Minimal Mechanisms of Microtubule Length Regulation in Living Cells.

The microtubule cytoskeleton is responsible for sustained, long-range intracellular transport of mRNAs, proteins, and organelles in neurons. Neuronal microtubules must be stable enough to ensure reliable transport, but they also undergo dynamic instability, as their plus and minus ends continuously switch between growth and shrinking. This process allows for continuous rebuilding of the cytoskeleton and for flexibility in injury settings. Motivated by in vivo experimental data on microtubule behavior in Drosophila neurons, we propose a mathematical model of dendritic microtubule dynamics, with a focus on understanding microtubule length, velocity, and state-duration distributions. We find that limitations on microtubule growth phases are needed for realistic dynamics, but the type of limiting mechanism leads to qualitatively different responses to plausible experimental perturbations. We therefore propose and investigate two minimally-complex length-limiting factors: limitation due to resource (tubulin) constraints and limitation due to catastrophe of large-length microtubules. We combine simulations of a detailed stochastic model with steady-state analysis of a mean-field ordinary differential equations model to map out qualitatively distinct parameter regimes. This provides a basis for predicting changes in microtubule dynamics, tubulin allocation, and the turnover rate of tubulin within microtubules in different experimental environments.

Understanding and Quantifying Network Robustness to Stochastic Inputs.

A variety of biomedical systems are modeled by networks of deterministic differential equations with stochastic inputs. In some cases, the network output is remarkably constant despite a randomly fluctuating input. In the context of biochemistry and cell biology, chemical reaction networks and multistage processes with this property are called robust. Similarly, the notion of a forgiving drug in pharmacology is a medication that maintains therapeutic effect despite lapses in patient adherence to the prescribed regimen. What makes a network robust to stochastic noise? This question is challenging due to the many network parameters (size, topology, rate constants) and many types of noisy inputs. In this paper, we propose a summary statistic to describe the robustness of a network of linear differential equations (i.e. a first-order mass-action system). This statistic is the variance of a certain random walk passage time on the network. This statistic can be quickly computed on a modern computer, even for complex networks with thousands of nodes. Furthermore, we use this statistic to prove theorems about how certain network motifs increase robustness. Importantly, our analysis provides intuition for why a network is or is not robust to noise. We illustrate our results on thousands of randomly generated networks with a variety of stochastic inputs.

Computational modelling of articular joints with biphasic cartilage: recent advances, challenges and opportunities.

Biphasic models have been widely used to simulate the time-dependent biomechanical response of soft tissues. Modelling techniques of joints with biphasic weight-bearing soft tissues have been markedly improved over the last decade, enhancing our understanding of the function, degenerative mechanism and outcomes of interventions of joints. This paper reviews the recent advances, challenges and opportunities in computational models of joints with biphasic weight-bearing soft tissues. The review begins with an introduction of the function and degeneration of joints from a biomechanical aspect. Different constitutive models of articular cartilage, in particular biphasic materials, are illustrated in the context of the study of contact mechanics in joints. Approaches, advances and major findings of biphasic models of the hip and knee are presented, followed by a discussion of the challenges awaiting to be addressed, including the convergence issue, high computational cost and inadequate validation. Finally, opportunities and clinical insights in the areas of subject-specific modeling and tissue engineering are provided and discussed.

The Lymphatic Vascular System: Does Nonuniform Lymphangion Length Limit Flow-Rate?

