Benchmark problems for dynamic modeling of intracellular processes

A mathematical model for the within-host (re)infection dynamics of SARS-CoV-2

Interactions between SARS-CoV-2 and the immune system during infection are complex. However, understanding the within-host SARS-CoV-2 dynamics is of enormous importance for clinical and public health outcomes. Current mathematical models focus on describing the within-host SARS-CoV-2 dynamics during the acute infection phase. Thereby they ignore important long-term post-acute infection effects. We present a mathematical model, which not only describes the SARS-CoV-2 infection dynamics during the acute infection phase, but extends current approaches by also recapitulating clinically observed long-term post-acute infection effects, such as the recovery of the number of susceptible epithelial cells to an initial pre-infection homeostatic level, a permanent and full clearance of the infection within the individual, immune waning, and the formation of long-term immune capacity levels after infection. Finally, we used our model and its description of the long-term post-acute infection dynamics to explore reinfection scenarios differentiating between distinct variant-specific properties of the reinfecting virus. Together, the model's ability to describe not only the acute but also the long-term post-acute infection dynamics provides a more realistic description of key outcomes and allows for its application in clinical and public health scenarios.

Posterior marginalization accelerates Bayesian inference for dynamical models of biological processes

Bayesian inference is an important method in the life and natural sciences for learning from data. It provides information about parameter and prediction uncertainties. Yet, generating representative samples from the posterior distribution is often computationally challenging. Here, we present an approach that lowers the computational complexity of sample generation for dynamical models with scaling, offset, and noise parameters. The proposed method is based on the marginalization of the posterior distribution. We provide analytical results for a broad class of problems with conjugate priors and show that the method is suitable for a large number of applications. Subsequently, we demonstrate the benefit of the approach for applications from the field of systems biology. We report an improvement up to 50 times in the effective sample size per unit of time. As the scheme is broadly applicable, it will facilitate Bayesian inference in different research fields.

Uncovering specific mechanisms across cell types in dynamical models

Ordinary differential equations are frequently employed for mathematical modeling of biological systems. The identification of mechanisms that are specific to certain cell types is crucial for building useful models and to gain insights into the underlying biological processes. Regularization techniques have been proposed and applied to identify mechanisms specific to two cell types, e.g., healthy and cancer cells, including the LASSO (least absolute shrinkage and selection operator). However, when analyzing more than two cell types, these approaches are not consistent, and require the selection of a reference cell type, which can affect the results. To make the regularization approach applicable to identifying cell-type specific mechanisms in any number of cell types, we propose to incorporate the clustered LASSO into the framework of ordinary differential equation modeling by penalizing the pairwise differences of the logarithmized fold-change parameters encoding a specific mechanism in different cell types. The symmetry introduced by this approach renders the results independent of the reference cell type. We discuss the necessary adaptations of state-of-the-art numerical optimization techniques and the process of model selection for this method. We assess the performance with realistic biological models and synthetic data, and demonstrate that it outperforms existing approaches. Finally, we also exemplify its application to published biological models including experimental data, and link the results to independent biological measurements.

Profile-Wise Analysis: A profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical models

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Developing mechanistic insight by combining mathematical models and experimental data is especially critical in mathematical biology as new data and new types of data are collected and reported. Key steps in using mechanistic mathematical models to interpret data include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. Here we present a systematic, computationally-efficient workflow we call Profile-Wise Analysis (PWA) that addresses all three steps in a unified way. Recently-developed methods for constructing 'profile-wise' prediction intervals enable this workflow and provide the central linkage between different workflow components. These methods propagate profile-likelihood-based confidence sets for model parameters to predictions in a way that isolates how different parameter combinations affect model predictions. We show how to extend these profile-wise prediction intervals to two-dimensional interest parameters. We then demonstrate how to combine profile-wise prediction confidence sets to give an overall prediction confidence set that approximates the full likelihood-based prediction confidence set well. Our three case studies illustrate practical aspects of the workflow, focusing on ordinary differential equation (ODE) mechanistic models with both Gaussian and non-Gaussian noise models. While the case studies focus on ODE-based models, the workflow applies to other classes of mathematical models, including partial differential equations and simulation-based stochastic models. Open-source software on GitHub can be used to replicate the case studies.

