- Email: [email protected]
Biomechanics of virus-to-cell and cell-to-cell fusion
BIOMECHANICS CELL FUSION
OF VIRUS-TO-CELL
AND
CELL-TO-
N. AkkaS
ABSTRACT A biomechanical model of virus-to-cell and cell-t&cellbsion is presented. Virus and the cells are modelled as initially spherical membranes of nonlinear elastic material thut undergo large deformations. The membranes are connected by a Lytoplasmic bridge which expands, resulting in the formation of a larger single sphere. In virus-to-cell&ion, Keywords:
Biomechanics,
is connected to a much larger one. Fusion membrane is completely incorporated into results predict the existence of a ring force It is possible that the microjluments of the band which may provide this ring force.
cell, virus, cell fusion
INTRODUCTION The study of virus-to-cell and cell-to-cell fusion is an area of research that has been developed extensively within the last ten years. Cell fusion is a common cytopathic or cytotoxic response to infection by a wide range of viruses’. Lipid enveloped viruses infect cells by a mechanism during which fusion between the virus envelope and the cellular plasma membrane takes place. This first stage of membrane fusion between the virus envelope and the cellular plasma membrane is frequently followed by the second stage of cellto-cell fusior?. The problem of whether the cell-tocell fusion occurs directly between cells, or whether an intermediate step of virus-to-cell fusion is necessary, has not been totally resolved yet’. Determining the precise mechanism of virus-to-cell and cell-to-cell fusion at the molecular level has been the topic of numerous ultrastructural studies. Very comprehensive review articles on this subject are available in the literature2”. In the present work we will look at the virus-to-cell and cell-to-cell fusion problem from a purely mechanical point of view. This mechanistic approach will hopefully prove to be helpful in a better understanding and provide a more satisfactory interpretation of some of the complex events taking place during either kind of fusion. The biomechanical model to be presented will never be claimed to be the most complete or most accurate one. However, it is hoped that the relatively simple model of the present work will stimulate further and better developments in case the mechanistic approach proves to be rewarding. ULTRASTRUCTURE Sendai virus has been the agent most frequently used to induce cell fusion in uitro3. Sendai virus particles are released from the plasma membrane of infected cells by a ‘budding’ mechanism6. They are roughly spherical in shape and most of them are 150-250 nm in diameter’. The particles are Middle East Technical
a smaller spherical membrane is completed when the smaller the larger one. The numerical in the so-calledfirrow plane. cortical layer form the furrow
University,
Ankara., Turkey
0 1984 Butteworth & Co (Publishers) 0141-5425/84/040257~08 $03.00
enveloped by a bilayer membrane which consists of lipids derived from the plasma membrane of the host cell’. The lipid bilayer proper is 5-7 nm in thickness. A helical nucleocapsid, which is required for infectivity but not for fusion activity, is enclosed by the lipid bilayer. The external surface of the virus envelope is covered with - 12 nm long spikelike projections which are composed of glycoproteins. The inner surface of the lipid bilayer of the virus envelope is coated with the so-called ‘M protein’. It is suggestesd that the M protein plays a major role in maintaining the structure of the virus envelops. It is a fibrous protein* and, apparently, it is not essential for cell fusion activity9. The spikes that cover the external surface of the lipid bilayer are believed to be required for both haemagglutination and cell fusion activitieslO. A diagrammatic representation of possible organization of structural components in Sendai virus is shown in Figure la. A much more detailed description of the ultrastructure of Sendai virus can be found in Knutton3 and in the references cited therein. The necessarily concise description given above is sufftcient for our purposes. In many studies directed towards understanding the mechanism of virus-induced cell fusion, the fusion of erythrocytes has been used as a model system although He-La and Lettre cells are also common. These cells differ in size and shape but they all show common characteristics in the ultrastructural organization. It is this common ultrastructure that will be discussed now. The cell can simply be defined as a discrete mass of cytoplasm enveloped in a retentive membrane. This membrane is, interchangeably, called the cytoplasmic or plasma membrane. According to the contemporary fluid mosaic model of Singer and Nicolson’t, the plasma membrane is composed of lipids, glycolipids, proteins and glycoproteins. The matrix of the fluid lipid bilayer has a thickness of about 7.5 nm. The cell is not merely bounded by this lipid bilayer. There is also a relatively thick and elastic layer underlying the former. This thick layer is called the cortex or the cortical layer and
Ltd J. Biomed Eng. 1984, Vol. 6, October
257
Cell biomechunics: N. Akkas PLASMA NUCLfOCAPSIO
MEMBRANE I
IO)
tb) NOT
DRAWN
TO
SCALE
Figure 1 Diagrammatic representations of possible organization of structural components in (a) Sendai virus, (b) an animal cell.
there is no definite inner boundary defining it. Various investigators used different values (ranging between 1 pm and 3 pm) for the thickness of the cortex12*13. Within the cortex there are microtubules and microfilament bundles which are functionally linked to plasma membrane glycoproteins I4 . The microtubules and microfilaments are both protein polymers and they may run very close to the inside of the lipid bilayer forming a lattice or network of filamentous structures. In erythrocytes the lipid bilayer is supported by the submembraneous spectrin network15. The plasma membrane, in its natural state, is not a tightly stretched layer, but rather loose and wrinkled. It covers fingerlike cytoplasmic processes called microvilli, and also lines deep depressions on the cell surface. A diagrammatic representation of the essential features of the cell ultrastructure is shown in Figure lb. A more detailed description of this ultrastructure can be found in Akkas and Engin16. MECHANISM Virus-to-cell fusion and cell-to-cell fusion are best considered as involving three separate stages3. In the first stage the lipid bilayers of virus and cell (or of two cells) must be brought into close contact. In the second stage the two membranes which envelope two separate compartments fuse. Finally, in the third stage the site of fusion expands, the two compartments form a single one and fusion is completed. Diagrammatic representations of these three stages during virus-to-cell and cell-to-cell fusions are shown in Figures 2a and 26, respectively.
aiQSTAGE
I
(-&44-v---STAGE
II
STAGE Ill
b)
_8-
8+-j-=> STPGE
I
STAG&Z II
?JAGE
Ill
Figure 2 Diagrammatic representations of three stages during (a) virus-to-cell fusion, (b) cell-to-cell fusion.
258
J. Biomed
Eng. 1984, Vol. 6, October
The shape change taking place during virus-to-cell fusion is exactly the opposite of that occurring during ‘budding’ of vi% particles from the plasma membrane of infected cells. For this reason it is perhaps reasonable to expect that the conclusions of the present work on the mechanics of virus-tocell fusion may be applicable also to virus budding. Moreover, the sha e change occurring in cell-tocell fusion is exac jPy the opposite of that occurring during cytokinesi?. It is hoped that our previous experience on cytokinesis in animal cells17’18will be helpful in the interpretation of the results on cellto-cell fusion. In the present work we will be concerned only with the third stage of the fusion processes (see Figure 2). In other words, it will be assumed that the lipid bilayers from two distinct membranes (of the virus and the cell or of the two cells) have already coalesced to form one continuous new membrane. The two initially separate components have become connected via the so-called ‘cytoplasmic bridge’. The mechanics of the expansion stage leading to complete fusion is the topic of the present work However, a concise presentation of the essential details of the previous stages also would appear to be helpful. The attractive and repulsive forces and the associated energies occurring between two membrane bounded spherical compartments approaching each other can be studied via the theory of colloid particle interaction. A detailed biophysical modelling of this phenomenon can be found in Gingell and Ginsberg5. The attachment of Sendai virus to the cell surface is achieved by the former’s binding via its spikes to the latter’s receptors3. This binding of virus to the cell surface results in cell agglutinationi9. The attachment of viral spikes to the cell surface deforms the cell causing deep invaginations in the cell surface. Otherwise, the cell essentially retains its original shape2O. Binding does not automatically result in the fusion of virus envelope with the cell membrane. Apparently fusion requires that phospholipids in the attached membranes be in a ‘fluid’ stated. Before the virus-to-cell fusion takes place, the surface of the virus changes from a spherical to a highly convoluted form. Durin fusion no global structural changes-in the ccl f: membrane are seen although localized changes at the fusion site may take plac3. The localized changes may push the intramembraneous protein particles ‘swimming’ in the lipid matrix away from the fusion site, thus generating a protein-free region in the cell and virus membranes. It has been suggested by many investigators that membrane fusion occurs via lipid-lipid interactions between protein-depleted regions of adjacent opposed membranes 21-23. Stated differently, fusion requires the introduction of local disorder or perturbation into membranes24. The movements of the intramembraneous protein particles in the lipid matrix are very likely controlled by the cytoskeletal elements (microfilaments and microtubules) of the cortex*~*~.
