Understanding the impact of numerical solvers on inference for differential equation models.

Most ordinary differential equation (ODE) models used to describe biological or physical systems must be solved approximately using numerical methods. Perniciously, even those solvers that seem sufficiently accurate for the forward problem, i.e. for obtaining an accurate simulation, might not be sufficiently accurate for the inverse problem, i.e. for inferring the model parameters from data. We show that for both fixed step and adaptive step ODE solvers, solving the forward problem with insufficient accuracy can distort likelihood surfaces, which might become jagged, causing inference algorithms to get stuck in local 'phantom' optima. We demonstrate that biases in inference arising from numerical approximation of ODEs are potentially most severe in systems involving low noise and rapid nonlinear dynamics. We reanalyse an ODE change point model previously fit to the COVID-19 outbreak in Germany and show the effect of the step size on simulation and inference results. We then fit a more complicated rainfall run-off model to hydrological data and illustrate the importance of tuning solver tolerances to avoid distorted likelihood surfaces. Our results indicate that, when performing inference for ODE model parameters, adaptive step size solver tolerances must be set cautiously and likelihood surfaces should be inspected for characteristic signs of numerical issues.

Corrigendum: Heterologous vaccination with inactivated vaccine and mRNA vaccine augments antibodies against both spike and nucleocapsid proteins of SARS-CoV-2: a local study in Macao.

[This corrects the article DOI: 10.3389/fimmu.2023.1131985.].

Classification of MLH1 Missense VUS Using Protein Structure-Based Deep Learning-Ramachandran Plot-Molecular Dynamics Simulations Method.

Pathogenic variation in DNA mismatch repair (MMR) gene MLH1 is associated with Lynch syndrome (LS), an autosomal dominant hereditary cancer. Of the 3798 MLH1 germline variants collected in the ClinVar database, 38.7% (1469) were missense variants, of which 81.6% (1199) were classified as Variants of Uncertain Significance (VUS) due to the lack of functional evidence. Further determination of the impact of VUS on MLH1 function is important for the VUS carriers to take preventive action. We recently developed a protein structure-based method named "Deep Learning-Ramachandran Plot-Molecular Dynamics Simulation (DL-RP-MDS)" to evaluate the deleteriousness of MLH1 missense VUS. The method extracts protein structural information by using the Ramachandran plot-molecular dynamics simulation (RP-MDS) method, then combines the variation data with an unsupervised learning model composed of auto-encoder and neural network classifier to identify the variants causing significant change in protein structure. In this report, we applied the method to classify 447 MLH1 missense VUS. We predicted 126/447 (28.2%) MLH1 missense VUS were deleterious. Our study demonstrates that DL-RP-MDS is able to classify the missense VUS based solely on their impact on protein structure.

The impact of uncertainty in hERG binding mechanism on in silico predictions of drug-induced proarrhythmic risk.

BACKGROUND AND PURPOSE: Drug-induced reduction of the rapid delayed rectifier potassium current carried by the human Ether-à-go-go-Related Gene (hERG) channel is associated with increased risk of arrhythmias. Recent updates to drug safety regulatory guidelines attempt to capture each drug's hERG binding mechanism by combining in vitro assays with in silico simulations. In this study, we investigate the impact on in silico proarrhythmic risk predictions due to uncertainty in the hERG binding mechanism and physiological hERG current model. EXPERIMENTAL APPROACH: Possible pharmacological binding models were designed for the hERG channel to account for known and postulated small molecule binding mechanisms. After selecting a subset of plausible binding models for each compound through calibration to available voltage-clamp electrophysiology data, we assessed their effects, and the effects of different physiological models, on proarrhythmic risk predictions. KEY RESULTS: For some compounds, multiple binding mechanisms can explain the same data produced under the safety testing guidelines, which results in different inferred binding rates. This can result in substantial uncertainty in the predicted torsade risk, which often spans more than one risk category. By comparison, we found that the effect of a different hERG physiological current model on risk classification was subtle. CONCLUSION AND IMPLICATIONS: The approach developed in this study assesses the impact of uncertainty in hERG binding mechanisms on predictions of drug-induced proarrhythmic risk. For some compounds, these results imply the need for additional binding data to decrease uncertainty in safety-critical applications.

