Approximation of reaction-diffusion dynamics via Markov state modeling

Accurate modeling and numerical simulation of reaction kinetics is a topic of steady interest. We consider the spatiotemporal chemical master equation (ST- CME) as a model for stochastic reaction-diffusion systems that exhibit properties of metastability. The space of motion is decomposed into metastable compartments, and diffusive motion is approximated by jumps between these compartments. Treating these jumps as first-order reactions, simulation of the resulting stochastic system is possible by the Gillespie method. We present the theory of Markov state models as a theoretical foundation of this intuitive approach. By means of Markov state modeling, both the number and shape of compartments and the transition rates between them can be determined. We consider the ST-CME for two reaction-diffusion systems and compare it to more detailed models. Moreover, a rigorous formal justification of the ST-CME by Galerkin projection methods is presented

Accurate reduced models for the pH oscillations in the urea-urease reaction confined to giant lipid vesicles

This theoretical study concerns a pH oscillator based on the urea-urease reaction confined to giant lipid vesicles. Under suitable conditions, differential transport of urea and hydrogen ion across the unilamellar vesicle membrane periodically resets the pH clock that switches the system from acid to basic, resulting in self-sustained oscillations. We analyse the structure of the phase flow and of the limit cycle, which controls the dynamics for giant vesicles and dominates the pronouncedly stochastic oscillations in small vesicles of submicrometer size. To this end, we derive reduced models, which are amenable to analytic treatments that are complemented by numerical solutions, and obtain the period and amplitude of the oscillations as well as the parameter domain, where oscillatory behavior persists. We show that the accuracy of these predictions is highly sensitive to the employed reduction scheme. In particular, we suggest an accurate two-variable model and show its equivalence to a three-variable model that admits an interpretation in terms of a chemical reaction network. The faithful modeling of a single pH oscillator appears crucial for rationalizing experiments and understanding communication of vesicles and synchronization of rhythms.Comment: submitted J. Phys. Chem.

Optimal Treatment Strategies in the Context of ‘Treatment for Prevention’ against HIV-1 in Resource-Poor Settings

An estimated 2.7 million new HIV-1 infections occurred in 2010. `Treatment- for-prevention’ may strongly prevent HIV-1 transmission. The basic idea is that immediate treatment initiation rapidly decreases virus burden, which reduces the number of transmittable viruses and thereby the probability of infection. However, HIV inevitably develops drug resistance, which leads to virus rebound and nullifies the effect of `treatment-for-prevention’ for the time it remains unrecognized. While timely conducted treatment changes may avert periods of viral rebound, necessary treatment options and diagnostics may be lacking in resource-constrained settings. Within this work, we provide a mathematical platform for comparing different treatment paradigms that can be applied to many medical phenomena. We use this platform to optimize two distinct approaches for the treatment of HIV-1: (i) a diagnostic-guided treatment strategy, based on infrequent and patient-specific diagnostic schedules and (ii) a pro-active strategy that allows treatment adaptation prior to diagnostic ascertainment. Both strategies are compared to current clinical protocols (standard of care and the HPTN052 protocol) in terms of patient health, economic means and reduction in HIV-1 onward transmission exemplarily for South Africa. All therapeutic strategies are assessed using a coarse-grained stochastic model of within-host HIV dynamics and pseudo-codes for solving the respective optimal control problems are provided. Our mathematical model suggests that both optimal strategies (i)-(ii) perform better than the current clinical protocols and no treatment in terms of economic means, life prolongation and reduction of HIV-transmission. The optimal diagnostic-guided strategy suggests rare diagnostics and performs similar to the optimal pro-active strategy. Our results suggest that ‘treatment-for-prevention’ may be further improved using either of the two analyzed treatment paradigms

Learning Interpretable Collective Variables of the Noisy Voter Model

We present a data-driven method to learn and understand collective variables for noisy voter model dynamics on networks. A collective variable (CV) is a projection of the high-dimensional system state into a low-dimensional space that preserves the essential dynamical information. Thus, CVs can be used to improve our understanding of complex emergent behaviors and to enable an easier analysis and prediction. We demonstrate our method using three example networks: the stochastic block model, a ring-shaped graph, and a scale-free network generated by the Albert--Barab\'asi model. Our method combines the recent transition manifold approach with a linear regression step to produce interpretable CVs that describe the role and importance of each network node

Partial mean-field model for neurotransmission dynamics

This article addresses reaction networks in which spatial and stochastic effects are of crucial importance. For such systems, particle-based models allow us to describe all microscopic details with high accuracy. However, they suffer from computational inefficiency if particle numbers and density get too large. Alternative coarse-grained-resolution models reduce computational effort tremendously, e.g., by replacing the particle distribution by a continuous concentration field governed by reaction-diffusion PDEs. We demonstrate how models on the different resolution levels can be combined into hybrid models that seamlessly combine the best of both worlds, describing molecular species with large copy numbers by macroscopic equations with spatial resolution while keeping the stochastic-spatial particle-based resolution level for the species with low copy numbers. To this end, we introduce a simple particle-based model for the binding dynamics of ions and vesicles at the heart of the neurotransmission process. Within this framework, we derive a novel hybrid model and present results from numerical experiments which demonstrate that the hybrid model allows for an accurate approximation of the full particle-based model in realistic scenarios.Comment: 16 pages + 2 pages appendix, 5 figures. Submitted to Mathematical Bioscience

Chemical diffusion master equation: formulations of reaction--diffusion processes on the molecular level

