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Paleobiology, 33(3), 2007, pp. 382–396

Growth modes of 2-Ga microfossils

Steven Bennett, David Boal, and Hanna Ruotsalainen

Abstract.—By digitally imaging colonies with more than a hundred cells, the distributions of cellsize and shape are determined for four examples of 2-Ga microfossils: bacillus-shaped Eosynecho-coccus moorei and three dyads or diplococci (Sphaerophycus parvum and two forms of Eoentophysalisbelcherensis). By assuming that each colony obeys steady-state growth, the measured distributionscan be inverted to infer the time evolution of the individual cell shape. The time evolution can alsobe predicted analytically from rate-based models of cell growth, permitting the data to distinguishamong different postulates for the physical principles governing growth. The cell cycles are foundto be best described by the exponential growth of cell volume, although linear volume growth isnot ruled out. However, the measured dyad cycles are inconsistent with several growth modelsbased on surface area or the behavior of the septum at the division plane. Where they have beenmeasured, modern bacilli obey exponential growth whereas eukaryotics obey linear growth, whichimplies that these 2-Ga microfossils are likely prokaryotic.

Steven Bennett, David Boal,* and Hanna Ruotsalainen. Department of Physics, Simon Fraser University,Burnaby, British Columbia V5A 1S6, Canada. E-mail: boal@sfu.ca

*Corresponding author

Accepted: 6 April 2007

Introduction

The record of the emergence of life on Earthincludes isotopic ratios typical of biologicalprocesses (Schidlowski 1988; Mojzsis et al.1996; Rosing 1999; Shen et al. 2001; for a recentcritique, see van Zuilen et al. 2002) and fossil-ized cell colonies and other remnants, ofwhich some have been dated at 3.5–3.0 Ga(Knoll and Barghoorn 1977; Walsh and Lowe1985; Schopf and Packer 1987; Schopf 1993;Rasmussen 2000; Furnes et al. 2004) and oth-ers at 2.5–2.0 Ga (Barghoorn and Tyler 1965;Hofmann 1976; Knoll and Barghoorn 1976; Ti-mofeev 1979). Individual microfossils displayshapes such as ellipsoids, dyads (diplococci,or pairs of joined spheres), rodlike bacilli, andfilamentous tubes and rods; collectively, theymay be found in loose colonies or dense mats.Lost during the fossilization process, their in-ternal architecture is largely unknown com-pared to the design and construction of mod-ern cells (Boal 2002). What, then, can belearned about their mechanical features or cellcycle when so few of the techniques for prob-ing modern cells are applicable?

Many taxa of microfossils have been imagedphotographically to permit the determinationof ensemble-averaged quantities such as themean cell width or eccentricity. Valuable as

these averaged quantities are for characteriza-tion purposes, the underlying probability dis-tributions of cell sizes and shapes are substan-tially more informative because they may beused to infer the cell cycle of the measuredpopulation as described below. At least threeconditions need to be satisfied for such ananalysis to succeed:

1. the shapes must be accurately determined;2. for conventional microscopy, the random

orientation of cells must be taken into ac-count;

3. the cell colonies must be sufficiently largeas to obey steady-state growth, such thatthe observed cells provide a statistically ap-propriate sample of the cell cycle.

Digital imaging with a CCD camera lends it-self to the application of numerical algorithmsthat resolve item (1), as will be described be-low. The formalism needed to accommodateitem (2) is straightforward and described inthe Appendix. As for item (3), there are sev-eral examples of microfossils in the 2–2.5 Gaage range where colonies of cells have notbeen subject to large-scale shear stress andcan be imaged without excessive background.In summary, items 1–3 can be addressed, per-mitting the study of microfossil shape to help

383GROWTH MODES OF MICROFOSSILS

FIGURE 1. Examples of Belcher Island microfossils.Clockwise from the upper left: bacillus-like E. mooreiand dyads EB, S. parvum, and E. belcherensis capsulata.Scale bar, 5 microns.

identify relationships between the architec-ture and cell cycle of microfossil taxa and theirmodern analogues.

A theoretical framework is needed to inter-pret the inferred cell cycle and expose thephysical and biological principles that drive it.One starting point for the mathematical de-scription of a cell cycle would be a set of rateequations governing the production of molec-ular building blocks such as lipids and pro-teins, and their incorporation into the cell’smechanical components such as the plasmamembrane or cell wall. In this paper, we de-scribe these manufacturing and assembly pro-cesses through a set of empirical rate equa-tions (for example, various postulates for thegrowth rates of the cell area or volume), al-though we cannot directly establish the actualmagnitudes of the rate constants appearing inthe differential equations. Each rate equation(such as dA/dt � An, where A is the surfacearea of the cell and n is a tunable exponent)leads to a prediction for the time-dependenceof a geometrical attribute such as the cell sizeor shape. In other words, although this theo-retical approach cannot predict the absolutetime scale for the cell cycle, it can predict theprobability distributions for measurable attri-butes, allowing experiment to select amongvarious models and principles for cell growth.

Not all geometrical quantities are useful de-scriptors of the cell cycle. Much of what isknown of the cycle of modern bacteria comesfrom studies of uniform cylindrical cells,whose shapes provide only limited informa-tion about the cell cycle because their length,surface area and volume are all roughly line-arly proportional. That is, a cylindrical shapecannot discriminate easily among models inwhich the cell cycle is driven by one of celllength, area, or volume, because all threequantities have similar time dependence.However, this is not true of dyads, whose vol-ume is not linearly proportional to cell area orlength. Fortunately, both dyadic and rodlikecell shapes are common among microfossils,providing a good laboratory for investigatingthe evolution of the cell cycle over several bil-lion years. What is known so far from moderncells is that cylindrical bacteria are observedto extend exponentially with time along their

symmetry axis (reviewed in Bramhill 1997);examples include Bacillus cereus (Collins andRichmond 1962), Escherichia coli (Koppes andNanninga 1980), and Bacillus subtilis (Sharpeet al. 1998). In contrast, the mass of moderneukaryotic cells increases linearly with time(Killander and Zetterberg 1965). The reader isreferred to Murray and Hunt (1993) or Hall etal. (2004) for broader treatments of the cellgrowth process.

