Original Paper Forma, 24, 87–92, 2009

A Simple Mathematical Model for Chamber Arrangementof Planktic Foraminifera

Takashi Yoshino1∗, Katsunori Kimoto2, Naoko Kishimoto3, Atsushi Matsuoka4, Toshiyuki Kurihara5,Naoto Ishida5 and Shu Matsuura6

1Department of Mechanical Engineering, Toyo University, Kujirai 2100, Kawagoe, Saitama 350-8585, Japan2Research Institute for Global Change (RIGC), Japan Agency for Marine-Earth Science and Technology (JAMSTEC), Japan

3Graduate School of Engineering, Kyoto Univiersity, Japan4Department of Geology, Faculty of Science, Niigata University, Japan

5Graduate School of Science and Technology, Niigata University, Japan6Faculty of Education, Tokyo Gakugei University, Japan

∗E-mail address: tyoshino@toyonet.toyo.ac.jp

(Received March 16, 2009; Accepted January 12, 2010)

We propose a simple mathematical model for chamber arrangement of planktic foraminifera. The arrangementis approximated using connection of spheres. It is represented by combination of four sequences: they correspondto radius of chamber, horizontal/vertical distance of connected chambers, and angle of horizontally projected twolines constructed by connection of centers of adjacent spheres. We assumed that the sequences of the radius andhorizontal/vertical distances are geometric series and that of the angle is constant. We succeeded in reproducingthe chamber configuration of all recent planktic foraminifera at the level of family in taxonomy. We also pointout that some other parameters are required in order to classify the chamber arrangement in detail. The modelenables us to consider an optimization problem of the volume-surface area ratio which corresponds to the costof chamber forming materials. Using the simplest case of our models, we obtain an optimized radius-distancerelation.Key words: Planktic Foraminifera, Chamber Arrangement, Mathematical Model

1. IntroductionForaminifers, a kind of single-celled marine Protista, live

in the sea all over the world. At present, the number ofspecies of foraminifers is known up to a few thousandsincluding fossils in the geologic ages. Foraminifers havetwo prominent types of lifestyles: one is planktic, whichlives in surface water through the life, and the other is ben-thic, that lives in surface sediments on the sea floor. Theygrow up by preying small marine zooplankton, phytoplank-ton, and/or organic detritus in the seawater and sediments,and build calcareous (CaCO3; carbonate) skeleton to protecttheir cells. Since most of foraminifers are less than 1 mmin size, we cannot recognize their detailed forms by nakedeyes. Observations by optical microscopes have revealednumerous polymorphic variations in the calcareous skele-tons, which are called “chambers”. Classification schemesof foraminifers have been constructed based on their micro-scopic morphological features.

In general, the skeletons of modern planktic foraminifersare simplified coils. Figures 1a–d show their representativeshapes. Each chamber is spherical or ovoid, and is inflatedand porous. Most of planktic foraminifers have trochoid-spires (named “trochospire”) and their coiling heights areone of the important criteria for classification.

Benthic foraminifera exhibit larger variations in theircoiling morphology than planktic ones as shown inFigs. 1e–g. On the other hand, the variation of benthicforaminifer’s morphology and coiling variations are huge

and complicated compared with planktic ones. Benthicforaminifers have large, rigid and thicker skeletons. Theshape of chambers and its arrangement are also variable:low and high trocospire (Figs. 1e and 1f), and bilocu-laris (Fig. 1g) forms are popular in benthic foraminifers.Other different morphotypes are also frequently observed;e.g. uniserial (adding chambers to one direction with-out coiling), plano-spire (ammonoid-like coiling named“planispire”).

It has been believed that benthic foraminifera has ex-isted since 0.5 billion years ago (Cambrian Period) on theearth according to some evidences of fossil records. Onthe contrary, planktic foraminifera appeared on 0.17 billionyears ago (middle Jurassic Period). It is believed that theyhad evolved from one and/or some benthic foraminiferallineages by the results of morphological analysis (e.g.Pessagno, 1967). Planktic foraminifera adapted to all oceanenvironments from surface to deeper (ca. ∼1,000 m) wa-ter depths throughout the geological time. Throughout suchan adaptation, they achieved some morphological variationsaccompanying adaptations (Fig. 2). It seems that these vari-ations are considered tightly related to the ambient environ-ments including their mechanical properties and their eco-logical functions in evolutionary histories. However, theirfunctional meaning of morphology has not been consid-ered until today. From the taxonomic standpoint, the clas-sifications of lives are generally based only on their mor-phological features. Therefore it is also important to de-

