Interaction of double-stranded DNA with a nanosphere: a coarse-grained molecular dynamics simulation study

Abstract

Using coarse-grained molecular dynamics simulations, we study the behavior of a DNA-nanosphere complex in the absence and presence of an external stretching force exerted on two ends of DNA chain. In this work, we use an accurate coarse-grained model for double-stranded DNA chain recently developed by Savelyev and Papoian [Biophys. J. 96, 4044 (2009)]. Charged particles are uniformly distributed on the surface of the sphere. Without a stretching force, an ordered or disordered complex is formed depending on the surface charge density and the salt concentration. It is found that DNA wraps randomly around the sphere only at an intermediate salt concentration and high surface charge density. Additionally, the DNA folding around the sphere induces a reduced distance between DNA monomers close to the spherical surface. When an external force is applied, the force-extension relation reveals a discontinuous transition of DNA stretching during the unwrapping process. Moreover, the discrete change becomes more obvious for a higher salt concentration.

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1. Introduction

In most eukaryotic organisms, one nucleosome consists of 146 base pairs of DNA wrapped around a histone octamer in 1.67 turns of a left-handed superhelix. Each histone octamer contains a single H3–H4 tetramer and two histone H2A–H2B dimers. The repeating nucleosome cores are further organized by the linker DNA in a nucleoprotein complex. This complex is called chromatin which plays a crucial role in living organisms because its dynamic organization is a key factor for controlling the regulation of gene expression such as replication, transcription, repair and recombination.¹ To understand the hierarchical structure and dynamics of chromatin, studying structural characteristics and dynamic properties of its lowest-level structure (or the nucleosome) possesses extremely important significance.

On the other hand, complexes of DNA with oppositely charged objects in vitro seem to be promising as model systems to reveal fundamental mechanisms of the natural packing of DNA by histone octamers. Examples include DNA compaction by nanoparticles,^2,3proteins⁴ and dendrimers.⁵ Several critical results have been obtained for the complexation of DNA with charged objects. In studies on compaction of long DNA by cationic nanoparticles, the salt concentration and the properties of nanoparticles—like the size, charge and concentration—were demonstrated to play an important role, and at high salt concentration a strong decrease in compaction amount was observed due to screened electrostatic repulsion.⁶ It was also observed that complexes of λ-DNA with like-charged silica particles exhibit core-shell structures and absorbed DNA on particle surfaces adapts coil-like conformation.³ Anther important feature is charge inversion or overcharging of oppositely charged objects by wrapped DNA. Theoretically, the overcharging of histone octamers or nanoparticles is caused by repulsion of non-compensated charges on the outer DNA surface.⁷

In recent years, a significant increase in the ability of computational approaches has been achieved to help us understand the structure and conformation of biological macromolecules.^8,9 Depending on the computational accuracy and efficiency, different types of biological systems are simulated using different approaches, mainly including Brownian dynamics (BD),^10–14 coarse-grained molecular dynamics (CGMD)^15–18 and all-atom molecular dynamics (AAMD).^19–21 The AAMD is the most suitable method for describing the detailed information of the nucleosome at the atomistic scale.^19,20 However, the AAMD can only mimic the formation of a single nucleosome at a very short timescale, because it consists of a large number of atoms (i.e., the crystal structure of a single nucleosome contains about 16850 atoms). To obtain structural and dynamics characteristics of the nucleosome at a longer time scale, a considerably coarse model, consisting of a nanosphere and a semiflexible charged chain as the model of histone octamer and that of DNA, is applied in BD simulations.¹² The inner structure of the nucleosome is very complex itself. Moreover, various kinds of interactions, including hydrogen bond, covalent bond, electrostatic interaction and short-range nonbonded interaction, further lead to complicated mechanical properties of the nucleosome. Nevertheless, in most BD simulations, the double-stranded DNA is usually modeled as a single charged polymer chain and the histone octamer is approximated by a nanosphere with charge in its center. Additionally, some parameters in these models are chosen empirically. Thus, these models are so coarse that they can only obtain some general properties at the large scale.

