Stability selection enables robust learning of differential equations from limited noisy data

We present a statistical learning framework for robust identification of differential equations from noisy spatio-temporal data. We address two issues that have so far limited the application of such methods, namely their robustness against noise and the need for manual parameter tuning, by proposing stability-based model selection to determine the level of regularization required for reproducible inference. This avoids manual parameter tuning and improves robustness against noise in the data. Our stability selection approach, termed PDE-STRIDE, can be combined with any sparsity-promoting regression method and provides an interpretable criterion for model component importance. We show that the particular combination of stability selection with the iterative hard-thresholding algorithm from compressed sensing provides a fast and robust framework for equation inference that outperforms previous approaches with respect to accuracy, amount of data required, and robustness. We illustrate the performance of PDE-STRIDE on a range of simulated benchmark problems, and we demonstrate the applicability of PDE-STRIDE on real-world data by considering purely data-driven inference of the protein interaction network for embryonic polarization in Caenorhabditis elegans. Using fluorescence microscopy images of C. elegans zygotes as input data, PDE-STRIDE is able to learn the molecular interactions of the proteins.

Face masking and COVID-19: potential effects of variolation on transmission dynamics

Face masks do not completely prevent transmission of respiratory infections, but masked individuals are likely to inhale fewer infectious particles. If smaller infectious doses tend to yield milder infections, yet ultimately induce similar levels of immunity, then masking could reduce the prevalence of severe disease even if the total number of infections is unaffected. It has been suggested that this effect of masking is analogous to the pre-vaccination practice of variolation for smallpox, whereby susceptible individuals were intentionally infected with small doses of live virus (and often acquired immunity without severe disease). We present a simple epidemiological model in which mask-induced variolation causes milder infections, potentially with lower transmission rate and/or different duration. We derive relationships between the effectiveness of mask-induced variolation and important epidemiological metrics (the basic reproduction number and initial epidemic growth rate, and the peak prevalence, attack rate and equilibrium prevalence of severe infections). We illustrate our results using parameter estimates for the original SARS-CoV-2 wild-type virus, as well as the Alpha, Delta and Omicron variants. Our results suggest that if variolation is a genuine side-effect of masking, then the importance of face masks as a tool for reducing healthcare burdens from COVID-19 may be under-appreciated.

On the propagation of nonlinear waves in the atmosphere

Starting from the general equations of fluid dynamics that describe the atmosphere, and using asymptotic methods, we present the derivation of the leading-order equations for nonlinear wave propagation in the troposphere. The only simplifying assumption is that the flow in the atmosphere exists in a thin shell over a sphere. The systematic approach adopted here enables us to find a consistent balance of terms describing the propagation, and to identify the temperature and pressure gradients that drive the motion, as well as the heat sources required. This produces a new nonlinear propagation equation that is then examined in some detail. With the morning glory in mind, we construct a few exact solutions, which, separately, describe breezes, bores and oscillatory motion.

On the propagation of waves in the atmosphere

The leading-order equations governing the unsteady dynamics of large-scale atmospheric motions are derived, via a systematic asymptotic approach based on the thin-shell approximation applied to the ellipsoidal model of the Earth’s geoid. We present some solutions of this single set of equations that capture properties of specific atmospheric flows, using field data to choose models for the heat sources that drive the motion. In particular, we describe standing-waves solutions, waves propagating towards the Equator, equatorially trapped waves and we discuss the African Easterly Jet/Waves. This work aims to show the benefits of a systematic analysis based on the governing equations of fluid dynamics.

A non-catalytic herpesviral protein reconfigures ERK-RSK signaling by targeting kinase docking systems in the host

The Kaposi's sarcoma associated herpesvirus protein ORF45 binds the extracellular signal-regulated kinase (ERK) and the p90 Ribosomal S6 kinase (RSK). ORF45 was shown to be a kinase activator in cells but a kinase inhibitor in vitro, and its effects on the ERK-RSK complex are unknown. Here, we demonstrate that ORF45 binds ERK and RSK using optimized linear binding motifs. The crystal structure of the ORF45-ERK2 complex shows how kinase docking motifs recognize the activated form of ERK. The crystal structure of the ORF45-RSK2 complex reveals an AGC kinase docking system, for which we provide evidence that it is functional in the host. We find that ORF45 manipulates ERK-RSK signaling by favoring the formation of a complex, in which activated kinases are better protected from phosphatases and docking motif-independent RSK substrate phosphorylation is selectively up-regulated. As such, our data suggest that ORF45 interferes with the natural design of kinase docking systems in the host.

