Four Ways to Fit an Ion Channel Model

© 2019 Biophysical Society Mathematical models of ionic currents are used to study the electrophysiology of the heart, brain, gut, and several other organs. Increasingly, these models are being used predictively in the clinic, for example, to predict the risks and results of genetic mutations, pharmacological treatments, or surgical procedures. These safety-critical applications depend on accurate characterization of the underlying ionic currents. Four different methods can be found in the literature to fit voltage-sensitive ion channel models to whole-cell current measurements: method 1, fitting model equations directly to time-constant, steady-state, and I-V summary curves; method 2, fitting by comparing simulated versions of these summary curves to their experimental counterparts; method 3, fitting to the current traces themselves from a range of protocols; and method 4, fitting to a single current trace from a short and rapidly fluctuating voltage-clamp protocol. We compare these methods using a set of experiments in which hERG1a current was measured in nine Chinese hamster ovary cells. In each cell, the same sequence of fitting protocols was applied, as well as an independent validation protocol. We show that methods 3 and 4 provide the best predictions on the independent validation set and that short, rapidly fluctuating protocols like that used in method 4 can replace much longer conventional protocols without loss of predictive ability. Although data for method 2 are most readily available from the literature, we find it performs poorly compared to methods 3 and 4 both in accuracy of predictions and computational efficiency. Our results demonstrate how novel experimental and computational approaches can improve the quality of model predictions in safety-critical applications

A cell cycle model for somitogenesis: mathematical formulation and numerical simulation

After many years of research, the mechanisms that generate a periodic pattern of repeated elements (somites) along the length of the embryonic body axis is still one of the major unresolved problems in developmental biology. Here we present a mathematical formulation of the cell cycle model for somitogenesis proposed in Development105 (1989), 119–130. Somite precursor cells in the node are asynchronous, and therefore, as a population, generate continuously pre-somite cells which enter the segmental plate. The model makes the hypothesis that there exists a time window within the cell cycle, making up one-seventh of the cycle, which gates the pre-somite cells so that they make somites discretely, seven per cycle. We show that the model can indeed account for the spatiotemporal patterning of somite formation during normal development as well as the periodic abnormalities produced by heat shock treatment. We also relate the model to recent molecular data on the process of somite formation

Soft tissue modelling of cardiac fibres for use in coupled mechano–electric simulations

The numerical solution of the coupled system of partial differential and ordinary differential equations that model the whole heart in three dimensions is a considerable computational challenge. As a consequence, it is not computationally practical--either in terms of memory or time--to repeat simulations on a finer computational mesh to ensure that convergence of the solution has been attained. In an attempt to avoid this problem while retaining mathematical rigour, we derive a one dimensional model of a cardiac fibre that takes account of elasticity properties in three structurally defined axes within the myocardial tissue. This model of a cardiac fibre is then coupled with an electrophysiological cell model and a model of cellular electromechanics to allow us to simulate the coupling of the electrical and mechanical activity of the heart. We demonstrate that currently used numerical methods for coupling electrical and mechanical activity do not work in this case, and identify appropriate numerical techniques that may be used when solving the governing equations. This allows us to perform a series of simulations that: (i) investigate the effect of some of the assumptions inherent in other models; and (ii) reproduce qualitatively some experimental observations

Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries

In this study we investigate the equations governing the transport of oxygen in pulmonary capillaries. We use a mathematical model consisting of a red blood cell completely surrounded by plasma within a cylindrical pulmonary capillary. This model takes account of convection and diffusion of oxygen through plasma, diffusion of oxygen through the red blood cell, and the reaction between oxygen and haemoglobin molecules. The velocity field within the plasma is calculated by solving the slow flow equations. We investigate the effect on the solution of the governing equations of: (i) mixed-venous blood oxygen partial pressure (the initial conditions); (ii) alveolar gas oxygen partial pressure (the boundary conditions); (iii) neglecting the convection term; and (iv) assuming an instantaneous reaction between the oxygen and haemoglobin molecules. It is found that: (a) equilibrium is reached much more rapidly for high values of mixed-venous blood and alveolar gas oxygen partial pressure; (b) the convection term has a negligible effect on the time taken to reach a prescribed degree of equilibrium; and (c) an instantaneous reaction may be assumed. Explanations are given for each of these results

A mathematical evaluation of the multiple breath nitrogen washout (MBNW) technique and the multiple inert gas elimination technique (MIGET)

We consider two and 50 compartment lung models for use with two techniques used to investigate the efficiency of the lungs: the Multiple Breath Nitrogen Washout (MBNW) technique used for investigating the ventilation-volume distribution; and the Multiple Inert Gas Elimination Technique (MIGET) used for investigating the ventilation-perfusion distribution. In each of these techniques pulmonary respiratory gas exchange is described by conservation of mass equations which may be written in identical form, and in each the underlying distributions of ventilation to volume and ventilation to perfusion are assume