Empirical and mechanistic mathematical models of temporal evolution of milk production in ruminants.

In the various sectors of animal science there has been little exploration of the theoretical mathematical aspects of data analysis and modelling. The dominant statistical methods used for the analysis of experimental data are rarely valuable for developing a deeper understanding of the problem. In addition they do not take account of the evolution over time of those variables of major interest to be studied. Only recently have more sophisticated methods of mathematical modelling begun to be used. Nonetheless attention tends to be focused exclusively on empirical models. Mathematical models with greater explanatory power, in particular those which use differential equations, are as yet little used. This work develops a mathematical approach to a problem that is of great interest in animal science: the development over time of milk production in economically important ruminant species.     

Towards mechanistic models of plant organ growth

Modelling and simulation are increasingly used as tools in the study of plant growth and developmental processes. By formulating experimentally obtained knowledge as a system of interacting mathematical equations, it becomes feasible for biologists to gain a mechanistic understanding of the complex behaviour of biological systems. In this review, the modelling tools that are currently available and the progress that has been made to model plant development, based on experimental knowledge, are described. In terms of implementation, it is argued that, for the modelling of plant organ growth, the cellular level should form the cornerstone. It integrates the output of molecular regulatory networks to two processes, cell division and cell expansion, that drive growth and development of the organ. In turn, these cellular processes are controlled at the molecular level by hormone signalling. Therefore, combining a cellular modelling framework with regulatory modules for the regulation of cell division, expansion, and hormone signalling could form the basis of a functional organ growth simulation model. The current state of progress towards this aim is that the regulation of the cell cycle and hormone transport have been modelled extensively and these modules could be integrated. However, much less progress has been made on the modelling of cell expansion, which urgently needs to be addressed. A limitation of the current generation models is that they are largely qualitative. The possibilities to characterize existing and future models more quantitatively will be discussed. Together with experimental methods to measure crucial model parameters, these modelling techniques provide a basis to develop a Systems Biology approach to gain a fundamental insight into the relationship between gene function and whole organ behaviour.     

Quantitative techniques for steady-state calculation and dynamic integrated modelling of membrane potential and intracellular ion concentrations

The membrane potential (E_m) is a fundamental cellular parameter that is primarily determined by the transmembrane permeabilities and concentration gradients of various ions. However, ion gradients are themselves profoundly influenced by E_m due to its influence upon transmembrane ion fluxes and cell volume (V_c). These interrelationships between E_m, V_c and intracellular ion concentrations make computational modelling useful or necessary in order to guide experimentation and to achieve an integrated understanding of experimental data, particularly in complex, dynamic, multi-compartment systems such as skeletal and cardiac myocytes. A variety of quantitative techniques exist that may assist such understanding, from classical approaches such as the Goldman–Hodgkin–Katz equation and the Gibbs–Donnan equilibrium, to more recent “current-summing” models as exemplified by cardiac myocyte models including those of DiFrancesco & Noble, Luo & Rudy and Puglisi & Bers, or the “charge-difference” modelling technique of Fraser & Huang so far applied to skeletal muscle. In general, the classical approaches provide useful and important insights into the relationships between E_m, V_c and intracellular ion concentrations at steady state, providing their core assumptions are fully understood, while the more recent techniques permit the modelling of changing values of E_m, V_c and intracellular ion concentrations. The present work therefore reviews the various approaches that may be used to calculate E_m, V_c and intracellular ion concentrations with the aim of establishing the requirements for an integrated model that can both simulate dynamic systems and recapitulate the key findings of classical techniques regarding the cellular steady state. At a time when the number of cellular models is increasing at an unprecedented rate, it is hoped that this article will provide a useful and critical analysis of the mathematical techniques fundamental to each of them.     

WGCNA: an R package for weighted correlation network analysis

Background

Modelling the time-related behaviour of biological systems is essential for understanding their dynamic responses to perturbations. In metabolic profiling studies, the sampling rate and number of sampling points are often restricted due to experimental and biological constraints.

