ARTICLE IN PRESS
Journal of Electrostatics 63 (2005) 309–327
www.elsevier.com/locate/elstat
Electrostatic force generation in chromosome
motions during mitosis
L. John Gagliardi
Department of Physics, Rutgers–the State University, Camden Campus, Camden, NJ 08102, USA
Received 29 June 2004; accepted 30 September 2004
Available online 5 November 2004
Abstract
In previous work, I have proposed that nanoscale electrostatics plays a significant role in
aster (spindle) assembly and motion, and in force generation at kinetochores and chromosome
arms for prometaphase, metaphase and anaphase-A motions during mitosis. I have also
discussed the possible role of electrostatics in anaphase-B cell elongation. Recent experimental
studies have revealed that force production at spindle poles dominates in some cell types. The
present work extends the model for motion producing electrostatic interactions during
prometaphase, metaphase, and anaphase-A to include force generation at spindle poles.
Microtubule heterodimer subunits are electric dipolar structures that can act as intermediaries,
extending electric fields over cellular distances in spite of ionic screening. This enables
nanoscale electrostatic interactions to provide the force, localized at kinetochores, spindle
poles, and chromosome arms, to move chromosomes during mitosis. It will be argued that
such Debye-screened nanoscale electrostatic interactions can provide a minimal assumptions,
comprehensive model for post-attachment chromosome motions during mitosis consistent
with experimental observations.
r 2004 Elsevier B.V. All rights reserved.
Keywords: Electrostatic force; Cells; Mitosis; Chromosome motion; Nucleus; Prometaphase; Metaphase;
Anaphase-A; Tubulin
Tel.: +1 856 225 6159; fax: +1 856 225 6624.
E-mail address: gagliard@camden.rutgers.edu (L.J. Gagliardi).
0304-3886/$ - see front matter r 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.elstat.2004.09.007
ARTICLE IN PRESS
310
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
1. Introduction
Primitive eukaryotic cells had to divide prior to the evolution of very many
biological mechanisms, and it is reasonable to assume that basic physics and
chemistry played dominant roles in both mitosis (nuclear division) and cytokinesis
(cytoplasmic division). It is proposed in this series that electrostatic force, a
component of the electromagnetic interaction, played a major role in the dynamics
of chromosomes during cell division in primitive cells, and that the fundamental
solutions to the problem of cell division that were found by primitive cells may
persist in modern eukaryotic cells.
The mitotic spindle is responsible for the segregation of sister chromatids during
cell division. Chromosomes are attached to the spindle with their kinetochores [1]
attached to the ‘‘plus’’ ends of microtubules [2,3]. Chromosome movement is
dependent on kinetochore–microtubule dynamics: a chromosome can move towards
a pole only when its kinetochore is connected to microtubules emanating from that
pole [4]. A number of experimental studies have been undertaken to obtain
information regarding microtubule dynamics, force production, and kinetochore
function in mitotic cells. These experiments have revealed that the spindle can
produce more force than is actually required to move a chromosome at the observed
speeds for post-attachment movements, and that the force for the poleward motion
of chromosomes can be localized at or near kinetochores [5–11] or at spindle poles
[12–14]. Quite some time ago, Cooper addressed a possible link between endogenous
electrostatic fields and the eukaryotic cell cycle [15]. An early review by Jaffe and
Nuccitelli [16] focused on the possible influence of relatively steady electric fields on
the control of growth and development in cells and tissues.
In the cytoplasmic medium (cytosol) that exists in biological cells, electrostatic
fields are subject to strong attenuation by screening with oppositely charged ions,
and decrease rapidly over a distance of several Debye lengths. The Debye length
within cells is typically 1 nm, see for example [17], and since cells of interest in the
present work (i.e., eukaryotic) can be taken to have dimensions between 10 and
30 mm; one would be tempted to conclude that electrostatic force could not be a
major factor in providing the cause for motion in biological cells. However, the
presence of microtubules changes the picture completely. Microtubules can be
thought of as intermediaries that extend the reach of the electrostatic interaction
over cellular distances, making this potent force available to cells in spite of their
ionic nature. A number of investigations have focused on the electrostatic properties
of microtubule dimer subunits [18–21]. Recent studies [22,23] have shown that the
net charge depends strongly on pH. The dipole moment has been calculated to be
between 1200 and 1800 debye [22].
The aster’s pincushion-like appearance is consistent with electrostatics, since
electric dipolar subunits will align radially outward about a central charge, with the
geometry of the resulting configuration resembling the electric field of a point charge.
From this it seems quite probable that the pericentriolar material-centriole complex,
the centrosome about which the microtubule dimer dipolar subunits assemble to
form the aster, carries a net charge. This is consistent with ultramicroscopic
ARTICLE IN PRESS
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
311
observations that the microtubules appear to start in the pericentriolar material
region (centrosome matrix), see for instance [24], aligning radially outward.
Since there is no direct experimental information regarding the sign of this charge,
it will be assumed negative. This assumption is consistent with studies showing that
g-tubulin nucleates the assembly of microtubules by binding to b-tubulin at the
positively charged free ends of microtubules [18,25–27]. In addition, experiments [28]
have shown that mitotic spindles can assemble around DNA-coated beads incubated
in Xenopus egg extracts. The phosphate groups of the DNA will manifest a net
negative charge at the pH of this experimental system.
Studies [29] have shown that in vivo microtubule assembly (polymerization) is
favored by higher pH values. It should be noted that in vitro studies of the role of pH
in regulating microtubule assembly indicate a pH optimum for microtubule assembly
in the range of 6.3–6.4. The disagreement between in vitro and in vivo studies
regarding microtubule polymerization has been analyzed in relation to the
nucleation potential of microtubule organizing centers (MTOCs) [29], and it has
been suggested that intracellular pH (pHi ) regulates the nucleation potential of
MTOCs [30–32]. This favors the more complex physiology characteristic of in vivo
studies to resolve this question. It will, therefore, be assumed in this paper that in
vivo experimental design is more appropriate for experiments relating to conditions
affecting microtubule assembly.
