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The tetrapod limb skeleton is initiated in unpatterned limb bud mesenchyme by the formation of precartilage condensations. Here, based on time-lapse videographic analysis of a forming condensation in a high-density culture of chicken limb bud mesenchyme, we observe a phase transition to a more fluidized state for cells within spatial compacted foci (protocondensations that will progress to condensations), as reflected in their spatial confinement, cell-substratum interaction and speed of motion. Previous work showed that galectin-8 and galectin-1A, two proteins of the galactoside-binding galectin family, are the earliest determinants of this process in the chicken limb bud, and that their interactions in forming skeletogenic patterns of condensations can be interpreted mathematically through a reaction-diffusion-adhesion framework. Based on this framework, we use an ordinary differential equation-based approach to analyze the core switching modality of the galectin reaction network and characterize the states of the network independent of the diffusive and adhesive arms of the patterning mechanism. We identify two steady states where the concentrations of both galectins are respectively, negligible, and very high. An explicit Lyapunov function shows that there are no periodic solutions. For sigmoidal galectin production terms, the model exhibits a bistable switch that arises from a monostable state via saddle-node bifurcation. Our model therefore predicts that the galectin network exists in low and high expression states separated in space or time without any intermediate states. This provides a causal basis for the observed outside vs. inside transition observed in the in vitro video data. We performed a quantitative analysis of the distribution of galectin-1A in cultures of condensing chick limb mesenchymal cells and found that the interior of the protocondensations had concentrations of this protein (compared to the immediate exterior) over and above that expected from its higher cell density, consistent with the model’s predictions. The galectin-based patterning network is thus suggested, on theoretical grounds, to incorporate a core switch independent of any spatial or temporal dynamics, that drives the chondrogenic cell state transition in limb skeletogenesis.

We review the current state of mathematical modeling of cartilage pattern formation in vertebrate limbs. We place emphasis on several reaction-diffusion type models that have been proposed in the last few years. These models are grounded in more detailed knowledge of the relevant regulatory processes than previous ones but generally refer to different molecular aspects of these processes. Considering these models in light of comparative phylogenomics permits framing of hypotheses on the evolutionary order of appearance of the respective mechanisms and their roles in the fin-to-limb transition. This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Mechanistic Models Developmental Biology > Developmental Processes in Health and Disease Analytical and Computational Methods > Analytical Methods.

We propose a numerical approach that combines a radial basis function (RBF) meshless approximation with a finite difference discretization to solve a nonlinear system of integro-differential equations. The equations are of advection-reaction-diffusion type modeling the formation of pre-cartilage condensations in embryonic chicken limbs. The computational domain is four dimensional in the sense that the cell density depends continuously on two spatial variables as well as two structure variables, namely membrane-bound counterreceptor densities. The biologically proper Dirichlet boundary conditions imposed in the semi-infinite structure variable region is in favor of a meshless method with Gaussian basis functions. Coupled with WENO5 finite difference spatial discretization and the method of integrating factors, the time integration via method of lines achieves optimal complexity. In addition, the proposed scheme can be extended to similar models with more general boundary conditions. Numerical results are provided to showcase the validity of the scheme.

The tetrapod appendicular skeleton is initiated as spatially patterned mesenchymal condensations. The size and spacing of these condensations in avian limb buds are mediated by a reaction-diffusion-adhesion network consisting of galectins Gal-1A, Gal-8 and their cell surface receptors. In cell cultures, the appearance of condensations is synchronized across distances greater than the characteristic wavelength of their spatial pattern. We explored the possible role of observed oscillations of the transcriptional co-regulator Hes1 in this phenomenon. Treatment of micromass cultures with DAPT, a γ-secretase inhibitor, damped Hes1 oscillations, elevated Gal-1A and -8 mRNA levels, and led to irregularly-sized proto-condensations that subsequently fused. In developing limb buds, DAPT led to spatially non-uniform Hes1 expression and fused, truncated and misshapen digits. Periodicity in adhesive response to Gal-1A, a plausible Hes1-dependent function, was added to a previously tested mathematical model for condensation patterning by the two-galectin network. The enhanced model predicted regularization of patterning due to synchronization of Hes1 oscillations and resulting spatiotemporal coordination of its expression. The model also predicted changes in galectin expression and patterning in response to suppression of Hes1 expression, which were confirmed in in vitro experiments. Our results indicate that the two-galectin patterning network is regulated by Hes1 dynamics, the synchronization of which refines and regularizes limb skeletogenesis.

