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Surface viscoelastic effects in cell cleavage
SURFACE
Department
VISCOELASTIC
of Chemical
Abstract-The hvdrodbnamlc Ybrk (l’Y87)]. to the passive material. The
EFFECTS
DANIEL
ZINEMANAS
Engineering.
Technion,
IN CELL CLEAVAGE
and AVINOASI Israel Institute
NIR
of Technology.
Haifa 3X00.
lsrncl
etkct of passive surface traction on the cleavage of cells is incorporated in the cytokinesis model of Zinemanas and Nir [Biomrchunic~s ofcell Dirision. pp. 281-305. Plenum Press. New DitTerent rheological behaviours were examined to model the surface tensions which arise due deformation of the cortex: a Mooney-Rivlin material. an STZC material and a viscoelastic calculations show [hat passive surface tensions may play a significant role in determining the
local surface deformations as well as in the modulation of the surface forces. Varying the rheological model has limited efTecton the overall deformations. The latter appear IO be affected mostly by contractile filament interactions.
relation
IIVTRODtiCTIO;V
between the force in the CR and the deforma-
tion of the cell. Their Cytokinesis
is the division
of the cytoplasm
follows the nuclear mitosis and culminates ductive cycle of an animal biochemical
which
the repro-
cell. The biophysical
processes which
control
enon are not well understood;
and
this phenom-
initially
spherical
internal
mations. i.e. the formation
bchaviour
an equatorial g&d
cleavage furrow, are confined to a thin-
layer
(Ritppitport,
beneath 19X6).
the
plasma
of muscle-like
the orientation
oriented
distribution
conliguration
1978) to a well defined
as the cortex,
contractila
observed to change from an initial dom
memhrane
Iaycr, known
This
contains a network filaments.
progress ol’
of which
uniform
(Opas
bundle
microis
and ran-
and
Soltynska,
of filaments
force. The
pressure
model
satisfactorily
did
not
follow
1968).
which were not taken into account in the
model
which
and
presumably
change in the mcmbranc A possible mechanism the CR
bundle,
a dynamic
for the formation
was given by Greenspan instability
produce
properties. or a CR
(19773. b, 197X). who sug-
is formed
by a hydrodynamic
which causes the microlilamcnts
reorient due to surface contractions.
lilamcntous
Akkas
phenomena
edge
This
to
(198 I). these discrepancies arise due to other biological
elements
1975).
predicts
the cxpcrimcntal
According
parallel to the cleavage plane under the leading furrow (Schroeder.
of an
due LO an equatorial
but the constricting force and the
(Hiramoto,
gested that
aligned
membrane
the cell deformations
that the active forces responsible for the surface deforand continuous
mainly on the second aspect
and is based on the study of the deformation constricting
however, it is accepted
work, as well as that of Akkus
(1980. 1981). concentrates
to agglomerate
or tensile
at the cell’s equator
and is
This instability
named the contractile
ring (CR), is an essential organ-
assumed to be triggered by a decrease in the concen-
elle which
the major
tration
provides
active
force
in the
cleavage process. Although
cytokincsis
is visually manifested only at
the late stages of cell division, i.c. late anaphase. the appearance
cal processes leading ultimately ized and
diflerentiated
structure
start at earlier
biochemical atus (MA)
stimulus originating (Rappaport,
cortical
layer remain hitherto
cytokinetic
provide
the nature
agent or its exact effect on the unknown.
a better
process and
which could
of the
Greenspan’s
agglomeration
at the furrow
region.
model assumes that the surface tractions
are isotropic and depend on the concentration tensile
elements.
However,
based on this model isotropic
numerical
of the deformation
driven
tensions do not show a significant
formation
(Sapir
and Nir,
of the
calculations
1985; White
by
furrow
and Borisy,
1983). a new model
which
and Nir (1987.1988)
presented
is based on some of the ideas
a mechanism
considered
to the experi-
physical description of the surface Iaycr and forces, and
models were pro-
and Lardncr
of the microtilamcnts and
from the poles toward the equator are responsible for
corroborate
(1979). the
aim of such a model has to be first, the explanation ring
The surface tension
gradient thus created and the consequent surface flow
Recently. Zinemanas
understanding
mental findings, various theoretical
contractile
at the poles or an
an explanation
posed. As stated by Pujara the alignment
a
at the mitotic appar-
1986). However,
the stimulating to attain
cortical
midphase
the cortex and alters its
ofeither
In order
forces and
stages. During
of contractile
the filaments’
to the highly organ-
surface
spreads toward
composition
with
of the cleavage furrow, the biochemi-
elements
increase of these at the equator.
second.
of
into an equatorial
a description
of the
above. The
incorporates
possible
model
biochemical
occur there. The evaluation ting the formation
uses a more suitable processes which
was successful in prcdic-
ofa contractile
ring and parameters
such as the surface forces, surface concentrations internal similarly
and
pressure and flow fields are found to behave to
the experimental
1968, 1971; Koppel
et al., 1982).
patterns
(Hiramoto,
418
D.
ZINEMANASand A. NIR
The calculations of Zinemanas and Nir (1987.1988) assume that the cell can be idealized as a Newtonian viscous droplet and the surface forces as an anisotropic tension. Consequently, relatively high deformations are predicted in the polar regions and the longitudinal dimension of the dividing all appears higher than observed. Also in that model the surface tension was calculated considering the active components of the interfacial layer only. The purpose of this communication is to incorporate more realistic rheological cortical properties in the biomechanical model. The interfacial forces will constitute both the active part due to the contractile microfilaments network as well as the forces which arise when passive components of the cortical matrix deform. Various rheological bchaviours of this cortical layer passive contribution are examined with relation to the overall cell deformation during cytokinesis.
RIOMECHANlCAL
;(I +i)v(x)+(I-i)
FORMJLATION
Consider the cell to be a viscous droplet, B. of viscosity /( which is immersed in a second fluid, B’, of
K(C):v(y)tiy)dS,
x, y&B
J(<).f(y)dS,
(1)
where v is the velocity field, n is a unit vector normal to dB pointing into B’. f denotes the surface tractions and vm is the velocity field far from the drop which must also satisfy the creeping flow equations. In our case v-=0. J and K are single and double layer potentials defined, respectively, by ’ ” J(<)=e+F.
MODEL
Consider the following biomechanical model (Zinemanas and Nir. 1987, 1988). (a) The cortical layer is approximated by a network of filaments confined to a continuous thin matrix. The surface forces are a combination of the contribution of the active filaments network and the passive reactions to surface deformations of the continuous matrix. (b) Active filaments exert a time-dependent contractilc force parallel to their symmetry axis; thcrcfore. their contribution to the surface tractions depend both on their local concentration and orientation distribution. (c) Reorientation of lilaments follows the surface motion and is enhanced by mutual interactions. (d) The initial uniform composition of the cortical layer is altered by a stimulus. This stimulatory agent originates at the MA asters and diffuses. during a given time interval, toward the surface. The stimulant is assumed to inhibit the formation of active filaments through a very simple kinetic scheme; hena. it atTectsthe surface mainly at the poles where its flux is expected to be higher. (e) The rheological properties of the continuous matrix, neglected in the previous calculations (Zinemanas and Nir, 1987, 1988). are now incorporated. Constitutive equations for a Mooney-Rivlin material (Green and Adkins. 1970). an STZC material (Skalak et al., 1973) and a viscoelastic material are examined. A detailed discussion of the biophysical and biochemical considerations and simplifying assumptions used in the derivation and solution of this model is given in Zinemanas and Nir (1987, 1988). hlATliEMATICAI.
viscosity cc’ and is surrounded by a thin layer dB. When both inner and outer fluids are assumed Newtonian and inertial efTects are neglected the momentum equation may be conveniently converted into a boundary equation for the surface velocities (Rallison and Acrivos, 1978):
K(<)=;-$
(2)
where 4=x-y
and
<=I<[
and 1 denotes the inner to outer viscosity ratio p/I/. For 1= I. equation (I), with v”(x)=O, is simplilied to the form
J (0 * f(y) dS,
(3)
which also holds everywhere in B and, B’. Similarly, the pressure lield is given by the expressions P(x)-
i 38
Q(C) * a(v(y)) * n(y) dS, G(C):v(y)n(y)dS,
-RF
I
W=PW+
xe& yedB
(4)
Q(C)*4W)~WW,
3n
+P
c
J&V
C(C): v(y)n(y)dS,
xaE’, ycdB
(3
where
QW=
--$.
