The self-organization of cortical microtubules within plant cells is an emergent phenomenon with important consequences for the synthesis of the cell wall, cell shape, and subsequently the structure of plants. Mathematical modelling and experiments have elucidated the underlying processes involved. There has been recent interest in the influence of geometric cues on array orientation, be it direct (cell shape) or indirect (tension in the membrane). However, the mechanical influence of membrane curvature on these elastic filaments has largely been ignored. A previous model was proposed to describe how the anchoring process may control the shape of individual microtubules seeking to minimize bending on a cylindrical cell. We implement this process into a model of interacting microtubules and find the cell curvature influence should be significant: the array favours orientations parallel to the direction of elongation rather than the expected transverse direction. Even without elasticity, the geometry of large cells hinders robust microtubule organization. These results suggest the necessity of additional processes to overcome these factors. We propose a simple model of orientation-dependent catastrophe in the context of cellulose microfibrils impeding microtubule polymerization and find a moderate impedance to be sufficient to generate transverse arrays despite the geometric influences. ### Competing Interest Statement The authors have declared no competing interest.

The patterns formed both in vivo and in vitro by the Min protein system have attracted much interest because of the complexity of their dynamic interactions given the apparent simplicity of the component parts. Despite both the experimental and theoretical attention paid to this system, the details of the biochemical interactions of MinD and MinE, the proteins responsible for the patterning, are still unclear. For example, no model consistent with the known biochemistry has yet accounted for the observed dual role of MinE in the membrane stability of MinD. Until now, a statistical comparison of models to the time course of Min protein concentrations on the membrane has not been carried out. Such an approach is a powerful way to test existing and novel models that are difficult to test using a purely experimental approach. Here, we extract time series from previously published fluorescence microscopy time lapse images of in vitro experiments and fit two previously described and one novel mathematical model to the data. We find that the novel model, which we call the Asymmetric Activation with Bridged Stability Model, fits the time-course data best. It is also consistent with known biochemistry and explains the dual MinE role via MinE-dependent membrane stability that transitions under the influence of rising MinE to membrane instability with positive feedback. Our results reveal a more complex network of interactions between MinD and MinE underlying Min-system dynamics than previously considered.

We study the dynamics of a two-variable mechanochemical model for GTPase signaling and cell tension using numerical bifurcation analysis, providing insight to the dynamics throughout parameter space. The model exhibits a wide range of local and global bifurcations, including three codimension-two bifurcations occurring along a locus of homoclinics. We use numerical bifurcation analysis and simulation to investigate these bifurcations. This analysis provides evidence for two rarely seen bifurcations: a neutral saddle homoclinic bifurcation and a non-central saddle-node homoclinic bifurcation. We expand the understanding of the dynamics of the mechanochemical model and provide a pedagogically useful example of a realistic but relatively simple model that exhibits a wide range of bifurcations.

The organization of cortical microtubule arrays play an important role in the development of plant cells. Until recently, the direct mechanical influence of cell geometry on the constrained microtubule (MT) trajectories have been largely ignored in computational models. Modelling MTs as thin elastic rods constrained on a surface, a previous study examined the deflection of MTs using a fixed number of segments and uniform segment lengths between MT anchors. It is known that the resulting MT curves converge to geodesics as the anchor spacing approaches zero. In the case of long MTs on a cylinder, buckling has been found for transverse trajectories. There is a clear interplay between two factors in the problem of deflection: curvature of the membrane and the lengths of MT segments. Here, we examine the latter in detail, in the backdrop of a circular cylinder. In reality, the number of segments are not predetermined and their lengths are not uniform. We present a minimal, realistic model treating the anchor spacing as a stochastic process and examine the net effect on deflection. We find that, by tuning the ratio of growth speed to anchoring rate, it is possible to mitigate MT deflection and even prevent buckling for lengths significantly larger than the previously-derived critical buckling length. We suggest that this mediation of deflection by anchoring might provide cells with a means of preventing arrays from deflecting away from the transverse orientation.