It is reasonable to conclude that the electric dipole nature of dimer subunits
greatly assists in their self-assembly into microtubules. In particular, their dipolar
nature would allow them (over the short distances consistent with Debye shielding)
to be attracted to, and align around, any net charge distribution within cells. This
may account for the self-assembly of the asters during prophase, when microtubule
polymerization and MTOC nucleation is favored because of the higher pHi at this
time. Thus, we may envision that electrostatic fields organize and align the electric
dipole dimer subunits, thereby facilitating their assembly into the microtubules that
form the aster [33]. The attraction between oppositely charged ends of the dipolar
subunits takes place over the short distances allowed by Debye shielding. An
electrostatic component to the biochemistry of the microtubules in the assembling
asters is consistent with experimental observations of pH effects on microtubule
assembly [29], as well as the sensitivity of microtubule stability to calcium ion
concentrations [34,35]. In addition, the mutual electrostatic repulsion of the
negatively charged free ends of microtubules in the assembling asters could provide
the driving force for their poleward migration in the forming spindle [33,36].
According to existing convention, these negatively charged microtubule ends are
designated ‘‘plus’’ ends because of their more rapid growth, there being no reference
to charge in the use of this nomenclature.
Microtubules continually assemble and disassemble, so the turnover of tubulin is
ongoing. The characteristics of microtubule lengthening (polymerization) and
shortening (depolymerization) follow a pattern known as ‘‘dynamic instability’’;
that is, at any given instant some of the microtubules are growing, while others are
undergoing rapid breakdown. In general, the rate at which microtubules undergo net
assembly, or disassembly, varies with mitotic stage; for example, during prophase the
ARTICLE IN PRESS
312
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
rates of microtubule polymerization and depolymerization change quite dramatically, see for example [37].
Poleward and antipoleward chromosome movements occur intermittently during
prometaphase and metaphase. Antipoleward motions dominate during the congressional movement of chromosomes to the cell equator. A more sustained poleward
motion of chromosomes is observed during anaphase-A. In present terminology
metaphase denotes the relatively brief period during which chromosomes are lined up
at the center of the cell (the equator) and are fully attached to both poles by the
microtubules of the spindle. The term prometaphase is used to encompass a much
wider time period during which most of the complex motions in this stage of mitosis
occur. Two events that are of major significance during prometaphase are (1) the
capture and attachment of chromatid pairs by microtubules, and (2) chromosome
movement to, and alignment at, the cell equator. The latter is comprised of several
distinguishable motions.
Regarding the first event, experiments [38] have shown that each pair of sister
chromatids attaches by a kinetochore to the outside walls of a single microtubule,
resulting in a rapid microtubule sidewall sliding movement toward a pole. This
motion is postulated to be driven by dynein-based molecular motors, since dynein
has been found at kinetochores. A molecular motor powered microtubule sidewall
sliding model for this prometaphase movement would appear to be widely accepted.
In particular, the speed (20–50 mm=min) [39] of kinetochores along microtubule walls
is consistent with known molecular motor behavior. Consequently, I agree that a
molecular motor model for the microtubule sidewall capture motion is supported by
the experimental observations. However, I propose that all of the subsequent (postattachment) prometaphase, metaphase and anaphase-A poleward and antipoleward
chromosome motions are based on nanoscale electrostatic microtubule disassembly
and assembly force mechanisms.
The material of kinetochores is proteinaceous, and could manifest a net positive
charge at the lower pHi levels during prometaphase [40]. In addition, kinetochores
self-assemble onto the highly condensed negatively charged DNA at centromeres, see
for instance [41], indicating that they may be positively charged.
As a result of the sliding capture motion described above, the approach to the
poles will result in the movement of a kinetochore to within several Debye lengths of
the ends of other microtubules emanating from the closer pole. The resulting
proximity, in conjunction with (1) an electrostatic attraction between positively
charged kinetochores and the negatively charged ends of these microtubules, and (2)
an electrostatic repulsion between negatively charged chromosome arms in the
chromatid pair and other microtubule ends, could be critical in the orientation and
attachment of kinetochores to the free ends of microtubules [42].
Following this monovalent attachment to one pole, chromosomes are observed to
move at considerably slower speeds, a few micrormeters per minute, in subsequent
motions throughout prometaphase [39]. In particular, a period of slow motions
toward and away from a pole will ensue, until close proximity of the negatively
charged end of a microtubule from the opposite pole with the other kinetochore in
the chromatid pair results in an attachment to both poles (a bivalent attachment).
ARTICLE IN PRESS
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
313
Attachments of additional microtubules from both poles will follow. (There may
have been additional attachments to the first pole before any attachment to the
second.) After the sister kinetochore becomes attached to microtubules from the
opposite pole, the chromosomes perform a slow (1–2 mm= min) congressional motion
to the spindle equator, resulting in the well-known metaphase alignment of
chromatid pairs. In addition to the mechanism facilitating attachment just discussed,
all of the above mentioned experimentally observed post-attachment poleward and
antipoleward prometaphase motions, as well as the oscillatory metaphase motion,
can be understood in terms of electrostatic interactions.
Chromosome motion during anaphase has two components, designated as
anaphase-A and -B. Anaphase-A is concerned with a relatively sustained poleward
motion of chromosomes, accompanied by the shortening of microtubules at
kinetochores and/or spindle poles. The second component, referred to as
anaphase-B, involves the separation of the poles. Both components contribute to
the increased separation of chromosomes during mitosis. An electrostatic force
mechanism for anaphase-B motion within the context of the present work is given
elsewhere [33,36]. A number of experiments have revealed that poleward motion of
chromosomes proceeds by kinetochore microtubule disassembly primarily in the
vicinity of kinetochores [5,7]. In some cell types, disassembly is observed to take
place primarily at poles [12–14]. Based on experiments centering on observations
near kinetochores, it has been proposed that the poleward force to move
chromosomes in some cell types is generated at kinetochores [43].
