scholarly journals Generation of multicellular spatiotemporal models of population dynamics from ordinary differential equations, with applications in viral infection

BMC Biology ◽  
2021 ◽  
Vol 19 (1) ◽  
Author(s):  
T. J. Sego ◽  
Josua O. Aponte-Serrano ◽  
Juliano F. Gianlupi ◽  
James A. Glazier
Keyword(s):  
Population Dynamics ◽  
Viral Infection ◽  
Spatial Information ◽  
Formal Method ◽  
Multiple Scales ◽  
Biological Systems ◽  
Spatial Models ◽  
Model Parameters ◽  
Ode Model ◽  
Ode Models

Abstract Background The biophysics of an organism span multiple scales from subcellular to organismal and include processes characterized by spatial properties, such as the diffusion of molecules, cell migration, and flow of intravenous fluids. Mathematical biology seeks to explain biophysical processes in mathematical terms at, and across, all relevant spatial and temporal scales, through the generation of representative models. While non-spatial, ordinary differential equation (ODE) models are often used and readily calibrated to experimental data, they do not explicitly represent the spatial and stochastic features of a biological system, limiting their insights and applications. However, spatial models describing biological systems with spatial information are mathematically complex and computationally expensive, which limits the ability to calibrate and deploy them and highlights the need for simpler methods able to model the spatial features of biological systems. Results In this work, we develop a formal method for deriving cell-based, spatial, multicellular models from ODE models of population dynamics in biological systems, and vice versa. We provide examples of generating spatiotemporal, multicellular models from ODE models of viral infection and immune response. In these models, the determinants of agreement of spatial and non-spatial models are the degree of spatial heterogeneity in viral production and rates of extracellular viral diffusion and decay. We show how ODE model parameters can implicitly represent spatial parameters, and cell-based spatial models can generate uncertain predictions through sensitivity to stochastic cellular events, which is not a feature of ODE models. Using our method, we can test ODE models in a multicellular, spatial context and translate information to and from non-spatial and spatial models, which help to employ spatiotemporal multicellular models using calibrated ODE model parameters. We additionally investigate objects and processes implicitly represented by ODE model terms and parameters and improve the reproducibility of spatial, stochastic models. Conclusion We developed and demonstrate a method for generating spatiotemporal, multicellular models from non-spatial population dynamics models of multicellular systems. We envision employing our method to generate new ODE model terms from spatiotemporal and multicellular models, recast popular ODE models on a cellular basis, and generate better models for critical applications where spatial and stochastic features affect outcomes.

scholarly journals Generating Agent-Based Multiscale Multicellular Spatiotemporal Models from Ordinary Differential Equations of Biological Systems, with Applications in Viral Infection

2021 ◽  
Author(s):  
T.J. Sego ◽  
Josua O. Aponte-Serrano ◽  
Juliano F. Gianlupi ◽  
James A. Glazier
Keyword(s):  
Differential Equations ◽  
Viral Infection ◽  
Spatial Information ◽  
Biological Systems ◽  
Spatial Models ◽  
Model Parameters ◽  
Ode Model ◽  
Agent Based ◽  
Ode Models ◽  
Computationally Expensive

