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Flexible Histone Tails in a New Mesoscopic Oligonucleosome Model

Department of Chemistry and Courant Institute of Mathematical Sciences, New York University, New York, New York 10012

Correspondence: Address reprint requests to Tamar Schlick, Tel.: 212-998-3116; E-mail: schlick{at}nyu.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION FLEXIBLE-TAIL MODEL SIMULATION DETAILS RESULTS SUMMARY AND FUTURE APPLICATIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
We describe a new mesoscopic model of oligonucleosomes that incorporates flexible histone tails. The nucleosome cores are modeled using the discrete surface-charge optimization model, which treats the nucleosome as an electrostatic surface represented by hundreds of point charges; the linker DNAs are treated using a discrete elastic chain model; and the histone tails are modeled using a bead/chain hydrodynamic approach as chains of connected beads where each bead represents five protein residues. Appropriate charges and force fields are assigned to each histone chain so as to reproduce the electrostatic potential, structure, and dynamics of the corresponding atomistic histone tails at different salt conditions. The dynamics of resulting oligonucleosomes at different sizes and varying salt concentrations are simulated by Brownian dynamics with complete hydrodynamic interactions. The analyses demonstrate that the new mesoscopic model reproduces experimental results better than its predecessors, which modeled histone tails as rigid entities. In particular, our model with flexible histone tails: correctly accounts for salt-dependent conformational changes in the histone tails; yields the experimentally obtained values of histone-tail mediated core/core attraction energies; and considers the partial shielding of electrostatic repulsion between DNA linkers as a result of the spatial distribution of histone tails. These effects are crucial for regulating chromatin structure but are absent or improperly treated in models with rigid histone tails. The development of this model of oligonucleosomes thus opens new avenues for studying the role of histone tails and their variants in mediating gene expression through modulation of chromatin structure.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION FLEXIBLE-TAIL MODEL SIMULATION DETAILS RESULTS SUMMARY AND FUTURE APPLICATIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The hierarchical process through which nuclear double-stranded DNA packs itself into micrometer-sized nuclei of cells while maintaining its template-directed gene expression activities is remarkable (1). At the first level of compaction, the DNA wraps itself around approximately cylindrical-shaped protein aggregates known as nucleosomes. This DNA/nucleosome array then folds itself into the chromatin fiber, which has a nominal diameter of 30 nm. The chromatin fiber undergoes several higher levels of folding thereafter, culminating in the highly compact chromosomes.

The crystal structure of the nucleosome and the DNA wrapped around it was first solved in 1997 (2), and more recently determined at high resolution (3,4). The nucleosome comprises two copies each of the H2A, H2B, H3, and H4 histones, a single copy of either an H1 or H5 linker histone, and the DNA double-helix which makes 1.75 turns around the curved face of the nucleosome core (see Fig. 1). The central domains of the H2A, H2B, H3, and H4 histones are fairly rigid and define the nucleosome core, while their terminal domains, the histone tails, are much more dynamic and extend outward. The linker histone resides in the triangular space formed between the nucleosome core and the entering and exiting linker DNAs.

View larger version (60K): [in this window] [in a new window]   FIGURE 1  Nucleosome core modeling using DiSCO. The top figure shows the crystal structure of the nucleosome without the histone tail residues (nucleosome core). The bottom figure shows our model nucleosome core with discretized charges. The charges on the nucleosome core are deliberately shown smaller than their excluded volume for clarity, and they are color-coded according to their magnitude relative to the electronic charge (e), as shown in the color chart. The surface of the nucleosome core has been displaced inwards by 2 Å to allow visibility of the charges.

Although the core histone domains clearly maintain the tightly wound DNA supercoil around the nucleosome, the positively charged linker histones are crucial for compacting chromatin by reducing the separation angle between the incoming and outgoing linker DNAs (10,11). Moreover, the histone tails critically regulate chromatin structure and function by charge modification. Namely, the tails' positive charge and highly flexible nature allows them to extend and electrostatically interact with the negatively charged regions of neighboring nucleosomes and proteins. This shielding can bring neighboring nucleosomes into closer spatial proximity. Altering the positive charge on the histone tails can thus substantially modify this attraction between nucleosomes. Indeed, acetylation of certain residues on the histone tails (which partially neutralizes the histone tails' charge) is believed to be the primary cause for the partial unfolding of the chromatin fiber resulting in an enhanced transcription of genes (12,13). Other covalent modifications of histone tails involve methylation, phosphorylation, and ubiquitination (14,15). Collectively, various residue-specific modifications play important roles in selectively modulating the structure of chromatin either through direct modification of the physical histone tail interactions or via the recruitment of chromatin modifying proteins; this is known as the histone-code hypothesis.