.
05)
These equations also simplify, for 1- 1, and P”(x)=O, to the form P(x)= I
a8
Q(C) * f(y) dS,
(7)
which is also valid throughout. The continuitykquation for an incompressible fluid may also be integrated to the form v*ndS=O. I
a0
(8)
419
Surfaceviscoelasticeffectsin cell cleavage Evaluation of the flow equations (1). (4) and (5) thus require only the knowledge of the instantaneous intcrfacial forces. 1. and the boundary shape S5. The latter, though a priori unknown, follows dynamic variations described by the kinematic condition dR -_=v dt
w(p)=&l,p’p4(v’IC-a’Pb,pv’)n+w(p)
x&B
where d/dt denotes a material derivative and R is the radius vector that specifies the location of the interface, 23. The interracial forces. in the absence of surface inertia, may be expressedin general form using a set of the surface coordinates u, by (Waxman, 1984) f=(y’bIp-hfM461p)R,1+(h,qy14-_M””J,p)n
(10)
where 7’S is the surface stresstensor, MzP the moment tensor and b,, the second fundamental form of the surface. A comma and a bar indicate a common and a covariant derivative. respectively,and R., are the basis vectors tangential to the surface. Although bending moments may arise and be important in the deformation of thin surface layers these effectsare neglected in the present study. As stated in the biomechanical model, the stress tensor 7“’ is composed of active and passive components and thus is described by an expression of the form
whcrc the active component y$, stems from the activities of the iilamentous network and dcpcnds on the local concentration and orientation distribution of the filaments. On the other hand, the passive reactions of the matrix to the surface deformations. yz,, depend on the rheological properties of the cortical material through the strain c,,, and rate of strain G,, or other related variables. The local concentration, r. and orientation distribution function, N, follow the mass and orientation balances which are of the form
dc ; + dN x
+
w, =
-c(Y’aJPb,p+Y”I,)+07alPCl,p+~
V’CI, =
DaVfp+”+ V,,,(wA’ 1
scale of an individual filament is much smaller than the distance typical for the surface velocity variations. the flow field around a filament may be considered linear. thus, the angular velocity is given by the expression
(12)
_ +;(N-N)+!+‘NI,). Here u.# is the surface metric tensor. D, and D, are, respectively. the rotational and translational diffusion coefficients, p is a unit vector along the filament axis and A denotes the orientation distribution function of filaments produced at a rate R‘. V’ and v3 are the velocity contravariant components tangential and normal to the interface, w is the filament angular velocity and Vo,, is an operator with respect to the filament orientation. Note that the LHS of equations (12) and (13) are also material derivatives. Since the
(14)
where E,,, is the permutation tensor and co(p) is an additional angular velocity arising from mutual filament interactions as suggested by Schroeder (1975). These interactions were conveniently assumed to be of the form ~(O)=Qsin?Ol~arctan(~)-11
(15)
which allows for a closed form solution of the orientation balance (13) and. at the same time, captures the physical requirement that the interaction between two parallel or perpendicular filaments does not cause a rotational motion and the assertion that this motion is enhanced as order increases. In equation (15) R is a proportionality constant, y,, and y2r are the principal stressesand 0 is the angle formed between the filament axis and the surface tangential velocity. The explicit expression that relates the active stresscs and the concentration and orientation distribution function of the tilamcnts is given by (Zinemanas and Nir. 1987) Y JP
(16)
where yba’ are the principal values of the stresstensor and the basis vectors coincide with the principal directions. F is the filament contractile force and L its length. The parentheses in the superscript indicate the physical value of the stresscomponent. The product of the force by the length is assumed to follow dynamic changes described by an expression of the form d(FL) -&-&c+
(17)
which approximately simulates a muscle-like behaviour. The constants C, and C, are chosen to match the process time scale. The cell cortex is not a well defined organelle since the thickness and composition of this denser layer, composed in part of a microfilamentous network, vary widely among ditferent types of cells (Bray et al.. 1986). Therefore, there have not been many attempts to derive constitutive equations based on the ultrastructure of this layer. The rheological models used in nonhydrodynamic simulations ofcell cleavage (e.g. Akkas. 1980) were based on a homogeneous approach and assumed, as stated above, the behaviour of a Mooney-Rivlin material. an STZC material or a viscoeslatic material with good successin predicting the cell deformations in all cases. It follows from these simulations in which material properties were dynamically modified, as well as other experimental data
420
D. ZI~EMAHAS
(Conrad and Rappaport. 1981). that the ceil surface may be active not only at the furrow region. Furthermore, since activity depends on the amount of stimulation that reaches the cortex, the surface may exhibit different activities in different zones. It is therefore clear that a completely passive description of most of the cortical layer is insufficient. In the present approach we have separated active from passive events; thus, in addition to the active forces emerging from the filamentous network as described above, the rheological properties of the continuous cortical matrix are considered. These passive tractions appear from other components of the layer as well as from non-active filaments. Since the properties of all these have not been measured separately and the composition of the layer is still unknown we will study the possible importance of passive surface tractions in the cleaving process using the aforementioned three models. i.e. Mooney-Rivlin, STZC and viscoelastic materials. The constitutive equation for a Mooney-Rivilin material is given by Y~11)=2hZ(t-~)(l+~i:1.
(18)
while for an STZC material it is y’l
I)
(: i.,
=,i[r(n:-I)+i:(i:E.:-l)]. ‘I
(19)
ifcrc < and f arc material constantsand h is the initial layer thickness. The second principal tension is obtained in both cases by interchanging the indicts I and 2. Similarly, for a viscoelastic material of the Kelvin
and A.
NIR
AXISY%t%tETRICAL NLJlERICAL
CLEAI’AGE
A\D
PRDCEDCRE
The general equations presented in the previous section were solved for the axisymmetrical case with fore and aft symmetry which corresponds to the most common pattern of cell cleavage. Thus. a set of cylindrical coordinates (r, c. 4) is used together with the surface coordinates, (:. 4). selected for the twodimensional equations while the cell shape is defined by the even function r=R(c) for -I<:
(24)
A@-Y)‘f(Y)dY.
I -I The normal and tangential components of fare
type Y ‘11’=-P~+(~)(j.i-l)+2~~(~).
(20)
This constitutive equation was derived by Evans and Hochmuth (1976) for red blood cell membranes and was used in the study of membrane relaxations. Here r7 and qc are a shear modulus and a surface viscosity, respectively. In these equations, i., denote the principal stretches which are related to the principal strains and rate of strain through e,,,,=G-,‘-
1)
&,) = II.).,.
(21)
(23)
which provides the instantaneous strain rate at any material point in terms of the surface velocities. As in Pujara and Lardner (1979). the passive tensions were set to 0 when the calculated stretch ratio is lower than I, i.e. when the surface folds. This phenomenon occurs at the furrow region as the surface contracts there.
(Y**- Yor,) R’(y) R(Y)
1
(26)
where the radii of curvature are given by I -=R:
1
I
R”(Y) [t+~‘~(y)]~‘*’
~=R(y)[I+R”(y)l”‘(27)
(b) Internal pressure I
(22)
Moreover, these surface strains are related to the surface motion by the expression
. e=b=~(V,lb+vpI,)--b,pv’
I /.=[l+R’20,),*,~
P(r. :) =
P(z-y).f(y)dy.