Therefore, it is essential to develop a new coarse-grained model bridging the all-atom model and the coarse-grained model applied in BD simulations and more close to the realistic or atomistic model for understanding the equilibrium structure and the unwrapping of the nucleosome. For this, a major task to be resolved is how to accurately model double-stranded DNA in nucleosome simulations. Although single chain models of DNA can be used to analyse the static and dynamic properties of DNA^22,23 as well as interactions of DNA with histone proteins¹² or nanoparticles⁶ to some extent, they still can not explain some important biological and mechanical characteristics caused by the double helix structure of DNA, such as DNA melting and effects of helical charge distribution.²⁴ Recently, coarse-grained multiscale methods have been applied widely in studying the properties of biological macromolecules, such as protein,⁹double-stranded DNA¹⁶ and DNA-protein complex.¹⁵ In these models, the choice of the force field and the determination of parameters are essentially important. The parameters of the coarse-grained force field are often obtained from AAMD simulations. This can avoid the uncertainty of the parameters obtained from experiments which may be influenced by many physical factors, just as in BD simulations using a very coarse-grained model. A novel coarse-grained model of the nucleosome developed by Voltz et al. has been used to study the nucleosomal dynamics.¹⁵ In their model, double-stranded DNA is represented by single beads with the mass centered on the phosphorus for nucleotides, corresponding to a coarse-grained double-stranded model of DNA. This model can reproduce equilibrium structural properties obtained from AAMD simulations and largely speed up nucleosome simulations. However, it is difficult to apply their model to the study of the folding and unfolding of DNA around histone octamer. Recently, Savelyev and Papoian proposed an accurate coarse-grained representative of double-stranded DNA derived from AAMD simulations.¹⁶ Compared to the model from Voltz et al.,¹⁵ this double-stranded model of DNA can be used to investigate the equilibrium structure and nonequilibrium dynamics of complexes of DNA with oppositely charged objects, i.e., nanoparticles and histone octamers. In the present paper, we apply the double-stranded model from Savelyev and Papoian to study the wrapping and unwrapping of DNA around a nanosphere with charged particles evenly distributed on its surface. The nanosphere can also be considered as a coarse-grained model of histone octamer.

Although the histone octamer has a very complex structure, it is approximated by a sphere or a cylinder with a net positive charge in many coarse-grained models of nucleosome.^12,25 Such coarse-grained models still can reproduce some important conformational and dynamic properties of the nucleosome, which is independent of its structural detail to some extent. Generally speaking, the true charge of the histone octamer is not known precisely. This is due to the fact that many factors, such as pH and salt concentration, affect the dissociation of amino acids, accordingly leading to the variation of the charge amount and distribution of the histone octamer. It is difficult to assign the charge on and within the sphere close to the charge distribution of the histone octamer with an irregular surface. In our simulations, all the charge is uniformly distributed on the spherical surface and the distribution of the charge in the inner region of the sphere is not considered. Thus, the present model is more suitable to mimic surface-charged nanospheres. Meanwhile, it is also expected that the inner charge can be compensated by varying the surface charge density for the purpose of simulating the histone octamer.

In addition to studying the wrapping of DNA around a nanosphere, we also pay attention to the unfolding of DNA-nanosphere complexes under an external force. Exploring the structural characteristics and dynamics of DNA are an important subject in the biological area. However, access to DNA requires disrupting higher order chromatin structures and unwrapping of the DNA from the histone octamer. An applied tension-based mechanism has been developed into an effective means to remove histones and attracted a great deal of interest in the last decade.^25–31 Meanwhile, understanding the relationship between the force and displacement at a molecular level becomes a key issue. Additionally, it is known that the force generated by molecular motors affects the wrapping behavior of nucleosomes in the cell. Experimentally, the stretching of chromatin is implemented by different approaches including nanoconfinement,²⁶ magnetic²⁷ and optical tweezers.^28–30 To our best knowledge, there are few experiments on stretching a single nucleosome. Recently, in the experiment carried out by Pope et al., three individual energy barriers were observed during the stretching of single chromatin fiber.²⁹ Similarly, in other experiments, a sawtooth-shape profile of force-extension, corresponding to a sudden release of DNA from histone cores, indicates the unwrapping of DNA needs to overcome a significant energy barrier.^28,30 The results obtained by computer simulations, in particular BD simulations,^25,31 are qualitatively consistent with experimental results. So far, the simulations have been performed to study the unwinding of DNA from an isolated nucleosome unlike the experiments carried out to pull an array of nucleosomes. Thus, as a complement to the experiments, computer simulations can reveal well the force-extension characteristics at different unwrapping stages and visualize the detailed unwinding process. In this work, it is expected that some essential features on the tension-induced release of DNA from nucleosomes can be captured through stretching the DNA/nanosphere complex. Pulling the DNA from oppositely charged objects can be achieved under constant stretching force or velocity. Our simulations are performed under a constant external force exerted on two ends of double-stranded DNA in entire stretching process.

