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To explore the influence of state changes on brucellosis, a stochastic brucellosis model with semi-Markovian switchings and diffusion is proposed in this paper. When there is no switching, we introduce a critical value R s and obtain the exponential stability in mean square when R s < 1 by using the stochastic Lyapunov function method. Sudden climate changes can drive changes in transmission rate of brucellosis, which can be modelled by a semi-Markov process. We study the influence of stationary distribution of semi-Markov process on extinction of brucellosis in switching environment including both stable states, during which brucellosis dies out, and unstable states, during which brucellosis persists. The results show that increasing the frequencies and average dwell times in stable states to certain extent can ensure the extinction of brucellosis. Finally, numerical simulations are given to illustrate the analytical results. We also suggest that herdsmen should reduce the densities of animal habitation to decrease the contact rate, increase slaughter rate of animals and apply disinfection measures to kill brucella., (© 2024. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

Even in large systems, the effect of noise arising from when populations are initially small can persist to be measurable on the macroscale. A deterministic approximation to a stochastic model will fail to capture this effect, but it can be accurately approximated by including an additional random time-shift to the initial conditions. We present a efficient numerical method to compute this time-shift distribution for a large class of stochastic models. The method relies on differentiation of certain functional equations, which we show can be effectively automated by deriving rules for different types of model rates that arise commonly when mass-action mixing is assumed. Explicit computation of the time-shift distribution can be used to build a practical tool for the efficient generation of macroscopic trajectories of stochastic population models, without the need for costly stochastic simulations. Full code is provided to implement the calculations and we demonstrate the method on an epidemic model and a model of within-host viral dynamics., (© 2024. The Author(s).)

During cell division, the mitotic spindle moves dynamically through the cell to position the chromosomes and determine the ultimate spatial position of the two daughter cells. These movements have been attributed to the action of cortical force generators which pull on the astral microtubules to position the spindle, as well as pushing events by these same microtubules against the cell cortex and plasma membrane. Attachment and detachment of cortical force generators working antagonistically against centring forces of microtubules have been modelled previously (Grill et al. in Phys Rev Lett 94:108104, 2005) via stochastic simulations and mean-field Fokker-Planck equations (describing random motion of force generators) to predict oscillations of a spindle pole in one spatial dimension. Using systematic asymptotic methods, we reduce the Fokker-Planck system to a set of ordinary differential equations (ODEs), consistent with a set proposed by Grill et al., which can provide accurate predictions of the conditions for the Fokker-Planck system to exhibit oscillations. In the limit of small restoring forces, we derive an algebraic prediction of the amplitude of spindle-pole oscillations and demonstrate the relaxation structure of nonlinear oscillations. We also show how noise-induced oscillations can arise in stochastic simulations for conditions in which the mean-field Fokker-Planck system predicts stability, but for which the period can be estimated directly by the ODE model and the amplitude by a related stochastic differential equation that incorporates random binding kinetics., (© 2024. The Author(s).)

The dynamics that govern disease spread are hard to model because infections are functions of both the underlying pathogen as well as human or animal behavior. This challenge is increased when modeling how diseases spread between different spatial locations. Many proposed spatial epidemiological models require trade-offs to fit, either by abstracting away theoretical spread dynamics, fitting a deterministic model, or by requiring large computational resources for many simulations. We propose an approach that approximates the complex spatial spread dynamics with a Gaussian process. We first propose a flexible spatial extension to the well-known SIR stochastic process, and then we derive a moment-closure approximation to this stochastic process. This moment-closure approximation yields ordinary differential equations for the evolution of the means and covariances of the susceptibles and infectious through time. Because these ODEs are a bottleneck to fitting our model by MCMC, we approximate them using a low-rank emulator. This approximation serves as the basis for our hierarchical model for noisy, underreported counts of new infections by spatial location and time. We demonstrate using our model to conduct inference on simulated infections from the underlying, true spatial SIR jump process. We then apply our method to model counts of new Zika infections in Brazil from late 2015 through early 2016., (© The Author(s) 2024. Published by Oxford University Press on behalf of The International Biometric Society.)

In this paper, we introduce the numerical strategy for mixed uncertainty propagation based on probability and Dempster-Shafer theories, and apply it to the computational model of peristalsis in a heart-pumping system. Specifically, the stochastic uncertainty in the system is represented with random variables while epistemic uncertainty is represented using non-probabilistic uncertain variables with belief functions. The mixed uncertainty is propagated through the system, resulting in the uncertainty in the chosen quantities of interest (QoI, such as flow volume, cost of transport and work). With the introduced numerical method, the uncertainty in the statistics of QoIs will be represented using belief functions. With three representative probability distributions consistent with the belief structure, global sensitivity analysis has also been implemented to identify important uncertain factors and the results have been compared between different peristalsis models. To reduce the computational cost, physics constrained generalized polynomial chaos method is adopted to construct cheaper surrogates as approximations for the full simulation., (© 2024. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

Noise activity is known to affect neural networks, enhance the system response to weak external signals, and lead to stochastic resonance phenomenon that can effectively amplify signals in nonlinear systems. In most treatments, channel noise has been modeled based on multi-state Markov descriptions or the use stochastic differential equation models. Here we probe a computationally simple approach based on a minor modification of the traditional Hodgkin-Huxley approach to embed noise in neural response. Results obtained from numerous simulations with different excitation frequencies and noise amplitudes for the action potential firing show very good agreement with output obtained from well-established models. Furthermore, results from the Mann-Whitney U Test reveal a statistically insignificant difference. The distribution of the time interval between successive potential spikes obtained from this simple approach compared very well with the results of complicated Fox and Lu type methods at much reduced computational cost. This present method could also possibly be applied to the analysis of spatial variations and/or differences in characteristics of random incident electromagnetic signals., (Creative Commons Attribution license.)