87

88 T. Yoshino et al.

Fig. 1. Representative morphologic variations of planktic and benthic foraminifera. Samples from a to b correspond to planktic foraminifera and otherscorrespond to benthic one. a) Globigerinella, b) Globigerina, c) Candeina, d) Globorotalia, e) Uvigerina, f) Cibicidoides, and g) Quinqueloculina.

scribe quantitatively their morphologies quantitatively us-ing mathematical expression. If their morphological fea-tures are described mathematically, we can test their me-chanical properties with their boundary conditions usingnumerical analysis. Furthermore it might be understoodtheir morphological meanings in terms of mechanical func-tions through evolutionary processes over the earth history.In this study, we investigate a simple mathematical modelfor chamber arrangement of planktic foraminifera in orderto consider the parametric description of their forms sys-tematically.

Here, we construct a discrete growth model forforaminifera and examine the reproducibility of their forms.The arrangement of foraminifera chamber can also be re-lated to shells of snails and ammonoids. Thompson (1942)insisted that the chamber arrangement of foraminifera wasqualitatively approximated to logarithmic spirals. Okamoto(1988) proposed a differential geometrical model of am-monoids called growing tube model. His model describedmany types of ammonoids and reproduced the formationof an irregularly arranged coiling of Eubostrychocerasjaponicum. Both of Thompson’s and Okamoto’s modelsare based on continuous function. Their methods are notdirectly applicable to chamber formation of foraminiferasince foraminifers have discrete chambers. We also discussthe optimization of simple chamber arrangement which formaximizing the ratio of volume to surface area as an appli-cation of the mathematical model of the chamber arrange-ment.

2. Model2.1 Chamber arrangement

We choose a sphere as a representative shape of a cham-ber for simplicity and applicability to analyses such as opti-mization problem described below. Construction of spherearrangement is successive addition of given radii of spheres.So there is an initial sphere at the beginning of construc-tion. The radius of the initial sphere is set to be unity and

Fig. 2. Classification of modern planktic foraminifera. Right side pho-tographs correspond to typical morphospecies of each family.

the radii of other spheres are given by the relative lengthsto the initial one. We allocate the initial sphere at the ori-gin of coordinate. The locations of the other spheres aredetermined by relative vectors with the center of the initialsphere. Locations of the other spheres are determined byrelative vectors with their beginning points at the center ofthe initial sphere. Because we focused on chamber arrange-ment, not formation process, the radii are kept constant once

A Simple Mathematical Model for Chamber Arrangement of Planktic Foraminifera 89

a) b)

Fig. 3. Schematic sketches of chamber arrangement model. a) Vertical view. b) Horizontal view.

they added to previous sphere.The configuration sequence of chamber spheres, which

corresponds to the temporal chamber formation as well, isdescribed by the series of four variables: radius of chamber,horizontal and vertical distances of connected chambers,and angle of two lines connecting the centers of adjacentspheres on horizontal projection. Figure 3a shows a verticalview of the model with the chamber radius ri , the horizontaldistance di , and the angle of centerlines θi . Figure 3bshows a horizontal view with the vertical distance hi . Inaddition to these variables, the total number of chambers,i.e. the total growth steps of chamber generation, is denotedby n. A chamber arrangement of planktic foraminifer isdetermined by these five variables. We choose the z-axis asthe axis of rotation called pivot since we defined the angleof centerlines on the horizontal x-y plane. Note that anycoordinate axis does not specify the direction of gravity.

There is another selection of three variables concerningwith location of sphere such as distance between centers ofneighboring spheres and two angles which denote the di-rection of relative vectors of neighboring centers. Althoughsuch a selection is better than ours from a viewpoint of in-tuitive understanding of foraminifera’s morphogenesis, wepreferred our definition from a viewpoint of descriptive lan-guage.

We assume that the configuration of a chamber is a func-tion of that of the previous chamber, what is called recur-rence formula, throughout the sequence of chamber forma-tion. Therefore, arrangement of chambers can be describedby,

ri = fr (ri−1), di = fd(di−1),

hi = fh(hi−1), θi = fθ (θi−1). (1)

Furthermore, we assume that the variables ri , di , and hi aregeometric series and θi is constant such that,

ri = r i−1, di = dr i−1, hi = hr i−1, θi = θ, (2)

where r , d, h, and θ are parameters which determine thechamber arrangement of a foraminifer. We call these pa-rameters radius ratio, initial horizontal/vertical distance,

Fig. 4. Definition of variables for chamber connection.

and initial angle, respectively. These geometric series ofvariables provide a discrete version of Thompson’s loga-rithmic spiral model, which we can extend by changing thefunctions in Eq. (1) if necessary.