Likelihood-ratio test statistic for the finite-sample case in nonlinear ordinary differential equation models

Likelihood ratios are frequently utilized as basis for statistical tests, for model selection criteria and for assessing parameter and prediction uncertainties, e.g. using the profile likelihood. However, translating these likelihood ratios into p-values or confidence intervals requires the exact form of the test statistic's distribution. The lack of knowledge about this distribution for nonlinear ordinary differential equation (ODE) models requires an approximation which assumes the so-called asymptotic setting, i.e. a sufficiently large amount of data. Since the amount of data from quantitative molecular biology is typically limited in applications, this finite-sample case regularly occurs for mechanistic models of dynamical systems, e.g. biochemical reaction networks or infectious disease models. Thus, it is unclear whether the standard approach of using statistical thresholds derived for the asymptotic large-sample setting in realistic applications results in valid conclusions. In this study, empirical likelihood ratios for parameters from 19 published nonlinear ODE benchmark models are investigated using a resampling approach for the original data designs. Their distributions are compared to the asymptotic approximation and statistical thresholds are checked for conservativeness. It turns out, that corrections of the likelihood ratios in such finite-sample applications are required in order to avoid anti-conservative results.

Realistic simulation of time-course measurements in systems biology

In systems biology, the analysis of complex nonlinear systems faces many methodological challenges. For the evaluation and comparison of the performances of novel and competing computational methods, one major bottleneck is the availability of realistic test problems. We present an approach for performing realistic simulation studies for analyses of time course data as they are typically measured in systems biology. Since the design of experiments in practice depends on the process of interest, our approach considers the size and the dynamics of the mathematical model which is intended to be used for the simulation study. To this end, we used 19 published systems biology models with experimental data and evaluated the relationship between model features (e.g., the size and the dynamics) and features of the measurements such as the number and type of observed quantities, the number and the selection of measurement times, and the magnitude of measurement errors. Based on these typical relationships, our novel approach enables suggestions of realistic simulation study designs in the systems biology context and the realistic generation of simulated data for any dynamic model. The approach is demonstrated on three models in detail and its performance is validated on nine models by comparing ODE integration, parameter optimization, and parameter identifiability. The presented approach enables more realistic and less biased benchmark studies and thereby constitutes an important tool for the development of novel methods for dynamic modeling.

Bayesian estimation reveals that reproducible models in Systems Biology get more citations

The Systems Biology community has taken numerous actions to develop data and modeling standards towards FAIR data and model handling. Nevertheless, the debate about incentives and rewards for individual researchers to make their results reproducible is ongoing. Here, we pose the specific question of whether reproducible models have a higher impact in terms of citations. Therefore, we statistically analyze 328 published models recently classified by Tiwari et al. based on their reproducibility. For hypothesis testing, we use a flexible Bayesian approach that provides complete distributional information for all quantities of interest and can handle outliers. The results show that in the period from 2013, i.e., 10 years after the introduction of SBML, to 2020, the group of reproducible models is significantly more cited than the non-reproducible group. We show that differences in journal impact factors do not explain this effect and that this effect increases with additional standardization of data and error model integration via PEtab. Overall, our statistical analysis demonstrates the long-term merits of reproducible modeling for the individual researcher in terms of citations. Moreover, it provides evidence for the increased use of reproducible models in the scientific community.

Efficient computation of adjoint sensitivities at steady-state in ODE models of biochemical reaction networks

Dynamical models in the form of systems of ordinary differential equations have become a standard tool in systems biology. Many parameters of such models are usually unknown and have to be inferred from experimental data. Gradient-based optimization has proven to be effective for parameter estimation. However, computing gradients becomes increasingly costly for larger models, which are required for capturing the complex interactions of multiple biochemical pathways. Adjoint sensitivity analysis has been pivotal for working with such large models, but methods tailored for steady-state data are currently not available. We propose a new adjoint method for computing gradients, which is applicable if the experimental data include steady-state measurements. The method is based on a reformulation of the backward integration problem to a system of linear algebraic equations. The evaluation of the proposed method using real-world problems shows a speedup of total simulation time by a factor of up to 4.4. Our results demonstrate that the proposed approach can achieve a substantial improvement in computation time, in particular for large-scale models, where computational efficiency is critical.

A Practical Guide for the Efficient Formulation and Calibration of Large, Energy- and Rule-Based Models of Cellular Signal Transduction

Aberrant signal transduction leads to complex diseases such as cancer. To rationally design treatment strategies with small molecule inhibitors, computational models have to be employed. Energy- and rule-based models allow the construction of mechanistic ordinary differential equation models based on structural insights. The detailed, energy-based description often generates large models, which are difficult to calibrate on experimental data. In this chapter, we provide a detailed, interactive protocol for the programmatic formulation and calibration of such large, energy- and rule-based models of cellular signal transduction based on an example model describing the action of RAF inhibitors on MAPK signaling. An interactive version of this chapter is available as Jupyter Notebook at github.com/FFroehlich/energy_modeling_chapter .

