Experimental
Cell Research 32, 59- 75 (1963)
MECHANICAL 1. SURFACE
59
PROPERTIES
FORCE AND ELASTIC
OF SEA URCHIN
MODULUS
EGGS
OF THE CELL MEMBRANE
Y. HIRAMOTOZ Zoological
Institute, Faculty of Science, University of Tokyo, Tokyo, and Misaki Biological Station, Miura-ski, Kanagawa-ken, Japan
Marine
Received November 17, 1962
DETAILED knowledge of the mechanical properties of the cell is important in studies of cellular phenomena, especially cell division, because any deformations of the cell including cleavage must be due to the forces developed in different parts of the cell. The experiments described in this and following papers were initiated to determine mechanical properties of the sea urchin egg, which is one of the experimental materials frequently used in the study of cell division. Measurements on mechanical properties of cells have been reviewed by Harvey and Danielli [7]. Among various methods, Cole’s compression one [l ] is most satisfactory. He calculated the surface forces of sea urchin eggs from the forces required to compress the eggs between two parallel plates and their compressed forms. Cole and Michaelis [2] found that there was no marked change in the surface force during fertilization. This result was rather unexpected in view of the fact that many other physical and morphological properties change during fertilization in sea urchin eggs, which suggests structural changes of the egg. Therefore, it was desired to repeat their experiment in detail. In the present series of investigations, mechanical properties of sea urchin eggs before, during and after fertilization and during cleavage were measured by a compression method similar to that used by Cole [ 11. In the present paper, surface forces, internal pressures of the cell and elastic moduli of the cell membranes in unfertilized eggs and fertilized eggs at various stages before the first cleavage are described. In the following paper [15], changes in the stiffness of the egg as a whole during fertilization, during one cell stage and during cleavage will be reported. 1 Most of the experiments in this and following papers were carried out in 1951 and 1952, and the results were read before the 23rd Annual Meeting of the Zoological Society of Japan, Oct. 1952 [ll]. * Present address: Misaki Marine Riological Station, University of Tokyo, Miura-shi, Kanagawaken, Japan. Experimental
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Y. Hiramoto
While this manuscript was being prepared, remarkable investigations on mechanical properties of egg cells were reported by British investigators [16, 17, 18, 19, 231. They determined relations between deformation of the cell surface and negative pressure applied to a portion of the cell with a fine pipette in sea urchin and amphibian eggs, and calculated elastic moduli of the cell membranes from the relations. In the present papers, the results of their experiments will be discussed, together with those of the author’s experiments. MATERIALS
AND
METHODS
The materials used were the eggs of the sea urchin, Hemicentrotus pulcherrimus. All the experiments were carried out at 15°C. At this temperature, eggs began to cleave 90-100 min after insemination. Six kinds of eggs were used: (1) Unfertilized eggs in normal sea water; the jelly of the eggs had been removed by acidified sea water following Dan’s procedure [3]. (2) Fertilized eggs at monaster stage with hyaline layer in normal sea water; the fertilization membranes had been removed by sucking the eggs into a fine pipette. (3) Fertilized eggs at monaster stage without hyaline layer in normal sea water; the fertilization membranes and hyaline layers had been removed by treatment with 1 M urea solution shortly after insemination. (4) Fertilized eggs at monaster stage without hyaline layer in Ca-free sea water. (5) Fertilized eggs at streak stage without hyaline layer in Ca-free sea water. (6) Fertilized eggs at diaster stage without hyaline layer in Ca-free sea water. The fertilization membranes of eggs (4), (5) and (6) had been removed by sucking the P
Fig. l.-Diagram showing the method for compressing sea B, Glass beam fixed at one end (e) to the stage of horizontal not shown in the figure. D, Displacement of glass plate (P) E, Egg. P, P’, Glass plates for compression. z, Thickness of Experimental
Cell Research 32
urchin egg with two parallel plates. microscope for observation which is owing to the bending of beam (I?). the compressed egg.
