THE EFFECT OF CZLL DTVISION OS THE SIIAl’l!; AND SIZE O F HKXAGONAL CELLS “ L ’avenir de l’histologie et sp&cialenient de la cytologie est tout enticr dans la niicrochimie. ”-Opinion prt‘vulontt . Division of tctrakaidecalietlral cells, when ai*ranged with hexagonal siirfaces above aiid below, OCCUI‘S in the equatorial plane, a i d , perpendicular to that plane, in three vertical planes a t angles of 60” with each 0ther.l 111 a simple epitlielium the ecyuat orial divisions are eliminated. If they occurred they would start stratification. There remain, then, 0111~7the three vertical plaiies sliown in fignrc 1 a t a, b, aiid c ; and the effect of division in these plaues upoii the shape and size of cells in simple epithelium is the primary object of this report. If the ontline of the cell is a regular hexagon, with sides of unit Ieiigth, its perimeter measures 6 units and its area 2.?598sq. units. Division in the plane a, 6 , or c, Tvliich bisects the area, atlds t o the perimeter; each resultiiig cell is a very irregular pentagon with x perimeter of 4.732 for a n area of 1.299. 111 cells with plastic W W ~ Isnrfnce R, tciision promptly rcdnces the rclatively great length of this perimeter by making the pentagons as regular a s possihle. If the area remains the same, the pentagon can be made regular by shortciiiiig its sidm to a iuiiform length of 0.869. But this would mmii a ~ ~ o i . I ’ ~ ~ s i ) o irecliiction ~ ( l i ~ i g in the coiitiguous sides of Thc tl11 isioir of tetl.:tk:iitlr(’;ihrdral ~ c ~ l lhsa q alread) An1c.r. Acnd., 1923, \n1. j8, p. ii-$(i ct s i q . :+.31 1, J t ? ( O R I ) TI \ I B F 1 < LO1 1926 I 1 SO i hcw1 ilc.sciil,c,d--l’loc. 332 F R E D E R I C T. LEWIS ati?joirlilig (.ells, and a consequent increase of their areas relas. To meet this situation the peiitagons their sides of nearly unit lengtli. If absolilt ely regular arid of unit sides, their t t r ~ a s would he& iiicrcasccl by 32.4 per cent arid would become 1.720. Tliejthen would l i a v ~ relatively, , the shortest perimc.tcrs possiblc f o r (~111sof peiitagoiial form. If they grew still larger in aii clffort to attain the size of the parent cell, it would aff’ect 1111l’i~vorablythe iteiglilmi~iiig cells by makiiig their prrimtltc1r.s exceed the minima. for the arcas cnclosed. (’011scqueritly growth may he expected to stop a t that point, rriakiiig tlic end result of tlic division of ilic hexagon the p * o tIuc*tiori of a pair of iicarlj7 regular pentagons of nearly miit siclcs. ‘l’hcp arc small cells, with a comhiried area seldom r of t h e 1)ureiit wll ; attaiiiiiig a 32.4 pcr cont iiicrensc o v ~ that t1ic.y by n o mtwis double its sizc. I)ivisiori of a hexagonal (bell in the plane tl of figure 1 \vould i i o t lie expected f o r several re‘asoiis. It ~ v o u l dplacc tlic p o l c n of tlitl mitotic spindle in a short axis of the cell. With t l 1 ~adjoining cells it would give rise to unstable tetrahedral i~iiglcsat either end. It would be contrary t o Errera’s rulc fliiit the ‘incipient partition-wall of a dividing cc.11 tciiiis t o l)e such that its area is the least p o s s i l k hy which the sp~ice-cont~~iit (*mi 1)e e~iclosccl.’ ’ 2 ‘I’etrahcdral aiiglcs a r e i i o t aiikiio~viiin tissues: the mitotic spindle by a skewiiig of tlic ~ i ~ ~ i s i o i i - r ) lmay ~ ~ i t occupy ti a short axis of the cell ; and F:rrcra’s law may hc flagrantly evaded, as is regularly tllc ciise in the cambial cclls ~vlii(*IiBailey studied,3 where (.Iongated cells split Icngthwise. Nevertheless, if thc axes of a ( ~ 1 1 1a1.e equal arid its outline is a regular hexagon, there are i cliaiices f o r division iii any one of the planes a, h, and c, with the probabilities against such planes as d. I f in a sheet of liexagonal cells, a single one divides in f lic plane a, l), o r c, it will protlucc n pair of pentagonal cells ‘Sir Il’Arcy Thompson: Growth a n d Form, ILIli, p. 318. IZnlle\, I. W. The c:ririhium, etc., Am. .JouIn. Rot., 1900, x o l . 7, p1). 417-4.31; J ~ i i r i i .( k n . 1’11) siol., 1920, xol. 2, pp. 51!)-5:13. aiid ~ v i l canse l t v o a d j o i i ~ h gcells to become heptagoiial (fig. 3 ) . If thc six cells surrouiidiiig the dividing cell should also divide in the plmes u, b , anti +-two in each plane and ZITraiigecl as in figure 4-there ~wmldbe altogcther eight new Fig 5 E'igs. 1 t o 1 1)i:igranis illustrating tlw r'fcct ~b w11 division on liesagonal ecI Is : equal nuiiibers of pmtagoas and heptagons are 1mduwd, a n oetagon sonictimcs replacing two heptagons. Fig. 5 A binuclcate dodecagonal cell from t h e pig~ncnted layer of t h r retina of n human tmibryu of S f months. x 700 dialn. All cells other are miarkcd w i t h figurcs i n d i c a t i n g t h e i r nunrher of sides. The :I 31 cells sho\vii is exactly F. liexagoiiul aild six lie\\- pelltagoiial cells ; two of the NITrouiidiiig cells woiild be made heptagonal and two octagonal. The average number of sides f o r these six pentagons, two heptagons, and two ociagons remains exactly six. Extending this procedure so that the 36 cells s ~ ~ r r o i i i ~ lthe i n g cen- tral one divide (and t.wclvc? of them in eac.11 of tlic three t!lierc may be produced 43 iiex liesagoiis plaiies a, $1, aiid a11d 22 ea.ch of pentagons and heptagons. ‘I.11~planes in the t.c?sttlmployeci were plalced a t ra.iidom, except that the plane i u any cell was iiever made the exteiision through i.t of the l . ) I i t l i ~clioseii ~ in ail ad,joi.ning cell. I n tissues t.lierc is evidcncc. tlia t. t,he ext.eiision of a division-plane through. ail adjoiiiiiig cell is not always avoided. Some irregularity of thcb t~t!l.ls,or sliglit obliquity of tlic pl.anes, usually prevcti1t.s the formation of t.he tetrahedral aiigles which are iiievi.t.ablc i.n a tliagram. But the cliagrams reveal the importa1i.t facts that. division of bhe hexagons produces mi equal number of pentagons aiid. heptagons--Hii oct,agoii somct.imcs replacing two .heptagons; aiid that., as division