Robust and efficient parameter estimation in dynamic models of biological systems

DataXflow: Synergizing data-driven modeling with best parameter fit and optimal control - An efficient data analysis for cancer research

Building data-driven models is an effective strategy for information extraction from empirical data. Adapting model parameters specifically to data with a best fitting approach encodes the relevant information into a mathematical model. Subsequently, an optimal control framework extracts the most efficient targets to steer the model into desired changes via external stimuli. The DataXflow software framework integrates three software pipelines, D2D for model fitting, a framework solving optimal control problems including external stimuli and JimenaE providing graphical user interfaces to employ the other frameworks lowering the barriers for the need of programming skills, and simultaneously automating reoccurring modeling tasks. Such tasks include equation generation from a graph and script generation allowing also to approach systems with many agents, like complex gene regulatory networks. A desired state of the model is defined, and therapeutic interventions are modeled as external stimuli. The optimal control framework purposefully exploits the model-encoded information by providing those external stimuli that effect the desired changes most efficiently. The implementation of DataXflow is available under https://github.com/MarvelousHopefull/DataXflow. We showcase its application by detecting specific drug targets for a therapy of lung cancer from measurement data to lower proliferation and increase apoptosis. By an iterative modeling process refining the topology of the model, the regulatory network of the tumor is generated from the data. An application of the optimal control framework in our example reveals the inhibition of AURKA and the activation of CDH1 as the most efficient drug target combination. DataXflow paves the way to an agile interplay between data generation and its analysis potentially accelerating cancer research by an efficient drug target identification, even in complex networks.

Building Digital Twins for Cardiovascular Health: From Principles to Clinical Impact

The past several decades have seen rapid advances in diagnosis and treatment of cardiovascular diseases and stroke, enabled by technological breakthroughs in imaging, genomics, and physiological monitoring, coupled with therapeutic interventions. We now face the challenge of how to (1) rapidly process large, complex multimodal and multiscale medical measurements; (2) map all available data streams to the trajectories of disease states over the patient's lifetime; and (3) apply this information for optimal clinical interventions and outcomes. Here we review new advances that may address these challenges using digital twin technology to fulfill the promise of personalized cardiovascular medical practice. Rooted in engineering mechanics and manufacturing, the digital twin is a virtual representation engineered to model and simulate its physical counterpart. Recent breakthroughs in scientific computation, artificial intelligence, and sensor technology have enabled rapid bidirectional interactions between the virtual-physical counterparts with measurements of the physical twin that inform and improve its virtual twin, which in turn provide updated virtual projections of disease trajectories and anticipated clinical outcomes. Verification, validation, and uncertainty quantification builds confidence and trust by clinicians and patients in the digital twin and establishes boundaries for the use of simulations in cardiovascular medicine. Mechanistic physiological models form the fundamental building blocks of the personalized digital twin that continuously forecast optimal management of cardiovascular health using individualized data streams. We present exemplars from the existing body of literature pertaining to mechanistic model development for cardiovascular dynamics and summarize existing technical challenges and opportunities pertaining to the foundation of a digital twin.

Microbiome modeling: a beginner's guide

Microbiomes, comprised of diverse microbial species and viruses, play pivotal roles in human health, environmental processes, and biotechnological applications and interact with each other, their environment, and hosts via ecological interactions. Our understanding of microbiomes is still limited and hampered by their complexity. A concept improving this understanding is systems biology, which focuses on the holistic description of biological systems utilizing experimental and computational methods. An important set of such experimental methods are metaomics methods which analyze microbiomes and output lists of molecular features. These lists of data are integrated, interpreted, and compiled into computational microbiome models, to predict, optimize, and control microbiome behavior. There exists a gap in understanding between microbiologists and modelers/bioinformaticians, stemming from a lack of interdisciplinary knowledge. This knowledge gap hinders the establishment of computational models in microbiome analysis. This review aims to bridge this gap and is tailored for microbiologists, researchers new to microbiome modeling, and bioinformaticians. To achieve this goal, it provides an interdisciplinary overview of microbiome modeling, starting with fundamental knowledge of microbiomes, metaomics methods, common modeling formalisms, and how models facilitate microbiome control. It concludes with guidelines and repositories for modeling. Each section provides entry-level information, example applications, and important references, serving as a valuable resource for comprehending and navigating the complex landscape of microbiome research and modeling.

Cell-specific models of hiPSC-CMs developed by the gradient-based parameter optimization method fitting two different action potential waveforms

Parameter optimization (PO) methods to determine the ionic current composition of experimental cardiac action potential (AP) waveform have been developed using a computer model of cardiac membrane excitation. However, it was suggested that fitting a single AP record in the PO method was not always successful in providing a unique answer because of a shortage of information. We found that the PO method worked perfectly if the PO method was applied to a pair of a control AP and a model output AP in which a single ionic current out of six current species, such as IKr, ICaL, INa, IKs, IKur or IbNSC was partially blocked in silico. When the target was replaced by a pair of experimental control and IKr-blocked records of APs generated spontaneously in human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs), the simultaneous fitting of the two waveforms by the PO method was hampered to some extent by the irregular slow fluctuations in the Vm recording and/or sporadic alteration in AP configurations in the hiPSC-CMs. This technical problem was largely removed by selecting stable segments of the records for the PO method. Moreover, the PO method was made fail-proof by running iteratively in identifying the optimized parameter set to reconstruct both the control and the IKr-blocked AP waveforms. In the lead potential analysis, the quantitative ionic mechanisms deduced from the optimized parameter set were totally consistent with the qualitative view of ionic mechanisms of AP so far described in physiological literature.

Bottom-up parameterization of enzyme rate constants: Reconciling inconsistent data

Kinetic models of metabolism are promising platforms for studying complex metabolic systems and designing production strains. Given the availability of enzyme kinetic data from historical experiments and machine learning estimation tools, a straightforward modeling approach is to assemble kinetic data enzyme by enzyme until a desired scale is reached. However, this type of 'bottom up' parameterization of kinetic models has been difficult due to a number of issues including gaps in kinetic parameters, the complexity of enzyme mechanisms, inconsistencies between parameters obtained from different sources, and in vitro-in vivo differences. Here, we present a computational workflow for the robust estimation of kinetic parameters for detailed mass action enzyme models while taking into account parameter uncertainty. The resulting software package, termed MASSef (the Mass Action Stoichiometry Simulation Enzyme Fitting package), can handle standard 'macroscopic' kinetic parameters, including Km, kcat, Ki, Keq, and nh, as well as diverse reaction mechanisms defined in terms of mass action reactions and 'microscopic' rate constants. We provide three enzyme case studies demonstrating that this approach can identify and reconcile inconsistent data either within in vitro experiments or between in vitro and in vivo enzyme function. We further demonstrate how parameterized enzyme modules can be used to assemble pathway-scale kinetic models consistent with in vivo behavior. This work builds on the legacy of knowledge on kinetic behavior of enzymes by enabling robust parameterization of enzyme kinetic models at scale utilizing the abundance of historical literature data and machine learning parameter estimates.

