A Quantitative Systems Pharmacology Perspective on the Importance of Parameter Identifiability.

There is an inherent tension in Quantitative Systems Pharmacology (QSP) between the need to incorporate mathematical descriptions of complex physiology and drug targets with the necessity of developing robust, predictive and well-constrained models. In addition to this, there is no "gold standard" for model development and assessment in QSP. Moreover, there can be confusion over terminology such as model and parameter identifiability; complex and simple models; virtual populations; and other concepts, which leads to potential miscommunication and misapplication of methodologies within modeling communities, both the QSP community and related disciplines. This perspective article highlights the pros and cons of using simple (often identifiable) vs. complex (more physiologically detailed but often non-identifiable) models, as well as aspects of parameter identifiability, sensitivity and inference methodologies for model development and analysis. The paper distills the central themes of the issue of identifiability and optimal model size and discusses open challenges.

Using Purely Sinusoidal Voltammetry for Rapid Inference of Surface-Confined Electrochemical Reaction Parameters.

Alternating current (AC) voltammetric techniques are experimentally powerful as they enable Faradaic current to be isolated from non-Faradaic contributions. Finding the best global fit between experimental voltammetric data and simulations based on reaction models requires searching a substantial parameter space at high resolution. In this paper, we estimate parameters from purely sinusoidal voltammetry (PSV) experiments, investigating the redox reactions of a surface-confined ferrocene derivative. The advantage of PSV is that a complete experiment can be simulated relatively rapidly, compared to other AC voltammetric techniques. In one example involving thermodynamic dispersion, a PSV parameter inference effort requiring 7,500,000 simulations was completed in 7 h, whereas the same process for our previously used technique, ramped Fourier transform AC voltammetry (ramped FTACV), would have taken 4 days. Using both synthetic and experimental data with a surface confined diazonium substituted ferrocene derivative, it is shown that the PSV technique can be used to recover the key chemical and physical parameters. By applying techniques from Bayesian inference and Markov chain Monte Carlo methods, the confidence, distribution, and degree of correlation of the recovered parameters was visualized and quantified.

A Monte Carlo method to estimate cell population heterogeneity from cell snapshot data.

Variation is characteristic of all living systems. Laboratory techniques such as flow cytometry can probe individual cells, and, after decades of experimentation, it is clear that even members of genetically identical cell populations can exhibit differences. To understand whether variation is biologically meaningful, it is essential to discern its source. Mathematical models of biological systems are tools that can be used to investigate causes of cell-to-cell variation. From mathematical analysis and simulation of these models, biological hypotheses can be posed and investigated, then parameter inference can determine which of these is compatible with experimental data. Data from laboratory experiments often consist of "snapshots" representing distributions of cellular properties at different points in time, rather than individual cell trajectories. These data are not straightforward to fit using hierarchical Bayesian methods, which require the number of cell population clusters to be chosen a priori. Nor are they amenable to standard nonlinear mixed effect methods, since a single observation per cell is typically too few to estimate parameter variability. Here, we introduce a computational sampling method named "Contour Monte Carlo" (CMC) for estimating mathematical model parameters from snapshot distributions, which is straightforward to implement and does not require that cells be assigned to predefined categories. The CMC algorithm fits to snapshot probability distributions rather than raw data, which means its computational burden does not, like existing approaches, increase with the number of cells observed. Our method is appropriate for underdetermined systems, where there are fewer distinct types of observations than parameters to be determined, and where observed variation is mostly due to variability in cellular processes rather than experimental measurement error. This may be the case for many systems due to continued improvements in resolution of laboratory techniques. In this paper, we apply our method to quantify cellular variation for three biological systems of interest and provide Julia code enabling others to use this method.

Accounting for variability in ion current recordings using a mathematical model of artefacts in voltage-clamp experiments.

