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Enhanced polymerization of polar macromolecules by an applied electric field with application to mitosis
J. theor. Biol. (1990) 145, 245-255
Enhanced Polymerization of Polar Macromolecules by an Applied Electric Field with Application to Mitosis WILLIAM J. M E G G S
Department of Medicine, East Carolina University, Greenville, N C 27858, U.S.A. (Received on 21 September 1988, Accepted in revised form on 3 January 1990) Numerous cellular processes are characterized by the rapid polymerization of protein molecules to form rod-like structures. Examples include the formation of spindle fibers from tubulin during cell division and the polymerization of actin into the actin filaments of the pseudopod in chemotaxis. It has been proposed that these proteins possess an electric dipole moment and that the onset of an internal electric field triggers polymerization. In this theoretical study, the relative probability of polymerization of a polar protein species is calculated in the presence and absence of an electric field. There is a significant enhancement of polymerization in the presence of an electric field, which increases as the size of the attachment site decreases. We conclude that a cytoplasmic pool of suitable proteins will rapidly polymerize if an electric field is applied, while remaining in a random configuration in the absence of a field. This mechanism is applied to the mitotic spindle structure, and by assuming that the spindle poles become oppositely charged during mitosis, a finite difference method is used to calculate the spindle structure at metaphase. Good agreement is obtained with experiment data.
Introduction There are a n u m b e r of cellular processes in which cytoplasmic pools of a protein m o n o m e r rapidly polymerize to form intricate structures which are necessary for a dynamic cellular function. During mitosis, tubulin molecules rapidly polymerize into microtubules forming bundles which converge at the spindle poles (Inoue & Sato, 1967). Actin filaments form in the direction of a chemotactic gradient in macrophages, leading to the development o f a p s e u d o p o d (Drutz & Mills, 1984). Actin filaments are also involved in the formation of the acrosomal process of inveterbrate sperm as an egg is a p p r o a c h e d (Tilney et al., 1973). It has been suggested that the c o m m o n mechanism explaining processes such as these is the development o f an endogenous internal electric field in cells. This interacts with the dipole m o m e n t of the protein monomers, causing them to align in the direction of the electric field. The m o n o m e r s would then form polymers which are parallel to the applied electric field (Meggs, 1988). An attractive feature of the theory is that simple charge distributions in the cell lead to observed structures. For example, if the spindle poles in mitosis are oppositely charged, the electric field lines resemble the distribution of microtubules observed experimentally (Meggs, 1988). I f chemotactic molecules interacting with cell m e m b r a n e receptors on 2,15 0022-5193/90/014245+ 11 $03.00/0
~ 1990 Academic Press Limited
246
w.J. MEGGS
phagocytes induce an electric charge, the membrane would become charged preferentially in the direction of the gradient, setting up an electric field in that direction, leading to the formation of actin filaments pointing in the direction of the chemotactic gradient. This theory is consistent with certain experimental observations. Gallin et al. (1974) have shown by cell electrophoresis in the presence and absence of chemotactic factor that the net charge of human granulocytes is changed by the addition of a chemotactic factor. Orida & Feldman (1982) have demonstrated that extracellularlyapplied electric fields cause macrophages to migrate toward the positive pole of the electric field. The existence of constant electric fields in cells has been verified in numerous experiments, as reviewed (Jape & Nuccitelli, 1977). Constant external fields have been shown to induce shape changes in fibroblasts and disrupt microfilament bundles (Yang et al., 1984). The important question, Will an applied electric field enhance the polymerization of proteins possessing an electric dipole moment? must be addressed. In order for this mechanism to be operative, the relative rates of polymerization in the presence and absence of an electric field must be such that no polymerization takes place in the absence of a field, and the molecules rapidly form microtubules or other structures when a field is applied. In this paper, we calculate the relative polymerization rates in the absence and presence of an electric field subject to the assumption that the electric field interacts with the dipole moments of the monomers to align them in the direction of the field. While this assumption is true for an isolated dipole in an electric field, thermal effects, dipole-dipole interactions, and other complications may disrupt this alignment in certain ranges of temperature, density, and applied fields (Onsager, 1936; Kirkwood, 1939; Frohlich, 1986). We feel that our assumption is a good one for proteins in solution over the range of variables of interest because measured static dielectric increments of proteins in solution are positive numbers (Oncley, 1943). It should be emphasized that the results of this paper do not apply to those solutions in which polar molecules do not align in the direction of the field. Dimer formation
To calculate the probability of a binding collision, we consider the molecules to be spherically shaped with binding sites depicted as a "cap" defined by an angle ct about the axis of the molecule, as depicted in Fig. 1. Binding can occur if the "north" cap of one molecule, denoted N, intersects the "south" cap of the other molecule, denoted S. The binding efficiency ~7 is the probability that binding will occur if the opposite caps of two molecules make contact. To simplify the calculations, spherical molecules and conical caps are used. Below it will be shown that the general conclusions are independent of the detailed shape of the molecule. First, consider the case when no electric field is applied. The proteins will be randomly oriented. We work in the rest frame of one of the molecules, where the other approaches with impact parameter b, defined as the perpendicular distance from the center of one molecule to the linear trajectory of the moving molecule, as depicted in Fig. l(a). The number of collisions per unit time is pvA, where p = the
POLYMERIZATION
OF
POLAR
MACROMOLECULES
247
(8 ,¢~) E=O
(a)
(O,~)
(b)
Fro. 1. Geometry of a collision between two monomers for (a) E = 0, with attachment caps randomly oriented, and (b) E ~ 0, with caps oriented in the direction of the field. Q denotes the point of contact between the spheres, and (Oi, tbi) are the spherical co-ordinates of the pole. density o f the molecule, v = the m e a n velocity, and A = the cross sectional area. This factor arises b e c a u s e a molecule o f cross sectional area A and velocity v sweeps out a cylinder o f v o l u m e vA in 1 sec. Let f)~, = ( O i , ~ ) a n d ~q2 = ( 0 2 , ~k2) be the p o l a r co-ordinate o f the north pole o f m o l e c u l e 1 and n o r t h pole o f molecule 2. T h e rate o f b i n d i n g will be: the probability that the north pole o f molecule 1 intersects the point o f contact,
f ((oill'~' d~/ 4 7ra2,
(1)
times the probability that the s o u t h pole o f molecule 2 intersects the point o f contact, 2~r)
f(("'
d~2/4~ra 2,
(2)
0.0)
times the b i n d i n g efficiency ~(12~, 122), times
puA. The radius
is d e n o t e d by a 2. T h a t
248
w . J . MEGGS
is, f2adbf''2") dO, f(~.2~)d122 ~ a o ( o . o ) 4 ~ a 2 J ( o . o ) 4 e r a 2r/(D''122)'
P~22°(°t)=2pVAjo
(3)
where d12i = a 2 sin Oi dOi d~bi is a differential element of area defined by about 12~. The superscript 2 denotes a chain length of two monomers. The factor of 2 arises because binding can also occur if the south pole of molecule 1 contacts the north pole of molecule 2. Performing the integrals, assuming 7/ to be constant, gives:
N=2 PE=0(a) =½pvArl(1-cos a ) 2.