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

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

Calibrating COVID-19 susceptible-exposed-infected-removed models with time-varying effectivecontact rates

We describe the population-based susceptible-exposed-infected-removed (SEIR) model developed by the Irish Epidemiological Modelling Advisory Group (IEMAG), which advises the Irish government on COVID-19 responses. The model assumes a time-varying effective contact rate (equivalently, a time-varying reproduction number) to model the effect of non-pharmaceutical interventions. A crucial technical challenge in applying such models is their accurate calibration to observed data, e.g. to the daily number of confirmed new cases, as the history of the disease strongly affects predictions of future scenarios. We demonstrate an approach based on inversion of the SEIR equations in conjunction with statistical modelling and spline-fitting of the data to produce a robust methodology for calibration of a wide class of models of this type. This article is part of the theme issue 'Data science approaches to infectious disease surveillance'.

Cubature Kalman Filter Based Training of Hybrid Differential Equation Recurrent Neural Network Physiological Dynamic Models

Modeling biological dynamical systems is challenging due to the interdependence of different system components, some of which are not fully understood. To fill existing gaps in our ability to mechanistically model physiological systems, we propose to combine neural networks with physics-based models. Specifically, we demonstrate how we can approximate missing ordinary differential equations (ODEs) coupled with known ODEs using Bayesian filtering techniques to train the model parameters and simultaneously estimate dynamic state variables. As a study case we leverage a well-understood model for blood circulation in the human retina and replace one of its core ODEs with a neural network approximation, representing the case where we have incomplete knowledge of the physiological state dynamics. Results demonstrate that state dynamics corresponding to the missing ODEs can be approximated well using a neural network trained using a recursive Bayesian filtering approach in a fashion coupled with the known state dynamic differential equations. This demonstrates that dynamics and impact of missing state variables can be captured through joint state estimation and model parameter estimation within a recursive Bayesian state estimation (RBSE) framework. Results also indicate that this RBSE approach to training the NN parameters yields better outcomes (measurement/state estimation accuracy) than training the neural network with backpropagation through time in the same setting.

Falkner-Skan Flow with Stream-Wise Pressure Gradient and Transfer of Mass over a Dynamic Wall

In this work, an important model in fluid dynamics is analyzed by a new hybrid neurocomputing algorithm. We have considered the Falkner-Skan (FS) with the stream-wise pressure gradient transfer of mass over a dynamic wall. To analyze the boundary flow of the FS model, we have utilized the global search characteristic of a recently developed heuristic, the Sine Cosine Algorithm (SCA), and the local search characteristic of Sequential Quadratic Programming (SQP). Artificial neural network (ANN) architecture is utilized to construct a series solution of the mathematical model. We have called our technique the ANN-SCA-SQP algorithm. The dynamic of the FS system is observed by varying stream-wise pressure gradient mass transfer and dynamic wall. To validate the effectiveness of ANN-SCA-SQP algorithm, our solutions are compared with state-of-the-art reference solutions. We have repeated a hundred experiments to establish the robustness of our approach. Our experimental outcome validates the superiority of the ANN-SCA-SQP algorithm.

Local Stability of McKean-Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies

A family of heterogeneous mean-field systems with jumps is analyzed. These systems are constructed as a Gibbs measure on block graphs. When the total number of particles goes to infinity, the law of large numbers is shown to hold in a multi-class context, resulting in the weak convergence of the empirical vector towards the solution of a McKean–Vlasov system of equations. We then investigate the local stability of the limiting McKean–Vlasov system through the construction of a local Lyapunov function. We first compute the limit of adequately scaled relative entropy functions associated with the explicit stationary distribution of the N-particles system. Using a Laplace principle for empirical vectors, we show that the limit takes an explicit form. Then we demonstrate that this limit satisfies a descent property, which, combined with some mild assumptions shows that it is indeed a local Lyapunov function.