proceeds, both pentagons a i d heptagons are being restored to the prima.ry liexagoiial form. l’hough proba.bly never rea.lizcd in tissues, the a.rra.ngemeiit of‘ planes in figure 2 permit,s restoration of regulai: liexagond oiitliiies throughout. (.!ells with more t1ia.n eight. or less t.haii five sides are not iincommon. A n undivided hexagonal cell, if surrounded by six w h i ~ hbisected radially, would acquire twelve sides. It wonld ht? enclosed by twelve pent,agoiis, which would have made six periphera.1 cells heptagons. Again the t.ottil loss and gain hi sides is equal-one dodecagon, six heptagons and t\velve pent.agoiis averaging si.x sides apiece. I n the pigmeiit layer of the retina I caii recwrd a dodecagon less simple in origin (fig. 5). .It is surrounded by seven hexagons, .fonr ~)cntagoas,a.iid a quadrilat.era1. T h e excess of six sides in t.he c?t.iitralcell is met by R deficiency of six in the cells in coiltact wit,li it.; and the second circ,le of investing cclls-ten hcsi~goiis, four pcliit.agons mid four hepta.gons-is another iiistaiice of thc! prcvailiiig baltmce. .Dodecagons in the retina have h e e ~ iiot~etlby Rouin,-’ and possibly cells with eveii more sides OCCUI’, sirice oiie of the bordering cells in 0111. case h a s two nnclei, iiiciic.at.iveof incomp1et.e division. F o r less t.han fire sides a hexagon presumably divides uneyuall\-. The divisioli ’ Priauaiit ct Houim, Trait6 cl’histologie, T. 2, 191.1, 1). 6 i 4 . DIVISION O F H E X A G O N A L CELLS 335 of a peiitagoii is an altcriiative explanation. I n oiie or the other of these ways doubtless most of the quadrilaterals are produced, aiid they a r e not rare, though triangular cells a r e almost uiikiio.v\.ii. Thus cells with more than seven or less than five sides a r e iii different categories. The former arise from exceptioiial groupiiigs of normal divisions ; hut tlie latter cannot he interprotctl in that wi;\-. The maiiiicr iii which pentagons are produc~?daiid why they a r e small havc already been explained. The hcptagons have becii S ~ J O W T Hto be merely Iievagoiial cells along one side of which cell division has taken place. A t first tlieir perimeters measure six units aiicl their areas a r e those of regular hexagons-2.598. If, Iiowever, these cells become regular l i e p tagons without inci.easc in area, each side is reduced t o 0.846 : should these sides be madc of unit leiigth, the cell must eiilarge hy 39.9 per cent aiid acquire an area of 3.633.. I n a simple epithelium in \vliicli the cells, after dividing, return to their original size, the elid result of a single clivisioii of every cell is clearly both to double the area of the layer a s a whole, aiid to produce from every hexagon another of equal size. For geometrical reasons it happens, curiously, that when the division of R single cell (as in fig. 3 ) introduces a iiew element, although the total increase is closely equiralent to adding a new hexagon of staiidard size, only oiie third of this increment is provided hy the two pciitagonal daughtercells-two thirds come from the formation of tlie two heptagons. Considering tlie cells as regular polygons, three hexagons, instead of adding a fourth and maliing their combined area 4 X 2.598, o r 10.4, become resolved into two pentaGoons arid two heptagons with a total area of 10.7-that is, 2 X 1.720 2 X 3.639. But i t is impossible to bring together regular polygoiis of these shapes without leaving interstices. Tlie problem of correcting the decimals to fit the actual conditions should interest some mathematician of leisure. The extraordinary liiological conclusion is, that ~-1ieiia cell with plastic walls receives a new side tlirough division of ail adjoining cell, it gro~7sas it never would liave done had i t s + ii(ligli1)or rc~rriaiiicclclniescctit. TVlic~na cell divides beside a i~.gularpciitagoii, Tvliicli tliei~eafterbecomes a rcgiilar liesagon of tlic original lciigtli of side, the growth of the pentagoil is 51 1wr t w i l ! ‘I’llc. gro\vll1 w11ich lras bccw : igiietl t o thc~several forms of’ pol?-gons depends npon tlirir acquisition of sides of equal l ~ ~ g t lxit, the same 1 ime 1)ecomiiig regular aiid of minimal pcirimctcr. The cAx-tcliit to n.1iic.h this is true of ichtual cells can l x ~mcasnrccl clii*ectIy. All thts x alls of thcb cclls ontlincd in figurc 1 2 (p.31.5)wero mcasured in a convenieiit pliotographic tiiilargement. rl’heir average length mas made tlie liiiear unit, ~ i i dits square tlie uiiil of surface. The area of every cell was measured wit11 a plaiiirrieter, aiid tlie resnlts a r c presented in t21lde 1. TABLE 1 IVe have assumctl that the pentagons arise from tlic b i s w tioii of liesagoris of average size, wliich means that at the time of thoii* oi.igin tlicir areas a r e 1.3 and tlicir sides arcrage 0.946; and that tlicy grow until nearly regular mid of unit sitlc, tfins iiicreasing by 32 per cent. ‘L’he seven pentagons in figure 12 have grown only 15 per cent,; they are still small, hi1 t Iinvc become approximately regular f o r their short pclrimeters as sliowii in tlie last column of the table. The four l i e p t agoiis, ho~vever,have exceeded the expected growth of 40 per cent, being 4(i per (wit larger than the hexagons, and having sides of more th;m unit length. The small size of the ptiitagoiis and large size of the heptagons in this little group is ilouhtless a chance occurrc’iice. By including more cells- DIS’ISION O F H E X A G O N A L CELLS 337 making fifty in all-the average length of side of the pentagons is raised to 0.996, and that of the hexagons a i d heptagons lowered to 1.006 and 1.009, respectively, indicating that i n general, the average length of side for these three kinds of polygons is the same. Within this average the range is very great-from 0.5 to 1.5-and extremes may be combined in a single perimeter; yet, as shown in the last column of the table, surface tension brings order out of these complcxities by making every perimeter