Journal articles: 'Molecular dynamics Mathematical models'

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Gruzdev, Roman, and Arkady Soloviev. "Polarizable Models in Molecular Dynamics." Solid State Phenomena 258 (December 2016): 202–5. http://dx.doi.org/10.4028/www.scientific.net/ssp.258.202.

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Current work is devoted to the problems of mathematical modeling of electrically polarized nanomaterials using LAMMPS software. There are next methods in this software for modeling of such kind: the fluctuating charge method; the adiabatic core-shell method; the thermalized Drude dipole method. This work provides information on advantages and disadvantages of each method; well-structured scripts for LAMMPS software. As our primary research is devoted to the crystalline elastic materials, much attention is given to the 1st and 2nd methods. Main purpose of research is to build models for zinc oxide (ZnO) for identification of elastic and piezoelectric constants and behavior of nanostructures in different fields. Results for analysis are given in figures and tables.

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SANCHEZ-OSORIO, ISMAEL, FERNANDO RAMOS, PEDRO MAYORGA, and EDGAR DANTAN. "FOUNDATIONS FOR MODELING THE DYNAMICS OF GENE REGULATORY NETWORKS: A MULTILEVEL-PERSPECTIVE REVIEW." Journal of Bioinformatics and Computational Biology 12, no. 01 (January 28, 2014): 1330003. http://dx.doi.org/10.1142/s0219720013300037.

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A promising alternative for unraveling the principles under which the dynamic interactions among genes lead to cellular phenotypes relies on mathematical and computational models at different levels of abstraction, from the molecular level of protein-DNA interactions to the system level of functional relationships among genes. This review article presents, under a bottom–up perspective, a hierarchy of approaches to modeling gene regulatory network dynamics, from microscopic descriptions at the single-molecule level in the spatial context of an individual cell to macroscopic models providing phenomenological descriptions at the population-average level. The reviewed modeling approaches include Molecular Dynamics, Particle-Based Brownian Dynamics, the Master Equation approach, Ordinary Differential Equations, and the Boolean logic abstraction. Each of these frameworks is motivated by a particular biological context and the nature of the insight being pursued. The setting of gene network dynamic models from such frameworks involves assumptions and mathematical artifacts often ignored by the non-specialist. This article aims at providing an entry point for biologists new to the field and computer scientists not acquainted with some recent biophysically-inspired models of gene regulation. The connections promoting intuition between different abstraction levels and the role that approximations play in the modeling process are highlighted throughout the paper.

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Curcio, Luciano, Laura D'Orsi, and Andrea De Gaetano. "Seven Mathematical Models of Hemorrhagic Shock." Computational and Mathematical Methods in Medicine 2021 (June 3, 2021): 1–34. http://dx.doi.org/10.1155/2021/6640638.

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Although mathematical modelling of pressure-flow dynamics in the cardiocirculatory system has a lengthy history, readily finding the appropriate model for the experimental situation at hand is often a challenge in and of itself. An ideal model would be relatively easy to use and reliable, besides being ethically acceptable. Furthermore, it would address the pathogenic features of the cardiovascular disease that one seeks to investigate. No universally valid model has been identified, even though a host of models have been developed. The object of this review is to describe several of the most relevant mathematical models of the cardiovascular system: the physiological features of circulatory dynamics are explained, and their mathematical formulations are compared. The focus is on the whole-body scale mathematical models that portray the subject’s responses to hypovolemic shock. The models contained in this review differ from one another, both in the mathematical methodology adopted and in the physiological or pathological aspects described. Each model, in fact, mimics different aspects of cardiocirculatory physiology and pathophysiology to varying degrees: some of these models are geared to better understand the mechanisms of vascular hemodynamics, whereas others focus more on disease states so as to develop therapeutic standards of care or to test novel approaches. We will elucidate key issues involved in the modeling of cardiovascular system and its control by reviewing seven of these models developed to address these specific purposes.

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Shain, Kenneth H. "Mathematical Models of Cancer Evolution and Cure." Blood 126, no. 23 (December 3, 2015): SCI—55—SCI—55. http://dx.doi.org/10.1182/blood.v126.23.sci-55.sci-55.

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You cannot cure what you do not understand. So how can mathematical modeling address this pressing issue? The advances in therapeutic success in multiple myeloma over the last decades have hinged on an an army of researchers identifying a critical genetic, epigenetic and biochemical signaling factors within of MM cells as well as the tumor microenvironment (TME). Unfortunately, despite these large scale efforts we do not yet offer our patients curative intent therapy. The inability to provide curative therapy, especially in the setting of HRMM, is characterized by evolving resistance to lines of sequential therapy as a result of alternating clonal dynamics following the failure of initial therapy to eradicate minimal residual disease (MRD). Recent results underline the importance of tumor heterogeneity, in the form of pre-existing genotypically (and phenotypically) distinct sub-populations that translate to drug-resistant phenotypes leading to treatment failure. This phenomenon of “clonal tides”, has been well characterized using contemporary molecular techniques demonstrating that clonal evolution progresses by different evolutionary patterns across patients. Thus, resistance to therapy is a consequence of Darwinian dynamics- influenced by tumor heterogeneity, genomic instability, the TME (ecosystem), and selective pressures induced by therapy. Such evolutionary principles can be analyzed and exploited by mathematical models to personalize therapeutic options for patients with MM. Currently available clinical decision support tools and physician acumen are not able to account for the shear amount of information available. Mathematical models, however, provide a critical mechanism(s) to account of the large number of aspects to help predict and manage MM- accounting for what we do not know. Models can be designed with the specific intent of characterizing intra-tumoral heterogeneity, changing ecosystems, and clinical parameters over time to create patient-specific clinical predictions much like hurricane prediction models. This can only be achieved by creating mathematical models parameterized by longitudinal data of a number of parameters. The novel application of mathematical models based on Darwinian dynamics can be imputed with data to 1) predict progression events (risk of progression to from smoldering to active MM), 2) relapse, and 3) predictions of clinical response of MM patients for the optimizing therapeutics for cure or optimal control of MM; thus, providing invaluable clinical decision support tools. Disclosures: Shain: Celgene: Consultancy , Speakers Bureau ; Amgen/Onyx: Consultancy , Speakers Bureau ; Takeda: Consultancy , Speakers Bureau ; Signal Genetics: Consultancy , Research Funding.

