The mechanism of MinD stability modulation by MinE in Min protein dynamics.

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.

A Stochastic Model of Cortical Microtubule Anchoring and Mechanics Provides Regulatory Control of Microtubule Shape.

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.

A multiscale computational model of YAP signaling in epithelial fingering behavior.

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.

Correction to: A Mathematical Model for the Kinetics of the MalFGK2 Maltose Transporter.

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.

A Mathematical Model for the Kinetics of the MalFGK[Formula: see text] Maltose Transporter.

The MalFGK[Formula: see text] 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.

Universal mechanisms for self-termination of rapid cardiac rhythm.

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.

Double-wave reentry in excitable media.

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.

A Model of Chloroplast Growth Regulation in Mesophyll Cells.

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.

Centering of a radial microtubule array by translocation along microtubules spontaneously nucleated in the cytoplasm.

Positioning of a radial array of microtubules (MTs) in the cell centre is crucial for cytoplasmic organization, but the mechanisms of such centering are difficult to study in intact cells that have pre-formed radial arrays. Here, we use cytoplasmic fragments of melanophores, and cytoplasts of BS-C-1 cells to study MT centering mechanisms. Using live imaging and computer modelling, we show that the MT aster finds a central location in the cytoplasm by moving along spontaneously nucleated non-astral MTs towards a point at which MT nucleation events occur equally on all sides. We hypothesize that similar mechanisms, in the presence of the centrosome, contribute to this centering mechanism and ensure the robustness of cytoplasmic organization.

Force generation by a dynamic Z-ring in Escherichia coli cell division.

FtsZ, a bacterial homologue of tubulin, plays a central role in bacterial cell division. It is the first of many proteins recruited to the division site to form the Z-ring, a dynamic structure that recycles on the time scale of seconds and is required for division to proceed. FtsZ has been recently shown to form rings inside tubular liposomes and to constrict the liposome membrane without the presence of other proteins, particularly molecular motors that appear to be absent from the bacterial proteome. Here, we propose a mathematical model for the dynamic turnover of the Z-ring and for its ability to generate a constriction force. Force generation is assumed to derive from GTP hydrolysis, which is known to induce curvature in FtsZ filaments. We find that this transition to a curved state is capable of generating a sufficient force to drive cell-wall invagination in vivo and can also explain the constriction seen in the in vitro liposome experiments. Our observations resolve the question of how FtsZ might accomplish cell division despite the highly dynamic nature of the Z-ring and the lack of molecular motors.

Beta-cell specific Insr deletion promotes insulin hypersecretion and improves glucose tolerance prior to global insulin resistance.

Insulin receptor (Insr) protein is present at higher levels in pancreatic ß-cells than in most other tissues, but the consequences of ß-cell insulin resistance remain enigmatic. Here, we use an Ins1cre knock-in allele to delete Insr specifically in ß-cells of both female and male mice. We compare experimental mice to Ins1cre-containing littermate controls at multiple ages and on multiple diets. RNA-seq of purified recombined ß-cells reveals transcriptomic consequences of Insr loss, which differ between female and male mice. Action potential and calcium oscillation frequencies are increased in Insr knockout ß-cells from female, but not male mice, whereas only male ßInsrKO islets have reduced ATP-coupled oxygen consumption rate and reduced expression of genes involved in ATP synthesis. Female ßInsrKO and ßInsrHET mice exhibit elevated insulin release in ex vivo perifusion experiments, during hyperglycemic clamps, and following i.p. glucose challenge. Deletion of Insr does not alter ß-cell area up to 9 months of age, nor does it impair hyperglycemia-induced proliferation. Based on our data, we adapt a mathematical model to include ß-cell insulin resistance, which predicts that ß-cell Insr knockout improves glucose tolerance depending on the degree of whole-body insulin resistance. Indeed, glucose tolerance is significantly improved in female ßInsrKO and ßInsrHET mice compared to controls at 9, 21 and 39 weeks, and also in insulin-sensitive 4-week old males. We observe no 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 ß-cell specific insulin resistance. The propensity for hyperinsulinemia is associated with mildly reduced fasting glucose and increased body weight. We further validate our main in vivo findings using an Ins1-CreERT transgenic line and find that male mice have improved glucose tolerance 4 weeks after tamoxifen-mediated Insr deletion. Collectively, our data show that ß-cell insulin resistance in the form of reduced ß-cell Insr contributes to hyperinsulinemia in the context of glucose stimulation, thereby improving glucose homeostasis in otherwise insulin sensitive sex, dietary and age contexts.