In this paper, we measure the shapes of fourexamples of 2-Ga microfossils collected fromCanada’s Belcher Islands (Hofmann 1976): ba-cillus-like Eosynechococcus moorei Hofmann,1976, and three dyads, Sphaerophycus parvumSchopf 1968, Eoentophysalis belcherensis capsu-lata Hofmann, 1976, and a cell type that we la-bel EB (not cataloged in the collection but re-sembling Eoentophysalis belcherensis punctataHofmann, 1976, which is a degradational var-iant of Eoentophysalis belcherensis). The cellshapes in this collection appear to have suf-fered little deformation by geological eventsfollowing fossilization; although the cells mayhave suffered plasmolysis and degradation,we assume that this has affected all cells in asimilar way. Figure 1 shows examples of eachcell colony, where the individual cell dimen-sions are most commonly in the 2–9 � range.These cells grow primarily along their sym-metry axis, which appears to be randomly ori-

384 STEVEN BENNETT ET AL.

ented in the colonies studied. In some cases,the cell interiors contain dark regions whichmight be adventurously interpreted as rem-nants of nuclei but are almost certainly degra-dational features (Golubic and Hofmann1976); with an age of about 2 Ga, these cellsoriginate from the epoch when eukaryoteswere thought to increase in abundance alongwith the rise in atmospheric oxygen. We willshow that the cell cycle inferred from the dis-tribution of cell shapes most closely resemblesthe growth of modern prokaryotes.

Containing results from several experimen-tal, analytical, and numerical techniques, thispaper is organized as follows:

1. Following an introduction to the source ofthe microfossil colonies, the image analysisalgorithm is described at length because ofits importance for obtaining accurate shapedistributions.

2. With this imaging technique, the cell shapecharacteristics are reported for the four celltypes of interest; all taxa exhibit relativelyconstant surface curvature during cellgrowth.

3. Five different growth-rate models for thecell cycle are introduced and solved ana-lytically. The method for angle-averagingthe cell orientations to permit comparisonwith experiment is straightforward, al-though it must be done numerically insome situations. We argue that dyadic taxaare best suited for distinguishing amongvarious models for cell growth.

4. We show that the cycles of individual cellsin all taxa are consistent with exponentialvolume growth, meaning that the rate atwhich the cell volume increases is propor-tional to the cell’s volume (that is, dV/dt �V). Most other growth models are clearlyinconsistent with the observed distribu-tions of cell shape, although linear volumegrowth is not ruled out.

5. Last, we illustrate how the data can beprobed further by means of a computersimulation of the cell cycle that could in-corporate a two-step growth process fordyads as well as some aspects of experi-mental uncertainties, etc. As it stands,more accurate data are needed to make

quantitative use of the simulation ap-proach.

Although the focus of this paper is the elu-cidation of the microfossil cell cycle, neverthe-less the techniques described are generic andapplicable to any population of cells obeyingsteady-state growth.

Material

The microfossils analyzed here are fromcherts collected in the Belcher Islands of Hud-son Bay by Hans Hofmann (1976) and are cu-rated at the Geological Survey of Canada(GSC). First proposed to contain the fossilizedremains of ancient cells by Moore (1918), theformations are roughly two billion years old:the overlying strata are 1.760 � 0.037 Ga (Fry-er 1972; recalculated) and the underlyingbasement is older than 2.5 Ga. The samples aredrawn mainly from the McLeary and Kase-galik Formations at several different strati-graphic levels, as described in more detail byHofmann (1976). Found in black chert in stro-matolitic dolostones, the sedimentary struc-tures indicate deposition in supratidal, inter-tidal and subtidal environments undergoinggradual subsidence. Modern analogues are in-tertidal and subtidal algal mats and mounds(Golubic and Hofmann 1976). Hofmann (1975)argues that these stromatolites are formed bypermineralization of algal mats by amorphousor gelatinous silica and carbonate, followed bycrystallization and recrystallization. In the 18thin sections of the collection, Hofmann (1976)has identified and cataloged examples of 24cell taxa, some of which may be degradationalproducts of others.

For the purposes of this analysis, we havescanned the collection manually for looselyassociated colonies of undistorted cells; largegrowth fronts of more than 104 cells were notincluded in our analysis, in part because thecells are distorted by contact with their neigh-bors. The coordinates of each colony present-ed here, and the number n of cells or cell pairsimaged from it, are as follows: Eosynechococcusmoorei (n � 224; GSC42770 at 26.6x, 14.5y);Sphaerophycus parvum (n � 99; GSC42773 at17.9x, 10.0y); Eoentophysalis belcherensis capsu-lata (n � 156; GSC42773 at 17.1x, 8.3y); and an

385GROWTH MODES OF MICROFOSSILS

uncataloged colony which we will refer to asEB, similar to Eoentophysalis belcherensis punc-tata (n � 184; GSC42769 at 38.4x, 20.6y). Thefive-digit number is the GSC identification ofthe microscope slide, and the xy coordinatesare displacements in millimeters from its up-per right-hand corner. To gauge the variationof cell shape with local environment, we im-aged additional colonies of E. moorei from dif-ferent locations: (n � 271; GSC42769 at 32.4x,7.3y) and (n � 76; GSC43589 at 26.2x, 9.9y).Where the cells in a colony display an axis ofcylindrical symmetry, their orientation ap-pears to be random; that is, they are not ar-ranged radially with respect to the center ofthe cell colony. The taxa examined appearcommonly in the Belcher Island samples (Hof-mann 1976) and should not be regarded as un-usual or exotic; our particular choice of colonywas motivated only by numerical size andclarity of images.

Methods

Our determination of cell shape distribu-tions requires an image analysis method thatyields accurate and numerically stable mea-surements of cell dimensions. Each cell colonyis examined microscopically (Olympus BX51)with oil immersion objectives of 50� or 100�magnification at 1 �m steps in the z-direction,perpendicular to the focal plane. As the colo-nies may be hundreds of microns across, theyare subdivided into 50–100 �m regions for im-age capture with a CoolSnap cf � CCD cam-era (Roper Scientific) having pixels of 4.65 �mto the side. From these regional domains, in-dividual cell images are extracted, having di-mensions of 60–250 pixels in a given direction.To reduce their interference with algorithmsfor determining the cell boundary, segmentsof nearby cells appearing in an image are re-moved manually using Adobe Photoshop�image manipulation software. Obviously in-complete cells and those near the boundary ofthe thin section are discarded to reduce sam-ple bias (Smith 1968 outlines the problem ofdetermining the mean dimensions of 200–400�m diameter grains embedded in thin sec-tions). Grayscale values of the Cartesian CCDimage are then translated, by using a MonteCarlo sampling approach, into a polar coor-

dinate representation with origin at the cell’svisual center of mass. Commonly, a cell ap-pears as a dark ring with a light interior, or adark disk on a light background; in either case,the somewhat diffuse outer boundary is about5–8 pixels thick, across which the grayscalevalue decreases away from the cell center(here, black and white have grayscale valuesof 255 and 0, respectively).

