A multi-cell, multi-scale model of vertebrate segmentation and somite formation

Somitogenesis, the formation of the body's primary segmental structure common to all vertebrate development, requires coordination between biological mechanisms at several scales. Explaining how these mechanisms interact across scales and how events are coordinated in space and time is necessary for a complete understanding of somitogenesis and its evolutionary flexibility. So far, mechanisms of somitogenesis have been studied independently. To test the consistency, integrability and combined explanatory power of current prevailing hypotheses, we built an integrated clock-and-wavefront model including submodels of the intracellular segmentation clock, intercellular segmentation-clock coupling via Delta/Notch signaling, an FGF8 determination front, delayed differentiation, clock-wavefront readout, and differential-cell-cell-adhesion-driven cell sorting. We identify inconsistencies between existing submodels and gaps in the current understanding of somitogenesis mechanisms, and propose novel submodels and extensions of existing submodels where necessary. For reasonable initial conditions, 2D simulations of our model robustly generate spatially and temporally regular somites, realistic dynamic morphologies and spontaneous emergence of anterior-traveling stripes of Lfng. We show that these traveling stripes are pseudo-waves rather than true propagating waves. Our model is flexible enough to generate interspecies-like variation in somite size in response to changes in the PSM growth rate and segmentation-clock period, and in the number and width of Lfng stripes in response to changes in the PSM growth rate, segmentation-clock period and PSM length.

Computer simulation of cellular patterning within the Drosophila pupal eye

We present a computer simulation and associated experimental validation of assembly of glial-like support cells into the interweaving hexagonal lattice that spans the Drosophila pupal eye. This process of cell movements organizes the ommatidial array into a functional pattern. Unlike earlier simulations that focused on the arrangements of cells within individual ommatidia, here we examine the local movements that lead to large-scale organization of the emerging eye field. Simulations based on our experimental observations of cell adhesion, cell death, and cell movement successfully patterned a tracing of an emerging wild-type pupal eye. Surprisingly, altering cell adhesion had only a mild effect on patterning, contradicting our previous hypothesis that the patterning was primarily the result of preferential adhesion between IRM-class surface proteins. Instead, our simulations highlighted the importance of programmed cell death (PCD) as well as a previously unappreciated variable: the expansion of cells' apical surface areas, which promoted rearrangement of neighboring cells. We tested this prediction experimentally by preventing expansion in the apical area of individual cells: patterning was disrupted in a manner predicted by our simulations. Our work demonstrates the value of combining computer simulation with in vivo experiments to uncover novel mechanisms that are perpetuated throughout the eye field. It also demonstrates the utility of the Glazier-Graner-Hogeweg model (GGH) for modeling the links between local cellular interactions and emergent properties of developing epithelia as well as predicting unanticipated results in vivo.

Coarsening foams robustly reach a self-similar growth regime

Dry liquid foams coarsen like other diphasic systems governed by interfacial energy: gas slowly diffuses across liquid films, resulting in large bubbles growing at the expense of smaller ones which eventually shrink and disappear. A foam scatters light very effectively, preventing direct optical observation of bubble sizes and shapes in large foams. Using high speed x-ray tomography, we have produced 4D movies (i.e., 3D + time) of up to 30,000 bubbles. After a transient regime, the successive images look alike, except that the average bubble size increases as the square root of time: This scaling state is the long sought self-similar growth regime. The bubble size and face-number distributions in this regime are compared with experimental distributions for grains in crystals and with numerical simulations of foams.

