- Email: [email protected]
A mathematical model for germ cell determination process and effect of ultraviolet light on the process
J. theor. Biol. (1976) 58, 15-32
A Mathematical Model for Germ Cell Determination Process and Effect of Ultraviolet Light on the Process KIJIRI
AND N.EGAMI
Zoological Institute, Faculty of Science, University of Tokyo, Hongo, Tokyo 113, Japan (Received 26 December 1974) The processof germcell determinationmay be regardedasone of the best examplesof cell determination processin general.The germ cellscan be readily traced through its developmentalstagesby their possessionof “germinal plasm”. During cleavage, the germinal plasm is distributed among cellsof vegetal blastomeres,and the germinal plasm-bearingcells later becomegermcells.Tadpoleslacking germcellscould be obtained by irradiating the vegetal hemisphereof egg with ultraviolet light (UV). The presentexperimental data revealed that there is a numerical relationship betweenthe germ cells formed and irradiating W dose. In this article, we present a mathematical model of the germ cell determination process.At a critical period, germinal plasmis supposed to exert somereaction to the nucleusof a cell, and the amount of this reaction will decidewhether this cell differentiates into a germ cell or a somaticcell. Following the introduction of the concept of the distribution function, germinalplasmpartition model is presented.Our basicassumption is that germinal plasm partition follows the probabilistic rule governed only by geometricrelation betweenarea of the cell surfaceand the germinal plasm contained. After obtaining the generalshapeof the distribution function from the partition model, we draw the mathematical equation which describesthe relationship betweensurvival of germ cells and UV dose. This equation showsa good agreementwith the actual experimental data. Finally, we point out that it is possibleto get a clue to what kind of molecule the germ cell determinant.is, through the analysisof actual data using this mathematicalequation.
1. Introdnction One of the great problems in developmental biology is the determination process or mechanism of the fate of the cells during the development. After extensive studies in the past, it now seemsalmost certain that cell differentiation during development is initiated by an effect of the heterogeneous egg cytoplasm upon originally identical nuclei. A wide range of vertebrate
16
K.
IJIRI
AND
N.
EGAMI
and invertebrate eggs are believed to contain gradients or localized regions of cytoplasm. However, except for very few cases, it has been impossible to identify the morphological basis for this localized information in the egg cytoplasm. Since the germinal plasm is so clearly recognizable during the early stage of development (see Plate I), a further study of their behavior and characteristics during early embryonic development should be helpful in understanding how this one case of embryonic determination occurs. Moreover, it may hopefully indicate ways in which other types of localized information of egg may function in cell determination process. Usually the problem of cell determination or cyto-differentiation process in embryonic development, involves so many factors and interactions. In contrast to most cells, germ cells are supposed to receive little or no influences from other cells during early embryonic development, until they reach the future gonadal area or genital ridges. This simplicity of influences from other environmental cells and tissues, is another reason that we hope studies of germinal plasm and its role in germ cell determination may bring us a clear-cut picture about the localized cytoplasmic information in general, and its role in cell determination process. Since Weismann (1883, 1892) and Nussbaum (1880) first discussed the continuity of the germ cells from one generation to the next, study in this field of embryology has resulted in a measure of agreement that the definitive germ cells of the vertebrates originate in certain large embryonic cells which make their appearance at an early period of embryonic development. In amphibian eggs, the germ cell line can be readily traced through its developmental stages by their possession of a particular kind of cytoplasm (Bounoure, 1934). Since then, several authors have studied the behavior of the germinal plasm during the normal course of embryonic development (Blackler, 1958; DiBerardino, 1961; Gipouloux, 1962). The first appearance of the germinal plasm was noticed in fertilized unsegmented egg, where small distinctly staining islands of cytoplasm are localized just under the cell membrane in an area around the vegetative pole of the egg. During cleavage the germinal plasm is distributed among the vegetal blastomeres. These germinal plasm-bearing cells migrate later into the genital ridges, where they eventually become the primordial germ cells. That the vegetal cytoplasmic localizations in the uncleavaged eggs later act as germ cell determinants was indicated by experiments of Bounoure (1939) and Bounoure, Aubry & Huck (1954), in which the vegetal hemisphere of Rana temporaria eggs were irradiated with ultraviolet light (UV). Eggs treated in this way developed into sterile animals. Recently the effect of W on the germ cell production was confirmed in R. pipiens (Smith, 1966)
PLATE I
(a)-(d). [facingp. 16
PLATE I (e)-(h). PLATE I. Behavior of germinal plasm in embryonic development. (a). Tangential section of the vegetative pole region at two-cell stage. Fused islands of germinal plasm (arrows) are situated parallel to the cell membrane. x 220. (b) Blastula stage. Germinal plasm (arrows)containing cells are situated in the endodermal mass. x 68. (c) In late blastula to gastrula, cells which contain germinal plasm show the intracellular migration of germinal plasm (gp) from the cell periphery to the juxta-nuclear position. n. nucleus. x870. (d) By the time gastrulation is under way, every cell possessing germinal plasm exhibits it in the nuclear position. x 870. (e) Endodermal mass in the trunk region at an early larval stage. x 63. (f) Higher magnification of the arrowed region in (e). Note the lobed nucleus containing two nucleoli, and the germinal plasm surrounding the nucleus. x790. (g) A cross section of a stage 47 tadpole. x60. (h) Higher magnification of the gonadal area arrowed in (g). Note the large germ cells (arrows). Direct count of germ cell number was carried out from such serial sections. x 845.
GERM
CELL
DETERMINATION
PROCESS
17
Ijiri, and XenopusZueuis(Tanabe& Kotani, 1974;Ijiri & Egami, 1975~~; 1976),in a more quantitativeway. In this article, we presenta mathematicalmodel on the determinationof germcells. 2. BiologicalObservations We tracedthe behaviorof this germinalplasmin development of X. laevis (SouthAfrican clawedtoad) embryo. During the first cleavagethe position of germinalplasm hardly changed [Plate I(a)]. During further cleavage,germinalplasm was found in some of the large blastomeresaround the vegetativepole, still situatedin the subcorticalplasm.With further divisions,more cells which did not contain germinalplasmwerecreated.At blastula,the vegetativeblastomeres moved closelyupwardsalong the animal-vegetative axis of the egg,and acquired a more central position in the endodermalmassat an advancedblastula stage [Plate I(b)]. Such migration of cells was ascribedto the general morphogeneticmovement, and the germinal plasm-bearingcells also followedthis movementas endodermalcells. In late blastulato gastrula,cellswhich containedgerminalplasm showed the intracellularmigration of the germinalplasm from the cell periphery to the juxta-nuclearposition [Plate I(c), (d)]. By the time gastrulationwas really under way, every cell possessing germinalplasmexhibitedit in the nuclearposition.The further displacement of the germinalplasm-containing cells evidentlyfollowedthe gastrulationmovements,so that it wasfound in the endodermalmass in the trunk region at an early larval stage[Plate I(e), (f)]. Later at tadpole stage,thesegerminalplasm-bearingcells were observedto migrate into the genital ridges,as the primordial germ cells [Plate I(g), (WI. It is generallyacceptedthat gastrulationis the time when somaticcells start differentiatingin their specificways through mitosis and cell-to-cell interactionsbetweenthem. Blackler (1970)suggeststhat just before this important eventtakesplace,thesegerminalplasm-bearing cells keepthemselvesin a stateof beingmitotically inhibited, so that they receiveno cellto-cell interactionsand can protecttheir unspecialized embryoniccharacter. Sofar, this kind of viewis a merespeculation.But it seemsquitereasonable to supposethat at least someinfluenceis exertedon the nucleusby the investinggerminalplasmat a certainperiod,and this influenceto the nucleus may play an importantrole in the determinationof this cell into a germcell. With theseobservationsand assumptionsin mind, we now proceedto the mathematicalformulation of the germ cell determinationprocessand the effectof UV on this process. T.B.