Here, we report the effects of salt concentration and surface charge density on the structural and dynamics properties of the DNA/nanosphere complex. Compared to the customary worm-like chain model of DNA, the results presented in this paper provide a deeper insight into the mechanisms of electrostatic-driven wrapping and tension-induced unwrapping of DNA around oppositely charged objects. Our work is partially motivated by the problem of nucleosome assembly. Hopefully, the present coarse-grained simulation is capable of serving as a valuable reference for studying the dynamic wrapping/unwrapping process of DNA around histones. The remainder of this paper is organized as follows. In the next section, we describe the model system and the simulation method. Following that, the results are presented and discussed. Finally, conclusions are given in Section 4.

2. Simulation model and method

In this paper, we use an accurate coarse-grained model for double-stranded DNA developed by Savelyev and Papoian.¹⁶Double-stranded DNA is represented as single spherical beads centered on the positions of phosphorus atoms not considering the details of other atoms (Fig. 1a). Therefore, the number of atoms in the simulated system is considerably reduced. In the coarse-grained model, all four types of DNA nucleotides are replaced with identical monomer units. Interbead interaction potentials include four terms as follows:

U_DNA = U_bond + U_angle + U_helix + U_coul (1)

Fig. 1 (a) Atomistic (solid beads) and coarse-grained (transparent beads) representation of double-stranded DNA. For the coarse-grained model, the residues are represented by single beads centered on the positions of phosphorus atoms. (b) Model of the nanosphere. Charged particles (cyan beads) of Z = 164 are uniformly distributed on the spherical surface (red).

The individual energetic contributions to this potential are given below:

U_bond = K^b₂(l − l₀)² + K^b₃(l − l₀)³ + K^b₄(l − l₀)⁴ (2)

U_helix = K^h₂(l − l₀)² + K^h₃(l − l₀)³ + K^h₄(l − l₀)⁴ (3)

U_angle = K^a₂(θ − θ₀)² + K^a₃(θ − θ₀)³ + K^a₄(θ − θ₀)⁴ (4)

Here, U_bond and U_angle are bond and bending angle potential energies, respectively. These two terms denote intrastrand interactions and reflect connectivity of each DNA strand. U_helix is responsible for maintenance of the double helix structure of DNA formed by two polynucleotides. l₀ is the bond equilibrium length and θ₀ is the equilibrium angle. The interaction parameters, K, are calculated by matching correlators obtained from AAMD and CGMD simulations. We refer the readers to Ref. 16 for a more detail description including the force field and model parameters.

The last term U_coul represents the electrostatic interactions between monomer units. In the expression of electrostatic energy derived by Savelyev and Papoian,¹⁶ the bare DNA nucleotide charge is replaced with an effective charge. However, when taking into account electrostatic interactions between DNA beads and charged particles on the spherical surface (described below), how to define the effective value of surface charge is hard to establish. Therefore, we use a simple Debye–Hückel potential to describe electrostatic interactions between all pairs of charged particles carrying a bare charge. In eqn (5), κ ≈ (8πl_BC_s)^1/2 is the inverse Debye length where C_s is the salt concentration and l_B is the Bjerrum length (l_B ≈ 0.71nm in water at room temperature). In the present simulations, the double-stranded DNA consists of N_s = 216 base pairs corresponding to 432 beads in the coarse-gained model. Each bead carries a unit negative charge q_i = −e. The stiffness of DNA is not only related to steric hindrance but also the salt concentration. The salt concentration influences electrostatic screening effects between charges in DNA chain, further varying the chain stiffness. It should be pointed out that given high charge density of DNA, the Debye–Hückel interaction potential is expected to be at best semi-quantitative, although qualitative trends might potentially be reproducible.^32,33

In our model, the charged particles are uniformly distributed on the surface of the sphere as shown in Fig. 1b. The charge distribution is different from that in previous simulation model, in which charged particles are not distributed explicitly on the surface of the oppositely charged sphere only considered as a spherical object with a homogenous surface charge density. The number Z of charged particles varies from 16 to 240 and the distance R of them from the spherical center is set to 3.2nm. Additionally, their relative positions during simulations always keep constant. To avoid DNA penetrating into the spherical inner, the repulsive interaction between the sphere and DNA is expressed by

U_w(r) = −k_w(r − R′)³/3 (6)

where r is the distance of the DNA beads from the spherical center, k_w is the repulsive constant. We adopt k_w = 20kcal mol⁻¹ Å⁻³ and R′ = R + 0.5nm (F_w(r) = −∇U_w(r) = 0, when r > R′).

