Self-organization of waves and pulse trains by molecular motors in cellular protrusions

Model of Processive Catalysis with Site Clustering and Blocking and Its Application to Cellulose Hydrolysis

Interactions between particles moving on a linear track and their possible blocking by obstacles can lead to crowding, impeding the particles' transport kinetics. When the particles are enzymes processively catalyzing a reaction along a linear polymeric substrate, these crowding and blocking effects may substantially reduce the overall catalytic rate. Cellulose hydrolysis by exocellulases processively moving along cellulose chains assembled into insoluble cellulose particles is an example of such a catalytic transport process. The details of the kinetics of cellulose hydrolysis and the causes of the often observed reduction of hydrolysis rate over time are not yet fully understood. Crowding and blocking of enzyme particles are thought to be one of the important factors affecting the cellulose hydrolysis, but its exact role and mechanism are not clear. Here, we introduce a simple model based on an elementary transport process that incorporates the crowding and blocking effects in a straightforward way. This is achieved by making a distinction between binding and non-binding sites on the chain. The model reproduces a range of experimental results, mainly related to the early phase of cellulose hydrolysis. Our results indicate that the combined effects of clustering of binding sites together with the occupancy pattern of these sites by the enzyme molecules play a decisive role in the overall kinetics of cellulose hydrolysis. It is suggested that periodic desorption and rebinding of enzyme molecules could be a basis of a strategy to partially counter the clustering of and blocking by the binding sites and so enhance the rate of cellulose hydrolysis. The general nature of the model means that it could be applicable also to other transport processes that make a distinction between binding and non-binding sites, where crowding and blocking are expected to be relevant.

Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling

We investigate a generic two-field Cahn-Hilliard model with variational and nonvariational coupling. It describes, for instance, passive and active ternary mixtures, respectively. Already a linear stability analysis of the homogeneous mixed state shows that activity not only allows for the usual large-scale stationary (Cahn-Hilliard) instability of the well-known passive case but also for small-scale stationary (Turing) and large-scale oscillatory (Hopf) instabilities. In consequence of the Turing instability, activity may completely suppress the usual coarsening dynamics. In a fully nonlinear analysis, we first briefly discuss the passive case before focusing on the active case. Bifurcation diagrams and selected direct time simulations are presented that allow us to establish that nonvariational coupling (i) can partially or completely suppress coarsening and (ii) may lead to the emergence of drifting and oscillatory states. Throughout, we emphasize the relevance of conservation laws and related symmetries for the encountered intricate bifurcation behavior.

Propagation of F-actin disassembly via Myosin15-Mical interactions

The F-actin cytoskeleton drives cellular form and function. However, how F-actin-based changes occur with spatiotemporal precision and specific directional orientation is poorly understood. Here, we identify that the unconventional class XV myosin [Myosin 15 (Myo15)] physically and functionally interacts with the F-actin disassembly enzyme Mical to spatiotemporally position cellular breakdown and reconstruction. Specifically, while unconventional myosins have been associated with transporting cargo along F-actin to spatially target cytoskeletal assembly, we now find they also target disassembly. Myo15 specifically positions this F-actin disassembly by associating with Mical and using its motor and MyTH4-FERM cargo-transporting functions to broaden Mical's distribution. Myo15's broadening of Mical's distribution also expands and directionally orients Mical-mediated F-actin disassembly and subsequent cellular remodeling, including in response to Semaphorin/Plexin cell surface activation signals. Thus, we identify a mechanism that spatiotemporally propagates F-actin disassembly while also proposing that other F-actin-trafficked-cargo is derailed by this disassembly to directionally orient rebuilding.

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.

Flow Induced Symmetry Breaking in a Conceptual Polarity Model

Important cellular processes, such as cell motility and cell division, are coordinated by cell polarity, which is determined by the non-uniform distribution of certain proteins. Such protein patterns form via an interplay of protein reactions and protein transport. Since Turing’s seminal work, the formation of protein patterns resulting from the interplay between reactions and diffusive transport has been widely studied. Over the last few years, increasing evidence shows that also advective transport, resulting from cytosolic and cortical flows, is present in many cells. However, it remains unclear how and whether these flows contribute to protein-pattern formation. To address this question, we use a minimal model that conserves the total protein mass to characterize the effects of cytosolic flow on pattern formation. Combining a linear stability analysis with numerical simulations, we find that membrane-bound protein patterns propagate against the direction of cytoplasmic flow with a speed that is maximal for intermediate flow speed. We show that the mechanism underlying this pattern propagation relies on a higher protein influx on the upstream side of the pattern compared to the downstream side. Furthermore, we find that cytosolic flow can change the membrane pattern qualitatively from a peak pattern to a mesa pattern. Finally, our study shows that a non-uniform flow profile can induce pattern formation by triggering a regional lateral instability.

