J. R. Soc. Interface (2012) 9, 1051–1062

*Author for c

Electronic sup10.1098/rsif.2

doi:10.1098/rsif.2011.0516Published online 12 October 2011

Received 2 AAccepted 14 S

Pattern formation in stromatolites:insights from mathematical modellingR. Cuerno1, C. Escudero2,*, J. M. Garcıa-Ruiz3 and M. A. Herrero4

1Departamento de Matematicas and Grupo Interdisciplinar de Sistemas Complejos (GISC),Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganes, Spain

2Departamento de Economıa Cuantitativa and Instituto de Ciencias Matematicas(CSIC-UAM-UCM-UC3M), Universidad Autonoma de Madrid, Ciudad Universitaria

de Cantoblanco, 28049 Madrid, Spain3Laboratorio de Estudios Cristalograficos, CSIC, 18100 Armilla, Granada, Spain

4IMI and Departamento de Matematica Aplicada, Facultad de Matematicas, UniversidadComplutense de Madrid, 28040 Madrid, Spain

To this day, computer models for stromatolite formation have made substantial use ofthe Kardar–Parisi–Zhang (KPZ) equation. Oddly enough, these studies yielded mutuallyexclusive conclusions about the biotic or abiotic origin of such structures. We show in thispaper that, at our current state of knowledge, a purely biotic origin for stromatolites canneither be proved nor disproved by means of a KPZ-based model. What can be shown, how-ever, is that whatever their (biotic or abiotic) origin might be, some morphologies found inactual stromatolite structures (e.g. overhangs) cannot be formed as a consequence of a processmodelled exclusively in terms of the KPZ equation and acting over sufficiently large times.This suggests the need to search for alternative mathematical approaches to model thesestructures, some of which are discussed in this paper.

Keywords: pattern formation; stromatolites; Kardar–Parisi–Zhang equation;origin of life

1. INTRODUCTION

The detection of the oldest remnants of life on the Earthis an important scientific problem. Indeed, it is crucial toset the timing of life in the only planet where matter isknown to have self-organized to yield complex shapesand behaviour. The search for remnants of the earliestlife is conducted along four different paths, namely: (i)the identification of carbonaceous decorated microstruc-tures displaying shapes characteristic of primitive life andunable to form solely by mineral precipitation [1,2]; (ii)the search for reduced carbon or sulphur with an isotopicsignature characteristic of life [3,4]; (iii) the detection oforganic molecules that may be exclusively derived frombiochemical degradation of once living organisms [5,6];and (iv) the study of sedimentary structures, namedstromatolites [7–10] that are thought to be built bymicrobial communities. Each of these approaches meetsconsiderable difficulties [11,12]. For instance, very oldmicrostructures with complex shapes are not easilydistinguished, neither morphologically nor chemically,from self-ordered mineral assemblies [13]. On the otherhand, the isotopic signatures of carbon compoundsformed by purely mineral reactions such as Fischer–Tropp cycles are very similar to those appearing in

orrespondence (carlos.escudero@uam.es).

plementary material is available at http://dx.doi.org/011.0516 or via http://rsif.royalsocietypublishing.org.

ugust 2011eptember 2011 1051

biological reactions [14]. To this day, the so-called mol-ecular fossils are the most solid evidence for life, but thetechnology available for their recognition can only beapplied when the geological context is well-known andtherefore the singenicity of the fossils can be assured[15,16]. All these concerns and limitations have madethe laminated, accretionary structures called stromato-lites important pieces in the search for life’s infancy[17,18]. Indeed, as some stromatolites are thought todate back as far as about 3.5 Ga [9,19–22], their possiblebiotic origin would have deep consequences for our cur-rent views on how, and when, life started on the Earth.

The relevance of stromatolites in the context of thesearch for life’s origin originates in their complexshape and structure, which in modern extant examplesis, undoubtedly, the result of a cooperative microbiolo-gical behaviour [23–27]. They appear as sedimentarylaminated structures whose sizes range from a few milli-metres for an individual to a kilometre for reefs made ofmany individuals, and displaying corrugated surfacesand sometimes columnar shapes with a rich variety ofpatterns (e.g. [17,22,23] and figure 1). But, interest-ingly, stromatolites cannot be considered as properfossils. Like termite mounds, see figure 2, stromatolitesbeing currently formed are not actual biological organ-isms, but petrological structures built by livingorganisms and rarely preserve detectable organismalremnants of their putative builders or users. However,while the complexity of termite mounds structures

This journal is q 2011 The Royal Society

(a) (b)

(c) (d)

Figure 1. Stromatolitic pattern from (a), (b) ca 2.7 Ga Tumbiana formation; (c) ca 1.8 Ga Duck Creek Dolomite; (d) ca 3.49 GaDresser formation.

Figure 2. In the biotic scenario, ancient stromatolites areviewed not as biological entities, but as the result of biologicalactivity from organisms that are no longer present, and fromwhich no biological record remains. The situation would bemuch like a termite mound, as depicted above, once insectshad left.

1052 Pattern formation in stromatolites R. Cuerno et al.

rules out a non-biotic origin, the much simpler struc-tures of stromatolites require that the possibility of anon-biological origin has to be explored, particularlyfor the case of many Archean and some Proterozoicstromatolites.

