SAN formulated the biological questions, provided the in vitro data and, with MA, designed the multipixel, multiscale, quasi–3-D representation. MA and SC designed the discrete stochastic model. SC performed the simulations and, with SAN and MA, interpreted the results. SC, MA, and SAN wrote the paper.
The authors have declared that no competing interests exist.
Cells of the embryonic vertebrate limb in high-density culture undergo chondrogenic pattern formation, which results in the production of regularly spaced “islands” of cartilage similar to the cartilage primordia of the developing limb skeleton. The first step in this process, in vitro and in vivo, is the generation of “cell condensations,” in which the precartilage cells become more tightly packed at the sites at which cartilage will form. In this paper we describe a discrete, stochastic model for the behavior of limb bud precartilage mesenchymal cells in vitro. The model uses a biologically motivated reaction–diffusion process and cell-matrix adhesion (haptotaxis) as the bases of chondrogenic pattern formation, whereby the biochemically distinct condensing cells, as well as the size, number, and arrangement of the multicellular condensations, are generated in a self-organizing fashion. Improving on an earlier lattice-gas representation of the same process, it is multiscale (i.e., cell and molecular dynamics occur on distinct scales), and the cells are represented as spatially extended objects that can change their shape. The authors calibrate the model using experimental data and study sensitivity to changes in key parameters. The simulations have disclosed two distinct dynamic regimes for pattern self-organization involving transient or stationary inductive patterns of morphogens. The authors discuss these modes of pattern formation in relation to available experimental evidence for the in vitro system, as well as their implications for understanding limb skeletal patterning during embryonic development.
The development of an organism from embryo to adult includes processes of pattern formation that involve the interactions over space and time of independent cells to form multicellular structures. Computational models permit exploration of possible alternative mechanisms that reproduce biological patterns and thereby provide hypotheses for empirical testing. In this article, we describe a biologically motivated discrete stochastic model that shows that the patterns of spots and stripes of tightly packed cells observed in cultures derived from the embryonic vertebrate limb can occur by a mechanism that uses only cell–cell signaling via diffusible molecules (morphogens) and cell substratum adhesion (haptotaxis). Moreover, similar-looking patterns can arise both from stable stationary dynamics and unstable transient dynamics of the same underlying core molecular–genetic mechanism. Simulations also show that spot and stripe patterns (which also correspond to the nodules and bars of the developing limb skeleton in vivo) are close in parameter space and can be generated in multiple ways with single-parameter variations. An important implication is that some developmental processes do not require a strict progression from one stable dynamic regime to another, but can occur by a succession of transient dynamic regimes tuned (e.g., by natural selection) to achieve a particular morphological outcome.
Skeletal pattern formation in the developing vertebrate limb depends on interactions of precartilage mesenchymal cells with factors that control the spatiotemporal differentiation of cartilage. The most fundamental skeletogenic processes involve the spatial separation of precartilage mesenchyme into chondrogenic and nonchondrogenic domains [
(A) Progress of limb skeletal development in chicken forelimb (wing) between 3 and 7 d of embryogenesis. Gray represents precartilage condensation, and black represents definitive cartilage. The developing limb, or limb bud, is paddle-shaped, being flatter in the back-to-front (dorsoventral) dimension than in the thumb-to-little finger (anteroposterior) dimension, or the shoulder-to-fingertips (proximodistal) direction in which it mainly grows. The cartilages that prefigure the bones first arise as stripe-like (e.g., long bones, digits) or spot-like (e.g., wrist bones shown here, or ankle bones in the hindlimb) mesenchymal condensations. The apical zone of the 5-d chicken wing bud (indicated by the arrowheads) or leg bud provide a source of not-yet-condensed mesenchymal cells that when grown in high-density “micromass” culture will form precartilage condensations.