Due to the geometrical restriction, the following relation-ships fold for the above four parameters: radius ratio r > 0,initial horizontal distance 0 < d < 1 + r , initial verticaldistance 0 < h < d, and initial angle −π < θ ≤ π . Theranges −π < θ < 0 and 0 < θ < π correspond to rightand left of foraminifeal coiling directions, respectively.

Real growth of foraminifers is initiated with a pair ofchambers. Observational studies revealed that the firstchamber, called proloculus, was so special in its featuresthat it is not consistent with its following arrangement se-quence described by Eqs. (1) and (2). For example, cham-ber radius of proloculus is about 5 percent larger thanthe other chambers. Therefore, we omit proloculus fromour model and construct the arrangement from the secondchamber.2.2 Optimization analysis

We apply our mathematical model of chamber arrange-ment to an optimization problem of the initial horizontaldistance d to make the volume-surface area ratios maxi-mized. This optimizes the cost of chamber forming ma-terial (carbonate). As described above, the value of initialdistance d is restricted between 1 and 1 + r because of thetriangle inequality. As the simplest but not trivial case, we

90 T. Yoshino et al.

a) c) e)

d)b) f)

Fig. 5. Comparison of real species and results of our model. Upper parts (a, c, and e) are simulated forms by our model. Lower parts are ScanningElectron MIcroscopes (SEM) images of real species. b) Globigerinita glutinata, d) Neogloboquadrina incompta, and f) Gallitellia vivans.

Fig. 6. Initial angle dependence of foraminiferal models. Other parameters are fixed.

consider the case of θ = 0 at which the chambers are con-nected on a straight line. Results of this linear connectioncondition can be applied to the cases of other θ values as faras the chambers overlap only with neighboring ones. Thistype of sequential form corresponds to an uniserial form ofbenthic foraminifers.

The objective function can be expressed as,

p(d|r, n)

=

4

3πr3

1 − v(l ′1, r1) +n∑

i=2

(4

3πr3

i − v(li , ri ) − v(l ′i , ri )

)

4πr21 − s(l ′1, r1) +

n∑i=2

(4πr2

i − s(li , ri ) − s(l ′i , ri )) ,

(3)

where ln and l ′n are

ln = rn cos αn = r2n + d2

n − r2n+1

2dn,

l ′n = rn cos α′n−1 = r2

n + d2n−1 − r2

n−1

2dn. (4)

Figure 4 shows the definitions of these variables. The func-tions v(x, r) and s(x, r) correspond to volume and surfacearea of spherical cap of a sphere whose radius and radius ofcross section are x and r , respectively. These functions arewritten as,

v(x, r) = π

(2

3r3 − r2x + 1

3x3

), (5)

and,s(x, r) = 2πx(r − x). (6)

The optimized value of d with fixed r and n can be foundnumerically using the steepest descend method (e.g. Presset al., 2002).

3. ResultsUsing some parameter set, the model realized the cham-

ber forms that are nearly identical to the real foraminifera.Figure 5 shows the comparison of model results and theSEM images of real foraminifers. The form of Fig. 5awas obtained by the parameter set of r = 1.3, d = 1.4,θ = −1.26, and h = 0.3 at n = 12, which was identical to

A Simple Mathematical Model for Chamber Arrangement of Planktic Foraminifera 91

Fig. 7. Examples of optimized form.

Fig. 8. Radius ratio dependence of the maximized volume surface area ratio (left). Radius ratio dependence of optimized initial distance (right).

the form of Globigerinita glutinata shown in Fig. 5b. Fur-thermore, the conditions r = 1.2, d = 1.3, θ = −1.28,h = 0.25 and n = 11 in Fig. 5c well reproduced the formof Neogloboquadrina incompta shown in Fig. 5d. Figure5e was obtained using r = 1.15, d = 1.3, θ = −1.88,h = 0.4 and n = 12, and corresponded to Gallitellia vivansshown in Fig. 5f. Changing the parameters adequately, wealso acquired other types of chamber configurations.

Figure 6 shows an example of θ dependence of the formwith other parameters fixed. Chamber arrangement of all re-cent planktic foraminifera can be produced using our modelat the level of family in taxonomy. Ranges of parametervalues for present planktic foraminifera are compiled as fol-lows: 1.2 ≤ r ≤ 1.3, d ≤ 1.9, 1.0 ≤ |θ | ≤ π , 0 ≤ h ≤ 0.5,and n ≤ 20.