18
K. IJIRI
AND
N. EGAMI
3. Germ Cell Determination
Fmwtion G
Although we are not certain at which stage the germinal plasm exerts its influence on the nucleus, we suppose at least the existence of such stage, and name this period the “critical period”. At this critical period, let us suppose that germinal plasm exerts some active chemical reaction P to the nucleus of the cell and this amount of active reaction P will decide whether this cell becomes a germ cell or a somatic cell. If the amount of chemical reaction P exceeds a certain threshold value PO, then this cell will be destined to become a germ cell. We represent the fate of a cell in the following way: G = l[P-PO] where l[x] denotes the Heaviside unit step-function, l[x]
=
1
1, x 2.0 0, x < 0.
(1) defined as: (2)
Let us call G the “germ cell determination function”, admitting only the value 1 and 0. It is clear that G = 1 corresponds to a germ cell and G = 0 to a somatic cell. We now consider the effect of W irradiation on the determination of germ cells. The actual experimental methods are as follows. The vegetal pole of a fertilized egg is irradiated by UV, and after irradiation the egg is allowed to develop to the stage at which we can determine the number of germ cells. This experiment will give us the relation between the UV dose
FIG. 1. At the critical period, germinal plasm (gp) of volume v exerts some active chemical reaction P to the nucleus (N) in a cell. This amount of P is assumed to determine whether this cell becomes a germ cell or a somatic cell according to equation (1).
GERM
CELL
DETERMINATION
PROCESS
19
and its effect on the germ cell determination process expressed in a quantitative way, such as germ cell number or survival of germ cells. To connect our treatment from now on with the event on the molecular level, we suppose that the molecular level reaction of P, expressed in p, obeys the hit theory with respect to UV dose, A brief explanation and the result of hit theory are given here. Suppose material irradiated contains a large number of identical units each capable of some function which can be measured in a quantitative way. These units may be molecules such as enzymes capable of catalyzing a specific reaction, m RNAs going to synthesize certain proteins, and others. If a single structure in each unit must be intact for its future functioning, then the hit theory predicts that the measured function still remaining after irradiation will be an exponential form with respect to the dose. A wide variety of biological molecules are inactivated by UV, according to this exponential law. It is therefore quite reasonable to suppose the relation between the molecular level reaction p and UV dose D as: p = p. eeaD.
(3) Here 0 is a constant, often termed an “inactivation cross-section”. Then the macro-reaction P in one cell is in proportion to the product of p and the volume of germinal plasm z, contained in this cell. Thus, P is given by P=A1esuDv (4) where Ai is a positive proportionality constant. From equations (1) and (4), the function G can be written as follows. G(D, v) = l[A, emuD v-P,& (5) Having set v&l)
= A; ‘PO e”” = A ebD
(6)
and utilizing the fact that division of the argument of Heaviside function by a positive value does not change its response, we can write equation (5) as: G(D, v) = l[v-v,(D)]. (7) Equation (7) implies that determination of a certain cell whether to differentiate into a germ cell or a somatic cell, depends upon the comparison between the germinal plasm volume v in that cell and the threshold t+,(D). 4. Survival of Germ Cells
We now introduce another important function f(v), which expresses the number of cells that contain certain amount of germinal plasm v at the critical period. With this distribution function f(v) and equation (7), the
20
K. IJIRI
AND
N. EGAMI
number of germ cells formed when irradiated at UV dose D is given as follows : N =or G(D, v)f(v) da = OO~Djj(v)dv. Here, vO is the maximum implies.
(8)
value of v. Figure 2 illustrates what equation (8)
FIG. 2. Number of germ cells N at UV irradiation of dose D, can be obtained as the number of cells contain germinal plasm volume larger than vti(D). This figure illustrates what equation (8) implies.
Survival of germ cells irradiated at a certain UV dose is defined as the ratio of the number of germ cells formed in an irradiated embryo to the number of non-irradiated one. Thus, as survival S, we obtain
It is obvious that if we can get the shape of the distribution function f(v), then survival of germ cells at a certain UV dose can be calculated by means of equation (9). 5. Germinal Plasm Partition Model and Shape off(v)
In this section, we consider the shape of the distribution function j(v) from the model of germinal plasm partition during cleavage. From the observations through histological sections of embryos, we have concluded that the general appearance of germinal plasm and its position in a cell can be simplified as follows. A cell is taken to be a sphere and germinal plasm as making a round sheet of curved surface located parallel to the cell membrane, rather than a three-dimensional mass.
GERM
CELL
DETERMINATION
PROCESS
21
Nothing mysterious is proposed here for the behavior of germinal plasm, and we assume that it is completely a matter of chance, during cleavage, how the germinal plasm in one cell is distributed between the two daughter cells. Thus it is our basic assumption here that germinal plasm partition follows the probabilistic rule governed only by geometric relation between area of the cell surface and the germinal plasm contained. To understand the thinking procedure more easily, we first consider a two-dimensional model of germinal plasm partition, then proceed to a three-dimensional model which represents the actual case more closely and gives almost the same result as that of two-dimensional model. (A)
TWO-DIMENSIONAL
MODEL
Initial values of the radius of the cell and the length of germinal plasm are set as r. and lo respectively. After the nth division, cell radius becomes r,, = 2-““r0 by conservation of total area. (Since this is a two-dimensional model, total area is preserved throughout cell division.) As the length of germinal plasm, the normalized value to the initial value Z, is taken. Let us suppose a cell that contains the germinal plasm of normalized length a (0 r a 5 1) after the nth division. At the nf lth division, this germinal plasm of the length a will be distributed between the two daughter cells according to the probability shown in Fig. 3(b). Here 2”“/3a expresses the probability that the cell division furrow splits the germinal plasm into two fragments. It is derived as the ratio of germinal plasm length to half the circumference of the cell, /I being put as Z&w,. The probability shown in Fig. 2(b) is calculated so that the total would become two cells, for quite obvious reasons. With this result in mind, we proceed to decide the distribution function f(x). Suppose the number of cells that contain the normalized length of germinal plasm x is f,(x) after the nth division. Then at the next n+ lth division, the distribution functionf,, 1(x) will become
where 61 is small. Equation (10) is replaced by equation (ll), which is quite a reasonable condition.
when x P 0,
f,,+ 1(x) = s’f,,(a)2”‘” + ‘p da -t-f,(x)(l - 2”‘2fix).
(11)
Taking derivative with resiect to X, and multiplying
both sides by x2, we get
22
K. IJIRI
AND
N. EGAMI
(b) FIG. 3. (a), (b). A cell containing germinal plasm of normalized length a after the nth division, and its division into two cells. In (a), both the length of germinal plasm and ceil radius are normalized to the initial germinal plasm length I,,. (b) shows the probability of the way of partition, in which germinal plasm of a in one cell will be distributed between two daughter cells. Here, total probability is taken as two cells.