Genotypic variation, environmental variation, and their interaction may produce variation in the developmental process and cause phenotypic differences among individuals. Developmental noise, which arises during development from stochasticity in cellular and molecular processes when genotype and environment are fixed, also contributes to phenotypic variation. While evolutionary biology has long focused on teasing apart the relative contribution of genes and environment to phenotypic variation, our understanding of the role of developmental noise has lagged due to technical difficulties in directly measuring the contribution of developmental noise. The influence of developmental noise is likely underestimated in studies of phenotypic variation due to intrinsic mechanisms within organisms that stabilize phenotypes and decrease variation. Since we are just beginning to appreciate the extent to which phenotypic variation due to stochasticity is potentially adaptive, the contribution of developmental noise to phenotypic variation must be separated and measured to fully understand its role in evolution. Here, we show that variation in the component of the developmental process corresponding to environmental and genetic factors (here treated together as a unit called the LALI-type) versus the contribution of developmental noise, can be distinguished for leopard gecko (Eublepharis macularius) head color patterns using mathematical simulations that model the role of random variation (corresponding to developmental noise) in patterning. Specifically, we modified the parameters of simulations corresponding to variation in the LALI-type to generate the full range of phenotypic variation in color pattern seen on the heads of eight leopard geckos. We observed that over the range of these parameters, variation in color pattern due to LALI-type variation exceeds that due to developmental noise in the studied gecko cohort. However, the effect of developmental noise on patterning is also substantial. Our approach addresses one of the major goals of evolutionary biology: to quantify the role of stochasticity in shaping phenotypic variation., Author summary The observable characteristics of an organism make up its phenotype. Variation among phenotypes is due to genetic differences, environmental factors and developmental noise (effects due to inherent stochasticity) during development. We used mathematical models to investigate the contributions of variation of the developmental process due to genetic and environmental factors (treated in this work as a single unit) versus developmental noise (unavoidable variation within the developmental program) to the development of pigment patterns on gecko heads. We found that for our cohort, the proportion of phenotypic variation due to variation in the unit composed of genotypic and environmental variation is larger than that due to developmental noise. Furthermore, by allowing the parameters of the mathematical model to vary, we generated the full extent of potential phenotypic pattern variation that could occur on the head of geckos. This serves to further study the influence of the buffering mechanisms (canalization, selection, and developmental stability) limiting phenotypic variation. This approach can be applied to any regular morphological trait that results from self-organized processes such as reaction-diffusion mechanisms, including the frequently found striped and spotted patterns of animal pigmentation patterning, patterning of bones in vertebrate limbs, and body segmentation in segmented animals.

The paired appendages (fins or limbs) of jawed vertebrates contain an endoskeleton consisting of nodules, bars and, in some groups, plates of cartilage, or bone arising from replacement of cartilaginous templates. The generation of the endoskeletal elements occurs by processes involving production and diffusion of morphogens, with, variously, positive and negative feedback circuits, adhesion, and receptor dynamics with similarities to the mechanism for chemical pattern formation proposed by Alan Turing. This review presents a unified interpretation of the evolution and functioning of these mechanisms. Studies are described indicating that protocondensations, compacted mesenchymal cell aggregates that prefigure the appendicular skeleton, arise through the adhesive activity of galectin-1, a matricellular protein with skeletogenic homologs in all jawed vertebrates. In the cartilaginous and lobe-finned fishes (and to a variable extent in ray-finned fishes) it additionally cooperates with an isoform of galectin-8 to constitute a self-organizing network capable of generating arrays of preskeletal nodules, bars and plates. Further, in the tetrapods, a putative galectin-8 control module was acquired that may have enabled proximodistal increase in the number of protocondensations. In parallel to this, other self-organizing networks emerged that acted, via Bmp, Wnt, Sox9 and Runx2, as well as transforming factor-β and fibronectin, to convert protocondensations into skeletal tissues. The progressive appearance and integration of these skeletogenic networks over evolution occurred in the context of an independently evolved system of Hox protein and Shh gradients that interfaced with them to tune the spatial wavelengths and refine the identities of the resulting arrays of elements.