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Frisman, E. Ya, O. L. Zhdanova, M. P. Kulakov, G. P. Neverova, and O. L. Revutskaya. "Mathematical Modeling of Population Dynamics Based on Recurrent Equations: Results and Prospects. Part I." Biology Bulletin 48, no. 1 (January 2021): 1–15. http://dx.doi.org/10.1134/s1062359021010064.

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Abstract Approaches to modeling population dynamics using discrete-time models are described in this two-part review. The development of the scientific ideas of discrete time models, from the Malthus model to modern population models that take into account many factors affecting the structure and dynamics, is presented. The most important and interesting results of recurrent equation application to biological system analysis obtained by the authors are given. In the first part of this review, the population dynamic effects that result from density-dependent regulation of population, the age and sex structures, and the influence of external factors are considered.

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Neelagandan, Nagammal, Irene Lamberti, Hugo J. F. Carvalho, Cédric Gobet, and Felix Naef. "What determines eukaryotic translation elongation: recent molecular and quantitative analyses of protein synthesis." Open Biology 10, no. 12 (December 2020): 200292. http://dx.doi.org/10.1098/rsob.200292.

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Protein synthesis from mRNA is an energy-intensive and tightly controlled cellular process. Translation elongation is a well-coordinated, multifactorial step in translation that undergoes dynamic regulation owing to cellular state and environmental determinants. Recent studies involving genome-wide approaches have uncovered some crucial aspects of translation elongation including the mRNA itself and the nascent polypeptide chain. Additionally, these studies have fuelled quantitative and mathematical modelling of translation elongation. In this review, we provide a comprehensive overview of the key determinants of translation elongation. We discuss consequences of ribosome stalling or collision, and how the cells regulate translation in case of such events. Next, we review theoretical approaches and widely used mathematical models that have become an essential ingredient to interpret complex molecular datasets and study translation dynamics quantitatively. Finally, we review recent advances in live-cell reporter and related analysis techniques, to monitor the translation dynamics of single cells and single-mRNA molecules in real time.

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Erban, Radek. "From molecular dynamics to Brownian dynamics." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2167 (July 8, 2014): 20140036. http://dx.doi.org/10.1098/rspa.2014.0036.

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Three coarse-grained molecular dynamics (MD) models are investigated with the aim of developing and analysing multi-scale methods which use MD simulations in parts of the computational domain and (less detailed) Brownian dynamics (BD) simulations in the remainder of the domain. The first MD model is formulated in one spatial dimension. It is based on elastic collisions of heavy molecules (e.g. proteins) with light point particles (e.g. water molecules). Two three-dimensional MD models are then investigated. The obtained results are applied to a simplified model of protein binding to receptors on the cellular membrane. It is shown that modern BD simulators of intracellular processes can be used in the bulk and accurately coupled with a (more detailed) MD model of protein binding which is used close to the membrane.

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Carson, Ewart R. "The Role of Dynamic Mathematical Models." Alternatives to Laboratory Animals 13, no. 4 (June 1985): 295–98. http://dx.doi.org/10.1177/026119298501300407.

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Weron, Aleksander. "Mathematical Models for Dynamics of Molecular Processes in Living Biological Cells. A Single Particle Tracking Approach." Annales Mathematicae Silesianae 32, no. 1 (September 1, 2018): 5–41. http://dx.doi.org/10.1515/amsil-2017-0019.

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Abstract In this survey paper we present a systematic methodology of how to identify origins of fractional dynamics. We consider three models leading to it, namely fractional Brownian motion (FBM), fractional Lévy stable motion (FLSM) and autoregressive fractionally integrated moving average (ARFIMA) process. The discrete-time ARFIMA process is stationary, and when aggregated, in the limit, it converges to either FBM or FLSM. In this sense it generalizes both models. We discuss three experimental data sets related to some molecular biology problems described by single particle tracking. They are successfully resolved by means of the universal ARFIMA time series model with various noises. Even if the finer details of the estimation procedures are case specific, we hope that the suggested checklist will still have been of great use as a practical guide. In Appendices A-F we describe useful fractional dynamics identification and validation methods.

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Koelle, Katia, and David A. Rasmussen. "Rates of coalescence for common epidemiological models at equilibrium." Journal of The Royal Society Interface 9, no. 70 (September 14, 2011): 997–1007. http://dx.doi.org/10.1098/rsif.2011.0495.

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Coalescent theory provides a mathematical framework for quantitatively interpreting gene genealogies. With the increased availability of molecular sequence data, disease ecologists now regularly apply this body of theory to viral phylogenies, most commonly in attempts to reconstruct demographic histories of infected individuals and to estimate parameters such as the basic reproduction number. However, with few exceptions, the mathematical expressions at the core of coalescent theory have not been explicitly linked to the structure of epidemiological models, which are commonly used to mathematically describe the transmission dynamics of a pathogen. Here, we aim to make progress towards establishing this link by presenting a general approach for deriving a model's rate of coalescence under the assumption that the disease dynamics are at their endemic equilibrium. We apply this approach to four common families of epidemiological models: standard susceptible-infected-susceptible/susceptible-infected-recovered/susceptible-infected-recovered-susceptible models, models with individual heterogeneity in infectivity, models with an exposed but not yet infectious class and models with variable distributions of the infectious period. These results improve our understanding of how epidemiological processes shape viral genealogies, as well as how these processes affect levels of viral diversity and rates of genetic drift. Finally, we discuss how a subset of these coalescent rate expressions can be used for phylodynamic inference in non-equilibrium settings. For the ones that are limited to equilibrium conditions, we also discuss why this is the case. These results, therefore, point towards necessary future work while providing intuition on how epidemiological characteristics of the infection process impact gene genealogies.

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Puwal, Steffan, Bradley J. Roth, and Serge Kruk. "Automating phase singularity localization in mathematical models of cardiac tissue dynamics." Mathematical Medicine and Biology: A Journal of the IMA 22, no. 4 (December 1, 2005): 335–46. http://dx.doi.org/10.1093/imammb/dqi013.

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BEREC, LUDĚK. "MODELS OF ALLEE EFFECTS AND THEIR IMPLICATIONS FOR POPULATION AND COMMUNITY DYNAMICS." Biophysical Reviews and Letters 03, no. 01n02 (April 2008): 157–81. http://dx.doi.org/10.1142/s1793048008000678.

