Mathematical modelling in physiology

The article illustrates the method of mathematical modelling in physiology as a unique tool to study physiological processes. A number of demonstrated examples appear as a result of long-term experience in mathematical modelling of electrical and mechanical phenomena in the heart muscle. These examples are presented here to show that the modelling provides insight into mechanisms underlying these phenomena and is capable to predict new ones that were previously unknown. While potentialities of the mathematical modelling are analyzed with regard to the myocardium, they are quite universal to deal with any physiological processes.     

Multi-scale computational modelling in biology and physiology

Recent advances in biotechnology and the availability of ever more powerful computers have led to the formulation of increasingly complex models at all levels of biology. One of the main aims of systems biology is to couple these together to produce integrated models across multiple spatial scales and physical processes. In this review, we formulate a definition of multi-scale in terms of levels of biological organisation and describe the types of model that are found at each level. Key issues that arise in trying to formulate and solve multi-scale and multi-physics models are considered and examples of how these issues have been addressed are given for two of the more mature fields in computational biology: the molecular dynamics of ion channels and cardiac modelling. As even more complex models are developed over the coming few years, it will be necessary to develop new methods to model them (in particular in coupling across the interface between stochastic and deterministic processes) and new techniques will be required to compute their solutions efficiently on massively parallel computers. We outline how we envisage these developments occurring.     

Integrating modelling of biodiversity composition and ecosystem function

There is increasing reliance on ecological models to improve our understanding of how ecological systems work, to project likely outcomes under alternative global change scenarios and to help develop robust management strategies. Two common types of spatiotemporally explicit ecological models are those focussed on biodiversity composition and those focussed on ecosystem function. These modelling disciplines are largely practiced separately, with separate literature, despite growing evidence that natural systems are shaped by the interaction of composition and function. Here we call for the development of new modelling approaches that integrate composition and function, accounting for the important interactions between these two dimensions, particularly under rapid global change. We examine existing modelling approaches that have begun to combine elements of composition and function, identifying their potential contribution to fully integrated modelling approaches. The development and application of integrated models of composition and function face a number of important challenges, including biological data limitations, system knowledge and computational constraints. We suggest a range of promising avenues that could help researchers overcome these challenges, including the use of virtual species, macroecological relationships and hybrid correlative‐mechanistic modelling. Explicitly accounting for the interactions between composition and function within integrated modelling approaches has the potential to improve our understanding of ecological systems, provide more accurate predictions of their future states and transform their management. Synthesis There is increasing attention from researchers and policy makers around the world on both assessing and projecting the state of the planet's biodiversity, its ecosystems and the essential services they provide to society. However, existing modelling approaches largely ignore the interactions between biodiversity composition and ecosystem function. We highlight the key challenges and potential solutions to developing integrated models of composition and function. Such models will require a new effort and focus from ecologists, yet the benefits are likely to be substantial, including better informing the management of natural systems at regional, national and international scales.     

Integrating recent advances in neuroscience into undergraduate neuroscience and physiology courses

Neuroscience has enjoyed tremendous growth over the past 20 years, including a substantial increase in the number of neuroscience departments, programs, and courses at the undergraduate level. To meet the need of new neuroscience courses, there has also been growth in the number of introductory neuroscience textbooks designed for undergraduates. However, textbooks typically trail current knowledge by five to ten years, especially in neuroscience where our understanding is increasing rapidly. Consequently, it is often important to supplement neuroscience and physiology textbooks with information about recent findings in neuroscience. To design supplementary educational material, it is essential first to identify the educational objectives of the program and the characteristics of the learners, which can differ dramatically between undergraduate and graduate or professional students. Four principles that may serve the selection and design of supplementary material for undergraduate neuroscience and physiology courses are that (1) material must be interesting to the undergraduates, (2) material should reinforce previously learned concepts, (3) students must be adequately prepared, and (4) the teacher and student must have sufficient appropriate resources.     

Integrating stress physiology,environmental change,and behavior in free-living sparrows

As weather deteriorates, breeding animals have a diverse array of options to ensure survival. Because of their mobility, birds can easily abandon territories to seek out benign conditions away from the breeding site. The timing of abandonment, however, may have repercussions for territory size, mate quality, reproductive success, and survival. There is a large body of evidence indicating that the adrenocortical response to stress plays a role in mediating the onset and maintenance of this behavioral switch. Here we develop a model describing the interactions of weather, food availability, body condition, and stress physiology in initiating departure from the breeding site. We tested the model using a population of white-crowned sparrows breeding at high elevation in the Sierra Nevada, where severe weather at the beginning of the breeding season often induces temporary abandonment of breeding territories and facultative altitudinal migration to lower elevation refugia. The data show that (1). during inclement weather, exogenous corticosterone delays return to the breeding site after territory abandonment; (2). during good weather, exogenous corticosterone alone does not induce territory abandonment, but does increase activity range around the breeding site; and (3). the magnitude of the corticosteroid response to stress is inversely related to body condition of the sparrow.     