A numerical algorithm provides a system-atic quantification of cell shape and reducesthe likelihood of subjective biases from the ob-server. Working in the polar coordinate rep-resentation, the numerical derivative di� of aseries of grayscale density values gi� is deter-mined simply from di� � (gi�1,� gi1,�)/2,where i is a radial pixel index at fixed polarangle � (measured with respect to the verticalaxis of the image). Searching from the coor-dinate origin, the location of the radial pixel mwith the most negative derivative dm� is deter-mined for each �, corresponding to the outeredge of the cell. From m, the radial location ofthe boundary r� at angle � is constructed froma weighted average of derivative pixelsaround m, namely r� � j rj dj�/j dj� , wherethe radial index j satisfies m 2 � j � m � 2and only negative values of dj� are included inthe sum. Consistent with the dimensions ofthe boundary mentioned above, these five pix-els cover most of the diffuse outer edge of thecell image. Lastly, the suite of values {r�} foreach cell image is fitted by a Legendre poly-nomial expansion R(�) � l cl Pl (cos �), wherethe polynomial index satisfies 0 � l � 4 andwhere cl are coefficients with the dimension oflength. The angle � is defined with respect toone of the cell’s symmetry axes (almost alwaysprolate) obtained by diagonalizing the ‘‘iner-tia tensor’’ of the cell boundary generatedfrom {r�}. That is, the representation R(�) is acontinuous function of angle �, which permitsthe cell’s shape to be determined more accu-rately than the discrete set of points {r�}.

The main sources of error in the algorithminclude (1) small irregularities in the cellboundary on the submicron length scale aris-ing from the fossilization process, and (2)slight smearing of the boundary by the CCDdiscretization and conversion to the polar rep-resentation. The extracted boundary typically

386 STEVEN BENNETT ET AL.

FIGURE 2. Definition of projected lengths L⊥ and L� usedin the shape analysis of a dyad. The top view is the ap-pearance of an overlapping cell pair in a dyad as seenthrough a microscope. The side view is the same pair asseen in the plane of the thin section, perpendicular tothe cylindrical symmetry axis of the dyad. The separa-tion between centers of the spherical caps is s, while �is the angle between the symmetry axis and the direc-tion of observation. The side view shows that the pro-jected length L� is equal to 2R � s sin �; the projectedwidth is L⊥ � 2R, independent of �. The thick line in-dicates the junction of the two caps; referred to as theneck, this junction region is circular with a radius r.

drifts by �1% when the size of the polar binsis varied, although changes up to �2% are oc-casionally observed. A change in the coordi-nate bin size tends to shift the sizes of all cellsin a colony by a similar scale factor, so that therelative distribution of sizes is less sensitive tothis uncertainty. We find for these images that32 angular bins and 40 to 50 radial bins pro-duce robust fits.

Results and Discussion

Our approach to the study of microfossilcell cycles involves three different toolkits, soto speak—image analysis, analytical model-ing, and computer simulation—all reported inthis one section of the paper. To clarify the pre-sentation, our results are organized into thefollowing subsections:

a. distributions of cell sizes and shapes as ob-tained from microfossil images,

b. analytical predictions for these distribu-tions from rate-based models for cellgrowth,

c. determination of cell cycles consistent withthe measured cell colonies, and

d. simulation of correlations in cell shapewhich could probe multistep cell cycles.

Of these, subsections (b) and (d) are of generalapplicability to equilibrium populations ofcells.

Size and Shape Distributions. As observedmicroscopically for the colonies analyzed, thethree-dimensional shape of each cell or cellpair appears to have an axis of rotational sym-metry. Figure 2 shows two views of the samedyad: the ‘‘top view’’ is its appearance as apair of overlapping spheres (as seen througha microscope), whereas the ‘‘side view’’ isdrawn perpendicular both to the symmetryaxis of the dyad and to the viewing direction(that is, within the plane of a thin section). Thesymmetry axis makes an angle � with respectto the viewing axis. Now, the symmetry axisseems to be randomly oriented in the samplesstudied, such that an average must be per-formed over � when comparing measure-ments with analytical models; this will betreated in the following subsection. Generallyspeaking, it is easier to predict analytically theproperties of projected shapes of the whole

cell rather than slices through it, particularlywhen angle-averaging must be performed. Asa result, we focus on the projected lengths L⊥

and L� , perpendicular and parallel to the cell’sprojected symmetry axis as defined in Figure2. For a dyad pair, where each subunit has ra-dius R and their centers of curvature are sep-arated by a distance s, one can see that L⊥ �2R independent of s or �, and L� � 2R �s·sin �.

Turning first to the dyadic taxa, a large frac-tion of cells in these colonies have the appear-ance of two spherical caps joined at a ring; thepercentage of visible dyads is roughly 45% ormore for all dyadic taxa reported in Table 1.Dyads with their symmetry axis lying close tothe direction of observation will not appear aslinked cells, but rather as one mildly ellipticalobject with a thick boundary, so the measuredfraction underestimates the true fraction by asmall percentage. That these cells spend 50%or more of their life as dyads indicates thatthey are not described by a cell cycle in whichthe cell slowly inflates radially like a balloonbefore it rapidly contracts along an equatorialarc to form two daughters; cells in such a cyclewould be dyads for only a small fraction oftime.

We now examine the projected shapes of allfour cell types of interest. The images are ob-tained by using a microscope objective with a

387GROWTH MODES OF MICROFOSSILS

TABLE 1. Percentage of visible dyads and lengths L⊥ and L� of projected cell shapes for four microfossils colonieswith n analyzable cells or pairs. Data are quoted as mean� � (� � standard deviation). The mean values are ob-tained from specific colonies, and may vary from one colony to the next depending on growth conditions.