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Allee effects are broadly defined as a decline in individual fitness at low population sizes or densities. Although the roots of the concept go back at least to 1920's, until recently, Allee effects eked out on the periphery of ecological theory, in the shade of the prominently discussed negative density dependence. The situation has changed dramatically in the last ten years or so, and we can find an ever increasing number of studies considering Allee effects from an ever increasing range of disciplines. Mathematical models have always been an important tool by which to assess impacts of Allee effects for population and community dynamics. Actually, much of what we know about Allee effects comes from mathematical models. Up to now, Allee effects have been examined in the context of most existing model structures, and significantly altered our picture of population and community dynamics based on assuming negative density dependence only.

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VINET, ALAIN, DANTE R. CHIALVO, and JOSE JALIFE. "Irregular Dynamics of Excitation in Biologic and Mathematical Models of Cardiac Cells." Annals of the New York Academy of Sciences 601, no. 1 Electrocardio (September 1990): 281–98. http://dx.doi.org/10.1111/j.1749-6632.1990.tb37307.x.

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Zagkos, Loukas, Mark Mc Auley, Jason Roberts, and Nikos I. Kavallaris. "Mathematical models of DNA methylation dynamics: Implications for health and ageing." Journal of Theoretical Biology 462 (February 2019): 184–93. http://dx.doi.org/10.1016/j.jtbi.2018.11.006.

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Packer, Aaron, Jonathan Forde, Sarah Hews, and Yang Kuang. "Mathematical models of the interrelated dynamics of hepatitis D and B." Mathematical Biosciences 247 (January 2014): 38–46. http://dx.doi.org/10.1016/j.mbs.2013.10.004.

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Stolarska, Magdalena A., Yangjin Kim, and Hans G. Othmer. "Multi-scale models of cell and tissue dynamics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367, no. 1902 (September 13, 2009): 3525–53. http://dx.doi.org/10.1098/rsta.2009.0095.

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Cell and tissue movement are essential processes at various stages in the life cycle of most organisms. The early development of multi-cellular organisms involves individual and collective cell movement; leukocytes must migrate towards sites of infection as part of the immune response; and in cancer, directed movement is involved in invasion and metastasis. The forces needed to drive movement arise from actin polymerization, molecular motors and other processes, but understanding the cell- or tissue-level organization of these processes that is needed to produce the forces necessary for directed movement at the appropriate point in the cell or tissue is a major challenge. In this paper, we present three models that deal with the mechanics of cells and tissues: a model of an arbitrarily deformable single cell, a discrete model of the onset of tumour growth in which each cell is treated individually, and a hybrid continuum–discrete model of the later stages of tumour growth. While the models are different in scope, their underlying mechanical and mathematical principles are similar and can be applied to a variety of biological systems.

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Sreenath, Sree N., Kwang-Hyun Cho, and Peter Wellstead. "Modelling the dynamics of signalling pathways." Essays in Biochemistry 45 (September 30, 2008): 1–28. http://dx.doi.org/10.1042/bse0450001.

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In the present chapter we discuss methodologies for the modelling, calibration and validation of cellular signalling pathway dynamics. The discussion begins with the typical range of techniques for modelling that might be employed to go from the chemical kinetics to a mathematical model of biochemical pathways. In particular, we consider the decision-making processes involved in selecting the right mechanism and level of detail of representation of the biochemical interactions. These include the choice between (i) deterministic and stochastic chemical kinetics representations, (ii) discrete and continuous time models and (iii) representing continuous and discrete state processes. We then discuss the task of calibrating the models using information available in web-based databases. For situations in which the data are not available from existing sources we discuss model calibration based upon measured data and system identification methods. Such methods, together with mathematical modelling databases and computational tools, are often available in standard packages. We therefore make explicit mention of a range of popular and useful sites. As an example of the whole modelling and calibration process, we discuss a study of the cross-talk between the IL-1 (interleukin-1)-stimulated NF-κB (nuclear factor κB) pathway and the TGF-β (transforming growth factor β)-stimulated Smad2 pathway.

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Negrean, Iuliu, Claudiu Schonstein, Kalman Kacso, Calin Negrean, and Adina Duca. "Formulations about Dynamics of Mobile Robots." Solid State Phenomena 166-167 (September 2010): 309–14. http://dx.doi.org/10.4028/www.scientific.net/ssp.166-167.309.

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In this paper the dynamics equations for a mobile robot, named PatrolBot, will be developed, using new concepts in advanced mechanics, based on important scientific researches of the main author, concerning the kinetic energy. In keeping the fact that the mathematical models of the mobile platforms are different besides the other robots types, due to nonholonomic constraints, these dynamic control functions, will be computed, according to these restrictions for robot motion.

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Lakhova, T. N., F. V. Kazantsev, S. A. Lashin, and Yu G. Matushkin. "The finding and researching algorithm for potentially oscillating enzymatic systems." Vavilov Journal of Genetics and Breeding 25, no. 3 (June 2, 2021): 318–30. http://dx.doi.org/10.18699/vj21.035.

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Many processes in living organisms are subject to periodic oscillations at different hierarchical levels of their organization: from molecular-genetic to population and ecological. Oscillatory processes are responsible for cell cycles in both prokaryotes and eukaryotes, for circadian rhythms, for synchronous coupling of respiration with cardiac contractions, etc. Fluctuations in the numbers of organisms in natural populations can be caused by the populations’ own properties, their age structure, and ecological relationships with other species. Along with experimental approaches, mathematical and computer modeling is widely used to study oscillating biological systems. This paper presents classical mathematical models that describe oscillatory behavior in biological systems. Methods for the search for oscillatory molecular-genetic systems are presented by the example of their special case – oscillatory enzymatic systems. Factors influencing the cyclic dynamics in living systems, typical not only of the molecular-genetic level, but of higher levels of organization as well, are considered. Application of different ways to describe gene networks for modeling oscillatory molecular-genetic systems is considered, where the most important factor for the emergence of cyclic behavior is the presence of feedback. Techniques for finding potentially oscillatory enzymatic systems are presented. Using the method described in the article, we present and analyze, in a step-by-step manner, first the structural models (graphs) of gene networks and then the reconstruction of the mathematical models and computational experiments with them. Structural models are ideally suited for the tasks of an automatic search for potential oscillating contours (linked subgraphs), whose structure can correspond to the mathematical model of the molecular-genetic system that demonstrates oscillatory behavior in dynamics. At the same time, it is the numerical study of mathematical models for the selected contours that makes it possible to confirm the presence of stable limit cycles in them. As an example of application of the technology, a network of 300 metabolic reactions of the bacterium Escherichia coli was analyzed using mathematical and computer modeling tools. In particular, oscillatory behavior was shown for a loop whose reactions are part of the tryptophan biosynthesis pathway.