Mechanical
properties of sea urchin eggs. I
61
eggs into a fine pipette, and then the eggs had been washed with Ca-free sea water. The eggs of monaster stage [(2), (3) and (4)], of streak stage (5) and of diaster stage (6) were represented by those eggs 14-26 min (mean: 20 min), 51-63 min (mean: 55 min) and 86-93 min (mean: 90 min) respectively after insemination.
‘b
Fig. 2.-Compression of sea urchin egg. (a) Before compression. N,, Surface force. P,, Internal pressure. R,, Radius of the egg. (b, c) During compression (b, side view; c, plane figure). d, Diameter of the portion of the egg surface being in contact with compression plates. F, Compressing force. NO, Surface force in direction of the meridian. Ne. Surface force in direction of the equator. P, internal pressure. R,, R,, Principal radii of curvature at the equatorial surface. z, Thickness of the compressed egg.
As shown in Fig. 1, an egg was put on a minute glass plate (P) attached to one end of a glass beam (B), the other end (e) being fixed to the stage of horizontal microscope for observation. The egg was compressed between plate (P) and another glass plate parallel to P(P), which was movable in a vertical direction with a fine adjustment device. When the egg was compressed, beam (B) bent, and consequently plate (P) was displaced downwards. Since the beam was sufficiently long as compared with the displacement of plate P(D), the two plates (P, P’) were kept almost parallel to each other. Before experimentation, the relation between the displacement of plate (P) and the force applied to it had been determined by putting minute drops of mercury on the plate (P). As a result, it was found that the displacement of the plate (P) was proportional to the applied force calculated from the size and the density of the drops. As described in the following paper [15], deformation of the sea urchin egg depends both on the applied force and on the duration of compression. In the present investigaExperimental
Cell Research 32
Y. Hiramoto tion, the egg was compressed applying four kinds of constant forces, 0.5 x 10-3, I x 10-3, 2.5 x IO-3 and 4 x 10-s dynes, each one of which was added to the egg by controlling the movement of P’ so as to keep the displacement of P(D) definite during compression. The eggs were photographed before and 5 set, 30 set and 5 min after onset of the compression. The form of the compressed egg is schematically shown in Fig. 2. If it is assumed that there is no structure resisting deformation within the endoplasm and bending stresses of the cell membrane can be omitted, the membrane theory (cf. [25]) should be applicable in calculation of stresses within the egg. In this case, as the compressed egg is regarded as a shell in the form of revolution and loaded symmetrically with respect to its axis, the equilibrium of the forces acting in an element cut from the equatorial portion of the membrane by two adjacent meridian planes and two sections parallel to the equator requires an equation (cf. [25]),
in which No and N$ are the force (per unit length) in direction of the equator and that in direction of the meridian, respectively, R, and R, are corresponding principal radii of curvature as shown in Fig. 2, and Z is the external force (per unit area) normal to the surface. Because Z is the pressure-difference between inside and outside of the membrane (P), No/R, t N6/Rz = P.
(2)
This equation is similar to eq. (1) in Cole’s paper [1], although the surface forces are expressed by two symbols (Ne, Nd) different in their directions in the present paper while they were assumed to be constant (T) regardless of their directions by Cole [l]. At the poles, applied force (F) balances with internal pressure, P= F/A,
(3)
in which A is the area of the cell surface being in contact with the plate (P or P’). The equation of the equilibrium of the forces acting in the portion of the cell membrane above the equatorial plane is 2nR,N+ t F = xR, =P.
