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How the Leopard Gets Its SpotsA single patternformation mechanism could underlie the wide variety of animal coat markings found in nature. Results fromthe mathematical model open lines of inquiry for the biologist
by James D. Murray
ammals exhibit a remarkablevariety of coat patterns; thevariety has elicited a compa
rable variety of explanations—manyof them at the level of cogency thatprevails in Rudyard Kipling's delightful "How the Leopard Got Its Spots."Although genes control the processes involved in coat pattern formation, the actual mechanisms that create the patterns are still not known. Itwould be attractive from the viewpoint of both evolutionary and developmental biology if a single mechanism were found to produce theenormous assortment of coat patterns found in nature.
I should like to suggest that a singlepatternformation mechanism couldin fact be responsible for most if notall of the observed coat markings. Inthis article I shall briefly describe asimple mathematical model for howthese patterns may be generated inthe course of embryonic development. An important feature of themodel is that the patterns it generates bear a striking resemblance tothe patterns found on a wide varietyof animals such as the leopard, thecheetah, the jaguar, the zebra andthe giraffe. The simple model is alsoconsistent with the observation thatalthough the distribution of spots onmembers of the cat family and ofstripes on zebras varies widely andis unique to an individual, each kindof distribution adheres to a generaltheme. Moreover, the model also predicts that the patterns can take onlycertain forms, which in turn impliesthe existence of developmental constraints and begins to suggest howcoat patterns may have evolved.
It is not clear as to precisely whathappens during embryonic development to cause the patterns. There arenow several possible mechanismsthat are capable of generating suchpatterns. The appeal of the simple
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model comes from its mathematicalrichness and its astonishing abilityto create patterns that correspond towhat is seen. I hope the model willstimulate experimenters to pose relevant questions that ultimately willhelp to unravel the nature of the biological mechanism itself.
ome facts, of course, are knownabout coat patterns. Physically,spots correspond to regions of
differently colored hair. Hair color isdetermined by specialized pigmentcells called melanocytes, which arefound in the basal, or innermost,layer of the epidermis. Themelanocytes generate a pigmentcalled melanin that then passes intothe hair. In mammals there areessentially only two kinds ofmelanin: eumelanin, from theGreek words eu (good) and melas(black), which results in black orbrown hairs, and phaeomelanin,from phaeos (dusty), which makeshairs yellow or reddish orange.
It is believed that whether or notmelanocytes produce melanin depends on the presence or absence ofchemical activators and inhibitors.Although it is not yet known whatthose chemicals are, each observedcoat pattern is thought to reflectan underlying chemical prepattern.The prepattern, if it exists, should reside somewhere in or just under theepidermis. The melanocytes arethought to have the role of "reading out" the pattern. The model Ishall describe could generate such aprepattern.
My work is based on a model developed by Alan M. Turing (the inventor of the Turing machine and thefounder of modern computing science). In 1952, in one of the most important papers in theoretical biology,Turing postulated a chemical mechanism for generating coat patterns. Hesuggested that biological form fol
lows a prepattern in the concentration of chemicals he called morphogens. The existence of morphogensis still largely speculative, exceptfor circumstantial evidence, but Turing's model remains attractive because it appears to explain a largenumber of experimental results withone or two simple ideas.
Turing began with the assumptionthat morphogens can react with oneanother and diffuse through cells. Hethen employed a mathematical model to show that if morphogens reactand diffuse in an appropriate way,spatial patterns of morphogen concentrations can arise from an initialuniform distribution in an assemblage of cells. Turing's model hasspawned an entire class of modelsthat are now referred to as reactiondiffusion models. These models areapplicable if the scale of the pattern islarge compared with the diameter ofan individual cell. The models are applicable to the leopard's coat, for instance, because the number of cellsin a leopard spot at the time the pattern is laid down is probably on theorder of 100.