The landscape of computational modeling in cancer systems biology is diverse, offering a spectrum of models and frameworks, each with its own trade-offs and advantages. Ideally, models are meant to be useful in refining hypotheses, to sharpen experimental procedures and, in the longer run, even for applications in personalized medicine. One of the greatest challenges is to balance model realism and detail with experimental data to eventually produce useful data-driven models. We contribute to this quest by developing a transparent, highly parsimonious, first principle in silico model of a growing avascular tumor. We initially formulate the physiological considerations and the specific model within a stochastic cell-based framework. We next formulate a corresponding mean-field model using partial differential equations which is amenable to mathematical analysis. Despite a few notable differences between the two models, we are in this way able to successfully detail the impact of all parameters in the stability of the growth process and on the eventual tumor fate of the stochastic model. This facilitates the deduction of Bayesian priors for a given situation, but also provides important insights into the underlying mechanism of tumor growth and progression. Although the resulting model framework is relatively simple and transparent, it can still reproduce the full range of known emergent behavior. We identify a novel model instability arising from nutrient starvation and we also discuss additional insight concerning possible model additions and the effects of those. Thanks to the framework's flexibility, such additions can be readily included whenever the relevant data become available., (© 2024. The Author(s).)

Vesicles carry out many essential functions within cells through the processes of endocytosis, exocytosis, and passive and active transport. This includes transporting and delivering molecules between different parts of the cell, and storing and releasing neurotransmitters in neurons. To date, computational simulation of these key biological players has been rather limited and has not advanced at the same pace as other aspects of cell modeling, restricting the realism of computational models. We describe a general vesicle modeling tool that has been designed for wide application to a variety of cell models, implemented within our software STochastic Engine for Pathway Simulation (STEPS), a stochastic reaction-diffusion simulator that supports realistic reconstructions of cell tissue in tetrahedral meshes. The implementation is validated in an extensive test suite, parallel performance is demonstrated in a realistic synaptic bouton model, and example models are visualized in a Blender extension module., (© 2024. The Author(s).)

In this article, we have constructed a stochastic SIR model with healthcare resources and logistic growth, aiming to explore the effect of random environment and healthcare resources on disease transmission dynamics. We have showed that under mild extra conditions, there exists a critical parameter, i.e., the basic reproduction number $ R&#95;0^s $, which completely determines the dynamics of disease: when $ R&#95;0^s < 1 $, the disease is eradicated; while when $ R&#95;0^s > 1 $, the disease is persistent. To validate our theoretical findings, we conducted some numerical simulations using actual parameter values of COVID-19. Both our theoretical and simulation results indicated that (1) the white noise can significantly affect the dynamics of a disease, and importantly, it can shift the stability of the disease-free equilibrium; (2) infectious disease resurgence may be caused by random switching of the environment; and (3) it is vital to maintain adequate healthcare resources to control the spread of disease.

We study an extension of the stochastic SIS (Susceptible-Infectious-Susceptible) model in continuous time that accounts for variation amongst individuals. By examining its limiting behaviour as the population size grows we are able to exhibit conditions for the infection to become endemic.

Cells mechanically interact with their environment to sense, for example, topography, elasticity and mechanical cues from other cells. Mechano-sensing has profound effects on cellular behaviour, including motility. The current study aims to develop a mathematical model of cellular mechano-sensing on planar elastic substrates and demonstrate the model's predictive capabilities for the motility of individual cells in a colony. In the model, a cell is assumed to transmit an adhesion force, derived from a dynamic focal adhesion integrin density, that locally deforms a substrate, and to sense substrate deformation originating from neighbouring cells. The substrate deformation from multiple cells is expressed as total strain energy density with a spatially varying gradient. The magnitude and direction of the gradient at the cell location define the cell motion. Cell-substrate friction, partial motion randomness, and cell death and division are included. The substrate deformation by a single cell and the motility of two cells are presented for several substrate elasticities and thicknesses. The collective motility of 25 cells on a uniform substrate mimicking the closure of a circular wound of 200 µm is predicted for deterministic and random motion. Cell motility on substrates with varying elasticity and thickness is explored for four cells and 15 cells, the latter again mimicking wound closure. Wound closure by 45 cells is used to demonstrate the simulation of cell death and division during migration. The mathematical model can adequately simulate the mechanically induced collective cell motility on planar elastic substrates. The model is suitable for extension to other cell and substrates shapes and the inclusion of chemotactic cues, offering the potential to complement in vitro and in vivo studies., (© 2023. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

We propose a data-driven stochastic method that allows the simulation of a complex system's long-term evolution. Given a large amount of historical data on trajectories in a multi-dimensional phase space, our method simulates the time evolution of a system based on a random selection of partial trajectories in the data without detailed knowledge of the system dynamics. We apply this method to a large data set of time evolution of approximately one million business firms for a quarter century. Accordingly, from simulations starting from arbitrary initial conditions, we obtain a stationary distribution in three-dimensional log-size phase space, which satisfies the allometric scaling laws of three variables. Furthermore, universal distributions of fluctuation around the scaling relations are consistent with the empirical data.

Pre-exposure prophylaxis (PrEP) is an important pillar to prevent HIV transmission. Because of experimental and clinical shortcomings, mathematical models that integrate pharmacological, viral- and host factors are frequently used to quantify clinical efficacy of PrEP. Stochastic simulations of these models provides sample statistics from which the clinical efficacy is approximated. However, many stochastic simulations are needed to reduce the associated sampling error. To remedy the shortcomings of stochastic simulation, we developed a numerical method that allows predicting the efficacy of arbitrary prophylactic regimen directly from a viral dynamics model, without sampling. We apply the method to various hypothetical dolutegravir (DTG) prophylaxis scenarios. The approach is verified against state-of-the-art stochastic simulation. While the method is more accurate than stochastic simulation, it is superior in terms of computational performance. For example, a continuous 6-month prophylactic profile is computed within a few seconds on a laptop computer. The method's computational performance, therefore, substantially expands the horizon of feasible analysis in the context of PrEP, and possibly other applications., Competing Interests: The authors have declared that no competing interests exist.

