e106 Nucleic Acids Research, 2007, Vol. 35, No. 16 which is available for keeping distance between the nucleosomes, thus depends on the size of a and the conformation of the linkers (open or closed). In the following, only the geometry of ﬁlaments with negatively closed linkers will be considered, if not stated otherwise. During uncoiling of the ﬁlament the two stacks of every second nucleosome rotate in opposite direction of the ﬁlament, arranging themselves in stacks parallel to the ﬁber axis at every value of a < 608 that equals the angle of a regular star, and at every value of a 5 608, which equals the angle of a regular polygon. With reference to this diﬀerence in symmetry ﬁbers will be called star ﬁbers for a < 608 and polygon ﬁbers for a 5 608 (compare Figure 1-II, a = 36 and 1088). The helical symmetry of ﬁbers formed by 08 < a < 1808 is characterized by the number of 3608 coils of the nucleosomes at the ﬁber periphery (in other words: how many nucleosomes it takes to traverse the ﬁber periphery 3608), as deﬁned by the helical twist of the nucleosomes (coils/nucleosome). As one nucleosome is placed at the periphery of the ﬁber for every turn (d) of the linkers, the ﬁbers are also characterized by the number of turns of the linkers, i.e. how many times the linkers have to turn 3608 to place all the nucleosomes in a coil (turns/coil). Star and polygon ﬁbers of the same axial symmetry have the same number of coils/nucleosome, but diﬀer in the number of turns/coil. Thus, while ﬁlaments with a = 368 (d = 1448) and a = 1088 (d = 728) both have 1/5 coil/nucleosome, the former has two turns/coil (5 1448 = 2 3608) and the latter has only one (5 728 = 3608). The effect of b on the conformation of the filament The rotational angle (b) between the ﬂat faces of conse- cutive nucleosomes varies with the length of the inter- connecting linker because the position of the core particle is ﬁxed relative to the minor and major grooves of the nucleosomal DNA. Insertion or deletion of base pairs in the linker therefore makes the nucleosome follow the right-handed helix of the linker DNA, rotating clockwise [b(+)] or counter-clockwise [b( )], respectively, in steps corresponding to the twist angle between the base pairs. The number of sterically allowed values of b depends on whether or not every second nucleosome is able to pass each other side-by-side at b = 1808 (Figure 1-III). When this is possible, the conformation of the ﬁlament is changed by clockwise rotation from a helix at b = 08 to a ribbon at b = 1808 with a concomitant 908 change in direction, because the pitch (p) of the dinucleosomes is reduced to zero. The conformation at b = 1808 is possible only for combinations of a and linker lengths, which allow for alignment of every second nucleosome side-by-side, requiring a NRL of at least 224 bp for a = 608 and 200 bp for a = 858. Clockwise rotation of the nucleo- somes beyond 1808 makes the ﬁlament coil left-handed in the opposite direction, but a full 3608 rotation is not PAGE 4 OF 15 possible because of the change from left- to right-handed coiling, except for ﬁlaments shorter than a single coil (Figure 1-IIIA). The size of b at which the change between right- and left-handed coiling takes place depends on a and the linker length. A star ﬁber with a = 368 and a NRL of 200 bp will only be able of right-handed coiling (Figure 1-IIIB). Conformations of filaments caused by homogeneous variations in b The conformational changes of the ﬁlament induced by rotation of the nucleosomes were studied by wire models, prefolded into pentagonal star ﬁber and polygon ﬁbers at b = 08, followed by rotation of the nucleosomes in steps of 368 (Figure 2). In the following, rotations are denoted according to the number of steps from b( 1) to b( 4) (+ for 08 < b < 1808 and-for 1808 < b < 3608) and b(0) and b(5) for 0 and 1808, respectively. The star ﬁber (a = 368) is unable to attain conforma- tions between b(+3) and b( 4), as this requires a NRL of at least 275 bp (Figure 2A). Conformations of a star ﬁber with linker lengths that are suﬃciently long to allow the nucleosomes to pass each other at b(5) have been studied by a computer model (12). In the polygon ﬁber rotation beyond 1808 is allowed because of the larger value of a, causing a shift from right- to left-handed coiling (Figure 2B). At b(5) the polygon ﬁber has the form of a ribbon with a0 = Z1 because the direction of the ribbon is perpendicular to the coiling of the DNA around the core particles. The ribbon is therefore not planar but coils right-handed. (For Z1 = 2g = 98, the ribbon will have 1/20 coil/nucleosome and 9.5 turns/coil). As b gets near to 3608 the polygon ﬁber approaches the so-called interdigitated model of the chromatin ﬁber, in which every ﬁfth nucleosome stacks upon each other, forming ﬁve left-handed coils of stacked nucleosomes (18) (except for the linkers not being bent in the present model). A NRL > 200 bp is required for the conformation at b( 1), as shown for two coils of the ﬁlament in Figure 2C. Provided that the linkers are long enough, a can be decreased in the polygon ﬁber at b(5) to generate a ribbon of stacked nucleosomes in which the orientation of the linkers alternates between positive and negative confor- mation (Figure 2D). This type of ribbon can be wrapped on the surface of a cylinder to form a ﬁber in which linker DNA zigzags up and down the helical axis, and such conformations have been described by two models as the twisted ribbon (19) and the helical ribbon (20), which are derived from ﬁlaments having an open and closed linker conﬁguration at b(5), respectively (data not shown). The axial symmetry of the ﬁber is changed when |b| is increased from 08 due to the decreasing size of a0. The axial symmetry of a polygon ﬁber is thus changed into that of a star ﬁber at a0 < 608. If, furthermore, the size of a is reduced, this will cause the ﬁber to become more compressed. The pentagonal polygon ﬁber can thus be of a is decreased below 1808 by bending of the linkers close to their entry/exit sites, causing the nucleosomes to rotate away from each other (arrows). (C) Rotation of the nucleosomes until a = 08, forming a stem conformation, which occupies 25–30 bp of the linker. (D–F) Bending of the linker may introduce a twist angle ( ) between the ﬂat faces of the nucleosomes. 146 bp nucleosomal DNA (gray); 54 bp connecting linker DNA (blue). The angle (758) between the terminals of the 146 bp nucleosomal DNA is shown by the white sectors on the core particles.