Elasticity Measurements Show the Existence of Thin Rigid Cores Inside Mitotic Chromosomes

Mitotic Xenopus egg extracts were isolated essentially as described by Hirano and Mitchison (1993). In brief, the dejellied eggs were crushed in EB (80 mM β-glycerophosphate, pH 7.3, 15 mM MgCl₂, 20 mM EGTA, and 1 mM DTT) supplemented with 10 μg/ml leupeptin and pepstatin, by centrifugation for 20 min at 20,000 g in an SW41 rotor (Beckman Instruments). The cytoplasmic fraction was collected and further fractionated by ultracentrifugation at 250,000 g for 2 h at 4°C by using the TLS-55 rotor (Beckman Instruments). The lipid layer was sucked very carefully under vacuum, the soluble fraction was removed and recentrifuged for an additional 30 min at 250,000 g in order to get rid of the residual membranes. The extract was aliquoted in 25-μl fractions and immediately frozen and stored at −80°C. Demembranated Xenopus sperm nuclei were prepared following the procedure of Smythe and Newport (1991) and stored at −80°C. Chromosome were assembled essentially as described by Hirano and Mitchison (1993).

Observations were made by an inverted microscope, using 60× phase-contrast objectives (NA = 0.6 or 1.3). Images were acquired through a CCD camera, and recorded on a VCR. When digitized, the resolution of images is 0.21 or 0.4 μm/pixel, depending on the objective used. Two micromanipulators (Sutter MPC-100 and WPI DC30001) were used to direct micropipettes and grab a chromosome by its ends.

Micropipettes were formed to a final inner diameter of 1 μm using a puller (Sutter P-97). To have cylindrical pipette tips, their last 100-μm length was cut by a laboratory-made forge and fire polished.

The deflection of micropipette tips was used to measure forces. A pipette acts like a normal spring, and the force F it applies is proportional to its tip deflection x (for small deflection compared with its length): F = kx, where k is the spring constant. The calibration of pipettes was done in two steps. First, a spring of known rigidity (0.84 × 10⁻² N/m) was used under the microscope to calibrate an intermediate pipette (rigidity constant 2.4 × 10⁻³ N/m) which in turn calibrates micropipettes used for elasticity measurements (2–3 × 10⁻⁴ N/m). For each pipette, the linearity of the spring was checked over 100 μm deflection, and the error was estimated to be <10%. Deflection of the micropipettes was measured by laboratory-made image correlation techniques and the minimum detectable displacement was ∼80 nm.

A thin rod of section S and length L submitted to a force F along its axis would be elongated to L + ΔL. The strain was defined as ε = ΔL/L and was dimensionless. The Young modulus is the proportionality factor between the force per unit of area and the strain: F/S = Yε. Y has the dimension of a pressure, or energy per unit of volume.

The energy E needed to bend the thin rod of length L along a circle of radius R reads E = BL/R². B has the dimension of an energy multiplied by a length.

The resistance to bending of small object like polymers, actin filament, etc., is measured in units of the energy of thermal noise available in the bath, i.e., KT, where K is the Boltzmann constant and T the temperature (at room temperature 300 K, KT = 4.1 × 10⁻²¹ J). Thermal noise induces random bending of the object on a scale called the persistence length, L_p = B/KT. If the filament is shorter than L_P, it will look like a straight rod (for example, L_P is in the range of few millimeters for microtubules, thus microtubules of 10-μm length seem straight under the microscope). On the other hand, if the filament is much longer than L_P, many random bends could be observed along its length. The measurement of these random bends is a powerful way of measuring the bending rigidity of a filament (Gittes et al., 1993; Ott et al., 1993) without any mechanical manipulation.

Consider tangents at two points belonging to the filaments (see Fig. 2 b). If the points are close to each other (compared with L_P), their tangent will point to nearly the same direction, and the angle θ between them will be close to 0. On the other hand, if the distance between the two points is much higher than L_P, there is no correlation between the tangents, and on average, <θ\> = π/2. One can show the following relation for the average angle between tangents at two points: <cos(θ(s))\> = exp(−s/L_p), where s is the arc length between the two points. <cos(θ)\>, which is called the tangent autocorrelation function, means the average of cos(θ) over all points along the filament separated by an arc length s. If the filament is confined to two dimensions, <cos(θ)\> = exp(−s/2L_p) (Doi and Edwards, 1986).

