Influence of Microscopic Interactions on the Flexible Mechanical Properties of Viral DNA

Abstract

During the packaging and ejection of viral DNA, its mechanical properties play an essential role in viral infection. Some of these mechanical properties originate from different microscopic interactions of the encapsulated DNA in the capsid. Based on an updated mesoscopic model of the interaction potential by Parsegian et al., an alternative continuum elastic model of the free energy of the confined DNA in the capsid is developed in this work. With this model, we not only quantitatively identify the respective contributions from hydration repulsion, electrostatic repulsion, entropy and elastic bending but also predict the ionic effect of viral DNA’s mechanical properties during the packaging and ejection. The relevant predictions are quantitively or qualitatively consistent with the existing experimental results. Furthermore, the nonmonotonous or monotonous changes in the respective contributions of microscopic interactions to the ejection force and free energy at different ejection stages are revealed systematically. Among these, the nonmonotonicity in the entropic contribution implies a transition of viral DNA structure from order to disorder during the ejection.

Introduction

The virus is one of the simplest biological objects (1, 2) but the most abundant biological entity on earth (3). A virus is generally composed of a protective protein shell, termed the capsid, and a nucleic acid genome contained within the capsid (4). Despite the simple structure of the virus, viral genes have higher mutation rates and faster evolution than the genes of cellular organisms (5). Some therapies intend to inhibit viral genome replication and protein expression by destroying encoded enzymes, but they can fail when gene mutation occurs (2). Therefore, an alternative therapy is required.

Recently, there is a new approach aiming to utilize the mechanical properties of viruses to control biological phenomena during viral infection. These mechanical properties include the packaging force, packaging velocity, ejection efficiency, ejection velocity, the rigidity of viral particle, the pressure in the viral capsid, the broken force of the capsid, and so on. Hence, the characterization and identification of these mechanical properties are important. The images and geometrical sizes of virus or phage particles could be obtained by the jumping mode of scanning-force microscopy (6); the stiffness and breaking force of phage λ and human herpes simplex virus type 1 (HSV-1) were measured by atomic force microscopy nanoindentation (7, 8, 9); DNA enthalpy was garnered by isothermal titration calorimetry (10); the configuration of DNA in viral capsids was detected by small angle x-ray scattering (11, 12, 13). The cryo-electron microscopy (cryo-EM) reconstruction of phages showed that the ordered and disordered states of viral DNA coexist (10, 14), and this phenomenon was also observed by small angle x-ray scattering (11, 13). Optical tweezers were exploited to observe the steps of viral DNA packaging into the capsid and measure the packaging force provided by rotary motors (15, 16); fluorescence microscopy was exploited to observe the DNA ejection of some phages and archaeal viruses (10, 17, 18). In the meantime, changes in ionic conditions such as ionic valence and concentration can lead to variations in the stiffness, packaging rate, and ejection fraction of viral DNA (10, 13, 17, 19, 20, 21, 22). Therefore, the study of mechanical properties of viral genes will give us a more fundamental understanding of virus infection and prevention.

The relevant theories about the mechanical behavior of double-stranded DNA (dsDNA) viruses can be traced back to the last century (23). However, in the last two decades, more systematic and diversified theoretical methods have been proposed. Gelbart et al. (23, 24) assumed the cross-sectional profiles of dsDNA chain in a bacteriophage capsid as a spool-like structure by optimizing the energy of the DNA condensation. Based on the continuum elastic theories, they divided the total energy of DNA into three contributions: bulk, surface, and bending elasticity. Finally, they estimated the required loading force and average pressure on the capsid’s internal wall. Then they reconsidered that the free energy of the confined dsDNA chain was composed of two parts: its own elastic bending and interactions between adjacent chains. Furthermore, with the help of the repulsive osmotic pressure model of Parsegian et al. (25, 26), Purohit and his collaborators established an exponential function of repulsive energy and specific formulations of DNA bending energy for different geometrical capsids (27, 28, 29). Almost in the same period, Evilevitch et al. obtained the ejection-driven force about DNA releasing from the capsid, also based on the above-mentioned osmotic pressure model (25, 30). However, the contribution of DNA bending was neglected (31). Thereafter, Grayson et al. used the free-energy model in the work of Purohit et al. to estimate the ejection force, and their calculations were consistent with their ejection experimental results of λ bacteriophages (32). Considering the attractive interaction between DNA strands in bulk solutions (33) and the toroidal structure of the DNA chain in the capsid (34) due to multivalent counterions, de Pablo and his collaborators calculated a reversible influence of spermidine on the stiffness and internal pressure of phage phi29 (35). Evilevitch et al. predicted the stiffness of λ bacteriophages filled with full-length DNA, and their estimations were qualitatively consistent with the nanoindentation measurements (8). Wang et al. improved the free energy model of Purohit et al.; after additionally considering the elastic deformation of the capsid, they determined the optimal size of bacteriophage λ to guarantee its infectivity and stability by the maximal shear stress model (36). In addition, in a “capstan model,” DNA was considered as a “coiled spring” with elastic bending and friction energy, and the Coulomb-Amonton law was used to elucidate the exponential relationship between the viral DNA mobility and the DNA length in the capsid during DNA ejection (37).

