§
¶
* Centre National de la Recherche Scientifique (CNRS), Laboratoire de Spectrométrie Physique, 38402 Saint-Martin-d'Hères
Cedex, France; Center for Studies in Physics and Biology, The Rockefeller University, New York 10021-6399; § Department of
Physics, The University of Illinois at Chicago, Chicago, Illinois 60607-7059;
Université Louis Pasteur, CNRS, Institut de
Physique, 67000 Strasbourg, France; and ¶ NEC Research Institute, Princeton, New Jersey 08540
The structure of mitotic chromosomes in cultured newt lung cells was investigated by a quantitative study of their deformability, using micropipettes. Metaphase chromosomes are highly extensible objects that return to their native shape after being stretched up to 10 times their normal length. Larger deformations of 10 to 100 times irreversibly and progressively transform the chromosomes into a "thin filament," parts of which display a helical organization. Chromosomes break for elongations of the order of 100 times, at which time the applied force is around 100 nanonewtons. We have also observed that as mitosis proceeds from nuclear envelope breakdown to metaphase, the native chromosomes progressively become more flexible. (The elastic Young modulus drops from 5,000 ± 1,000 to 1,000 ± 200 Pa.) These observations and measurements are in agreement with a helix-hierarchy model of chromosome structure. Knowing the Young modulus allows us to estimate that the force exerted by the spindle on a newt chromosome at anaphase is roughly one nanonewton.
MITOSIS involves gross physical reorganization of
chromosomes; the duplicated chromatids are
condensed, resolved, and finally segregated.
These processes can be expected to change the material properties of chromosomes, notably their elasticity. Elasticity indicates the nature and strength of the interactions
holding materials together, and thus can be used to probe
chromosome structure. Given the poor state of understanding of chromosome structure, it is therefore remarkable how little this subject has been studied. In a pioneering work, Nicklas (1983) An often-discussed model is one in which the "thick"
metaphase chromosome is composed of a "thin filament"
of diameter 200-300 nm (Sedat and Manuelidis, 1978 The proposal for a helical structure of metaphase chromosomes is old. Observations of "spiral chromatonema"
during meiotic metaphase I date to at least 1926. Ohnuki
(1968) In this paper, we report a simple mechanical study of mitotic chromosomes in living cultured newt lung cells using
micropipette aspiration and manipulation. First, we find
that chromosomes display remarkable elasticity, returning
to their initial shape after being extended by up to 10 times. For larger deformations, the chromosome no longer
returns to its initial length. Instead, the thick native chromosome is progressively converted into a thin filament 15 times the length of the original chromosome. This thin filament is itself elastic; it can be stretched six times before
breaking. After the filament is released, it takes on an irregular but unmistakably helical form. Furthermore, by
measuring force versus deformation, we have determined
the Young elastic modulus of the metaphase chromosome,
the force at which the metaphase chromosome begins to
be converted to thin fiber, and the force required to break
the thin fiber. These measurements reveal the strength of
interactions that stabilize the different levels of structure.
Finally, we have observed that the Young modulus drops by about fivefold during the interval from nuclear envelope breakdown to metaphase.
Our results lead to a simple unifying picture of chromosome elasticity and structure: Our conclusion is that
metaphase chromosomes are composed of an underlying
thin filament. The large range over which the metaphase
chromosome is elastic, the scale of its Young modulus, and
the helical structure of the filament all argue in favor of its
helical folding. By the same line of reasoning, the fact that
the thin filament is elastic over a large range of extensions suggests that it also has a folded or helical structure. We
also show how the force exerted by the mitotic spindle on
a chromosome and the resistance of the cytoplasm to chromosome movement can be deduced from the Young modulus measurement and chromosome shape at anaphase.
Tissue Culture and Solutions
Newt lung cultures were prepared in a Rose chamber following standard
procedures (Rieder and Hard, 1990 Microscopy and Micromanipulation
All microscopy was done with an inverted microscope (Carl Zeiss, Inc.,
Thornwood, NY) using a 60×, 1.4 NA differential interference contrast
objective and an XY motorized stage. Glass micropipettes were prepared
using a puller (model 87; Sutter Instrument Co., Novato, CA) and were
then cut using a forge to obtain sharp edges. A hydraulic micromanipulator was used to move the pipettes. All mechanical experiments were carried out using a combination of stage and manipulator movement. When
strong adhesion of the chromosome to the micropipette was required, the
micropipette was filled with culture buffer. For elasticity measurements
using aspiration, 10 g/liter BSA (A3156; Sigma Chemical Co.) was added
to the filling solution to prevent sticking of the chromosome to the glass
micropipette.