The principle of persistence length measurement is the same as described in Ott et al. (1993) for actin filament.

Condensed chromosomes were spread between two coverslips and the reservoir of ∼4-μm thickness was sealed. Images of freely fluctuating chromosomes were recorded for few minutes on a tape recorder. A laboratory-developed computer program digitized the images, and a geometrical curve (1 pixel thick) describing the chromosome was obtained by classical skeleton finding algorithms (Chassery and Montanvert, 1991) (this curve is highlighted on chromosome images of Fig. 2 a). Each curve has been reparametrized in order to have constant arc length between consecutive points. The tangent autocorrelation function (see above) has been calculated for each curve and averaged over images of one fluctuating chromosome. The result has been averaged over 16 different chromosomes (∼5,000 images have been used for averaging). The process is summarized in Fig. 2, a and b.

After completion of the chromosome assembly, 5 μl of the solution was deposited in a small reservoir of 20 mm in diameter and 4 mm high, and diluted in 300 μl of EB. A chromosome was grabbed at both ends by two micropipettes. The displacement of one micropipette deformed the chromosome, which in turn applied force to the fixed micropipette. Deflection of the fixed micropipette (which is calibrated at the end of the experiment) measured directly the force applied to the chromosomes (see above). The simultaneous measurement of the length of the chromosome and the deflection of the fixed pipette allowed the determination of the Young modulus (see Fig. 3).

Chromosomes were assembled by incubating demembranated Xenopus sperm in mitotic high speed extract isolated from Xenopus eggs. The kinetics of chromosome assembly is shown in Fig. 1. As seen, sperm nuclei undergo a series of well-defined structural changes and after 3 h of incubation individual, fully condensed and well-separated chromosomes are formed.

We have performed two independent measurements in order to determine the flexural rigidity and the elongation deformability of these chromosomes (for definitions, see Materials and Methods). The first measurement uses the random bends induced by thermal fluctuations. The second one is performed using two micropipettes, the first to deform the chromosome, and the second to measure the force applied to it. The Young modulus can be deduced when deformations are small. The high deformation regime gives additional critical information on the inner structure of chromosomes.

The in vitro assembled mitotic chromosomes allow for precise measurement of various elastic responses, since they are condensed in cell-free extracts and are easy to manipulate. Their use is critical for the measurement of the flexural rigidity: as they are in a cell-free extract, their random bends are induced by thermal fluctuation. On the other hand, the in vivo chromosomes are constrained by the presence of the cytoskeleton. Moreover, during mitosis, the curvature fluctuations of in vivo chromosomes are due to the nonthermal length fluctuations of microtubules and cannot be assumed to be thermal. Applying force inside the cytoplasm to measure directly the bending modulus presents the same difficulties. All these considerations make the in vitro assembled mitotic chromosomes very suitable for elastic measurements. We will show below that we can reasonably assume the similarity between in vitro and in vivo mitotic chromosomes.

The flexibility of an object is characterized by its bending modulus B. The energy needed to bend the object is proportional to B, and to the square of the induced curvature. We will show below that the measure of this material property shed light on the inner structure of chromosome. This information is also important for a better understanding of the processes taking place at anaphase: during anaphase, sister chromosomes are separated and pulled toward the poles. The force the cell needs to exert in order to pull the chromosome depends on the bending modulus of the chromosomes. If chromosomes can be easily bent, the force which resists their movement (due to the viscosity of the cytoplasm and the constraints imposed by the cytoskeleton network) will be small. On the other hand, if chromosomes are stiff, when pulled toward the poles, they will remain straight and perpendicular to the direction of motion. In this case, pulling them through the cytoskeleton network would need a much higher force.

