INTRODUCTION

Top Abstract Introduction Results & Discussion References

The structure of condensed chromosomes on scales beyond the level of the basic 10-nm beads-on-a-string fiber is still controversial, in part because chromatin is dense yet labile and easily disturbed (van Holde, 1989). In this article, we use polymer statistical mechanics to elaborate the equilibrium properties of bottle-brush models of meiotic chromosomes and to outline how the kinetics of mitotic chromosome condensation directs the resolution of entanglements. Without any assumptions about well-ordered structures other than that of the chromatin fiber, we show how semiquantitative predictions about experiments can still be made. These predictions concern the morphology and mechanical properties of chromosomes on scales that can be studied via light microscopy in buffer or, in some cases, in vivo. If these predictions are borne out (several already have qualitative support), then it will lend credence to the starting assumptions and will allow inference of mechanical and kinetic parameters not obtainable by imaging fixed samples.

The phenomenological approach to polymer solutions has proved very powerful, because nontrivial and experimentally testable consequences can be derived with weak assumptions under general conditions (de Gennes, 1979; Doi and Edwards, 1986; Grosberg, 1994). Following Flory (de Gennes, 1979, chapter 4), all the chemical details can be lumped into three essential parameters, the effective monomer size a, the number of monomers per chain N, and a dimensionless interaction parameter  that expresses the tendency for monomers to adhere to (or repel) one another. Monomer in this context is the smallest unit that can substantially reorient in thermal equilibrium and so naturally, in the context of chromatin, includes both the DNA and bound protein (e.g., several successive nucleosomes). The strongest predictions from this class of theories are frequently cast as scaling laws, valid for large N to within dimensionless order-unity constants. Chromosomes are without question the longest polymers known, so it is natural to ask whether aspects of their morphology can be explained using polymer phenomenology.

The first dichotomy to be faced in any discussion of polymers in solution is whether they are in good ( <1/2) or bad ( >1/2) solvent. By solvent we mean the water, ions, and other small (<10 nm) moieties in the cytosol that make up the medium surrounding chromatin fibers and chromosomes. A good solvent wets the monomers, making the polymers dissolve and disperse; a bad solvent is one in which the monomers and, therefore, the polymers aggregate, leading to phase separation between concentrated polymers and essentially polymer-free solvent. We will work exclusively in good solvent conditions (chromosomes in meiotic prophase) or marginal theta-solvent conditions ( ~ 1/2), which with several caveats we apply to mitotic chromosomes.

Brush models, in which chromatin loops are attached to a central axis but otherwise free to fluctuate, have been part of the debates over chromosome structure for many years. From the microscopic evidence, such a model is most clearly relevant to chromosomes in meiotic prophase [Moens, 1987; applications of the model to mitotic chromosomes are due to Laemmli and coworkers (Paulson, 1988)]. Once we add the assumption that the bristles behave as if in good solvent (for our purposes it is immaterial that they are loops rather than single fibers), the basic physical theory can be taken from the polymer brush literature (Li and Witten, 1994). The existing theory has not found many applications due to the difficulty of preparing synthetic brush-like polymers; the process by which chromosomes condense into brushes is therefore of interest to materials scientists as well as to biologists.

The most striking pictures of brushes have been obtained with certain solvent washes and drying that serve to comb out the bristles of an otherwise more compact structure (Miller and Hamkalo, 1972). The Christmas trees formed by the nascent RNA transcripts along certain heavily transcribed genes are classical, and similar pictures of flattened extended brushes have been obtained for meiotic prophase chromosomes (Moens and Pearlman, 1988). These preparations are a good assay for the arc length of individual bristles and their axial spacing, but by themselves do not rule out the possibility of proteinaceous attachments between bristles in vivo or that the bristles are collapsed or self-adhering. Similar caveats have been expressed about the high salt treatments of mitotic chromosomes that have been interpreted as evidence for a radial loop model (Paulson, 1988; Jackson et al., 1990).