Mathematical models of ion channels, which constitute indispensable components of action potential models, are commonly constructed by fitting to whole-cell patch-clamp data. In a previous study, we fitted cell-specific models to hERG1a (Kv11.1) recordings simultaneously measured using an automated high-throughput system, and studied cell-cell variability by inspecting the resulting model parameters. However, the origin of the observed variability was not identified. Here, we study the source of variability by constructing a model that describes not just ion current dynamics, but the entire voltage-clamp experiment. The experimental artefact components of the model include: series resistance, membrane and pipette capacitance, voltage offsets, imperfect compensations made by the amplifier for these phenomena, and leak current. In this model, variability in the observations can be explained by either cell properties, measurement artefacts, or both. Remarkably, by assuming that variability arises exclusively from measurement artefacts, it is possible to explain a larger amount of the observed variability than when assuming cell-specific ion current kinetics. This assumption also leads to a smaller number of model parameters. This result suggests that most of the observed variability in patch-clamp data measured under the same conditions is caused by experimental artefacts, and hence can be compensated for in post-processing by using our model for the patch-clamp experiment. This study has implications for the question of the extent to which cell-cell variability in ion channel kinetics exists, and opens up routes for better correction of artefacts in patch-clamp data. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Four Ways to Fit an Ion Channel Model.

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.

Rapid Characterization of hERG Channel Kinetics I: Using an Automated High-Throughput System.

Predicting how pharmaceuticals may affect heart rhythm is a crucial step in drug development and requires a deep understanding of a compound's action on ion channels. In vitro hERG channel current recordings are an important step in evaluating the proarrhythmic potential of small molecules and are now routinely performed using automated high-throughput patch-clamp platforms. These machines can execute traditional voltage-clamp protocols aimed at specific gating processes, but the array of protocols needed to fully characterize a current is typically too long to be applied in a single cell. Shorter high-information protocols have recently been introduced that have this capability, but they are not typically compatible with high-throughput platforms. We present a new 15 second protocol to characterize hERG (Kv11.1) kinetics, suitable for both manual and high-throughput systems. We demonstrate its use on the Nanion SyncroPatch 384PE, a 384-well automated patch-clamp platform, by applying it to Chinese hamster ovary cells stably expressing hERG1a. From these recordings, we construct 124 cell-specific variants/parameterizations of a hERG model at 25°C. A further eight independent protocols are run in each cell and are used to validate the model predictions. We then combine the experimental recordings using a hierarchical Bayesian model, which we use to quantify the uncertainty in the model parameters, and their variability from cell-to-cell; we use this model to suggest reasons for the variability. This study demonstrates a robust method to measure and quantify uncertainty and shows that it is possible and practical to use high-throughput systems to capture full hERG channel kinetics quantitatively and rapidly.

Rapid Characterization of hERG Channel Kinetics II: Temperature Dependence.

Ion channel behavior can depend strongly on temperature, with faster kinetics at physiological temperatures leading to considerable changes in currents relative to room temperature. These temperature-dependent changes in voltage-dependent ion channel kinetics (rates of opening, closing, inactivating, and recovery) are commonly represented with Q10 coefficients or an Eyring relationship. In this article, we assess the validity of these representations by characterizing channel kinetics at multiple temperatures. We focus on the human Ether-à-go-go-Related Gene (hERG) channel, which is important in drug safety assessment and commonly screened at room temperature so that results require extrapolation to physiological temperature. In Part I of this study, we established a reliable method for high-throughput characterization of hERG1a (Kv11.1) kinetics, using a 15-second information-rich optimized protocol. In this Part II, we use this protocol to study the temperature dependence of hERG kinetics using Chinese hamster ovary cells overexpressing hERG1a on the Nanion SyncroPatch 384PE, a 384-well automated patch-clamp platform, with temperature control. We characterize the temperature dependence of hERG gating by fitting the parameters of a mathematical model of hERG kinetics to data obtained at five distinct temperatures between 25 and 37°C and validate the models using different protocols. Our models reveal that activation is far more temperature sensitive than inactivation, and we observe that the temperature dependency of the kinetic parameters is not represented well by Q10 coefficients; it broadly follows a generalized, but not the standardly-used, Eyring relationship. We also demonstrate that experimental estimations of Q10 coefficients are protocol dependent. Our results show that a direct fit using our 15-s protocol best represents hERG kinetics at any given temperature and suggests that using the Generalized Eyring theory is preferable if no experimental data are available to derive model parameters at a given temperature.

Representation of Multiple Cellular Phenotypes Within Tissue-Level Simulations of Cardiac Electrophysiology.

Distinct electrophysiological phenotypes are exhibited by biological cells that have differentiated into particular cell types. The usual approach when simulating the cardiac electrophysiology of tissue that includes different cell types is to model the different cell types as occupying spatially distinct yet coupled regions. Instead, we model the electrophysiology of well-mixed cells by using homogenisation to derive an extension to the commonly used monodomain or bidomain equations. These new equations permit spatial variations in the distribution of the different subtypes of cells and will reduce the computational demands of solving the governing equations. We validate the homogenisation computationally, and then use the new model to explain some experimental observations from stem cell-derived cardiomyocyte monolayers.