A mechanochemical model explains interactions between cortical microtubules in plants.

Microtubules anchored to the two-dimensional cortex of plant cells collide through plus-end polymerization. Collisions can result in rapid depolymerization, directional plus-end entrainment, or crossover. These interactions are believed to give rise to cellwide self-organization of plant cortical microtubules arrays, which is required for proper cell wall growth. Although the cell-wide self-organization has been well studied, less emphasis has been placed on explaining the interactions mechanistically from the molecular scale. Here we present a model for microtubule-cortex anchoring and collision-based interactions between microtubules, based on a competition between cross-linker bonding, microtubule bending, and microtubule polymerization. Our model predicts a higher probability of entrainment at smaller collision angles and at longer unanchored lengths of plus-ends. This model addresses observed differences between collision resolutions in various cell types, including Arabidopsis cells and Tobacco cells.

Estimating the bending modulus of a FtsZ bacterial-division protein filament.

FtsZ, a cytoskeletal protein homologous to tubulin, is the principle constituent of the division ring in bacterial cells. It is known to have force-generating capacity in vitro and has been conjectured to be the source of the constriction force in vivo. Several models have been proposed to explain the generation of force by the Z ring. Here we re-examine data from in vitro experiments in which Z rings formed and constricted inside tubular liposomes, and we carry out image analysis on previously published data with which to better estimate important model parameters that have proven difficult to measure by direct means. We introduce a membrane-energy-based model for the dynamics of multiple Z rings moving and colliding inside a tubular liposome and a fluid model for the drag of a Z ring as it moves through the tube. Using this model, we estimate an effective membrane bending modulus of 500-700 pN nm. If we assume that FtsZ force generation is driven by hydrolysis into a highly curved conformation, we estimate the FtsZ filament bending modulus to be 310-390 pN nm(2). If we assume instead that force is generated by the non-hydrolysis-dependent intermediate curvature conformation, we find that B(f)>1400 pN nm(2). The former value sits at the lower end of the range of previously estimated values and, if correct, may raise challenges for models that rely on filament bending to generate force.

A CLASP-modulated cell edge barrier mechanism drives cell-wide cortical microtubule organization in Arabidopsis.

It is well known that the parallel order of microtubules in the plant cell cortex defines the direction of cell expansion, yet it remains unclear how microtubule orientation is controlled, especially on a cell-wide basis. Here we show through 4D imaging and computational modelling that plant cell polyhedral geometry provides spatial input that determines array orientation and heterogeneity. Microtubules depolymerize when encountering sharp cell edges head-on, whereas those oriented parallel to those sharp edges remain. Edge-induced microtubule depolymerization, however, is overcome by the microtubule-associated protein CLASP, which accumulates at specific cell edges, enables microtubule growth around sharp edges and promotes formation of microtubule bundles that span adjacent cell faces. By computationally modelling dynamic 'microtubules on a cube' with edges differentially permissive to microtubule passage, we show that the CLASP-edge complex is a 'tuneable' microtubule organizer, with the inherent flexibility to generate the numerous cortical array patterns observed in nature.

Mechanisms of self-organization of cortical microtubules in plants revealed by computational simulations.

Microtubules confined to the two-dimensional cortex of elongating plant cells must form a parallel yet dispersed array transverse to the elongation axis for proper cell wall expansion. Some of these microtubules exhibit free minus-ends, leading to migration at the cortex by hybrid treadmilling. Collisions between microtubules can result in plus-end entrainment ("zippering") or rapid depolymerization. Here, we present a computational model of cortical microtubule organization. We find that plus-end entrainment leads to self-organization of microtubules into parallel arrays, whereas catastrophe-inducing collisions do not. Catastrophe-inducing boundaries (e.g., upper and lower cross-walls) can tune the orientation of an ordered array to a direction transverse to elongation. We also find that changes in dynamic instability parameters, such as in mor1-1 mutants, can impede self-organization, in agreement with experimental data. Increased entrainment, as seen in clasp-1 mutants, conserves self-organization, but delays its onset and fails to demonstrate increased ordering. We find that branched nucleation at acute angles off existing microtubules results in distinctive sparse arrays and infer either that microtubule-independent or coparallel nucleation must dominate. Our simulations lead to several testable predictions, including the effects of reduced microtubule severing in katanin mutants.