(4)
Next, consider the case in which an electric field is present. The molecules will align so that their axes are parallel, as depicted in Fig. l(b). Since the direction o f the velocity vector will be randomly distributed with respect to the electric field, 12 = (O, ~b) will also be randomly distributed. Hence, the probability o f binding is just the probability that the cap o f molecule 1 intersects the point o f contact. If the north pole of molecule 1 intersects, the south pole of 2 will, and binding can occur. Hence, p NE#o~ = 2 / o t j-2~lpvA a_
foadf
(a" 2~) d122 ~7(f~l 121)
v(o, o)
4"n'a2
'
(5)
= pvArl(1 - c o s a). The ratio o f the binding probability in the presence o f an electric field to that in the absence o f a field,
N=2 N=2 P~#o(a)/PE=o(a) = 2(1 - c o s a ) - ' ,
(6)
is plotted as a function of a in Fig. 2, in the curve labeled N = 2. There is a dramatic rise in the relative probability of dimer formation as the size of the attachment cap decreases. These arguments can easily be extended to molecules of arbitrary shape and attachment site. Let aN and as be the areas of the north and south attachment sites and A be the surface area o f the protein. Then, with no electric fluid, the probability o f attachment is: pN=2, E=ot aNaS), : 2 p w l A ( a ~N ) . (_~)
(7)
= 2pvrlaNas/A. The factors o f aN/A and as/A are the probabilities that the north attachment cap of molecule one and south attachment cap of molecule two intersect the contact point o f the molecules. The factor of two arises as before from the interchange of north and south. In the presence o f an electric field, if the north and south attachment sites are at opposite poles, the north pole o f one will intersect the contact point if, and only if, the south pole o f the other intersects. Then,
P N=2 ~ #o( aN) = 2pwlA ( aN/ A ) = 2pwlaN
(8)
POLYMERIZATION
OF
POLAR
MACROMOLECULES
249
l0 7
iO 6 %%
E % IO s o
~
. - ~: E_
10 4
~, ~
10 3
~.~ o
o
8~ t~
I0'
i0 ° (45 o)
~- (9o °)
Angle subtended by (~rI1ochmenf cop in rodion (de?rees)
FIG. 2. A graph is given of the ratio of the probability of forming a polymer of length N in the presence and absence of a constant electric field E, as a function of the size of the attachment cap.
and, N =2
N=2
Pe,~o(aN)/ Pe,~o(aN) = Alas-> oo,
(9)
as as-> 0. So, in the general case, there is also a marked enhancement of dimer formation provided the north and south attachment sites are at opposite poles.
Growth of Polymer Chain In the above section, it was shown that there is an enhancement of dimer formation o f polar molecules in the presence of an electric field, so long as the attachment sites symmetrically aligned at opposite poles. In this section, we consider the relative rates of a polymer chain adding a m o n o m e r to increase its length by one monomer. Figure 4(a) and (b) shows the m o n o m e r impacting the end of polymer chain in the cases o f E = 0 and E # 0, respectively.
250
W.
J.
MEGGS
ta)
E~O
FIG. 3. Geometry of a collision between a m o n o m e r and the end of a polymer chain for (a) E = 0, with attachment caps randomly oriented, and (b) E ~ 0, with caps oriented in the direction of the field.
Let ~¢ be the probability that the m o n o m e r impacts the terminal attachment cap of the polymer, which is identical in the two cases. Then pN E=O~,
the probability of forming a p o l y m e r of length N by adding a m o n o m e r to a polymer o f length N - 1, is, p NE = O - _- 2 E~ a r~t, E N=- O i //¢ ' I,t,
where a and A are the area o f the attachment cap and the surface area, respectively. The factor of 2 arises because the polymer can grow at both ends. For E ~ 0, we have, P~,o
= 2s~PE ,v-i ¢o
(10)
and,
For the case o f spherical molecules with an attachment cap defined by an angle a, a plot is given o f the relative enhancement o f the formation o f polymers o f lengths N = 2, 3, 4, 5 and 6 in Fig. 3. The enhancement factor grows exponentially as N increases and has an asymptote at a = 0.