Cell biomechunics: N. Akkas
Under conditions of virus-to-cell fusion, cells can be seen to swell. Knutton3 attributes this to the increase in membrane permeability which results in the osmotic cell swelling. The question of whether cell-to-cell fusion is initiated by the simultaneous fusion of a virus particle with the plasma membrane of two cells has not been resolved yet. There is evidence that both supports and refutes this hypothesis2. There have been other mechanisms proposed to explain virus-induced cell fusion3. They will not be discussed here since we are interested in what happens after the cytoplasmic bridge is formed. What is generally accepted is the fact that the bridge formation is followed by cell swelling. Indeed, the completion of cell-to-cell fusion requires this process of cell swelling. If cell swelling is somehow inhibited, the formation of completely fused cells is prevented but cytoplasmic bridges are still seen. The expansion of the bridge is apparently caused by the cell swelling. During swelling both erythrocytes and HeLa cells increase in volume and assume an essentially smooth spherical shape2O. Erythrocytes do not initially have any microvilli; therefore, swelling usually results in their haemolysis. On the other hand, HeLa cells are initially covered with microvilli. During swelling microvilli are lost and HeLa cells can increase in volume significantly without lysis. Further details on the mechanism of virus-to-cell and cell-to-cell fusion can be found in Knutton et a120, Knutton31i9 and Poste and Pasternak2. MATHEMATICAL
MODEL
From a mechanical point of view the third stages of virus-to-cell fusion and cell-to-cell fusion shown in Figure 2 can both be studied using the same mathematical model. Let us examine the deformation pattern of an initially spherical membrane for prescribed displacement at the equator. The spherical, uninflated configuration is shown as A in Figure 3 and it will be our reference configuration. All the quantities occurring in the governing equations will be expressed in terms of those measured on the uninflated reference configuration A. Now assume that the spherical membrane in configuration A is slightly inflated and then subjected to an increasing equatorial ring load till configuration B of Figure 3 is attained. The transiCl
-9
L-
C2 FUSION
COMPLETED
FVSKION COMPLETED
REFERENCE COWFlG”RATIDN
CONFIGuRATwN
0,
4
Figure 3 Mathematical model indicating various stages of the virus-to-cell and cell-to-cell fusion processes.
tion from configuration A to configuration B has actually been studied previously as a model of cytokinesis in animal cells”. In the present work, the process will be reversed. Configuration B will be our starting configuration, but recall that the reference configuration is still A of Figure 3. By considering the deformation pattern of only the upper half of the starting configuration B, we can model virus-to-cell fusion. This is depicted by configurauons C, and C, in Figure 3. B considering the deformation pattern o fythe two halves of the starting configuration B together, cellto-cell fusion can be modelled. Configurations D, and D, depict this case. Here it is inherently assumed that the two fusing cells have the same size. In both models the governing equations are exactly the same, the only difference being in the interpretation of the numerical results. The problem has thus been reduced to the large deformation analysis of a membrane, which is spherical in its reference configuration, for prescribed displacement at the equator. The membrane material is assumed to be nonlinearly elastic. The viscous properties of the contents of the cell or virus are ignored; hence, the internal pressure is uniform. The problem is a quasi-static one. The governing equations of the problem are derived below in a concise manner. Derivations of similar equations can be found in, for instance Fliigge and Chou26, Yang and Fene, and Pujara and LardnerZs. The equilibrium equations are obtained by considering the equilibrium of the deformed element. Then, all quantities of the deformed element are expressed in terms of the corresponding ones of the undeformed element. The equilibrium of forces parallel the symmetry axis of the membrane and in the direction of the radius p of a latitude circle yields the following equations: (ti@ r Sir@)’ -
j Cos*rTi
X, X, = 0,
(~~;Cos*)‘---*r,+pSin*;r,~Xg=O
(2)
In these equations all barred quantities are dimensional. Referring to Figure 3, it is seen that (p’, Pi, @) and (t T,, @) define the deformed and reference configurations, respectively. The normal pressure acting on the deformed membrane is p r, is the radius of the spherical reference configuration. The derivatives, ( )‘, are with respect to the meridian-al angle_@ of the reference configuration. iV@ and Ne are the membrane forces per unit length of t_he reference configuration. X, = $r’and & = pi d$lri d@ are the stretch ratios in the circumferential and meridional d@ directions, respective1 . The total surface area and the volume ef the de ryormed membrane will be denoted by S and V, respectively. The initial uniform thickness of the spherical membrane is h and the axial force in the equatorial ring is denoted
J. Biomed Eng. 1984, Vol. 6, October
259
cell &mechanics:
N. Akh
by FR. The membrane material, which is assumed to be nonlinearly elastic, can be modelled as either the Mooney-Rivlinz9 or the so-called STZC material’O. The material constants for the latter will be denoted by C (force/length) and l? (dimensionless). The dimensional quantities described above will be nondimensionalized through the following relations: T1 = 7,/h,
S = g/h 2, V = v/h3,
N+ = ii$/C,Ne = &/C, FR =
p =Fh/C,
2. -.-
in’6,18. It should be noted, however, that both the Mooney-Rivlin and the STZC representations yield results which are in qualitative agreement as expected 31. There are, of course, quantitative differences among the results. However, the state of the art at present does not necessitate concern over such differences. The constitutive relations for the STZC material, in their nondimensional form, are
NG = ;
(3)
x, [I’@$ -l)+X;(gX$,
-I)],
~ Ne = i b uv-3 - 1) +q(% q - I)]
The equilibrium equations (1) and (Z), after some simple manipulations, can be brought to the following nondimensional form: N& = (No Costi - N@ Cos@)/Sin@, \1’=P!!
Ne A,+%
Sin+ W’
(4) NUMERICAL (5)
PROCEDirRE
When the constitutive relations (10) and (11) are substituted into the equilibrium equations (4) and (5), the latter, together with the compatibility condition (6), can be brought to the following form:
Xgl = (X, Costi -X0 Cos@)/Sin@.
(124
The total surface area and the volume of the deformed membrane, in nondimensional form, are
I/ = 27rrT
42 b & Sin@ d&
s
(7)
0
[ n’2 q h
Sin2@ Sin* d@.
(11)
The problem description is now completed. The numerical method of solution is explained in the 1 following section.
The compatibility condition gives an additional relation among the variables:
S = 4lrrj
(lo)
(8)
Xgr = fe(& @9xe, &5>p),
(1w
V = f&h JI, Xe, A+).
(12c)
Now, one has three equations to determine the three unknowns (A#, X,, JI). The independent variable is +, and p is the loading parameter. Equations (12) can be solved without any difficulty using the well-known Runge-Kutta method32.
J 0
The nondimensional ring is
axial force in the equatorial
FR = 2r, N+ Cos(lr-Q) at @I= 1~12.
(9)
Here, S, V and FR are all for the cell-to-cell fusion case. For the virus-to-cell f&ion case only one half of the deforming membrane is considered. Accordingly, in the latter case S, V, and FR are half of those given by equations (7), (8) and (9), respectively. The statement of the problem will be completed by describing the constitutive relations. In the present work the analysis has been undertaken for both the Mooney-Rivlin material29 and the STZC materia130. However, for the sake of brevity, the numerical results will be presented for the STZC material only. The STZC form essentially yields an exponential stress-strain law. Skalak et al 30. use this exponential form in their studies concerned with the properties of red blood cell membrane. Critical discussions of what kind of material representation can better model cell membrane material are given
280 J. Biomed Eng. 1984, Vol. 6, October
The limits of the integrations in equations (12) are Q = 0 and @ = n/2. The problem is actually a twopoint boundary value problem. The numerical integration could start at $ = n/2; however, there only X6 can be specified. Accordingly, it is more convenient to treat the problem as a one-point boundary value (initial value) problem for which the Runge-Kutta method is very suitable. The numerical integration should start at @ = 0 because, at this point, it is known that JI = 0 and xe = A+ = & 2 1.0. & will be called the apex stretch ratio. Once a value for h, is selected, the numerical integration can be completed and (&, &,, +) can be evaluated at any other location #, provided that the loading parameterp is specified also. Stated differently, one can freely vary X, andp and, thus, can obtain a new equilibrium configuration for the deformed membrane for each new set of (&,,,p). A mong these different equilibrium configurations one selects those that satisfy the constraint condition(s) to be specified. For instance, for cleavage in animal cells, the commonly used constraint condition has been that the cell volume remains constant during division’7*33. For virus-tocell and cell-to-cell fusion there is evidence that the
Cell biomechunics: N. Ahkas
volume does not remain constant, so one has to come up with some other constraint conditions and the possibilities will be discussed in the following section. In the numerical solution, the angular increment in the meridional direction was taken to be three degrees. Finally, it is assumed that the membrane can not be subjected to compressive stresses. Accordingly, whenever the stress resultant Ne turned out to be compressive, which may be the case near the equator in the early part of the fusion, it was set equal to zero. NUMERICAL
RESULTS
AND DISCUSSION
The numerical results presented in Figures 4, 5, and 6 are all nondimensional because the effective thickness of the virus envelope and the cell membrane is open to question. It is known that both virus and the cell have an outer lipid bilayer as shown in Figure I. The thickness of this bilayer is somewhere between 5 and 8 nm. However, as mentioned previously, in both cases this lipid bilayer is connected to, and apparently supported by, a fibrous protein layer (M protein for virus and cortex for the cell). At this time it is not clear whether these protein layers contribute to the
_ _ -0
-_--
+
02
0.4
0.6
08
1.0
1.2
1.4
1.6
I.8
2.0
2.2
2.4
Ff /T, Figure 5 Variations of membrane surface area (solid curves) and membrane volume (dashed curves) during fusion for various values of the apex stretch ratio.
