Pattern formation mechanisms of self-organizing reaction-diffusion systems

Self-Organization of Cellular Units

The purpose of this review is to explore self-organizing mechanisms that pattern microtubules (MTs) and spatially organize animal cell cytoplasm, inspired by recent experiments in frog egg extract. We start by reviewing conceptual distinctions between self-organizing and templating mechanisms for subcellular organization. We then discuss self-organizing mechanisms that generate radial MT arrays and cell centers in the absence of centrosomes. These include autocatalytic MT nucleation, transport of minus ends, and nucleation from organelles such as melanosomes and Golgi vesicles that are also dynein cargoes. We then discuss mechanisms that partition the cytoplasm in syncytia, in which multiple nuclei share a common cytoplasm, starting with cytokinesis, when all metazoan cells are transiently syncytial. The cytoplasm of frog eggs is partitioned prior to cytokinesis by two self-organizing modules, protein regulator of cytokinesis 1 (PRC1)-kinesin family member 4A (KIF4A) and chromosome passenger complex (CPC)-KIF20A. Similar modules may partition longer-lasting syncytia, such as early Drosophila embryos. We end by discussing shared mechanisms and principles for the MT-based self-organization of cellular units.

Systems of pattern formation within developmental biology

Applications of mathematical models to developmental biology have provided helpful insight into various subfields, ranging from the patterning of animal skin to the development of complex organ systems. Systems involved in patterning within morphology present a unique path to explain self-organizing systems. Current efforts show that patterning systems, notably Reaction-Diffusion and specific signaling pathways, provide insight for explaining morphology and could provide novel applications revolving around the formation of biological systems. Furthermore, the application of pattern formation provides a new perspective on understanding developmental biology and pathology research to study molecular mechanisms. The current review is to cover and take a more in-depth overlook at current applications of patterning systems while also building on the principles of patterning of future research in predictive medicine.

Patterning with clocks and genetic cascades: Segmentation and regionalization of vertebrate versus insect body plans

Oscillatory and sequential processes have been implicated in the spatial patterning of many embryonic tissues. For example, molecular clocks delimit segmental boundaries in vertebrates and insects and mediate lateral root formation in plants, whereas sequential gene activities are involved in the specification of regional identities of insect neuroblasts, vertebrate neural tube, vertebrate limb, and insect and vertebrate body axes. These processes take place in various tissues and organisms, and, hence, raise the question of what common themes and strategies they share. In this article, we review 2 processes that rely on the spatial regulation of periodic and sequential gene activities: segmentation and regionalization of the anterior-posterior (AP) axis of animal body plans. We study these processes in species that belong to 2 different phyla: vertebrates and insects. By contrasting 2 different processes (segmentation and regionalization) in species that belong to 2 distantly related phyla (arthropods and vertebrates), we elucidate the deep logic of patterning by oscillatory and sequential gene activities. Furthermore, in some of these organisms (e.g., the fruit fly Drosophila), a mode of AP patterning has evolved that seems not to overtly rely on oscillations or sequential gene activities, providing an opportunity to study the evolution of pattern formation mechanisms.

Model comparison via simplicial complexes and persistent homology

In many scientific and technological contexts, we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models a priori. Here, we develop and illustrate two such approaches that allow us to compare model structures in a systematic way by representing models as simplicial complexes. Using well-developed concepts from simplicial algebraic topology, we define a distance between models based on their simplicial representations. Employing persistent homology with a flat filtration provides for alternative representations of the models as persistence intervals, which represent model structure, from which the model distances are also obtained. We then expand on this measure of model distance to study the concept of model equivalence to determine the conceptual similarity of models. We apply our methodology for model comparison to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.