POLYMERIZATION
OF
POLAR
251
MACROMOLECULES
-1-6 ~-1-8 20 -1"4 \ \ ~-Cu -3"0.q5._~.( °,
;'g -0.6 0.4
-20
- IO
O -
IO
20
50
x
20
0 -002
(b)
0.009
20
I0
0
I0
20
x
FIG. 4. (a) Numerical integration of the differential equation reqmring that the polymers be parallel to the field lines gives a family of curves determined by the slope at the origin for oppositely charged spindle poles a = 30 ~ apart. Curves are labeled by the slope at the origin. (b) Contours give constant values of the electric field. Curves are labeled by 41reE/Q i n / ~ - 2 .
It has been shown that if a protein molecule can polymerize to form chains and possesses an electric dipole moment, there is an enhancement of polymerization in the presence of an electric field. This enhancement factor grows exponentially with chain length and increases as the size of the attachment cap decreases. The result is independent of the strength of the electric field, depending only on the tendency of an electric dipole to rotate parallel to the field, after a transient period following switching on the field. We assume that the field is strong enough to orient the molecules. The length of the transient period will decrease with increasing field strength. The result is independent of the detailed geometry of the molecule, so long as the attachment sites are at opposite poles and aligned with the dipole moment. This result gives strong support to the hypothesis (Meggs, 1988) that electric fields are important in both triggering the formation, and determining the structure, of microtubules and actin filaments. Direct experimental study of this hypothesis is therefore justified.
Application to Mitosis The proposal that intracellular electric fields determine the spatial organization of the mitotic spindle structure is tested by comparing theoretically calculated mitotic structures with observations. It is assumed that the spindle poles become oppositely
252
w . J . MEGGS
charged, leading to alignment of tubulin molecules in the direction of the resulting field, which enhances polymerization and leads to microtubules which parallel the field lines. Solving the equation d y / d x = E y / E x numerically, where (x, y) are spatial co-ordinates and (Ex, Ey) the components of the electric field, which states that the slope of the microtubules is parallel to the field lines, leads to specific predictions of microtubule shape which is compared to phase contrast microscopic (Salmon & Segall, 1980) and election microscopic data (McIntosh et al., 1976). Consider spindle poles located at (0, 0) and (a, 0), where a, the distance between the poles, is on the order of 10 ~ for eukaryotic cells. Let the poles be oppositely charged, with charges of q and - q , respectively. The electric field E will be given by Coulomb's law. 4~re
3 "
(12)
where e is the dielectric constant in the cell, r is the position vector locating a point in space, and (0, 0) and a = (a, 0) locate the spindle poles. The assumption that polymers form parallel to the elective field lines can be written, y'(x, y) = d__~y= Ey dx Ex'
(13)
where E = ( E y , Ex) is the electric field vector and y = y ( x ) is the equation of an electric field line, and hence of a microtubule. The spindle fiber is of course a three-dimensional structure, but the axial symmetry of the problem allows us to work in a two-dimensional plane. Equation (13) may be solved numerically using a simple finite difference method, y, = y'(xn-1, Yn-I) . Ax +y~_l,
(14)
where A is the increment size for the numerical integration. This first order differential equation requires one parameter to specify a solution, which we take to be the slope of the trajectory at the origin. It can be seen from eqn (2) that d y / d x is undefined (0/0) at the origin, but as one approaches the origin along a line y = mx, y'(x) -->m as x ~ 0. Hence, by beginning the numerical iteration of (13) at (0, 0) with y'(0, 0 ) = m, a specified parameter, one generates a distinct curve for each value of m, as shown in Fig. 4(a) with a = 30 W- Convergence of the method was verified by changing the step size, noting y ( a ) = 0 in the numerical solution, and observing that the numerical solution preserves the symmetry about the line x = a/2. Figure 4(b) gives contours of equal electric field strengths, IEI = constant. These curves were calculated by evaluating the electric field at various points in the plane, and connecting points where the field strength is equal. In this theory, the spindle poles become oppositely charged, and polymerization will take place out to a threshold value of the field strength Eth, below which the monomers will not become aligned with the field due to competition from thermal effects and other forces.
253
P O L Y M E R I Z A T I O N OF P O L A R M A C R O M O L E C U L E S
4WCt=E" 0
(a)
> 0.009
W-=
\
|
I
I0~
(b)
4'r~"E < 0.002 ~-= o
I
I Io/~.
FIG. 5. (a) Computer simulation of a mitotic spindle structure if polymerization takes place for electric fields greater than 0.009 Q/4~re. (b) Computer simulation of a mitotic spindle if polymerization takes place for electric fields greater than 0-002 O/4rre. The insert shows an isolated mitotic spindle from Salmon & Segall (1980). Used with permission of Rockefeller University Press and Dr E. D. Salmon.