A minimal model for gene expression dynamics of bacterial type II toxin-antitoxin systems

Toxin-antitoxin (TA) modules are part of most bacteria's regulatory machinery for stress responses and general aspects of their physiology. Due to the interplay of a long-lived toxin with a short-lived antitoxin, TA modules have also become systems of interest for mathematical modelling. Here we resort to previous modelling efforts and extract from these a minimal model of type II TA system dynamics on a timescale of hours, which can be used to describe time courses derived from gene expression data of TA pairs. We show that this model provides a good quantitative description of TA dynamics for the 11 TA pairs under investigation here, while simpler models do not. Our study brings together aspects of Biophysics with its focus on mathematical modelling and Computational Systems Biology with its focus on the quantitative interpretation of 'omics' data. This mechanistic model serves as a generic transformation of time course information into kinetic parameters. The resulting parameter vector can, in turn, be mechanistically interpreted. We expect that TA pairs with similar mechanisms are characterized by similar vectors of kinetic parameters, allowing us to hypothesize on the mode of action for TA pairs still under discussion.

Data Management and Modeling in Plant Biology

The study of plant-environment interactions is a multidisciplinary research field. With the emergence of quantitative large-scale and high-throughput techniques, amount and dimensionality of experimental data have strongly increased. Appropriate strategies for data storage, management, and evaluation are needed to make efficient use of experimental findings. Computational approaches of data mining are essential for deriving statistical trends and signatures contained in data matrices. Although, current biology is challenged by high data dimensionality in general, this is particularly true for plant biology. Plants as sessile organisms have to cope with environmental fluctuations. This typically results in strong dynamics of metabolite and protein concentrations which are often challenging to quantify. Summarizing experimental output results in complex data arrays, which need computational statistics and numerical methods for building quantitative models. Experimental findings need to be combined by computational models to gain a mechanistic understanding of plant metabolism. For this, bioinformatics and mathematics need to be combined with experimental setups in physiology, biochemistry, and molecular biology. This review presents and discusses concepts at the interface of experiment and computation, which are likely to shape current and future plant biology. Finally, this interface is discussed with regard to its capabilities and limitations to develop a quantitative model of plant-environment interactions.

Modeling the effects of EMT-immune dynamics on carcinoma disease progression

During progression from carcinoma in situ to an invasive tumor, the immune system is engaged in complex sets of interactions with various tumor cells. Tumor cell plasticity alters disease trajectories via epithelial-to-mesenchymal transition (EMT). Several of the same pathways that regulate EMT are involved in tumor-immune interactions, yet little is known about the mechanisms and consequences of crosstalk between these regulatory processes. Here we introduce a multiscale evolutionary model to describe tumor-immune-EMT interactions and their impact on epithelial cancer progression from in situ to invasive disease. Through simulation of patient cohorts in silico, the model predicts that a controllable region maximizes invasion-free survival. This controllable region depends on properties of the mesenchymal tumor cell phenotype: its growth rate and its immune-evasiveness. In light of the model predictions, we analyze EMT-inflammation-associated data from The Cancer Genome Atlas, and find that association with EMT worsens invasion-free survival probabilities. This result supports the predictions of the model, and leads to the identification of genes that influence outcomes in bladder and uterine cancer, including FGF pathway members. These results suggest new means to delay disease progression, and demonstrate the importance of studying cancer-immune interactions in light of EMT.