The system temperature T is controlled by a Langevin thermostat based on fluctuation-dissipation theorem.³⁴ In this case, dynamic evolution of the system is described by

where F_i is the deterministic force acting on the particle i from other particles. The damping rate γ is used to control the relaxation rate of temperature. F^r_i is the random force that is sampled from the Gaussian distribution

〈F^r_i(t)〉 = 0 (8)

〈F^r_i(t)·F^r_j(t′)〉 = 6mγk_BTδ_ijδ(t − t′) (9)

Here, γ and T are set to 10fs⁻¹ and 300 K, respectively. Three cases of inverse Debye length are included to investigate the wrapping of DNA and its stretching under an applied external force: κ = 0.03nm⁻¹ (C_s = 0.1mM), 0.33nm⁻¹ (10mM), 0.74nm⁻¹ (50mM). Additionally, a high salt concentration C_s = 1M (κ = 3.28nm⁻¹) is also included to reveal the spontaneous unwrapping of DNA-nanosphere complexes. The positions and velocities of the particles are calculated using the velocity Verlet algorithm. All simulations are conducted with a time step Δt = 10fs. After the systems are equilibrated, each production run is at least 2 × 10⁷ time steps. The equilibrium time required depends on the salt concentration and the surface charge density as well as the force under the external stretching. How to make the system reach equilibrium state more rapidly is discussed in next section.

3. Results and discussion

Before studying the formation of the complex between double-stranded DNA and a single nanosphere with varying number of surface charges, Z, or the surface charge density, we first examine the evolutions of the total electrostatic energy with simulation time. To make the complex reach the equilibrium state more rapidly, the mass of single DNA beads is reduced to 1u. After achieving the equilibrium state, the bead mass is reset to the original value 330u. Then, we continue to simulate the system to produce the equilibrium conformations. Initially, double-stranded DNA assumes a fully extended conformation and the center of the sphere is placed at a distance r = 4.7nm from the axis of DNA chain. Fig. 2 gives the profiles of electrostatic energy in cases of different Z at κ = 0.33nm⁻¹. It is observed that during 0–30ns, corresponding to the case of reduced mass, the electrostatic energy decreases with time and its equilibrium value remains constant after 25ns. After 30ns, we restore the bead mass to the true value. Note that the electrostatic energy no more varies with time. Moreover, its fluctuation magnitude becomes smaller than that for the case of reduced mass because of the stronger thermal motion for the latter. However, the equilibrium values of electrostatic energy are almost equal in both cases. This is due to the fact that changing individual atom masses influence the dynamic properties of the system, but not its equilibrium distribution.³⁵ In the following, unless otherwise stated, initially the system is simulated under the condition of reduced mass. After reaching the equilibrium state, the DNA bead mass is reset to the original value. For the initial conformation of the complex, the larger the surface charge density, the higher the electrostatic energy becomes. Nevertheless, after a sufficiently long time an opposite change is observed. It reveals that more DNA beads are adsorbed on the spherical surface upon increasing Z, leading to an increased attractive energy.

Fig. 2 Total electrostatic energy curves is shown in dependence on time. The direction of the arrow corresponds to Z = 16, 64, 100, 164 and 240, respectively. Simulations are carried out under reduced mass (left side of dashed line) and true mass (right side of dashed line) of DNA beads.

Fig. 3 gives typical simulation snapshots of the complex at κ = 0.33nm⁻¹ with a varying number of surface charges. It can be found that the conformation of the complex, or the degree of DNA wrapping around the sphere, is sensitive to the number of surface charges. At low surface densities, i.e., Z = 32, DNA is in an extended state and only a slight bending occurs near the sphere. At Z = 64, DNA wraps in a half turn around the sphere. Upon a further increase in the surface charge density, the wrapping degree increases significantly. At Z = 240, DNA chain is entirely adsorbed on the nanospherical surface (Fig. 3f). These typical conformations have been also found in other numerical studies.²DNA wraps in about two turns around the sphere at Z = 164. Such morphology can be considered as a nucleosomelike structure. In many experiments on DNA compaction in the presence of multivalent counterions^36,37 or nanoparticles,⁶ it is observed that increasing the particle concentration leads to an enhanced shrinking of the DNA chain. In the present work, regardless of double-stranded DNA conformations the increase of the surface charge density seems to have the same effect on DNA compaction as those cases of increasing ion concentration. The main difference is that increasing ion concentration induces a compact and random coil form of DNA, but an increased number of charges uniformly distributed on the spherical surface leads to a regular wrapping of DNA. Certainly, the ordering degree of DNA wrapping is dependent on the surface charge density and salt concentration (or Debye length κ⁻¹), as discussed below.