254
w . J . MEGGS
Figure 5(a) and (b) show a calculated microtubule distribution with a = 30 for two values of Eth, normalized to charge and dielectric constant, using the values o f slope depicted in Fig. 4. The insert in Fig. 5(b) is a reproduction from Salmon & Segall (1980) of an isolated mitotic spindle at metaphase viewed with phase contrast microscopy, and is in qualitative agreement with Fig. 5(b). The dynamical development of the spindle fiber has been elucidated by fluorescence microscopy o f dye-tagged tubulin injected intracetlularly (Mclntosh et al., 1986; Wadsworth & Salmon, 1986). In a sequence o f photographs, fluorescence tagged microtubules form about the spindle pole, as in Fig. 5(a) the progress and meet as in Fig. 5(b). Since the threshold field for polymerization is proportional to the charge, an increasing charge on the spindle poles would lead to a temporal development o f the spindle structure in agreement with observation, though the mechanism o f this increasing charge remains unexplained at present. Electron microscopy data (Mclntosh et al., 1976) allows one to visualize individual microtubules over a limited range. Some tubules are warped and S-shaped, which may be an artifact of the fixing and cutting process. The data points in Fig. 6 are taken from smooth microtubules from this data, and smooth curves are theoretical curves calculated numerically, using eqns (12-14). There is reasonable agreement between the data points and experimental curves. Here a = 10 ~, the difference being that a smaller cell was used in this reference from that of the data of Fig. 5(b). A large amount of experimental work has been done on mitosis, but little attention has been given to the forces that move the tubulin molecules into position and determine the spatial organization o f the mitotic spindle structure. In 1986, Deery
~.~y,.,m=0. 1-~ 8 re=l-4
i
t
|
I
t
I
I
f-8
~
m=2"0
FIG. 6. Calculated curves for an isolated spindle fiber are compared to data points taken from an electron micrograph of a mitotic spindle from McIntosh et al. (1976), with a = 10 p..
POLYMERIZATION
OF P O L A R M A C R O M O L E C U L E S
255
wrote, "The factors responsible for the precise spatial distribution of microtubules a n d t h e i n t r a c e l l u l a r l o c a l i z a t i o n o f t h e s e f a c t o r s r e m a i n s to b e e l u c i d a t e d " . P r e c i s e l y this p r o b l e m is a d d r e s s e d in this p a p e r . M i t o s i s is a c o m p l i c a t e d p r o c e s s , a n d a s i m p l e m o d e l s u c h as this o n e c a n n o t a c c o u n t f o r t h e c o m p l e x d y n a m i c s o f m i t o s i s . T h e g o a l o f this w o r k is l i m i t e d , to e x p l a i n t h e o r g a n i z a t i o n o f t h e m i t o t i c s p i n d l e at m e t a p h a s e . T h e g o o d a g r e e m e n t w i t h o b s e r v e d s t r u c t u r e s s u p p o r t s t h e n o t i o n t h a t i n t r a c e l l u l a r e l e c t r i c fields m a y p l a y a r o l e in this p r o c e s s , a n d h o p e f u l l y will e n c o u r a g e f u r t h e r e x p e r i m e n t a l a n d t h e o r e t i c a l w o r k in this d i r e c t i o n . The author greatly acknowledges permission from Dr E. D. Salmon and The Rockefeller University Press to reproduce a photograph from Salmon & Segall (1980). REFERENCES DEERY, W. J. (1986). Control of microtubule assembly-disassembly in lysed cell models. Ann. N.Y. Acad. Sci. 466, 593-608. DRUTZ, D. & MXLLS, J. (1984). Immunity and infection. In: Basic and Clinical Immunology (Stites, D. P. et aL, eds) p. 202, 5th edn., Los Altos: Lange. FROHLICH, H. (1986). Theory o f Dielectrics 2nd edn. Oxford: Oxford University Press. GALLIN, J., DUROCHER, J. & KAPLAN, A. (1974). Interaction of leukocyte chemotactic factors with the cell surface. Jr. clin. Invest. 55, 94-96. INOUE, S. & SATO, H. (1967). Cell motility by labile association of molecules: the nature of the mitotic spindle fibers and their rote in chromosome movement. J. gen. Physiol. 50, 259-292. JAFFE, L. F. & NUCC~TELLI, R. (1977). Electrical controls of development. Ann. Rev. biophys. Bioeng. 6, 445-476. KIRKWOOD, J. G. (1939). The dielectric polarization o polar liquids. J. chem. Phys. 7, 911-919. McI NTOSH, J. R., HEPLER, R. K. & VAN CIE, D. G. (1976). Model for mitosis. Nature, Lond. 224, 659-663. MCINTOSH, J. R., SAXTON, W. M., STEMPLE, D. L. & WELSH, M. J. (1986). Dynamics of tubulin and calmodulin in the mammalian mitotic spindle. Ann. N.Y. Acad. Sci. 466, 580-592. MEOGS, W. J. (1988). Electric fields determine the spatial organization of microtubules and actin filaments. Med. Hypotheses 26, 165-170. ONCLEY, J. L. (1943). In: Proteins, amino acids, and peptides as ions and dipolar ions. (Cohn, E. J. & Edsalt, J. P., eds) chapter 22. New York: Rheinhold. ONSAGER, L. (1936). Electric moments of molecules in liquids. J. Am. Chem. Soc. 58, 1486-1493. ORIDA, N. & FELDMAN, J. D. (1982). Directional protrusive pseudopodial activity and motility in macrophages induced by extracellular electric fields. Cell Motility 2, 243-255. SALMON, E. D. & SEGALL, R. R. (1980). Calcium-labile mitotic spindles isolated from sea urchin eggs. J. Cell Biol. 86, 355-365. TILNEY, L., HATANO, S., ISHIKAWA, H. & MOOSEKAR, M. S. (1973). The polymerization of actin: its role in the generation of the acrosomal process of certain echinoderm sperm. J. Cell Biol. 59, 109-126. WADSWORTH, P.& SALMON, E. O. (1986). Microtubule dynamics in mitotic spindles of living cells. Ann. N. Y.. Acad. Sci. 446, 580-592. YANG, W. P., ONUMA, E. K. & HUI, S. W. (1984). Response o f C 3 H / I O T 1 / 2 fibroblasts to an external steady field stimulation. Expl. Cell Res. 155, 92-104.