2. Poleward microtubule disassembly force
2.1. Nanoscale electrostatic force at a centrosome
The above observations on post-attachment chromosome movements, including
the motive force at spindle poles, are explained in the context of the present model as
follows. Microtubules invariably assemble or disassemble at their ends; that is, at
some discontinuity in their structure. Furthermore, they are known to be in a
constant condition of dynamic instability at the balanced state [44]. According to the
aster self-assembly model referred to above [33], the charge on the free ends of
microtubules at a centrosome matrix is positive. A g-tubulin molecule, embedded in
the fibrous matrix, takes the form of a ring from which a microtubule appears to
emerge, see for example [45]. This could allow the electric field of the negatively
charged g-tubulin rings to draw the positively charged ends of microtubules through
the centrosome matrix, with the resulting rapid change of field strength destabilizing
the microtubules as they pass through the charge distribution. Thus, g-tubulin rings
may be regarded as forming a firmly anchored negative charge distribution near the
surface of the centrosome matrix through which microtubules pass, disassembling in
the passage, as depicted schematically in Fig. 1. As also mentioned earlier,
observations on a number of cell types have shown that disassembly of microtubules
at spindle poles accompanies chromosome poleward movement. Accordingly, within
ARTICLE IN PRESS
314
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
Fig. 1. Nanoscale electrostatic disassembly force at a centrosome. A poleward force results from an
electrostatic attraction between positively charged free ends of microtubules and an oppositely charged
centrosome matrix.
the context of the present model, force generation at a spindle pole for prometaphase
post-attachment, metaphase, and anaphase-A poleward chromosome motions can
be attributed to an electrostatic attraction between the positively charged free ends of
disassembling kinetochore microtubules and a negatively charged centrosome matrix
at a spindle pole.
We now calculate the magnitude of the maximum force produced in this manner
by a single non-penetrating microtubule. It will be shown later in a similar
calculation at a kinetochore that the average force on a penetrating microtubule has
approximately the same value as a non-penetrating microtubule. Since the outer
diameter of a centrosome matrix is considerably larger than the diameter of a
microtubule, we may model it as a large, approximately planar, slab with negative
surface charge density of magnitude s: From the well-known Debye–Hückel result
ARTICLE IN PRESS
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
315
for a charged surface with charge density s immersed in an electrolyte [46], we have
for the electrostatic potential
Ds x=D
e
;
(1)
2e
where D is the Debye length and x is the distance from the surface. For a dielectric
constant of 71, the cytosol permittivity e is 71e0 ; where e0 is the permittivity of free
space. The room temperature permittivity of water is 80e0 ; the value of 71e0
incorporates corrections for the temperature and ionic depression of the dielectric
constant [47] appropriate to the cytosol of mammalian cells.
As indicated earlier, based on recent calculations, a tubulin heterodimer has a
dipole moment between 1200 and 1800 debye. Assuming a midrange value of
1500 debye, a calculation of the force per microtubule may be carried out with a
dipole charge magnitude q of 6 electron charges on a tubulin dimer at each of the free
ends of the 13 protofilaments in a microtubule interacting with a centrosome. The
electric field EðxÞ; obtained from the negative gradient of the electrostatic potential,
multiplied by the charge q gives the magnitude of the attractive force F ðxÞ between
the charge on a dimer subunit at the end of a protofilament and the centrosome. This
results in
sq x=D
F ðxÞ ¼
e
:
(2)
2e
It is well established in electrochemistry [48] that the permittivity of the first few
water layers outside a charged surface is an order of magnitude smaller than that of
the bulk phase, and the Debye shielding of the electric field begins just beyond the
water layers, at a distance of approximately 0.5 nm. The effective permittivity of
water as a function of distance from a charged surface has been determined by
atomic force microscopy [49] to increase monotonically from 4 to 6e0 at the interface
to 78e0 at a distance of 25 nm from the interface. The experiment was carried out
with mica, which is known to have a surface charge density that varies from 1 to
50 mC=m2 ; in the same range as biological surfaces [50,51]. Thus, the expression for
the force between the charge at the free end of a protofilament and a centrosome may
be written
sq ðx0:5Þ=D
e
F ðxÞ ¼
;
(3)
2eðxÞ
fðxÞ ¼
where eðxÞ is obtained from the experimental results referred to above, q is 6 electron
charges [42] and the Debye shielding begins at a distance of 0.5 nm. There are 13
protofilaments arranged circularly in a microtubule, with an axial shift of 0.92 nm for
each protofilament as one moves around the circumference of a B lattice microtubule
[52]. Based on this axial shift, a comparison with experimental values for the
maximum force exerted by a microtubule may be obtained by assuming that the
positive charge centers for dimers at the free ends of every fifth protofilament are at
the closest distances of 1, 1.28, and 1.6 nm. As indicated above, experimental values
of surface charge density s for biological surfaces range from 1 to 50 mC=m2 : For a
Debye length of 1 nm and a conservative value for s of 10 mC=m2 ; we find that the
ARTICLE IN PRESS
316
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
electrostatic force on the dimers at the free ends of the 13 protofilaments of a
microtubule sums to 70 pN. This value compares quite favorably with the
experimentally measured range of 1–74 pN per microtubule [10]; however, this
calculation is primarily intended to demonstrate that electrostatic interactions are
able to produce a maximum force per microtubule within the experimental range.