For elongated objects on the micron and submicron scale, the thermal noise randomly bends the object on a scale called the persistence length L_p, which is proportional to B (B = KTL_p, where K is the Boltzmann constant and T is the temperature) (see Materials and Methods). Hence, the analysis of the curvature fluctuation spectrum is a straightforward way of measuring the persistence length.

In brief, in vitro assembled chromosomes were observed under the microscope and their thermal fluctuation was video recorded (Fig. 2 a). Video records of 16 freely fluctuating chromosomes (representing ∼5,000 images) were analyzed, and the tangent autocorrelation function was computed. This function shows a perfect exponential decay over approximately one order of magnitude and the deduced chromosome persistence length was found to be 2.7 ± 0.1 μm (B = 1.2 × 10⁻²⁶ J · m) (Fig. 2 c). Note that this represents only a few times the diameter of the chromosome (0.8 μm) and it is comparable to the persistence length of actin filaments (∼10 μm) which are 100 times thinner. The small value of L_p (i.e., high chromosome flexibility) is likely necessary for the achievement of the anaphase.

Chromosomes were visualized by phase-contrast or fluorescent microscopy with Hoechst 258 used for labeling. No difference has been observed due to the presence of the dye at concentrations up to 10⁻⁶ M. An independent end to end distance measurement of 30 nonlabeled chromosomes gave similar result: L_p = 3 ± 0.5 μm (data not shown).

The extensibility of an object upon the action of a force is characterized by its Young modulus Y. The Young modulus depends on the underlying structure of a material. A high Young modulus reflects the fact that a high force is needed to elongate the object. The Young modulus of metals is in the range of 10¹⁰ to 10¹¹ Pa, with those of gels lays in the 10⁶ to 10⁷ Pa interval. Microtubules and actin filaments have a Young modulus of 10⁹ Pa. During mitosis, chromosomes are submitted to elongation stress due to microtubules and motors. A careful measurement of chromosome elongation at different times and the knowledge of their Young modulus should allow the determination of the force exerted on them at different stages of mitosis, a subject still submitted to debate after the article published by Nicklas (1983).

In this work, we have measured the chromosome Young modulus, and we will discuss below the relevance of this elastic constant for the chromosome underlying structure.

For a thin rod of section S and length L submitted to a force F along its axis, the Young modulus is defined as: Y = F/S(ΔL/L)⁻¹, where ΔL is the elongation induced by the force (see Materials and Methods). We determined the chromosome Young modulus from its force–extension curve by using the micropipette technique. A chromosome was suspended between two micropipettes, using a small amount of aspiration. The chromosome was then stretched by moving one of the micropipettes at a constant speed, while the length of the chromosome and the deflection of the other micropipette (used as a nanodynanometer) were simultaneously measured after recording the images on a video recorder (Fig. 3). For a small deformation (ΔL/L < 1) and stretching rate (<1 μm/s), the curve of force versus deformation is reversible over many cycles. Fig. 4 shows a typical measurement for two different chromosomes. The Young modulus of chromosomes (measurements performed on 11 different chromosomes) is found to be in the 800–1,350 Pa range, with an average value of 1,100 Pa.

Chromosomes can be elongated manifolds. For higher deformations, hysteresis is observed (Fig. 5). The elasticity ceases to be linear and the behavior is similar to that observed for long biological polymers such as titin and tenascin (Kellermayer et al., 1997; Rief et al., 1997; Tskhovrebova et al., 1997; Oberhauser et al., 1998). Successive deformation cycles, where the final extension is gradually elevated, induce also an irreversible transition in chromosome structure: after each cycle, the chromosome is softened and the next stretch necessitates a lower force in order to reach the same elongation. The maximum length to which chromosomes can be extended is variable: disruption of different chromosomes is observed for elongation between 12 and 100 times their original length. The force– extension curve of chromosomes that did not break after being stretched ∼15 times their original length show a plateau ∼5–10 nN (Fig. 6 a). This plateau is associated with the partial unraveling of the chromosomes: chromosome parts exhibiting similar diameter as the nondeformed ones are found separated from each other and are connected by thin filaments.