Stochastic models of biomolecular reaction networks are commonly employed in systems and synthetic biology to study the effects of stochastic fluctuations emanating from reactions involving species with low copy-numbers. For such models, the Kolmogorov's forward equation is called the chemical master equation (CME), and it is a fundamental system of linear ordinary differential equations (ODEs) that describes the evolution of the probability distribution of the random state-vector representing the copy-numbers of all the reacting species. The size of this system is given by the number of states that are accessible by the chemical system, and for most examples of interest this number is either very large or infinite. Moreover, approximations that reduce the size of the system by retaining only a finite number of important chemical states (e.g. those with non-negligible probability) result in high-dimensional ODE systems, even when the number of reacting species is small. Consequently, accurate numerical solution of the CME is very challenging, despite the linear nature of the underlying ODEs. One often resorts to estimating the solutions via computationally intensive stochastic simulations. The goal of the present paper is to develop a novel deep-learning approach for computing solution statistics of high-dimensional CMEs by reformulating the stochastic dynamics using Kolmogorov's backward equation. The proposed method leverages superior approximation properties of Deep Neural Networks (DNNs) to reliably estimate expectations under the CME solution for several user-defined functions of the state-vector. This method is algorithmically based on reinforcement learning and it only requires a moderate number of stochastic simulations (in comparison to typical simulation-based approaches) to train the "policy function". This allows not just the numerical approximation of various expectations for the CME solution but also of its sensitivities with respect to all the reaction network parameters (e.g. rate constants). We provide four examples to illustrate our methodology and provide several directions for future research., Competing Interests: The authors have declared that no competing interests exist.

Biochemical systems consist of numerous elementary reactions governed by the law of mass action. However, experimentally characterizing all the elementary reactions is nearly impossible. Thus, over a century, their deterministic models that typically contain rapid reversible bindings have been simplified with non-elementary reaction functions (e.g., Michaelis-Menten and Morrison equations). Although the non-elementary reaction functions are derived by applying the quasi-steady-state approximation (QSSA) to deterministic systems, they have also been widely used to derive propensities for stochastic simulations due to computational efficiency and simplicity. However, the validity condition for this heuristic approach has not been identified even for the reversible binding between molecules, such as protein-DNA, enzyme-substrate, and receptor-ligand, which is the basis for living cells. Here, we find that the non-elementary propensities based on the deterministic total QSSA can accurately capture the stochastic dynamics of the reversible binding in general. However, serious errors occur when reactant molecules with similar levels tightly bind, unlike deterministic systems. In that case, the non-elementary propensities distort the stochastic dynamics of a bistable switch in the cell cycle and an oscillator in the circadian clock. Accordingly, we derive alternative non-elementary propensities with the stochastic low-state QSSA, developed in this study. This provides a universally valid framework for simplifying multiscale stochastic biochemical systems with rapid reversible bindings, critical for efficient stochastic simulations of cell signaling and gene regulation. To facilitate the framework, we provide a user-friendly open-source computational package, ASSISTER, that automatically performs the present framework., Competing Interests: The authors have declared that no competing interests exist.

Mathematical models are powerful tools in HIV epidemiology, producing quantitative projections of key indicators such as HIV incidence and prevalence. In order to improve the accuracy of predictions, such models need to incorporate a number of behavioural and biological heterogeneities, especially those related to the sexual network within which HIV transmission occurs. An individual-based model, which explicitly models sexual partnerships, is thus often the most natural type of model to choose. In this paper we present PopART-IBM, a computationally efficient individual-based model capable of simulating 50 years of an HIV epidemic in a large, high-prevalence community in under a minute. We show how the model calibrates within a Bayesian inference framework to detailed age- and sex-stratified data from multiple sources on HIV prevalence, awareness of HIV status, ART status, and viral suppression for an HPTN 071 (PopART) study community in Zambia, and present future projections of HIV prevalence and incidence for this community in the absence of trial intervention., Competing Interests: The authors have declared that no competing interests exist.

Cells polarize their movement or growth toward external directional cues in many different contexts. For example, budding yeast cells grow toward potential mating partners in response to pheromone gradients. Directed growth is controlled by polarity factors that assemble into clusters at the cell membrane. The clusters assemble, disassemble, and move between different regions of the membrane before eventually forming a stable polarity site directed toward the pheromone source. Pathways that regulate clustering have been identified but the molecular mechanisms that regulate cluster mobility are not well understood. To gain insight into the contribution of chemical noise to cluster behavior we simulated clustering using the reaction-diffusion master equation (RDME) framework to account for molecular-level fluctuations. RDME simulations are a computationally efficient approximation, but their results can diverge from the underlying microscopic dynamics. We implemented novel concentration-dependent rate constants that improved the accuracy of RDME-based simulations, allowing us to efficiently investigate how cluster dynamics might be regulated. Molecular noise was effective in relocating clusters when the clusters contained low numbers of limiting polarity factors, and when Cdc42, the central polarity regulator, exhibited short dwell times at the polarity site. Cluster stabilization occurred when abundances or binding rates were altered to either lengthen dwell times or increase the number of polarity molecules in the cluster. We validated key results using full 3D particle-based simulations. Understanding the mechanisms cells use to regulate the dynamics of polarity clusters should provide insights into how cells dynamically track external directional cues., Competing Interests: The authors have declared that no competing interests exist.

During the last decade, a consensus has emerged that the stochastic triggering of an excitable system drives pseudopod formation and subsequent migration of amoeboid cells. The presence of chemoattractant stimuli alters the threshold for triggering this activity and can bias the direction of migration. Though noise plays an important role in these behaviors, mathematical models have typically ignored its origin and merely introduced it as an external signal into a series of reaction-diffusion equations. Here we consider a more realistic description based on a reaction-diffusion master equation formalism to implement these networks. In this scheme, noise arises naturally from a stochastic description of the various reaction and diffusion terms. Working on a three-dimensional geometry in which separate compartments are divided into a tetrahedral mesh, we implement a modular description of the system, consisting of G-protein coupled receptor signaling (GPCR), a local excitation-global inhibition mechanism (LEGI), and signal transduction excitable network (STEN). Our models implement detailed biochemical descriptions whenever this information is available, such as in the GPCR and G-protein interactions. In contrast, where the biochemical entities are less certain, such as the LEGI mechanism, we consider various possible schemes and highlight the differences between them. Our simulations show that even when the LEGI mechanism displays perfect adaptation in terms of the mean level of proteins, the variance shows a dose-dependence. This differs between the various models considered, suggesting a possible means for determining experimentally among the various potential networks. Overall, our simulations recreate temporal and spatial patterns observed experimentally in both wild-type and perturbed cells, providing further evidence for the excitable system paradigm. Moreover, because of the overall importance and ubiquity of the modules we consider, including GPCR signaling and adaptation, our results will be of interest beyond the field of directed migration., Competing Interests: The authors have declared that no competing interests exist.

