Monte Carlo simulation of chromatin stretching

We present Monte Carlo (MC) simulations of the stretching of a single $30\mathrm{nm}$ chromatin fiber. The model approximates the DNA by a flexible polymer chain with Debye-Hückel electrostatics and uses a two-angle zigzag model for the geometry of the linker DNA connecting the nucleosomes. The latter are represented by flat disks interacting via an attractive Gay-Berne potential. Our results show that the stiffness of the chromatin fiber strongly depends on the linker DNA length. Furthermore, changing the twisting angle between nucleosomes from 90° to 130° increases the stiffness significantly. An increase in the opening angle from 22° to 34° leads to softer fibers for small linker lengths. We observe that fibers containing a linker histone at each nucleosome are stiffer compared to those without the linker histone. The simulated persistence lengths and elastic moduli agree with experimental data. Finally, we show that the chromatin fiber does not behave as an isotropic elastic rod, but its rigidity depends on the direction of deformation: Chromatin is much more resistant to stretching than to bending.

Figure 1

The geometry of the implemented fiber model. Incoming and outgoing linker DNA of length $l$ build the opening angle $\alpha$ and are set a distance $d$ off each other. Succeeding nucleosomes are twisted by the torsion angle $\beta$. Interactions between the linker segments are described by a Debye-Hückel potential, whereas nucleosomes interact via a Gay-Berne potential.

Figure 2

Mean effective opening angle ${\alpha }_{\mathrm{eff}}$ as a function of the initial opening angle ${\alpha }_{\mathrm{init}}$. During relaxation the initial opening angle ${\alpha }_{\mathrm{init}}$ converges to the effective opening angle ${\alpha }_{\mathrm{eff}}$ due to electrostatic repulsion and thermal fluctuations. For ${\alpha }_{\mathrm{init}}=26°$ the effective opening angle ${\alpha }_{\mathrm{eff}}$ is in the range of experimental values between 35° and 45° (Ref. 26).

Figure 3

(Color online) Stretching simulation of a fiber consisting of 100 nucleosomes (red), linker segments (blue) of repeat length $l=205\mathrm{nm}$, an opening angle ${\alpha }_{\mathrm{init}}=26°$ and a twisting angle $\beta =90°$. Starting the relaxation of the chromatin fiber from a twisted (a) and linear (b) initial conformation the chromatin fiber is equilibrated. (c) is a typical example of an equilibrated structure. In a second step an external pulling force of $F_{\mathrm{pull}}=5\mathrm{pN}$ is applied and further equilibration initiated. The application of the stretching force leads to an extension of the fiber as shown in (d) and (e).

Figure 4

Persistence length (a) and stretching modulus (b) as function of the nucleosome window length for different linker lengths and a torsion angle of $\beta =110°$, no stem. All functions show a plateau beginning at a window length of ten nucleosomes. We chose a nucleosome window of 14 nucleosomes for the calculation of the contour of the simulated fibers to ensure a nearly constant behavior in this region; open diamond: repeat length $l=192$ bp; filled circles: repeat length $l=190$ bp; open triangles: repeat length $l=187$ bp; filled diamonds: repeat length $l=185$ bp; open circles: repeat length $l=182$ bp.

Figure 5

Persistence length $L_p$ calculated by the fluctuation of the tangent angles ($A1$, open symbols, solid lines) and the fluctuation of the squared end-to-end-distance ($A2$, closed symbols, dashed lines line) as a function of different linker lengths $l$ and twisting angles $\beta$ without (a) and with (b) stem. All methods show the same general trend: With increasing linker length and decreasing torsion angles, the persistence lengths of the fibers decrease and the fiber becomes easier to bend. The fit of the extensible WLC [$A2$, Eq. (2b)], which agrees with all other methods within less than 10% and is used to calculate the persistence length $L_p$. $엯$: $A2$, inextensible WLC by numerical solution of Eq. (2a); ◇: $A2$, inextensible WLC by fitting Eq. (2a); $\rho$: $A2$, extensible WLC by fitting Eq. (2b); $●$: $A1$, fitting exponential decay Eq. (1).

Figure 6

Fit of Eq. (1) describing the exponential decay of the correlation of the tangent angles to the data of a 100 nucleosome chromatin fiber with 187 bp, $\beta =110°$, and no stem. For the first data points, the segments are too long compared to the contour length and as a result the correlation is unrealistically high. Thus these points are excluded from the fit and the peak cannot be reflected in the exponential fit. This fit also shows that the decay length of the fitted curve is slightly lower than can be expected from the data. $엯$: 187 bp, $\beta =110°$, no stem; solid line: $A1$, fit of exponential decay Eq. (1).

Figure 7

Fit of the Kratky-Porod Eq. (2a) describing an inextensible WLC and fit of Eq. (2b) describing an extensible WLC to the data of a 100 nucleosome chromatin chain with 187 bp repeat length, $\beta =110°$, and no stem. Both fits show an excellent agreement with the data. The ${\chi }^2$ of the extensible WLC is slightly lower, reflecting the more realistic description of the fiber. $엯$: 187 bp, $\beta =110°$, no stem; $●$: extensible WLC, Eq. (2b); solid line: inextensible WLC, Eq. (2a).