Enhanced Polymerization of Polar Macromolecules by an Applied Electric Field with Application to Mitosis WILLIAM J. M E G G S
Department of Medicine, East Carolina University, Greenville, N C 27858, U.S.A. (Received on 21 September 1988, Accepted in revised form on 3 January 1990) Numerous cellular processes are characterized by the rapid polymerization of protein molecules to form rod-like structures. Examples include the formation of spindle fibers from tubulin during cell division and the polymerization of actin into the actin filaments of the pseudopod in chemotaxis. It has been proposed that these proteins possess an electric dipole moment and that the onset of an internal electric field triggers polymerization. In this theoretical study, the relative probability of polymerization of a polar protein species is calculated in the presence and absence of an electric field. There is a significant enhancement of polymerization in the presence of an electric field, which increases as the size of the attachment site decreases. We conclude that a cytoplasmic pool of suitable proteins will rapidly polymerize if an electric field is applied, while remaining in a random configuration in the absence of a field. This mechanism is applied to the mitotic spindle structure, and by assuming that the spindle poles become oppositely charged during mitosis, a finite difference method is used to calculate the spindle structure at metaphase. Good agreement is obtained with experiment data.
Introduction There are a n u m b e r of cellular processes in which cytoplasmic pools of a protein m o n o m e r rapidly polymerize to form intricate structures which are necessary for a dynamic cellular function. During mitosis, tubulin molecules rapidly polymerize into microtubules forming bundles which converge at the spindle poles (Inoue & Sato, 1967). Actin filaments form in the direction of a chemotactic gradient in macrophages, leading to the development o f a p s e u d o p o d (Drutz & Mills, 1984). Actin filaments are also involved in the formation of the acrosomal process of inveterbrate sperm as an egg is a p p r o a c h e d (Tilney et al., 1973). It has been suggested that the c o m m o n mechanism explaining processes such as these is the development o f an endogenous internal electric field in cells. This interacts with the dipole m o m e n t of the protein monomers, causing them to align in the direction of the electric field. The m o n o m e r s would then form polymers which are parallel to the applied electric field (Meggs, 1988). An attractive feature of the theory is that simple charge distributions in the cell lead to observed structures. For example, if the spindle poles in mitosis are oppositely charged, the electric field lines resemble the distribution of microtubules observed experimentally (Meggs, 1988). I f chemotactic molecules interacting with cell m e m b r a n e receptors on 2,15 0022-5193/90/014245+ 11 $03.00/0
~ 1990 Academic Press Limited
246
w.J. MEGGS
phagocytes induce an electric charge, the membrane would become charged preferentially in the direction of the gradient, setting up an electric field in that direction, leading to the formation of actin filaments pointing in the direction of the chemotactic gradient. This theory is consistent with certain experimental observations. Gallin et al. (1974) have shown by cell electrophoresis in the presence and absence of chemotactic factor that the net charge of human granulocytes is changed by the addition of a chemotactic factor. Orida & Feldman (1982) have demonstrated that extracellularlyapplied electric fields cause macrophages to migrate toward the positive pole of the electric field. The existence of constant electric fields in cells has been verified in numerous experiments, as reviewed (Jape & Nuccitelli, 1977). Constant external fields have been shown to induce shape changes in fibroblasts and disrupt microfilament bundles (Yang et al., 1984). The important question, Will an applied electric field enhance the polymerization of proteins possessing an electric dipole moment? must be addressed. In order for this mechanism to be operative, the relative rates of polymerization in the presence and absence of an electric field must be such that no polymerization takes place in the absence of a field, and the molecules rapidly form microtubules or other structures when a field is applied. In this paper, we calculate the relative polymerization rates in the absence and presence of an electric field subject to the assumption that the electric field interacts with the dipole moments of the monomers to align them in the direction of the field. While this assumption is true for an isolated dipole in an electric field, thermal effects, dipole-dipole interactions, and other complications may disrupt this alignment in certain ranges of temperature, density, and applied fields (Onsager, 1936; Kirkwood, 1939; Frohlich, 1986). We feel that our assumption is a good one for proteins in solution over the range of variables of interest because measured static dielectric increments of proteins in solution are positive numbers (Oncley, 1943). It should be emphasized that the results of this paper do not apply to those solutions in which polar molecules do not align in the direction of the field. Dimer formation
To calculate the probability of a binding collision, we consider the molecules to be spherically shaped with binding sites depicted as a "cap" defined by an angle ct about the axis of the molecule, as depicted in Fig. 1. Binding can occur if the "north" cap of one molecule, denoted N, intersects the "south" cap of the other molecule, denoted S. The binding efficiency ~7 is the probability that binding will occur if the opposite caps of two molecules make contact. To simplify the calculations, spherical molecules and conical caps are used. Below it will be shown that the general conclusions are independent of the detailed shape of the molecule. First, consider the case when no electric field is applied. The proteins will be randomly oriented. We work in the rest frame of one of the molecules, where the other approaches with impact parameter b, defined as the perpendicular distance from the center of one molecule to the linear trajectory of the moving molecule, as depicted in Fig. l(a). The number of collisions per unit time is pvA, where p = the
POLYMERIZATION
OF
POLAR
MACROMOLECULES
247
(8 ,¢~) E=O
(a)
(O,~)
(b)
Fro. 1. Geometry of a collision between two monomers for (a) E = 0, with attachment caps randomly oriented, and (b) E ~ 0, with caps oriented in the direction of the field. Q denotes the point of contact between the spheres, and (Oi, tbi) are the spherical co-ordinates of the pole. density o f the molecule, v = the m e a n velocity, and A = the cross sectional area. This factor arises b e c a u s e a molecule o f cross sectional area A and velocity v sweeps out a cylinder o f v o l u m e vA in 1 sec. Let f)~, = ( O i , ~ ) a n d ~q2 = ( 0 2 , ~k2) be the p o l a r co-ordinate o f the north pole o f m o l e c u l e 1 and n o r t h pole o f molecule 2. T h e rate o f b i n d i n g will be: the probability that the north pole o f molecule 1 intersects the point o f contact,
f ((oill'~' d~/ 4 7ra2,
(1)
times the probability that the s o u t h pole o f molecule 2 intersects the point o f contact, 2~r)
f(("'
d~2/4~ra 2,
(2)
0.0)
times the b i n d i n g efficiency ~(12~, 122), times
puA. The radius
is d e n o t e d by a 2. T h a t
248
w . J . MEGGS
is, f2adbf''2") dO, f(~.2~)d122 ~ a o ( o . o ) 4 ~ a 2 J ( o . o ) 4 e r a 2r/(D''122)'
P~22°(°t)=2pVAjo
(3)
where d12i = a 2 sin Oi dOi d~bi is a differential element of area defined by about 12~. The superscript 2 denotes a chain length of two monomers. The factor of 2 arises because binding can also occur if the south pole of molecule 1 contacts the north pole of molecule 2. Performing the integrals, assuming 7/ to be constant, gives:
N=2 PE=0(a) =½pvArl(1-cos a ) 2.
