e106 Nucleic Acids Research, 2007, Vol. 35, No. 16 Table 1. Loops of a ﬁlament with random variation in b Loop number Nucleosome position 1 Nucleosome position 2 Number of nucleosomes in loop 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 6 7=8 7 8 11 16 17 21 30 31 31 32 46 48 50 52 52 53 7 12 12 13 14 16 22 20 25 42 36 40 44 50 51 54 58 61 57 8 7 5/6 7 7 6 7 4 5 12 6 10 13 5 4 4 7 10 5 The loops are for the ﬁlament shown in Figure 7A–F. Nucleosomes in relatively stable clusters are marked with a black background. formed by the ﬁrst ﬁve nucleosomes of the ﬁlament, analogous to the zip-motive of the double ﬁber shown in Figure 5A. Given the irregular conformation of ﬁbers with random variation in b, it was ﬁnally asked whether such ﬁbers could be transformed into more regular ﬁbers, especially the type of ﬁber (333)n, which can be compacted to a high density (Figure 4F). Two properties of the ﬁlament made this possible. First, the displacement of core particles along the DNA changes the two ﬂanking b values in opposite direction by the same amount. Secondly, twist constraints of the linkers are negligible for variations of b corresponding to maximally 2 bp (28). In Figure 7G–I and Figure 8 it is shown how the ﬁber can be changed by displacements of core particles to generate a b sequence which, by twisting of the linkers maximally 2 368, can attain the b sequence (333)n, which can be compacted. Changing the conformation in this way step-by-step from one end of the ﬁlament to the other provides a simple model for the spread of heterochromatin. As (333)n is also the sequence required for formation of one type of double ﬁber (Figure 5A), combinations of core particle displace- ments (chromatin remodeling) and limited linker twisting may thus provide the b sequences necessary for stable inter ﬁber associations. DISCUSSION A major result of the present study is the demonstration of the signiﬁcance of a for the conformational freedom of the nucleosomal ﬁlament. When a is smaller than 608 the ﬁlament thus remains a helix with little space for variations in the b sequence, while larger values of a provide it with a freedom to attain conformations, which are not dependent on a helical symmetry, by allowing for an increased variation in b sequence. The size of a in chromatin is not known with certainty, but a has been PAGE 12 OF 15 shown by cryo-EM imaging of native chromatin ﬁbers (24,29) to decrease from 85 to 358 when the salt concen- tration is increased. These relatively small DNA entry–exit angles have been suggested to be caused by the stem motive, which has not been observed by scanning force microscopy (SFM). Studies by SFM of chromatin ﬁbers ﬁxed at low ionic strength have yielded average values of a of 100 408 in the presence and 130 408 in the absence of linker histones (15), and unﬁxed mononucleosomes at 10 mM MgCl2 showed two maxima of a at around 80 and 1368 in the absence of H1 and one maximum at 85–888 in the presence of H1 (16) (mean values diﬀered slightly for samples scanned in air and liquid). These large values of a thus correspond to a polygon ﬁber geometry. An important property of the polygon ﬁber is the length expansion at 08 < b < 1808 (in contrast to the star ﬁber, which contracts) and the more than 908 change in direction at b > 1808. These properties are essential for creation of swivel-linkers and for transformation of helical coils into loops by reversing the pitch of the ﬁlament. Swivel-linkers can make space for close contacts between two ﬁbers by separating two coils in a ﬁlament, thereby forming a pocket of 5–6 nucleosomes in which a nucleo- some from another ﬁlament can bind. Tandem repeats of the b sequences of such pockets allow for the side-by-side association of two ﬁbers like a zip, and the separation of helical coils by swivel-linkers can also provide space for the formation of double helices. It is tempting to speculate that tandem repeated DNA sequences, which often contain regularly positioned nucleosomes, might be stabi- lized and compacted by such types of ﬁber interactions. Another type of double ﬁber, which does not involve regularly spaced nucleosomes, is represented by the left- handed coiling of two polygon ﬁbers around each other by intercalation of every second nucleosome (Figure 5C). The positions of close contacts between the intercalated nucleosomes in this type of ﬁber appear to ﬁt well with the positions of alternating asymmetric protection of nucleo- somes in isolated nuclei against digestion by DNase I (30), which were shown to give rise to a dinucleosomal DNA repeat pattern (31–34). However, this type of double ﬁber may not provide the high degree of protection of the linkers, as inferred from analysis of the cleavage pattern, which was explained as a result of intercalation of nucleo- somes in helical crosslinker model with linkers being buried in the interior of the ﬁber (35). Although the inter- digitated model may not be able to fully explain these observations, the polygon ﬁber geometry has the potential to provide the chromatin with architectural properties by facilitating the formation of double ﬁbers, thus enabling the ﬁlaments to form 3D networks. A special property of the helical conformation formed by b sequence (333)n is the ability to compact to a high density by a decrease in a, followed by interdigitation. The extent of compaction ( 6 nucleosomes/11 nm), as well as the radius of gyration (32 nm), agrees with results of neutron scattering and scanning electron microscopy of isolated chromatin at elevated ionic strength (36), but the volume of the ﬁber is smaller because the cross-section area is smaller, and the shape of the ﬁber allows for interdigitation into sheets, which can be packed