GERM
Suppose f,‘+l(x) (x3f,(4>’ = 0,
CELL
DETERMINATION
resembles f,‘(x)
23
PROCESS
at a large IZ in equation
fn(4= 3
(12), then (13)
where q is a constant independent of x. Thus simple shape of distribution function was obtained, stating that f,(x) is inversely proportional to the cube of X, at a large FZ. (B> THREE-DIMENSIONAL
MODEL
In the case of a three-dimensional model, almost the same procedure can be applied. Initial values of the radius of the cell and the half of the apical angle of the germinal plasm are set as r0 and & respectively. After the nth division, cell radius becomes r, = 2-“‘3r,,, this time with the condition that total volume is unchanged. When cell radius r, and the angle 8 are given, the surface area of germinal plasm denoted by S, is expressed as s = 2741 -cos e>. Thus, as for the area of germinal plasm, the normalized value S, is given by x = 2-2”‘3p-1[1 -cos e(x)] where /?El-cos&.
(14) value to the initial (1%
(16) Suppose a cell that contains the germinal plasm of normalized area s after the nth division. At the n + 1th division, this germinal plasm of area s will be distributed between the two daughter cells according to the probability, which takes the value of 1 -sin 13at x = 0 and x = s, and a uniform value of 2 sin 6/s over the interval (0, s). Here 1 -sin 6 was obtained as the probability that the cell division furrow does not split the germinal plasm into two fragments. Let us suppose the number of cells that contain the normalized area of germinal plasm x is f,(x) after the nth division. Then at the next nf lth division, the distribution function f, + 1(x) will become f,+ &c) = i f,(s) 2 ““, ‘6) ds + f,(x)[l -sin 0(x)] (17) x with the relation between x and 8(x) expressed as equation (15). Taking derivative with respect to x, and multiplying both sides by X, we get (.&(x)x sin f?(x)} = -x(X+
h) - fXx>> -fXd
sin W.
(18)
24
K.
IJIRI
AND
N.
EGAMI
FIG. 4. The half of apical angIe of germinal plasm (gp) is denoted by 8. The area of germinal plasm S is then expressed as S = 27~r,,~(1 -cos e), where r,, is the cell radius after the nth division.
Suppose f.‘+ I(x) resembles f,‘(x) at a large 12in equation (18), then {J;I(x)x sin t!?(x)}’ = -f,(x)
sin 19(x).
This differential equation is solved as
f,(x)= 2 sinc16(x)’ Setting h = 22n’3p in equation (15), we get 112
sin@)
= hx cx - 1 ( >
,
which gives us
f,(x) = x3(2/hx q-
1p2
(21)
where q is a constant independent of x. Let us define the slope off,(x) on the log-log plot as --a. That is
--cI= d PnfnWl - d [In x] ’ Then, from equation (21), we will have the following relation: CL=3-2-hx.
1
(22)
GERM
CELL
DETERMINATION
PROCESS
25
Since the interval (0, l/h) is the possible domain of x [see equation (A3) in Appendix A], the range of a can be obtained as between 2 and 2.5, from such simplified treatment of equation (17). The computer simulation was performed using equation (A4) in Appendix A, which is a realistic modification of equation (17). The result is shown in Fig. 5. As n approaches 6, 7, 8, the slope on the log-log plot has already become a straight line and takes its value as about -3, which means f,(x) has become inversely proportional to the cube of x. The less steep region far apart from the maximum value of x is negligible, since such part does not actually participate in the determination of germ cells.
05
I.0
X FIG. 5. The result of computer simulation performed by equation (A4) in Appendix A. In this log-log plot, as n approaches 6, 7, 8, within the range of properly large x, the slope has already become around -3, which shows the relation fn(x) = q/x3. Here, /%,fO(l) were chosen as fi = 0.05, and&(l) = 1.
26
From above, function -a on
K.
IJIRI
AND
(C)
SHAPE
N.
EGAMI
OFf(V)
the result of the mathematical treatment and computer simulation we now assume that within the range of [A, a,] the distribution at the critical period takes its form as almost a straight line of slope the fog-log plot. Thus, the distribution function can be written as
where a is a constant. We know, from our treatment so far, that a is nearly equal to 3. To get a precise value of M at a certain ith division, we have to depend on the computer simulation with the knowledge of the value of the initial parameter p. But, to a certain extent, we can take the value of M as about 3, and we now are sure that it is a fairly good estimate (Appendix C).
6. Survival Curve S Now that the shape of the function f(v) is obtained, following the discussions in sections 3 and 4, we can easily draw the equations for germ cell number N and survival S. From equations (6), (8) and (23), we get N = T f(v) dv =,ID$ w(D)
dv = ---
(e-(~-l)~D-c)
(24)
where c1--
A
a-l .
(25)
0 vo N takes the value 0 for the value of D beyond a certain dose I),, where Do is determined as the dose when v,(D) exceeds the value vo. Thus A euDo = vo, which is rearranged to give:
As for the survival S, in the same manner, we get,
(27) i
s = 0,
D > Do.
)
GERM
CELL
DETERMINATION
7. Experimental
27
PROCESS
Data and Discussion
Fertilized eggs just before the first cleavage were placed on a quartz dish and the vegetal hemisphere was irradiated by UV light (254 nm). After irradiation, the eggs were allowed to develop to the tadpoles of stage 47 at which we can determine the number of germ cells (Ijiri & Egami, 1975b). All tadpoles were fixed and serially sectioned transversely in paraffin. After staining, direct count on the number of germ cells were performed. The result is shown in Fig. 6 in the form of dose-survival relationship curve. As can be seen, the experimental results show quite a good agreement with the theoretical equation of survival curve. The dose DsurE denotes the irradiating UV light dose arrived at the vegetal surface of the egg. In order to decide the inactivation cross-section c of the molecular level reaction p, it is necessary to know the actual dose D arriving at the germinal plasm
0 I 0
100
300 J/m’
200 I OS
I I
7surf
(a-lb0
FIG. 6. Dose-survival relationship expressed in a semi-log plot. The real line expresses the experimental survival curve: fertilized eggs just before the first cleavage were irradiated, and the germ cell number was counted from the serial sections of stage 47 tadpoles (data from Ijiri & Egami, unpublished). A theoretical curve of equation (27) with c = 0.25 is shown in a dotted line, taking the value (a-l)aD as an abscissa.
28
K.
IJIRI
AND
N.