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.

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.

Valorization of organic wastes using bioreactors for polyhydroxyalkanoate production: Recent advancement, sustainable approaches, challenges, and future perspectives

Microbial polyhydroxyalkanoates (PHA) are encouraging biodegradable polymers, which may ease the environmental problems caused by petroleum-derived plastics. However, there is a growing waste removal problem and the high price of pure feedstocks for PHA biosynthesis. This has directed to the forthcoming requirement to upgrade waste streams from various industries as feedstocks for PHA production. This review covers the state-of-the-art progress in utilizing low-cost carbon substrates, effective upstream and downstream processes, and waste stream recycling to sustain entire process circularity. This review also enlightens the use of various batch, fed-batch, continuous, and semi-continuous bioreactor systems with flexible results to enhance the productivity and simultaneously cost reduction. The life-cycle and techno-economic analyses, advanced tools and strategies for microbial PHA biosynthesis, and numerous factors affecting PHA commercialization were also covered. The review includes the ongoing and upcoming strategies viz. metabolic engineering, synthetic biology, morphology engineering, and automation to expand PHA diversity, diminish production costs, and improve PHA production with an objective of "zero-waste" and "circular bioeconomy" for a sustainable future.

A genome-scale dynamic constraint-based modelling (gDCBM) framework predicts growth dynamics, medium composition and intracellular flux distributions in CHO clonal variations

Optimizing mammalian cell growth and bioproduction is a tedious task. However, due to the inherent complexity of eukaryotic cells, heuristic experimental approaches such as, metabolic engineering and bioprocess design, are frequently integrated with mathematical models of cell culture to improve biological process efficiency and find paths for improvement. Constraint-based metabolic models have evolved over the last two decades to be used for dynamic modelling in addition to providing a linear description of steady-state metabolic systems. Formulation and implementation of the underlying optimization problems require special attention to the model's performance and feasibility, lack of defects in the definition of system components, and consideration of optimal alternate solutions, in addition to processing power limitations. Here, the time-resolved dynamics of a genome-scale metabolic network of Chinese hamster ovary (CHO) cell metabolism are shown using a genome-scale dynamic constraint-based modelling framework (gDCBM). The metabolic network was adapted from a reference model of CHO genome-scale metabolic model (GSMM), iCHO_DG44_v1, and dynamic restrictions were imposed to its exchange fluxes based on experimental results. We used this framework for predicting physiological changes in CHO clonal variants. Because of the methodical creation of the components for the flux balance analysis optimization problem and the integration of a switch time, this model can generate sequential predictions of intracellular fluxes during growth and non-growth phases (per hour of culture time) and transparently reveal the shortcomings in such practice. As a result of the differences exploited by various clones, we can understand the relevance of changes in intracellular flux distribution and exometabolomics. The integration of various omics data into the given gDCBM framework, as well as the reductionist analysis of the model, can further help bioprocess optimization.

Data driven model discovery and interpretation for CAR T-cell killing using sparse identification and latent variables

In the development of cell-based cancer therapies, quantitative mathematical models of cellular interactions are instrumental in understanding treatment efficacy. Efforts to validate and interpret mathematical models of cancer cell growth and death hinge first on proposing a precise mathematical model, then analyzing experimental data in the context of the chosen model. In this work, we present the first application of the sparse identification of non-linear dynamics (SINDy) algorithm to a real biological system in order discover cell-cell interaction dynamics in in vitro experimental data, using chimeric antigen receptor (CAR) T-cells and patient-derived glioblastoma cells. By combining the techniques of latent variable analysis and SINDy, we infer key aspects of the interaction dynamics of CAR T-cell populations and cancer. Importantly, we show how the model terms can be interpreted biologically in relation to different CAR T-cell functional responses, single or double CAR T-cell-cancer cell binding models, and density-dependent growth dynamics in either of the CAR T-cell or cancer cell populations. We show how this data-driven model-discovery based approach provides unique insight into CAR T-cell dynamics when compared to an established model-first approach. These results demonstrate the potential for SINDy to improve the implementation and efficacy of CAR T-cell therapy in the clinic through an improved understanding of CAR T-cell dynamics.

Minimal Out-of-Equilibrium Metabolism for Synthetic Cells: A Membrane Perspective

Life-like systems need to maintain a basal metabolism, which includes importing a variety of building blocks required for macromolecule synthesis, exporting dead-end products, and recycling cofactors and metabolic intermediates, while maintaining steady internal physical and chemical conditions (physicochemical homeostasis). A compartment, such as a unilamellar vesicle, functionalized with membrane-embedded transport proteins and metabolic enzymes encapsulated in the lumen meets these requirements. Here, we identify four modules designed for a minimal metabolism in a synthetic cell with a lipid bilayer boundary: energy provision and conversion, physicochemical homeostasis, metabolite transport, and membrane expansion. We review design strategies that can be used to fulfill these functions with a focus on the lipid and membrane protein composition of a cell. We compare our bottom-up design with the equivalent essential modules of JCVI-syn3a, a top-down genome-minimized living cell with a size comparable to that of large unilamellar vesicles. Finally, we discuss the bottlenecks related to the insertion of a complex mixture of membrane proteins into lipid bilayers and provide a semiquantitative estimate of the relative surface area and lipid-to-protein mass ratios (i.e., the minimal number of membrane proteins) that are required for the construction of a synthetic cell.

Elucidating yeast glycolytic dynamics at steady state growth and glucose pulses through kinetic metabolic modeling

Microbial cell factories face changing environments during industrial fermentations. Kinetic metabolic models enable the simulation of the dynamic metabolic response to these perturbations, but their development is challenging due to model complexity and experimental data requirements. An example of this is the well-established microbial cell factory Saccharomyces cerevisiae, for which no consensus kinetic model of central metabolism has been developed and implemented in industry. Here, we aim to bring the academic and industrial communities closer to this consensus model. We developed a physiology informed kinetic model of yeast glycolysis connected to central carbon metabolism by including the effect of anabolic reactions precursors, mitochondria and the trehalose cycle. To parametrize such a large model, a parameter estimation pipeline was developed, consisting of a divide and conquer approach, supplemented with regularization and global optimization. Additionally, we show how this first mechanistic description of a growing yeast cell captures experimental dynamics at different growth rates and under a strong glucose perturbation, is robust to parametric uncertainty and explains the contribution of the different pathways in the network. Such a comprehensive model could not have been developed without using steady state and glucose perturbation data sets. The resulting metabolic reconstruction and parameter estimation pipeline can be applied in the future to study other industrially-relevant scenarios. We show this by generating a hybrid CFD-metabolic model to explore intracellular glycolytic dynamics for the first time. The model suggests that all intracellular metabolites oscillate within a physiological range, except carbon storage metabolism, which is sensitive to the extracellular environment.