Stretching Single Chromosomes Using a Micropipette
The micropipette diameter (2 µm) must be chosen to be slightly less than
the total chromosome diameter (3 µm) to ensure strong chromosome-
pipette contact. After insertion of the pipette into a mitotic cell while positive pressure difference (relative to atmospheric pressure) of 100 Pa was
maintained, the first 2-4 µm of a chromosome was aspirated using 100 Pa
of suction. After 30 s, the chromosome was stuck to the glass, and further
aspiration was not necessary. The pipette could then be moved from the
cell to stretch the chromosome, while it was anchored at its other end to
the mass of other chromosomes.
The bending of the pipette, when translated perpendicular to its axis,
measured the force it exerted on the chromosome. However, since one
uses a stiff pipette to penetrate the cell, this measurement is not very sensitive. The elasticity of the pipette is 10 Measurements of Young Modulus
Chromosome elasticity for small deformations was measured using aspiration into the micropipette. Contrary to the stretching experiment described above, the pipette was this time fixed and maintained inside the
cell, and pressure difference inside was used to extend the chromosome.
As before, a pipette is introduced inside the mitotic cell, and weak suction
is used to grab a chromosome. At some point as the chromosome slides in,
a seal is made at the end of the pipette. From this point on, the only part of
the chromosome that can be deformed is the portion inside the micropipette. The length of that portion is then measured as a function of pressure. Note that to avoid adhesion of the chromosome inside the pipette, the medium filling the pipette contained BSA.
The length measurement was made by digitizing images on a computer
and has a precision of 5%. The uncertainty on the pressure measurement
is also in the range of 5%, mainly because of drift problems. Finally, possible sticking of chromosomes to the pipette glass added another uncertainty. After the release of pressure, if the measured length was different
from the original one by more than 10%, the measurement was discarded.
Elasticity Definitions and Relations
Strain.
The strain used in this paper is longitudinal deformation and is defined as Young Modulus.
The Young modulus Y is the proportionality factor between the force per unit area (or pressure) exerted and the resulting dimensionless deformation: P = Y Poisson Ratio.
A longitudinal deformation of a solid is always accompanied by a change (almost always a decrease) in its thickness. The Poisson ratio Bending Modulus.
The bending of a thin rod is characterized by one
elastic coefficient, called the bending modulus B (Love, 1944 Polymer Definitions and Relations
Persistence Length.
A thin rod can be bent by thermal fluctuations. The
persistence length of a thin rod is defined as Lp = B/kBT, where kB = 1.4 × 10 End-to-End Size of Thermally Bent Rod.
Consider a rod of length L Elastic Response of a Single Polymer.
Consider a single polymer subjected to a force F between its ends. Knowing that the spontaneous fluctuation of the end-to-end distance is Le indicates that if we force the ends of
the chain to be a distance X apart, there will be a free energy cost of ~kBT(X/Le)2 (assuming that X is small compared to L). This free energy
cost is due to there being fewer configurations available to the chain (its
entropy is reduced) if its ends are pulled apart. The derivative of the free
energy with X is just the force required to obtain extension X, or F = kB
TX/Le2. A single flexible polymer thus displays linear entropic elasticity that is strictly of thermal origin. Since the unperturbed overall size of the
polymer is Le, we define its strain as Polymer Gel and Young's Modulus.
A polymer gel is a network of cross-linked polymers, swollen in a solvent that has affinity for the chains.
(Think of a three-dimensional jungle gym with flexible links.) From the
point of view of this paper, it is immaterial whether the gel is composed of
many polymers cross-linked together (e.g., an agarose gel as prepared for
electrophoresis), or a single tremendously long fiber cross-linked to itself
many times (a conceivable model for a condensed chromosome). The relevant polymer length is the amount of chain between successive
crosslinks. If this is more than a persistence length, each chain segment in
the gel will display entropic elasticity, and therefore the network as a
whole will be an elastic medium.
measured that the force applied to grasshopper chromosomes during anaphase was 700 piconewtons, from which he inferred the chromosome stiffness. More recently, Claussen et al. (1994)
stretched human
metaphase chromosomes spread on a cover glass. They
found that after stretching of up to 10 times, the chromosomes returned to their original shape. However, these
studies did not address the question of chromosome architecture.