(4)
Next, consider the case in which an electric field is present. The molecules will align so that their axes are parallel, as depicted in Fig. l(b). Since the direction o f the velocity vector will be randomly distributed with respect to the electric field, 12 = (O, ~b) will also be randomly distributed. Hence, the probability o f binding is just the probability that the cap o f molecule 1 intersects the point o f contact. If the north pole of molecule 1 intersects, the south pole of 2 will, and binding can occur. Hence, p NE#o~ = 2 / o t j-2~lpvA a_
foadf
(a" 2~) d122 ~7(f~l 121)
v(o, o)
4"n'a2
'
(5)
= pvArl(1 - c o s a). The ratio o f the binding probability in the presence o f an electric field to that in the absence o f a field,
N=2 N=2 P~#o(a)/PE=o(a) = 2(1 - c o s a ) - ' ,
(6)
is plotted as a function of a in Fig. 2, in the curve labeled N = 2. There is a dramatic rise in the relative probability of dimer formation as the size of the attachment cap decreases. These arguments can easily be extended to molecules of arbitrary shape and attachment site. Let aN and as be the areas of the north and south attachment sites and A be the surface area o f the protein. Then, with no electric fluid, the probability o f attachment is: pN=2, E=ot aNaS), : 2 p w l A ( a ~N ) . (_~)
(7)
= 2pvrlaNas/A. The factors o f aN/A and as/A are the probabilities that the north attachment cap of molecule one and south attachment cap of molecule two intersect the contact point o f the molecules. The factor of two arises as before from the interchange of north and south. In the presence o f an electric field, if the north and south attachment sites are at opposite poles, the north pole o f one will intersect the contact point if, and only if, the south pole o f the other intersects. Then,
P N=2 ~ #o( aN) = 2pwlA ( aN/ A ) = 2pwlaN
(8)
POLYMERIZATION
OF
POLAR
MACROMOLECULES
249
l0 7
iO 6 %%
E % IO s o
~
. - ~: E_
10 4
~, ~
10 3
~.~ o
o
8~ t~
I0'
i0 ° (45 o)
~- (9o °)
Angle subtended by (~rI1ochmenf cop in rodion (de?rees)
FIG. 2. A graph is given of the ratio of the probability of forming a polymer of length N in the presence and absence of a constant electric field E, as a function of the size of the attachment cap.
and, N =2
N=2
Pe,~o(aN)/ Pe,~o(aN) = Alas-> oo,
(9)
as as-> 0. So, in the general case, there is also a marked enhancement of dimer formation provided the north and south attachment sites are at opposite poles.
Growth of Polymer Chain In the above section, it was shown that there is an enhancement of dimer formation o f polar molecules in the presence of an electric field, so long as the attachment sites symmetrically aligned at opposite poles. In this section, we consider the relative rates of a polymer chain adding a m o n o m e r to increase its length by one monomer. Figure 4(a) and (b) shows the m o n o m e r impacting the end of polymer chain in the cases o f E = 0 and E # 0, respectively.
250
W.
J.
MEGGS
ta)
E~O
FIG. 3. Geometry of a collision between a m o n o m e r and the end of a polymer chain for (a) E = 0, with attachment caps randomly oriented, and (b) E ~ 0, with caps oriented in the direction of the field.
Let ~¢ be the probability that the m o n o m e r impacts the terminal attachment cap of the polymer, which is identical in the two cases. Then pN E=O~,
the probability of forming a p o l y m e r of length N by adding a m o n o m e r to a polymer o f length N - 1, is, p NE = O - _- 2 E~ a r~t, E N=- O i //¢ ' I,t,
where a and A are the area o f the attachment cap and the surface area, respectively. The factor of 2 arises because the polymer can grow at both ends. For E ~ 0, we have, P~,o
= 2s~PE ,v-i ¢o
(10)
and,
For the case o f spherical molecules with an attachment cap defined by an angle a, a plot is given o f the relative enhancement o f the formation o f polymers o f lengths N = 2, 3, 4, 5 and 6 in Fig. 3. The enhancement factor grows exponentially as N increases and has an asymptote at a = 0.