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Cheffer, Augusto, and Marcelo A. Savi. "Random effects inducing heart pathological dynamics: An approach based on mathematical models." Biosystems 196 (October 2020): 104177. http://dx.doi.org/10.1016/j.biosystems.2020.104177.

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Bardhan, Jaydeep P. "Gradient models in molecular biophysics: progress, challenges, opportunities." Journal of the Mechanical Behavior of Materials 22, no. 5-6 (December 1, 2013): 169–84. http://dx.doi.org/10.1515/jmbm-2013-0024.

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AbstractIn the interest of developing a bridge between researchers modeling materials and those modeling biological molecules, we survey recent progress in developing nonlocal-dielectric continuum models for studying the behavior of proteins and nucleic acids. As in other areas of science, continuum models are essential tools when atomistic simulations (e.g., molecular dynamics) are too expensive. Because biological molecules are essentially all nanoscale systems, the standard continuum model, involving local dielectric response, has basically always been dubious at best. The advanced continuum theories discussed here aim to remedy these shortcomings by adding nonlocal dielectric response. We begin by describing the central role of electrostatic interactions in biology at the molecular scale, and motivate the development of computationally tractable continuum models using applications in science and engineering. For context, we highlight some of the most important challenges that remain, and survey the diverse theoretical formalisms for their treatment, highlighting the rigorous statistical mechanics that support the use and improvement of continuum models. We then address the development and implementation of nonlocal dielectric models, an approach pioneered by Dogonadze, Kornyshev, and their collaborators almost 40 years ago. The simplest of these models is just a scalar form of gradient elasticity, and here we use ideas from gradient-based modeling to extend the electrostatic model to include additional length scales. The review concludes with a discussion of open questions for model development, highlighting the many opportunities for the materials community to leverage its physical, mathematical, and computational expertise to help solve one of the most challenging questions in molecular biology and biophysics.

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Gillespie, John H. "Could natural selection account for molecular evolution and polymorphism?" Genome 31, no. 1 (January 1, 1989): 311–15. http://dx.doi.org/10.1139/g89-049.

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A model of molecular evolution is presented that is based on the combined action of natural selection, genetic drift, and mutation. The mathematical description of the model uses strong-selection, weak-mutation limits to approximate the dynamics of multidimensional diffusion processes with one dimensional Markov chains. This approach leads to a great simplification of the dynamics and provides a unified method for describing many different mechanisms of natural selection. In this paper two models are examined, one based on selection in a randomly fluctuating environment, the other on overdominance. Both models exhibit similar dynamics, with a rapid buildup phase that introduces new alleles into the population, followed by a relatively quiescent phase where new alleles may enter and leave the population at a low rate. If occasional extreme environmental changes occur that favor particular alleles, the resulting dynamics turn out to be in remarkable agreement with many of the observations on molecular evolution and polymorphism. Thus the model is at least as successful as the neutral theory in accounting for evolutionary events at the molecular level.Key words: molecular evolution, natural selection, neutral theory.

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Wolkenhauer, O., S. N. Sreenath, P. Wellstead, M. Ullah, and K. H. Cho. "A systems- and signal-oriented approach to intracellular dynamics." Biochemical Society Transactions 33, no. 3 (June 1, 2005): 507–15. http://dx.doi.org/10.1042/bst0330507.

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A mathematical understanding of regulation, and, in particular, the role of feedback, has been central to the advance of the physical sciences and technology. In this article, the framework provided by systems biology is used to argue that the same can be true for molecular biology. In particular, and using basic modular methods of mathematical modelling which are standard in control theory, a set of dynamic models is developed for some illustrative cell signalling processes. These models, supported by recent experimental evidence, are used to argue that a control theoretical approach to the mechanisms of feedback in intracellular signalling is central to furthering our understanding of molecular communication. As a specific example, a MAPK (mitogen-activated protein kinase) signalling pathway is used to show how potential feedback mechanisms in the signalling process can be investigated in a simulated environment. Such ‘what if’ modelling/simulation studies have been an integral part of physical science research for many years. Using tools of control systems analysis, as embodied in the disciplines of systems biology, similar predictive modelling/simulation studies are now bearing fruit in cell signalling research.

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Michelson, Seth. "Multidrug Resistance and Its Reversal: Mathenatical Models." Journal of Theoretical Medicine 1, no. 2 (1997): 103–15. http://dx.doi.org/10.1080/10273669708833011.

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Classic multidrug resistance (MDR) is a phenomenon by which cells nonspecifically extrude noxious agents from the cutoplasm before lethal concentrations buils up. Some chemotherapeutically treated tumors exhibit these same dynamics. In tumor systems, the most common mechanism of facilitating MDR is the upregulation of the P-glycoprotein pump. This protein forms a transmembrance channel, and agter binding the chemotherapeutic agent and 2ATP molecules, forces the noxius agent through the channel. Hydrolysis of ATP to ADP provides the energy component of this reaction. General mathematical models describing drug resistamce are reviewed in this article. One model describing the molecular function of the P-glycoprotein pump in MDR cell lines is developed and presented in detail. The pump is modeled as an energy-dependent facilitated diffusion process. A partial differential equation is linked to a pair of ordinary differential equations to form the core of the model. To describe MDR reversal, the model is extended by additing an inhibitor to the equation system. Equations for competitive, one-site non-competitive, and allosteric non-competitive inhibition are then derived. Numerical simulations have been run to describe P-glycoprotein dynamics both in the presence and absence of inhibition, and these results are briefly reviewed. The character of the pump and its response to inhibition are discussed within the comtext of the models.

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Rinkeviciene, Roma, Algimantas Juozas Poška, and Alvydas Slepikas. "Dynamics of Dust Explosion Localizing System." Solid State Phenomena 164 (June 2010): 79–84. http://dx.doi.org/10.4028/www.scientific.net/ssp.164.79.