A previously developed model of a lymphatic vessel as a chain of lymphangions was investigated to determine whether lymphangions of unequal length reduce pumping relative to a similar chain of equal-length ones. The model incorporates passive elastic and active contractile properties taken from ex vivo measurements, and intravascular lymphatic valves as transvalvular pressure-dependent resistances to flow with hysteresis and transmural pressure-dependent bias to the open state as observed experimentally. Coordination of lymphangion contractions is managed by marrying an autonomous transmural pressure-dependent pacemaker for each lymphangion with bidirectional transmission of activation signals between lymphangions, qualitatively matching empirical observations. With eight lymphangions as used here and many nonlinear constraints, the model is capable of complex outcomes. The expected flow-rate advantage conferred by longer lymphangions everywhere was confirmed. However, the anticipated advantage of uniform lymphangions over those of unequal length, compared in chains of equal overall length, was not found. A wide variety of dynamical outcomes was observed, with the most powerful determinant being the adverse pressure difference, rather than the arrangement of long and short lymphangions. This work suggests that the wide variation in lymphangion length which is commonly observed in collecting lymphatic vessels does not confer disadvantage in pumping lymph.

Optimal Rotation Age in Fast Growing Plantations: A Dynamical Optimization Problem.

Forest plantations are economically and environmentally relevant, as they play a key role in timber production and carbon capture. It is expected that the future climate change scenario affects forest growth and modify the rotation age for timber production. However, mathematical models on the effect of climate change on the rotation age for timber production remain still limited. We aim to determine the optimal rotation age that maximizes the net economic benefit of timber volume in a negative scenario from the climatic point of view. For this purpose, a bioeconomic optimal control problem was formulated from a system of Ordinary Differential Equations (ODEs) governed by the state variables live biomass volume, intrinsic growth rate, and area affected by fire. Then, four control variables were associated to the system, representing forest management activities, which are felling, thinning, reforestation, and fire prevention. The existence of optimal control solutions was demonstrated, and the solutions of the optimal control problem were also characterized using Pontryagin's Maximum Principle. The solutions of the model were approximated numerically by the Forward-Backward Sweep method. To validate the model, two scenarios were considered: a realistic scenario that represents current forestry activities for the exotic species Pinus radiata D. Don, and a pessimistic scenario, which considers environmental conditions conducive to a higher occurrence of forest fires. The optimal solution that maximizes the net benefit of timber volume consists of a strategy that considers all four control variables simultaneously. For felling and thinning, regardless of the scenario considered, the optimal strategy is to spend on both activities depending on the amount of biomass in the field. Similarly, for reforestation, the optimal strategy is to spend as the forest is harvested. In the case of fire prevention, in the realistic scenario, the optimal strategy consists of reducing the expenses in fire prevention because the incidence of fires is lower, whereas in the pessimistic scenario, the opposite is true. It is concluded that the optimal rotation age that maximizes the net economic benefit of timber volume in P. radiata plantations is 24 and 19 years for the realistic and pessimistic scenarios, respectively. This corroborates that the presence of fires influences the determination of the optimal rotation age, and as a consequence, the net economic benefit.

Logic programming-based Minimal Cut Sets reveal consortium-level therapeutic targets for chronic wound infections.

Minimal Cut Sets (MCSs) identify sets of reactions which, when removed from a metabolic network, disable certain cellular functions. The traditional search for MCSs within genome-scale metabolic models (GSMMs) targets cellular growth, identifies reaction sets resulting in a lethal phenotype if disrupted, and retrieves a list of corresponding gene, mRNA, or enzyme targets. Using the dual link between MCSs and Elementary Flux Modes (EFMs), our logic programming-based tool aspefm was able to compute MCSs of any size from GSMMs in acceptable run times. The tool demonstrated better performance when computing large-sized MCSs than the mixed-integer linear programming methods. We applied the new MCSs methodology to a medically-relevant consortium model of two cross-feeding bacteria, Staphylococcus aureus and Pseudomonas aeruginosa. aspefm constraints were used to bias the computation of MCSs toward exchanged metabolites that could complement lethal phenotypes in individual species. We found that interspecies metabolite exchanges could play an essential role in rescuing single-species growth, for instance inosine could complement lethal reaction knock-outs in the purine synthesis, glycolysis, and pentose phosphate pathways of both bacteria. Finally, MCSs were used to derive a list of promising enzyme targets for consortium-level therapeutic applications that cannot be circumvented via interspecies metabolite exchange.

