On structural and practical identifiability

Mechanism-based and data-driven modeling in cell-free synthetic biology

Cell-free systems have emerged as a versatile platform in synthetic biology, finding applications in various areas such as prototyping synthetic circuits, biosensor development, and biomanufacturing. To streamline the prototyping process, cell-free systems often incorporate a modeling step that predicts the outcomes of various experimental scenarios, providing a deeper insight into the underlying mechanisms and functions. There are two recognized approaches for modeling these systems: mechanism-based modeling, which models the underlying reaction mechanisms; and data-driven modeling, which makes predictions based on data without preconceived interactions between system components. In this highlight, we focus on the latest advancements in both modeling approaches for cell-free systems, exploring their potential for the design and optimization of synthetic genetic circuits.

Nonlinear compartmental modeling to monitor ovarian follicle population dynamics on the whole lifespan

In this work, we introduce a compartmental model of ovarian follicle development all along lifespan, based on ordinary differential equations. The model predicts the changes in the follicle numbers in different maturation stages with aging. Ovarian follicles may either move forward to the next compartment (unidirectional migration) or degenerate and disappear (death). The migration from the first follicle compartment corresponds to the activation of quiescent follicles, which is responsible for the progressive exhaustion of the follicle reserve (ovarian aging) until cessation of reproductive activity. The model consists of a data-driven layer embedded into a more comprehensive, knowledge-driven layer encompassing the earliest events in follicle development. The data-driven layer is designed according to the most densely sampled experimental dataset available on follicle numbers in the mouse. Its salient feature is the nonlinear formulation of the activation rate, whose formulation includes a feedback term from growing follicles. The knowledge-based, coating layer accounts for cutting-edge studies on the initiation of follicle development around birth. Its salient feature is the co-existence of two follicle subpopulations of different embryonic origins. We then setup a complete estimation strategy, including the study of structural identifiability, the elaboration of a relevant optimization criterion combining different sources of data (the initial dataset on follicle numbers, together with data in conditions of perturbed activation, and data discriminating the subpopulations) with appropriate error models, and a model selection step. We finally illustrate the model potential for experimental design (suggestion of targeted new data acquisition) and in silico experiments.

Inferring Stochastic Rates from Heterogeneous Snapshots of Particle Positions

Many imaging techniques for biological systems-like fixation of cells coupled with fluorescence microscopy-provide sharp spatial resolution in reporting locations of individuals at a single moment in time but also destroy the dynamics they intend to capture. These snapshot observations contain no information about individual trajectories, but still encode information about movement and demographic dynamics, especially when combined with a well-motivated biophysical model. The relationship between spatially evolving populations and single-moment representations of their collective locations is well-established with partial differential equations (PDEs) and their inverse problems. However, experimental data is commonly a set of locations whose number is insufficient to approximate a continuous-in-space PDE solution. Here, motivated by popular subcellular imaging data of gene expression, we embrace the stochastic nature of the data and investigate the mathematical foundations of parametrically inferring demographic rates from snapshots of particles undergoing birth, diffusion, and death in a nuclear or cellular domain. Toward inference, we rigorously derive a connection between individual particle paths and their presentation as a Poisson spatial process. Using this framework, we investigate the properties of the resulting inverse problem and study factors that affect quality of inference. One pervasive feature of this experimental regime is the presence of cell-to-cell heterogeneity. Rather than being a hindrance, we show that cell-to-cell geometric heterogeneity can increase the quality of inference on dynamics for certain parameter regimes. Altogether, the results serve as a basis for more detailed investigations of subcellular spatial patterns of RNA molecules and other stochastically evolving populations that can only be observed for single instants in their time evolution.

Evolving Improved Sampling Protocols for Dose-Response Modelling Using Genetic Algorithms with a Profile-Likelihood Metric

Practical limitations of quality and quantity of data can limit the precision of parameter identification in mathematical models. Model-based experimental design approaches have been developed to minimise parameter uncertainty, but the majority of these approaches have relied on first-order approximations of model sensitivity at a local point in parameter space. Practical identifiability approaches such as profile-likelihood have shown potential for quantifying parameter uncertainty beyond linear approximations. This research presents a genetic algorithm approach to optimise sample timing across various parameterisations of a demonstrative PK-PD model with the goal of aiding experimental design. The optimisation relies on a chosen metric of parameter uncertainty that is based on the profile-likelihood method. Additionally, the approach considers cases where multiple parameter scenarios may require simultaneous optimisation. The genetic algorithm approach was able to locate near-optimal sampling protocols for a wide range of sample number (n = 3-20), and it reduced the parameter variance metric by 33-37% on average. The profile-likelihood metric also correlated well with an existing Monte Carlo-based metric (with a worst-case r > 0.89), while reducing computational cost by an order of magnitude. The combination of the new profile-likelihood metric and the genetic algorithm demonstrate the feasibility of considering the nonlinear nature of models in optimal experimental design at a reasonable computational cost. The outputs of such a process could allow for experimenters to either improve parameter certainty given a fixed number of samples, or reduce sample quantity while retaining the same level of parameter certainty.

Dynamic Model Selection and Optimal Batch Design for Polyhydroxyalkanoate (PHA) Production by Cupriavidus necator

Mathematical modelling of microbial polyhydroxyalkanoates (PHAs) production is essential to develop optimal bioprocess design. Though the use of mathematical models in PHA production has increased over the years, the selection of kinetics and model identification strategies from experimental data remains largely heuristic. In this study, PHA production from Cupriavidus necator utilizing sucrose and urea was modelled using a parametric discretization approach. Product formation kinetics and relevant parameters were established from urea-free experimental sets, followed by the selection of growth models from a batch containing both sucrose and urea. Logistic growth and Luedeking-Piret model for PHA production was selected based on regression coefficient (R2: 0.941), adjusted R2 (0.930) and AICc values (-42.764). Model fitness was further assessed through cross-validation, confidence interval and sensitivity analysis of the parameters. Model-based optimal batch startup policy, incorporating multi-objective desirability, suggests an accumulation of 2.030 g l-1 of PHA at the end of 120 h. The modelling framework applied in this study can be used not only to avoid over-parameterization and identifiability issues but can also be adopted to design optimal batch startup policies.

Structural and practical identifiability of contrast transport models for DCE-MRI

Contrast transport models are widely used to quantify blood flow and transport in dynamic contrast-enhanced magnetic resonance imaging. These models analyze the time course of the contrast agent concentration, providing diagnostic and prognostic value for many biological systems. Thus, ensuring accuracy and repeatability of the model parameter estimation is a fundamental concern. In this work, we analyze the structural and practical identifiability of a class of nested compartment models pervasively used in analysis of MRI data. We combine artificial and real data to study the role of noise in model parameter estimation. We observe that although all the models are structurally identifiable, practical identifiability strongly depends on the data characteristics. We analyze the impact of increasing data noise on parameter identifiability and show how the latter can be recovered with increased data quality. To complete the analysis, we show that the results do not depend on specific tissue characteristics or the type of enhancement patterns of contrast agent signal.