Autocorrelated measurement processes and inference for ordinary differential equation models of biological systems

Ordinary differential equation models are used to describe dynamic processes across biology. To perform likelihood-based parameter inference on these models, it is necessary to specify a statistical process representing the contribution of factors not explicitly included in the mathematical model. For this, independent Gaussian noise is commonly chosen, with its use so widespread that researchers typically provide no explicit justification for this choice. This noise model assumes ‘random’ latent factors affect the system in the ephemeral fashion resulting in unsystematic deviation of observables from their modelled counterparts. However, like the deterministically modelled parts of a system, these latent factors can have persistent effects on observables. Here, we use experimental data from dynamical systems drawn from cardiac physiology and electrochemistry to demonstrate that highly persistent differences between observations and modelled quantities can occur. Considering the case when persistent noise arises owing only to measurement imperfections, we use the Fisher information matrix to quantify how uncertainty in parameter estimates is artificially reduced when erroneously assuming independent noise. We present a workflow to diagnose persistent noise from model fits and describe how to remodel accounting for correlated errors.

Crosstalk between biochemical signalling network architecture and trafficking governs AMPAR dynamics in synaptic plasticity

Synaptic plasticity involves modification of both biochemical and structural components of neurons. Many studies have revealed that the change in the number density of the glutamatergic receptor AMPAR at the synapse is proportional to synaptic weight update; an increase in AMPAR corresponds to strengthening of synapses while a decrease in AMPAR density weakens synaptic connections. The dynamics of AMPAR are thought to be regulated by upstream signalling, primarily the calcium-CaMKII pathway, trafficking to and from the synapse, and influx from extrasynaptic sources. Previous work in the field of deterministic modelling of CaMKII dynamics has assumed bistable kinetics, while experiments and rule-based modelling have revealed that CaMKII dynamics can be either monostable or ultrasensitive. This raises the following question: how does the choice of model assumptions involving CaMKII dynamics influence AMPAR dynamics at the synapse? To answer this question, we have developed a set of models using compartmental ordinary differential equations to systematically investigate contributions of different signalling and trafficking variations, along with their coupled effects, on AMPAR dynamics at the synaptic site. We find that the properties of the model including network architecture describing different stability features of CaMKII and parameters that capture the endocytosis and exocytosis of AMPAR significantly affect the integration of fast upstream species by slower downstream species. Furthermore, we predict that the model outcome, as determined by bound AMPAR at the synaptic site, depends on (1) the choice of signalling model (bistable CaMKII or monostable CaMKII dynamics), (2) trafficking versus influx contributions and (3) frequency of stimulus. KEY POINTS: The density of AMPA receptors (AMPARs) at the postsynaptic density of the synapse provides a readout of synaptic plasticity, which involves crosstalk between complex biochemical signalling networks including CaMKII dynamics and trafficking pathways including exocytosis and endocytosis. Here we build a model that integrates CaMKII dynamics and AMPAR trafficking to explore this crosstalk. We compare different models of CaMKII that result in monostable or bistable kinetics and their impact on AMPAR dynamics. Our results show that AMPAR density depends on the coupling between aspects of biochemical signalling and trafficking. Specifically, assumptions regarding CaMKII dynamics and its stability features can alter AMPAR density at the synapse. Our model also predicts that the kinetics of trafficking versus influx of AMPAR from the extrasynaptic space can further impact AMPAR density. Thus, the contributions of both signalling and trafficking should be considered in computational models.

Using Kinetic Modelling to Infer Adaptations in Saccharomyces cerevisiae Carbohydrate Storage Metabolism to Dynamic Substrate Conditions

Microbial metabolism is strongly dependent on the environmental conditions. While these can be well controlled under laboratory conditions, large-scale bioreactors are characterized by inhomogeneities and consequently dynamic conditions for the organisms. How Saccharomyces cerevisiae response to frequent perturbations in industrial bioreactors is still not understood mechanistically. To study the adjustments to prolonged dynamic conditions, we used published repeated substrate perturbation regime experimental data, extended it with proteomic measurements and used both for modelling approaches. Multiple types of data were combined; including quantitative metabolome, ¹³C enrichment and flux quantification data. Kinetic metabolic modelling was applied to study the relevant intracellular metabolic response dynamics. An existing model of yeast central carbon metabolism was extended, and different subsets of enzymatic kinetic constants were estimated. A novel parameter estimation pipeline based on combinatorial enzyme selection supplemented by regularization was developed to identify and predict the minimum enzyme and parameter adjustments from steady-state to dynamic substrate conditions. This approach predicted proteomic changes in hexose transport and phosphorylation reactions, which were additionally confirmed by proteome measurements. Nevertheless, the modelling also hints at a yet unknown kinetic or regulation phenomenon. Some intracellular fluxes could not be reproduced by mechanistic rate laws, including hexose transport and intracellular trehalase activity during substrate perturbation cycles.

Geometric analysis enables biological insight from complex non-identifiable models using simple surrogates

An enduring challenge in computational biology is to balance data quality and quantity with model complexity. Tools such as identifiability analysis and information criterion have been developed to harmonise this juxtaposition, yet cannot always resolve the mismatch between available data and the granularity required in mathematical models to answer important biological questions. Often, it is only simple phenomenological models, such as the logistic and Gompertz growth models, that are identifiable from standard experimental measurements. To draw insights from complex, non-identifiable models that incorporate key biological mechanisms of interest, we study the geometry of a map in parameter space from the complex model to a simple, identifiable, surrogate model. By studying how non-identifiable parameters in the complex model quantitatively relate to identifiable parameters in surrogate, we introduce and exploit a layer of interpretation between the set of non-identifiable parameters and the goodness-of-fit metric or likelihood studied in typical identifiability analysis. We demonstrate our approach by analysing a hierarchy of mathematical models for multicellular tumour spheroid growth experiments. Typical data from tumour spheroid experiments are limited and noisy, and corresponding mathematical models are very often made arbitrarily complex. Our geometric approach is able to predict non-identifiabilities, classify non-identifiable parameter spaces into identifiable parameter combinations that relate to features in the data characterised by parameters in a surrogate model, and overall provide additional biological insight from complex non-identifiable models.