Artificial Neural Network Prediction of COVID-19 Daily Infection Count.

This study addresses COVID-19 testing as a nonlinear sampling problem, aiming to uncover the dependence of the true infection count in the population on COVID-19 testing metrics such as testing volume and positivity rates. Employing an artificial neural network, we explore the relationship among daily confirmed case counts, testing data, population statistics, and the actual daily case count. The trained artificial neural network undergoes testing in in-sample, out-of-sample, and several hypothetical scenarios. A substantial focus of this paper lies in the estimation of the daily true case count, which serves as the output set of our training process. To achieve this, we implement a regularized backcasting technique that utilize death counts and the infection fatality ratio (IFR), as the death statistics and serological surveys (providing the IFR) as more reliable COVID-19 data sources. Addressing the impact of factors such as age distribution, vaccination, and emerging variants on the IFR time series is a pivotal aspect of our analysis. We expect our study to enhance our understanding of the genuine implications of the COVID-19 pandemic, subsequently benefiting mitigation strategies.

Mathematical Modelling of Parasite Dynamics: A Stochastic Simulation-Based Approach and Parameter Estimation via Modified Sequential-Type Approximate Bayesian Computation.

The development of mathematical models for studying newly emerging and re-emerging infectious diseases has gained momentum due to global events. The gyrodactylid-fish system, like many host-parasite systems, serves as a valuable resource for ecological, evolutionary, and epidemiological investigations owing to its ease of experimental manipulation and long-term monitoring. Although this system has an existing individual-based model, it falls short in capturing information about species-specific microhabitat preferences and other biological details for different Gyrodactylus strains across diverse fish populations. This current study introduces a new individual-based stochastic simulation model that uses a hybrid τ -leaping algorithm to incorporate this essential data, enhancing our understanding of the complexity of the gyrodactylid-fish system. We compare the infection dynamics of three gyrodactylid strains across three host populations. A modified sequential-type approximate Bayesian computation (ABC) method, based on sequential Monte Carlo and sequential importance sampling, is developed. Additionally, we establish two penalised local-linear regression methods (based on L1 and L2 regularisations) for ABC post-processing analysis to fit our model using existing empirical data. With the support of experimental data and the fitted mathematical model, we address open biological questions for the first time and propose directions for future studies on the gyrodactylid-fish system. The adaptability of the mathematical model extends beyond the gyrodactylid-fish system to other host-parasite systems. Furthermore, the modified ABC methodologies provide efficient calibration for other multi-parameter models characterised by a large set of correlated or independent summary statistics.

Partial differential equation models for invasive species spread in the presence of spatial heterogeneity.

Models of invasive species spread often assume that landscapes are spatially homogeneous; thus simplifying analysis but potentially reducing accuracy. We extend a recently developed partial differential equation model for invasive conifer spread to account for spatial heterogeneity in parameter values and introduce a method to obtain key outputs (e.g. spread rates) from computational simulations. Simulations produce patterns of spatial spread which appear qualitatively similar to observed patterns in grassland ecosystems invaded by exotic conifers, validating our spatially explicit strategy. We find that incorporating spatial variation in different parameters does not significantly affect the evolution of invasions (which are characterised by a long quiescent period followed by rapid evolution towards to a constant rate of invasion) but that distributional assumptions can have a significant impact on the spread rate of invasions. Our work demonstrates that spatial variation in site-suitability or other parameters can have a significant impact on invasions and must be considered when designing models of invasive species spread.

The Demographic-Wealth model for cliodynamics.

