Cortical geometry may influence placement of interface between Par protein domains in early Caenorhabditis elegans embryos

Delineating the mechanisms and design principles of Caenorhabditis elegans embryogenesis using in toto high-resolution imaging data and computational modeling

The nematode (roundworm) Caenorhabditis elegans is one of the most popular animal models for the study of developmental biology, as its invariant development and transparent body enable in toto cellular-resolution fluorescence microscopy imaging of developmental processes at 1-min intervals. This has led to the development of various computational tools for the systematic and automated analysis of imaging data to delineate the molecular and cellular processes throughout the embryogenesis of C. elegans, such as those associated with cell lineage, cell migration, cell morphology, and gene activity. In this review, we first introduce C. elegans embryogenesis and the development of techniques for tracking cell lineage and reconstructing cell morphology during this process. We then contrast the developmental modes of C. elegans and the customized technologies used for studying them with the ones of other animal models, highlighting its advantage for studying embryogenesis with exceptional spatial and temporal resolution. This is followed by an examination of the physical models that have been devised-based on accurate determinations of developmental processes afforded by analyses of imaging data-to interpret the early embryonic development of C. elegans from subcellular to intercellular levels of multiple cells, which focus on two key processes: cell polarization and morphogenesis. We subsequently discuss how quantitative data-based theoretical modeling has improved our understanding of the mechanisms of C. elegans embryogenesis. We conclude by summarizing the challenges associated with the acquisition of C. elegans embryogenesis data, the construction of algorithms to analyze them, and the theoretical interpretation.

Biophysical Models of PAR Cluster Transport by Cortical Flow in C. elegans Early Embryogenesis

The clustering of membrane-bound proteins facilitates their transport by cortical actin flow in early Caenorhabditis elegans embryo cell polarity. PAR-3 clustering is critical for this process, yet the biophysical processes that couple protein clusters to cortical flow remain unknown. We develop a discrete, stochastic agent-based model of protein clustering and test four hypothetical models for how clusters may interact with the flow. Results show that the canonical way to assess transport characteristics from single-particle tracking data used thus far in this area, the Péclet number, is insufficient to distinguish these hypotheses and that all models can account for transport characteristics quantified by this measure. However, using this model, we demonstrate that these different cluster-cortex interactions may be distinguished using a different metric, namely the scalar projection of cluster displacement on to the flow displacement vector. Our results thus provide a testable way to use existing single-particle tracking data to test how endogenous protein clusters may interact with the cortical flow to localize during polarity establishment. To facilitate this investigation, we also develop both improved simulation and semi-analytic methodologies to quantify motion summary statistics (e.g., Péclet number and scalar projection) for these stochastic models as a function of biophysical parameters.

Long time behavior and stable patterns in high-dimensional polarity models of asymmetric cell division

Asymmetric cell division is one of the fundamental processes to create cell diversity in the early stage of embryonic development. During this process, the polarity formation in the cell membrane has been considered as a key process by which the entire polarity formation in the cytosol is controlled, and it has been extensively studied in both experiments and mathematical models. Nonetheless, a mathematically rigorous analysis of the polarity formation in the asymmetric cell division has been little explored, particularly for bulk-surface models. In this article, we deal with polarity models proposed for describing the PAR polarity formation in the asymmetric cell division of a C. elegans embryo. Using a simpler but mathematically consistent model, we exhibit the long time behavior of the polarity formation of a bulk-surface cell. Moreover, we mathematically prove the existence of stable polarity solutions of the model equation in an arbitrary high-dimensional domain and analyse how the boundary position of polarity domain is determined. Our results propose that the existence and dynamics of the polarity in the asymmetric cell division can be understood universally in terms of basic mathematical structures.

The Role of Cytoplasmic MEX-5/6 Polarity in Asymmetric Cell Division

In the process of asymmetric cell division, the mother cell induces polarity in both the membrane and the cytosol by distributing substrates and components asymmetrically. Such polarity formation results from the harmonization of the upstream and downstream polarities between the cell membrane and the cytosol. MEX-5/6 is a well-investigated downstream cytoplasmic protein, which is deeply involved in the membrane polarity of the upstream transmembrane protein PAR in the Caenorhabditis elegans embryo. In contrast to the extensive exploration of membrane PAR polarity, cytoplasmic polarity is poorly understood, and the precise contribution of cytoplasmic polarity to the membrane PAR polarity remains largely unknown. In this study, we explored the interplay between the cytoplasmic MEX-5/6 polarity and the membrane PAR polarity by developing a mathematical model that integrates the dynamics of PAR and MEX-5/6 and reflects the cell geometry. Our investigations show that the downstream cytoplasmic protein MEX-5/6 plays an indispensable role in causing a robust upstream PAR polarity, and the integrated understanding of their interplay, including the effect of the cell geometry, is essential for the study of polarity formation in asymmetric cell division.