Open problems in mathematical biology

Biology is data-rich, and it is equally rich in concepts and hypotheses. Part of trying to understand biological processes and systems is therefore to confront our ideas and hypotheses with data using statistical methods to determine the extent to which our hypotheses agree with reality. But doing so in a systematic way is becoming increasingly challenging as our hypotheses become more detailed, and our data becomes more complex. Mathematical methods are therefore gaining in importance across the life- and biomedical sciences. Mathematical models allow us to test our understanding, make testable predictions about future behaviour, and gain insights into how we can control the behaviour of biological systems. It has been argued that mathematical methods can be of great benefit to biologists to make sense of data. But mathematics and mathematicians are set to benefit equally from considering the often bewildering complexity inherent to living systems. Here we present a small selection of open problems and challenges in mathematical biology. We have chosen these open problems because they are of both biological and mathematical interest.

Gradient-based parameter optimization method to determine membrane ionic current composition in human induced pluripotent stem cell-derived cardiomyocytes

Premature cardiac myocytes derived from human induced pluripotent stem cells (hiPSC-CMs) show heterogeneous action potentials (APs), probably due to different expression patterns of membrane ionic currents. We developed a method for determining expression patterns of functional channels in terms of whole-cell ionic conductance (G_x) using individual spontaneous AP configurations. It has been suggested that apparently identical AP configurations can be obtained using different sets of ionic currents in mathematical models of cardiac membrane excitation. If so, the inverse problem of G_x estimation might not be solved. We computationally tested the feasibility of the gradient-based optimization method. For a realistic examination, conventional 'cell-specific models' were prepared by superimposing the model output of AP on each experimental AP recorded by conventional manual adjustment of G_xs of the baseline model. G_xs of 4-6 major ionic currents of the 'cell-specific models' were randomized within a range of ± 5-15% and used as an initial parameter set for the gradient-based automatic G_xs recovery by decreasing the mean square error (MSE) between the target and model output. Plotting all data points of the MSE-G_x relationship during optimization revealed progressive convergence of the randomized population of G_xs to the original value of the cell-specific model with decreasing MSE. The absence of any other local minimum in the global search space was confirmed by mapping the MSE by randomizing G_xs over a range of 0.1-10 times the control. No additional local minimum MSE was obvious in the whole parameter space, in addition to the global minimum of MSE at the default model parameter.

Reformulating the susceptible-infectious-removed model in terms of the number of detected cases: well-posedness of the observational model

Compartmental models are popular in the mathematics of epidemiology for their simplicity and wide range of applications. Although they are typically solved as initial value problems for a system of ordinary differential equations, the observed data are typically akin to a boundary value-type problem: we observe some of the dependent variables at given times, but we do not know the initial conditions. In this paper, we reformulate the classical susceptible-infectious-recovered system in terms of the number of detected positive infected cases at different times to yield what we term the observational model. We then prove the existence and uniqueness of a solution to the boundary value problem associated with the observational model and present a numerical algorithm to approximate the solution. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.

pH dependencies of glycolytic enzymes of yeast under in vivo-like assay conditions

Under carbon source transitions, the intracellular pH of Saccharomyces cerevisiae is subject to change. Dynamics in pH modulate the activity of the glycolytic enzymes, resulting in a change in glycolytic flux and ultimately cell growth. To understand how pH affects the global behavior of glycolysis and ethanol fermentation, we measured the activity of the glycolytic and fermentative enzymes in S. cerevisiae under in vivo-like conditions at different pH. We demonstrate that glycolytic enzymes exhibit differential pH dependencies, and optima, in the pH range observed during carbon source transitions. The forward reaction of GAPDH shows the highest decrease in activity, 83%, during a simulated feast/famine regime upon glucose removal (cytosolic pH drop from 7.1 to 6.4). We complement our biochemical characterization of the glycolytic enzymes by fitting the V_max to the progression curves of product formation or decay over time. The fitting analysis shows that the observed changes in enzyme activities require changes in V_max , but changes in K_m cannot be excluded. Our study highlights the relevance of pH as a key player in metabolic regulation and provides a large set of quantitative data that can be explored to improve our understanding of metabolism in dynamic environments.

Discrete Biochemical Systems Theory

Almost every biomedical systems analysis requires early decisions regarding the choice of the most suitable representations to be used. De facto the most prevalent choice is a system of ordinary differential equations (ODEs). This framework is very popular because it is flexible and fairly easy to use. It is also supported by an enormous array of stand-alone programs for analysis, including many distinct numerical solvers that are implemented in the main programming languages. Having selected ODEs, the modeler must then choose a mathematical format for the equations. This selection is not trivial as nearly unlimited options exist and there is seldom objective guidance. The typical choices include ad hoc representations, default models like mass-action or Lotka-Volterra equations, and generic approximations. Within the realm of approximations, linear models are typically successful for analyses of engineered systems, but they are not as appropriate for biomedical phenomena, which often display nonlinear features such as saturation, threshold effects or limit cycle oscillations, and possibly even chaos. Power-law approximations are simple but overcome these limitations. They are the key ingredient of Biochemical Systems Theory (BST), which uses ODEs exclusively containing power-law representations for all processes within a model. BST models cover a vast repertoire of nonlinear responses and, at the same time, have structural properties that are advantageous for a wide range of analyses. Nonetheless, as all ODE models, the BST approach has limitations. In particular, it is not always straightforward to account for genuine discreteness, time delays, and stochastic processes. As a new option, we therefore propose here an alternative to BST in the form of discrete Biochemical Systems Theory (dBST). dBST models have the same generality and practicality as their BST-ODE counterparts, but they are readily implemented even in situations where ODEs struggle. As a case study, we illustrate dBST applied to the dynamics of the aryl hydrocarbon receptor (AhR), a signal transduction system that simultaneously involves time delays and stochasticity.

Parameter Estimation for Dynamical Systems Using a Deep Neural Network

The deep neural network (DNN) was applied for estimating a set of unknown parameters of a dynamical system whose measured data are given for a set of discrete time points. We developed a new vectorized algorithm that takes the number of unknowns (state variables) and number of parameters into consideration. The algorithm, first, trains the network to determine weights and biases. Next, the algorithm solves the systems of algebraic equations to estimate the parameters of the system. If the right hand side function of the system is smooth and the system have equal numbers of unknowns and parameters, the algorithm solves the algebraic equation at the discrete point where absolute error between the neural network solutions and the measured data is minimum. This improves the accuracy and reduces computational time. Several tests were carried out in linear and non-linear dynamical systems. Last, we showed that the DNN approach is more successful in terms of computational time as the number of hidden layers increases.

