Nonlinear delay differential equations and their application to modeling biological network motifs

Temporal multiscale modeling of biochemical regulatory networks: Calcium-regulated hepatocyte lipid and glucose metabolism

Hepatocyte lipid and glucose metabolism is regulated not only by major hormones like insulin and glucagon but also by many other factors, including calcium ions. Recently, mitochondria-associated membrane (MAM) dysfunction combined with incorrect IP3-receptor regulation has been shown to result in abnormal calcium signaling in hepatocytes. This dysfunction could further lead to hepatic metabolism pathology. However, the exact contribution of MAM dysfunction, incorrect IP3-receptor regulation and insulin resistance to the calcium-insulin-glucagon interplay is not understood yet. In this work, we analyze the role of abnormal calcium signaling and insulin dysfunction in hepatocytes by proposing a model of hepatocyte metabolic regulatory network with a detailed focus on the model construction details besides the biological aspect. In this work, we analyze the role of abnormal calcium signaling and insulin dysfunction in hepatocytes by proposing a model of hepatocyte metabolic regulatory network. We focus on the model construction details, model validation, and predictions. We describe the dynamic regulation of signaling processes by sigmoid Hill function. In particular, we study the effect of both the Hill function slope and the distance between Hill function extremes on metabolic processes in hepatocytes as a model of nonspecific insulin dysfunction. We also address the significant time difference between characteristic time of glucose hepatic processing and a typical calcium oscillation period in hepatocytes. Our modeling results show that calcium signaling dysfunction results in an abnormal increase in postprandial glucose levels, an abnormal glucose decrease in fasting, and a decreased amount of stored glycogen. An insulin dysfunction of glucose phosphorylation, glucose dephosphorylation, and glycogen breakdown also cause a noticeable effect. We also get some insight into the so-called hepatic insulin resistance paradox, confirming the hypothesis regarding indirect insulin action on hepatocytes via dysfunctional adipocyte lipolysis.

From biological data to oscillator models using SINDy

Periodic changes in the concentration or activity of different molecules regulate vital cellular processes such as cell division and circadian rhythms. Developing mathematical models is essential to better understand the mechanisms underlying these oscillations. Recent data-driven methods like SINDy have fundamentally changed model identification, yet their application to experimental biological data remains limited. This study investigates SINDy's constraints by directly applying it to biological oscillatory data. We identify insufficient resolution, noise, dimensionality, and limited prior knowledge as primary limitations. Using various generic oscillator models of different complexity and/or dimensionality, we systematically analyze these factors. We then propose a comprehensive guide for inferring models from biological data, addressing these challenges step by step. Our approach is validated using glycolytic oscillation data from yeast.

Modulating autophagy to treat diseases: A revisited review on in silico methods

BACKGROUND

Autophagy refers to the conserved cellular catabolic process relevant to lysosome activity and plays a vital role in maintaining the dynamic equilibrium of intracellular matter by degrading harmful and abnormally accumulated cellular components. Accumulating evidence has recently revealed that dysregulation of autophagy by genetic and exogenous interventions may disrupt cellular homeostasis in human diseases. In silico approaches as powerful aids to experiments have also been extensively reported to play their critical roles in the storage, prediction, and analysis of massive amounts of experimental data. Thus, modulating autophagy to treat diseases by in silico methods would be anticipated.

AIM OF REVIEW

Here, we focus on summarizing the updated in silico approaches including databases, systems biology network approaches, omics-based analyses, mathematical models, and artificial intelligence (AI) methods that sought to modulate autophagy for potential therapeutic purposes, which will provide a new insight into more promising therapeutic strategies.

KEY SCIENTIFIC CONCEPTS OF REVIEW

Autophagy-related databases are the data basis of the in silico method, storing a large amount of information about DNA, RNA, proteins, small molecules and diseases. The systems biology approach is a method to systematically study the interrelationships among biological processes including autophagy from a macroscopic perspective. Omics-based analyses are based on high-throughput data to analyze gene expression at different levels of biological processes involving autophagy. mathematical models are visualization methods to describe the dynamic process of autophagy, and its accuracy is related to the selection of parameters. AI methods use big data related to autophagy to predict autophagy targets, design targeted small molecules, and classify diverse human diseases for potential therapeutic applications.