2.2. Nanoscale electrostatic force at a kinetochore
Experimental observations on force generation at kinetochores may also be
explained by the present model. It has been accepted for some time that electron
microscope studies show kinetochore microtubules running uninterrupted between
poles and kinetochores, terminating in the outer plate of the kinetochores [2]. As I
have previously proposed—given observations (cited earlier) that microtubule
disassembly at or near kinetochores (experimental resolution not being able to
determine precisely where) accompanies chromosome poleward movement in some
cell types—one could assume that microtubules disassemble near kinetochores with
microtubule stubs remaining fixed to the kinetochores [42]. The motive force for
prometaphase post-attachment, metaphase and anaphase-A poleward chromosome
motions could then be attributed to a Debye-screened electrostatic attraction
between the positive ends of the microtubule stubs attached to kinetochores and the
negative ends of the remaining intact kinetochore microtubules.
Based on this model, the ab initio computed magnitude of the maximum force on
a chromosome due to one microtubule was found to be 24 pN [42], consistent with
the above calculation at a spindle pole, and within the experimentally observed range
of 1–74 pN. Also based on this model, a computer simulation for anaphase-A
motion incorporating the geometry of microtubules and a numerical integration of
Newton’s second law with typical values of chromosome mass [53] and cytosol
viscosity [10] shows that electrostatic force is robust enough to sustain chromosome
motion within a wide range of microtubule disassembly modes [42].
As mentioned earlier, kinetochores may manifest a net positive charge at the lower
pHi levels during prometaphase [40,54]. Additionally, kinetochores assemble onto
highly condensed negatively charged DNA at centromeres, see for instance [41],
indicating that kinetochores may be positively charged. Assuming a positive charge
on kinetochores, we may envision an additional mechanism for electrostatic force
generation at kinetochores. As referred to above, it is generally believed that
kinetochore microtubules penetrate the outer plates of kinetochores [2]. It is also
assumed that this kinetochore microtubule–kinetochore association is the locus of
force generation by molecular motors acting between kinetochores and kinetochore
microtubules. Consequently, not much attention has been focused on the possibility
that kinetochore microtubules may be generating force in non-contact interactions
such as those arising from electrostatics. As a result, ultrastructural studies of
kinetochore–microtubule associations have concentrated on the microtubules that
are apparently penetrating the outer plate of kinetochores, and possibly being pulled
into kinetochores by molecular motors to generate force for the poleward motion of
chromosomes, and non-penetrating microtubules in close proximity to kinetochores
ARTICLE IN PRESS
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
317
have been ignored regarding possible force generation. However, the exact role of
the postulated molecular motors has not been established. The relatively constant
speed and abrupt reversals of direction would require a coordinated switching on
and off of many motor molecules located at kinetochores separated by micrometer
distances. In addition, the dynamics of the microtubules on sister kinetochores
would also need to be coordinated. These difficulties do not arise in the nanoscale
electrostatics model presented in this paper.
Since kinetochore plate diameters are large compared to the diameters of
microtubules (500 nm vs. 25 nm) we may model the kinetochore–microtubule
interaction by assuming a large approximately planar slab—of uniform positive
charge density with thickness a parallel to the x axis—for the outer kinetochore
plate interacting with the negatively charged free ends of microtubules, as depicted
in Fig. 2.
A standard result from an application of Gauss’s law gives the following result for
the electric field inside a large, uniformly charged slab of positive charge
EðxÞ ¼ rx= e;
(4)
where r is the volume charge density, e is the dielectric permittivity of the slab, and
x ¼ 0 at the plane of symmetry in the center of the slab. (Note that previously in (3),
x ¼ 0 at the right boundary of the centrosome matrix.) Making use of the uniform
charge relation s ¼ r a; this result may be expressed in terms of the surface charge
density s as
EðxÞ ¼ sx=ea:
(5)
At the left face of the slab, x ¼ a=2; E ¼ s=2e; and the force in the positive x
direction on a protofilament with negative charge of absolute value q at its free end
has magnitude sq=2e:
Electron microscopic studies reveal that there are three kinetochore plates, firmly
anchored to each other and to the chromosome, with electron translucent layers in
between, and that kinetochore microtubules penetrate only the outer (polewardfacing) plate on each kinetochore [2]. The force on a protofilament of negative
charge magnitude q at its free end a distance x from the plane of symmetry
is given by
F ¼ qsx=ea:
(6)
Using the conservative value s ¼ 10 mC=m2 in carrying out a calculation for a
microtubule with protofilament ends at an average distance x ¼ a=4 from the
symmetry plane, x ¼ 0 (where the force is 0), we find that the force sums to 50 pN.
This represents the average force on a microtubule in the slab. The reversal of field
direction at the plane of symmetry can destabilize the protofilaments in the
microtubules; as in the case for the centrosome matrix, this could cause microtubules
to disassemble in passing through the outer kinetochore plate as force is generated,
in agreement with experiment. The observation that kinetochore microtubules are
confined to the outer plate has a simple interpretation in terms of the model. Since it
seems likely that the electron translucent regions between plates contain cytosol,
ARTICLE IN PRESS
318
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
Fig. 2. Nanoscale electrostatic disassembly force at a charged kinetochore. A poleward force results from
an electrostatic attraction between negatively charged free ends of microtubules and an oppositely charged
kinetochore.
Debye shielding would effectively prevent the fields of the other two plates from
competing with the field of the outer plate.
As discussed above for the charged centrosome matrix, non-penetrating
microtubules could disassemble in the region of high Debye-screened field gradient
just outside the outer plate of the charged kinetochore, also generating a poleward
force, as shown in Fig. 2. Because of the similarity in geometry, a calculation of the
maximum force per microtubule for non-penetrating kinetochores will yield
essentially the same result as the above calculation at a spindle pole. As at the
centrosome matrix, a force calculation with (3) is carried out with a value for s of
10 mC=m2 ; yielding 70 pN as the nanoscale electrostatic microtubule disassembly
force at a kinetochore. This approximate equality in the calculations for electrostatic
ARTICLE IN PRESS
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
319
force generation by non-penetrating and penetrating kinetochore microtubules is
also demonstrable for microtubules at the centrosome matrix.