Front instabilities and invasiveness of simulated 3D avascular tumors

We use the Glazier-Graner-Hogeweg model to simulate three-dimensional (3D), single-phenotype, avascular tumors growing in an homogeneous tissue matrix (TM) supplying a single limiting nutrient. We study the effects of two parameters on tumor morphology: a diffusion-limitation parameter defined as the ratio of the tumor-substrate consumption rate to the substrate-transport rate, and the tumor-TM surface tension. This initial model omits necrosis and oxidative/hypoxic metabolism effects, which can further influence tumor morphology, but our simplified model still shows significant parameter dependencies. The diffusion-limitation parameter determines whether the growing solid tumor develops a smooth (noninvasive) or fingered (invasive) interface, as in our earlier two-dimensional (2D) simulations. The sensitivity of 3D tumor morphology to tumor-TM surface tension increases with the size of the diffusion-limitation parameter, as in 2D. The 3D results are unexpectedly close to those in 2D. Our results therefore may justify using simpler 2D simulations of tumor growth, instead of more realistic but more computationally expensive 3D simulations. While geometrical artifacts mean that 2D sections of connected 3D tumors may be disconnected, the morphologies of 3D simulated tumors nevertheless correlate with the morphologies of their 2D sections, especially for low-surface-tension tumors, allowing the use of 2D sections to partially reconstruct medically-important 3D-tumor structures.

Modeling gastrulation in the chick embryo: formation of the primitive streak

The body plan of all higher organisms develops during gastrulation. Gastrulation results from the integration of cell proliferation, differentiation and migration of thousands of cells. In the chick embryo gastrulation starts with the formation of the primitive streak, the site of invagination of mesoderm and endoderm cells, from cells overlaying Koller's Sickle. Streak formation is associated with large-scale cell flows that carry the mesoderm cells overlying Koller's sickle into the central midline region of the embryo. We use multi-cell computer simulations to investigate possible mechanisms underlying the formation of the primitive streak in the chick embryo. Our simulations suggest that the formation of the primitive streak employs chemotactic movement of a subpopulation of streak cells, as well as differential adhesion between the mesoderm cells and the other cells in the epiblast. Both chemo-attraction and chemo-repulsion between various combinations of cell types can create a streak. However, only one combination successfully reproduces experimental observations of the manner in which two streaks in the same embryo interact. This finding supports a mechanism in which streak tip cells produce a diffusible morphogen which repels cells in the surrounding epiblast. On the other hand, chemotactic interaction alone does not reproduce the experimental observation that the large-scale vortical cell flows develop simultaneously with streak initiation. In our model the formation of large scale cell flows requires an additional mechanism that coordinates and aligns the motion of neighboring cells.

Bulk elastic properties of chicken embryos during somitogenesis

We present measurements of the bulk Young's moduli of early chick embryos at Hamburger-Hamilton stage 10. Using a micropipette probe with a force constant k ~0.025 N/m, we applied a known force in the plane of the embryo in the anterior-posterior direction and imaged the resulting tissue displacements. We used a two-dimensional finite-element simulation method to model the embryo as four concentric elliptical elastic regions with dimensions matching the embryo's morphology. By correlating the measured tissue displacements to the displacements calculated from the in-plane force and the model, we obtained the approximate short time linear-elastic Young's moduli: 2.4 ± 0.1 kPa for the midline structures (notocord, neural tube, and somites), 1.3 ± 0.1 kPa for the intermediate nearly acellular region between the somites and area pellucida, 2.1 ± 0.1 kPa for the area pellucida, and 11.9 ± 0.8 kPa for the area opaca.

Microfluidic devices integrating microcavity surface-plasmon-resonance sensors: glucose oxidase binding-activity detection

We have developed miniature (approximately 1 microm diameter) microcavity surface-plasmon-resonance sensors (MSPRS), integrated them with microfluidics, and tested their sensitivity to refractive-index changes. We tested their biosensing capability by distinguishing the interaction of glucose oxidase (M(r) 160 kDa) with its natural substrate (beta-D-glucose, M(r) 180 Da) from its interactions with nonspecific substrates (L-glucose, D-mannose, and 2-deoxy-D-glucose). We ran the identical protocol we had used with the MSPRS on a Biacore 3000 instrument using their bare gold chip. Only the MSPRS was able to detect beta-D-glucose binding to glucose oxidase. Each MSPRS can detect the binding to its surface of fewer than 35,000 glucose oxidase molecules (representing 9.6 fg or 60 zmol of protein), about 10(6) times fewer than classical surface-plasmon-resonance biosensors.