The dynamic-process characterization and prediction of synthetic gene circuits by dynamic delay model

Differential equation models are widely used to describe genetic regulations, predict multicomponent regulatory circuits, and provide quantitative insights. However, it is still challenging to quantitatively link the dynamic behaviors with measured parameters in synthetic circuits. Here, we propose a dynamic delay model (DDM) which includes two simple parts: the dynamic determining part and the doses-related steady-state-determining part. The dynamic determining part is usually supposed as the delay time but without a clear formula. For the first time, we give the detail formula of the dynamic determining function and provide a method for measuring all parameters of synthetic elements (include 8 activators and 5 repressors) by microfluidic system. Three synthetic circuits were built to show that the DDM can notably improve the prediction accuracy and can be used in various synthetic biology applications.

Nonlinear differential equations and their application to evaluating the integrated impacts of multiple parameters on the biochemical safety of drinking water

The present study aimed to narrow such gaps by applying nonlinear differential equations to biostability in drinking water. Biostability results from the integrated dynamics of nutrients and disinfectants. The linear dynamics of biostability have been well studied, while there remain knowledge gaps concerning nonlinear effects. The nonlinear effects are explained by phase plots for specific scenarios in a drinking water system, including continuous nutrient release, flush exchange with the adjacent environment, periodic pulse disinfection, and periodic biofilm development. The main conclusions are, (1) The correlations between the microbial community and nutrients go through phases of linear, nonlinear, and chaotic dynamics. Disinfection breaks the chaotic phase and returns the system to the linear phase, increasing the microbial growth potential. (2) Post-disinfection after multiple microbial peaks produced via metabolism can increase disinfection efficiency and decrease the risks associated with disinfectant byproduct risks. This can provide guidelines for optimizing the disinfection strategy, according to the long-term water safety target or a short management. Limited disinfection and ultimate disinfection may be more effective and have low chemical risk, facing longer stagnant conditions. (3) Periodic biofilm formation and biofilm detachment increase the possibility of uncertainty in the chaotic phase. For future study, nonlinear differential equation models can accordingly be applied at the molecular and ecological levels to further explore more nonlinear regulation mechanisms.

Chaos in gene regulatory networks: Effects of time delays and interaction structure

In biological system models, gene expression levels are typically described by regulatory feedback mechanisms. Many studies of gene network models focus on dynamical interactions between components, but often overlook time delays. Here we present an extended model for gene regulatory networks with time delayed negative feedback, which is described by delay differential equations. We analyze nonlinear properties of the model in terms of chaos and compare the conditions with the benchmark homogeneous gene regulatory network model. Chaotic dynamics depend strongly on the inclusion of time delays, but the minimum motifs that show chaos differ when both original and extended models are considered. Our results suggest that, for a particular higher order extension of the gene network, it is possible to observe chaotic dynamics in a two-gene system without adding any self-inhibition. This finding can be explained as a result of modification of the original benchmark model induced by previously unmodeled dynamics. We argue that the inclusion of additional parameters in regulatory gene circuit models substantially enhances the likelihood of observing non-periodic dynamics.

Plasmodium RON11 triggers biogenesis of the merozoite rhoptry pair and is essential for erythrocyte invasion