Fig. 3 Typical snapshots of the DNA-sphere complex at κ = 0.33nm⁻¹ and various number of surface charges: (a) Z = 32, (b) 64, (c) 100, (d) 132, (e) 164 and (f) 240.

Increasing the number of surface charges not only leads to an increase of DNA beads adsorbed on the spherical surface, but also may cause a change in the wrapping state. As seen from Fig. 3, an ordered wrapping of DNA chain can be observed in cases of Z from 64 to 164. For a large surface charge density of Z = 240, we give the simulation snapshots of the complex at κ = 0.33nm⁻¹ obtained from other two view angles (Fig. 4). One can see clearly that unlike other cases, DNA chain exhibits a misfolding on the spherical surface. When the surface charge density is not too large, i.e., Z < 200 at κ = 0.33nm⁻¹, corresponding to relatively weak attraction between DNA chain and the sphere, DNA chain spontaneously tends to an ordering wrapping around the sphere due to the thermal fluctuation. A qualitative analysis is present below. However, at sufficiently high surface charge densities, strong attraction confines DNA chain tightly on the spherical surface, that is, DNA chain stays in a kinetically trapped conformation. In BD simulations of a single nucleosome, the probability of misfolding significantly increases when the attractive interaction becomes sufficiently strong.¹² This is qualitatively consistent with our simulation results. It reveals that general conformational characteristics of DNA chain adsorbed on a nanosphere or histone core can be reproduced using more coarse-grained DNA model, i.e., a polymer chain model. Nevertheless, some local details and more quantitative descriptions can not be addressed by such DNA model.

Fig. 4 Snapshots of the complex of DNA wrapped around the sphere with Z = 240 at κ = 0.33nm⁻¹ observed from two different view angles. The direction of arrows refers to the wrapping direction of DNA around the sphere.

To quantify the ordered degree of DNA wrapping, a parameter η is introduced as a measure of the proper ordering¹²

where r̄_i,i+1 = r′_i,i+1/r′_i,i + 1, r′_i,i+1 = r′_i+1 − r′_i, r′_i = (r_i + r_i+Ns)/2. N_d denotes the number of DNA bead pairs in the vicinity of the spherical surface (r′_i ≤ 4.5nm). If the DNA chain wraps orderly around the sphere, the direction of each vector product between r̄_i,i+1 and its adjacent vector is nearly parallel to the helical axis of adsorbed DNA. We note that disordered complexes are formed when η < 0.05. Fig. 5a and b give the average order parameter η and the fraction ω of adsorbed DNA beads as a function of Z for three different κ. It is clear from Fig. 5a that at low values of Z, η decreases significantly upon increasing the surface charge density and then shows a very slow decrease, except for a sudden drop at Z = 240 and κ = 0.33nm⁻¹. At κ = 0.74nm⁻¹, the electrostatic attraction or repulsion is relatively weak inducing a reduced stiffness of DNA. At low Z, the adsorption fraction ω for κ = 0.74nm⁻¹ is obviously smaller as shown in Fig. 5b. At κ = 0.03nm⁻¹, corresponding to a very weak electrostatic screening, the change of η is small in the entire range of Z investigated and the adsorption fraction is also lower compared to the case of κ = 0.33nm⁻¹ when Z > 64. It indicates that although the attraction between DNA and the sphere become very strong, the strong repulsion between DNA beads also leads to an enhanced stiffness of DNA chain, which suppresses the bending of DNA and favors an ordered arrangement of DNA beads on the adsorbed surface. However, at Z = 64, the larger ω for κ = 0.03nm⁻¹ indicates that the electrostatic attraction becomes overwhelming. For a high surface charge density of Z = 240, one can observe two features: first, ω for κ = 0.33nm⁻¹ is larger compared to the other two cases; second, the misfolding of DNA on the sphere does not occur at a high or low salt concentration but at an intermediate salt concentration, i.e., κ = 0.33nm⁻¹ in the present work. Experimentally, the nucleosome structure has failed to be reconstituted when DNA and histone are directly mixed under physiological salt conditions. As demonstrated in Ref. 12, the misfolding of DNA under direct mixing condition is induced by a rather strong electrostatic attraction at physiological salt concentration. The standard approach requires a first mixing of DNA and histone at a high salt concentration then gradually decreasing the salt concentration. Our results show that increasing the salt concentration from a low value to the physiological value also may lead to a well-reconstituted nucleosome structure. Thus, the salt concentration modulates the electrostatic screening effect, further controlling the DNA stiffness and the attractive strength between DNA and the sphere. Additionally, combined with the surface charge density, they together control the adsorption amount and the folding state of DNA.