3D multi-cell simulation of tumor growth and angiogenesis

We present a 3D multi-cell simulation of a generic simplification of vascular tumor growth which can be easily extended and adapted to describe more specific vascular tumor types and host tissues. Initially, tumor cells proliferate as they take up the oxygen which the pre-existing vasculature supplies. The tumor grows exponentially. When the oxygen level drops below a threshold, the tumor cells become hypoxic and start secreting pro-angiogenic factors. At this stage, the tumor reaches a maximum diameter characteristic of an avascular tumor spheroid. The endothelial cells in the pre-existing vasculature respond to the pro-angiogenic factors both by chemotaxing towards higher concentrations of pro-angiogenic factors and by forming new blood vessels via angiogenesis. The tumor-induced vasculature increases the growth rate of the resulting vascularized solid tumor compared to an avascular tumor, allowing the tumor to grow beyond the spheroid in these linear-growth phases. First, in the linear-spherical phase of growth, the tumor remains spherical while its volume increases. Second, in the linear-cylindrical phase of growth the tumor elongates into a cylinder. Finally, in the linear-sheet phase of growth, tumor growth accelerates as the tumor changes from cylindrical to paddle-shaped. Substantial periods during which the tumor grows slowly or not at all separate the exponential from the linear-spherical and the linear-spherical from the linear-cylindrical growth phases. In contrast to other simulations in which avascular tumors remain spherical, our simulated avascular tumors form cylinders following the blood vessels, leading to a different distribution of hypoxic cells within the tumor. Our simulations cover time periods which are long enough to produce a range of biologically reasonable complex morphologies, allowing us to study how tumor-induced angiogenesis affects the growth rate, size and morphology of simulated tumors.

An object-oriented modelling framework for the arterial wall

An object-oriented modelling framework for the arterial wall is presented. The novelty of the framework is the possibility to generate customizable artery models, taking advantage of imaging technology. In our knowledge, this is the first object-oriented modelling framework for the arterial wall. Existing models do not allow close structural mapping with arterial microstructure as in the object-oriented framework. In the implemented model, passive behaviour of the arterial wall was considered and the tunica adventitia was the objective system. As verification, a model of an arterial segment was generated. In order to simulate its deformation, a matrix structural mechanics simulator was implemented. Two simulations were conducted, one for an axial loading test and other for a pressure-volume test. Each simulation began with a sensitivity analysis in order to determinate the best parameter combination and to compare the results with analogue controls. In both cases, the simulated results closely reproduced qualitatively and quantitatively the analogue control plots.

Front instabilities and invasiveness of simulated avascular tumors

We study the interface morphology of a 2D simulation of an avascular tumor composed of identical cells growing in an homogeneous healthy tissue matrix (TM), in order to understand the origin of the morphological changes often observed during real tumor growth. We use the Glazier-Graner-Hogeweg model, which treats tumor cells as extended, deformable objects, to study the effects of two parameters: a dimensionless diffusion-limitation parameter defined as the ratio of the tumor consumption rate to the substrate transport rate, and the tumor-TM surface tension. We model TM as a nondiffusing field, neglecting the TM pressure and haptotactic repulsion acting on a real growing tumor; thus, our model is appropriate for studying tumors with highly motile cells, e.g., gliomas. We show that the diffusion-limitation parameter determines whether the growing tumor develops a smooth (noninvasive) or fingered (invasive) interface, and that the sensitivity of tumor morphology to tumor-TM surface tension increases with the size of the dimensionless diffusion-limitation parameter. For large diffusion-limitation parameters, we find a transition (missed in previous work) between dendritic structures, produced when tumor-TM surface tension is high, and seaweed-like structures, produced when tumor-TM surface tension is low. This observation leads to a direct analogy between the mathematics and dynamics of tumors and those observed in nonbiological directional solidification. Our results are also consistent with the biological observation that hypoxia promotes invasive growth of tumor cells by inducing higher levels of receptors for scatter factors that weaken cell-cell adhesion and increase cell motility. These findings suggest that tumor morphology may have value in predicting the efficiency of antiangiogenic therapy in individual patients.

