A biofilm and organomineralisation model for the growth and limiting size of ooids

Ooids are typically spherical sediment grains characterised by concentric layers encapsulating a core. There is no universally accepted explanation for ooid genesis, though factors such as agitation, abiotic and/or microbial mineralisation and size limitation have been variously invoked. Here we examine the possible influence of microbial organomineralisation on the formation of some naturally occurring ooids. We develop a mathematical model for ooid growth, inspired by work on avascular brain tumours, that assumes mineralisation in a biofilm to form a central core which then nucleates the progressive growth of concentric laminations. The model predicts a limiting size with the sequential width variation of growth rings comparing favourably with those observed in experimentally grown ooids generated from biomicrospheres. In reality, this model pattern may be complicated during growth by syngenetic aggrading neomorphism of the unstable mineral phase, followed by diagenetic recrystallisation that further complicates the structure. Our model provides a potential key to understanding the genetic archive preserved in the internal structures of some ooids.


Results
Mathematical model. We assume that the biofilm contains microbial organisms that require nutrients to survive and to multiply 11 . If the nutrient level falls below a critical concentration then the microbes cannot grow. It is assumed that the nutrients come from the surrounding media and diffuse into the biofilm, similar to the diffusion of nutrients in the tumour model of Greenspan 16 . In Greenspan's model for avascular tumour growth, nutrients diffuse in towards a growing tumour and are taken up by the tumour at a constant rate per unit time.
The key predictions that have been made in the mathematical analysis of this model are: (i) The tumor will develop a necrotic core. This occurs after the tumour reaches a critical size, when the nutrients are taken up in the outer parts of the tumour before they have time to diffuse into the centre of the tumour. The morphology of the tumour is then a solid necrotic core surrounded by an outer layer that is still receiving nutrients. (ii) The size of the necrotic core increases until, after long times, the width of the outer layer, and the overall size of the tumour reach constant limiting sizes. The width of the outer layer depends on the background nutrient level, the rate of take up of nutrients by the tumour, and the threshold nutrient level required by the tumour not to necrotise.
In our model we hypothesise that the overall growth of ooids is similar to that of avascular tumours, where growth occurs in a nutrient limited environment. We suppose that ooids form in a biofilm whose growth is dependent on the take up of diffusing nutrients at a constant rate per unit time. Following the analysis of Greenspan, under this hypothesis, the biofilm will be characterised by two regions: An outer layer that is supplied by diffusing nutrients, and an inner zone that nutrients cannot diffuse into before they are taken up by the biofilm in the outer layer. Furthermore, similar to Greenspan's model for avascular tumour growth, the overall size of the biofilm, and the width of the active region that is supplied by nutrients, will approach constant limits.
For simplification we assume that the growth is radially symmetric. The growth with these model assumptions is characterised by a threshold size beyond which there will be two distinct regions, an outer region in which the biotic organisms grow and an inner region in which they do not. In the inner region we will suppose that the microbes will either decay or become overwhelmed by mineralisation. In the inner region there will therefore be an increase in volume due to mineralisation and a loss of volume due to decay. We assume these are two competing processes. In the outer region there will be an increase in volume due to microbial colonisation.
