Physical model of contractile ring initiation in dividing cells

Membrane rigidity regulates E. coli proliferation rates

Combining single cell experiments, population dynamics and theoretical methods of membrane mechanics, we put forward that the rate of cell proliferation in E. coli colonies can be regulated by modifiers of the mechanical properties of the bacterial membrane. Bacterial proliferation was modelled as mediated by cell division through a membrane constriction divisome based on FtsZ, a mechanically competent protein at elastic interaction against membrane rigidity. Using membrane fluctuation spectroscopy in the single cells, we revealed either membrane stiffening when considering hydrophobic long chain fatty substances, or membrane softening if short-chained hydrophilic molecules are used. Membrane stiffeners caused hindered growth under normal division in the microbial cultures, as expected for membrane rigidification. Membrane softeners, however, altered regular cell division causing persistent microbes that abnormally grow as long filamentous cells proliferating apparently faster. We invoke the concept of effective growth rate under the assumption of a heterogeneous population structure composed by distinguishable individuals with different FtsZ-content leading the possible forms of cell proliferation, from regular division in two normal daughters to continuous growing filamentation and budding. The results settle altogether into a master plot that captures a universal scaling between membrane rigidity and the divisional instability mediated by FtsZ at the onset of membrane constriction.

More than just a barrier: using physical models to couple membrane shape to cell function

The correct execution of many cellular processes, such as division and motility, requires the cell to adopt a specific shape. Physically, these shapes are determined by the interplay of the plasma membrane and internal cellular driving factors. While the plasma membrane defines the boundary of the cell, processes inside the cell can result in the generation of forces that deform the membrane. These processes include protein binding, the assembly of protein superstructures, and the growth and contraction of cytoskeletal networks. Due to the complexity of the cell, relating observed membrane deformations back to internal processes is a challenging problem. Here, we review cell shape changes in endocytosis, cell adhesion, cell migration and cell division and discuss how by modeling membrane deformations we can investigate the inner working principles of the cell.

On the Role of Curved Membrane Nanodomains, and Passive and Active Skeleton Forces in the Determination of Cell Shape and Membrane Budding

Biological membranes are composed of isotropic and anisotropic curved nanodomains. Anisotropic membrane components, such as Bin/Amphiphysin/Rvs (BAR) superfamily protein domains, could trigger/facilitate the growth of membrane tubular protrusions, while isotropic curved nanodomains may induce undulated (necklace-like) membrane protrusions. We review the role of isotropic and anisotropic membrane nanodomains in stability of tubular and undulated membrane structures generated or stabilized by cyto- or membrane-skeleton. We also describe the theory of spontaneous self-assembly of isotropic curved membrane nanodomains and derive the critical concentration above which the spontaneous necklace-like membrane protrusion growth is favorable. We show that the actin cytoskeleton growth inside the vesicle or cell can change its equilibrium shape, induce higher degree of segregation of membrane nanodomains or even alter the average orientation angle of anisotropic nanodomains such as BAR domains. These effects may indicate whether the actin cytoskeleton role is only to stabilize membrane protrusions or to generate them by stretching the vesicle membrane. Furthermore, we demonstrate that by taking into account the in-plane orientational ordering of anisotropic membrane nanodomains, direct interactions between them and the extrinsic (deviatoric) curvature elasticity, it is possible to explain the experimentally observed stability of oblate (discocyte) shapes of red blood cells in a broad interval of cell reduced volume. Finally, we present results of numerical calculations and Monte-Carlo simulations which indicate that the active forces of membrane skeleton and cytoskeleton applied to plasma membrane may considerably influence cell shape and membrane budding.