EGAMI
itself. Measurements of the transmittance of the vegetal subcortical plasm of a X. laevis egg and the depth of germinal plasm from the egg surface at the stage of UV irradiation are now under way by the authors. From these measurements, we will be able to tell the real inactivation cross section of the molecular reaction, and we hope its value gives us a clue to the biological nature of the germ cell determinant. As stated before, during cleavage the germinal plasm is distributed among the vegetative blastomeres. In our model we have postulated that it is a matter of chance how the germinal plasm is distributed among these cells, and we have found out that the final result of this model shows quite a good agreement with the experimental data we obtained. This suggests the validity of our probabilistic model of germ cell determination process discussed well in the previous sections. In Drosophila, cytochemical techniques have shown the germinal plasm to be rich in RNA (Paulson & Waterhouse, 1960; Mahowald, 1962; Counce, 1963). Based on this, Mahowald (1968, 1971) has postulated that the germinal plasm of Drosophila contains messenger RNA(m RNA), which codes for the synthesis of specific proteins involved in the determination of germ eels. The same postulation is reasonable for the amphibian germinal plasm, since cytochemical studies have also demonstrated the existence of RNA within germinal plasm of amphibian eggs (Blackler, 1958; Czolowska, 1969). Many workers have demonstrated the presence of “masked” maternal m RNA in eggs of a variety of species (reviewed by Spirin, 1966), although it is not yet clarified whether this “masked” m RNA has a role in such embryonic determination process. On the other hand, the germinal plasm region is obviously rich in ribosomes, both in insect and amphibian eggs, and this alone could account for positive RNA staining. Moreover, cytochemical methods only reveal what kinds of substances exist there, and do not tell their actual participation in germ cell determination process. As Williams & Smith (1971) write, the most direct identification of the biochemical nature of the germinal plasm and a definite confirmation that they represent the germ cell determinants can be obtained, only if it becomes possible to isolate them from the remainder of the egg. At present, no one has ever succeeded in isolating this substance because of the technical difficulties which come from its small quantity and its specific localization in the whole egg. With such a background, in this report we have pointed out that through the analysis of the quantitative study of UV irradiation experiments, it is possible to find out the value of molecular inactivation cross-section d of this germ cell determinant, of which value may offer an access to its biochemical nature.
PROCESS
29
We are indebted to the criticisms of Dr J. Sakamoto, Dr colleagues in our laboratory who have suffered through the treatments performed in this article. Also, thanks are offered President of Atomi University, Niiza, Saitama, for supplying toads.
Y. Saimi, and other development of the to Dr. S. Kambara, us mature Xenopus
GERM
CELL
DETERMINATION
REFERENCES (1958). J. Embryol. exp. Morph. 6, 491. (1970). In Current Topics in Developmental Biology (A. A. Moscona & A. Monroy, eds) Vol. 5, p. 71. New York: Academic Press. BOIJNO~RE, L. (1934). Annls. Sci. nat. 10, 67. BOUNOURE, L. (1939). In L’Origine des Cellules Reproductrices et le Problime deala Lignge Germinale. Paris: Gauthier-Villars. BOUNOURE, L., AUBRY, R. & HUCK, M. L. (1954). J. Embryol. exp. Morph. 2,245. Cowce, S. J. (1963). J. Morph. 112, 129. CZOLOWSKA, R. (1969). J. Embryol. exp. Morph. 22,229. DIBERARDG-GO, M. (1961). J. Embryol. exp. Morph. 9, 507. Gtpou~oux, J. D. (1962). C. r. hebd. Sianc. Acad. Sci., Paris 254,2433. IJIRI, K. (1976). J. Embryol. exp. Morph. (in press). IJIRI, K. & EGAMI, N. (1975a). J. Radiat. Res., Tokyo 16, 71. IJIRI, K. & E~AMI, N. (19756). J. Embryo/. exp. Morph. 34, 687. MAHOWALD, A. P. (1962). J. exp. Zoo/. 151,201. MAHOWALD, A. P. (1968). J. exp. Zool. 167, 237. MAHOWALD, A. P. (1971). J. exp. Zool. 176, 345. NIEUWKOOP, P. D. &FABER, J. (1956). Normal Table of Xenopus laevis (Daud.). Amsterdam: North Holland Publishing Co. Nussbaum, M. (1880). Arch. mikrosk. Anat. EntwMech. 80, 1. POULSON, D. F. &WATERHOUSE, D. F. (1960). Aust. J. biol. Sci. 13, 541. SMITH. L. D. (1966). Devel. Biol. 14. 330. SPIRI$ A. S. (1966). In Current Toaics in Developmental Biology (A. A. Moscona & A. Monroy, eds). Vol. 1, p. 1. New York: Academic Press. BLACKLER, BLACKLER,
A. W. A. W.
TANABE, K. & KOTANI, WEISMANN, A. (1883).
Fischer. WEISMANN, WILLIAMS,
M. (1974). J. Embryol. Die Entstehung der
exp. Morph. Sexualzellen
31, 89. bei den
A. (1892). Das Keimplasma. Eine Theorie der Vererbung. M. A. & SMITH, L. D. (1971). Devel. Biol. 25, 568. APPENDIX
Hydromedusen.
Jena:
kna.
A
Computer Simulation
(i) Two-dimensional
model
The more the cell divisions take place, the smaller the cell radius becomes. So it is quite obvious that as the cleavage progresses it becomes impossible
for a cell to contain the germinal plasm of its original length Z,-,being intact from splitting. This means the maximum value of x, the normalized length of germinal plasm, shifts smaller than one after a certain number of divisions. Figure 7(a)-(c) represents the relation between the length of germinal plasm and cell radius, which shows one-to-one correspondence to the sign of l -2”‘*j?x. At the next n i- lth division, germinal plasm of (c) has no
K. IJIRI
30
AND N. EGAMI
I--&>O
(a)
(b)
FIG. 7. The relation between germinal plasm length x and cell radius. The cases (a), (b), (c) correspond to the signs of 1 - 2”j2. fix, i.e. >O, =0,
way to escape being split. The length of (b) will be the maximum remaining length of germinal plasm, thus the value of x satisfying the equation 1-2’@/3x = 0, which we denote by a&+ l), gives the maximum limit. c&z + 1) = pzp* -1 Avoiding the somewhat complicated splitting case of (c), we take a rough approximation, regarding all the case of(c) as equal to that of(b). Following rough correction to equation (11) from this bold approximation will be enough to serve our present purpose. When x S a&+ I), ao(n+l) fn+ 114 = j f1,W”‘2 + ‘P da x
and when x > ~(n + l), Ll(X)
= 0.
I
GERM
CELL
DETERMINATION
31
PRQCESS
(ii) Three dimensional model The correction to equation (17) can be presented in the same manner as was proposed in two-dimensional model. The value of x satisfying equation (15) with B(x) = n/2, which we denote by s&f l), represents the maximum limit. s&-i 1) = 2+‘3p-l. 6’4 Then, correction to equation (17) becomes when x 5 s,(rt+l) \ so(n+1) f,+ 1(x) = f f,(s) ‘F ds -Ix + f,(x)( 1 - sin 8(x)) + s&
;j@) f&S ds ‘(A4) son-b
when x > s,,(n + 1) f,+ l(“4 = 0.
APPENDIX
I
B
Application of Multi-targets
Hit Theory
In section 3, we assumed that the dose-response curve of molecular level reaction p as p = p. emuD. (A3 Sometimes this relation can take a little more complex form such as p = pO(l -(l -e-oD)k). W) This generalized hit theory is called “multi-targets” hit theory, and of course includes equation (A5) as a special case of k = 1. For this generalized relation, the same procedure so far performed can be applicable, and the results come out as follows. s=
P--(1
-,-~D)ky-LC
) OsDgDD,
l-c cz-
A a-1
0 vo
(A71
.
Here Do is the value which satisfies the equation 1-(1-e -aDo ) k = ,+(a-1) . WI However, our experimental result shown in Fig. 6 rather supports the case of single-target hit theory, i.e. the special case of k = 1.
32
K.
IJIRI
AND
APPENDIX
N.
EGAMI
C
Instead of computer simulations on equations (A2) and (A4) which contain rough approximations, more generalized equations together with their analytical treatments have been developed. They are to be discwssed elsewhere (Ogihara & Ijiri, in preparation).