The calculation for penetrating microtubules shows that nanoscale electrostatics is
able to explain force production at kinetochores for penetrating microtubules, and
that the molecular motor models for force production at kinetochores that dominate
the current literature are not essential. Although there is a fair amount of
experimental work reported on microtubule flux at poles accompanying chromosome
dynamics, there has not been much discussion in the literature regarding models for
force generation at spindle poles associated with this flux. The present work unifies
force generation at both kinetochores and spindle poles within a minimal
assumptions, comprehensive model.
3. Antipoleward microtubule assembly force
Since chromosome arms are negatively charged, following chromosome attachment they will be repelled from the negatively charged free ends of the shorter astral
microtubules in the polar region. As discussed above, this force will be effective for
the nanoscale distances allowed by Debye screening. As chromosomes move farther
from the poles, there will be a filling in of dipolar subunits as the microtubules
assemble. The interaction between astral microtubules and chromosome arms is
depicted in Fig. 3.
Polymerization will take place in the gaps as chromosomes drift farther from the
poles, and chromosomes will be continuously repelled from the poles. This
mechanism may account for the antipoleward ‘‘astral exclusion force’’ or ‘‘polar
wind’’, the nature of which has remained unclear since it was first observed [55]. Very
short range entropic forces associated with growing microtubules [56] would
complement the electrostatic repulsive interaction at small microtubule–chromosome arm separations, adding to the total astral exclusion force. Although the
complex geometry precludes a theoretical calculation of the magnitude of these
forces, a model calculation of the repulsive force between two like-charged parallel
surfaces with an electrolyte in between shows that entropic forces must be included
for separations of less than 2 nm; at greater separations electrostatic theory fits the
data well [57,58].
As a chromatid pair moves farther from a pole, the electrostatic repulsive force
between the negatively charged free ends of astral microtubules and chromosomes
will decrease as the microtubules fan radially outward. At a surface defined by the
microtubule ends, the charge density and therefore the force, will decrease according
to an ‘‘inverse square law’’ as we can see from the following. Given that the repulsive
force on a chromosome arm depends on the total number N of negatively charged
free ends of microtubules from which it is repelled, we have F Nq; where q is the
charge at the end of a microtubule. For N microtubules fanning radially outward
from a pole, the total charge Nq is distributed over an area that increases as the
distance from the pole r2 ; and s; the effective charge per unit area at a surface defined
by the microtubule ends, decreases as r2 : This results in an electrostatic
ARTICLE IN PRESS
320
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
Fig. 3. Antipoleward electrostatic interaction between microtubules and chromosome arms. An
antipoleward force results from electrostatic repulsion between free ends of charged microtubules and
like-charged chromosome arms.
antipoleward force that decays with an inverse square dependence on the polar
distance.
The falloff is expected to be even more pronounced than inverse square due to the
decreased number of free ends of microtubules at greater polar distances, as shown
schematically in Fig. 4.
4. Operation of the model
4.1. Prometaphase and metaphase chromosome motions
The possibility that microtubule polymerization or depolymerization can occur, in
combination with a repulsive electrostatic antipoleward astral exclusion force and an
attractive electrostatic poleward-directed force acting at kinetochores and spindle
poles is sufficient to explain the observed motion of monovalently attached
chromosomes toward and away from poles. Because of statistical fluctuations both
in the number of kinetochore microtubules interacting with kinetochores and
centrosomes, and in the number of assembling astral microtubules responsible for
ARTICLE IN PRESS
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
321
Fig. 4. Antipoleward inverse square repulsive force. Two chromatid pairs at differing polar distances are
shown depicting the inverse square dependence of the nanoscale antipoleward force.
the antipoleward astral exclusion force, the interaction of these opposing forces
could result in a ‘‘tug of war’’, consistent with the experimentally observed series of
movements toward and away from a pole for a monovalently attached chromatid
pair. Microtubule assembly at kinetochores and poles is possible; however, because
the necessary inverse square dependence of the antipoleward force cannot be derived
from microtubule assembly at kinetochores and spindle poles, it is assumed in this
work that assembly at either location is in passive stochastic response to assembly at
chromosome arms.
After a bivalent attachment has been established, the attractive force to the far
(distal) pole will be in opposition to the attractive force to the near (proximal) pole.
The inverse square nature of the antipoleward force, along with a growing number of
kinetochore attachments to microtubules from the distal pole tending to equalize the
poleward forces would result in a relatively sustained congressional motion away
from the proximal pole, as observed.
ARTICLE IN PRESS
322
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
As a chromatid pair moves farther from the proximal pole, there will be a growing
number of attachments to both poles. Following approximately equal numbers of
attachments to both poles, and comparable distances of chromatid pairs from the
two poles, the forces exerted by both sets of polar attractive disassembly and
antipoleward repulsive assembly forces will approach equality. Thus, as a chromatid
pair congresses to the midcell region, the number of attachments to both poles will
tend to be the same, as will the number of microtubules interacting with
chromosome arms, and equilibrium of poleward-directed forces and antipoleward
astral exclusion forces will be approached. Without specifying their precise nature,
such balanced pairs of attractive and repulsive forces have previously been
postulated for the metaphase alignment of chromatid pairs, see for instance [59].
An explanation of experimentally observed metaphase oscillations about the cell
equator just prior to anaphase-A provides still another example of the predictability
and minimal assumptions nature of the present model. In agreement with experiment
[60], the model predicts that the poleward force at a kinetochore depends on the total
number of microtubules interacting with kinetochores. At the metaphase ‘‘plate’’,
the bivalent attachment of chromatid pairs ensures that the poleward-directed
electrostatic disassembly force at one kinetochore at a given moment could be
greater than that at the sister chromatid’s kinetochore attached to the opposite pole.
An imbalance of these forces would result from statistical fluctuations in the number
of interacting microtubules at sister kinetochores and at poles.
This situation, coupled with similar fluctuations in the number of microtubules
responsible for the antipoleward astral exclusion force, can result in a momentary
motion toward a pole in the direction of the instantaneous net electrostatic force.