Cliodynamics is a still a relatively new research area with the purpose of investigating and modelling historical processes. One of its first important mathematical models was proposed by Turchin and called "Demographic-Fiscal Model" (DFM). This DFM was one of the first and is one of a few models that link population with state dynamics. In this work, we propose a possible alternative to the classical Turchin DFM, which contributes to further model development and comparison essential for the field of cliodynamics. Our "Demographic-Wealth Model" (DWM) aims to also model link between population and state dynamics but makes different modelling assumptions, particularly about the type of possible taxation. As an important contribution, we employ tools from nonlinear dynamics, e.g., existence theory for periodic orbits as well as analytical and numerical bifurcation analysis, to analyze the DWM. We believe that these tools can also be helpful for many other current and future models in cliodynamics. One particular focus of our analysis is the occurrence of Hopf bifurcations. Therefore, a detailed analysis is developed regarding equilibria and their possible bifurcations. Especially noticeable is the behavior of the so-called coexistence point. While changing different parameters, a variety of Hopf bifurcations occur. In addition, it is indicated, what role Hopf bifurcations may play in the interplay between population and state dynamics. There are critical values of different parameters that yield periodic behavior and limit cycles when exceeded, similar to the "paradox of enrichment" known in ecology. This means that the DWM provides one possible avenue setup to explain in a simple format the existence of secular cycles, which have been observed in historical data. In summary, our model aims to balance simplicity, linking to the underlying processes and the goal to represent secular cycles.

Advice to a Young Mathematical Biologist.

This paper offers advice to early-mid career researchers in Mathematical Biology from ten past and current Presidents of the Society for Mathematical Biology. The topics covered include deciding if a career in academia is right for you; finding and working with a mentor; building collaborations and working with those from other disciplines; formulating a research question; writing a paper; reviewing papers; networking; writing fellowship or grant proposals; applying for faculty positions; and preparing and giving lectures. While written with mathematical biologists in mind, it is hoped that this paper will be of use to early and mid career researchers across the mathematical, physical and life sciences, as they embark on careers in these disciplines.

Mathematical Models of Early Hepatitis B Virus Dynamics in Humanized Mice.

Analyzing the impact of the adaptive immune response during acute hepatitis B virus (HBV) infection is essential for understanding disease progression and control. Here we developed mathematical models of HBV infection which either lack terms for adaptive immune responses, or assume adaptive immune responses in the form of cytolytic immune killing, non-cytolytic immune cure, or non-cytolytic-mediated block of viral production. We validated the model that does not include immune responses against temporal serum hepatitis B DNA (sHBV) and temporal serum hepatitis B surface-antigen (HBsAg) experimental data from mice engrafted with human hepatocytes (HEP). Moreover, we validated the immune models against sHBV and HBsAg experimental data from mice engrafted with HEP and human immune system (HEP/HIS). As expected, the model that does not include adaptive immune responses matches the observed high sHBV and HBsAg concentrations in all HEP mice. By contrast, while all immune response models predict reduction in sHBV and HBsAg concentrations in HEP/HIS mice, the Akaike Information Criterion cannot discriminate between non-cytolytic cure (resulting in a class of cells refractory to reinfection) and antiviral block functions (of up to 99 % viral production 1-3 weeks following peak viral load). We can, however, reject cytolytic killing, as it can only match the sHBV and HBsAg data when we predict unrealistic levels of hepatocyte loss.

Putting theory to the test: An integrated computational/experimental chemostat model of the tragedy of the commons.

Cooperation via shared public goods is ubiquitous in nature, however, noncontributing social cheaters can exploit the public goods provided by cooperating individuals to gain a fitness advantage. Theory predicts that this dynamic can cause a Tragedy of the Commons, and in particular, a 'Collapsing' Tragedy defined as the extinction of the entire population if the public good is essential. However, there is little empirical evidence of the Collapsing Tragedy in evolutionary biology. Here, we experimentally demonstrate this outcome in a microbial model system, the public good-producing bacterium Pseudomonas aeruginosa grown in a continuous-culture chemostat. In a growth medium that requires extracellular protein digestion, we find that P. aeruginosa populations maintain a high density when entirely composed of cooperating, protease-producing cells but completely collapse when non-producing cheater cells are introduced. We formulate a mechanistic mathematical model that recapitulates experimental observations and suggests key parameters, such as the dilution rate and the cost of public good production, that define the stability of cooperative behavior. We combine model prediction with experimental validation to explain striking differences in the long-term cheater trajectories of replicate cocultures through mutational events that increase cheater fitness. Taken together, our integrated empirical and theoretical approach validates and parametrizes the Collapsing Tragedy in a microbial population, and provides a quantitative, mechanistic framework for generating testable predictions of social behavior.