Sinusoidal voltage protocols for rapid characterisation of ion channel kinetics.

KEY POINTS: Ion current kinetics are commonly represented by current-voltage relationships, time constant-voltage relationships and subsequently mathematical models fitted to these. These experiments take substantial time, which means they are rarely performed in the same cell. Rather than traditional square-wave voltage clamps, we fitted a model to the current evoked by a novel sum-of-sinusoids voltage clamp that was only 8 s long. Short protocols that can be performed multiple times within a single cell will offer many new opportunities to measure how ion current kinetics are affected by changing conditions. The new model predicts the current under traditional square-wave protocols well, with better predictions of underlying currents than literature models. The current under a novel physiologically relevant series of action potential clamps is predicted extremely well. The short sinusoidal protocols allow a model to be fully fitted to individual cells, allowing us to examine cell-cell variability in current kinetics for the first time. ABSTRACT: Understanding the roles of ion currents is crucial to predict the action of pharmaceuticals and mutations in different scenarios, and thereby to guide clinical interventions in the heart, brain and other electrophysiological systems. Our ability to predict how ion currents contribute to cellular electrophysiology is in turn critically dependent on our characterisation of ion channel kinetics - the voltage-dependent rates of transition between open, closed and inactivated channel states. We present a new method for rapidly exploring and characterising ion channel kinetics, applying it to the hERG potassium channel as an example, with the aim of generating a quantitatively predictive representation of the ion current. We fitted a mathematical model to currents evoked by a novel 8 second sinusoidal voltage clamp in CHO cells overexpressing hERG1a. The model was then used to predict over 5 minutes of recordings in the same cell in response to further protocols: a series of traditional square step voltage clamps, and also a novel voltage clamp comprising a collection of physiologically relevant action potentials. We demonstrate that we can make predictive cell-specific models that outperform the use of averaged data from a number of different cells, and thereby examine which changes in gating are responsible for cell-cell variability in current kinetics. Our technique allows rapid collection of consistent and high quality data, from single cells, and produces more predictive mathematical ion channel models than traditional approaches.

Comparing individual-based approaches to modelling the self-organization of multicellular tissues.

The coordinated behaviour of populations of cells plays a central role in tissue growth and renewal. Cells react to their microenvironment by modulating processes such as movement, growth and proliferation, and signalling. Alongside experimental studies, computational models offer a useful means by which to investigate these processes. To this end a variety of cell-based modelling approaches have been developed, ranging from lattice-based cellular automata to lattice-free models that treat cells as point-like particles or extended shapes. However, it remains unclear how these approaches compare when applied to the same biological problem, and what differences in behaviour are due to different model assumptions and abstractions. Here, we exploit the availability of an implementation of five popular cell-based modelling approaches within a consistent computational framework, Chaste (http://www.cs.ox.ac.uk/chaste). This framework allows one to easily change constitutive assumptions within these models. In each case we provide full details of all technical aspects of our model implementations. We compare model implementations using four case studies, chosen to reflect the key cellular processes of proliferation, adhesion, and short- and long-range signalling. These case studies demonstrate the applicability of each model and provide a guide for model usage.

The role of the Hes1 crosstalk hub in Notch-Wnt interactions of the intestinal crypt.

The Notch pathway plays a vital role in determining whether cells in the intestinal epithelium adopt a secretory or an absorptive phenotype. Cell fate specification is coordinated via Notch's interaction with the canonical Wnt pathway. Here, we propose a new mathematical model of the Notch and Wnt pathways, in which the Hes1 promoter acts as a hub for pathway crosstalk. Computational simulations of the model can assist in understanding how healthy intestinal tissue is maintained, and predict the likely consequences of biochemical knockouts upon cell fate selection processes. Chemical reaction network theory (CRNT) is a powerful, generalised framework which assesses the capacity of our model for monostability or multistability, by analysing properties of the underlying network structure without recourse to specific parameter values or functional forms for reaction rates. CRNT highlights the role of ß-catenin in stabilising the Notch pathway and damping oscillations, demonstrating that Wnt-mediated actions on the Hes1 promoter can induce dynamic transitions in the Notch system, from multistability to monostability. Time-dependent model simulations of cell pairs reveal the stabilising influence of Wnt upon the Notch pathway, in which ß-catenin- and Dsh-mediated action on the Hes1 promoter are key in shaping the subcellular dynamics. Where Notch-mediated transcription of Hes1 dominates, there is Notch oscillation and maintenance of fate flexibility; Wnt-mediated transcription of Hes1 favours bistability akin to cell fate selection. Cells could therefore regulate the proportion of Wnt- and Notch-mediated control of the Hes1 promoter to coordinate the timing of cell fate selection as they migrate through the intestinal epithelium and are subject to reduced Wnt stimuli. Furthermore, mutant cells characterised by hyperstimulation of the Wnt pathway may, through coupling with Notch, invert cell fate in neighbouring healthy cells, enabling an aberrant cell to maintain its neighbours in mitotically active states.