Stability and robustness properties of bioelectric networks: A computational approach

Morphogenesis during development and regeneration requires cells to communicate and cooperate toward the construction of complex anatomical structures. One important set of mechanisms for coordinating growth and form occurs via developmental bioelectricity-the dynamics of cellular networks driving changes of resting membrane potential which interface with transcriptional and biomechanical downstream cascades. While many molecular details have been elucidated about the instructive processes mediated by ion channel-dependent signaling outside of the nervous system, future advances in regenerative medicine and bioengineering require the understanding of tissue, organ, or whole body-level properties. A key aspect of bioelectric networks is their robustness, which can drive correct, invariant patterning cues despite changing cell number and anatomical configuration of the underlying tissue network. Here, we computationally analyze the minimal models of bioelectric networks and use the example of the regenerating planarian flatworm, to reveal important system-level aspects of bioelectrically derived patterns. These analyses promote an understanding of the robustness of circuits controlling regeneration and suggest design properties that can be exploited for synthetic bioengineering.

Mechanics and self-organization in tissue development

Self-organization is an all-important feature of living systems that provides the means to achieve specialization and functionality at distinct spatio-temporal scales. Herein, we review this concept by addressing the packing organization of cells, the sorting/compartmentalization phenomenon of cell populations, and the propagation of organizing cues at the tissue level through traveling waves. We elaborate on how different theoretical models and tools from Topology, Physics, and Dynamical Systems have improved the understanding of self-organization by shedding light on the role played by mechanics as a driver of morphogenesis. Altogether, by providing a historical perspective, we show how ideas and hypotheses in the field have been revisited, developed, and/or rejected and what are the open questions that need to be tackled by future research.

Self-organized protein patterns: The MinCDE and ParABS systems

Self-organized protein patterns are of tremendous importance for biological decision-making processes. Protein patterns have been shown to identify the site of future cell division, establish cell polarity, and organize faithful DNA segregation. Intriguingly, several key concepts of pattern formation and regulation apply to a variety of different protein systems. Herein, we explore recent advances in the understanding of two prokaryotic pattern-forming systems: the MinCDE system, positioning the FtsZ ring precisely at the midcell, and the ParABS system, distributing newly synthesized DNA along with the cell. Despite differences in biological functionality, these two systems have remarkably similar molecular components, mechanisms, and strategies to achieve biological robustness.

Chemically controlled pattern formation in self-oscillating elastic shells

Patterns and morphology develop in living systems such as embryos in response to chemical signals. To understand and exploit the interplay of chemical reactions with mechanical transformations, chemomechanical polymer systems have been synthesized by attaching chemicals into hydrogels. In this work, we design autonomous responsive elastic shells that undergo morphological changes induced by chemical reactions. We couple the local mechanical response of the gel with the chemical processes on the shell. This causes swelling and deswelling of the gel, generating diverse morphological changes, including periodic oscillations. We further introduce a mechanical instability and observe buckling-unbuckling dynamics with a response time delay. Moreover, we investigate the mechanical feedback on the chemical reaction and demonstrate the dynamic patterns triggered by an initial deformation. We show the chemical characteristics that account for the shell morphology and discuss the future designs for autonomous responsive materials.

Reaction-Diffusion Model-Based Research on Formation Mechanism of Neuron Dendritic Spine Patterns

The pattern abnormalities of dendritic spine, tiny protrusions on neuron dendrites, have been found related to multiple nervous system diseases, such as Parkinson's disease and schizophrenia. The determination of the factors affecting spine patterns is of vital importance to explore the pathogenesis of these diseases, and further, search the treatment method for them. Although the study of dendritic spines is a hot topic in neuroscience in recent years, there is still a lack of systematic study on the formation mechanism of its pattern. This paper provided a reinterpretation of reaction-diffusion model to simulate the formation process of dendritic spine, and further, study the factors affecting spine patterns. First, all four classic shapes of spines, mushroom-type, stubby-type, thin-type, and branched-type were reproduced using the model. We found that the consumption rate of substrates by the cytoskeleton is a key factor to regulate spine shape. Moreover, we found that the density of spines can be regulated by the amount of an exogenous activator and inhibitor, which is in accordance with the anatomical results found in hippocampal CA1 in SD rats with glioma. Further, we analyzed the inner mechanism of the above model parameters regulating the dendritic spine pattern through Turing instability analysis and drew a conclusion that an exogenous inhibitor and activator changes Turing wavelength through which to regulate spine densities. Finally, we discussed the deep regulation mechanisms of several reported regulators of dendritic spine shape and densities based on our simulation results. Our work might evoke attention to the mathematic model-based pathogenesis research for neuron diseases which are related to the dendritic spine pattern abnormalities and spark inspiration in the treatment research for these diseases.