We also found out several types of forms that are simi-lar to some kinds of benthic foraminifers. They were ob-tained using |θ | values out of the above range of actualplanktic foraminifera. Particularly, simulated forms with0 ≤ |θ | ≤ 0.62 indicate uniserial and relatively loosercoiling (i.e. large number of chambers in outermost or fi-nal whorl, see Fig. 6). Actually, these forms are frequentlyobserved in present benthic foraminifers. The uniserial ar-

rangements of chambers are common in modern benthicforaminiferal morphologies. Such uniserial form had neverappeared in the past planktic foraminifers in the geologichistory. In other words, chamber arrangements of plankticforaminifers are certainly composed of at least 3 chamberswhich are connected each other in nature. For this reason,it can be considered that parameter θ provides morphologi-cal and ecological “boundary” between planktic and benthicforaminifers.

Figure 7 shows the examples of the optimized chamberarrangements, and Fig. 8 shows the numerical results ofthe optimization problem. The examples in Fig. 7 indi-cate that the optimal shapes are not similar to real cham-bers. As shown in Fig. 8, the solution d is a monotonicincrease function of r for each n. This function convergeswith an increase of n. Because the adult foraminifera con-sist of more than 10 chambers, the solution for foraminiferacan be regarded as almost the same function. The obtainedvolume/surface area ratio is also monotonic increase func-tion of r . Radial dependence of the ratio becomes sensitivewith an increase of n. We can conclude that there exists theoptimized solution of d for all r and n for our simple caseand that the solution converges when n goes to infinity.

92 T. Yoshino et al.

4. DiscussionsAlthough we have not yet examined that our mathemati-

cal model reproduced all kind of shapes of foraminifers, asseen in right hand side of Fig. 1, θ is considered to be one ofthe most sensitive parameters for classification of form be-tween planktic and benthic foraminifers. At this moment, itis not clear why the separation occurs sensitively by chang-ing θ . However, θ seems to be the universal parameter forconsidering foraminiferal morphologic variability.

To give a possible answer for this question, we considera proposition about the phylogeny of planktic foraminifera.Based on the morphologic analysis, the early plankticforaminifers are thought to have possibly evolved from agroup of benthic foraminifer (Tappan and Loeblich, 1988).Recent phylogenetic techniques using small subunit of ri-bosomal DNA gene (SSU rDNA) of foraminifera revealedthat almost recent planktic foraminifera are the polyphyleticin origin, and derived at least three ancestral benthic lin-eages (Darling et al., 1999). It is possible that plank-tic foraminifer had been changed above five parameters toadapt their lifestyles from benthic to planktic in the geolog-ical history. In other words, morphologic changes shouldbe tightly connected to their surrounding environments. Weinfer that these five parameters, especially parameter θ , arethe key to interpret functional, morphologic variations andevolutions of foraminifers.

Overlapping of chambers of not-successive generation isalso an important factor for chamber arrangement from aviewpoint of the optimization problem of chamber formingmaterial. It may cause the difference of d value betweenobservation and numerical solution. In order to argue themeaning of the extra overlapping further, we have to de-

velop a method for considering mechanical aspect of thechamber structure: location of centroid and strength of thechamber structure, for example.

We have other problems left unsolved, relating to com-plete description of foraminifera. The first one is the treat-ment of proloculus. As described above, proloculus is notconsistent with the sequence of chamber arrangement sothat we have to describe the location of the special cham-ber apart from the following chambers. The second one isthat the locations and directions of their aperture (openingof chambers). Although we do not discuss them in this ar-ticle, they are also critical factors for classification becausethey are conservative traits in the evolution of foraminifera.The last one is that the descriptions of species whose growthcannot be described by our geometrical series model and/orspherical chambers. Further improvement of the modelis required for reconstructing unconventional species ofplanktic foraminifera.

ReferencesDarling, K. F., Wade, C. M., Kroon, D., Leigh Brown, A. J. and Bijma, J.

(1999) The diversity and distribution of modern planktic foraminiferalsmall subunit ribosomal RNA genotypes and their potential as tracers ofpresent and past ocean circulations, Paleoceanography, 14, 3–12.

Okamoto, T. (1988) Analysis of heteromorph ammonoids by differentialgeometry, Paleontology, 31, 35–52.

Pessagno, E. A., Jr. (1967) Upper Cretaceous planktonic foraminifera fromthe western Gulf Coastal Plain, Paleontogr. Americana, 5, 245–445.

Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery B. P. (2002)Numerical Recipes in C++: The Art of Scientific Computing, Cam-bridge.

Tappan, H. and Loeblich, A. R. (1988) Foraminiferal evolution, diversifi-cation and extinction, Journal of Paleontology, 62, 695–714.

Thompson, D. (1942) On Growth and Form, Cambridge.