The tetrapod appendicular skeleton is initiated as spatially patterned mesenchymal condensations. The size and spacing of these condensations in avian limb buds are mediated by a reaction-diffusion-adhesion network consisting of galectins Gal-1A, Gal-8 and their cell surface receptors. In cell cultures, the appearance of condensations is synchronized across distances greater than the characteristic wavelength of their spatial pattern. We explored the possible role of observed oscillations of the transcriptional co-regulator Hes1 in this phenomenon. Treatment of micromass cultures with DAPT, a γ-secretase inhibitor, damped Hes1 oscillations, elevated Gal-1A and -8 mRNA levels, and led to irregularly-sized and fused condensations. In developing limb buds, DAPT led to spatially non-uniform Hes1 expression and fused and misshapen digits. Periodicity in adhesive response to Gal-1A, a plausible Hes1-dependent function, was added to a previously tested mathematical model for condensation patterning by the two-galectin network. The enhanced model predicted regularization of patterning due to synchronization of Hes1 oscillations and resulting spatiotemporal coordination of its expression. The model also predicted changes in galectin expression and patterning in response to suppression of Hes1 expression, which were confirmed in in vitro experiments. Our results indicate that the two-galectin patterning network is regulated by Hes1 dynamics, the synchronization of which refines and regularizes limb skeletogenesis.

We present a mathematical model for the morphogenesis and patterning of the mesenchymal condensations that serve as primordia of the avian limb skeleton. The model is based on the experimentally established dynamics of a multiscale regulatory network consisting of two glycan-binding proteins expressed early in limb development: CG (chicken galectin)-1A, CG-8 and their counterreceptors that determine the formation, size, number and spacing of the “protocondensations” that give rise to the condensations and subsequently the cartilaginous elements that serve as the templates of the bones. The model, a system of partial differential and integro-differential equations containing a flux term to represent local adhesion gradients, is simulated in a “full” and a “reduced” form to confirm that the system has pattern-forming capabilities and to explore the nature of the patterning instability. The full model recapitulates qualitatively and quantitatively the experimental results of network perturbation and leads to new predictions, which are verified by further experimentation. The reduced model is used to demonstrate that the patterning process is inherently morphodynamic, with cell motility being intrinsic to it. Furthermore, subtle relationships between cell movement and the positive and negative interactions between the morphogens produce regular patterns without the requirement for activators and inhibitors with widely separated diffusion coefficients. The described mechanism thus represents an extension of the category of activator–inhibitor processes capable of generating biological patterns with repetitive elements beyond the morphostatic mechanisms of the Turing/Gierer–Meinhardt type.

Background A multiscale network of two galectins Galectin-1 (Gal-1) and Galectin-8 (Gal-8) patterns the avian limb skeleton. Among vertebrates with paired appendages, chondrichthyan fins typically have one or more cartilage plates and many repeating parallel endoskeletal elements, actinopterygian fins have more varied patterns of nodules, bars and plates, while tetrapod limbs exhibit tandem arrays of few, proximodistally increasing numbers of elements. We applied a comparative genomic and protein evolution approach to understand the origin of the galectin patterning network. Having previously observed a phylogenetic constraint on Gal-1 structure across vertebrates, we asked whether evolutionary changes of Gal-8 could have critically contributed to the origin of the tetrapod pattern. Results Translocations, duplications, and losses of Gal-8 genes in Actinopterygii established them in different genomic locations from those that the Sarcopterygii (including the tetrapods) share with chondrichthyans. The sarcopterygian Gal-8 genes acquired a potentially regulatory non-coding motif and underwent purifying selection. The actinopterygian Gal-8 genes, in contrast, did not acquire the non-coding motif and underwent positive selection. Conclusion These observations interpreted through the lens of a reaction-diffusion-adhesion model based on avian experimental findings can account for the distinct endoskeletal patterns of cartilaginous, ray-finned, and lobe-finned fishes, and the stereotypical limb skeletons of tetrapods. Electronic supplementary material The online version of this article (doi:10.1186/s12862-016-0729-6) contains supplementary material, which is available to authorized users.

We investigate a simple generic model of a reaction–diffusion system consisting of an activator and an inhibitor molecule in the presence of a linear morphogen gradient. We assume that this morphogen gradient is established independently of the reaction–diffusion system and acts by increasing the production of the activator proportional to the morphogen concentration. The model is motivated by several existing models in developmental biology in which a Turing patterning mechanism is proposed and various chemical gradients are known to be important for development. Mathematically, this leads to reaction–diffusion equations with explicit spatial dependence. We investigate how the Turing pattern is affected, if it exists. We also show that in the parameter range where a Turing pattern is not possible, the system may nevertheless produce 'Turing-like' patterns.