However, because of the inverse square dependence of the astral exclusion force and
the approximate equality of poleward-directed forces for chromatid pairs near the
midcell region, electrostatic repulsion from the slightly nearer pole would eventually
reverse the direction of motion, resulting in stable equilibrium midcell metaphase
oscillations, as observed experimentally.
4.2. Anaphase-A chromosome motion
A number of experimental studies have shown that intracellular free calcium
concentration (½Ca2þ Þ increases are associated with anaphase-A chromosome
movement [61–63]. It is well known that increased calcium facilitates the
depolymerization of spindle microtubules both in vitro [64] and in vivo [65]. Studies
have also shown that changes in ½Ca2þ can modulate the speed of chromosome
motion [63,66]. In living cells, intracellular calcium releases show a remarkably close
temporal correlation with the onset of anaphase-A [63]. Significantly, there appears
to be an optimum concentration for maximizing the speed of chromosome motions
during anaphase-A. If the ½Ca2þ is increased to a micromolar level, anaphase-A
chromosome motion is increased two-fold above the control rate; however, if the
concentration is further increased beyond a few micromolar, the chromosomes will
slow down, and possibly stop [63]. It has long been recognized that one way elevated
½Ca2þ could increase the speed of chromosome motion during anaphase-A is by
ARTICLE IN PRESS
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
323
facilitating microtubule depolymerization [34,64–67], and it is commonly believed
that the breakdown of microtubules, if not the motor for chromosome motion, is at
least the rate-determining step [68–71]. However, the slowing or stopping of
chromosome motion associated with small increases beyond an optimum ½Ca2þ is
more difficult to interpret since the microtubule network of the spindle is not
compromised to the extent that anaphase-A chromosome motion should be slowed
or stopped; this would require considerably higher concentrations [63,72].
These experimental observations have a direct interpretation within the framework of the model presented in this paper. As discussed above, antipoleward
electrostatic forces compete stochastically with poleward electrostatic forces during
prometaphase and metaphase. For example, as discussed above, after a bivalent
attachment is established, the action of poleward-directed forces from both poles, in
conjunction with the inverse square nature of the antipoleward force, is sufficient for
congressional motion to the cell equator. Stable metaphase midcell chromosome
oscillations were seen as resulting from inverse square antipoleward nanoscale
electrostatic microtubule assembly forces at chromosome arms acting in conjunction
with approximately balanced electrostatic poleward forces. With the experimentally
observed increase of ½Ca2þ at the onset of anaphase-A and the resultant instability
of microtubules, the probability for microtubule assembly is decreased significantly
while the probability of disassembly is increased significantly, effectively switching
off the antipoleward forces from both poles. This allows microtubule electrostatic
disassembly poleward forces to dominate, and anaphase-A motion ensues.
The observation that an increase in calcium levels beyond micromolecular levels
results in a slowing or stopping of anaphase-A motion is also a direct consequence of
the present model. Higher concentrations of doubly charged calcium ions could
effectively swamp the negative charge at the free ends of disassembling kinetochore
microtubules and at the centrosome matrix, shutting down the poleward-directed
nanoscale electrostatic disassembly force. Since this happens at concentrations that
do not seriously compromise the spindle’s microtubule network [63,72], it is
reasonable to interpret these results as an experimentally consistent prediction of the
present model. This experimental observation does not appear to be addressed by
any of the other current models. Thus, anaphase-A motion could result from the
increased instability of microtubules and the resulting predominance of microtubule
disassembly over assembly following the increase in intracellular free calcium
concentration that is concurrent with anaphase-A movement. Finally, it is
reasonable to assume that the sudden shutting down of antipoleward forces in
combination with an enhancement of poleward forces is integral to the separation of
sister chromatids that heralds the onset of anaphase-A.
5. Conclusions
It would appear that molecular motors are involved in the sliding microtubule
sidewall capture motion of chromatid pairs, and I submit that kinetochore dynein
ARTICLE IN PRESS
324
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
may be present for this purpose, but not for the other (post-attachment) motions of
prometaphase, metaphase, and anaphase-A.
According to leading molecular motor models of anaphase-A motion, the
experimentally observed shortening of spindle fibers at a kinetochore is believed to
be accompanied by molecular motors that are associated with the kinetochore, and
are thought to provide the motive force to move the kinetochore–chromosome
assembly. However, there is as yet no agreement on a model which can describe how
a molecular motor associated with a kinetochore can be operating while
microtubules are disassembling at that kinetochore.
Models based on molecular motors appear to require substantial regulation. The
relatively constant speed and abrupt reversals of chromosome direction would
require a coordinated switching on and off of many motor molecules separated by
micrometer distances. The dynamics of motors on sister kinetochores would also
need to be coordinated. These difficulties do not arise in the nanoscale electrostatics
model presented in this paper.
It is significant that anaphase-A has been observed to proceed in isolated spindles
in the absence of ATP if conditions in the experimental system are set up to promote
microtubule disassembly [43,73]. These results are difficult to explain within a
molecular motor model, but are completely consistent with the present model.
In a key experimental study with grasshopper spermatocytes [74], it was found
that both anaphase-A and -B, as well as cytokinesis, proceeded independently of
chromosomes. The authors of this study concluded that chromosomes, when
present, might migrate to the poles by having their kinetochores latch onto the ends
of shortening microtubules, a scenario that is completely consistent with the present
work. Why would cells need to utilize molecular motors when the requisite
mechanism for anaphase-A poleward motion is already present?
It is not clear within the context of a molecular motor model why the velocity of
the poleward motion during anaphase-A should be governed by the relatively slow
(compared to known molecular motor behavior) shortening rate of microtubules
with attached chromosomes [75]. As indicated above, proponents of molecular
motor models assume that microtubule disassembly is the rate-determining step for
the motion, necessitating additional assumptions and models within the framework
of the molecular motor models to account for prometaphase post-attachment and
anaphase-A chromosome velocities. No such additional assumptions are needed in
the present model.