Retuning the Catalytic Bias and Overpotential of a [NiFe]-Hydrogenase via a Single Amino Acid Exchange at the Electron Entry/Exit Site.

The redox chemistry of the electron entry/exit site in Escherichia coli hydrogenase-1 is shown to play a vital role in tuning biocatalysis. Inspired by nature, we generate a HyaA-R193L variant to disrupt a proposed Arg-His cation-π interaction in the secondary coordination sphere of the outermost, "distal", iron-sulfur cluster. This rewires the enzyme, enhancing the relative rate of H2 production and the thermodynamic efficiency of H2 oxidation catalysis. On the basis of Fourier transformed alternating current voltammetry measurements, we relate these changes in catalysis to a shift in the distal [Fe4S4]2+/1+ redox potential, a previously experimentally inaccessible parameter. Thus, metalloenzyme chemistry is shown to be tuned by the second coordination sphere of an electron transfer site distant from the catalytic center.

Analysis of HypD Disulfide Redox Chemistry via Optimization of Fourier Transformed ac Voltammetric Data.

Rapid disulfide bond formation and cleavage is an essential mechanism of life. Using large amplitude Fourier transformed alternating current voltammetry (FTacV) we have measured previously uncharacterized disulfide bond redox chemistry in Escherichia coli HypD. This protein is representative of a class of assembly proteins that play an essential role in the biosynthesis of the active site of [NiFe]-hydrogenases, a family of H2-activating enzymes. Compared to conventional electrochemical methods, the advantages of the FTacV technique are the high resolution of the faradaic signal in the higher order harmonics and the fact that a single electrochemical experiment contains all the data needed to estimate the (very fast) electron transfer rates (both rate constants ≥ 4000 s-1) and quantify the energetics of the cysteine disulfide redox-reaction (reversible potentials for both processes approximately -0.21 ± 0.01 V vs SHE at pH 6). Previously, deriving such data depended on an inefficient manual trial-and-error approach to simulation. As a highly advantageous alternative, we describe herein an automated multiparameter data optimization analysis strategy where the simulated and experimental faradaic current data are compared for both the real and imaginary components in each of the 4th to 12th harmonics after quantifying the charging current data using the time-domain response.

Uncertainty and variability in models of the cardiac action potential: Can we build trustworthy models?

Cardiac electrophysiology models have been developed for over 50years, and now include detailed descriptions of individual ion currents and sub-cellular calcium handling. It is commonly accepted that there are many uncertainties in these systems, with quantities such as ion channel kinetics or expression levels being difficult to measure or variable between samples. Until recently, the original approach of describing model parameters using single values has been retained, and consequently the majority of mathematical models in use today provide point predictions, with no associated uncertainty. In recent years, statistical techniques have been developed and applied in many scientific areas to capture uncertainties in the quantities that determine model behaviour, and to provide a distribution of predictions which accounts for this uncertainty. In this paper we discuss this concept, which is termed uncertainty quantification, and consider how it might be applied to cardiac electrophysiology models. We present two case studies in which probability distributions, instead of individual numbers, are inferred from data to describe quantities such as maximal current densities. Then we show how these probabilistic representations of model parameters enable probabilities to be placed on predicted behaviours. We demonstrate how changes in these probability distributions across data sets offer insight into which currents cause beat-to-beat variability in canine APs. We conclude with a discussion of the challenges that this approach entails, and how it provides opportunities to improve our understanding of electrophysiology.

Multiparameter Estimation in Voltammetry When an Electron Transfer Process Is Coupled to a Chemical Reaction.