This paper deals with the problem of dissipativity-based filtering for switched genetic regulatory networks (GRNs) with stochastic perturbation and time-varying delays. By choosing an appropriate piecewise Lyapunov function and using the average dwell time method, we propose a new set of sufficient conditions in terms of Linear matrix inequalities (LMIs) for the existence of dissipative filter, which ensures that the resulting filtering error system is mean-square exponentially stable with dissipativity performance. The filter gains are provided by solving feasible solutions to a certain set of LMIs. A simulation example is given to demonstrate the effectiveness of the desired dissipativity-based filter design approach.

Purpose: Given the high level of expertise required for navigation and interpretation of ultrasound images, computational simulations can facilitate the training of such skills in virtual reality. With ray-tracing based simulations, realistic ultrasound images can be generated. However, due to computational constraints for interactivity, image quality typically needs to be compromised., Methods: We propose herein to bypass any rendering and simulation process at interactive time, by conducting such simulations during a non-time-critical offline stage and then learning image translation from cross-sectional model slices to such simulated frames. We use a generative adversarial framework with a dedicated generator architecture and input feeding scheme, which both substantially improve image quality without increase in network parameters. Integral attenuation maps derived from cross-sectional model slices, texture-friendly strided convolutions, providing stochastic noise and input maps to intermediate layers in order to preserve locality are all shown herein to greatly facilitate such translation task., Results: Given several quality metrics, the proposed method with only tissue maps as input is shown to provide comparable or superior results to a state-of-the-art that uses additional images of low-quality ultrasound renderings. An extensive ablation study shows the need and benefits from the individual contributions utilized in this work, based on qualitative examples and quantitative ultrasound similarity metrics. To that end, a local histogram statistics based error metric is proposed and demonstrated for visualization of local dissimilarities between ultrasound images., Conclusion: A deep-learning based direct transformation from interactive tissue slices to likeness of high quality renderings allow to obviate any complex rendering process in real-time, which could enable extremely realistic ultrasound simulations on consumer-hardware by moving the time-intensive processes to a one-time, offline, preprocessing data preparation stage that can be performed on dedicated high-end hardware.

Interacting Particle Systems (IPSs) are used to model spatio-temporal stochastic systems in many disparate areas of science. We design an algorithmic framework that reduces IPS simulation to simulation of well-mixed Chemical Reaction Networks (CRNs). This framework minimizes the number of associated reaction channels and decouples the computational cost of the simulations from the size of the lattice. Decoupling allows our software to make use of a wide class of techniques typically reserved for well-mixed CRNs. We implement the direct stochastic simulation algorithm in the open source programming language Julia. We also apply our algorithms to several complex spatial stochastic phenomena. including a rock-paper-scissors game, cancer growth in response to immunotherapy, and lipid oxidation dynamics. Our approach aids in standardizing mathematical models and in generating hypotheses based on concrete mechanistic behavior across a wide range of observed spatial phenomena., Competing Interests: The authors have declared that no competing interests exist.

We study the dynamic growth behaviors of Bacillus subtilis biofilms by using a stochastic delay differential equation similar to the Mackey-Glass equation. We observe a wealth of dynamic behavior from a mathematical perspective. Firstly, the change of the time delay (i.e. the time that the growth state of the periphery to affect the stress of the interior cells needs and the time required for the stress signal coming from the interior to reach the peripheral cells) and the external excitation (i.e. the change in temperature) frequency will influence the size of Bacillus subtilis biofilm and the collective growth of the cell community. We also found the resonance phenomena in mathematics, which is of great help in understanding the behavior of stress levels in the biofilm periphery. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form theory and the center manifold reduction. Numerical simulations are carried out to ensure theoretical results. Finally, the effects of noise on stress levels in the biofilm periphery are analyzed and reveal more complex dynamics of the system., (Copyright © 2020 Elsevier B.V. All rights reserved.)

Evolutionary branching occurs when a population with a unimodal phenotype distribution diversifies into a multimodally distributed population consisting of two or more strains. Branching results from frequency-dependent selection, which is caused by interactions between individuals. For example, a population performing a social task may diversify into a cooperator strain and a defector strain. Branching can also occur in multi-dimensional phenotype spaces, such as when two tasks are performed simultaneously. In such cases, the strains may diverge in different directions: possible outcomes include division of labor (with each population performing one of the tasks) or the diversification into a strain that performs both tasks and another that performs neither. Here we show that the shape of the population's phenotypic distribution plays a role in determining the direction of branching. Furthermore, we show that the shape of the distribution is, in turn, contingent on the direction of approach to the evolutionary branching point. This results in a distribution-selection feedback that is not captured in analytical models of evolutionary branching, which assume monomorphic populations. Finally, we show that this feedback can influence long-term evolutionary dynamics and promote the evolution of division of labor., Competing Interests: The authors have declared that no competing interests exist.

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.

The gradual accumulation of damage and dysregulation during the aging of living organisms can be quantified. Even so, the aging process is complex and has multiple interacting physiological scales - from the molecular to cellular to whole tissues. In the face of this complexity, we can significantly advance our understanding of aging with the use of computational models that simulate realistic individual trajectories of health as well as mortality. To do so, they must be systems-level models that incorporate interactions between measurable aspects of age-associated changes. To incorporate individual variability in the aging process, models must be stochastic. To be useful they should also be predictive, and so must be fit or parameterized by data from large populations of aging individuals. In this perspective, we outline where we have been, where we are, and where we hope to go with such computational models of aging. Our focus is on data-driven systems-level models, and on their great potential in aging research., (Copyright © 2020 Elsevier B.V. All rights reserved.)

