Physics behind the mechanical nucleosome positioning code

The positions along DNA molecules of nucleosomes, the most abundant DNA-protein complexes in cells, are influenced by the sequence-dependent DNA mechanics and geometry. This leads to the “nucleosome positioning code”, a preference of nucleosomes for certain sequence motives. Here we introduce a simplified model of the nucleosome where a coarse-grained DNA molecule is frozen into an idealized superhelical shape. We calculate the exact sequence preferences of our nucleosome model and find it to reproduce qualitatively all the main features known to influence nucleosome positions. Moreover, using well-controlled approximations to this model allows us to come to a detailed understanding of the physics behind the sequence preferences of nucleosomes.

Figure 1

(a) Schematic view of one half of the symmetric nucleosome; the vertical dashed line indicates the dyad axis. Key dinucleotides that position the nucleosome along DNA are indicated with their location inside the nucleosome constituting the “nucleosome positioning code”. (b) Visual representation of our model for nucleosomal DNA. Base pairs represented by rigid plates are frozen in an idealized superhelical shape.

Figure 2

The rotational degrees of freedom between neighboring bp in the rigid base pair model. Each base pair has a coordinate system (a), which can be used to describe the relative orientation between two plates. In our model we account only for energy contributions from (b) roll and (c) tilt but neglect contributions from (d) twist. Also the translational degrees are not considered.

Figure 3

(a) The probability to find AA, TA, or TT, and the probability to encounter GC at the full range of dinucleotide positions is shown. The solid and dashed vertical lines indicate minor and major groove bending sites (maximum negative and positive roll, respectively). The probabilities are in qualitative agreement with the well-known nucleosome positioning rules [5]. (b) Same as (a) but showing all four dinucleotide probabilities individually.

Figure 4

(a, b) Upper and lower bounds on the probabilities to have the dinucleotide AA, TA, TT, or GC at several dinucleotide positions on a nucleosome. Specifically, (a) depicts the first-order bounds and (b) the second-order bounds of the probability. The upper and lower bounds of the same dinucleotide have the same color (line style). (c) Difference between the upper and lower bounds of the probabilities to encounter AA, TA, TT, and GC at position 79 at increasing order. The difference, and thereby the effect of the neighbors $k$ steps away from the dinucleotide of interest, decreases exponentially as the order $k$ increases.

Figure 5

The probability of dinucleotide TA, its energy, and the average of the energies of its possible neighbors are shown for 10 different positions along the nucleosome, i.e., for one full DNA helical repeat. The numbers give absolute values, whereas the colors indicate how the corresponding value of the TA step compares with the values of all other possible dinucleotides at the same position. Yellow (light gray) colors represent relatively favorable values, red (dark gray) indicates unfavorable values. The probability follows mainly from a “mixing” of the colors of the corresponding dinucleotide energies and average neighbor energies. The table also provides subdivisions of the TA energies into roll and tilt contributions.

Figure 6

Same as Fig. 5 but for GC.

Figure 7

(a) Probability, (b) dinucleotide energy, and (c) average neighbor energy of all 16 dinucleotides for one DNA helical repeat. Yellow (light gray) denotes high probability and low energy, red (dark gray) low probability and high energy (relative to all other dinucleotides at the corresponding location). In addition, provided are subdivisions of dinucleotide energies into (d) roll and (e) tilt, and of neighbor energies into (f) roll and (g) tilt. The colors representing the probabilities can be seen as a “sum” of the colors of the dinucleotide energies and the average neighbor energies. The colors corresponding to the dinucleotide energies are the “sum” of the colors for roll and tilt energies. The same holds for the neighbor energies.

Figure 8

Original dinucleotide energy costs [left; same as Fig. 7], and energy cost with all stiffnesses set to 1 (right) are shown side by side. The strong similarity between the two tables reveals that stiffnesses play only a minor role for the dinucleotide positional preferences.

Figure 9

Same as Fig. 3 but with including the cross-terms [see Eq. (A3)]. The positioning rules (i.e., the relative behavior of the dinucleotide probabilities at different position) has stayed the same.

Figure 10

The exact probability and its average neighbor energy approximation to find AA steps at all dinucleotide positions. The approximation introduces an error that is nowhere larger than $3.5\%$.

Figure 11

The standard deviation [Eq. (B1)] divided by the mean [Eq. (B2)] of $C_p(x,y)$. As this ratio is very small at all positions $p$ the function $C_p(x,y)$ is nearly constant, explaining the high accuracy of the average neighbor energy approximation.

Figure 12

The probability to obtain AA at all dinucleotide positions at several temperatures.

Figure 13

The first-order bounds of the probability of encountering an AA step are shown at five different temperatures. At low temperatures, the bounds become significantly far apart from each other and only provide a qualitative description of the behavior of the probability as a function of position.

Figure 14

The second-order bounds of the probability of encountering an AA step are shown at five different temperatures. At all temperatures the method provides a quantitative description of the probability, clearly outperforming the second-order bounds.

Figure 15

(a) First-order and (b) second-order bounds on the probability to find dinucleotide AA at several dinucleotide positions on a nucleosome, in the limit of zero temperature. Higher-order bounds get increasingly sharper. At zero temperature the only possible DNA sequences are ground-state sequences, hence the bounds provide us statistics on the ground states of our system. When the lower and upper bounds are 0 and 1, AA is part of an unknown number of ground states at this position. If AA is part of all possible ground states, then the bounds are 1 and 1, and if it is not a part of the ground state, then they are 0 and 0.