Visible dyads L⊥� (�m) �⊥/ L⊥� L�� (�m) ��/ L�� L�/L⊥�

Sphaerophycus parvum (dyads, n � 99)45 � 7% 2.82 � 0.28 0.10 3.62 � 0.60 0.17 1.28 � 0.18

Eoentophysalis belcherensis capsulata (dyads, n � 156)72 � 7% 4.96 � 0.59 0.12 6.68 � 1.16 0.17 1.35 � 0.18

EB (dyads, n � 184)43 � 5% 5.90 � 0.54 0.092 7.63 � 1.24 0.16 1.30 � 0.19

Eosynechococcus moorei (rods, n � 224)no meaning 2.71 � 0.31 0.12 3.85 � 0.67 0.17 1.43 � 0.24

large enough depth of field to capture theshape as projected onto the viewing plane(100� for the small E. moorei and S. parvumand 50� for the larger cells). The observationsare summarized in Table 1, where � is thestandard deviation of a distribution, and n isthe number of objects in the analysis. Themean projected width L⊥� varies consider-ably, ranging from just under 3 �m for E.moorei and S. parvum to 5 �m for E. belcherensiscapsulata and 6 �m for EB. Even though thecolonies represent cells of moderately differ-ent mean sizes, the ratio of the standard de-viation of the distribution of L⊥ to its mean L⊥�is quite constant: �⊥/ L⊥� � 0.12, 0.10, 0.12,and 0.092 for the four colonies, in order of in-creasing L⊥�. However, the observed distri-bution is broadened by the fuzzy biologicalboundary of the original living cell, the effectsof degradation during fossilization, and theuncertainties of the measurement process, sothese ratios are an overestimate of the native ra-tios before cell death. Lastly, we note that themean values of these geometric quantitiesmay drift by perhaps 10% from one colony ofcells to the next for the same taxon, judgingfrom the behavior we observed for three dif-ferent colonies of E. moorei. Such differencesare presumably attributable to variation of thelocal environments in which the cells grew.

To probe the distributions of L⊥ further, werepeat the shape analysis on the individualsubunits of dyads using the same colonies asTable 1. Now, the subunits are captured by us-ing an objective (100�) having a narrowdepth of field; following image capture, theboundaries of any connected subunits are re-

moved from the image manually. Comparingthe results with Table 1, we find L⊥� � �⊥ ofthe subunits is: 2.80 � 0.28 �m for S. parvum(n � 78), 4.57 � 0.51 �m for E. belcherensis cap-sulata (n � 279) and 5.89 � 0.47 �m for EB (n� 155), where L⊥ and �⊥ have the same mean-ing as Table 1 but n includes individual sub-units as well as complete cells for those situ-ations where resolution into subunits is im-possible. We first observe that L⊥� and �⊥ arefairly similar for both the separated subunitsand the complete dyads of Table 1. This givesus confidence that our imaging techniquesand analysis are robust, as the two data setsfor each colony were imaged, extracted, andnumerically analyzed independently. Impor-tant for their interpretation, the distributionsin L⊥ are even narrower than those of the in-tact cells of Table 1: for the subunits, �⊥/ L⊥�� 0.10, 0.11, and 0.080. As for Table 1, thesenumbers overestimate the native ratios beforecell death.

A more informative probe of cell width isthe probability density ⊥ for the projectedPwidth L⊥ , where ⊥ has units of inverse lengthPand is normalized to unity via # ⊥ dL⊥ � 1.PFigure 3 compares ⊥ of small E. moorei (solidPline) and larger EB (dashed line). Clearly, ⊥Pdoes not correspond to a uniform distributionin L⊥ , even allowing for some smearing aris-ing from fossilization and measurement ef-fects. Rather, ⊥ resembles a sharply peakedPnative distribution, which has been broadenedby a Gaussian function with � � 0.1. Thus, theratio �⊥/ L⊥� of the subunits and the distri-bution ⊥ of all taxa both point toward a ratherPnarrow distribution in cell width during the

388 STEVEN BENNETT ET AL.

FIGURE 3. Measured probability density ⊥ of the trans-Pverse dimension L⊥ for small bacillus-shaped E. moorei(solid line, n � 224) and large dyads EB (dashed line, n� 184). The peaks have about 30 cells per bin, resultingin a statistical uncertainty of about 20% for ⊥ at its peak.PDimensions are microns for L⊥ and inverse micronsfor ⊥.P

FIGURE 4. Combined data for the probability density� of the dyads S. parvum (diamonds), E. belcherensis cap-P

sulata (circles), and EB (triangles) compared with the ex-pectations of growth at constant curvature and expo-nential increase in volume (solid curve) or area (dashedcurve). The largest values of � have about 30 cells perPdata bin for a statistical uncertainty of about 20% perindividual datum.

cell cycle for dyads and for rodlike E. moorei.This is consistent with �⊥/ L⊥� � 0.10 in mod-ern B. subtilis, which grows by elongation, notradial inflation (Sharpe et al. 1998). Measure-ments of seven taxa of modern diplococcaland rodlike cyanobacteria yield �⊥/ L⊥� � 0.07to 0.10 (C. Forde and D. H. Boal unpublisheddata).

Cells of the microfossil taxa studied here arenot generally spherical in shape, and this is re-flected in the values of the mean projectedlength L�� compared to L⊥�. As reported inTable 1, L�� is 30–40% higher than L⊥� for thedyads and E. moorei. A slightly more sensitiveratio than L��/ L⊥� is the ensemble average L�/L⊥� which has the experimental advantageof lower systematic uncertainties through theuse of the dimensionless ratio � � L�/L⊥ on acell-by-cell basis. As seen in Table 1, �� is ap-proximately 1.3 for dyads and somewhat morethan 1.4 for rodlike cells; �� is expected to belarger for rodlike cells because the cylindricalsection of the cell has a non-zero minimumvalue (whereas dyads are pairs of truncatedspheres). Measurements of several modern cy-anobacteria yield �� � 1.3 to 1.5 for diplo-cocci and 1.8 to 2.4 for long rodlike cells (C.Forde and D. H. Boal unpublished data). Giv-en that the averaging process includes orien-tations with small values of �, the mean pro-jected length L�� is obviously less than the ac-

tual mean length, and this is particularly truefor rodlike cells. For both dyads and bacilli,the upper bound on L�/L⊥ is at least 2, so thereshould be a broader distribution in L� than inL⊥. This behavior can be seen in Table 1, wherethe relative width ��/ L�� is about 0.17 for allexamples, compared with �⊥/ L⊥� of about0.11.