Estimation of thermodynamic and kinetic parameters in electrochemical studies is usually undertaken via comparison of the experimental results with theory based on a model that mimics the experiment. The present study examines the credibility of transient d.c. and a.c. voltammetric theory-experiment comparisons for recovery of the parameters needed to model the ubiquitous mechanism when an electron transfer (E) reaction is followed by a chemical (C) step in the EC process ([Formula: see text]). The data analysis has been undertaken using optimization methods facilitated in some cases by grid computing. These techniques have been applied to the simulated (5% noise added) and experimental (reduction of trans-stilbene) voltammograms to assess the capabilities of parameter recovery of E(0) (reversible potential for the E step), k(0) (heterogeneous electron transfer rate constant at E(0)), α (charge transfer coefficient for the E step), and k(f) and k(b) (forward and backward rate constants for the C step) under different kinetic regimes. The advantages provided by the use of a.c. instead of d.c. voltammetry and data optimization methods over heuristic approaches to "experiment"-theory comparisons are discussed, as are the limitations in the efficient recovery of a unique set of parameters for the EC mechanism. In the particular experimental case examined herein, results for the protonation of the electrochemically generated stilbene dianion demonstrate that, notwithstanding significant advances in experiment and theory of voltammetric analysis, reliable recovery of the parameters for the EC mechanism with a fast chemical process remains a stiff problem.

Cardiac tissue slices: preparation, handling, and successful optical mapping.

Cardiac tissue slices are becoming increasingly popular as a model system for cardiac electrophysiology and pharmacology research and development. Here, we describe in detail the preparation, handling, and optical mapping of transmembrane potential and intracellular free calcium concentration transients (CaT) in ventricular tissue slices from guinea pigs and rabbits. Slices cut in the epicardium-tangential plane contained well-aligned in-slice myocardial cell strands ("fibers") in subepicardial and midmyocardial sections. Cut with a high-precision slow-advancing microtome at a thickness of 350 to 400 µm, tissue slices preserved essential action potential (AP) properties of the precutting Langendorff-perfused heart. We identified the need for a postcutting recovery period of 36 min (guinea pig) and 63 min (rabbit) to reach 97.5% of final steady-state values for AP duration (APD) (identified by exponential fitting). There was no significant difference between the postcutting recovery dynamics in slices obtained using 2,3-butanedione 2-monoxime or blebistatin as electromechanical uncouplers during the cutting process. A rapid increase in APD, seen after cutting, was caused by exposure to ice-cold solution during the slicing procedure, not by tissue injury, differences in uncouplers, or pH-buffers (bicarbonate; HEPES). To characterize intrinsic patterns of CaT, AP, and conduction, a combination of multipoint and field stimulation should be used to avoid misinterpretation based on source-sink effects. In summary, we describe in detail the preparation, mapping, and data analysis approaches for reproducible cardiac tissue slice-based investigations into AP and CaT dynamics.

Theoretical Analysis of the Relative Significance of Thermodynamic and Kinetic Dispersion in the dc and ac Voltammetry of Surface-Confined Molecules.

Commonly, significant discrepancies are reported in theoretical and experimental comparisons of dc voltammograms derived from a monolayer or close to monolayer coverage of redox-active surface-confined molecules. For example, broader-than-predicted voltammetric wave shapes are attributed to the thermodynamic or kinetic dispersion derived from distributions in reversible potentials (E(0)) and electrode kinetics (k(0)), respectively. The recent availability of experimentally estimated distributions of E(0) and k(0) values derived from the analysis of data for small numbers of surface-confined modified azurin metalloprotein molecules now allows more realistic modeling to be undertaken, assuming the same distributions apply under conditions of high surface coverage relevant to voltammetric experiments. In this work, modeling based on conventional and stochastic kinetic theory is considered, and the computationally far more efficient conventional model is shown to be equivalent to the stochastic one when large numbers of molecules are present. Perhaps unexpectedly, when experimentally determined distributions of E(0) and k(0) are input into the model, thermodynamic dispersion is found to be unimportant and only kinetic dispersion contributes significantly to the broadening of dc voltammograms. Simulations of ac voltammetric experiments lead to the conclusion that the ac method, particularly when the analysis of kinetically very sensitive higher-order harmonics is undertaken, are far more sensitive to kinetic dispersion than the dc method. ac methods are therefore concluded to provide a potentially superior strategy for addressing the inverse problem of determining the k(0) distribution that could give rise to the apparent anomalies in surface-confined voltammetry.

The free energy landscape of dimerization of a membrane protein, NanC.