(28)
-1 The components of A and P obtained from the analytical circumferential integration of the kernels J and F are given by Zinemanas and Nir (1988). (c) Mass balance
I
+ R(:)[l
+
R”(i)]“*
c?(s, R)
8:
1+d.
(29)
421
Surface viscoelasric et7ectsin cell cleavage 2.0)
Here v, and v, are the normal and meridional tangential velocity components, respectively. (d) Orientation balance dS -=p dr
S( w(0l.l;) (30)
20
1
1.5.
where the angular velocity is given by w(O)= -isin 1.0, 0
+ o(0).
. , . . 0.5
.
.
.
. 0
(31)
(e) Deformation
sv,
I
I (32)
e(x’)=[, +R"(-_)]t/2~+R,vn “, 4,,
. .
=
R(.-)[I+R’2(;)]“z~+R,v”’
dR
I
Fig. I. A comparison of the change of the non-dimensional polar length. 1, with nondimensional furrow radius, R,. for various surface rheologies used in the model. (---) SurfaLp rheology neglected; (- - - - -) Mooney-Rivlin; (- - -) STZC: vi&elastic material. Experimental data Or G-4 Hiramoto (1958) and Yoneda and Dan (1972).
(33)
Since the equation for the surface velocity is highly non-linear due to the strong coupling which arises between the momentum. mass and orientation balances through the intcrfacial forces, a numerical approach is required to solve this moving boundary problem. The integral equation may beconvcrted into a linear set of algebraic equations by discrctizing the boundary into a number of clcmcnts and employing a finite ditTerencemethod (Rallison and Acrivos, 1978). Consequently. the equations for the cell shape. concentration, orientation distribution and deformation may be integrated in time using a simple Newtonian integration and an updated Lagrangian algorithm. Some mathematical difficulties arise from the singular nature, as y approaches x, of the single layer potential, J, but these may be overcome by subtracting the leading orders of the singular integrand and adding an analytical integration of the asymptotic singular expansion around x (Rallison and Acrivos, 1978). For a detailed account of the numerical procedure the reader is referred to Zinemanas and Nir (1988).
RFSULTS
. The model of cell division presented in the previous sections was evaluated for a number of casesinvolving diKerent surface rheologies. In Fig. I the dynamics of the overall deformation are presented in terms of the change of the polar length, 1. as a function of the furrow radius, R,, for four different cases; a Mooney-Rivlin fluid with Zhd/y,=O.I and r=O.l,an SZTC material with 6/2y,=O.O5 and r=O.l. a viscoelastic material with P,=0.)1~~,,=0.6 and q,/yO =O.IS, and a case with no surface rheology. The variation in the overall deformation was not significant for the various surface rheologies and a slight deviation from the experimental observations of
1
0
0.5
1.0
4 Fig. 2. The ekt of vatiation in material Constant of the Mooney-Rivlin model on the dynamics of the nondimensional polar length. i, and nondimensional furrow radius, R,. ( ---) r-0; (-----) r-0.1; (---) r-=0.2; (---) l-=0.5.
Hiramoto (1958) and Yoneda and Dan (1972) was noticed. The sensitivity of the deformation to the material constants of a surface rheology of the Mooney-Rivlin type is shown in Fig. 2. Again. variations of the overall deformation due to changes in the material constant, r. are observed to be small. However, the incorporation of the surface rheology into the model results in a dilTercntqualitative behaviour of the surface tension dynamics. The evolution of the circumferential tension at the furrow region, depicted in Fig. 3, indicated that when surface rheology is included in the model the tension grows and decays monotonically, as is also evident in experimental observations. When surface rheology is not included the final level of the circumferential tension remains unreasonably high. The changes of the stretch ratio in the meridional direction at the equator and at the poles are shown in
422
D. ZINEMANAS and A. NIR
\
\
\
‘*. \ \\ \
‘_
ol......-..,.....“.. 0
5
1.0
10
.
.
.
.
.
.
.
(
-
.
.
.
.
.
.
1.0
0.5
0
Tlma
R1 Fig. 3. The dynamic development of the circumferential tension at the furrow, yW. (-) Surface rehology neglected;
(- - - - -) Mooney-Rivlin material.
Fig. 5. A comparison of the change of the polar stretch ratios, I,. with non-dimensional furrow radius. R,. for the various surface rheologies. (-) Surface rheology neglected: (----) Mooney-Rivlin; (---) STZC: (--) viscoelastic material. ( x ) Experimental data of Hiramoto (1968).
10 2.0.
.
---------.- . . . .._I’-. -‘\ -... *... *-. -. -.. ....‘. ‘..::\
1 1.5.
a
O~....‘,‘..,“‘......l 0
0.S Rl
1.0
Fig. 4. A comparison of the change of the meridional equatorial stretch ratio. 1,. with non-dimensional furrow radius, R,, for the various surface rheologics used. (-) Surface rheology neglected; (- - - - -) Mooney-Rivlin: (- - -) STZC; (- -) viscoelartic material. ( x ) Exprrimemal data of Hiramoto (1968).
Figs 4 and 5. respectively. and are compared to the measured values of Hiramoto (1958). The values of the stretch ratio obtained with the present model are in qualitative agreement with the experimental observations and do not vary considerably for the different surface rheologies used in the calculations. The most signiticant influence on the overall deformation was found to be due to variations in the filaments’ interaction level. These are observed in the results shown in Fig. 6 where, with f2 assuming the values 1.2, 1.3 and 1.4. a considerable variation in overall cell deformation is encountered. Finally the trajectories of a number of particles on the cell surface are followed during the furrowing process and are shown in Fig. 7. DISCUSSION
It is convenient to start this discussion with a short summary of the results obtained by the hydrodynam-
l.OC. 0
.
,
.
.
1
.
.
I
0.5
.
.
.
“:>,
.
1
1
.
.
’
\ 1.0
Rf Fig. 6. The effect of filamcnl interaction as manifested by changes of the non-dimensional polar length. f. with nondimensional furrow radius_ H,. for various tilamcn~ interaction constams. (-) R=l.4; (-----) R=l.3; (----)Q=l.2.
ical model in the absence of surface rheological effects (Zinemanas and Nir, 1987, 1988). The formation of a contractile ring apparatus was successfullyexplained in terms of an initial disturbance of the cortical layer composition, caused by the mitotic apparatus, which produces a surface tension gradient. Lower tensions prevail at the polar regions due to an assumed decrease in the concentration of active filaments in these zones. This initial tension gradient induces a motion from the polar regions towards the cell equator which consequently causes an agglomeration of filaments at the leading edge of the cleaving furrow. The Row thus produced and the interactions between filaments cause, simultaneously, a reorientation of the filaments parallel to the cleavage plane. This effect, together with the agglomeration, forms the organelle known as the contractile ring which exerts the major active force in the cleaving process.,The occurrence of filament interactions as postulated by Schroeder (1975) is also found to be essential in the reorientation of filaments
Surface viscoelastic effects in cell cleavage
0
09
1e
Fig. 7. Consecutive shapes during cleavage showing trajectories of surface trace particles. R and I denote radius and axial coordinates.
