scholarly journals Ordinary Differential Equations in Cancer Biology

2016 ◽  
Author(s):  
Margaret P. Chapman ◽  
Claire J. Tomlin
Keyword(s):  
Experimental Data ◽  
Differential Equations ◽  
Qualitative Analysis ◽  
Ordinary Differential Equations ◽  
Cancer Biology ◽  
Biological Systems ◽  
Short Paper ◽  
Ode Model ◽  
Modeling Framework ◽  
Biological Insight

AbstractOrdinary differential equations (ODEs) provide a classical framework to model the dynamics of biological systems, given temporal experimental data. Qualitative analysis of the ODE model can lead to further biological insight and deeper understanding compared to traditional experiments alone. Simulation of the model under various perturbations can generate novel hypotheses and motivate the design of new experiments. This short paper will provide an overview of the ODE modeling framework, and present examples of how ODEs can be used to address problems in cancer biology.

scholarly journals Symmetries and Conservation Laws for Some Compacton Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
M. L. Gandarias ◽  
M. S. Bruzón ◽  
M. Rosa
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Dynamical Behaviour ◽  
Point Of View ◽  
Nonlinear Dispersion ◽  
Multipliers Method

We consider some equations with compacton solutions and nonlinear dispersion from the point of view of Lie classical reductions. The reduced ordinary differential equations are suitable for qualitative analysis and their dynamical behaviour is described. We derive by using the multipliers method some nontrivial conservation laws for these equations.

scholarly journals Improved constraints increase the predictivity and applicability of a linear programming-based dynamic metabolic modeling framework

2021 ◽  
Author(s):  
Justin Y Lee ◽  
Mark P Styczynski
Keyword(s):  
Experimental Data ◽  
Linear Programming ◽  
Biological Systems ◽  
Metabolic Modeling ◽  
Training Data ◽  
Modeling Framework ◽  
Modeling Tools ◽  
Regulatory Constraints ◽  
Simplifying Assumptions ◽  
Optimal Approach

Background: Current metabolic modeling tools suffer from a variety of limitations, from scalability to simplifying assumptions, that preclude their use in many applications. We recently created a modeling framework, LK-DFBA, that addresses a key gap: capturing metabolite dynamics and regulation while retaining a potentially scalable linear programming structure. Key to this framework's success are the linear kinetics and regulatory constraints imposed on the system. Here, we present improvements to these constraints to improve the predictivity of LK-DFBA models and their applicability to biological systems. Method: Three new constraint approaches were created to better model interactions between metabolites and the reactions they regulate. These new approaches (and the original LK-DFBA approach) were tested on several synthetic and biological systems to determine their performance when using both noiseless and noisy data. To validate our framework, we compared experimental data to metabolite dynamics predicted by LK-DFBA. Results: There was no single optimal type of constraints across all synthetic and biological systems; rather, any one of the four approaches could perform best for a given system. The optimal approach for fitting to wildtype data of a given model was consistently the best approach when predicting new phenotypes for that model. Furthermore, many of LK-DFBA's predictions qualitatively agreed with experimental data. Conclusions: LK-DFBA can be improved by using several kinetics constraint approaches, with the ideal one selected based on wild-type training data. LK-DFBA's ability to predict metabolic trends in experimental data further supports its potential for modeling metabolite dynamics in systems of all sizes.

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 Dynamics of an Imperfect Microbeam Considering its Exact Shape

2014 ◽  
Author(s):  
Ahmad M. Bataineh ◽  
Mohammad I. Younis
Keyword(s):  
Experimental Data ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Reduced Order Model ◽  
Effective Length ◽  
Mode Shapes ◽  
Nonlinear Phenomena ◽  
Nonlinear Ordinary Differential Equations ◽  
Order Model ◽  
Reduced Order

We study the static and dynamic behavior of electrically actuated micromachined arches. First, we conduct experiments on micromachined polysilicon beams by driving them electrically and varying their amplitude and frequency of voltage loads. The results reveal several interesting nonlinear phenomena of jumps, hysteresis, and softening behaviors. Next, we conduct analytical and theoretical investigation to understand the experiments. First, we solve the Eigen value problem analytically. We study the effect of the initial rise on the natural frequency and mode shapes, and use a Galerkin-based procedure to derive a reduced order model, which is then used to solve both the static and dynamic responses. We use two symmetric modes in the reduced order model to have accurate and converged results. We use long time integration to solve the nonlinear ordinary differential equations, and then modify our model using effective length to match experimental results. To further improve the matching with the experimental data, we curve-fit the exact profile of the microbeam to match the experimentally measured profile and use it in the reduced-order model to generate frequency-response curves. Finally, we use another numerical technique, the shooting technique, to solve the nonlinear ordinary differential equations. By using shooting and the curve fitted function, we found that we get good agreement with the experimental data.