Lampbrush chromosomes isolated from amphibian oocytes provide the best evidence for an open brush morphology suggestive of good solvent conditions in vivo and do so on two scales (Callan, 1986). The prominent DNA loops are themselves brushes with a DNA core and ribonucleoprotein (RNP) bristles and form in turn the bristles of a larger brush that is the chromosome itself. The loops appear to undergo Brownian motion when viewed free in buffer and the chromosomes themselves unfold when they emerge from a punctured nucleus (Macgregor and Varley, 1988). The theory to be developed below shows that the retraction of the loops when the RNA transcripts are snipped off and the ubiquitous oval shape of diplotene bivalents between chiasmata indicate that the bristles are in good solvent conditions.

Meiotic chromosomes in a variety of other species during leptotene through pachytene and in diplotene also evidence a brush-like morphology (Comings and Okada, 1970; Rattner et al., 1980, 1981; Heng et al., 1994). Preparations in these studies involve surface speads, fixation, and degrees of squashing, so that some distortion is occurring, though less than with classic Miller spreads. Microscopy on whole nuclei probably yields the most faithful measures of dimensions (Dawe et al., 1994). In all cases, an axial core is readily visible surrounded by a chromatin halo. For mammals a prominent protein constituent of the core has been identified and fluorescent antibodies are available (Dobson et al., 1994; Moens, 1994; Pearlman et al., 1992). We have interpreted a number of observations on fixed specimens, as evidence for good solvent conditions.

Mitotic chromosomes are typically denser than their meiotic (prophase) counterparts and the proteins that condense along with the chromatin fiber are only now being identified with the aid of cell extracts and genetics (Gasser, 1995; Hirano et al., 1995; Strunnikov et al., 1995; Koshland and Strunnikov, 1996). We accept the prevalent hypothesis that chromosome condensation plays an essential driving role in the resolution of entanglements and the separation of duplicate sister chromatid arms.

For entanglement resolution to occur by physical-chemical means (i.e., by a local mechanism rather than by an external process such as motors along tracks), we develop the hypothesis that the cell in early prophase tunes solvent quality (or equally well, physical properties of chromatin) toward theta-solvent conditions, i.e.,  = 1/2. Nonhistone proteins that coprecipitate with the chromatin are taken to act as a gentler and regulated version of the well-known polyanion condensing agents and come into play when 1/2   is sufficiently small (but still positive). We insist that the solvent conditions for the chromatin must never become truly bad: if poor solvent conditions prevailed, all the chromosomes would condense together into a ball. In poor solvent, sister chromatid arms would remain intertwined until pulled apart by microtubule-based motors. Instead, in mitotically arrested cells, some delineation between the arms of the chromatids occurs in the absence of any known motor proteins and in the absence of microtubules (Shamu and Murray, 1992; Miyazaki and Orr-Weaver, 1994).

We stress that our discussion of mitotic chromosomes focuses on the thermodynamics of chromatin packing and the kinetics of its disentanglement along the chromatid arms and has nothing to say about how the chromatids are joined at the centromere. These attachments are known to be precisely regulated in response to tension across the kinetochore (Bickel and Orr-Weaver, 1996).

	THEORY

In this section we rederive by elementary means some of the theoretical results on polymer brushes in good solvent as a way of summarizing the physics involved and making comprehensible the extensions needed to describe chromosomes. Two ingredients are needed. The first just formalizes the notion that two monomers cannot occupy the same region of space, which is expressed as an interaction contribution to the free energy of order Twn per monomer, where n is the local monomer density (units of number/volume) and w ~ (1  2) a³ is the excluded volume per monomer with length a. For good solvent, we will set  = 0, so as not to carry a parameter we have no hope of determining from the data. (For bad solvents,  > 1/2 and w < 0, which corresponds to attraction between the monomers and their condensation into a dense phase.) The energy scale is set by the thermal energy T = 4.1 × 10²¹ J or 4.1 pN·nm (the work done by a force of 4.1 × 10¹² N during a displacement of 1 nm).

The second ingredient is intuitively less obvious but is just a restatement of the second law of thermodynamics in terms of an entropic force. Namely, to distort a flexible polymer (treated as a random walk and hence without an internal energy) from its most probable Gaussian configuration (so-called because the displacement between the ends of a random walk follows a Gaussian distribution) requires work that decreases the entropy and increases the free energy of the polymer. For forces < T/a, the free energy as a function of extension R is ~ TR²/Na², where numerical factors of order unity are suppressed. Alternatively, this force law is just a restatement of what is meant by good solvent conditions, ignoring for the moment that the polymer is constrained not run into itself.