denote the volume of biotic organism in inner region I, and let V t ( ) b II denote the volume of biotic organism in outer region II. We assume growth rates as follows: (i) the rate of growth of the mineralisation volume is proportional to the volume of biotic organisms in the inner zone, (ii) the rate of decay of biotic volume is proportional to the volume of microbes in the inner zone, (iii) the rate of growth of biotic volume in the outer region is proportional to the volume of microbes in the outer zone, The assumption underlying equation (1) is that the organomineralisation occurs in the inner region and it is dependent on the microbial organisms in that region. The simplest dependence is that it is proportional to the amount of microbial organisms in that region. Note that equation (2) for the breakdown of microbial organisms also assumes a proportional dependence on the amount of microbial organisms in the region. This might be expected to be a reasonable assumption for microbes that no longer have access to nutrients. The constants of proportionality in equations (1) and (2) are taken to be different. The assumption underlying equation (3) is that there is a constant per capita growth rate of microbes if they have sufficient nutrients. In addition to the above we assume that microbes from region II will transition into region I at a rate k b proportional to the volume of microbes in region II. The model equations are then given by Finally we assume that the volume of the outer region is proportional to the volume of microbes in the outer zone, II . The volume of the inner region is the volume of mineralisation plus a volume that is proportional to the volume of microbes in the inner zone, We now consider predictions from the model equations (4)- (7), under the overriding hypothesis that the overall growth is limited, similar to the Greenspan model for avascular tumours. The meanings of variables and parameters in our model equations are summarised in Table 1.
The ooid will reach a limiting size if = 0 dV dt . From the above we have For the limiting size we note, as in the avascular tumour model 16 , that The constant width depends on the threshold nutrient concentration required by microbes to survive, c I , the background nutrient concentration in the outer region, c II , the diffusivity of the nutrients, D, and the rate of take up of nutrients by microbes k. In the case of spherical growth this is given by 16 The limiting size thus follows from the equation It is constructive to write this equation as This equation is of the form −AR 3 + BR 2 + C = 0 with exactly one sign change in the coefficients, thus by Descartes rule of signs there is exactly one positive root R. It is possible to write down an explicit algebraic solution for R but we can get a better physical understanding of the solution by considering upper and lower bounds, and scaling with w. First we note that the limiting radius R increases with increasing V m . We can thus obtain a lower bound for R by setting V m = 0, then , so that an upper bound for R can be found by setting which shows that the limiting size, R max , grows linearly with w. From equation (11) we can deduce the sensitivity of the limiting size to the diffusivity, ∼ R D max 1/2 , and to the nutrient consumption rate, ∼ − R k max 1/2 . A faster diffusivity will result in larger ooids and a faster nutrient consumption rate will result in smaller ooids.
Ooids typically contain concentric layers, or laminations, around a nucleus. The laminations may be characteristic of seasonal growth cycles, if there was seasonal variability in the nutrient concentration levels, or longer term environmental events, depending on the age of the laminations. Data from modern ooids 17 suggests that the age of ooids scales linearly with mass. Assuming constant density then age scales linearly with volume or R 3 .  This means that the radius R scales with time as t 1/3 . Of course this scaling cannot continue at t 1/3 or ooids would become arbitrarily large.
The different stages of growth as described by the model ooid are depicted in Fig. 2.
Spacing of laminations and comparison with experimentally grown ooids. Our mathematical model for nutrient limited ooid growth can be used to provide information on the spacing in laminations, by sampling the solution of the mineralisation process at constant time intervals. The radially symmetric assumption results in smooth laminations. In order to study the roughness of laminations, a different type of model, such as involving the radial version of the Kardar-Parisi-Zhang equation 18 , could be employed. One test of our model would be whether the spacing between laminations, in a controlled environment, match the spacings predicted by the model. Here we suppose that the mineralisation is being produced by a non-replenishing source of necrotising microbes in region I. In this case k b = 0 and the model equations reduce to (1)- (3). Suppose that R is the limiting radius before the onset of mineralisation and this radius is reached at time τ. We can integrate equation (2) Substituting this result into (1) and integrating from τ to t, with V m (τ) = 0 gives and thus the radius of the mineralisation front is given by for the radius of the mineralisation front. The result (20) can also be obtained in a different way. In order for the overall size to remain constant we need to have the microbes in the outer proliferating layer replacing those taken up by mineralisation or decay in the inner layer. This effect can be included directly by writing where V is the fixed volume equal to that at time τ. This is saying the volume of microbes in the inner layer is being reduced as the mineralisation takes up more volume. We would then have the mineralisation rate equation, with the rate proportional to the volume of microbes, Solving this equation gives precisely the same dimensionless result (20). Initially the growth rate is seen from equation (20) to scale as t 1/3 , which is the characteristic growth rate in the mathematical theory of Ostwald ripening 19,20 . As already remarked above, such a growth rate is unsustainable, as ooids are observed to be size-limited. The result of sampling equation (20) at constant time intervals, over long times, is shown in Fig. 3b. The characteristic features are a large central region surrounded by concentric laminations whose spacing decreases over time, resulting in a size-limited growth.