;
Manuelidis, 1990
). In fact, Bak et al. (1977)
reported that
as isolated human metaphase chromosomes disintegrate,
they can change into a thin filament of diameter 400 nm,
five times the original chromosome length. They suggested that metaphase chromosomes were formed by helical
wrapping of this thin fiber. On the basis of electron microscopy, they further proposed that the thin fiber had a
helical structure.
established that hypotonic treatment stabilized spiral structure in human mitotic metaphase chromosomes.
Boy de la Tour and Laemmli (1988)
observed that fluorescent anti-topoisomerase II was helically organized when
bound to histone H1-depleted chromosomes. Recent
work by Hirano and Mitchison (1994)
revealed that a protein heterodimer required for chromosome condensation
in vitro (XCAP-C/E) was localized along a helical track
along the metaphase-like chromatids. These and other
studies (Belmont et al., 1987
, 1989
; Saitoh and Laemmli, 1993
) suggest a chromosome with an internal structure
made of a coiled or folded fiber. However, the spiral structures observed may be the result of chemical treatments of
the chromosomes (Cook, 1995
).
MATERIALS AND METHODS
). Newts (Notophthalamus viridescens,
Connecticut Valley L500) were killed by immersion for 20 min in 1 mg/ml
Tricaine (A-5040; Sigma Chemical Co., St. Louis, MO) in distilled water.
The lungs were immediately dissected and cut into 1-mm2 pieces under
sterile conditions and then soaked for 24 h in culture buffer (50% L-15,
L-5520 [Sigma Chemical Co.]; 42% distilled water; 8% FBS, F-2442
[Sigma Chemcial Co.]; 40 U/ml Pen-strep, 15075-013 [GIBCO BRL, Gaithersburg, MD]; 1 µg/ml Fungizone, 0437-60 [Bristol-Myers Squibb, New
York]). The lung fragments were then lightly squashed between the lower
glass of the Rose chamber and a dialysis filter (SpectraPor 12-14 kD, 08-667E [Fisher Scientific, Pittsburgh, PA]); the Rose chamber was filled
with culture buffer. A monolayer of epithelial cells formed after 3 or 4 d.
The dialysis filter was removed after 7 d, leaving the monolayer in the bottom of a shallow dish suitable for micromanipulation. Mitotic activity was
usually most intense over the next 48 h. Almost all experiments were done
before the cultures were 12 d old.
9 ± 30% Newton/µm; this was determined by a two-step calibration procedure. First, the elasticity of a thin
wire was measured using a balance. This wire was then used, under the microscope, to bend the micropipette.
=
L/L, where L is the original length of the chromosome and
L is the added length due to the application of tension. (Note that the
precise definition of the strain includes a term proportional to (
L)2,
which may be ignored for measurements of the Young modulus as long as
data for
L/L < 1 are used; see Landau and Lifshitz [1986].)
. Typical hard materials (e.g., metals and
glass) have a Young modulus of the order of 1010 Pa. (1 Pascal = 1 Newton/m2 is the unit of pressure). For synthetic polymer gels, Y is in the
range of 104-105 Pa (Horkay et al., 1989
).
relates relative change in the thickness,
thick, to relative change in
length:
thick =
. A typical value is
= 0.3.
). The elastic energy E per length of rod is proportional to the square of its curvature
:
E/L = B
2/2. Curvature is just the inverse of the radius of the circle that
describes the bend. If the curvature varies with arc length s along the rod,
the total energy is just given by the integral E = (B/2)
2ds. Thus, the
minimum-energy shape is straight (
= 0). The bending modulus in terms
of the Young modulus of the rod and the rod radius R is B = (
/4)YR4
(Love, 1944
). B has dimensions of energy times length.
23 J/K is Boltzmann's constant, and T is absolute temperature.
To see what this means, consider that a typical thermal fluctuation has energy kBT; thus, given the bending energy above, the length of rod L that
can be bent through an angle of one radian by thermal fluctuation satisfies
kBT
B/L. So the persistence length is the length of rod that is typically
deflected by about a radian by thermal fluctuations. Roughly speaking, Lp
is the smallest length scale over which bending fluctuations can be observed; a rod is essentially straight on scales smaller than its Lp. For double-stranded DNA, Lp
0.05 µm; for polymerized actin, Lp
0.5 µm; for
microtubules Lp
5 mm.