c--,
0
IO
0
‘effective’ thicknesses of the virus envelope and the cell membrane during fusion activities. The author has proposed in a previous work that it is reasonable to treat the cortex, rather than the lipid bilayer proper only, of dividing cells as the ‘membrane’ enveloping the cytoplasm’*. Whether this proposition can be extended to the case of fusing cells and viruses is not known. For this reason, here we have preferred to present the
_t<-, 0__-_
-9--
6
0.8 0.6 0.40.2 0
0.2
04
06
0.6
I0
I2
I4
16
I8
2.0
2.2
2.4
T /?, f
Figure 4 Variations of internal pressure (solid curves) and furrow ring force (dashed curves) during fusion for various values of the apex stretch ratio.
0
0.2
0.4
0.6
0.8
1.0
1.2 1.4
1.6
1.8 2.0
2.2
q/‘i Figure 6 Variation of polar length during fusion for various values of the apex stretch ratio.
J. Biomed Eng. 1984, Vol. 6, October
26’1
Cellbimnechanics: N. Akhns
results in their nondimensional forms. The nondimensional material constant for the STZC material is taken to be r = 0.25. The numerical results in Figures 4, 5, and 6 are all given as functions of the furrow radius to reference radius ratio. The furrow radius Fr and the reference radius Ti are depicted in Figure 3. Moreover, the deformed configurations of the fusing cells (or virus) are schematically shown in the upper part of each figure. For instance, G& E 0.1 corresponds to the beginning of the fusion and the larger values of Ffii correspond to progressing fusion. We will, first of all, discuss the results in general and will, later, note their possible implications in the specific fusion problem. The solid curves in Figure 4 give the intracellular (or intraviral) pressure, p, as a function of the stage of fusion for various values of the apex stretch ratio, &,,. The dashed curves in the same figure give the ring force, FR, that must exist in the furrow plane for equilibrium. It is seen from these curves that, for a fured value of &,, bothp and FR must decrease monotonically for fusion to be completed. It should be emphasized immediately that, during an actual fusion process, it is very likely not necessary that the apex stretch ratio remain constant. We will come back to this point later. It is apparent from the dashed curves in Figure 4 that, for a fixed a, the furrow plane ring force remains practically constant during the early stages of fusion (roughly for FdFi 5 0.7). The small circles on each curve in Figure 4 indicate the final stage of the cell-to-cell fusion. At this stage, the two cells have completely fused to form a single spherical cell. In other words, for cell-to-cell fusion, only those parts of the curves (solid or dashed) remaining to the left of the small circles must be considered. The small circles on the dashed curves of Figure 4, all correspond to FR = 0. This is expected because it is known from simple equilibrium considerations that the furrow plane ring force is zero if the equilibrium configuration has a spherical shape. The curves in Figure 4 must be studied in their entirety if virus-tocell fusion is considered, because the final stage of the latter fusion process corresponds to an almost flat configuration withp * 0 rather than a hemispherical one. In other words, the furrow plane ring force changes its sign during the virusto-cell fusion process. At early stages of fusion, the ring force is contractile but later it becomes a stretching force to complete fusion. Recall that for cell-to-cell fusion no stretching is required and the ring force must always be contractile for equilibrium up until it reduces to zero at the end of fusion, The solid curves in Figure 5 give the membrane surface area, S, as a function of the stage of fusion for values of the apex stretch ratio, &,. The dashed curves in the same figure give the volume of the deforming membrane. For a fixed value of X, the surface area increases monotonically during the entire fusion process and it attains its maximum value at the end of the fusion process (whether it is
262
J. Biomed Eng. 1984, Vol. 6, October
cell-to-cell or virus-to-cell fusion). The situation is different for I/. Recall that the small circles on the curves mark the end of the cell-to-cell fusion process. Accordingly, during the cell-to-cell fusion process, for a fixed &,, the membrane volume increases monotonically (almost linearly) until it attains its maximum value at the end of fusion which corresponds to a spherical configuration. For virus-to-cell fusion, on the other hand, the membrane volume starts decreasing as soon as the hemispherical configuration is reached and the membrane is now being stretched. At the completion of the virus-to-cell fusion, the volume enclosed by the virus membrane approaches zero as expected, because the terminating configuration of virus-to-cell fusion is assumed to be an almost flat membrane. The curves in_F@re 6 give the nondimensional polar length, I$-,, as a function of the stage of fusion for various values of the apex stretch ratio, &,. The definition of the polar length & is shown in Figure 3. Such !P versus 5 curves are probably the easiest to obtam during experiments and they may prove to be useful in predicting the apex stretch ratio variation during a fusion process. The figure is self explanatory. For a fixed &,, the polar length increases slightly during the early stages of fusion (roughly for YdS, _( 0.5) and this is followed by continuous decrease during the later stages. The virus-to-cell fusion is completed when the polar length approaches zero. At the end of the cell-to-cell fusion process (the small circles in Figure 6) the polar length is equal to the furrow radius as expected. The above discussion of Figures 4, 5, and 6 was limited to the cases in which X, is assumed to remain constant during fusion. As stated previously, this is probably not the case in actual virus-to-cell and cell-to-cell fusion processes. In his literature survey, the author was unable to find any explicit remark concerning the possible change in the apex stretch ratio during fusion. For that matter, we could not find any numerical values for the polar length at various stages of fusion either. In case such results are or become available, the following discussion can be modified accordingly and for this reason, it should be considered as discussion of various possibilities rather than that of well established facts. Although there are no data available at present on the apex stretch ratio change during fusion, there are experimental observations reported which clearly state that fusion leads to cell swelling. This occurs during both virus-to-cell and cell-to-cell fusion. Swelling probably continues during the entire fusion process because, as stated in Knutton), haemolysis takes place in the case of erythrocytes. In the case of HeLa cells, swelling involves an ‘unfolding’ of surface microvilli and the formation of large, smooth-surfaced fused cells*(‘. It has been suggested that cell swelling is the driving
Cell biomechanics:N. Akh
force which results in the complete incorporation of the viral envelopes into the cell membrane. A very detailed discussion of swelling during fusion can be found in the review article by Knutton’. Let us first consider cell-to-cell fusion. Cell swelling during fusion implies a simultaneous increase in the intracellular pressure, the surface area, the volume, and the apex stretch ratio. For a physically acceptable fusion process, the furrow radius must also increase during swelling. However, there is no immediate reason for the polar length to increase also because, as can be seen from Figures 5 and 6, volume increase may occur although polar length is decreasing. Refer to F&we 4 and, purely for the sake of demonstration, assume that our starting configuration corresponds to point A on the solid & = 1.1 curve. Assume also that cell-to-cell fusion is completed at point B on the solid X, = I.5 curve, The corresponding points on the dashed curves in Figure 4 are A’ and B’. The corresponding points are marked in Figures 5 and 6 also. As fusion progresses from point A to point B (or from A’ to B’) in these figures, the path followed implies the following conclusions (whichever physically reasonable path we follow). The intracellular pressure, the apex stretch ratio, the surface area, the volume, and the polar length all increase monotonically during fusion which is accompanied by swelling. In other words, the numerical predictions agree with the implications of the experimental observations. The numerical results also predict that, from the