Diagrammatic representations for modelling biological knowledge

The contemporary research and development context in multidisciplinary biology has a serious requirement for integrating knowledge from disparate sources, and facilitating much-needed inter- and intra-disciplinary dialogue. A multiplicity of models arises when pluralistic approaches to modelling are followed, and also when there is not only a requirement to model systems and data, but also knowledge of systems and data. The challenges of addressing this multiplicity do not only include articulating the structure of complex systems, but also placing modelling within the framework of a process as well as a product. The graph representations presented here facilitate dialogue, modelling, clarification and specification of concepts, and the sharing of terms. This paper explores relationships between collections of graph representations. It is hoped that in future, when readers look at a node or a process in a graph, they will have a much deeper appreciation of relationships and context.     

Intensity generalization: physiology and modelling of a neglected topic

I briefly review empirical data about the generalization of acquired behaviour to novel stimuli, showing that variations in stimulus intensity affect behaviour differently from variations in characteristics such as, for instance, visual shape or sound frequency. I argue that such differences can be seen already in how the sense organs react to changes in intensity compared to changes in other stimulus characteristics. I then evaluate a number of models of generalization with respect to their ability to reproduce intensity generalization. I reach three main conclusions. First, realistic stimulus representations, based on knowledge of the sense organs, are necessary to account for intensity effects. Models employing stimulus representations too remote from the sense organs are unable to reproduce the data. Second, the intuitive notion that generalization is based on similarities between stimuli, possibly modelled as distances in an appropriate representation space, is difficult to reconcile with data about intensity generalization. Third, several simple models, in conjunction with realistic stimulus representations, can account for a wide array of generalization phenomena along both intensity and non-intensity stimulus dimensions. The paper also introduces concepts which may be generally useful to evaluate and compare different models of behaviour.     

Integrating life history and physiology to understand latitudinal size variation in a damselfly

Our understanding of latitudinal life history patterns may benefit by jointly considering age and mass at maturity and growth rate. Additional insight may be gained by exploring potential constraints through pushing growth rates to their maximum and scoring physiological cost‐related variables. Therefore, we reared animals of a univoltine Spanish and Belgian population and of a semivoltine Swedish population of the damselfly Enallagma cyathigerum (spanning a latitude gradient of ca 2350 km) in a common environment from the eggs until adult emergence and exposed them to a transient starvation period to induce compensatory growth. Besides age and mass at maturity and growth rate we also scored investment in energy storage (i.e. triglycerides) and immune function (i.e. total activity of phenoloxidase). At emergence, body mass was greater in Spain and Sweden and lower in Belgium, suggesting a genetic component for the U‐shaped latitudinal pattern that was found also in a previous study based on field‐collected adults. The mass difference between univoltine populations can be explained by the shorter development time in the Belgian population, and this despite a higher growth rate, a pattern consistent with undercompensating countergradient variation. In line with the assumed shorter growth seasons, Belgian and Swedish animals showed higher routine growth rates and compensatory growth after transient starvation. Despite a strong link with metabolic rates (as measured by oxygen consumption) populations with higher routine growth rates had no lower fat content and had higher immune function (i.e. immune function decreased from Sweden to Spain), which was unexpected. Rapid compensatory growth did, however, result in a lowered immune function. This may contribute to the absence of perfect compensating countergradient variation in the Belgian population and the lowest routine growth rates in the Spanish population. Our results underscore the importance of integrating key life historical with physiological traits for understanding latitudinal population differentiation.     

Ologs: a categorical framework for knowledge representation

In this paper we introduce the olog, or ontology log, a category-theoretic model for knowledge representation (KR). Grounded in formal mathematics, ologs can be rigorously formulated and cross-compared in ways that other KR models (such as semantic networks) cannot. An olog is similar to a relational database schema; in fact an olog can serve as a data repository if desired. Unlike database schemas, which are generally difficult to create or modify, ologs are designed to be user-friendly enough that authoring or reconfiguring an olog is a matter of course rather than a difficult chore. It is hoped that learning to author ologs is much simpler than learning a database definition language, despite their similarity. We describe ologs carefully and illustrate with many examples. As an application we show that any primitive recursive function can be described by an olog. We also show that ologs can be aligned or connected together into a larger network using functors. The various methods of information flow and institutions can then be used to integrate local and global world-views. We finish by providing several different avenues for future research.     