Fig. 5 (a) Order parameter η, (b) adsorption fraction ω and (c) net charge β as a function of the number of surface charges for different κ.

To study the overcharging of the complex by DNA, we give the net charge β = (Z − 2N_d) of the complex as a function of the surface charge density in Fig. 5c. If the net charge becomes negative, the complex is overcharged. Single nucleosomal particles consist of 146 base pairs of DNA (charge −292e) wrapped in 1.67 turns around the histone octamer (charge 220e). Thus, the wrapped DNA overcharges the histone octamer considerably. It has been suggested that the repulsion of non-compensated charges on the outer DNA surface provokes a spontaneous wrapping of DNA and results in an overcharging of nucleosomes.⁷ Clearly, this conclusion also can be applied to the complex of DNA with a nanosphere. A drop of the net charge profiles is observed upon increasing Z as shown by the plot of β. At κ = 0.03nm⁻¹ and 0.33nm⁻¹, a negative net charge is observed over the whole range of Z. It is also found that a charge inversion occurs for the cases of κ = 0.74nm⁻¹ when Z exceeds a certain critical value. Theoretical and numerical investigations indicate that the degree of overcharging rises with a decrease in the chain intrinsic rigidity and an increase in the macroion diameter.^38–40 For flexible polyelectrolyte chains, Chodanowski and Stoll concluded that charge inversion increases with the salt concentration using Monte Carlo simulations.⁴¹ However, our simulations show that the effect of increasing the salt concentration or κ on charge inversion is more complicated. Two features should be pointed out: first, for a low surface charge density of Z = 64, the degree of overcharging at κ = 0.03nm⁻¹ is higher than that at κ = 0.33nm⁻¹; second, there is a higher degree of overcharging at κ = 0.33nm⁻¹ but not at κ = 0.74nm⁻¹. This is a consequence of interplay of the surface charge density and the DNA rigidity, as discussed above for η and ω.

The folding of DNA on the sphere can cause a deformation of its local structure. One remarkable characteristic is local stretching and contracting of DNA. B-DNA in solution contains 10.5 base pairs per helical turn. In our simulations, to characterize local deformation of DNA we measure the distance D between two DNA beads separated by nine beads in one single chain. Fig. 6a gives the probability distribution P of D at κ = 0.33nm⁻¹. In the absence of the charged nanosphere, the pronounced maximum at D ≈ 3.6nm is approximately equal to the equilibrium pitch of B-DNA. The profiles shift towards smaller values of D as Z increases and one relatively low peak occurs at low D. This demonstrates that the wrapping of DNA results in a reduced distance between DNA beads close to the spherical surface. Nevertheless, the profiles do not shift towards larger values of D, indicating that the stretching of DNA at the scale of one helical turn never takes place. One possible explanation is that the electrostatic attraction between DNA and surface charges can compensate the stretching of outer DNA segments due to the bending of inner DNA segments. Here, inner DNA segments and outer ones refer to those DNA segments close to the spherical surface and correspondingly complementary ones, respectively. Thus, the folding of DNA on the sphere is implemented by the contracting of inner DNA segments. Additionally, Fig. 6b shows the probability distribution P of D for cases of three different κ in the absence of the charged nanosphere. As expected, upon decreasing κ the peak value slightly offsets towards larger values of Z because of enhanced electrostatic repulsion.

Fig. 6 Probability distribution P of the distance D between two DNA beads separated nine beads in one single chain for (a) various number of surface charges at κ = 0.33nm⁻¹ (b) different κ in the absence of the charged sphere.