dashed curves in Figure 4, the furrow plane ring force may increase initially but must go down to zero towards the end of fusion. The author was unable to find any data concerning the ring force during fusion. Indeed, the possible existence of a ring is merely implied by Poste and PasternakZ (p. 344) as follows: “ . . . the two fusing cells are in a ‘dumb-bell shape. This raises the possiblity that it is the gross morphological change from this shape to that of a single, double-sized sphere, that involves microfilaments and is sensitive to cytochalasin B. Since the latter shape change is exactly the opposite of that occurring during cytokinesis, which is known to involve a contractile ring of microfilaments . . .” Accordingly, it is not surprising that our numerical results indicate the existence of a furrow plane ring force which approaches zero as the cell-to-cell fusion is completed. Microfilaments existing in the cortical layer of the fusing cells at focal areas of contact are the reasonable agents that could possibly form the ring. As stated above, the intramembraneous protein particles ‘swimming’ in the lipid matrix appear to be pushed away from the fusion site. The cytoskeletal elements (microfilaments and microtubules) are connected to these protein particles. From a mechanical point of view, it is possible for the microfilaments connected to the protein particles surrounding the protein depleted region to provide the ring force necessary
for equilibrium. The ring force occurring in cytokinesis has been measured3’; but, to the author’s knowledge, no such measurement is available for the ring force possibly occuring during fusion. For the virus-to-cell fusion case, the situation is the same until the predicted ring force changes sign. Again the microfilamentous ring is capable of producing the expanding force that is necessary for flattening the virus envelope. Now, as seen in Figure 4, this force must increase in magnitude. It is possible that this increase in the ring force is passively controlled by the cell swelling which can push the intramembraneous protein particles further apart, causing further stretching of the furrow ring. Although it is possible, and tempting, to make additional conjectures concerning the mechanics of the fusion processes discussed, it is probably better to wait till some concrete data become available concerning the points raised in the present work. Only then will it be realistic to discuss the biomechanics of virus-to-cell and cell-to-cell fusion and its implications further. It is merely hoped that the present preliminary discussion of the phenomena will draw the attention of cytologists to those mechanical aspects which have so far been apparently overlooked. SUMMARY
AND CONCLUSIONS
It appears that a better understanding of the biomechanical model of fusion has been presented. be helpful in shedding some additional light on the phenomena. With this purpose in mind, a biomechnical model of fusion has been presented. Virus and the cells are modelled as initially spherical membranes of nonlinear elastic material that undergo large deformations. The two spherical membranes are connected by a cytoplasmic bridge which expands resulting in the formation of a larger single sphere. This is cell-to-cell fusion. In virus-to-cell fusion, a smaller spherical membrane is connected to a much larger one. Fusion is completed when the smaller membrane is completely incorporated into the membrane of the larger sphere. The numerical results predict the existence of a ring force in the furrow plane. It is possible that the microfilaments of the cortical layer form the furrow band which may provide this ring force. REFERENCES Poste, G. Mechanisms of virus-induced cell fusion. Int. Rev. Cytd, 1972 33, 157-252. Postes, G., Pasternak, CA. Virus-induced cell fusion. Membrane Fusion, (Eds G. Poste and G.L. Nicolson), Elsevier/North-Holland Biomedical Press, 1978, 305-367. Knutton, S. The mechanism of virus-induced cell fusion. Micron, 1978, 9, 133-154. Lucy, J.A. Mechanisms of chemically induced cell fusion. Membrane Fusion, (Eds G. Poste and G.L. Nicolson), ElseviedNonh-Holland Biomedical Press, 1978, 267-304.
J. Biomed Eng. 1984, Vol. 6, October
263
Cell biomechanics: N. Ahkas 5
6
7
8
9
10
11 12
13 14
15
16
17 18
264
Gingell, D., Ginsberg, L. Problems in the physical interpretation of membrane interaction and fusion. Membrane Fusion, (Eds G. Poste and G.L. Nicolson) Elsevier/North-Holland Biomedical Press, 1978, 791-833. Berkaloff, A. Etude au microscope electronique de la morphogenese de la particule du virus Sendai. J Microscop. 1963, 2, 633-640. Hosaka, Y., Nishi, Y. and Fukai, K. The structure of HVJ II. The fine structure of the subunits. Biken’sJ. 1961, 4, 243-254. Hewitt, J.A. and Nermut, M.V. A morphological study of the M-protein of Sendai virus. J Gen. ViroL 1977, 34, 127-136. Hosaka, Y. and Shimizu, Y.K. Artificial assembly of envelope particles of Sendai virus (HVJ). I. Assembly of hemolytic and fusion factors from envelopes solubilized by Nonidet P40. Virology, 1972, 49, 627-639. Hosaka, Y. and Shimizu, Y.K. Cell fusion by Sendai virus. Virus Infection and the Cell Surface, (Eds G. Poste and G.L. Nicolson), North-Holland, Amsterdam, 1977 129-155 Singer, S.J., Nicolson, G.L. The fluid mosaic mode1 of the structure of cell membranes. Science, 1972, 175, 720. Mitchison, J.M. and Swann, M.M. The mechanical properties of the cell surface, III. The sea urchin egg from fertilization to c1eavage.J. Exp BioL 1955, 32, 734-750. Hiramoto, Y. Rheological properties of sea urchin eggs. Biorheo. 1970, 6, 201-234. Nicolson, G.L., Poste, G. and Ji, T.H. The dynamics of cell membrane organization. Dynamic Aspects of Cell Surface Organization, (Eds G. Poste and G.L. Nicolson), Elsevied North-Holland Biomedical Press, 1977 l-73 Evans, E.A. and Skalak, R. Mechanics and thermodynamics of biomembranes: Parts 1 and 2. CRC Critical Reviews in Bioengng. 1979, 3, 181-330 and 331-418. Akkas, N. and Engin, A.E. Ultrastructure of animal cells and its role during cytokinesis from a structural mechanics viewpoint, Mechanics of Structured Media, (Ed A.P.S. Selvadurai). Studies in Applied Mechanics 5A. Elsevier Scientific Publ. Corn Amsterdam, 1981, Pan A, 187-207. Akkas, N. On the biomechanics of cytokinesis in animal ce11s.J. Biomechantis, 1980, 13, 977-988. Akkas, N. A viscoelastic model for cytokinesis in animal ce11s.J. Biomechanics, 1981, 14, 621-631.
J. Biomed Eng.
1984, Vol. 6,
October
19
20
21
22
23
Knutton, S. Studies of membrane fusion. II. Fusion of human erythrocytes by Sendai virus.J. Cell Sci 1977, 28, 189-210. Knutton, S., Jackson, D. and Ford, M. Studies of membrane fusion. I. Paramyxovirus-induced cell fusion, a scanning electron-microscope study. J. Cell Sci. 1977, 28, 179-188. Satir, B., Schooley, C. and Satir, P. Membrane fusion in a mode1 system. Mucocyst secretion in Tetrahymena. J Cell BioL 1973, 56, 153-176. Orci, L., Perrelet, A. and Friend, D.S. Freeze-fracture of membrane fusions during exocytosis in pancreatic B-Cells. J. Cell BioL 1977, 15, 23-30. Lawson, D., Raff, M.C., Gomperts, B., Fewtrell, C. and Gilula, N.B. Molecular events during membrane fusion. A study of exocytosis in rat peritonal mast cells. J Cell BioL 1977,12,242-259.
24 25
26 27
28
29 30
31 32 33 34
Postes, G. and Allison, A.C. Membrane fusion, Biochim. Biophys. Acta, 1973, 300, 421-465. Orci, L. and Perrelet, A. Ultrastructural aspects of exocytotic membrane fusion. Membrane Fusion, (Eds G. Poste and G.L. Nicolson), Elsevier/Nonh-Holland Biomedical Press, 1978 629-656 Fhigge, W. and Chou, XC. Large-deformation theory of shells of revolution. J. A/I/IL Mech 1967, 34, 56-58. Yang, W.H., Feng, W.W. On axisymmetrical deformations of nonlinear membranes. J AppL Mech 1970, 37, 1002-1001. Pujara, P. and Lardner, T.J. Deformations of elastic membranes - effect of different constitutive relations. ZAMP, 1978,29, 315-327. Treloar, L. R. G. The Physics of Rubber Elasticity. Clarendon Press, Oxford, 3rd edn. 1975. Skalak, R., Tozeren, A., Zarda, R. and Chien, S. Strain energy function of red blood cell membranes. Biophys. J 1970, 13, 245-264. Akkas, N. and Engin, A.E. Nonlinear mechanics and cell division. ASME 1980 Advances in Bioengineering. 1980, 27 l-273 mson, A. and Wilf, H. Mathematical Methodsfor Digital Computers. John Wiley, New York, 1960, 110-120. Pujara, P. and Lardner, T.J. A mode1 for cell division. J. Biomechanics, 1979, 12, 293-299. Hiramoto, Y. Force exerted by the cleavage furrow of sea urchin eggs. Dev. Growth and Dijer. 1975, 11, 27-38.
