Viewpoint 3

Abstract

Pattern formation in nature is best thought of as a process of symmetry breaking. That is, an initially homogeneous system becomes spatially, and sometimes temporally, inhomogeneous. Examples include the wind-dependent generation of sand dunes, the abrupt appearance of hexagonal convection cells in a thin layer of oil heated from below and a remarkable phenomenon known as the Belusov–Zhabotinsky reaction where a chemical reaction in a shallow dish can spontaneously form a chemical concentration pattern (1). These latter patterns can be stationary, manifest as unchanging spots or stripes, or wave like, in which a chemical concentration profile can propagate, producing macroscopically visible expanding concentric circles or spirals. In biology, although symmetry breaking is ubiquitous, understanding the mechanisms involved has met with limited success. Embryogenesis is a paradigm of pattern formation in nature and is still poorly understood. What makes the heart localize to the left and the liver to the right? In development the heart does not remain a midline structure; nature makes a choice between left and right and the axial symmetry is broken. The correct outcome is not always guaranteed – witness dextrocardia – but there are extremely robust mechanisms in place to ensure such anomalies are rare. In some cases, symmetry breaking may not occur where it should; an example is the uniformly pigmented zebra, or it may occur where it should not; an example from human dermatology is the unilateral icthyosiform eruption found in the CHILD naevus (2). What can be learnt from the study of pattern formation in relatively simple physical systems, such as the Belusov–Zhabotinsky reaction, and how can such knowledge increase understanding of the more complex processes occurring in biology? First, inanimate systems can undergo a process of self-organization provided they exchange energy and/or matter with their environment (3). Second, biological systems may exploit physical constraints, such as surface tension, to generate structure without the need for an explicit genetic message (4). Finally, the analogies between inanimate pattern formation and biological pattern formation offer the tantalizing prospect that the latter may be approached and understood in a quantitative manner. Indeed, Alan Turing, the British mathematician, wrote down a series of reaction–diffusion equations in 1952 under the title ‘The chemical basis of morphogenesis’ that show how chemicals that react and diffuse can form spontaneous patterns in solution (5). This is perhaps the simplest mathematical model that can exhibit self-organization. As such, it should not be interpreted in a literal sense, rather it should be considered a paradigm model. Other models for self-organization have been proposed based on different biological hypotheses, but, intriguingly, many of these models make similar predictions, suggesting possible developmental constraints which are independent of the exact details of the biological processes involved (6). It is reaction–diffusion theory that will be used here to explain symmetry breaking in dermatology. Consider naevoid patterns. How do the patterns arise? The answer remains unresolved despite many attempts at an explanation. Most authors have suggested that the patterns are due to the clonal outgrowth of abnormal cell lines during embryogenesis. For example, the morphology of quadrant naevi has been considered the result of a postzygotic mutation such that the destination of abnormal clones reflects the patterning (7). Similar arguments have been applied to bathing trunk naevi. Likewise, the lines of Blaschko have usually been attributed to clonal outgrowths of clones of cells either from the primitive streak (8) or from the neural crest (9). Yet there is evidence that suggests these explanations are inadequate. Lineage studies on embryonic mice show that early postzygotic mutated cell clones become widely dispersed throughout the body; they do not remain localized (10). In humans, evidence supporting these results is provided by studying the McCune–Albright syndrome, a genodermatosis thought to be due to a postzygotic mutation, and human chimaerism. Although both conditions exhibit large patches of uniformly brown skin over a background of normal skin, it has been shown that fine-grained mosaicism is present throughout the skin in the former (11), and present throughout all tissues studied in the latter (12). Patterns following the lines of Blaschko are occasionally observed in the McCune–Albright syndrome, but are more commonly seen in X-linked conditions where there is a random inactivation, termed lyonization, of one of the X chromosomes in each cell early in embryogenesis. In contrast to the previous example, it has been demonstrated that clonality may exist in tissue specimens of lesions following the lines of Blaschko (9,13) despite the expectation that fine-grained mosaicism secondary to lyonization should be present. The issue is confounded by a lack of data. It is not known whether lesional tissue from all naevi following the lines of Blaschko