A novel within-host model of HIV and nutrition

In this paper we develop a four compartment within-host model of nutrition and HIV. We show that the model has two equilibria: an infection-free equilibrium and infection equilibrium. The infection free equilibrium is locally asymptotically stable when the basic reproduction number $ \mathcal{R}_0 < 1 $, and unstable when $ \mathcal{R}_0 > 1 $. The infection equilibrium is locally asymptotically stable if $ \mathcal{R}_0 > 1 $ and an additional condition holds. We show that the within-host model of HIV and nutrition is structured to reveal its parameters from the observations of viral load, CD4 cell count and total protein data. We then estimate the model parameters for these 3 data sets. We have also studied the practical identifiability of the model parameters by performing Monte Carlo simulations, and found that the rate of clearance of the virus by immunoglobulins is practically unidentifiable, and that the rest of the model parameters are only weakly identifiable given the experimental data. Furthermore, we have studied how the data frequency impacts the practical identifiability of model parameters.

Leveraging continuous glucose monitoring for personalized modeling of insulin-regulated glucose metabolism

Continuous glucose monitoring (CGM) is a promising, minimally invasive alternative to plasma glucose measurements for calibrating physiology-based mathematical models of insulin-regulated glucose metabolism, reducing the reliance on in-clinic measurements. However, the use of CGM glucose, particularly in combination with insulin measurements, to develop personalized models of glucose regulation remains unexplored. Here, we simultaneously measured interstitial glucose concentrations using CGM as well as plasma glucose and insulin concentrations during an oral glucose tolerance test (OGTT) in individuals with overweight or obesity to calibrate personalized models of glucose-insulin dynamics. We compared the use of interstitial glucose with plasma glucose in model calibration, and evaluated the effects on model fit, identifiability, and model parameters' association with clinically relevant metabolic indicators. Models calibrated on both plasma and interstitial glucose resulted in good model fit, and the parameter estimates associated with metabolic indicators such as insulin sensitivity measures in both cases. Moreover, practical identifiability of model parameters was improved in models estimated on CGM glucose compared to plasma glucose. Together these results suggest that CGM glucose may be considered as a minimally invasive alternative to plasma glucose measurements in model calibration to quantify the dynamics of glucose regulation.

LIES of omission: complex observation processes in ecology

Advances in statistics mean that it is now possible to tackle increasingly sophisticated observation processes. The intricacies and ambitious scale of modern data collection techniques mean that this is now essential. Methodological research to make inference about the biological process while accounting for the observation process has expanded dramatically, but solutions are often presented in field-specific terms, limiting our ability to identify commonalities between methods. We suggest a typology of observation processes that could improve translation between fields and aid methodological synthesis. We propose the LIES framework (defining observation processes in terms of issues of Latency, Identifiability, Effort and Scale) and illustrate its use with both simple examples and more complex case studies.

How robust are estimates of key parameters in standard viral dynamic models?

Mathematical models of viral infection have been developed, fitted to data, and provide insight into disease pathogenesis for multiple agents that cause chronic infection, including HIV, hepatitis C, and B virus. However, for agents that cause acute infections or during the acute stage of agents that cause chronic infections, viral load data are often collected after symptoms develop, usually around or after the peak viral load. Consequently, we frequently lack data in the initial phase of viral growth, i.e., when pre-symptomatic transmission events occur. Missing data may make estimating the time of infection, the infectious period, and parameters in viral dynamic models, such as the cell infection rate, difficult. However, having extra information, such as the average time to peak viral load, may improve the robustness of the estimation. Here, we evaluated the robustness of estimates of key model parameters when viral load data prior to the viral load peak is missing, when we know the values of some parameters and/or the time from infection to peak viral load. Although estimates of the time of infection are sensitive to the quality and amount of available data, particularly pre-peak, other parameters important in understanding disease pathogenesis, such as the loss rate of infected cells, are less sensitive. Viral infectivity and the viral production rate are key parameters affecting the robustness of data fits. Fixing their values to literature values can help estimate the remaining model parameters when pre-peak data is missing or limited. We find a lack of data in the pre-peak growth phase underestimates the time to peak viral load by several days, leading to a shorter predicted growth phase. On the other hand, knowing the time of infection (e.g., from epidemiological data) and fixing it results in good estimates of dynamical parameters even in the absence of early data. While we provide ways to approximate model parameters in the absence of early viral load data, our results also suggest that these data, when available, are needed to estimate model parameters more precisely.

A review of mechanistic learning in mathematical oncology

Mechanistic learning refers to the synergistic combination of mechanistic mathematical modeling and data-driven machine or deep learning. This emerging field finds increasing applications in (mathematical) oncology. This review aims to capture the current state of the field and provides a perspective on how mechanistic learning may progress in the oncology domain. We highlight the synergistic potential of mechanistic learning and point out similarities and differences between purely data-driven and mechanistic approaches concerning model complexity, data requirements, outputs generated, and interpretability of the algorithms and their results. Four categories of mechanistic learning (sequential, parallel, extrinsic, intrinsic) of mechanistic learning are presented with specific examples. We discuss a range of techniques including physics-informed neural networks, surrogate model learning, and digital twins. Example applications address complex problems predominantly from the domain of oncology research such as longitudinal tumor response predictions or time-to-event modeling. As the field of mechanistic learning advances, we aim for this review and proposed categorization framework to foster additional collaboration between the data- and knowledge-driven modeling fields. Further collaboration will help address difficult issues in oncology such as limited data availability, requirements of model transparency, and complex input data which are embraced in a mechanistic learning framework.

Structural and practical identifiability analysis in bioengineering: a beginner's guide

Advancements in digital technology have brought modelling to the forefront in many disciplines from healthcare to architecture. Mathematical models, often represented using parametrised sets of ordinary differential equations, can be used to characterise different processes. To infer possible estimates for the unknown parameters, these models are usually calibrated using associated experimental data. Structural and practical identifiability analyses are a key component that should be assessed prior to parameter estimation. This is because identifiability analyses can provide insights as to whether or not a parameter can take on single, multiple, or even infinitely or countably many values which will ultimately have an impact on the reliability of the parameter estimates. Also, identifiability analyses can help to determine whether the data collected are sufficient or of good enough quality to truly estimate the parameters or if more data or even reparameterization of the model is necessary to proceed with the parameter estimation process. Thus, such analyses also provide an important role in terms of model design (structural identifiability analysis) and the collection of experimental data (practical identifiability analysis). Despite the popularity of using data to estimate the values of unknown parameters, structural and practical identifiability analyses of these models are often overlooked. Possible reasons for non-consideration of application of such analyses may be lack of awareness, accessibility, and usability issues, especially for more complicated models and methods of analysis. The aim of this study is to introduce and perform both structural and practical identifiability analyses in an accessible and informative manner via application to well established and commonly accepted bioengineering models. This will help to improve awareness of the importance of this stage of the modelling process and provide bioengineering researchers with an understanding of how to utilise the insights gained from such analyses in future model development.

