Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry.

Conditions for self-organisation via Turing's mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a spatially homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further.

Mathematical-based morphological classification of skin eruptions corresponding to the pathophysiological state of chronic spontaneous urticaria.

BACKGROUND: Chronic spontaneous urticaria (CSU) is one of the most intractable human-specific skin diseases. However, as no experimental animal model exists, the mechanism underlying disease pathogenesis in vivo remains unclear, making the establishment of a curative treatment challenging. METHODS: A novel approach combining mathematical modelling, in vitro experiments and clinical data analysis was used to infer the pathological state of CSU patients from geometric features of the skin eruptions. RESULTS: Based on our hierarchical mathematical modelling, the eruptions of CSU were classified into five categories, each with distinct histamine, basophils, mast cells and coagulation factors network signatures. The analysis of 105 real CSU patients with this classification by six individual dermatologists achieved 87.6% agreement. Furthermore, our network analysis revealed that the coagulation status likely determines boundary/area pattern of wheals, while the state of spontaneous histamine release from mast cells may contribute to the divergence of size and outline of the eruptions. CONCLUSIONS: Our multi-faceted approach was accurate in defining pathophysiological states of disease based on geometric features offering the potential to improve the accuracy of CSU diagnosis and better management of the disease in the clinic.

Chronic spontaneous urticaria (CSU) is a persistent skin disease that causes red itchy skin eruptions, called wheals, that are of various shapes. These wheals repeatedly appear and disappear daily for up to weeks or even decades, severely impacting the quality of life of those affected. The causes, consequences or disease processes are largely understudied. Here, we developed a novel approach using mathematical modelling to analyse the geometric measurements of patients' wheals alongside laboratory experiments on human skin samples with eruptions to improve CSU diagnosis and clinical management. We find that geometric measurements of these skin eruptions can be classified into five categories, which could facilitate accurate diagnosis and consequently better clinical management of CSU and may be more widely applied to other skin diseases.

The optimal strategy of incompatible insect technique (IIT) using Wolbachia and the application to malaria control.

For decades, techniques to control vector population with low environmental impact have been widely explored in both field and theoretical studies. The incompatible insect technique (IIT) using Wolbachia, based on cytoplasmic incompatibility, is a technique that Wolbachia-infected male mosquitoes are incapable of producing viable offspring after mating with wild-type female mosquitoes. While the IIT method experimentally ensured its effectiveness in several field works, the failure of female mosquito population control by replacement owing to the accidental contamination of Wolbachia-infected female mosquitoes has been a concern and an obstacle in implementing the IIT method in nature. In this study, we develop a population-based IIT mathematical model using cytoplasmic incompatibility and evaluate the effectiveness of the IIT method in scenarios where contamination is present or absent. In addition, by extending the model to assess the disease infection status of the human population with malaria, we evaluate the optimal release strategy and cost for successful disease control. Our study proves that IIT could be a promising method to control mosquito-borne diseases without perfect suppression of vector mosquito population regardless of contamination.

The extra-embryonic space and the local contour are crucial geometric constraints regulating cell arrangement.

In multicellular systems, cells communicate with adjacent cells to determine their positions and fates, an arrangement important for cellular development. Orientation of cell division, cell-cell interactions (i.e. attraction and repulsion) and geometric constraints are three major factors that define cell arrangement. In particular, geometric constraints are difficult to reveal in experiments, and the contribution of the local contour of the boundary has remained elusive. In this study, we developed a multicellular morphology model based on the phase-field method so that precise geometric constraints can be incorporated. Our application of the model to nematode embryos predicted that the amount of extra-embryonic space, the empty space within the eggshell that is not occupied by embryonic cells, affects cell arrangement in a manner dependent on the local contour and other factors. The prediction was validated experimentally by increasing the extra-embryonic space in the Caenorhabditis elegans embryo. Overall, our analyses characterized the roles of geometrical contributors, specifically the amount of extra-embryonic space and the local contour, on cell arrangements. These factors should be considered for multicellular systems.

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.

CDC-42 Interactions with Par Proteins Are Critical for Proper Patterning in Polarization.