Neural Network Differential Equations For Ion Channel Modelling

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

Evolution of opposing regulatory interactions underlies the emergence of eukaryotic cell cycle checkpoints

In eukaryotes the entry into mitosis is initiated by activation of cyclin-dependent kinases (CDKs), which in turn activate a large number of protein kinases to induce all mitotic processes. The general view is that kinases are active in mitosis and phosphatases turn them off in interphase. Kinases activate each other by cross- and self-phosphorylation, while phosphatases remove these phosphate groups to inactivate kinases. Crucial exceptions to this general rule are the interphase kinase Wee1 and the mitotic phosphatase Cdc25. Together they directly control CDK in an opposite way of the general rule of mitotic phosphorylation and interphase dephosphorylation. Here we investigate why this opposite system emerged and got fixed in almost all eukaryotes. Our results show that this reversed action of a kinase-phosphatase pair, Wee1 and Cdc25, on CDK is particularly suited to establish a stable G2 phase and to add checkpoints to the cell cycle. We show that all these regulators appeared together in LECA (Last Eukaryote Common Ancestor) and co-evolved in eukaryotes, suggesting that this twist in kinase-phosphatase regulation was a crucial step happening at the emergence of eukaryotes.

Dynamical Modeling of the Core Gene Network Controlling Transition to Flowering in Pisum sativum

Transition to flowering is an important stage of plant development. Many regulatory modules that control floral transition are conservative across plants. This process is best studied for the model plant Arabidopsis thaliana. The homologues of Arabidopsis genes responsible for the flowering initiation in legumes have been identified, and available data on their expression provide a good basis for gene network modeling. In this study, we developed several dynamical models of a gene network controlling transition to flowering in pea (Pisum sativum) using two different approaches. We used differential equations for modeling a previously proposed gene regulation scheme of floral initiation in pea and tested possible alternative hypothesis about some regulations. As the second approach, we applied neural networks to infer interactions between genes in the network directly from gene expression data. All models were verified on previously published experimental data on the dynamic expression of the main genes in the wild type and in three mutant genotypes. Based on modeling results, we made conclusions about the functionality of the previously proposed interactions in the gene network and about the influence of different growing conditions on the network architecture. It was shown that regulation of the PIM, FTa1, and FTc genes in pea does not correspond to the previously proposed hypotheses. The modeling suggests that short- and long-day growing conditions are characterized by different gene network architectures. Overall, the results obtained can be used to plan new experiments and create more accurate models to study the flowering initiation in pea and, in a broader context, in legumes.

Learning differential equation models from stochastic agent-based model simulations

Agent-based models provide a flexible framework that is frequently used for modelling many biological systems, including cell migration, molecular dynamics, ecology and epidemiology. Analysis of the model dynamics can be challenging due to their inherent stochasticity and heavy computational requirements. Common approaches to the analysis of agent-based models include extensive Monte Carlo simulation of the model or the derivation of coarse-grained differential equation models to predict the expected or averaged output from the agent-based model. Both of these approaches have limitations, however, as extensive computation of complex agent-based models may be infeasible, and coarse-grained differential equation models can fail to accurately describe model dynamics in certain parameter regimes. We propose that methods from the equation learning field provide a promising, novel and unifying approach for agent-based model analysis. Equation learning is a recent field of research from data science that aims to infer differential equation models directly from data. We use this tutorial to review how methods from equation learning can be used to learn differential equation models from agent-based model simulations. We demonstrate that this framework is easy to use, requires few model simulations, and accurately predicts model dynamics in parameter regions where coarse-grained differential equation models fail to do so. We highlight these advantages through several case studies involving two agent-based models that are broadly applicable to biological phenomena: a birth-death-migration model commonly used to explore cell biology experiments and a susceptible-infected-recovered model of infectious disease spread.

Optimal control of the COVID-19 pandemic: controlled sanitary deconfinement in Portugal

The COVID-19 pandemic has forced policy makers to decree urgent confinements to stop a rapid and massive contagion. However, after that stage, societies are being forced to find an equilibrium between the need to reduce contagion rates and the need to reopen their economies. The experience hitherto lived has provided data on the evolution of the pandemic, in particular the population dynamics as a result of the public health measures enacted. This allows the formulation of forecasting mathematical models to anticipate the consequences of political decisions. Here we propose a model to do so and apply it to the case of Portugal. With a mathematical deterministic model, described by a system of ordinary differential equations, we fit the real evolution of COVID-19 in this country. After identification of the population readiness to follow social restrictions, by analyzing the social media, we incorporate this effect in a version of the model that allow us to check different scenarios. This is realized by considering a Monte Carlo discrete version of the previous model coupled via a complex network. Then, we apply optimal control theory to maximize the number of people returning to "normal life" and minimizing the number of active infected individuals with minimal economical costs while warranting a low level of hospitalizations. This work allows testing various scenarios of pandemic management (closure of sectors of the economy, partial/total compliance with protection measures by citizens, number of beds in intensive care units, etc.), ensuring the responsiveness of the health system, thus being a public health decision support tool.