Parameter estimation in behavioral epidemic models with endogenous societal risk-response

Behavioral epidemic models incorporating endogenous societal risk-response, where changes in risk perceptions prompt adjustments in contact rates, are crucial for predicting pandemic trajectories. Accurate parameter estimation in these models is vital for validation and precise projections. However, few studies have examined the problem of identifiability in models where disease and behavior parameters must be jointly estimated. To address this gap, we conduct simulation experiments to assess the effect on parameter estimation accuracy of a) delayed risk response, b) neglecting behavioral response in model structure, and c) integrating disease and public behavior data. Our findings reveal systematic biases in estimating behavior parameters even with comprehensive and accurate disease data and a well-structured simulation model when data are limited to the first wave. This is due to the significant delay between evolving risks and societal reactions, corresponding to the duration of a pandemic wave. Moreover, we demonstrate that conventional SEIR models, which disregard behavioral changes, may fit well in the early stages of a pandemic but exhibit significant errors after the initial peak. Furthermore, early on, relatively small data samples of public behavior, such as mobility, can significantly improve estimation accuracy. However, the marginal benefits decline as the pandemic progresses. These results highlight the challenges associated with the joint estimation of disease and behavior parameters in a behavioral epidemic model.

Modeling cancer progression: an integrated workflow extending data-driven kinetic models to bio-mechanical PDE models

Computational modeling of cancer can help unveil dynamics and interactions that are hard to replicate experimentally. Thanks to the advancement in cancer databases and data analysis technologies, these models have become more robust than ever. There are many mathematical models which investigate cancer through different approaches, from sub-cellular to tissue scale, and from treatment to diagnostic points of view. In this study, we lay out a step-by-step methodology for a data-driven mechanistic model of the tumor microenvironment. We discuss data acquisition strategies, data preparation, parameter estimation, and sensitivity analysis techniques. Furthermore, we propose a possible approach to extend mechanistic ordinary differential equation models to PDE models coupled with mechanical growth. The workflow discussed in this article can help understand the complex temporal and spatial interactions between cells and cytokines in the tumor microenvironment and their effect on tumor growth.

Practical Identifiability of Plant Growth Models: A Unifying Framework and Its Specification for Three Local Indices

Amid the rise of machine learning models, a substantial portion of plant growth models remains mechanistic, seeking to capture an in-depth understanding of the underlying phenomena governing the system's dynamics. The development of these models typically involves parameter estimation from experimental data. Ensuring that the estimated parameters align closely with their respective "true" values is crucial since they hold biological interpretation, leading to the challenge of uniqueness in the solutions. Structural identifiability analysis addresses this issue under the assumption of perfect observations of system dynamics, whereas practical identifiability considers limited measurements and the accompanying noise. In the literature, definitions for structural identifiability vary only slightly among authors, whereas the concept and quantification of practical identifiability lack consensus, with several indices coexisting. In this work, we provide a unified framework for studying identifiability, accommodating different definitions that need to be instantiated depending on each application case. In a more applicative second step, we focus on three widely used methods for quantifying practical identifiability: collinearity indices, profile likelihood, and average relative error. We show the limitations of their local versions, and we propose a new risk index built on the profile likelihood-based confidence intervals. We illustrate the usefulness of these concepts for plant growth modeling using a discrete-time individual plant growth model, LNAS, and a continuous-time plant population epidemics model. Through this work, we aim to underline the significance of identifiability analysis as a complement to any parameter estimation study and offer guidance to the modeler.

Parameter-by-parameter estimation method for adsorption isotherm in hydrophobic interaction chromatography

Hydrophobic interaction chromatography (HIC) is used as a critical polishing step in the downstream processing of biopharmaceuticals. Normally the process development of HIC is a cumbersome and time-consuming task, and the mechanical models can provide a powerful tool to characterize the process, assist process design and accelerate process development. However, the current estimation of model parameters relies on the inverse method, which lacks an efficient and logical parameter estimation strategy. In this study, a parameter-by-parameter (PbP) method based on the theoretical derivation and simplifying assumptions was proposed to estimate the Mollerup isotherm parameters for HIC. The method involves three key steps: (1) linear regression (LR) to estimate the salt-protein interaction parameter and the equilibrium constant; (2) linear approximation (LA) to estimate the stoichiometric parameter and the maximum binding capacity; and (3) inverse method to estimate the protein-protein interaction parameter and the kinetic coefficient. The results indicated that the LR step should be used for dilution condition (loading factor below 5%), while the LA step should be conducted when the isotherm is in the transition or nonlinear regions. Six numerical experiments were conducted to implement the PbP method. The results demonstrated that the PbP method developed allows for the systematic estimation of HIC parameters one-by-one, effectively reducing the number of parameters required for inverse method estimation from six to two. This helps prevent non-identifiability of structural parameters. The feasibility of the PbP-HIC method was further validated by real-world experiments. Moreover, the PbP method enhances the mechanistic understanding of adsorption behavior of HIC and shows a promising application to other stoichiometric displacement model-derived isotherms.

System identifiability in a time-evolving agent-based model

Mathematical models are a valuable tool for studying and predicting the spread of infectious agents. The accuracy of model simulations and predictions invariably depends on the specification of model parameters. Estimation of these parameters is therefore extremely important; however, while some parameters can be derived from observational studies, the values of others are difficult to measure. Instead, models can be coupled with inference algorithms (i.e., data assimilation methods, or statistical filters), which fit model simulations to existing observations and estimate unobserved model state variables and parameters. Ideally, these inference algorithms should find the best fitting solution for a given model and set of observations; however, as those estimated quantities are unobserved, it is typically uncertain whether the correct parameters have been identified. Further, it is unclear what 'correct' really means for abstract parameters defined based on specific model forms. In this work, we explored the problem of non-identifiability in a stochastic system which, when overlooked, can significantly impede model prediction. We used a network, agent-based model to simulate the transmission of Methicillin-resistant staphylococcus aureus (MRSA) within hospital settings and attempted to infer key model parameters using the Ensemble Adjustment Kalman Filter, an efficient Bayesian inference algorithm. We show that even though the inference method converged and that simulations using the estimated parameters produced an agreement with observations, the true parameters are not fully identifiable. While the model-inference system can exclude a substantial area of parameter space that is unlikely to contain the true parameters, the estimated parameter range still included multiple parameter combinations that can fit observations equally well. We show that analyzing synthetic trajectories can support or contradict claims of identifiability. While we perform this on a specific model system, this approach can be generalized for a variety of stochastic representations of partially observable systems. We also suggest data manipulations intended to improve identifiability that might be applicable in many systems of interest.