POLYMERIZATION
OF
POLAR
251
MACROMOLECULES
-1-6 ~-1-8 20 -1"4 \ \ ~-Cu -3"0.q5._~.( °,
;'g -0.6 0.4
-20
- IO
O -
IO
20
50
x
20
0 -002
(b)
0.009
20
I0
0
I0
20
x
FIG. 4. (a) Numerical integration of the differential equation reqmring that the polymers be parallel to the field lines gives a family of curves determined by the slope at the origin for oppositely charged spindle poles a = 30 ~ apart. Curves are labeled by the slope at the origin. (b) Contours give constant values of the electric field. Curves are labeled by 41reE/Q i n / ~ - 2 .
It has been shown that if a protein molecule can polymerize to form chains and possesses an electric dipole moment, there is an enhancement of polymerization in the presence of an electric field. This enhancement factor grows exponentially with chain length and increases as the size of the attachment cap decreases. The result is independent of the strength of the electric field, depending only on the tendency of an electric dipole to rotate parallel to the field, after a transient period following switching on the field. We assume that the field is strong enough to orient the molecules. The length of the transient period will decrease with increasing field strength. The result is independent of the detailed geometry of the molecule, so long as the attachment sites are at opposite poles and aligned with the dipole moment. This result gives strong support to the hypothesis (Meggs, 1988) that electric fields are important in both triggering the formation, and determining the structure, of microtubules and actin filaments. Direct experimental study of this hypothesis is therefore justified.
Application to Mitosis The proposal that intracellular electric fields determine the spatial organization of the mitotic spindle structure is tested by comparing theoretically calculated mitotic structures with observations. It is assumed that the spindle poles become oppositely
252
w . J . MEGGS
charged, leading to alignment of tubulin molecules in the direction of the resulting field, which enhances polymerization and leads to microtubules which parallel the field lines. Solving the equation d y / d x = E y / E x numerically, where (x, y) are spatial co-ordinates and (Ex, Ey) the components of the electric field, which states that the slope of the microtubules is parallel to the field lines, leads to specific predictions of microtubule shape which is compared to phase contrast microscopic (Salmon & Segall, 1980) and election microscopic data (McIntosh et al., 1976). Consider spindle poles located at (0, 0) and (a, 0), where a, the distance between the poles, is on the order of 10 ~ for eukaryotic cells. Let the poles be oppositely charged, with charges of q and - q , respectively. The electric field E will be given by Coulomb's law. 4~re
3 "
(12)
where e is the dielectric constant in the cell, r is the position vector locating a point in space, and (0, 0) and a = (a, 0) locate the spindle poles. The assumption that polymers form parallel to the elective field lines can be written, y'(x, y) = d__~y= Ey dx Ex'
(13)
where E = ( E y , Ex) is the electric field vector and y = y ( x ) is the equation of an electric field line, and hence of a microtubule. The spindle fiber is of course a three-dimensional structure, but the axial symmetry of the problem allows us to work in a two-dimensional plane. Equation (13) may be solved numerically using a simple finite difference method, y, = y'(xn-1, Yn-I) . Ax +y~_l,
(14)
where A is the increment size for the numerical integration. This first order differential equation requires one parameter to specify a solution, which we take to be the slope of the trajectory at the origin. It can be seen from eqn (2) that d y / d x is undefined (0/0) at the origin, but as one approaches the origin along a line y = mx, y'(x) -->m as x ~ 0. Hence, by beginning the numerical iteration of (13) at (0, 0) with y'(0, 0 ) = m, a specified parameter, one generates a distinct curve for each value of m, as shown in Fig. 4(a) with a = 30 W- Convergence of the method was verified by changing the step size, noting y ( a ) = 0 in the numerical solution, and observing that the numerical solution preserves the symmetry about the line x = a/2. Figure 4(b) gives contours of equal electric field strengths, IEI = constant. These curves were calculated by evaluating the electric field at various points in the plane, and connecting points where the field strength is equal. In this theory, the spindle poles become oppositely charged, and polymerization will take place out to a threshold value of the field strength Eth, below which the monomers will not become aligned with the field due to competition from thermal effects and other forces.
253
P O L Y M E R I Z A T I O N OF P O L A R M A C R O M O L E C U L E S
4WCt=E" 0
(a)
> 0.009
W-=
\
|
I
I0~
(b)
4'r~"E < 0.002 ~-= o
I
I Io/~.
FIG. 5. (a) Computer simulation of a mitotic spindle structure if polymerization takes place for electric fields greater than 0.009 Q/4~re. (b) Computer simulation of a mitotic spindle if polymerization takes place for electric fields greater than 0-002 O/4rre. The insert shows an isolated mitotic spindle from Salmon & Segall (1980). Used with permission of Rockefeller University Press and Dr E. D. Salmon.