Lp. Thermal fluctuations produce random bends along the rod. Thus, the
rod becomes a "random walk" of L/Lp steps, each of length Lp. The average distance between ends of a random walk grows as the square root of
the number of steps in the walk. Thus the mean distance between the ends
of the rod will be Le
Lp
. (The exact result for the average of Le2
is 2LLp; note that the average end-to-end displacement in any direction is
zero since the polymer may reorient in any direction.) Applied to a polymer (e.g., a DNA longer than a few kilobuses), Le corresponds to the size
of a typical "random coil" conformation (de Gennes, 1988
).
= X/Le; giving us F = (kBT/Le)
.
= L/Le =
(the ratio of total
length of a typical segment to its random walk size).
Effect of Entanglements on a Gel.
Two chain segments in a gel may be
entangled together, meaning that they may not be individually free to explore all random walk conformations thanks to unremovable links or
knots. This is a poorly understood area of polymer statistical mechanics, but entanglements may be very roughly taken into account by considering each entanglement to contribute an additional cross-link to the network.
If we suppose that there are n entanglements per chain segment in our gel,
a very rough estimate of the gel modulus is Y kBT
(1 + n)/(
LR2). This
line of argument indicates that increased entanglement boosts the modulus of a gel (stiffens it); resolution of entanglements in a gel will reduce its
modulus.
The large size of newt cells (~100 µm) and of their mitotic chromosomes (~20 µm long) make them ideal for micromanipulation. In the following, the term "chromosome" always refers to a pair of sister chromatids; all stretching experiments were performed on the two sisters together.
Two types of experiments are described below. The first was used to study large deformations and forces by stretching the chromosomes between the cell and a micropipette. The micropipette, when moved perpendicular to the stretching direction, was observed to bend. Calibration of this bending allowed simultaneous measurement of force and strain. The second was used to measure small deformation and forces by stretching the chromosome under suction inside the micropipette, the tip of which was corked by the chromosome. The elastic (Young) modulus was thus deduced.
Large Deformation of Chromosomes and Thin Fiber Formation
Large deformations were studied by exploiting the very
strong adhesion of the chromosome to the pipette at one
end and to the mass of other chromosomes at the other
end. The chromosome is suspended between the pipette
and the cell, in the culture buffer. The pipette is then
moved to deform the chromosome by a given amount and
brought back (Fig. 1).
The resulting shape of the chromosome depends on this deformation. We define the length of the chromosome after each cycle as the minimal length over which the chromosome is straight. (If the pipette is pushed closer to the cell, one observes bends or helicies along the chromosome.) We first observe that the final shape of the chromosome depends only on the imposed deformation. Exposure to the buffer for a long time (1 h) or restretching the chromosome by the same amount does not change the chromosome morphology.
The different chromosome morphologies obtained after different deformations can be categorized as follows:
Elastic Regime. For < 10, the chromosome is highly
elastic, relaxing to its initial "native" length and appearance (Fig. 2 a). Such a large range of elasticity is rare (most
materials are elastic only for
< 0.01) but is characteristic
of polymer gels (Horkay et al., 1989
) or of extensible elastic objects, such as a helical spring (Love, 1944
). The elastic Poisson coefficient can be evaluated directly from the
images (the diameter of the chromosome decreases as
increases) giving 0.20 ± 0.05. This regime of perfect elasticity indicates that there is a well-defined elastic modulus for
a chromosome.
Plastic Regime.
When is larger than 10, the relaxed
chromosome is longer than its original length; it no longer
returns to its native state (Fig. 2, b-d). For example, a
chromosome with native length of 20 µm, after stretching
to 300 µm (
= 14), relaxes to a length of about 30 µm.
This increase in length is inhomogeneous: Part of the
chromosome remains thick while part of it is thinned. The
border between these two regions is usually at the kinetochore. In contrast to the elastic regime, the length and diameter after plastic deformation vary with . The thinned
chromosome diameter decreases from 3 to ~0.8 µm as
is
increased from 10 to 25-35 (depending on the rate of deformation).
Fig. 3 shows an interesting aspect of the plastic behavior
for
20. When the micropipette tip is brought back near
the cell so as to slightly compress the chromosome, transient undulations develop along parts of the chromosome.