(B) Discrete spot-like cartilage nodules that have formed after 6 d in a micromass culture of 5-d leg bud apical zone limb mesenchymal cells, visualized by staining with Alcian blue. The cells in these cultures are initially plated as a densely packed monolayer (“micromass”) and rearrange over short distances in the 2-D plane of the ∼3 mm diameter culture during the indicated period. Each nodule arises from a condensation containing approximately 30–50 cells. As indicated by the parallel lines, the spatial scale of the spot-like nodules (and the precartilage condensations from which they arise) in the micromass cultures is comparable to the diameter of the precartilage and cartilaginous skeletal primordia in the developing limb. The left panel is adapted, with changes, from [
In certain developmental processes, such as angiogenesis (sprouting of capillaries) and invasion by cancer cells of surrounding tissues, pre-existing multicellular structures become more elaborate. Precartilage condensation, by contrast, is an example of a developmental process in which cells that start out as independent entities interact to form multicellular structures. Others in this second category include vasculogenesis (the initial formation of blood vessels), the formation of feather germs, and the aggregation of social amoebae into streams and fruiting bodies. Both continuous [
In a previous study [
In the earlier model, each of the limb precartilage mesenchymal cells, and each molecule from a “core” subset of the molecules they secrete (the diffusible activator morphogen TGF-β, a diffusible inhibitor of TGF-β's effects, the extracellular matrix [ECM] protein fibronectin), was represented as a single particle (pixel) on a common grid. Default motion of the cell particles was random, but cell movement was also biased by the presence of fibronectin particles produced and deposited by the cells according to a set of rules involving TGF-β and inhibitor particles. The latter in turn were produced in a cell-dependent fashion according to a reaction–diffusion scheme, the network structure of which was suggested by in vitro experiments [
The ability of the model of Kiskowski et al. [
Despite the successes of the model of Kiskowski et al. [
We have found that not only does this improved model reproduce the experimental data accounted for by the model of Kiskowski et al., but that additional morphogenetic features of the micromass culture system are simulated as well. Moreover, potential dynamic properties of the developmental process not seen in the earlier simulations, and not capable of being distinguished on the basis of existing experimental data, were disclosed in simulations using the new model, which has therefore provided motivation for further empirical tests.
We represented each model cell as an extended object on a 2-D spatial grid. The rate and probability at which cells move (by random walk) and change shape are parameterized separately from movement of molecules so that they can be calibrated to the scale of actual biological cells. Each model cell behaves according to a predefined set of experimentally motivated rules involving morphogen dynamics controlling the production and deposition of fibronectin (see
We chose the simplest multipixel representation of limb mesenchymal cells subject to the following biological constraints: (1) cells have essentially isotropic geometry (i.e., they do not elongate in the direction of migration, but rather probe their environment by extending short randomly oriented projections); (2) the cell nucleus is also isotropic but is relatively unchanging in shape and comprises on the order of half the cell volume; and (3) cells in fibronectin-rich, condensing areas of the micromass round up such that their cross-section in the plane of the culture is significantly reduced [
(A) Three cells on the spatial grid each occupying seven pixels.
(B) Cell changes shape. The region of the cell that contains the nucleus, indicated by the four gray pixels, is structurally maintained; two border pixels move to new locations, and one border pixel (top right) displaces a nucleus pixel, which gets shifted to the right.
(C) Cell rounding-up on fibronectin. The surface area in the presence of suprathreshold amounts of fibronectin is reduced with two border pixels moving into a quasi-third dimension above the cell.
The degrees of freedom built into our model allowed us to calibrate some of the simulation parameters with experimentally determined values obtained in related or analogous systems. In particular, the diffusion rate of the activator morphogen and that of mesenchymal cells correspond well to experimental values, and they both play an important role in the resultant behavior of the model.
Lander et al. [
The effective “diffusion” rate for cells is considerably slower than that of morphogens, and cells do not move significant distances over the time period of precartilage condensation formation [
Using the experimental values for activator and cell diffusion coefficients greatly facilitated choosing other parameters so that appropriately sized and spaced condensations formed in silico. This contrasted with parameter searches performed with nonbiological choices of activator and cell diffusion coefficients. In those cases, no realistic patterns formed in scores of simulations.
The inhibitor morphogen, elicited when cells in incipient condensations are exposed to one or more ectodermally produced fibroblast growth factors [
Variation of Average Peak Interval and Average Island Size over a Range of Diffusion Ratios
Experimental evidence indicated that limb mesenchymal cells in vitro respond to transient elevation in TGF-β concentration early during the culture period by upregulating fibronectin production for at least a day [
Consistent with the experimental constraints described above, we searched for a parameter set in the model that reproduces the formation of precartilage condensation patterns. We calculated the average interval of the centroids (“peak interval”) [
(A) Discrete spot-like precartilage condensations that have formed after 72 h in a micromass culture of 5-d leg bud apical zone limb mesenchymal cells, visualized by Hoffman Contrast Modulation optics. Actual size of the microscopic field is 1 × 1.4 mm, and each condensation contains approximately 30–50 tightly packed cells.