(4)
In the present investigation, A, R, and R, were measured on the photographs of eggs and the values obtained from eggs in the same experimental conditions (i.e. the eggs which were in the same stage of development, in the presence or the absence of hyaline layer, in the presence or the absence of calcium ions in the medium, and in magnitude and duration of applied force) were respectively averaged, and Nd, N0 and P were calculated from averaged A, R, and R, and F using eqs (2), (3) and (4). Elastic (Young’s) modulus of the cell membrane was calculated as follows. Let us consider a square element of the cell membrane on the equator cut by two adjacent meridian planes and two sections parallel to the equator. It is supposed that the element is stretched even in state without deformation, because there is internal pressure within the cell. If the initial (linear) stretch of the membrane is denoted by E,,,the length of the sides of the element is a(1 +E,,), in which a is the length of the Experimental
Cell Research 32
Mechanical
63
properties of sea urchin eggs. I
sides of the element in resting (unstretched) state. If, in the compressed egg, a(1 + ~4) is the length of the sides of the element along the meridian, a(1 +Q) is the length of the sides of the element parallel to the equator, N+ and Ns are the surface forces per unit length in direction of the meridian and in direction of the equator, respectively, o6 and oe are the stresses in respective directions, h is the thickness of the membrane in resting state, and E is Young’s modulus of the membrane, they should satisfy the equations (cf. [25]), Uc$ = N,#,(1 + Ed) (1 + “&-I,
(5)
ue = Ne (1 + 4 Cl+ d/h,
(6)
E+=(a+-0.5as)/E,
(7)
Eg=(ue -0.5 q/E,
(8)
in which the Poisson’s ratio of the membrane is assumed to be 0.5. As the length of the sides of the element parallel to the equator changes in proportion to the radius of the equator during compression (cf. Fig. 2), 1 + E. - R,. 0.3
0.2 5 -2 e 6 .E z" -0 f i?
0.1
OL 0
I
I
I
I
1
2
3
4
Force(F)
in lo-'dynes
Fig. 3.-Relations between applied force and surface forces (N4, N,g) in the unfertilized
egg. 0, Results when the duration of compression is 5 sec. o, Results when the duration of compression is 30 sec. 0, Results when the duration of compression is 5 min. N,, Surface force of the egg without deformation. Experimental
Cell Research 32
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Y. Hiramoto
If h, is the thickness of the membrane of the egg without
deformation,
h,=h/(l+E,)~. Eliminating
(10)
04, be, ~4, ~0 and h from eqs (5), (6), (7), (8), (9) and (IO),
4 (I+ 4* h,R,
1.5 (No - AV~) + $
(i +
E,)
(N+h - 0.5 Nfj)
0
(11)
It is clear from the left side of this equation that, if values of the equation are plotted against RJR,, they should lie on a straight line the slope of which equals to Young’s modulus of the membrane. According to Mitchison and Swann [18, 191, the initial stretch of the membrane of sea urchin eggs is less than 14 per cent before fertilization, less than 3.7 per cent at sperm aster stage and less than 9.7 per cent at late anaphase stage. In the present paper, 5 per cent was taken for the initial stretch (E& both in unfertilized eggs and in fertilized eggs. For the thickness of the membrane (h,), values obtained by the author [14], namely, 3.1 p in unfertilized egg, 3.5 ,u in monaster eggs, 3.7 ,Uin streak eggs and 4.0 p in diaster eggs were taken. Young’s modulus of the membrane was estimated from the slope of the curve indicating the relation between RJR, and the values of eq. (11) calculated from h,, Ne, N4, R,, R, and E,,( =0.05).
0% 0
1
2
3
4
Force (F) in lob3 dynes
Fig. 4.--Relations between applied force and internal pressure of the cell in the unfertilized Symbols as in Fig. 3. P,, Internal pressure of the egg without deformation. Experimental
Cell Research 32
egg.
Mechanical
65
properties of sea urchin eggs. I RESULTS
Relations between surface forces (N,, NO) and applied forces (F) in the unfertilized egg are shown in Fig. 3. As shown in the figure, the surface force increases with increase in the applied force (and consequently, with increase in deformation of the cell membrane). This fact suggests the existence of an elastic force in the membrane rather than an interfacial force alone. The lines for various durations of compression intersect the vertical axis at almost the same point (NJ, which indicates surface force of the egg without deformation. Surface forces of the fertilized egg without deformation were obtained in the same way and they are collectively shown in Table I. As shown in the table, the surface force increases upon fertilization and before the first cleavage. Relations between applied forces (F) and internal pressures (P) in the unfertilized egg are shown in Fig. 4. As shown in the figure, internal pressure
0' 1.0
I
1
I
1.1
1.2
1.3
b/R,
Fig. 5.-Relations between relative radius of the equator and the value of eq. (11) in the unfertilized egg. The slope of the lines indicates Young’s modulus of the cell membrane. Symbols as in Fig. 3. 5-631815
Experimental
Cell Research 32
66
Y. Hiramofo
of the egg without deformation (PO) obtained by extrapolating the curves is almost independent of the duration of compression. Internal pressures of the unfertilized egg and of the fertilized egg are shown in Table I. As expected from the membrane theory, the internal pressure of the cell changes in parallel with the surface force. I. Internal pressure, surface force and Young’s modulus of the cell membrane of the eggs of the sea urchin, Hemicentrotus pulcherrimus.