The formation of DNA-nanosphere complexes at three different salt concentrations is investigated above. For a high salt concentration, DNA can not wrap around the sphere to form a complex, just as DNA is released from histone cores at about C_s = 750mM.⁴² We take the equilibrium conformation of the complex with Z = 240 at κ = 0.03nm⁻¹ as the starting point of CGMD simulation. Then, the simulation is carried out at κ = 3.28nm⁻¹ (C_s = 1M) and under reduced mass of DNA beads. Initially, a compact DNA-sphere complex is observed (Fig. 7a). As the simulation proceeds in time, DNA winds loosely around the sphere (Fig. 7b). Finally, DNA unfolds completely (Fig. 7c). This reveals that at high salt concentrations the intrinsic stiffness of DNA is sufficiently strong to overwhelm the electrostatic attraction between DNA and the sphere. Fig. 8 gives the total electrostatic energy E_coul as a function of time. E_coul is growing sharply indicating that DNA detaches rapidly from the spherical surface. After E_coul reaches the maximum value, a drop is followed up to a steady energy. This is due to the attractive energy releasing more quickly than the repulsive energy within DNA chain. The slow decrease in E_coul indicates the release of the repulsive energy exceeding that of the attractive energy, corresponding to a spontaneous stretching of DNA after the adsorbed DNA detaches from the surface.

Fig. 7 Typical snapshots of the complex of DNA with the sphere with Z = 240 at different simulation times (a) t = 0, (b) 6 and (c) 60ns. The equilibrium conformation at κ = 0.03nm⁻¹ is taken as the initial conformation. Simulation results are obtained at κ = 3.28nm⁻¹ and under reduced mass of DNA beads.

Fig. 8 Total electrostatic energy versus time for the DNA-sphere complex. All conditions are the same as those in Fig. 7.

We discuss in the remainder of this paper, the structural characteristics of the complex, in the presence of an external stretching force. Two constant forces being equal in magnitude and opposite in direction, both of which are parallel to the axis of initially extended DNA, are exerted on the two ends of the DNA chain (or four end beads in double-stranded DNA). Fig. 9a gives the average extension L_as (or the end-to-end distance of DNA) as a function of the external force F_s at a fixed surface charge density of Z = 200 and different inverse Debye length κ. When an external force is not applied, the DNA chain forms an ordered complex with the sphere. In our simulations, the unwrapping of DNA is driven by a constant external force. We calculate the average extension of DNA until the stretching length almost no longer varies with time. Thus, a stretching force corresponds to a steady conformation of the complex (solid symbols in Fig. 9a). The force-extension curves obtained at constant external force are apparently different from those obtained at constant stretching velocities. Under constant stretching velocities, the force exerted on the ends of DNA increases with the extension until the inner DNA turn unwraps, then there is a sudden drop of the force.²⁵

Fig. 9 (a) Force-extension and (b) force-adsorption fraction relations of the DNA-sphere complex with Z = 200 for different κ.

It is clearly seen from Fig. 9a that the extension increases with the force until DNA eventually reaches complete stretching. One notes that the curves exhibit different force-extension stages depending on κ. For the complex with κ = 0.74nm⁻¹, a relatively slow increase in the extension is observed at weak stretching force and then the extension rapidly reaches the maximum of DNA stretching. This slow increase corresponds to the unwrapping of the outer DNA turn. An unwrapping transition of the inner DNA turn with an adsorption fraction ω ≈ 0.4 occurs at F_s ≈ 30pN. Clearly, one can find in Fig. 9b that the adsorption fraction undergoes an opposite change. An obvious fact is that weak electrostatic attraction for κ = 0.74nm⁻¹ makes DNA release more easily from the sphere compared to the other two cases. For stiffer DNA chain (κ = 0.03nm⁻¹and 0.33nm⁻¹), a large part of the outer DNA turn releases in initial stretching stage upon increasing the force until ω≈0.5 as seen in Fig. 9. Subsequently, the stretching of DNA enters the second stage. Unlike the case of κ = 0.74nm⁻¹, at this stage the inner DNA turn is not opened immediately. At κ = 0.33nm⁻¹, Fig. 10b gives a typical snapshot at F_s ≈ 30pN corresponding to the turning point of two different stretching regimes.

Fig. 10 Typical snapshots of the DNA-sphere complex with Z = 200 for κ = 0.33nm⁻¹ under different stretching force (a) F_s = 0, (b) 30, (c) 70 and (d) 80pN.