The thin filaments are not observable under the microscope, but their presence can be demonstrated upon moving the micropipette which induces displacement of the whole structure (Fig. 6 b).

Chromosomes were assembled from Xenopus sperm nuclei. Since these structures showed unusual elastic properties, we have been further interested to understand whether this behavior was specific to mitotic chromosomes only, or if it could be observed also on the “parental” structure, i.e., the sperm nucleus.

In contrast to mitotic chromosomes, demembranated sperm nuclei appear as very rigid objects for both flexural and longitudinal stress. The Young modulus Y of sperm nuclei has been measured by the same micropipette technique as described above. The value of Y, measured on 11 different sperm nuclei, was found to be 200 ± 50 kPa. Note that this value is two orders of magnitude higher than those of mitotic chromosomes.

The bending rigidity B of the sperm nucleus is too high to allow measurable thermal fluctuations. Thus, we used the micropipette directly to induce bending of sperms, and measured the force–curvature relation. The value of the bending rigidity obtained from three different samples was found to be 1 ± 0.5 × 10⁻²⁰ J · m, which corresponds to a persistence length of 2.5 m! This is six orders of magnitude higher than for mitotic chromosome.

In this work we have studied the elastic behavior of in vitro assembled chromosomes in Xenopus egg extracts. Let us summarize the results obtained: we have measured a chromosome Young modulus of 1,100 Pa, and a persistence length of 2.7 μm. The relations between these values and the nonlinear high deformation behavior have profound consequences on the underlying chromosome structure and lead us to propose a model for the inner organization of mitotic chromosomes.

Below, we will first discuss the fact that in vitro assembled chromosomes have similar substructures as their somatic counterparts. Next, we will show that our data strongly suggest the existence of thin rigid chromosome axes essential for mitotic chromosome organization. Finally, we will demonstrate that the nonlinear force–extension relationship observed for high deformations requires these rigid axes to be built similarly to an ensemble of highly elastic proteins.

An important question to be addressed regarding the data presented is to what extent in vitro assembled chromosomes are related to the in vivo ones. Both types of chromosomes possess a similar overall shape and size as well as biochemical compositions (Hirano and Mitchison, 1993). But their internal structures are not really known and the subject is still controversial (Hirano and Mitchison, 1993; Poljak and Käs, 1995). Determination of their material properties is well-adapted to answer this question. The Young modulus of chromosomes for small deformations has been measured by Nicklas (1983) for grasshopper anaphase chromosomes (500 Pa) and by Houchmandzadeh et al. (1997) for metaphase newt lung chromosomes (1,000 Pa). Moreover, the last authors showed that chromosomes can be stretched up to 100 times their initial length, and if the deformation is less than 10, their original length can be restored. Compared with these results, in vitro assembled Xenopus chromosomes are very similar: their Young modulus is within the same range as somatic ones (1,100 Pa), their original length can be restored for 5–10 times deformations, and they can be plastically deformed up to 100 times. All these similarities lead us to conclude that the internal structures of both assembled in vivo and Xenopus egg extract chromosomes are essentially identical.

Let us summarize the results presented in the previous sections: the persistence length L_p (which characterizes the resistance to bending) of chromosomes is 2.7 μm, while their Young's modulus Y (which is a measure of their resistance to stretching) is of the order of 1,000 Pa. No information about the underlying structure can be obtained from each value taken independently, but their relation has profound consequences. For a large class of material organizations, the relation between L_p and Y obeys the following relation (Landau and Lifshitz, 1986; Love, 1992):

where r is the radius of the object, C a numerical constant of order of unity (π/4 for a solid cylinder), T the absolute temperature, and K the Boltzmann factor. This relation is due to the fact that bending (with elastic response proportional to L_p) induces elongation/compression above/below a central neutral plane, with an elastic response that is proportional to Y. Eq. 1 would hold, for example, if the chromosome were formed of a crystalline polymer, if it were a gel-like material (the chromatin fiber cross-linked to itself by some molecular agent at random intervals), or, most importantly, if it were formed of a thinner fiber which folded helically to make the whole chromosome. Examples of biological objects obeying Eq. 1 are microtubules and actin filaments (Gittes et al., 1993). In fact, Eq. 1 does not hold for the chromosome: as the radius of the chromosome is 0.4 μm, the right hand of Eq. 1 equals ∼5 mm, which is 2,000 times higher than the measured value of the persistence length. Roughly speaking, chromosomes appear to be hard when stretched, but very soft when bent.