Empirical Quantification of Predictive Uncertainty Due to Model Discrepancy by Training with an Ensemble of Experimental Designs: An Application to Ion Channel Kinetics.

When using mathematical models to make quantitative predictions for clinical or industrial use, it is important that predictions come with a reliable estimate of their accuracy (uncertainty quantification). Because models of complex biological systems are always large simplifications, model discrepancy arises-models fail to perfectly recapitulate the true data generating process. This presents a particular challenge for making accurate predictions, and especially for accurately quantifying uncertainty in these predictions. Experimentalists and modellers must choose which experimental procedures (protocols) are used to produce data used to train models. We propose to characterise uncertainty owing to model discrepancy with an ensemble of parameter sets, each of which results from training to data from a different protocol. The variability in predictions from this ensemble provides an empirical estimate of predictive uncertainty owing to model discrepancy, even for unseen protocols. We use the example of electrophysiology experiments that investigate the properties of hERG potassium channels. Here, 'information-rich' protocols allow mathematical models to be trained using numerous short experiments performed on the same cell. In this case, we simulate data with one model and fit it with a different (discrepant) one. For any individual experimental protocol, parameter estimates vary little under repeated samples from the assumed additive independent Gaussian noise model. Yet parameter sets arising from the same model applied to different experiments conflict-highlighting model discrepancy. Our methods will help select more suitable ion channel models for future studies, and will be widely applicable to a range of biological modelling problems.

Leak current, even with gigaohm seals, can cause misinterpretation of stem cell-derived cardiomyocyte action potential recordings.

AIMS: Human-induced pluripotent stem cell-derived cardiomyocytes (iPSC-CMs) have become an essential tool to study arrhythmia mechanisms. Much of the foundational work on these cells, as well as the computational models built from the resultant data, has overlooked the contribution of seal-leak current on the immature and heterogeneous phenotype that has come to define these cells. The aim of this study is to understand the effect of seal-leak current on recordings of action potential (AP) morphology. METHODS AND RESULTS: Action potentials were recorded in human iPSC-CMs using patch clamp and simulated using previously published mathematical models. Our in silico and in vitro studies demonstrate how seal-leak current depolarizes APs, substantially affecting their morphology, even with seal resistances (Rseal) above 1 GΩ. We show that compensation of this leak current is difficult due to challenges with obtaining accurate measures of Rseal during an experiment. Using simulation, we show that Rseal measures (i) change during an experiment, invalidating the use of pre-rupture values, and (ii) are polluted by the presence of transmembrane currents at every voltage. Finally, we posit that the background sodium current in baseline iPSC-CM models imitates the effects of seal-leak current and is increased to a level that masks the effects of seal-leak current on iPSC-CMs. CONCLUSION: Based on these findings, we make recommendations to improve iPSC-CM AP data acquisition, interpretation, and model-building. Taking these recommendations into account will improve our understanding of iPSC-CM physiology and the descriptive ability of models built from such data.

Model-driven optimal experimental design for calibrating cardiac electrophysiology models.

BACKGROUND AND OBJECTIVE: Models of the cardiomyocyte action potential have contributed immensely to the understanding of heart function, pathophysiology, and the origin of heart rhythm disturbances. However, action potential models are highly nonlinear, making them difficult to parameterise and limiting to describing 'average cell' dynamics, when cell-specific models would be ideal to uncover inter-cell variability but are too experimentally challenging to be achieved. Here, we focus on automatically designing experimental protocols that allow us to better identify cell-specific maximum conductance values for each major current type. METHODS AND RESULTS: We developed an approach that applies optimal experimental designs to patch-clamp experiments, including both voltage-clamp and current-clamp experiments. We assessed the models calibrated to these new optimal designs by comparing them to the models calibrated to some of the commonly used designs in the literature. We showed that optimal designs are not only overall shorter in duration but also able to perform better than many of the existing experiment designs in terms of identifying model parameters and hence model predictive power. CONCLUSIONS: For cardiac cellular electrophysiology, this approach will allow researchers to define their hypothesis of the dynamics of the system and automatically design experimental protocols that will result in theoretically optimal designs.