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The problem of dust explosion in corn processing enterprises is analyzed. System for localizing of dust explosions with dampers driven by linear induction motors (LIM) is considered. The paper presents the developed mathematical and computer models of a damper drive together with results of simulations.

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Perkins, Theodore J., Roy Wilds, and Leon Glass. "Robust dynamics in minimal hybrid models of genetic networks." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, no. 1930 (November 13, 2010): 4961–75. http://dx.doi.org/10.1098/rsta.2010.0139.

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Many gene-regulatory networks necessarily display robust dynamics that are insensitive to noise and stable under evolution. We propose that a class of hybrid systems can be used to relate the structure of these networks to their dynamics and provide insight into the origin of robustness. In these systems, the genes are represented by logical functions, and the controlling transcription factor protein molecules are real variables, which are produced and destroyed. As the transcription factor concentrations cross thresholds, they control the production of other transcription factors. We discuss mathematical analysis of these systems and show how the concepts of robustness and minimality can be used to generate putative logical organizations based on observed symbolic sequences. We apply the methods to control of the cell cycle in yeast.

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de Sousa, Bruno C., and Celso Cunha. "Development of Mathematical Models for the Analysis of Hepatitis Delta Virus Viral Dynamics." PLoS ONE 5, no. 9 (September 16, 2010): e12512. http://dx.doi.org/10.1371/journal.pone.0012512.

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Bhuju, G., G. R. Phaijoo, and D. B. Gurung. "Fuzzy Approach Analyzing SEIR-SEI Dengue Dynamics." BioMed Research International 2020 (October 14, 2020): 1–11. http://dx.doi.org/10.1155/2020/1508613.

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Dengue fever is a mosquito-borne infectious disease threatening more than a hundred tropical countries of the world. The heterogeneity of mosquito bites of human during the spread of dengue virus is an important factor that should be considered while modeling the dynamics of the disease. However, traditional models assumed homogeneous transmission between host and vectors which is inconsistent with reality. Mathematically, we can describe the heterogeneity and uncertainty of the transmission of the disease by introducing fuzzy theory. In the present work, we study transmission dynamics of dengue with the fuzzy SEIR-SEI compartmental model. The transmission rate and recovery rate of the disease are considered as fuzzy numbers. The dynamical behavior of the system is discussed with different amounts of dengue viruses. Also, the fuzzy basic reproduction number for a group of infected individuals with different virus loads is calculated using Sugeno integral. Simulations are made to illustrate the mathematical results graphically.

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LIM, HWA A. "ELECTROPHORESIS OF TOPOLOGICALLY NONTRIVIAL MACROMOLECULES: MATHEMATICAL AND COMPUTATIONAL STUDIES." International Journal of Modern Physics C 07, no. 02 (April 1996): 217–71. http://dx.doi.org/10.1142/s0129183196000193.

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Mathematical and numerical models for studying the electrophoresis of topologically nontrivial molecules in two and three dimensions are presented. The molecules are modeled as polygons residing on a square lattice and a cubic lattice whereas the electrophoretic media of obstacle network are simulated by removing vertices from the lattices at random. The dynamics of the polymeric molecules are modeled by configurational readjustments of segments of the polygons. Configurational readjustments arise from thermal fluctuations and they correspond to piecewise reptation in the simulations. A Metropolis algorithm is introduced to simulate these dynamics, and the algorithms are proven to be reversible and ergodic. Monte Carlo simulations of steady field random obstacle electrophoresis are performed and the results are presented.

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Vasieva, Olga, Manan'Iarivo Rasolonjanahary, and Bakhtier Vasiev. "Mathematical modelling in developmental biology." REPRODUCTION 145, no. 6 (June 2013): R175—R184. http://dx.doi.org/10.1530/rep-12-0081.

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In recent decades, molecular and cellular biology has benefited from numerous fascinating developments in experimental technique, generating an overwhelming amount of data on various biological objects and processes. This, in turn, has led biologists to look for appropriate tools to facilitate systematic analysis of data. Thus, the need for mathematical techniques, which can be used to aid the classification and understanding of this ever-growing body of experimental data, is more profound now than ever before. Mathematical modelling is becoming increasingly integrated into biological studies in general and into developmental biology particularly. This review outlines some achievements of mathematics as applied to developmental biology and demonstrates the mathematical formulation of basic principles driving morphogenesis. We begin by describing a mathematical formalism used to analyse the formation and scaling of morphogen gradients. Then we address a problem of interplay between the dynamics of morphogen gradients and movement of cells, referring to mathematical models of gastrulation in the chick embryo. In the last section, we give an overview of various mathematical models used in the study of the developmental cycle of Dictyostelium discoideum, which is probably the best example of successful mathematical modelling in developmental biology.

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Schütte, Christof, Stefanie Winkelmann, and Carsten Hartmann. "Optimal control of molecular dynamics using Markov state models." Mathematical Programming 134, no. 1 (May 24, 2012): 259–82. http://dx.doi.org/10.1007/s10107-012-0547-6.

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Duffin, R. Paul, and Richard H. Tullis. "Mathematical Models of the Complete Course of HIV Infection and AIDS." Journal of Theoretical Medicine 4, no. 4 (2002): 215–21. http://dx.doi.org/10.1080/1027366021000051772.

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Mathematical models of HIV infection are important to our understanding of AIDS. However, most models do not predict both the decrease in CD4+ T cells and the increase in viral load seen over the course of infection. By including terms for continuous loss of CD4+ T cells and incorporating alteration in viral clearance and viral production, two new models have been created that accurately predict the dynamics of the disease. The first model is a clearance rate reduction model and is based on a 10% per year decrease in both viral clearance and CD4+ T cell levels. A macrophage reservoir model incorporating the observation that macrophage viral production increases up to 1000 fold in the presence of opportunistic infections that become increasingly common as disease progresses. Both viral clearance and macrophage reservoir models predict the expected decrease in T cell levels and rise in viral load observed at the onset of AIDS.

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ZAVADSKY, SERGEY V., DMITRI A. OVSYANNIKOV, and SHENG-LUEN CHUNG. "PARAMETRIC OPTIMIZATION METHODS FOR THE TOKAMAK PLASMA CONTROL PROBLEM." International Journal of Modern Physics A 24, no. 05 (February 20, 2009): 1040–47. http://dx.doi.org/10.1142/s0217751x09044486.