Parameter estimation and uncertainty quantification using information geometry

In this work, we: (i) review likelihood-based inference for parameter estimation and the construction of confidence regions; and (ii) explore the use of techniques from information geometry, including geodesic curves and Riemann scalar curvature, to supplement typical techniques for uncertainty quantification, such as Bayesian methods, profile likelihood, asymptotic analysis and bootstrapping. These techniques from information geometry provide data-independent insights into uncertainty and identifiability, and can be used to inform data collection decisions. All code used in this work to implement the inference and information geometry techniques is available on GitHub.

Puzzle Out Machine Learning Model-Explaining Disintegration Process in ODTs

Tablets are the most common dosage form of pharmaceutical products. While tablets represent the majority of marketed pharmaceutical products, there remain a significant number of patients who find it difficult to swallow conventional tablets. Such difficulties lead to reduced patient compliance. Orally disintegrating tablets (ODT), sometimes called oral dispersible tablets, are the dosage form of choice for patients with swallowing difficulties. ODTs are defined as a solid dosage form for rapid disintegration prior to swallowing. The disintegration time, therefore, is one of the most important and optimizable critical quality attributes (CQAs) for ODTs. Current strategies to optimize ODT disintegration times are based on a conventional trial-and-error method whereby a small number of samples are used as proxies for the compliance of whole batches. We present an alternative machine learning approach to optimize the disintegration time based on a wide variety of machine learning (ML) models through the H2O AutoML platform. ML models are presented with inputs from a database originally presented by Han et al., which was enhanced and curated to include chemical descriptors representing active pharmaceutical ingredient (API) characteristics. A deep learning model with a 10-fold cross-validation NRMSE of 8.1% and an R² of 0.84 was obtained. The critical parameters influencing the disintegration of the directly compressed ODTs were ascertained using the SHAP method to explain ML model predictions. A reusable, open-source tool, the ODT calculator, is now available at Heroku platform.

GAMES: A Dynamic Model Development Workflow for Rigorous Characterization of Synthetic Genetic Systems

Mathematical modeling is invaluable for advancing understanding and design of synthetic biological systems. However, the model development process is complicated and often unintuitive, requiring iteration on various computational tasks and comparisons with experimental data. Ad hoc model development can pose a barrier to reproduction and critical analysis of the development process itself, reducing the potential impact and inhibiting further model development and collaboration. To help practitioners manage these challenges, we introduce the Generation and Analysis of Models for Exploring Synthetic Systems (GAMES) workflow, which includes both automated and human-in-the-loop processes. We systematically consider the process of developing dynamic models, including model formulation, parameter estimation, parameter identifiability, experimental design, model reduction, model refinement, and model selection. We demonstrate the workflow with a case study on a chemically responsive transcription factor. The generalizable workflow presented in this tutorial can enable biologists to more readily build and analyze models for various applications.

Kinetic Modeling of Saccharomyces cerevisiae Central Carbon Metabolism: Achievements, Limitations, and Opportunities

Central carbon metabolism comprises the metabolic pathways in the cell that process nutrients into energy, building blocks and byproducts. To unravel the regulation of this network upon glucose perturbation, several metabolic models have been developed for the microorganism Saccharomyces cerevisiae. These dynamic representations have focused on glycolysis and answered multiple research questions, but no commonly applicable model has been presented. This review systematically evaluates the literature to describe the current advances, limitations, and opportunities. Different kinetic models have unraveled key kinetic glycolytic mechanisms. Nevertheless, some uncertainties regarding model topology and parameter values still limit the application to specific cases. Progressive improvements in experimental measurement technologies as well as advances in computational tools create new opportunities to further extend the model scale. Notably, models need to be made more complex to consider the multiple layers of glycolytic regulation and external physiological variables regulating the bioprocess, opening new possibilities for extrapolation and validation. Finally, the onset of new data representative of individual cells will cause these models to evolve from depicting an average cell in an industrial fermenter, to characterizing the heterogeneity of the population, opening new and unseen possibilities for industrial fermentation improvement.

Mathematical Modelling in Plant Synthetic Biology

Mathematical modelling techniques are integral to current research in plant synthetic biology. Modelling approaches can provide mechanistic understanding of a system, allowing predictions of behaviour and thus providing a tool to help design and analyse biological circuits. In this chapter, we provide an overview of mathematical modelling methods and their significance for plant synthetic biology. Starting with the basics of dynamics, we describe the process of constructing a model over both temporal and spatial scales and highlight crucial approaches, such as stochastic modelling and model-based design. Next, we focus on the model parameters and the techniques required in parameter analysis. We then describe the process of selecting a model based on tests and criteria and proceed to methods that allow closer analysis of the system's behaviour. Finally, we highlight the importance of uncertainty in modelling approaches and how to deal with a lack of knowledge, noisy data, and biological variability; all aspects that play a crucial role in the cooperation between the experimental and modelling components. Overall, this chapter aims to illustrate the importance of mathematical modelling in plant synthetic biology, providing an introduction for those researchers who are working with or working on modelling techniques.

Relationship Between Dimensionality and Convergence of Optimization Algorithms: A Comparison Between Data-Driven Normalization and Scaling Factor-Based Methods Using PEPSSBI

Ordinary differential equation models are used to represent intracellular signaling pathways in silico, aiding and guiding biological experiments to elucidate intracellular regulation. To construct such quantitative and predictive models of intracellular interactions, unknown parameters need to be estimated. Most of biological data are expressed in relative or arbitrary units, raising the question of how to compare model simulations with data. It has recently been shown that for models with large number of unknown parameters, fitting algorithms using a data-driven normalization of the simulations (DNS) performs best in terms of the convergence time and parameter identifiability. DNS approach compares model simulations and corresponding data both normalized by the same normalization procedure, without requiring additional parameters to be estimated, as necessary for widely used scaling factor-based methods. However, currently there is no parameter estimation software that directly supports DNS. In this chapter, we show how to apply DNS to dynamic models of systems and synthetic biology using PEPSSBI (Parameter Estimation Pipeline for Systems and Synthetic Biology). PEPSSBI is the first software that supports DNS, through algorithmically supported data normalization and objective function construction. PEPSSBI also supports model import using SBML and repeated parameter estimation runs executed in parallel either on a personal computer or a multi-CPU cluster.

