Tail-induced attraction between nucleosome core particles

We study a possible electrostatic mechanism underlying the compaction of DNA inside the nuclei of eucaryotes: the tail-bridging effect between nucleosomes, the fundamental DNA packaging units of the chromatin complex. As a simple model of the nucleosome we introduce the eight-tail colloid, a charged sphere with eight oppositely charged, flexible, grafted chains that represent the terminal histone tails. We show that our complexes attract each other via the formation of chain bridges and contrast this to the effect of attraction via charge patches. We demonstrate that the attraction between eight-tail colloids can be tuned by changing the fraction of charged monomers on the tails. This suggests a physical mechanism of chromatin compaction where the degree of DNA condensation is controlled via biochemical means, namely the acetylation and deacetylation of lysines in the histone tails.

Figure 1

(Color online) The eight-tail colloid. Eight polyelectrolyte chains are grafted onto the surface of an oppositely charged sphere. Both pictures show the identical colloid but highlight a different set of stiff harmonic bonds. The first set on the left is given by eight bonds from the colloidal center to the grafted monomers that enforce them to lie on the surface of the colloidal sphere. The second set of bonds is shown on the right that act between neighboring grafted monomers so that they sit on the vertices of a cube.

Figure 2

(Color online) Average diameter $D_{\mathit{max}}$ of the eight-tail complex—defined in Eq. (10)—as a function of $\kappa a$ and $Z$. For a neutral sphere, $Z=0$, the colloid is extended in the absence of salt, $\kappa a=0$ (cf. the example configuration on the upper right corner) and decreases with increasing salt to a structure like the one depicted on the left. A highly charged complex, e.g., $Z=300$, features collapsed tails for no salt (cf. the colloid on the lower right) and shows for high salt, i.e., strong screening, also configurations like the one shown on the left.

Figure 3

(Color online) The radial monomer densities of the tails grafted on a sphere with $Z=150$ as a function of distance above its surface. The different curves correspond to different salt concentrations leading to $\kappa \sigma =0$ (solid line), $\kappa \sigma =0.3$ (dashed line), $\kappa \sigma =0.4$ (dashed-dotted line), $\kappa \sigma =0.5$ (dotted line), and $\kappa \sigma =1.0$ (solid line).

Figure 4

(Color online) Mean force and resulting pair potential between two complexes. Top: force as a function of surface-surface separation of the cores for $\kappa \sigma =0.4$. The symbols represent the measured mean force, the line shows a mean square fit. Bottom: pair potential between two complexes obtained via integration and averaging over a sample of fitted force functions. For comparison also depicted are the force-distance and potential-distance curves (red) for two charged spheres without tails that carry the net charge $Z=70$ of the eight-tail colloid.

Figure 5

$A_2$ of the multi chain complex. The inlay shows the pair potential of two multichain complexes as a function of the surface distance for different values of $\kappa$. The values for $\kappa$ are $0.2{\sigma }^{-1}$ (black solid), $0.3{\sigma }^{-1}$ (black dashed), $0.4{\sigma }^{-1}$ (black dot-dashed), and $0.6{\sigma }^{-1}$ (gray solid).

Figure 6

(Color online) $A_2$ of the patch model. The inlay shows the pair potential of two patch complexes as a function of the surface distance for different values of $\kappa$. The values for $\kappa$ are $0.2{\sigma }^{-1}$ (black solid), $0.3{\sigma }^{-1}$ (black dashed), $0.4{\sigma }^{-1}$ (black dot-dashed), and $0.6{\sigma }^{-1}$ (gray solid).

Figure 7

(Color online) $A_2$ of the single chain model. The inlay shows the pair potential of two single chain complexes as a function of the surface distance for different values of $\kappa$. The values for $\kappa$ are $0.23{\sigma }^{-1}$ (black solid), $0.40{\sigma }^{-1}$ (black dashed), $0.51{\sigma }^{-1}$ (black dot-dashed), and $0.81{\sigma }^{-1}$ (gray solid).

Figure 8

(Color online) Range of the interaction. The upper line of plots represent the attractive part of the pair interaction of the full chain model on a logarithmic scale for different values of $\kappa$ as indicated by the labels. The red line indicates the impact of the screening and is proportional to $-C\exp (-\kappa d)$. The plots in the middle and at the bottom represent the patch model and the single chain model, respectively.

Figure 9

Monomer density of bridges for different values of the surface-surface separation. The values are $d=0$ (solid black line), $d=4\sigma$ (black dot dashed line), $d=7\sigma$ (gray solid line), and $d=9\sigma$ (gray dashed line). For $d=9\sigma$ the distribution shows two maxima, indicating the preference of the monomers to stay on the surface of one of the charged cores. The inlay shows the probability distribution of the closest approach of a monomers to the alien core for $d=6\sigma$ (solid line), $d=8\sigma$ (black dashed line), $d=9\sigma$ (grey solid line), and $d=13\sigma$ (gray dashed line); see text for details.

Figure 10

Separation of the total average of the interaction force (circle) into the part stemming from configurations that feature bridges (squares) and nonbridging configurations (diamonds).

Figure 11

(Color online) Frequency of bridges as a function of the surface-surface separation for different values of $\kappa$.

Figure 12

(Color online) Comparison of the decay parameters $m_{\mathit{sim}}$ obtained from simulated data to our theoretical estimate $m_{\mathit{theo}}$. See text for details.

Figure 13

(Color online) Interaction potential as a function of the surface-surface separation for $\kappa \sigma =0.4$ and different charge fractions $f$ of the tails. Also shown is a typical configuration for weakly charged tails, $f=0.17$.

Figure 14

(Color online) Equilibrium surface-surface separation of two eight-tail colloids as a function of the charge fraction $f$ of the tails. Also shown are example configurations at those optimal distances.

Figure 15

Pair interaction potential of the floating chain model (dashed line) and that of the eight-tail colloid with fixed grafting points (gray line). Both models feature a minimum of about $-5\sigma$ at approximately $4\sigma$ surface-surface separation. The minimum of the eight-tail colloid is slightly deeper.

Figure 16

Correlation function of bridge-forming tails for the two type of models as a function of the LJ time $\tau$ for the core-core distance $9\sigma$ at $\kappa =0.4{\sigma }^{-1}$. The correlation times for the fixed chain system is considerably larger than for floating chains.