Benchmarking of numerical integration methods for ODE models of biological systems

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

Human Movement Representation on Multivariate Time Series for Recognition of Professional Gestures and Forecasting Their Trajectories

Human-centered artificial intelligence is increasingly deployed in professional workplaces in Industry 4.0 to address various challenges related to the collaboration between the operators and the machines, the augmentation of their capabilities, or the improvement of the quality of their work and life in general. Intelligent systems and autonomous machines need to continuously recognize and follow the professional actions and gestures of the operators in order to collaborate with them and anticipate their trajectories for avoiding potential collisions and accidents. Nevertheless, the recognition of patterns of professional gestures is a very challenging task for both research and the industry. There are various types of human movements that the intelligent systems need to perceive, for example, gestural commands to machines and professional actions with or without the use of tools. Moreover, the interclass and intraclass spatiotemporal variances together with the very limited access to annotated human motion data constitute a major research challenge. In this paper, we introduce the Gesture Operational Model, which describes how gestures are performed based on assumptions that focus on the dynamic association of body entities, their synergies, and their serial and non-serial mediations, as well as their transitioning over time from one state to another. Then, the assumptions of the Gesture Operational Model are translated into a simultaneous equation system for each body entity through State-Space modeling. The coefficients of the equation are computed using the Maximum Likelihood Estimation method. The simulation of the model generates a confidence-bounding box for every entity that describes the tolerance of its spatial variance over time. The contribution of our approach is demonstrated for both recognizing gestures and forecasting human motion trajectories. In recognition, it is combined with continuous Hidden Markov Models to boost the recognition accuracy when the likelihoods are not confident. In forecasting, a motion trajectory can be estimated by taking as minimum input two observations only. The performance of the algorithm has been evaluated using four industrial datasets that contain gestures and actions from a TV assembly line, the glassblowing industry, the gestural commands to Automated Guided Vehicles as well as the Human-Robot Collaboration in the automotive assembly lines. The hybrid approach State-Space and HMMs outperforms standard continuous HMMs and a 3DCNN-based end-to-end deep architecture.

Model-based analysis on social acceptability and feasibility of a focused protection strategy against the COVID-19 pandemic

This paper studies the social acceptability and feasibility of a focused protection strategy against coronavirus disease 2019 (COVID-19). We propose a control scheme to develop herd immunity while satisfying the following two basic requirements for a viable policy option. The first requirement is social acceptability: the overall deaths should be minimized for social acceptance. The second is feasibility: the healthcare system should not be overwhelmed to avoid various adverse effects. To exploit the fact that the disease severity increases considerably with age and comorbidities, we assume that some focused protection measures for those high-risk individuals are implemented and the disease does not spread within the high-risk population. Because the protected population has higher severity ratios than the unprotected population by definition, the protective measure can substantially reduce mortality in the whole population and also avoid the collapse of the healthcare system. Based on a simple susceptible-infected-recovered model, social acceptability and feasibility of the proposed strategy are summarized into two easily computable conditions. The proposed framework can be applied to various populations for studying the viability of herd immunity strategies against COVID-19. For Japan, herd immunity may be developed by the proposed scheme if [Formula: see text] and the severity rates of the disease are 1/10 times smaller than the previously reported value, although as high mortality as seasonal influenza is expected.

