoglobin are reported.}, language = {en} } @article{ArmasMerbisMeylahnetal.2025, author = {Armas, Jay and Merbis, Wout and Meylahn, Janusz M and Rafiee Rad, Soroush and del Razo, Mauricio J}, title = {Risk aversion can promote cooperation}, volume = {6}, journal = {Journal of Physics: Complexity}, number = {1}, arxiv = {http://arxiv.org/abs/2306.05971}, doi = {10.1088/2632-072X/adb234}, year = {2025}, abstract = {Cooperative dynamics are central to our understanding of many phenomena in living and complex systems. However, we lack a universal mechanism to explain the emergence of cooperation. We present a novel framework for modelling social dilemma games with an arbitrary number of players by combining reaction networks, methods from quantum mechanics applied to stochastic complex systems, game theory and stochastic simulations of molecular reactions. Using this framework, we propose a novel and robust mechanism for cooperation based on risk aversion that leads to cooperative behaviour in population games. Rather than individuals seeking to maximise payouts in the long run, individuals seek to obtain a minimum set of resources with a given level of confidence and in a limited time span. We show that this mechanism can lead to the emergence of new equilibria in a range of social dilemma games.}, language = {en} } @article{delRazoDelleSite2025, author = {del Razo, Mauricio and Delle Site, Luigi}, title = {Dynamics of systems with varying number of particles: from Liouville equations to general master equations for open systems}, volume = {18, 001}, journal = {SciPost Physics}, doi = {10.21468/SciPostPhys.18.1.001}, year = {2025}, abstract = {A varying number of particles is one of the most relevant characteristics of systems of interest in nature and technology, ranging from the exchange of energy and matter with the surrounding environment to the change of particle number through internal dynamics such as reactions. The physico-mathematical modeling of these systems is extremely challenging, with the major difficulty being the time dependence of the number of degrees of freedom and the additional constraint that the increment or reduction of the number and species of particles must not violate basic physical laws. Theoretical models, in such a case, represent the key tool for the design of computational strategies for numerical studies that deliver trustful results. In this manuscript, we review complementary physico-mathematical approaches of varying number of particles inspired by rather different specific numerical goals. As a result of the analysis on the underlying common structure of these models, we propose a unifying master equation for general dynamical systems with varying number of particles. This equation embeds all the previous models and can potentially model a much larger range of complex systems, ranging from molecular to social agent-based dynamics.}, language = {en} } @article{delRazoKostre2025, author = {del Razo, Mauricio and Kostr{\´e}, Margarita}, title = {Open reaction-diffusion systems: bridging probabilistic theory and simulations across scales}, volume = {58}, journal = {Journal of Physics A: Mathematical and Theoretical}, number = {14}, doi = {10.1088/1751-8121/adc520}, year = {2025}, abstract = {Reaction-diffusion processes are the foundational model for a diverse range of complex systems, ranging from biochemical reactions to social agent-based phenomena. The underlying dynamics of these systems occur at the individual particle/agent level, and in realistic applications, they often display interaction with their environment through energy or material exchange with a reservoir. This requires intricate mathematical considerations, especially in the case of material exchange since the varying number of particles/agents results in ``on-the-fly'' modification of the system dimension. In this work, we first overview the probabilistic description of reaction-diffusion processes at the particle level, which readily handles varying number of particles. We then extend this model to consistently incorporate interactions with macroscopic material reservoirs. Based on the resulting expressions, we bridge the probabilistic description with macroscopic concentration-based descriptions for linear and nonlinear reaction-diffusion systems, as well as for an archetypal open reaction-diffusion system. Using these mathematical bridges across scales, we finally develop numerical schemes for open reaction-diffusion systems, which we implement in two illustrative examples. This work establishes a methodological workflow to bridge particle-based probabilistic descriptions with macroscopic concentration-based descriptions of reaction-diffusion in open settings, laying the foundations for a multiscale theoretical framework upon which to construct theory and simulation schemes that are consistent across scales.