Biotic or abiotic in origin, stromatolite pattern forma-tion needs to be better understood. By this, we meanthe issue of ascertaining the way in which the three-dimensional topographies and textures of these rockstructures are built up. Modern stromatolites providethe only dynamic models available for experimentation[28–31]. Such limitations emphasize the importance ofinsights gained from mathematical models of stromato-lite morphogenesis. To provide useful information,mathematical models should be able to represent thosephysical processes likely to be involved in stromatoliteformation (mainly associated with precipitation and

J. R. Soc. Interface (2012)

sedimentation) and the influence (if any) of the growthof mats on these processes. Moreover, analyses and simu-lation of the corresponding models should shed light intothe manner in which stromatolites are formed. To thebest of our knowledge, mathematical models of stromato-lite formation have been proposed and discussed onlyrecently [32–34]. The choice of equations selected tomodel their formation basically reduces to the so-calledKardar–Parisi–Zhang (KPZ) stochastic equation [35]

@h@t¼ nr2h þ l

2jrhj2 þ hðx; tÞ þ F ; ð1:1Þ

which provides a classical model to describe the growth ofan interface, whose position is identified by a height func-tion h, separating a growing medium which advances intoa receding one under the effect of random fluctuations h[35–37]. The KPZ equation has been postulated for manydifferent types of systems, from polymers to vortex lines,domain walls, thin films, to biophysical systems. In ourcontext, the interface position h is identified with theboundaries of the laminae displayed in stromatolites.For instance, in Grotzinger & Rothman [32], the authorspropose that a model based on a KPZ equation is able toreproduce some morphological features (fractal dimen-sion, power spectra, . . .) observed in some stromatolitespecimens and they claim a possible purely abioticorigin for stromatolite formation. On the other hand, aparticular limit of the deterministic KPZ equation(where noise effects are ignored) has been studied[33,34], where a biotic origin for stromatolite structuresis proposed. The use of similar equations to supportmutually exclusive conclusions raises at once the ques-tion: what features of stromatolite formation canactually be inferred from the analysis of simple math-ematical models of which KPZ is an example? In thispaper, we claim that, at our current state of knowledge,a purely biotic origin for stromatolites cannot be proved

Figure 3. A stromatolitic sample from 2.7 Ga Tumbiana for-mation (Andalusian Hill). The arrow indicates the locationof an overhang that cannot be described by the KPZ equation.

Pattern formation in stromatolites R. Cuerno et al. 1053

(nor disproved) by means of a KPZ-based mathematicalmodel. The reason is that each term in equation (1.1) canbe shown to correspond to a physical process (diffusion,convection, sedimentation, precipitation . . .) which canbe of a biotic or abiotic nature, both cases being indistin-guishable in the absence of further information. Whatcan be shown, however, is that whatever their (biotic orabiotic) origin may be, some stromatolite morphologiescannot be formed as a consequence of a process modelledexclusively in terms of equation (1.1) and acting oversufficiently large times.

We conclude this introduction by describing the planof this paper. We first recall in §2 some basic facts con-cerning the KPZ equation. In particular, the physicalmeaning of each term in that equation is described. Wethen address in §3, the main issue in this work. More pre-cisely, we compare the behaviour of solutions to KPZwith the laminar morphologies observed in some stroma-tolite samples. In particular, it is observed that some ofthe former are not compatible with a formation processbasically described by a KPZ equation. Section 4 con-tains a discussion on the relevance of KPZ in ourcurrent context, and some directions to model stromato-lite formation are suggested to gain insight intostromatolite pattern formation. Finally, a number oftechnical results related to the physical derivation ofKPZ in the context of kinetic roughening, as well as onthe mathematical properties of that equation, are gath-ered in electronic supplementary material, appendicesA and B at the end of the paper.

2. THE MEANING OF KARDAR–PARISI–ZHANG EQUATION

The KPZ equation (1.1) is a classical model for kineticroughening. This last term denotes the general problemof describing how microscopic fluctuations, which arepresent in virtually any interface displacement process,eventually develop into large-scale behaviours withuniversal, observable properties [37]. As mentioned,in equation (1.1), h ¼ h(x,t) denotes the position of aninterface separating an advancing medium which isinvading a receding one; x [ Rd represents a spacecoordinate in a d-dimensional space (d � 1), t standsfor time, n , l and F are positive parameters to be dis-cussed below; and h(x,t) is a Gaussian noisecharacterized by its two first moments

khðx; tÞl ¼ 0 ð2:1Þ

and

khðx; tÞhðx 0; t0Þl ¼ Ddðx � x 0Þdðt � t0Þ; ð2:2Þ

where D is a positive parameter known as the noiseamplitude and d(.) is Dirac’s delta function, so thatd(x) ¼ 0 for x = 0, and

ÐRd dðxÞdx ¼ 1, where d �1 is

the dimension of the space where the previous integralis considered. Conditions (2.1) and (2.2) are usuallyrephrased by stating that the noise term h(x,t) inequation (1.1) has zero mean, a variance given by Dand correlations are considered negligible.

At this juncture, we should stress that equation (1.1)is a continuous equation, by which we mean that x can

J. R. Soc. Interface (2012)

be any point in Rd and t can be any positive realnumber. While particularly convenient for mathemat-ical treatment, continuous equations are generallyunderstood as suitable approximations to discreteones. The latter are characterized by being formulatedin discrete domains, with given time and space unitlengths Dt and Dx.

Modelling particular aspects of stromatolite for-mation by means of the KPZ equation raises at once anumber of questions, as for instance:

(i) How can equation (1.1) be derived from firstprinciples, and which physical assumptionsneed to be considered to that end?

(ii) What is the physical meaning of parameters n, land F in equation (1.1), and how can they berelated to experimental measurements?

(iii) Which types of behaviour do solutions to equa-tion (1.1) display, and what relation (if any)exists between such behaviours and stromatolitemorphologies.