The effect of normalisation and error model choice on the distribution of the maximum likelihood estimator for a biochemical reaction

Sparse and noisy measurements make parameter estimation for biochemical reaction networks difficult and might lead to ill-posed optimisation problems. This is potentiated if the data has to be normalised, and only fold changes rather than absolute amounts are available. Here, the authors consider the propagation of measurement noise to the distribution of the maximum likelihood (ML) estimator in an in silico study. Therefore, a model of a reversible reaction is considered, for which reaction rate constants using fold changes is estimated. Noise propagation is analysed for different normalisation strategies and different error models. In particular, accuracy, precision, and asymptotic properties of the ML estimator is investigated. Results show that normalisation by the mean of a time series outperforms normalisation by a single time point in the example provided by the authors. Moreover, the error model with a heavy-tail distribution is slightly more robust to large measurement noise, but, beyond this, the choice of the error model did not have a significant impact on the estimation results provided by the authors.

SBbadger: biochemical reaction networks with definable degree distributions

MOTIVATION

An essential step in developing computational tools for the inference, optimization and simulation of biochemical reaction networks is gauging tool performance against earlier efforts using an appropriate set of benchmarks. General strategies for the assembly of benchmark models include collection from the literature, creation via subnetwork extraction and de novo generation. However, with respect to biochemical reaction networks, these approaches and their associated tools are either poorly suited to generate models that reflect the wide range of properties found in natural biochemical networks or to do so in numbers that enable rigorous statistical analysis.

RESULTS

In this work, we present SBbadger, a python-based software tool for the generation of synthetic biochemical reaction or metabolic networks with user-defined degree distributions, multiple available kinetic formalisms and a host of other definable properties. SBbadger thus enables the creation of benchmark model sets that reflect properties of biological systems and generate the kinetics and model structures typically targeted by computational analysis and inference software. Here, we detail the computational and algorithmic workflow of SBbadger, demonstrate its performance under various settings, provide sample outputs and compare it to currently available biochemical reaction network generation software.

AVAILABILITY AND IMPLEMENTATION

SBbadger is implemented in Python and is freely available at https://github.com/sys-bio/SBbadger and via PyPI at https://pypi.org/project/SBbadger/. Documentation can be found at https://SBbadger.readthedocs.io.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Altered expression response upon repeated gene repression in single yeast cells

Cells must continuously adjust to changing environments and, thus, have evolved mechanisms allowing them to respond to repeated stimuli. While faster gene induction upon a repeated stimulus is known as reinduction memory, responses to repeated repression have been less studied so far. Here, we studied gene repression across repeated carbon source shifts in over 1,500 single Saccharomyces cerevisiae cells. By monitoring the expression of a carbon source-responsive gene, galactokinase 1 (Gal1), and fitting a mathematical model to the single-cell data, we observed a faster response upon repeated repressions at the population level. Exploiting our single-cell data and quantitative modeling approach, we discovered that the faster response is mediated by a shortened repression response delay, the estimated time between carbon source shift and Gal1 protein production termination. Interestingly, we can exclude two alternative hypotheses, i) stronger dilution because of e.g., increased proliferation, and ii) a larger fraction of repressing cells upon repeated repressions. Collectively, our study provides a quantitative description of repression kinetics in single cells and allows us to pinpoint potential mechanisms underlying a faster response upon repeated repression. The computational results of our study can serve as the starting point for experimental follow-up studies.

Cells have to continuously adjust to their environment and cope with changing temperature, stress conditions, or metabolic resources. Yeast cells exposed to repeated carbon source shifts have shown to be “primed” by their first exposure, exhibiting enhanced gene expression of specific genes later on. However, how cells respond to a repeated repressive stimulus, e.g., withdrawal of metabolic resources, has been so far much less studied.

For this, we investigated the expression kinetics of a carbon source-responsive gene across repeated repressions. We measured single-cell expression and used mathematical modeling to evaluate potential causes underlying an observed faster repression response upon a repeated stimulus. Specifically, we investigated whether i) an increased dilution due to e.g., proliferation, ii) an increased fraction of repressing cells, or iii) different kinetic parameters in the repeated repression cause the observed faster response in the second repression. Leveraging quantitative mathematical model comparison, we discovered that the faster response is mediated by a shortened estimated time between carbon source shift and protein production termination at the single-cell level.

Our study provides a quantitative description of repression kinetics in single cells and allows us to pinpoint potential mechanisms underlying a faster response upon repeated repression.

Merits and Demerits of ODE Modeling of Physicochemical Systems for Numerical Simulations

In comparison with the first-principles calculations mostly using partial differential equations (PDEs), numerical simulations with modeling by ordinary differential equations (ODEs) are sometimes superior in that they are computationally more economical and that important factors are more easily traced. However, a demerit of ODE modeling is the need of model validation through comparison with experimental data or results of the first-principles calculations. In the present review, examples of ODE modeling are reviewed such as sonochemical reactions inside a cavitation bubble, oriented attachment of nanocrystals, dynamic response of flexoelectric polarization, ultrasound-assisted sintering, and dynamics of a gas parcel in a thermoacoustic engine.