The need for speed - Does the force-velocity property significantly alter strain distributions within skeletal muscle?

Skeletal muscles are complex structures with nonlinear constitutive properties. This complexity often requires finite element (FE) modeling to better understand muscle behavior and response to activation, especially the fiber strain distributions that can be difficult to measure in vivo. However, many FE muscle models designed to study fiber strain do not include force-velocity behavior. To investigate force-velocity property impact on strain distributions within skeletal muscle, we modified a muscle constitutive model with active and passive force-length properties to include force-velocity properties. We implemented the new constitutive model as a plugin for the FE software FEBio and applied it to four geometries: 1) a single element, 2) a multiple-element model representing a single fiber, 3) a model of tapering fibers, and 4) a model representing the bicep femoris long head (BFLH) morphology. Maximum fiber velocity and boundary conditions of the finite element models were varied to test their influence on fiber strain distribution. We found that force-velocity properties in the constitutive model behaved as expected for the single element and multi-element conditions. In the tapered fiber models, fiber strain distributions were impacted by changes in maximum fiber velocity; the range of strains increased with maximum fiber velocity, which was most noted in isometric contraction simulations. In the BFLH model, maximum fiber velocity had minimal impact on strain distributions, even in the context of sprinting. Taken together, the combination of muscle model geometry, activation, and displacement parameters play a critical part in determining the magnitude of impact of force-velocity on strain distribution.

Development and applications of genome-scale metabolic network models.

The genome-scale metabolic network model is an effective tool for characterizing the gene-protein-response relationship in the entire metabolic pathway of an organism. By combining various algorithms, the genome-scale metabolic network model can effectively simulate the influence of a specific environment on the physiological state of cells, optimize the culture conditions of strains, and predict the targets of genetic modification to achieve targeted modification of strains. In this review, we summarize the whole process of model building, sort out the various tools that may be involved in the model building process, and explain the role of various algorithms in model analysis. In addition, we also summarized the application of GSMM in network characteristics, cell phenotypes, metabolic engineering, etc. Finally, we discuss the current challenges facing GSMM.

Bounds on the Ultrasensitivity of Biochemical Reaction Cascades.

The ultrasensitivity of a dose response function can be quantifiably defined using the generalized Hill coefficient of the function. We examined an upper bound for the Hill coefficient of the composition of two functions, namely the product of their individual Hill coefficients. We proved that this upper bound holds for compositions of Hill functions, and that there are instances of counterexamples that exist for more general sigmoidal functions. Additionally, we tested computationally other types of sigmoidal functions, such as the logistic and inverse trigonometric functions, and we provided computational evidence that in these cases the inequality also holds. We show that in large generality there is a limit to how ultrasensitive the composition of two functions can be, which has applications to understanding signaling cascades in biochemical reactions.

On hierarchical competition through reduction of individual growth.

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition. However, the model can also be formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the delay formulation, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and analyse its stability using the principle of linearised stability for delay equations. Finally, we relate the results to the alternative, partial differential equation formulation of the model.

Untangling the Molecular Interactions Underlying Intracellular Phase Separation Using Combined Global Sensitivity Analyses.