Malaria is a global and deadly human disease caused by the apicomplexan parasites of the genus Plasmodium. Parasite proliferation within human red blood cells (RBC) is associated with the clinical manifestations of the disease. This asexual expansion within human RBCs, begins with the invasion of RBCs by P. falciparum, which is mediated by the secretion of effectors from two specialized club-shaped secretory organelles in merozoite-stage parasites known as rhoptries. We investigated the function of the Rhoptry Neck Protein 11 (RON11), which contains seven transmembrane domains and calcium-binding EF-hand domains. We generated conditional mutants of the P. falciparum RON11. Knockdown of RON11 inhibits parasite growth by preventing merozoite invasion. The loss of RON11 did not lead to any defects in processing of rhoptry proteins but instead led to a decrease in the amount of rhoptry proteins. We utilized ultrastructure expansion microscopy (U-ExM) to determine the effect of RON11 knockdown on rhoptry biogenesis. Surprisingly, in the absence of RON11, fully developed merozoites had only one rhoptry each. The single rhoptry in RON11 deficient merozoites were morphologically typical with a bulb and a neck oriented into the apical polar ring. Moreover, rhoptry proteins are trafficked accurately to the single rhoptry in RON11 deficient parasites. These data show that in the absence of RON11, the first rhoptry is generated during schizogony but upon the start of cytokinesis, the second rhoptry never forms. Interestingly, these single-rhoptry merozoites were able to attach to host RBCs but are unable to invade RBCs. Instead, RON11 deficient merozoites continue to engage with RBC for prolonged periods eventually resulting in echinocytosis, a result of secreting the contents from the single rhoptry into the RBC. Together, our data show that RON11 triggers the de novo biogenesis of the second rhoptry and functions in RBC invasion.

Chaos in a bacterial stress response

Cellular responses to environmental changes are often highly heterogeneous and exhibit seemingly random dynamics. The astonishing insight of chaos theory is that such unpredictable patterns can, in principle, arise without the need for any random processes, i.e., purely deterministically without noise. However, while chaos is well understood in mathematics and physics, its role in cell biology remains unclear because the complexity and noisiness of biological systems make testing difficult. Here, we show that chaos explains the heterogeneous response of Escherichia coli cells to oxidative stress. We developed a theoretical model of the gene expression dynamics and demonstrate that chaotic behavior arises from rapid molecular feedbacks that are coupled with cell growth dynamics and cell-cell interactions. Based on theoretical predictions, we then designed single-cell experiments to show we can shift gene expression from periodic oscillations to chaos on demand. Our work suggests that chaotic gene regulation can be employed by cell populations to generate strong and variable responses to changing environments.

Time-delayed Duffing oscillator in an active bath

During recent decades active particles have attracted an incipient attention as they have been observed in a broad class of scenarios, ranging from bacterial suspension in living systems to artificial swimmers in nonequilibirum systems. The main feature of these particles is that they are able to gain kinetic energy from the environment, which is widely modeled by a stochastic process due to both (Gaussian) white and Ornstein-Uhlenbeck noises. In the present work, we study the nonlinear dynamics of the forced, time-delayed Duffing oscillator subject to these noises, paying special attention to their impact upon the maximum oscillations amplitude and characteristic frequency of the steady state for different values of the time delay and the driving force. Overall, our results indicate that the role of the time delay is substantially modified with respect to the situation without noise. For instance, we show that the oscillations amplitude grows with increasing noise strength when the time delay acts as a damping term in absence of noise, whereas the trajectories eventually become aperiodic when the oscillations are sustained by the time delay. In short, the interplay among the noises, forcing, and time delay gives rise to a rich dynamics: a regular and periodic motion is destroyed or restored owing to the competition between the noise and the driving force depending on time delay values, whereas an erratic motion insensitive to the driving force emerges when the time delay makes the motion aperiodic. Interestingly, we also show that, for a sufficient noise strength and forcing amplitude, an approximately periodic interwell motion is promoted by means of stochastic resonance.

Inferring delays in partially observed gene regulation processes

MOTIVATION

Cell function is regulated by gene regulatory networks (GRNs) defined by protein-mediated interaction between constituent genes. Despite advances in experimental techniques, we can still measure only a fraction of the processes that govern GRN dynamics. To infer the properties of GRNs using partial observation, unobserved sequential processes can be replaced with distributed time delays, yielding non-Markovian models. Inference methods based on the resulting model suffer from the curse of dimensionality.

RESULTS

We develop a simulation-based Bayesian MCMC method employing an approximate likelihood for the efficient and accurate inference of GRN parameters when only some of their products are observed. We illustrate our approach using a two-step activation model: an activation signal leads to the accumulation of an unobserved regulatory protein, which triggers the expression of observed fluorescent proteins. With prior information about observed fluorescent protein synthesis, our method successfully infers the dynamics of the unobserved regulatory protein. We can estimate the delay and kinetic parameters characterizing target regulation including transcription, translation, and target searching of an unobserved protein from experimental measurements of the products of its target gene. Our method is scalable and can be used to analyze non-Markovian models with hidden components.