Tools for computational analysis of moving boundary problems in cellular mechanobiology

A cell's ability to change shape is one of the most fundamental biological processes and is essential for maintaining healthy organisms. When the ability to control shape goes awry, it often results in a diseased system. As such, it is important to understand the mechanisms that allow a cell to sense and respond to its environment so as to maintain cellular shape homeostasis. Because of the inherent complexity of the system, computational models that are based on sound theoretical understanding of the biochemistry and biomechanics and that use experimentally measured parameters are an essential tool. These models involve an inherent feedback, whereby shape is determined by the action of regulatory signals whose spatial distribution depends on the shape. To carry out computational simulations of these moving boundary problems requires special computational techniques. A variety of alternative approaches, depending on the type and scale of question being asked, have been used to simulate various biological processes, including cell motility, division, mechanosensation, and cell engulfment. In general, these models consider the forces that act on the system (both internally generated, or externally imposed) and the mechanical properties of the cell that resist these forces. Moving forward, making these techniques more accessible to the non-expert will help improve interdisciplinary research thereby providing new insight into important biological processes that affect human health. This article is categorized under: Cancer > Cancer>Computational Models Cancer > Cancer>Molecular and Cellular Physiology.

Towards Design of Self-Organizing Biomimetic Systems

The ability of designing biosynthetic systems with well-defined functional biomodules from scratch is an ambitious and revolutionary goal to deliver innovative, engineered solutions to future challenges in biotechnology and process systems engineering. In this work, several key challenges including modularization, functional biomodule identification, and assembly are discussed. In addition, an in silico protocell modeling approach is presented as a foundation for a computational model-based toolkit for rational analysis and modular design of biomimetic systems.

Classical and Emerging Regulatory Mechanisms of Cytokinesis in Animal Cells

The primary goal of cytokinesis is to produce two daughter cells, each having a full set of chromosomes. To achieve this, cells assemble a dynamic structure between segregated sister chromatids called the contractile ring, which is made up of filamentous actin, myosin-II, and other regulatory proteins. Constriction of the actomyosin ring generates a cleavage furrow that divides the cytoplasm to produce two daughter cells. Decades of research have identified key regulators and underlying molecular mechanisms; however, many fundamental questions remain unanswered and are still being actively investigated. This review summarizes the key findings, computational modeling, and recent advances in understanding of the molecular mechanisms that control the formation of the cleavage furrow and cytokinesis.

Guided by curvature: shaping cells by coupling curved membrane proteins and cytoskeletal forces

Eukaryote cells have flexible membranes that allow them to have a variety of dynamical shapes. The shapes of the cells serve important biological functions, both for cells within an intact tissue, and during embryogenesis and cellular motility. How cells control their shapes and the structures that they form on their surface has been a subject of intensive biological research, exposing the building blocks that cells use to deform their membranes. These processes have also drawn the interest of theoretical physicists, aiming to develop models based on physics, chemistry and nonlinear dynamics. Such models explore quantitatively different possible mechanisms that the cells can employ to initiate the spontaneous formation of shapes and patterns on their membranes. We review here theoretical work where one such class of mechanisms was investigated: the coupling between curved membrane proteins, and the cytoskeletal forces that they recruit. Theory indicates that this coupling gives rise to a rich variety of membrane shapes and dynamics, while experiments indicate that this mechanism appears to drive many cellular shape changes.This article is part of the theme issue 'Self-organization in cell biology'.

Self-organized shape dynamics of active surfaces

Mechanochemical processes in thin biological structures, such as the cellular cortex or epithelial sheets, play a key role during the morphogenesis of cells and tissues. In particular, they are responsible for the dynamical organization of active stresses that lead to flows and deformations of the material. Consequently, advective transport redistributes force-generating molecules and thereby contributes to a complex mechanochemical feedback loop. It has been shown in fixed geometries that this mechanism enables patterning, but the interplay of these processes with shape changes of the material remains to be explored. In this work, we study the fully self-organized shape dynamics using the theory of active fluids on deforming surfaces and develop a numerical approach to solve the corresponding force and torque balance equations. We describe the spontaneous generation of nontrivial surface shapes, shape oscillations, and directed surface flows that resemble peristaltic waves from self-organized, mechanochemical processes on the deforming surface. Our approach provides opportunities to explore the dynamics of self-organized active surfaces and can help to understand the role of shape as an integral element of the mechanochemical organization of morphogenetic processes.