is clonal. It is also unclear in some naevi as to which cell types are involved, so the relevant cell type may have been overlooked. Some authors have suggested the lines of Blaschko may represent the paths of migrating melanocytes (14,15), but this is unlikely. Mintz's allophenic mice (mosaic for black or white coat colour where the relevant genes are expressed by melanocytes) do not exhibit patterns that look like the lines of Blaschko in humans (16). In addition, it is difficult (although not impossible) to see how a gene expressed by a melanocyte could cause, for example, the blistering and hyperkeratosis seen in incontinentia pigmenti. Here it is suggested that the patterning found in all naevi, ranging from the simple to the complex, can be accounted for by assuming that they are secondary to chemical prepatterns laid down early in embryogenesis (Fig. 1) (17). A chemical prepattern is a spatially varying chemical concentration gradient that remains fixed. Its generation from uniformity is a striking example of symmetry breaking and in the example shown it is produced by a reaction–diffusion process. Given the presence of the prepattern, there are three possible ways in which macroscopic pattern can subsequently develop. First, when the concentration reaches a threshold, spatially dependent gene activation may occur. It is then possible to see how the abnormal clone becomes activated in regions that follow the prepattern. The distribution of the naevus over the skin will be independent of the presence of the abnormal clone, but not of the presence of the activated gene of the abnormal clone. Second, the prepattern may act as a chemo-attractant gradient for the abnormal clone (18). In naevi following the lines of Blaschko abnormal clones may aggregate as they migrate laterally within the surface ectoderm [one ectodermal cell, as it proliferates and migrates laterally cannot be expected to produce a macroscopic band with the exclusion of the normal cell type (19)]. Finally, the chemical prepattern may be the trigger for a spatially dependent selective proliferation of one cell type over another. An interesting consequence of the chemical prepattern hypothesis is that the pattern-forming process itself may be pathological, so that in people unaffected by naevoid skin disease the lines of Blaschko do not exist. The spontaneous generation of chemical concentration patterns over the dorsum of the early human embryo. Here the evolution of the ‘Brusselator’ (22), a simple chemical scheme involving six distinct compounds, is modelled using two coupled non-linear partial differential equations and solved using a finite element scheme over the domain as shown. This domain is a representation of the shape of the ectodermal surface of the day 25 tri-laminar embryo (a)–(e) and the day 24 embryo (f). The bottom quarter of the surface of the day 25 embryo is the region associated with the developing thoracic somites. In (a)–(c), the temporal evolution of pattern is shown culminating in a pattern resembling the narrow-banded lines of Blaschko. Images (d), (e) and (f) represent, in the order of decreasing complexity, the broad banded lines of Blaschko, quadrant naevi and the unilateral CHILD naevus, respectively. Next consider erythema gyratum repens. Here a pattern-forming process must account for the appearance of the rash, including expanding rings and spirals, and collision fronts, as well as its dynamic evolution. Hitherto, the rapid spread of the bands of inflammatory skin across the skin surface (up to 1 cm per day) has defied explanation. Some authors have suggested it is due to diffusion, but a simple calculation shows that the rate of diffusion for even small macromolecules is orders of magnitude too small (17). Once again, consider the chemical prepattern hypothesis as a possible solution to the problem. In contrast to the stationary patterns described above, in this case, the chemical gradient is a wave that propagates across the skin surface. Many features of the rash, including its morphology and rapid evolution, emerge naturally as a consequence of this reaction–diffusion mechanism (Fig. 2) (20,21). Modelling erythema gyratum repens. In (a) the one-dimensional solution to a reaction–diffusion scheme is shown (21). This chemical concentration waveform is repeated end-to-end and propagates to the left with a speed of the order 1 cm per day and with a wavelength approximately 1 cm. In (b)–(d), the model shown in (a) is approximated by a cellular automaton simulation known as the Hodgepodge machine (21,23). Note the morphology and evolution of the rash is accurately represented; the features include growing arcs, spirals and collision fronts. Here the concept of symmetry breaking found in dynamical systems theory is applied to pattern formation in some examples of human skin disease. Although as yet unproven, the mathematics show that the mechanisms are physically well founded. Quantitative approaches to the problem of pattern formation in skin disease result in falsifiable predictions and offer the prospect of new and counterintuitive insights into pathogenesis.