Undoubtedly, understanding the physical mechanism through which histone tail modifications lead to modulation of chromatin structure is important from the point of view of transcriptional activation and repression of genes. Due to the natural variability in the length, position, and constitution of the histone tails, as well as the anisotropy in the position and orientation of nucleosomes within the chromatin fiber, we anticipate tail-specific roles. Thus, elucidating the positional distribution of individual histone tails and the role of each unit within chromatin is crucial. Indeed, innovative experimental studies attempting to systematically dissect the role of each histone tail are only beginning to appear (16–19). Still, due to the highly dynamic nature of histone tails, only time and configuration-averaged properties of the histone tails are generally obtained, rather than transient dynamics of each histone tail. Thus, theoretical and computational techniques subject to the usual limitations and approximations have great potential to contribute to an understanding of the function and properties of histone tails.

Several models of chromatin structure and dynamics have been developed and examined. The existing theoretical models can be broadly classified into two categories:

1. Simple but insightful wire-frame, mechanical models (9,20,21), which relate the structure of chromatin only to a few geometrical parameters such as the linker DNA length, linker DNA entry/exit angle, and the nucleosome/nucleosome twisting angle; and
   
2. Coarse-grained dynamic models of oligonucleosomes (22–30), which include stretching, bending, and twisting of linker DNA and also account for distance- and orientation-dependent interactions among linker DNA and nucleosomes using effective energetic potentials.
   

These representations are typically coupled to computational sampling techniques such as Monte Carlo and Brownian dynamics to determine the structural and dynamical properties of the oligonucleosomes. The models range in complexity from nucleosomes treated as spherical particles (22) to models that treat both the excluded volume surface as well as the electrostatic potential of nucleosomes to high accuracy (24–30).

Our group has recently begun to incorporate the effect of histone tails into a macroscopic chromatin model (30) developed several years ago (24), where the nucleosome cores were treated as rigid bodies and the linker DNAs as elastic chains. Even though the histone tails were approximated as rigid charged entities that protrude outwards from the rigid nucleosome core, the study nonetheless demonstrated theoretically the role of tails in mediating the salt-dependent folding of oligonucleosomes via electrostatic interactions with neighboring nucleosome cores.

Here, we extend the recent coarse-grained model of oligonucleosomes (30) to incorporate flexibility into each histone tail. This is accomplished by representing the histone tails as a set of connected charged beads with optimized salt-dependent charges at the bead centers that reproduce the electric field surrounding the charged tails. The nucleosome cores and linker DNA are modeled as in earlier studies (27,30), where the nucleosome core is represented as a set of discrete optimized charges on its surface with appropriate excluded volumes, while the linker DNA is modeled using a discrete elastic chain model with physically derived parameters (31). The effect of linker histone H1 or H5 is not yet considered. Brownian dynamics simulations with complete hydrodynamic interactions among the DNA, nucleosome cores, and histone tails are employed for capturing the dynamics and structure of oligonucleosomes of different sizes under different salt conditions.

We demonstrate that the new model of oligonucleosomes reproduces well a range of available experimental data, including salt-dependent variation of maximal extension of mononucleosomes, self-diffusion coefficients of dinucleosome and trinucleosome, and sedimentation coefficients of 12-unit nucleosomal arrays, and better than that of the previous model with fixed histone tails (30). Predictions concerning the positional distribution of each histone tail at low and high salt are also presented, illustrating the highly dynamic and salt-dependent nature of histone tails, which allows them to mediate internucleosomal interactions to a moderate degree and, in the case of H3 tails, partially shield the electrostatic repulsion between the linker DNA emerging from a nucleosome core. We conclude by suggesting how the new model may be used to study the role of histone tails and their modifications and variants in modulating chromatin structure and dynamics.

	FLEXIBLE-TAIL MODEL

TOP ABSTRACT INTRODUCTION FLEXIBLE-TAIL MODEL SIMULATION DETAILS RESULTS SUMMARY AND FUTURE APPLICATIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Our coarse-grained model of chromatin (oligonucleosomes) consists of three components: nucleosome core, DNA linker, and histone tails. Each component requires a different modeling strategy. Building on our earlier model of chromatin (30), we regard the histone tails as flexible entities rather than rigid protrusions from the nucleosome core. Details of the linker DNA and nucleosome core modeling are given elsewhere (24,25,27,30), so their modeling is only briefly described. The histone tail modeling is provided in greater detail.