AbstractThe biophysics of an organism span scales from subcellular to organismal and include spatial processes like diffusion of molecules, cell migration, and flow of intravenous fluids. Mathematical biology seeks to explain biophysical processes in mathematical terms at, and across, all relevant spatial and temporal scales. While non-spatial, ordinary differential equation (ODE) models are often used and readily calibrated to experimental data, they do not explicitly represent spatial and stochastic features of a biological system, limiting their insights and applications. Spatial models describe biological systems with spatial information but are mathematically complex and computationally expensive, which limits the ability to calibrate and deploy them. In this work we develop a formal method for deriving cell-based, spatial, multicellular models from ODE models of population dynamics in biological systems, and vice-versa. We provide examples of generating spatiotemporal, multicellular models from ODE models of viral infection and immune response. In these models the determinants of agreement of spatial and non-spatial models are the degree of spatial heterogeneity in viral production and rates of extracellular viral diffusion and decay. We show how ODE model parameters can implicitly represent spatial parameters, and cell-based spatial models can generate uncertain predictions through sensitivity to stochastic cellular events, which is not a feature of ODE models. Using our method, we can test ODE models in a multicellular, spatial context and translate information to and from non-spatial and spatial models, which help to employ spatiotemporal multicellular models using calibrated ODE model parameters, investigate objects and processes implicitly represented by ODE model terms and parameters, and improve the reproducibility of spatial, stochastic models. We hope to employ our method to generate new ODE model terms from spatiotemporal, multicellular models, recast popular ODE models on a cellular basis, and generate better models for critical applications where spatial and stochastic features affect outcomes.Statement of SignificanceOrdinary differential equations (ODEs) are widely used to model and efficiently simulate multicellular systems without explicit spatial information, while spatial models permit explicit spatiotemporal modeling but are mathematically complicated and computationally expensive. In this work we develop a method to generate stochastic, agent-based, multiscale models of multicellular systems with spatial resolution at the cellular level according to non-spatial ODE models. We demonstrate how to directly translate model terms and parameters between ODE and spatial models and apply non-spatial model terms to boundary conditions using examples of viral infection modeling, and show how spatial models can interrogate implicitly represented biophysical mechanisms in non-spatial models. We discuss strategies for co-developing spatial and non-spatial models and reconciling disagreements between them.

scholarly journals A Multiscale Multicellular Spatiotemporal Model of Local Influenza Infection and Immune Response

2021 ◽  
Author(s):  
T.J. Sego ◽  
Ericka D. Mochan ◽  
G. Bard Ermentrout ◽  
James A. Glazier
Keyword(s):  
Immune Response ◽  
Viral Infection ◽  
Viral Infections ◽  
Influenza Infection ◽  
Ode Model ◽  
Spatiotemporal Model ◽  
Spatially Resolved ◽  
Ode Models ◽  
Respiratory Viral Infections ◽  
Spatial Cell

AbstractRespiratory viral infections pose a serious public health concern, from mild seasonal influenza to pandemics like those of SARS-CoV-2. Spatiotemporal dynamics of viral infection impact nearly all aspects of the progression of a viral infection, like the dependence of viral replication rates on the type of cell and pathogen, the strength of the immune response and localization of infection. Mathematical modeling is often used to describe respiratory viral infections and the immune response to them using ordinary differential equation (ODE) models. However, ODE models neglect spatially-resolved biophysical mechanisms like lesion shape and the details of viral transport, and so cannot model spatial effects of a viral infection and immune response. In this work, we develop a multiscale, multicellular spatiotemporal model of influenza infection and immune response by combining non-spatial ODE modeling and spatial, cell-based modeling. We employ cellularization, a recently developed method for generating spatial, cell-based, stochastic models from non-spatial ODE models, to generate much of our model from a calibrated ODE model that describes infection, death and recovery of susceptible cells and innate and adaptive responses during influenza infection, and develop models of cell migration and other mechanisms not explicitly described by the ODE model. We determine new model parameters to generate agreement between the spatial and original ODE models under certain conditions, where simulation replicas using our model serve as microconfigurations of the ODE model, and compare results between the models to investigate the nature of viral exposure and impact of heterogeneous infection on the time-evolution of the viral infection. We found that using spatially homogeneous initial exposure conditions consistently with those employed during calibration of the ODE model generates far less severe infection, and that local exposure to virus must be multiple orders of magnitude greater than a uniformly applied exposure to all available susceptible cells. This strongly suggests a prominent role of localization of exposure in influenza A infection. We propose that the particularities of the microenvironment to which a virus is introduced plays a dominant role in disease onset and progression, and that spatially resolved models like ours may be important to better understand and more reliably predict future health states based on susceptibility of potential lesion sites using spatially resolved patient data of the state of an infection. We can readily integrate the immune response components of our model into other modeling and simulation frameworks of viral infection dynamics that do detailed modeling of other mechanisms like viral internalization and intracellular viral replication dynamics, which are not explicitly represented in the ODE model. We can also combine our model with available experimental data and modeling of exposure scenarios and spatiotemporal aspects of mechanisms like mucociliary clearance that are only implicitly described by the ODE model, which would significantly improve the ability of our model to present spatially resolved predictions about the progression of influenza infection and immune response.