The mathematically oriented reader will find a discus-sion on points (i) and (ii) (respectively, (iii)) above inelectronic supplementary material, appendix A (respect-ively, in electronic supplementary material, appendixB). Here, we merely describe the main conclusions thatresult from electronic supplementary material, appendixA concerning the physical meaning of the terms arisingin the KPZ equation.

To begin with, equation (1.1) represents an approxi-mation, obtained under a number of simplifyinghypotheses, to a large class of non-equilibrium pro-cesses. These are characterized by the coexistence oftwo different media (or phases), one of which is advan-cing into the other. Both media are separated by acomparatively thin, moving interface. The location of

1054 Pattern formation in stromatolites R. Cuerno et al.

such interface at any space coordinate x and time t isgiven by a function h(x,t), sometimes termed the inter-face height, whose evolution in time is governed byequation (1.1). Parameter l represents the interface vel-ocity along its normal direction. Such value is thusassumed to be constant in time. On the other hand,n . 0 is the (material) diffusion coefficient along theinterface and F is an external forcing term that hasdimensions of velocity. Denoting by [u] the dimensionsof a quantity u, we thus have

½l� ¼ lengthtime

¼ lt

and ½n� ¼ ðlengthÞ2

time¼ l2

t: ð2:3Þ

On its turn, the effect of noise on the interface motionis encoded in the term h(x,t) whose properties (zeromean noise, amplitude given by D . 0) have beenrecalled before. As a matter of fact, in the particular set-ting discussed in electronic supplementary material,appendix A, n, l and F can be related to some materialproperties of the interface as the interfacial tension s andthe effective interface mobility G (that is the interfacespeed per unit of bending force measured along thenormal direction to the interface), as well as to an exter-nal driving potential (necessary for the interfacemotion), m0 by means of the relations

n ¼ sG and F ¼ l ¼ Gm0 ð2:4Þ

hence

½s� ¼ energylength

¼ force ¼ m � lt2 ; ½m� ¼ ½m0� ¼

½s�l

and ½G � ¼ l � tm

: ð2:5Þ

Note that equation (2.3) is consistent with equations(2.4) and (2.5). In view of our previous discussion, wemay say that equation (1.1) describes interface motionunder the action of four superposing effects. These arerepresented by the four terms on the right-hand side(r.h.s.) of equation (1.1) and correspond to diffusion,growth, external forcing and noise. Diffusion at theinterface is characterized by the diffusion coefficient n

given in equation (2.4), whereas the mean interfacevelocity in equation (2.4), which is the key player in theinterface motion, is given by l. Except for the noiseterm h(x,t), no additional effect is considered to be rel-evant for interface motion. At this modelling level, theparticular physico-chemical properties of any given pro-cess under consideration are contained in the materialparameters s and G, the external potential at work m0,and the noise term h(x,t). In the absence of further speci-fications, it is impossible to assign to any of them a bioticor abiotic origin. Therefore, the use of the KPZ equationin itself does not provide any hint about the biotic (orabiotic) nature of the process being modelled.

3. A TEST FOR MATHEMATICALMODELLING: STROMATOLITE BANDINTERFACES

In this section, we address the issue of discussing whichmorphological features (if any) of stromatolites can be

J. R. Soc. Interface (2012)

efficiently modelled by means of the KPZ equation(1.1). Following earlier studies [32–34], we shall concen-trate on the formation of laminae, and more precisely onthat of their separating interfaces, which constitute acharacteristic feature of stromatolites. To be morespecific, we will select a sample recently collected fromthe 2.724 Ga Tumbiana formation in the FortescueGroup (Australia), to be referred henceforth as theAndalusian Hill (AH) specimen, a picture of which isshown in figure 3.

A central point in our analysis consists of ascertain-ing whether the band interfaces observed in theprevious figure can be somehow associated with thesequential profiles of a function h(x,t) which solvesequation (1.1) for a suitable choice of the parametersn, l and D therein. This is the basic assumption made[32,33]. For instance, in an earlier study [32] authorsassert that, under suitable assumptions on the equationparameters ‘numerical solutions of equation (1.1) (ournotation) yield interfaces that evolve to a statisticalsteady state with fractal dimension Df (our notation)ffi 2.6 independent of the initial conditions. Assumingthat our measurement of Df ffi 2.5 is not significantlydifferent from the theoretical prediction of Df ffi 2.6,our theoretical description is consistent with ourquantitative measurements’.

We have quoted in some detail the previousexcerpt from the study of Grotzinger & Rothman[32] as it clearly illustrates an implicit assumptionoften made at the modelling stage. More precisely, itis widely assumed that the bands actually observedin rock samples correspond to what is called in math-ematics an asymptotic pattern. By this, we mean asurface morphology to which any solution resembles,irrespective of its initial profile, after a sufficientlylarge time has elapsed. However, before an asymptoticpattern (assumed to exist) unfolds, solutions gothrough what is called a transient period. Duringsuch a period, solutions may display features quitedifferent from their asymptotic morphology. Inciden-tally, the transient period may be quite long andthere seems to be no general technique to a prioriestimate its duration.

Actually, when looking for asymptotic patterns inequations similar to equation (1.1), serious mathemati-cal difficulties are encountered which reflectunderlying, deep scientific problems. In particular,and as a consequence of the presence of noise termslike h(x,t) in these equations, no general existencetheory for solutions of nonlinear stochastic partialdifferential equations is available mainly for highspace dimensions d (cf. [38]). This fact notwithstand-ing a wealth of results, mainly numerical simulations,have been obtained for equation (1.1) [37,39,40], a sol-ution to the one-dimensional case having beenachieved only very recently (see [41] and referencestherein). Although results suggested by these simu-lations are not always compatible among themselves,altogether they provide significant information aboutprocesses governed by the KPZ equation, as will benoticed below. On the other hand, if the noise termh(x,t) in equation (1.1) is dropped, a convenient exist-ence theory is at hand for the deterministic equation

Pattern formation in stromatolites R. Cuerno et al. 1055

thus obtained [42,43]. Furthermore, in these papers, acomplete characterization of the solutions to this equationis presented, and the conditions under which non-unique-ness takes place are found.