Fides: Reliable trust-region optimization for parameter estimation of ordinary differential equation models

Ordinary differential equation (ODE) models are widely used to study biochemical reactions in cellular networks since they effectively describe the temporal evolution of these networks using mass action kinetics. The parameters of these models are rarely known a priori and must instead be estimated by calibration using experimental data. Optimization-based calibration of ODE models on is often challenging, even for low-dimensional problems. Multiple hypotheses have been advanced to explain why biochemical model calibration is challenging, including non-identifiability of model parameters, but there are few comprehensive studies that test these hypotheses, likely because tools for performing such studies are also lacking. Nonetheless, reliable model calibration is essential for uncertainty analysis, model comparison, and biological interpretation.

We implemented an established trust-region method as a modular Python framework (fides) to enable systematic comparison of different approaches to ODE model calibration involving a variety of Hessian approximation schemes. We evaluated fides on a recently developed corpus of biologically realistic benchmark problems for which real experimental data are available. Unexpectedly, we observed high variability in optimizer performance among different implementations of the same mathematical instructions (algorithms). Analysis of possible sources of poor optimizer performance identified limitations in the widely used Gauss-Newton, BFGS and SR1 Hessian approximation schemes. We addressed these drawbacks with a novel hybrid Hessian approximation scheme that enhances optimizer performance and outperforms existing hybrid approaches. When applied to the corpus of test models, we found that fides was on average more reliable and efficient than existing methods using a variety of criteria. We expect fides to be broadly useful for ODE constrained optimization problems in biochemical models and to be a foundation for future methods development.

Integrating knowledge and omics to decipher mechanisms via large-scale models of signaling networks

Signal transduction governs cellular behavior, and its dysregulation often leads to human disease. To understand this process, we can use network models based on prior knowledge, where nodes represent biomolecules, usually proteins, and edges indicate interactions between them. Several computational methods combine untargeted omics data with prior knowledge to estimate the state of signaling networks in specific biological scenarios. Here, we review, compare, and classify recent network approaches according to their characteristics in terms of input omics data, prior knowledge and underlying methodologies. We highlight existing challenges in the field, such as the general lack of ground truth and the limitations of prior knowledge. We also point out new omics developments that may have a profound impact, such as single‐cell proteomics or large‐scale profiling of protein conformational changes. We provide both an introduction for interested users seeking strategies to study cell signaling on a large scale and an update for seasoned modelers.

Deciphering signal transduction networks in the liver by mechanistic mathematical modelling

In health and disease, liver cells are continuously exposed to cytokines and growth factors. While individual signal transduction pathways induced by these factors were studied in great detail, the cellular responses induced by repeated or combined stimulations are complex and less understood. Growth factor receptors on the cell surface of hepatocytes were shown to be regulated by receptor interactions, receptor trafficking and feedback regulation. Here, we exemplify how mechanistic mathematical modelling based on quantitative data can be employed to disentangle these interactions at the molecular level. Crucial is the analysis at a mechanistic level based on quantitative longitudinal data within a mathematical framework. In such multi-layered information, step-wise mathematical modelling using submodules is of advantage, which is fostered by sharing of standardized experimental data and mathematical models. Integration of signal transduction with metabolic regulation in the liver and mechanistic links to translational approaches promise to provide predictive tools for biology and personalized medicine.

A protocol for dynamic model calibration

Ordinary differential equation models are nowadays widely used for the mechanistic description of biological processes and their temporal evolution. These models typically have many unknown and nonmeasurable parameters, which have to be determined by fitting the model to experimental data. In order to perform this task, known as parameter estimation or model calibration, the modeller faces challenges such as poor parameter identifiability, lack of sufficiently informative experimental data and the existence of local minima in the objective function landscape. These issues tend to worsen with larger model sizes, increasing the computational complexity and the number of unknown parameters. An incorrectly calibrated model is problematic because it may result in inaccurate predictions and misleading conclusions. For nonexpert users, there are a large number of potential pitfalls. Here, we provide a protocol that guides the user through all the steps involved in the calibration of dynamic models. We illustrate the methodology with two models and provide all the code required to reproduce the results and perform the same analysis on new models. Our protocol provides practitioners and researchers in biological modelling with a one-stop guide that is at the same time compact and sufficiently comprehensive to cover all aspects of the problem.