Figure 8

The persistence length for different sets of repeat lengths and twisting angles as a function of the initial opening angle ${\alpha }_{\mathrm{init}}$. An increase in the opening angle leads to a decrease in the persistence length corresponding to softer fibers. This tendency is stronger for short linker lengths, where electrostatic and nucleosomal interactions are more intense than for longer linker lengths. $엯$: repeat $\mathrm{length}=188$, twisting angle $\beta =110°$, no stem; ◇: repeat $\mathrm{length}=192$, twisting angle $\beta =120°$, no stem; $\rho$: repeat $\mathrm{length}=199$, twisting angle $\beta =110°$, no stem.

Figure 9

Total energy of a 100 nucleosome chromatin chain for a repeat length of 190 bp, $\beta =110°$, and no stem as function of the chain contour length. The elastic force constant $D$ is obtained by fitting Hooke’s Law $E=\frac{1}{2}D(L-L_0{)}^2+\mathrm{const}$ to the data.

Figure 10

(a) shows the stretching force as a function of the end-to-end distance for a 100 nucleosome chromatin chain of 190 bp repeat length, $\beta =110°$, and no stem. The linear behavior for intermediate forces shows that entropic contributions in the force range from $5–20\mathrm{pN}$ can be neglected. In the lower force range a nonlinear behavior can be seen due to the entropic energy contributions, which is in agreement with recent stretching experiments. For forces less than $0.5\mathrm{pN}$, the fiber shows a linear behavior in first order. (b) is a magnification of the force regime $0–0.5\mathrm{pN}$ for the chromatin chain as described in (a) and allows us to calculate the stretching modulus $ϵ=1.1\pm 0.3\mathrm{pN}$ for the entropic dominated part of the force regime, which is less than 2% of the stretching modulus of the intermediate force regime.

Figure 11

Stretching force plotted vs the contour length of a 100 nucleosome chromatin fiber for a repeat of 190 bp, $\beta =110°$, and no stem. The stretching modulus $ϵ$ is calculated from the fit of $F=\frac{ϵ}{L_0}L-ϵ$ to the data of the force region between 5 and $20\mathrm{pN}$ and is in agreement with data derived from Fig. 10a.

Figure 12

Stretching modules $ϵ$ calculated by a linear fit to the force-extension curve (open symbols, solid lines) compared to the method of using the theorem of equipartition ($B2$, closed symbols, dashed lines) for simulation runs without stem (a) and with stem motif (b) as a function of different linker lengths $l$ and torsion angles $\beta$. Both methods show a good agreement, which supports the assumption that the entropic contribution to the total energy can be neglected in this force region. Circles: $\beta =130°$; diamonds: $\beta =110°$; triangles: $\beta =90°$.

Figure 13

Stretching modules $ϵ$ calculated by fitting Hooke’s Law ($B1$, open symbols, solid lines) and by using the theorem of equipartition ($B2$, filled symbols, dashed lines) for simulation runs without stem (a) and with stem motif (b) as a function of different linker lengths $l$ and torsion angles $\beta$. Both methods show a good agreement except for small repeat lengths and high torsion angles. Independent of the presence of a stem the stretching modulus $ϵ$ of the fibers decreases with increasing linker length $l$ and decreasing torsion angle $\beta$ in agreement with Fig. 12. Symbols and lines as in Fig. 12.

Figure 14

Examination of the directional dependence of the fiber stiffness by comparing the persistence lengths for different linker lengths $l$ and twisting angles $\beta$. The solid lines represent the persistence lengths $L_{p,A1}$ calculated directly by fluctuation of the tangent angles ($A1$, open symbols, solid lines) whereas the dashed lines show the persistence lengths $L_{p,B1}$ derived under the assumption that the fiber behaves like a homogeneous isotropic elastic rod with circular cross section using the fit of Hooke’s Law to the stretching data ($B1$, closed symbols, dashed lines). $L_{p,B1}$ is six to ten times higher compared to $L_{p,A1}$ for simulations without stem [(a) and (c)] and with stem [(b) and (d)]. Symbols in (a) and (b) circles: $\beta =130°$, diamonds: $\beta =110°$; triangles: $\beta =90°$; (c) circles: repeat length 188 bp, $\beta =110°$, diamonds: repeat length 192 bp, $\beta =120°$; triangles: repeat length 199 bp, $\beta =110°$; (d): circles: repeat length 201 bp, $\beta =110°$, diamonds: repeat length 205 bp, $\beta =120°$; triangles: repeat length 212 bp, $\beta =110°$.

Figure 15

Examination of the stem influence for simulations with equal repeat length. For these simulations the twisting angle $\beta$ is adjusted to its corresponding linker length: One helical turn of $ds\mathrm{DNA}$ corresponds to a length of 10.5 bp. (a) and (b) show that the persistence lengths of fibers with stem (closed circles, dashed lines) are higher than for fibers without stem (open squares, solid lines). This effect is stronger for short repeats and weakens with increasing repeat length. The peaks show that the twisting angle strongly influences the stiffness of the fiber and can lead to stiffer fibers, although for longer linker lengths a softer fiber could be expected.