To explain our results, one has to imagine a model of chromosome organization in which the two apparent modes of elasticity (bending and stretching of the whole structure) are not related by Eq. 1. A straightforward explanation is to suppose that the chromosome is formed of a thin rigid (and elastic) core, surrounded by a soft envelope. As the core is thin and the persistence length varies as the fourth power of the radius, the whole object can be bent easily, while its stretching implies large forces. As a commonplace example, this is similar to the case of electric wires, which can be bent easily because they are formed of many thin filaments of copper, but cannot be elongated.

In such a model, chromosomes would possess a core with a Young's modulus higher than 10⁶ Pa and a diameter <20 nm, while the Young's modulus of the surrounding soft envelope would be <1 Pa. In the limit case where the diameter of the core is 20 nm, the resistance to bending and stretching of the whole chromosome is ensured only by the thin axis and there is no contribution from the soft envelope.

If, on the other hand, the diameter of the axis is much <20 nm, the resistance to bending of the thin axis becomes negligible: the persistence length of the whole chromosome is due only to the soft envelope, while its resistance to stretching is due only to the thin axis. In this latter case, bending and stretching of the whole chromosome involve independent elastic elements (see  Appendix) and each elastic element can be treated separately. Moreover, if we suppose that the diameter of the axis lies in the molecular range (and has negligible persistence length), one does not even need to assume that the core itself obeys classical elasticity, the only important attribute of the core being its resistance to elongation. This resistance, for the whole range of extension, can be computed using classical polymer theory, as for proteins such as titin and tenascin (Kellermayer et al., 1997; Oberhauser et al., 1998).

Both of the above limit cases can consistently explain the relation between the measured values of the elastic constants of chromosomes. We will show below that deforming chromosomes by a large amount favors the second hypothesis.

For simplicity in our calculations we have considered only one rigid axis. However, one can envisage several such axes distributed through the whole section of the chromosome, connected to each other by the soft envelope and constituting a columnar phase. From the measurement of L_p and Y it is not possible to estimate the number and the space distribution of these axes. However, by using the same calculation, one can prove that the thickness of each axis should be <20 nm, and that their Young's modulus should be of the order of Y_in/N, where N is the number of axes and Y_in the Young's modulus of the inner structure for the one-axis case.

Before going further, note that Eq. 1 holds for the sperm nucleus, the other very compact organization of DNA we have measured. Thus, sperm nuclei should have a homogeneous or helical substructure. The remodeling of the very condensed sperm nucleus in the Xenopus extract is associated with the removal of the protamine-like proteins and the uptake of numerous histone and nonhistone proteins (Dimitrov et al., 1994; Hirano, 1995). Our data show that these changes in protein composition result in dramatic changes in the underlying structure.

How are the rigid axes built? As shown above (Fig. 5), chromosomes can be stretched by many times their length; therefore, the axes store a large reservoir of length. Moreover, the chromosome force–extension curve shows a strong hysteresis and a gradual softening. Recently, similar force–extension behavior has been reported for a class of proteins that include titin and tenascin (Kellermayer et al., 1997; Rief et al., 1997; Tskhovrebova et al., 1997; Oberhauser et al., 1998), and the presence of titin inside mitotic chromosomes has also been demonstrated (Machado et al., 1998).

Native titin is a long (1 μm) polymer, formed of ∼250 Ig domains. Each domain, in its native form, has a length of ∼4 nm. Upon stretching, each domain can unfold and reach an overall length of ∼30 nm. When submitted to a gradually increasing force, the titin molecule first straightens from a coiled form, and when the force reaches a critical value, domains begin to unfold gradually. The elasticity of titin (and tenascin) is well-described by a modified worm-like chain model (Kellermayer et al., 1997; Rief et al., 1997; Tskhovrebova et al., 1997).