Shaping up synthetic cells

How do the cells in our body reconfigure their shape to achieve complex tasks like migration and mitosis, yet maintain their shape in response to forces exerted by, for instance, blood flow and muscle action? Cell shape control is defined by a delicate mechanical balance between active force generation and passive material properties of the plasma membrane and the cytoskeleton. The cytoskeleton forms a space-spanning fibrous network comprising three subsystems: actin, microtubules and intermediate filaments. Bottom-up reconstitution of minimal synthetic cells where these cytoskeletal subsystems are encapsulated inside a lipid vesicle provides a powerful avenue to dissect the force balance that governs cell shape control. Although encapsulation is technically demanding, a steady stream of advances in this technique has made the reconstitution of shape-changing minimal cells increasingly feasible. In this topical review we provide a route-map of the recent advances in cytoskeletal encapsulation techniques and outline recent reports that demonstrate shape change phenomena in simple biomimetic vesicle systems. We end with an outlook toward the next steps required to achieve more complex shape changes with the ultimate aim of building a fully functional synthetic cell with the capability to autonomously grow, divide and move.

Mathematical Model of Contractile Ring-Driven Cytokinesis in a Three-Dimensional Domain

In this paper, a mathematical model of contractile ring-driven cytokinesis is presented by using both phase-field and immersed-boundary methods in a three-dimensional domain. It is one of the powerful hypotheses that cytokinesis happens driven by the contractile ring; however, there are only few mathematical models following the hypothesis, to the author's knowledge. I consider a hybrid method to model the phenomenon. First, a cell membrane is represented by a zero-contour of a phase-field implicitly because of its topological change. Otherwise, immersed-boundary particles represent a contractile ring explicitly based on the author's previous work. Here, the multi-component (or vector-valued) phase-field equation is considered to avoid the emerging of each cell membrane right after their divisions. Using a convex splitting scheme, the governing equation of the phase-field method has unique solvability. The numerical convergence of contractile ring to cell membrane is proved. Several numerical simulations are performed to validate the proposed model.

Mechanics of active surfaces

We derive a fully covariant theory of the mechanics of active surfaces. This theory provides a framework for the study of active biological or chemical processes at surfaces, such as the cell cortex, the mechanics of epithelial tissues, or reconstituted active systems on surfaces. We introduce forces and torques acting on a surface, and derive the associated force balance conditions. We show that surfaces with in-plane rotational symmetry can have broken up-down, chiral, or planar-chiral symmetry. We discuss the rate of entropy production in the surface and write linear constitutive relations that satisfy the Onsager relations. We show that the bending modulus, the spontaneous curvature, and the surface tension of a passive surface are renormalized by active terms. Finally, we identify active terms which are not found in a passive theory and discuss examples of shape instabilities that are related to active processes in the surface.

On the role of external force of actin filaments in the formation of tubular protrusions of closed membrane shapes with anisotropic membrane components

Biological membranes are composed of different components and there is no a priori reason to assume that all components are isotropic. It was previously shown that the anisotropic properties of membrane components may explain the stability of membrane tubular protrusions even without the application of external force. Our theoretical study focuses on the role of anisotropic membrane components in the stability of membrane tubular structures generated or stabilized by actin filaments. We show that the growth of the actin cytoskeleton inside the vesicle can induce the partial lateral segregation of different membrane components. The entropy of mixing of membrane components hinders the total lateral segregation of the anisotropic and isotropic membrane components. Self-assembled aggregates formed by anisotropic membrane components facilitate the growth of long membrane tubular protrusions. Protrusive force generated by actin filaments favors strong segregation of membrane components by diminishing the opposing effect of mixing entropy.