Membrane proteins are frequently present in crowded environments, which favour lateral association and, on occasions, two-dimensional crystallization. To better understand the non-specific lateral association of a membrane protein we have characterized the free energy landscape for the dimerization of a bacterial outer membrane protein, NanC, in a phospholipid bilayer membrane. NanC is a member of the KdgM-family of bacterial outer membrane proteins and is responsible for sialic acid transport in E. coli. Umbrella sampling and coarse-grained molecular dynamics were employed to calculate the potentials of mean force (PMF) for a variety of restrained relative orientations of two NanC proteins as the separation of their centres of mass was varied. We found the free energy of dimerization for NanC to be in the range of -66 kJ mol(-1) to -45 kJ mol(-1). Differences in the depths of the PMFs for the various orientations are related to the shape of the proteins. This was quantified by calculating the lipid-inaccessible buried surface area of the proteins in the region around the minimum of each PMF. The depth of the potential well of the PMF was shown to depend approximately linearly on the buried surface area. We were able to resolve local minima in the restrained PMFs that would not be revealed using conventional umbrella sampling. In particular, these features reflected the local organization of the intervening lipids between the two interacting proteins. Through a comparison with the distribution of lipids around a single freely-diffusing NanC, we were able to predict the location of these restrained local minima for the orientational configuration in which they were most pronounced. Our ability to make this prediction highlights the important role that lipid organization plays in the association of two NanCs in a bilayer.

Inappropriate use of the quasi-reversible electrode kinetic model in simulation-experiment comparisons of voltammetric processes that approach the reversible limit.

Many electrode processes that approach the "reversible" (infinitely fast) limit under voltammetric conditions have been inappropriately analyzed by comparison of experimental data and theory derived from the "quasi-reversible" model. Simulations based on "reversible" and "quasi-reversible" models have been fitted to an extensive series of a.c. voltammetric experiments undertaken at macrodisk glassy carbon (GC) electrodes for oxidation of ferrocene (Fc(0/+)) in CH3CN (0.10 M (n-Bu)4NPF6) and reduction of [Ru(NH3)6](3+) and [Fe(CN)6](3-) in 1 M KCl aqueous electrolyte. The confidence with which parameters such as standard formal potential (E(0)), heterogeneous electron transfer rate constant at E(0) (k(0)), charge transfer coefficient (α), uncompensated resistance (Ru), and double layer capacitance (CDL) can be reported using the "quasi-reversible" model has been assessed using bootstrapping and parameter sweep (contour plot) techniques. Underparameterization, such as that which occurs when modeling CDL with a potential independent value, results in a less than optimal level of experiment-theory agreement. Overparameterization may improve the agreement but easily results in generation of physically meaningful but incorrect values of the recovered parameters, as is the case with the very fast Fc(0/+) and [Ru(NH3)6](3+/2+) processes. In summary, for fast electrode kinetics approaching the "reversible" limit, it is recommended that the "reversible" model be used for theory-experiment comparisons with only E(0), Ru, and CDL being quantified and a lower limit of k(0) being reported; e.g., k(0) ≥ 9 cm s(-1) for the Fc(0/+) process.

Chaste: an open source C++ library for computational physiology and biology.

Chaste - Cancer, Heart And Soft Tissue Environment - is an open source C++ library for the computational simulation of mathematical models developed for physiology and biology. Code development has been driven by two initial applications: cardiac electrophysiology and cancer development. A large number of cardiac electrophysiology studies have been enabled and performed, including high-performance computational investigations of defibrillation on realistic human cardiac geometries. New models for the initiation and growth of tumours have been developed. In particular, cell-based simulations have provided novel insight into the role of stem cells in the colorectal crypt. Chaste is constantly evolving and is now being applied to a far wider range of problems. The code provides modules for handling common scientific computing components, such as meshes and solvers for ordinary and partial differential equations (ODEs/PDEs). Re-use of these components avoids the need for researchers to 're-invent the wheel' with each new project, accelerating the rate of progress in new applications. Chaste is developed using industrially-derived techniques, in particular test-driven development, to ensure code quality, re-use and reliability. In this article we provide examples that illustrate the types of problems Chaste can be used to solve, which can be run on a desktop computer. We highlight some scientific studies that have used or are using Chaste, and the insights they have provided. The source code, both for specific releases and the development version, is available to download under an open source Berkeley Software Distribution (BSD) licence at http://www.cs.ox.ac.uk/chaste, together with details of a mailing list and links to documentation and tutorials.