Let us consider the question of describing what mor-phologies can be identified, for sufficiently large times,in the interface height h(x,t) of a process governed byequation (1.1). As it turns out, in order to obtain a mean-ingful answer, the previous question needs to be carefullyreformulated. First of all, different morphologies may beobtained, each of them being an appropriate approxi-mation over different periods of time. The latter may bequite large, reaching up to the scale of 104–106 yearscharacteristic of carbonate platform deposition. Suchdifferent transient periods are separated by crossovertimes, for which only rough estimates can be provided.Moreover, as a consequence of the stochastic natureof equation (1.1), the asymptotic morphologies forwhich a theory exists correspond to some average inter-face properties, but not to the interface height itself.Fortunately enough, these interface-related quantitieson which patterns can be shown to unfold have a clearmorphological meaning as discussed below.

3.1. Evolution of the interface width in Kardar–Parisi–Zhang equation: crossover times

Consider first the simpler idealized case d ¼ 1, in whichthe interface position depends on a single spatial coordi-nate, and on time. Then solutions are known to developdifferent features as time passes, as recalled next.

Bearing in mind the morphologies actually observed instromatolite samples, we shall focus herein on a quantityassociated with solutions of equation (1.1), the interfacewidth W(t), which is defined as follows. Consider amedium of length L . 0, divided into N equal intervalsIi(1 � i � N), each having a size Dx. Let h(xi,t) be theaverage value of h(x,t) on each interval Ii. Then, theaverage value of h(x,t) at time t, �hðtÞ is given by

�hðtÞ ¼ 1N

XNi¼1

hðxi; tÞ; ð3:1Þ

and the interface width W(t) is defined as follows:

W ðtÞ2 ¼ 1N

XNi¼1

ðhðxi; tÞ � �hðtÞÞ2: ð3:2Þ

Therefore, W(t) represents the mean quadratic devi-ation of the interface height around its average value, ameasure of the roughness of the interface itself. Supposenow that at t ¼ 0, the interface is flat so that

hðx; 0Þ ¼ 0: ð3:3Þ

Then for sufficiently small times, the corresponding sol-ution is expected to remain almost flat as well, so that thefirst and second terms on the r.h.s. of equation (1.1) arenegligible, and equation (1.1) can be effectively replacedby the so-called random deposition (RD) equation

@h@t¼ hðx; tÞ þ F : ð3:4Þ

J. R. Soc. Interface (2012)

Notice that the term F in equation (3.4) can be dis-pensed with upon replacing h by h 2 Ft. For equation(3.4), it is known that, after some regularization [37]

W ðtÞ � t1=2 as t increases; ð3:5Þ

where W(t) � f(t) for suitable times t is to be under-stood as follows: there exist positive constants C1, C2

such that C1f(t) �W(t) � C2f(t) in the range of timesunder consideration.

As time passes, solutions to equations (1.1) and (3.3)begin to develop non-flat profiles, so that the terms ofequation (1.1) which were discarded to obtain equation(3.4) become relevant. In view of the assumptions madein electronic supplementary material, appendix A toderive equation (1.1), the new term to be retained is thefirst on the r.h.s. of equation (1.1). One thus obtainsthe so-called Edwards–Wilkinson (EW) equation

@h@t¼ nr2 h þ hðx; tÞ: ð3:6Þ

The transition from the range of validity of equation(3.4) to that of equation (3.6) occurs at a first crossovertime, t1C that can be estimated as follows

t1C � ðDxÞ2

n; ð3:7Þ

where Dx is the length scale corresponding to the dis-crete process of which equation (3.7) represents acontinuous approximation. Equation (3.6) is linearand can be explicitly solved. It can be shown that forsuch equation

W ðtÞ � t1=4 as time increases: ð3:8Þ

Finally, when t . t1C becomes large enough, allterms in equation (1.1) become relevant, and the fullequation (1.1) needs to be considered. Transition fromEW equation (3.6) to the KPZ equation (1.1) takesplace at a second crossover time t2C, which is suchthat [36]

t2C � n5D�2l�4: ð3:9Þ

For convenience of the reader, let us compute t2C in ahypothetical situation, where

n � 10�6 cm2 s�1; l � 10 cm yr�1

and D � 103 cm3 yr�1;

then equation (3.9) yields t2C � 1023 years, that is, afew hours. This fact is consistent with the assumptionsmade that correspond to a fast-growing process. On theother hand, for the same value of n as before, taking

l � 1 cm yr�1 and D � 1 cm3 yr�1;

gives t2C � 107 years. Note the strong dependence offormula (3.19) on the values of the parameters involved.

Once the full equation (1.1) has been developed, theinterface width W(t) continues to exhibit a power-likegrowth first.

W ðtÞ � t1=3; ð3:10Þ

2.5

2.0

1.5

1.0

0.5

0 1 2

t1C

t2C

ts

3 4log t (arb. units)

log

W (

arb.

uni

ts)

5 6 7 8

Figure 4. Time evolution of the surface width for the KPZ equation in dimension d ¼ 1. The slopes of the straight lines are 1/2,1/4, 1/3, left to right. Crossover times t1C, t2C and saturation time ts are indicated by arrows.