Rather than probing the shapes further us-ing the probability density � (with units of in-Pverse length), we work with the unitless prob-ability density � generated from � � L�/L⊥.PIn Figure 4, � for the three colonies of dyadsPis seen to rise rapidly to a maximum exceed-ing � � 2.5 at � near 1 before falling morePgently as � approaches its upper limit of 2 forlinked spheres. In contrast, � for the bacillus-Plike E. moorei in Figure 5 does not exhibit sucha pronounced peak, rising only to � � 1.5;Pfurther, � for this rodlike shape persistsPabove the dyad data at larger � and continuesbeyond � � 2 as permitted by geometry. Inboth figures, � is normalized to unity byP# � d� � 1. Comparing Figure 3 with eitherPof Figures 4 and 5 demonstrates the breadthof the distribution in L� , and confirms the ob-servation that values for ��/ L�� are larger thanthose of �⊥/ L⊥� obtained from the much nar-rower distribution ⊥. We now use � to dis-P Pcriminate among models for cell growth; we

389GROWTH MODES OF MICROFOSSILS

FIGURE 5. Measured probability density � for the ba-Pcillus-like E. moorei compared with predictions from ex-ponential (solid curve) and linear (dashed curve) vol-ume growth for spherocylinders (�0 � 0.2). Statisticaluncertainties as in Figure 4.

return to correlations in L� and L⊥ in a latersubsection.

Models for the Cell Cycle. In an ideal world,the measured probability densities from im-Page analysis would be sufficiently accuratethat they could be numerically integrated toobtain the time evolution of the cell shape—the geometrical description of the cell cycle.Two aspects of the measurement serve to con-found this approach:

1. the distributions are limited in their accu-racy, in part because the colonies suitablefor imagery contain only a few hundredcells; and

2. by measuring projected shapes, one isforced to include orientational averaging.

The method of choice, then, is to work fromthe opposite direction by proposing analyticalmodels for the cell cycle from which the func-tions can be predicted, including orienta-Ptional averaging.

To model the cell cycle, we hypothesize arate equation for a geometrical quantity (forexample, dV/dt � V, where V is the cell vol-ume) and solve it to determine the time evo-lution of the quantity. For an arbitrary cellshape, the length, surface area, and enclosedvolume each can obey a different rate equationand hence display different time dependenceduring cell growth. However, by imposing

constraints on the general cell shape, thesequantities become related so that the time evo-lution of the cell shape depends on the behav-ior of only one or two independent geometri-cal characteristics. Specifically, we interpret�⊥/ L⊥� � 0.10 of dyad subunits to mean thatthe curvature of the cell boundary is fairlyconstant during cell growth; this constraint issufficient to reduce the number of indepen-dent variables to just one for both dyads andspherocylinders. That is, once the time evolu-tion of just one of separation s(t), area A(t), orvolume V(t) is known, where s is defined onFigure 2, the remainder are determined by theconstraint of growth at constant curvature.

Mathematically, our approach is to

1. propose a simple rate equation for one of s,A, V or some other shape characteristic;

2. solve for the time-dependence of that char-acteristic using the rate equation from (1);

3. determine s(t) using the solution in (2) un-der the assumption that surface curvatureis constant during growth.

Knowing s(t) from (3), the probability densitys in s under steady-state conditions can beP

calculated from

P � (dt/ds)/T ,s 2 (1)

where T2 is the doubling time of the cycle. Asan example, the simplest case for dyads is lin-ear growth in s, or s(t) � 2Rt/T2; this givesds/dt � 2R/T2 and hence s � (2R)1 fromPequation (1). The assumption of steady-stategrowth is central to the analysis, as it relatesthe probability density to the time evolution ofthe cell shape. Most readers encountered aform of equation (1) when introduced to sim-ple harmonic motion, where the probability ofobserving an object at a given displacementduring an oscillation is inversely proportionalto its speed at that point: an object is morelikely to be observed at a turning point whereits speed is low than at zero displacementwhere its speed is high. Equation (1) can beintegrated to confirm the normalization of s.P

For dyads, s and A are linearly proportionalto each other (and thus are not independent),but not to the volume nor to the radius r of thecircular neck at the division plane. Hence, it isredundant to consider separate models for s

390 STEVEN BENNETT ET AL.

TABLE 2. Calculated probability density P� for fivemodels of dyad growth. To compare these functionswith data from randomly oriented cells, P� must be av-eraged using equation (2).

Case Rate equation P�

LinearI ds/dt � 2R/T2

or dA/dt � 4�R2/T2

1

II dV/dt � (4�R3/3)/T2 3(1 �2)/2III dr/dt � R/T2 �/(1 �2)1/2

ExponentialIV ds/dt � s ln 2/T2

or dA/dt � A ln 2/T2

1/[(� � 1)ln 2]

V dV/dt � V ln 2/T2 [3(1 �2)/ln 2]� [2 � �(3 �2)]

and A; rather, we propose several growthmodels each of which treats as an independentvariable just one of cell area, cell volume, orradius r of the septum defining the divisionplane. This approach assumes that division issymmetric and occurs at a unique time T2; asdiscussed below, our data are not sufficientlyaccurate to isolate and identify the effects ofdistributions in division time or daughter size,as considered by Koch and Schaechter (1962),Rosenberger et al. (1978) or Sharpe et al.(1998), for example. With their limited abilityto distinguish among rate models for s, A, orV, only two growth models are considered forbacillus-like shapes: either linear or exponen-tial increase in the volume with time.

Care must be taken to average over the an-gle � of the cell symmetry axis with respect tothe direction of observation, a process that re-duces the mean projected length. As a simpleexample, consider a set of identical thin rodsof length LR and negligible radius. The meanprojected length of a randomly oriented col-lection of such rods is L�� � { LR sin �1#0

d cos �}/{ d cos �} � (�/4)LR; that is, L�� is1#0

about 20% smaller than LR. Although impor-tant, this reduction is modest because, in arandom distribution, there are far fewer ‘‘ver-tical’’ orientations (toward the viewer) thanorientations close to the focal plane. As estab-lished in the Appendix, the angle-averagedprobability density � for the dimensionlessPprojected length � can be obtained from

[(� 1)/�]P � P d�, (2)� � � 2 2 1/2[� (� 1) ]

where the dimensionless integration variable� is defined by � � s/2R such that � � 2R s.P PThe integration limits are provided in the Ap-pendix. In other words, once s has been ob-Ptained for a given model, it leads to a predic-tion for � on the basis of equation (2) via �.P P

Analytical expressions for � � 2R s are giv-P Pen in Table 2 for five different rate-basedgrowth models, each based on a rate equationfor one of A, V, or r, with growth occurring atconstant curvature (please see the Appendixfor details of the derivation). Two forms forthe rates of change are investigated: (1) con-stant rate of change, leading to linear time de-pendence of the observable; and (2) a rate pro-

portional to the characteristic itself, leading toexponential time dependence. Any of theselinear or exponential approaches could beplausible under certain conditions; however,we omit a model with linearly decreasingneck radius r, as is justified below. In general,the angle-averaged � does not have a simplePanalytical form and must be evaluated nu-merically by using equation (2). An exceptionis linear growth in s or A (ds/dt � 2R/T2 forboth) leading from s � (2R)1 to � � 2R s �P P P1 as discussed as an example following equa-tion (1); as a result

P � �/2 arcsin(� 1), [case I]� (3)

according to equation (2). In the next subsec-tion, the expressions for � from Table 2 willPbe integrated to obtain � and then comparedPwith measured distributions.