Predicting gene-level sensitivity to JAK-STAT signaling perturbation using a mechanistic-to-machine learning framework

The Janus kinase (JAK)-signal transducer and activator of transcription (STAT) pathway integrates complex cytokine signals via a limited number of molecular components, inspiring numerous efforts to clarify the diversity and specificity of STAT transcription factor function. We developed a computational framework to make global cytokine-induced gene predictions from STAT phosphorylation dynamics, modeling macrophage responses to interleukin (IL)-6 and IL-10, which signal through common STATs, but with distinct temporal dynamics and contrasting functions. Our mechanistic-to-machine learning model identified cytokine-specific genes associated with late pSTAT3 time frames and a preferential pSTAT1 reduction upon JAK2 inhibition. We predicted and validated the impact of JAK2 inhibition on gene expression, identifying genes that were sensitive or insensitive to JAK2 variation. Thus, we successfully linked STAT signaling dynamics to gene expression to support future efforts targeting pathology-associated STAT-driven gene sets. This serves as a first step in developing multi-level prediction models to understand and perturb gene expression outputs from signaling systems. A record of this paper's transparent peer review process is included in the supplemental information.

Minimally sufficient experimental design using identifiability analysis

Mathematical models are increasingly being developed and calibrated in tandem with data collection, empowering scientists to intervene in real time based on quantitative model predictions. Well-designed experiments can help augment the predictive power of a mathematical model but the question of when to collect data to maximize its utility for a model is non-trivial. Here we define data as model-informative if it results in a unique parametrization, assessed through the lens of practical identifiability. The framework we propose identifies an optimal experimental design (how much data to collect and when to collect it) that ensures parameter identifiability (permitting confidence in model predictions), while minimizing experimental time and costs. We demonstrate the power of the method by applying it to a modified version of a classic site-of-action pharmacokinetic/pharmacodynamic model that describes distribution of a drug into the tumor microenvironment (TME), where its efficacy is dependent on the level of target occupancy in the TME. In this context, we identify a minimal set of time points when data needs to be collected that robustly ensures practical identifiability of model parameters. The proposed methodology can be applied broadly to any mathematical model, allowing for the identification of a minimally sufficient experimental design that collects the most informative data.

Accuracy and precision of fixed and random effects in meta-analyses of randomized control trials for continuous outcomes

Meta-analyses of treatment effects in randomized control trials are often faced with the problem of missing information required to calculate effect sizes and their sampling variances. Particularly, correlations between pre- and posttest scores are frequently not available. As an ad-hoc solution, researchers impute a constant value for the missing correlation. As an alternative, we propose adopting a multivariate meta-regression approach that models independent group effect sizes and accounts for the dependency structure using robust variance estimation or three-level modeling. A comprehensive simulation study mimicking realistic conditions of meta-analyses in clinical and educational psychology suggested that imputing a fixed correlation 0.8 or adopting a multivariate meta-regression with robust variance estimation work well for estimating the pooled effect but lead to slightly distorted between-study heterogeneity estimates. In contrast, three-level meta-regressions resulted in largely unbiased fixed effects but more inconsistent prediction intervals. Based on these results recommendations for meta-analytic practice and future meta-analytic developments are provided.

Cluster Gauss-Newton method for a quick approximation of profile likelihood: With application to physiologically-based pharmacokinetic models

Physiologically-based pharmacokinetic (PBPK) models can be challenging to work with because they can have too many parameters to identify from observable data. The profile likelihood method can help solve this issue by determining parameter identifiability and confidence intervals, but it involves repetitive parameter optimizations that can be time-consuming. The Cluster Gauss-Newton method (CGNM) is a parameter estimation method that efficiently searches through a wide range of parameter space. In this study, we propose a method that approximates the profile likelihood by reusing intermediate computation results from CGNM, allowing us to obtain the upper bounds of the profile likelihood without conducting additional model evaluation. This method allows us to quickly draw approximate profile likelihoods for all unknown parameters. Additionally, the same approach can be used to draw two-dimensional profile likelihoods for all parameter combinations within seconds. We demonstrate the effectiveness of this method on three PBPK models.

A yeast cell cycle pulse generator model shows consistency with multiple oscillatory and checkpoint mutant datasets

Modeling biological systems holds great promise for speeding up the rate of discovery in systems biology by predicting experimental outcomes and suggesting targeted interventions. However, this process is dogged by an identifiability issue, in which network models and their parameters are not sufficiently constrained by coarse and noisy data to ensure unique solutions. In this work, we evaluated the capability of a simplified yeast cell-cycle network model to reproduce multiple observed transcriptomic behaviors under genomic mutations. We matched time-series data from both cycling and checkpoint arrested cells to model predictions using an asynchronous multi-level Boolean approach. We showed that this single network model, despite its simplicity, is capable of exhibiting dynamical behavior similar to the datasets in most cases, and we demonstrated the drop in severity of the identifiability issue that results from matching multiple datasets.

Structural and practical identifiability of contrast transport models for DCE-MRI

Compartment models are widely used to quantify blood flow and transport in dynamic contrast-enhanced magnetic resonance imaging. These models analyze the time course of the contrast agent concentration, providing diagnostic and prognostic value for many biological systems. Thus, ensuring accuracy and repeatability of the model parameter estimation is a fundamental concern. In this work, we analyze the structural and practical identifiability of a class of nested compartment models pervasively used in analysis of MRI data. We combine artificial and real data to study the role of noise in model parameter estimation. We observe that although all the models are structurally identifiable, practical identifiability strongly depends on the data characteristics. We analyze the impact of increasing data noise on parameter identifiability and show how the latter can be recovered with increased data quality. To complete the analysis, we show that the results do not depend on specific tissue characteristics or the type of enhancement patterns of contrast agent signal.

Nonself RNA rewires IFN-β signaling: A mathematical model of the innate immune response

Type I interferons (IFNs) are key coordinators of the innate immune response to viral infection, which, through activation of the transcriptional regulators STAT1 and STAT2 (STAT1/2) in bystander cells, induce the expression of IFN-stimulated genes (ISGs). Here, we showed that in cells transfected with poly(I:C), an analog of viral RNA, the transcriptional activity of STAT1/2 was terminated because of depletion of the interferon-β (IFN-β) receptor, IFNAR. Activation of RNase L and PKR, products of two ISGs, not only hindered the replenishment of IFNAR but also suppressed negative regulators of IRF3 and NF-κB, consequently promoting IFNB transcription. We incorporated these findings into a mathematical model of innate immunity. By coupling signaling through the IRF3-NF-κB and STAT1/2 pathways with the activities of RNase L and PKR, the model explains how poly(I:C) switches the transcriptional program from being STAT1/2 induced to being IRF3 and NF-κB induced, which converts IFN-β-responding cells to IFN-β-secreting cells.

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.

Fitting parameters and therapies of ODE tumor models with senescence and immune system

This work is devoted to introduce and study two quasispecies nonlinear ODE systems that model the behavior of tumor cell populations organized in different states. In the first model, replicative, senescent, extended lifespan, immortal and tumor cells are considered, while the second also includes immune cells. We fit the parameters regulating the transmission between states in order to approximate the outcomes of the models to a real progressive tumor invasion. After that, we study the identifiability of the fitted parameters, by using two sensitivity analysis methods. Then, we show that an adequate reduced fitting process (only accounting to the most identifiable parameters) gives similar results but saving computational cost. Three different therapies are introduced in the models to shrink (progressively in time) the tumor, while the replicative and senescent cells invade. Each therapy is identified to a dimensionless parameter. Then, we make a fitting process of the therapies' parameters, in various scenarios depending on the initial tumor according to the time when the therapies started. We conclude that, although the optimal combination of therapies depends on the size of initial tumor, the most efficient therapy is the reinforcement of the immune system. Finally, an identifiability analysis allows us to detect a limitation in the therapy outcomes. In fact, perturbing the optimal combination of therapies under an appropriate therapeutic vector produces virtually the same results.

