Learning interpretable collective variables for spreading processes on networks.

Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They are used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this Letter, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. We demonstrate our method using four example networks: the stochastic block model, a ring-shaped graph, a random regular graph, and a scale-free network generated by the Albert-Barabási model. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.

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 spatial-stochastic 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.

Rate-limiting recovery processes in neurotransmission under sustained stimulation.

At active zones of chemical synapses, an arriving electric signal induces the fusion of vesicles with the presynaptic membrane, thereby releasing neurotransmitters into the synaptic cleft. After a fusion event, both the release site and the vesicle undergo a recovery process before becoming available for reuse again. Of central interest is the question which of the two restoration steps acts as the limiting factor during neurotransmission under high-frequency sustained stimulation. In order to investigate this problem, we introduce a non-linear reaction network which involves explicit recovery steps for both the vesicles and the release sites, and includes the induced time-dependent output current. The associated reaction dynamics are formulated by means of ordinary differential equations (ODEs), as well as via the associated stochastic jump process. While the stochastic jump model describes the dynamics at a single active zone, the average over many active zones is close to the ODE solution and shares its periodic structure. The reason for this can be traced back to the insight that recovery dynamics of vesicles and release sites are statistically almost independent. A sensitivity analysis on the recovery rates based on the ODE formulation reveals that neither the vesicle nor the release site recovery step can be identified as the essential rate-limiting step but that the rate-limiting feature changes over the course of stimulation. Under sustained stimulation, the dynamics given by the ODEs exhibit transient changes leading from an initial depression of the postsynaptic response to an asymptotic periodic orbit, while the individual trajectories of the stochastic jump model lack the oscillatory behavior and asymptotic periodicity of the ODE-solution.

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 analyze 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.

Modelling altered signalling of G-protein coupled receptors in inflamed environment to advance drug design.

We previously reported the successful design, synthesis and testing of the prototype opioid painkiller NFEPP that does not elicit adverse side effects. The design process of NFEPP was based on mathematical modelling of extracellular interactions between G-protein coupled receptors (GPCRs) and ligands, recognizing that GPCRs function differently under pathological versus healthy conditions. We now present an additional and novel stochastic model of GPCR function that includes intracellular dissociation of G-protein subunits and modulation of plasma membrane calcium channels and their dependence on parameters of inflamed and healthy tissue (pH, radicals). The model is validated against in vitro experimental data for the ligands NFEPP and fentanyl at different pH values and radical concentrations. We observe markedly reduced binding affinity and calcium channel inhibition for NFEPP at normal pH compared to lower pH, in contrast to the effect of fentanyl. For increasing radical concentrations, we find enhanced constitutive G-protein activation but reduced ligand binding affinity. Assessing the different effects, the results suggest that, compared to radicals, low pH is a more important determinant of overall GPCR function in an inflamed environment. Future drug design efforts should take this into account.

The Impact of Membrane Protein Diffusion on GPCR Signaling.

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.

Variance of filtered signals: Characterization for linear reaction networks and application to neurotransmission dynamics.

Neurotransmission at chemical synapses relies on the calcium-induced fusion of synaptic vesicles with the presynaptic membrane. The distance of the synaptic vesicle to the calcium channels determines the release probability and consequently the postsynaptic signal. Suitable models of the process need to capture both the mean and the variance observed in electrophysiological measurements of the postsynaptic current. In this work, we propose a method to directly compute the exact first- and second-order moments for signals generated by a linear reaction network under convolution with an impulse response function, rendering computationally expensive numerical simulations of the underlying stochastic counting process obsolete. We show that the autocorrelation of the process is central for the calculation of the filtered signal's second-order moments, and derive a system of PDEs for the cross-correlation functions (including the autocorrelations) of linear reaction networks with time-dependent rates. Finally, we employ our method to efficiently compare different spatial coarse graining approaches for a specific model of synaptic vesicle fusion. Beyond the application to neurotransmission processes, the developed theory can be applied to any linear reaction system that produces a filtered stochastic signal.

Stochastic pH Oscillations in a Model of the Urea-Urease Reaction Confined to Lipid Vesicles.

The urea-urease clock reaction is a pH switch from acid to basic that can turn into a pH oscillator if it occurs inside a suitable open reactor. We numerically study the confinement of the reaction to lipid vesicles, which permit the exchange with an external reservoir by differential transport, enabling the recovery of the pH level and yielding a constant supply of urea molecules. For microscopically small vesicles, the discreteness of the number of molecules requires a stochastic treatment of the reaction dynamics. Our analysis shows that intrinsic noise induces a significant statistical variation of the oscillation period, which increases as the vesicles become smaller. The mean period, however, is found to be remarkably robust for vesicle sizes down to approximately 200 nm, but the periodicity of the rhythm is gradually destroyed for smaller vesicles. The observed oscillations are explained as a canard-like limit cycle that differs from the wide class of conventional feedback oscillators.

Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading.

Agent based models (ABMs) are a useful tool for modeling spatio-temporal population dynamics, where many details can be included in the model description. Their computational cost though is very high and for stochastic ABMs a lot of individual simulations are required to sample quantities of interest. Especially, large numbers of agents render the sampling infeasible. Model reduction to a metapopulation model leads to a significant gain in computational efficiency, while preserving important dynamical properties. Based on a precise mathematical description of spatio-temporal ABMs, we present two different metapopulation approaches (stochastic and piecewise deterministic) and discuss the approximation steps between the different models within this framework. Especially, we show how the stochastic metapopulation model results from a Galerkin projection of the underlying ABM onto a finite-dimensional ansatz space. Finally, we utilize our modeling framework to provide a conceptual model for the spreading of COVID-19 that can be scaled to real-world scenarios.

