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Modeling conditions of pigmented integument by dissipative structures

2013; Elsevier BV; Volume: 73; Issue: 1 Linguagem: Inglês

10.1016/j.jdermsci.2013.08.012

1873-569X

Stefan Kippenberger, August Bernd, Matthias Hofmann, Nadja Zöller, Eva Valesky, Roland Kaufmann, Markus Meißner,

Diffusion and Search Dynamics

Dating back to the middle of the last century Alan Turing, the famous pioneer of computer sciences, established a mathematical model, the reaction–diffusion systems (RDS), allowing to model many patterns found in animate and unanimate nature [[1]Turing A.M. The chemical basis of morphogenesis.Phil Trans Roy Soc, Series B Biol Sci. 1952; 237: 37-72Crossref Google Scholar]. In his theoretical paper two components, an activator and an inhibitor, named morphogens, with different diffusion coefficients and kinetic constants are sufficient to produce non-random configurations out of an initial setting (Fig. S1). According to this model the components organize by themselves the pattern progression in a self-referential fashion. Utilizing RDS was successfully applied to model patterns of many organisms including fish skin [[2]Kondo S. The reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skin.Genes Cells. 2002; 7: 535-541Crossref PubMed Scopus (88) Google Scholar] and animal furs including hair follicle development [[3]Mooney J.R. Nagorcka B.N. Spatial patterns produced by a reaction-diffusion system in primary hair follicles.J Theor Biol. 1985; 115: 299-317Crossref PubMed Scopus (26) Google Scholar]. Recently, also molecular equivalents of RDS have been characterized [4Watanabe M. Kondo S. Changing clothes easily: connexin41.8 regulates skin pattern variation.Pigment Cell Melanoma Res. 2012; 25: 326-330Crossref PubMed Scopus (45) Google Scholar, 5Sick S. Reinker S. Timmer J. Schlake T. WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism.Science. 2006; 314: 1447-1450Crossref PubMed Scopus (423) Google Scholar]. This concept was amended by showing that also non-diffusible elements such as cellular receptors have impact on pattern formation [[6]Klika V. Baker R.E. Headon D. Gaffney E.A. The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organisation.Bull Math Biol. 2012; 74: 935-957Crossref PubMed Scopus (50) Google Scholar]. However, besides the RDS there are also many other spatiotemporal pattern-forming systems featuring amazing similarities with patterns found in nature [[7]Cross M.C. Hohenberg P.C. Pattern formation outside of equilibrium.Rev Mod Phys. 1993; 65: 851-1112Crossref Scopus (6523) Google Scholar]. Common to all these systems is pattern formation far from the chemical or physical equilibrium. Due to the self-organizing properties of these systems we hypothesize similar mechanisms at work generating the plethora of patterns present in human skin. In particular we want to focus on the Rayleigh–Bénard convection (RBC) and the Belousov–Zhabotinsky-Reaction (BZR) and compare their dynamic course to pigment patterns in human skin. The RBC dates back to the beginning of the last century and is the prototypical example of spatial pattern formation in a non-equilibrium fluid [[8]Rayleigh On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side.Phil Mag. 1916; 32: 529-546Crossref Google Scholar]. A thin fluid layer is brought away from the thermal equilibrium by applying heat to a plane horizontal bottom. As a result, a temperature gradient is established in the vertical direction. The fluid close the bottom expands due to the heat applied and becomes less dense. The buoyancy force counteracts gravity causing a fluid lift from the bottom. Due to the vertical temperature gradient the fluid temperature drops with increasing distance from the bottom. Above a critical temperature the system becomes sensitive to slight perturbations in fluid density interfering with the homogenous fluid conductivity allowing self-organized formation of fluid molecules. This process can be monitored macroscopically by the presence of large-scale structures known as Bénard cells. The property of these structures depends on many structural fluid parameters including density and conductivity [[7]Cross M.C. Hohenberg P.C. Pattern formation outside of equilibrium.Rev Mod Phys. 1993; 65: 851-1112Crossref Scopus (6523) Google Scholar]. In the present approach we took regular coffee with a dash of milk to produce a RBC (Fig. 1a). When heat is applied to a 2 mm high fluid layer, self-organized patterns emerge within minutes according to the aforementioned mechanism (Fig. 1b). These patterns appear as a reticular mesh with a “cell-like” structure. Interestingly, similar phenotypes can be found in animal furs, as shown for a Connemara pony (Fig. 1c). Likewise, conditions in human skin such as Erythema ab igne, Livedo racemosa, and on a smaller scale, compound nevi (Fig. 1d–f) show a surprising similarity with structures shown in Fig. 1a. As stated above, the characteristics of RBC depend on the physical parameters of the fluid. Thus, it was tested if a binary mixture containing a more dense fluid (glycerine) together with insoluble Chelex®-100 pearls produces patterns matching with skin conditions (Fig. S2). Driven by thermal convection the Chelex®-100 pearls become arranged, first in concentric circles, progressing to an irregular and spotty pattern with partly