These models can be extended without dif-ficulty from dyads to spherocylinders, an ide-alized shape that contains a cylindrical sec-tion of initial length s0, capped at each end byhemispheres of radius R. We assume that thecylinder first extends from s0 to 2s0, afterwhich the neck constricts while s rises to 2s0

� 2R at the division point. As the fixed pa-rameter of this shape s0/R becomes large, thetime dependence of s, A, and V become evermore similar, such that the behavior of � de-Ppends mainly on the form of the growth rate(i.e., linear or exponential), not the choice of s,A, or V for its basis. Thus, we narrow our dis-cussion to volume-based growth, which isalso a good approximation to length- or area-

391GROWTH MODES OF MICROFOSSILS

based growth. For linear volume growth, onecan easily establish that for spherocylinders:

1P � � � � � 2� (4a)� 0 0(2/3 � � )0

2(1 �� )P � 2� � � � 2� � 1, (4b)� 0 0(2/3 � � )0

where �0 � s0/2R and �� � � 2�0. For ex-ponential volume growth, one obtains

1P �� [(ln 2) · (2/3 � �)]

� � � � 2� (5a)0 0

2(1 �� )P �� 22 ln 2 · {[1 � ��(3 �� )/2]/3 � � }0

2� � � � 2� � 1. (5b)0 0

Both � and its first derivative are continuousPat � � 2�0.

Comparisons with Data. The dimensionlessprobability density � is now used for com-Pparing our measurements against the predic-tions of Table 2 and equations (4) and (5), afteraveraging over observation angle with equa-tion (2). Because of their better ability to dis-criminate among cell growth models, we be-gin with dyadic cells, as shown in Figure 4.The scatter in the data roughly approximatesthe statistical uncertainties in the individualdata points. The most notable feature of Fig-ure 4 is the strong peak in � at small �, ex-Pceeding � � 2.5. A successful model mustPslightly overpredict the maximum value of thepeak in order to accommodate its reductiononce the smearing effects of experimental un-certainties are taken into account. As � is nor-Pmalized to unity, if a model underpredicts thedata in one range of �, it must overpredict thedata elsewhere. The discussion proceeds caseby case: the first three cases represent growththat is linear in time, whereas the remainingtwo cases are exponential.

Case I: linear growth in area or separation.Linear time dependence in s or A yields thesame �. Although it slowly decreases with �,Pthe predicted � is much flatter than the data;Pfor example, the � � 1 value of equation (3),namely �(1) � �/2 is well below the data ofP

Figure 4 at small �, and necessarily above itas � → 2. The data do not support the model.

Case II: linear growth in volume. At � � 1,this model predicts that � approaches 3�/4P� 2.36, which is barely sufficient for small �;as in cases I and IV, � then exceeds the dataPas � → 2.

Case III: linear shrinkage of septum radius.Here, the expression for � in Table 2 is peakedPat � � 1 (or s � 2R), the termination of the cellcycle. Even when angle-averaged, it predictsthat � increases with �, certainly in disagree-Pment with all taxa on Figure 4. Exponentialdecrease in radius is also in qualitative dis-agreement with the data, but is not reportedin Table 2.

Case IV: exponential growth in area. Asshown on Figure 4, this model does not pos-sess the strong peak observed in � and isPsomewhat higher than the data as � → 2. Aswith cases I and III, the data do not supportthis model.

Case V: exponential growth in volume.Only this model possesses both the quantita-tive and qualitative features of � , as seen inPFigure 4. It predicts �(1) � 3.34; incorporat-Ping experimental uncertainties will reduce thepredicted peak and shift its location above �� 1.

Summarizing the results so far for dyads,the data of Figure 4 agree with case V, do notrule out case II, but do not support cases I, III,and IV.

What physical growth mechanisms under-lie the equations in Table 2? Linear models as-sume that change occurs at the same ratethroughout the cell cycle no matter what thecontents of the cell. Examples of linear modelscan be found in eukaryotic cells, where thecontractile ring may shrink at a constant rate(Biron et al. 2005) and the cell mass grows lin-early with time (Killander and Zetterberg1965). Here, we find that the only linear modelnot immediately ruled out by data is case II,linear rise in volume, for which agreementwith data is marginal at best. Exponentialgrowth may arise from several different mech-anistic origins. Case IV corresponds to newsurface being created at a rate proportional tothe area available to absorb new material—a

392 STEVEN BENNETT ET AL.

FIGURE 6. Scatter plot of (L⊥, L�) for 224 cells in the E.moorei colony reported in Table 1. Dimensions are quot-ed in microns.

FIGURE 7. Monte Carlo simulation of a rodlike cell col-ony with exponential volume growth parametrized byR � 1.3 �m, s0 � 0.5 �m, and experimental uncertainty� � 0.3 �m. The vertical arrow is the idealized growthtrajectory. Dimensions are quoted in microns.

logical possibility but not supported by Fig-ure 4. Lastly, case V arises if new volume iscreated at a rate proportional to the cell’s con-tents, which is the only scenario to describethe data comfortably.

Cylindrical cells cannot select amonggrowth models to the same extent as can dy-ads, but they are a distinct morphology andcould in principle obey a different mechanismfor growth. Here, quantities such as � can dis-Ptinguish between linear and exponentialgrowth only for high statistics data, being gen-erally insensitive to whether growth is drivenby s, A, or V. The angle-averaged prediction for

� is displayed in Figure 5 for linear and ex-Pponential growth in volume for �0 � 0.2. Com-pared with the �0 � 0 (dyad) curve in Figure4, the peak in � at � � 1 is suppressed forPspherocylinders in Figure 5, and the proba-bility density is spread to regions both belowthe peak, and above � � 2. Accounting for ex-perimental uncertainties will smear the peakfurther. Although the agreement with the datadisplayed for E. moorei is not perfect for �0 �0.2, nevertheless the predictions are consis-tent. The quality of the agreement is degradedif �0 is raised to 0.5. Exponential elongationhas been established with good statistics for avariety of modern bacilli: B. subtilis (Sharpe etal. 1998), B. cereus (Collins and Richmond1962), and E. coli (Koppes and Nanninga

1980), indicating that the exponential growthmode may have characterized rodlike bacteriafor several billion years.