Distilling identifiable and interpretable dynamic models from biological data

Mechanistic dynamical models allow us to study the behavior of complex biological systems. They can provide an objective and quantitative understanding that would be difficult to achieve through other means. However, the systematic development of these models is a non-trivial exercise and an open problem in computational biology. Currently, many research efforts are focused on model discovery, i.e. automating the development of interpretable models from data. One of the main frameworks is sparse regression, where the sparse identification of nonlinear dynamics (SINDy) algorithm and its variants have enjoyed great success. SINDy-PI is an extension which allows the discovery of rational nonlinear terms, thus enabling the identification of kinetic functions common in biochemical networks, such as Michaelis-Menten. SINDy-PI also pays special attention to the recovery of parsimonious models (Occam's razor). Here we focus on biological models composed of sets of deterministic nonlinear ordinary differential equations. We present a methodology that, combined with SINDy-PI, allows the automatic discovery of structurally identifiable and observable models which are also mechanistically interpretable. The lack of structural identifiability and observability makes it impossible to uniquely infer parameter and state variables, which can compromise the usefulness of a model by distorting its mechanistic significance and hampering its ability to produce biological insights. We illustrate the performance of our method with six case studies. We find that, despite enforcing sparsity, SINDy-PI sometimes yields models that are unidentifiable. In these cases we show how our method transforms their equations in order to obtain a structurally identifiable and observable model which is also interpretable.

Kinetic Modeling and Parameter Estimation of a Prebiotic Peptide Reaction Network

Although our understanding of how life emerged on Earth from simple organic precursors is speculative, early precursors likely included amino acids. The polymerization of amino acids into peptides and interactions between peptides are of interest because peptides and proteins participate in complex interaction networks in extant biology. However, peptide reaction networks can be challenging to study because of the potential for multiple species and systems-level interactions between species. We developed and employed a computational network model to describe reactions between amino acids to form di-, tri-, and tetra-peptides. Our experiments were initiated with two of the simplest amino acids, glycine and alanine, mediated by trimetaphosphate-activation and drying to promote peptide bond formation. The parameter estimates for bond formation and hydrolysis reactions in the system were found to be poorly constrained due to a network property known as sloppiness. In a sloppy model, the behavior mostly depends on only a subset of parameter combinations, but there is no straightforward way to determine which parameters should be included or excluded. Despite our inability to determine the exact values of specific kinetic parameters, we could make reasonably accurate predictions of model behavior. In short, our modeling has highlighted challenges and opportunities toward understanding the behaviors of complex prebiotic chemical experiments.

Bayesian parameter estimation for the inclusion of uncertainty in progressive damage simulation of composites

Despite gradual progress over the past decades, the simulation of progressive damage in composite laminates remains a challenging task, in part due to inherent uncertainties of material properties. This paper combines three computational methods - finite element analysis (FEA), machine learning and Markov Chain Monte Carlo - to estimate the probability density of FEA input parameters while accounting for the variation of mechanical properties. First, 15,000 FEA simulations of open-hole tension tests are carried out with randomly varying input parameters by applying continuum damage mechanics material models. This synthetically-generated data is then used to train and validate a neural network consisting of five hidden layers and 32 nodes per layer to develop a highly efficient surrogate model. With this surrogate model and the incorporation of statistical test data from experiments, the application of Markov Chain Monte Carlo algorithms enables Bayesian parameter estimation to learn the probability density of input parameters for the simulation of progressive damage evolution in fibre reinforced composites. This methodology is validated against various open-hole tension test geometries enabling the determination of virtual design allowables.

An adaptive information-theoretic experimental design procedure for high-to-low fidelity calibration of prostate cancer models

The use of mathematical models to make predictions about tumor growth and response to treatment has become increasingly prevalent in the clinical setting. The level of complexity within these models ranges broadly, and the calibration of more complex models requires detailed clinical data. This raises questions about the type and quantity of data that should be collected and when, in order to maximize the information gain about the model behavior while still minimizing the total amount of data used and the time until a model can be calibrated accurately. To address these questions, we propose a Bayesian information-theoretic procedure, using an adaptive score function to determine the optimal data collection times and measurement types. The novel score function introduced in this work eliminates the need for a penalization parameter used in a previous study, while yielding model predictions that are superior to those obtained using two potential pre-determined data collection protocols for two different prostate cancer model scenarios: one in which we fit a simple ODE system to synthetic data generated from a cellular automaton model using radiotherapy as the imposed treatment, and a second scenario in which a more complex ODE system is fit to clinical patient data for patients undergoing intermittent androgen suppression therapy. We also conduct a robust analysis of the calibration results, using both error and uncertainty metrics in combination to determine when additional data acquisition may be terminated.

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.

Combining a parsimonious mathematical model with infection data from tailor-made experiments to understand environmental transmission

Although most infections are transmitted through the environment, the processes underlying the environmental stage of transmission are still poorly understood for most systems. Improved understanding of the environmental transmission dynamics is important for effective non-pharmaceutical intervention strategies. To study the mechanisms underlying environmental transmission we formulated a parsimonious modelling framework including hypothesised mechanisms of pathogen dispersion and decay. To calibrate and validate the model, we conducted a series of experiments studying distance-dependent transmission of Campylobacter jejuni in broilers. We obtained informative simultaneous estimates for all three model parameters: the parameter of C. jejuni inactivation, the diffusion coefficient describing pathogen dispersion, and the transmission rate parameter. The time and distance dependence of transmission in the fitted model is quantitatively consistent with marked spatiotemporal patterns in the experimental observations. These results, for C. jejuni in broilers, show that the application of our modelling framework to suitable transmission data can provide mechanistic insight in environmental pathogen transmission.

Structural identifiability of biomolecular controller motifs with and without flow measurements as model output

Controller motifs are simple biomolecular reaction networks with negative feedback. They can explain how regulatory function is achieved and are often used as building blocks in mathematical models of biological systems. In this paper we perform an extensive investigation into structural identifiability of controller motifs, specifically the so-called basic and antithetic controller motifs. Structural identifiability analysis is a useful tool in the creation and evaluation of mathematical models: it can be used to ensure that model parameters can be determined uniquely and to examine which measurements are necessary for this purpose. This is especially useful for biological models where parameter estimation can be difficult due to limited availability of measureable outputs. Our aim with this work is to investigate how structural identifiability is affected by controller motif complexity and choice of measurements. To increase the number of potential outputs we propose two methods for including flow measurements and show how this affects structural identifiability in combination with, or in the absence of, concentration measurements. In our investigation, we analyze 128 different controller motif structures using a combination of flow and/or concentration measurements, giving a total of 3648 instances. Among all instances, 34% of the measurement combinations provided structural identifiability. Our main findings for the controller motifs include: i) a single measurement is insufficient for structural identifiability, ii) measurements related to different chemical species are necessary for structural identifiability. Applying these findings result in a reduced subset of 1568 instances, where 80% are structurally identifiable, and more complex/interconnected motifs appear easier to structurally identify. The model structures we have investigated are commonly used in models of biological systems, and our results demonstrate how different model structures and measurement combinations affect structural identifiability of controller motifs.