50

40

30

20

10

01 3 5 7 9 11 13

x

h

15 17 19 21 23–10

Figure 5. Simulated profiles at different times for solutionsof the RD equation for which the initial condition is asmall-amplitude sinusoidal profile. Here F ¼ 10 and D ¼ 1.Snapshots correspond to times: dark blue, t ¼ 0; red, t ¼ 1;green, t ¼ 2; purple, t ¼ 3; light blue, t ¼ 4.

1056 Pattern formation in stromatolites R. Cuerno et al.

to eventually saturate to an L-dependent value

W � La ; L1=2: ð3:11Þ

This happens for times t . ts, where ts is thesaturation time [37]

ts � Lz ; L3=2: ð3:12Þ

The set of parameters a, z and b ¼ a/z (whosemeaning will be explained in electronic supplementarymaterial, appendix B) are called the critical exponentsfor the KPZ equation in one space dimension. Theyare also said to characterize the dynamical scalingcorresponding to that equation. The dependence ofW(t) on the subsequent crossover times is sketchedin figure 4.

To conclude this paragraph, we remark that resultssimilar to those previously described continue to holdwhen, instead of equation (3.3), we consider initial datathat are close enough to a flat profile. For instance, thisis the case when we take h(x,0) in the form of a small oscil-lation with small amplitude, a situation appropriate as astarting point for stromatolite formation.

3.2. Interface patterns in different transientregions

We next provide some simulations to illustrate the mor-phologies that can be generated by means of KPZ-typeequations. Consider first the earliest stages of a processgoverned by equation (1.1) with h(x,0) being a small-amplitude sinusoidal function. As observed before,that transient is governed by an RD equation (3.4). Aplot of the unfolding profiles is given in figure 5.

Concerning our previous figure, some remarks are inorder. First, some memory of the initial condition is pre-served during the times considered, although the initialprofile is increasingly distorted owing to stochastic

J. R. Soc. Interface (2012)

effects. At the technical level, what is shown infigure 5 is a (explicit) realization of the stochasticprocess that can be shown to provide a solution to theproblem under consideration. Such realization can beshown to be discontinuous almost everywhere, althoughthe plots actually given above have been made continu-ous for ease of visualization. Namely, in the numericalcomputation, we have discretized the equation in space,and have solved it on a finite number of nodes. Then,we have plotted the values of the solution in the differentnodes, linking the values of first neighbours withstraight lines.

We next consider the full KPZ equation (1.1) withsame initial value as before, and simulate the solutionprofiles in each of the subsequent regimes (RD, EW andKPZ). Without loss of generality, we may set F ¼ 0, asthis amounts to replacing h(x,t) by (h(x,t) 2 Ft) there.The corresponding results are illustrated in figure 6.Figure 6a displays the evolution starting out from aninitial sinusoidal shape of small amplitude, whilefigure 6b shows snapshots of the dynamics corresponding

10

20

30

(a)

(b)

h(x)

0 100 200 300 400 500x

0

20

40

h(x)

Figure 6. (a) Simulation of the KPZ equation with sinusoidalinitial data corresponding to the RD transient (t ¼ 0.01,t ¼ 0.1), EW transient (t ¼ 1) and the full KPZ stage (t ¼100, 1000). Parameter values: l ¼ 3, g ¼ 1, D ¼ 1. (b) Simu-lation for a different initial sinusoidal data, whose amplitudehas been taken of the same order as that of the profile forthe full KPZ regime. Parameter values are as in (a): black,t ¼ 0; red, t ¼ 0.01; green, t ¼ 0.1; blue, t ¼ 1; brown, t ¼100; purple, t ¼ 1000.

–0.5

0

0.5

1.0

1.5(a)

(b)

h(x)

0 100 200 300 400 500x

0

0.5

1.0

h(x)

Figure 7. (a) Solution profiles for the deterministic KPZequation when l ¼ 10, n ¼ 1. (b) Profiles corresponding tothe same equation and same values of l and n in the casewhere h(x,0) is an uncorrelated white noise of amplitudeD ¼ 0.01 (black, t ¼ 0; pink, t ¼ 11.7).

8

6

4h

2

5 10x

15 20

Figure 8. Evolution of a sinusoidal profile under equation (20).Notice the existence of an initial transient period, where thesolution remains as smooth as its initial value (snapshots 1and 2) after which cusps begin to unfold at the localmaxima of the solution (snapshots 3 and 4). The profile willeventually converge to zero everywhere for a sufficientlylong time. The initial profile is h(x,0) ¼ cos(px/5), F ¼ 2and n ¼ 1/2. Snapshots correspond to times t ¼ 0, 1, 2, 3, 4.

Pattern formation in stromatolites R. Cuerno et al. 1057

to an initial sinusoidal shape with the same wavelength,but with a larger amplitude that is comparable withthe roughness of the stationary state. Comparing thetwo panels, we see that the long time morphologies(e.g. t ¼ 100, 1000 in the plots) are disordered andrough, independent of the initial condition, while thereare intermediate times (e.g. t ¼ 1) at which the period-icity of the initial condition may still exist if itsamplitude was large enough (figure 6b).

We now consider the case of the deterministicequation obtained when the term h(x,t) in equation(1.1) is dropped and the constant forcing term isrescaled upon the transformation h! (h 2 Ft).

@h@t¼ nr2 h þ l

2jrhj2: ð3:13Þ

We show in figure 7, how an initially sinusoidalprofile evolves as time passes, and consider also the situ-ation where the initial value itself is of a stochasticnature. Notice that in both cases considered infigure 7, the corresponding solution approaches towardsa nearly constant profile as time passes.