TABLE
Hyaline layer
Stage
Before fertilization
Monaster stage
Streak Diaster
stage stage
Calcium ions in medium
present
Duration of compression (se@
Internal pressure (PCJ (dw/cma)
Surface force (No) (dyn/cm)
5 30 300 5 30 5 30
13
0.03
25
0.06
30
0.08
present
present
Gabsent
present
absent
absent
5 30
65
0.15
absent
absent
65
0.15
absent
absent
5 30 5 30
80
0.2
Young’s modulus (E) (lo3 dyn/cm2)
4.0 2.2 1.2 9 6 3 2 4 1.5 10 5 200 50
In Fig. 5, values of eq. (11) in the unfertilized egg are plotted against the radius of the equator of the compressed egg divided by the radius of the egg before compression (RJR,). It must be noted in the figure that the relations are almost linear. This result indicates that the membrane of the unfertilized egg obeys Hooke’s law, because it is expected from eq. (11) for Hookian elastic shell that the relation becomes linear. As the slope of the curve indicates Young’s modulus of the membrane, the modulus could be obtained and is shown in Table I. In fertilized eggs, relations between the values of eq. (11) and RI/R, were, in general, not linear but concave toward the RJR,-axis. Therefore, the values of Young’s modulus of the membrane in the fertilized egg shown in Table I, which were estimated from the slopes of the curves in range of small deformations, merely give the order of the magnitudes of the modulus. Nevertheless, increase in the modulus before cleavage is evident. Experimental
Cell Research 32
Mechanical properties of sea urchin eggs. I
67
In the present calculation, the modulus was obtained assuming that the surface force is due solely to the structural cell membrane (cortex) and that forces exerted by hyaline layer, transparent surface layer [la, 131 or halolayer [lo, 201 were omitted. Actually, however, forces exerted by hyaline layer may not be negligible, since calculated modulus of the membrane of the egg with hyaline layer was unmistakably larger than that of the egg without hyaline layer at the same stage (cf. Table I).
DISCUSSION
In the present study, mechanical characteristics of the cell membrane have been determined assuming that any viscous and elastic resistances of the endoplasm to deformation and bending stresses of the cell membrane may be neglected. According to Harvey and Shapiro’s calculation [9], the protoplasmic viscosity of sea urchin eggs is too small to affect the rate of rounding-up of the egg which have been elongated by passing through capillaries. Because the rate of deformation of the egg when the form of the egg was recorded by photography in the present experiment was by far smaller than the rate of rounding-up used in their calculation, the effect of viscosity may be negligible in the determination of mechanical characteristics of the membrane in the present study. There is a possibility that not only the cell membrane but also the structures in the endoplasm such as monaster and diaster resist deformation of the egg. However, the stiffness of the egg as a whole does not always change in accordance with development of monaster and a sharp rise in the stiffness prior to the first cleavage starts at prophase at which the size of the diaster is too small to resist deformation of the egg (cf. [ 151). Recently Yoneda (unpublished) found by a compression method similar to author’s that the stiffness of the sea urchin egg during cleavage was scarcely changed by treatment of the egg with colchicine-sea water which is believed to destroy the mitotic apparatus [24]. According to the author’s experience, the shape of sea urchin egg was hardly changed even when the mitotic apparatus was greatly deformed by microneedles inserted into the egg. These facts may suggest that resistance to deformation is mostly due to the cell membranes rather than structures within the endoplasm. It has been shown (Appendix) that membrane theory is applicable to rubber shell models of sea urchin eggs. Therefore, the theory is applicable to sea urchin eggs if it is assumed that the cell membrane is mechanically isoExperimental
Cell Research 32