A threaded Java concurrent implementation of the Monte-Carlo Metropolis Ising model

This paper describes a concurrent Java implementation of the Metropolis Monte-Carlo algorithm that is used in 2D Ising model simulations. The presented method uses threads, monitors, shared variables and high level concurrent constructs that hide the low level details. In our algorithm we assign one thread to handle one spin flip attempt at a time. We use special lattice site selection algorithm to avoid two or more threads working concurently in the region of the lattice that "belongs" to two or more different spins undergoing spin-flip transformation. Our approach does not depend on the current platform and maximizes concurrent use of the available resources.

Self-similar mitochondrial DNA

We show that repeated sequences, like palindromes (local repetitions) and homologies between two different nucleotide sequences (motifs along the genome), compose a self-similar (fractal) pattern in mitochondrial DNA. This self-similarity comes from the looplike structures distributed along the genome. The looplike structures generate scaling laws in a pseudorandom DNA walk constructed from the sequence, called a Lévy flight. We measure the scaling laws from the generalized fractal dimension and singularity spectrum for mitochondrial DNA walks for 35 different species. In particular, we report characteristic loop distributions for mammal mitochondrial genomes.

Multicell simulations of development and disease using the CompuCell3D simulation environment

Mathematical modeling and computer simulation have become crucial to biological fields from genomics to ecology. However, multicell, tissue-level simulations of development and disease have lagged behind other areas because they are mathematically more complex and lack easy-to-use software tools that allow building and running in silico experiments without requiring in-depth knowledge of programming. This tutorial introduces Glazier-Graner-Hogeweg (GGH) multicell simulations and CompuCell3D, a simulation framework that allows users to build, test, and run GGH simulations.

Contact-inhibited chemotaxis in de novo and sprouting blood-vessel growth

Blood vessels form either when dispersed endothelial cells (the cells lining the inner walls of fully formed blood vessels) organize into a vessel network (vasculogenesis), or by sprouting or splitting of existing blood vessels (angiogenesis). Although they are closely related biologically, no current model explains both phenomena with a single biophysical mechanism. Most computational models describe sprouting at the level of the blood vessel, ignoring how cell behavior drives branch splitting during sprouting. We present a cell-based, Glazier-Graner-Hogeweg model (also called Cellular Potts Model) simulation of the initial patterning before the vascular cords form lumens, based on plausible behaviors of endothelial cells. The endothelial cells secrete a chemoattractant, which attracts other endothelial cells. As in the classic Keller-Segel model, chemotaxis by itself causes cells to aggregate into isolated clusters. However, including experimentally observed VE-cadherin-mediated contact inhibition of chemotaxis in the simulation causes randomly distributed cells to organize into networks and cell aggregates to sprout, reproducing aspects of both de novo and sprouting blood-vessel growth. We discuss two branching instabilities responsible for our results. Cells at the surfaces of cell clusters attempting to migrate to the centers of the clusters produce a buckling instability. In a model variant that eliminates the surface-normal force, a dissipative mechanism drives sprouting, with the secreted chemical acting both as a chemoattractant and as an inhibitor of pseudopod extension. Both mechanisms would also apply if force transmission through the extracellular matrix rather than chemical signaling mediated cell-cell interactions. The branching instabilities responsible for our results, which result from contact inhibition of chemotaxis, are both generic developmental mechanisms and interesting examples of unusual patterning instabilities.