Mini-batch optimization enables training of ODE models on large-scale datasets

Quantitative dynamic models are widely used to study cellular signal processing. A critical step in modelling is the estimation of unknown model parameters from experimental data. As model sizes and datasets are steadily growing, established parameter optimization approaches for mechanistic models become computationally extremely challenging. Mini-batch optimization methods, as employed in deep learning, have better scaling properties. In this work, we adapt, apply, and benchmark mini-batch optimization for ordinary differential equation (ODE) models, thereby establishing a direct link between dynamic modelling and machine learning. On our main application example, a large-scale model of cancer signaling, we benchmark mini-batch optimization against established methods, achieving better optimization results and reducing computation by more than an order of magnitude. We expect that our work will serve as a first step towards mini-batch optimization tailored to ODE models and enable modelling of even larger and more complex systems than what is currently possible.

Systems biology–the transformative approach to integrate sciences across disciplines

Life science is the study of living organisms, including bacteria, plants, and animals. Given the importance of biology, chemistry, and bioinformatics, we anticipate that this chapter may contribute to a better understanding of the interdisciplinary connections in life science. Research in applied biological sciences has changed the paradigm of basic and applied research. Biology is the study of life and living organisms, whereas science is a dynamic subject that as a result of constant research, new fields are constantly emerging. Some fields come and go, whereas others develop into new, well-recognized entities. Chemistry is the study of composition of matter and its properties, how the substances merge or separate and also how substances interact with energy. Advances in biology and chemistry provide another means to understand the biological system using many interdisciplinary approaches. Bioinformatics is a multidisciplinary or rather transdisciplinary field that encourages the use of computer tools and methodologies for qualitative and quantitative analysis. There are many instances where two fields, biology and chemistry have intersection. In this chapter, we explain how current knowledge in biology, chemistry, and bioinformatics, as well as its various interdisciplinary domains are merged into life sciences and its applications in biological research.

A Modular Workflow for Model Building, Analysis, and Parameter Estimation in Systems Biology and Neuroscience

Neuroscience incorporates knowledge from a range of scales, from single molecules to brain wide neural networks. Modeling is a valuable tool in understanding processes at a single scale or the interactions between two adjacent scales and researchers use a variety of different software tools in the model building and analysis process. Here we focus on the scale of biochemical pathways, which is one of the main objects of study in systems biology. While systems biology is among the more standardized fields, conversion between different model formats and interoperability between various tools is still somewhat problematic. To offer our take on tackling these shortcomings and by keeping in mind the FAIR (findability, accessibility, interoperability, reusability) data principles, we have developed a workflow for building and analyzing biochemical pathway models, using pre-existing tools that could be utilized for the storage and refinement of models in all phases of development. We have chosen the SBtab format which allows the storage of biochemical models and associated data in a single file and provides a human readable set of syntax rules. Next, we implemented custom-made MATLAB® scripts to perform parameter estimation and global sensitivity analysis used in model refinement. Additionally, we have developed a web-based application for biochemical models that allows simulations with either a network free solver or stochastic solvers and incorporating geometry. Finally, we illustrate convertibility and use of a biochemical model in a biophysically detailed single neuron model by running multiscale simulations in NEURON. Using this workflow, we can simulate the same model in three different simulators, with a smooth conversion between the different model formats, enhancing the characterization of different aspects of the model.

Efficient gradient-based parameter estimation for dynamic models using qualitative data

MOTIVATION

Unknown parameters of dynamical models are commonly estimated from experimental data. However, while various efficient optimization and uncertainty analysis methods have been proposed for quantitative data, methods for qualitative data are rare and suffer from bad scaling and convergence.

RESULTS

Here, we propose an efficient and reliable framework for estimating the parameters of ordinary differential equation models from qualitative data. In this framework, we derive a semi-analytical algorithm for gradient calculation of the optimal scaling method developed for qualitative data. This enables the use of efficient gradient-based optimization algorithms. We demonstrate that the use of gradient information improves performance of optimization and uncertainty quantification on several application examples. On average, we achieve a speedup of more than one order of magnitude compared to gradient-free optimization. In addition, in some examples, the gradient-based approach yields substantially improved objective function values and quality of the fits. Accordingly, the proposed framework substantially improves the parameterization of models from qualitative data.

AVAILABILITY AND IMPLEMENTATION

The proposed approach is implemented in the open-source Python Parameter EStimation TOolbox (pyPESTO). pyPESTO is available at https://github.com/ICB-DCM/pyPESTO. All application examples and code to reproduce this study are available at https://doi.org/10.5281/zenodo.4507613.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Efficient gradient-based parameter estimation for dynamic models using qualitative data

Motivation

Unknown parameters of dynamical models are commonly estimated from experimental data. However, while various efficient optimization and uncertainty analysis methods have been proposed for quantitative data, methods for qualitative data are rare and suffer from bad scaling and convergence.