Quantitative analysis of tumour spheroid structure

Tumour spheroids are common in vitro experimental models of avascular tumour growth. Compared with traditional two-dimensional culture, tumour spheroids more closely mimic the avascular tumour microenvironment where spatial differences in nutrient availability strongly influence growth. We show that spheroids initiated using significantly different numbers of cells grow to similar limiting sizes, suggesting that avascular tumours have a limiting structure; in agreement with untested predictions of classical mathematical models of tumour spheroids. We develop a novel mathematical and statistical framework to study the structure of tumour spheroids seeded from cells transduced with fluorescent cell cycle indicators, enabling us to discriminate between arrested and cycling cells and identify an arrested region. Our analysis shows that transient spheroid structure is independent of initial spheroid size, and the limiting structure can be independent of seeding density. Standard experimental protocols compare spheroid size as a function of time; however, our analysis suggests that comparing spheroid structure as a function of overall size produces results that are relatively insensitive to variability in spheroid size. Our experimental observations are made using two melanoma cell lines, but our modelling framework applies across a wide range of spheroid culture conditions and cell lines.

Learning Brain Dynamics With Coupled Low-Dimensional Nonlinear Oscillators and Deep Recurrent Networks

Many natural systems, especially biological ones, exhibit complex multivariate nonlinear dynamical behaviors that can be hard to capture by linear autoregressive models. On the other hand, generic nonlinear models such as deep recurrent neural networks often require large amounts of training data, not always available in domains such as brain imaging; also, they often lack interpretability. Domain knowledge about the types of dynamics typically observed in such systems, such as a certain type of dynamical systems models, could complement purely data-driven techniques by providing a good prior. In this work, we consider a class of ordinary differential equation (ODE) models known as van der Pol (VDP) oscil lators and evaluate their ability to capture a low-dimensional representation of neural activity measured by different brain imaging modalities, such as calcium imaging (CaI) and fMRI, in different living organisms: larval zebrafish, rat, and human. We develop a novel and efficient approach to the nontrivial problem of parameters estimation for a network of coupled dynamical systems from multivariate data and demonstrate that the resulting VDP models are both accurate and interpretable, as VDP's coupling matrix reveals anatomically meaningful excitatory and inhibitory interactions across different brain subsystems. VDP outperforms linear autoregressive models (VAR) in terms of both the data fit accuracy and the quality of insight provided by the coupling matrices and often tends to generalize better to unseen data when predicting future brain activity, being comparable to and sometimes better than the recurrent neural networks (LSTMs). Finally, we demonstrate that our (generative) VDP model can also serve as a data-augmentation tool leading to marked improvements in predictive accuracy of recurrent neural networks. Thus, our work contributes to both basic and applied dimensions of neuroimaging: gaining scientific insights and improving brain-based predictive models, an area of potentially high practical importance in clinical diagnosis and neurotechnology.

Valency of HER2 targeting antibodies influences tumor cell internalization and penetration

T cell Dependent Bispecific antibodies (TDBs) have been a major advancement in the the treatment of cancer that allow for improved targeting and efficacy for large molecule therapeutics. TDBs are comprised of one arm targeting a surface antigen on a cancer cell and another targeting an engaging surface antigen on a cytotoxic T cell. In order to impart this function, the antibody must be in a bispecific format as opposed to the more conventional bivalent format. Through in vitro and in vivo studies, we sought to determine the impact of changing antibody valency on solid tumor distribution and catabolism. A bivalent anti-HER2 antibody exhibited higher catabolism than its full-length monovalent binding counterpart in vivo by both invasive tissue harvesting and non-invasive SPECT-CT imaging despite similar systemic exposures for the two molecules. In order to determine what molecular factors drove in vivo distribution and uptake, we developed a mechanistic model for binding and catabolism of monovalent and bivalent HER2 antibodies in KPL4 cells. This model suggests that observed differences in cellular uptake of monovalent and bivalent antibodies are caused by the change in apparent affinity conferred by avidity as well as differences in internalization and degradation rates of receptor bound antibodies. To our knowledge, this is the first study to directly compare the targeting abilities of monovalent and bivalent full-length antibodies. These findings may inform diverse antibody therapeutic modalities including T cell redirecting therapies and drug delivery strategies relying upon receptor internalization.

Metabolic Modeling Combined With Machine Learning Integrates Longitudinal Data and Identifies the Origin of LXR-Induced Hepatic Steatosis

Temporal multi-omics data can provide information about the dynamics of disease development and therapeutic response. However, statistical analysis of high-dimensional time-series data is challenging. Here we develop a novel approach to model temporal metabolomic and transcriptomic data by combining machine learning with metabolic models. ADAPT (Analysis of Dynamic Adaptations in Parameter Trajectories) performs metabolic trajectory modeling by introducing time-dependent parameters in differential equation models of metabolic systems. ADAPT translates structural uncertainty in the model, such as missing information about regulation, into a parameter estimation problem that is solved by iterative learning. We have now extended ADAPT to include both metabolic and transcriptomic time-series data by introducing a regularization function in the learning algorithm. The ADAPT learning algorithm was (re)formulated as a multi-objective optimization problem in which the estimation of trajectories of metabolic parameters is constrained by the metabolite data and refined by gene expression data. ADAPT was applied to a model of hepatic lipid and plasma lipoprotein metabolism to predict metabolic adaptations that are induced upon pharmacological treatment of mice by a Liver X receptor (LXR) agonist. We investigated the excessive accumulation of triglycerides (TG) in the liver resulting in the development of hepatic steatosis. ADAPT predicted that hepatic TG accumulation after LXR activation originates for 80% from an increased influx of free fatty acids. The model also correctly estimated that TG was stored in the cytosol rather than transferred to nascent very-low density lipoproteins. Through model-based integration of temporal metabolic and gene expression data we discovered that increased free fatty acid influx instead of de novo lipogenesis is the main driver of LXR-induced hepatic steatosis. This study illustrates how ADAPT provides estimates for biomedically important parameters that cannot be measured directly, explaining (side-)effects of pharmacological treatment with LXR agonists.

Potential of Integrating Model-Based Design of Experiments Approaches and Process Analytical Technologies for Bioprocess Scale-Down

Typically, bioprocesses on an industrial scale are dynamic systems with a certain degree of variability, system inhomogeneities, and even population heterogeneities. Therefore, the scaling of such processes from laboratory to industrial scale and vice versa is not a trivial task. Traditional scale-down methodologies consider several technical parameters, so that systems on the laboratory scale tend to qualitatively reflect large-scale effects, but not the dynamic situation in an industrial bioreactor over the entire process, from the perspective of a cell. Supported by the enormous increase in computing power, the latest scientific focus is on the application of dynamic models, in combination with computational fluid dynamics to quantitatively describe cell behavior. These models allow the description of possible cellular lifelines which in turn can be used to derive a regime analysis for scale-down experiments. However, the approaches described so far, which were for a very few process examples, are very labor- and time-intensive and cannot be validated easily. In parallel, alternatives have been developed based on the description of the industrial process with hybrid process models, which describe a process mechanistically as far as possible in order to determine the essential process parameters with their respective variances. On-line analytical methods allow the characterization of population heterogeneity directly in the process. This detailed information from the industrial process can be used in laboratory screening systems to select relevant conditions in which the cell and process related parameters reflect the situation in the industrial scale. In our opinion, these technologies, which are available in research for modeling biological systems, in combination with process analytical techniques are so far developed that they can be implemented in industrial routines for faster development of new processes and optimization of existing ones.