Departures between the width and spacing of laminations of actual ooids and the laminations predicted by our model, in controlled conditions, could provide evidence for different environmental conditions, with different   Figure 4 shows an indicative fit of the laminations from periodic time sampling to the laminations on an experimentally grown ooid in a controlled environment 14 . The fit to the laminations is remarkable given that the only parameters to fit are the overall magnification of the pattern, and the constant time sampling interval.
Laminations with replenishment of microbes. In the model laminations described above, the microbes in region I are dying out, without replenishment, and mineralisation is occurring in this region. We now consider the more general case with replenishment as biotic material from region II transitions to region I. This is described by the model equations (4)-(6) with k b > 0. A more general version of the model taking into account that region I expands as the biofilm expands is defined by these equations. It is possible to solve this more general model but for simplification it suffices to consider the special case when When α = 0 this recovers the previous result (20) for R m (t). The temporal scaling behaviour ∼ R t t ( ) m 1/3 for ⪆ t 1 holds for all α. Moreover, the pattern of laminations does not differ significantly when α ≠ 0.
Diagenetic modification of the ideal ooid structure. It has been clearly shown that ooids are very susceptible to authigenic and diagenetic change (see, e.g., Fig. 1.2 in ref. 21 ). In reality the ideal ooid structure may be complicated by syngenetic aggrading neomorphism of the mineral phase 6 with Ostwald ripening 22,23 being the likely driver of this recrystallisation (Figs 1d and 3a). Organic matter and other impurities are rejected by the growing crystallites and form a boundary layer ahead of the growth front 24 . Davies et al. 25 have suggested a modified version of Sorby's theory in which the evolving ooid alternates between 'suspension' and 'resting' growth phases. We suggest instead that ooids are created by the interaction of the mineralising results of biofilm accretion, superimposed by the effects of syngenetic mineral growth. Here we illustrate an example of how the original ooid fabric may be modified by diagenetic changes. Kalkowsky 26 described details of several diagenetic structures that have modified the original concentric ooid. Figure 5 shows ooids of approximately 0.5 to 1 cm in diameter from the Triassic Rogenstein of the Heeseberg Quarry 27,28 . These were originally spherically concentric ooids, as described by our model. The apparent branching is probably caused by the effects of syngenetic and diagenetic mineralisation superimposed on the primary concentric structures. From the observed patterns it is tempting to suggest that the 'growth' of the branches was controlled from the outset by the radial version 18 of the KPZ equation 29 . Branching can occur when anisotropic effects (e.g., noise or amplification of a small bump through an instability mechanism) compete with surface tension. In fact a model has been developed for conical stromatolites in which diffusive gradients, extending over the thickness of an overlying microbial mat, causes mineral precipitation to be faster in regions of high curvature 30 . The mathematical model is similar to that for growth of stalactites 31 and icicles 32 .