Fragmentation-tree density representation for crystallographic modelling of bound ligands

The identification and modelling of ligands into macromolecular models is important for understanding molecule's function and for designing inhibitors to modulate its activities. We describe new algorithms for the automated building of ligands into electron density maps in crystal structure determination. Location of the ligand-binding site is achieved by matching numerical shape features describing the ligand to those of density clusters using a "fragmentation-tree" density representation. The ligand molecule is built using two distinct algorithms exploiting free atoms with inter-atomic connectivity and Metropolis-based optimisation of the conformational state of the ligand, producing an ensemble of structures from which the final model is derived. The method was validated on several thousand entries from the Protein Data Bank. In the majority of cases, the ligand-binding site could be correctly located and the ligand model built with a coordinate accuracy of better than 1 ?. We anticipate that the method will be of routine use to anyone modelling ligands, lead compounds or even compound fragments as part of protein functional analyses or drug design efforts.     

A hybrid representation approach for modelling complex dynamic bioprocesses

This paper considers the use of hybrid models to represent the dynamic behaviour of biotechnological processes. Each hybrid model consists of a set of non linear differential equations and a neural model. The set of differential equations attempts to describe as much as possible the phenomenology of the process whereas neural networks model predict some key parameters that are an essential part of the phenomenological model. The neural model is obtained indirectly, that is, using the prediction errors of one or more state variables to adjust its weights instead of successive presentations of input-output data of the neural network. This approach allows to use actual measurements to derive a suitable neural model that not only represents the variation of some key parameters but it is also able to partly include dynamic behaviour unaccounted for by the phenomenological model. The approach is described in detail using three test cases: (1) the fermentation of glucose to gluconic acid by the micro-organism Pseudomonas ovalis, (2) the growth of filamentous fungi in a solid state fermenter, and (3) the propagation of filamentous fungi growing on a 2-D solid substrate. Results for the three applications clearly demon- strate that using a hybrid model is a viable alternative for modelling complex biotechnological bioprocesses.     

From inverse problems in mathematical physiology to quantitative differential diagnoses

The improved capacity to acquire quantitative data in a clinical setting has generally failed to improve outcomes in acutely ill patients, suggesting a need for advances in computer-supported data interpretation and decision making. In particular, the application of mathematical models of experimentally elucidated physiological mechanisms could augment the interpretation of quantitative, patient-specific information and help to better target therapy. Yet, such models are typically complex and nonlinear, a reality that often precludes the identification of unique parameters and states of the model that best represent available data. Hypothesizing that this non-uniqueness can convey useful information, we implemented a simplified simulation of a common differential diagnostic process (hypotension in an acute care setting), using a combination of a mathematical model of the cardiovascular system, a stochastic measurement model, and Bayesian inference techniques to quantify parameter and state uncertainty. The output of this procedure is a probability density function on the space of model parameters and initial conditions for a particular patient, based on prior population information together with patient-specific clinical observations. We show that multimodal posterior probability density functions arise naturally, even when unimodal and uninformative priors are used. The peaks of these densities correspond to clinically relevant differential diagnoses and can, in the simplified simulation setting, be constrained to a single diagnosis by assimilating additional observations from dynamical interventions (e.g., fluid challenge). We conclude that the ill-posedness of the inverse problem in quantitative physiology is not merely a technical obstacle, but rather reflects clinical reality and, when addressed adequately in the solution process, provides a novel link between mathematically described physiological knowledge and the clinical concept of differential diagnoses. We outline possible steps toward translating this computational approach to the bedside, to supplement today's evidence-based medicine with a quantitatively founded model-based medicine that integrates mechanistic knowledge with patient-specific information.     

Integrating genealogical and dynamical modelling to infer escape and reversion rates in HIV epitopes

The rates of escape and reversion in response to selection pressure arising from the host immune system, notably the cytotoxic T-lymphocyte (CTL) response, are key factors determining the evolution of HIV. Existing methods for estimating these parameters from cross-sectional population data using ordinary differential equations (ODEs) ignore information about the genealogy of sampled HIV sequences, which has the potential to cause systematic bias and overestimate certainty. Here, we describe an integrated approach, validated through extensive simulations, which combines genealogical inference and epidemiological modelling, to estimate rates of CTL escape and reversion in HIV epitopes. We show that there is substantial uncertainty about rates of viral escape and reversion from cross-sectional data, which arises from the inherent stochasticity in the evolutionary process. By application to empirical data, we find that point estimates of rates from a previously published ODE model and the integrated approach presented here are often similar, but can also differ several-fold depending on the structure of the genealogy. The model-based approach we apply provides a framework for the statistical analysis and hypothesis testing of escape and reversion in population data and highlights the need for longitudinal and denser cross-sectional sampling to enable accurate estimate of these key parameters.