Comprehensive estimation of input signals and dynamics in biochemical reaction networks

Data-driven prediction of COVID-19 cases in Germany for decision making

Background

The COVID-19 pandemic has led to a high interest in mathematical models describing and predicting the diverse aspects and implications of the virus outbreak. Model results represent an important part of the information base for the decision process on different administrative levels. The Robert-Koch-Institute (RKI) initiated a project whose main goal is to predict COVID-19-specific occupation of beds in intensive care units: Steuerungs-Prognose von Intensivmedizinischen COVID-19 Kapazitäten (SPoCK). The incidence of COVID-19 cases is a crucial predictor for this occupation.

Methods

We developed a model based on ordinary differential equations for the COVID-19 spread with a time-dependent infection rate described by a spline. Furthermore, the model explicitly accounts for weekday-specific reporting and adjusts for reporting delay. The model is calibrated in a purely data-driven manner by a maximum likelihood approach. Uncertainties are evaluated using the profile likelihood method. The uncertainty about the appropriate modeling assumptions can be accounted for by including and merging results of different modelling approaches. The analysis uses data from Germany describing the COVID-19 spread from early 2020 until March 31st, 2021.

Results

The model is calibrated based on incident cases on a daily basis and provides daily predictions of incident COVID-19 cases for the upcoming three weeks including uncertainty estimates for Germany and its subregions. Derived quantities such as cumulative counts and 7-day incidences with corresponding uncertainties can be computed. The estimation of the time-dependent infection rate leads to an estimated reproduction factor that is oscillating around one. Data-driven estimation of the dark figure purely from incident cases is not feasible.

Conclusions

We successfully implemented a procedure to forecast near future COVID-19 incidences for diverse subregions in Germany which are made available to various decision makers via an interactive web application. Results of the incidence modeling are also used as a predictor for forecasting the need of intensive care units.

Supplementary Information

The online version contains supplementary material available at (10.1186/s12874-022-01579-9).

SEEDS: data driven inference of structural model errors and unknown inputs for dynamic systems biology

SUMMARY

Dynamic models formulated as ordinary differential equations can provide information about the mechanistic and causal interactions in biological systems to guide targeted interventions and to design further experiments. Inaccurate knowledge about the structure, functional form and parameters of interactions is a major obstacle to mechanistic modeling. A further challenge is the open nature of biological systems which receive unknown inputs from their environment. The R-package SEEDS implements two recently developed algorithms to infer structural model errors and unknown inputs from output measurements. This information can facilitate efficient model recalibration as well as experimental design in the case of misfits between the initial model and data.

AVAILABILITY AND IMPLEMENTATION

For the R-package seeds, see the CRAN server https://cran.r-project.org/package=seeds.

Benchmark problems for dynamic modeling of intracellular processes

Motivation

Dynamic models are used in systems biology to study and understand cellular processes like gene regulation or signal transduction. Frequently, ordinary differential equation (ODE) models are used to model the time and dose dependency of the abundances of molecular compounds as well as interactions and translocations. A multitude of computational approaches, e.g. for parameter estimation or uncertainty analysis have been developed within recent years. However, many of these approaches lack proper testing in application settings because a comprehensive set of benchmark problems is yet missing.

Results

We present a collection of 20 benchmark problems in order to evaluate new and existing methodologies, where an ODE model with corresponding experimental data is referred to as problem. In addition to the equations of the dynamical system, the benchmark collection provides observation functions as well as assumptions about measurement noise distributions and parameters. The presented benchmark models comprise problems of different size, complexity and numerical demands. Important characteristics of the models and methodological requirements are summarized, estimated parameters are provided, and some example studies were performed for illustrating the capabilities of the presented benchmark collection.

Availability and implementation

The models are provided in several standardized formats, including an easy-to-use human readable form and machine-readable SBML files. The data is provided as Excel sheets. All files are available at https://github.com/Benchmarking-Initiative/Benchmark-Models, including step-by-step explanations and MATLAB code to process and simulate the models.

Supplementary information

Supplementary data are available at Bioinformatics online.

Hierarchical optimization for the efficient parametrization of ODE models

Motivation

Mathematical models are nowadays important tools for analyzing dynamics of cellular processes. The unknown model parameters are usually estimated from experimental data. These data often only provide information about the relative changes between conditions, hence, the observables contain scaling parameters. The unknown scaling parameters and corresponding noise parameters have to be inferred along with the dynamic parameters. The nuisance parameters often increase the dimensionality of the estimation problem substantially and cause convergence problems.

Results

In this manuscript, we propose a hierarchical optimization approach for estimating the parameters for ordinary differential equation (ODE) models from relative data. Our approach restructures the optimization problem into an inner and outer subproblem. These subproblems possess lower dimensions than the original optimization problem, and the inner problem can be solved analytically. We evaluated accuracy, robustness and computational efficiency of the hierarchical approach by studying three signaling pathways. The proposed approach achieved better convergence than the standard approach and required a lower computation time. As the hierarchical optimization approach is widely applicable, it provides a powerful alternative to established approaches.