in the furrow region and in the accomplishment of complete divisions. As stated before, a good qualitative agreement was also found for various other variables such as the internal pressure.the surface tensions at the poles and at the furrow region and the internal Row patterns. The overall geometric dimensions of the cell. however. wcrc found to dilfer slightly from those observed experimentally with polar lengths larger than those rcportcd by Hiramoto (1958) and Yoncda and Dan (1972). It was assumed that these dilTcrenccs can be attributed to ncglccting surface forces which arise from the passive deformations of components of the cortical layer other than the active filamcntous network. In the absence of surface rheological elfccts the active forces rcsponsiblc for the cytoplasmic division are balanced solely by the viscous dissipation in the inner and outer fluids. Including these effects will therefore produce additional surface forces which oppose the active forces exerted by the ftlamcntous network and may play an important role in the cytokinctic process,especially in the polar region and the equatorial zones where the stretch is large. The model presented here has some limitations which stem from the lack of a comprehensive understanding of the biochemical processes at the cell surface. Thus, assumptions were made with regard to the dynamics of filament production and interactions and the ability of separating the effects of active and passive tensions at the cortex. In addition we have regarded the cytoplasm as a Newtonian fluid and have neglected the etfcct of the adjacent cells on the cleavage process. This has simplified considerably the mathematical procedures without atTectingthe major characteristics of the observed process. In order to elucidate the importance of these surface rheological effects on the overall geometrical parameters and on the local deformations, various rheological behaviours were assumed to model these cortical passive tractions. The results of the overall deformations are presented in terms of the evolution of the polar length with the stage of division which is expressed by the corresponding furrow radius. The
calculated deformations of the cell, shown in Fig. I, depend only mildly on the type of rheology assumed. Comparison of these with the experimental findings of Hiramoto (1958) and Yoneda and Dan (1972) show that the calculated values of the polar length are larger by 20% than those measured experimentally. The sensitivity of these calculations is examined by studying their dependence on the particular material constants used in the respective rheological models. The results obtained for the Mooney-Rivlin material and shown in Fig. 2 indicate that the overall deformation is insensitive to the value of the material constant r in the range studied. It should be noted that as the surface matrix stretches further and departs from linearity the passive surface tractions are respectively augmented and the process is halted since the active forces, at a certain point. are unable to overcome these restoring tensions. Similar results are obtained if the constant r? is increased. The influence of the surface rheology on the geometry of the overall cell deformations is limited. However, it may be of critical importance in the modulation of the surface tensions. This latter elfect is shown in Fig. 3 in which the tension at the furrow in the circumferential direction is plotted versus time. Here we compare a case in which the passive surface tractions arc neglected and a cast where the surface follows a Mooney-Rivlin rheology. The incorporation of the passive tensions has two cfTccts.It products it retarding effect and brings the tension at the end of the process close to the initial Ievcl, which is in agrccmcnt with the cxpcrimcntal findings of tiiramoto (1968). While surface rheology has only minor influence on the overall geometry it may be important in determining local surface deformations. Such an evaluation is presented in Figs 4 and 5 where the local stretch ratios in the meridional direction at the poles and at the equator, respectively, are plotted against the stage of cleavage, also as a function of the particular rheology assumed. Comparison is made with the experimental data of Hiramoto (1958). who followed the motion of small carbon particles attached to the ccl1 surface. A similar procedure using other inert particles as markers was employed by Rappaport and Rappaport (1976) and by Koppel et 01. (1982) to observe surface motions at the onset of and during cytokinesis. The results presented by Rappaport and Rappaport (1976) dilfer from those of Hiramoto (1958) while the work of Koppel er al. (1982) gives an indication of the change in concentration of the attached particles, mainly at the furrow region where a large agglomeration of particles is observed. The values reported correspond to surface contractions at the equatorial region in the range of 25-50% as compared to only 5-10% measured by Hiramoto. The results obtained with the hydrodynamical model presented here, thcrcfore, agree preferably with the values reported by Rappaport and Rappaport (1976) and the results of Koppel ef cl. (1982). These are also compatible with the view that surface contraction of the equatorial
424
D. ZIF~EMANASand A. NIR
region may play an active role in the formation of the CR. Moreover, the stretch ratios calculated at the poles and at the furrow region are shown to follow the same qualitative behaviour for the various constitutive equations used and the reported values (Hiramoto, 1958). In view of the discrepancy among the experimental data, a quantitative comparison is difficult at this stage and more data will be desirable in order to verify our calculations. Since the surface rheology seems to have little importance in determining the geometry of the dividing cell. the latter may be controlled by other factors. One of these is the interaction between filaments which, in the lack of knowledge on their nature, were assumed to follow dynamic changes given by an expression of the type of equation (15). Since the forces which primarily determine the cell deformation are those exerted by the filamentous network and the expression used in the calculation of the interaction is only an approximation of the real processes, any deviation in the estimate of the local orientation distribution of filaments may have some effect on the cytokinetic simulations and the cell deformations (see Fig. 6). A variation of the c0nstant.R. is shown to have a significant etfect on the overall cell deformation. This result indicates a need for a better understanding of these interactions. which are probably induced by steric effects, electrical and molecular forces and hydrodynamic interactions. Finally the trajectories of various material points on the cell surface during cell cleavage arc close to those reported by lliramoto (1958, 1968). except for the higher initial strctchcs of the surface at the polar regions predicted by the model which later approach the observed values (Fig. 7). Similarly, points in the zone where all shapes intersect were found to remain almost static, as is found in the experimental data.
CONCLUSIONS
Inclusion of surface rheological effects in the hydrodynamical model of cytokinesis proposed by Zinemanas and Nir (1987, 1988) is found to be important, especially concerning the prediction of the local surface deformation and the modulation of the surface forces. The effect on the overall deformation is found to be limited, The model, then, is basically able to predict the major phenomenological characteristics of the process; however, more experimental data are needed as well as a better understanding and knowledge of the cortical layer composition and structure and the nature of filament interactions, in order to attain more accurate predictions of the cell geometry.
Ackno~r/~*~~ml~~rs-This work wassupportedin part by the Fund for Promotion of Research at the Tcchnion. A. N. also acknowledges the hospices of the Benjamin Levich Institute for Physicochemical Hydrodynamics during the preparation of the manuscript.
REFERESCES
Akkas, N. (1980) On the biomechanics of cytokinesis in animal cells. J. Biomechunics 13. 977-988. Akkas. N. (1981) A viscoelastic model for cytokinesis in animal cells. J. Biomechonics II. 621-631. Bray, D.. Heath, J. and Moss, D. (1986) The membraneassociated ‘cortex’ of animal cells: its structure and mechanical properties. J. Cell Sci. Suppl. 4, 71-88. Conrad, G. W. and Rappaport, R. (1981) Mechanisms of cvtokinesis in animal cells. MitosisiCvrokinesis (Edited by Zimmerman. A. M. and Forer, A.). pp. 363-396. Academic Press. New York. Evans, E. A. and Hochmuth. R. M. (1976) Membrane viscoelasticity. Biophys. 1. 16. I-l I. Green, A. E. and Adkins. J. E. (1970) Laryv EkusticDejormutions. 2nd Edn. Oxford University Press. London. Greenspan. H. P. (1977a) On the dynamics of cell cleavage. J. rheor. Biol. 65. 79-99. Greenspan. H. P. (I977b) On the deformation of a viscous droplet caused by variable surface tension. Srud. uppl. Mu;h.
.
57.45-58.
Greenspan. H. P. (1978) On fluid-mechanical simulations of cell division and movement. J. theor. Biol. 70. I?S-134. Hiramoto. Y. (1958) A quantitative description of protoplasmic movement during cleavage in the sea-urchin egg. Exp.
Bid.
35.407-424.
Hiramoto. Y. (1968) The mechanics and mechanism of cleavage in the sea urchin egg. Symp. Svc. np. Bid. 22, 31 l-327.
Hiramoto, Y. (1971) A photographic analysis of protoplasmic movement during cleavage in the sea urchin egg. Deul.
Growth
Difir.
13. 19 I-200.