To conclude this paragraph, we next consider a par-ticular case of the deterministic KPZ equation (3.12),which has been used [33,34] to support the biotic

J. R. Soc. Interface (2012)

origin of a class of stromatolites. More precisely, weshall deal with the following equation

@h@t¼ njrhj2 þ F ; ð3:14Þ

where F . 0 is constant, and h(x,0) is a prescri-bed sinusoid. Figure 8 provides information on thebehaviour in time of the corresponding solutions.

1058 Pattern formation in stromatolites R. Cuerno et al.

It is interesting to compare figure 8 with figure 1 inthe study of Batchelor et al. [33]. In both cases, onemakes use of explicit formulae to represent the sol-utions. However, the authors in [33] are able to derivetheir plots through formulae (2.1) and (2.2) thereinbecause they consider initial values with a single maxi-mum. In our case, the initial value has several maxima,so that characteristics originating from them may inter-sect in time, which eventually results in loss ofregularity and cusp formation. Therefore, the coexis-tence of several adjacent paraboloid-like structures asthose represented in figure 2 in the study of Batcheloret al. [34] can be maintained only for sufficiently smalltimes (see electronic supplementary material, appendixB for further details).

3.3. Is Kardar–Parisi–Zhang a suitable model ofstromatolite growth?

Consider again the AH specimen in figure 3. One readilysees there a system of three undulations that runfrom bottom to top, the one on the left eventuallyundergoing a tip-splitting process. It is not a prioriclear whether a precise wavelength can be associatedwith such undulations, as their number is too small toperform a reliable statistical analysis. This could bedone, however, if larger samples displaying about 10–20 of such shapes were available. In any case, if sucha wavelength could be identified, this would excludethat band formation had occurred at times corres-ponding to the last asymptotic regime for the KPZequation. The reason is that the large-time behaviourof solutions to equation (1.1) is known to be incompati-ble with the persistence of a characteristic length [37],as illustrated in figure 6a. However, characteristiclengths can be preserved for comparatively long times,as seen in figure 6b. For instance, they are compatiblewith the first time regime (t , ts in equation (3.12)),which in practice may last for a long while. Note forinstance that, as observed before, ts . t2C, and thelatter quantity might be of the order of 10 Myr underassumptions on l and D as those discussed in ourprevious paragraph.

Another point to be noticed is that neither over-hangs (as that visible on the right of the firstundulation at the bottom left in figure 3) nor increas-ingly steep interface profiles (the rightmost system inthe same figure) are compatible with the asymptoticKPZ regime [37].

A final remark in this paragraph is concerned withthe effect of noise on the KPZ equation. This isknown to be crucial for non-zero patterns to evolvefrom an initially flat interface h(x,0) ¼ 0. Indeed, if wedrop the noise term h(x,t) in equation (1.1), then theonly solution in the whole space of the resulting equa-tion corresponding to an initial value h(x,0) ¼ 0 ish(x,t) ¼ 0. Moreover, in that case, solutions correspond-ing to data which are not too large (see electronicsupplementary material, appendix B for details) areknown to approach flat profiles as time goes to infinity.Therefore, the patterns depicted earlier [33,34] eitherdevelop from sufficiently large initial values, or willeventually be damped out as time passes.

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3.4. More on kinetic roughening: beyond theKardar–Parisi–Zhang equation

We have already illustrated some of the problems thatarise when using equation (1.1) to model stromatolitemorphologies. We should add in this respect that, tothis day, there are very few experiments on rough inter-faces where the scaling behaviour corresponding to theKPZ equation had been unambiguously observed [44].While the reasons for that discrepancy remain stillopen for discussion, partial explanations thereof canbe provided. Some of them are related to the mannerin which equation (1.1) is obtained. For instance, thegeneral arguments on symmetry assumptions to berecalled in electronic supplementary material, appendixA are fully meaningful at the largest time and spacescales only. For smaller scales, however, mechanismsthat are not compatible with the KPZ formalism (aris-ing for instance from specific microscopic details) mayplay a significant role. This may result in the onsetof a KPZ scaling being delayed, and even completelyhindered for practical purposes. To illustrate thispoint, a specific example is shortly discussed below.

Consider a growth problem in which particles diffusewithin a vapour phase until they reach a solid substrateonto which they attach through chemical reactions,forming an aggregate with a rough surface. If the chemi-cal reactions resulting in that attachment are not toofast, it has been shown that the aggregate interfaceevolves according to the noisy Kuramoto–Shivashinskyequation [45]

@h@t¼ �nr2h �Kr4h þ l

2jrhj2 þ h: ð3:15Þ

Here n, K and l are positive constants and h is as inequation (1.1).

As in the KPZ case, equation (3.15) is symmetricunder reflection of the space coordinates, x! 2x, whilenot having up-down symmetry (h!2h) either. Basedon standard arguments on universality [37], this mightlead to expect a similar dynamical behaviour as describedby both equations. However, the sign of the diffusionterm in equation (3.15) is the opposite to that in theKPZ equation (1.1). This introduces important differ-ences in the behaviour of solutions to both equations,particularly at small time scales. For instance, solutionsto equation (3.15) may develop a periodic pattern ofcells for times t . 0. This morphological instability iscertainly compatible with the presence of undulatorypatterns as those already discussed for the (AH) speci-men. In the case of equation (3.15), such a periodicpattern subsequently evolves in a chaotic manner toeventually yield, for a suitable parameter range, a fullKPZ scaling for sufficiently large times [45,46]. Thus,KPZ scaling can be viewed as an emergent property ofequation (3.15), although that equation itself is stronglydifferent from equation (1.1). Moreover, for some (finite)system sizes, and some parameter choices in equation(3.15), the asymptotic KPZ scaling may not be obser-vable at all in equation (3.15) [44]. Incidentally, thisseems to be the case in many experimental systemswhere diffusive instabilities seem to be a norm ratherthan an exception [44].