A bistable autoregulatory module in the developing embryo commits cells to binary expression fates

Bistable autoactivation has been proposed as a mechanism for cells to adopt binary fates during embryonic development. However, it is unclear whether the autoactivating modules found within developmental gene regulatory networks are bistable, unless their parameters are quantitatively determined. Here, we combine in vivo live imaging with mathematical modeling to dissect the binary cell fate dynamics of the fruit fly pair-rule gene fushi tarazu (ftz), which is regulated by two known enhancers: the early (non-autoregulating) element and the autoregulatory element. Live imaging of transcription and protein concentration in the blastoderm revealed that binary Ftz fates are achieved as Ftz expression rapidly transitions from being dictated by the early element to the autoregulatory element. Moreover, we discovered that Ftz concentration alone is insufficient to activate the autoregulatory element, and that this element only becomes responsive to Ftz at a prescribed developmental time. Based on these observations, we developed a dynamical systems model and quantitated its kinetic parameters directly from experimental measurements. Our model demonstrated that the ftz autoregulatory module is indeed bistable and that the early element transiently establishes the content of the binary cell fate decision to which the autoregulatory module then commits. Further in silico analysis revealed that the autoregulatory element locks the Ftz fate quickly, within 35 min of exposure to the transient signal of the early element. Overall, our work confirms the widely held hypothesis that autoregulation can establish developmental fates through bistability and, most importantly, provides a framework for the quantitative dissection of cellular decision-making.

Predictive power of non-identifiable models

Resolving practical non-identifiability of computational models typically requires either additional data or non-algorithmic model reduction, which frequently results in models containing parameters lacking direct interpretation. Here, instead of reducing models, we explore an alternative, Bayesian approach, and quantify the predictive power of non-identifiable models. We considered an example biochemical signalling cascade model as well as its mechanical analogue. For these models, we demonstrated that by measuring a single variable in response to a properly chosen stimulation protocol, the dimensionality of the parameter space is reduced, which allows for predicting the measured variable's trajectory in response to different stimulation protocols even if all model parameters remain unidentified. Moreover, one can predict how such a trajectory will transform in the case of a multiplicative change of an arbitrary model parameter. Successive measurements of remaining variables further reduce the dimensionality of the parameter space and enable new predictions. We analysed potential pitfalls of the proposed approach that can arise when the investigated model is oversimplified, incorrect, or when the training protocol is inadequate. The main advantage of the suggested iterative approach is that the predictive power of the model can be assessed and practically utilised at each step.

Bayesian analysis dissects kinetic modulation during non-stationary gene expression

Labelling of nascent stem loops with fluorescent proteins has fostered the visualization of transcription in living cells. Quantitative analysis of recorded fluorescence traces can shed light on kinetic transcription parameters and regulatory mechanisms. However, existing methods typically focus on steady state dynamics. Here, we combine a stochastic process transcription model with a hierarchical Bayesian method to infer global as well locally shared parameters for groups of cells and recover unobserved quantities such as initiation times and polymerase loading of the gene. We apply our approach to the cyclic response of the yeast CUP1 locus to heavy metal stress. Within the previously described slow cycle of transcriptional activity on the scale of minutes, we discover fast time-modulated bursting on the scale of seconds. Model comparison suggests that slow oscillations of transcriptional output are regulated by the amplitude of the bursts. Several polymerases may initiate during a burst.

A plausible identifiable model of the canonical NF-κB signaling pathway

An overwhelming majority of mathematical models of regulatory pathways, including the intensively studied NF-κB pathway, remains non-identifiable, meaning that their parameters may not be determined by existing data. The existing NF-κB models that are capable of reproducing experimental data contain non-identifiable parameters, whereas simplified models with a smaller number of parameters exhibit dynamics that differs from that observed in experiments. Here, we reduced an existing model of the canonical NF-κB pathway by decreasing the number of equations from 15 to 6. The reduced model retains two negative feedback loops mediated by IκBα and A20, and in response to both tonic and pulsatile TNF stimulation exhibits dynamics that closely follow that of the original model. We carried out the sensitivity-based linear analysis and Monte Carlo-based analysis to demonstrate that the resulting model is both structurally and practically identifiable given measurements of 5 model variables from a simple TNF stimulation protocol. The reduced model is capable of reproducing different types of responses that are characteristic to regulatory motifs controlled by negative feedback loops: nearly-perfect adaptation as well as damped and sustained oscillations. It can serve as a building block of more comprehensive models of the immune response and cancer, where NF-κB plays a decisive role. Our approach, although may not be automatically generalized, suggests that models of other regulatory pathways can be transformed to identifiable, while retaining their dynamical features.

Improved Jacobian matrix estimation applied to snake robots

Two manipulator Jacobian matrix estimators for constrained planar snake robots are developed and tested, which enables the implementation of Jacobian-based obstacle-aided locomotion (OAL) control schemes. These schemes use obstacles in the robot's vicinity to obtain propulsion. The devised estimators infer manipulator Jacobians for constrained planar snake robots in situations where the positions and number of surrounding obstacle constraints might change or are not precisely known. The first proposed estimator is an adaptation of contemporary research in soft robots and builds on convex optimization. The second estimator builds on the unscented Kalman filter. By simulations, we evaluate and compare the two devised algorithms in terms of their statistical performance, execution times, and robustness to measurement noise. We find that both algorithms lead to Jacobian matrix estimates that are similarly useful to predict end-effector movements. However, the unscented filter approach requires significantly lower computing resources and is not poised by convergence issues displayed by the convex optimization-based method. We foresee that the estimators may have use in other fields of research, such as soft robotics and visual servoing. The estimators may also be adapted for use in general non-planar snake robots.

Informative and adaptive distances and summary statistics in sequential approximate Bayesian computation

Calibrating model parameters on heterogeneous data can be challenging and inefficient. This holds especially for likelihood-free methods such as approximate Bayesian computation (ABC), which rely on the comparison of relevant features in simulated and observed data and are popular for otherwise intractable problems. To address this problem, methods have been developed to scale-normalize data, and to derive informative low-dimensional summary statistics using inverse regression models of parameters on data. However, while approaches only correcting for scale can be inefficient on partly uninformative data, the use of summary statistics can lead to information loss and relies on the accuracy of employed methods. In this work, we first show that the combination of adaptive scale normalization with regression-based summary statistics is advantageous on heterogeneous parameter scales. Second, we present an approach employing regression models not to transform data, but to inform sensitivity weights quantifying data informativeness. Third, we discuss problems for regression models under non-identifiability, and present a solution using target augmentation. We demonstrate improved accuracy and efficiency of the presented approach on various problems, in particular robustness and wide applicability of the sensitivity weights. Our findings demonstrate the potential of the adaptive approach. The developed algorithms have been made available in the open-source Python toolbox pyABC.