Boolean logic and arithmetic through DNA excision (BLADE) is a recently developed platform for implementing inducible and logical control over gene expression in mammalian cells, which has the potential to revolutionise cell engineering for therapeutic applications. This 2-input 2-output platform can implement 256 different logical circuits that exploit the specificity and stability of DNA recombination. Here, we develop the first mechanistic mathematical model of the 2-input BLADE platform based on Cre- and Flp-mediated DNA excision. After calibrating the model on experimental data from two circuits, we demonstrate close agreement between model outputs and data on the other 111 circuits that have so far been experimentally constructed using the 2-input BLADE platform. Model simulations of the remaining 143 circuits that have yet to be tested experimentally predict excellent performance of the 2-input BLADE platform across the range of possible circuits. Circuits from both the tested and untested subsets that perform less well consist of a disproportionally high number of STOP sequences. Model predictions suggested that circuit performance declines with a decrease in recombinase expression and new experimental data was generated that confirms this relationship., Competing Interests: The authors have declared that no competing interests exist.

In this paper, we show some dynamical and biological mechanisms of the short-term memory (the fixed point attractor) through the toggle switch in the FitzHugh-Nagumo model (FN). Firstly, we obtain the bistable conditions, show the effect of Gaussian noise on the toggle switch, and explain the short-term memory's switch mechanism by mean first passage time (MFPT). Then, we obtain a Fokker-Planck equation and illustrate the meaning of the monostable and bistable state in the short-term memory. Furthermore, we study the toggle switch under the interaction of network and noise. Meanwhile, we show that network structure and noise play a vital role in the toggle switch based on network mean first passage time (NMFPT). And we illustrate that the modest clustering coefficient and noise are necessary to maintain memories. Finally, the numerical simulation shows that the analytical results agree with it., Competing Interests: The authors declare that there is no conflict of interests., (Copyright © 2020 Qianqian Zheng et al.)

Background: Antibodies revolutionized cancer treatment over the past decades. Despite their successfully application, there are still challenges to overcome to improve efficacy, such as the heterogeneous distribution of antibodies within tumors. Tumor microenvironment features, such as the distribution of tumor and other cell types and the composition of the extracellular matrix may work together to hinder antibodies from reaching the target tumor cells. To understand these interactions, we propose a framework combining in vitro and in silico models. We took advantage of in vitro cancer models previously developed by our group, consisting of tumor cells and fibroblasts co-cultured in 3D within alginate capsules, for reconstruction of tumor microenvironment features., Results: In this work, an experimental-computational framework of antibody transport within alginate capsules was established, assuming a purely diffusive transport, combined with an exponential saturation effect that mimics the saturation of binding sites on the cell surface. Our tumor microenvironment in vitro models were challenged with a fluorescent antibody and its transport recorded using light sheet fluorescence microscopy. Diffusion and saturation parameters of the computational model were adjusted to reproduce the experimental antibody distribution, with root mean square error under 5%. This computational framework is flexible and can simulate different random distributions of tumor microenvironment elements (fibroblasts, cancer cells and collagen fibers) within the capsule. The random distribution algorithm can be tuned to follow the general patterns observed in the experimental models., Conclusions: We present a computational and microscopy framework to track and simulate antibody transport within the tumor microenvironment that complements the previously established in vitro models platform. This framework paves the way to the development of a valuable tool to study the influence of different components of the tumor microenvironment on antibody transport.

We study continuous-time multi-agent models, where agents interact according to a network topology. At any point in time, each agent occupies a specific local node state. Agents change their state at random through interactions with neighboring agents. The time until a transition happens can follow an arbitrary probability density. Stochastic (Monte-Carlo) simulations are often the preferred-sometimes the only feasible-approach to study the complex emerging dynamical patterns of such systems. However, each simulation run comes with high computational costs mostly due to updating the instantaneous rates of interconnected agents after each transition. This work proposes a stochastic rejection-based, event-driven simulation algorithm that scales extremely well with the size and connectivity of the underlying contact network and produces statistically correct samples. We demonstrate the effectiveness of our method on different information spreading models., Competing Interests: The authors have declared that no competing interests exist.

For over a century, the Michaelis-Menten (MM) rate law has been used to describe the rates of enzyme-catalyzed reactions and gene expression. Despite the ubiquity of the MM rate law, it accurately captures the dynamics of underlying biochemical reactions only so long as it is applied under the right condition, namely, that the substrate is in large excess over the enzyme-substrate complex. Unfortunately, in circumstances where its validity condition is not satisfied, especially so in protein interaction networks, the MM rate law has frequently been misused. In this review, we illustrate how inappropriate use of the MM rate law distorts the dynamics of the system, provides mistaken estimates of parameter values, and makes false predictions of dynamical features such as ultrasensitivity, bistability, and oscillations. We describe how these problems can be resolved with a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA). Furthermore, we show that the tQSSA can be used for accurate stochastic simulations at a lower computational cost than using the full set of mass-action rate laws. This review describes how to use quasi-steady state approximations in the right context, to prevent drawing erroneous conclusions from in silico simulations., Competing Interests: The authors have declared that no competing interests exist.

FLASH radiotherapy delivers a high dose (≥10 Gy) at a high rate (≥40 Gy/s). In this way, particles are delivered in pulses as short as a few nanoseconds. At that rate, intertrack reactions between chemical species produced within the same pulse may affect the heterogeneous chemistry stage of water radiolysis. This stochastic process suits the capabilities of the Monte Carlo method, which can model intertrack effects to aid in radiobiology research, including the design and interpretation of experiments. In this work, the TOPAS-nBio Monte Carlo track-structure code was expanded to allow simulations of intertrack effects in the chemical stage of water radiolysis. Simulation of the behavior of radiolytic yields over a long period of time (up to 50 s) was verified by simulating radiolysis in a Fricke dosimeter irradiated by 60Co γ rays. In addition, LET-dependent G values of protons delivered in single squared pulses of widths, 1 ns, 1 µs and 10 µs, were obtained and compared to simulations using no intertrack considerations. The Fricke simulation for the calculated G value of Fe3+ ion at 50 s was within 0.4% of the accepted value from ICRU Report 34. For LET-dependent G values at the end of the chemical stage, intertrack effects were significant at LET values below 2 keV/µm. Above 2 keV/µm the reaction kinetics remained limited locally within each track and thus, effects of intertrack reactions remained low. Therefore, when track structure simulations are used to investigate the biological damage of FLASH irradiation, these intertrack reactions should be considered. The TOPAS-nBio framework with the expansion to intertrack chemistry simulation provides a useful tool to assist in this task., (©2020 by Radiation Research Society. All rights of reproduction in any form reserved.)