Turing's initial work has beendeveloped by a number of investigators, including me, into a morecomplete mathematical theory. In atypical reactiondiffusion model onestarts with two morphogens that canreact with each other and diffuse atvarying rates. In the absence of diffusion—in a wellstirred reaction,for example—the two morphogenswould react and reach a steady uniform state. If the morphogens arenow allowed to diffuse at equal rates,any spatial variation from that steadystate will be smoothed out. If, however, the diffusion rates are not equal,
LEOPARD reposes. Do mathematical aswell as genetic rules produce its spots?
M
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diffusion can be destabilizing: the reaction rates at any given point may not be able to adjust quickly enough toreach equilibrium. If the conditions are right, a small spatial disturbance can become unstable and a pattern beginsto grow. Such an instability is said to be diffusion driven.
n reactiondiffusion models it is assumed that one of the morphogens is an activator that causes the melanocytesto produce one kind of melanin, say black, and the other is an inhibitor that results in the pigment cells'producing no melanin. Suppose the reactions are such that the activator increases its concentration locally and
simultaneously generates the inhibitor. If the inhibitor diffuses faster than the activator, an island of highactivator concentration will be created within a region of high inhibitor concentration.
One can gain an intuitive notion of how such an activatorinhibitormechanism can give rise to spatial patterns of morphogen concentrations from the following, albeit somewhatunrealistic, example. The analogy involves a very dry forest—a situation ripe for forest fires. In an attempt tominimize potential damage, a number of fire fighters with helicopters and firefighting equipment have beendispersed throughout the forest. Now imagine that a fire (the activator) breaks out. A fire front starts to propagateoutward. Initially there are not enough fire fighters (the inhibitors) in the vicinity of the fire to put it out. Flying intheir helicopters, however, the fire fighters can outrun the fire front and spray fireresistant chemicals on trees;when the fire reaches the sprayed trees, it is extinguished. The front is stopped.
If fires break out spontaneously in random parts of the forest, over the course of time several fire fronts (activation waves) will propagate outward. Each front in turn causes thefire fighters in their helicopters (inhibition waves) to travel out faster and quench the front at some distance aheadof the fire. The final result of this scenario is a forest with blackened patches of burned trees interspersed withpatches of green, unburned trees. In effect, the outcome mimics the outcome of reactiondiffusion mechanisms thatare diffusion driven. The type of pattern that results depends on the various parameters of the model and can beobtained from mathematical analysis.
Many specific reactiondiffusion models have been proposed, based on plausible or real biochemical reactions,and their patternformation properties have been examined. These mechanisms involve several parameters,including the rates at which the reactions proceed, the rates at which the chemicals diffuse and—of crucialimportance—the geometry and scale of the tissue. A fascinating property of reactiondiffu
MATHEMATICAL MODEL called a reactiondiffusionmechanism generates patterns that bear a strikingresemblance to those found on certain animals. Herethe patterns on the tail of
the leopard (left), the jaguar and the cheetah {middle)and the genet (right) are shown, along with the patternsfrom the model for tapering cylinders of varying width(right side of each panel).
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I
ZEBRA STRIPES at the junction of the
foreleg and body (left) can be produced bya reactiondiffusion mechanism (above).
sion models concerns the outcomeof beginning with a uniform steadystate and holding all the parametersfixed except one, which is varied. Tobe specific, suppose the scale of thetissue is increased. Then eventuallya critical point called a bifurcationvalue is reached at which the uniform steady state of the morphogensbecomes unstable and spatial patterns begin to grow.
The most visually dramatic example of reactiondiffusion pattern formation is the colorful class of chemical reactions discovered by the Soviet investigators B. P. Belousov andA. M. Zhabotinsky in the late 1950's[see "Rotating Chemical Reactions,"by Arthur T. Winfree; SCIENTIFICAMERICAN, June, 1974]. The reactionsvisibly organize themselves in spaceand time, for example as spiralwaves. Such reactions can oscillatewith clocklike precision, changingfrom, say, blue to orange and back toblue again twice a minute.
