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

Close encounters of the sticky kind: Brownian motion at absorbing boundaries

Encounter-based models of diffusion provide a probabilistic framework for analyzing the effects of a partially absorbing reactive surface, in which the probability of absorption depends upon the amount of surface-particle contact time. In this paper we develop a class of encounter-based models that deal with absorption at sticky boundaries. Sticky boundaries occur in a diverse range of applications, including cell biology, colloidal physics, finance, and human crowd dynamics. They also naturally arise in active matter, where confined active particles tend to spontaneously accumulate at boundaries even in the absence of any particle-particle interactions. We begin by constructing a one-dimensional encounter-based model of sticky Brownian motion (BM), which is based on the zero-range limit of nonsticky BM with a short-range attractive potential well near the origin. In this limit, the boundary-contact time is given by the amount of time (occupation time) that the particle spends at the origin. We calculate the joint probability density or propagator for the particle position and the occupation time, and then identify an absorption event as the first time that the occupation time crosses a randomly generated threshold. We illustrate the theory by considering diffusion in a finite interval with a partially absorbing sticky boundary at one end. We show how various quantities, such as the mean first passage time (MFPT) for single-particle absorption and the relaxation to steady state at the multiparticle level, depend on moments of the random threshold distribution. Finally, we determine how sticky BM can be obtained by taking a particular diffusion limit of a sticky run-and-tumble particle (RTP).

Multi-spike solutions of a hybrid reaction–transport model

Simulations of classical pattern-forming reaction–diffusion systems indicate that they often operate in the strongly nonlinear regime, with the final steady state consisting of a spatially repeating pattern of localized spikes. In activator–inhibitor systems such as the two-component Gierer–Meinhardt (GM) model, one can consider the singular limit D a  ≪  D h , where D a and D h are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyse the existence and linear stability of multi-spike solutions. In this paper, we analyse multi-spike solutions in a hybrid reaction–transport model, consisting of a slowly diffusing activator and an actively transported inhibitor that switches at a rate α between right-moving and left-moving velocity states. Such a model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in Caenorhabditis elegans . We exploit the fact that the hybrid model can be mapped onto the classical GM model in the fast switching limit α  → ∞, which establishes the existence of multi-spike solutions. Linearization about the multi-spike solution yields a non-local eigenvalue problem that is used to investigate stability of the multi-spike solution by combining analytical results for α  → ∞ with a graphical construction for finite α .

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.

Transcriptomic and proteomic analysis reveals mechanisms of embryo abortion during chrysanthemum cross breeding

Embryo abortion is the main cause of failure in chrysanthemum cross breeding, and the genes and proteins associated with embryo abortion are poorly understood. Here, we applied RNA sequencing and isobaric tags for relative and absolute quantitation (iTRAQ) to analyse transcriptomic and proteomic profiles of normal and abortive embryos. More than 68,000 annotated unigenes and 700 proteins were obtained from normal and abortive embryos. Functional analysis showed that 140 differentially expressed genes (DEGs) and 41 differentially expressed proteins (DEPs) were involved in embryo abortion. Most DEGs and DEPs associated with cell death, protein degradation, reactive oxygen species scavenging, and stress-response transcriptional factors were significantly up-regulated in abortive embryos relative to normal embryos. In contrast, most genes and proteins related to cell division and expansion, the cytoskeleton, protein synthesis and energy metabolism were significantly down-regulated in abortive embryos. Furthermore, abortive embryos had the highest activity of three executioner caspase-like enzymes. These results indicate that embryo abortion may be related to programmed cell death and the senescence- or death-associated genes or proteins contribute to embryo abortion. This adds to our understanding of embryo abortion and will aid in the cross breeding of chrysanthemum and other crops in the future.

Actin cable distribution and dynamics arising from cross-linking, motor pulling, and filament turnover

A computational model of actin cables in fission yeast is presented that includes polymerization, severing, cross-linking, and motor pulling. Results reproduce observations in wild-type cells and cells lacking myosin V and are compared to images of cells overexpressing α-actinin. Formin clustering at cell tips is predicted to promote cable formation.

The growth of fission yeast relies on the polymerization of actin filaments nucleated by formin For3p, which localizes at tip cortical sites. These actin filaments bundle to form actin cables that span the cell and guide the movement of vesicles toward the cell tips. A big challenge is to develop a quantitative understanding of these cellular actin structures. We used computer simulations to study the spatial and dynamical properties of actin cables. We simulated individual actin filaments as semiflexible polymers in three dimensions composed of beads connected with springs. Polymerization out of For3p cortical sites, bundling by cross-linkers, pulling by type V myosin, and severing by cofilin are simulated as growth, cross-linking, pulling, and turnover of the semiflexible polymers. With the foregoing mechanisms, the model generates actin cable structures and dynamics similar to those observed in live-cell experiments. Our simulations reproduce the particular actin cable structures in myoVΔ cells and predict the effect of increased myosin V pulling. Increasing cross-linking parameters generates thicker actin cables. It also leads to antiparallel and parallel phases with straight or curved cables, consistent with observations of cells overexpressing α-actinin. Finally, the model predicts that clustering of formins at cell tips promotes actin cable formation.