An overarching theory of chromatography capable of modeling all analyte-stationary phase interactions would enable predictive design of pharmaceutically relevant separations. The stochastic theory of chromatography has been postulated as a suitable basis to achieve this goal. Here, we implement Dondi and Cavazzini's Monte Carlo framework that utilizes experimentally accessible single molecule kinetics and use it to correlate heterogenous adsorption statistics at the stationary phase to shifts in asymmetry. The contributions cannot be captured or modeled through ensemble chemometrics. Simulations reveal that peak asymmetry scales non-linearly with longer analyte-stationary phase interactions and migrates towards symmetry across the column length, even without column overloading., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2020. Published by Elsevier B.V.)

More than eighty percent of pancreatic cancer involves ductal adenocarcinoma with an abundant desmoplastic extracellular matrix surrounding the solid tumor entity. This aberrant tumor microenvironment facilitates a strong resistance of pancreatic cancer to medication. Although various therapeutic strategies have been reported to be effective in mice with pancreatic cancer, they still need to be tested quantitatively in wider animal-based experiments before being applied as therapies. To aid the design of experiments, we develop a cell-based mathematical model to describe cancer progression under therapy with a specific application to pancreatic cancer. The displacement of cells is simulated by solving a large system of stochastic differential equations with the Euler-Maruyama method. We consider treatment with the PEGylated drug PEGPH20 that breaks down hyaluronan in desmoplastic stroma followed by administration of the chemotherapy drug gemcitabine to inhibit the proliferation of cancer cells. Modeling the effects of PEGPH20 + gemcitabine concentrations is based on Green's fundamental solutions of the reaction-diffusion equation. Moreover, Monte Carlo simulations are performed to quantitatively investigate uncertainties in the input parameters as well as predictions for the likelihood of success of cancer therapy. Our simplified model is able to simulate cancer progression and evaluate treatments to inhibit the progression of cancer.

Measles is an awfully contagious acute viral infection. It can be fatal, causing cough, red eyes, followed by a fever and skin rash with signs of respiratory infection. In this paper, we propose and analyze a model describing the transmission dynamics of a measles epidemic in the human population using the stability theory of differential equations. The model proposed undergoes a backward bifurcation for some parameter values. Sensitivity analysis is carried out on the model parameters in order to determine their impact on the disease dynamics. We extend the model to an optimal control problem by including time-dependent control variables: prevention, treatment of infected people and vaccination of the susceptible humans. In an attempt to minimize the infected people and the cost applied we design the cost functional. Next, we show that optimal control exists for the system, and the Pontryagin maximum principle is employed to characterize the continuous controls. Numerical simulation is performed to justify the analytical results and discussed quantitatively., (Copyright © 2020. Published by Elsevier B.V.)

Although double-precision floating-point arithmetic currently dominates high-performance computing, there is increasing interest in smaller and simpler arithmetic types. The main reasons are potential improvements in energy efficiency and memory footprint and bandwidth. However, simply switching to lower-precision types typically results in increased numerical errors. We investigate approaches to improving the accuracy of reduced-precision fixed-point arithmetic types, using examples in an important domain for numerical computation in neuroscience: the solution of ordinary differential equations (ODEs). The Izhikevich neuron model is used to demonstrate that rounding has an important role in producing accurate spike timings from explicit ODE solution algorithms. In particular, fixed-point arithmetic with stochastic rounding consistently results in smaller errors compared to single-precision floating-point and fixed-point arithmetic with round-to-nearest across a range of neuron behaviours and ODE solvers. A computationally much cheaper alternative is also investigated, inspired by the concept of dither that is a widely understood mechanism for providing resolution below the least significant bit in digital signal processing. These results will have implications for the solution of ODEs in other subject areas, and should also be directly relevant to the huge range of practical problems that are represented by partial differential equations. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'.

The choice of the state space representation of a system can turn into a prominent advantage or burden in any endeavour to mathematically model dynamical systems since it entails the analytical tractability of the related modelling formalism and the efficiency of the numerical computation. The Reaction-Based Model (RBM) of the state space, which is presented in this article, is a novel formalization of the kinetics of a system of interacting molecules. According to our representation, the state S μ of a system of M reactions and N molecular species, is identified with the occurrence of the reaction R μ ( μ = 1, ..., M). The transition between any two states S μ and S ν is modelled as a first-order reaction S μ → S ν and described by mass action-like equation for the partial time derivative of the variables P(S μ , t) and P(S ν , t), which denote the probabilities that the system lies in the two states, respectively. The rate equations for the state probabilities are coupled with those for the abundance of molecular species. Altogether, the rate equations along with the specification of the initial conditions define the Cauchy problem whose solution describes the time-evolution of the system. The RBM has been applied to a typical severely stiff biological case study. The numerical solutions of the system's dynamics turned out to be computationally more efficient and in agreement with the results of the stochastic and hybrid stochastic/deterministic simulation algorithms.

Computational modelling of in vivo protein synthesis is highly complicated, as it requires the simulation of ribosomal movement over the entire transcriptome, as well as consideration of the concentration effects from 40+ different types of tRNAs and numerous other protein factors. Here I report on the development of a stochastic model for protein translation that is capable of simulating the dynamical process of in vivo protein synthesis in a prokaryotic cell containing several thousand unique mRNA sequences, with explicit nucleotide information for each, and report on a number of biological predictions which are beyond the scope of existing models. In particular, I show that, when the complex network of concentration dependent interactions between elongation factors, tRNAs, ribosomes, and other factors required for protein synthesis are included in full detail, several biological phenomena, such as the increasing peptide elongation rate with bacterial growth rate, are predicted as emergent properties of the model. The stochastic model presented here demonstrates the importance of considering the translational process at this level of detail, and provides a platform to interrogate various aspects of translation that are difficult to study in more coarse-grained models., Competing Interests: The author has declared that no competing interests exist.