A Mathematical Model for Germ Cell Determination Process and Effect of Ultraviolet Light on the Process KIJIRI
AND N.EGAMI
Zoological Institute, Faculty of Science, University of Tokyo, Hongo, Tokyo 113, Japan (Received 26 December 1974) The processof germcell determinationmay be regardedasone of the best examplesof cell determination processin general.The germ cellscan be readily traced through its developmentalstagesby their possessionof “germinal plasm”. During cleavage, the germinal plasm is distributed among cellsof vegetal blastomeres,and the germinal plasm-bearingcells later becomegermcells.Tadpoleslacking germcellscould be obtained by irradiating the vegetal hemisphereof egg with ultraviolet light (UV). The presentexperimental data revealed that there is a numerical relationship betweenthe germ cells formed and irradiating W dose. In this article, we present a mathematical model of the germ cell determination process.At a critical period, germinal plasmis supposed to exert somereaction to the nucleusof a cell, and the amount of this reaction will decidewhether this cell differentiates into a germ cell or a somaticcell. Following the introduction of the concept of the distribution function, germinalplasmpartition model is presented.Our basicassumption is that germinal plasm partition follows the probabilistic rule governed only by geometricrelation betweenarea of the cell surfaceand the germinal plasm contained. After obtaining the generalshapeof the distribution function from the partition model, we draw the mathematical equation which describesthe relationship betweensurvival of germ cells and UV dose. This equation showsa good agreementwith the actual experimental data. Finally, we point out that it is possibleto get a clue to what kind of molecule the germ cell determinant.is, through the analysisof actual data using this mathematicalequation.
1. Introdnction One of the great problems in developmental biology is the determination process or mechanism of the fate of the cells during the development. After extensive studies in the past, it now seemsalmost certain that cell differentiation during development is initiated by an effect of the heterogeneous egg cytoplasm upon originally identical nuclei. A wide range of vertebrate
16
K.
IJIRI
AND
N.
EGAMI
and invertebrate eggs are believed to contain gradients or localized regions of cytoplasm. However, except for very few cases, it has been impossible to identify the morphological basis for this localized information in the egg cytoplasm. Since the germinal plasm is so clearly recognizable during the early stage of development (see Plate I), a further study of their behavior and characteristics during early embryonic development should be helpful in understanding how this one case of embryonic determination occurs. Moreover, it may hopefully indicate ways in which other types of localized information of egg may function in cell determination process. Usually the problem of cell determination or cyto-differentiation process in embryonic development, involves so many factors and interactions. In contrast to most cells, germ cells are supposed to receive little or no influences from other cells during early embryonic development, until they reach the future gonadal area or genital ridges. This simplicity of influences from other environmental cells and tissues, is another reason that we hope studies of germinal plasm and its role in germ cell determination may bring us a clear-cut picture about the localized cytoplasmic information in general, and its role in cell determination process. Since Weismann (1883, 1892) and Nussbaum (1880) first discussed the continuity of the germ cells from one generation to the next, study in this field of embryology has resulted in a measure of agreement that the definitive germ cells of the vertebrates originate in certain large embryonic cells which make their appearance at an early period of embryonic development. In amphibian eggs, the germ cell line can be readily traced through its developmental stages by their possession of a particular kind of cytoplasm (Bounoure, 1934). Since then, several authors have studied the behavior of the germinal plasm during the normal course of embryonic development (Blackler, 1958; DiBerardino, 1961; Gipouloux, 1962). The first appearance of the germinal plasm was noticed in fertilized unsegmented egg, where small distinctly staining islands of cytoplasm are localized just under the cell membrane in an area around the vegetative pole of the egg. During cleavage the germinal plasm is distributed among the vegetal blastomeres. These germinal plasm-bearing cells migrate later into the genital ridges, where they eventually become the primordial germ cells. That the vegetal cytoplasmic localizations in the uncleavaged eggs later act as germ cell determinants was indicated by experiments of Bounoure (1939) and Bounoure, Aubry & Huck (1954), in which the vegetal hemisphere of Rana temporaria eggs were irradiated with ultraviolet light (UV). Eggs treated in this way developed into sterile animals. Recently the effect of W on the germ cell production was confirmed in R. pipiens (Smith, 1966)
PLATE I
(a)-(d). [facingp. 16
PLATE I (e)-(h). PLATE I. Behavior of germinal plasm in embryonic development. (a). Tangential section of the vegetative pole region at two-cell stage. Fused islands of germinal plasm (arrows) are situated parallel to the cell membrane. x 220. (b) Blastula stage. Germinal plasm (arrows)containing cells are situated in the endodermal mass. x 68. (c) In late blastula to gastrula, cells which contain germinal plasm show the intracellular migration of germinal plasm (gp) from the cell periphery to the juxta-nuclear position. n. nucleus. x870. (d) By the time gastrulation is under way, every cell possessing germinal plasm exhibits it in the nuclear position. x 870. (e) Endodermal mass in the trunk region at an early larval stage. x 63. (f) Higher magnification of the arrowed region in (e). Note the lobed nucleus containing two nucleoli, and the germinal plasm surrounding the nucleus. x790. (g) A cross section of a stage 47 tadpole. x60. (h) Higher magnification of the gonadal area arrowed in (g). Note the large germ cells (arrows). Direct count of germ cell number was carried out from such serial sections. x 845.
GERM
CELL
DETERMINATION
PROCESS
17
Ijiri, and XenopusZueuis(Tanabe& Kotani, 1974;Ijiri & Egami, 1975~~; 1976),in a more quantitativeway. In this article, we presenta mathematicalmodel on the determinationof germcells. 2. BiologicalObservations We tracedthe behaviorof this germinalplasmin development of X. laevis (SouthAfrican clawedtoad) embryo. During the first cleavagethe position of germinalplasm hardly changed [Plate I(a)]. During further cleavage,germinalplasm was found in some of the large blastomeresaround the vegetativepole, still situatedin the subcorticalplasm.With further divisions,more cells which did not contain germinalplasmwerecreated.At blastula,the vegetativeblastomeres moved closelyupwardsalong the animal-vegetative axis of the egg,and acquired a more central position in the endodermalmassat an advancedblastula stage [Plate I(b)]. Such migration of cells was ascribedto the general morphogeneticmovement, and the germinal plasm-bearingcells also followedthis movementas endodermalcells. In late blastulato gastrula,cellswhich containedgerminalplasm showed the intracellularmigration of the germinalplasm from the cell periphery to the juxta-nuclearposition [Plate I(c), (d)]. By the time gastrulationwas really under way, every cell possessing germinalplasmexhibitedit in the nuclearposition.The further displacement of the germinalplasm-containing cells evidentlyfollowedthe gastrulationmovements,so that it wasfound in the endodermalmass in the trunk region at an early larval stage[Plate I(e), (f)]. Later at tadpole stage,thesegerminalplasm-bearingcells were observedto migrate into the genital ridges,as the primordial germ cells [Plate I(g), (WI. It is generallyacceptedthat gastrulationis the time when somaticcells start differentiatingin their specificways through mitosis and cell-to-cell interactionsbetweenthem. Blackler (1970)suggeststhat just before this important eventtakesplace,thesegerminalplasm-bearing cells keepthemselvesin a stateof beingmitotically inhibited, so that they receiveno cellto-cell interactionsand can protecttheir unspecialized embryoniccharacter. Sofar, this kind of viewis a merespeculation.But it seemsquitereasonable to supposethat at least someinfluenceis exertedon the nucleusby the investinggerminalplasmat a certainperiod,and this influenceto the nucleus may play an importantrole in the determinationof this cell into a germcell. With theseobservationsand assumptionsin mind, we now proceedto the mathematicalformulation of the germ cell determinationprocessand the effectof UV on this process. T.B.