}, language = {en} } @article{delRazoFroembergStraubeetal.2022, author = {del Razo, Mauricio and Fr{\"o}mberg, Daniela and Straube, Arthur and Sch{\"u}tte, Christof and H{\"o}fling, Felix and Winkelmann, Stefanie}, title = {A probabilistic framework for particle-based reaction-diffusion dynamics using classical Fock space representations}, volume = {112}, journal = {Letters in Mathematical Physics}, number = {49}, arxiv = {http://arxiv.org/abs/arXiv:2109.13616}, doi = {10.1007/s11005-022-01539-w}, year = {2022}, language = {en} } @article{delRazoDibakSchuetteetal.2021, author = {del Razo, Mauricio J. and Dibak, Manuel and Sch{\"u}tte, Christof and No{\´e}, Frank}, title = {Multiscale molecular kinetics by coupling Markov state models and reaction-diffusion dynamics}, volume = {155}, journal = {The Journal of Chemical Physics}, number = {12}, doi = {10.1063/5.0060314}, year = {2021}, language = {en} } @article{delRazoWinkelmannKleinetal.2023, author = {del Razo, Mauricio and Winkelmann, Stefanie and Klein, Rupert and H{\"o}fling, Felix}, title = {Chemical diffusion master equation: formulations of reaction-diffusion processes on the molecular level}, volume = {64}, journal = {Journal of Mathematical Physics}, number = {1}, arxiv = {http://arxiv.org/abs/2210.02268}, doi = {10.1063/5.0129620}, year = {2023}, abstract = {The chemical diffusion master equation (CDME) describes the probabilistic dynamics of reaction--diffusion systems at the molecular level [del Razo et al., Lett. Math. Phys. 112:49, 2022]; it can be considered the master equation for reaction--diffusion processes. The CDME consists of an infinite ordered family of Fokker--Planck equations, where each level of the ordered family corresponds to a certain number of particles and each particle represents a molecule. The equations at each level describe the spatial diffusion of the corresponding set of particles, and they are coupled to each other via reaction operators --linear operators representing chemical reactions. These operators change the number of particles in the system, and thus transport probability between different levels in the family. In this work, we present three approaches to formulate the CDME and show the relations between them. We further deduce the non-trivial combinatorial factors contained in the reaction operators, and we elucidate the relation to the original formulation of the CDME, which is based on creation and annihilation operators acting on many-particle probability density functions. Finally we discuss applications to multiscale simulations of biochemical systems among other future prospects.}, language = {en} } @article{delRazoLammaMerbis2026, author = {del Razo, Mauricio and Lamma, Tommaso and Merbis, Wout}, title = {Field theories and quantum methods for stochastic reaction-diffusion systems}, volume = {98}, journal = {Rev. Mod. Phys.}, arxiv = {http://arxiv.org/abs/2409.13377}, doi = {10.1103/9qlw-gyd7}, year = {2026}, abstract = {Complex systems are composed of many particles or agents that move and interact with one another. In most real-world applications, these systems involve a varying number of particles/agents that change due to interactions with the environment or their internal dynamics. The underlying mathematical framework to model these systems must incorporate the spatial transport of particles/agents and their interactions, as well as changes to their copy numbers, all of which can be formulated in terms of stochastic reaction-diffusion processes. However, the standard probabilistic representation of these processes can be overly complex because of the combinatorial aspects arising due to the non-linear interactions and varying particle numbers. In this manuscript, we review the main field theory representations of stochastic reaction-diffusion systems, which handle these issues "under-the-hood''. First, we focus on bringing techniques familiar to theoretical physicists —such as second quantization, Fock space, path integrals and quantum field theory— back into the classical domain of reaction-diffusion systems. We demonstrate how various field theory representations, which have evolved historically, can all be unified under a single basis-independent representation. We then extend existing quantum-based methods and notation to work directly on the level of the unifying representation, and we illustrate how they can be used to consistently obtain previous known results in a more straightforward manner, such as numerical discretizations and relations between model parameters at multiple scales. Throughout the work, we contextualize how these representations mirror well-known models of chemical physics depending on their spatial resolution, as well as the corresponding macroscopic (large copy number) limits. The framework presented here may find applications in a diverse set of scientific fields, including physical chemistry, theoretical ecology, epidemiology, game theory and socio-economical models of complex systems, specifically in the modeling and multi-scale simulation of complex systems with varying numbers of particles/agents. The presentation is done in a self-contained educational and unifying manner such that it can be followed by researchers across several fields.}, language = {en} }