Koppcl. D. E.. Oliver, J. M. and Berlin. R. D. (1982) Quantitative analysis of ligand-receptor movement into the cleavage furrow: diffusion vs. flow. 1. Cell Biol. 93. 950-960. Opas. J. and Soltynska. M. (1978) Reorganization of the cortical layer during cytokincsis in mouse blastomcrcs. Exp. Cd/ Res. 113, 208-21 I. Pujara, P. and Lardncr, T. J. (1979) A model for cell division. J. Biomechunics
12, 293-299.
Rallison. J. M. and Acrivos. A. (1978) A numerical study of the deformation and burst of a viscous drop in an extcnsional flow. 1. Fluid Much. 89, I9 l-200. Rappaport. R. (1986) Establishment of the mechanism of cytokinesis in animal cells. Inr. Reu. Cyrd. 105. 245-281. Rappaport. R. and Rappaport, B. N. (1976) Prefurrow behavior of the equatorial surface in Aruuciu lixulu eggs.Drd. Growth Dlfir. 18, 189-193. Lpir. T. and Nir, A. (1985) A hydrodynamic study of the furrowing stage during cleavage. Physicochrm. Nydrodyn. 6,803-S
14.
Schroeder, T. E. (1975) Dynamics of the contractile ring. Ma/ecu/es oncl Cell Movement (Edited by Inouc, S. and Stephens, R. E.), pp. 305-334. Raven Press, New York. Skalak. R.. Tozeren. A., Zarda. R. P. and Chien. S. (1973) Strain energy function of red blood cell membranes. Biophys.
J. 13.245-264.
Waxman, A. M. (1984) Dynamics of a couple-stress fluid membrane. Sfud. appl. Murh. 70, 63-86. White, J. G. and Borisy, G. G. (1983) On the mechanism of cvtokinesis in animal cells. 1. theor. Bid. 101. 289-316. Yoieda. M. and Dan, K. (1972) Tension at the surface of the dividing sea-urchin egg. J. cxp. Bib/. 57, 575-587. Zinemanas. D. and Nir. A. (1987) Fluid mechanical simulations cell furrowing due to anisotropic surface forces. Biomechonics of Ce// Division (Edited by Akkas, N.), pp. 281-305. Plenum Press, New York. Zinemanas, D. and Nir, A. (1988) On the viscousdeformation of biological cells under anisotropic surface tension. J. Fluid Mech. 193, 217-241.
of
Department
VISCOELASTIC
of Chemical
Abstract-The hvdrodbnamlc Ybrk (l’Y87)]. to the passive material. The
EFFECTS
DANIEL
ZINEMANAS
Engineering.
Technion,
IN CELL CLEAVAGE
and AVINOASI Israel Institute
NIR
of Technology.
Haifa 3X00.
lsrncl
etkct of passive surface traction on the cleavage of cells is incorporated in the cytokinesis model of Zinemanas and Nir [Biomrchunic~s ofcell Dirision. pp. 281-305. Plenum Press. New DitTerent rheological behaviours were examined to model the surface tensions which arise due deformation of the cortex: a Mooney-Rivlin material. an STZC material and a viscoelastic calculations show [hat passive surface tensions may play a significant role in determining the
local surface deformations as well as in the modulation of the surface forces. Varying the rheological model has limited efTecton the overall deformations. The latter appear IO be affected mostly by contractile filament interactions.
relation
IIVTRODtiCTIO;V
between the force in the CR and the deforma-
tion of the cell. Their Cytokinesis
is the division
of the cytoplasm
follows the nuclear mitosis and culminates ductive cycle of an animal biochemical
which
the repro-
cell. The biophysical
processes which
control
enon are not well understood;
and
this phenom-
initially
spherical
internal
mations. i.e. the formation
bchaviour
an equatorial g&d
cleavage furrow, are confined to a thin-
layer
(Ritppitport,
beneath 19X6).
the
plasma
of muscle-like
the orientation
oriented
distribution
conliguration
1978) to a well defined
as the cortex,
contractila
observed to change from an initial dom
memhrane
Iaycr, known
This
contains a network filaments.
progress ol’
of which
uniform
(Opas
bundle
microis
and ran-
and
Soltynska,
of filaments
force. The
pressure
model
satisfactorily
did
not
follow
1968).
which were not taken into account in the
model
which
and
presumably
change in the mcmbranc A possible mechanism the CR
bundle,
a dynamic
for the formation
was given by Greenspan instability
produce
properties. or a CR
(19773. b, 197X). who sug-
is formed
by a hydrodynamic
which causes the microlilamcnts
reorient due to surface contractions.
lilamcntous
Akkas
phenomena
edge
This
to
(198 I). these discrepancies arise due to other biological
elements
1975).
predicts
the cxpcrimcntal
According
parallel to the cleavage plane under the leading furrow (Schroeder.
of an
due LO an equatorial
but the constricting force and the
(Hiramoto,
gested that
aligned
membrane
the cell deformations
that the active forces responsible for the surface deforand continuous
mainly on the second aspect
and is based on the study of the deformation constricting
however, it is accepted
work, as well as that of Akkus
(1980. 1981). concentrates
to agglomerate
or tensile
at the cell’s equator
and is
This instability
named the contractile
ring (CR), is an essential organ-
assumed to be triggered by a decrease in the concen-
elle which
the major
tration
provides
active
force
in the
cleavage process. Although
cytokincsis
is visually manifested only at
the late stages of cell division, i.c. late anaphase. the appearance
cal processes leading ultimately ized and
diflerentiated
structure
start at earlier
biochemical atus (MA)
stimulus originating (Rappaport,
cortical
layer remain hitherto
cytokinetic
provide
the nature
agent or its exact effect on the unknown.
a better
process and
which could
of the
Greenspan’s
agglomeration
at the furrow
region.
model assumes that the surface tractions
are isotropic and depend on the concentration tensile
elements.
However,
based on this model isotropic
numerical
of the deformation
driven
tensions do not show a significant
formation
(Sapir
and Nir,
of the
calculations
1985; White
by
furrow
and Borisy,
1983). a new model
which
and Nir (1987.1988)
presented
is based on some of the ideas
a mechanism
considered
to the experi-
physical description of the surface Iaycr and forces, and
models were pro-
and Lardncr
of the microtilamcnts and
from the poles toward the equator are responsible for
corroborate
(1979). the
aim of such a model has to be first, the explanation ring
The surface tension
gradient thus created and the consequent surface flow
Recently. Zinemanas
understanding
mental findings, various theoretical
contractile
at the poles or an
an explanation
posed. As stated by Pujara the alignment
a
at the mitotic appar-
1986). However,
the stimulating to attain
cortical
midphase
the cortex and alters its
ofeither
In order
forces and
stages. During
of contractile
the filaments’
to the highly organ-
surface
spreads toward
composition
with
of the cleavage furrow, the biochemi-
elements
increase of these at the equator.
second.
of
into an equatorial
a description
of the
above. The
incorporates
possible
model
biochemical
occur there. The evaluation ting the formation
uses a more suitable processes which
was successful in prcdic-
ofa contractile
ring and parameters
such as the surface forces, surface concentrations internal similarly
and
pressure and flow fields are found to behave to
the experimental
1968, 1971; Koppel
et al., 1982).
patterns
(Hiramoto,
418
D.
ZINEMANASand A. NIR
The calculations of Zinemanas and Nir (1987.1988) assume that the cell can be idealized as a Newtonian viscous droplet and the surface forces as an anisotropic tension. Consequently, relatively high deformations are predicted in the polar regions and the longitudinal dimension of the dividing all appears higher than observed. Also in that model the surface tension was calculated considering the active components of the interfacial layer only. The purpose of this communication is to incorporate more realistic rheological cortical properties in the biomechanical model. The interfacial forces will constitute both the active part due to the contractile microfilaments network as well as the forces which arise when passive components of the cortical matrix deform. Various rheological bchaviours of this cortical layer passive contribution are examined with relation to the overall cell deformation during cytokinesis.