Roadmap for the multiscale coupling of biochemical and mechanical signals during development

The way in which interactions between mechanics and biochemistry lead to the emergence of complex cell and tissue organization is an old question that has recently attracted renewed interest from biologists, physicists, mathematicians and computer scientists. Rapid advances in optical physics, microscopy and computational image analysis have greatly enhanced our ability to observe and quantify spatiotemporal patterns of signalling, force generation, deformation, and flow in living cells and tissues. Powerful new tools for genetic, biophysical and optogenetic manipulation are allowing us to perturb the underlying machinery that generates these patterns in increasingly sophisticated ways. Rapid advances in theory and computing have made it possible to construct predictive models that describe how cell and tissue organization and dynamics emerge from the local coupling of biochemistry and mechanics. Together, these advances have opened up a wealth of new opportunities to explore how mechanochemical patterning shapes organismal development. In this roadmap, we present a series of forward-looking case studies on mechanochemical patterning in development, written by scientists working at the interface between the physical and biological sciences, and covering a wide range of spatial and temporal scales, organisms, and modes of development. Together, these contributions highlight the many ways in which the dynamic coupling of mechanics and biochemistry shapes biological dynamics: from mechanoenzymes that sense force to tune their activity and motor output, to collectives of cells in tissues that flow and redistribute biochemical signals during development.

Evolution of pigment cells and patterns: recent insights from teleost fishes

Skin pigment patterns of vertebrates are stunningly diverse, and nowhere more so than in teleost fishes. Several species, including relatives of zebrafish, recently evolved cichlid fishes of East Africa, clownfishes, deep sea fishes, and others are providing insights into pigment pattern evolution. This overview describes recent advances in understanding periodic patterns, like stripes and spots, the loss of patterns, and the role of cell-type diversification in generating pigmentation phenotypes. Advances in this area are being facilitated by the application of modern methods of gene editing, genomics, computational analysis, and other approaches to non-traditional model organisms having interesting pigmentary phenotypes. Several topics worthy of future attention are outlined as well.

Community effects allow bioelectrical reprogramming of cell membrane potentials in multicellular aggregates: Model simulations

Bioelectrical patterns are established by spatiotemporal correlations of cell membrane potentials at the multicellular level, being crucial to development, regeneration, and tumorigenesis. We have conducted multicellular simulations on bioelectrical community effects and intercellular coupling in multicellular aggregates. The simulations aim at establishing under which conditions a local heterogeneity consisting of a small patch of cells can be stabilized against a large aggregate of surrounding identical cells which are in a different bioelectrical state. In this way, instructive bioelectrical information can be persistently encoded in spatiotemporal patterns of separated domains with different cell polarization states. The multicellular community effects obtained are regulated both at the single-cell and intercellular levels, and emerge from a delicate balance between the degrees of intercellular coupling in: (i) the small patch, (ii) the surrounding bulk, and (iii) the interface that separates these two regions. The model is experimentally motivated and consists of two generic voltage-gated ion channels that attempt to establish the depolarized and polarized cell states together with coupling conductances whose individual and intercellular different states permit a dynamic multicellular connectivity. The simulations suggest that community effects may allow the reprogramming of single-cell bioelectrical states, in agreement with recent experimental data. A better understanding of the resulting electrical regionalization can assist the electroceutical correction of abnormally depolarized regions initiated in the bulk of normal tissues as well as suggest new biophysical mechanisms for the establishment of target patterns in multicellular engineering.