AVAILABILITY AND IMPLEMENTATION

Our code is implemented in R and is freely available with a simple example data at https://github.com/Mathbiomed/SimMCMC.

Time delays modulate the stability of complex ecosystems

What drives the stability, or instability, of complex ecosystems? This question sits at the heart of community ecology and has motivated a large body of theoretical work exploring how community properties shape ecosystem dynamics. However, the overwhelming majority of current theory assumes that species interactions are instantaneous, meaning that changes in the abundance of one species will lead to immediate changes in the abundances of its partners. In practice, time delays in how species respond to one another are widespread across ecological contexts, yet the impact of these delays on ecosystems remains unclear. Here we derive a new body of theory to comprehensively study the impact of time delays on ecological stability. We find that time delays are important for ecosystem stability. Large delays are typically destabilizing but, surprisingly, short delays can substantially increase community stability. Moreover, in stark contrast to delay-free systems, delays dictate that communities with more abundant species can be less stable than ones with less abundant species. Finally, we show that delays fundamentally shift how species interactions impact ecosystem stability, with communities of mixed interaction types becoming the most stable class of ecosystem. Our work demonstrates that time delays can be critical for the stability of complex ecosystems.

Generalized neural closure models with interpretability

Improving the predictive capability and computational cost of dynamical models is often at the heart of augmenting computational physics with machine learning (ML). However, most learning results are limited in interpretability and generalization over different computational grid resolutions, initial and boundary conditions, domain geometries, and physical or problem-specific parameters. In the present study, we simultaneously address all these challenges by developing the novel and versatile methodology of unified neural partial delay differential equations. We augment existing/low-fidelity dynamical models directly in their partial differential equation (PDE) forms with both Markovian and non-Markovian neural network (NN) closure parameterizations. The melding of the existing models with NNs in the continuous spatiotemporal space followed by numerical discretization automatically allows for the desired generalizability. The Markovian term is designed to enable extraction of its analytical form and thus provides interpretability. The non-Markovian terms allow accounting for inherently missing time delays needed to represent the real world. Our flexible modeling framework provides full autonomy for the design of the unknown closure terms such as using any linear-, shallow-, or deep-NN architectures, selecting the span of the input function libraries, and using either or both Markovian and non-Markovian closure terms, all in accord with prior knowledge. We obtain adjoint PDEs in the continuous form, thus enabling direct implementation across differentiable and non-differentiable computational physics codes, different ML frameworks, and treatment of nonuniformly-spaced spatiotemporal training data. We demonstrate the new generalized neural closure models (gnCMs) framework using four sets of experiments based on advecting nonlinear waves, shocks, and ocean acidification models. Our learned gnCMs discover missing physics, find leading numerical error terms, discriminate among candidate functional forms in an interpretable fashion, achieve generalization, and compensate for the lack of complexity in simpler models. Finally, we analyze the computational advantages of our new framework.

The ups and downs of biological oscillators: a comparison of time-delayed negative feedback mechanisms

Many biochemical oscillators are driven by the periodic rise and fall of protein concentrations or activities. A negative feedback loop underlies such oscillations. The feedback can act on different parts of the biochemical network. Here, we mathematically compare time-delay models where the feedback affects production and degradation. We show a mathematical connection between the linear stability of the two models, and derive how both mechanisms impose different constraints on the production and degradation rates that allow oscillations. We show how oscillations are affected by the inclusion of a distributed delay, of double regulation (acting on production and degradation) and of enzymatic degradation.

Insect visuomotor delay adjustments in group flight support swarm cohesion

Flying insects routinely demonstrate coordinated flight in crowded assemblies despite strict communication and processing constraints. This study experimentally records multiple flying insects tracking a moving visual stimulus. System identification techniques are used to robustly identify the tracking dynamics, including a visuomotor delay. The population delay distributions are quantified for solo and group behaviors. An interconnected visual swarm model incorporating heterogeneous delays is developed, and bifurcation analysis and swarm simulation are applied to assess swarm stability under the delays. The experiment recorded 450 insect trajectories and quantified visual tracking delay variation. Solitary tasks showed a 30ms average delay and standard deviation of 50ms, while group behaviors show a 15ms average and 8ms standard deviation. Analysis and simulation indicate that the delay adjustments during group flight support swarm formation and center stability, and are robust to measurement noise. These results quantify the role of visuomotor delay heterogeneity in flying insects and their role in supporting swarm cohesion through implicit communication.