Why a Large-Scale Mode Can Be Essential for Understanding Intracellular Actin Waves

During the last decade, intracellular actin waves have attracted much attention due to their essential role in various cellular functions, ranging from motility to cytokinesis. Experimental methods have advanced significantly and can capture the dynamics of actin waves over a large range of spatio-temporal scales. However, the corresponding coarse-grained theory mostly avoids the full complexity of this multi-scale phenomenon. In this perspective, we focus on a minimal continuum model of activator-inhibitor type and highlight the qualitative role of mass conservation, which is typically overlooked. Specifically, our interest is to connect between the mathematical mechanisms of pattern formation in the presence of a large-scale mode, due to mass conservation, and distinct behaviors of actin waves.

Excitable solitons: Annihilation, crossover, and nucleation of pulses in mass-conserving activator-inhibitor media

Excitable pulses are among the most widespread dynamical patterns that occur in many different systems, ranging from biological cells to chemical reactions and ecological populations. Traditionally, the mutual annihilation of two colliding pulses is regarded as their prototypical signature. Here we show that colliding excitable pulses may exhibit solitonlike crossover and pulse nucleation if the system obeys a mass conservation constraint. In contrast to previous observations in systems without mass conservation, these alternative collision scenarios are robustly observed over a wide range of parameters. We demonstrate our findings using a model of intracellular actin waves since, on time scales of wave propagations over the cell scale, cells obey conservation of actin monomers. The results provide a key concept to understand the ubiquitous occurrence of actin waves in cells, suggesting why they are so common, and why their dynamics is robust and long-lived.

Length regulation of multiple flagella that self-assemble from a shared pool of components

The single-celled green algae Chlamydomonas reinhardtii with its two flagella-microtubule-based structures of equal and constant lengths-is the canonical model organism for studying size control of organelles. Experiments have identified motor-driven transport of tubulin to the flagella tips as a key component of their length control. Here we consider a class of models whose key assumption is that proteins responsible for the intraflagellar transport (IFT) of tubulin are present in limiting amounts. We show that the limiting-pool assumption is insufficient to describe the results of severing experiments, in which a flagellum is regenerated after it has been severed. Next, we consider an extension of the limiting-pool model that incorporates proteins that depolymerize microtubules. We show that this 'active disassembly' model of flagellar length control explains in quantitative detail the results of severing experiments and use it to make predictions that can be tested in experiments.

Self-organization principles of intracellular pattern formation

Dynamic patterning of specific proteins is essential for the spatio-temporal regulation of many important intracellular processes in prokaryotes, eukaryotes and multicellular organisms. The emergence of patterns generated by interactions of diffusing proteins is a paradigmatic example for self-organization. In this article, we review quantitative models for intracellular Min protein patterns in Escherichia coli, Cdc42 polarization in Saccharomyces cerevisiae and the bipolar PAR protein patterns found in Caenorhabditis elegans. By analysing the molecular processes driving these systems we derive a theoretical perspective on general principles underlying self-organized pattern formation. We argue that intracellular pattern formation is not captured by concepts such as ‘activators’, ‘inhibitors’ or ‘substrate depletion’. Instead, intracellular pattern formation is based on the redistribution of proteins by cytosolic diffusion, and the cycling of proteins between distinct conformational states. Therefore, mass-conserving reaction–diffusion equations provide the most appropriate framework to study intracellular pattern formation. We conclude that directed transport, e.g. cytosolic diffusion along an actively maintained cytosolic gradient, is the key process underlying pattern formation. Thus the basic principle of self-organization is the establishment and maintenance of directed transport by intracellular protein dynamics.

This article is part of the theme issue ‘Self-organization in cell biology’.

Bidirectional transport model of morphogen gradient formation via cytonemes

Morphogen protein gradients play an important role in the spatial regulation of patterning during embryonic development. The most commonly accepted mechanism for gradient formation is diffusion from a source combined with degradation. Recently, there has been growing interest in an alternative mechanism, which is based on the direct delivery of morphogens along thin, actin-rich cellular extensions known as cytonemes. In this paper, we develop a bidirectional motor transport model for the flux of morphogens along cytonemes, linking a source cell to a one-dimensional array of target cells. By solving the steady-state transport equations, we show how a morphogen gradient can be established, and explore how the mean velocity of the motors affects properties of the morphogen gradient such as accumulation time and robustness. In particular, our analysis suggests that in order to achieve robustness with respect to changes in the rate of synthesis of morphogen, the mean velocity has to be negative, that is, retrograde flow or treadmilling dominates. Thus the potential targeting precision of cytonemes comes at an energy cost. We then study the effects of non-uniformly allocating morphogens to the various cytonemes projecting from a source cell. This competition for resources provides a potential regulatory control mechanism not available in diffusion-based models.