Results

Here, we propose an efficient and reliable framework for estimating the parameters of ordinary differential equation models from qualitative data. In this framework, we derive a semi-analytical algorithm for gradient calculation of the optimal scaling method developed for qualitative data. This enables the use of efficient gradient-based optimization algorithms. We demonstrate that the use of gradient information improves performance of optimization and uncertainty quantification on several application examples. On average, we achieve a speedup of more than one order of magnitude compared to gradient-free optimization. In addition, in some examples, the gradient-based approach yields substantially improved objective function values and quality of the fits. Accordingly, the proposed framework substantially improves the parameterization of models from qualitative data.

Availability and implementation

The proposed approach is implemented in the open-source Python Parameter EStimation TOolbox (pyPESTO). pyPESTO is available at https://github.com/ICB-DCM/pyPESTO. All application examples and code to reproduce this study are available at https://doi.org/10.5281/zenodo.4507613.

Supplementary information

Supplementary data are available at Bioinformatics online.

Benchmarking of numerical integration methods for ODE models of biological systems

Ordinary differential equation (ODE) models are a key tool to understand complex mechanisms in systems biology. These models are studied using various approaches, including stability and bifurcation analysis, but most frequently by numerical simulations. The number of required simulations is often large, e.g., when unknown parameters need to be inferred. This renders efficient and reliable numerical integration methods essential. However, these methods depend on various hyperparameters, which strongly impact the ODE solution. Despite this, and although hundreds of published ODE models are freely available in public databases, a thorough study that quantifies the impact of hyperparameters on the ODE solver in terms of accuracy and computation time is still missing. In this manuscript, we investigate which choices of algorithms and hyperparameters are generally favorable when dealing with ODE models arising from biological processes. To ensure a representative evaluation, we considered 142 published models. Our study provides evidence that most ODEs in computational biology are stiff, and we give guidelines for the choice of algorithms and hyperparameters. We anticipate that our results will help researchers in systems biology to choose appropriate numerical methods when dealing with ODE models.

PEtab-Interoperable specification of parameter estimation problems in systems biology

Reproducibility and reusability of the results of data-based modeling studies are essential. Yet, there has been-so far-no broadly supported format for the specification of parameter estimation problems in systems biology. Here, we introduce PEtab, a format which facilitates the specification of parameter estimation problems using Systems Biology Markup Language (SBML) models and a set of tab-separated value files describing the observation model and experimental data as well as parameters to be estimated. We already implemented PEtab support into eight well-established model simulation and parameter estimation toolboxes with hundreds of users in total. We provide a Python library for validation and modification of a PEtab problem and currently 20 example parameter estimation problems based on recent studies.

H4K20 Methylation Is Differently Regulated by Dilution and Demethylation in Proliferating and Cell-Cycle-Arrested Xenopus Embryos

DNA replication during cell division leads to dilution of histone modifications and can thus affect chromatin-mediated gene regulation, raising the question of how the cell-cycle shapes the histone modification landscape, particularly during embryogenesis. We tackled this problem by manipulating the cell cycle during early Xenopus laevis embryogenesis and analyzing in vivo histone H4K20 methylation kinetics. The global distribution of un-, mono-, di-, and tri-methylated histone H4K20 was measured by mass spectrometry in normal and cell-cycle-arrested embryos over time. Using multi-start maximum likelihood optimization and quantitative model selection, we found that three specific biological methylation rate constants were required to explain the measured H4K20 methylation state kinetics. While demethylation is essential for regulating H4K20 methylation kinetics in non-cycling cells, demethylation is very likely dispensable in rapidly dividing cells of early embryos, suggesting that cell-cycle-mediated dilution of H4K20 methylation is an essential regulatory component for shaping its epigenetic landscape during early development. A record of this paper's transparent peer review process is included in the Supplemental Information.

Compromised Humoral Functional Evolution Tracks with SARS-CoV-2 Mortality

The urgent need for an effective SARS-CoV-2 vaccine has forced development to progress in the absence of well-defined correlates of immunity. While neutralization has been linked to protection against other pathogens, whether neutralization alone will be sufficient to drive protection against SARS-CoV-2 in the broader population remains unclear. Therefore, to fully define protective humoral immunity, we dissected the early evolution of the humoral response in 193 hospitalized individuals ranging from moderate to severe. Although robust IgM and IgA responses evolved in both survivors and non-survivors with severe disease, non-survivors showed attenuated IgG responses, accompanied by compromised Fcɣ receptor binding and Fc effector activity, pointing to deficient humoral development rather than disease-enhancing humoral immunity. In contrast, individuals with moderate disease exhibited delayed responses that ultimately matured. These data highlight distinct humoral trajectories associated with resolution of SARS-CoV-2 infection and the need for early functional humoral immunity.