RIOMECHANlCAL
;(I +i)v(x)+(I-i)
FORMJLATION
Consider the cell to be a viscous droplet, B. of viscosity /( which is immersed in a second fluid, B’, of
K(C):v(y)tiy)dS,
x, y&B
J(<).f(y)dS,
(1)
where v is the velocity field, n is a unit vector normal to dB pointing into B’. f denotes the surface tractions and vm is the velocity field far from the drop which must also satisfy the creeping flow equations. In our case v-=0. J and K are single and double layer potentials defined, respectively, by ’ ” J(<)=e+F.
MODEL
Consider the following biomechanical model (Zinemanas and Nir. 1987, 1988). (a) The cortical layer is approximated by a network of filaments confined to a continuous thin matrix. The surface forces are a combination of the contribution of the active filaments network and the passive reactions to surface deformations of the continuous matrix. (b) Active filaments exert a time-dependent contractilc force parallel to their symmetry axis; thcrcfore. their contribution to the surface tractions depend both on their local concentration and orientation distribution. (c) Reorientation of lilaments follows the surface motion and is enhanced by mutual interactions. (d) The initial uniform composition of the cortical layer is altered by a stimulus. This stimulatory agent originates at the MA asters and diffuses. during a given time interval, toward the surface. The stimulant is assumed to inhibit the formation of active filaments through a very simple kinetic scheme; hena. it atTectsthe surface mainly at the poles where its flux is expected to be higher. (e) The rheological properties of the continuous matrix, neglected in the previous calculations (Zinemanas and Nir, 1987, 1988). are now incorporated. Constitutive equations for a Mooney-Rivlin material (Green and Adkins. 1970). an STZC material (Skalak et al., 1973) and a viscoelastic material are examined. A detailed discussion of the biophysical and biochemical considerations and simplifying assumptions used in the derivation and solution of this model is given in Zinemanas and Nir (1987, 1988). hlATliEMATICAI.
viscosity cc’ and is surrounded by a thin layer dB. When both inner and outer fluids are assumed Newtonian and inertial efTects are neglected the momentum equation may be conveniently converted into a boundary equation for the surface velocities (Rallison and Acrivos, 1978):
K(<)=;-$
(2)
where 4=x-y
and
<=I<[
and 1 denotes the inner to outer viscosity ratio p/I/. For 1= I. equation (I), with v”(x)=O, is simplilied to the form
J (0 * f(y) dS,
(3)
which also holds everywhere in B and, B’. Similarly, the pressure lield is given by the expressions P(x)-
i 38
Q(C) * a(v(y)) * n(y) dS, G(C):v(y)n(y)dS,
-RF
I
W=PW+
xe& yedB
(4)
Q(C)*4W)~WW,
3n
+P
c
J&V
C(C): v(y)n(y)dS,
xaE’, ycdB
(3
where
QW=
--$.
.
05)
These equations also simplify, for 1- 1, and P”(x)=O, to the form P(x)= I
a8
Q(C) * f(y) dS,
(7)
which is also valid throughout. The continuitykquation for an incompressible fluid may also be integrated to the form v*ndS=O. I
a0
(8)
419
Surfaceviscoelasticeffectsin cell cleavage Evaluation of the flow equations (1). (4) and (5) thus require only the knowledge of the instantaneous intcrfacial forces. 1. and the boundary shape S5. The latter, though a priori unknown, follows dynamic variations described by the kinematic condition dR -_=v dt
w(p)=&l,p’p4(v’IC-a’Pb,pv’)n+w(p)
x&B
where d/dt denotes a material derivative and R is the radius vector that specifies the location of the interface, 23. The interracial forces. in the absence of surface inertia, may be expressedin general form using a set of the surface coordinates u, by (Waxman, 1984) f=(y’bIp-hfM461p)R,1+(h,qy14-_M””J,p)n
(10)
where 7’S is the surface stresstensor, MzP the moment tensor and b,, the second fundamental form of the surface. A comma and a bar indicate a common and a covariant derivative. respectively,and R., are the basis vectors tangential to the surface. Although bending moments may arise and be important in the deformation of thin surface layers these effectsare neglected in the present study. As stated in the biomechanical model, the stress tensor 7“’ is composed of active and passive components and thus is described by an expression of the form
whcrc the active component y$, stems from the activities of the iilamentous network and dcpcnds on the local concentration and orientation distribution of the filaments. On the other hand, the passive reactions of the matrix to the surface deformations. yz,, depend on the rheological properties of the cortical material through the strain c,,, and rate of strain G,, or other related variables. The local concentration, r. and orientation distribution function, N, follow the mass and orientation balances which are of the form
dc ; + dN x
+
w, =
-c(Y’aJPb,p+Y”I,)+07alPCl,p+~
V’CI, =
DaVfp+”+ V,,,(wA’ 1
scale of an individual filament is much smaller than the distance typical for the surface velocity variations. the flow field around a filament may be considered linear. thus, the angular velocity is given by the expression
(12)
_ +;(N-N)+!+‘NI,). Here u.# is the surface metric tensor. D, and D, are, respectively. the rotational and translational diffusion coefficients, p is a unit vector along the filament axis and A denotes the orientation distribution function of filaments produced at a rate R‘. V’ and v3 are the velocity contravariant components tangential and normal to the interface, w is the filament angular velocity and Vo,, is an operator with respect to the filament orientation. Note that the LHS of equations (12) and (13) are also material derivatives. Since the
(14)
where E,,, is the permutation tensor and co(p) is an additional angular velocity arising from mutual filament interactions as suggested by Schroeder (1975). These interactions were conveniently assumed to be of the form ~(O)=Qsin?Ol~arctan(~)-11
(15)
which allows for a closed form solution of the orientation balance (13) and. at the same time, captures the physical requirement that the interaction between two parallel or perpendicular filaments does not cause a rotational motion and the assertion that this motion is enhanced as order increases. In equation (15) R is a proportionality constant, y,, and y2r are the principal stressesand 0 is the angle formed between the filament axis and the surface tangential velocity. The explicit expression that relates the active stresscs and the concentration and orientation distribution function of the tilamcnts is given by (Zinemanas and Nir. 1987) Y JP
(16)
where yba’ are the principal values of the stresstensor and the basis vectors coincide with the principal directions. F is the filament contractile force and L its length. The parentheses in the superscript indicate the physical value of the stresscomponent. The product of the force by the length is assumed to follow dynamic changes described by an expression of the form d(FL) -&-&c+
(17)
which approximately simulates a muscle-like behaviour. The constants C, and C, are chosen to match the process time scale. The cell cortex is not a well defined organelle since the thickness and composition of this denser layer, composed in part of a microfilamentous network, vary widely among ditferent types of cells (Bray et al.. 1986). Therefore, there have not been many attempts to derive constitutive equations based on the ultrastructure of this layer. The rheological models used in nonhydrodynamic simulations ofcell cleavage (e.g. Akkas. 1980) were based on a homogeneous approach and assumed, as stated above, the behaviour of a Mooney-Rivlin material. an STZC material or a viscoeslatic material with good successin predicting the cell deformations in all cases. It follows from these simulations in which material properties were dynamically modified, as well as other experimental data
420
D. ZI~EMAHAS
(Conrad and Rappaport. 1981). that the ceil surface may be active not only at the furrow region. Furthermore, since activity depends on the amount of stimulation that reaches the cortex, the surface may exhibit different activities in different zones. It is therefore clear that a completely passive description of most of the cortical layer is insufficient. In the present approach we have separated active from passive events; thus, in addition to the active forces emerging from the filamentous network as described above, the rheological properties of the continuous cortical matrix are considered. These passive tractions appear from other components of the layer as well as from non-active filaments. Since the properties of all these have not been measured separately and the composition of the layer is still unknown we will study the possible importance of passive surface tractions in the cleaving process using the aforementioned three models. i.e. Mooney-Rivlin, STZC and viscoelastic materials. The constitutive equation for a Mooney-Rivilin material is given by Y~11)=2hZ(t-~)(l+~i:1.