Y. Hiramoto tropic and its volume does not change by deformation (Poisson’s ratio is 0.5), because distribution of stresses and strains within the egg must be similar to those in the models. Absolute values of surface forces have been determined on various egg cells (dynes/cm): by compression method, 0.08 [l], 0.03-0.05 [2] in Arbacia; by centrifuge method, 0.19 in Arbacia, 0.33 in Chaetopterus, 1.1 in Illyanassa, 0.54 in Cumingia [4]; by sessile drop method, lo-40 in Paracentrotus [26], 0.16 (unfertilized egg) and 0.1 (fertilized egg) in Tritrus [S], 0.34 in Busycon [5], 0.18 (unfertilized egg) and 0.07-1.34 (fertilized egg) in Oryzias [22]. Among these values, that of Paracentrotus may be erroneous because sea urchin eggs are not appreciably deformed by their own gravity (cf. IS]), and the values obtained by the sessile drop method may be somewhat larger than true values according to the result of Harvey’s model experiment [6]. Because the eggs in centrifuge experiments and those in sessile drop ones are deformed by centrifugation or by their own gravity, surface forces of the egg without deformation may be smaller than the above values. Therefore, it may be safely said that surface forces of many egg cells are of the order of 0.1 dynes/cm, which is similar to that obtained in the present investigation. Vies [27] obtained a value of about lo3 dynes/cm2 for Young’s modulus of unfertilized sea urchin egg. If the elasticity is assumed to be attributed to the cell membrane of 3 ,D thick, the modulus of the membrane is about lo4 dynes/cm2. Norris [al] determined the relation between the force and the elongation of the cell when unfertilized sea urchin eggs were stretched with microneedles. Young’s modulus of the membrane is calculated to be 0.8 X lo3 dynes/cm2 from his result (Fig. 4 of his paper [al]), if it is assumed that the membrane is 3 ,U in thickness and presence of internal pressure can be omitted in the range where the force-elongation curve is linear. These values are comparable with those in the present investigaton. Mitchison and Swann [17-191 and Mitchison [16] determined relations between the negative pressure applied to a portion of egg surface with a pipette (cell elastimeter) being closely in contact with the surface and the deformation by the negative pressure. They found that the relations were linear in sea urchin eggs. This fact, together with the result of their model experiment [17], made them conclude that the resistance to deformation of the cell membrane is due mostly to the flexural rigidity rather than tangential (tensile or compressive) stresses. This conclusion was, however, criticized by Wolpert [28], who pointed out the fact that the “stiffness” (the slope of the negative pressure-deformation curve) is directly proportional to the thickness of the wall in Mitchison and Swann’s model experiment [17] which Experimental
Cell Research 32
Mechanical suggested the than bending The author of Mitchison In elastimeter
69
properties of sea urchin eggs. I
resistance to deformation is due to tangential stresses rather ones. independently noticed this fact and tried to explain the results and Swann’s experiment by membrane theory as follows: experiments, the surface of the spheres (eggs or rubber balls) without
4
15.3x104 dynes/cm2 internal pressure
5
15
10 Deformation
20
25
(x) in mm
b Fig. 6.-Deformation with a pipette.
of a rubber ball when negative pressure is applied to a portion of the surface
$)-Deformation of the ball. d, Diameter of the pipette. Nin, Surface force inside the pipette. Out’ Surface force outsrde the pipette. r, Radius of curvature of the surface inside the pipette. R, Radius of curvature of the surface outside the pipette. I, Deformation of the surface by negative pressure. (b)-Relations between deformation and negative pressure. Curves: results derived from membrane theory. Circles connected with lines: experimental results obtained by Mitchison and Swann [17]. Diameter of the ball: 100 mm. Thickness of the wall: 1.5 mm. Young’s modulus of the rubber: 10’ dynes/cma. Diameter of the pipette: 49.5 mm. Solid lines indicate the results when the initial internal pressure is nil and broken lines the results when the initial internal pressure is 15.3 x lo4 dynes/cma.