Liquid-liquid phase separation is an intracellular mechanism by which molecules, usually proteins and RNAs, interact and then rapidly demix from the surrounding matrix to form membrane-less compartments necessary for cellular function. Occurring in both the cytoplasm and the nucleus, properties of the resulting droplets depend on a variety of characteristics specific to the molecules involved, such as valency, density, and diffusion within the crowded environment. Capturing these complexities in a biologically relevant model is difficult. To understand the nuanced dynamics between proteins and RNAs as they interact and form droplets, as well as the impact of these interactions on the resulting droplet properties, we turn to sensitivity analysis. In this work, we examine a previously published mathematical model of two RNA species competing for the same protein-binding partner. We use the combined analyses of Morris Method and Sobol' sensitivity analysis to understand the impact of nine molecular parameters, subjected to three different initial conditions, on two observable LLPS outputs: the time of phase separation and the composition of the droplet field. Morris Method is a screening method capable of highlighting the most important parameters impacting a given output, while the variance-based Sobol' analysis can quantify both the importance of a given parameter, as well as the other model parameters it interacts with, to produce the observed phenomena. Combining these two techniques allows Morris Method to identify the most important dynamics and circumvent the large computational expense associated with Sobol', which then provides more nuanced information about parameter relationships. Together, the results of these combined methodologies highlight the complicated protein-RNA relationships underlying both the time of phase separation and the composition of the droplet field. Sobol' sensitivity analysis reveals that observed spatial and temporal dynamics are due, at least in part, to high-level interactions between multiple (3+) parameters. Ultimately, this work discourages using a single measurement to extrapolate the value of any single rate or parameter value, while simultaneously establishing a framework in which to analyze and assess the impact of these small-scale molecular interactions on large-scale droplet properties.

Collective behavior from surprise minimization.

Collective motion is ubiquitous in nature; groups of animals, such as fish, birds, and ungulates appear to move as a whole, exhibiting a rich behavioral repertoire that ranges from directed movement to milling to disordered swarming. Typically, such macroscopic patterns arise from decentralized, local interactions among constituent components (e.g., individual fish in a school). Preeminent models of this process describe individuals as self-propelled particles, subject to self-generated motion and "social forces" such as short-range repulsion and long-range attraction or alignment. However, organisms are not particles; they are probabilistic decision-makers. Here, we introduce an approach to modeling collective behavior based on active inference. This cognitive framework casts behavior as the consequence of a single imperative: to minimize surprise. We demonstrate that many empirically observed collective phenomena, including cohesion, milling, and directed motion, emerge naturally when considering behavior as driven by active Bayesian inference-without explicitly building behavioral rules or goals into individual agents. Furthermore, we show that active inference can recover and generalize the classical notion of social forces as agents attempt to suppress prediction errors that conflict with their expectations. By exploring the parameter space of the belief-based model, we reveal nontrivial relationships between the individual beliefs and group properties like polarization and the tendency to visit different collective states. We also explore how individual beliefs about uncertainty determine collective decision-making accuracy. Finally, we show how agents can update their generative model over time, resulting in groups that are collectively more sensitive to external fluctuations and encode information more robustly.

Trapping and scattering of a multiflagellated bacterium by a hard surface.

Thiovulum majus, which is one of the fastest known bacteria, swims using hundreds of flagella. Unlike typical pusher cells, which swim in circular paths over hard surfaces, T. majus localize near hard boundaries by turning their flagella to exert a net force normal to the surface. To probe the torques that stabilize this hydrodynamically bound state, the trajectories of several thousand collisions between a T. majus cell and a wall of a quasi-two-dimensional microfluidic chamber are analyzed. Measuring the fraction of cells escaping the wall either to the left or to the right of the point of contact-and how this probability varies with incident angle and time spent in contact with the surface-maps the scattering dynamics onto a first passage problem. These measurements are compared to the prediction of a Fokker-Planck equation to fit the angular velocity of a cell in contact with a hard surface. This analysis reveals a bound state with a narrow basin of attraction in which cells orient their flagella normal to the surface. The escape angle predicted by matching these near field dynamics with the far-field hydrodynamics is consistent with observation. We discuss the significance of these results for the ecology of T. majus and their self-organization into active chiral crystals.