Predicting gene level sensitivity to JAK-STAT signaling perturbation using a mechanistic-to-machine learning framework

The JAK-STAT pathway integrates complex cytokine signals via a limited number of molecular components, inspiring numerous efforts to clarify the diversity and specificity of STAT transcription factor function. We developed a computational workflow to make global cytokine-induced gene predictions from STAT phosphorylation dynamics, modeling macrophage responses to IL-6 and IL-10, which signal through common STATs, but with distinct temporal dynamics and contrasting functions. Our mechanistic-to-machine learning model identified select cytokine-induced gene sets associated with late pSTAT3 timeframes and a preferential pSTAT1 reduction upon JAK2 inhibition. We predicted and validated the impact of JAK2 inhibition on gene expression, identifying dynamically regulated genes that were sensitive or insensitive to JAK2 variation. Thus, we successfully linked STAT signaling dynamics to gene expression to support future efforts targeting pathology-associated STAT-driven gene sets. This serves as a first step in developing multi-level prediction models to understand and perturb gene expression outputs from signaling systems.

Structural Identifiability and Observability of Microbial Community Models

Biological communities are populations of various species interacting in a common location. Microbial communities, which are formed by microorganisms, are ubiquitous in nature and are increasingly used in biotechnological and biomedical applications. They are nonlinear systems whose dynamics can be accurately described by models of ordinary differential equations (ODEs). A number of ODE models have been proposed to describe microbial communities. However, the structural identifiability and observability of most of them-that is, the theoretical possibility of inferring their parameters and internal states by observing their output-have not been determined yet. It is important to establish whether a model possesses these properties, because, in their absence, the ability of a model to make reliable predictions may be compromised. Hence, in this paper, we analyse these properties for the main families of microbial community models. We consider several dimensions and measurements; overall, we analyse more than a hundred different configurations. We find that some of them are fully identifiable and observable, but a number of cases are structurally unidentifiable and/or unobservable under typical experimental conditions. Our results help in deciding which modelling frameworks may be used for a given purpose in this emerging area, and which ones should be avoided.

A Bayesian approach to blood rheological uncertainties in aortic hemodynamics

Computational hemodynamics has received increasing attention recently. Patient-specific simulations require questionable model assumptions, for example, for geometry, boundary conditions, and material parameters. Consequently, the credibility of these simulations is much doubted, and rightly so. Yet, the matter may be addressed by a rigorous uncertainty quantification. In this contribution, we investigated the impact of blood rheological models on wall shear stress uncertainties in aortic hemodynamics obtained in numerical simulations. Based on shear-rheometric experiments, we compare the non-Newtonian Carreau model to a simple Newtonian model and a Reynolds number-equivalent Newtonian model. Bayesian Probability Theory treats uncertainties consistently and allows to include elusive assumptions such as the comparability of flow regimes. We overcome the prohibitively high computational cost for the simulation with a surrogate model, and account for the uncertainties of the surrogate model itself, too. We have two main findings: (1) The Newtonian models mostly underestimate the uncertainties as compared to the non-Newtonian model. (2) The wall shear stresses of specific persons cannot be distinguished due to largely overlapping uncertainty bands, implying that a more precise determination of person-specific blood rheological properties is necessary for person-specific simulations. While we refrain from a general recommendation for one rheological model, we have quantified the error of the uncertainty quantification associated with these modeling choices.

Mechanistic modeling of viral particle production

Viral systems such as wild-type viruses, viral vectors, and virus-like particles are essential components of modern biotechnology and medicine. Despite their importance, the commercial-scale production of viral systems remains highly inefficient for multiple reasons. Computational strategies are a promising avenue for improving process development, optimization, and control, but require a mathematical description of the system. This article reviews mechanistic modeling strategies for the production of viral particles, both at the cellular and bioreactor scales. In many cases, techniques and models from adjacent fields such as epidemiology and wild-type viral infection kinetics can be adapted to construct a suitable process model. These process models can then be employed for various purposes such as in-silico testing of novel process operating strategies and/or advanced process control.

Solid State Kinetics of Nitrosation Using Native Sources of Nitrite

While many reactive species are known to cause N-nitrosation, trace nitrite (NO₂^-), which may be present in several excipients, is a source of nitrosating agents in pharmaceutical formulations. In this study we have found that the salt form of NO₂^- can influence the favored nitrosation conditions and final amount of nitrosamine being formed. Using native levels of NO₂^-, most likely present as ammonium nitrite (NH₄NO₂), in microcrystalline cellulose, we have determined the kinetics of nitrosamine formation in solid state with dimethylamine substrate present in metformin, used as model compound. It was found that the competing degradation of NH₄NO₂ into N₂ and H₂O limited the amount of nitrosamine formation to a great extent. Empirically modelling the kinetic data predicted reaching at maximum 1.6% conversion over a hypothetical 3-year shelf-life. These results also showed that using other sources of NO₂^- as spiking reagents, such as NaNO₂, may lead to unrealistic worst-case situations when the main form of NO₂^- in the drug product (DP) under evaluation may be NH₄NO₂. As well, measuring NO₂^- in freshly manufactured excipients containing NO₂^- potentially as NH₄NO₂ may lead to biased high NO₂^- content, which is not representative of the actual amounts present at the time of DP manufacture.

Designing experimental conditions to use the Lotka-Volterra model to infer tumor cell line interaction types

The Lotka-Volterra model is widely used to model interactions between two species. Here, we generate synthetic data mimicking competitive, mutualistic and antagonistic interactions between two tumor cell lines, and then use the Lotka-Volterra model to infer the interaction type. Structural identifiability of the Lotka-Volterra model is confirmed, and practical identifiability is assessed for three experimental designs: (a) use of a single data set, with a mixture of both cell lines observed over time, (b) a sequential design where growth rates and carrying capacities are estimated using data from experiments in which each cell line is grown in isolation, and then interaction parameters are estimated from an experiment involving a mixture of both cell lines, and (c) a parallel experimental design where all model parameters are fitted to data from two mixtures (containing both cell lines but with different initial ratios) simultaneously. Each design is tested on data generated from the Lotka-Volterra model with noise added, to determine efficacy in an ideal sense. In addition to assessing each design for practical identifiability, we investigate how the predictive power of the model - i.e., its ability to fit data for initial ratios other than those to which it was calibrated - is affected by the choice of experimental design. The parallel calibration procedure is found to be optimal and is further tested on in silico data generated from a spatially-resolved cellular automaton model, which accounts for oxygen consumption and allows for variation in the intensity level of the interaction between the two cell lines. We use this study to highlight the care that must be taken when interpreting parameter estimates for the spatially-averaged Lotka-Volterra model when it is calibrated against data produced by the spatially-resolved cellular automaton model, since baseline competition for space and resources in the CA model may contribute to a discrepancy between the type of interaction used to generate the CA data and the type of interaction inferred by the LV model.