scholarly journals In Silico Assessment of 5-FU Therapy via A Mathematical Model with Fuzzy Uncertain Parameters

2020 ◽  
Author(s):  
Sajad Shafiekhani ◽  
Amir. H. Jafari ◽  
L. Jafarzadeh ◽  
N. Gheibi
Keyword(s):  
Differential Equation ◽  
Immune Cells ◽  
In Silico ◽  
Parametric Uncertainty ◽  
Testable Hypothesis ◽  
Model Parameters ◽  
Ode Model ◽  
Ode Models ◽  
Mathematical Oncology ◽  
Uncertainty Band

Abstract Background: Ordinary differential equation (ODE) models widely have been used in mathematical oncology to capture dynamics of tumor and immune cells and evaluate the efficacy of treatments. However, for dynamic models of tumor-immune system (TIS), some parameters are uncertain due to inaccurate, missing or incomplete data, which has hindered the application of ODEs that require accurate parameters. Methods: We extended an available ODE model of TIS interactions via fuzzy logic to illustrate the fuzzification procedure of an ODE model. Fuzzy ODE (FODE) models, in comparison with the stochastic differential equation (SDE) models, assigns a fuzzy number instead of a random number (from a specific probability density function) to the parameters, to capture parametric uncertainty. We used FODE model to predict tumor and immune cells dynamics and assess the efficacy of 5-FU. The present model is configurable for 5-FU chemotherapy injection timing and propose testable hypothesis in vitro/ in vivo experiments. Result: FODE model was used to explore the uncertainty of cells dynamics resulting from parametric uncertainty in presence and absence of 5-FU therapy. In silico experiments revealed that the frequent 5-FU injection created a beneficial tumor microenvironment that exerted detrimental effects on tumor cells by enhancing the infiltration of CD8+ T cells, and NK cells, and decreasing that of myeloid-derived suppressor (MDSC) cells. We investigate the effect of perturbation on model parameters on dynamics of cells through global sensitivity analysis (GSA) and compute correlation between model parameters and cell dynamics. Conclusion: ODE models with fuzzy uncertain kinetic parameters cope with insufficient experimental data in the field of mathematical oncology and can predict cells dynamics uncertainty band. In silico assessment of treatments considering parameter uncertainty and investigating the effect of the drugs on movement of cells dynamics uncertainty band may be more appropriate than in crisp setting.

scholarly journals Modeling biological oscillations: integration of short reaction pauses into a stationary model of a negative feedback loop generates sustained long oscillations

2015 ◽  
Author(s):  
Louis Yang ◽  
Ming Yang
Keyword(s):  
Negative Feedback ◽  
Feedback Loop ◽  
Biological Systems ◽  
Intrinsic Property ◽  
Oscillatory System ◽  
Feedback System ◽  
Negative Feedback Loop ◽  
Ode Model ◽  
Ode Models ◽  
Sustained Oscillations