Those undulating regions relax into a region of thicker
straight chromosome after 1 to 2 s. If the pipette is pushed
closer, stable helices appear (Fig. 4). Some but not all of the
plastic deformation can be reabsorbed by the chromosome.
Thin Filament.
For larger than ~ 30, the relaxed chromosome converts to a thin filament that is 15 times the
length of the original thick chromosome. Further elongation proceeds by elastic deformation of this thin filament.
The conversion of the chromosome to thin filament is not only irreversible, but it also depends on the rate of deformation. Most of our cycles of deformation-release are done at speeds of less than 20 µm/s. At higher speed, the transition is abrupt, with part of the chromosome suddenly relaxing while the remainder extends to thin filament.
Finally, for
100, the filament breaks. (The precise
value of
at breaking depends on rate of elongation, ranging from 60 to 100; Fig. 5.) The deflection of the micropipette at the breaking point indicates a force between 90 to
150 nanonewtons. It is not easy to deduce the Young modulus of the thin filament from the force measurement. The
thickness d of the thin filament when highly stretched is at
the resolution limit of the microscope, and its measurement is critical for the Young modulus [Ythin
F/(
d2)].
Our best estimate is Ythin = 1-5 × 105 Pa.
When the tension on the thin filament is released, following either partial or total conversion of the chromosome to this form, portions of it take on a helical shape.
Each helical turn involves 5 to 10 µm of relaxed thin filament. Both left- and right-handed curls were observed.
Fig. 6 a shows a thin filament after its breakage. If we then
perturb it with a flow, the helical filament behaves like a
spiral spring (Fig. 6, b-d).
Young Modulus Measurements
We now return to the elasticity regime where the deformations are relatively small ( < 2). Using aspiration of chromosomes, we measured the deformation as a function of
pressure inside the pipette (Fig. 7). Results for 10 metaphase chromosomes are shown in Fig. 8; at metaphase Ymeta = 1,000 ± 200 Pa. As mentioned above, the
Poisson ratio of metaphase chromosomes was measured to
be 0.20 ± 0.05. Thus, the metaphase chromosome has a
Young modulus about two orders of magnitude less than
that of the thin filament.
The same type of experiment was also carried out for cells at prometaphase, immediately following (<10 min) nuclear envelope breakdown (NEB).1 Data from three NEB chromosomes are shown in Fig. 8; YNEB = 5,000 ± 1,000 Pa. Between NEB and metaphase, the chromosome Young modulus ranges from 5,000 to 1,000 Pa, but it is difficult to follow this variation since the duration of mitosis varies from cell to cell.
Let us summarize our observations. Metaphase chromosomes have a Young modulus of 1,000 Pa and are elastically deformable by up to 10 times. Under higher stress, they are transformed into a thin fiber that is 15 times the native chromosome in length. The thin filament is itself elastically deformable by at least a factor of six, and its Young modulus is in the range of 1-5 × 105 Pa.
In the following, we present a model for the structure of thick chromosome and thin filament consistent with our observations. We also discuss the variation of the Young modulus during the cell cycle. We finally show that the shape of the anaphase chromosome allows us to deduce the resistance of the cytoplasm to the motion once the Young modulus is known.
Chromosome Structure
Remarkably, the metaphase chromosome returns to its original size after being lengthened by a factor of 10. There are two classes of materials that exhibit elasticity over such a large range of extension.
Chromatin Gel.The first possibility for the structure of
the metaphase chromosome is that of a polymer gel, i.e.,
where the length is stored in random thermal fluctuations
of a cross-linked network of flexible chains. Since each
chromatid is composed of a single contiguous chromatin
fiber, chromatin is known to be flexible (recent experiments indicate a persistence length of roughly Lp 50 nm
[Castro, 1994
]), and it is known that the chromatin is periodically constrained (i.e., to form the "loops" discussed by
many authors [Paulson and Laemmli, 1977
, Gasser et al.,
1986
; Cook et al., 1990
; Saitoh and Laemmli, 1993
]), it behooves us to consider whether chromosome elasticity can
be plausibly attributed to elasticity of a chromatin gel,
rather than to the rigidity of some internal "scaffold." Stretching a gel forces the chains to extend, reducing their
entropy, thus requiring work to be done. The maximum
extension possible for a polymer gel is roughly the ratio of
the length of the chains between cross-links to the random
walk size of the chains (see Materials and Methods).