If the effects of syngenetic mineral growth dominates the mineralising results of biofilm accretion, for example as nutrient availability becomes limited and surface tension is reduced, or during later diagenetic recrystallisation, structural complexity is superimposed on the pattern predicted by our model (Fig. 5), ultimately creating ooids exhibiting the spindelstruktur and kegelstruktur described by Kalkowsky 26 (tafel iv, Figs 2 and 3 therein) with surface protruberences giving rise to their description as cerebroid ooids 33 . We suggest that spindelstruktur and kegelstruktur provide evidence for the existence of competing processes of "Greenspan" biofilm accretion and "Ostwald" mineralisation operating simultaneously in ooid genesis. For ooids we assume radial symmetry when the influence of biofilm accretion dominates. A diagnostic characteristic of an ooid is that any irregularities in the nucleus are damped progressively in successive layers within the cortex until a spherical form is established. This observed behaviour can be explained by the fact that radially symmetric growth will occur when the surface tension is dominant in the growth process. In simulations involving the radial KPZ equation, irregular initial shapes grow to be either circular or spherical, depending on the dimensionality 18 . When surface tension dominates in the KPZ equation there is only diffusion. Mineralisation initially takes place within these concentric layers, but diffusion limited mineral growth gradually develops the radial spindelstruktur (Fig. 5a) and increasing mineral growth in the areas between will eventually lead to the development of kegelstruktur (Fig. 5b) forming outward projecting bumps that will grow faster through diffusion limited growth.

Discussion
Despite extensive research over more than a century there is still a lack of conclusive information on the genesis of ooids 4,34 . One limitation is that, although highly sophisticated geobiological analyses are now being made, they are undertaken on samples collected using 19th century techniques (e.g., refs 35,36 ). As pointed out by Fabricius 34 , it would be a mistake to assume that a similar process has formed all concentrically laminated grains. Clearly the structure of the typical ooid does not reflect successive surface accretion of carbonate mud on the surface of a rolling grain, as suggested by Sorby, but rather provides a detailed archive of organo-sedimentary concentric accumulation. Informed analysis of this archive may elucidate the detailed history of the growth of the ooids 12 . Such analysis is necessarily complicated by the task of discriminating whether growth corresponds to the model outlined in this paper, whether the original fabric has been overprinted by subsequent mineralisation, or even whether an entirely different process is responsible for the concentric structure (e.g., Fig. 6 in ref. 37 ). Although these discussions generally concentrate on carbonate ooids, our model will equally apply to ooids mineralised by chamosite, stevensite, phosphate and possibly some other mineral phases.
In the mathematical model developed here, for what could be described as the 'ideal ooid' , it has not been necessary to consider the particular biochemical processes involved in microbial organomineralisation. In simply assuming that there are such processes, this aspect of our approach could be described as a 'black box' . Rather the focus of our model is on capturing the essential features of ooid growth, which include the variation of growth rings observed in the experimentally grown ooids generated from biomicrospheres 14 , as demonstrated in Fig. 4. Another consequence of the model is that the resulting ooids grow to a limiting maximum size. Traditionally ooids have been arbitrarily regarded as having a diameter of 2 mm or less (Fig. 1a,b), but this was never specified originally 4,26 . It is clear that ooids can attain much larger sizes (Fig. 1). Such "giant" ooids have been explained in terms of Sorby's model by formation under conditions of high current velocity 10 , but Fig. 1c clearly shows an assemblage of poorly sorted "giant" ooids that clearly do not support the evidence of accumulation by a high velocity current. From the perspective of the model presented in this paper the only difference between "giant" ooids and conventional ooids is that the former grow under more favourable conditions reflected in the model parameters (e.g., availability of biomass and nutrients). The occurrence of large accumulations of well sorted ooids in units showing current-generated depositional structures (Fig. 1a,b) raises the question as to whether these ooids have been reworked and concentrated together, as Brückmann 4 suggested in 1721 when he wrote "a global wind (Aeolus macrocosmicus), ruling during the flood and stirring the waters, drove the eggs flowing in the waters of the flood to (concentrate in) certain places". Evidence for this reworking could include abraided, polished grain surfaces (Fig. 1a) and evidence of microbial boring within the ooids 17 . Our model thus provides a potential key to understanding the genetic information preserved in the internal structures of some naturally occurring ooids that might reflect environmental conditions in complete contrast to those operating at the site of final deposition, for example, the ooids of Kalij el-Arab 38 .