SINDy for delay-differential equations: application to model bacterial zinc response

We extend the data-driven method of sparse identification of nonlinear dynamics (SINDy) developed by Brunton et al. , Proc. Natl Acad. Sci. USA 113 (2016) to the case of delay differential equations (DDEs). This is achieved in a bilevel optimization procedure by first applying SINDy for fixed delay and then subsequently optimizing the error of the reconstructed SINDy model over delay times. We test the SINDy-delay method on a noisy short dataset from a toy DDE and show excellent agreement. We then apply the method to experimental data of gene expressions in the bacterium Pseudomonas aeruginosa subject to the influence of zinc. The derived SINDy model suggests that the increase in zinc concentration mainly affects the time delay and not the strengths of the interactions between the different agents controlling the zinc export mechanism.

Computational characterization of recombinase circuits for periodic behaviors

Recombinases are site-specific proteins found in nature that are capable of rearranging DNA. This function has made them promising gene editing tools in synthetic biology, as well as key elements in complex artificial gene circuits implementing Boolean logic. However, since DNA rearrangement is irreversible, it is still unclear how to use recombinases to build dynamic circuits like oscillators. In addition, this goal is challenging because a few molecules of recombinase are enough for promoter inversion, generating inherent stochasticity at low copy number. Here, we propose six different circuit designs for recombinase-based oscillators operating at a single copy number. We model them in a stochastic setting, leveraging the Gillespie algorithm for extensive simulations, and show that they can yield coherent periodic behaviors. Our results support the experimental realization of recombinase-based oscillators and, more generally, the use of recombinases to generate dynamic behaviors in synthetic biology.

Fractional transit compartment model for describing drug delayed response to tumors using Mittag-Leffler distribution on age-structured PKPD model

The response of a cell population is often delayed relative to drug injection, and individual cells in a population of cells have a specific age distribution. The application of transit compartment models (TCMs) is a common approach for describing this delay. In this paper, we propose a TCM in which damaged cells caused by a drug are given by a single fractional derivative equation. This model describes the delay as a single equation composed of fractional and ordinary derivatives, instead of a system of ODEs expressed in multiple compartments, applicable to the use of the PK concentration in the model. This model tunes the number of compartments in the existing model and expresses the delay in detail by estimating an appropriate fractional order. We perform model robustness, sensitivity analysis, and change of parameters based on the amount of data. Additionally, we resolve the difficulty in parameter estimation and model simulation using a semigroup property, consisting of a system with a mixture of fractional and ordinary derivatives. This model provides an alternative way to express the delays by estimating an appropriate fractional order without determining the pre-specified number of compartments.

Computational modeling of protracted HCMV replication using genome substrates and protein temporal profiles

Human cytomegalovirus (HCMV) is a major cause of illness in immunocompromised individuals. The HCMV lytic cycle contributes to the clinical manifestations of infection. The lytic cycle occurs over ∼96 h in diverse cell types and consists of viral DNA (vDNA) genome replication and temporally distinct expression of hundreds of viral proteins. Given its complexity, understanding this elaborate system can be facilitated by the introduction of mechanistic computational modeling of temporal relationships. Therefore, we developed a multiplicity of infection (MOI)-dependent mechanistic computational model that simulates vDNA kinetics and late lytic replication based on in-house experimental data. The predictive capabilities were established by comparison to post hoc experimental data. Computational analysis of combinatorial regulatory mechanisms suggests increasing rates of protein degradation in association with increasing vDNA levels. The model framework also allows expansion to account for additional mechanisms regulating the processes. Simulating vDNA kinetics and the late lytic cycle for a wide range of MOIs yielded several unique observations. These include the presence of saturation behavior at high MOIs, inefficient replication at low MOIs, and a precise range of MOIs in which virus is maximized within a cell type, being 0.382 IU to 0.688 IU per fibroblast. The predicted saturation kinetics at high MOIs are likely related to the physical limitations of cellular machinery, while inefficient replication at low MOIs may indicate a minimum input material required to facilitate infection. In summary, we have developed and demonstrated the utility of a data-driven and expandable computational model simulating lytic HCMV infection.