Nucleosome core and DNA linker
Model
Fig. 1 illustrates the basic nucleosome core comprising the eight core histones H2A, H2B, H3, and H4 (excluding their terminal regions) accompanied by 1.75 turns of DNA wrapped around it. Our model is based on the recent crystal structure of the nucleosome by Davey et al. (3) with PDB code 1KX5. Table 1 lists the protein residues that constitute our definition of the histone tails. Briefly, these include N-terminal portions of all the core histones and short C-terminal portions of H2A. Since the nucleosome core is fairly rigid, the motion of the nucleosome core is effectively modeled using rigid-body dynamics. For hydrodynamic purposes, the nucleosome cores are considered as spheres with a hydrodynamic radii R_c (see full parameter values in Table 2).

The linker DNA connecting two adjacent nucleosome cores is modeled using the discrete elastic chain model of Allison et al. (31,40), as sketched in Fig. 2. Thus, double-stranded DNA is modeled as a chain of charged beads where each bead represents a 3-nm-long strand of relaxed DNA. The hydrodynamic radius associated with each linker bead is represented by R_l, and each linker bead is assigned a salt concentration-dependent negative charge according to the procedure of Stigter (41). The linker DNA is governed by stretching, bending, and twisting potential energy terms.

View larger version (36K): [in this window] [in a new window]   FIGURE 2  Discrete elastic bead model for linker DNA. The top figure shows the atomistic linker DNA while the bottom figure shows our model.

View larger version (28K): [in this window] [in a new window]   FIGURE 3  (a) Schematic representation of our model nucleosomal arrays with a total of N nucleosome cores and linker DNA beads. Each linker DNA is represented by six beads in red for this illustration. For clarity, nucleosome cores are drawn as gray cylinders while only one out of 10 histone tails with five beads is shown in blue. Missing portions between the second nucleosome and the last are shown as thick dots. (b) Schematic representation of the nucleosome core without histone tails showing the wound DNA supercoil and the relative positions of the entering and leaving linker DNA. (c) Model geometry showing the coordinate systems adopted for modeling linker DNA-nucleosome mechanics.

The bending and twisting potential energies of the nucleosome-linker DNA complex are expressed in terms of Euler angles {_i, ß_i, _i} and , which transform one coordinate system to the next. The two Euler angles and their transformations are given below.

(1)

To ensure that no torsion is introduced in the Euler transformation , we set . The mathematical details of computing the coordinate system vectors the associated Euler angles is provided elsewhere (25).

Energetics
The potential energies associated with linker DNA stretching, bending, and twisting are respectively given by

(2)

(3)

(4)

where h, g, and s are the stretching, bending, and torsional force constants; l_i and l₀ are the linker DNA segment lengths and the corresponding equilibrium lengths, respectively; _i, ß_i, _i, and are Euler angles.

The linker DNA beads and the discrete charges of the nucleosome cores interact via a combination of electrostatic and excluded volume interactions, which are respectively modeled using the Debye-Hückel () and Lennard-Jones () potentials given by

(5)

(6)

where q_i and q_j are the charges on two interacting beads separated by a distance r_i,j in a medium with a dielectric constant of and an inverse Debye length of ; ₀ is the electric permittivity of vacuum; is the effective diameter of the two interacting beads; and k_ev is an energy parameter that controls the steepness of the excluded volume potential. The electrostatic energy of the nucleosome core and linker DNA system is given by the superposition of three electrostatic interactions: linker/linker, linker/core, and core/core interactions, as given by

(7)

where q_L represents the effective charges on the linker DNA beads, and q_Ck represents the k^th discrete charge on the nucleosome core. The excluded volume energy of the nucleosome-linker DNA system is given by

(8)

where the two terms represent excluded volume energies for linker/nucleosome and nucleosome nucleosome interactions, respectively. The parameters _lc and _cc represent the excluded volume parameters for the two types of interactions, respectively. No excluded volume interactions are considered for linker/linker interactions since they remain separated due to strong electrostatic repulsions. The values for all parameters mentioned in this section are given in Table 2.

Histone tails
Our model for the histone tails, which we term the protein-bead model, began with the thesis work of Qing Zhang (42). It is obtained in two steps (see Fig. 4). First, the fully atomistic histone tails are simplified using the subunit model of Levitt and Warshel (43–45). Second, we build the protein bead model from the subunit model via a matching procedure where the force field parameters of the protein bead model are adjusted to mimic the Brownian dynamics of the subunit model corresponding to that histone tail. This additional level of coarse-graining avoids severe size inconsistency between the tail subunits and the other units, and also expedites computations substantially.