Many cells rearrange proteins and other components into spatially distinct domains in a process called polarization. This asymmetric patterning is required for a number of biological processes including asymmetric division, cell migration, and embryonic development. Proteins involved in polarization are highly conserved and include members of the Par and Rho protein families. Despite the importance of these proteins in polarization, it is not yet known how they interact and regulate each other to produce the protein localization patterns associated with polarization. In this study, we develop and analyse a biologically based mathematical model of polarization that incorporates interactions between Par and Rho proteins that are consistent with experimental observations of CDC-42. Using minimal network and eFAST sensitivity analyses, we demonstrate that CDC-42 is predicted to reinforce maintenance of anterior PAR protein polarity which in turn feedbacks to maintain CDC-42 polarization, as well as supporting posterior PAR protein polarization maintenance. The mechanisms for polarity maintenance identified by these methods are not sufficient for the generation of polarization in the absence of cortical flow. Additional inhibitory interactions mediated by the posterior Par proteins are predicted to play a role in the generation of Par protein polarity. More generally, these results provide new insights into the role of CDC-42 in polarization and the mutual regulation of key polarity determinants, in addition to providing a foundation for further investigations.

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.

A single reaction-diffusion equation for the multifarious eruptions of urticaria.

Urticaria is a common skin disorder characterized by the rapid appearance and disappearance of local skin edema and flares with itching. It is characterized by various macroscopic skin eruptions unique to patients and/or subtypes of urticaria with respect to shape, size, color, and/or duration of eruptions. Nevertheless, the mechanism underlying multifarious eruptions in urticaria is largely unknown. The eruptions are believed to be evoked by histamine release from mast cells in the skin. However, the majority of visible characteristics of urticaria cannot be explained by a simple injection of histamine to the skin. To explain the multifarious eruptions of urticaria, we developed a single reaction-diffusion model suggesting the self-activation and self-inhibition regulation of histamine release from mast cells. Using the model, we found that various geometrical shapes of eruptions typically observed in patients can be explained by the model parameters and randomness or strength of the initial stimuli to mast cells. Furthermore, we verified that the wheal-expanding speed of urticaria, which is shown to be much smaller than that of the intradermal injection experimental system may be explained by our model and a simple diffusion equation. Our study suggests that the simple reaction-diffusion dynamics, including the independent self-activating and -inhibitory regulation of histamine release, may account for the essential mechanism underlying the formation of multifarious eruptions in urticaria.

Role of dynamic nuclear deformation on genomic architecture reorganization.

Higher-order genomic architecture varies according to cell type and changes dramatically during differentiation. One of the remarkable examples of spatial genomic reorganization is the rod photoreceptor cell differentiation in nocturnal mammals. The inverted nuclear architecture found in adult mouse rod cells is formed through the reorganization of the conventional architecture during terminal differentiation. However, the mechanisms underlying these changes remain largely unknown. Here, we found that the dynamic deformation of nuclei via actomyosin-mediated contractility contributes to chromocenter clustering and promotes genomic architecture reorganization during differentiation by conducting an in cellulo experiment coupled with phase-field modeling. Similar patterns of dynamic deformation of the nucleus and a concomitant migration of the nuclear content were also observed in rod cells derived from the developing mouse retina. These results indicate that the common phenomenon of dynamic nuclear deformation, which accompanies dynamic cell behavior, can be a universal mechanism for spatiotemporal genomic reorganization.

Role of domain in pattern formation.

Pattern formation during development is one of the elegant self-organized phenomena that allow cells to regulate their functions. At all levels, from DNA to a tissue or organ, many developmental processes include the determination of cellular functions through pattern formation. To elucidate the mechanism underlying pattern formation, numerous mathematical models have been developed and applied. However, model simplification has resulted in the role of domains not being seriously considered in pattern formation. Here, we introduce a novel application of the phase-field method for analysis of chromatin dynamics, and a mathematical approach that includes domain information into a biochemical model of pattern formation. Using this new modeling method, here, we consider the role of nuclear and cellular cell shapes on pattern formation.

Lateral inhibition-induced pattern formation controlled by the size and geometry of the cell.