(B) Spatial grid of equal physical size to (A) containing over 6,000 cells produced by simulation using the parameter values in
(C) Spatial grid of same simulation as (B) showing fibronectin-rich patches (black) produced by the differentiated cells.
(D) Spatial grid of same simulation as (B) showing activator concentrations at time slightly after the initial onset of cell differentiation. The color bar indicates the range, with magenta for high concentration and light blue for low concentration.
Averages are shown for 12 experimental (circle) and five simulation (square) points using parameter values in
Calibrated Simulation Parameters to Known Physical Values
Different views of one simulation with parameters chosen within the “standard” range are shown in
The simulated distribution of fibronectin (
We explored the robustness of the parameter set by varying key parameters independently (±5%); results can be seen in
Average peak interval versus average island size for variations in the some of the key parameters are shown: +5% (diamond) and −5% (filled diamond) for
Consistent with observations of limb precartilage development in vitro and in vivo, our simulation results indicate that cells can form condensation patterns by undergoing small displacements of less than a cell diameter, packing more closely by changing their shapes, while maintaining a relatively uniform cell density across the entire spatial domain.
Given the possibility that choices of spatial domain and boundary conditions could lead to simulation artifacts, we sampled various alternatives in combination and investigated changes in the resulting condensation patterns.
With respect to the spatial domain, we ran simulations with rectangular grids of various widths and heights (unpublished data); this produced no noticeable effects on the size, shape, or distribution of the condensations. We conclude that the total area of the spatial domain determines only the number of condensations.
We also ran simulations with periodic and no-flux conditions. In periodic conditions, grid boundaries are connected together simulating a continuous space, whereas the no-flux boundary acts as a barrier. Both types of boundary conditions produced similar results for the size, shape, and distribution of the condensation patterns apart from the expected pattern truncations under no-flux conditions (unpublished data).
Our simulations disclosed two regimes of behavior in the reaction–diffusion system of morphogens (
(A) Oscillatory regime produces transient patterns that repeat over time but are spatially stochastic.
(B) Stationary regime produces stable patterns with minor stochastic fluctuations around an equilibrium concentration. Graphs show the maximum concentration value for a single pixel across the entire molecular grid (that pixel lies within an activator peak as in
Both regimes for the reaction–diffusion system can produce condensations patterns in the range of experimental values for size and distribution. (See
The limits on morphogen production (
Variations in the limits on morphogen production in the oscillatory regime produced minimal changes in the average peak interval and average island size of the fibronectin patch distribution (
Robustness of Average Peak Interval and Average Island Size over a Range of Production Maximums
The oscillatory regime is robust to a noisy threshold level for cell differentiation. Simulations where each cell's threshold is randomly assigned from a normal distribution,
The formation of patterns in the stationary regime is sensitive to the period that cells are exposed to activator morphogen and to the threshold level for cell differentiation. If the exposure time is too short, small, irregularly spaced condensations are produced. If the exposure is too long, irregularly shaped condensations are produced. Although the stationary regime produces stable activator peaks, those peaks tend to wander spatially over time due to the underlying cell diffusion. The oscillatory regime is less sensitive to the threshold level for cell differentiation, and a single transient pulse provides a well-defined exposure period.
While the focus of our model has been on producing the spot patterns typically seen in leg-cell cultures [
(A) Stripe-like precartilage condensations.
(B) Spatial grid containing more than 6,000 cells produced by simulation showing stripes of fibronectin-producing differentiated cells (white), nondifferentiated cells (blue-gray), and empty space between cells (black).
(C) Spatial grid of same simulation as (B) showing fibronectin-rich stripes (black) produced by the differentiated cells.
(D) Spatial grid of same simulation as (B) showing activator concentrations at time slightly after the initial onset of cell differentiation. The color bar indicates the range, with magenta for high concentration and light blue for low concentration.