Building kinetic models for metabolic engineering

Kinetic formalisms of metabolism link metabolic fluxes to enzyme levels, metabolite concentrations and their allosteric regulatory interactions. Though they require the identification of physiologically relevant values for numerous parameters, kinetic formalisms uniquely establish a mechanistic link across heterogeneous omics datasets and provide an overarching vantage point to effectively inform metabolic engineering strategies. Advances in computational power, gene annotation coverage, and formalism standardization have led to significant progress over the past few years. However, careful interpretation of model predictions, limited metabolic flux datasets, and assessment of parameter sensitivity remain as challenges. In this review we highlight fundamental considerations which influence model quality and prediction, advances in methodologies, and success stories of deploying kinetic models to guide metabolic engineering.

Recent advances in constraint and machine learning-based metabolic modeling by leveraging stoichiometric balances, thermodynamic feasibility and kinetic law formalisms

Understanding the governing principles behind organisms' metabolism and growth underpins their effective deployment as bioproduction chassis. A central objective of metabolic modeling is predicting how metabolism and growth are affected by both external environmental factors and internal genotypic perturbations. The fundamental concepts of reaction stoichiometry, thermodynamics, and mass action kinetics have emerged as the foundational principles of many modeling frameworks designed to describe how and why organisms allocate resources towards both growth and bioproduction. This review focuses on the latest algorithmic advancements that have integrated these foundational principles into increasingly sophisticated quantitative frameworks.

Walking Drosophila navigate complex plumes using stochastic decisions biased by the timing of odor encounters

How insects navigate complex odor plumes, where the location and timing of odor packets are uncertain, remains unclear. Here we imaged complex odor plumes simultaneously with freely-walking flies, quantifying how behavior is shaped by encounters with individual odor packets. We found that navigation was stochastic and did not rely on the continuous modulation of speed or orientation. Instead, flies turned stochastically with stereotyped saccades, whose direction was biased upwind by the timing of prior odor encounters, while the magnitude and rate of saccades remained constant. Further, flies used the timing of odor encounters to modulate the transition rates between walks and stops. In more regular environments, flies continuously modulate speed and orientation, even though encounters can still occur randomly due to animal motion. We find that in less predictable environments, where encounters are random in both space and time, walking flies navigate with random walks biased by encounter timing.

CaliPro: A Calibration Protocol That Utilizes Parameter Density Estimation to Explore Parameter Space and Calibrate Complex Biological Models

Introduction

Mathematical and computational modeling have a long history of uncovering mechanisms and making predictions for biological systems. However, to create a model that can provide relevant quantitative predictions, models must first be calibrated by recapitulating existing biological datasets from that system. Current calibration approaches may not be appropriate for complex biological models because: 1) many attempt to recapitulate only a single aspect of the experimental data (such as a median trend) or 2) Bayesian techniques require specification of parameter priors and likelihoods to experimental data that cannot always be confidently assigned. A new calibration protocol is needed to calibrate complex models when current approaches fall short.

Methods

Herein, we develop CaliPro, an iterative, model-agnostic calibration protocol that utilizes parameter density estimation to refine parameter space and calibrate to temporal biological datasets. An important aspect of CaliPro is the user-defined pass set definition, which specifies how the model might successfully recapitulate experimental data. We define the appropriate settings to use CaliPro.

Results

We illustrate the usefulness of CaliPro through four examples including predator-prey, infectious disease transmission, and immune response models. We show that CaliPro works well for both deterministic, continuous model structures as well as stochastic, discrete models and illustrate that CaliPro can work across diverse calibration goals.

Conclusions

We present CaliPro, a new method for calibrating complex biological models to a range of experimental outcomes. In addition to expediting calibration, CaliPro may be useful in already calibrated parameter spaces to target and isolate specific model behavior for further analysis.

Extended Robust Boolean Network of Budding Yeast Cell Cycle

Background:

How to explore the dynamics of transition probabilities between phases of budding yeast cell cycle (BYCC) network based on the dynamics of protein activities that control this network? How to identify the robust structure of protein interactions of BYCC Boolean network (BN)? Budding yeast allows scientists to put experiments into effect in order to discover the intracellular cell cycle regulating structures which are well simulated by mathematical modeling.

Methods:

We extended an available deterministic BN of proteins responsible for the cell cycle to a Markov chain model containing apoptosis besides G1, S, G2, M, and stationary G1. Using genetic algorithm (GA), we estimated the kinetic parameters of the extended BN model so that the subsequent transition probabilities derived using Markov chain model of cell states as normal cell cycle becomes the maximum while the structure of chemical interactions of extended BN of cell cycle becomes more stable.

Results:

Using kinetic parameters optimized by GA, the probability of the subsequent transitions between cell cycle phases is maximized. The relative basin size of stationary G1 increased from 86% to 96.48% while the number of attractors decreased from 7 in the original model to 5 in the extended one. Hence, an increase in the robustness of the system has been achieved.

Conclusion:

The structure of interacting proteins in cell cycle network affects its robustness and probabilities of transitions between different cell cycle phases. Markov chain and BN are good approaches to study the stability and dynamics of the cell cycle network.

Calibration of ionic and cellular cardiac electrophysiology models

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

It's about time: Analysing simplifying assumptions for modelling multi-step pathways in systems biology

Thoughtful use of simplifying assumptions is crucial to make systems biology models tractable while still representative of the underlying biology. A useful simplification can elucidate the core dynamics of a system. A poorly chosen assumption can, however, either render a model too complicated for making conclusions or it can prevent an otherwise accurate model from describing experimentally observed dynamics. Here, we perform a computational investigation of sequential multi-step pathway models that contain fewer pathway steps than the system they are designed to emulate. We demonstrate when such models will fail to reproduce data and how detrimental truncation of a pathway leads to detectable signatures in model dynamics and its optimised parameters. An alternative assumption is suggested for simplifying such pathways. Rather than assuming a truncated number of pathway steps, we propose to use the assumption that the rates of information propagation along the pathway is homogeneous and, instead, letting the length of the pathway be a free parameter. We first focus on linear pathways that are sequential and have first-order kinetics, and we show how this assumption results in a three-parameter model that consistently outperforms its truncated rival and a delay differential equation alternative in recapitulating observed dynamics. We then show how the proposed assumption allows for similarly terse and effective models of non-linear pathways. Our results provide a foundation for well-informed decision making during model simplifications.