18
K. IJIRI
AND
N. EGAMI
3. Germ Cell Determination
Fmwtion G
Although we are not certain at which stage the germinal plasm exerts its influence on the nucleus, we suppose at least the existence of such stage, and name this period the “critical period”. At this critical period, let us suppose that germinal plasm exerts some active chemical reaction P to the nucleus of the cell and this amount of active reaction P will decide whether this cell becomes a germ cell or a somatic cell. If the amount of chemical reaction P exceeds a certain threshold value PO, then this cell will be destined to become a germ cell. We represent the fate of a cell in the following way: G = l[P-PO] where l[x] denotes the Heaviside unit step-function, l[x]
=
1
1, x 2.0 0, x < 0.
(1) defined as: (2)
Let us call G the “germ cell determination function”, admitting only the value 1 and 0. It is clear that G = 1 corresponds to a germ cell and G = 0 to a somatic cell. We now consider the effect of W irradiation on the determination of germ cells. The actual experimental methods are as follows. The vegetal pole of a fertilized egg is irradiated by UV, and after irradiation the egg is allowed to develop to the stage at which we can determine the number of germ cells. This experiment will give us the relation between the UV dose
FIG. 1. At the critical period, germinal plasm (gp) of volume v exerts some active chemical reaction P to the nucleus (N) in a cell. This amount of P is assumed to determine whether this cell becomes a germ cell or a somatic cell according to equation (1).
GERM
CELL
DETERMINATION
PROCESS
19
and its effect on the germ cell determination process expressed in a quantitative way, such as germ cell number or survival of germ cells. To connect our treatment from now on with the event on the molecular level, we suppose that the molecular level reaction of P, expressed in p, obeys the hit theory with respect to UV dose, A brief explanation and the result of hit theory are given here. Suppose material irradiated contains a large number of identical units each capable of some function which can be measured in a quantitative way. These units may be molecules such as enzymes capable of catalyzing a specific reaction, m RNAs going to synthesize certain proteins, and others. If a single structure in each unit must be intact for its future functioning, then the hit theory predicts that the measured function still remaining after irradiation will be an exponential form with respect to the dose. A wide variety of biological molecules are inactivated by UV, according to this exponential law. It is therefore quite reasonable to suppose the relation between the molecular level reaction p and UV dose D as: p = p. eeaD.
(3) Here 0 is a constant, often termed an “inactivation cross-section”. Then the macro-reaction P in one cell is in proportion to the product of p and the volume of germinal plasm z, contained in this cell. Thus, P is given by P=A1esuDv (4) where Ai is a positive proportionality constant. From equations (1) and (4), the function G can be written as follows. G(D, v) = l[A, emuD v-P,& (5) Having set v&l)
= A; ‘PO e”” = A ebD
(6)
and utilizing the fact that division of the argument of Heaviside function by a positive value does not change its response, we can write equation (5) as: G(D, v) = l[v-v,(D)]. (7) Equation (7) implies that determination of a certain cell whether to differentiate into a germ cell or a somatic cell, depends upon the comparison between the germinal plasm volume v in that cell and the threshold t+,(D). 4. Survival of Germ Cells
We now introduce another important function f(v), which expresses the number of cells that contain certain amount of germinal plasm v at the critical period. With this distribution function f(v) and equation (7), the
20
K. IJIRI
AND
N. EGAMI
number of germ cells formed when irradiated at UV dose D is given as follows : N =or G(D, v)f(v) da = OO~Djj(v)dv. Here, vO is the maximum implies.
(8)
value of v. Figure 2 illustrates what equation (8)
FIG. 2. Number of germ cells N at UV irradiation of dose D, can be obtained as the number of cells contain germinal plasm volume larger than vti(D). This figure illustrates what equation (8) implies.
Survival of germ cells irradiated at a certain UV dose is defined as the ratio of the number of germ cells formed in an irradiated embryo to the number of non-irradiated one. Thus, as survival S, we obtain
It is obvious that if we can get the shape of the distribution function f(v), then survival of germ cells at a certain UV dose can be calculated by means of equation (9). 5. Germinal Plasm Partition Model and Shape off(v)
In this section, we consider the shape of the distribution function j(v) from the model of germinal plasm partition during cleavage. From the observations through histological sections of embryos, we have concluded that the general appearance of germinal plasm and its position in a cell can be simplified as follows. A cell is taken to be a sphere and germinal plasm as making a round sheet of curved surface located parallel to the cell membrane, rather than a three-dimensional mass.
GERM
CELL
DETERMINATION
PROCESS
21
Nothing mysterious is proposed here for the behavior of germinal plasm, and we assume that it is completely a matter of chance, during cleavage, how the germinal plasm in one cell is distributed between the two daughter cells. Thus it is our basic assumption here that germinal plasm partition follows the probabilistic rule governed only by geometric relation between area of the cell surface and the germinal plasm contained. To understand the thinking procedure more easily, we first consider a two-dimensional model of germinal plasm partition, then proceed to a three-dimensional model which represents the actual case more closely and gives almost the same result as that of two-dimensional model. (A)
TWO-DIMENSIONAL
MODEL
Initial values of the radius of the cell and the length of germinal plasm are set as r. and lo respectively. After the nth division, cell radius becomes r,, = 2-““r0 by conservation of total area. (Since this is a two-dimensional model, total area is preserved throughout cell division.) As the length of germinal plasm, the normalized value to the initial value Z, is taken. Let us suppose a cell that contains the germinal plasm of normalized length a (0 r a 5 1) after the nth division. At the nf lth division, this germinal plasm of the length a will be distributed between the two daughter cells according to the probability shown in Fig. 3(b). Here 2”“/3a expresses the probability that the cell division furrow splits the germinal plasm into two fragments. It is derived as the ratio of germinal plasm length to half the circumference of the cell, /I being put as Z&w,. The probability shown in Fig. 2(b) is calculated so that the total would become two cells, for quite obvious reasons. With this result in mind, we proceed to decide the distribution function f(x). Suppose the number of cells that contain the normalized length of germinal plasm x is f,(x) after the nth division. Then at the next n+ lth division, the distribution functionf,, 1(x) will become
where 61 is small. Equation (10) is replaced by equation (ll), which is quite a reasonable condition.
when x P 0,
f,,+ 1(x) = s’f,,(a)2”‘” + ‘p da -t-f,(x)(l - 2”‘2fix).
(11)
Taking derivative with resiect to X, and multiplying
both sides by x2, we get
22
K. IJIRI
AND
N. EGAMI
(b) FIG. 3. (a), (b). A cell containing germinal plasm of normalized length a after the nth division, and its division into two cells. In (a), both the length of germinal plasm and ceil radius are normalized to the initial germinal plasm length I,,. (b) shows the probability of the way of partition, in which germinal plasm of a in one cell will be distributed between two daughter cells. Here, total probability is taken as two cells.