Heterologous vaccination with inactivated vaccine and mRNA vaccine augments antibodies against both spike and nucleocapsid proteins of SARS-CoV-2: a local study in Macao.

The mRNA vaccines (RVs) can reduce the severity and mortality of severe acute respiratory syndrome coronavirus (SARS-CoV-2). However, almost only the inactivated vaccines (IVs) but no RVs had been used in mainland China until most recently, and the relaxing of its anti-pandemic strategies in December 2022 increased concerns about new outbreaks. In comparison, many of the citizens in Macao Special Administrative Region of China received three doses of IV (3IV) or RV (3RV), or 2 doses of IV plus one booster of RV (2IV+1RV). By the end of 2022, we recruited 147 participants with various vaccinations in Macao and detected antibodies (Abs) against the spike (S) protein and nucleocapsid (N) protein of the virus as well as neutralizing antibodies (NAb) in their serum. We observed that the level of anti-S Ab or NAb was similarly high with both 3RV and 2IV+1RV but lower with 3IV. In contrast, the level of anti-N Ab was the highest with 3IV like that in convalescents, intermediate with 2IV+1RV, and the lowest with 3RV. Whereas no significant differences in the basal levels of cytokines related to T-cell activation were observed among the various vaccination groups before and after the boosters. No vaccinees reported severe adverse events. Since Macao took one of the most stringent non-pharmaceutical interventions in the world, this study possesses much higher confidence in the vaccination results than many other studies from highly infected regions. Our findings suggest that the heterologous vaccination 2IV+1RV outperforms the homologous vaccinations 3IV and 3RV as it induces not only anti-S Ab (to the level as with 3RV) but also anti-N antibodies (via the IV). It combines the advantages of both RV (to block the viral entry) and IV (to also intervene the subsequent pathological processes such as intracellular viral replication and interference with the signal transduction and hence the biological functions of host cells).

Importance of modelling hERG binding in predicting drug-induced action potential prolongations for drug safety assessment.

Reduction of the rapid delayed rectifier potassium current (I Kr) via drug binding to the human Ether-à-go-go-Related Gene (hERG) channel is a well recognised mechanism that can contribute to an increased risk of Torsades de Pointes. Mathematical models have been created to replicate the effects of channel blockers, such as reducing the ionic conductance of the channel. Here, we study the impact of including state-dependent drug binding in a mathematical model of hERG when translating hERG inhibition to action potential changes. We show that the difference in action potential predictions when modelling drug binding of hERG using a state-dependent model versus a conductance scaling model depends not only on the properties of the drug and whether the experiment achieves steady state, but also on the experimental protocols. Furthermore, through exploring the model parameter space, we demonstrate that the state-dependent model and the conductance scaling model generally predict different action potential prolongations and are not interchangeable, while at high binding and unbinding rates, the conductance scaling model tends to predict shorter action potential prolongations. Finally, we observe that the difference in simulated action potentials between the models is determined by the binding and unbinding rate, rather than the trapping mechanism. This study demonstrates the importance of modelling drug binding and highlights the need for improved understanding of drug trapping which can have implications for the uses in drug safety assessment.

Integration of deep learning with Ramachandran plot molecular dynamics simulation for genetic variant classification.

Functional classification of genetic variants is a key for their clinical applications in patient care. However, abundant variant data generated by the next-generation DNA sequencing technologies limit the use of experimental methods for their classification. Here, we developed a protein structure and deep learning (DL)-based system for genetic variant classification, DL-RP-MDS, which comprises two principles: 1) Extracting protein structural and thermodynamics information using the Ramachandran plot-molecular dynamics simulation (RP-MDS) method, 2) combining those data with an unsupervised learning model of auto-encoder and a neural network classifier to identify the statistical significance patterns of the structural changes. We observed that DL-RP-MDS provided higher specificity than over 20 widely used in silico methods in classifying the variants of three DNA damage repair genes: TP53, MLH1, and MSH2. DL-RP-MDS offers a powerful platform for high-throughput genetic variant classification. The software and online application are available at https://genemutation.fhs.um.edu.mo/DL-RP-MDS/.