Brush Equilibrium Free Energy and Radius

To describe a brush, we ignore the obvious variation in monomer density as a function of distance from the axis and express the free energy per bristle G_b as a function of brush radius R, as the sum of the two terms above (Li and Witten, 1994):

(1)

where  is the distance along the axis of successive bristles (1/ is the number of bristles per length) and N is the number of monomers per bristle. The optimal R and G_b (denoted with a*) are obtained by minimizing Eq. 1 with respect to R; hence,

(2a)

and

(2b)

Note that the radius increases with N more rapidly than for a random walk (Figure 1), because of the excluded volume of the other bristles; i.e., the polymer must not run into itself. The equilibrium structure is just a balance between intermonomer excluded volume and the entropic stretching. Decreasing  increases the monomer density, and R* increases as a result. For reasonable parameters, the free energy is always larger than T (we assume  << R; for  ~ R, each bristle is an isolated coil), reflecting the energetic cost of confining the bristles to a common axis. Alternatively, the constraint that each bristle is anchored on the axis gives rise to a chromatin pressure, which stretches the bristles.

View larger version (10K): [in this window] [in a new window]   Figure 1.   An isolated chromatin bristle will be a random coil (left), but when attached to an axis and spaced by << R, interchain repulsion stretches it (right).

When one properly treats the variable density, the radius and free energy change by no more than 20% in the above formulas (Li and Witten, 1994). However, the detailed density profile is quite interesting: the free ends are found essentially only between 0.5R and R; the monomer density is very peaked around the core (as expected just for geometric reasons) and quite tenuous beyond 2R/3. The monomer density goes smoothly to zero at R.

Several other properties of brushes pertinent to chromosomes can be derived by similar dimensional reasoning. A given bristle will sample a cone-like region of space with its apex on the axis. The base (outer edge) has a radius characteristic of a random walk (in contrast with Eq. 2a)

(3)

Thinking of the polymer as a random walker starting out from the axis, one realizes that the radial steps will be biased outward by the chromatin pressure, but a finite fraction of the total steps will be transverse to the radius and otherwise unbiased, giving rise to a random walk transverse to the radial direction.

The R is pertinent to chromosomes because it can be used to qualitatively assess the entanglement of adjacent bristles in one brush and of bristles in two overlapping brushes (e.g., paired chromatids at an early stage of entanglement resolution). From the study of both experiments on and simulations of flexible polymers, it is known that when eight random-walk polymers share the same volume, the chains will be constrained by entanglements (Kavassalis and Noolandi, 1989). Adopting this criterion, from the number of segments per volume in our brush [n* = N/(R*2 )] and the volume occupied by a chain (R*R²), the number of chains that share the same volume follows as R*R²n*/N = [a³/(³)]1/4N1/4. Because of the weak N dependence, this number will not reach 8 for typical chromosome parameters; shorter bristles can be considered as essentially unentangled once equilibrium has been reached.

Repulsive Force between Two Brushes

When two brushes overlap, they will push apart to lower the chromatin density; this is another manifestation of chromatin pressure (Figure 2). To calculate the order of magnitude of this repulsive force, note that if two brushes coincided the free energy per bristle would increase by (take  /2 in Eq. 2). This free energy could also be derived by integrating the inter brush force from a spacing of 2R, where it vanishes to zero. The dependence of the force on the radial spacing is nontrivial and could be calculated following established methods (Li and Witten, 1994) but its order of magnitude of this force per unit of axial length F_a is:

(4)

View larger version (15K): [in this window] [in a new window]   Figure 2.   Chromatin pressure develops when the bristles are compressed, either when two brushes overlap (left) or when a brush is bent (right). Arrows indicate forces.