Pattern formation in development biology is one of the fundamental processes by which cells change their functions. It is based on the communication of cells via intra- and intercellular dynamics of biochemicals. Thus, the cell is directly involved in biochemical interactions. However, many theoretical approaches describing biochemical pattern formation have usually neglected the cell's role or have simplified the subcellular process without considering cellular aspects despite the cell being the environment where biochemicals interact. On the other hand, recent experimental observations suggest that a change in the physical conditions of cell-to-cell contact can result in a change in cell fate and tissue patterning in a lateral inhibition system. Here we develop a mathematical model by which biochemical dynamics can be directly observed with explicitly expressed cell structure and geometry in higher dimensions, and reconsider pattern formation by lateral inhibition of the Notch-Delta signaling pathway. We explore how the physical characteristic of cell, such as cell geometry or size, influences the biochemical pattern formation in a multi-cellular system. Our results suggest that a property based on cell geometry can be a novel mechanism for symmetry breaking inducing cell asymmetry. We show that cell volume can critically influence cell fate determination and pattern formation at the tissue level, and the surface area of the cell-to-cell contact can directly affect the spatial range of patterning.

Positioning of polarity formation by extracellular signaling during asymmetric cell division.

Anterior-posterior (AP) polarity formation of cell membrane proteins plays a crucial role in determining cell asymmetry, which ultimately generates cell diversity. In Caenorhabditis elegans, a single fertilized egg cell (P0), its daughter cell (P1), and the germline precursors (P2 and P3 cells) form two exclusive domains of different PAR proteins on the membrane along the anterior-posterior axis. However, the phenomenon of polarity reversal has been observed in which the axis of asymmetric cell division of the P2 and P3 cells is formed in an opposite manner to that of the P0 and P1 cells. The extracellular signal MES-1/SRC-1 has been shown to induce polarity reversal, but the detailed mechanism remains elusive. Here, using a mathematical model, I explore the mechanism by which MES-1/SRC-1 signaling can induce polarity reversal and ultimately affect the process of polarity formation. I show that a positive correlation between SRC-1 and the on-rate of PAR-2 is the essential mechanism underlying polarity reversal, providing a mathematical basis for the orientation of cell polarity patterns.

Self-organization and advective transport in the cell polarity formation for asymmetric cell division.

Anterior-Posterior (AP) polarity formation of cell membrane proteins plays a crucial role in determining cell asymmetry, which depends not only on the several genetic process but also biochemical and biophysical interactions. The mechanism of AP formation of Caenorhabditis elegans embryo is characterized into the three processes: (i) membrane association and dissociation of posterior and anterior proteins, (ii) diffusion into the membrane and cytosol, and (iii) active cortical and cytoplasmic flows induced by the contraction of the acto-myosin cortex. We explored the mechanism of symmetry breaking and AP polarity formation using self-recruitment model of posterior proteins. We found that the AP polarity pattern is established over wide range in the total mass of polarity proteins and the diffusion ratio in the cytosol to the membrane. We also showed that the advective transport in both membrane and cytosol during the establishment phase affects optimal time interval of establishment and positioning of the posterior domain, and plays a role to increase the robustness in the AP polarity formation by reducing the number of posterior domains for the sensitivity of initial conditions. We also demonstrated that a proper ratio of the total mass to cell size robustly regulate the length scale of the posterior domain.

Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems.

Cellular gene expression is a complex process involving many steps, including the transcription of DNA and translation of mRNA; hence the synthesis of proteins requires a considerable amount of time, from ten minutes to several hours. Since diffusion-driven instability has been observed to be sensitive to perturbations in kinetic delays, the application of Turing patterning mechanisms to the problem of producing spatially heterogeneous differential gene expression has been questioned. In deterministic systems a small delay in the reactions can cause a large increase in the time it takes a system to pattern. Recently, it has been observed that in undelayed systems intrinsic stochasticity can cause pattern initiation to occur earlier than in the analogous deterministic simulations. Here we are interested in adding both stochasticity and delays to Turing systems in order to assess whether stochasticity can reduce the patterning time scale in delayed Turing systems. As analytical insights to this problem are difficult to attain and often limited in their use, we focus on stochastically simulating delayed systems. We consider four different Turing systems and two different forms of delay. Our results are mixed and lead to the conclusion that, although the sensitivity to delays in the Turing mechanism is not completely removed by the addition of intrinsic noise, the effects of the delays are clearly ameliorated in certain specific cases.