The solid lines of Fig. 5 represent numerical resolution of several cycles of chromosome elongation, assuming 45 titin-like molecules per chromosome. The equation describing the force–elongation relation of these polymers is detailed in Kellermayer et al. (1997) and Rief et al. (1997). Note that the persistence length of each titin molecule in this model is 2 nm. The total contribution of titin-like axes to the persistence length of the whole chromosome is <100 nm. Thus, the persistence length of the chromosome (3 μm) is mostly due to the chromatin matrix. The chromatin matrix itself does not contribute to the resistance to elongation. This model belongs to the limit case discussed above, where the resistance to bending and stretching of the whole chromosome is supposed to be due to independent elastic substructures.

To explain the softening of the chromosome after a deformation cycle, we have assumed that, after each cycle, a fraction of unfolded domains fails to refold. Similar behavior has been observed for the elasticity of titin (Kellermayer et al., 1997). As seen, the model is in impressive agreement with our experimental data and this implies that the rigid axes are built like titin molecules. Thus, proteins or protein complexes possessing titin-like elastic properties should be the main components of the chromosome axes. What are these proteins? At present we are not in a position to give a definite answer. Obviously, a good candidate may be titin itself, since it has been found to be associated with mitotic chromosomes (Machado et al., 1998). Other plausible candidates are the proteins from the SMC family (Hirano et al., 1995). These proteins exist in Xenopus egg extract as high molecular complexes called condensins (Kimura et al., 1998). Condensin immunodepleted extract is no longer able to assemble chromosomes, and thus condensins are essential for chromosome condensation and architecture (Strunnikov, 1998). In addition, it was shown that bacterial SMC proteins have their coiled-coil domains organized around a hinge (Melby et al., 1998), which seems to be structurally flexible. Since the SMC proteins are very conservative, condesins could conceivably exhibit similar hinge structure. If this is really the case, the chromosome resistance to elongation will be determined by the elastic properties of the hinge. Recently, a model for chromosome condensation based on the “scissoring” action of the SMC proteins was proposed by Hirano (1999).

The presence of a single backbone of nonhistone proteins (a scaffold), responsible for the organization of metaphase chromosomes, has been postulated in the past (Paulson and Laemmli, 1977). However, the existence of the scaffold structure in intact mitotic chromosomes has remained controversial (Earnshaw, 1991). Indeed, the scaffold was observed only upon treatment of mitotic chromosomes with different detergents and salts and thus it was not clear whether the observed structure was not an artifact due to the precipitation of high molecular weight chromosomal proteins (Okada and Comings, 1980; Hirano and Mitchison, 1993). For example, the first biochemically defined component of the scaffold was topo II, but depletion of topo II from in vitro assembled chromosomes (using salt extraction, as in Hirano and Mitchison, 1993) did not change their elastic properties (data not shown). Additionally, within the most sophisticated version of this model, the scaffold and the chromatin loops form an ∼200-nm fiber (Rattner, 1992; Saitoh and Laemmli, 1994), which further folds helically to assemble the mitotic chromosome. However, such a structure is not compatible with the measured chromosome elasticity: for this model, the diameter of the object (the outer diameter of the supposed helix) which ensures the elastic responses is of the order of the chromosome thickness. Hence, if the measured Young modulus is 1,000 Pa, the persistence length would be in the millimeter range, three orders of magnitude larger than the measured value. The same argument can be made for other model of chromosome structure (homogeneous gel-like; McDowall et al., 1986), hierarchical helical folding (Sedat and Manuelidis, 1978; Manuelidis, 1990), which cannot explain the apparent discrepancy between the measured values of chromosome elasticity.