Another example of reactiondiffusion patterns in nature was discovered and studied by the Frenchchemist Daniel Thomas in 1975. Thepatterns are produced during reactions between uric acid and oxygenon a thin membrane within whichthe chemicals can diffuse. Althoughthe membrane contains an immobilized enzyme that catalyzes the reaction, the empirical model for describing the mechanism involves only thetwo chemicals and ignores the en
| zyme. In addition, since the membrane is thin, one can assume correctly that the mechanism takes
I place in a twodimensional space.I should like to suggest that a good
candidate for the universal mecha
nism that generates the prepatternfor mammalian coat patterns is a reactiondiffusion system that exhibits diffusiondriven spatial patterns.Such patterns depend strongly onthe geometry and scale of the domain where the chemical reactiontakes place. Consequently the sizeand shape of the embryo at the timethe reactions are activated shoulddetermine the ensuing spatial patterns. (Later growth may distort theinitial pattern.)
ny reactiondiffusion mechanismLcapable of generating diffusiondriven spatial patterns would
provide a plausible model foranimal coat markings. Thenumerical and mathematical resultsI present in this article are based onthe model that grew out of Thomas'work. Employing typical values forthe parameters, the time to form coatpatterns during embryogenesiswould be on the order of a day or so.
Interestingly, the mathematicalproblem of describing the initial stages of spatial pattern formation by reactiondiffusion mechanisms (whendepartures from uniformity are minute) is similar to the mathematicalproblem of describing the vibrationof thin plates or drum surfaces. Theways in which pattern growth depends on geometry and scale cantherefore be seen by consideringanalogous vibrating drum surfaces.
If a surface is very small, it simplywill not sustain vibrations; the disturbances die out quickly. A minimumsize is therefore needed to drive anysustainable vibration. Suppose thedrum surface, which corresponds tothe reactiondiffusion domain, is a
rectangle. As the size of the rectangle is increased, a set of increasingly complicated modes of possible vibration emerge.
An important example of howthe geometry constrains the possiblemodes of vibration is found whenthe domain is so narrow that onlysimple—essentially onedimensional—modes can exist. Genuine twodimensional patterns require the domain to have enough breadth as wellas length. The analogous requirement for vibrations on the surfaceof a cylinder is that the radius cannotbe too small, otherwise only quasionedimensional modes can exist;only ringlike patterns can form, inother words. If the radius is largeenough, however, twodimensionalpatterns can exist on the surface. As aconsequence, a tapering cylinder canexhibit a gradation from a twodimensional pattern to simple stripes[see illustration on opposite page].
Returning to the actual twomorphogen reactiondiffusion mechanism I considered, I chose a set of reaction and diffusion parameters thatcould produce a diffusiondriven instability and kept them fixed for allthe calculations. I varied only thescale and geometry of the domain. Asinitial conditions for my calculations,which I did on a computer, I choserandom perturbations about the uniform steady state. The resulting patterns are colored dark and light in regions where the concentration of oneof the morphogens is greater than orless than the concentration in the homogeneous steady state. Even withsuch limitations on the parametersand the initial conditions the wealthof possible patterns is remarkable.
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A
EXAMPLES OF DRAMATIC PATTERNS occurring naturally arefound in the anteater (left) and the Valais goat, Capra aegagrus
hircus (right). Such patterns can be accounted for by the author'sreactiondiffusion mechanism (see bottom illustration onthese
How do the results of the model cheetah (Acinonyx jubatus), the jag relatively short, and so one wouldcompare with typical coat markings uar (Panthem onca) and the genet expect that it could support spots toand general features found on ani {Genetta genetta) provide good exam the very tip. (The adult leopard tail ismals? I started by employing taper pies of such pattern behavior. The long but has the same number of vering cylinders to model the patterns spots of the leopard reach almost to tebrae.)The tail of the genet embryo,on the tails and legs of animals. The the tip of the tail. The tails of the at the other extreme, has a remarkresults are mimicked by the results cheetah and the jaguar have distinct ably uniform diameter that is quitefrom the vibratingplate analogue, ly striped parts, and the genet has thin. The genet tail should thereforenamely, if a twodimensional region a totally striped tail. These obser not be able to support spots,marked by spots is made sufficiently vations are consistent with what is The model also provides an inthin, the spots will eventually change known about the embryonic struc stance of a developmental conto stripes, ture of the four animals. The prenatal straint, documented examples of
The leopard (Panthera pardus), the leopard tail is sharply tapered and which are exceedingly rare. If the
SCALE AFFECTS PATTERNS generated within the constraints of a generic animal shape in the author's model. Increasing thescale and holding all other parameters fixed produces a remarkable variety of patterns. The model agrees with the fact that
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two pages). The drawing of the anteater was originally published by G. and W. B. Whittaker in February, 1824, and the photograph was made by Avi Baron and Paul Munro.