In epithelial-mesenchymal transition (EMT), cells organized into sheets break away and become motile mesenchymal cells. EMT plays a crucial role in wound healing, embryonic development, and cancer metastasis. Intracellular signaling in response to mechanical, topographic, or chemical stimuli can promote EMT. We present a multiscale model for EMT downstream of the protein YAP, which suppresses the cell-cell adhesion protein E-cadherin and activates the GTPase Rac1 that enhances cell migration. We first propose an ordinary differential equation (ODE) model for intracellular YAP/Rac1/E-cadherin interactions. The ODE model dynamics are bistable, accounting for both motile loose cells and adherent slower cells. We incorporate this model into a cellular Potts model simulation of two-dimensional wound healing using the open-source platform Morpheus. We show that, under suitable stimuli representing topographic cues, the sheet exhibits finger-like projections and EMT. Morphological differences and quantitative differences in YAP levels as well as variations in cell speed across the sheet are consistent with previous experimental observations of epithelial sheets grown on topographic features in vitro. The simulation is also consistent with experiments that knock down or overexpress YAP, inhibit Rac1, or block E-cadherin.

AbstractInsulin receptor (Insr) protein can be found at higher levels in pancreatic β-cells than in most other tissues, but the consequences of β-cell insulin resistance remain enigmatic. Ins1cre allele was used to delete Insr specifically in β-cells of both female and male mice. Experimental mice were compared to Ins1cre-containing littermate controls at multiple ages and on multiple diets. RNA-seq of purified recombined β-cells revealed transcriptomic consequences of Insr loss, which differed between female and male mice. Action potential and calcium oscillation frequencies were increased in Insr knockout β- cells from female, but not male mice, whereas only male βInsrKO mice had reduced ATP-coupled oxygen consumption rate and reduced expression of genes involved in ATP synthesis. Female βInsrKO and βInsrHET mice exhibited elevated insulin release in perifusion experiments, during hyperglycemic clamps, and following i.p. glucose challenge. Deletion of Insr did not alter β-cell area up to 9 months of age, nor did it impair hyperglycemia-induced proliferation. Based on our data, we adapted a mathematical model to include β-cell insulin resistance, which predicted that β-cell Insr knockout would improve glucose tolerance depending on the degree of whole-body insulin resistance. Indeed, glucose tolerance was significantly improved in female βInsrKO and βInsrHET mice when compared to controls at 9, 21 and 39 weeks, and also in insulin-sensitive 4-week old males. We did not observe improved glucose tolerance in older male mice or in high fat diet-fed mice, corroborating the prediction that global insulin resistance obscures the effects of β-cell specific insulin resistance. The propensity for hyperinsulinemia was associated with mildly reduced fasting glucose and increased body weight. We further validated our main in vivo findings using the Ins1-CreERT transgenic line and found that male mice had improved glucose tolerance 4 weeks after tamoxifen-mediated Insr deletion. Collectively, our data show that loss of β-cell Insr contributes to glucose-induced hyperinsulinemia, thereby improving glucose homeostasis in otherwise insulin sensitive dietary and age contexts.

The fossil record contains dinosaur jaws with rows of unerupted successional teeth that are arranged in a variety of elegant patterns. The remnants of these patterns are visible in modern dentate reptiles but the mechanism for generating and maintaining these rows of teeth is unknown. The biology underlying the tooth replacement pattern was hypothesized to either be stimuli transmitted across tooth families in the jaws (Edmund) or secretion of local inhibitory molecules that would stagger development of adjacent tooth families (Osborn). To test these hypotheses and generate new ones, we completed a study on 6 treated adult geckos in which one side of the jaw had teeth removed. Wax bites were used to record the maxillary teeth 2 times a week. Tooth presence or absence was recorded and transformed mathematically. The time between eruption at each tooth position was measured as was the relative phase compared to the immediate adjacent teeth over successive bites. The period between eruption events at each tooth position was approximately 30 days with some lengthening over time. The average relative phase showed there was a tilt in the data that fit the observation that alternating teeth were being shed. This tilt was opposite on the left and right sides of the jaw. The asymmetry of the right and left sides was consistent across the dentition. After plucking, the pattern recovers after 3 periods fitting with the consistent finding that there are 3 teeth in each tooth family. Ablated areas did not recover tooth formation even after 14 months. The plucked animals showed evidence of fixed, local signaling that restores the pattern. Two models based on Osborn’s concept of a “zone of inhibition” deviate from the observed data. The ablated animals show no change in patterns of tooth eruption anterior and posterior to the gap. Thus there is no support for the Wave stimulus theory of Edmund. Finally, we propose a new Phase Inhibition Model. This model assumes fixed initiation sites at which teeth are initiated at some phase within a month-long cycle and that, as a tooth is initiated, the cycles of nearby initiation sites are inhibited in their progress. This inhibition causes nearest neighbours to erupt in anti-synchrony. This model best maintained the tilt, spacing timing of the real biological data. Mathematical modeling was sensitive enough to measure the normal developmental instability and the resilience of the gecko to restore homeostasis after tooth removal.