General neck condition for the limit shape of budding vesicles

The shape equation and linking conditions for a vesicle with two phase domains are derived. We refine the conjecture on the general neck condition for the limit shape of a budding vesicle proposed by Jülicher and Lipowsky [Phys. Rev. Lett. 70, 2964 (1993)PRLTAO0031-900710.1103/PhysRevLett.70.2964; Phys. Rev. E 53, 2670 (1996)1063-651X10.1103/PhysRevE.53.2670], and then we use the shape equation and linking conditions to prove that this conjecture holds not only for axisymmetric budding vesicles, but also for asymmetric ones. Our study reveals that the mean curvature at any point on the membrane segments adjacent to the neck satisfies the general neck condition for the limit shape of a budding vesicle when the length scale of the membrane segments is much larger than the characteristic size of the neck but still much smaller than the characteristic size of the vesicle.

Actin Cytoskeleton-Mediated Constriction of Membrane Organelles via Endoplasmic Reticulum Scaffolding

Intracellular organelles constantly undergo fission to facilitate turnover, transport, and functional changes. The cytoskeleton has long been understood to play a role in these events, and recent work strongly suggests that several conserved molecular players cooperate with the cytoskeleton to mediate the fission process. Membrane curvature-inducing, membrane scission proteins, and force-inducing cytoskeletal proteins all cooperate to drive the fission process. Recent work suggests that the endoplasmic reticulum serves as the linchpin that orchestrates and spatially organizes fission via these curvature-inducing, scission, and force-producing molecules. This all leads us to postulate a "universal theory" of organelle fission with distinct biophysical and biochemical features mediated by a finite number of physical and molecular constraints. This new physical paradigm deserves special attention from those who wish to model these processes, since previous theoretical and experimental attempts to elucidate these fission mechanisms have not included the organizing factor of the endoplasmic reticulum. Here we review the basic concepts of this new model for organelle fission, and explore the implications thereof. Previous studies that didn't include this component can now be interpreted in light of these new data and serve as a useful guide for understanding how this process happens in vivo. Thus, this review provides direction for future modeling and experimental efforts to better understand how these complex systems and processes are regulated in both healthy and diseased biological systems.

Still and rotating myosin clusters determine cytokinetic ring constriction

The cytokinetic ring is essential for separating daughter cells during division. It consists of actin filaments and myosin motors that are generally assumed to organize as sarcomeres similar to skeletal muscles. However, direct evidence is lacking. Here we show that the internal organization and dynamics of rings are different from sarcomeres and distinct in different cell types. Using micro-cavities to orient rings in single focal planes, we find in mammalian cells a transition from a homogeneous distribution to a periodic pattern of myosin clusters at the onset of constriction. In contrast, in fission yeast, myosin clusters rotate prior to and during constriction. Theoretical analysis indicates that both patterns result from acto-myosin self-organization and reveals differences in the respective stresses. These findings suggest distinct functional roles for rings: contraction in mammalian cells and transport in fission yeast. Thus self-organization under different conditions may be a generic feature for regulating morphogenesis in vivo.

A theoretical model of cytokinesis implicates feedback between membrane curvature and cytoskeletal organization in asymmetric cytokinetic furrowing

Furrow ingression is asymmetric in cytokinesis in the Caenorhabditis elegans zygote. A combination of quantitative high-resolution live-cell microscopy and theoretical modeling revealed a mechanistic basis for asymmetry: feedback among membrane curvature, cytoskeletal alignment, and contractility. The model also suggests that asymmetry promotes energy efficiency.

During cytokinesis, the cell undergoes a dramatic shape change as it divides into two daughter cells. Cell shape changes in cytokinesis are driven by a cortical ring rich in actin filaments and nonmuscle myosin II. The ring closes via actomyosin contraction coupled with actin depolymerization. Of interest, ring closure and hence the furrow ingression are nonconcentric (asymmetric) within the division plane across Metazoa. This nonconcentricity can occur and persist even without preexisting asymmetric cues, such as spindle placement or cellular adhesions. Cell-autonomous asymmetry is not explained by current models. We combined quantitative high-resolution live-cell microscopy with theoretical modeling to explore the mechanistic basis for asymmetric cytokinesis in the Caenorhabditis elegans zygote, with the goal of uncovering basic principles of ring closure. Our theoretical model suggests that feedback among membrane curvature, cytoskeletal alignment, and contractility is responsible for asymmetric cytokinetic furrowing. It also accurately predicts experimental perturbations of conserved ring proteins. The model further suggests that curvature-mediated filament alignment speeds up furrow closure while promoting energy efficiency. Collectively our work underscores the importance of membrane–cytoskeletal anchoring and suggests conserved molecular mechanisms for this activity.