Asymmetric cell division from a cell to cells: Shape, length, and location of polarity domain

Asymmetric cell division is one of the most elegant biological systems by which cells create daughter cells with different functions and increase cell diversity. In particular, PAR polarity in the cell membrane plays a critical role in regulating the whole process of asymmetric cell division. Numerous studies have been conducted to determine the underlying mechanism of PAR polarity formation using both experimental and theoretical approaches in the last 10 years. However, they have mostly focused on answering the fundamental question of how this exclusive polarity is established but the precise dynamics of polarity domain have been little notified. In this review, I focused on studies on the shape, length, and location of PAR polarity from a theoretical perspective that may be important for an integrated understanding of the entire process of asymmetric cell division.

Geometric cues stabilise long-axis polarisation of PAR protein patterns in C. elegans

In the Caenorhabditis elegans zygote, PAR protein patterns, driven by mutual anatagonism, determine the anterior-posterior axis and facilitate the redistribution of proteins for the first cell division. Yet, the factors that determine the selection of the polarity axis remain unclear. We present a reaction-diffusion model in realistic cell geometry, based on biomolecular reactions and accounting for the coupling between membrane and cytosolic dynamics. We find that the kinetics of the phosphorylation-dephosphorylation cycle of PARs and the diffusive protein fluxes from the cytosol towards the membrane are crucial for the robust selection of the anterior-posterior axis for polarisation. The local ratio of membrane surface to cytosolic volume is the main geometric cue that initiates pattern formation, while the choice of the long-axis for polarisation is largely determined by the length of the aPAR-pPAR interface, and mediated by processes that minimise the diffusive fluxes of PAR proteins between cytosol and membrane.

The importance of mechanical constraints for proper polarization and psuedo-cleavage furrow generation in the early Caenorhabditis elegans embryo

Intracellular polarization, where a cell specifies a spatial axis by segregation of specific factors, is a fundamental biological process. In the early embryo of the nematode worm Caenorhabditis elegans (C. elegans), polarization is often accompanied by deformations of the cortex, a highly contractile structure consisting of actin filaments cross-linked by the motor protein myosin (actomyosin). It has been suggested that the eggshell surrounding the early embryo plays a role in polarization although its function is not understood. Here we develop a mathematical model which couples a reaction-diffusion model of actomyosin dynamics with a phase field model of the cell cortex to implicitly track cell shape changes in the early C. elegans embryo. We investigate the potential rigidity effect of the geometric constraint imposed by the presence and size of the eggshell on polarization dynamics. Our model suggests that the geometric constraint of the eggshell is essential for proper polarization and the size of the eggshell also affects the dynamics of polarization. Therefore, we conclude that geometric constraint on a cell might affect the dynamics of a biochemical process.

Polarization, whereby molecules and proteins are asymmetrically distributed throughout the cell, is a vital process for many cellular functions. In the early C. elegans embryo the asymmetric distribution of cell cytoskeleton during the initiation of polarization leads to asymmetric contractions which are higher in the anterior and lower in the posterior of a cell. The C. elegans embryo is surrounded by a rigid body, the eggshell, which functions in numerous cell processes. We investigate the structural support of eggshell during the establishment phase by tracking the moving cell surface. We incorporate protein dynamics involved in polarization into the membrane evolution. We conclude that eggshell might have a role in cell polarization by preventing the distortion of cell surface.