In our previous work on in vivo chromosomes (Houchmandzadeh et al., 1997), based on a limited amount of information, we had favored a model of chromosome organization based on helical folding of a thin filament. The diameter of the thin filament was supposed to be four times smaller than that of the mitotic chromosome. The argument was based on the huge extensibility of chromosomes, on the knowledge of their Young's modulus, and on geometrical considerations. Two critical data were missing at that time: the value of the persistence length of chromosomes and the force–elongation relation for high chromosome deformation, which are, as stressed above, very difficult to measure in vivo. These data have been obtained in this work, and as shown above, are not compatible with the model of helical organization.

In conclusion, we have developed a novel approach for studying chromosome structure. Our experiments show that mitotic chromosomes exhibit specific elastic responses. Therefore, we propose a model for chromosome structure based on a single or several thin elastic axes surrounded by a soft envelope (Fig. 7). The axes are proposed to consist of elastic titin-like molecules, while the envelope contains chromatin. The chromatin material is attached to the axes and is responsible for keeping the axes close together. The resistance of mitotic chromosomes to bending is ensured by the chromatin matrix, where the axes are responsible for the resistance of chromosome to elongation.

The condensation state of the chromosome is determined by the interplay between the entropic forces exerted by the axes which tend to collapse them into a random coil and the excluded volume effects due to chromatin (Marko and Siggia, 1997). Moreover, local decondensation of chromosomes can be achieved by a simple unfolding of axis protein domains.

We thank D. Dimitrov, J. Hayes, M. Vallade, B. Berge, E. Katz, and M. Creaven for fruitful discussions and careful reading of the manuscript. We also thank A. Libchaber and J.-J. Lawrence for support throughout the course of this work.

As noted above, many material organizations obey the relation:

where B is the bending modulus of the object, Y its Young modulus, r the radius, and C a numerical constant of order of unity. The persistence length L_p of the chromosome is B/KT.

Mitotic chromosomes do not obey this relation. As the bending modulus varies as the fourth power of the diameter, a straightforward explanation is to suppose that the diameter of the object which ensures the resistance to stretching is much smaller than the apparent diameter of the chromosome, and its Young modulus much higher. We can estimate the order of magnitude of such an object. Suppose for simplicity that the chromosome is formed of one rigid axis of radius r_in and Young modulus Y_in, surrounded by a soft envelope of radius r (the radius of the whole chromosome) and Young modulus Y_out. The measured values of elastic constants of the whole chromosome are related to the elastic constants of these two substructures through:

where γ = r_in/r. We will note below Y* = 4B/πr⁴. For in vitro assembled chromosomes, r = 0.4 μm, Y = 1,000 Pa, and Y* = 0.6 Pa. Note that there are three variables (r_in, Y_in, Y_out) and only two equations, so exact values for the variables cannot be obtained. However, an upper bound for γ is easily found from Eq. 3 and Eq. 4:

as γ < 1, and thus

which implies

where r_max is the maximum radius of the elastic axis. As γ ≪ 1 and Y* ≪ Y, the elastic constant of the two substructures are: Y_in ≈ Y/γ² and Y_out ≈ Y* − Yγ².

There are two interesting limit cases. First, if r_in ≈ r_max, Y_out ≈ 0 and Y_in ≈ 1.5 × 10⁶ Pa. In this case, the soft envelope does not contribute to resistance to stretching, nor to resistance to bending and the elasticity of the whole chromosome reduces to the elasticity of the thin axis. The other limit case is when r_in ≪ r_max, the resistance to bending of the thin axis becomes negligible, and the persistence length of the chromosome is only due to the soft envelope of chromatin (Y_out ≈ 0.6 Pa). The contribution to stretching on the other hand is only due to the rigid axis. In this case, bending and stretching involve independent elastic elements. Note that if we suppose the radius of the axis to be much <10 nm, it becomes difficult to suppose that it still obeys classical elastic theory. The axis will still resist stretching like classical polymers or titin-like molecule. As the bending and the stretching of the whole chromosome are uncoupled, one can treat each substructure independently: the resistance to stretching through the modified worm-like chain model (as for titin; Kellermayer et al., 1997) for the thin axis, and the resistance to bending using elasticity for the soft envelope (a theoretical ground for this calculation is proposed by Marko and Siggia, 1997).