ings of the zebra. It is not difficult to
generate a series of stripes with ourmechanism. The junction of the foreleg with the body is more complicated, but the mathematical modelpredicts the typical pattern of legbody scapular stripes [see illustrationon page 83}.
In order to study the effect of scalein a more complicated geometry, wecomputed the patterns for a genericanimal shape consisting of a body,a head, four appendages and a tail
small animals such as the mouse have uniform coats, intermediatesize ones such asthe leopard have patterned coats and large animals such as the elephant are uniform.
[see bottom illustration on these twopages}. We started with a very smallshape and gradually increased itssize, keeping all the parts in proportion. We found several interestingresults. If the domain is too small,no pattern can be generated. As thesize of the domain is increased successive bifurcations occur: differentpatterns suddenly appear and disappear. The patterns show morestructure and more spots as the sizeof the domain is increased. Slenderextremities still retain their stripedpattern, however, even for domainsthat are quite large. When the domainis very large, the pattern structure isso fine that it becomes almost uniform in color again.
he effects of scale on pattern suggest that if the reactiondiffusionmodel is correct, the time at
which the patternformingmechanism is activated duringembryogenesis is of the utmostimportance. There is an implicitassumption here, namely that therate constants and diffusioncoefficients in the mechanism areroughly similar in different animals.If the mechanism is activated earlyin development by a genetic switch,say, most small animals that haveshort periods of gestation should beuniform in color. This is generallythe case. For larger surfaces, at thetime of activation there is the possibility that animals will be half blackand half white. The honey badger(Mellivora capensis) and the dramatically patterned Valais goat (Capra aegagrus hircus) are two examples [seetop illustration on these two pages}. Asthe size of the domain increases, soshould the extent of patterning. Infact, there is a progression in complexity from the Valais goat to certainanteaters, through the zebra and onto the leopard and the cheetah. At theupper end of the size scale the spotsof giraffes are closely spaced. Finally, very large animals should be uniform in color again, which indeed isthe case with the elephant, the rhinoceros and the hippopotamus.
We expect that the time at whichthe patternforming mechanism is activated is an inherited trait, and so, atleast for animals whose survival depends to a great extent on pattern,the mechanism is activated when theembryo has reached a certain size. Ofcourse, the conditions on the embryo's surface at the time of activation exhibit a certain randomness.The reactiondiffusion model produces patterns that depend uniquely
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prepatternforming mechanism foranimal coat markings is a reactiondiffusion process (or any process that issimilarly dependent on scale andgeometry), the constraint would develop from the effects of the scaleand geometry of the embryos. Specifically, the mechanism shows that it ispossible for a spotted animal to have astriped tail but impossible for a stripedanimal to have a spotted tail.
We have also met with success inour attempts to understand the mark
T
on the initial conditions, the geometry and the scale. An important aspect of the mechanism is that, for agiven geometry and scale, the patterns generated for a variety of random initial conditions are qualitatively similar. In the case of a spottedpattern, for example, only the distribution of spots varies. The findingis consistent with the individuality
of an animal's markings within aspecies. Such individuality allows forkin recognition and also for generalgroup recognition.