Sustained oscillations are frequently observed in biological systems consisting of a negative feedback loop, but a mathematical model with two ordinary differential equations (ODE) that has a negative feedback loop structure fails to produce sustained oscillations. Only when a time delay is introduced into the system by expanding to a three-ODE model, transforming to a two-DDE model, or introducing a bistable trigger do stable oscillations present themselves. In this study, we propose another mechanism for producing sustained oscillations based on periodic reaction pauses of chemical reactions in a negative feedback system. We model the oscillatory system behavior by allowing the coefficients in the two-ODE model to be periodic functions of time – called pulsate functions – to account for reactions with go-stop pulses. We find that replacing coefficients in the two-ODE system with pulsate functions with micro-scale (several seconds) pauses can produce stable system-wide oscillations that have periods of approximately one to several hours long. We also compare our two-ODE and three-ODE models with the two-DDE, three-ODE, and three-DDE models without the pulsate functions. Our numerical experiments suggest that sustained long oscillations in biological systems with a negative feedback loop may be an intrinsic property arising from the slow diffusion-based pulsate behavior of biochemical reactions.

scholarly journals Modeling biological oscillations: integration of short reaction pauses into a stationary model of a negative feedback loop generates sustained long oscillations

2015 ◽  
Author(s):  
Louis Yang ◽  
Ming Yang
Keyword(s):  
Negative Feedback ◽  
Feedback Loop ◽  
Biological Systems ◽  
Intrinsic Property ◽  
Oscillatory System ◽  
Feedback System ◽  
Negative Feedback Loop ◽  
Ode Model ◽  
Ode Models ◽  
Sustained Oscillations

Sustained oscillations are frequently observed in biological systems consisting of a negative feedback loop, but a mathematical model with two ordinary differential equations (ODE) that has a negative feedback loop structure fails to produce sustained oscillations. Only when a time delay is introduced into the system by expanding to a three-ODE model, transforming to a two-DDE model, or introducing a bistable trigger do stable oscillations present themselves. In this study, we propose another mechanism for producing sustained oscillations based on periodic reaction pauses of chemical reactions in a negative feedback system. We model the oscillatory system behavior by allowing the coefficients in the two-ODE model to be periodic functions of time – called pulsate functions – to account for reactions with go-stop pulses. We find that replacing coefficients in the two-ODE system with pulsate functions with micro-scale (several seconds) pauses can produce stable system-wide oscillations that have periods of approximately one to several hours long. We also compare our two-ODE and three-ODE models with the two-DDE, three-ODE, and three-DDE models without the pulsate functions. Our numerical experiments suggest that sustained long oscillations in biological systems with a negative feedback loop may be an intrinsic property arising from the slow diffusion-based pulsate behavior of biochemical reactions.

scholarly journals Demographic Dynamics in Multitype Populations with Migrations

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 246
Author(s):  
Manuel Molina-Fernández ◽  
Manuel Mota-Medina
Keyword(s):  
Mathematical Modeling ◽  
Population Dynamics ◽  
Branching Process ◽  
Probability Model ◽  
Biological Systems ◽  
Research Work ◽  
General Setting ◽  
Process Theory ◽  
Multitype Branching Process ◽  
Demographic Dynamics

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies.

scholarly journals Coarse, Medium or Fine? A Quantum Mechanics Approach to Single Species Population Dynamics

2020 ◽  
Author(s):  
Olcay Akman ◽  
Leon Arriola ◽  
Aditi Ghosh ◽  
Ryan Schroeder
Keyword(s):  
Population Dynamics ◽  
Quantum Mechanics ◽  
Single Species ◽  
Stable Equilibrium Point ◽  
Ode Model ◽  
Deterministic Models ◽  
Species Population ◽  
Quantum Approach ◽  
Single Species Population ◽  
Quantum Framework

AbstractStandard heuristic mathematical models of population dynamics are often constructed using ordinary differential equations (ODEs). These deterministic models yield pre-dictable results which allow researchers to make informed recommendations on public policy. A common immigration, natural death, and fission ODE model is derived from a quantum mechanics view. This macroscopic ODE predicts that there is only one stable equilibrium point . We therefore presume that as t → ∞, the expected value should be . The quantum framework presented here yields the same standard ODE model, however with very unexpected quantum results, namely . The obvious questions are: why isn’t , why are the probabilities ≈ 0.37, and where is the missing probability of 0.26? The answer lies in quantum tunneling of probabilities. The goal of this paper is to study these tunneling effects that give specific predictions of the uncertainty in the population at the macroscopic level. These quantum effects open the possibility of searching for “black–swan” events. In other words, using the more sophisticated quantum approach, we may be able to make quantitative statements about rare events that have significant ramifications to the dynamical system.