(18)
while for an STZC material it is y’l
I)
(: i.,
=,i[r(n:-I)+i:(i:E.:-l)]. ‘I
(19)
ifcrc < and f arc material constantsand h is the initial layer thickness. The second principal tension is obtained in both cases by interchanging the indicts I and 2. Similarly, for a viscoelastic material of the Kelvin
and A.
NIR
AXISY%t%tETRICAL NLJlERICAL
CLEAI’AGE
A\D
PRDCEDCRE
The general equations presented in the previous section were solved for the axisymmetrical case with fore and aft symmetry which corresponds to the most common pattern of cell cleavage. Thus. a set of cylindrical coordinates (r, c. 4) is used together with the surface coordinates, (:. 4). selected for the twodimensional equations while the cell shape is defined by the even function r=R(c) for -I<:
(24)
A@-Y)‘f(Y)dY.
I -I The normal and tangential components of fare
type Y ‘11’=-P~+(~)(j.i-l)+2~~(~).
(20)
This constitutive equation was derived by Evans and Hochmuth (1976) for red blood cell membranes and was used in the study of membrane relaxations. Here r7 and qc are a shear modulus and a surface viscosity, respectively. In these equations, i., denote the principal stretches which are related to the principal strains and rate of strain through e,,,,=G-,‘-
1)
&,) = II.).,.
(21)
(23)
which provides the instantaneous strain rate at any material point in terms of the surface velocities. As in Pujara and Lardner (1979). the passive tensions were set to 0 when the calculated stretch ratio is lower than I, i.e. when the surface folds. This phenomenon occurs at the furrow region as the surface contracts there.
(Y**- Yor,) R’(y) R(Y)
1
(26)
where the radii of curvature are given by I -=R:
1
I
R”(Y) [t+~‘~(y)]~‘*’
~=R(y)[I+R”(y)l”‘(27)
(b) Internal pressure I
(22)
Moreover, these surface strains are related to the surface motion by the expression
. e=b=~(V,lb+vpI,)--b,pv’
I /.=[l+R’20,),*,~
P(r. :) =
P(z-y).f(y)dy.
(28)
-1 The components of A and P obtained from the analytical circumferential integration of the kernels J and F are given by Zinemanas and Nir (1988). (c) Mass balance
I
+ R(:)[l
+
R”(i)]“*
c?(s, R)
8:
1+d.
(29)
421
Surface viscoelasric et7ectsin cell cleavage 2.0)
Here v, and v, are the normal and meridional tangential velocity components, respectively. (d) Orientation balance dS -=p dr
S( w(0l.l;) (30)
20
1
1.5.
where the angular velocity is given by w(O)= -isin 1.0, 0
+ o(0).
. , . . 0.5
.
.
.
. 0
(31)
(e) Deformation
sv,
I
I (32)
e(x’)=[, +R"(-_)]t/2~+R,vn “, 4,,
. .
=
R(.-)[I+R’2(;)]“z~+R,v”’
dR
I
Fig. I. A comparison of the change of the non-dimensional polar length. 1, with nondimensional furrow radius, R,. for various surface rheologies used in the model. (---) SurfaLp rheology neglected; (- - - - -) Mooney-Rivlin; (- - -) STZC: vi&elastic material. Experimental data Or G-4 Hiramoto (1958) and Yoneda and Dan (1972).
(33)
Since the equation for the surface velocity is highly non-linear due to the strong coupling which arises between the momentum. mass and orientation balances through the intcrfacial forces, a numerical approach is required to solve this moving boundary problem. The integral equation may beconvcrted into a linear set of algebraic equations by discrctizing the boundary into a number of clcmcnts and employing a finite ditTerencemethod (Rallison and Acrivos, 1978). Consequently. the equations for the cell shape. concentration, orientation distribution and deformation may be integrated in time using a simple Newtonian integration and an updated Lagrangian algorithm. Some mathematical difficulties arise from the singular nature, as y approaches x, of the single layer potential, J, but these may be overcome by subtracting the leading orders of the singular integrand and adding an analytical integration of the asymptotic singular expansion around x (Rallison and Acrivos, 1978). For a detailed account of the numerical procedure the reader is referred to Zinemanas and Nir (1988).
RFSULTS
. The model of cell division presented in the previous sections was evaluated for a number of casesinvolving diKerent surface rheologies. In Fig. I the dynamics of the overall deformation are presented in terms of the change of the polar length, 1. as a function of the furrow radius, R,, for four different cases; a Mooney-Rivlin fluid with Zhd/y,=O.I and r=O.l,an SZTC material with 6/2y,=O.O5 and r=O.l. a viscoelastic material with P,=0.)1~~,,=0.6 and q,/yO =O.IS, and a case with no surface rheology. The variation in the overall deformation was not significant for the various surface rheologies and a slight deviation from the experimental observations of
1
0
0.5
1.0
4 Fig. 2. The ekt of vatiation in material Constant of the Mooney-Rivlin model on the dynamics of the nondimensional polar length. i, and nondimensional furrow radius, R,. ( ---) r-0; (-----) r-0.1; (---) r-=0.2; (---) l-=0.5.
Hiramoto (1958) and Yoneda and Dan (1972) was noticed. The sensitivity of the deformation to the material constants of a surface rheology of the Mooney-Rivlin type is shown in Fig. 2. Again. variations of the overall deformation due to changes in the material constant, r. are observed to be small. However, the incorporation of the surface rheology into the model results in a dilTercntqualitative behaviour of the surface tension dynamics. The evolution of the circumferential tension at the furrow region, depicted in Fig. 3, indicated that when surface rheology is included in the model the tension grows and decays monotonically, as is also evident in experimental observations. When surface rheology is not included the final level of the circumferential tension remains unreasonably high. The changes of the stretch ratio in the meridional direction at the equator and at the poles are shown in
422
D. ZINEMANAS and A. NIR
\
\
\
‘*. \ \\ \
‘_
ol......-..,.....“.. 0
5
1.0
10
.
.
.
.
.
.
.
(
-
.
.
.
.
.
.
1.0
0.5
0
Tlma
R1 Fig. 3. The dynamic development of the circumferential tension at the furrow, yW. (-) Surface rehology neglected;
(- - - - -) Mooney-Rivlin material.
Fig. 5. A comparison of the change of the polar stretch ratios, I,. with non-dimensional furrow radius. R,. for the various surface rheologies. (-) Surface rheology neglected: (----) Mooney-Rivlin; (---) STZC: (--) viscoelastic material. ( x ) Experimental data of Hiramoto (1968).
10 2.0.
.
---------.- . . . .._I’-. -‘\ -... *... *-. -. -.. ....‘. ‘..::\
1 1.5.
a
O~....‘,‘..,“‘......l 0
0.S Rl
1.0
Fig. 4. A comparison of the change of the meridional equatorial stretch ratio. 1,. with non-dimensional furrow radius, R,, for the various surface rheologics used. (-) Surface rheology neglected; (- - - - -) Mooney-Rivlin: (- - -) STZC; (- -) viscoelartic material. ( x ) Exprrimemal data of Hiramoto (1968).
Figs 4 and 5. respectively. and are compared to the measured values of Hiramoto (1958). The values of the stretch ratio obtained with the present model are in qualitative agreement with the experimental observations and do not vary considerably for the different surface rheologies used in the calculations. The most signiticant influence on the overall deformation was found to be due to variations in the filaments’ interaction level. These are observed in the results shown in Fig. 6 where, with f2 assuming the values 1.2, 1.3 and 1.4. a considerable variation in overall cell deformation is encountered. Finally the trajectories of a number of particles on the cell surface are followed during the furrowing process and are shown in Fig. 7. DISCUSSION
It is convenient to start this discussion with a short summary of the results obtained by the hydrodynam-
l.OC. 0
.