slips around the edge of the pipette, and the degree of the slip depends on the friction between the surface and the edge. If there is no friction, the value of the surface force may be uniform all over the surface and be proportional to the product of the thickness of the wall and the (linear) increase in the surface of the membrane. On the other hand, if there is no slip of the surface around the edge, the surface force of the portion of the membrane Experimental
Cell Research 32
70
Y. Hiramoto
inside the pipette may be different from that of the portion outside the pipette and the values may be proportional to the product of the thickness and the increase in the surface of respective portions of the membrane. If P is the sapplied negative pressure, N,, and Nout are the surface force of the portion of the membrane inside the pipette and that outside respectively, and r and R are the radius of curvature of the surface inside the pipette and that outside respectively (cf. Fig. 6a), they should satisfy, Deformation
P= V%&
(5) is
N,,t/R).
z=r-vrz-d2/4
(12)
,
(13)
in which d is the diameter of the pipette (cf. Fig. 6~). r was calculated for various values of n: using eq. (13) and R was calculated from the equation indicating the constancy of the total volume of the sphere, V=
n{
z
ff+d2
(6
8)
+
R”+d”
(3
vJR2+#+iR3
24) ___
}
=
con&.
(14)
by successive approximation. The surface area and the thickness of the membrane inside the pipette and those outside could be calculated from d, r and R when initial conditions were given. Surface forces (Ni, and N,,,,) were calculated from Young’s modulus, the increase of surface area and the thickness of the membrane. Finally, negative pressure (P) was calculated using eq. (12). The result in the case of a rubber ball with 100 mm diameter, 1.5 mm wall-thickness, 10’ dynes/cm2 Young’s modulus and nil or 15.3 X lo4 dynes/cm2 initial internal pressure are shown by curves in Fig. 6 b. As shown in the figure, the shape of the curve depends not only on the wall-thickness and the internal pressure, but also on the degree of slip of the surface around the edge of the pipette, and therefore, linearity of the curve in sea urchin eggs is not a solid foundation for their conclusion [ 171 that the cell membrane is sufficiently thick to resist deformation by virtue of its rigidity (resistance to flexure or bending). The results of the experiment in the conditions mentioned above which appeared in Fig. 9 of Mitchison and Swann’s paper [ 171 are shown by circles connected with lines in Fig. 6 b. It is expected from the theory that the observed negative pressure for a given deformation is larger than the theoretical value in the case of “slip” and smaller than that in the case of “without slip” since the surface of the ball neither completely slips nor never slips around the edge of the pipette. Therefore, the agreement between theoretical and experimental results is considered to be fairly satisfactory. Experimental
Cell Research 32
Mechanical properties of sea urchin eggs. I
71
This fact, together with the fact that the stiffness is proportional to the thickness of the wall mentioned above, indicates that the resistance to deformation in elastimeter experiments is mostly due to tangential stresses. Values of Young’s modulus obtained by Mitchison and Swann [ 17-191 may be acceptable if it is assumed that the degree of slip of the surface around the edge of the pipette in sea urchin eggs was similar to that in the rubber ball models and the thickness of the cell membrane was 1.6 ,u, Since the thickness of the membrane is considered to be larger than the above-mentioned value (cf. [14]), true value of the modulus may be smaller, i.e. 3-5 x lo3 dynes/cm2 in unfertilized eggs. Selman and Waddington [23] determined elastic properties of the cell membrane of amphibian eggs with cell elastimeters, and they obtained 1.5 x lo6 dynes/cm2 for the modulus of the membrane. This value may not be acceptable because the author obtained 3 X lo3 dynes/cm2 from the data given in Selman and Waddington’s paper (Fig. 11 of [23]) by membrane theory assuming that the thickness of the membrane is 2 ,u and that there is no slip of the membrane around the edge of the pipette (cf. [23]). Surface forces were calculated to be 0.03 dynes/cm if the initial stretch was assumed to be 5 per cent and to be 0.06 dynes/cm if the initial stretch was 10 per cent. These values are in right order because the surface force of amphibian eggs may be somewhat smaller than 0.1 dynes/cm obtained by sessile drop method [8] as already mentioned. By the discussion mentioned above and the result of the present investigation, it may be concluded that the Young’s moduli of the egg cell membranes of sea urchin and amphibia are of order of 103-lo* dynes/cm2, which are as low as that of weak table jelly. Cole and Michaelis [2] calculated surface forces of fertilized sea urchin eggs from relations between applied force and deformation of the egg assuming that the form of compressed fertilized egg without fertilization membrane was the same as that of compressed unfertilized egg when the degree of deformation (z in Figs. 1, 2) was the same. As a result, they concluded that there was no marked change in surface force upon fertilization. It has, however, been found in the present experiment that, the forms are not exactly the same, i.e., the area of contact in the fertilized egg is, in general, smaller than that in the unfertilized egg under the same degree of deformation, although the force-deformation curves are not much different [15]. Consequently, the surface force and the internal pressure of the fertilized egg have been calculated to be larger than those of the unfertilized egg in the present study. Experimental