Breeding programs for different species aim to improve performance by testing members of full-sib (FS) and half-sib (HS) families in different environments. When genotypes respond differently to changes in the environment, this is defined as genotype by environment (G × E) interaction. The presence of common environmental effects within families generates covariance between siblings, and these effects should be taken into account when estimating a genetic correlation. Therefore, an optimal design should be established to accurately estimate the genetic correlation between environments in the presence of common environmental effects. We used stochastic simulation to find the optimal population structure using a combination of FS and HS groups with different levels of common environmental effects. Results show that in a population with a constant population size of 2,000 individuals per environment, ignoring common environmental effects when they are present in the population will lead to an upward bias in the estimated genetic correlation of on average 0.3 when the true genetic correlation is 0.5. When no common environmental effects are present in the population, the lowest standard error (SE) of the estimated genetic correlation was observed with a mating ratio of one dam per sire, and 10 offspring per sire per environment. When common environmental effects are present in the population and are included in the model, the lowest SE is obtained with mating ratios of at least 5 dams per sire and with a minimum number of 10 offspring per sire per environment. We recommend that studies that aim to estimate the magnitude of G × E in pigs, chicken, and fish should acknowledge the potential presence of common environmental effects and adjust the mating ratio accordingly., (© The Author(s) 2020. Published by Oxford University Press on behalf of the American Society of Animal Science.)

In this article, we present an overview of simulation strategies in the context of subcellular domains where calcium-dependent signaling plays an important role. The presentation follows the spatial and temporal scales involved and represented by each algorithm. As an exemplary cell type, we will mainly cite work done on striated muscle cells, i.e. skeletal and cardiac muscle. For these cells, a wealth of ultrastructural, biophysical and electrophysiological data is at hand. Moreover, these cells also express ubiquitous signaling pathways as they are found in many other cell types and thus, the generalization of the methods and results presented here is straightforward.The models considered comprise the basic calcium signaling machinery as found in most excitable cell types including Ca 2+ ions, diffusible and stationary buffer systems, and calcium regulated calcium release channels. Simulation strategies can be differentiated in stochastic and deterministic algorithms. Historically, deterministic approaches based on the macroscopic reaction rate equations were the first models considered. As experimental methods elucidated highly localized Ca 2+ signaling events occurring in femtoliter volumes, stochastic methods were increasingly considered. However, detailed simulations of single molecule trajectories are rarely performed as the computational cost implied is too large. On the mesoscopic level, Gillespie's algorithm is extensively used in the systems biology community and with increasing frequency also in models of microdomain calcium signaling. To increase computational speed, fast approximations were derived from Gillespie's exact algorithm, most notably the chemical Langevin equation and the τ-leap algorithm. Finally, in order to integrate deterministic and stochastic effects in multiscale simulations, hybrid algorithms are increasingly used. These include stochastic models of ion channels combined with deterministic descriptions of the calcium buffering and diffusion system on the one hand, and algorithms that switch between deterministic and stochastic simulation steps in a context-dependent manner on the other. The basic assumptions of the listed methods as well as implementation schemes are given in the text. We conclude with a perspective on possible future developments of the field.

Large scale biological responses are inherently uncertain, in part as a consequence of noisy systems that do not respond deterministically to perturbations and measurement errors inherent to technological limitations. As a result, they are computationally difficult to model and current approaches are notoriously slow and computationally intensive (multiscale stochastic models), fail to capture the effects of noise across a system (chemical kinetic models), or fail to provide sufficient biological fidelity because of broad simplifying assumptions (stochastic differential equations). We use a new approach to modeling multiscale stationary biological processes that embraces the noise found in experimental data to provide estimates of the parameter uncertainties and the potential mis-specification of models. Our approach models the mean stationary response at each biological level given a particular expected response relationship, capturing variation around this mean using conditional Monte Carlo sampling that is statistically consistent with training data. A conditional probability distribution associated with a biological response can be reconstructed using this method for a subset of input values, which overcomes the parameter identification problem. Our approach could be applied in addition to dynamical modeling methods (see above) to predict uncertain biological responses over experimental time scales. To illustrate this point, we apply the approach to a test case in which we model the variation associated with measurements at multiple scales of organization across a reproduction-related Adverse Outcome Pathway described for teleosts., Competing Interests: The authors have declared that no competing interests exist.

We suggest a new method for building data-driven dynamical models from observed multidimensional time series. The method is based on a recurrent neural network with specific structure, which allows for the joint reconstruction of both a low-dimensional embedding for dynamical components in the data and an evolution operator. The key link of the method is a Bayesian optimization of both model structure and the hypothesis about the data generating law, which is needed for constructing the cost function for model learning. First, the performance of the method is successfully tested in the situation when a signal from a low-dimensional dynamical system is hidden in noisy multidimensional observations. Second, the method is used for building the data-driven model of the low frequency variability (LFV) in the quasigeostrophic model of the Earth's midlatitude atmosphere-a high-dimensional chaotic system. It is demonstrated that the key regimes of the atmospheric LFV are reproduced correctly in data simulations by means of the obtained model.

Stochastic actor-oriented models (SAOMs) can be used to analyse dynamic network data, collected by observing a network and a behaviour in a panel design. The parameters of SAOMs are usually estimated by the method of moments (MoM) implemented by a stochastic approximation algorithm, where statistics defining the moment conditions correspond in a natural way to the parameters. Here, we propose to apply the generalized method of moments (GMoM), using more statistics than parameters. We concentrate on statistics depending jointly on the network and the behaviour, because of the importance of their interdependence, and propose to add contemporaneous statistics to the usual cross-lagged statistics. We describe the stochastic algorithm developed to approximate the GMoM solution. A small simulation study supports the greater statistical efficiency of the GMoM estimator compared to the MoM.

Spleen colony-forming unit (CFU-s) growth in spleen colonies is a stochastic process in which CFU-s, with each cell division, can either self-renew or differentiate, but not both. The fundamental parameter governing this process is p, or the probability of CFU-s self-renewing. Previously, when CFU-s growth was modeled by Monte Carlo simulations, p was kept constant during the 20 cell cycles required for the modeling. However, it is known that CFU-s self-renewal undergoes decline with proliferation. In the present study, this was taken into consideration, such that p was forced to undergo a small decline with each cell division. These new Monte Carlo calculations give an improved fit to CFU-s cumulative growth curves as compared with those calculations using fixed p. This new model, referred to as the variable p model, offers an explanation as to how large mammals can amplify marrow output from stem cell compartments that are no larger than those found in small mammals. It is a model in which small changes in active stem cell aging generate disproportionally large increases in the size of active stem cell clones., (Copyright © 2019 ISEH -- Society for Hematology and Stem Cells. Published by Elsevier Inc. All rights reserved.)