A Bayesian nonparametric method for detecting rapid changes in disease transmission.

Whether an outbreak of infectious disease is likely to grow or dissipate is determined through the time-varying reproduction number, Rt. Real-time or retrospective identification of changes in Rt following the imposition or relaxation of interventions can thus contribute important evidence about disease transmission dynamics which can inform policymaking. Here, we present a method for estimating shifts in Rt within a renewal model framework. Our method, which we call EpiCluster, is a Bayesian nonparametric model based on the Pitman-Yor process. We assume that Rt is piecewise-constant, and the incidence data and priors determine when or whether Rt should change and how many times it should do so throughout the series. We also introduce a prior which induces sparsity over the number of changepoints. Being Bayesian, our approach yields a measure of uncertainty in Rt and its changepoints. EpiCluster is fast, straightforward to use, and we demonstrate that it provides automated detection of rapid changes in transmission, either in real-time or retrospectively, for synthetic data series where the Rt profile is known. We illustrate the practical utility of our method by fitting it to case data of outbreaks of COVID-19 in Australia and Hong Kong, where it finds changepoints coinciding with the imposition of non-pharmaceutical interventions. Bayesian nonparametric methods, such as ours, allow the volume and complexity of the data to dictate the number of parameters required to approximate the process and should find wide application in epidemiology. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

A hackable, multi-functional, and modular extrusion 3D printer for soft materials.

Three-dimensional (3D) printing has emerged as a powerful tool for material, food, and life science research and development, where the technology's democratization necessitates the advancement of open-source platforms. Herein, we developed a hackable, multi-functional, and modular extrusion 3D printer for soft materials, nicknamed Printer.HM. Multi-printhead modules are established based on a robotic arm for heterogeneous construct creation, where ink printability can be tuned by accessories such as heating and UV modules. Software associated with Printer.HM were designed to accept geometry inputs including computer-aided design models, coordinates, equations, and pictures, to create prints of distinct characteristics. Printer.HM could further perform versatile operations, such as liquid dispensing, non-planar printing, and pick-and-place of meso-objects. By 'mix-and-match' software and hardware settings, Printer.HM demonstrated printing of pH-responsive soft actuators, plant-based functional hydrogels, and organ macro-anatomical models. Integrating affordability and open design, Printer.HM is envisaged to democratize 3D printing for soft, biological, and sustainable material architectures.

Learning transmission dynamics modelling of COVID-19 using comomodels.

The COVID-19 epidemic continues to rage in many parts of the world. In the UK alone, an array of mathematical models have played a prominent role in guiding policymaking. Whilst considerable pedagogical material exists for understanding the basics of transmission dynamics modelling, there is a substantial gap between the relatively simple models used for exposition of the theory and those used in practice to model the transmission dynamics of COVID-19. Understanding these models requires considerable prerequisite knowledge and presents challenges to those new to the field of epidemiological modelling. In this paper, we introduce an open-source R package, comomodels, which can be used to understand the complexities of modelling the transmission dynamics of COVID-19 through a series of differential equation models. Alongside the base package, we describe a host of learning resources, including detailed tutorials and an interactive web-based interface allowing dynamic investigation of the model properties. We then use comomodels to illustrate three key lessons in the transmission of COVID-19 within R Markdown vignettes.

Nicotinamide promotes cardiomyocyte derivation and survival through kinase inhibition in human pluripotent stem cells.

Nicotinamide, the amide form of Vitamin B3, is a common nutrient supplement that plays important role in human fetal development. Nicotinamide has been widely used in clinical treatments, including the treatment of diseases during pregnancy. However, its impacts during embryogenesis have not been fully understood. In this study, we show that nicotinamide plays multiplex roles in mesoderm differentiation of human embryonic stem cells (hESCs). Nicotinamide promotes cardiomyocyte fate from mesoderm progenitor cells, and suppresses the emergence of other cell types. Independent of its functions in PARP and Sirtuin pathways, nicotinamide modulates differentiation through kinase inhibition. A KINOMEscan assay identifies 14 novel nicotinamide targets among 468 kinase candidates. We demonstrate that nicotinamide promotes cardiomyocyte differentiation through p38 MAP kinase inhibition. Furthermore, we show that nicotinamide enhances cardiomyocyte survival as a Rho-associated protein kinase (ROCK) inhibitor. This study reveals nicotinamide as a pleiotropic molecule that promotes the derivation and survival of cardiomyocytes, and it could become a useful tool for cardiomyocyte production for regenerative medicine. It also provides a theoretical foundation for physicians when nicotinamide is considered for treatments for pregnant women.