It is proposed in this paper that post-attachment chromosome motions during
prometaphase and metaphase can be explained by statistical fluctuations in
nanoscale electrostatic microtubule antipoleward assembly forces acting between
astral microtubules and chromosome arms, combined with similar fluctuations in
nanoscale electrostatic microtubule poleward disassembly forces acting at kinetochores and spindle poles. A mathematical model for the kinematics of such
fluctuations in the context of a different dynamical model for chromosome motions
has recently been given [76]. Increased kinetochore attachments to the distal pole,
along with an approximate inverse square falloff of the antipoleward force are
sufficient to ensure congressional motion to the cell equator.
ARTICLE IN PRESS
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
325
There does not appear to be any consensus on a model for the generation of force
at cell poles. Experimental observations regarding the microtubule disassembly force
at poles, with an associated microtubule flux are explained consistently within the
model presented in this work by the same electrostatic force mechanism as that
operating at kinetochores.
Midcell metaphase oscillations result from the near equality of poleward-directed
microtubule electrostatic disassembly forces and inverse square antipoleward
microtubule electrostatic assembly forces.
The observed abrupt intracellular increase in ½Ca2þ that occurs at the onset of
anaphase-A significantly decreases the probability for microtubule assembly,
switching off antipoleward nanoscale microtubule assembly forces from both poles.
The sudden dominance of unopposed poleward nanoscale microtubule disassembly
forces could be integral in the initial separation of sister chromatids and responsible
for anaphase-A chromosome motion.
The slowing or stopping of anaphase-A chromosome motion with free calcium
concentration increases beyond an optimum concentration range for maximum
anaphase-A chromosome speed—but not reaching concentration levels that
compromise the mitotic apparatus—is completely consistent with an electrostatic
disassembly motor; this experimental observation has not been addressed within the
context of any of the other current models for anaphase-A motion.
The calculated force per microtubule falls within the experimentally measured range,
and represents the only successful ab initio derivation of the magnitude of this force. In
agreement with experiment [77], the model presented in this paper satisfies the
requirement that the maximum force per microtubule be the same for all prometaphase
post-attachment, metaphase, and anaphase-A kinetochore–microtubule interactions.
The electrostatic force model presented here encompasses the dynamics, timing,
and sequencing of prometaphase, metaphase, and anaphase-A chromosome
motions. Electrostatic force could also be integral in the assembly and motion of
asters and the dynamics of prometaphase attachment [33,36].
Anaphase-B elongation chromosome speeds follow directly from electrostatic
interactions consistent with the model presented in this paper [33].
All experimentally observed post-attachment chromosome velocities proceed at
the relatively slow rate of a few micrometers per minute, the speed at which
microtubules with attached chromosomes lengthen or shorten. These speeds are a
direct consequence of the model.
Finally, based on current separate models for prometaphase post-attachment,
metaphase and anaphase-A chromosome motions, there does not seem to be any
possibility to relate their timing and sequencing, a situation that has been remedied
by the comprehensive model proposed in this paper.
References
[1] U. Euteneur, J.R. McIntosh, J. Cell Biol. 89 (1981) 338.
[2] C.L. Rieder, Int. Rev. Cytol. 79 (1982).
[3] L.G. Bergen, R. Kuriyama, G.G. Borisy, in vitro J. Cell Biol. 84 (1980) 151.
ARTICLE IN PRESS
326
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
R.B. Nicklas, D.F. Kubai, Chromosoma 92 (1985) 313.
G.J. Gorbsky, P.J. Sammak, G.G. Borisy, J. Cell Biol. 104 (1987) 9.
T.J. Mitchison, J. Cell Biol. 109 (1989) 637.
T.J. Mitchison, et al., Cell 45 (1986) 515.
R.B. Nicklas, J. Cell Biol. 97 (1983) 542.
R.B. Nicklas, J. Cell Biol. 109 (1989) 2245.
S.P. Alexander, C.L. Rieder, J. Cell Biol. 113 (1991) 805.
S. Inoue, E.D. Salmon, Mol. Biol. Cell 6 (1995) 1619.
T.J. Mitchison, E.D. Salmon, J. Cell Biol. 119 (1992) 569.
P. Maddox, Curr. Biol. 12 (2002) 1670.
D. Zhang, W. Chen, Dynamics of severed kinetochore fiber stubs in the cytoplasm of anaphase
grasshopper spermatocytes, presented at the 48th Annual Meeting of the American Society for Cell
Biology, San Francisco, December 13–17, 2003.
M.S. Cooper, Physiol. Chem. Phys. 11 (1979) 435.
L.F. Jaffe, R. Nuccitelli, Ann. Rev. Biophys. Bioeng. 6 (1977) 445.
G.B. Benedek, F.M.H. Villars, Physics: With Illustrative Examples From Medicine and Biology:
Electricity and Magnetism, Springer, New York, 2000, p. 403.
M.V. Satarić, J.A. Tuszyński, R.B. Z̆akula, Phys. Rev. E 48 (1993) 589.
J.A. Brown, J.A. Tuszyński, Phys. Rev. E 56 (1997) 5834.
N.A. Baker, et al., Proc. Natl. Acad. Sci. 98 (2001) 10037.
J.A. Tuszyński, J.A. Brown, P. Hawrylak, Phil. Trans. R. Soc. London A 356 (1998) 1897.
J.A. Tuszyński, private communication, March 2002.
D. Sackett, private communication, February 2002.
S.L. Wolfe, Molecular and Cellular Biology, second ed. Wadsworth, Belmont, CA, 1993, p. 1012.
Y.H. Song, E. Mandelkow, J. Cell Biol. 128 (1995) 81.
C.F. Weil, C.E. Oakley, B.R. Oakley, Mol. Cell Biol. 6 (1986) 2963.
B.R. Oakley, Trends Cell Biol. 2 (1992) 1.