254
w . J . MEGGS
Figure 5(a) and (b) show a calculated microtubule distribution with a = 30 for two values of Eth, normalized to charge and dielectric constant, using the values o f slope depicted in Fig. 4. The insert in Fig. 5(b) is a reproduction from Salmon & Segall (1980) of an isolated mitotic spindle at metaphase viewed with phase contrast microscopy, and is in qualitative agreement with Fig. 5(b). The dynamical development of the spindle fiber has been elucidated by fluorescence microscopy o f dye-tagged tubulin injected intracetlularly (Mclntosh et al., 1986; Wadsworth & Salmon, 1986). In a sequence o f photographs, fluorescence tagged microtubules form about the spindle pole, as in Fig. 5(a) the progress and meet as in Fig. 5(b). Since the threshold field for polymerization is proportional to the charge, an increasing charge on the spindle poles would lead to a temporal development o f the spindle structure in agreement with observation, though the mechanism o f this increasing charge remains unexplained at present. Electron microscopy data (Mclntosh et al., 1976) allows one to visualize individual microtubules over a limited range. Some tubules are warped and S-shaped, which may be an artifact of the fixing and cutting process. The data points in Fig. 6 are taken from smooth microtubules from this data, and smooth curves are theoretical curves calculated numerically, using eqns (12-14). There is reasonable agreement between the data points and experimental curves. Here a = 10 ~, the difference being that a smaller cell was used in this reference from that of the data of Fig. 5(b). A large amount of experimental work has been done on mitosis, but little attention has been given to the forces that move the tubulin molecules into position and determine the spatial organization o f the mitotic spindle structure. In 1986, Deery
~.~y,.,m=0. 1-~ 8 re=l-4
i
t
|
I
t
I
I
f-8
~
m=2"0
FIG. 6. Calculated curves for an isolated spindle fiber are compared to data points taken from an electron micrograph of a mitotic spindle from McIntosh et al. (1976), with a = 10 p..
POLYMERIZATION
OF P O L A R M A C R O M O L E C U L E S
255
wrote, "The factors responsible for the precise spatial distribution of microtubules a n d t h e i n t r a c e l l u l a r l o c a l i z a t i o n o f t h e s e f a c t o r s r e m a i n s to b e e l u c i d a t e d " . P r e c i s e l y this p r o b l e m is a d d r e s s e d in this p a p e r . M i t o s i s is a c o m p l i c a t e d p r o c e s s , a n d a s i m p l e m o d e l s u c h as this o n e c a n n o t a c c o u n t f o r t h e c o m p l e x d y n a m i c s o f m i t o s i s . T h e g o a l o f this w o r k is l i m i t e d , to e x p l a i n t h e o r g a n i z a t i o n o f t h e m i t o t i c s p i n d l e at m e t a p h a s e . T h e g o o d a g r e e m e n t w i t h o b s e r v e d s t r u c t u r e s s u p p o r t s t h e n o t i o n t h a t i n t r a c e l l u l a r e l e c t r i c fields m a y p l a y a r o l e in this p r o c e s s , a n d h o p e f u l l y will e n c o u r a g e f u r t h e r e x p e r i m e n t a l a n d t h e o r e t i c a l w o r k in this d i r e c t i o n . The author greatly acknowledges permission from Dr E. D. Salmon and The Rockefeller University Press to reproduce a photograph from Salmon & Segall (1980). REFERENCES DEERY, W. J. (1986). Control of microtubule assembly-disassembly in lysed cell models. Ann. N.Y. Acad. Sci. 466, 593-608. DRUTZ, D. & MXLLS, J. (1984). Immunity and infection. In: Basic and Clinical Immunology (Stites, D. P. et aL, eds) p. 202, 5th edn., Los Altos: Lange. FROHLICH, H. (1986). Theory o f Dielectrics 2nd edn. Oxford: Oxford University Press. GALLIN, J., DUROCHER, J. & KAPLAN, A. (1974). Interaction of leukocyte chemotactic factors with the cell surface. Jr. clin. Invest. 55, 94-96. INOUE, S. & SATO, H. (1967). Cell motility by labile association of molecules: the nature of the mitotic spindle fibers and their rote in chromosome movement. J. gen. Physiol. 50, 259-292. JAFFE, L. F. & NUCC~TELLI, R. (1977). Electrical controls of development. Ann. Rev. biophys. Bioeng. 6, 445-476. KIRKWOOD, J. G. (1939). The dielectric polarization o polar liquids. J. chem. Phys. 7, 911-919. McI NTOSH, J. R., HEPLER, R. K. & VAN CIE, D. G. (1976). Model for mitosis. Nature, Lond. 224, 659-663. MCINTOSH, J. R., SAXTON, W. M., STEMPLE, D. L. & WELSH, M. J. (1986). Dynamics of tubulin and calmodulin in the mammalian mitotic spindle. Ann. N.Y. Acad. Sci. 466, 580-592. MEOGS, W. J. (1988). Electric fields determine the spatial organization of microtubules and actin filaments. Med. Hypotheses 26, 165-170. ONCLEY, J. L. (1943). In: Proteins, amino acids, and peptides as ions and dipolar ions. (Cohn, E. J. & Edsalt, J. P., eds) chapter 22. New York: Rheinhold. ONSAGER, L. (1936). Electric moments of molecules in liquids. J. Am. Chem. Soc. 58, 1486-1493. ORIDA, N. & FELDMAN, J. D. (1982). Directional protrusive pseudopodial activity and motility in macrophages induced by extracellular electric fields. Cell Motility 2, 243-255. SALMON, E. D. & SEGALL, R. R. (1980). Calcium-labile mitotic spindles isolated from sea urchin eggs. J. Cell Biol. 86, 355-365. TILNEY, L., HATANO, S., ISHIKAWA, H. & MOOSEKAR, M. S. (1973). The polymerization of actin: its role in the generation of the acrosomal process of certain echinoderm sperm. J. Cell Biol. 59, 109-126. WADSWORTH, P.& SALMON, E. O. (1986). Microtubule dynamics in mitotic spindles of living cells. Ann. N. Y.. Acad. Sci. 446, 580-592. YANG, W. P., ONUMA, E. K. & HUI, S. W. (1984). Response o f C 3 H / I O T 1 / 2 fibroblasts to an external steady field stimulation. Expl. Cell Res. 155, 92-104.