The patterns generated by themodel mechanism are thought tocorrespond to spatial patterns ofmorphogen concentrations. If theconcentration is high enough, melanocytes will produce the melanin
pigments. For simplicity we assumedthat the uniform steady state is thethreshold concentration, and we reasoned that melanin will be generatedif the value is equal to or greater thanthat concentration. The assumptionis somewhat arbitrary, however. It isreasonable to expect that the threshold concentration may vary, evenwithin species. To investigate such
DIFFERENT GIRAFFES have different kinds of markings. The subspecies Giraffa camelopardalis tippelskirchi is characterizedby rather small spots separated by wide spaces (top left); G. camelopardalis reticulata, in contrast, is covered by large, closelyspaced spots (top right). Both kinds of pattern can be accounted for by the author's reactiondiffusion model (bottom left and
bottom right). The assumption is that at the time the pattern is laid down the embryo is between 35 and 45 days old and has alength of roughly eight to 10 centimeters. (The gestation period of the giraffe is about 457 days.)
effects, we considered the variouskinds of giraffe. For a given typeof pattern, we varied the parameterthat corresponds to the morphogenthreshold concentration for melanocyte activity. By varying the parameter, we found we could produce patterns that closely resemble those oftwo different kinds of giraffe [see illustration on opposite page].
ecently the results of our modelhave been corroborateddramatically by Charles M. Vest
and Youren Xu of the Universityof Michigan. They generatedstandingwave patterns on avibrating plate and changed thenature of the patterns by changingthe frequency of vibration. Thepatterns were made visible by aholographic technique in which theplate was bathed in laser light. Lightreflected from the plate interferedwith a reference beam, so that crestsof waves added to crests, troughsadded to troughs, and crests andtroughs canceled, and the resultingpattern was recorded on a piece ofphotographic emulsion [see illustration at right].
Vest and Youren found that low frequencies of vibration produce simplepatterns and high frequencies of vibration produce complex patterns.The observation is interesting, because it has been shown that if a pattern forms on a plate vibrating at agiven frequency, the pattern formedon the same plate vibrated at a higher frequency is identical with thepattern formed on a proportionally larger plate vibrated at the original frequency. In other words, Vestand Youren's data support our conclusion that more complex patternsshould be generated as the scale of
! the reactiondiffusion domain is increased. The resemblance betweenour patterns and the patterns subsequently produced by the Michiganworkers is striking.
I should like to stress again thatall the patterns generated were produced by varying only the scale andgeometry of the reaction domain; allthe other parameters were held fixed(with the exception of the differentthreshold concentrations in the caseof the giraffe). Even so, the diversity of pattern is remarkable. The mod
I el also suggests a possible explanation for the various pattern anomalies seen in some animals. Undersome circumstances a change inthe
[ value of one of the parameters can result in a marked change in the pattern obtained. The size of theeffect
STANDINGWAVE PATTERNS generated on a thin vibrating plate resemble coat patterns and confirm the author's work. More complex patterns correspond to higher frequencies of vibration. The experiments were done by Charles M. Vest and Youren Xu.
depends on how close the value ofthe parameter is to a bifurcation value: the value at which a qualitativechange in the pattern is generated.
If one of the parameters, say a rateconstant in the reaction kinetics, isvaried continuously, the mechanismpasses from a state in which no spatial pattern can be generated to apatterned state and finally back toa state containing no patterns. Thefact that such small changes in aparameter near a bifurcation valuecan result in such large changes inpattern is consistent with thepunctuatedequilibrium theory ofevolution. This theory holds thatlong periods of little evolutionarychange are punctuated by shortbursts of sudden and rapid change.
any factors, of course, affect animal coloration. Temperature,humidity, diet, hormones and
meta
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bolic rate are among some of them.Although the effects of such factorsprobably could be mimicked by manipulating various parameters, thereis little point in doing so until moreis known about how the patterns reflected in the melanin pigments areactually produced. In the meantimeone cannot help but note the wide variety of patterns that can be generated with a reactiondiffusion model byvarying only the scale and geometry.The considerable circumstantial evidence derived from comparison withspecific animalpattern features is encouraging. I am confident that mostof the observed coat patterns canbe generated by a reactiondiffusionmechanism. The fact that many general and specific features of mammalian coat patterns can be explainedby this simple theory, however, doesnot make it right. Only experimentalobservation can confirm the theory.
M