enclose more than 92 per cent of t,he area wliich can possibly be brought within its bounds. The average f o r the twenty-four cells is 96.6 per cent. Since regular polygons caiinot be fitted together, 100 per cent is unattainable, so that the actual percentages a r e distinctly nearer mathematical perfection than they appear. The a p plication of the purely geometrical coiiclusions of our introductory paragraphs to cellular tissue seems therefore fully justified. Figure 12, which we have examined in some detail, is from a valuable paper by Dr. Georg Wetzel--‘ ‘Zur entwicklungsmechanischen Analyse des einf achen pri smatischeii Epithels”5-kindly brought to my attention by my associate, Mr. Weatherford. It is a study of the cell outlines in the pigmented layer of the retina-‘ein zierliches Jlosaik’--u-here cells, as Jones described them i n ’33, a r c “very minute plates of a n hexagonal form, accurately joined by their edge^."^ Covering much of the approximately spherical surface of the eyeball, this layer grows a s nearly uniformly in all directions as perhaps any layer in the body, and therefore is especially favorable for this investigation. Here, as Thompson remarks, “conditions all point and lead to a perfectly symmetrical result. ” Wetzcl counted the number of sides presented by the pigment cells in seven fields of the microscope, and found that the various polygons occurred a s in table 2. ‘Arch. f . Entw. uiechxnik, 1926, U d . 107, pp. 177-183. ‘Thomas Wh:irton Junes: Yotire relatirc t o thc pigmmtum nigruin of t h e eye. Edinburgh Jled. and Surg. Jourii., 1833, 101. 40, p. 81. 338 FREDERIC T. LEWIS He is impressed with the fact that pentagons and heptagons are present in nearly equal numbers,-in some fields exactly even; but in seeking the cause, he does not so much as mention the partitioning effects of cell division, and thus overlooks the simple explanation which accounts f o r the case in hand. Wetzel stresses the geometrical proposition that if a plane is to be covered completely with approximately regular polygons with three sides always meeting at a point, and if with hexagons only pentagons and heptagons are present, then the two latter must occur in equal numbers, however they arise. This is important,-more so than the mathematical formula to express it, supplied him by Reinhardt. As a student of topology, Reinhardt had published his inaugural dissertation “Uber die Zerlegung der Ebene in Polygone,”7 but that he can write also simple equations will appear presently. Let n8,n4,etc. represent the number of octagons, quadrilaterals, etc. which are present, the number of hexagons being omitted as variable and immaterial ; then Reinhardt ’s formula “welche alle Falle umfasst,” as Wetzel expresses it, is as follows: 2n8 m7 =n5 2n4 + + [To include nonagons, should it not be,- + 2x8 + 3%n, %; =‘ n 5 + 2%, ?] Wetzel, verifying the formula as to equal numbers of pentagons and heptagons in the retina, does not consider it profitable to test it further, merely noting that in his preparations the number of quadrilaterals and octagons is very small. Disregarding them, he counted the sides of 456 pigment cells-hexagons, heptagons, and pentagons-for which the average number of sides proves to be 5.98. Including the two quadrilaterals, three octagons and one nonagon which he would disregard, the average becomes 5.99. If we consider the polygons as independent of each other, discounting the fact that they have sides in common, then for mathematically ’ 1naug.-Diss., Frankfurt a.M., 1918, pp. 1-85. perfect results tlie 462 cells should have liad three more sides than were found-2772 instead of 2769. This is perhaps tlie most perfect demonstration of tlie geometrical form of cells ever recorded. Greater pi-ecisioii is liardly expected in the realm of pare physics.* Iiisteacl of seeing iii the large size of the many-sidcd cells ail effect of the iiumber of sides assigned to them, arid vice versa with the small cells, which is our view of tlie matter, TVctzel takes the reverse position. Any vital activity which makes some of the cells large thereby increases their number of sides, lie suggests. “And these larger cells, howwrer they may liave arisen, must occasion in their neighborhood a rearrangement of cell forms, with tlie production of a coi-respoiitling iiumher of cells with fewer sides. ” But G raper recogiiizes that a hexagonal cell “must hecome very large t o reach a single additional cell ;” and later lie writes,-“it must double its size t o come into relation with a seventh cell, and trehlc it for contact with an eighth.”9 It is not clear how those proportions mere cletermiiied, whicli certainly differ from the expected and actnal dimensions of such cclls, a s shown in table 1. What Wetzel descarihes in this general way may actually take place in stratified epithelium. A hexagon may acquire tTnirnprcssed W i t h his omn fiiidings, \vetzcl turns to R L(shrbuch-figure of the r e t i n a l ctslli;, f r o m which he derires t h e shapes of 29 morcz to add to 111s list. Thcse do not better the a i c m g e , and i i s there stwris to be n o r(’ason f o r including them, wc have disregnrrlrd them altogether. G I aper, L u h i g . Eiiie iieuc Arisc~lrnuangiilier pl~ysiologisclicZ c 1 l : ~ u s s c l i ~ l t u i i ~ . Arch. f . Zellforscliung, 1914, Btl. 12, p. 387. Mec.lianische Eetrac~litungc~n untl Vcrsuthc uhrr Zcllforrri und Z ~ l l g r o s s e . Arch. f . Entw.-mcchanlk, 1919, Bd. 45, p. 44i. a scveiitli side 1):- ciilargemcnt, as in figure 6A n may meet 6, rnakiiig both of them hcytagons, and extending between c ant1 Snch a group, in figure ti' \rliicli : ~ J X ' 1.cniierct1 p a n t a g o i d . 