Understanding how cells proliferate, migrate and die in various environments is essential in determining how organisms develop and repair themselves. Continuum mathematical models, such as the logistic equation and the Fisher-Kolmogorov equation, can describe the global characteristics observed in commonly used cell biology assays, such as proliferation and scratch assays. However, these continuum models do not account for single-cell-level mechanics observed in high-throughput experiments. Mathematical modelling frameworks that represent individual cells, often called agent-based models, can successfully describe key single-cell-level features of these assays but are computationally infeasible when dealing with large populations. In this work, we propose an agent-based model with crowding effects that is computationally efficient and matches the logistic and Fisher-Kolmogorov equations in parameter regimes relevant to proliferation and scratch assays, respectively. This stochastic agent-based model allows multiple agents to be contained within compartments on an underlying lattice, thereby reducing the computational storage compared to existing agent-based models that allow one agent per site only. We propose a systematic method to determine a suitable compartment size. Implementing this compartment-based model with this compartment size provides a balance between computational storage, local resolution of agent behaviour and agreement with classical continuum descriptions.

Background: The challenges of glycaemic control in critically ill patients have been debated for 20 years. While glycaemic control shows benefits inter- and intra-patient metabolic variability results in increased hypoglycaemia and glycaemic variability, both increasing morbidity and mortality. Hence, current recommendations for glycaemic control target higher glycaemic ranges, guided by the fear of harm. Lately, studies have proven the ability to provide safe, effective control for lower, normoglycaemic, ranges, using model-based computerised methods. Such methods usually identify patient-specific physiological parameters to personalize titration of insulin and/or nutrition. The Stochastic-Targeted (STAR) glycaemic control framework uses patient-specific insulin sensitivity and a stochastic model of its future variability to directly account for both inter- and intra-patient variability in a risk-based insulin-dosing approach., Results: In this study, a more personalized and specific 3D version of the stochastic model used in STAR is compared to the current 2D stochastic model, both built using kernel-density estimation methods. Fivefold cross validation on 681 retrospective patient glycaemic control episodes, totalling over 65,000 h of control, is used to determine whether the 3D model better captures metabolic variability, and the potential gain in glycaemic outcome is assessed using validated virtual trials. Results show that the 3D stochastic model has similar forward predictive power, but provides significantly tighter, more patient-specific, prediction ranges, showing the 2D model over-conservative > 70% of the time. Virtual trial results show that overall glycaemic safety and performance are similar, but the 3D stochastic model reduced median blood glucose levels (6.3 [5.7, 7.0] vs. 6.2 [5.6, 6.9]) with a higher 61% vs. 56% of blood glucose within the 4.4-6.5 mmol/L range., Conclusions: This improved performance is achieved with higher insulin rates and higher carbohydrate intake, but no loss in safety from hypoglycaemia. Thus, the 3D stochastic model developed better characterises patient-specific future insulin sensitivity dynamics, resulting in improved simulated glycaemic outcomes and a greater level of personalization in control. The results justify inclusion into ongoing clinical use of STAR.

For many biological applications, exploration of the massive parametric space of a mechanism-based model can impose a prohibitive computational demand. To overcome this limitation, we present a framework to improve computational efficiency by orders of magnitude. The key concept is to train a neural network using a limited number of simulations generated by a mechanistic model. This number is small enough such that the simulations can be completed in a short time frame but large enough to enable reliable training. The trained neural network can then be used to explore a much larger parametric space. We demonstrate this notion by training neural networks to predict pattern formation and stochastic gene expression. We further demonstrate that using an ensemble of neural networks enables the self-contained evaluation of the quality of each prediction. Our work can be a platform for fast parametric space screening of biological models with user defined objectives.

We study stochastic Amari-type neural field equations, which are mean-field models for neural activity in the cortex. We prove that under certain assumptions on the coupling kernel, the neural field model can be viewed as a gradient flow in a nonlocal Hilbert space. This makes all gradient flow methods available for the analysis, which could previously not be used, as it was not known, whether a rigorous gradient flow formulation exists. We show that the equation is well-posed in the nonlocal Hilbert space in the sense that solutions starting in this space also remain in it for all times and space-time regularity results hold for the case of spatially correlated noise. Uniqueness of invariant measures, ergodic properties for the associated Feller semigroups, and several examples of kernels are also discussed.

This paper studies a nonlinear dynamical phenomenon called the multiple firing event (MFE) in a spatially heterogeneous stochastic neural field model, which is extended from that in our previous paper (Li et al. in J Math Biol 78:83-115, 2018). MFEs are a partially synchronized spiking barrages that are believed to be responsible for the Gamma oscillation. Rigorous results about the stochastic stability and the law of large numbers are proved, which further imply the well-definedness and computability of many quantities related to MFEs. Then we devote to study spatial and temporal properties of MFEs. Our key finding is that MFEs are spatially correlated but the spatial correlation decays quickly. Detailed mathematical justifications are made based on our qualitative models that aim to demonstrate the mechanism of MFEs.

Deciphering the key mechanisms of morphogenesis during embryonic development is crucial to understanding the guiding principles of the body plan and promote applications in biomedical research fields. Although several computational tissue reconstruction methods using cellular gene expression data have been proposed, those methods are insufficient with regard to arranging cells in their correct positions in tissues or organs unless spatial information is explicitly provided. Here, we report SPRESSO, a new in silico three-dimensional (3D) tissue reconstruction method using stochastic self-organizing map (stochastic-SOM) clustering, to estimate the spatial domains of cells in tissues or organs from only their gene expression profiles. With only five gene sets defined by Gene Ontology (GO), we successfully demonstrated the reconstruction of a four-domain structure of mid-gastrula mouse embryo (E7.0) with high reproducibility (success rate = 99%). Interestingly, the five GOs contain 20 genes, most of which are related to differentiation and morphogenesis, such as activin A receptor and Wnt family member genes. Further analysis indicated that Id2 is the most influential gene contributing to the reconstruction. SPRESSO may provide novel and better insights on the mechanisms of 3D structure formation of living tissues via informative genes playing a role as spatial discriminators.