Model of fission yeast cell shape driven by membrane-bound growth factors and the cytoskeleton

Fission yeast serves as a model for how cellular polarization machinery consisting of signaling molecules and the actin and microtubule cytoskeleton regulates cell shape. In this work, we develop mathematical models to investigate how these cells maintain a tubular shape of approximately constant diameter. Many studies identify active Cdc42, found in a cap at the inner membrane of growing cell tips, as an important regulator of local cell wall remodeling, likely through control of exocyst tethering and the targeting of other polarity-enhancing structures. First, we show that a computational model with Cdc42-dependent local cell wall remodeling under turgor pressure predicts a relationship between spatial extent of growth signal and cell diameter that is in agreement with prior experiments. Second, we model the consequences of feedback between cell shape and distribution of Cdc42 growth signal at cell tips. We show that stability of cell diameter over successive cell divisions places restrictions on their mutual dependence. We argue that simple models where the spatial extent of the tip growth signal relies solely on geometrical alignment of confined microtubules might lead to unstable width regulation. Third, we study a computational model that combines a growth signal distributed over a characteristic length scale (as, for example, by a reaction-diffusion mechanism) with an axis-sensing microtubules system that places landmarks at positions where microtubule tips touch the cortex. A two-dimensional implementation of this model leads to stable cell diameter for a wide range of parameters. Changes to the parameters of this model reproduce straight, bent, and bulged cell shapes, and we discuss how this model is consistent with other observed cell shapes in mutants. Our work provides an initial quantitative framework for understanding the regulation of cell shape in fission yeast, and a scaffold for understanding this process on a more molecular level in the future.

Fission yeast is a rod-shaped organism that is studied, in part, as a model for how cells develop and regulate their shape. Despite extensive work identifying effects of genetic mutations and pharmacological treatments on the shape of these cells, there is a lack of mathematical and computational models examining how internal cell signals and the cytoskeleton organize to remodel the cell wall, direct growth at cell tips, and maintain tubular shape. In this work we describe how the spatial distribution of regulatory protein signal at growing cell tips relates to cell diameter. Further, we describe the consequences of this signal depending on the shape of the cell, namely its length and diameter. Finally, we propose a computational model for understanding growth and shape that includes an axis-sensing microtubule system, landmarks delivered to cell tips along those microtubules, and a growth zone signal that moves around but is attracted to the landmarks. This picture explains a large number of reported abnormal shapes in terms of only a few modular components.

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.

Mathematical model for growth regulation of fission yeast Schizosaccharomyces pombe

Regulation of polarised cell growth is essential for many cellular processes including spatial coordination of cell morphology changes during the division cycle. We present a mathematical model of the core mechanism responsible for the regulation of polarised growth dynamics during the fission yeast cell cycle. The model is based on the competition of growth zones localised at the cell tips for a common substrate distributed uniformly in the cytosol. We analyse the bifurcations in this model as the cell length increases, and show that the growth activation dynamics provides an explanation for the new-end take-off (NETO) as a saddle-node bifurcation at which the cell sharply switches from monopolar to bipolar growth. We study the parameter sensitivity of the bifurcation diagram and relate qualitative changes of the growth pattern, e.g. delayed or absent NETO, to previously observed mutant phenotypes. We investigate the effects of imperfect asymmetric cell division, and show that this leads to distinct growth patterns that provide experimentally testable predictions for validating the presented competitive growth zone activation model. Finally we discuss extension of the model for describing mutant cells with more than two growth zones.

Oscillatory dynamics of Cdc42 GTPase in the control of polarized growth

Cells promote polarized growth by activation of Rho-family protein Cdc42 at the cell membrane. We combined experiments and modeling to study bipolar growth initiation in fission yeast. Concentrations of a fluorescent marker for active Cdc42, Cdc42 protein, Cdc42-activator Scd1, and scaffold protein Scd2 exhibited anticorrelated fluctuations and oscillations with a 5-minute average period at polarized cell tips. These dynamics indicate competition for active Cdc42 or its regulators and the presence of positive and delayed negative feedbacks. Cdc42 oscillations and spatial distribution were sensitive to the amounts of Cdc42-activator Gef1 and to the activity of Cdc42-dependent kinase Pak1, a negative regulator. Feedbacks regulating Cdc42 oscillations and spatial self-organization appear to provide a flexible mechanism for fission yeast cells to explore polarization states and to control their morphology.

A systems-biology approach to yeast actin cables

We focus on actin cables in yeast as a model system for understanding cytoskeletal organization and the workings of actin itself. In particular, we highlight quantitative approaches on the kinetics of actin-cable assembly and methods of measuring their morphology by image analysis. Actin cables described by these studies can span greater lengths than a thousand end-to-end actin-monomers. Because of this difference in length scales, control of the actin-cable system constitutes a junction between short-range interactions - among actin-monomers and nucleating, polymerization-facilitating, side-binding, severing, and cross-linking proteins - and the emergence of cell-scale physical form as embodied by the actin cables themselves.

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.