Brownian dynamics simulations reveal regulatory properties of higher-order chromatin structures

Abstract

A present model of the higher-order chromosome organization suggests the organization of chromosome built up by loops. Here we focus on a single rosette-like part of the fiber and analyse the diffusion behaviour of small particles (corresponding to single proteins/protein complexes) and the accessibility of such particles in relation to the dynamic rosette structure. Surprisingly, although the diffusion pattern of the diffusing particles revealed free diffusion, an area of about 6–12 kbp in the innermost part of these domains becomes visible which is inaccessible even for small particles (corresponding to single proteins/protein complexes). A localisation of a promotor sequence in this area might silence the respective gene by the physical inaccessibility of this area for transcription factors. We conclude that the compartmentalisation of chromatin in domains of a specific dynamical three-dimensional (3D) structure might be of high functional importance.

Simulation techniques

Polymer model

To model the polymer backbone of the chromatin fiber, the continuous backbone mass model was used (Schöppe and Heermann 1999). In contrast to the commonly used bead-spring model, here non-spherical force fields are applied for the non-bonded interaction. By this procedure the possible anisotropy of a group of atoms which has to be course-grained in a construction unit can be taken into account. The simplest anisotropic geometrical object one can think of is an ellipsoid of rotational symmetric form, which is used also for the present simulations. Besides the bonded interaction between adjacent segments, also non-bonded interactions between different segments have to be taken into account. Here we apply the repulsive part (or, respectively, the attractive and the repulsive part) of the Lennard–Jones force field to model the non-bonded interactions between repulsive (respectively, attractive) segments. Debye-type electrostatic interactions are expected to be limited to a range <10 nm (compare Münkel and Langowski 1998) and can therefore be neglected for such large-scale simulations.

In the following a list of parameters and model parameterisations for the simulations is given:

• Segment diameter: 30 nm

• Kuhn length of 15 kbp corresponds to a segment length of 150 nm

• The harmonic bond potential is taken to be

  $$ U_{\text{bond}} (l) = {\frac{{k_{b} T}}{{2\delta^{2} }}}(l - l_{0} )^{2} $$

  (2)

  with δ = 0.1 and l ₀ = 150 nm at 310.15 K

• The angular and torsional potentials are taken to be 0. On this scale the chain is flexible.

• Repulsive segment potential:

  $$ U_{\text{rep}} (r) = \varepsilon \left( {{\frac{\sigma }{{r - r_{\text{segment}} }}}} \right)^{6} $$

  (3)

  with ε = 0.14k _B T at body temperature, σ = 15 nm, and r _segment = 15 nm being the fiber radius

• Cutoff for the repulsive potential is r _c = 8 nm (after the 15 nm fiber radius).

• Attraction segment potential

  $$ U_{\text{attr}} (r) = 4\varepsilon \left[ {\left( {{\frac{\sigma }{{r - r_{\text{segment}} }}}} \right)^{12} - \left( {{\frac{\sigma }{{r - r_{\text{segment}} }}}} \right)^{6} } \right] $$

  with ε = 7k _B T at body temperature and σ was taken to be 2.5, 10.0, 21.1 or 27.3 nm.

• Cutoff for the Lennard–Jones potential is r _c = 80 nm (after the 30 nm fiber diameter)

The spring constant δ = 0.1 was chosen such that the 150-nm segment was reasonably stiff and at the same time soft enough to ensure a reasonable integration time step. For the Lennard–Jones potential ε = 7k _B T was chosen because this proved to be the smallest potential depth for which the segments remain attractive at body temperature. The analogous reasoning applies to ε = 0.14k _B T for the repulsive potential.

Relaxation

A starting configuration for the Brownian dynamics simulation run is obtained by molecular dynamics (MD) relaxation as described in Odenheimer et al. (2005). After an initial rise, this is due to the random and hence mostly unphysical starting configuration, the maximum extension of the initial structure decays over a time of about 10,000 MD steps. After this time the distance drops no more, hence all attractive segments have found each other. The fully equilibrated structure is shown to be a rosette (compare Fig. 6).

Fig. 6

Left a starting configuration of a 60-segment chromatin fiber. The spheres represent the attractive sites. Centre: an intermediary configuration. This state of mainly two clusters of approximately equal size turns out to be a metastable state. Right: in the final state all attractive segments are concentrated in the centre. A rosette has formed

Brownian dynamics

The 1.2-Mbp chromatin rosette was put in a simulation box of 1 × 1 × 1 μm³ with periodic boundary conditions. The chromatin rosette had a diameter of about 800 nm. Thus the average distance between rosettes was about 200 nm. A closer packing of rosettes, i.e. a smaller simulation box, was not possible because of the necessity to generate a feasible starting configuration. The starting configuration was generated as follows. First an already condensed rosette (see previous section) was moved to the centre of the simulation box. Then a spherical shell with a radius of 900 nm was tessellated into a 1,024-on. At each vertex of the 1,024-on a particle was placed. This starting configuration was then equilibrated with dissipative particle dynamics (DPD) until a random distribution was achieved. This configuration was then the new starting configuration for one DPD run. For such simulation techniques the mass of the diffusing particles is immaterial; only for an estimation of the integration time step is the mass of the smallest diffusing protein (here streptavidin) regarded. The detailed justification of this method and its superiority over regular Brownian dynamics, especially for block copolymers, is described by Frenkel and Smit (2002).

Statistics

One simulation consists of one substance at one intra-rosette spacing. Ten runs were computed for each simulation; each run consists of 7,000 uncorrelated configurations. Simulations were done for all substances at all intra-rosette spacings. For the streptavidin and ribosomes, a run without the rosette structure was performed first to obtain simulation data in water only. The simulation box and duration was the same as for the latter runs with the rosette. The streptavidin water-only run was used to calibrate the time scale.