View larger version (38K): [in this window] [in a new window]   FIGURE 4  Two-step modeling of histone tails. The top figure shows the atomistic description of the H3 histone tail. The middle figure portrays the subunit model corresponding to that tail. The bottom figure shows the protein-bead model developed in this study derived from the subunit model.

For the protein bead model, five adjacent beads of the subunit model are represented by a single macro bead with an effective hydrodynamic radius of R_t. Thus, 50 tail beads per nucleosome are employed to model the 250 or so histone tail residues that comprise each nucleosome. The center of each macro bead is placed at the C^ß atom of the third amino acid. For the matching procedure, each bead is assigned a charge equal to the sum of the charges on the five amino acids it represents, as tabulated in Table 1. Later, we will show how these charges are further adjusted for our final oligonucleosome simulations. The total intramolecular energy of the protein-bead histone tail is given by four components: electrostatics, excluded volume, bond-stretching, and bond-angle bending, which are described in detail next.

The excluded volume interactions are given by the Lennard-Jones potential with fixed parameters of k_evt and _tt. The electrostatic interactions between protein beads are modeled via Coulombic terms for the matching procedure only. In all our subsequent simulations of oligonucleosomes, we employ the Debye-Hückel potential to model the histone tail electrostatics. The bond-stretching and bond-bending potentials are given by quadratic functions of the deviations in the bond-length and bond-angle from their equilibrium values, respectively. For a histone tail consisting of N_b protein beads, the adjustable parameters in our model are: the equilibrium bond distance between beads, l_i0, and the associated force constants k_bi, where i = 1, ···, N_b – 1; and the equilibrium bond angle, _i0, and the associated force constants, k_i, where i = 1, ···, N_b – 2.

For our protein bead model to realistically represent the fully atomistic histone tails, we require it to closely reproduce the dynamical and configurational properties of the subunit model (assuming that the subunit model reasonably represents polypeptide flexibility in solution). To this end, we seek the most suitable values of l_i0, , _i0, and for the protein bead model to yield the same bond-length and bond-angle distributions as those observed between every fifth bead in the Brownian dynamics simulations of the subunit model. Accordingly, l_i0 is taken as the average distance between the subunit beads (5i – 2) and (5i + 3) observed in the simulations, and _0i is taken as the average angle between the (5i – 2), (5i + 3), and (5i + 8) subunit beads observed in the simulations. The chosen values of l_i0 and _i0 are shown in bold in the third column of Tables 3 and 4, respectively.

k_i

We next compute the electric field surrounding a histone tail from the superposition of Debye-Hückel potentials arising from the bead charges in the protein bead model of a histone tail at different salt conditions. Recall that each protein bead carries a charge equal to the sum of the charges of its constituents. We found that the electric field of the protein bead model consistently underestimated the electric field computed for the fully atomistic histone tail (by solving the Poisson-Boltzmann equation with QNIFFT 1.2 (37)) at high salt conditions. On the other hand, the protein bead model overestimated the electric field of the histone tails at low salt conditions. This trend can be explained by the variation in the magnitude of the effective charges on the nucleosome core and linker DNA beads with salt concentration; effective charges higher in magnitude than the actual charges are required to accurately reproduce the surrounding electrostatic potential at high salt, and vice versa. It was therefore deemed necessary to rescale the charges on the histone tail beads. We found that rescaling the bead charges by factors of 0.75, 0.80, 0.90, 1.05, and 1.2 corresponding to salt concentrations of 0.01, 0.02, 0.05, 0.1, and 0.2 M yielded the best possible matches between the electric fields of each histone tail and its corresponding protein bead model.

Energetics
Once the protein bead model for each histone tail has been built and parameterized, the histone tails must be properly attached to the nucleosome core. We use a simple harmonic spring that attaches the first bead of each histone tail (as given in Table 1) to its idealized position in the nucleosome crystal structure (see Fig. 3). The stretching energy of histone tail beads is therefore composed of two terms: stretching of tail beads with respect to its immediate neighbors within the same tail, and an additional contribution due to the displacement of the histone tail bead from its assigned attachment site, as given by

(9)

In the first term, N_t represents the number of histone tails per nucleosome core, N_bj is the number of beads in tail j, is the stretching constant of the bond between the k and k + 1 beads of the j^th histone tail, and l_ijk and l_jk0 represent the distance between consecutive tail beads k and k + 1, and their equilibrium separation distance, respectively.