Dendritic spine geometry can localize GTPase signaling in neurons

Dendritic spines are the postsynaptic terminals of most excitatory synapses in the mammalian brain. Learning and memory are associated with long-lasting structural remodeling of dendritic spines through an actin-mediated process regulated by the Rho-family GTPases RhoA, Rac, and Cdc42. These GTPases undergo sustained activation after synaptic stimulation, but whereas Rho activity can spread from the stimulated spine, Cdc42 activity remains localized to the stimulated spine. Because Cdc42 itself diffuses rapidly in and out of the spine, the basis for the retention of Cdc42 activity in the stimulated spine long after synaptic stimulation has ceased is unclear. Here we model the spread of Cdc42 activation at dendritic spines by means of reaction-diffusion equations solved on spine-like geometries. Excitable behavior arising from positive feedback in Cdc42 activation leads to spreading waves of Cdc42 activity. However, because of the very narrow neck of the dendritic spine, wave propagation is halted through a phenomenon we term geometrical wave-pinning. We show that this can account for the localization of Cdc42 activity in the stimulated spine, and, of interest, retention is enhanced by high diffusivity of Cdc42. Our findings are broadly applicable to other instances of signaling in extreme geometries, including filopodia and primary cilia.

Protein abundance may regulate sensitivity to external cues in polarized cells

Cell polarization is a ubiquitous process which results in cellular constituents being organized into discrete intracellular spatial domains. It occurs in a variety of cell types, including epithelial cells, immune system cells and neurons. A key player in this process is the Par protein family whose asymmetric localization to anterior and posterior parts of the cell is crucial for proper division and cell fate specification. In this paper, we explore a stochastic analogue of the temporal model of Par protein interactions first developed in Dawes & Munro (Dawes and Munro 2011 Biophys. J. 101, 1412-1422. (doi:10.1016/j.bpj.2011.07.030)). We focus on how protein abundance influences the behaviour of both the deterministic and stochastic versions of the model. In Dawes & Munro (2011), it was found that bistable behaviour in the temporal model of Par protein led to the existence of complementary domains in the corresponding spatio-temporal model. Here, we find that the corresponding temporal stochastic model permits switching behaviour (the model solution 'jumps' between steady states) for lower protein abundances, whereas for higher protein abundances the stochastic and deterministic models are in good agreement (the model solution evolves to one of two steady states). This led us to the testable hypothesis that cells with lower abundances of Par protein may be more sensitive to external cues, whereas cells with higher abundances of Par protein may be less sensitive to external cues. In order to gain more control over the precise abundance of Par protein, we proposed and explored a second model (again, examining both deterministic and stochastic versions) in which the total number of Par molecules is conserved. We found that this model required an additional dimerization reaction in the cytoplasm in order for bistable and switching behaviour to be found. Once this additional reaction was included, we found that both the first and second models gave qualitatively similar results but in different regions of the parameter space, suggesting a further regulatory mechanism that cells could potentially use to modulate their response to external signals.

Actomyosin regulation and symmetry breaking in a model of polarization in the early Caenorhabditis elegans embryo : symmetry breaking in cell polarization

Polarization, whereby a cell defines a spatial axis by segregating specific determinants to distinct regions, is an essential and highly conserved biological process. The process of polarization is initiated by a cue that breaks an initially symmetric distribution of determinants, allowing for a spatially asymmetric redistribution. The nature of this cue is currently not well understood. Utilizing the conservation of polarization process and its determinants, we theoretically investigate the nature of the cue and the regulation of contractility that enables the establishment of polarity in early embryos of the nematode worm Caenorhabditis elegans (C. elegans). Our biologically based model, which consists of coupled partial differential equations, suggests that a biochemical but not mechanical cue is sufficient for symmetry breaking, and inhibition of contractile elements by specific determinants is needed for sustained spatial redistribution.

Mechanotransduction in C. elegans morphogenesis and tissue function

Mechanobiology is an emerging field that investigates how living cells sense and respond to their physical surroundings. Recent interest in the field has been sparked by the finding that stem cells differentiate along different lineages based on the stiffness of the cell surroundings (Engler et al., 2006), and that metastatic behavior of cancer cells is strongly influenced by the mechanical properties of the surrounding tissue (Kumar and Weaver, 2009). Many questions remain about how cells convert mechanical information, such as viscosity, stiffness of the substrate, or stretch state of the cells, into the biochemical signals that control tissue function. Caenorhabditis elegans researchers are making significant contributions to the understanding of mechanotransduction in vivo. This review summarizes recent insights into the role of mechanical forces in morphogenesis and tissue function. Examples of mechanical regulation across length scales, from the single-celled zygote, to the intercellular coordination that enables cohesive tissue function, to the mechanical influences between tissues, are considered. The power of the C. elegans system as a gene discovery and in vivo quantitative bioimaging platform is enabling an important discoveries in this exciting field.