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Mathematical models of the structural parametric optimization of plasma dynamics are discussed. Optimization approach to plasma dynamic is based on the consideration of trajectory ensemble. This ensemble describes transient process in tokamak subject to the initial data and external disturbances. In the framework of this approach the optimization of dynamics of the trajectory ensemble in ITER tokamak is given. The trajectories of this ensemble are perturbed at the initial point set and the set of external disturbances.

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Iwasaki, Tsuyoshi, Ryo Takiguchi, Takumi Hiraiwa, Takahiro G. Yamada, Kazuto Yamazaki, Noriko F. Hiroi, and Akira Funahashi. "Neural Differentiation Dynamics Controlled by Multiple Feedback Loops in a Comprehensive Molecular Interaction Network." Processes 8, no. 2 (February 2, 2020): 166. http://dx.doi.org/10.3390/pr8020166.

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Mathematical model simulation is a useful method for understanding the complex behavior of a living system. The construction of mathematical models using comprehensive information is one of the techniques of model construction. Such a comprehensive knowledge-based network tends to become a large-scale network. As a result, the variation of analyses is limited to a particular kind of analysis because of the size and complexity of the model. To analyze a large-scale regulatory network of neural differentiation, we propose a contractive method that preserves the dynamic behavior of a large network. The method consists of the following two steps: comprehensive network building and network reduction. The reduction phase can extract network loop structures from a large-scale regulatory network, and the subnetworks were combined to preserve the dynamics of the original large-scale network. We confirmed that the extracted loop combination reproduced the known dynamics of HES1 and ASCL1 before and after differentiation, including oscillation and equilibrium of their concentrations. The model also reproduced the effects of the overexpression and knockdown of the Id2 gene. Our model suggests that the characteristic change in HES1 and ASCL1 expression in the large-scale regulatory network is controlled by a combination of four feedback loops, including a large loop, which has not been focused on. The model extracted by our method has the potential to reveal the critical mechanisms of neural differentiation. The method is applicable to other biological events.

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Louca, Stilianos, Angela McLaughlin, Ailene MacPherson, Jeffrey B. Joy, and Matthew W. Pennell. "Fundamental Identifiability Limits in Molecular Epidemiology." Molecular Biology and Evolution 38, no. 9 (May 19, 2021): 4010–24. http://dx.doi.org/10.1093/molbev/msab149.

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Abstract Viral phylogenies provide crucial information on the spread of infectious diseases, and many studies fit mathematical models to phylogenetic data to estimate epidemiological parameters such as the effective reproduction ratio (Re) over time. Such phylodynamic inferences often complement or even substitute for conventional surveillance data, particularly when sampling is poor or delayed. It remains generally unknown, however, how robust phylodynamic epidemiological inferences are, especially when there is uncertainty regarding pathogen prevalence and sampling intensity. Here, we use recently developed mathematical techniques to fully characterize the information that can possibly be extracted from serially collected viral phylogenetic data, in the context of the commonly used birth-death-sampling model. We show that for any candidate epidemiological scenario, there exists a myriad of alternative, markedly different, and yet plausible “congruent” scenarios that cannot be distinguished using phylogenetic data alone, no matter how large the data set. In the absence of strong constraints or rate priors across the entire study period, neither maximum-likelihood fitting nor Bayesian inference can reliably reconstruct the true epidemiological dynamics from phylogenetic data alone; rather, estimators can only converge to the “congruence class” of the true dynamics. We propose concrete and feasible strategies for making more robust epidemiological inferences from viral phylogenetic data.

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Belfiore, Mariagrazia, Marzio Pennisi, Giuseppina Aricò, Simone Ronsisvalle, and Francesco Pappalardo. "In Silico Modeling of the Immune System: Cellular and Molecular Scale Approaches." BioMed Research International 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/371809.

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The revolutions in biotechnology and information technology have produced clinical data, which complement biological data. These data enable detailed descriptions of various healthy and diseased states and responses to therapies. For the investigation of the physiology and pathology of the immune responses, computer and mathematical models have been used in the last decades, enabling the representation of biological processes. In this modeling effort, a major issue is represented by the communication between models that work at cellular and molecular level, that is, multiscale representation. Here we sketch some attempts to model immune system dynamics at both levels.

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Unni, Pranav, and Padmanabhan Seshaiyer. "Mathematical Modeling, Analysis, and Simulation of Tumor Dynamics with Drug Interventions." Computational and Mathematical Methods in Medicine 2019 (October 8, 2019): 1–13. http://dx.doi.org/10.1155/2019/4079298.

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Over the last few decades, there have been significant developments in theoretical, experimental, and clinical approaches to understand the dynamics of cancer cells and their interactions with the immune system. These have led to the development of important methods for cancer therapy including virotherapy, immunotherapy, chemotherapy, targeted drug therapy, and many others. Along with this, there have also been some developments on analytical and computational models to help provide insights into clinical observations. This work develops a new mathematical model that combines important interactions between tumor cells and cells in the immune systems including natural killer cells, dendritic cells, and cytotoxic CD8+ T cells combined with drug delivery to these cell sites. These interactions are described via a system of ordinary differential equations that are solved numerically. A stability analysis of this model is also performed to determine conditions for tumor-free equilibrium to be stable. We also study the influence of proliferation rates and drug interventions in the dynamics of all the cells involved. Another contribution is the development of a novel parameter estimation methodology to determine optimal parameters in the model that can reproduce a given dataset. Our results seem to suggest that the model employed is a robust candidate for studying the dynamics of tumor cells and it helps to provide the dynamic interactions between the tumor cells, immune system, and drug-response systems.

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Martin, Benjamin T., Stephan B. Munch, and Andrew M. Hein. "Reverse-engineering ecological theory from data." Proceedings of the Royal Society B: Biological Sciences 285, no. 1878 (May 16, 2018): 20180422. http://dx.doi.org/10.1098/rspb.2018.0422.