Extended transit compartment model to describe tumor delay using Coxian distribution

The measured response of cell population is often delayed relative to drug injection, and individuals in a population have a specific age distribution. Common approaches for describing the delay are to apply transit compartment models (TCMs). This model reflects that all damaged cells caused by drugs suffer transition processes, resulting in death. In this study, we present an extended TCM using Coxian distribution, one of the phase-type distributions. The cell population attacked by a drug is described via age-structured models. The mortality rate of the damaged cells is expressed by a convolution of drug rate and age density. Then applying to Erlang and Coxian distribution, we derive Erlang TCM, representing the existing model, and Coxian TCMs, reflecting sudden death at all ages. From published data of drug and tumor, delays are compared after parameter estimations in both models. We investigate the dynamical changes according to the number of the compartments. Model robustness and equilibrium analysis are also performed for model validation. Coxian TCM is an extended model considering a realistic case and captures more diverse delays.

Turing pattern design principles and their robustness

Turing patterns have morphed from mathematical curiosities into highly desirable targets for synthetic biology. For a long time, their biological significance was sometimes disputed but there is now ample evidence for their involvement in processes ranging from skin pigmentation to digit and limb formation. While their role in developmental biology is now firmly established, their synthetic design has so far proved challenging. Here, we review recent large-scale mathematical analyses that have attempted to narrow down potential design principles. We consider different aspects of robustness of these models and outline why this perspective will be helpful in the search for synthetic Turing-patterning systems. We conclude by considering robustness in the context of developmental modelling more generally. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

A frequency-amplitude coordinator and its optimal energy consumption for biological oscillators

Biorhythm including neuron firing and protein-mRNA interaction are fundamental activities with diffusive effect. Their well-balanced spatiotemporal dynamics are beneficial for healthy sustainability. Therefore, calibrating both anomalous frequency and amplitude of biorhythm prevents physiological dysfunctions or diseases. However, many works were devoted to modulate frequency exclusively whereas amplitude is usually ignored, although both quantities are equally significant for coordinating biological functions and outputs. Especially, a feasible method coordinating the two quantities concurrently and precisely is still lacking. Here, for the first time, we propose a universal approach to design a frequency-amplitude coordinator rigorously via dynamical systems tools. We consider both spatial and temporal information. With a single well-designed coordinator, they can be calibrated to desired levels simultaneously and precisely. The practical usefulness and efficacy of our method are demonstrated in representative neuronal and gene regulatory models. We further reveal its fundamental mechanism and optimal energy consumption providing inspiration for biorhythm regulation in future.

Neural closure models for dynamical systems

Complex dynamical systems are used for predictions in many domains. Because of computational costs, models are truncated, coarsened or aggregated. As the neglected and unresolved terms become important, the utility of model predictions diminishes. We develop a novel, versatile and rigorous methodology to learn non-Markovian closure parametrizations for known-physics/low-fidelity models using data from high-fidelity simulations. The new neural closure models augment low-fidelity models with neural delay differential equations (nDDEs), motivated by the Mori–Zwanzig formulation and the inherent delays in complex dynamical systems. We demonstrate that neural closures efficiently account for truncated modes in reduced-order-models, capture the effects of subgrid-scale processes in coarse models and augment the simplification of complex biological and physical–biogeochemical models. We find that using non-Markovian over Markovian closures improves long-term prediction accuracy and requires smaller networks. We derive adjoint equations and network architectures needed to efficiently implement the new discrete and distributed nDDEs, for any time-integration schemes and allowing non-uniformly spaced temporal training data. The performance of discrete over distributed delays in closure models is explained using information theory, and we find an optimal amount of past information for a specified architecture. Finally, we analyse computational complexity and explain the limited additional cost due to neural closure models.