Availability and implementation

The code is included in the MATLAB toolbox PESTO which is available at http://github.com/ICB-DCM/PESTO.

Supplementary information

Supplementary data are available at Bioinformatics online.

Full observability and estimation of unknown inputs, states and parameters of nonlinear biological models

In this paper, we address the system identification problem in the context of biological modelling. We present and demonstrate a methodology for (i) assessing the possibility of inferring the unknown quantities in a dynamic model and (ii) effectively estimating them from output data. We introduce the term Full Input-State-Parameter Observability (FISPO) analysis to refer to the simultaneous assessment of state, input and parameter observability (note that parameter observability is also known as identifiability). This type of analysis has often remained elusive in the presence of unmeasured inputs. The method proposed in this paper can be applied to a general class of nonlinear ordinary differential equations models. We apply this approach to three models from the recent literature. First, we determine whether it is theoretically possible to infer the states, parameters and inputs, taking only the model equations into account. When this analysis detects deficiencies, we reformulate the model to make it fully observable. Then we move to numerical scenarios and apply an optimization-based technique to estimate the states, parameters and inputs. The results demonstrate the feasibility of an integrated strategy for (i) analysing the theoretical possibility of determining the states, parameters and inputs to a system and (ii) solving the practical problem of actually estimating their values.

Estimation of time-varying growth, uptake and excretion rates from dynamic metabolomics data

Motivation

Technological advances in metabolomics have made it possible to monitor the concentration of extracellular metabolites over time. From these data, it is possible to compute the rates of uptake and excretion of the metabolites by a growing cell population, providing precious information on the functioning of intracellular metabolism. The computation of the rate of these exchange reactions, however, is difficult to achieve in practice for a number of reasons, notably noisy measurements, correlations between the concentration profiles of the different extracellular metabolites, and discontinuties in the profiles due to sudden changes in metabolic regime.

Results

We present a method for precisely estimating time-varying uptake and excretion rates from time-series measurements of extracellular metabolite concentrations, specifically addressing all of the above issues. The estimation problem is formulated in a regularized Bayesian framework and solved by a combination of extended Kalman filtering and smoothing. The method is shown to improve upon methods based on spline smoothing of the data. Moreover, when applied to two actual datasets, the method recovers known features of overflow metabolism in Escherichia coli and Lactococcus lactis, and provides evidence for acetate uptake by L. lactis after glucose exhaustion. The results raise interesting perspectives for further work on rate estimation from measurements of intracellular metabolites.

Availability and implementation

The Matlab code for the estimation method is available for download at https://team.inria.fr/ibis/rate-estimation-software/, together with the datasets.

Supplementary information

Supplementary data are available at Bioinformatics online.

Robust parameter estimation for dynamical systems from outlier-corrupted data

Motivation

Dynamics of cellular processes are often studied using mechanistic mathematical models. These models possess unknown parameters which are generally estimated from experimental data assuming normally distributed measurement noise. Outlier corruption of datasets often cannot be avoided. These outliers may distort the parameter estimates, resulting in incorrect model predictions. Robust parameter estimation methods are required which provide reliable parameter estimates in the presence of outliers.

Results

In this manuscript, we propose and evaluate methods for estimating the parameters of ordinary differential equation models from outlier-corrupted data. As alternatives to the normal distribution as noise distribution, we consider the Laplace, the Huber, the Cauchy and the Student's t distribution. We assess accuracy, robustness and computational efficiency of estimators using these different distribution assumptions. To this end, we consider artificial data of a conversion process, as well as published experimental data for Epo-induced JAK/STAT signaling. We study how well the methods can compensate and discover artificially introduced outliers. Our evaluation reveals that using alternative distributions improves the robustness of parameter estimates.

Availability and Implementation

The MATLAB implementation of the likelihood functions using the distribution assumptions is available at Bioinformatics online.

Contact

jan.hasenauer@helmholtz-muenchen.de.

Supplementary information

Supplementary material are available at Bioinformatics online.

Data-driven mechanistic analysis method to reveal dynamically evolving regulatory networks

Motivation: Mechanistic models based on ordinary differential equations provide powerful and accurate means to describe the dynamics of molecular machinery which orchestrates gene regulation. When combined with appropriate statistical techniques, mechanistic models can be calibrated using experimental data and, in many cases, also the model structure can be inferred from time–course measurements. However, existing mechanistic models are limited in the sense that they rely on the assumption of static network structure and cannot be applied when transient phenomena affect, or rewire, the network structure. In the context of gene regulatory network inference, network rewiring results from the net impact of possible unobserved transient phenomena such as changes in signaling pathway activities or epigenome, which are generally difficult, but important, to account for.

Results: We introduce a novel method that can be used to infer dynamically evolving regulatory networks from time–course data. Our method is based on the notion that all mechanistic ordinary differential equation models can be coupled with a latent process that approximates the network structure rewiring process. We illustrate the performance of the method using simulated data and, further, we apply the method to study the regulatory interactions during T helper 17 (Th17) cell differentiation using time–course RNA sequencing data. The computational experiments with the real data show that our method is capable of capturing the experimentally verified rewiring effects of the core Th17 regulatory network. We predict Th17 lineage specific subnetworks that are activated sequentially and control the differentiation process in an overlapping manner.