Pattern formation in stromatolites R. Cuerno et al. 1059

As a matter of fact, morphological instabilities maybe driven by mechanisms of nature other than diffusive,and noise is not required to sustain them. A simpleexample is provided by the nonlinear equation

@h@t¼ n

@2 h@x2 �

@2 h@y2 �

@2 h@x@y

� �2" #

�Kr4 h; ð3:16Þ

which has been recently studied in the context ofnon-equilibrium growth [47,48]. Within the theory ofsurface growth, the second, linear term on the r.h.s.of equation (3.16) is usually associated with surface dif-fusion: it models how newly deposited particles diffusealong the growing surface. The nonlinear term inequation (3.16) is, as we explain in the following, tightlyrelated to the surface Gaussian curvature (which is anintrinsic measure of the curvature of the surface).Furthermore, it is one of the simplest nonlinearitiescompatible with the symmetry requirements detailedin electronic supplementary material, appendix A1.

Figure 9 presents a one-dimensional cut along the x ¼y-axis of a two-dimensional numerical simulation ofequation (3.16). What this figure shows is mass displace-ment from saddle points to the top of the hills by means ofa nonlinear instability mechanism. A complementaryeffect, displacement of mass from saddle points to thebottom of the valleys, can be observed along the perpen-dicular axis. This sort of behaviour requires the action ofa nonlinearity. Indeed, a linear instability would promotethe growth of both holes and mounds. That phenomenol-ogy can be explained by means of a simple geometricargument as follows. The surface Gaussian curvature @is given by

@ ¼ @2h@x2 �

@2h@y2 �

@2h@x@y

� �2" #

� 1þ @h@x

� �2

þ @h@y

� �2" #�1

; ð3:17Þ

and in the small gradient limit (jr hj 1), @ reduces to

@ @2h@x2 �

@2h@y2 �

@2h@x@y

� �2" #

; ð3:18Þ

which is precisely the drift in equation (3.16). Thus, thedynamics of equation (3.16) favours the growth of pat-terns with positive Gaussian curvature (as is the case ofholes and mounds) with respect to those with negativeGaussian curvature (as saddles).

4. DISCUSSION

In this work, we have addressed some aspects of patternformation in stromatolites. In particular, we have beenconcerned with the question of whether the interfacesappearing on such structures can be modelled bymeans of the KPZ equation (1.1). This issue has beenraised as a tool to prove (or disprove) the biogenicityof such structures, and consequently on the way inwhich life was started in our planet.

We have focused into a particular aspect of the mod-elling question described before. More precisely, an

J. R. Soc. Interface (2012)

effort has been made to clarify the following points:(i) what does any term in the KPZ equation representsin terms of physical mechanisms involved? (ii) Whatconclusions can be drawn from comparison of the prop-erties of solutions to the KPZ equation with the profiles(particularly band interfaces) observed in actualstromatolite samples, and in particular in the AH speci-men depicted in figure 3. A pressing motivation toascertain (i) and (ii) is given by the fact that [32,33]different versions of the KPZ equation have been usedto sustain seemingly contradicting views, namely apossibly abiotic (in the first case) and biotic (in thesecond one) origin of stromatolites. We now summarizeour results, and put forward some research directionsthat can be pursued to understand pattern formationin such structures.

To begin with, we have observed that KPZ is a sto-chastic partial differential equation. This simple modelhas been largely used to describe the evolution of an inter-face separating a medium which advances into another.Three important aspects to retain such formulation arethat (i) KPZ is a continuous equation, therefore obtainedby means of an approximation process to a discretephenomenon, (ii) noise effects are crucial, and (iii) atany given time, the interface is not considered to be inequilibrium: by assumption, there is an external forcethat drives the growth problem.

Interestingly enough, and in spite of the generality ofthe arguments leading to KPZ derivation (some ofwhich are recalled in electronic supplementary material,appendix A), a first obstacle to be reckoned with is itsscarce experimental relevance. Indeed, very few pro-cesses have been clearly shown to be governed by theKPZ equation. This admittedly inconvenient aspecthas been linked to the very generality of the modellingassumptions leading to equation (1.1). For instance, ithas been suggested [44] that in any particular caseconsidered, the microscopic aspects of the underlyingdynamics could have a strong impact on the dynamicsof that process, and that those might have been ignored(at least in part) when deriving the continuous equation(1.1). We should bear in mind that any modellingequation as that of equation (1.1) is arrived at undera number of simplifying assumptions, some of whichmay be quite heavy (or utterly unacceptable) for ourcurrent problem.

While the previous remark has been specifically tai-lored to the KPZ equation, the following one is of ageneral nature. When using any particular equationfor modelling purposes, one has to pay particular atten-tion to the physical meaning of the various terms andcoefficients retained. These encode information aboutthe mechanisms that are assumed to be mainly, if notsolely, responsible for the process we intend to model.However, rarely (if ever) do such terms and coefficientsprovide information about the biotic or abiotic origin ofthe forces that set such terms in action.