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

From interacting agents to density-based modeling with stochastic PDEs

Many real-world processes can naturally be modeled as systems of interacting agents. However, the long-term simulation of such agent-based models is often intractable when the system becomes too large. In this paper, starting from a stochastic spatio-temporal agent-based model (ABM), we present a reduced model in terms of stochastic PDEs that describes the evolution of agent number densities for large populations. We discuss the algorithmic details of both approaches; regarding the SPDE model, we apply Finite Element discretization in space which not only ensures efficient simulation but also serves as a regularization of the SPDE. Illustrative examples for the spreading of an innovation among agents are given and used for comparing ABM and SPDE models

Identification of candidate genes associated with tolerance to apple replant disease by genome-wide transcriptome analysis

Apple replant disease (ARD) is a worldwide economic risk in apple cultivation for fruit tree nurseries and fruit growers. Several studies on the reaction of apple plants to ARD are documented but less is known about the genetic mechanisms behind this symptomatology. RNA-seq analysis is a powerful tool for revealing candidate genes that are involved in the molecular responses to biotic stresses in plants. The aim of our work was to find differentially expressed genes in response to ARD in Malus. For this, we compared transcriptome data of the rootstock ‘M9’ (susceptible) and the wild apple genotype M. ×robusta 5 (Mr5, tolerant) after cultivation in ARD soil and disinfected ARD soil, respectively. When comparing apple plantlets grown in ARD soil to those grown in disinfected ARD soil, 1,206 differentially expressed genes (DEGs) were identified based on a log2 fold change, (LFC) ≥ 1 for up– and ≤ −1 for downregulation (p < 0.05). Subsequent validation revealed a highly significant positive correlation (r = 0.91; p < 0.0001) between RNA-seq and RT-qPCR results indicating a high reliability of the RNA-seq data. PageMan analysis showed that transcripts of genes involved in gibberellic acid (GA) biosynthesis were significantly enriched in the DEG dataset. Most of these GA biosynthesis genes were associated with functions in cell wall stabilization. Further genes were related to detoxification processes. Genes of both groups were expressed significantly higher in Mr5, suggesting that the lower susceptibility to ARD in Mr5 is not due to a single mechanism. These findings contribute to a better insight into ARD response in susceptible and tolerant apple genotypes. However, future research is needed to identify the defense mechanisms, which are most effective for the plant to overcome ARD

Identification of Candidate Genes Associated With Tolerance to Apple Replant Disease by Genome-Wide Transcriptome Analysis

Apple replant disease (ARD) is a worldwide economic risk in apple cultivation for fruit tree nurseries and fruit growers. Several studies on the reaction of apple plants to ARD are documented but less is known about the genetic mechanisms behind this symptomatology. RNA-seq analysis is a powerful tool for revealing candidate genes that are involved in the molecular responses to biotic stresses in plants. The aim of our work was to find differentially expressed genes in response to ARD in Malus. For this, we compared transcriptome data of the rootstock ‘M9’ (susceptible) and the wild apple genotype M. ×robusta 5 (Mr5, tolerant) after cultivation in ARD soil and disinfected ARD soil, respectively. When comparing apple plantlets grown in ARD soil to those grown in disinfected ARD soil, 1,206 differentially expressed genes (DEGs) were identified based on a log2 fold change, (LFC) ≥ 1 for up– and ≤ −1 for downregulation (p < 0.05). Subsequent validation revealed a highly significant positive correlation (r = 0.91; p < 0.0001) between RNA-seq and RT-qPCR results indicating a high reliability of the RNA-seq data. PageMan analysis showed that transcripts of genes involved in gibberellic acid (GA) biosynthesis were significantly enriched in the DEG dataset. Most of these GA biosynthesis genes were associated with functions in cell wall stabilization. Further genes were related to detoxification processes. Genes of both groups were expressed significantly higher in Mr5, suggesting that the lower susceptibility to ARD in Mr5 is not due to a single mechanism. These findings contribute to a better insight into ARD response in susceptible and tolerant apple genotypes. However, future research is needed to identify the defense mechanisms, which are most effective for the plant to overcome ARD. Copyright © 2022 Reim, Winkelmann, Cestaro, Rohr and Flachowsky

The impact of membrane protein diffusion on GPCR signaling

This research was carried out as part of the Math-+ excellence cluster (DFG EXC 2046, Project A01-11 [HHB, PA]) and was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the following grants: Project 421152132 SFB1423 subproject C03 (PA), SFB 1470 subproject A01 (PA) and SFB 1114/2 (SW).Spatiotemporal signal shaping in G protein-coupled receptor (GPCR) signaling is now a well-established and accepted notion to explain how signaling specificity can be achieved by a superfamily sharing only a handful of downstream second messengers. Dozens of Gs-coupled GPCR signals ultimately converge on the production of cAMP, a ubiquitous second messenger. This idea is almost always framed in terms of local concentrations, the differences in which are maintained by means of spatial separation. However, given the dynamic nature of the reaction-diffusion processes at hand, the dynamics, in particular the local diffusional properties of the receptors and their cognate G proteins, are also important. By combining some first principle considerations, simulated data, and experimental data of the receptors diffusing on the membranes of living cells, we offer a short perspective on the modulatory role of local membrane diffusion in regulating GPCR-mediated cell signaling. Our analysis points to a diffusion-limited regime where the effective production rate of activated G protein scales linearly with the receptor–G protein complex’s relative diffusion rate and to an interesting role played by the membrane geometry in modulating the efficiency of coupling.Publisher PDFPeer reviewe