A Salutary Role of Reactive Oxygen Species in Intercellular Tunnel-Mediated Communication

The reactive oxygen species, generally labeled toxic due to high reactivity without target specificity, are gradually uncovered as signaling molecules involved in a myriad of biological processes. But one important feature of ROS roles in macromolecule movement has not caught attention until recent studies with technique advance and design elegance have shed lights on ROS signaling for intercellular and interorganelle communication. This review begins with the discussions of genetic and chemical studies on the regulation of symplastic dye movement through intercellular tunnels in plants (plasmodesmata), and focuses on the ROS regulatory mechanisms concerning macromolecule movement including small RNA-mediated gene silencing movement and protein shuttling between cells. Given the premise that intercellular tunnels (bridges) in mammalian cells are the key physical structures to sustain intercellular communication, movement of macromolecules and signals is efficiently facilitated by ROS-induced membrane protrusions formation, which is analogously applied to the interorganelle communication in plant cells. Although ROS regulatory differences between plant and mammalian cells exist, the basis for ROS-triggered conduit formation underlies a unifying conservative theme in multicellular organisms. These mechanisms may represent the evolutionary advances that have enabled multicellularity to gain the ability to generate and utilize ROS to govern material exchanges between individual cells in oxygenated environment.

Application of Quasi-Steady-State Methods to Nonlinear Models of Intracellular Transport by Molecular Motors

Molecular motors such as kinesin and dynein are responsible for transporting material along microtubule networks in cells. In many contexts, motor dynamics can be modelled by a system of reaction-advection-diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the full PDE system. Here, we extend this QSS reduction methodology to certain nonlinear reaction models. The QSS method relies on the assumption that the nonlinear binding and unbinding interactions of the cellular motors occur on a faster timescale than the spatial diffusion and advection processes. The full system dynamics are shown to be well approximated by the dynamics on the slow manifold. The slow manifold is parametrized by a single scalar quantity that satisfies a scalar nonlinear PDE, called the QSS PDE. We apply the QSS method to several specific nonlinear models for the binding and unbinding of molecular motors, and we use the resulting approximations to draw conclusions regarding the parameter dependence of the spatial distribution of motors for these models.

Filopodia formation and endosome clustering induced by mutant plus-end-directed myosin VI

Myosin VI (MYO6) is the only myosin known to move toward the minus end of actin filaments. It has roles in numerous cellular processes, including maintenance of stereocilia structure, endocytosis, and autophagosome maturation. However, the functional necessity of minus-end-directed movement along actin is unclear as the underlying architecture of the local actin network is often unknown. To address this question, we engineered a mutant of MYO6, MYO6+, which undergoes plus-end-directed movement while retaining physiological cargo interactions in the tail. Expression of this mutant motor in HeLa cells led to a dramatic reorganization of cortical actin filaments and the formation of actin-rich filopodia. MYO6 is present on peripheral adaptor protein, phosphotyrosine interacting with PH domain and leucine zipper 1 (APPL1) signaling endosomes and MYO6+ expression causes a dramatic relocalization and clustering of this endocytic compartment in the cell cortex. MYO6+ and its adaptor GAIP interacting protein, C terminus (GIPC) accumulate at the tips of these filopodia, while APPL1 endosomes accumulate at the base. A combination of MYO6+ mutagenesis and siRNA-mediated depletion of MYO6 binding partners demonstrates that motor activity and binding to endosomal membranes mediated by GIPC and PI(4,5)P₂ are crucial for filopodia formation. A similar reorganization of actin is induced by a constitutive dimer of MYO6+, indicating that multimerization of MYO6 on endosomes through binding to GIPC is required for this cellular activity and regulation of actin network structure. This unique engineered MYO6+ offers insights into both filopodia formation and MYO6 motor function at endosomes and at the plasma membrane.

Exclusion and Hierarchy of Time Scales Lead to Spatial Segregation of Molecular Motors in Cellular Protrusions

Molecular motors that carry cargo along biopolymer filaments within cells play a crucial role in the functioning of the cell. In particular, these motors are essential for the formation and maintenance of the cellular protrusions that play key roles in motility and specific functionalities, such as the stereocilia in hair cells. Typically, there are several species of motors, carrying different cargos, that share the same track. Furthermore, it was observed that in the mature stereocilia, the different motors occupy well-segregated bands as a function of distance from the tip. We use a totally asymmetric exclusion process model with two- and three-motor species, to study the conditions that give rise to such spatial patterns. We find that the well-segregated bands appear for motors with a strong hierarchy of attachment or detachment rates. This is a striking example of pattern formation in nonequilibrium, low-dimensional systems.

Closed membrane shapes with attached BAR domains subject to external force of actin filaments

Membrane deformations induced by attached BAR superfamily domains could trigger or facilitate the growth of plasma membrane protrusions. The BAR domain family consists of BAR, F-BAR and I-BAR domains, each enforcing a different local curvature when attached to the membrane surface. Our theoretical study mainly focuses on the role of I-BAR in the membrane tubular deformations generated or stabilised by actin filaments. The influence of the area density of membrane attached BAR domains and their intrinsic curvature on the closed membrane shapes (vesicles) was investigated numerically. We derived an analytical approximative expression for the critical relative area density of BARs at which the membrane tubular protrusions on vesicles are most prominent. We have shown that the BARs with a higher intrinsic curvature induce thinner and longer cylindrical protrusions. The average orientation of the membrane attached BARs is altered when the vesicle shape is subjected to external force of growing actin rod-like structure inside a vesicle. The average orientation angle of membrane attached BARs may indicate whether the actin filaments are just stabilising the protrusion or generating it by stretching the vesicle.