The original version of this article unfortunately contained a mistake. The co-author Dr. Franck Duong Van Hoa first name and last name were misinterpreted in the original publication.

The MalFGK 2 transporter regulates the movement of maltose across the inner membrane of E. coli and serves as a model system for bacterial ATP binding cassette (ABC) importers. Despite the wealth of biochemical and structural data available, a general model describing the various translocation pathways is still lacking. In this study, we formulate a mathematical model with the goal of determining the transporter reaction pathway, specifically looking at the order of binding events and conformation changes by which transport proceeds. Fitting our mathematical model to equilibrium binding data, we estimate the unknown equilibrium parameters of the system, several of which are key determinants of the transport process. Using these estimates along with steady-state ATPase rate data, we determine which of several possible reaction pathways is dominant, as a function of five underdetermined kinetic parameter values. Because neither experimental measurements nor estimates of certain kinetic rate constants are available, the problem of deciding which of the reaction pathways is responsible for transport remains unsolved. However, using the mathematical framework developed here, a firmer conclusion regarding the dominant reaction pathway as a function of MalE and maltose concentration could be drawn once these unknown kinetic parameters are determined.

We model the extent to which age-targeted protective sequestration can be used to reduce ICU admissions caused by novel coronavirus COVID-19. Using demographic data from New Zealand, we demonstrate that lowering the age threshold to 50 years of age reduces ICU admissions drastically and show that for sufficiently strict isolation protocols, sequestering one-third of the countries population for a total of 8 months is sufficient to avoid overwhelming ICU capacity throughout the entire course of the epidemic. Similar results are expected to hold for other countries, though some minor adaption will be required based on local age demographics and hospital facilities.

Excitable media sustain circulating waves. In the heart, sustained circulating waves can lead to serious impairment or even death. To investigate factors affecting the stability of such waves, we have used optogenetic techniques to stimulate a region at the apex of a mouse heart at a fixed delay after the detection of excitation at the base of the heart. For long delays, rapid circulating rhythms can be sustained, whereas for shorter delays, there are paroxysmal bursts of activity that start and stop spontaneously. By considering the dependence of the action potential and conduction velocity on the preceding recovery time using restitution curves, as well as the reduced excitability (fatigue) due to the rapid excitation, we model prominent features of the dynamics including alternation of the duration of the excited phases and conduction times, as well as termination of the bursts for short delays. We propose that this illustrates universal mechanisms that exist in biological systems for the self-termination of such activities.

Chemical, physical, and ecological systems passing through a saddle-node bifurcation will, momentarily, find themselves balanced at a semistable steady state. If perturbed by noise, such systems will escape from the zero-steady state, with escape time sensitive to noise. When the model is extended to include space, this leads to different points in space "escaping from zero" at different times, and uniform initial conditions nucleate into sharp peaks spreading randomly across a nearly uniform background, a phenomenon closely resembling nucleation during phase transition. We use Large Deviation Theory to determine burst shape and temporal scaling with respect to noise amplitude. These results give a prototype for a particular form of patternless symmetry breaking in the vicinity of a stability boundary and demonstrate how microscopic noise can lead to macroscopic effects in such a region.

In growing plant cells, parallel ordering of microtubules (MTs) along the inner surface of the cell membrane influences the direction of cell expansion and thereby plant morphology. For correct expansion of organs that primarily grow by elongating, such as roots and stems, MTs must bend in the high-curvature direction along the cylindrically shaped cell membrane in order to form the required circumferential arrays. Computational studies, which have recapitulated the self-organization of these arrays, ignored MT mechanics and assumed MTs follow geodesics of the cell surface. Here, we show, through analysis of a derived Euler-Lagrange equation, that an elastic MT constrained to a cylindrical surface will deflect away from geodesics and toward low curvature directions to minimize bending energy. This occurs when the curvature of the cell surface is relatively high for a given anchor density. In the limit of infinite anchor density, MTs always follow geodesics. We compare our analytical predictions to measured curvatures and anchor densities and find that the regime in which cells are forming these cortical arrays straddles the region of parameter space in which arrays must form under the antagonistic influence of this mechanically induced deflection. Although this introduces a potential obstacle to forming circumferentially orientated arrays that needs to be accounted for in the models, it also raises the question of whether plants use this mechanical phenomenon to regulate array orientation. The model also constitutes an elegant generalization of the classical Euler-buckling instability along with an intrinsic unfolding of the associated pitchfork bifurcation.