reticular portions (Fig. S2a). Particularly, the latter compares well to patterns found on the integument of sand lizards (Fig. S2b). Moreover, similar patterns can be found in Vitiligo subsets with follicular repigmentation (Fig. S2c). Also in pigmented basal cell carcinomas (BCC) (Fig. S2d) and dysplastic compound nevi (Fig. S2e), reticular and partly spotty areas can be identified matching well with self-organized patterns produced by convection. It could be speculated that the thermal gradient across human skin may trigger fluid convection analog to RBC. By coincidence the Russian chemist Boris Pavlovič Belousov discovered in the late 1950s a prolonged oscillating reaction [[9]Belousov B.P. A periodic reaction and its mechanism.in: Field R. Burger M. Oscillations and traveling waves in chemical systems. John Wiley, New York1985: 605-613Google Scholar] that attracted much interest from scientists of different areas. At first, chemists were sceptical about chemical oscillators assuming a violation of the second law of thermodynamics. Further studies revealed that the BZR is another example of a dissipative system, a term coined by the Nobel Prize laureate Ilya Prigogine, generating higher ordered structures far from thermodynamic equilibrium. The composition of ingredients used for the BZR shown in Fig. 2a was adapted from the classical work of Richard J. Field [[10]Field R.J. Das Experiment: Eine oszillierende Reaktion.Chem unserer Zeit. 1973; 7: 171-176Crossref Scopus (10) Google Scholar]. The basic principle of the BZR is schematically shown in Fig. 2b. Concentric chemical waves generated by the BZR form a spatial pattern showing some similarity to pigmented patterns found on zebra fur (Fig. 2c) or in lesions of epidermal nevi (Fig. 2d). A slight change in chemical composition namely, halving of all concentration, damped self-perpetuating reaction waves. After starting the reaction with a hot metal rod, a concentric area expands quickly turning into a margin-accented circle before vanishing (Fig. S3a). Again there are conditions of the integument that partially depict such patterns. Examples are eye-like rings found on some butterfly wings (Fig. S3b) and also on human skin, circular and margin-accented lesions are present e.g. in subsets of Vitiligo and Lupus erythmatosus (Fig. S3c–e). The use of autocatalytic systems to model various aspects of nature is a relatively young discipline. Since the milestone set by Alan Turings’ work in 1952, related approaches were shown to depict biological, physical and economical processes. In the context of dermatology, such methods were recently applied to produce patterns that match well with Erythema characteristics [[11]Kippenberger S. Bernd A. Thaçi D. Kaufmann R. Meissner M. Modeling pattern formation in skin diseases by a cellular automaton.J Invest Dermatol. 2013; 133: 567-571Crossref PubMed Scopus (6) Google Scholar]. In the present approach we show that dissipative self organization as described for non-equilibrium thermodynamics could be suitable to model pigment patterns found in skin. More generally, the formation of patterns can be understood as an inherently mechanical process leading to uneven distribution of morphogens [[12]Howard J. Grill S.W. Bois J.S. Turing's next steps: the mechanochemical basis of morphogenesis.Nat Rev Mol Cell Biol. 2011; 12: 392-398Crossref PubMed Scopus (175) Google Scholar]. In skin, lesions following the lines of Blaschko such as epidermal nevi (Fig. 2d) are assumed to reflect embryogenic tissue mechanics. With regard to BZR it seems likely that mechanochemical interaction of molecules produces such patterns in a spatiotemporal manner. Considering the viscoelastic properties of body tissues, mechanical forces can generate uneven material distribution as shown for thermal convection. Friction and viscosity may be critical parameters in this context. The authors are grateful to Dr. Adrian Sewell for critically reading the manuscript. We like to thank Lennart Kippenberger for providing the photograph of Antherina suraka. This work is dedicated to the late Ruprecht Kippenberger who opened my eyes to the beauty of nature (S.K.). The following are Supplementary data to this article:Fig. S2The Rayleigh-Bénard-convection of non-soluble components mimics skin patterns with irregular distributed agglomerations. (a) Glycerine mixed with Chelex®-100-pearls heated on a plate forms spotty patterns. (t0 to t5 indicates the progression of time.) (b) Spots on the integument of a sand lizard (Lacerta agilis). (c) Vitiligo with partial follicular repigmentation. (d) Surface skin microscopy of a pigmented basal cell carcinoma and (e) a dysplastic compound nevus with reticular and partly spotty areas.View Large Image Figure ViewerDownload Hi-res image Download (PPT)Fig. S3The Belousov–Zhabotinsky-Reaction mimics margin-accented patterns. (a) BZR reaction with lower concentrated ingredients (for details see text) started with a hot metal wire. (t0 to t5 indicates the progression of time.) (b) Eye-like pattern on the hind wing of Antherina suraka, a silk-moth of Madagascar. (c) Sutton nevus with typical hypopigmented halo. (d) Subacute cutaneous lupus erythematosus (SCLE) with annular plaques. (e) Vitiligo of the face with hyperpigmented margins, an iatrogenic artifact after PUVA treatment.View Large Image Figure ViewerDownload Hi-res image Download (PPT)