We consider three-dimensional Turing patterns and their isoconcentration surfaces corresponding to the equilibrium concentration of the reaction kinetics. We call these surfaces equilibrium concentration surfaces (EC surfaces). They are the interfaces between the regions of "high" and "low" concentrations in Turing patterns. We give alternate characterizations of EC surfaces by means of two variational principles, one of them being that they are optimal for diffusive transport. Several examples of EC surfaces are considered. Remarkably, they are often very well approximated by certain minimal surfaces. We give a dynamical explanation for the emergence of Scherk's surface in certain cases, a structure that has been observed numerically previously in [De Wit et al., 1997].

In this paper we present the foundation of a unified, object-oriented, three-dimensional biomodelling environment, which allows us to integrate multiple submodels at scales from subcellular to those of tissues and organs. Our current implementation combines a modified discrete model from statistical mechanics, the Cellular Potts Model, with a continuum reaction–diffusion model and a state automaton with well-defined conditions for cell differentiation transitions to model genetic regulation. This environment allows us to rapidly and compactly create computational models of a class of complex-developmental phenomena. To illustrate model development, we simulate a simplified version of the formation of the skeletal pattern in a growing embryonic vertebrate limb.

The stability of Turing patterns in an n-dimensional cube (0 ,π ) n is studied, where n 2. It is shown by using a generalization of a classical result of Ermentrout concerning spots and stripes in two dimensions that under appropriate assumptions only sheet-like or nodule-like structures can be stable in an n-dimensional cube. Other patterns can also be stable in regions comprising products of lower-dimensional cubes and intervals of appropriate length. Stability results are applied to a new model of skeletal pattern formation in the vertebrate limb.

We consider the problem of designing a reflector that transforms a spherical wave front with a given intensity into an output front illuminating a prespecified region of the far-sphere with prescribed intensity. In earlier approaches, it was shown that in the geometric optics approximation this problem is reduced to solving a second order nonlinear elliptic partial differential equation of Monge–Ampere type. We show that this problem can be solved as a variational problem within the framework of Monge–Kantorovich mass transfer problem. We develop the techniques used by the authors in their work “Optical Design of Two-Reflector Systems, the Monge–Kantorovich Mass Transfer Problem and Fermat's Principle” [Preprint, 2003], where the design problem for a system with two reflectors was considered. An important consequence of this approach is that the design problem can be solved numerically by tools of linear programming. A known convergent numerical scheme for this problem was based on the construction of very special approximate solutions to the corresponding Monge–Ampere equation. Bibliography: 14 titles.

We investigate the stability of Turing-type patterns in one spatial dimension in a system of reaction–diffusion equations with a term depending linearly on the spatial position. The system is a generic model of two interacting chemical species where production rates are dependent on a linear external gradient. This is motivated by mathematical models in developmental biology. In a previous paper, we found analytic approximations of Turing-like steady state patterns. In the present article, we derive conditions for the stability of these patterns and show bifurcation diagrams in two small parameters related to the slope of the external gradient and the ratio of the diffusion coefficients.

The production of cartilage (chondrogenic patterning) in the limb is one of the best-studied examples of the emergence of form in developmental biology. At the core of the theoretical study is an effort to understand the mechanism that establishes the characteristic distribution of cartilage in the embryonic limb, which defines the future sites and shapes of bones that will be present in the mature limb. This review article gives an overview of the history and current state of a rich literature of mathematical and computational models that seek to contribute to this problem. We describe models for the mechanisms of limb growth and shaping via interaction with various chemical fields, as well as models addressing the intrinsic self-organization capabilities of the embryonic mesenchymal tissue, such as reaction-diffusion and mechanochemical models. We discuss the contributions of these models to the current understanding of chondrogenesis in vertebrate limbs, as well as their relation to the varied conceptual models that have been proposed by experimentalists. Birth Defects Research (Part C) 96:176–192, 2012. © 2012 Wiley Periodicals, Inc.