Tension between Bristles Inside Brush

There is also an axial force F_r exerted on whatever glue fastens the bristles together. Its magnitude is just G_b^* expressed per unit length of brush or

(5)

Brushes Have a Long Bending Persistence Length

The last property of interest is the thermal persistence length of the brush or the length of brush that can be deflected by an angle of one radian at an energetic cost of T. Assume whatever constitutes the axis is completely flexible. Forcing the brush to bend with a radius of curvature  >> R will increase (decrease) the volume accessible to the outer (inner) bristles by a factor ~ 1/(1 ± R/). Redo the argument leading from Eq. 1 to Eq. 2 separately for the two populations of bristles to find Converting to units of per length of brush and extracting the persistence length yields

(6)

This is typically a long distance compared with brush radius R. A polymer bottle brush is, therefore, a rather stiff object, on the scale of its radius, even if the axial core has no intrinsic stiffness. Chromatin pressure alone is sufficient to stiffen the brush. On the other hand, it should be noted that chromatin pressure provides no resistance to shear: twisting elasticity purely probes the stiffness of the axial attachments or of bristle cross-linking.

Brush in Theta Solvent

The above theory can be extended to the case of theta solvents, but we will merely sketch how the various scaling laws change, because when evaluated for parameters appropriate for meiotic chromosomes, the numerical results change by no more than 50%. The definition of a theta solvent is that the interaction energy per monomer is no longer proportional to n but rather to n². Such a situation arises when monomer-monomer avoidance and sticking approximately cancel one another (i.e.,  = 1/2) ; one is left with only three-body interactions that contribute a free energy per monomer proportional to n².

Hence Eq. 1 is replaced by

(7)

and Eq. 2 becomes

(8)

Thus the monomer density is still lower than in a collapsed state (i.e., a melt of densely packed monomers where n  1/a³) by a factor of ~ (N /a).

Gel Model of Chromosome

An alternative to a brush model where the chromatin attachments are all to a central axis, is a gel, where the cross linkages are uniform through out the volume (note that these two pictures are not mutually exclusive: cross-linked radial loops give rise to a gel). If there are N >> 1 monomers between successive cross-links, the resulting network can be elastic over a large range of extensions (i.e., five or more). A gel can be immersed in a good solvent and swollen so that a large part of its volume is liquid; alternately, if the solvent quality is poor the chains will collapse and the gel will be dense, like rubber.

If we assume the monomers in a gel are free to move, then the elasticity of a gel comes from the entropic elasticity of the constituent chains: to extend each chain a distance R, a force of TR/(Na²) must be applied. A key point is that synthetic polymer gels (e.g., cross-linked polystyrene chains) have a much smaller monomer size (a  0.5 nm) than chromatin (a  30 nm) and, therefore, a gel of chromatin will have intrinsically rather weak elasticity.

The elasticity of a gel is described by an elastic modulus E that has the dimension of a pressure. Its value reflects the pressure, or force per area, that must be applied to a gel to increase its length by a factor of two. If we consider a gel in good solvent, the inter-cross-link chains will be swollen and separated from one another. If before stretching, the radius of each chain is R₀, then the number of chains per cross-sectional area is just 1/R₀²; the force per area as a function of extension R of each chain is (1/R₀²)TR/(Na²) = (R/R₀)T/(NR₀a²). Therefore the elastic modulus is E = T/(Na²R₀). Because before stretching each chain is of radius R₀  a, the elastic modulus can also be expressed as E = T/R₀³ or roughly one thermal unit of energy per cross-link volume (de Gennes, 1979). Measurement of gel elasticity therefore leads to an estimate of cross-link density. In poor solvent, one must consider more carefully how the chains are packed and the effects of surface tension, but a value of a thermal unit per cross-link volume is in a good starting estimate for E.

Finally, we note that a gel can be deformed either while keeping its volume fixed, or while allowing its volume to change. The former type of deformation measures the shear modulus; the latter is sensitive also to the bulk modulus. Most mechanical measurements (e.g., stretching, bending, or twisting an elastic rod) are primarily sensitive to the shear modulus. On the other hand, the change in volume of a gel in response to a change in hydrostatic or osmotic pressure is dependent on its bulk modulus. For swollen gels, the two moduli are usually comparable.