Consider for instance, the first term on the r.h.s. inequation (1.1). As remarked in §2, it can be interpretedas a diffusion term, and the corresponding coefficienthas dimensions of a diffusion coefficient. One of themany possible ways in that such a term can be obtainedis represented by formula (2.4), which provides a link

2.0

1.5

1.0h

s

0.5

0.2 0.4 0.6 0.8 1.0

Figure 9. One-dimensional cut of the two-dimensionalnumerical solution of equation (3.16) shown along thex ¼ y-axis at time t ¼ 3.05 � 1024 (red line). The variables ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p=ffiffiffi2p

parametrizes this axis. The values of theparameters are l ¼ 1 and n ¼ 0.1. We have assumed periodicboundary conditions and the symmetric initial condition h0 ¼

sin(2px)sin(2py) (green line). One can clearly see how themass is displaced from the saddle points to the top of thehills as a consequence of the nonlinear instability mechanism.

1060 Pattern formation in stromatolites R. Cuerno et al.

between diffusivity and two material properties of themoving interface (interfacial tension and effective mobi-lity). However, there is no way to clarify at that stagewhether such physical mechanisms are the result of abiotic or abiotic process. What is more, no clear linkbetween the underlying microscopic mechanisms andthe averaged, effective coefficients appearing inequation (2.4) has yet been established. Clearly,equation (2.4) provides some insight into the natureof the laws at work at the interface. The knowledgethus obtained is, however, too scarce to draw any con-clusion, not only on the biogenicity question, but alsoon the nature of the chemistry leading to interface for-mation. For this reason, we are left in almost completeignorance of the physical forces that shaped the inter-face and determined its motion. To gain insight aboutsuch issues, additional information is needed. Forinstance, a direction that we consider as promising isthat of deriving the effective coefficients appearing inequation (2.4) from the actual grain properties(volume fraction, composition, mechanisms of for-mation, and so on) of the structures that are presentin an actual interface (that is, in the boundary of aband) of a living young stromatolite sample. Thisrequires going beyond the representation of the inter-face as a comparatively thin line, to take instead intoaccount its internal structure. Any significant result inthat direction will be instrumental to improve ourunderstanding on the way these bands were formed.

Concerning the remaining terms in equation (1.1),the second one in its r.h.s. has a particularly precisemeaning. It postulates that interface velocity is nearlyconstant, a fact that might be perhaps compared withgeological estimates (assumed available) on band for-mation. Finally, the presence of noise is crucial toproduce non-trivial patterns out of nearly flat initialprofiles. This makes a strong case for stochasticequations versus deterministic ones as models todescribe stratified structures arising out of gentle sandripples. The latter is widely assumed to be a starting

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point for currently forming stromatolites in shallowwater environments.

A point to be stressed is that the KPZ equation canbe shown to be incompatible with some particular inter-face morphologies. We have remarked on this issue in§3. In particular, we have noticed that overhangs andsteep profiles are incompatible with solutions to KPZ.This is not a reason to exclude altogether KPZ fromthat scenario, but certainly hints at least at the pres-ence of additional mechanisms at work. We have alsonoticed that, in our view, wave-like structures as thatapparent in the AH specimen deserve further study.Indeed, if such undulations were common (a fact thatthe size of AH sample does not allow to conclude) thecorresponding wavelength could possibly be identified.In that case, KPZ should be discarded as a guide todescribe band formation in comparatively long times.Other equations, which take into account different driv-ing mechanisms should then be considered, a fact thathas been briefly discussed in §3.4.

A highly relevant question is often ignored at themodelling stage: it is widely assumed that, out of amanifold variety of patterns that a given equation cansustain, only a few of them will be preserved in thelong run. In some cases, such robust, observablepatterns unfold very quickly. However, for a givenequation, different time regimes may occur, separatedby suitable crossover times, and the behaviour of sol-utions might differ sharply from one region toanother. We have addressed that issue in §3.1 wherewe have recalled that any of such time regions may bequite long, and that a given pattern might have beenformed, and then stabilized by some additional mechan-isms changing the sedimentary and/or precipitationconditions, such as stronger sedimentation owing tolocal storms or transgressive conditions, changes of pHowing to hydrothermal activity, or fast evaporationinducing evaporitic precipitation. In this way, theobserved pattern may look incompatible with the be-haviour which eventually sets in for very long times.The conclusion that emerges is that a given process(for instance, band formation in stromatolites) mighthave occurred according to a, say, KPZ mechanism,and still display morphologies that are incompatiblewith the behaviour of KPZ solutions in the long run(as time goes to infinity). This in turn raises anotherresearch direction: identifying the crossover times of agiven equation in terms of measurable quantities. Inour case, a possible strategy could be to relate suchcrossover times to the grain properties mentionedbefore. Incidentally, a remark made earlier [32] (seefourth line after formula (2.2) therein) may be relatedto the consideration of any of such crossover times,although the precise form of their assumption maynot agree with estimates (3.7), (3.9) and (3.11) in §3.

Summing up, we believe that linking interface evol-ution with internal interface structure, searching forwave-like regularities in larger samples than thoseexamined so far, and estimating the actual size of cross-over times (as well as relating such times with materialproperties of stromatolites) are particular issues whosestudy might be helpful to obtain suitable interfacelamina formation models for stromatolite samples.

Pattern formation in stromatolites R. Cuerno et al. 1061

Gaining insight into such issues, as well as on the gen-eral question of lithification and on how accretionproceeds [17], will improve our current knowledge ofpattern formation in such geological structures.

We thank Martin van Kranendok and Malcolm Walter fortheir critical reading of the manuscript. We would also liketo acknowledge the two anonymous referees for theirdetailed and insightful remarks on our original manuscript.This work has been supported in part by MICINN grantnos FIS2009-12964-C05-01 (R.C.), CGL2010-12099-E(J.M.G.R.) and MTM2008-01867 (M.A.H.).

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