Dynamical model fitting to a synthetic positive feedback circuit in E. coli

Applying the principles of engineering to Synthetic Biology relies on the development of robust and modular genetic components, as well as underlying quantitative dynamical models that closely predict their behaviour. This study looks at a simple positive feedback circuit built by placing filamentous phage secretin pIV under a phage shock promoter. A single-equation ordinary differential equation model is developed to closely replicate the behaviour of the circuit, and its response to inhibition by TetR. A stepwise approach is employed to fit the model's parameters to time-series data for the circuit. This approach allows the dissection of the role of different parameters and leads to the identification of dependencies and redundancies between parameters. The developed genetic circuit and associated model may be used as a building block for larger circuits with more complex dynamics, which require tight quantitative control or tuning.

A dual-parameter identification approach for data-based predictive modeling of hybrid gene regulatory network-growth kinetics in Pseudomonas putida mt-2

Data integration to model-based description of biological systems incorporating gene dynamics improves the performance of microbial systems. Bioprocess performance, typically predicted using empirical Monod-type models, is essential for a sustainable bioeconomy. To replace empirical models, we updated a hybrid gene regulatory network-growth kinetic model, predicting aromatic pollutants degradation and biomass growth in Pseudomonas putida mt-2. We modeled a complex biological system including extensive information to understand the role of the regulatory elements in toluene biodegradation and biomass growth. The updated model exhibited extra complications such as the existence of oscillations and discontinuities. As parameter estimation of complex biological models remains a key challenge, we used the updated model to present a dual-parameter identification approach (the 'dual approach') combining two independent methodologies. Approach I handled the complexity by incorporation of demonstrated biological knowledge in the model-development process and combination of global sensitivity analysis and optimisation. Approach II complemented Approach I handling multimodality, ill-conditioning and overfitting through regularisation estimation, global optimisation, and identifiability analysis. To systematically quantify the biological system, we used a vast amount of high-quality time-course data. The dual approach resulted in an accurately calibrated kinetic model (NRMSE: 0.17055) efficiently handling the additional model complexity. We tested model validation using three independent experimental data sets, achieving greater predictive power (NRMSE: 0.18776) than the individual approaches (NRMSE I: 0.25322, II: 0.25227) and increasing model robustness. These results demonstrated data-driven predictive modeling potentially leading to bioprocess' model-based control and optimisation.

Bridging the gap between complexity science and clinical practice by formalizing idiographic theories: a computational model of functional analysis

BACKGROUND

The past decades of research have seen an increase in statistical tools to explore the complex dynamics of mental health from patient data, yet the application of these tools in clinical practice remains uncommon. This is surprising, given that clinical reasoning, e.g., case conceptualizations, largely coincides with the dynamical system approach. We argue that the gap between statistical tools and clinical practice can partly be explained by the fact that current estimation techniques disregard theoretical and practical considerations relevant to psychotherapy. To address this issue, we propose that case conceptualizations should be formalized. We illustrate this approach by introducing a computational model of functional analysis, a framework commonly used by practitioners to formulate case conceptualizations and design patient-tailored treatment.

METHODS

We outline the general approach of formalizing idiographic theories, drawing on the example of a functional analysis for a patient suffering from panic disorder. We specified the system using a series of differential equations and simulated different scenarios; first, we simulated data without intervening in the system to examine the effects of avoidant coping on the development of panic symptomatic. Second, we formalized two interventions commonly used in cognitive behavioral therapy (CBT; exposure and cognitive reappraisal) and subsequently simulated their effects on the system.

RESULTS

The first simulation showed that the specified system could recover several aspects of the phenomenon (panic disorder), however, also showed some incongruency with the nature of panic attacks (e.g., rapid decreases were not observed). The second simulation study illustrated differential effects of CBT interventions for this patient. All tested interventions could decrease panic levels in the system.

CONCLUSIONS

Formalizing idiographic theories is promising in bridging the gap between complexity science and clinical practice and can help foster more rigorous scientific practices in psychotherapy, through enhancing theory development. More precise case conceptualizations could potentially improve intervention planning and treatment outcomes. We discuss applications in psychotherapy and future directions, amongst others barriers for systematic theory evaluation and extending the framework to incorporate interactions between individual systems, relevant for modeling social learning processes. With this report, we hope to stimulate future efforts in formalizing clinical frameworks.

Bridging the gap between complexity science and clinical practice by formalizing idiographic theories: a computational model of functional analysis

BACKGROUND

The past decades of research have seen an increase in statistical tools to explore the complex dynamics of mental health from patient data, yet the application of these tools in clinical practice remains uncommon. This is surprising, given that clinical reasoning, e.g., case conceptualizations, largely coincides with the dynamical system approach. We argue that the gap between statistical tools and clinical practice can partly be explained by the fact that current estimation techniques disregard theoretical and practical considerations relevant to psychotherapy. To address this issue, we propose that case conceptualizations should be formalized. We illustrate this approach by introducing a computational model of functional analysis, a framework commonly used by practitioners to formulate case conceptualizations and design patient-tailored treatment.

METHODS

We outline the general approach of formalizing idiographic theories, drawing on the example of a functional analysis for a patient suffering from panic disorder. We specified the system using a series of differential equations and simulated different scenarios; first, we simulated data without intervening in the system to examine the effects of avoidant coping on the development of panic symptomatic. Second, we formalized two interventions commonly used in cognitive behavioral therapy (CBT; exposure and cognitive reappraisal) and subsequently simulated their effects on the system.

RESULTS

The first simulation showed that the specified system could recover several aspects of the phenomenon (panic disorder), however, also showed some incongruency with the nature of panic attacks (e.g., rapid decreases were not observed). The second simulation study illustrated differential effects of CBT interventions for this patient. All tested interventions could decrease panic levels in the system.

CONCLUSIONS

Formalizing idiographic theories is promising in bridging the gap between complexity science and clinical practice and can help foster more rigorous scientific practices in psychotherapy, through enhancing theory development. More precise case conceptualizations could potentially improve intervention planning and treatment outcomes. We discuss applications in psychotherapy and future directions, amongst others barriers for systematic theory evaluation and extending the framework to incorporate interactions between individual systems, relevant for modeling social learning processes. With this report, we hope to stimulate future efforts in formalizing clinical frameworks.