Simulation of single-species bacterial-biofilm growth using the Glazier-Graner-Hogeweg model and the CompuCell3D modeling environment

The CompuCell3D modeling environment provides a convenient platform for biofilm simulations using the Glazier-Graner-Hogeweg (GGH) model, a cell-oriented framework designed to simulate growth and pattern formation due to biological cells' behaviors. We show how to develop such a simulation, based on the hybrid (continuum-discrete) model of Picioreanu, van Loosdrecht, and Heijnen (PLH), simulate the growth of a single-species bacterial biofilm, and study the roles of cell-cell and cell-field interactions in determining biofilm morphology. In our simulations, which generalize the PLH model by treating cells as spatially extended, deformable bodies, differential adhesion between cells, and their competition for a substrate (nutrient), suffice to produce a fingering instability that generates the finger shapes of biofilms. Our results agree with most features of the PLH model, although our inclusion of cell adhesion, which is difficult to implement using other modeling approaches, results in slightly different patterns. Our simulations thus provide the groundwork for simulations of medically and industrially important multispecies biofilms.

Coordinated action of N-CAM, N-cadherin, EphA4, and ephrinB2 translates genetic prepatterns into structure during somitogenesis in chick

During gastrulation in vertebrates, mesenchymal cells at the anterior end of the presomitic mesoderm (PSM) periodically compact, transiently epithelialize and detach from the posterior PSM to form somites. In the prevailing clock-and-wavefront model of somitogenesis, periodic gene expression, particularly of Notch and Wnt, interacts with an FGF8-based thresholding mechanism to determine cell fates. However, this model does not explain how cell determination and subsequent differentiation translates into somite morphology. In this paper, we use computer simulations of chick somitogenesis to show that experimentally-observed temporal and spatial patterns of adhesive N-CAM and N-cadherin and repulsive EphA4-ephrinB2 pairs suffice to reproduce the complex dynamic morphological changes of somitogenesis in wild-type and N-cadherin (-/-) chick, including intersomitic separation, boundary-shape evolution and sorting of misdifferentiated cells across compartment boundaries. Since different models of determination yield the same, experimentally-observed, distribution of adhesion and repulsion molecules, the patterning is independent of the details of this mechanism.

Compact microfluidic structures for generating spatial and temporal gradients

We present an improved microfluidic design for generating spatial and temporal gradients. The basic functional elements are bifurcated and trifurcated channels used to split flow between two and three channels, respectively. We use bifurcated channels on the exterior of the channel manifold and trifurcated channels in the interior with mixing tees to recombine flows. For N gradient-forming levels, the number of discrete steps in the gradient is 2(N) + 1, allowing a compact gradient-forming structure that is only 1.6 mm long and 0.5 mm wide. Control of the relative sample concentration at the inlets enables generation of gradients with varying slopes and offsets. The small total channel length allows faster switching (only 2.6 s) between gradients of different compositions than did previous designs, allowing complex temporal sequences and reducing total displacement volume and reagent use. The design permits opposing-gradient experiments and generation of complex nonlinear gradients. We fabricated and tested three channel designs with either three or four gradient-forming levels, 20- or 40-microm channel widths, 60- or 120-microm center-to-center channel spacings, and 9 or 17 output steps. These devices produced essentially identical high-quality linear gradients using both pressure-driven and electrokinetic flow.

Experimental growth law for bubbles in a moderately "wet" 3D liquid foam

We used x-ray tomography to characterize the geometry of all bubbles in a liquid foam of average liquid fraction phi(l) approximately 17% and to follow their evolution, measuring the normalized growth rate G=V(-1/3) dV/dt for 7000 bubbles. While G does not depend only on the number of faces of a bubble, its average over f-faced bubbles scales as G(f) approximately f - f(0) for large f's at all times. We discuss the dispersion of G and the influence of V and phi(l) on G.

From Genes to Organisms Via the Cell A Problem-Solving Environment for Multicellular Development

To gain performance, developers often build scientific applications in procedural languages, such as C or Fortran, which unfortunately reduces flexibility. To address this imbalance, the authors present CompuCell3D, a multitiered, flexible, and scalable problem-solving environment for morphogenesis simulations that's written in C++ using object-oriented design patterns.