As for the influence of microscopic interactions, Purohit et al. have realized that interactions between DNA chains include at least two parts: electrostatic interaction and hydration interaction (8, 29, 38). However, there has been a controversy for the entropic contribution. In molecular dynamics (MD) simulations, the entropy provides a major contribution to free energy, and its numerical value is strongly dependent on the effective chain diameter (39, 40, 41). But in a continuum model, it seems that entropy has disappeared. Actually, the existing continuum models did not explicitly separate microscopic interactions (28, 42); in these models, the microscopic interaction is treated as an exponential function of interstrand spacing divided by a decay length. Nevertheless, the hydration repulsion is predominant at a short separation (≤30 Å), but the electrostatic repulsion overshadows the hydration part at a larger spacing (28, 43), and they have different sensitivities at different levels of ionic conditions (44). Moreover, considering the semiflexibility of DNA chains, there should be fluctuation-enhanced parts due to entropy (44), but it is not explicitly characterized in those classical continuum elastic theories. Parsegian and his collaborators presented an updated state equation of DNA liquid crystals in bulk solutions; they divided interactions into electrostatic, hydrational, and fluctuation-enhanced repulsions. The relevant fitting parameters were obtained from osmotic pressure and x-ray experiments (44, 45). Since then, Podgornik et al. used a polymer nematic droplet model to establish the state equation of DNA in vivo (46, 47). They divided the osmotic pressure caused by DNA into four parts, i.e., elastic bending, hydrational, electrostatic, and entropic parts, and they also evaluated the encapsidated fraction and spatial density distribution of DNA inside the viral capsid by measured osmotic pressure (46, 47).

In this work, we aim to identify the respective microscopic interaction contributions to viral DNA properties in the dynamic process based on an updated model of dsDNA repulsion potential of Parsegian et al. (44, 45) and the DNA geometric model of Purohit et al. (27, 28, 29). First, we present an alternative continuum elastic model for free energy of the confined DNA in the capsid in which the free energy is induced by the four microscale interactions mentioned above. The current model can explicitly reflect the variation in the entropic part during the dynamic process via a statistical mechanics. Then we obtain the internal forces of the confined DNA by taking the derivative of this DNA free energy with respect to the encapsidated DNA length under variable ionic conditions. To validate the current models, we use this analytic model to calculate the packaging force and the work done by the packaging force for phage phi29, then compare our predictions with phi29 DNA-packaging experiment results (21). The analytic results are comparable to these experimental results. In addition, we study the influence of external pressure and ionic conditions on ejection length for phage phi29 and virus HSV-1. We focus on the respective contributions of microscopic interactions to the ejection force and internal free energy during DNA ejection of phage phi29 and virus HSV-1. In fact, their monotonic or nonmonotonic variations reflect the competition between DNA interchain repulsion and its own bending. The foremost insight of the nonmonotonicity in the entropic contribution change is helpful to make clear the controversy on the entropic contribution.

Materials and Methods

The dsDNA could be coarse grained as a wormlike chain because of its intrinsic stiffness (48, 49, 50). As for the structure of DNA in the capsid, we adopt the previous assumption of a coaxial inverse spool (24, 51), i.e., each hoop of DNA finds itself at the center of a hexagon formed by nearly parallel DNA hoops (28). Fig. 1 shows the DNA structure in the capsid of virus or bacteriophage. Purohit et al. simplified various capsids into three geometries: sphere, cylinder, and capped cylinder (27). They proposed the specific geometrical relationships of DNA chains for various capsid geometries (27). Here, we consider the capsid as a rigid body (14, 21, 29). In the DNA spool, the inner and outer radii are R_in and R_out, respectively. The radius of an arbitrary circular strand is R.

Figure 1.

(A) Arranged DNA in a cylindrical capsid in 3D. (B) The cross section of the DNA spool is shown.

The free energy stored in the viral DNA chain is composed of two parts (23, 24, 28): the interaction energy between adjacent DNA strands and the elastic bending energy of DNA itself. The free energy could be expressed with the interchain distance d as follows:

$$E_{\text{total}}=G_{\text{in}}{L}_{\text{in}}+\frac{2\text{π}{k}_{\text{c}}}{\sqrt{3}d}{\int }_{R_{\text{in}}}^{R_{\text{out}}}\frac{N\left(R\right)}{R}\text{d}R,$$ (1)

in which

$$G_{\text{in}}=G_0+c{k}_{B}T\frac{{\text{π}}^{5/2}}{20\text{π}\sqrt{3}}{k_{\text{c}}}^{-1/4}\sqrt[4]{\frac{{\partial }^2{G}_0}{{\partial }^{2}d}-\frac{1}{d}\frac{\partial G_0}{\partial d}},G_{\text{0}}=a\sqrt{\frac{\text{π}}{2}}\frac{e^{-d/{\lambda }_{\text{H}}}}{\sqrt{d/{\lambda }_{\text{H}}}}+b\sqrt{\frac{\text{π}}{2}}\frac{e^{-d/{\lambda }_{\text{D}}}}{\sqrt{d/{\lambda }_{\text{D}}}},$$ (2)