613, has been carefiilly outliiied from an actual specimen: o x + h of the four (~11sw a s sectioned through its nucleus. It niakcs a clefinite pattern, hnt one nol characteristic of the retina, and iii fact not ahunclant in the stratified epithelium. For the pentagonal ccllls do not remain quiescent. They, t 00, enlarge, wedging their way Iatcrally, and hecoming hcxagoiial as the\- deprive the heptagons of their extra sides and lift the overlying strata.l" That this has not occurred in the gronp in figure 6 is perhaps clue to the small size of the pciitagons, which a r e but one-half the area of the heptagons, @a A B Fig. 6 1)iagran~ ( A ) a n d canicrit outline ( K ) of f o u r cells of 1ium:in oral r p i t l i c ~ l ~ u i ~F~r.o m a n cml)ryo o f 5 inonthx. X &SO cliain. Extensio~i of c.clls n :ind b until the! nicet produrcs two Iieptagons arid two pentagons. aiitl so happcn to be of the proportion which Griiper specified. If tlie retinal hexagons are made pentagons by the enlargemcnt of acl jacent cells which become heptagons, then the pentagons should have tlie area of the hexagons. They ~voulcl iiot he reduced in size hy this process. The fact that the retiiial pentagons are smaller tliaii the hexagons, and the ahserice of tlie characteristic grouping, lead to tlie rejection of Wetzel's hypotliesis of uriequal gromf h. I t explaiiis rncircly the Tariation in size of polygons of a given iium1)er of sidcs . Another way of accouiitiiig for the equal number of heptagous a r i d pentagons is based upon Griiper 's studies of cell l') 'Phis is furtlicr t1cwrit)ctl 111 t h c €'roc.. Ainer. Arad., 1925, x o l . 61, pl). 30, 31. ' 1 ' 1 1 ~ ct,lls srctionc~(l111 figure (i arc not piiwvi, but tctrak:i~tiec~nhrtl~n. DIVISION O F IIEXAGONBL C E L T S 341 fusion. His premise is that in the adult, when an organ invested by a simple epithelium has ceased to grow and neither enlarges nor diminishes in bulli, there must be a cell lost for every cell added through mitosis, provided the size of the cells remains constant.ll I n regressive orgaris, like the yolksac of the dogfish, which he studied particularly, elimination of cells outstrips production. H e fouiid that this elimination occurred through fusion of hexagonal cells in a strictly simple epithelium, and very interestingly he has described its effects upon the cell pattern. In a hexagoiial mosaic he points out that two adjacent cells a r e always enclosed by eight others, arid should the two fuse and become one binucleate cell, they would still be surrounded hy the eight; whereas should the binucleate condition arise from the division of a nucleus in a single cell, that cell would be surrounded by six only.12 One important feature he omits to mention, namely that with the formation of this large octagonal cell hy fusion (cell a,fig. 7 ) , two adjacent hexagonal cells become pentagoiis (fig. 7, h and c). I n fact, erroneously in his diagram (which is here reproduced in fig. 8) these cells a r e shown as hexagons. It happens, then, that in cell fusion, which is the reverse of cell division, the total loss arid gain in sides is also equal. This outcome, to which apparently Griiper nowhere refers, is further illustrated in the following arrangements. If the fused octagonal cell becomes reduced to normal size, Graper shows (fig. 8) that i t may become hexagonal, but will then be in relation with two more pentagons, sometimes at opposite poles, as in figure 8 at d and e. But these join the group at unstable tetrahedral angles. Should cell f in figure 8 acquire contact with 9, and cell h with i, those hexagons mould become heptagons-the unstable angles would be avoided-and the elid result of this symmetrical fusion would be the loss of one hexagon arid the formation of four pentagons and four heptagons. See footnote 9, page 339. This assunirs that tlic nuclear division occurs in a hexagonal cell. Griiiper does not mention that if nuclear dirision takes place in a n octagonal cell, the conditions which he ascribes t o fusion would he duplicated. la T H E A N A T O M I C A L RECORD, VOL. 3 3 , N O . 5 342 FREDERIC T. LEWIS A variant of the pattern shown in figure 7 is presented in our figure 9, in which the two hexagons are considered to have fused asymmetrically, that on the left draining, as it were, the substance of the one on the right, and so forming the irregular octagon a. Should cell a in figure 9 shrink still further, it might become one of the six approximately pentagonal elements in the ‘rosettes’ (fig. 10) which Graper both observed in tissues and interpreted as evidence of fusion. The same thing would happen if any cell in the hexagonal mosaic should be forced up and out, and the surrounding six should come together below it, as in forming a stratified epithelium. But the cells do not quite meet ‘in einem Punkte ’ as in our diagram and in Graper’s description. Instead, as his figures show, the central meeting place is unstable territory. The group may resolve itself in three ways.13 One of them is by retaining more or less of the contacts shown in figure 9. A second is the arrangement in figure 11, which would come from that in figure 10 if cell b formed contacts with c and i, and cell i with b and h. Instead of a cluster of six pentagons, there would then be two each of pentagons, hexagons, and heptagons, with the average of six sides maintained. The third pattern arises when, in figure 10, cells a, h, and i all acquire contacts with each other, becoming three heptagons, and shutting out h, c, and e as three pentagons. Graper does not analyze these groups even as far as we have done, being content t o say of the arrangement in figure 8,“Allmahlich wird naturlich auch das ursprungliche Sechseckmosaik wiederhergestellt . . . .,, In the pigment layer Wetzel found no rosettes or other unequivocal evidence of f~si0n.l~ Thompson: Growth and Form, p. 375. 