A parallel implementation of the Cellular Potts Model for simulation of cell-based morphogenesis

The Cellular Potts Model (CPM) has been used in a wide variety of biological simulations. However, most current CPM implementations use a sequential modified Metropolis algorithm which restricts the size of simulations. In this paper we present a parallel CPM algorithm for simulations of morphogenesis, which includes cell-cell adhesion, a cell volume constraint, and cell haptotaxis. The algorithm uses appropriate data structures and checkerboard subgrids for parallelization. Communication and updating algorithms synchronize properties of cells simulated on different processor nodes. Tests show that the parallel algorithm has good scalability, permitting large-scale simulations of cell morphogenesis (10(7) or more cells) and broadening the scope of CPM applications. The new algorithm satisfies the balance condition, which is sufficient for convergence of the underlying Markov chain.

Adhesion between cells, diffusion of growth factors, and elasticity of the AER produce the paddle shape of the chick limb

A central question in developmental biology is how cells interact to organize into tissues? In this paper, we study the role of mesenchyme-ectoderm interaction in the growing chick limb bud using Glazier and Graner's cellular Potts model, a grid-based stochastic framework designed to simulate cell interactions and movement. We simulate cellular mechanisms including cell adhesion, growth, and division and diffusion of morphogens, to show that differential adhesion between the cells, diffusion of growth factors through the extracellular matrix, and the elastic properties of the apical ectodermal ridge together can produce the proper shape of the limb bud.

Cell movement during chick primitive streak formation

Gastrulation in amniotes begins with extensive re-arrangements of cells in the epiblast resulting in the formation of the primitive streak. We have developed a transfection method that enables us to transfect randomly distributed epiblast cells in the Stage XI-XIII chick blastoderms with GFP fusion proteins. This allows us to use time-lapse microscopy for detailed analysis of the movements and proliferation of epiblast cells during streak formation. Cells in the posterior two thirds of the embryo move in two striking counter-rotating flows that meet at the site of streak formation at the posterior end of the embryo. Cells divide during this rotational movement with a cell cycle time of 6-7 h. Daughter cells remain together, forming small clusters and as result of the flow patterns line up in the streak. Expression of the cyclin-dependent kinase inhibitor, P21/Waf inhibits cell division and severely limits embryo growth, but does not inhibit streak formation or associated flows. To investigate the role off cell-cell intercalation in streak formation we have inhibited the Wnt planar-polarity signalling pathway by expression of a dominant negative Wnt11 and a Dishevelled mutant Xdd1. Both treatments do not result in an inhibition of streak formation, but both severely affect extension of the embryo in later development. Likewise inhibition of myosin II which as been shown to drive cell-cell intercalation during Drosophila germ band extension, has no effect on streak formation, but also effectively blocks elongation after regression has started. These experiments make it unlikely that streak formation involves known cell-cell intercalation mechanisms. Expression of a dominant negative FGFR1c receptor construct as well as the soluble extracellular domain of the FGFR1c receptor both effectively block the cell movements associated with streak formation and mesoderm differentiation, showing the importance of FGF signalling in these processes.

Dynamic mechanisms of blood vessel growth

The formation of a polygonal configuration of proto-blood-vessels from initially dispersed cells is the first step in the development of the circulatory system in vertebrates. This initial vascular network later expands to form new blood vessels, primarily via a sprouting mechanism. We review a range of recent results obtained with a Monte Carlo model of chemotactically migrating cells which can explain both de novo blood vessel growth and aspects of blood vessel sprouting. We propose that the initial network forms via a percolation-like instability depending on cell shape, or through an alternative contact-inhibition of motility mechanism which also reproduces aspects of sprouting blood vessel growth.

Viscous instabilities in flowing foams: a Cellular Potts Model approach

The Cellular Potts Model (CPM) successfully simulates drainage and shear in foams. Here we use the CPM to investigate instabilities due to the flow of a single large bubble in a dry, monodisperse two-dimensional flowing foam. As in experiments in a Hele-Shaw cell, above a threshold velocity the large bubble moves faster than the mean flow. Our simulations reproduce analytical and experimental predictions for the velocity threshold and the relative velocity of the large bubble, demonstrating the utility of the CPM in foam rheology studies.