A random-walk/giant-loop model for interphase chromosomes

Proc. Natl. Acad. Sci. USA Vol. 92, pp. 2710-2714, March 1995 Biochemistry

A random-walk/giant-loop model for interphase chromosomes (chromatin/fluorescent in situ hybridization/cell nucleus ultrastructure/polymers)

R. K. SACHS*, G. VAN DEN ENGHt, B. TRASKt, H. YOKOTAt, AND J. E. HEARSTt Departments of *Mathematics and tChemistry, University of California, Berkeley, CA 94720; and tDepartment of Molecular Biotechnology, University of Washington, Seattle, WA 98195

Communicated by Alexandre J. Chorin, University of California, Berkeley, CA, November 21, 1994 (received for review October 17, 1994)

separations from 0.15 Mbp to 190 Mbp and located on human chromosomes 4, 5, or 19. These new data can be used to test the models of large-scale chromatin geometry. Our analysis will presuppose that the distributions observed in the fixed samples of Yokota et at (8) reflect the distribution of marker pairs in the living cell. On this assumption the data show there are two, and probably only two, levels of chromatin structure over the range 0.15-200 Mbp, with the chromatin at genomic separations greater than a few Mbp not spreading out as fast as one would expect from its behavior in the 0.15- to 1.5-Mbp range. We shall argue that the simplest interpretation of the data is to assume that there are flexible chromatin loops averaging -3 Mbp, with their base points along a random walk. The heart of the argument is given by Eqs. 10-12 in the Results section.

ABSTRACT Fluorescence in situ hybridization data on distances between defined genomic sequences are used to construct a quantitative model for the overall geometric structure of a human chromosome. We suggest that the large-scale geometry during the Go/G1 part of the cell cycle may consist of flexible chromatin loops, averaging -3 million bp, with a random-walk backbone. A fully explicit, threeparametric polymer model of this random-walk/giant-loop structure can account well for the data. More general models consistent with the data are briefly discussed.

A human chromosome is a very large molecule. Its DNA strand has in the order of 100 million base pairs (Mbp) arrayed along its contour and has a Mr of -1011 Da. Quantitative information on mammalian chromosome geometry during the interphase part of the cell cycle is very extensive for scales 100 probe pairs, having genomic

DATA Figs. 1 and 2 compare results of H.Y. et at (unpublished data) with a model to be introduced later. Relationships between mean-square physical distance and genomic separation fall into two markedly different regimes, with a sharp transition at several Mbp (Fig. 1). For values of genomic separation 0.1 Mbp, such as structure responsible for the transition at several Mbp in Fig. 1, can be modeled by interactions between beads that are not consecutive (see Results). We follow the notation of Doi and Edwards (10). Cartesian coordinates of the beads are denoted by Xg, i = 0, .. ., N (Fig. 3). The Z direction is taken perpendicular to the microscope slide. The data concern only the (X, Y) plane. Cartesian components and projections in the (X, Y) plane are written as follows:

xi (xi, Yi, Zi) (x; Zi), =

=

i. e., (Xj, Yj) xj.

Ill

Equilibrium statistical properties of polymers are often represented as properties of a mechanical system with negligible kinetic energy (10). The probability density G(Xo, .. ., XN) for having a bead configuration Xo, .. ., XN can then be regarded as a Boltzmann distribution (10): [2] G(XO, ... ., XN) = C exp(- U/kBTh.

r=

-II= -xjII.

[(Xi22-=X)2 + (Y -

[31

To analyze the data, one needs the distance probability density P(r) for r. P(r) can be obtained by first finding the joint probability density P2 (xi - xj) P2 (u, v) for the two Cartesian intervals u and v defined by u = Xi - Xj and v = Yi - Yj. P2 in turn can be obtained by integrating G in Eq. 2 over all variables except u and v, i.e.,

P2(xi - xj) = c

ds dw dZi dZj dX exp(- U/kBT), [4]

where s = '/2(Xi + Xj), w = '/2(Yi + Yj), and dX denotes integration over 3 x (N - 1) Cartesian variables. The distance probability density P(r) is then obtained by the Jacobian relation appropriate for two dimensions, i.e., dr = r dr ~~~~~P(r)

2)rf

d OP2(r coOS, r sinO).

[5]

RESULTS We now model chromosome geometry, first on intermediate scales of 0.1-1.5 Mbp and then on larger scales. Intermediate Scales. The model used by van den Engh et al (5) to analyze chromatin behavior at the 0.1- to 1.5-Mbp level is equivalent to taking U = UO in the probability density of Eq. 2, where N

Uo = (K/2) IIXj

-

Xj_112,

[6]

j=l

with K a nominal spring constant that elegantly summarizes all relevant smaller-scale chromatin structure (10). If U = Uo, the mean-square physical distance depends linearly on genomic separation (5, 10). The recent data of H.Y. et aL (unpublished data) confirm the earlier results that on scales of 0.1-1.5 Mbp this model is appropriate, in that points in the lower left of Fig.

Proc. Natt Acad. Sci USA 92 (1995)

Biochemistry: Sachs et aL

2712

N

i-I

N-1 %

%

ii +1 I

oN1 FIG. 3. Polymer models, showing the numbering of "beads," (solid circles), used to track the large-scale geometry of a chromosome. Beads 0 and N are at the chromosome ends (telomeres). N 1900 for human chromosome 4. Chromatin connections between consecutively numbered beads are shown schematically as straight lines but are actually tortuous paths with a contour length very much larger than the bead-to-bead distance. With a genomic separation of 0.1 Mbp between consecutive beads, their rms distance, determined by Eqs. 11 and 12, is 1/2 ,um. The intricate DNA-protein structure of the 30-nm chromatin fiber is too small-scale to appear (and appears in the mathematics of the text only via a single constant K). Higher-order chromatin structure is modeled by additional interactions (not shown)-e.g., interactions between nonconsecutive beads (Fig. 4) and/or interactions of the beads with an extrachromosomal structure. -

1A lie approximately on a straight line (whose slope is -2 Therefore we later take U = Uo + CT in Eq. 2, where CJis almost negligible at scales of