Correlations of (L�, L� ). To probe furtherinto the cell cycle we examine the correlationsin L⊥ and L� , which are removed when the di-mensionless variable � � L�/L⊥ is taken. Ascatter plot of (L⊥ , L�) from the bacillus-like E.moorei colony of Table 1 is displayed in Figure6, where the coordinate origin has been shift-ed to visually exaggerate the distribution. Thevertical trend is what one would expect froma model of elongation of a cylinder at constantwidth: for cells lying perpendicular to theviewing axis (� � �/2), one would expect(L⊥ , L�) to run from (2R, 2R � s0) to (2R, 4R �2s0). To compare these data quantitativelywith a model, one would have to incorporateboth the random distribution in �, which willshift the distribution toward the L� � L⊥ line,and experimental uncertainties, which tend tobroaden the distribution horizontally (al-though the behavior is more complex near L⊥

� L� because of the definitions of L⊥ and L�). AMonte Carlo simulation of the cell cycle in-cluding these effects is shown in Figure 7; thisis a model with exponential volume growthand R � 1.3 �m, s0 � 0.5 �m (�0 � 0.2), andexperimental uncertainty in (L⊥ , L�) governed

393GROWTH MODES OF MICROFOSSILS

FIGURE 8. Scatter plot of (L⊥, L�) for 156 cells in the E.belcherensis capsulata colony reported in Table 1. Visibledyads (doublets) are indicated by open circles, and sin-glets are displayed as solid disks. Dimensions are quot-ed in microns. For clarity of presentation, the scale of thex-axis is double that of the y-axis.

by a normal distribution with � � 0.3 �m. Thevertical arrow indicates the trajectory in theidealized � � 0 limit. The simulation is seento be in qualitative agreement with the data.Unfortunately, the interpretation of � is cloud-ed because it represents not just experimentaluncertainty but also effects arising from dis-tributions in daughter size or division time(Koch and Schaechter 1962; Rosenberger et al.1978; Sharpe et al. 1998; see also Koch 1983).

Compared with rodlike E. moorei, potential-ly more information can be extracted fromscatter plots of (L⊥ , L�) for dyads, where theadded feature is classification of the cell ac-cording to the presence of a visible neck orjunction ring. We will refer to cell pairs with avisible ring as doublets and without as singlets,recognizing that physical dyads may appearas singlets when the orientation angle � � 0.A scatter plot of the E. belcherensis capsulatadata is displayed in Figure 8; note that the xand y scales are inequivalent. The singletstend to be distributed along the x � y line, cor-responding to uniform expansion of the cellearly in its cycle. The doublets are distributedmore vertically, corresponding to the forma-tion of the symmetry axis and division plane.For such a two-step picture, the idealized tra-

jectory for dyads would be a straight linealong x � y starting at (2R, 2R) followed by astraight, non-vertical line to (2R, 4R). Al-though the data aren’t strong enough to sup-port this picture quantitatively, they are nev-ertheless suggestive. Each dyad colony of Ta-ble 1 has a different fraction of visible dyads,and the scatter plots of each colony suggestthat the larger this fraction, the less time isspent in the initial phase of uniform expan-sion. Given the ambiguity of identifying visi-ble dyads when � is near zero, we cannot yetquantitatively assign a time fraction for the ex-pansion phase. It’s worth mentioning thatpopulations of modern diplococcal cyanobac-teria display the same qualitative features asFigure 8 (C. Forde and D. H. Boal unpublisheddata): the fraction and distribution of visibledyads, and the distribution of singlets alongthe L� � L⊥ line are similar in both the ancientand modern cells.

Conclusions

Adopting a digital algorithm for character-izing cell shapes from good-resolution CCDimages, we have measured size and shape dis-tributions of two-billion-year-old dyads andbacillus-like cells. To interpret these distribu-tions, five different rate-based models weredeveloped for dyad growth, and two for ba-cillus growth; the models easily accommodatenumerical averaging over the cell’s orientationangle with respect to the observer. In this pa-per, more emphasis has been placed on dyadsthan on rodlike bacilli, because the geometryof the former is more suitable for distinguish-ing among growth models. For four types ofmicrofossils, the shape distributions are con-sistent with exponential volume growth atconstant surface curvature, obeying a rate dV/dt proportional to cell volume V: the greaterthe cell’s contents, the faster it grows. The datado not rule out linear volume growth, but theyare inconsistent with linear or exponentialgrowth in area (or separation s) or with lineardecrease in the radius of the septum. Scatterplots of (L⊥ , L�) demonstrate that dyad growthmay include a short initial phase of uniformexpansion before the axis of elongation ap-pears. We continue to investigate whether theextra information contained in scatter plots

394 STEVEN BENNETT ET AL.

may prove useful in establishing the identityof microfossils through comparison withmodern cyanobacteria.

Qualitatively, the most important termdriving the shape of these ancient cells is theincrease in the cell’s contents, which the sur-face area accommodates but does not direct.The behavior is consistent with exponentialelongation observed in modern cylindrical B.subtilis (Sharpe et al. 1998), B. cereus (Collinsand Richmond 1962) and E. coli (Koppes andNanninga 1980). Linear volume growth asseen in modern eukaryotes is not dismissedby the microfossil data, but is certainly lesslikely. Thus, the growth modes of these twomorphologically distinct groups of 2-Ga mi-crofossils (dyads and bacillus-like cells) arecloser to modern prokaryotes than moderneukaryotes, in spite of the presence of dark in-teriors in some taxa that might be suggestiveof nuclei.

We have attempted to identify the presenceof a wall at the interface between cell com-partments by measuring the optical intensityprofile along the symmetry axis of the dyad.Referring to the orientation of the dyad in Fig-ure 2 (top view), light traveling through onlyone of the subunits in the dyad passes throughtwo wall regions, whereas light travelingthrough the center passes through three. If thecentral wall has the same thickness as the ex-ternal cell wall, then an absorption intensityprofile taken along the symmetry axis of thedyad should display three minima in roughlya 2:3:2 ratio, ignoring the angle-dependentpath length of light through the walls. Wehave succeeded in analyzing only a handful ofE. belcherensis capsulata and found that the in-tensity minimum of the overlap region is 0–70% larger than that of the individual sub-units. While suggestive, this range is too largeto allow a conclusion to be drawn about thepresence of an internal wall; a technique withfiner resolution than optical microscopy willmost likely be needed to resolve this question.Further, the use of three-dimensional imagereconstruction would permit the cell shapes tobe determined directly, eliminating the angle-averaging procedure of the approach takenhere (for example, Grotzinger et al. 2000; Xiao2002).