,
.
.
1
.
.
I
0.5
.
.
.
“:>,
.
1
1
.
.
’
\ 1.0
Rf Fig. 6. The effect of filamcnl interaction as manifested by changes of the non-dimensional polar length. f. with nondimensional furrow radius_ H,. for various tilamcn~ interaction constams. (-) R=l.4; (-----) R=l.3; (----)Q=l.2.
ical model in the absence of surface rheological effects (Zinemanas and Nir, 1987, 1988). The formation of a contractile ring apparatus was successfullyexplained in terms of an initial disturbance of the cortical layer composition, caused by the mitotic apparatus, which produces a surface tension gradient. Lower tensions prevail at the polar regions due to an assumed decrease in the concentration of active filaments in these zones. This initial tension gradient induces a motion from the polar regions towards the cell equator which consequently causes an agglomeration of filaments at the leading edge of the cleaving furrow. The Row thus produced and the interactions between filaments cause, simultaneously, a reorientation of the filaments parallel to the cleavage plane. This effect, together with the agglomeration, forms the organelle known as the contractile ring which exerts the major active force in the cleaving process.,The occurrence of filament interactions as postulated by Schroeder (1975) is also found to be essential in the reorientation of filaments
Surface viscoelastic effects in cell cleavage
0
09
1e
Fig. 7. Consecutive shapes during cleavage showing trajectories of surface trace particles. R and I denote radius and axial coordinates.
in the furrow region and in the accomplishment of complete divisions. As stated before, a good qualitative agreement was also found for various other variables such as the internal pressure.the surface tensions at the poles and at the furrow region and the internal Row patterns. The overall geometric dimensions of the cell. however. wcrc found to dilfer slightly from those observed experimentally with polar lengths larger than those rcportcd by Hiramoto (1958) and Yoncda and Dan (1972). It was assumed that these dilTcrenccs can be attributed to ncglccting surface forces which arise from the passive deformations of components of the cortical layer other than the active filamcntous network. In the absence of surface rheological elfccts the active forces rcsponsiblc for the cytoplasmic division are balanced solely by the viscous dissipation in the inner and outer fluids. Including these effects will therefore produce additional surface forces which oppose the active forces exerted by the ftlamcntous network and may play an important role in the cytokinctic process,especially in the polar region and the equatorial zones where the stretch is large. The model presented here has some limitations which stem from the lack of a comprehensive understanding of the biochemical processes at the cell surface. Thus, assumptions were made with regard to the dynamics of filament production and interactions and the ability of separating the effects of active and passive tensions at the cortex. In addition we have regarded the cytoplasm as a Newtonian fluid and have neglected the etfcct of the adjacent cells on the cleavage process. This has simplified considerably the mathematical procedures without atTectingthe major characteristics of the observed process. In order to elucidate the importance of these surface rheological effects on the overall geometrical parameters and on the local deformations, various rheological behaviours were assumed to model these cortical passive tractions. The results of the overall deformations are presented in terms of the evolution of the polar length with the stage of division which is expressed by the corresponding furrow radius. The
calculated deformations of the cell, shown in Fig. I, depend only mildly on the type of rheology assumed. Comparison of these with the experimental findings of Hiramoto (1958) and Yoneda and Dan (1972) show that the calculated values of the polar length are larger by 20% than those measured experimentally. The sensitivity of these calculations is examined by studying their dependence on the particular material constants used in the respective rheological models. The results obtained for the Mooney-Rivlin material and shown in Fig. 2 indicate that the overall deformation is insensitive to the value of the material constant r in the range studied. It should be noted that as the surface matrix stretches further and departs from linearity the passive surface tractions are respectively augmented and the process is halted since the active forces, at a certain point. are unable to overcome these restoring tensions. Similar results are obtained if the constant r? is increased. The influence of the surface rheology on the geometry of the overall cell deformations is limited. However, it may be of critical importance in the modulation of the surface tensions. This latter elfect is shown in Fig. 3 in which the tension at the furrow in the circumferential direction is plotted versus time. Here we compare a case in which the passive surface tractions arc neglected and a cast where the surface follows a Mooney-Rivlin rheology. The incorporation of the passive tensions has two cfTccts.It products it retarding effect and brings the tension at the end of the process close to the initial Ievcl, which is in agrccmcnt with the cxpcrimcntal findings of tiiramoto (1968). While surface rheology has only minor influence on the overall geometry it may be important in determining local surface deformations. Such an evaluation is presented in Figs 4 and 5 where the local stretch ratios in the meridional direction at the poles and at the equator, respectively, are plotted against the stage of cleavage, also as a function of the particular rheology assumed. Comparison is made with the experimental data of Hiramoto (1958). who followed the motion of small carbon particles attached to the ccl1 surface. A similar procedure using other inert particles as markers was employed by Rappaport and Rappaport (1976) and by Koppel et 01. (1982) to observe surface motions at the onset of and during cytokinesis. The results presented by Rappaport and Rappaport (1976) dilfer from those of Hiramoto (1958) while the work of Koppel er al. (1982) gives an indication of the change in concentration of the attached particles, mainly at the furrow region where a large agglomeration of particles is observed. The values reported correspond to surface contractions at the equatorial region in the range of 25-50% as compared to only 5-10% measured by Hiramoto. The results obtained with the hydrodynamical model presented here, thcrcfore, agree preferably with the values reported by Rappaport and Rappaport (1976) and the results of Koppel ef cl. (1982). These are also compatible with the view that surface contraction of the equatorial
424
D. ZIF~EMANASand A. NIR
region may play an active role in the formation of the CR. Moreover, the stretch ratios calculated at the poles and at the furrow region are shown to follow the same qualitative behaviour for the various constitutive equations used and the reported values (Hiramoto, 1958). In view of the discrepancy among the experimental data, a quantitative comparison is difficult at this stage and more data will be desirable in order to verify our calculations. Since the surface rheology seems to have little importance in determining the geometry of the dividing cell. the latter may be controlled by other factors. One of these is the interaction between filaments which, in the lack of knowledge on their nature, were assumed to follow dynamic changes given by an expression of the type of equation (15). Since the forces which primarily determine the cell deformation are those exerted by the filamentous network and the expression used in the calculation of the interaction is only an approximation of the real processes, any deviation in the estimate of the local orientation distribution of filaments may have some effect on the cytokinetic simulations and the cell deformations (see Fig. 6). A variation of the c0nstant.R. is shown to have a significant etfect on the overall cell deformation. This result indicates a need for a better understanding of these interactions. which are probably induced by steric effects, electrical and molecular forces and hydrodynamic interactions. Finally the trajectories of various material points on the cell surface during cell cleavage arc close to those reported by lliramoto (1958, 1968). except for the higher initial strctchcs of the surface at the polar regions predicted by the model which later approach the observed values (Fig. 7). Similarly, points in the zone where all shapes intersect were found to remain almost static, as is found in the experimental data.
CONCLUSIONS
Inclusion of surface rheological effects in the hydrodynamical model of cytokinesis proposed by Zinemanas and Nir (1987, 1988) is found to be important, especially concerning the prediction of the local surface deformation and the modulation of the surface forces. The effect on the overall deformation is found to be limited, The model, then, is basically able to predict the major phenomenological characteristics of the process; however, more experimental data are needed as well as a better understanding and knowledge of the cortical layer composition and structure and the nature of filament interactions, in order to attain more accurate predictions of the cell geometry.
Ackno~r/~*~~ml~~rs-This work wassupportedin part by the Fund for Promotion of Research at the Tcchnion. A. N. also acknowledges the hospices of the Benjamin Levich Institute for Physicochemical Hydrodynamics during the preparation of the manuscript.
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