Results

A supervised multivariate modelling approach with the objective to model the time-related variation in the data for short and sparsely sampled time-series is described. A set of piecewise Orthogonal Projections to Latent Structures (OPLS) models are estimated, describing changes between successive time points. The individual OPLS models are linear, but the piecewise combination of several models accommodates modelling and prediction of changes which are non-linear with respect to the time course. We demonstrate the method on both simulated and metabolic profiling data, illustrating how time related changes are successfully modelled and predicted.

Conclusion

The proposed method is effective for modelling and prediction of short and multivariate time series data. A key advantage of the method is model transparency, allowing easy interpretation of time-related variation in the data. The method provides a competitive complement to commonly applied multivariate methods such as OPLS and Principal Component Analysis (PCA) for modelling and analysis of short time-series data.     

Bacteriorhodopsin in liposomes. II. Experimental evidence in support of a theoretical model

In the preceding article equations describing relevant ion flows in illuminated suspensions of bacteriorhodopsin liposomes have been derived. Here these equations are subjected to experimental tests. Changes in permeability characteristics of the liposomal membrane are brought about by addition of specific ionophores and change of medium composition. Using light-driven proton uptake and electrochemical potential differences for protons across the membrane as observation parameters, ridig attempts to falsify the derived equations are unsuccessful.Agreement between equations and experimental results is established on the point of: (i) the antagonistic effect of valinomycin and nigericin on the two components of the proton-motive force, (ii) the time dependence of the changes in transmembrane electrical and chemical potential differences after the onset of illumination.In three independent experimental systems evidence was obtained for the correctness of the postulated dependence of the turnover rate of the photochemical cycle on back pressure by the transmembrane electrochemical potential difference for protons.     

Analysis of red blood cell motion through cylindrical micropores: effects of cell properties.

Filtration through micropores is frequently used to assess red blood cell deformability, but the dependence of pore transit time on cell properties is not well understood. A theoretical model is used to simulate red cell motion through cylindrical micropores with diameters of 3.6, 5, and 6.3 microns, and 11-microns length, at driving pressures of 100-1000 dyn/cm2. Cells are assumed to have axial symmetry and to conserve surface area during deformation. Effects of membrane shear viscosity and elasticity are included, but bending resistance is neglected. A time-dependent lubrication equation describing the motion of the suspending fluid is solved, together with the equations for membrane equilibrium, using a finite difference method. Predicted transit times are consistent with previous experimental observations. Time taken for cells to enter pores represents more than one-half of the transit time. Predicted transit time increases with increasing membrane viscosity and with increasing cell volume. It is relatively insensitive to changes in internal viscosity and to changes in membrane elasticity except in the narrowest pores at low driving pressures. Elevating suspending medium viscosity does not increase sensitivity of transit time to membrane properties. Thus filterability of red cells is sensitively dependent on their resistance to transient deformations, which may be a key determinant of resistance to blood flow in the microcirculation.     

Bacteriorhodopsin in liposomes. II. Experimental evidence in support of a theoretical model.

In the preceding article equations describing relevant ion flows in illuminated suspensions of bacteriorhodopsin liposomes have been derived. Here these equations are subjected to experimental tests. Changes in permeability characteristics of the liposomal membrane are brought about by addition of specific ionophores and change of medium composition. Using light-driven proton uptake and electrochemical potential differences for protons across the membrane as observation parameters, ridig attempts to falsify the derived equations are unsuccessful. Agreement between equations and experimental results is established on the point of: (i) the antagonistic effect of valinomycin and nigericin on the two components of the proton-motive force, (ii) the time dependence of the changes in transmembrane electrical and chemical potential differences after the onset of illumination. In three independent experimental systems evidence was obtained for the correctness of the postulated dependence of the turnover rate of the photochemical cycle on back pressure by the transmembrane electrochemical potential difference for protons.     