Standardized method for mechanistic modeling of multimodal anion exchange chromatography in flow through operation

Multimodal chromatography offers an increased selectivity compared to unimodal chromatographic methods and is often employed for challenging separation tasks in industrial downstream processing (DSP). Unfortunately, the implementation of multimodal polishing into a generic downstream platform can be hampered by non-robust platform conditions leading to a time and cost intensive process development. Mechanistic modeling can assist experimental process development but readily applicable and easy to calibrate multimodal chromatography models are lacking. In this work, we present a mechanistic modeling aided approach that paves the way for an accelerated development of anionic mixed-mode chromatography (MMC) for biopharmaceutical purification. A modified multimodal isotherm model was calibrated using only three chromatographic experiments and was employed in the retention prediction of four antibody formats including a Fab, a bispecific, as well as an IgG1 and IgG4 antibody subtype at pH 5.0 and 6.0. The chromatographic experiments were conducted using the anionic mixed-mode resin Capto adhere at industrial relevant process conditions to enable flow through purification. An existing multimodal isotherm model was reduced to hydrophobic interactions in the linear range of the adsorption isotherm and successfully employed in the simulation of six chromatographic experiments per molecule in concert with the transport dispersive model (TDM). The model reduction to only three parameters did prevent structural parameter non-identifiability and enabled an analytical isotherm parameter determination that was further refined by incorporation of size exclusion effects of the selected multimodal resin. During the model calibration, three linear salt gradient elution experiments were performed for each molecule followed by an isotherm parameter uncertainty assessment. Lastly, each model was validated with a set of step and isocratic elution experiments. This standardized modeling approach facilitates the implementation of multimodal chromatography as a key unit operation for the biopharmaceutical downstream platform, while increasing the mechanistic insight to the multimodal adsorption behavior of complex biologics.

Benchmarking tools for a priori identifiability analysis

MOTIVATION

The theoretical possibility of determining the state and parameters of a dynamic model by measuring its outputs is given by its structural identifiability and its observability. These properties should be analysed before attempting to calibrate a model, but their a priori analysis can be challenging, requiring symbolic calculations that often have a high computational cost. In recent years, a number of software tools have been developed for this task, mostly in the systems biology community. These tools have vastly different features and capabilities, and a critical assessment of their performance is still lacking.

RESULTS

Here, we present a comprehensive study of the computational resources available for analysing structural identifiability. We consider 13 software tools developed in 7 programming languages and evaluate their performance using a set of 25 case studies created from 21 models. Our results reveal their strengths and weaknesses, provide guidelines for choosing the most appropriate tool for a given problem and highlight opportunities for future developments.

AVAILABILITY AND IMPLEMENTATION

https://github.com/Xabo-RB/Benchmarking_files.

Mechanistic model of MAPK signaling reveals how allostery and rewiring contribute to drug resistance

BRAF is prototypical of oncogenes that can be targeted therapeutically and the treatment of BRAF^V600E melanomas with RAF and MEK inhibitors results in rapid tumor regression. However, drug-induced rewiring generates a drug adapted state thought to be involved in acquired resistance and disease recurrence. In this article, we study mechanisms of adaptive rewiring in BRAF^V600E melanoma cells using an energy-based implementation of ordinary differential equation (ODE) modeling in combination with proteomic, transcriptomic and imaging data. We develop a method for causal tracing of ODE models and identify two parallel MAPK reaction channels that are differentially sensitive to RAF and MEK inhibitors due to differences in protein oligomerization and drug binding. We describe how these channels, and timescale separation between immediate-early signaling and transcriptional feedback, create a state in which the RAS-regulated MAPK channel can be activated by growth factors under conditions in which the BRAF^V600E -driven channel is fully inhibited. Further development of the approaches in this article is expected to yield a unified model of adaptive drug resistance in melanoma.

ASAS-NANP symposium: Mathematical Modeling in Animal Nutrition: The power of identifiability analysis for dynamic modeling in animal science:a practitioner approach

Constructing dynamic mathematical models of biological systems requires estimating unknown parameters from available experimental data, usually using a statistical fitting procedure. This procedure is usually called parameter identification, parameter estimation, model fitting, or model calibration. In animal science, parameter identification is often performed without analytic considerations on the possibility of determining unique values of the model parameters. These analytical studies are related to the mathematical property of structural identifiability, which refers to the theoretical ability to recover unique values of the model parameters from the measures defined in an experimental setup and use the model structure as the sole basis. The structural identifiability analysis is a powerful tool for model construction because it informs whether the parameter identification problem is well-posed (i.e., the problem has a unique solution). Structural identifiability analysis is helpful to determine which actions (e.g., model reparameterization, choice of new data measurements, and change of the model structure) are needed to render the model parameters identifiable (when possible). The mathematical technicalities associated with structural identifiability analysis are very sophisticated. However, the development of dedicated, freely available software tools enables the application of identifiability analysis without needing to be an expert in mathematics and computer programming. We refer to such a non-expert user as a practitioner for hands-on purposes. However, a practitioner should be familiar with the model construction and software implementation process. In this paper, we propose to adopt a practitioner approach that takes advantage of available software tools to integrate identifiability analysis in the modeling practice in the animal science field. The application of structural identifiability implies switching our regard of the parameter identification problem as a downstream process (after data collection) to an upstream process (before data collection) where experiment design is applied to guarantee identifiability. This upstream approach will substantially improve the workflow of model construction toward robust and valuable models in animal science. Illustrative examples with different levels of complexity support our work. The source codes of the examples were provided for learning purposes and to promote open science practices.

On Parameter Identifiability in Network-Based Epidemic Models

Modelling epidemics on networks represents an important departure from classical compartmental models which assume random mixing. However, the resulting models are high-dimensional and their analysis is often out of reach. It turns out that mean-field models, low-dimensional systems of differential equations, whose variables are carefully chosen expected quantities from the exact model provide a good approximation and incorporate explicitly some network properties. Despite the emergence of such mean-field models, there has been limited work on investigating whether these can be used for inference purposes. In this paper, we consider network-based mean-field models and explore the problem of parameter identifiability when observations about an epidemic are available. Making use of the analytical tractability of most network-based mean-field models, e.g. explicit analytical expressions for leading eigenvalue and final epidemic size, we set up the parameter identifiability problem as finding the solution or solutions of a system of coupled equations. More precisely, subject to observing/measuring growth rate and final epidemic size, we seek to identify parameter values leading to these measurements. We are particularly concerned with disentangling transmission rate from the network density. To do this, we give a condition for practical identifiability and we find that except for the simplest model, parameters cannot be uniquely determined, that is, they are practically unidentifiable. This means that there exist multiple solutions (a manifold of infinite measure) which give rise to model output that is close to the data. Identifying, formalising and analytically describing this problem should lead to a better appreciation of the complexity involved in fitting models with many parameters to data.