Modeling large-scale dynamic processes in the cell: polarization, waves, and division

The past decade has witnessed significant developments in molecular biology techniques, fluorescent labeling, and super-resolution microscopy, and together these advances have vastly increased our quantitative understanding of the cell. This detailed knowledge has concomitantly opened the door for biophysical modeling on a cellular scale. There have been comprehensive models produced describing many processes such as motility, transport, gene regulation, and chemotaxis. However, in this review we focus on a specific set of phenomena, namely cell polarization, F-actin waves, and cytokinesis. In each case, we compare and contrast various published models, highlight the relevant aspects of the biology, and provide a sense of the direction in which the field is moving.

Dynamic network morphology and tension buildup in a 3D model of cytokinetic ring assembly

During fission yeast cytokinesis, actin filaments nucleated by cortical formin Cdc12 are captured by myosin motors bound to a band of cortical nodes and bundled by cross-linking proteins. The myosin motors exert forces on the actin filaments, resulting in a net pulling of the nodes into a contractile ring, while cross-linking interactions help align actin filaments and nodes into a single bundle. We used these mechanisms in a three-dimensional computational model of contractile ring assembly, with semiflexible actin filaments growing from formins at cortical nodes, capturing of filaments by neighboring nodes, and cross-linking among filaments through attractive interactions. The model was used to predict profiles of actin filament density at the cell cortex, morphologies of condensing node-filament networks, and regimes of cortical tension by varying the node pulling force and strength of cross-linking among actin filaments. Results show that cross-linking interactions can lead to confinement of actin filaments at the simulated cortical boundary. We show that the ring-formation region in parameter space lies close to regions leading to clumps, meshworks or double rings, and stars/cables. Since boundaries between regions are not sharp, transient structures that resemble clumps, stars, and meshworks can appear in the process of ring assembly. These results are consistent with prior experiments with mutations in actin-filament turnover regulators, myosin motor activity, and changes in the concentration of cross-linkers that alter the morphology of the condensing network. Transient star shapes appear in some simulations, and these morphologies offer an explanation for star structures observed in prior experimental images. Finally, we quantify tension along actin filaments and forces on nodes during ring assembly and show that the mechanisms describing ring assembly can also drive ring constriction once the ring is formed.

The value of mechanistic biophysical information for systems-level understanding of complex biological processes such as cytokinesis

This review illustrates the value of quantitative information including concentrations, kinetic constants and equilibrium constants in modeling and simulating complex biological processes. Although much has been learned about some biological systems without these parameter values, they greatly strengthen mechanistic accounts of dynamical systems. The analysis of muscle contraction is a classic example of the value of combining an inventory of the molecules, atomic structures of the molecules, kinetic constants for the reactions, reconstitutions with purified proteins and theoretical modeling to account for the contraction of whole muscles. A similar strategy is now being used to understand the mechanism of cytokinesis using fission yeast as a favorable model system.

Interaction between dipolar lipid headgroups and charged nanoparticles mediated by water dipoles and ions

In this work, a theoretical model describing the interaction between a positively or negatively charged nanoparticle and neutral zwitterionic lipid bilayers is presented. It is shown that in the close vicinity of the positively charged nanoparticle, the zwitterionic lipid headgroups are less extended in the direction perpendicular to the membrane surface, while in the vicinity of the negatively charged nanoparticle, the headgroups are more extended. This result coincides with the calculated increase in the osmotic pressure between the zwitterionic lipid surface and positively charged nanoparticle and the decrease of osmotic pressure between the zwitterionic lipid surface and the negatively charged nanoparticle. Our theoretical predictions agree well with the experimentally determined fluidity of a lipid bilayer membrane in contact with positively or negatively charged nanoparticles. The prospective significance of the present work is mainly to contribute to better understanding of the interactions of charged nanoparticles with a zwitterionic lipid bilayer, which may be important in the efficient design of the lipid/nanoparticle nanostructures (like liposomes with encapsulated nanoparticles), which have diverse biomedical applications, including targeted therapy (drug delivery) and imaging of cancer cells.