Phase Separation of Bristles and Banding

So far we have assumed chemical homogeneity, but the well-known process of chromosome banding converts small differences in base composition to a pattern of nested stripes particular to the chromosome (Craig and Bickmore, 1993). We therefore note in this connection, one very characteristic feature of dense polymer solutions or melts, namely, their tendency to phase separate, i.e., spatially cluster together similar molecules. This can be understood semiquantitatively from a Flory theory because the entropy of mixing depends on the concentration of chains, each of which moves as a unit, while the van der Waals energy mismatch, which promotes phase separation, goes as the number of monomers. (van der Waals interactions may be overwhelmed by other interactions favoring separation, e.g., interactions between proteins that bind favorably to either A+T- or G+C-rich chromatin; however, simple van der Waals theory provides a starting point that may describe sequence-driven separation in purified DNA or chromatin fiber solutions)

A very crude estimate of this effect is (chapter 6 in Israelachvili, 1992).

(9)

where E_g ~ 10⁴ is the frequency range in temperature units where the molecular dielectric polarizabilities []_1,2 differ appreciably. The dielectric constant of water, _w, although ~80 for zero frequency, is a factor 10-40 smaller at the frequencies appropriate to the van der Waals interaction. Polarizabilities are of order the molecular volume, so the difference between []_A+T versus []_G+C could be ~10%. The contrast in the A+T:G+C density is as high as 10% and the volume fraction of nucleotide pairs in condensed chromatin ~10%. Hence, ([]₁  []₂)n ~ 10³. Larger contrasts could be produced by proteins or the dyes used to visualize the bands. Hence, individual bristles with N ~ 100 could phase separate.

For chromosomes with the bristles anchored to a central axis, the loop size limits the degree of spatial phase separation, and conversely observing phase separation would measure the transverse loop radius R Our prediction is therefore that the A+T:G+C ratio in chromosome bands will be enhanced over what would be predicted if the genome were layed down sequentially. Another consequence of chemical heterogeneity near the theta point is that certain regions may collapse while others remain extended.

Alternative TheoryBad Solvent

Sikorav and Jannink (1994) envision a chromosome as a polymer melt (bad solvent conditions) and then show that topoisomerase (topo) II is necessary for the final stages of chromatin compaction by allowing strand passages. We feel that experiments on both native and in vitro-assembled mitotic chromosomes argue in favor of good solvent conditions as regards the chromatin-buffer system and that a third glue component is necessary to understand mitotic chromosomes, which they did not postulate. Topo II will certainly link dense chromatin, but this seems to us parasitic rather than essential to the condensation process and we feel that plausible chemical condensing agents alone can give densities compatible with experiments. Jannink et al. (1996) considered the kinetics of chromosome separation under melt conditions and concluded that some outside force (i.e., from the spindle) is needed when there is more than 10⁵ bp per chromosome. This is at variance with the in vitro and in vivo experiments summarized below.

	RESULTS AND DISCUSSION

Top Abstract Introduction Results & Discussion References

Meiotic Prophase Chromosomes

Brush Model Parameters. The effective monomer size a is the length of the minimal unit that can reorient in thermal equilibrium. For chromatin in vivo, we will identify this with 30-nm-long segments of the 30-nm fiber seen via electron microscopy and x-ray scattering for a variety of preparations (Thoma et al., 1979; Widom, 1986, 1989). However, a density correlation on this scale does not imply thermally induced bending on the same scale. The interpretation of what is seen in these experiments as the monomer size for our purposes is suggested by atomic force microscopy experiments, sectioned cryoelectron microscopy, and modeling studies of interphase chromosomes (Woodcock et al., 1993; Horowitz et al., 1994; Leuba et al., 1994; Woodcock and Horowitz, 1995). These authors show how a random variation in the linker distance, plus the natural helicity of the DNA, can give rise to decorrelation in the direction of the fiber on the scale of 5-10 nucleosomes. Because the amount of linker DNA contained in such a segment is one or two thermal persistence lengths, thermal fluctuations should be comparable to those induced by the linker variability.