We consider the geometric optics problem of finding a system of two reflectors that transform a spherical wavefront into a beam of parallel rays with prescribed intensity distribution. Using techniques from optimal transportation theory, it has been shown before that this problem is equivalent to an infinite dimensional linear programming (LP) problem. We investigate techniques for constructing the two reflectors numerically. A straightforward discretization of this problem has the disadvantage that the number of constraints increases rapidly with the mesh size. So with this technique only very coarse meshes are practical. To address this well-known issue we propose an iterative solution scheme. In each step an LP problem is solved. Information from the previous iteration step is used to reduce the number of constraints necessary. As a proof of concept we apply our proposed scheme to solve a problem with synthetic data. We give evidence that the scheme converges. We also show that it allows for much finer meshes than a simple discretization scheme. There exists a growing literature for the application of optimal transportation theory to other beam shaping problems. Our proposed scheme is easy to adapt for these problems as well., submitted

We investigate a reaction–diffusion system consisting of an activator and an inhibitor in a two-dimensional domain. There is a morphogen gradient in the domain. The production of the activator depends on the concentration of the morphogen. Mathematically, this leads to reaction–diffusion equations with explicitly space-dependent terms. It is well known that in the absence of an external morphogen, the system can produce either spots or stripes via the Turing bifurcation. We derive first-order expansions for the possible patterns in the presence of an external morphogen and show how both stripes and spots are affected. This work generalizes previous one-dimensional results to two dimensions. Specifically, we consider the quasi-one-dimensional case of a thin rectangular domain and the case of a square domain. We apply the results to a model of skeletal pattern formation in vertebrate limbs. In the framework of reaction–diffusion models, our results suggest a simple explanation for some recent experimental findings in the mouse limb which are much harder to explain in positional-information-type models.

We consider the following geometric optics problem: Construct a system of two reflectors which transforms a spherical wavefront generated by a point source into a beam of parallel rays. This beam has a prescribed intensity distribution. We give a rigorous analysis of this problem. The reflectors we construct are (parts of) the boundaries of convex sets. We prove existence of solutions for a large class of input data and give a uniqueness result. To the author's knowledge, this is the first time that a rigorous mathematical analysis of this problem is given. The approach is based on optimal transportation theory. It yields a practical algorithm for finding the reflectors. Namely, the problem is equivalent to a constrained linear optimization problem., 5 Figures - pdf files attached to submission, but not shown in manuscript

Dynamical systems in which geometrically extended model cells produce and interact with diffusible (morphogen) and nondiffusible (extracellular matrix) chemical fields have proved very useful as models for developmental processes. The embryonic vertebrate limb is an apt system for such mathematical and computational modeling since it has been the subject of hundreds of experimental studies, and its normal and variant morphologies and spatiotemporal organization of expressed genes are well known. Because of its stereotypical proximodistally generated increase in the number of parallel skeletal elements, the limb lends itself to being modeled by Turing-type systems which are capable of producing periodic, or quasiperiodic, arrangements of spot- and stripe-like elements. This chapter describes several such models, including, (i) a system of partial differential equations in which changing cell density enters into the dynamics explicitly, (ii) a model for morphogen dynamics alone, derived from the latter system in the "morphostatic limit" where cell movement relaxes on a much slower time-scale than cell differentiation, (iii) a discrete stochastic model for the simplified pattern formation that occurs when limb cells are placed in planar culture, and (iv) several hybrid models in which continuum morphogen systems interact with cells represented as energy-minimizing mesoscopic entities. Progress in devising computational methods for handling 3D, multiscale, multimodel simulations of organogenesis is discussed, as well as for simulating reaction-diffusion dynamics in domains of irregular shape.

Dynamical systems in which geometrically extended model cells produce and interact with diffusible (morphogen) and nondiffusible (extracellular matrix) chemical fields have proved very useful as models for developmental processes. The embryonic vertebrate limb is an apt system for such mathematical and computational modeling since it has been the subject of hundreds of experimental studies, and its normal and variant morphologies and spatiotemporal organization of expressed genes are well known. Because of its stereotypical proximodistally generated increase in the number of parallel skeletal elements, the limb lends itself to being modeled by Turing-type systems which are capable of producing periodic, or quasiperiodic, arrangements of spot- and stripe-like elements. This chapter describes several such models, including, (i) a system of partial differential equations in which changing cell density enters into the dynamics explicitly, (ii) a model for morphogen dynamics alone, derived from the latter system in the "morphostatic limit" where cell movement relaxes on a much slower time-scale than cell differentiation, (iii) a discrete stochastic model for the simplified pattern formation that occurs when limb cells are placed in planar culture, and (iv) several hybrid models in which continuum morphogen systems interact with cells represented as energy-minimizing mesoscopic entities. Progress in devising computational methods for handling 3D, multiscale, multimodel simulations of organogenesis is discussed, as well as for simulating reaction-diffusion dynamics in domains of irregular shape.