We have demonstrated that parameter choices can be found for our quasi–3-D discrete model that reproduce the experimental distribution and size range of precartilage condensations in experimental micromass cultures. The performance of the model was equal to that of Kiskowski et al. [
The new model has allowed us to study the interplay between reaction–diffusion processes, fibronectin production, and cell–fibronectin interaction in greater detail than previously possible. In particular, our simulations disclosed two regimes in the interplay of the reaction–diffusion system of morphogens with fibronectin production and cell behavior. In one regime, stationary morphogen patterns were produced, followed by cell rearrangement into patterns of condensation. In the second regime, morphogen patterns were transient and oscillatory in time, and the induced fibronectin production (and consequent cell rearrangement) occurred with a delay. In addition, the dynamic characteristics of the second regime provide a natural explanation for apparent oscillatory effects of limb precartilage cell responses to TGF-β seen in previous experimental studies [
Our model generates realistic patterns of precartilage condensation in high-density culture without the need to postulate direct cell–cell adhesive interactions. This feature appears to reflect biological reality. First, although the separation of condensing from noncondensing cells superficially resembles sorting out by differential adhesion (see [
We note that in both the oscillatory and stationary cases, the region of parameter space that leads to realistic fibronectin patch and condensation patterns corresponds to activator morphogen peaks that are on the spatial scale of the condensations themselves. For the oscillatory regime, a small number of those peaks (see
The capacity of our model to generate both spots and stripes of precartilage condensation under slightly different parameter choices corresponds well to experimental results in which either morphotype may be generated under similar initial conditions. Because the developing limb itself generates its skeleton in the form of spots and stripes of precartilage condensation (
Cell cultures were prepared using precartilage mesenchymal tissue isolated from the myoblast-free distal 0.3 mm [
The spatial environment that cells and molecules occupy is modeled on a 2-D plane. The implementation provides support for multiple superimposed discrete grids of various spatial scales. In our current model, we use two scales: one for the cellular level and another finer-resolution scale for the molecular level. The coarsest resolution spatial scale is considered to be the base spatial scale, which is the cellular level for our model; all other grids are an integer ratio size of that base grid. The base spatial grid can be defined as a square or rectangular grid of any height and width, and all of the grids overlay one another and cover the same physical area.
Each cell is represented as a set of seven contiguous pixels operating on the base spatial grid as shown in
Cell diffusion was implemented as a random walk. If the cell moves, then all of its seven (or five) pixels move one pixel in the appropriate direction (up, down, left, right). Cells can also fluctuate in shape, yet such fluctuations maintain a structural representation of the central region containing the nucleus by preserving intact a two-by-two square block of pixels. Therefore, shape fluctuations are restricted to the motion of the three (or one) border pixels around the nucleus which either move to new border pixels or displace nucleus pixels;
In our discrete representation of the Turing mechanism, a discrete number of activator and inhibitor molecules occupy each pixel on the grid, and each molecule is considered to have a spatial representation of just one pixel. We modeled the reaction dynamics of the activator and inhibitor molecules at each pixel as follows: let
In our model, production of the activator and inhibitor molecules, as represented by the parameters
In keeping with the biology, we considered cells to respond to low concentrations of morphogens and thus represent morphogen molecules as discrete entities. Consequently, the morphogen concentrations (
In any physicochemical reaction there is limitation on how much reagent a single cell can realistically produce during any period of time. For this reason, our model provides separate parameters (
Molecular diffusion from any pixel can occur randomly toward any of the four neighboring pixels (up, down, left, right). The diffusion rates (
Fibronectin is a nondiffusing ECM molecule that forms the template for precartilage condensations. As the concentration levels of the activator morphogen increases in the presence of a cell, that cell produces fibronectin mRNA, which can then be translated into actual fibronectin protein molecules. The model supports a simple threshold level (
We assumed that there is a critical period during which exposure, or lack of exposure, of cells to activator morphogen, causes them to either differentiate into fibronectin-producing cells or follow an alternative differentiation pathway. For purposes of simulation, we disabled the reaction–diffusion dynamics after this critical period (adjustable by a parameter) and prevented additional cells from differentiating. For the oscillatory regime, a single transient pulse (
When a cell produces fibronectin, a single unit representing a multimolecular complex is secreted with random probability for each of the pixels on the molecular grid in the cell's spatial domain, and each unit is allowed to perform an initial small diffusion of at most one pixel [
In attempting to calibrate our model parameters with known empirical parameters, we wanted to correlate the spatial and temporal patterns produced by computer simulation results with in vivo and in vitro experiments. For spatial patterns, we considered the size, shape, and distribution of the fibronectin-rich spatial domains; for temporal patterns, we considered the reaction rates of activator and inhibitor production, the diffusion rates of both cells and molecules, the onset of fibronectin production, the production rate of fibronectin, and the shape and movement fluctuations of cells on fibronectin. The actual values for the set of key parameters used in the simulation and their corresponding physical measurements, if known, are shown in
Whereas the previous model of Kiskowski et al. [
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The authors acknowledge the technical assistance of Sovannary Tan and helpful discussions with Maria Kiskowski and Xuelian Zhu.
extracellular matrix