3D printed biomimetic cochleae and machine learning co-modelling provides clinical informatics for cochlear implant patients.

Cochlear implants restore hearing in patients with severe to profound deafness by delivering electrical stimuli inside the cochlea. Understanding stimulus current spread, and how it correlates to patient-dependent factors, is hampered by the poor accessibility of the inner ear and by the lack of clinically-relevant in vitro, in vivo or in silico models. Here, we present 3D printing-neural network co-modelling for interpreting electric field imaging profiles of cochlear implant patients. With tuneable electro-anatomy, the 3D printed cochleae can replicate clinical scenarios of electric field imaging profiles at the off-stimuli positions. The co-modelling framework demonstrated autonomous and robust predictions of patient profiles or cochlear geometry, unfolded the electro-anatomical factors causing current spread, assisted on-demand printing for implant testing, and inferred patients' in vivo cochlear tissue resistivity (estimated mean = 6.6 kΩcm). We anticipate our framework will facilitate physical modelling and digital twin innovations for neuromodulation implants.

Neural Network Differential Equations For Ion Channel Modelling.

Mathematical models of cardiac ion channels have been widely used to study and predict the behaviour of ion currents. Typically models are built using biophysically-based mechanistic principles such as Hodgkin-Huxley or Markov state transitions. These models provide an abstract description of the underlying conformational changes of the ion channels. However, due to the abstracted conformation states and assumptions for the rates of transition between them, there are differences between the models and reality-termed model discrepancy or misspecification. In this paper, we demonstrate the feasibility of using a mechanistically-inspired neural network differential equation model, a hybrid non-parametric model, to model ion channel kinetics. We apply it to the hERG potassium ion channel as an example, with the aim of providing an alternative modelling approach that could alleviate certain limitations of the traditional approach. We compare and discuss multiple ways of using a neural network to approximate extra hidden states or alternative transition rates. In particular we assess their ability to learn the missing dynamics, and ask whether we can use these models to handle model discrepancy. Finally, we discuss the practicality and limitations of using neural networks and their potential applications.

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'.

Considering discrepancy when calibrating a mechanistic electrophysiology model.

Uncertainty quantification (UQ) is a vital step in using mathematical models and simulations to take decisions. The field of cardiac simulation has begun to explore and adopt UQ methods to characterize uncertainty in model inputs and how that propagates through to outputs or predictions; examples of this can be seen in the papers of this issue. In this review and perspective piece, we draw attention to an important and under-addressed source of uncertainty in our predictions-that of uncertainty in the model structure or the equations themselves. The difference between imperfect models and reality is termed model discrepancy, and we are often uncertain as to the size and consequences of this discrepancy. Here, we provide two examples of the consequences of discrepancy when calibrating models at the ion channel and action potential scales. Furthermore, we attempt to account for this discrepancy when calibrating and validating an ion channel model using different methods, based on modelling the discrepancy using Gaussian processes and autoregressive-moving-average models, then highlight the advantages and shortcomings of each approach. Finally, suggestions and lines of enquiry for future work are provided. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

An audit of uncertainty in multi-scale cardiac electrophysiology models.

Models of electrical activation and recovery in cardiac cells and tissue have become valuable research tools, and are beginning to be used in safety-critical applications including guidance for clinical procedures and for drug safety assessment. As a consequence, there is an urgent need for a more detailed and quantitative understanding of the ways that uncertainty and variability influence model predictions. In this paper, we review the sources of uncertainty in these models at different spatial scales, discuss how uncertainties are communicated across scales, and begin to assess their relative importance. We conclude by highlighting important challenges that continue to face the cardiac modelling community, identifying open questions, and making recommendations for future studies. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.