Role of domain in pattern formation

Morphogen-directed cell fate boundaries: slow passage through bifurcation and the role of folded saddles

One of the fundamental mechanisms in embryogenesis is the process by which cells differentiate and create tissues and structures important for functioning as a multicellular organism. Morphogenesis involves diffusive process of chemical signalling involving morphogens that pre-pattern the tissue. These morphogens influence cell fate through a highly nonlinear process of transcriptional signalling. In this paper, we consider this multiscale process in an idealised model for a growing domain. We focus on intracellular processes that lead to robust differentiation into two cell lineages through interaction of a single morphogen species with a cell fate variable that undergoes a bifurcation from monostability to bistability. In particular, we investigate conditions that result in successful and robust pattern formation into two well-separated domains, as well as conditions where this fails and produces a pinned boundary wave where only one part of the domain grows. We show that successful and unsuccessful patterning scenarios can be characterised in terms of presence or absence of a folded saddle singularity for a system with two slow variables and one fast variable; this models the interaction of slow morphogen diffusion, slow parameter drift through bifurcation and fast transcription dynamics. We illustrate how this approach can successfully model acquisition of three cell fates to produce three-domain "French flag" patterning, as well as for a more realistic model of the cell fate dynamics in terms of two mutually inhibiting transcription factors.

Modeling and simulation of cell nuclear architecture reorganization process

We develop a special phase field/diffusive interface method to model the nuclear architecture reorganization process. In particular, we use a Lagrange multiplier approach in the phase field model to preserve the specific physical and geometrical constraints for the biological events. We develop several efficient and robust linear and weakly nonlinear schemes for this new model. To validate the model and numerical methods, we present ample numerical simulations which in particular reproduce several processes of nuclear architecture reorganization from the experiment literature.

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.

Boundary maintenance in the ancestral metazoan Hydra depends on histone acetylation

Much of boundary formation during development remains to be understood, despite being a defining feature of many animal taxa. Axial patterning of Hydra, a member of the ancient phylum Cnidaria which diverged prior to the bilaterian radiation, involves a steady-state of production and loss of tissue, and is dependent on an organizer located in the upper part of the head. We show that the sharp boundary separating tissue in the body column from head and foot tissue depends on histone acetylation. Histone deacetylation disrupts the boundary by affecting numerous developmental genes including Wnt components and prevents stem cells from entering the position dependent differentiation program. Overall, our results suggest that reversible histone acetylation is an ancient regulatory mechanism for partitioning the body axis into domains with specific identity, which was present in the common ancestor of cnidarians and bilaterians, at least 600 million years ago.

Fluctuations and phase transitions of uniaxial and biaxial liquid crystals using a theoretically informed Monte Carlo and a Landau free energy density

In this work, we explore fluctuations during phase transitions of uniaxial and biaxial liquid crystals using a phenomenological free energy functional. We rely on a continuum-level description of the liquid crystal ordering with a tensorial parameter and a temperature dependent Landau polynomial expansion of the tensor's invariants. The free energy functional, over a three-dimensional periodic domain, is integrated with a Gaussian quadrature and minimized with a theoretically informed Monte Carlo method. We reconstruct analytical phase diagrams, following Landau and Doi's notations, to verify that the free energy relaxation reaches the global minimum. Importantly, our relaxation method is able to follow the thermodynamic behavior provided by other non-phenomenological approaches; we predict the first order character of the isotropic-nematic transition, and we identify the uniaxial-biaxial transition as second order. Finally, we use a finite-size scaling method, using the nematic susceptibility, to calculate the transition temperatures for 4-Cyano-4'-pentylbiphenyl (5CB) and N-(4-methoxybenzylidene)-4-butylaniline (MBBA). Our results show good agreement with experimental values, thereby validating our minimization method. Our approach is an alternative towards the relaxation of temperature dependent continuum-level free energy functionals, in any geometry, and can incorporate complicated elastic and surface energy densities.