Compete or Coexist? Why the Same Mechanisms of Symmetry Breaking Can Yield Distinct Outcomes

Cellular morphogenesis is governed by the prepattern based on the symmetry-breaking emergence of dense protein clusters. Thus, a cluster of active GTPase Cdc42 marks the site of nascent bud in the baker's yeast. An important biological question is which mechanisms control the number of pattern maxima (spots) and, thus, the number of nascent cellular structures. Distinct flavors of theoretical models seem to suggest different predictions. While the classical Turing scenario leads to an array of stably coexisting multiple structures, mass-conserved models predict formation of a single spot that emerges via the greedy competition between the pattern maxima for the common molecular resources. Both the outcome and the kinetics of this competition are of significant biological importance but remained poorly explored. Recent theoretical analyses largely addressed these questions, but their results have not yet been fully appreciated by the broad biological community. Keeping mathematical apparatus and jargon to the minimum, we review the main conclusions of these analyses with their biological implications in mind. Focusing on the specific example of pattern formation by small GTPases, we speculate on the features of the patterning mechanisms that bypass competition and favor formation of multiple coexisting structures and contrast them with those of the mechanisms that harness competition to form unique cellular structures.

Wnt/β-catenin Signaling in Tissue Self-Organization

Across metazoans, animal body structures and tissues exist in robust patterns that arise seemingly out of stochasticity of a few early cells in the embryo. These patterns ensure proper tissue form and function during early embryogenesis, development, homeostasis, and regeneration. Fundamental questions are how these patterns are generated and maintained during tissue homeostasis and regeneration. Though fascinating scientists for generations, these ideas remain poorly understood. Today, it is apparent that the Wnt/β-catenin pathway plays a central role in tissue patterning. Wnt proteins are small diffusible morphogens which are essential for cell type specification and patterning of tissues. In this review, we highlight several mechanisms described where the spatial properties of Wnt/β-catenin signaling are controlled, allowing them to work in combination with other diffusible molecules to control tissue patterning. We discuss examples of this self-patterning behavior during development and adult tissues' maintenance. The combination of new physiological culture systems, mathematical approaches, and synthetic biology will continue to fuel discoveries about how tissues are patterned. These insights are critical for understanding the intricate interplay of core patterning signals and how they become disrupted in disease.

Engineering and modeling of multicellular morphologies and patterns

Synthetic multicellular (MC) systems have the capacity to increase our understanding of biofilms and higher organisms, and to serve as engineering platforms for developing complex products in the areas of medicine, biosynthesis and smart materials. Here we provide an interdisciplinary perspective and review on emerging approaches to engineer and model MC systems. We lay out definitions for key terms in the field and identify toolboxes of standardized parts which can be combined into various MC algorithms to achieve specific outcomes. Many essential parts and algorithms have been demonstrated in some form. As key next milestones for the field, we foresee the improvement of these parts and their adaptation to more biological systems, the demonstration of more complex algorithms, the advancement of quantitative modeling approaches and compilers to support rational MC engineering, and implementation of MC engineering for practical applications.

Stripes on finite domains: Why the zigzag instability is only a partial story

Stationary periodic patterns are widespread in natural sciences, ranging from nano-scale electrochemical and amphiphilic systems to mesoscale fluid, chemical, and biological media and to macro-scale vegetation and cloud patterns. Their formation is usually due to a primary symmetry breaking of a uniform state to stripes, often followed by secondary instabilities to form zigzag and labyrinthine patterns. These secondary instabilities are well studied under idealized conditions of an infinite domain; however, on finite domains, the situation is more subtle since the unstable modes depend also on boundary conditions. Using two prototypical models, the Swift-Hohenberg equation and the forced complex Ginzburg-Landau equation, we consider finite size domains with no flux boundary conditions transversal to the stripes and reveal a distinct mixed-mode instability that lies in between the classical zigzag and the Eckhaus lines. This explains the stability of stripes in the mildly zigzag unstable regime and, after crossing the mixed-mode line, the evolution of zigzag stripes in the bulk of the domain and the formation of defects near the boundaries. The results are of particular importance for problems with large timescale separation, such as bulk-heterojunction deformations in organic photovoltaic and vegetation in semi-arid regions, where early temporal transients may play an important role.