Hybrid models for chemical reaction networks: Multiscale theory and application to gene regulatory systems.

Well-mixed stochastic chemical kinetics are properly modeled by the chemical master equation (CME) and associated Markov jump processes in molecule number space. If the reactants are present in large amounts, however, corresponding simulations of the stochastic dynamics become computationally expensive and model reductions are demanded. The classical model reduction approach uniformly rescales the overall dynamics to obtain deterministic systems characterized by ordinary differential equations, the well-known mass action reaction rate equations. For systems with multiple scales, there exist hybrid approaches that keep parts of the system discrete while another part is approximated either using Langevin dynamics or deterministically. This paper aims at giving a coherent overview of the different hybrid approaches, focusing on their basic concepts and the relation between them. We derive a novel general description of such hybrid models that allows expressing various forms by one type of equation. We also check in how far the approaches apply to model extensions of the CME for dynamics which do not comply with the central well-mixed condition and require some spatial resolution. A simple but meaningful gene expression system with negative self-regulation is analysed to illustrate the different approximation qualities of some of the hybrid approaches discussed. Especially, we reveal the cause of error in the case of small volume approximations.

The spatiotemporal master equation: 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.

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.

Maximizing hydrogen production by cyanobacteria.

When incubated anaerobically, in the light, in the presence of C2H2 and high concentrations of H2, both Mo-grown Anabaena variabilis and either Mo- or V-grown Anabaena azotica produce large amounts of H2 in addition to the H2 initially added. In contrast, C2H2-reduction is diminished under these conditions. The additional H2-production mainly originates from nitrogenase with the V-enzyme being more effective than the Mo-protein. This enhanced H2-production in the presence of added H2 and C2H2 should be of interest in approaches to commercially exploit solar energy conversion by cyanobacterial photosynthesis for the generation of molecular hydrogen as a clean energy source.

An optimal radial profile order based on the Golden Ratio for time-resolved MRI.

In dynamic magnetic resonance imaging (MRI) studies, the motion kinetics or the contrast variability are often hard to predict, hampering an appropriate choice of the image update rate or the temporal resolution. A constant azimuthal profile spacing (111.246 degrees), based on the Golden Ratio, is investigated as optimal for image reconstruction from an arbitrary number of profiles in radial MRI. The profile order is evaluated and compared with a uniform profile distribution in terms of signal-to-noise ratio (SNR) and artifact level. The favorable characteristics of such a profile order are exemplified in two applications on healthy volunteers. First, an advanced sliding window reconstruction scheme is applied to dynamic cardiac imaging, with a reconstruction window that can be flexibly adjusted according to the extent of cardiac motion that is acceptable. Second, a contrast-enhancing k-space filter is presented that permits reconstructing an arbitrary number of images at arbitrary time points from one raw data set. The filter was utilized to depict the T1-relaxation in the brain after a single inversion prepulse. While a uniform profile distribution with a constant angle increment is optimal for a fixed and predetermined number of profiles, a profile distribution based on the Golden Ratio proved to be an appropriate solution for an arbitrary number of profiles.

Simultaneous imaging and R2* mapping using a radial multi-gradient-echo (rMGE) sequence.

PURPOSE: To demonstrate a rapid MR technique that combines imaging and R2* mapping based on a single radial multi-gradient-echo (rMGE) data set. The technique provides a fast method for online monitoring of the administration of (super-)paramagnetic contrast agents as well as image-guided drug delivery. MATERIALS AND METHODS: Data are acquired using an rMGE sequence, resulting in interleaved undersampled radial k-spaces representing different echo times (TEs). These data sets are reconstructed separately, yielding a series of images with different TEs used for pixelwise R2* mapping. A fast numerical algorithm implemented on a real-time reconstruction platform provides online estimation of the relaxation rate R2*. Simultaneously the images are summed for the computation of a high-resolution image. RESULTS: Convenient high-resolution R2* maps of phantoms and the liver of a healthy volunteer were obtained. In addition to stable intrinsic baseline maps, the proposed technique provides particularly accurate results for the high relaxation rates observed during the presence of (super-)paramagnetic contrast agents. Assuming that the change in R2* is proportional to the concentration of the agent, the technique offers a rough estimate for dynamic dosage. CONCLUSION: The simultaneous online display of morphological and parametric information permits convenient, quantitative surveillance of contrast-agent administration.

SNR enhancement in radial SSFP imaging using partial k-space averaging.

The steady-state free precessing (SSFP) sequences, widely used in MRI today, acquire data only during a short fraction of the repetition time (TR). Thus, they exhibit a poor scan efficiency. In this paper, a novel approach to extending the acquisition window for a given TR without considerably modifying the basic sequence is explored for radial SSFP sequences. The additional data are primarily employed to increase the signal-to-noise ratio, rather than to improve the temporal resolution of the imaging. The approach is analyzed regarding its effect on the image SNR (signal to noise ratio) and the reconstruction algorithm. Results are presented for phantom experiments and cardiac functions studies. The gain in SNR is most notable in rapid imaging, since SNR enhancement for a constant repetition time may be used to compensate for the increase in noise resulting from angular undersampling.