GERM
Suppose f,‘+l(x) (x3f,(4>’ = 0,
CELL
DETERMINATION
resembles f,‘(x)
23
PROCESS
at a large IZ in equation
fn(4= 3
(12), then (13)
where q is a constant independent of x. Thus simple shape of distribution function was obtained, stating that f,(x) is inversely proportional to the cube of X, at a large FZ. (B> THREE-DIMENSIONAL
MODEL
In the case of a three-dimensional model, almost the same procedure can be applied. Initial values of the radius of the cell and the half of the apical angle of the germinal plasm are set as r0 and & respectively. After the nth division, cell radius becomes r, = 2-“‘3r,,, this time with the condition that total volume is unchanged. When cell radius r, and the angle 8 are given, the surface area of germinal plasm denoted by S, is expressed as s = 2741 -cos e>. Thus, as for the area of germinal plasm, the normalized value S, is given by x = 2-2”‘3p-1[1 -cos e(x)] where /?El-cos&.
(14) value to the initial (1%
(16) Suppose a cell that contains the germinal plasm of normalized area s after the nth division. At the n + 1th division, this germinal plasm of area s will be distributed between the two daughter cells according to the probability, which takes the value of 1 -sin 13at x = 0 and x = s, and a uniform value of 2 sin 6/s over the interval (0, s). Here 1 -sin 6 was obtained as the probability that the cell division furrow does not split the germinal plasm into two fragments. Let us suppose the number of cells that contain the normalized area of germinal plasm x is f,(x) after the nth division. Then at the next nf lth division, the distribution function f, + 1(x) will become f,+ &c) = i f,(s) 2 ““, ‘6) ds + f,(x)[l -sin 0(x)] (17) x with the relation between x and 8(x) expressed as equation (15). Taking derivative with respect to x, and multiplying both sides by X, we get (.&(x)x sin f?(x)} = -x(X+
h) - fXx>> -fXd
sin W.
(18)
24
K.
IJIRI
AND
N.
EGAMI
FIG. 4. The half of apical angIe of germinal plasm (gp) is denoted by 8. The area of germinal plasm S is then expressed as S = 27~r,,~(1 -cos e), where r,, is the cell radius after the nth division.
Suppose f.‘+ I(x) resembles f,‘(x) at a large 12in equation (18), then {J;I(x)x sin t!?(x)}’ = -f,(x)
sin 19(x).
This differential equation is solved as
f,(x)= 2 sinc16(x)’ Setting h = 22n’3p in equation (15), we get 112
sin@)
= hx cx - 1 ( >
,
which gives us
f,(x) = x3(2/hx q-
1p2
(21)
where q is a constant independent of x. Let us define the slope off,(x) on the log-log plot as --a. That is
--cI= d PnfnWl - d [In x] ’ Then, from equation (21), we will have the following relation: CL=3-2-hx.
1
(22)
GERM
CELL
DETERMINATION
PROCESS
25
Since the interval (0, l/h) is the possible domain of x [see equation (A3) in Appendix A], the range of a can be obtained as between 2 and 2.5, from such simplified treatment of equation (17). The computer simulation was performed using equation (A4) in Appendix A, which is a realistic modification of equation (17). The result is shown in Fig. 5. As n approaches 6, 7, 8, the slope on the log-log plot has already become a straight line and takes its value as about -3, which means f,(x) has become inversely proportional to the cube of x. The less steep region far apart from the maximum value of x is negligible, since such part does not actually participate in the determination of germ cells.
05
I.0
X FIG. 5. The result of computer simulation performed by equation (A4) in Appendix A. In this log-log plot, as n approaches 6, 7, 8, within the range of properly large x, the slope has already become around -3, which shows the relation fn(x) = q/x3. Here, /%,fO(l) were chosen as fi = 0.05, and&(l) = 1.
26
From above, function -a on
K.
IJIRI
AND
(C)
SHAPE
N.
EGAMI
OFf(V)
the result of the mathematical treatment and computer simulation we now assume that within the range of [A, a,] the distribution at the critical period takes its form as almost a straight line of slope the fog-log plot. Thus, the distribution function can be written as
where a is a constant. We know, from our treatment so far, that a is nearly equal to 3. To get a precise value of M at a certain ith division, we have to depend on the computer simulation with the knowledge of the value of the initial parameter p. But, to a certain extent, we can take the value of M as about 3, and we now are sure that it is a fairly good estimate (Appendix C).
6. Survival Curve S Now that the shape of the function f(v) is obtained, following the discussions in sections 3 and 4, we can easily draw the equations for germ cell number N and survival S. From equations (6), (8) and (23), we get N = T f(v) dv =,ID$ w(D)
dv = ---
(e-(~-l)~D-c)
(24)
where c1--
A
a-l .
(25)
0 vo N takes the value 0 for the value of D beyond a certain dose I),, where Do is determined as the dose when v,(D) exceeds the value vo. Thus A euDo = vo, which is rearranged to give:
As for the survival S, in the same manner, we get,
(27) i
s = 0,
D > Do.
)
GERM
CELL
DETERMINATION
7. Experimental
27
PROCESS
Data and Discussion
Fertilized eggs just before the first cleavage were placed on a quartz dish and the vegetal hemisphere was irradiated by UV light (254 nm). After irradiation, the eggs were allowed to develop to the tadpoles of stage 47 at which we can determine the number of germ cells (Ijiri & Egami, 1975b). All tadpoles were fixed and serially sectioned transversely in paraffin. After staining, direct count on the number of germ cells were performed. The result is shown in Fig. 6 in the form of dose-survival relationship curve. As can be seen, the experimental results show quite a good agreement with the theoretical equation of survival curve. The dose DsurE denotes the irradiating UV light dose arrived at the vegetal surface of the egg. In order to decide the inactivation cross-section c of the molecular level reaction p, it is necessary to know the actual dose D arriving at the germinal plasm
0 I 0
100
300 J/m’
200 I OS
I I
7surf
(a-lb0
FIG. 6. Dose-survival relationship expressed in a semi-log plot. The real line expresses the experimental survival curve: fertilized eggs just before the first cleavage were irradiated, and the germ cell number was counted from the serial sections of stage 47 tadpoles (data from Ijiri & Egami, unpublished). A theoretical curve of equation (27) with c = 0.25 is shown in a dotted line, taking the value (a-l)aD as an abscissa.
28
K.
IJIRI
AND
N.