141n a portion o f Wetzel’s drawing here reproduced as figure 12, there is a group of six cells which we have marked with asterisks, exactly duplicating the pattern in figure 11. There are two pentagons, two hexagons, and two heptagons, all o f which may be made hexagonal by inserting a central hexagonal cell. We do not infer, however, that a central cell has dropped out, since the same arrangement, as will be seen, can be explained on a different and more probable assumption, l3 Three radically tlift’cwiit I of prodnciiig ail equal number of pciitagons aiitl licptagoiis iii a hcsngonal epithclium liavc iiow heen consicleiwl, i i a m ~ ~ l y1) , cell dii-isioii ; 2 ) iuiequal growth under special coiiclitioiis ; aiitl, 3 ) cell fusion or elimination. Wctzcl, leaviiig out of consideratioil the first alteriiati\.e, conltl not accouiit f o r the particular c1istr.il)ution of the various polygoiis. “1 t n.onlc1 certainly be ~ v o r t l i wliilc,” he writes, “ t o subject t h c ~cwiiclit ions of tlic prodiictioii of iliese clift’erent - Fig 7 fig. 9 Accordingly we haye i.eproclncet1 ii portioii of his figure s h o w ing an appareiitly Iiapliaznrd arixiigemciit of s e ~ e i ipeutiigoiis, twelve liexagoiis, f o u r Iicptagoiis, m i d one octago1l (fig.1 2 ) , and have aiialyzctl it in figure 13. If it is assumed that i i i a group of eigliteeii r~g:.ul:~r liexagoiis, six divide iii tlic 1)laiies intlicated iii figu1.e 1:3, they will yield three liesagoiis aiid iiiiic peiittngoiis. Seren of the latter bear the number 5 iii tlic figui*c;the other two we h a r e made hesagoris 11)- division of tlic mai*ginal cells b a i i d fj. Balaiiciiig the iiiiie pciitagoiis, seveii lieptagoiis aiicl o i i v 341 FlLE1)ERIC T. LEWIS c ~ t a g o i iarc producctl by the divisions specified. F o u r of these heptagons are not within the area. selected, being thc margiiial cells u, c, tl, aiitl f . Air adclitioiial heptagon withiii the group comes from the dix-isioii of the marginal cell 9. If we re-draw figure 13 making c:very cell as nearly equilateral ;ISmay be, figure 14 is tlie result. It looks like epithelium, but oiicl more in repose tliari that of figure 12, ~ ~ l i c r owing e, to somtl natural or artificial stress in the preparation, there is a pronouiiced trend of the cells obliquely up and down the page. 1Cvery cell of the twentyfour lias hecn accounted for, both as t o its num1)er of sidcs aiitl relative position. An occasional area which caii be aiialyzecl in this way does not prove the explanatioii correct. We know that a uniform layer of hexagons cannot be tlie real starting point, but ixtlier ;in area already including pentagons and heptagons with perhaps further deviations. Flour of the six cell divisions in figure 13, whicli have been described as tlie division of hexagons, can equally well be considered divisions of heptagons. Thns let c ~ l c, l which is perliaps a heptagon, divide first. It produces a lieptagon (beneath the octagon in the figure) wliich divides next, and gives rise to a heptagon which divides in turn (and so creates tlic octagon). It will readily he fouiid that tlie entire set of divisions in figarc 2 can como from the successive divisioit of heptagoiis. Since heptagons arc' large cells, siich as would be expected to divide if a unif o r m size is to be maintained, the yucstioii arises wlietlier, after tlie mosaic has become established, tlie division of lirlxagons is not an exceptional occurrence. The bisection of a regular heptagon is accomplished by a liricl passirig from the micldle of any side to the apex of the angle opposite, ~7TTlierc*,with adjoining cells i t would make a Ietralicdrnl angle. That would be avoitled by shiftiiig tlie clivision-plane to oiie side or tlrc other, as ill figure 15 to ah. i\ii unequal clirisioii of the lieptagon ensues, yielding a hexagon and a pentagon. In figure 15 tlie plane ab has been so placrd that tlie resulting cells sliall have the proportionate areas \vliic.li belong to regular hexagons and peiitagons re- 1)IVISION O F H E X A G O N A L C E L L S 345 spectively; 60.2 per cent of the area goes to the former and 39.8 per cent to the latter. The pentagonal product, with an area of 1.448, is 11.5 per cent larger than the pentagon derived from the bisection of the average hexagon. This considerable adrantage would facilitate its assimilation into the d .e Fig. 1 2 A n area of 2 1 ret,iiial pignient cells t,akcn from Wctzel’s “ R l ~ b . 1” (Arrli. f. Entw-ineclianik, 1926, Bd. 107, p. 180) originally drawn expressly to sliow the polygonal f o r m of the cells. Fig. 13, diagram showing how the number of sicles and the relations t o each other of all the cells in figure 12 may b e accounted for, through t,he division of regular hexagons. Fig. 14, diagram constructed by attempting to ninkc the perimeter of every cell niarked o u t in figure 13 niiniiiial f o r the area rnclost?d. I n all three drawings every polygon other tlian a hexagon cmrluscs a fignrc indicating the numlit~rof its sidc3s. mosaic. 'I!he hcxa.gozial product has still greater advantages ; its area is 2.186, and by a growth of 18.8 per cent it becomes R full-sized hexagonal ,cell. A pentagon derired from division of an average hexagon must enlarge by 32.4 per cent., and t,lien, after: receiving an acidit,ioiittl side from the c1ivisic)ii of a n adjaceiit cell, must grow 50.1 per cent more, to accomplish the same resnlt. Clearly, for inaint.aiziiiig tlic hexagona'l 1);i.t.t ~ r i itlie , division of heptagons has decided advantages. 11 typical iastaiice of its occurrence is shown in figure 16. Tlic t.woceiitra.1cells-a 1iexa.goiiand pentagon-have cert.aiiily come from thc division of a common parent as indicat.ed by the position of their nuclei arid t.lie thinness of t.he irit.ervening wall. The parent cell was a heptagon, and its divisiori mtde heptagons of the hexagons which were above and below it. Thc cciitral peiitagoii anti hexagon arc now 42 and 58 per c.ent i*espect.ively of their combined a.rea, as compared wit:ti the geometric 39.8 and 60.2 per cent marked off in figure 1.5. Should the upper 1i.ept.agoriin figure 16 now divide in thc plane cxwssing the cell from the line a, t.he ent.ire group would coiisist of t.hree Iiexagons, one pentagon, a~idone heptagon ; h i t if the plane of divkion i i i the upper cell shoiild be directed toward the licxagoii below, tho group would coiisist of one I1ex2zg011,t.wo pentagons, aiid two 1icpt.agons. The prevalence of 1iexagona.l cells iiidic.at.es l.liat tlie former altcmiative is likely to be chosen, and surface tension offers, perhaps, ii simple meeliartism for making the selection. For the pentagon, having a greater perimeter for its area than t.he hexagon, may have a. greater tcridcricy t:o expand, a.iid so may muse tlica c:mistr.ictioii tfuct to divisioii of tho adjaceiit cell to be tlircctetl towaid it. rl'liat, Iiowctver, is a purely Iiypotlietical ass ump tioii. 