R. Heald, et al., Nature 382 (1996) 420.
G. Schatten, et al., European J. Cell Biol. 36 (1985) 116.
M. Kirschner, J. Cell Biol. 86 (1980) 330.
M. De Brabander, et al., Cold Spring Harbor Quant. Biol. 45 (1982) 227.
W.J. Deery, B.R. Brinkley, J. Cell Biol. 96 (1983) 1631.
L.J. Gagliardi, J. Electrostat. 54 (2002) 219.
R.C. Weisenberg, Science 177 (1972) 1104.
G.G. Borisy, J.B. Olmsted, Science 177 (1972) 1196.
L.J. Gagliardi, Microscale electrostatics in mitosis and cytokinesis, presented at the Electrostatics Society
of America 2000 Conference, Brock University, St. Catharines, Ontario, Canada, June 18–21, 2000.
B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, J.D. Watson, Molecular Biology of the Cell,
third ed., Garland, New York, 1994.
C.L. Rieder, S.P. Alexander, J. Cell Biol. 110 (1990) 81.
A. Grancell, P.K. Sorger, Curr. Biol. 8 (1998) 382.
R.A. Steinhardt, M. Morisawa, in: R. Nuccitelli, D.W. Deamer (Eds.), Intracellular pH: Its
Measurement, Regulation, and Utilization in Cellular Functions, Alan R. Liss, New York, 1982,
pp. 361–374.
B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, J.D. Watson, Molecular Biology of the Cell,
third ed., Garland, New York, 1994, p. 1041.
L.J. Gagliardi, Phys. Rev. E 66 (2002) 011901.
D.E. Koshland, T.J. Mitchison, M.W. Kirschner, Nature 331 (1988) 499.
S.L. Wolfe, Molecular and Cellular Biology, second ed., Wadsworth, Belmont, CA, 1993.
B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, J.D. Watson, Molecular Biology of the Cell,
third ed., Garland, New York, 1994.
G.B. Benedek, F.M.H. Villars, Physics: With Illustrative Examples From Medicine and Biology:
Electricity and Magnetism, Springer, New York, 2000, p. 400.
ARTICLE IN PRESS
L.J. Gagliardi / Journal of Electrostatics 63 (2005) 309–327
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]
[75]
[76]
[77]
327
J.B. Hasted, D.M. Ritson, C.H. Collie, J. Chem. Phys. 16 (1948) 1.
J.O. Bockris, A.K.N. Reddy, Modern Electrochemistry, Plenum Press, New York, 1977.
O. Teschke, G. Ceotto, E.F. de Souza, Phys. Rev. E 64 (2001) 011605.
R.M. Pashley, J. Colloid Interface Sci. 83 (1981) 531.
W.F. Heinz, J.H. Hoh, Biophys. J. 76 (1999) 528.
J.A. Brown, J.A. Tuszyński, Phys. Rev. E 56 (1997) 5834.
B.M. Lewin, Gene Expression: Eucaryotic Chromosomes, vol. 2, Wiley, New York, 1978, pp. 34–36.
C. Amirand, et al., Biol. Cell 92 (2000) 409.
C.L. Rieder, et al., J. Cell Biol. 103 (1986) 581.
M. Dogterom, B. Yurke, Science 278 (1997) 856.
J.N. Israelachvili, Intermolecular and Surface Forces, second ed., Academic Press, London, 1991.
A.C. Cowley, et al., Biochemistry 17 (1978) 3163.
B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, J.D. Watson, Molecular Biology of the Cell, third
ed., Garland, New York, 1994, p. 926.
T.S. Hays, E.D. Salmon, J. Cell Biol. 110 (1990) 391.
P.K. Hepler, J. Cell Biol. 105 (1987) 2137.
P.K. Hepler, Membranes in the mitotic apparatus, in: J.S. Hyams, B.R. Brinkley (Eds.), Mitosis:
Molecules and Mechanisms, Academic Press, San Diego, 1989, pp. 241–271.
D.H. Zhang, D.A. Callaham, P.K. Hepler, J. Cell Biol. 111 (1990) 171.
E.D. Salmon, R.R. Segall, J. Cell Biol. 86 (1980) 355.
D.P. Kiehart, J. Cell Biol. 88 (1981) 604.
W.Z. Cande, Physiology of chromosome movement in lysed cell models, in: H.G. Schweiger (Ed.),
International Cell Biology, Springer, Berlin, 1981, pp. 382–391.
J.B. Olmsted, G.G. Borisy, Biochemistry 14 (1975) 2996.
R.B. Nicklas, Chromosome movement: current models and experiments on living cells, in: S. Inoue,
R.E. Stephens (Eds.), Molecules and Cell Movement, Raven Press, New York, 1975, pp. 97–117.
R.B. Nicklas, Chromosomes and kinetochores do more in mitosis than previously thought, in:
J.P. Gustafson, R. Appels, R.J. Kaufman (Eds.), Chromosome Structure and Function, Plenum, New
York, 1987, pp. 53–74.
E.D. Salmon, Ann. N.Y. Acad. Sci. 253 (1975) 383.
E.D. Salmon, Microtubule dynamics and chromosome movement, in: J.S. Hyams, B.R. Brinkley
(Eds.), Mitosis: Molecules and Mechanisms, Academic Press, San Diego, 1989, pp. 119–181.
S.L. Wolfe, Molecular and Cellular Biology, second ed., Wadsworth, Belmont, CA, 1993, p. 425.
S.L. Wolfe, Molecular and Cellular Biology, second ed., Wadsworth, Belmont, CA, 1993, p. 1025.
D. Zhang, R.B. Nicklas, Nature 382 (1996) 466.
M. Coue, V.A. Lombillo, J.R. McIntosh, in vitro J. Cell Biol. 112 (1991) 1165.
A.P. Joglekar, Alan J. Hunt, Biophys. J. 83 (2002) 42.
R.B. Nicklas, Ann. Rev. Biophys. Biophys. Chem. 17 (1988) 431.