Our measurement of the Young modulus (1,000 Pa)
rules out the possibility that the initial elasticity is that of a
chromatin gel. To see this, consider that the observed extensibility by 10 times requires that the length of fiber between cross-links be L = 200Lp 10 µm (Materials and
Methods). Such a gel will have a Young modulus of
roughly
kBT/(
LR2), where
is the fraction of volume
occupied by the polymer. (The remaining fraction 1
is
supposed to be occupied by "solvent," or more precisely
the fraction of cytosol that can freely enter and leave the
spaces between chromatin fibers.) Plugging in the required
L = 10 µm, setting
= 1 (this maximizes the modulus estimate) and using kBT = 4 × 10
21 J and R = 15 nm, we
find Ygel
0.6 Pa. The large L required by the observed
extensibility necessitates a tiny elastic modulus. The observed elasticity is not plausibly due to that of the constituent chromatin and therefore must be due to the springlike
elasticity of a coiled or folded internal fiber.
The second possibility is that of a fiber with permanent bends along its length. The permanent bends store a large amount of length that can be liberated by tension. The simplest example of such a structure is a regular helical spring, but one could envision a mixture of left- and right-handed helical turns, or a zigzag structure.
This case is compatible with the idea of a folded or
coiled chromosome scaffold (Paulson and Laemmli, 1977;
Gasser et al., 1986
; Boy de la Tour and Laemmli, 1988
; Saitoh and Laemmli, 1993
; Hirano and Mitchison, 1994
) and
is consistent with several of our observations. First, we observe a sharp transition from thick chromosome to thin fiber; the transition is sharpened by rapid extension. Second, after relaxation, the thin fiber is mechanically stiff
and has a permanent undulating shape. Third, a geometrical argument can be advanced to support the picture of a
thin filament that is bent or wrapped into the metaphase
chromosome, as follows. Suppose the thin filament of diameter d is wrapped into N successive solenoidal turns of
diameter D that form the thick chromosome. The length
of the thick chromosome is then L = Nd, while the length of the filament l = N
D, so l/L =
D/d. This is consistent
with our observed elongation l/L = 15 and diameter reduction D/d = 4, when the thick chromosome is converted
to thin filament.
Thin Filament Structure
For > 10, metaphase chromosomes were permanently
lengthened. Supposing that the metaphase chromosome is
composed of a folded or coiled filament, this plasticity and
the extraction of the thin filament is to be interpreted as
the progressive failure of structural elements that define
its intrinsic shape. Eventually, the entire chromosome is
converted into a thin filament that displays elasticity over
a sixfold range of extensions and a Young modulus of
Ythin = 1-5 × 105 Pa.
We are thus faced with a second case of a large range of
elasticity. Since Ythin 102 Ymeta, we can reuse the argument
of the previous section to again argue against the thin filament being a polymer gel. The thin filament itself should
therefore be constituted of a "basic fiber" that is wrapped
or folded so as to again provide a reservoir of length.
In the experiments described above, the two sister chromatids were always deformed together, while our discussion of their unfolding considers them as independent. In
the initial elastic regime, the main question is of the contribution of interchromatid entanglement to the Young modulus. In the next subsection, we will see that this is at most
a small part of the measured Young modulus. During the
plastic deformation, the permanent and irreversible bends
that form may indicate that the original compactions of the two underlying sister thin filaments are correlated. An
observation due to Boy de la Tour and Laemmli (1988) of
opposite helical handedness of sister chromatids indicates
the same conclusion. On the other hand, the helical shape
of the thin filament may reflect that the two chromatids
have been forced to uncoil during their extension (possibly
in conjunction with the constraint that they stay alongside one another) and when released have refolded in some
way unrelated to their native folding. This direction of argument again suggests that the helical shape of the thin fiber comes from the necessity that the chromatids be uncoiled to be lengthened. Further experiments are needed
to clarify this issue. It would be very interesting to study
single chromatids assembled from Xenopus egg extracts, for which there is some evidence for a helical "scaffold"
(Hirano and Mitchison, 1994
).
Decrease of Young Modulus during Mitosis
We observed that the chromosome Young modulus decreased from 5,000 Pa at NEB to 1,000 Pa at metaphase. This gradual weakening of the chromosome can simply be explained by supposing a structural change in the internal scaffold. Just to raise one possibility, it is known than 70% of the topoisomerase II present at NEB is progressively removed before anaphase (Sweldow et al., 1993). This removal may reduce the chromosome stiffness.