A sharp rise in force-extension profiles takes place after the stretching length of DNA undergoes a slow increase at κ = 0.03nm⁻¹ and 0.33nm⁻¹. It is clear that the variation of the extension with the stretching force is related to the adsorption fraction. As seen in Fig. 9b, ω shows a relatively slow decrease under intermediate forces. This stage corresponds to unwrapping of the remainder of the outer DNA turn and a small part of the inner DNA turn. The behavior can be observed from snapshots at κ = 0.33nm⁻¹ when the force varies from 30pN to 70pN (Fig. 10b and c). And then, a sharp decrease of ω means an unwrapping transition at certain critical forces. After the unwrapping transition, a very weak variation of the extension is mainly due to the harmonic contribution of the stretching energy. Under constant stretching velocities,²⁵ when the force reaches a critical value, further unwrapping of the inner DNA turn only needs a smaller force. Additionally, the same conclusion is also obtained under constant forces.³¹ In order to verify this fact, when DNA unwraps completely near certain critical forces, a reduced force is applied to two ends of DNA (hollow symbols in Fig. 9a). It is found that the extension only shows a small drop, revealing that DNA does not fold back around the sphere. The discontinuous transition of DNA stretching reflects the same phenomenon as sawtooth-shape profiles of force-extension observed during the unwrapping process of individual nucleosomes using optical tweezers.²⁸ This is also in agreement with previous BD simulations.³¹

Note that at this final stretching stage, DNA chain does not exhibit a completely straight conformation. At κ = 0.33nm⁻¹, the electrostatic attraction induces a considerable bending of DNA close to the sphere (Fig. 10d). Additionally, although the final extension near a critical force is slightly smaller at larger κ, a distinctly higher adsorption fraction is observed at smaller κ due to the stronger DNA-sphere interaction. It should be emphasized that the release of different DNA turns is accompanied by a rotation of the plane normal to the axis of the helical pathway of DNA chain on the sphere (Fig. 10). Recent theoretical analysis reveals that the toroidal geometry formed by DNA wrapped around protein plays a critical role in maintaining mechanical stabilization of DNA-protein complexes.⁴³ The BD simulations indicate that increasing the stiffness of DNA chain leads to a smaller force to pull different DNA turns away from protein and a more obviously discrete transition in force-extension profiles.³¹ However, if the chain stiffness is enhanced by reducing the salt concentration, an opposite conclusion can be obtained as shown in Fig. 9a. This is due to the stronger attraction between DNA and the sphere, accompanied by a reduction in the salt concentration, so it becomes dominant.

4. Conclusions

On basis of a coarse-grained model of double-stranded DNA proposed by Savelyev and Papoian,¹⁶ we performed a series of CGMD simulations to investigate the equilibrium conformations and the stretching characteristics of the DNA-sphere complex in the absence and presence of an external stretching force exerted on two ends of the DNA chain. The wrapping degree and the folding state of DNA around the sphere is significantly dependent on the surface charge density and the salt concentration (or inverse Debye length, κ). For an intermediate salt concentration (κ = 0.33nm⁻¹) and high surface charge density (Z = 240), DNA exhibits a disordered folding on the spherical surface and a high adsorption fraction is also observed. However, the adsorption fraction becomes larger at a low surface charge density (Z = 64) and a very low salt concentration (κ = 0.03nm⁻¹). These features are a consequence of the competition between the DNA rigidity and the electrostatic attraction between DNA and the sphere. At a high salt concentration (κ = 0.74nm⁻¹), the chain stiffness is reduced due to a strong electrostatic screening effect. Nevertheless, a disordered folding state does not occur even at high surface charge densities. This reveals that at a high salt concentration, the intrinsic stiffness of DNA plays a dominant role in controlling the folding state as well as the adsorption fraction in the range of surface charge densities investigated. Additionally, at a higher salt concentration (κ = 3.28nm⁻¹), DNA releases rapidly and detaches completely from the spherical surface as a result of the intrinsic chain stiffness and quite weak attraction. By analyzing the local structure at the scale of about one pitch, it is possibly true that the attraction between DNA and surface charges can compensate the stretching of outer DNA segments due to the bending of inner DNA segments. When an external force is applied to two ends of DNA chain, different force-extension stages are observed in the stretching process accompanying an unwrapping transition. This behavior also has been observed in experiments and other simulations on the stretching of chromatin. For this, the toroidal geometry formed by DNA wrapped around oppositely charged objects can give an insightful understanding. Furthermore, with decreasing the salt concentration the stronger attraction leads to a weaker unwrapping transition. Compared to single chain models, the coarse-grained model for double-stranded DNA describes more details and is more suitable to study the nonequilibrium dynamics of DNA. However, some local properties for the coarse-grained DNA model may not be detected in the present work. In forthcoming works, we will focus on developing a coarse-grained nucleosome model instead of simple spherical or cylindrical models to further study the structural characteristics of the nucleosome as well as the unwrapping process.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 30770501).

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