In the second term, h_tc is the stretching bond constant of the spring attaching the histone tail to the nucleosome core, r_ij is the position vector of the first tail bead in the coordinate system of its parent nucleosome, and r_ij0 is its ideal position vector in the crystal configuration. The intramolecular bending contribution to the histone tail energies, E_tB, is given by

(10)

where _ijk and _jk0 represent the angle between three consecutive tail beads k, k + 1, and k + 2, and their equilibrium angle, and is the corresponding bending force constant.

The total electrostatic energy of oligonucleosomes due to the histone tails is given by the superposition of various Debye-Hückel potentials energy terms arising from the interaction of histone tail charges among themselves and with nucleosome and linker bead charges, as given by

(11)

where q_Tij represents the magnitude of the charge (in terms of the electronic charge e) on the j^th protein bead in the i^th histone tail.

The first term in Eq. 11 represents electrostatic energy arising from histone tail/linker DNA bead interactions. The second term in the equation arises from interactions between histone tail charges and the charges on its parent-nucleosome. Note that the first bead of each histone tail, which is attached to the nucleosome core, does not interact with that nucleosome core's charges. The third term represents interactions between charges on the histone tails and non-parent nucleosomes' charges. The fourth term represents intermolecular electrostatic interactions across different histone tails belonging to different nucleosome cores, and the fifth term represents intermolecular interactions between histone tails belonging to the same nucleosome cores. The last term represents intramolecular interactions between charges within the same histone tails; three consecutive charges within the histone tails do not interact electrostatically with each other.

The excluded volume interaction energy between the histone tail beads and the rest of the oligonucleosome components is given similarly by

(12)

where _tl, _tc, and _tt are excluded volume size parameters for tail/linker, tail/core, and tail/tail interactions, and k_evt is excluded volume energy parameter for tail/tail interactions. As in Eq. 11, the six terms in Eq. 12 respectively represent the van der Waals energies of histone tail beads interacting with: the linker DNA beads, parent nucleosomes charges, non-parent nucleosome charges, histone tail beads associated with non-parent nucleosome cores, beads of other histone tails belonging to the parent nucleosome core, and beads within the same histone tail. Again, three consecutive beads within a histone tail do not interact with each other, and the histone tail bead attached to the nucleosome cores does not interact with that core. Fig. 5 sketches our model integrating all the three components of the system.

View larger version (48K): [in this window] [in a new window]   FIGURE 5  Repeating motif of an oligonucleosome containing 51-bp linker DNA. The top figure shows its atomistic representation, while the bottom figure shows its coarse-grained representation via flexible-tail model. The histone tails are color-coded as follows: H3 (blue), H4 (green), H2A (yellow), and H2B (red); the nucleosome cores and linker beads are colored gray and red, respectively.

Forces and torques
The total energy of the oligonucleosome is given by the sum of all the different interaction energies above:

(13)

The forces on the system components are defined by the relation

(14)

where r_i and F_i are the position vector and deterministic force acting on component i, respectively. Expressions for forces on the DNA linker and the rigid nucleosome core due to stretching, bending, and twisting of DNA, and electrostatic and excluded volume interactions, are derived elsewhere (24). The internal forces arising within the oligonucleosome as a result of stretching, bending, electrostatic, and excluded volume interactions of the histone tails may also be derived in an identical fashion. The torque on the linker DNA beads, which arises due to the twisting potential (Eq. 4), acts only in the longitudinal direction a and is given by

(15)

The torque on each nucleosome core (_i), which acts along all three coordinate axes, is given as a sum of the three terms

(16)

where is the torque acting on the nucleosome core due to forces applied at a positional vector r_i away from the center of mass. The next two torque terms are associated with the bending and torsional potentials of DNA linkers entering or leaving the core. We again refer readers to Beard and Schlick (24) for more details on the two additional contributions. Note that no torque acts on the histone tail beads.

	SIMULATION DETAILS

TOP ABSTRACT INTRODUCTION FLEXIBLE-TAIL MODEL SIMULATION DETAILS RESULTS SUMMARY AND FUTURE APPLICATIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Brownian dynamics algorithm
We employ Brownian dynamics (BD) with complete hydrodynamics to simulate the dynamics of oligonucleosomes. A second-order, Runge-Kutta-based Brownian dynamics algorithm following the approach of Iniesta and de la Torre (48) is employed to update the rotational and position vectors of the various components of the oligonucleosomes. This approach is based on predicting the value of an arbitrary time-varying vector p at time t + t from its value at time t using the average of the time derivatives of p at times t and t + t, as given by

(17)

where p* is the predicted p(t + t). The procedures for obtaining p* and p(t + t) (using Eq. 17) are referred to as first and second-order updates, respectively.