scholarly journals NetPyNE, a tool for data-driven multiscale modeling of brain circuits

eLife ◽  
2019 ◽  
Vol 8 ◽  
Author(s):  
Salvador Dura-Bernal ◽  
Benjamin A Suter ◽  
Padraig Gleeson ◽  
Matteo Cantarelli ◽  
Adrian Quintana ◽  
...  
Keyword(s):  
Multiple Scales ◽  
Network Models ◽  
Brain Regions ◽  
Data Driven ◽  
Model Parameters ◽  
Biophysical Modeling ◽  
Neuron Network ◽  
Graphical Interfaces ◽  
Information Theoretic Measures ◽  
High Level

Biophysical modeling of neuronal networks helps to integrate and interpret rapidly growing and disparate experimental datasets at multiple scales. The NetPyNE tool (www.netpyne.org) provides both programmatic and graphical interfaces to develop data-driven multiscale network models in NEURON. NetPyNE clearly separates model parameters from implementation code. Users provide specifications at a high level via a standardized declarative language, for example connectivity rules, to create millions of cell-to-cell connections. NetPyNE then enables users to generate the NEURON network, run efficiently parallelized simulations, optimize and explore network parameters through automated batch runs, and use built-in functions for visualization and analysis – connectivity matrices, voltage traces, spike raster plots, local field potentials, and information theoretic measures. NetPyNE also facilitates model sharing by exporting and importing standardized formats (NeuroML and SONATA). NetPyNE is already being used to teach computational neuroscience students and by modelers to investigate brain regions and phenomena.

scholarly journals Benchmark problems for dynamic modeling of intracellular processes

2019 ◽  
Vol 35 (17) ◽  
pp. 3073-3082 ◽  
Author(s):  
Helge Hass ◽  
Carolin Loos ◽  
Elba Raimúndez-Álvarez ◽  
Jens Timmer ◽  
Jan Hasenauer ◽  
...  
Keyword(s):  
Dynamic Models ◽  
Supplementary Information ◽  
Benchmark Problems ◽  
Ode Model ◽  
Cellular Processes ◽  
Ode Models ◽  
Estimated Parameters ◽  
Readable Form ◽  
Machine Readable ◽  
Molecular Compounds

Abstract Motivation Dynamic models are used in systems biology to study and understand cellular processes like gene regulation or signal transduction. Frequently, ordinary differential equation (ODE) models are used to model the time and dose dependency of the abundances of molecular compounds as well as interactions and translocations. A multitude of computational approaches, e.g. for parameter estimation or uncertainty analysis have been developed within recent years. However, many of these approaches lack proper testing in application settings because a comprehensive set of benchmark problems is yet missing. Results We present a collection of 20 benchmark problems in order to evaluate new and existing methodologies, where an ODE model with corresponding experimental data is referred to as problem. In addition to the equations of the dynamical system, the benchmark collection provides observation functions as well as assumptions about measurement noise distributions and parameters. The presented benchmark models comprise problems of different size, complexity and numerical demands. Important characteristics of the models and methodological requirements are summarized, estimated parameters are provided, and some example studies were performed for illustrating the capabilities of the presented benchmark collection. Availability and implementation The models are provided in several standardized formats, including an easy-to-use human readable form and machine-readable SBML files. The data is provided as Excel sheets. All files are available at https://github.com/Benchmarking-Initiative/Benchmark-Models, including step-by-step explanations and MATLAB code to process and simulate the models. Supplementary information Supplementary data are available at Bioinformatics online.

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