'l'he division of ail octagoii offers still greater atlva.iitagcs t.lian t.liat of tlic heptagon. If of full size, it cmi divide int.o li It. is evident that a st.at.istical study of t.he incidenee of niit.ot.ie figurc-s in smie suit,able siniple epithelium, for the purpose of finding in what shapes enil s i m s of cells thcy oeeur most frequently, is a very promising program. Tlit? edls must br oxamintd in surface view or horizontal sections. In stratifid cpitlitlliuni R i tni1:tr olisnrvations would liw of litt.lr significance u n l ( w the ent,irv cc+lls w w ~ llll)~lt~l~~d. 347 DIVISION O F HEXAGONAL CELLS a pair of hexagons which iieecl grow oiily 7.6 per cent t o acquire the average area of hexagons. Rut octagons and cells with nine o r more sides are nsually not of full size. Formed in tlie way that lias been describetl, it is not surprising that often t,hey hare riot grown to tlic great extent necessary to make their sides of unit length. The octagon in figure 12 is no larger than an average heptagon, arid the dodecagon in figlire 5 is uiiahle to fill out its territory and eliminate the iiivadiiig mciiisci of tlie surrounding cells. The mechanical a Fig. 15 Fig. 16 Fig. 1.5 1111ision of a regular 1iclitag1111i n t o a n irregular hexagon and an irregular pvnt:rgon, tlie awas of xhwh a r c t o eat11 otliei as t l i r arms of regulal h c x ~ g o n t o rcgulnr pentagon of unit length of sirltl. Prom calculations 11% Dr. H. W. Rrinkmann. Fig. 16 A n instaricc of the d i on of a Iic~ptagon into 11t~x;igonand pentagon with t h r convcmion of two :uljoining hexagons into heptagons. Pignicnt Iapcr of retina, hunian embryo of t5$ months. X 1000 diani. Note that division of the upper hcptsgon i n the plane a xonld make tlic cntirc group c.onsist of tlilcc. hcxngons, one p e n t a g o n , and one hrptagon. readjustment wlieii sncli a cell divides is f ormidalole, and division is apt to stop with the nuclens. If coristrictioii in the division-plane is prevciitctl, tlie 11ew cxll wall must, be more than double the unit length, and siicli an extensive membrane is riot formed. Thus the retiiial epithelium becomes dotted with large many-sided hiniicleate cells. “111 tlie large cells of tlie human retina, geiierally two riuclei may be recognized” ( TTTetzel). The small cells-quadrilaterals and all the pentamoiionucleate. gons which I liappen to have observed-are 348 FBlmERIC ‘r. I,R\\’TS (iriiper considers that in ‘most organs’ there arc’ cells with t \vo nuclei due to fusion. Otlicw regal-d birincleate cells as oedcmatous or hypertrophic clegenerations, dividing throngli amitosis. Rut if cell division is a method of maintainiiig uniform size in droplets of protoplasm, tlieri these Ininucleatch cells are just such as should he in process of normal division. ‘L’licy presumably are in the telopliase of mitosis. Cptoplasmic division is retarded or prevented since the constriction which it irivolves is mechanically difficnlt. Thnis far, tlie pigmciit cells have heen coilsidered a s plane siirfaccs. I n taking accwuiit of the liciglit of the cells, I have tlependecl on calculations wliicli mere begun by Professor IFT. C. Graustein; most of them, however, wire made during his absence hy his assistant, Dr. H. W. Briilkmann, aiid the following section of this report is largely theirs. Tlie height of a prismatic cell of minimal surface is fomid by dividing four times tlie area of the base by the perimeter,- P If’ tlrr l~asesa r e rcyqlar polygons with sides of unit) leiigtli (\vhicli is made the unit> of measurement) their heights arc t licreby fixed, and also their volumes ; and the possibility of having a small surface for the volume decreases a s tlie sides hecome fewer. In comparisoii with the surface of a sphere of cqual volume ( C ) , their surfaces ( 8 ) meas1m as shown in table 3 ; the quotient S / C may conveniently be called the surfmc. iiitlex of tlie cell or prism. T o assist in visualizing these differences, projections of the prisms of minimal surface have been drawn in figure 17. Interest centers in tlic hexagoiial prism, which h a s the lowest iiiclex of any form whicli can he joined to others of its kind without interstices, anti consequently is the shape assumed by cwlls in simple epithelium. A sigiiificaiit feature, not shown in tfie projection, is the fact that the axes a, b , and c of figure I exactly equal the height, in case the surface is minimal. The name ‘cuboidal epithelium’ is justified, therefore, to the extcrrt that vertical sections of its cells in these plaiics sliould he perfect squares ; aiid if the epithelium is neither stretched nor compressed but rests a t ease, this condition is realized, save that instead of being flat above and below, the cells may be more or less rounded. TliornpsonlG has calculated that tlie height of the rounded cap, in the case of cylindrical drops or cells, should be 0.27 of the radius of the cylinder, and that the entire disc would be that part of a spherical surface subterided by a cone of 60”. Some rounding of the retinal pigment cells has been generally recognized, and sometimes the lateral walls arc not vertical but oblique,-which may be important; hut in this further account it will he assumed that the ends are flat aiitl the sides vertical. TABLE 3 I I ,201 ’ 1.183 1.172 1.165 I On examining vertical sections of the pigmented epithelium, the