First-order rotational updates of the coordinate frames of reference a_i, b_i, and c_i are given by

(18)

where _xi(t) represents the change in the rotational state of the i^th bead about its original coordinate system x_i = {a_i, b_i, c_i}, _xi(t) is the instantaneous torque on bead i along vector x_i at time t, t is the BD timestep, and is the rotational diffusion matrix. For hydrodynamic purposes, the nucleosome core is treated as a sphere with a rotational diffusion coefficient given by

(19)

where is the solvent viscosity. For linker DNA beads, only rotation about the a_i axis is allowed and the rotational diffusion coefficient along that axis is

(20)

The term in Eq. 17 represents random (Brownian) rotations applied to the core along all three reference axes, or along the a_i axis only for the linker DNA beads. The random rotations obey the fluctuation-dissipation relation given by

(21)

where _ij is the Kronecker delta-function. First-order updates of the position vectors are now computed by using the Ermak-McCammon algorithm (49), as given by

(22)

where D_ij(t) is the configuration-dependent Rotne-Prager hydrodynamic diffusion tensor, t is the timestep, and represents the random fluctuations obeying the fluctuation-dissipation relation

(23)

Determination of requires factorization of the matrix D such that LL^T, where L is a lower triangular matrix. We employ the standard Cholesky decomposition method to achieve this (50). The vectors a_i, b_i, c_i, , and are updated according the procedure described in Beard and Schlick (51).

The second-order rotational updates are computed as

(24)

where *_xi is the torque acting on the system components after the first-order translational and rotational updates. Second-order updates of the position vectors are computed as (52)

(25)

where F_i* represents the force on bead i computed according to the temporary position vectors {r*} obtained after the first-order translational and rotational updates. To save computational time, the hydrodynamic tensor is computed only every 100 timesteps and is thus assumed to be fixed within that time frame. Our results remain unaffected with a change in this frequency as long as the frequency of recalculations is not decreased further, as found in simulations of supercoiled DNA (52).

Hydrodynamic interactions
The hydrodynamic diffusion tensor employed within our BD code is given by

(26)

where D_ij is a 3 x 3 matrix representing the interactions between beads i and j. Each D_ij can be calculated using modified forms of the Rotne-Prager tensor for unequal size beads (53–55). We consider two cases, nonoverlapping and overlapping beads. In the former,

(27)

where no overlap means 2r_ij > (_i + _j), and _i and _j are the sizes of the two beads; is the solvent viscosity; r_ij = r_i – r_j is the vector joining the center of mass of the two beads i and j; r_ij is the distance between the two beads; and I is the 3 x 3 unit tensor. For overlapping beads (2r_ij < (_i + _j)), the hydrodynamic tensor is given by

(28)

Note that the top expression in Eq. 28 has been theoretically proven for equal-sized beads (53). No expression is available for the case of non-equal-sized beads, but we have found that with a choice of _eff = , the tensor remains positive-definite as it should. Other researchers have likewise proposed similar expressions (55). In our simulations, _i equals R_c, R_l, or R_t, depending upon whether the component is a nucleosome core, linker DNA bead, or histone tail protein bead.

	RESULTS

TOP ABSTRACT INTRODUCTION FLEXIBLE-TAIL MODEL SIMULATION DETAILS RESULTS SUMMARY AND FUTURE APPLICATIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
We show here how the flexible-tail model for oligonucleosomes reproduces existing experimental data, notably better than its predecessor, which considered the histone tails as rigid bodies (fixed-tail model). We also present predictive data on tail distribution as a function of salt. All simulations were performed at 20°C to be consistent with the experiments conducted at 20–23°C (56–60).

Histone tail configurations
Recently, Livolant and co-workers (17,56) undertook an exhaustive experimental study on the conformation of histone tails and their dependence on salt (NaCl) concentration of the surrounding buffer. They employed nucleosome core particles with intact histone tails carrying 146 ± 3 bp DNA and performed small-angle X-ray scattering experiments to determine their radius of gyration, R_g, and maximum extension of the particle, D_max. The radius of gyration of nucleosome particles, which was obtained from the scattering intensity according to the Guinier approximation, is defined as

(29)

where (r) is the mass (electron) density of the nucleosome at a distance r from its center of mass and is the volume of the nucleosome. The other quantity of interest, D_max, is defined as the distance at which the function describing the distribution of the intramolecular distance r between scattering elements within the nucleosome becomes zero. In essence, D_max captures the maximum observable separation distance between histone tails and hence provides information on whether the histone tails are collapsed onto the nucleosome core or extended.