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Ecologists have long sought to understand the dynamics of populations and communities by deriving mathematical theory from first principles. Theoretical models often take the form of dynamical equations that comprise the ecological processes (e.g. competition, predation) believed to govern system dynamics. The inverse of this approach—inferring which processes and ecological interactions drive observed dynamics—remains an open problem in ecology. Here, we propose a way to attack this problem using a machine learning method known as symbolic regression, which seeks to discover relationships in time-series data and to express those relationships using dynamical equations. We found that this method could rapidly discover models that explained most of the variance in three classic demographic time series. More importantly, it reverse-engineered the models previously proposed by theoretical ecologists to describe these time series, capturing the core ecological processes these models describe and their functional forms. Our findings suggest a potentially powerful new way to merge theory development and data analysis.

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Alfsnes, Kristian, Xavier Raynaud, Tone Tønjum, and Ole Herman Ambur. "Mathematical and Live Meningococcal Models for Simple Sequence Repeat Dynamics – Coherent Predictions and Observations." PLoS ONE 9, no. 7 (July 7, 2014): e101637. http://dx.doi.org/10.1371/journal.pone.0101637.

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Cody, Jonathan W., Amy L. Ellis-Connell, Shelby L. O’Connor, and Elsje Pienaar. "Mathematical modeling of N-803 treatment in SIV-infected non-human primates." PLOS Computational Biology 17, no. 7 (July 28, 2021): e1009204. http://dx.doi.org/10.1371/journal.pcbi.1009204.

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Immunomodulatory drugs could contribute to a functional cure for Human Immunodeficiency Virus (HIV). Interleukin-15 (IL-15) promotes expansion and activation of CD8+ T cell and natural killer (NK) cell populations. In one study, an IL-15 superagonist, N-803, suppressed Simian Immunodeficiency Virus (SIV) in non-human primates (NHPs) who had received prior SIV vaccination. However, viral suppression attenuated with continued N-803 treatment, partially returning after long treatment interruption. While there is evidence of concurrent drug tolerance, immune regulation, and viral escape, the relative contributions of these mechanisms to the observed viral dynamics have not been quantified. Here, we utilize mathematical models of N-803 treatment in SIV-infected macaques to estimate contributions of these three key mechanisms to treatment outcomes: 1) drug tolerance, 2) immune regulation, and 3) viral escape. We calibrated our model to viral and lymphocyte responses from the above-mentioned NHP study. Our models track CD8+ T cell and NK cell populations with N-803-dependent proliferation and activation, as well as viral dynamics in response to these immune cell populations. We compared mathematical models with different combinations of the three key mechanisms based on Akaike Information Criterion and important qualitative features of the NHP data. Two minimal models were capable of reproducing the observed SIV response to N-803. In both models, immune regulation strongly reduced cytotoxic cell activation to enable viral rebound. Either long-term drug tolerance or viral escape (or some combination thereof) could account for changes to viral dynamics across long breaks in N-803 treatment. Theoretical explorations with the models showed that less-frequent N-803 dosing and concurrent immune regulation blockade (e.g. PD-L1 inhibition) may improve N-803 efficacy. However, N-803 may need to be combined with other immune therapies to countermand viral escape from the CD8+ T cell response. Our mechanistic model will inform such therapy design and guide future studies.

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Tilocca, Antonio. "Structural models of bioactive glasses from molecular dynamics simulations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2104 (January 13, 2009): 1003–27. http://dx.doi.org/10.1098/rspa.2008.0462.

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The bioactive mechanism, by which living tissues attach to and integrate with an artificial implant through stable chemical bonds, is at the core of many current medical applications of biomaterials, as well as of novel promising applications in tissue engineering. Having been employed in these applications for almost 40 years, soda-lime phosphosilicate glasses such as 45S5 represent today the paradigm of bioactive materials. Despite their strategical importance in the field, the relationship between the structure and the activity of a glass composition in a biological environment has not been studied in detail. This fundamental gap negatively affects further progress, for instance, to improve the chemical durability and tailor the biodegradability of these materials for specific applications. This paper reviews recent advances in computer modelling of bioactive glasses based on molecular dynamics simulations, which are starting to unveil key structural features of these materials, thus contributing to improve our fundamental understanding of how bioactive materials work.

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Bellomo, Nicola, Elena De Angelis, and Luigi Preziosi. "Multiscale Modeling and Mathematical Problems Related to Tumor Evolution and Medical Therapy." Journal of Theoretical Medicine 5, no. 2 (2003): 111–36. http://dx.doi.org/10.1080/1027336042000288633.

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This paper provides a survey of mathematical models and methods dealing with the analysis and simulation of tumor dynamics in competition with the immune system. The characteristic scales of the phenomena are identified and the mathematical literature on models and problems developed on each scale is reviewed and critically analyzed. Moreover, this paper deals with the modeling and optimization of therapeutical actions. The aim of the critical analysis and review consists in providing the background framework towards the development of research perspectives in this promising new field of applied mathematics.

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Ajelli, Marco, Mimmo Iannelli, Piero Manfredi, and Marta L. Ciofi degli Atti. "Basic mathematical models for the temporal dynamics of HAV in medium-endemicity Italian areas." Vaccine 26, no. 13 (March 2008): 1697–707. http://dx.doi.org/10.1016/j.vaccine.2007.12.058.

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Alameddine, Abdallah K., Frederick Conlin, and Brian Binnall. "An Introduction to the Mathematical Modeling in the Study of Cancer Systems Biology." Cancer Informatics 17 (January 2018): 117693511879975. http://dx.doi.org/10.1177/1176935118799754.

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Background: Frequently occurring in cancer are the aberrant alterations of regulatory onco-metabolites, various oncogenes/epigenetic stochasticity, and suppressor genes, as well as the deficient mismatch repair mechanism, chronic inflammation, or those deviations belonging to the other cancer characteristics. How these aberrations that evolve overtime determine the global phenotype of malignant tumors remains to be completely understood. Dynamic analysis may have potential to reveal the mechanism of carcinogenesis and can offer new therapeutic intervention. Aims: We introduce simplified mathematical tools to model serial quantitative data of cancer biomarkers. We also highlight an introductory overview of mathematical tools and models as they apply from the viewpoint of known cancer features. Methods: Mathematical modeling of potentially actionable genomic products and how they proceed overtime during tumorigenesis are explored. This report is intended to be instinctive without being overly technical. Results: To date, many mathematical models of the common features of cancer have been developed. However, the dynamic of integrated heterogeneous processes and their cross talks related to carcinogenesis remains to be resolved. Conclusions: In cancer research, outlining mathematical modeling of experimentally obtained data snapshots of molecular species may provide insights into a better understanding of the multiple biochemical circuits. Recent discoveries have provided support for the existence of complex cancer progression in dynamics that span from a simple 1-dimensional deterministic system to a stochastic (ie, probabilistic) or to an oscillatory and multistable networks. Further research in mathematical modeling of cancer progression, based on the evolving molecular kinetics (time series), could inform a specific and a predictive behavior about the global systems biology of vulnerable tumor cells in their earlier stages of oncogenesis. On this footing, new preventive measures and anticancer therapy could then be constructed.