Patterning of polar active filaments on a tense cylindrical membrane

We study the dynamics and patterning of polar contractile filaments on the surface of a cylindrical cell using active hydrodynamic equations that incorporate couplings between curvature and filament orientation. Cables and rings spontaneously emerge as steady state configurations on the cylinder, and can be stationary or moving, helical or tilted segments moving along helical trajectories. We observe phase transitions in the steady state patterns upon changing cell diameter or motor-driven activity and make several testable predictions. Our results are relevant to the dynamics and patterning of a variety of active biopolymers in cylindrical cells.

Localization of cytokinesis factors to the future cell division site by microtubule-dependent transport

The mechanism by which spindle microtubules (MTs) determine the site of cell division in animal cells is still highly controversial. Putative cytokinesis "signals" have been proposed to be positioned by spindle MTs at equatorial cortical regions to increase cortical contractility and/or at polar regions to decrease contractility [Rappaport, 1986; von Dassow, 2009]. Given the relative paucity of MTs at the future division site, it has not been clear how MTs localize cytokinesis factors there. Here, we test cytokinesis models using computational and experimental approaches. We present a simple lattice-based model in which signal-kinesin complexes move by transient plus-end directed movements on MTs interspersed with occasions of uniform diffusion in the cytoplasm. In simulations, complexes distribute themselves initially at the spindle midzone and then move on astral MTs to accumulate with time at the equatorial cortex. Simulations accurately predict cleavage patterns of cells with different geometries and MT arrangements and elucidate several experimental observations that have defied easy explanation by previous models. We verify this model with experiments on indented sea urchin zygotes showing that cells often divide perpendicular to the spindle at sites distinct from the indentations. These studies support an equatorial stimulation model and provide a simple mechanism explaining how cytokinesis factors localize to the future division site.

Cytokinesis: cells go back and forth about division

An elegant quantitative model to explain cellular oscillations during cytokinesis reveals a novel function for polar blebbing and raises the question as to why cells live right on the edge.

Deconvolution of the cellular force-generating subsystems that govern cytokinesis furrow ingression

Cytokinesis occurs through the coordinated action of several biochemically-mediated stresses acting on the cytoskeleton. Here, we develop a computational model of cellular mechanics, and using a large number of experimentally measured biophysical parameters, we simulate cell division under a number of different scenarios. We demonstrate that traction-mediated protrusive forces or contractile forces due to myosin II are sufficient to initiate furrow ingression. Furthermore, we show that passive forces due to the cell's cortical tension and surface curvature allow the furrow to complete ingression. We compare quantitatively the furrow thinning trajectories obtained from simulation with those observed experimentally in both wild-type and myosin II null Dictyostelium cells. Our simulations highlight the relative contributions of different biomechanical subsystems to cell shape progression during cell division.

Cytokinesis, the physical separation of a mother cell into two daughter cells, requires force to deform the cell. Though there is ample evidence in many systems that myosin II provides some of this force, it is also well known that some cell types can divide in the absence of myosin II. To elucidate the mechanisms by which cells control furrow ingression, we developed a computational model of cellular dynamics during cytokinesis in the social amoeba, Dictyostelium discoideum. We took advantage of a large number of experimentally measured parameters and well-characterized furrow ingression dynamics for a number of different strains. Our simulations demonstrate that there are distinct phases of cytokinesis. Myosin II plays a role providing the stress that initiates furrow ingression. In its absence, however, this force can be supplied by a combination of adhesion and protrusion-mediated stresses. Thereafter, Laplace-like pressures take over and provide stresses that enable the cell to divide. Overall, we show how various mechanical parameters quantitatively impact furrow ingression kinetics, accounting for the cytokinesis dynamics of wild type and mutant cell-lines.