impression is gained that the cells are not of the several heights shown in figure 17, even if it is assumed that the cells are so aligned as to divide the differences in elevation between the top and bottom surfaces. There are slight irregularities, but unfortunately there is no ready way to tell whether the taller cells are many-sided aiid the shorter ones pentagons o r quaclrilaterals. They all seem t o be very much of the same height. Thompson says of them that they are “flattened to a uniform thickness by the fluid pressure [of the vitreous body] acting radially.” But in cuboidal epithelia subject to no such pressure, it has not yet been shown that tlie cells vary in height to accord with their several shapes in cross section. If the pentagonal cells are not shorter than the minimal hexagons, they have increased in volume against surface tension ; their l6 Growth and Form, page 2213. hu~*face iiiclclx, alrc~atlyhigh ( 1.201) hecomcs higher (1.208). r 1 1 l i e Iieptagons m t l octagons, if rctlucetl to tlie liciglit of licsagoiis, also increasc their surface iiiclic~,-to 1.17(iO a r i d 1. l i M ~ ~ ~ s p e c t i v c ~This l y . is apparently tlie actual coiiditicm in c.nhoida1 epitlielia, tliougli a further study of the heights of i1itlivichal cells is nectlctl. It means that the contacts of the c~c~lls with foreign substances, at their f rec and basal siirfaces, a y e made minimal. 111 stmtified epithelium, d i e r e tlic upper siii*facc of a definite basal layer of cells is c o v e r ~ l1,)- cells ctf similar substance, tlie upper surface of the basal layer is 110 longer flat. Its (~11s map vary in heiglit, aiid arc facetetl i i i m n t n d efforts toward minimal snrface. 1t woultl intc~rcst El D C D cytologist to kiiow the conformation of the upper and nii(lc.r siurfaccs of mi oily emiilsioii, cwiitaining drops of TTari011s sizes, a s it floats 0x1 some watery fluid, but the physicists c~oiisultctlas t o this 11i~vebeen somenkit cvasire. T'niform hc~ipht,l i o \ v t ~ e r;itt:tincd, makes tlie volumes of the cells tlirectly proportional t o their basal areas, ant1 so simplifies tlw followiiig roliimeti~icstudy of their divisioii. T h c ~hcsagoiial prism of minimal siirfaw is shoivii i n p v .jcc*tioii in figurt. 18,c'. Let it divide in halves along the plane ( I , 11, or c of figure 1 , anti the resiiltiiig cells ma>?retain their oi.igiiiu1 liciglits by redwing their sides to 0.8fiN (fig. 18, B ) ; or t Iiep may hecome rtlgular pc~iit:igoiialprisms of iinit sides h? retluciiig thcir heiglits from 1.7321 to 1.2078 (fig. 18,B ) . Rnt ctills of iicithcr of tliese types have hecii obsen-ccl. 111- tlich ~ 1 ~ 7 1 OF ~ 1 HEXAGONAL 0 ~ 351 CELLS stead, the irregular pentagons formed by such a division seem to fill out to approximate regularity while retaining their original heights, which means a growth in volume of 32.4 per cent. There is little doubt that much of this growth is accomplished before division. Division of a heptagon of average size produces a pentagon which need grow only 19 per cent to be of unit sides mid full height; and the largest hexagons, which are the ones likeliest to divide, may be cut into a pair of pentagons already full grown. If the pentagonal cell is of the height of the hexagon of minimal surface, it can reduce its surface index to the most favorable figure possible for a pentagonal prism by increasing the width of its sides to 1.26. This would raise the surface A B C D E Fig. 18 Projections to illustrate possible results of bisecting the hexagonal prisni of minimal surfacc, C , having unit sides and a height of 1.7321. A and B rrgular pentagonal prisms of half the voluinr of C: A, height 1.7321, sides 0.8691; B, sides 1.0, height 1.30%. and E, regular heptagonal prisms of the voluinr of C: D, sides 1.0, height 1.2383; E, height 1.7321, sides 0.84556. index of neighboring cells, however, and is perhaps never fully realized. The largest pentagon of fourteen measured had sides of 1.19; the smallest, of 0.89. I n part this range is due to the varying size of the cells from which the pentagons are cut off; in part, also, it represents growth-the large cells being older than small ones-since there are no cells at hand capable of producing the largest pentagons by simple division. When thc hexagonal cell, figure 18, 6, becomes heptagonal by the division of an adjacent cell, it may bccome regular and retain its height by reducing its width of side to 0.8456 (fig. 18, E ) ; or it may have unit sides by reducing its height to 1.238 (fig. 18, D ) . Again, neither alternative is adopted, but the cell grovvs so that it has the cross section of D and the height of E , with surface index of 1.176. I have 110 eviclcncc that it reclnces this index by increasing its lieight above the general level axid acquiring the optimum proportions of fiq- 17, C. rl’hc surface indices thus f a r recorded a r e those of geometrical Itrisms. A n approximate estimate of the indices f o r tlic twenty-four cells included in figure 12 may be obtained as follows. F r o m the area of each cell in surface view, its hciglit may bc calculated, assuming that its surface is minimal. The avorage of these heights may tlieri be taken as the iuniform thicliness of the epithelium. With the average leiigtli of side of the cells in this group a s tlic unit of measarement. t lie arerage of their heights proves to he 1.66-somewhat shorter tlinii that of the regular hexagonal prism of minimal surface, which is 1.73. After the volume of evei*y cell had been determined on the arbitrary assumption of a uniform height of 1.66, Doctor Brinkmann calculated their sui.face iiidices with the results shown in table 4. For tlie tliicltriess post nlatccl, which is such as may actuall? lie foiund uiider favorahlc conditions, tweiity-four cells of tlic shapes outliiied in figure 13 have surface areas averaging hut 1.5 per cent more tliari the theoretical minimal. Xiiice the minimal prisms caniiot exist in combination