Availability and Implementation: An implementation of the method is available at http://research.ics.aalto.fi/csb/software/lem/.

Contacts:jukka.intosalmi@aalto.fi or harri.lahdesmaki@aalto.fi

Learning (from) the errors of a systems biology model

Mathematical modelling is a labour intensive process involving several iterations of testing on real data and manual model modifications. In biology, the domain knowledge guiding model development is in many cases itself incomplete and uncertain. A major problem in this context is that biological systems are open. Missed or unknown external influences as well as erroneous interactions in the model could thus lead to severely misleading results. Here we introduce the dynamic elastic-net, a data driven mathematical method which automatically detects such model errors in ordinary differential equation (ODE) models. We demonstrate for real and simulated data, how the dynamic elastic-net approach can be used to automatically (i) reconstruct the error signal, (ii) identify the target variables of model error, and (iii) reconstruct the true system state even for incomplete or preliminary models. Our work provides a systematic computational method facilitating modelling of open biological systems under uncertain knowledge.

Studying Cellular Signal Transduction with OMIC Technologies

In the gulf between genotype and phenotype exists proteins and, in particular, protein signal transduction systems. These systems use a relatively limited parts list to respond to a much longer list of extracellular, environmental, and/or mechanical cues with rapidity and specificity. Most signaling networks function in a highly non-linear and often contextual manner. Furthermore, these processes occur dynamically across space and time. Because of these complexities, systems and "OMIC" approaches are essential for the study of signal transduction. One challenge in using OMIC-scale approaches to study signaling is that the "signal" can take different forms in different situations. Signals are encoded in diverse ways such as protein-protein interactions, enzyme activities, localizations, or post-translational modifications to proteins. Furthermore, in some cases, signals may be encoded only in the dynamics, duration, or rates of change of these features. Accordingly, systems-level analyses of signaling may need to integrate multiple experimental and/or computational approaches. As the field has progressed, the non-triviality of integrating experimental and computational analyses has become apparent. Successful use of OMIC methods to study signaling will require the "right" experiments and the "right" modeling approaches, and it is critical to consider both in the design phase of the project. In this review, we discuss common OMIC and modeling approaches for studying signaling, emphasizing the philosophical and practical considerations for effectively merging these two types of approaches to maximize the probability of obtaining reliable and novel insights into signaling biology.

Quantitative control of intracellular signaling activity through chimeric receptors incorporating multiple identical tyrosine motifs

Controlling activation levels and durations of native signaling molecules is important for efficiently controlling cellular fates. Previously we developed single-chain Fv (scFv)/cytokine receptor chimeras incorporating tyrosine motifs in the intracellular domain, which artificially control the activation of specific intracellular signaling proteins. In this study, to quantitatively control the activation levels of signaling molecules with an extended dynamic range, we constructed scFv/receptor chimeras incorporating multiple identical motifs at the different positions in the intracellular domain. We used retroviral transduction to express chimeric receptors with multiple STAT3 binding motifs connected with or without flexible linkers in a murine IL-3-dependent pro-B cell line, Ba/F3. Our results showed that the chimeric receptors can control the activation levels of STAT3 depending on ligand concentration and the number of motifs. The existence of linkers between the motifs also affected the signal intensity. Furthermore, the STAT3 activation levels significantly depended on the number of motifs rather than the distance from the JAK-binding region to the tyrosine motif.

Profile likelihood in systems biology

Inferring knowledge about biological processes by a mathematical description is a major characteristic of Systems Biology. To understand and predict system's behavior the available experimental information is translated into a mathematical model. Since the availability of experimental data is often limited and measurements contain noise, it is essential to appropriately translate experimental uncertainty to model parameters as well as to model predictions. This is especially important in Systems Biology because typically large and complex models are applied and therefore the limited experimental knowledge might yield weakly specified model components. Likelihood profiles have been recently suggested and applied in the Systems Biology for assessing parameter and prediction uncertainty. In this article, the profile likelihood concept is reviewed and the potential of the approach is demonstrated for a model of the erythropoietin (EPO) receptor.

Maximum profile likelihood estimation of differential equation parameters through model based smoothing state estimates

Statistical inference for biochemical models often faces a variety of characteristic challenges. In this paper we examine state and parameter estimation for the JAK-STAT intracellular signalling mechanism, which exemplifies the implementation intricacies common in many biochemical inference problems. We introduce an extension to the Generalized Smoothing approach for estimating delay differential equation models, addressing selection of complexity parameters, choice of the basis system, and appropriate optimization strategies. Motivated by the JAK-STAT system, we further extend the generalized smoothing approach to consider a nonlinear observation process with additional unknown parameters, and highlight how the approach handles unobserved states and unevenly spaced observations. The methodology developed is generally applicable to problems of estimation for differential equation models with delays, unobserved states, nonlinear observation processes, and partially observed histories.