To compute D_max and R_g, Brownian dynamics simulations were performed on mononucleosomes without the linker DNA at varying monovalent salt concentrations in the range C_s = 0.01–0.3 M. The wound DNA length utilized in the experiments exactly matches the values in the crystal structure of the nucleosome on which our model is based (3). The simulations were performed over 50 µs at six different salt concentrations within the same concentration range investigated by the experimentalists, and D_max and R_g were computed every 100 timesteps and averaged over the second half of the simulation run. D_max was calculated as the maximum observable distance between two histone tail beads, and R_g was computed from the formula

(30)

where N_cry is the number of atoms constituting the nucleosome core crystal structure excluding those belonging to the histone tails, m_i is the molecular weight of the individual atoms, r_i is the distance of the atoms from the center of mass of the nucleosome, M is the mass assigned to each histone tail bead (equal to the total molecular weight of the 10 histone tails divided by the total number of protein beads representing them), and R_i,j is the distance of the histone tails protein beads from the center of mass of the nucleosome. The first and second terms in the numerator and denominator of Eq. 29 therefore compute the contribution to R_g from the tail-less nucleosome core and the histone tails, respectively.

Fig. 6 compares our computed salt-dependence of D_max values to those obtained by Livolant and co-workers (56). Both studies predict a moderate increase in the magnitude of D_max with salt concentration. The magnitude of this increase in D_max from the lowest to the highest salt concentrations explored here (D_max 2 nm) also agrees well with experiments. The data thus suggest that the histone tails maintain a fairly compact conformation at low salt conditions (relatively small values of D_max) and gradually extend outwards as the salt concentration increases.

View larger version (12K): [in this window] [in a new window]   FIGURE 6  Variation of the maximum nucleosome extension (top) and the radius of gyration (bottom) versus the monovalent salt concentration. Simulation results are represented by circles and experimental data from Bertin et al. (56) as squares. The dashed line for the simulation results serves as a guide to the eye.

View larger version (35K): [in this window] [in a new window]   FIGURE 7  Positional distribution of histone tail beads projected onto the a-b plane of the nucleosome. Top and bottom figures correspond to monovalent salt concentrations of 0.01 M and 0.2 M, respectively. Color-coding for the histone tails is as follows: H3 (blue), H4 (green), H2A (yellow), and H2B (red). The nucleosome core in the background is colored black.

The variation in the radii of gyration with salt as obtained via simulations and experiments is also compared in Fig. 6. Again, both the experiments and the simulations predict a moderate increase in R_g with salt; R_g increases by 0.1–0.2 nm as the salt concentration is increased from 0.01 M to 0.3 M. The consistently larger experimental values of R_g relative to simulations (by 0.3–0.4 nm) might be explained by the apparent swelling of the nucleosome cores in the experiments when compared to their structure in the crystal state (1KX5.pdb). Namely, we observed that the radius of gyration of the nucleosome core (without the tails) computed from the crystal structure (3.8 nm, using only the first summation terms of the denominator and numerator in Eq. 29) is smaller than that obtained experimentally (4.2–4.4 nm).

In summary, the flexible-tail model predicts correctly the trends and their respective magnitudes in the conformational changes that the histone tails undergo as the salt concentration is varied. The overall high degree of conformational freedom that each histone tail possesses, in addition to their dependence on salt conditions, plays a key role in mediating internucleosomal interactions within folded chromatin at physiological salt concentrations, and thus the current model improves upon the fixed tail representation, which completely neglects such conformational effects.

Diffusion coefficients
We now compare the flexible-tail model's self-diffusion coefficients (D_s) for mononucleosomes, dinucleosomes, and trinucleosomes at infinite dilution to values obtained experimentally. Mononucleosomes without linker DNA were employed in our simulations to model the experimental mononucleosomes of Yao et al. (57). Dinucleosomes containing two linkers and trinucleosomes containing three linkers were employed to mimic the experimental dinucleosomes (58) and trinucleosomes (59), respectively. The linker DNA length in both these models was fixed at seven linker beads to model the 22-nm-long linker DNA in the dinucleosome and trinucleosome experiments. All three experimental systems lack linker histones, as in our basic model. Multiple Brownian dynamics simulations on individual oligonucleosomes (to mimic infinite dilution), each starting from a different starting configuration, were performed. Each simulation run was 50-µs long. The self-diffusion coefficients of the oligonucleosomes were computed from the mean-square deviation of their center of mass as follows

(31)

where denotes an ensemble average, and r(t) denotes the center of mass of the nucleosomal array at time t.