On the role of membrane anisotropy and BAR proteins in the stability of tubular membrane structures

Recent studies have demonstrated that actin filaments are not crucial for the short-term stability of tubular membrane protrusions originating from the cell surface. It has also been demonstrated that prominin nanodomains and curvature inducing I-BAR proteins could account for the stability of the membrane protrusion. Here we constructed an axisymmetric model of a membrane protrusion that excludes actin filaments in order to investigate the contributions of prominin nanodomains (rafts) and I-BAR proteins to the membrane protrusion stability. It was demonstrated that prominin nanodomains and I-BAR proteins can stabilize the membrane protrusion only over a specific range of spontaneous curvature. On the other hand, high spontaneous curvature and/or high density of I-BAR proteins could lead to system instability and to non-uniform contraction in the radial direction of the membrane protrusion. In agreement with previous studies, it was also shown that the isotropic bending energy of lipids is not sufficient to explain the stability of the observed tubular membrane protrusion without actin filaments.

Model of myosin node aggregation into a contractile ring: the effect of local alignment

Actomyosin bundles frequently form through aggregation of membrane-bound myosin clusters. One such example is the formation of the contractile ring in fission yeast from a broad band of cortical nodes. Nodes are macromolecular complexes containing several dozens of myosin-II molecules and a few formin dimers. The condensation of a broad band of nodes into the contractile ring has been previously described by a search, capture, pull and release (SCPR) model. In SCPR, a random search process mediated by actin filaments nucleated by formins leads to transient actomyosin connections among nodes that pull one another into a ring. The SCPR model reproduces the transport of nodes over long distances and predicts observed clump-formation instabilities in mutants. However, the model does not generate transient linear elements and meshwork structures as observed in some wild-type and mutant cells during ring assembly. As a minimal model of node alignment, we added short-range aligning forces to the SCPR model representing currently unresolved mechanisms that may involve structural components, cross-linking and bundling proteins. We studied the effect of the local node alignment mechanism on ring formation numerically. We varied the new parameters and found viable rings for a realistic range of values. Morphologically, transient structures that form during ring assembly resemble those observed in experiments with wild-type and cdc25-22 cells. Our work supports a hierarchical process of ring self-organization involving components drawn together from distant parts of the cell followed by progressive stabilization.

Kinetics of myosin node aggregation into a contractile ring

We study a stochastic aggregation model for the assembly of the contractile ring from a broad band of nodes during cytokinesis in fission yeast. We found that bands of nodes condense into rings when the range of node interactions is larger than the width of the band. Wide bands are unstable to clump formation due to Poisson density fluctuations. We derive expressions for node kinetics and times for ring vs clump formation and test them using numerical simulations. These results suggest clump formation mechanisms in mutant cells.

Cytoskeletal dynamics in fission yeast: a review of models for polarization and division

We review modeling studies concerning cytoskeletal activity of fission yeast. Recent models vary in length and time scales, describing a range of phenomena from cellular morphogenesis to polymer assembly. The components of cytoskeleton act in concert to mediate cell-scale events and interactions such as polarization. The mathematical models reduce these events and interactions to their essential ingredients, describing the cytoskeleton by its bulk properties. On a smaller scale, models describe cytoskeletal subcomponents and how bulk properties emerge.

Mechanical stability of membrane nanotubular protrusions influenced by attachment of flexible rod-like proteins

It is indicated that nonhomogeneous lateral distribution of membrane attached and flexible rod-like proteins (MRPs) may stabilize nanotubular membrane protrusions. We have shown that curvature induced accumulation of MRPs in the nanotubular membrane protrusion and the corresponding reduction of the membrane free energy are possible if the decrease of the deviatoric free energy of MRPs in the nanotubular protrusions is large enough to overcome the increase of the free energy due to decrease of configurational entropy in the process of lateral sorting of MRPs. The decrease of isotropic curvature energy of MRPs in the region of membrane protrusion is usually not sufficient for substantial MRPs sorting and consequent stabilization of the nanotubular membrane protrusions.