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Hudson, Lawrence N., and Daniel C. Reuman. "A cure for the plague of parameters: constraining models of complex population dynamics with allometries." Proceedings of the Royal Society B: Biological Sciences 280, no. 1770 (November 7, 2013): 20131901. http://dx.doi.org/10.1098/rspb.2013.1901.

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A major goal of ecology is to discover how dynamics and structure of multi-trophic ecological communities are related. This is difficult, because whole-community data are limited and typically comprise only a snapshot of a community instead of a time series of dynamics, and mathematical models of complex system dynamics have a large number of unmeasured parameters and therefore have been only tenuously related to real systems. These are related problems, because long time-series, if they were commonly available, would enable inference of parameters. The resulting ‘plague of parameters’ means most studies of multi-species population dynamics have been very theoretical. Dynamical models parametrized using physiological allometries may offer a partial cure for the plague of parameters, and these models are increasingly used in theoretical studies. However, physiological allometries cannot determine all parameters, and the models have also rarely been directly tested against data. We confronted a model of community dynamics with data from a lake community. Many important empirical patterns were reproducible as outcomes of dynamics, and were not reproducible when parameters did not follow physiological allometries. Results validate the usefulness, when parameters follow physiological allometries, of classic differential-equation models for understanding whole-community dynamics and the structure–dynamics relationship.

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Otten, E. "Inverse and forward dynamics: models of multi–body systems." Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 358, no. 1437 (August 13, 2003): 1493–500. http://dx.doi.org/10.1098/rstb.2003.1354.

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Connected multi–body systems exhibit notoriously complex behaviour when driven by external and internal forces and torques. The problem of reconstructing the internal forces and/or torques from the movements and known external forces is called the ‘inverse dynamics problem’, whereas calculating motion from known internal forces and/or torques and resulting reaction forces is called the ‘forward dynamics problem’. When stepping forward to cross the street, people use muscle forces that generate angular accelerations of their body segments and, by virtue of reaction forces from the street, a forward acceleration of the centre of mass of their body. Inverse dynamics calculations applied to a set of motion data from such an event can teach us how temporal patterns of joint torques were responsible for the observed motion. In forward dynamics calculations we may attempt to create motion from such temporal patterns, which is extremely difficult, because of the complex mechanical linkage along the chains forming the multi–body system. To understand, predict and sometimes control multi–body systems, we may want to have mathematical expressions for them. The Newton–Euler, Lagrangian and Featherstone approaches have their advantages and disadvantages. The simulation of collisions and the inclusion of muscle forces or other internal forces are discussed. Also, the possibility to perform a mixed inverse and forward dynamics calculation are dealt with. The use and limitations of these approaches form the conclusion.

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Gonze, Didier. "Modeling circadian clocks: From equations to oscillations." Open Life Sciences 6, no. 5 (October 1, 2011): 699–711. http://dx.doi.org/10.2478/s11535-011-0061-5.

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AbstractCircadian rhythms are generated at the cellular level by a small but tightly regulated genetic network. In higher eukaryotes, interlocked transcriptional-translational feedback loops form the core of this network, which ensures the activation of the right genes (proteins) at the right time of the day. Understanding how such a complex molecular network can generate robust, self-sustained oscillations and accurately responds to signals from the environment (such as light and temperature) is greatly helped by mathematical modeling. In the present paper we review some mathematical models for circadian clocks, ranging from abstract, phenomenological models to the most detailed molecular models. We explain how the equations are derived, highlighting the challenges for the modelers, and how the models are analyzed. We show how to compute bifurcation diagrams, entrainment, and phase response curves. In the subsequent paper, we discuss, through a selection of examples, how modeling efforts have contributed to a better understanding of the dynamics of the circadian regulatory network.

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Nev, Olga A., Richard J. Lindsay, Alys Jepson, Lisa Butt, Robert E. Beardmore, and Ivana Gudelj. "Predicting microbial growth dynamics in response to nutrient availability." PLOS Computational Biology 17, no. 3 (March 18, 2021): e1008817. http://dx.doi.org/10.1371/journal.pcbi.1008817.

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Developing mathematical models to accurately predict microbial growth dynamics remains a key challenge in ecology, evolution, biotechnology, and public health. To reproduce and grow, microbes need to take up essential nutrients from the environment, and mathematical models classically assume that the nutrient uptake rate is a saturating function of the nutrient concentration. In nature, microbes experience different levels of nutrient availability at all environmental scales, yet parameters shaping the nutrient uptake function are commonly estimated for a single initial nutrient concentration. This hampers the models from accurately capturing microbial dynamics when the environmental conditions change. To address this problem, we conduct growth experiments for a range of micro-organisms, including human fungal pathogens, baker’s yeast, and common coliform bacteria, and uncover the following patterns. We observed that the maximal nutrient uptake rate and biomass yield were both decreasing functions of initial nutrient concentration. While a functional form for the relationship between biomass yield and initial nutrient concentration has been previously derived from first metabolic principles, here we also derive the form of the relationship between maximal nutrient uptake rate and initial nutrient concentration. Incorporating these two functions into a model of microbial growth allows for variable growth parameters and enables us to substantially improve predictions for microbial dynamics in a range of initial nutrient concentrations, compared to keeping growth parameters fixed.

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Gonze, Didier. "Modeling circadian clocks: Roles, advantages, and limitations." Open Life Sciences 6, no. 5 (October 1, 2011): 712–29. http://dx.doi.org/10.2478/s11535-011-0062-4.

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AbstractCircadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified.

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Jivulescu, M. A., E. Ferraro, A. Napoli, and A. Messina. "Exact dynamics of XX central spin models." Physica Scripta T135 (July 2009): 014049. http://dx.doi.org/10.1088/0031-8949/2009/t135/014049.

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Journal articles on the topic 'Molecular dynamics Mathematical models'

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