A second possibility is that resolution of entanglements of
the sister chromatids during NEB to metaphase might
weaken the chromosome elasticity. The maximum change in
Y that can be possibly obtained is (L/Lp)kBT/(LR2) = kBT/
(
LpR2). This supposes maximal entanglement (one per persistence length of chromatin, certainly a large overestimate)
at NEB and total disentanglement at metaphase. This extreme estimate only accounts for an ~100 Pa change in Y. We conclude that the observed change in Y is due to a structural change of the bent scaffold.
Young Modulus and the Shape of Anaphase Chromosomes
This paper was mainly dedicated to the elasticity of the chromosomes as a probe of their internal structure, but our results also allow us to solve an apparent paradox concerning the forces exerted during anaphase.
Nicklas (1983) measured that the spindle exerts a force
of ~700 piconewtons on a chromosome at anaphase. This
level of force was puzzling because previous calculations
(Nicklas, 1965
; Taylor, 1965
) had shown that even assuming a cytoplasm viscosity as large as 1 Poise, the force
needed to move a chromosome at 0.3 µm/min would be
only about 0.1 piconewton. The spindle is thus applying a
force 104 times larger than might be expected. To explain
this paradox, Nicklas considered a few mechanisms, including a feedback system to limit chromosome velocity.
The basic physical facts indicate that these large forces
are really required to move chromosomes. The shape of
the chromosome, its Young modulus, and the force exerted by the cytoplasm are closely related. An elastic cylinder pulled by its center and moving through any medium is
submitted to a drag force that tends to bend it. At constant
speed, the force exerted by the medium equals the force
that pulls the object. The bigger the force, the more severe
the bending. In the case of high drag force, a cylindrical
object takes a U shape, which is what is observed for chromosomes pulled at their kinetochore by the spindle. The distance X between the arms of the U, the exerted force
F0, the Young modulus Y, and the radius R of the chromosome are related by F0 = 6YR4/X2, which can be obtained using classical elasticity theory (see Appendix).
Note that this result can be estimated by dimensional
analysis: The bending modulus B YR4 and the distance
X are the only quantities in the problem, and the only way
to form a force from them is the ratio B/X2. To obtain this
relation, no assumption about the effective viscosity of the
cytoplasm, or even whether or not it has Newtonian behavior, is required; only the fact that there is a drag (friction) force in balance with the driving force is invoked.
For newt chromosomes, X 2 µm, R
0.7 µm, and Y
= 1,000 Pa, so the force exerted by the spindle, and therefore the total resistance presented by the surrounding medium, is ~1,000 piconewtons. This means that the resistance of the cytoplasm to the movement of the chromosome
is orders of magnitude higher than what would be encountered in a 1-Poise viscous fluid, as supposed by Nicklas and
Taylor. This is not surprising if one considers that a large object like a chromosome has to move by deformation of a
network of cytoplasmic filaments. The drag force on a
large object could consequently be much larger than the
drag inferred from motion of small molecules.
It should be remarked that the bending modulus of the
chromosome has not been directly measured, but rather
deduced from the Young modulus. The relation B = YR4/4 holds for uniform solids because bending causes
the same kind of local deformations as stretching. (Note
that on the outer edge of a bend, an object is stretched,
while on the inner edge, it is compressed.) However, one
can imagine a chromosome with a very flexible yet inextensible filament running down its center, or a chromosome that is predominantly liquid. Our observations of elastic and plastic behavior make these possibilities seem
unlikely, but of course they must be checked experimentally. We are planning a direct measurement of the bending modulus to definitively settle this point.
Received for publication 27 January 1997 and in revised form 28 April 1997.
1. Abbreviation used in this paper: NEB, nuclear envelope breakdown.Computation of the Resistive Force during Anaphase
The relation F0 = 6YR4/X2 introduced in the last section
relates drag forces to the U shape of the chromosome during anaphase movement. In this appendix, three routes to
this equation are described, in order of increasing detail.
We consider an elastic rod of length 2L , pulled at its central point (centromere) by a (spindle) force F0. The rod experiences resistive (drag) forces K distributed along its
length (K is a force per length of rod), which when
summed balance F0 (Fig. A1). Under the action of these forces, the rod bends to form a U shape.