Robust, quantitative tools for modelling ex-vivo expansion of haematopoietic stem cells and progenitors

Despite substantial research activity on bioreactor design and experiments, there are very few reports of modelling tools that can be used to generate predictive models describing how bioreactor parameters affect performance. New developments in mathematics, such as sparse Bayesian feature selection methods and nonlinear model-free modelling regression methods, offer considerable promise for modelling diverse types of data. The utility of these mathematical tools in stem cell biology are demonstrated by analysis of a large set of bioreactor data derived from the literature. In spite of the diversity of the data sources, and the inherent difficulty in representing bioreactor variables, these modelling methods were able to develop robust, quantitative, predictive models. These models relate bioreactor operational parameters to the degree of expansion of haematopoietic stem cells or their progenitors, and also identify the bioreactor variables that are most likely to affect performance across many experiments. These methods show substantial promise in assisting the design and optimisation of stem cell bioreactors.     

Computational analysis of CFSE proliferation assay

CFSE based tracking of the lymphocyte proliferation using flow cytometry is a powerful experimental technique in immunology allowing for the tracing of labelled cell populations over time in terms of the number of divisions cells undergone. Interpretation and understanding of such population data can be greatly improved through the use of mathematical modelling. We apply a heterogenous linear compartmental model, described by a system of ordinary differential equations similar to those proposed by Kendall. This model allows division number-dependent rates of cell proliferation and death and describes the rate of changes in the numbers of cells having undergone j divisions. The experimental data set that we specifically analyze specifies the following characteristics of the kinetics of PHA-induced human T lymphocyte proliferation assay in vitroL (1) the total number of live cells, (2) the total number of dead but not disintegrated cells and (3) the number of cells divided j times. Following the maximum likelihood approach for data fitting, we estimate the model parameters which, in particular, present the CTL birth- and death rate “functions”. It is the first study of CFSE labelling data which convincingly shows that the lymphocyte proliferation and death both in vitro and in vivo are division number dependent. For the first time, the confidence in the estimated parameter values is analyzed by comparing three major methods: the technique based on the variance–covariance matrix, the profile-likelihood-based approach and the bootstrap technique. We compare results and performance of these methods with respect to their robustness and computational cost. We show that for evaluating mathematical models of differing complexity the information-theoretic approach, based upon indicators measuring the information loss for a particular model (Kullback–Leibler information), provides a consistent basis. We specifically discuss methodological and computational difficulties in parameter identification with CFSE data, e.g. the loss of confidence in the parameter estimates starting around the sixth division. Overall, our study suggests that the heterogeneity inherent in cell kinetics should be explicitly incorporated into the structure of mathematical models.      

Soft tissue modelling for applications in virtual surgery and surgical robotics

Soft tissue modelling has gained a great deal of importance, for a large part due to its application in surgical training simulators for minimally invasive surgery (MIS). This article provides a structured overview of different continuum-mechanical models that have been developed over the years. It aims at facilitating model choice for specific soft tissue modelling applications. According to the complexity of the model, different features of soft biological tissue will be incorporated, i.e. nonlinearity, viscoelasticity, anisotropy, heterogeneity and finally, tissue damage during deformation. A brief summary of experimental methods for material characterisation and an introduction to methods for geometric modelling are also provided.

The overview is non-exhaustive, focusing on the most important general models and models with specific biological applications. A trade-off in complexity must be made for enabling real-time simulation, but still maintaining realistic representation of the organ deformation. Depending on the organ and tissue types, different models with emphasis on certain features will prove to be more appropriate, meaning the optimal model choice is organ and tissue-dependent.     

Concentration dependence of the cell membrane permeability to cryoprotectant and water and implications for design of methods for post-thaw washing of human erythrocytes

For more than fifty years the human red blood cell (RBC) has been a widely studied model for transmembrane mass transport. Existing literature spans myriad experimental designs with varying results and physiologic interpretations. In this review, we examine the kinetics and mechanisms of membrane transport in the context of RBC cryopreservation. We include a discussion of the pathways for water and glycerol permeation through the cell membrane and the implications for mathematical modeling of the membrane transport process. In particular, we examine the concentration dependence of water and glycerol transport and provide equations for estimating permeability parameters as a function of concentration based on a synthesis of literature data. This concentration-dependent transport model may allow for design of improved methods for post-thaw removal of glycerol from cryopreserved blood. More broadly, the consideration of the concentration dependence of membrane permeability parameters may be important for other cell types as well, especially for design of methods for equilibration with the highly concentrated solutions used for vitrification.     