The diffusion coefficients of the mononucleosomes, dinucleosomes, and trinucleosomes computed in this study for the fixed-tail and flexible-tail models as well as those obtained experimentally are plotted in Fig. 8. The quantitative agreement between the simulation results and the experiments is excellent. The flexible-tail model matches the experimental diffusion coefficients better than the fixed-tail model in the case of dinucleosomes and trinucleosomes. Both models indicate that the diffusion coefficients of the dinucleosomes and trinucleosomes increases slightly with the salt concentration, especially between 0.01 M and 0.02 M salt concentration, as in the experiments. For mononucleosomes, the fixed-tail model performs marginally better than the flexible-tail model in terms of agreement with experiments. Again, both models predict the same trend as the experiments, i.e., insensitivity to salt. These results demonstrate that Brownian dynamics simulations of the flexible-tail model reproduce the Brownian motion and dynamics of oligonucleosomes reasonably well.

View larger version (17K): [in this window] [in a new window]   FIGURE 8  Dependence of the diffusion constant of mononucleosome (solid symbols), dinucleosomes (gray symbols), and trinucleosomes (open symbols) on the salt concentration. Circular symbols represent experimental values from Yao et al. (57) (mononucleosomes), Yao et al. (58) (dinucleosomes), and Bednar et al. (59) (trinucleosomes). Square and triangular symbols represent results from Brownian dynamics simulations of flexible-tail and fixed-tail model of oligonucleosomes, respectively. The dashed lines, which represent the mean experimental diffusion values for the three array sizes, serve to guide the eye.

View larger version (60K): [in this window] [in a new window]   FIGURE 9  Two starting configurations for the 12-unit nucleosomal array simulations corresponding to a solenoidlike (solenoid with straight linkers) configuration (left) and zigzag (right). Insets show corresponding stacking patterns without histone tails. In the main figures, the histone tails are color-coded according to: H3 (blue), H4 (green), H2A (yellow), and H2B (red); the nucleosome cores and linker beads are shaded gray and red, respectively. In the insets, the nucleosome core is white and the wrapped + linker DNA is red.

View larger version (54K): [in this window] [in a new window]   FIGURE 10  Configuration of the 12-unit nucleosomal array at the end of 5-µs runs at 0.2 M (left) and 0.01 M (right) salt concentration. Insets show corresponding stacking patterns without histone tails. In the main figures, the histone tails are color-coded according to: H3 (blue), H4 (green), H2A (yellow), and H2B (red); the nucleosome cores and linker beads are shaded gray and red, respectively. In the insets, the nucleosome core is white and the wrapped + linker DNA is red.

(32)

where N' = 12 represents number of nucleosome units in the oligonucleosome, R is the effective radius of the nucleosomes (taken as the hydrodynamic radius R_c here), S₁ is the S_20,w of a mononucleosome taken as equal to 11.1 Svedberg (S), and R_ij is the distance between two nucleosomes. The reported values sedimentation coefficients have been averaged over all the different starting configurations once the evolution of the sedimentation coefficient with time has stabilized, typically after 3–4 µs.

The sedimentation coefficients computed from our Brownian dynamics simulations using the fixed and flexible-tail models are plotted versus salt concentration in Fig. 11 along with corresponding values obtained from experiments (60) and Monte Carlo simulations of the fixed-tail model (30). Both the experimental and the simulation data show an increase in S_20,w as the salt concentration is increased, indicating an increased compaction of the nucleosomal arrays with the salt concentration. The widely accepted explanation for this trend is that the increased electrostatic screening at high salt concentration decreases linker/linker electrostatic repulsions and allows the linker DNA to come closer to each other, which naturally results in a compaction of the arrays. Our calculations suggest that reducing the salt concentration from 0.2 M to 0.01 M results in a 15-fold increase in the total electrostatic repulsion energy between the linker DNAs for moderately folded oligonucleosomes with S_20,w 38. The data also show that the flexible-tail model reproduces the experimental S_20,w much better than the fixed-tail model. The fixed-tail model underestimates experimental S_20,w by 3 S at low salt concentration (0.01 M) and overestimates S_20,w by 2.5 S at high salt concentrations (0.2 M).

View larger version (9K): [in this window] [in a new window]   FIGURE 11  Sedimentation coefficients versus salt concentration obtained using Brownian dynamics of the oligonucleosome model developed in this study (open circles). Also shown are sedimentation coefficients obtained experimentally by Hansen et al. (60) (open triangles), and those obtained theoretically using the fixed-tail model (open squares and diamonds). The open squares represent results obtained by Sun et al. (30) via Monte Carlo simulations and the open diamonds represent results obtained via Brownian dynamics simulations in the current study.