Physical Properties of Meiotic Chromosomes and Comparison with Experiment Scale of Intracellular Forces. The observed brush parameters (Table 1) permit us to evaluate (Table 2) the physical parameters derived in the previous section. To put the force numbers in some context, note that a radial force of 1 pN/µm acting on a rod with a length-to-diameter ratio of 10-100 will drag it through water at a velocity of 0.1 mm/sec normal to its long axis. Second, an axial force of T/a ~ 0.1 pN will stretch chromatin fiber to about half its contour length of aN [which otherwise would be a random coil of radius ~ aN]. Forces in the range of 2 pN will cause the release of bound histone octamers from DNA because they are comparable to the binding energy of the nucleosome divided by the length of the wrapped DNA (Marko and Siggia, 1997). Even larger forces of 50-100 pN are capable of modifying DNA or protein secondary structure (Cluzel et al., 1996; Smith et al., 1996). These numbers are all much less than the force that the mitotic spindle is capable of exerting on a chromosome of ~700 pN (Nicklas, 1983).

View this table: [in this window] [in a new window]   Table 2.  Predictions for in vivo radius R, radial repulsive force per length Fr, the axial force Fa, and the thermal persistence length th for the three systems (a-c) defined in Table 1

Mitotic Chromosomes

Experimental Basis for a Physio-Chemical Model of Condensation. The chromatin organization within a mitotic chromosome is more difficult to infer from experiment than was the case for meiosis. Evidence for a radial loop model is surveyed by Paulson (1988) and refinements that relate it to banding patterns are presented in Saitoh and Laemmli (1994). This latter work proposes that each chromatid contains a helically wound thin fiber, which echoes a series of microscopy studies arguing in favor of such a coiled or perhaps folded structure (Ohnuki, 1968; Sedat and Manuelidis, 1978; Boy de la Tour and Laemmli, 1988). In any case, the chromatin must be somehow fastened to itself by the action of scaffold proteins (Paulson, 1988). The most prevalent nonhistone protein in the metaphase chromosome is topo II. Although required for chromosome resolution and condensation, recent work argues against topo II having an essential scaffold function (Hirano and Mitchison, 1993; Swedlow et al., 1993).

Implications of Model. Given our physio-chemical model (re-expressible in more formal terms as a phase diagram of the chromatin-buffer-SMC system), we can rationalize the following classical results: 1) chromosomes condense separately and are rod like in vivo, 2) sister chromatid arms separate in mitotically arrested cells without motors (see for instance Shamu and Murray, 1992, page 933; Miyazaki and Orr-Weaver, 1994; Bickel and Orr-Weaver, 1996) and are held together only by their centromeres, and 3) chromosomes thicken and shorten from prophase to metaphase. As well, we will reinterpret a number of more recent experiments.

View larger version (25K): [in this window] [in a new window]   Figure 3.   Two sister chromatids undergoing condensation and resolution. Interdigitated interchromatid loops will be under slightly more tension than the intrachromatid loops defining the arm. Bars represent condensing (possibly SMC) proteins.

Bulk Elasticity of Mitotic Chromosomes. There have been only a few experiments that measure the elastic properties of mitotic chromosomes, yet they do convey significant information about the how the chromatin is internally organized and rule out certain models. It has been noted that human metaphase chromosomes display rather impressive elasticity, returning to their native length after being stretched by a factor of 10 in length (Claussen et al., 1994). For anaphase grasshopper chromosomes, Nicklas (1983) has estimated an elastic modulus of about 500 Pa (1 Pa = 1 N/m²). This is very weak elasticity compared with that of a DNA molecule (5 × 10⁸ Pa) or some other chemically bonded structure. Recent work of Houchmandzadeh et al. (1997) on mitotic newt lung cells found an elastic modulus of 5000 Pa at the end of prophase and 1000 Pa at metaphase; the chromosomes were found to return to their native lengths after being stretched up to 10 times.

Banding of Mitotic Chromosomes. There are suggestions that the contrast in the A+T:G+C ratio that is picked up by the dyes used for chromosome banding is larger than would be expected by merely laying down the genome sequentially in the chromosome (Saitoh and Laemmli, 1994). This plus the appearance of bands within bands is indicative of thermodynamic phase separation. However, the parameters entering Eq. 9 are too poorly known to assess whether this is a realistic possibility. One can, however, reliably scale the kinetics from one set of parameters (e.g., overall DNA density, and A+T:G+C ratio contrast) to others, as well as predict how domain size grows in time.