Soft tissue modelling for applications in virtual surgery and surgical robotics

Soft tissue modelling has gained a great deal of importance, for a large part due to its application in surgical training simulators for minimally invasive surgery (MIS). This article provides a structured overview of different continuum-mechanical models that have been developed over the years. It aims at facilitating model choice for specific soft tissue modelling applications. According to the complexity of the model, different features of soft biological tissue will be incorporated, i.e. nonlinearity, viscoelasticity, anisotropy, heterogeneity and finally, tissue damage during deformation. A brief summary of experimental methods for material characterisation and an introduction to methods for geometric modelling are also provided. The overview is non-exhaustive, focusing on the most important general models and models with specific biological applications. A trade-off in complexity must be made for enabling real-time simulation, but still maintaining realistic representation of the organ deformation. Depending on the organ and tissue types, different models with emphasis on certain features will prove to be more appropriate, meaning the optimal model choice is organ and tissue-dependent.     

Mathematical modelling of glycolysis and adenine nucleotide metabolism of human erythrocytes. I. Reaction-kinetic statements, analysis of in vivo state and determination of starting conditions for in vitro experiments

A mathematical model of energy metabolism of human red cells is presented, which includes besides the glycolytic reactions the adenine nucleotide metabolism. The model is based on the network of chemical reactions, the thermodynamic equilibrium constants of fast reversible reactions and on the kinetic equations for irreversible enzyme reactions. The model consists of a system of 16 differential equations and allows the mathematical evaluation of metabolic levels in the steady state of energy metabolism corresponding to the in vivo state erythrocytes with the kinetic data for the enzymes derived from in vitro experiments. The dependence of the levels of metabolites in the steady state on the activity of some enzymes is analysed to characterize the regulatory properties of the system. The comparison of the steady state levels of the model with experimental data makes it possible to estimate values of some controversial enzyme parameters. Estimates of the kinetic parameters of the following intracellular processes are presented: 1) rate constant of AMP-phosphatase, 2) maximum rate of adenylate deaminase, 3) activity of adenine phosphoribosylpyrophosphate transferase and 4) adenosine transport through the cell membrane. The simulation of the preparatory phase before incubation of erythrocytes indicates, that the model also permits to compute the time course of changes of levels of metabolites. To solve the initial problem the stiff differential equation system is integrated numerically by an efficient program without the application of the quasi-steady-state approximation.     

Correlation and studies of habitat selection: problem,red herring or opportunity?

With the advent of new technologies, animal locations are being collected at ever finer spatio-temporal scales. We review analytical methods for dealing with correlated data in the context of resource selection, including post hoc variance inflation techniques, ‘two-stage’ approaches based on models fit to each individual, generalized estimating equations and hierarchical mixed-effects models. These methods are applicable to a wide range of correlated data problems, but can be difficult to apply and remain especially challenging for use–availability sampling designs because the correlation structure for combinations of used and available points are not likely to follow common parametric forms. We also review emerging approaches to studying habitat selection that use fine-scale temporal data to arrive at biologically based definitions of available habitat, while naturally accounting for autocorrelation by modelling animal movement between telemetry locations. Sophisticated analyses that explicitly model correlation rather than consider it a nuisance, like mixed effects and state-space models, offer potentially novel insights into the process of resource selection, but additional work is needed to make them more generally applicable to large datasets based on the use–availability designs. Until then, variance inflation techniques and two-stage approaches should offer pragmatic and flexible approaches to modelling correlated data.     