Predictions from a stochastic polymer model for the MinDE protein dynamics in Escherichia coli.

The spatiotemporal oscillations of the Min proteins in the bacterium Escherichia coli play an important role in cell division. A number of different models have been proposed to explain the dynamics from the underlying biochemistry. Here, we extend a previously described discrete polymer model from a deterministic to a stochastic formulation. We express the stochastic evolution of the oscillatory system as a map from the probability distribution of maximum polymer length in one period of the oscillation to the probability distribution of maximum polymer length half a period later and solve for the fixed point of the map with a combined analytical and numerical technique. This solution gives a theoretical prediction of the distributions of both lengths of the polar MinD zones and periods of oscillations--both of which are experimentally measurable. The model provides an interesting example of a stochastic hybrid system that is, in some limits, analytically tractable.

A multistranded polymer model explains MinDE dynamics in E. coli cell division.

In Escherichia coli, the location of the site for cell division is regulated by the action of the Min proteins. These proteins undergo a periodic pole-to-pole oscillation that involves polymerization and ATPase activity of MinD under the controlling influence of MinE. This oscillation suppresses division near the poles while permitting division at midcell. Here, we propose a multistranded polymer model for MinD and MinE dynamics that quantitatively agrees with the experimentally observed dynamics in wild-type cells and in several well-studied mutant phenotypes. The model also provides new explanations for several phenotypes that have never been addressed by previous modeling attempts. In doing so, the model bridges a theoretical gap between protein structure, biochemistry, and mutant phenotypes. Finally, the model emphasizes the importance of nonequilibrium polymer dynamics in cell function by demonstrating how behavior analogous to the dynamic instability of microtubules is used by E. coli to achieve a sufficiently rapid timescale in controlling division site selection.

Actin filament branching and protrusion velocity in a simple 1D model of a motile cell.

We formulate and analyse a 1D model for the spatial distribution of actin density at the leading edge of a motile cell. The model incorporates nucleation, capping, growth and decay of actin filaments, as well as retrograde flow of the actin meshwork and known parameter values based on the literature. Using a simplified geometry, and reasonable assumptions about the biochemical processes, we derive PDEs for the density of actin filaments and their tips. Analytic travelling wave solutions are used to predict how the speed of the cell depends on rates of nucleation, capping, polymerization and membrane resistance. Analysis and simulations agree with experimental profiles for measured actin distributions. Extended versions of the model are studied numerically. We find that our model produces stable travelling wave solutions with reasonable cell speeds. Increasing the rate of nucleation of filaments (by the actin related protein Arp2/3) or the rate of actin polymerization leads to faster cell speed, whereas increasing the rate of capping or the membrane resistance reduces cell speed. We consider several variants of nucleation (spontaneous, tip, and side branching) and find best agreement with experimentally measured spatial profiles of filament and tip density in the side branching case.

Periodic stimulus and the single cardiac cell-getting more out of 1D maps.

The response of an isolated cardiac cell to a periodic stimulus has traditionally been studied in terms of the duration of the action potential (APD) immediately following each stimulus. The APD approach offers explanations of several experimental observations, including the stability of the so-called 1:1 response which is thought to be relevant to the problem of spiral wave breakup and the onset of fibrillation. A discussion of some theoretical problems with the APD approach is given in order to motivate the derivation of a new type of map. This new one-dimensional map, which gives successive values of the prestimulus transmembrane potential instead of successive values of APD, relies on the presence of a one-dimensional slow manifold in the underlying dynamics. This slow manifold map extends the understanding offered by the APD approach to include an explanation of Wenckebach rhythms. In addition, the bifurcation structure of the map provides a unified description of the parameter dependence that agrees fairly well with experimental observation.