Cell Research 32
Y. Hiramoto SUMMARY
1. A method was described to compress eggs between two plates parallel to each other and to determine the force required for compression. 2. Surface forces and internal pressures of the cell were determined in the before and at various eggs of the sea urchin, Hemicentrotus pulcherrimus stages after fertilization, assuming that the cell is a spherical shell filled with liquid. 3. Young’s modulus of the cell membrane was calculated from the relation between surface force and extension of the membrane by compression. 4. Both the surface force and the internal pressure of the egg increased upon fertilization and before cleavage. Young’s modulus of the cell membrane increased before cleavage. 5. The results of previous investigations on the mechanical properties of egg cells were discussed and were compared with those of the present study. It was concluded that the surface forces are of order of 10-l dynes/cm and the Young’s moduli of the cell membranes are of order of 103-lo4 dynes/cm2 in many kinds of egg cells. 6. It was concluded that both the resistances to deformation of the endoplasm and those to bending of the cell membrane are negligible in analysis of mechanical properties of the cell membrane by the present compression method as well as by suction method using cell elastimeter. I am grateful to Professor H. Kinosita for his helpful advice and critical reading of the manuscript and to Professor K. Dan for his useful discussion on the present study. APPENDIX
Model Experiments In order to examine whether or not the membrane theory is applicable in analysis of stresses within sea urchin egg, model experiments were carried out using rubber balls. The ratio of wall-thickness to diameter in balls was similar to the ratio of membrane-thickness to diameter in sea urchin eggs, although the sizes were quite different (the diameters of the balls were 6-7 cm, while those of Hemicentrotus eggs are about 95 ,u). After the ball was filled with water, it was placed on a plate of plastics (P in Fig. 7). Before experimentation, internal pressure of the ball had been adjusted by means of a compressor (C) and a manometer (M) to be 0.1 atm. The diameter of the ball was increased about 5 per cent as a result of the Experimental
Cell Research 32
Mechanical
properties of sea urchin eggs. I
73
increase of internal pressure. The ball was compressed between two plates parallel to each other (P, P’), and the force was measured by means of a scale balance (S). During compression, the volume of the ball was kept unchanged by closing valve (V) and by keeping the meniscus of the manometer (m) at a fixed position. The ball was photographed before, during and
Fig. 7.-Diagram showing the methods for compressing rubber ball and measuring compressing force and internal pressure. B, Rubber ball filled with water. C, Compressor. M, Manometer. m, Meniscus of the manometer (M). S, Scale balance for measuring compressing force. V, Valve. z, Thickness of the compressed ball.
after compression. The area of the surface of the ball being in contact with the plates was directly measured through transparent plate (P’). Internal pressures calculated by dividing the applied force by the area of contact and those directly measured by the manometer in the same ball at various degrees of compression are shown in Table 11. It must be noted that the calculated pressures are always larger by several per cent than the directlymeasured pressures. This fact suggests that the force required to bend the wall is several per cent of the force balancing with internal pressure. The relation between the applied force and the deformation of a ball is shown by black circles in Fig. 8. The white circles in the figure indicate the relation between the applied force and the deformation in a similar ball in which the internal pressure was nil throughout the compression. The difference of these curves indicates that the resistance owing to bending stresses of the wall accounts for a quite small fraction of the total resistance to deforExperimental
Cell Research 32
74
Y. Hiramoto
mation in the case of compression of the ball without volume change because the resistance in the case of compression of the ball without internal pressure is considered to be due to bending stresses of the wall, although the conclusion is not a strict one since the forms of the ball during compression in these two cases were not exactly the same even when the z-values (cf. Fig. 7) were the same as each other. TABLE
II. Internal
pressure of a rubber ball at various degrees of compression.