Modelling cell signalling and differentiation in the urothelium

Urothelial cells line the bladder. If the urothelium is damaged, it is vital that it repairs itself quickly. Experimental results shedding light on how this repair process works are presented, revealing in particular the dependence of the response on the length of time for which the drug Troglitazone (TZ) is applied. A simple mathematical model for the basic mechanism (comprising ordinary differential equations) is then developed and analysed, seeking specifically to clarify and quantify the mechanisms governing the dependence of the cell differentiation response on the TZ administration time, rather than providing a comprehensive model of differentiation. Through biologically justified simplifications, analysis reveals that the model gives results in accord with the experimental observations, and suggests new experiments that may aid further understanding. Directions in which this preliminary modelling of the PPARγ (peroxisome proliferator activated receptor γ) pathway could be usefully extended are also indicated.     

Biology by numbers: mathematical modelling in developmental biology

In recent years, mathematical modelling of developmental processes has earned new respect. Not only have mathematical models been used to validate hypotheses made from experimental data, but designing and testing these models has led to testable experimental predictions. There are now impressive cases in which mathematical models have provided fresh insight into biological systems, by suggesting, for example, how connections between local interactions among system components relate to their wider biological effects. By examining three developmental processes and corresponding mathematical models, this Review addresses the potential of mathematical modelling to help understand development.     

Effects of cellular geometry on current flow during a propagated action potential.

An impulse propagating in a cell with nonuniform geometry sees an increased electrical load at regions of increasing diameter or at branch points with certain morphologies. We present here theoretical and experimental studies on the changes in membrane current and axial current associated with diameter changes. The theoretical studies were done with numerical solutions for cable equations that were generalized to include a varying diameter; the Hodgkin-Huxley equations were used to represent the membrane properties. The experimental studied were done using squid axons with the axial insertion of platinized platinum wires to create a localized region of increased electrical load. As an action potential approaches a region of increased electrical load, the action potential amplitude and rate of rise decrease, but there is a marked increase in the magnitude of the inward sodium current. The time integrals of the inward and outward currents are not constant along the fiber and indicate net inward charge movement at regions subjected to an increased electrical load. Changes in the waveform of the axial current at such a region help to explain the temperature dependence of propagation failure at regions of increasing electrical load.     

Stochastic and deterministic multiscale models for systems biology: an auxin-transport case study

Background  

Stochastic and asymptotic methods are powerful tools in developing multiscale systems biology models; however, little has been done in this context to compare the efficacy of these methods. The majority of current systems biology modelling research, including that of auxin transport, uses numerical simulations to study the behaviour of large systems of deterministic ordinary differential equations, with little consideration of alternative modelling frameworks.     

An Eulerian/XFEM formulation for the large deformation of cortical cell membrane

Most animal cells are surrounded by a thin layer of actin meshwork below their membrane, commonly known as the actin cortex (or cortical membrane). An increasing number of studies have highlighted the role of this structure in many cell functions including contraction and locomotion, but modelling has been limited by the fact that the membrane thickness (about 1?μm) is usually much smaller than the typical size of a cell (10-100?μm). To overcome theoretical and numerical issues resulting from this observation, we introduce in this paper a continuum formulation, based on surface elasticity, that views the cortex as an infinitely thin membrane that can resists tangential deformation. To accurately model the large deformations of cells, we introduced equilibrium equations and constitutive relations within the Eulerian viewpoint such that all quantities (stress, rate of deformation) lie in the current configuration. A solution procedure is then introduced based on a coupled extended finite element approach that enables a continuum solution to the boundary value problem in which discontinuities in both strain and displacement (due to cortical elasticity) are easily handled. We validate the approach by studying the effect of cortical elasticity on the deformation of a cell adhering on a stiff substrate and undergoing internal contraction. Results show very good prediction of the proposed method when compared with experimental observations and analytical solutions for simple cases. In particular, the model can be used to study how cell properties such as stiffness and contraction of both cytoskeleton and cortical membrane lead to variations in cell's surface curvature. These numerical results show that the proposed method can be used to gain critical insights into how the cortical membrane affects cell deformation and how it may be used as a means to determine a cell's mechanical properties by measuring curvatures of its membrane.