Acknowledgments

The authors wish to thank H. Hofmann forextensive conversations regarding Belcher Is-land microfossils. We are also indebted to J.Dougherty, curator of the microfossil collec-tion at the Geological Survey of Canada in Ot-tawa, both for facilitating access to the collec-tion and for many patient discussions. Wethank A. Knoll for his comments on the initialversion of this manuscript. This work is sup-ported in part by the Natural Sciences and En-gineering Research Council of Canada.

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Appendix

The purpose of this appendix is to provide a sketch of themathematical steps leading to the analytical distributions in Ta-ble 2. The starting point for the calculations is an expression forthe surface area or volume of the idealized geometries chosento represent the shapes of dyads or bacillus-like cells. Becauseit is analytically simpler, dyad geometry is discussed first.

Given the narrow distribution observed for the diameter ofdyads, their assumed geometry will be that of two sphericalcaps of fixed radius R intersecting at a ring (or neck) of varyingradius r, as in Figure 2. The separation between the centers ofthe caps is s, which can range from s � 0 at the beginning of thecell cycle to s � 2R at the division point. It is not difficult toshow that the total surface area A and total enclosed volume Vof the dyad are as follows:

2 2A � 4�R (1 � s/2R) � 4�R (1 � �) (dyad) (A1)

3 2V � (4�R /3){1 � (s/2R) · [3 (s/2R) ]/2}

3 2� (4�R /3)[1 � � · (3 � )/2]. (dyad) (A2)

In the limit s � 0, these expressions have their single-sphere val-ues, and at s � 2R the area and volume are doubled. The com-bination s/2R appears so frequently in what follows that it iseasier to work with the second representation in equations (A1)and (A2), where the dimensionless quantity

� � s/2R (A3)

has been introduced. The volume of rodlike cells will be pre-sented below.

From the viewpoint of measurement accuracy and interpre-tation, the geometrical attribute of greatest interest is the pro-jection of the separation s onto the observation plane of a mi-croscope. Thus, the measurement process yields information on

s, a probability density (per unit s). The simple example s �P P(2R)1 for linear growth in s is given in the text; here we describethe more relevant situation of exponential volume growth

V(t) � V exp(ln2 t/T ).0 2 (A4)

In this equation, the volume doubles from its initial value V0 ina time T2. To obtain s we assume steady-state conditions suchPthat s � [T2·(ds/dt)]1. As will become obvious momentarily, itPis less cumbersome to work with � than with s , for whichP P

1P � [T ·(d�/dt)] .� 2 (A5)

The two probability densities are related by � � (ds/d�) s �P P2R s , according to equation (A3).P

Armed with the time-dependence of the volume in equation(A4), one can invert either representation of equation (A2) to ob-tain the functions s(t) or �(t). However, equation (A2) is cubicin s or �, so the inversion is cumbersome; in fact, it’s unneces-sary, as all that is needed to solve equation (A5) is the time de-rivative d�/dt. Rearranging equation (A4) to read

t � (T /ln 2) ln(V/V ),2 0 (A6)

and replacing V(t) by equation (A2), one obtains

2 2dt/d� � (T /ln 2)·3(1 � )/[2 � �·(3 � )].2 (A7)

Equation (A7) then may be substituted into equation (A5) to ob-tain

2 2P � [3(1 � )/ln 2]/[2 � �·(3 � )],� (A8)

which is the result quoted in Table 2 for exponential volumegrowth of dyads.

The idealized shape we choose for representing bacillus-likecells is the spherocylinder: a uniform cylinder of radius Rcapped at each end by hemispheres of the same radius. If theminimum cylinder length at the start of the cell cycle is s0 (wheres is still the distance between the centers of curvature of the end-caps), then s runs from s0 to 2s0 � 2R by the time the cell hasdoubled at t � T2. Alternatively, � runs from �0 (� s0/2R) to 2�0

as the length of the cylindrical section doubles, and then to 2�0

� 1 as two new hemispheres are formed at the division plane ofthe cylinder. Expressed in terms of �, the volume of the spher-ocylinder is

3V � (4�R /3) · (1 � 3�/2)

� � � � 2� (A9a)0 0

3 2 3V � (4�R /3) · [1 � ��(3 �� )/2)] � 4�R �0

2� � � � 2� � 1 (A9b)0 0

where

�� � � 2� .0 (A10)

396 STEVEN BENNETT ET AL.

These expressions can be manipulated as above to yield � , al-Pthough the mathematics is slightly more cumbersome becauseof the change in functional form at 2�0. Again, it is not necessaryto solve for �(t) or s(t), simply finding their time derivative issufficient. However, here it is more convenient to use (d�/dt) �(d�/dV)·(dV/dt) to derive equation (5) for exponential volumegrowth.

In the lab, it may be possible to manipulate cells such thattheir symmetry axes lie in observational plane, even if they arenot parallel to each other. In such circumstances, � or s can beP Pmeasured directly. However, the microfossil cell colonies thatare studied here have random orientations in three dimensions,which forces us to average over the angle � in Figure 2 (the anglebetween the symmetry axis and the viewing axis) when pre-dicting the distribution of shapes as projected onto the obser-vational plane. Now, the projected length and width of the ide-alized shapes here are L� � 2R � s·sin � and L⊥ � 2R, respec-tively, such that the dimensionless ratio � � L�/L⊥ is given by

� � 1 � (s/2R) sin � � 1 � � sin �. (A11)

The probability density � of � is then obtained from s or � byP P Paveraging over �, as in

P d� � P d� sin � d�. (A12)� � �

Using equation (A11) to obtain (d�/d�) � � cos � as well as ex-pressions for sin � and cos �, we can rewrite equation (A12) as

[(� 1)/�]P � P d�, (A13)� � � 2 2 1/2[� (� 1) ]

which is equation (2). Note that the integration limits depend onthe value of � chosen for �: the lower limit is the larger of � P1 or �0 and the upper limit is 1 � 2�0.