OF VIRUS-TO-CELL
AND
CELL-TO-
N. AkkaS
ABSTRACT A biomechanical model of virus-to-cell and cell-t&cellbsion is presented. Virus and the cells are modelled as initially spherical membranes of nonlinear elastic material thut undergo large deformations. The membranes are connected by a Lytoplasmic bridge which expands, resulting in the formation of a larger single sphere. In virus-to-cell&ion, Keywords:
Biomechanics,
is connected to a much larger one. Fusion membrane is completely incorporated into results predict the existence of a ring force It is possible that the microjluments of the band which may provide this ring force.
cell, virus, cell fusion
INTRODUCTION The study of virus-to-cell and cell-to-cell fusion is an area of research that has been developed extensively within the last ten years. Cell fusion is a common cytopathic or cytotoxic response to infection by a wide range of viruses’. Lipid enveloped viruses infect cells by a mechanism during which fusion between the virus envelope and the cellular plasma membrane takes place. This first stage of membrane fusion between the virus envelope and the cellular plasma membrane is frequently followed by the second stage of cellto-cell fusior?. The problem of whether the cell-tocell fusion occurs directly between cells, or whether an intermediate step of virus-to-cell fusion is necessary, has not been totally resolved yet’. Determining the precise mechanism of virus-to-cell and cell-to-cell fusion at the molecular level has been the topic of numerous ultrastructural studies. Very comprehensive review articles on this subject are available in the literature2”. In the present work we will look at the virus-to-cell and cell-to-cell fusion problem from a purely mechanical point of view. This mechanistic approach will hopefully prove to be helpful in a better understanding and provide a more satisfactory interpretation of some of the complex events taking place during either kind of fusion. The biomechanical model to be presented will never be claimed to be the most complete or most accurate one. However, it is hoped that the relatively simple model of the present work will stimulate further and better developments in case the mechanistic approach proves to be rewarding. ULTRASTRUCTURE Sendai virus has been the agent most frequently used to induce cell fusion in uitro3. Sendai virus particles are released from the plasma membrane of infected cells by a ‘budding’ mechanism6. They are roughly spherical in shape and most of them are 150-250 nm in diameter’. The particles are Middle East Technical
a smaller spherical membrane is completed when the smaller the larger one. The numerical in the so-calledfirrow plane. cortical layer form the furrow
University,
Ankara., Turkey
0 1984 Butteworth & Co (Publishers) 0141-5425/84/040257~08 $03.00
enveloped by a bilayer membrane which consists of lipids derived from the plasma membrane of the host cell’. The lipid bilayer proper is 5-7 nm in thickness. A helical nucleocapsid, which is required for infectivity but not for fusion activity, is enclosed by the lipid bilayer. The external surface of the virus envelope is covered with - 12 nm long spikelike projections which are composed of glycoproteins. The inner surface of the lipid bilayer of the virus envelope is coated with the so-called ‘M protein’. It is suggestesd that the M protein plays a major role in maintaining the structure of the virus envelops. It is a fibrous protein* and, apparently, it is not essential for cell fusion activity9. The spikes that cover the external surface of the lipid bilayer are believed to be required for both haemagglutination and cell fusion activitieslO. A diagrammatic representation of possible organization of structural components in Sendai virus is shown in Figure la. A much more detailed description of the ultrastructure of Sendai virus can be found in Knutton3 and in the references cited therein. The necessarily concise description given above is sufftcient for our purposes. In many studies directed towards understanding the mechanism of virus-induced cell fusion, the fusion of erythrocytes has been used as a model system although He-La and Lettre cells are also common. These cells differ in size and shape but they all show common characteristics in the ultrastructural organization. It is this common ultrastructure that will be discussed now. The cell can simply be defined as a discrete mass of cytoplasm enveloped in a retentive membrane. This membrane is, interchangeably, called the cytoplasmic or plasma membrane. According to the contemporary fluid mosaic model of Singer and Nicolson’t, the plasma membrane is composed of lipids, glycolipids, proteins and glycoproteins. The matrix of the fluid lipid bilayer has a thickness of about 7.5 nm. The cell is not merely bounded by this lipid bilayer. There is also a relatively thick and elastic layer underlying the former. This thick layer is called the cortex or the cortical layer and
Ltd J. Biomed Eng. 1984, Vol. 6, October
257
Cell biomechunics: N. Akkas PLASMA NUCLfOCAPSIO
MEMBRANE I
IO)
tb) NOT
DRAWN
TO
SCALE
Figure 1 Diagrammatic representations of possible organization of structural components in (a) Sendai virus, (b) an animal cell.
there is no definite inner boundary defining it. Various investigators used different values (ranging between 1 pm and 3 pm) for the thickness of the cortex12*13. Within the cortex there are microtubules and microfilament bundles which are functionally linked to plasma membrane glycoproteins I4 . The microtubules and microfilaments are both protein polymers and they may run very close to the inside of the lipid bilayer forming a lattice or network of filamentous structures. In erythrocytes the lipid bilayer is supported by the submembraneous spectrin network15. The plasma membrane, in its natural state, is not a tightly stretched layer, but rather loose and wrinkled. It covers fingerlike cytoplasmic processes called microvilli, and also lines deep depressions on the cell surface. A diagrammatic representation of the essential features of the cell ultrastructure is shown in Figure lb. A more detailed description of this ultrastructure can be found in Akkas and Engin16. MECHANISM Virus-to-cell fusion and cell-to-cell fusion are best considered as involving three separate stages3. In the first stage the lipid bilayers of virus and cell (or of two cells) must be brought into close contact. In the second stage the two membranes which envelope two separate compartments fuse. Finally, in the third stage the site of fusion expands, the two compartments form a single one and fusion is completed. Diagrammatic representations of these three stages during virus-to-cell and cell-to-cell fusions are shown in Figures 2a and 26, respectively.
aiQSTAGE
I
(-&44-v---STAGE
II
STAGE Ill
b)
_8-
8+-j-=> STPGE
I
STAG&Z II
?JAGE
Ill
Figure 2 Diagrammatic representations of three stages during (a) virus-to-cell fusion, (b) cell-to-cell fusion.
258
J. Biomed
Eng. 1984, Vol. 6, October
The shape change taking place during virus-to-cell fusion is exactly the opposite of that occurring during ‘budding’ of vi% particles from the plasma membrane of infected cells. For this reason it is perhaps reasonable to expect that the conclusions of the present work on the mechanics of virus-tocell fusion may be applicable also to virus budding. Moreover, the sha e change occurring in cell-tocell fusion is exac jPy the opposite of that occurring during cytokinesi?. It is hoped that our previous experience on cytokinesis in animal cells17’18will be helpful in the interpretation of the results on cellto-cell fusion. In the present work we will be concerned only with the third stage of the fusion processes (see Figure 2). In other words, it will be assumed that the lipid bilayers from two distinct membranes (of the virus and the cell or of the two cells) have already coalesced to form one continuous new membrane. The two initially separate components have become connected via the so-called ‘cytoplasmic bridge’. The mechanics of the expansion stage leading to complete fusion is the topic of the present work However, a concise presentation of the essential details of the previous stages also would appear to be helpful. The attractive and repulsive forces and the associated energies occurring between two membrane bounded spherical compartments approaching each other can be studied via the theory of colloid particle interaction. A detailed biophysical modelling of this phenomenon can be found in Gingell and Ginsberg5. The attachment of Sendai virus to the cell surface is achieved by the former’s binding via its spikes to the latter’s receptors3. This binding of virus to the cell surface results in cell agglutinationi9. The attachment of viral spikes to the cell surface deforms the cell causing deep invaginations in the cell surface. Otherwise, the cell essentially retains its original shape2O. Binding does not automatically result in the fusion of virus envelope with the cell membrane. Apparently fusion requires that phospholipids in the attached membranes be in a ‘fluid’ stated. Before the virus-to-cell fusion takes place, the surface of the virus changes from a spherical to a highly convoluted form. Durin fusion no global structural changes-in the ccl f: membrane are seen although localized changes at the fusion site may take plac3. The localized changes may push the intramembraneous protein particles ‘swimming’ in the lipid matrix away from the fusion site, thus generating a protein-free region in the cell and virus membranes. It has been suggested by many investigators that membrane fusion occurs via lipid-lipid interactions between protein-depleted regions of adjacent opposed membranes 21-23. Stated differently, fusion requires the introduction of local disorder or perturbation into membranes24. The movements of the intramembraneous protein particles in the lipid matrix are very likely controlled by the cytoskeletal elements (microfilaments and microtubules) of the cortex*~*~.