Calculated pressure (kg wt./cm2) Directly-measured pressure (kg wt/cm2) Calculated pressure Directly-measured
pressure
0.121
0.131
0.147
0.155
0.194
0.197
0.280
0.117
0.126
0.141
0.148
0.191
0.196
0.272
1.03
1.04
1.04
1.05
1.02
1.01
1.03
After the compression experiment was finished, strips of the wall several centimeters in length and about one centimeter in breadth were dissected out of the ball. The relations between stresses and strains of the strips were determined by applying weights to them. As a result, it was found that the relations were linear, the Young’s modulus was uniform all over the surface
0
1
1
3 Force
4
5
6
7
8
in kg wt
Fig. S.-Relations between applied force and deformation in rubber balls. 0, Relation in a rubber ball without volume-change during compression. 0, Relation in a similar ball without internal pressure during compression. Experimental
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Mechanical
75
properties of sea urchin eggs. I
of the ball and the modulus was independent of the direction of stretching. The bending stresses at the equatorial surface of the ball were calculated from the Young’s modulus and bending strains measured on the photographs of the ball. The ratio of the bending stress to the tensile stress calculated by membrane theory in the same portion and in the same direction, which was different by the degree of compression, was less than 0.02 in direction of the equator and was less than 0.1 in direction of the meridian. From the results mentioned above, it may be concluded that membrane theory is applicable with approximation to rubber balls, the figures of which are similar to those of sea urchin eggs. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
COLE, K. S., J. Cell. Comp. Physiol. 1, 1 (1932). COLE, K. S. and MICHAELIS, E. M., ibid. 2, 121 (1932). DAN, K., Biol. Bull. 93, 259 (1947). HARVEY, E. N., BioZ. Bull. 61, 273 (1931). ~ J. Cell. Camp. Physiol. 4, 35 (1933). __ ibid. 8, 251 (1936). HARVEY, E. N. and DANIELLI, J. F., Biol. Rev. 13, 319 (1938). HARVEY, E. N. and FRANKHAUSER, G., J. Cell. Comp. Physiol. 3, 463 (1933). HARVEY, E. N. and SHAPIRO, H., ibid. 17, 135 (1941). HERBST, C., Wilhelm Roux’ Arch. Entwicklungsmech. Organ. 9, 424 (1900). HIRAMOTO, Y., Zool. Msg. (Tokyo) 62, 139 (1953) (in Japanese). ~ Japan. J. Zool. 11, 227 (1954). ~ ibid. 11, 333 (1955). ~ Embryologia 3, 361 (1957). ~ Exptl. Cell Res. In press. MITCH&N, J. M., J. Ex$l. BioI. 33, 524 (1956). MITCHISON. J. M. and SWANN. M. M.. J. ExDU. Biol. 31., 443 (19541. \ ’ __ ibid. il, 461 (1954). ~ ibid. 32, 734 (1955). MOTOMURA, I., Sci. Rep. Tohoku Imp. Uniu. IV, 16, 283 (1941). NORRIS, C. H., J. Celt. Comp. Physiol. 14, 117 (1939). SAKAI, Y. T., Embrgologia 5, 357 (1961). SELM~N, G. 6. ~~~WA&INGTON,‘C. fi., J. Exptl. Biol. 32, 700 (1955). SWANN, M. M. and MITCHISON, J. M., J. ExDtl. Biol. 30, 506 (1953). TIMOS&NKO, S., Theory of plates and shell;. McGraw-h, New York, 1940. VL&, F., Arch. phys. biol. 4, 263 (1926). ~ Arch. Zool. Exptl. Gen. 75, 421 (1933). WOLPERT, L., Intern. Rev. Cytol. 10, 163 (1960).
Experimental
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