On multiscale approaches to three-dimensional modelling of morphogenesis

Abstract

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.

Figure 1

Hierarchy of scales from molecule to organ, and the corresponding mechanisms and modelling approaches. Models/subsystems at coarser scales use information from finer scales. See table 1 for length and time-scales.

Figure 2

Skeletal pattern formation—time-series of chick limb-bud development in longitudinal section. For each figure, proximal is to the left, distal to the right, anterior up and posterior down. Black represents differentiated cartilage and stipple precartilage condensation. The single bone on the left of each figure is the humerus, the two bones in the mid-limb are the radius and ulna, and the three bones that form at the distal end, beginning on day 6, are the digits (based on Newman & Frisch 1979).

Figure 3

Schematic diagram of chick-limb organogenesis at mid-development (corresponding to day 5 in figure 2), showing primary axes. The earliest-developing region of the skeleton has differentiated into cartilage (black) by this stage. The region in which the skeletal pattern is forming is undergoing mesenchymal condensation (medium grey). The digits at the distal tip have not yet begun to form.

Figure 4

The CPM grid and representation of cells and ECM. The shading denotes the cell type. Different cells (e.g. cells 1 and 3) may be of the same cell type. We also show the fourth-neighbour interactions of voxel S on a two-dimensional grid.

Figure 5

Time-series of the concentration of the diffusible morphogen TGF-β for equations (4.12) (displayed along proximo-distal cross-sections) with time increasing along the distal direction (upwards). This first prepattern of cylindrically elongated parallel elements drives the final cell condensation through the mediating second prepattern of non-diffusing fibronectin. p denotes the proximal direction, v the ventral and a the anterior.

Figure 6

Fibronectin production corresponding to normal chondrogenesis. We show fibronectin accumulation along the distal direction, as time progresses and the limb grows. Haptotaxis of cells in response to fibronectin (the second prepattern) and cells continuing fibronectin secretion make the pattern robust and does not require a persistent activator (first) prepattern. The fibronectin pattern establishes itself faster (a) 400 Monte Carlo steps (half-formed limb); (b) 800 Monte Carlo steps (fully formed limb) than the final cell condensations (1040 Monte Carlo steps; figure 7), emphasizing fibronectin's role in pattern consolidation. Fibronectin accumulates at its secretion location: its concentration in the humerus region in (b) is larger than in (a).

Figure 7

Cell condensation into humerus, ulna and radius, and digits after 1040 Monte Carlo steps. Visualization using volume rendering. The axes correspond to the p, a and v of figure 5.

Figure 8

Time-series (successive transverse cross-sections in the distal direction, which is upward in the figure) of TGF-β concentration in a growing limb bud, corresponding to the pathology of extra digits (four digits form instead of three; see figure 9).

Figure 9

(a) Fibronectin distribution and (b) cell condensation, after 940 Monte Carlo steps, corresponding to the TGF-β (first) prepattern in figure 8.

Figure 10

(a) Time-series (successive transverse cross-sections in the distal direction) of the TGF-β concentration in a growing limb bud, corresponding to the pathology of Apert's syndrome. (b) The fibronectin concentration field after 500 Monte Carlo steps (limb not fully formed). The radius and ulna fuse and digits fail to form.