Cell biomechunics: N. Akkas
Under conditions of virus-to-cell fusion, cells can be seen to swell. Knutton3 attributes this to the increase in membrane permeability which results in the osmotic cell swelling. The question of whether cell-to-cell fusion is initiated by the simultaneous fusion of a virus particle with the plasma membrane of two cells has not been resolved yet. There is evidence that both supports and refutes this hypothesis2. There have been other mechanisms proposed to explain virus-induced cell fusion3. They will not be discussed here since we are interested in what happens after the cytoplasmic bridge is formed. What is generally accepted is the fact that the bridge formation is followed by cell swelling. Indeed, the completion of cell-to-cell fusion requires this process of cell swelling. If cell swelling is somehow inhibited, the formation of completely fused cells is prevented but cytoplasmic bridges are still seen. The expansion of the bridge is apparently caused by the cell swelling. During swelling both erythrocytes and HeLa cells increase in volume and assume an essentially smooth spherical shape2O. Erythrocytes do not initially have any microvilli; therefore, swelling usually results in their haemolysis. On the other hand, HeLa cells are initially covered with microvilli. During swelling microvilli are lost and HeLa cells can increase in volume significantly without lysis. Further details on the mechanism of virus-to-cell and cell-to-cell fusion can be found in Knutton et a120, Knutton31i9 and Poste and Pasternak2. MATHEMATICAL
MODEL
From a mechanical point of view the third stages of virus-to-cell fusion and cell-to-cell fusion shown in Figure 2 can both be studied using the same mathematical model. Let us examine the deformation pattern of an initially spherical membrane for prescribed displacement at the equator. The spherical, uninflated configuration is shown as A in Figure 3 and it will be our reference configuration. All the quantities occurring in the governing equations will be expressed in terms of those measured on the uninflated reference configuration A. Now assume that the spherical membrane in configuration A is slightly inflated and then subjected to an increasing equatorial ring load till configuration B of Figure 3 is attained. The transiCl
-9
L-
C2 FUSION
COMPLETED
FVSKION COMPLETED
REFERENCE COWFlG”RATIDN
CONFIGuRATwN
0,
4
Figure 3 Mathematical model indicating various stages of the virus-to-cell and cell-to-cell fusion processes.
tion from configuration A to configuration B has actually been studied previously as a model of cytokinesis in animal cells”. In the present work, the process will be reversed. Configuration B will be our starting configuration, but recall that the reference configuration is still A of Figure 3. By considering the deformation pattern of only the upper half of the starting configuration B, we can model virus-to-cell fusion. This is depicted by configurauons C, and C, in Figure 3. B considering the deformation pattern o fythe two halves of the starting configuration B together, cellto-cell fusion can be modelled. Configurations D, and D, depict this case. Here it is inherently assumed that the two fusing cells have the same size. In both models the governing equations are exactly the same, the only difference being in the interpretation of the numerical results. The problem has thus been reduced to the large deformation analysis of a membrane, which is spherical in its reference configuration, for prescribed displacement at the equator. The membrane material is assumed to be nonlinearly elastic. The viscous properties of the contents of the cell or virus are ignored; hence, the internal pressure is uniform. The problem is a quasi-static one. The governing equations of the problem are derived below in a concise manner. Derivations of similar equations can be found in, for instance Fliigge and Chou26, Yang and Fene, and Pujara and LardnerZs. The equilibrium equations are obtained by considering the equilibrium of the deformed element. Then, all quantities of the deformed element are expressed in terms of the corresponding ones of the undeformed element. The equilibrium of forces parallel the symmetry axis of the membrane and in the direction of the radius p of a latitude circle yields the following equations: (ti@ r Sir@)’ -
j Cos*rTi
X, X, = 0,
(~~;Cos*)‘---*r,+pSin*;r,~Xg=O
(2)
In these equations all barred quantities are dimensional. Referring to Figure 3, it is seen that (p’, Pi, @) and (t T,, @) define the deformed and reference configurations, respectively. The normal pressure acting on the deformed membrane is p r, is the radius of the spherical reference configuration. The derivatives, ( )‘, are with respect to the meridian-al angle_@ of the reference configuration. iV@ and Ne are the membrane forces per unit length of t_he reference configuration. X, = $r’and & = pi d$lri d@ are the stretch ratios in the circumferential and meridional d@ directions, respective1 . The total surface area and the volume ef the de ryormed membrane will be denoted by S and V, respectively. The initial uniform thickness of the spherical membrane is h and the axial force in the equatorial ring is denoted
J. Biomed Eng. 1984, Vol. 6, October
259
cell &mechanics:
N. Akh
by FR. The membrane material, which is assumed to be nonlinearly elastic, can be modelled as either the Mooney-Rivlinz9 or the so-called STZC material’O. The material constants for the latter will be denoted by C (force/length) and l? (dimensionless). The dimensional quantities described above will be nondimensionalized through the following relations: T1 = 7,/h,
S = g/h 2, V = v/h3,
N+ = ii$/C,Ne = &/C, FR =
p =Fh/C,
2. -.-
in’6,18. It should be noted, however, that both the Mooney-Rivlin and the STZC representations yield results which are in qualitative agreement as expected 31. There are, of course, quantitative differences among the results. However, the state of the art at present does not necessitate concern over such differences. The constitutive relations for the STZC material, in their nondimensional form, are
NG = ;
(3)
x, [I’@$ -l)+X;(gX$,
-I)],
~ Ne = i b uv-3 - 1) +q(% q - I)]
The equilibrium equations (1) and (Z), after some simple manipulations, can be brought to the following nondimensional form: N& = (No Costi - N@ Cos@)/Sin@, \1’=P!!
Ne A,+%
Sin+ W’
(4) NUMERICAL (5)
PROCEDirRE
When the constitutive relations (10) and (11) are substituted into the equilibrium equations (4) and (5), the latter, together with the compatibility condition (6), can be brought to the following form:
Xgl = (X, Costi -X0 Cos@)/Sin@.
(124
The total surface area and the volume of the deformed membrane, in nondimensional form, are
I/ = 27rrT
42 b & Sin@ d&
s
(7)
0
[ n’2 q h
Sin2@ Sin* d@.
(11)
The problem description is now completed. The numerical method of solution is explained in the 1 following section.
The compatibility condition gives an additional relation among the variables:
S = 4lrrj
(lo)
(8)
Xgr = fe(& @9xe, &5>p),
(1w
V = f&h JI, Xe, A+).
(12c)
Now, one has three equations to determine the three unknowns (A#, X,, JI). The independent variable is +, and p is the loading parameter. Equations (12) can be solved without any difficulty using the well-known Runge-Kutta method32.
J 0
The nondimensional ring is
axial force in the equatorial
FR = 2r, N+ Cos(lr-Q) at @I= 1~12.
(9)
Here, S, V and FR are all for the cell-to-cell fusion case. For the virus-to-cell f&ion case only one half of the deforming membrane is considered. Accordingly, in the latter case S, V, and FR are half of those given by equations (7), (8) and (9), respectively. The statement of the problem will be completed by describing the constitutive relations. In the present work the analysis has been undertaken for both the Mooney-Rivlin material29 and the STZC materia130. However, for the sake of brevity, the numerical results will be presented for the STZC material only. The STZC form essentially yields an exponential stress-strain law. Skalak et al 30. use this exponential form in their studies concerned with the properties of red blood cell membrane. Critical discussions of what kind of material representation can better model cell membrane material are given
280 J. Biomed Eng. 1984, Vol. 6, October
The limits of the integrations in equations (12) are Q = 0 and @ = n/2. The problem is actually a twopoint boundary value problem. The numerical integration could start at $ = n/2; however, there only X6 can be specified. Accordingly, it is more convenient to treat the problem as a one-point boundary value (initial value) problem for which the Runge-Kutta method is very suitable. The numerical integration should start at @ = 0 because, at this point, it is known that JI = 0 and xe = A+ = & 2 1.0. & will be called the apex stretch ratio. Once a value for h, is selected, the numerical integration can be completed and (&, &,, +) can be evaluated at any other location #, provided that the loading parameterp is specified also. Stated differently, one can freely vary X, andp and, thus, can obtain a new equilibrium configuration for the deformed membrane for each new set of (&,,,p). A mong these different equilibrium configurations one selects those that satisfy the constraint condition(s) to be specified. For instance, for cleavage in animal cells, the commonly used constraint condition has been that the cell volume remains constant during division’7*33. For virus-tocell and cell-to-cell fusion there is evidence that the
Cell biomechunics: N. Ahkas
volume does not remain constant, so one has to come up with some other constraint conditions and the possibilities will be discussed in the following section. In the numerical solution, the angular increment in the meridional direction was taken to be three degrees. Finally, it is assumed that the membrane can not be subjected to compressive stresses. Accordingly, whenever the stress resultant Ne turned out to be compressive, which may be the case near the equator in the early part of the fusion, it was set equal to zero. NUMERICAL
RESULTS
AND DISCUSSION
The numerical results presented in Figures 4, 5, and 6 are all nondimensional because the effective thickness of the virus envelope and the cell membrane is open to question. It is known that both virus and the cell have an outer lipid bilayer as shown in Figure I. The thickness of this bilayer is somewhere between 5 and 8 nm. However, as mentioned previously, in both cases this lipid bilayer is connected to, and apparently supported by, a fibrous protein layer (M protein for virus and cortex for the cell). At this time it is not clear whether these protein layers contribute to the
_ _ -0
-_--
+
02
0.4
0.6
08
1.0
1.2
1.4
1.6
I.8
2.0
2.2
2.4
Ff /T, Figure 5 Variations of membrane surface area (solid curves) and membrane volume (dashed curves) during fusion for various values of the apex stretch ratio.