witlioiit iiitersticcs, their cleviatioit from mathematical perfection may hc distinctly less than 1.5 per cent. To this extent the reqniremciiits of minimal surface may be met, even though the heights of the cells are made uniform. A n inevitahlc question at this point is the natnrc of the growth wliicli occurs when a cell receives ~ J Iadditional sicle 1)y division of its neighbor. There seems to be 110 evidence cells. Conscthat it is at tlie expense of the surro~iiidii~g qnently the growth mnst be due to the acquisition of subiiices from the tissue fluids beneath, and in this process osmosis donlitless plays a large part. Wetzel, who supposes that the turgor of the cells-their internal or osmotic pressure --controls their shape eve11 in cross section, declares that ‘‘\vith equal turgor in all cells and in all directions, 0111)- ecyiiilire lateral hexagonal prisms woulcl occur.” But in his figures, a s in actnal preparations, the interfaces between pentagons, hexagons a i d heptagons a r e neither coiivex nor concave but conspicnously flat, compatible with an isotonic condition. There a r e occasioiial exceptions, as in the dodecagon of figure 5 with its coilcave walls. Leduc in describing his artificial cells, which a r e drops of a potassium ferrocyaiiide solution spreading over a gelatin plate, states explicitly that “the lines of contact between the drops a r e straight when the solutions a r e isotonic ; they a r e the more curved, the greater the difference in osmotic teiisioii of the liquids; the convexity is found on the hvpertoiiic side, the concavity on the side of the h ~ p o - i pc~itagoiis. . . . . . . . . . . . . , 1 2 hexagons. . . . . . . . . . . . . . 4 hcptagoni: . . . . . . . . . . . . . 1 octagon . . . . . . . . . . . . . . . ~ I 1.211-1.24; A / . 1 . 2 % 1.19 1.21; 1.199 1 . 1 8 - 1 . 2 0 ; .I?. 1.190 1.182 1,201 1.183 l.lT2 1 .l(i.5 tonic liquid.’”7 The straightness or curvature of cell walls, though sometimes significant i n this way, does not localize growth in the cells which receive new sides, making it a n osmotic phenomenon. It is a problem in that field in which ‘the whole future of cytology lies. ’ Time is sometimes considered a fourth dimension. The morphologist may deal with it by estimating the rate at which new cells a r e added to the retinal epithelium. The posterior half of the eyeball may, without great error, be considered a perfect hemisphere. Its radius, i n a human embryo of 10 weeks (70 mm. C.-R.), wliich my assistant, Rlr. €3. J. Anson, kindly dissected and measured f o r me, is 2.2 mm., and its surface is therefore 30 sq.mm. Three moiiths later, in a n em- ’‘ Leduc, S. p. 1301. Diffusion tlans la gblatinc. (”. lt., Acad. (1. Sci., Paris, 1901, T. 132, 206 mm. (C.-R.), A h . Xiisoii fiiids the radius 6.2 mm. : the surface is tlieii 241 sq.mm. This eightfold iiicrease means Illat cvery pigment wll, during the three months in qurstioii, llns divided three times. Altlioiigli the ratti of oiie divisioii ppr month f o r every cell is it great dccliiie from the origiiid irnpcltus, it is still very high. Should it coiitiiiue unaBateil, a t tlie age of five years the e ~ - e l ~~voulcl ll exccetl the size of tlie globc. At birth, from fig:uYes rccorclcd by JYeiss,ls te1~c&i-ial tllo r:idius of the pigrnented layer is pei*liaps7.tj mm., aiicl the i1rc1a of tlie posterior hemisphere is thcn 353 sq.mm. If the (>ells1 l a 1 ~remained of the same size, 46 per cent of those present a t *j1 months have tliricled oiice. After hirtli tlic t ~ ~ - c ~ hgi*o\vs iIl rapidly diiriiig tlic first few )-c'ars, ant1 slo\vljtlici.raftei*, iuitil in the adult the area of this postcrior liemisplicrci is 830 sq.mrn. Assuming that cells a r e not lost ailti I-eylncd-aiid therci seems t o he no c ~ k I e i i (that ~ tliep arc riot as pcrmaiierit a s iierve cells,-then e r e i y pigment wll tlivitles OIICP after I)ii.tli, hut only a small proportion of them twice. 111this coiiclnsioii somc account is tak(w of theii. g ~ n ern1 increasci iii size. At 5 Imoiitlis, the ttvcrage cliameter of fifty hexagonal ccfls;, meas;uretl from m g l e t o angle, v a s foalid t o he 13.7 1-1, alicl the i i w a of such ;I liexagon (corrected t o 96 per cent of the area of a r*egular oiie) is 11'7 sq.1~. There arc, thrreforc, approsimately 2,050,000 pigment cells in the posterior half of thc ~j-cballat that stag:. if, a s stated in the preceding paragraph, 46 per cent 0-1' these cells divide diiriiig the 3; mouths 1)eforc birth, tlieii there is approximately m e cell tlivih i o i l per hour among 5500 cclls. If the process of mitosis takes two hours, a mitotic figure may lie expcctctl among ovclry 2i50 cells. After hirtli they become much rarer, and tliroug.11cytoplasmic growth the diameters of the pigment c ~ l l s Imorne illcreased to 12--17 p according to Ronin,-i2-18 11, Koellikcr. TYhether the mitotic figiii~ls are miiformly distribiitcd tliroi~gliout the Iiemispliere, or, as in so rnany organs, a i ~ 1 ) 1 . ~ of o , 13. CIm d:rs SVnchstum des nrcnscli~ichcw~ u g r s . L h i : i t . H c f t v , .il)th. I , 1x97, Utl. 8, 1'1'. 191-248. DIVISIOW OF H E X A G O S A L CELLS 355 limited to t.he periphery or other spccial regions, leaving large groups of cells undist,urbed, requires further iiivestigati.on. Apparently the gr0mt.h of thc subjaccnt layer of rods and cones gently stretches the pigmciit layer in a.11 dircetions. The pigment c.ells t,akc sustmancc from the uiiderlyiiig tissue fluids and grow, a.nd t.hc largcst of t.hesc protoplasmic droplets here and there divide, forming pciitagons and heptagons in the kaleidoscopic way which. this study may help to visualize. By the use of such rnet.liods as Voiiwiller has described,19 it may yet be possible, in some rapidly growiiig vegeta.hle tissue, to observe this process in action. uVonwiller, P. Xeue Wege der Gcwebelehrc, IT. Zeitschr. f . Anat. u. Entn., 19.5, Bd. 7G, pp. 497-533.

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