Mechanics of cytokinesis in eukaryotes

Research on eukaryotic cytokinesis using advantageous model systems is rapidly advancing our understanding of most aspects of the process. Cytokinesis is very complicated with more than 100 proteins participating. Both fungi and animal cells use proteins to mark the cleavage site for the assembly of a contractile ring of actin filaments and myosin-II. Formins nucleate and elongate the actin filaments and myosin-II helps to organize the filaments into a contractile ring. Much is still to be learned about the organization of the contractile ring and the mechanisms that disassemble the ring as it constricts. Although fungi and animals share many proteins that contribute to cytokinesis, the extent to which they share mechanisms for the location, assembly, constriction, and disassembly of their contractile rings is still in question.

Calcium-actin waves and oscillations of cellular membranes

We propose a mechanism for the formation of membrane oscillations and traveling waves, which arise due to the coupling between the actin cytoskeleton and the calcium flux through the membrane. In our model, the fluid cell membrane has a mobile but constant population of proteins with a convex spontaneous curvature, which act as nucleators of actin polymerization and adhesion. Such a continuum model couples the forces of cell-substrate adhesion, actin polymerization, membrane curvature, and the flux of calcium through the membrane. Linear stability analysis shows that sufficiently strong coupling among the calcium, membrane, and protein dynamics may induce robust traveling waves on the membrane. This result was checked for a reduced feedback scheme and is compared to the results without the effects of calcium, where permanent phase separation without waves or oscillations is obtained. The model results are compared to the published observations of calcium waves in cell membranes, and a number of testable predictions are proposed.

Simple modeling of FtsZ polymers on flat and curved surfaces: correlation with experimental in vitro observations

FtsZ is a GTPase that assembles at midcell into a dynamic ring that constricts the membrane to induce cell division in the majority of bacteria, in many archea and several organelles. In vitro, FtsZ polymerizes in a GTP-dependent manner forming a variety of filamentous flexible structures. Based on data derived from the measurement of the in vitro polymerization of Escherichia coli FtsZ cell division protein we have formulated a model in which the fine balance between curvature, flexibility and lateral interactions accounts for structural and dynamic properties of the FtsZ polymers observed with AFM. The experimental results have been used by the model to calibrate the interaction energies and the values obtained indicate that the filaments are very plastic. The extension of the model to explore filament behavior on a cylindrical surface has shown that the FtsZ condensates promoted by lateral interactions can easily form ring structures through minor modulations of either filament curvature or longitudinal bond energies. The condensation of short, monomer exchanging filaments into rings is shown to produce enough force to induce membrane deformations.PACS codes: 87.15.ak, 87.16.ka, 87.17.Ee.

Physical model for the width distribution of axons

The distribution of widths of axons was recently investigated, and was found to have a distinct peak at an optimized value. The optimized axon width at the peak may arise from the conflicting demands of minimizing energy consumption and assuring signal transmission reliability. The distribution around this optimized value is found to have a distinct non-Gaussian shape, with an exponential "tail". We propose here a mechanical model whereby this distribution arises from the interplay between the elastic energy of the membrane surrounding the axon core, the osmotic pressure induced by the neurofilaments inside the axon bulk, and active processes that remodel the microtubules and neurofilaments inside the axon. The axon's radius of curvature can be determined by the cell's control of the osmotic pressure difference across the membrane, the membrane tension or by changing the composition of the different components of the membrane. We find that the osmotic pressure, determined by the neurofilaments, seems to be the dominant control parameter.

Mathematical models and simulations of cellular processes based on actin filaments

Actin filaments help to maintain the physical integrity of cells and participate in many processes that produce cellular movements. Studies of the processes that depend on actin filaments have progressed to the point where mathematical models and computer simulations are an essential part of the experimental toolkit. These quantitative models integrate knowledge about the structures of the key proteins and the rate and equilibrium constants for the reactions for comparison with a growing body of quantitative measurements of dynamic processes in live cells. Models and simulations are essential because it is impossible to appreciate by intuition alone the properties that emerge from a network of coupled reactions, particularly when the system contains many components, and force is one of the parameters.