Domain Model Explains Propagation Dynamics and Stability of Histone H3K27 and H3K36 Methylation Landscapes

Chromatin states must be maintained during cell proliferation to uphold cellular identity and genome integrity. Inheritance of histone modifications is central in this process. However, the histone modification landscape is challenged by incorporation of new unmodified histones during each cell cycle, and the principles governing heritability remain unclear. We take a quantitative computational modeling approach to describe propagation of histone H3K27 and H3K36 methylation states. We measure combinatorial H3K27 and H3K36 methylation patterns by quantitative mass spectrometry on subsequent generations of histones. Using model comparison, we reject active global demethylation and invoke the existence of domains defined by distinct methylation endpoints. We find that H3K27me3 on pre-existing histones stimulates the rate of de novo H3K27me3 establishment, supporting a read-write mechanism in timely chromatin restoration. Finally, we provide a detailed quantitative picture of the mutual antagonism between H3K27 and H3K36 methylation and propose that it stabilizes epigenetic states across cell division.

Parameterization of mechanistic models from qualitative data using an efficient optimal scaling approach

Quantitative dynamical models facilitate the understanding of biological processes and the prediction of their dynamics. These models usually comprise unknown parameters, which have to be inferred from experimental data. For quantitative experimental data, there are several methods and software tools available. However, for qualitative data the available approaches are limited and computationally demanding. Here, we consider the optimal scaling method which has been developed in statistics for categorical data and has been applied to dynamical systems. This approach turns qualitative variables into quantitative ones, accounting for constraints on their relation. We derive a reduced formulation for the optimization problem defining the optimal scaling. The reduced formulation possesses the same optimal points as the established formulation but requires less degrees of freedom. Parameter estimation for dynamical models of cellular pathways revealed that the reduced formulation improves the robustness and convergence of optimizers. This resulted in substantially reduced computation times. We implemented the proposed approach in the open-source Python Parameter EStimation TOolbox (pyPESTO) to facilitate reuse and extension. The proposed approach enables efficient parameterization of quantitative dynamical models using qualitative data.

Calibration of ionic and cellular cardiac electrophysiology models

Cardiac electrophysiology models are among the most mature and well-studied mathematical models of biological systems. This maturity is bringing new challenges as models are being used increasingly to make quantitative rather than qualitative predictions. As such, calibrating the parameters within ion current and action potential (AP) models to experimental data sets is a crucial step in constructing a predictive model. This review highlights some of the fundamental concepts in cardiac model calibration and is intended to be readily understood by computational and mathematical modelers working in other fields of biology. We discuss the classic and latest approaches to calibration in the electrophysiology field, at both the ion channel and cellular AP scales. We end with a discussion of the many challenges that work to date has raised and the need for reproducible descriptions of the calibration process to enable models to be recalibrated to new data sets and built upon for new studies. This article is categorized under: Analytical and Computational Methods > Computational Methods Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Cellular Models.

Model-based analysis of response and resistance factors of cetuximab treatment in gastric cancer cell lines

Targeted cancer therapies are powerful alternatives to chemotherapies or can be used complementary to these. Yet, the response to targeted treatments depends on a variety of factors, including mutations and expression levels, and therefore their outcome is difficult to predict. Here, we develop a mechanistic model of gastric cancer to study response and resistance factors for cetuximab treatment. The model captures the EGFR, ERK and AKT signaling pathways in two gastric cancer cell lines with different mutation patterns. We train the model using a comprehensive selection of time and dose response measurements, and provide an assessment of parameter and prediction uncertainties. We demonstrate that the proposed model facilitates the identification of causal differences between the cell lines. Furthermore, our study shows that the model provides predictions for the responses to different perturbations, such as knockdown and knockout experiments. Among other results, the model predicted the effect of MET mutations on cetuximab sensitivity. These predictive capabilities render the model a basis for the assessment of gastric cancer signaling and possibly for the development and discovery of predictive biomarkers.

Unraveling the causal differences between drug responders and non-responders is an important challenge. The information can help to understand molecular mechanisms and to guide the selection and design of targeted therapies. Here, we approach this problem for cetuximab treatment for gastric cancer using mechanistic mathematical modeling. The proposed model describes responder and non-responder gastric cancer cell lines and can predict the response in several validation experiments. Our analysis provides a differentiated view on mutations and explains, for instance, the relevance of MET mutations and the insignificance of PIK3CA mutation in the considered cell lines. The model might potentially provide the basis for understanding the recent failure of several clinical studies.