Optimal experimental design for mathematical models of haematopoiesis

The haematopoietic system has a highly regulated and complex structure in which cells are organized to successfully create and maintain new blood cells. It is known that feedback regulation is crucial to tightly control this system, but the specific mechanisms by which control is exerted are not completely understood. In this work, we aim to uncover the underlying mechanisms in haematopoiesis by conducting perturbation experiments, where animal subjects are exposed to an external agent in order to observe the system response and evolution. We have developed a novel Bayesian hierarchical framework for optimal design of perturbation experiments and proper analysis of the data collected. We use a deterministic model that accounts for feedback and feedforward regulation on cell division rates and self-renewal probabilities. A significant obstacle is that the experimental data are not longitudinal, rather each data point corresponds to a different animal. We overcome this difficulty by modelling the unobserved cellular levels as latent variables. We then use principles of Bayesian experimental design to optimally distribute time points at which the haematopoietic cells are quantified. We evaluate our approach using synthetic and real experimental data and show that an optimal design can lead to better estimates of model parameters.

Design of efficacious somatic cell genome editing strategies for recessive and polygenic diseases

Compound heterozygous recessive or polygenic diseases could be addressed through gene correction of multiple alleles. However, targeting of multiple alleles using genome editors could lead to mixed genotypes and adverse events that amplify during tissue morphogenesis. Here we demonstrate that Cas9-ribonucleoprotein-based genome editors can correct two distinct mutant alleles within a single human cell precisely. Gene-corrected cells in an induced pluripotent stem cell model of Pompe disease expressed the corrected transcript from both corrected alleles, leading to enzymatic cross-correction of diseased cells. Using a quantitative in silico model for the in vivo delivery of genome editors into the developing human infant liver, we identify progenitor targeting, delivery efficiencies, and suppression of imprecise editing outcomes at the on-target site as key design parameters that control the efficacy of various therapeutic strategies. This work establishes that precise gene editing to correct multiple distinct gene variants could be highly efficacious if designed appropriately.

Tracing the footsteps of autophagy in computational biology

Autophagy plays a crucial role in maintaining cellular homeostasis through the degradation of unwanted materials like damaged mitochondria and misfolded proteins. However, the contribution of autophagy toward a healthy cell environment is not only limited to the cleaning process. It also assists in protein synthesis when the system lacks the amino acids’ inflow from the extracellular environment due to diet consumptions. Reduction in the autophagy process is associated with diseases like cancer, diabetes, non-alcoholic steatohepatitis, etc., while uncontrolled autophagy may facilitate cell death. We need a better understanding of the autophagy processes and their regulatory mechanisms at various levels (molecules, cells, tissues). This demands a thorough understanding of the system with the help of mathematical and computational tools. The present review illuminates how systems biology approaches are being used for the study of the autophagy process. A comprehensive insight is provided on the application of computational methods involving mathematical modeling and network analysis in the autophagy process. Various mathematical models based on the system of differential equations for studying autophagy are covered here. We have also highlighted the significance of network analysis and machine learning in capturing the core regulatory machinery governing the autophagy process. We explored the available autophagic databases and related resources along with their attributes that are useful in investigating autophagy through computational methods. We conclude the article addressing the potential future perspective in this area, which might provide a more in-depth insight into the dynamics of autophagy.

ROS networks: designs, aging, Parkinson's disease and precision therapies

How the network around ROS protects against oxidative stress and Parkinson's disease (PD), and how processes at the minutes timescale cause disease and aging after decades, remains enigmatic. Challenging whether the ROS network is as complex as it seems, we built a fairly comprehensive version thereof which we disentangled into a hierarchy of only five simpler subnetworks each delivering one type of robustness. The comprehensive dynamic model described in vitro data sets from two independent laboratories. Notwithstanding its five-fold robustness, it exhibited a relatively sudden breakdown, after some 80 years of virtually steady performance: it predicted aging. PD-related conditions such as lack of DJ-1 protein or increased α-synuclein accelerated the collapse, while antioxidants or caffeine retarded it. Introducing a new concept (aging-time-control coefficient), we found that as many as 25 out of 57 molecular processes controlled aging. We identified new targets for "life-extending interventions": mitochondrial synthesis, KEAP1 degradation, and p62 metabolism.