Reaction-diffusion models of development with state-dependent chemical diffusion coefficients

Reaction-diffusion models are widely used to model developmental processes. The great majority of current models invoke constant diffusion coefficients. However, the diffusion of metabolites or signals through tissues is frequently such that this assumption may reasonably be questioned. We consider several different physical mechanisms leading to effective diffusion coefficients in biological tissues which vary with the local conditions, including models in which juxtacrine signaling results in the diffusion of a signal in the absence of material transport. We develop a mathematical formalism for transforming local transport laws into diffusive terms. This procedure is appropriate when the typical length scale over which the concentrations change significantly is much greater than the dimensions of a cell. We review previous developmental models which considered the possibility of state-dependent diffusion coefficients. We also provide a few new motivating examples.     

Reaction-diffusion models of growing plant tips: bifurcations on hemispheres

We study two chemical models for pattern formation in growing plant tips. For hemisphere radius and parameter values together optimal for spherical surface harmonic patterns of index l = 3, the Brusselator model gives an 84% probability of dichotomous branching pattern and 16% of annular pattern, while the hyperchirality model gives 88% probability of dichotomous branching and 12% of annular pattern. The models are two-morphogen reaction-diffusion systems on the surface of a hemispherical shell, with Dirichlet boundary conditions. Bifurcation analysis shows that both models give possible mechanisms for dichotomous branching of the growing tips. Symmetries of the models are used in the analysis.     

Spatial patterning of pigmentation in evergreen leaves in response to freezing stress

Evergreen leaves of temperate climate plants are often subject to frosts. Changes in carbon gain patterns arise from freezing‐related tissue damage, and from interactions between light and temperature stress. We examined relationships between spatial patterns in freezing and concentrations of chlorophyll. Spatial patterns in pigmentation in leaves that had or had not been exposed to naturally occurring frosts were determined by conventional extraction techniques combined with high‐resolution hyperspectral imaging of reflectance from intact leaves. Predictive indices were developed to relate reflectance to chlorophyll content and chlorophyll a/b ratios within intact leaves. Leaves exposed to frosts had lower chlorophyll contents and more variable a/b ratios than protected leaves. In frost‐affected leaves, chlorophyll content was highest near leaf centres and decreased toward leaf tips and margins. Decline in chlorophyll content was associated with shifts in chlorophyll a/b ratios and increases in red pigmentation due to anthocyanin, with effects being greater on leaf sides exposed directly to the sun. These altered pigmentation patterns were consistent with patterns in freezing. The present results illustrate the fine scale of spatial variation in leaf response to freezing, and raise important questions about impacts of freezing on photosynthetic function in over‐wintering evergreens.     

Auxin transport-feedback models of patterning in plants

Many patterning events in plants are regulated by the phytohormone auxin. In fact, so many things are under the influence of auxin that it seems difficult to understand how a single hormone can do so much. Auxin moves throughout the plant via a network of specialized membrane-bound import and export proteins, which are often differentially expressed and polarized depending on tissue type. Here, we review simulation models of pattern formation that are based on the control of these transporters by auxin itself. In these transport-feedback models, diversity in patterning comes not from the addition of more morphogens, but rather by varying the mechanism that regulates the transporters.     

Epidermal patterning in Arabidopsis: models make a difference

The leaf and root epidermis in Arabidopsis provide ideal systems in which to explore the mechanisms that underlie the patterned assignment of cell fates during development. Extensive experimental studies have uncovered a complex interlocked feedback network that operates within the epidermis to coordinate the choice between hair and nonhair fates. A number of recent studies using mathematical models have begun to study this network, highlighting new mechanisms that have subsequently been confirmed in model-directed experiments. These studies illustrate the potential of integrated modeling and experimentation to shed new light on developmental processes. Moreover, these models enable systems-level comparative analyses that may help understand the origin and role of properties, such as robustness and redundancy in developmental systems and, concomitantly, the evolution of development itself.     

Classic limb patterning models and the work of Dennis Summerbell

Dennis Summerbell was a leading contributor to our understanding of limb patterning prior to the advent of molecular biology. He published several groundbreaking papers, including one that developed a key model for patterning the limb from the shoulder to the fingertips and another that presented the co-discovery of the effect of retinoids on limb morphogenesis. He brought detailed quantitative analyses to bear on these studies, as highlighted in two of his insightful papers published in the Journal of Embryology and Experimental Morphology, in which he provided elegant models that, today, remain relevant to limb patterning, as well as to many disciplines of developmental biology.     

Exploring alternative models of rostral-caudal patterning in the zebrafish neurectoderm with computer simulations

The StarLogo and NetLogo programming environments allow developmental biologists to build computer models of cell-cell interactions in an epithelium and visualize emergent properties of hypothetical genetic regulatory networks operating in the cells. These environments were used to explore alternative models that show how a posteriorizing morphogen gradient might define gene-expression domains along the rostral-caudal axis in the zebrafish neurectoderm. The models illustrate how a hypothetical genetic network based on auto-activation and cross-repression could lead to establishment of discrete non-overlapping gene-expression domains.     