Our first estimate will be the simplest and least satisfying. We consider the situation where the force is large
enough that the rod is tightly bent near its center, with its
nearly parallel arms lagging behind. The drag forces will
be spread nearly evenly along the arms, so we can estimate
that K F0/L in order for total drag force to balance the
spindle force. Arguing that the deformation and therefore
the elastic energy is concentrated at the hairpin bend and
the radius of the bend is approximately X, we estimate the
elastic energy to be E
B/X. The length of rod in the
bend is ~X and the curvature is ~1/X (see Materials and Methods). Since the bend is made by forcing the ends of
the bent region together by a distance ~X, the (drag)
force that must be applied to make such a bend should be
just E/X, or B/X2. Thus, we arrive at a relation between
the total drag force ~KL, which is in balance with the
spindle force F0, and X: F0
KL
YR4/X2.
The first estimate is essentially dimensional analysis and
does not address the complications that X is not precisely
given by the radius of curvature of the hairpin bend and
that both drag forces and the deformation are not uniformily spread along the entire length of chromosome.
The complete equations of elastic equilibrium for the rod
are (Landau and Lifshitz, 1986)
![]() |
(A1) |
![]() |
(A2) |
![]() |
(A3) |
![]() |
(A4) |
where s is the contour length along the rod (s = 0 corresponds to the middle of the rod, or centromere), is the
angle between the y-axis and the tangent to the rod at
point s, F(s) is the net external force acting on the rod
from point s to its free end L, B is the bending modulus
(for a cylinder of radius R, B =
YR4/4; see Materials and
Methods), K is the density of external forces acting on the
rod at point s, and
(see Fig. A1). Note that as the applied forces are symmetric, one needs only to solve these equations for one half of the rod (i.e., for 0 < s < L), with the equivalent boundary condition that the rod be "clamped" at s = 0.
Before going to the complicated case in which K is spread along the entire rod, we solve these equations for a simpler case which the drag force is concentrated at the lagging ends. Suppose that instead of resistive forces distributed along the rod, we have drag forces of F0/2 (note that the total drag must balance the spindle force F0) applied to the ends (s = ±L). This simple case is the classic problem of a rod clamped to a table and bent by a weight hung from its free end (Landau and Lifshitz, 1965; Sec. 19, problems 1 and 2), which has exact solution
![]() |
(A5) |
where (L) is the angle between the tangent to the rod at
the extremity (S = L). Replacing B and computing the integral,
![]() |
(A6) |
For strong pulling, the rod ends will be forced to be along
the y-axis, giving cos [(L)]
1. (The precise value of cos
[
(L)] can of course be computed from the exact solution.)
So, the magnitude of the resistive force can thus be deduced from the bending of the rod. As before, we find that
F0
YR4/X2.
Finally, consider the case of a chromosome pulled during anaphase toward the pole. As the migration speed is constant, the pulling force and the drag forces balance, and the chromosome bends. The amplitude of these forces can be inferred from the bending since we know the elasticity constant from other measurements. The problem is to choose a reasonable model for the drag forces. As the chromosome has to pass through a network of filaments, the frictional forces applied to a small segment of it plausibly depend on its projected length in the x direction, or
![]() |
(A7) |
This accounts for the effect that a segment of chromosome will experience a bigger force if it is parallel to the x-axis (with its tangent perpendicular to velocity) than if it is parallel to the y-axis (tangent parallel to velocity). The coefficient g depends on the deformability of the filament network, on the speed of the chromosome, and other shape-independent parameters.
Using this expression for the drag force density, integrating the differential equations A1 and A2 and using boundary conditions in Eqs. A3 and A4, we find
![]() |
(A8) |
![]() |
(A9) |
The distance X between the arms is
![]() |
(A10) |
which again reduces to F0 YR4/X2 for strong pulling (cos
[
(L)]
1).
The value of the drag force is not very sensitive to the
choice of the form of the force density function K(s); the
extreme choice of all drag force concentrated at the telomeres gave the same magnitude of spindle force and bending. It should be noted that Nicklas (1965) has observed
that the velocity of chromosomes during anaphase is independent of their length. This is in good agreement with our
model: The value of the spindle force required to move the
chromosomes is independent of the chromosome's length L, as long as L
2(B/F0)1/2
1 µm for newt chromosomes.
In summary, the force which resists chromosome movement during anaphase is also responsible for chromosome bending and can be deduced by measuring their deformation.
This work was supported by National Science Foundation grant PHY9408995 and the Mather Foundation. J. Marko also acknowledges support of the Meyer Foundation at Rockefeller University (New York).
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