c--,
0
IO
0
‘effective’ thicknesses of the virus envelope and the cell membrane during fusion activities. The author has proposed in a previous work that it is reasonable to treat the cortex, rather than the lipid bilayer proper only, of dividing cells as the ‘membrane’ enveloping the cytoplasm’*. Whether this proposition can be extended to the case of fusing cells and viruses is not known. For this reason, here we have preferred to present the
_t<-, 0__-_
-9--
6
0.8 0.6 0.40.2 0
0.2
04
06
0.6
I0
I2
I4
16
I8
2.0
2.2
2.4
T /?, f
Figure 4 Variations of internal pressure (solid curves) and furrow ring force (dashed curves) during fusion for various values of the apex stretch ratio.
0
0.2
0.4
0.6
0.8
1.0
1.2 1.4
1.6
1.8 2.0
2.2
q/‘i Figure 6 Variation of polar length during fusion for various values of the apex stretch ratio.
J. Biomed Eng. 1984, Vol. 6, October
26’1
Cellbimnechanics: N. Akhns
results in their nondimensional forms. The nondimensional material constant for the STZC material is taken to be r = 0.25. The numerical results in Figures 4, 5, and 6 are all given as functions of the furrow radius to reference radius ratio. The furrow radius Fr and the reference radius Ti are depicted in Figure 3. Moreover, the deformed configurations of the fusing cells (or virus) are schematically shown in the upper part of each figure. For instance, G& E 0.1 corresponds to the beginning of the fusion and the larger values of Ffii correspond to progressing fusion. We will, first of all, discuss the results in general and will, later, note their possible implications in the specific fusion problem. The solid curves in Figure 4 give the intracellular (or intraviral) pressure, p, as a function of the stage of fusion for various values of the apex stretch ratio, &,,. The dashed curves in the same figure give the ring force, FR, that must exist in the furrow plane for equilibrium. It is seen from these curves that, for a fured value of &,, bothp and FR must decrease monotonically for fusion to be completed. It should be emphasized immediately that, during an actual fusion process, it is very likely not necessary that the apex stretch ratio remain constant. We will come back to this point later. It is apparent from the dashed curves in Figure 4 that, for a fixed a, the furrow plane ring force remains practically constant during the early stages of fusion (roughly for FdFi 5 0.7). The small circles on each curve in Figure 4 indicate the final stage of the cell-to-cell fusion. At this stage, the two cells have completely fused to form a single spherical cell. In other words, for cell-to-cell fusion, only those parts of the curves (solid or dashed) remaining to the left of the small circles must be considered. The small circles on the dashed curves of Figure 4, all correspond to FR = 0. This is expected because it is known from simple equilibrium considerations that the furrow plane ring force is zero if the equilibrium configuration has a spherical shape. The curves in Figure 4 must be studied in their entirety if virus-tocell fusion is considered, because the final stage of the latter fusion process corresponds to an almost flat configuration withp * 0 rather than a hemispherical one. In other words, the furrow plane ring force changes its sign during the virusto-cell fusion process. At early stages of fusion, the ring force is contractile but later it becomes a stretching force to complete fusion. Recall that for cell-to-cell fusion no stretching is required and the ring force must always be contractile for equilibrium up until it reduces to zero at the end of fusion, The solid curves in Figure 5 give the membrane surface area, S, as a function of the stage of fusion for values of the apex stretch ratio, &,. The dashed curves in the same figure give the volume of the deforming membrane. For a fixed value of X, the surface area increases monotonically during the entire fusion process and it attains its maximum value at the end of the fusion process (whether it is
262
J. Biomed Eng. 1984, Vol. 6, October
cell-to-cell or virus-to-cell fusion). The situation is different for I/. Recall that the small circles on the curves mark the end of the cell-to-cell fusion process. Accordingly, during the cell-to-cell fusion process, for a fixed &,, the membrane volume increases monotonically (almost linearly) until it attains its maximum value at the end of fusion which corresponds to a spherical configuration. For virus-to-cell fusion, on the other hand, the membrane volume starts decreasing as soon as the hemispherical configuration is reached and the membrane is now being stretched. At the completion of the virus-to-cell fusion, the volume enclosed by the virus membrane approaches zero as expected, because the terminating configuration of virus-to-cell fusion is assumed to be an almost flat membrane. The curves in_F@re 6 give the nondimensional polar length, I$-,, as a function of the stage of fusion for various values of the apex stretch ratio, &,. The definition of the polar length & is shown in Figure 3. Such !P versus 5 curves are probably the easiest to obtam during experiments and they may prove to be useful in predicting the apex stretch ratio variation during a fusion process. The figure is self explanatory. For a fixed &,, the polar length increases slightly during the early stages of fusion (roughly for YdS, _( 0.5) and this is followed by continuous decrease during the later stages. The virus-to-cell fusion is completed when the polar length approaches zero. At the end of the cell-to-cell fusion process (the small circles in Figure 6) the polar length is equal to the furrow radius as expected. The above discussion of Figures 4, 5, and 6 was limited to the cases in which X, is assumed to remain constant during fusion. As stated previously, this is probably not the case in actual virus-to-cell and cell-to-cell fusion processes. In his literature survey, the author was unable to find any explicit remark concerning the possible change in the apex stretch ratio during fusion. For that matter, we could not find any numerical values for the polar length at various stages of fusion either. In case such results are or become available, the following discussion can be modified accordingly and for this reason, it should be considered as discussion of various possibilities rather than that of well established facts. Although there are no data available at present on the apex stretch ratio change during fusion, there are experimental observations reported which clearly state that fusion leads to cell swelling. This occurs during both virus-to-cell and cell-to-cell fusion. Swelling probably continues during the entire fusion process because, as stated in Knutton), haemolysis takes place in the case of erythrocytes. In the case of HeLa cells, swelling involves an ‘unfolding’ of surface microvilli and the formation of large, smooth-surfaced fused cells*(‘. It has been suggested that cell swelling is the driving
Cell biomechanics:N. Akh
force which results in the complete incorporation of the viral envelopes into the cell membrane. A very detailed discussion of swelling during fusion can be found in the review article by Knutton’. Let us first consider cell-to-cell fusion. Cell swelling during fusion implies a simultaneous increase in the intracellular pressure, the surface area, the volume, and the apex stretch ratio. For a physically acceptable fusion process, the furrow radius must also increase during swelling. However, there is no immediate reason for the polar length to increase also because, as can be seen from Figures 5 and 6, volume increase may occur although polar length is decreasing. Refer to F&we 4 and, purely for the sake of demonstration, assume that our starting configuration corresponds to point A on the solid & = 1.1 curve. Assume also that cell-to-cell fusion is completed at point B on the solid X, = I.5 curve, The corresponding points on the dashed curves in Figure 4 are A’ and B’. The corresponding points are marked in Figures 5 and 6 also. As fusion progresses from point A to point B (or from A’ to B’) in these figures, the path followed implies the following conclusions (whichever physically reasonable path we follow). The intracellular pressure, the apex stretch ratio, the surface area, the volume, and the polar length all increase monotonically during fusion which is accompanied by swelling. In other words, the numerical predictions agree with the implications of the experimental observations. The numerical results also predict that, from the