scholarly journals Scalable Nonlinear Programming Framework for Parameter Estimation in Dynamic Biological System Models

2018 ◽  
Author(s):  
Sungho Shin ◽  
Ophelia Venturelli ◽  
Victor M. Zavala
Keyword(s):  
Experimental Data ◽  
Parameter Estimation ◽  
Differential Equations ◽  
Parameter Uncertainty ◽  
Large Scale ◽  
Dynamic Models ◽  
Biological Systems ◽  
Computationally Efficient ◽  
Estimation Problems ◽  
Data Points

AbstractWe present a nonlinear programming (NLP) framework for the scalable solution of parameter estimation problems that arise in dynamic modeling of biological systems. Such problems are computationally challenging because they often involve highly nonlinear and stif differential equations as well as many experimental data sets and parameters. The proposed framework uses cutting-edge modeling and solution tools which are computationally efficient, robust, and easy-to-use. Specifically, our framework uses a time discretization approach that: i) avoids repetitive simulations of the dynamic model, ii) enables fully algebraic model implementations and computation of derivatives, and iii) enables the use of computationally efficient nonlinear interior point solvers that exploit sparse and structured linear algebra techniques. We demonstrate these capabilities by solving estimation problems for synthetic human gut microbiome community models. We show that an instance with 156 parameters, 144 differential equations, and 1,704 experimental data points can be solved in less than 3 minutes using our proposed framework (while an off-the-shelf simulation-based solution framework requires over 7 hours). We also create large instances to show that the proposed framework is scalable and can solve problems with up to 2,352 parameters, 2,304 differential equations, and 20,352 data points in less than 15 minutes. Competing methods reported in the computational biology literature cannot address problems of this level of complexity. The proposed framework is flexible, can be broadly applied to dynamic models of biological systems, and enables the implementation of sophisticated estimation techniques to quantify parameter uncertainty, to diagnose observability/uniqueness issues, to perform model selection, and to handle outliers.Author summaryConstructing and validating dynamic models of biological systems spanning biomolecular networks to ecological systems is a challenging problem. Here we present a scalable computational framework to rapidly infer parameters in complex dynamic models of biological systems from large-scale experimental data. The framework was applied to infer parameters of a synthetic microbial community model from large-scale time series data. We also demonstrate that this framework can be used to analyze parameter uncertainty, to diagnose whether the experimental data are sufficient to uniquely determine the parameters, to determine the model that best describes the data, and to infer parameters in the face of data outliers.

Docking Two Models of Insurgency Growth

2013 ◽  
Vol 4 (3) ◽  
pp. 19-30 ◽  
Author(s):  
Michael Jaye ◽  
Robert Burks
Keyword(s):  
Infectious Disease ◽  
Social Science ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Conceptual Model ◽  
The Other ◽  
Political Systems ◽  
Ode Model ◽  
Agent Based ◽  
Sirs Model

The use of agent-based simulations (ABS) in social science applications presents validation challenges. In this study, the authors use two theories for the growth of rebellion, one an ABS and the other implemented as a system of ordinary differential equations (ODEs). Epstein’s (2001) theory for the rise of rebellion serves as one conceptual model. The authors implement this theory in NetLogo, with several modifications. The second conceptual model likens the spread of an insurgency to that of an infectious disease, specifically the susceptible-infected-removed-susceptible (SIRS) model. The authors find that the similarity of the ODE model results to those obtained from certain parameters of the ABS implementation serves as a form of model validation. The term used for this type of validation is docking. In addition, other results obtained from the ABS – not directly attainable from the ODE model but which match observed phenomenon in socio-political systems – also demonstrates operational validity.

A method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters

1988 ◽  
Vol 416 (1851) ◽  
pp. 361-389 ◽  
Keyword(s):  
Differential Equations ◽  
Qualitative Analysis ◽  
Ordinary Differential Equations ◽  
Parameter Space ◽  
Bifurcation Theory ◽  
Parameter Plane ◽  
Phase Portraits ◽  
Relevant Parameter ◽  
Bifurcation Diagrams ◽  
Independent Parameters

In this paper, recent advances in bifurcation theory are specialized to systems describable by two coupled ordinary differential equations (ODEs) containing at most three independent parameters. For such systems, by plotting in the relevant parameter plane the locus of successively degenerate singular points, a complete description of all the qualitatively distinct behaviour of the system can be obtained. The description is in terms of phase portraits and bifurcation diagrams. Even though much use is made of existing results obtained via local analyses, the results of this technique cover the entire parameter space. Furthermore, because the information is built up in successive stages the question of whether the parameters universally unfold a given degeneracy does not arise. This can mean a major saving in effort, particularly for degenerate Hopf points. Finally if, as is often the case, the parameters appear in the system in a simple way, the procedure can be applied analytically because the variables (which will appear non-linearly) can be used to parametrize the relevant loci.

Sign in / Sign up

Export Citation Format

Share Document