Active learning techniques, such as peer instruction and group work, have been gaining a lot of traction in universities. Taking a natural next step in re-evaluating current practices, many institutions recently started experimenting student-centred group exams. In order to assess the feasibility and effectiveness of collaborative assessments, we implemented a study that focuses on lower-level and intermediate-level mathematics courses with large student populations. We collected and analyzed data relating to student material retention, their perceptions of the format, as well as the effects of the group composition (in terms of the strength of its individual members) and question types (multiple choice versus short answer) on the exam success. Although we establish that long-term learning is not strongly influenced by the group exams, students find them useful and enjoyable, with the majority of students preferring group exams to individual ones. Examining group composition and question type results, we conclude that multiple choice questions are more suitable for group exams. We also comment on the format in view of limited people and time resources available for test creation, administration, and marking. Finally, we consider instructor perspectives on the process.

A monolayer of chick embryo cardiac cells grown in an annular geometry supports two simultaneous reentrant excitation waves that circulate as a doublet. We propose a mechanism that can lead to such behavior. The velocity restitution gives the instantaneous velocity of a wave as a function of the time since the passage of the previous wave at a given point in space. Nonmonotonic restitution relationships will lead to situations in which various spacings between circulating waves are possible. In cardiology, the situation in which two waves travel in an anatomically defined circuit is referred to as double-wave reentry. Since double-wave reentry may arise as a consequence of pacing during cardiac arrhythmias, understanding the dynamic features of double-wave reentry may be helpful in understanding the physiological properties of cardiac tissue and in the design of therapy.

The FitzHugh-Nagumo (FHN) system of PDEs is a generic model for excitable media, often used to build a qualitative understanding of electrophysiological phenomena. A well-characterized traveling pulse solution to FHN serves as a model for action potentials in cardiac tissue and other contexts. The stability of the traveling pulse has been well studied, but the more global problem of predicting when an arbitrary initial condition will converge to the uniform rest solution and when it will converge to the traveling pulse remains unsolved. Here we prove the existence of an invariant winding number in an asymptotic limit of the FHN system called the singular FHN system (SFHN) that provides a crucial step toward a global convergence result. This result is valid for a relatively general class of nonlinearities. In addition, we provide evidence that our SFHN results extend with limitations to FHN and outline conditions under which the SFHN approximation fails. The invariant winding number provides explanations for several observations of physiological relevance. For example, it explains the requirements on stimulus protocols that allow the formation and elimination of reentrant rhythms in cardiac tissue.

ATP-binding cassette (ABC) transporters have evolved an ATP-dependent alternating-access mechanism to transport substrates across membranes. Despite important progress, especially in their structural analysis, it is still unknown how the substrate stimulates ATP hydrolysis, the hallmark of ABC transporters. In this study, we measure the ATP turnover cycle of MalFGK2 in steady and pre-steady state conditions. We show that (i) the basal ATPase activity of MalFGK2 is very low because the cleavage of ATP is rate-limiting, (ii) the binding of open-state MalE to the transporter induces ATP cleavage but leaves release of Pi limiting, and (iii) the additional presence of maltose stimulates release of Pi, and therefore increases the overall ATP turnover cycle. We conclude that open-state MalE stabilizes MalFGK2 in the outward-facing conformation until maltose triggers return to the inward-facing state for substrate and Pi release. This concerted action explains why ATPase activity of MalFGK2 depends on maltose, and why MalE is essential for transport.

Chloroplasts regulate their growth to optimize photosynthesis. Quantitative data show that the ratio of total chloroplast area to mesophyll cell area is constant across different cells within a single species and also across species. Wild-type chloroplasts exhibit little scatter around this trend; highly irregularly shaped mutant chloroplasts exhibit more scatter. Here we propose a model motivated by a bacterial quorum-sensing model consisting of a switch-like signaling network that turns off chloroplast growth. We calculated the dependence of the location of the relevant saddle-node bifurcation on the geometry of the chloroplasts. Our model exhibits a linear trend, with linearly growing scatter dependent on chloroplast shape, consistent with the data. When modeled chloroplasts are of a shape that grows with a constant area-to-volume ratio (disks, cylinders), we find a linear trend with minimal scatter. Chloroplasts with area and volume that do not grow proportionally (spheres) exhibit a linear trend with additional scatter.