dashed curves in Figure 4, the furrow plane ring force may increase initially but must go down to zero towards the end of fusion. The author was unable to find any data concerning the ring force during fusion. Indeed, the possible existence of a ring is merely implied by Poste and PasternakZ (p. 344) as follows: “ . . . the two fusing cells are in a ‘dumb-bell shape. This raises the possiblity that it is the gross morphological change from this shape to that of a single, double-sized sphere, that involves microfilaments and is sensitive to cytochalasin B. Since the latter shape change is exactly the opposite of that occurring during cytokinesis, which is known to involve a contractile ring of microfilaments . . .” Accordingly, it is not surprising that our numerical results indicate the existence of a furrow plane ring force which approaches zero as the cell-to-cell fusion is completed. Microfilaments existing in the cortical layer of the fusing cells at focal areas of contact are the reasonable agents that could possibly form the ring. As stated above, the intramembraneous protein particles ‘swimming’ in the lipid matrix appear to be pushed away from the fusion site. The cytoskeletal elements (microfilaments and microtubules) are connected to these protein particles. From a mechanical point of view, it is possible for the microfilaments connected to the protein particles surrounding the protein depleted region to provide the ring force necessary
for equilibrium. The ring force occurring in cytokinesis has been measured3’; but, to the author’s knowledge, no such measurement is available for the ring force possibly occuring during fusion. For the virus-to-cell fusion case, the situation is the same until the predicted ring force changes sign. Again the microfilamentous ring is capable of producing the expanding force that is necessary for flattening the virus envelope. Now, as seen in Figure 4, this force must increase in magnitude. It is possible that this increase in the ring force is passively controlled by the cell swelling which can push the intramembraneous protein particles further apart, causing further stretching of the furrow ring. Although it is possible, and tempting, to make additional conjectures concerning the mechanics of the fusion processes discussed, it is probably better to wait till some concrete data become available concerning the points raised in the present work. Only then will it be realistic to discuss the biomechanics of virus-to-cell and cell-to-cell fusion and its implications further. It is merely hoped that the present preliminary discussion of the phenomena will draw the attention of cytologists to those mechanical aspects which have so far been apparently overlooked. SUMMARY
AND CONCLUSIONS
It appears that a better understanding of the biomechanical model of fusion has been presented. be helpful in shedding some additional light on the phenomena. With this purpose in mind, a biomechnical model of fusion has been presented. Virus and the cells are modelled as initially spherical membranes of nonlinear elastic material that undergo large deformations. The two spherical membranes are connected by a cytoplasmic bridge which expands resulting in the formation of a larger single sphere. This is cell-to-cell fusion. In virus-to-cell fusion, a smaller spherical membrane is connected to a much larger one. Fusion is completed when the smaller membrane is completely incorporated into the membrane of the larger sphere. The numerical results predict the existence of a ring force in the furrow plane. It is possible that the microfilaments of the cortical layer form the furrow band which may provide this ring force. REFERENCES Poste, G. Mechanisms of virus-induced cell fusion. Int. Rev. Cytd, 1972 33, 157-252. Postes, G., Pasternak, CA. Virus-induced cell fusion. Membrane Fusion, (Eds G. Poste and G.L. Nicolson), Elsevier/North-Holland Biomedical Press, 1978, 305-367. Knutton, S. The mechanism of virus-induced cell fusion. Micron, 1978, 9, 133-154. Lucy, J.A. Mechanisms of chemically induced cell fusion. Membrane Fusion, (Eds G. Poste and G.L. Nicolson), ElseviedNonh-Holland Biomedical Press, 1978, 267-304.
J. Biomed Eng. 1984, Vol. 6, October
263
Cell biomechanics: N. Ahkas 5
6
7
8
9
10
11 12
13 14
15
16
17 18
264
Gingell, D., Ginsberg, L. Problems in the physical interpretation of membrane interaction and fusion. Membrane Fusion, (Eds G. Poste and G.L. Nicolson) Elsevier/North-Holland Biomedical Press, 1978, 791-833. Berkaloff, A. Etude au microscope electronique de la morphogenese de la particule du virus Sendai. J Microscop. 1963, 2, 633-640. Hosaka, Y., Nishi, Y. and Fukai, K. The structure of HVJ II. The fine structure of the subunits. Biken’sJ. 1961, 4, 243-254. Hewitt, J.A. and Nermut, M.V. A morphological study of the M-protein of Sendai virus. J Gen. ViroL 1977, 34, 127-136. Hosaka, Y. and Shimizu, Y.K. Artificial assembly of envelope particles of Sendai virus (HVJ). I. Assembly of hemolytic and fusion factors from envelopes solubilized by Nonidet P40. Virology, 1972, 49, 627-639. Hosaka, Y. and Shimizu, Y.K. Cell fusion by Sendai virus. Virus Infection and the Cell Surface, (Eds G. Poste and G.L. Nicolson), North-Holland, Amsterdam, 1977 129-155 Singer, S.J., Nicolson, G.L. The fluid mosaic mode1 of the structure of cell membranes. Science, 1972, 175, 720. Mitchison, J.M. and Swann, M.M. The mechanical properties of the cell surface, III. The sea urchin egg from fertilization to c1eavage.J. Exp BioL 1955, 32, 734-750. Hiramoto, Y. Rheological properties of sea urchin eggs. Biorheo. 1970, 6, 201-234. Nicolson, G.L., Poste, G. and Ji, T.H. The dynamics of cell membrane organization. Dynamic Aspects of Cell Surface Organization, (Eds G. Poste and G.L. Nicolson), Elsevied North-Holland Biomedical Press, 1977 l-73 Evans, E.A. and Skalak, R. Mechanics and thermodynamics of biomembranes: Parts 1 and 2. CRC Critical Reviews in Bioengng. 1979, 3, 181-330 and 331-418. Akkas, N. and Engin, A.E. Ultrastructure of animal cells and its role during cytokinesis from a structural mechanics viewpoint, Mechanics of Structured Media, (Ed A.P.S. Selvadurai). Studies in Applied Mechanics 5A. Elsevier Scientific Publ. Corn Amsterdam, 1981, Pan A, 187-207. Akkas, N. On the biomechanics of cytokinesis in animal ce11s.J. Biomechantis, 1980, 13, 977-988. Akkas, N. A viscoelastic model for cytokinesis in animal ce11s.J. Biomechanics, 1981, 14, 621-631.
J. Biomed Eng.
1984, Vol. 6,
October
19
20
21
22
23
Knutton, S. Studies of membrane fusion. II. Fusion of human erythrocytes by Sendai virus.J. Cell Sci 1977, 28, 189-210. Knutton, S., Jackson, D. and Ford, M. Studies of membrane fusion. I. Paramyxovirus-induced cell fusion, a scanning electron-microscope study. J. Cell Sci. 1977, 28, 179-188. Satir, B., Schooley, C. and Satir, P. Membrane fusion in a mode1 system. Mucocyst secretion in Tetrahymena. J Cell BioL 1973, 56, 153-176. Orci, L., Perrelet, A. and Friend, D.S. Freeze-fracture of membrane fusions during exocytosis in pancreatic B-Cells. J. Cell BioL 1977, 15, 23-30. Lawson, D., Raff, M.C., Gomperts, B., Fewtrell, C. and Gilula, N.B. Molecular events during membrane fusion. A study of exocytosis in rat peritonal mast cells. J Cell BioL 1977,12,242-259.
24 25
26 27
28
29 30
31 32 33 34
Postes, G. and Allison, A.C. Membrane fusion, Biochim. Biophys. Acta, 1973, 300, 421-465. Orci, L. and Perrelet, A. Ultrastructural aspects of exocytotic membrane fusion. Membrane Fusion, (Eds G. Poste and G.L. Nicolson), Elsevier/Nonh-Holland Biomedical Press, 1978 629-656 Fhigge, W. and Chou, XC. Large-deformation theory of shells of revolution. J. A/I/IL Mech 1967, 34, 56-58. Yang, W.H., Feng, W.W. On axisymmetrical deformations of nonlinear membranes. J AppL Mech 1970, 37, 1002-1001. Pujara, P. and Lardner, T.J. Deformations of elastic membranes - effect of different constitutive relations. ZAMP, 1978,29, 315-327. Treloar, L. R. G. The Physics of Rubber Elasticity. Clarendon Press, Oxford, 3rd edn. 1975. Skalak, R., Tozeren, A., Zarda, R. and Chien, S. Strain energy function of red blood cell membranes. Biophys. J 1970, 13, 245-264. Akkas, N. and Engin, A.E. Nonlinear mechanics and cell division. ASME 1980 Advances in Bioengineering. 1980, 27 l-273 mson, A. and Wilf, H. Mathematical Methodsfor Digital Computers. John Wiley, New York, 1960, 110-120. Pujara, P. and Lardner, T.J. A mode1 for cell division. J. Biomechanics, 1979, 12, 293-299. Hiramoto, Y. Force exerted by the cleavage furrow of sea urchin eggs. Dev. Growth and Dijer. 1975, 11, 27-38.