EGAMI
itself. Measurements of the transmittance of the vegetal subcortical plasm of a X. laevis egg and the depth of germinal plasm from the egg surface at the stage of UV irradiation are now under way by the authors. From these measurements, we will be able to tell the real inactivation cross section of the molecular reaction, and we hope its value gives us a clue to the biological nature of the germ cell determinant. As stated before, during cleavage the germinal plasm is distributed among the vegetative blastomeres. In our model we have postulated that it is a matter of chance how the germinal plasm is distributed among these cells, and we have found out that the final result of this model shows quite a good agreement with the experimental data we obtained. This suggests the validity of our probabilistic model of germ cell determination process discussed well in the previous sections. In Drosophila, cytochemical techniques have shown the germinal plasm to be rich in RNA (Paulson & Waterhouse, 1960; Mahowald, 1962; Counce, 1963). Based on this, Mahowald (1968, 1971) has postulated that the germinal plasm of Drosophila contains messenger RNA(m RNA), which codes for the synthesis of specific proteins involved in the determination of germ eels. The same postulation is reasonable for the amphibian germinal plasm, since cytochemical studies have also demonstrated the existence of RNA within germinal plasm of amphibian eggs (Blackler, 1958; Czolowska, 1969). Many workers have demonstrated the presence of “masked” maternal m RNA in eggs of a variety of species (reviewed by Spirin, 1966), although it is not yet clarified whether this “masked” m RNA has a role in such embryonic determination process. On the other hand, the germinal plasm region is obviously rich in ribosomes, both in insect and amphibian eggs, and this alone could account for positive RNA staining. Moreover, cytochemical methods only reveal what kinds of substances exist there, and do not tell their actual participation in germ cell determination process. As Williams & Smith (1971) write, the most direct identification of the biochemical nature of the germinal plasm and a definite confirmation that they represent the germ cell determinants can be obtained, only if it becomes possible to isolate them from the remainder of the egg. At present, no one has ever succeeded in isolating this substance because of the technical difficulties which come from its small quantity and its specific localization in the whole egg. With such a background, in this report we have pointed out that through the analysis of the quantitative study of UV irradiation experiments, it is possible to find out the value of molecular inactivation cross-section d of this germ cell determinant, of which value may offer an access to its biochemical nature.
PROCESS
29
We are indebted to the criticisms of Dr J. Sakamoto, Dr colleagues in our laboratory who have suffered through the treatments performed in this article. Also, thanks are offered President of Atomi University, Niiza, Saitama, for supplying toads.
Y. Saimi, and other development of the to Dr. S. Kambara, us mature Xenopus
GERM
CELL
DETERMINATION
REFERENCES (1958). J. Embryol. exp. Morph. 6, 491. (1970). In Current Topics in Developmental Biology (A. A. Moscona & A. Monroy, eds) Vol. 5, p. 71. New York: Academic Press. BOIJNO~RE, L. (1934). Annls. Sci. nat. 10, 67. BOUNOURE, L. (1939). In L’Origine des Cellules Reproductrices et le Problime deala Lignge Germinale. Paris: Gauthier-Villars. BOUNOURE, L., AUBRY, R. & HUCK, M. L. (1954). J. Embryol. exp. Morph. 2,245. Cowce, S. J. (1963). J. Morph. 112, 129. CZOLOWSKA, R. (1969). J. Embryol. exp. Morph. 22,229. DIBERARDG-GO, M. (1961). J. Embryol. exp. Morph. 9, 507. Gtpou~oux, J. D. (1962). C. r. hebd. Sianc. Acad. Sci., Paris 254,2433. IJIRI, K. (1976). J. Embryol. exp. Morph. (in press). IJIRI, K. & EGAMI, N. (1975a). J. Radiat. Res., Tokyo 16, 71. IJIRI, K. & E~AMI, N. (19756). J. Embryo/. exp. Morph. 34, 687. MAHOWALD, A. P. (1962). J. exp. Zoo/. 151,201. MAHOWALD, A. P. (1968). J. exp. Zool. 167, 237. MAHOWALD, A. P. (1971). J. exp. Zool. 176, 345. NIEUWKOOP, P. D. &FABER, J. (1956). Normal Table of Xenopus laevis (Daud.). Amsterdam: North Holland Publishing Co. Nussbaum, M. (1880). Arch. mikrosk. Anat. EntwMech. 80, 1. POULSON, D. F. &WATERHOUSE, D. F. (1960). Aust. J. biol. Sci. 13, 541. SMITH. L. D. (1966). Devel. Biol. 14. 330. SPIRI$ A. S. (1966). In Current Toaics in Developmental Biology (A. A. Moscona & A. Monroy, eds). Vol. 1, p. 1. New York: Academic Press. BLACKLER, BLACKLER,
A. W. A. W.
TANABE, K. & KOTANI, WEISMANN, A. (1883).
Fischer. WEISMANN, WILLIAMS,
M. (1974). J. Embryol. Die Entstehung der
exp. Morph. Sexualzellen
31, 89. bei den
A. (1892). Das Keimplasma. Eine Theorie der Vererbung. M. A. & SMITH, L. D. (1971). Devel. Biol. 25, 568. APPENDIX
Hydromedusen.
Jena:
kna.
A
Computer Simulation
(i) Two-dimensional
model
The more the cell divisions take place, the smaller the cell radius becomes. So it is quite obvious that as the cleavage progresses it becomes impossible
for a cell to contain the germinal plasm of its original length Z,-,being intact from splitting. This means the maximum value of x, the normalized length of germinal plasm, shifts smaller than one after a certain number of divisions. Figure 7(a)-(c) represents the relation between the length of germinal plasm and cell radius, which shows one-to-one correspondence to the sign of l -2”‘*j?x. At the next n i- lth division, germinal plasm of (c) has no
K. IJIRI
30
AND N. EGAMI
I--&>O
(a)
(b)
FIG. 7. The relation between germinal plasm length x and cell radius. The cases (a), (b), (c) correspond to the signs of 1 - 2”j2. fix, i.e. >O, =0,
way to escape being split. The length of (b) will be the maximum remaining length of germinal plasm, thus the value of x satisfying the equation 1-2’@/3x = 0, which we denote by a&+ l), gives the maximum limit. c&z + 1) = pzp* -1 Avoiding the somewhat complicated splitting case of (c), we take a rough approximation, regarding all the case of(c) as equal to that of(b). Following rough correction to equation (11) from this bold approximation will be enough to serve our present purpose. When x S a&+ I), ao(n+l) fn+ 114 = j f1,W”‘2 + ‘P da x
and when x > ~(n + l), Ll(X)
= 0.
I
GERM
CELL
DETERMINATION
31
PRQCESS
(ii) Three dimensional model The correction to equation (17) can be presented in the same manner as was proposed in two-dimensional model. The value of x satisfying equation (15) with B(x) = n/2, which we denote by s&f l), represents the maximum limit. s&-i 1) = 2+‘3p-l. 6’4 Then, correction to equation (17) becomes when x 5 s,(rt+l) \ so(n+1) f,+ 1(x) = f f,(s) ‘F ds -Ix + f,(x)( 1 - sin 8(x)) + s&
;j@) f&S ds ‘(A4) son-b
when x > s,,(n + 1) f,+ l(“4 = 0.
APPENDIX
I
B
Application of Multi-targets
Hit Theory
In section 3, we assumed that the dose-response curve of molecular level reaction p as p = p. emuD. (A3 Sometimes this relation can take a little more complex form such as p = pO(l -(l -e-oD)k). W) This generalized hit theory is called “multi-targets” hit theory, and of course includes equation (A5) as a special case of k = 1. For this generalized relation, the same procedure so far performed can be applicable, and the results come out as follows. s=
P--(1
-,-~D)ky-LC
) OsDgDD,
l-c cz-
A a-1
0 vo
(A71
.
Here Do is the value which satisfies the equation 1-(1-e -aDo ) k = ,+(a-1) . WI However, our experimental result shown in Fig. 6 rather supports the case of single-target hit theory, i.e. the special case of k = 1.
32
K.
IJIRI
AND
APPENDIX
N.
EGAMI
C
Instead of computer simulations on equations (A2) and (A4) which contain rough approximations, more generalized equations together with their analytical treatments have been developed. They are to be discwssed elsewhere (Ogihara & Ijiri, in preparation).