Reaction-diffusion pattern in shoot apical meristem of plants

A fundamental question in developmental biology is how spatial patterns are self-organized from homogeneous structures. In 1952, Turing proposed the reaction-diffusion model in order to explain this issue. Experimental evidence of reaction-diffusion patterns in living organisms was first provided by the pigmentation pattern on the skin of fishes in 1995. However, whether or not this mechanism plays an essential role in developmental events of living organisms remains elusive. Here we show that a reaction-diffusion model can successfully explain the shoot apical meristem (SAM) development of plants. SAM of plants resides in the top of each shoot and consists of a central zone (CZ) and a surrounding peripheral zone (PZ). SAM contains stem cells and continuously produces new organs throughout the lifespan. Molecular genetic studies using Arabidopsis thaliana revealed that the formation and maintenance of the SAM are essentially regulated by the feedback interaction between WUSHCEL (WUS) and CLAVATA (CLV). We developed a mathematical model of the SAM based on a reaction-diffusion dynamics of the WUS-CLV interaction, incorporating cell division and the spatial restriction of the dynamics. Our model explains the various SAM patterns observed in plants, for example, homeostatic control of SAM size in the wild type, enlarged or fasciated SAM in clv mutants, and initiation of ectopic secondary meristems from an initial flattened SAM in wus mutant. In addition, the model is supported by comparing its prediction with the expression pattern of WUS in the wus mutant. Furthermore, the model can account for many experimental results including reorganization processes caused by the CZ ablation and by incision through the meristem center. We thus conclude that the reaction-diffusion dynamics is probably indispensable for the SAM development of plants.     

Reconciling different models of forebrain induction and patterning: a dual role for the hypoblast

Several models have been proposed for the generation of the rostral nervous system. Among them, Nieuwkoop's activation/transformation hypothesis and Spemann's idea of separate head and trunk/tail organizers have been particularly favoured recently. In the mouse, the finding that the visceral endoderm (VE) is required for forebrain development has been interpreted as support for the latter model. Here we argue that the chick hypoblast is equivalent to the mouse VE, based on fate, expression of molecular markers and characteristic anterior movements around the time of gastrulation. We show that the hypoblast does not fit the criteria for a head organizer because it does not induce neural tissue from na?ve epiblast, nor can it change the regional identity of neural tissue. However, the hypoblast does induce transient expression of the early markers Sox3 and Otx2. The spreading of the hypoblast also directs cell movements in the adjacent epiblast, such that the prospective forebrain is kept at a distance from the organizer at the tip of the primitive streak. We propose that this movement is important to protect the forebrain from the caudalizing influence of the organizer. This dual role of the hypoblast is more consistent with the Nieuwkoop model than with the notion of separate organizers, and accommodates the available data from mouse and other vertebrates.     

Mechanism of skin pigmentation

Melanin is a pigment that plays an important role in providing coloration and protecting human skin from the harmful effects of UV light radiation. Human skin color is determined by the type and amount of melanins that are synthesized and deposited within the melanosomes. In addition, the transfer of these specialized membrane-bound organelles from melanocytes to surrounding keratinocytes also plays a role in dictating human skin color. In order to investigate the principle features of skin pigmentation, the origin, function, and production ability of melanin should be highly understood in terms of biological and pathophysiological aspects. Furthermore, a deep understanding of melanin synthesis will also contribute to cosmetics and drugs development. In this review, the processes of melanin biosynthesis, such as survival, proliferation, and differentiation of melanin cells, as well as the biological regulation of human pigmentation were described.     

Reaction-diffusion microtubule concentration patterns occur during biological morphogenesis.

Reaction-diffusion processes can lead to a macroscopic concentration pattern from an initially homogeneous solution, and thus provide a physical-chemical mechanism for biological pattern formation and morphogenesis. The central prediction of reaction-diffusion theory is that the patterns contain periodic concentration variations in some of the reactives. Microtubules assembled in vitro spontaneously self-organise and form stationary striped macroscopic structures. In agreement with reaction-diffusion theory. Here we show, in agreement with reaction-diffusion theory, that these preparations contain substantial microtubule concentration variations. Similar striped microtubule patterns arise during Drosophila embryogenesis. A characteristic of these patterns is their dependence on sample dimensions. In Drosophila eggs shortened by ligation, we found that the microtubule pattern varied with egg fragment length in the same way as the in vitro microtubule pattern varied with sample length, and as expected from theory. This is evidence that reaction-diffusion structures occur during Drosophila morphogenesis.     

混合拟单调系统行波解的存在性

运用单调迭代方法,证明了混合拟单调系统的行波解的存在性．当反应扩散系。统的反应函数是混合拟单调函数时,如果选取一对合适的耦合上下解作为迭代初值,则迭代序列将收敛到一对拟解．而且在这对拟解之间存在系统的行波解．     

Strategies for salmonid pigmentation

Well pigmented flesh is regarded second after freshness as one of the most important quality criteria for farmed Atlantic salmon ( Salmo salar ). The required redness of salmonid flesh has caused a steady increase of astaxanthin concentration in salmonid flesh from about 2–3 mg kg⁻¹ during the 1970s to above 6 mg kg⁻¹ today. An individual variation in flesh pigmentation of more than 100% is experienced under practical as well as experimental conditions. An individual variation of this magnitude makes the precision of carotenoid estimations low, and impairs comparisons of treatments. The present paper discuss the importance of standardisation of the experimental conditions in relation to the level of flesh pigmentation and to variation in populations with the main focus on the effect of fish size, flesh carotenoid concentration and environmental factors.