Guidelines for benchmarking of optimization-based approaches for fitting mathematical models

Insufficient performance of optimization-based approaches for the fitting of mathematical models is still a major bottleneck in systems biology. In this article, the reasons and methodological challenges are summarized as well as their impact in benchmark studies. Important aspects for achieving an increased level of evidence for benchmark results are discussed. Based on general guidelines for benchmarking in computational biology, a collection of tailored guidelines is presented for performing informative and unbiased benchmarking of optimization-based fitting approaches. Comprehensive benchmark studies based on these recommendations are urgently required for the establishment of a robust and reliable methodology for the systems biology community.

Parameter Estimation and Uncertainty Quantification for Systems Biology Models

Mathematical models can provide quantitative insights into immunoreceptor signaling, and other biological processes, but require parameterization and uncertainty quantification before reliable predictions become possible. We review currently available methods and software tools to address these problems. We consider gradient-based and gradient-free methods for point estimation of parameter values, and methods of profile likelihood, bootstrapping, and Bayesian inference for uncertainty quantification. We consider recent and potential future applications of these methods to systems-level modeling of immune-related phenomena.

PyBioNetFit and the Biological Property Specification Language

In systems biology modeling, important steps include model parameterization, uncertainty quantification, and evaluation of agreement with experimental observations. To help modelers perform these steps, we developed the software PyBioNetFit, which in addition supports checking models against known system properties and solving design problems. PyBioNetFit introduces Biological Property Specification Language (BPSL) for the formal declaration of system properties. BPSL allows qualitative data to be used alone or in combination with quantitative data. PyBioNetFit performs parameterization with parallelized metaheuristic optimization algorithms that work directly with existing model definition standards: BioNetGen Language (BNGL) and Systems Biology Markup Language (SBML). We demonstrate PyBioNetFit's capabilities by solving various example problems, including the challenging problem of parameterizing a 153-parameter model of cell cycle control in yeast based on both quantitative and qualitative data. We demonstrate the model checking and design applications of PyBioNetFit and BPSL by analyzing a model of targeted drug interventions in autophagy signaling.

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.

Efficient parameterization of large-scale dynamic models based on relative measurements

MOTIVATION

Mechanistic models of biochemical reaction networks facilitate the quantitative understanding of biological processes and the integration of heterogeneous datasets. However, some biological processes require the consideration of comprehensive reaction networks and therefore large-scale models. Parameter estimation for such models poses great challenges, in particular when the data are on a relative scale.

RESULTS

Here, we propose a novel hierarchical approach combining (i) the efficient analytic evaluation of optimal scaling, offset and error model parameters with (ii) the scalable evaluation of objective function gradients using adjoint sensitivity analysis. We evaluate the properties of the methods by parameterizing a pan-cancer ordinary differential equation model (>1000 state variables, >4000 parameters) using relative protein, phosphoprotein and viability measurements. The hierarchical formulation improves optimizer performance considerably. Furthermore, we show that this approach allows estimating error model parameters with negligible computational overhead when no experimental estimates are available, providing an unbiased way to weight heterogeneous data. Overall, our hierarchical formulation is applicable to a wide range of models, and allows for the efficient parameterization of large-scale models based on heterogeneous relative measurements.

AVAILABILITY AND IMPLEMENTATION

Supplementary code and data are available online at http://doi.org/10.5281/zenodo.3254429 and http://doi.org/10.5281/zenodo.3254441.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Extensions of ℓ1 regularization increase detection specificity for cell-type specific parameters in dynamic models

BACKGROUND

Ordinary differential equation systems are frequently utilized to model biological systems and to infer knowledge about underlying properties. For instance, the development of drugs requires the knowledge to which extent malign cells differ from healthy ones to provide a specific treatment with least side effects. As these cell-type specific properties may stem from any part of biochemical cell processes, systematic quantitative approaches are necessary to identify the relevant potential drug targets. An ℓ₁ regularization for the maximum likelihood parameter estimation proved to be successful, but falsely predicted cell-type dependent behaviour had to be corrected manually by using a Profile Likelihood approach.

RESULTS

The choice of extended ℓ₁ penalty functions significantly decreased the number of falsely detected cell-type specific parameters. Thus, the total accuracy of the prediction could be increased. This was tested on a realistic dynamical benchmark model used for the DREAM6 challenge. Among Elastic Net, Adaptive Lasso and a non-convex ℓ_q penalty, the latter one showed the best predictions whilst also requiring least computation time. All extended methods include a hyper-parameter in the regularization function. For an Erythropoietin (EPO) induced signalling pathway, the extended methods ℓ_q and Adaptive Lasso revealed an unpublished alternative parsimonious model when varying the respective hyper-parameters.

CONCLUSIONS

Using ℓ_q or Adaptive Lasso with an a-priori choice for the hyper-parameter can lead to a more specific and accurate result than ℓ₁. Scanning different hyper-parameters can yield additional pieces of information about the system.