The Cellular Potts Model (CPM) has been used at a cellular scale for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angiogenesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. Continuous models in the form of partial differential, integral or integro-differential equations are used for studying biological problems at large scale. It is crucial for developing multiscale biological models to establish a connection between discrete microscopic stochastic models, including CPM, and macroscopic continuous models. To demonstrate multiscale approach we derive in this paper continuous limit of a two-dimensional CPM with the chemotactic interactions in the form of a Fokker-Planck equation describing evolution of the cell probability density function. This equation is then reduced to the classical macroscopic Keller-Segel model. We demonstrate that CPM Monte Carlo simulations are in excellent agreement with the numerics for the continuous macroscopic model with different forms of the chemical field term.

We present COMPUCELL3D, a software framework for three-dimensional simulation of morphogenesis in different organisms. COMPUCELL3D employs biologically relevant models for cell clustering, growth, and interaction with chemical fields. COMPUCELL3D uses design patterns for speed, efficient memory management, extensibility, and flexibility to allow an almost unlimited variety of simulations. We have verified COMPUCELL3D by building a model of growth and skeletal pattern formation in the avian (chicken) limb bud. Binaries and source code are available, along with documentation and input files for sample simulations, at http:// compucell.sourceforge.net.

A recently proposed mathematical model of a “core” set of cellular and molecular interactions present in the developing vertebrate limb was shown to exhibit pattern-forming instabilities and limb skeleton-like patterns under certain restrictive conditions, suggesting that it may authentically represent the underlying embryonic process (Hentschel et al., Proc. R. Soc. B 271, 1713–1722, 2004). The model, an eight-equation system of partial differential equations, incorporates the behavior of mesenchymal cells as “reactors,” both participating in the generation of morphogen patterns and changing their state and position in response to them. The full system, which has smooth solutions that exist globally in time, is nonetheless highly complex and difficult to handle analytically or numerically. According to a recent classification of developmental mechanisms (Salazar-Ciudad et al., Development 130, 2027–2037, 2003), the limb model of Hentschel et al. is “morphodynamic,” since differentiation of new cell types occurs simultaneously with cell rearrangement. This contrasts with “morphostatic” mechanisms, in which cell identity becomes established independently of cell rearrangement. Under the hypothesis that development of some vertebrate limbs employs the core mechanism in a morphostatic fashion, we derive in an analytically rigorous fashion a pair of equations representing the spatiotemporal evolution of the morphogen fields under the assumption that cell differentiation relaxes faster than the evolution of the overall cell density (i.e., the morphostatic limit of the full system). This simple reaction–diffusion system is unique in having been derived analytically from a substantially more complex system involving multiple morphogens, extracellular matrix deposition, haptotaxis, and cell translocation. We identify regions in the parameter space of the reduced system where Turing-type pattern formation is possible, which we refer to as its “Turing space.” Obtained values of the parameters are used in numerical simulations of the reduced system, using a new Galerkin finite element method, in tissue domains with nonstandard geometry. The reduced system exhibits patterns of spots and stripes like those seen in developing limbs, indicating its potential utility in hybrid continuum-discrete stochastic modeling of limb development. Lastly, we discuss the possible role in limb evolution of selection for increasingly morphostatic developmental mechanisms.

The Cellular Potts Model (CPM) has been used for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angiogenesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. In this paper, we derive continuous limit of discrete one dimensional CPM with the chemotactic interactions between cells in the form of a Fokker-Planck equation for the evolution of the cell probability density function. This equation is then reduced to the classical macroscopic Keller-Segel model. In particular, all coefficients of the Keller-Segel model are obtained from parameters of the CPM. Theoretical results are verified numerically by comparing Monte Carlo simulations for the CPM with numerics for the Keller-Segel model., 15 pages, 7 figures

It is shown that the problem of designing a two-reflector system transforming a plane wave front with given intensity into an output plane front with prescribed output intensity can be formulated and solved as the Monge-Kantorovich mass transfer problem., 25 pages, 2 figures