where the first term on the right side of Eq. 1 represents the energy caused by the interactions between adjacent DNA loops and the second term is the elastic bending energy of a single DNA formulated by Purohit et al. (27). In Eq. 2, G_in represents the free energy per length caused by the interchain interactions, where the first term in G₀ is the bare interaction energy per length, including the contributions of hydration repulsion and electrostatic repulsion. The second term in G_in is the configurational entropy per length, representing the disorder level of DNA arrangement via the statistical average (44). In the previous continuum elastic models, the interaction energy was only assumed as an exponential function of the interchain distance divided by a decay length without distinguishing electrostatic and hydration (28, 32, 36). Here, the hydration amplitude a, the electrostatic amplitude b, the entropic amplitude c, and the Debye screening length λ_D are affected by the ionic conditions; however, the hydration decay length λ_H is insensitive to the ionic strength (44). It is notable that the values of coefficients a, b, and c are obtained by fitting the experimental data, so we regard them as fitting parameters as in (27). The Debye screening length λ_D has its own expression given below. In Eq. 1, k_c = ξ_pk_BT is the intrinsic bending stiffness of a single DNA chain, k_B is the Boltzmann constant, ξ_p is the persistence length of dsDNA chain, and T is the environmental temperature. N(R) is the number of hoops packaged at the location of R, reflecting the geometrical characteristics of different capsid shapes. The specific expressions of N(R) for three capsid geometries can be found in (27). As for the specific capsid geometry and given packaged DNA length, the relationship between the inner radius R_in and the packaged DNA length L_in will be obtained.

Substituting the geometrical functions into Eq. 1 yields a function of the free energy E_total(d, L_in). For the given packaged length L_in, minimizing the total energy E_total(d, L_in) with respect to the spacing d gives

$$\frac{\partial E_{\text{total}}\left(d,L_{\text{in}}\right)}{\partial d}=0.$$ (3)

In this way, the spacing d of the DNA spool’s stable configuration can be obtained. However, it is not expected to find an analytic function of spacing d expressed by the packaged length L_in because Eq. 3 is a transcendental equation. In calculations, the dichotomy method could be used to find a series of discrete spacing d corresponding to discrete packaged length L_in.

The related experiments show that there is a large internal force during DNA packaging into the viral capsid due to capsid confinement, so the portal motor needs to generate a great force to successfully drag the entire genome (51, 52, 53). Next, by using a similar method to Purohit et al., we obtain a theoretical prediction of the internal forces varying with the packaged length under different ionic conditions. Once the total free energy of the packaged DNA is obtained, the internal force can be deduced by differentiating the total energy E_total(d, L_in) with respect to L_in as follows:

$$F_{\mathrm{int}}=\frac{\partial E_{\text{total}}\left(d,L_{\text{in}}\right)}{\partial L_{\text{in}}}$$ (4)

During the packaging, the inertia force could be ignored because of a low packaging speed (13), so the internal force almost counteracts the packaging force generated by viral portal motor, namely F_pack ≈ F_int. However, during DNA ejection, the internal force will drive DNA ejecting from the capsid into the external solution or host-cell cytoplasm (32).

The related parameters and variables in Eqs. 2 and 4 (i.e., the persistence length ξ_p, the fitting parameters a, b, and c, and the Debye screening length λ_D) all vary with different ionic conditions. In the solution with the ionic strength I_i[M], the Debye screening length λ_D is given as

$${\lambda }_{\text{D}}={\left(\frac{{\epsilon }_0{\epsilon }_r\left(T\right)k_{\text{B}}T}{\sum _i{I}_i\left[M\right]N_{\text{A}}{q}_i^2{z}_i^2}\right)}^{1/2},$$

where ε₀ε_r(T) is the dielectric constant of a medium, N_A is Avogadro constant, q_i is the elementary charge, and z_i is the valence of cation. The fitting parameters a, b, and c are obtained from the related experimental data under different ionic conditions, as shown in Table 1. For example, Parsegian et al. fitted these parameters from the osmotic pressure experimental data in NaCl solutions (45). Here, we supplement the other values of these parameters from the osmotic pressure experimental data in 25 mM Mg²⁺ solution (25) and from the packaging force experimental data in two mixed solutions of Na⁺ and Mg²⁺ (21), and the goodness of all fits is more than 0.99. However, the hydration decay length λ_H seems relatively insensitive to ionic conditions; it is almost a constant (2.88 ± 0.2 Å) in low-salt-concentration solutions.

Table 1.

The Fitting Parameters in 1 M, 0.5 M, 0.1 M Na⁺, 25 mM Mg²⁺ Solutions from Osmotic Pressure Experiments and Two Mixed Solutions of Na⁺ and Mg²⁺ from Packaging Force Experiments

Experiment	Ionic Condition	a(× 10−7 J/m)	λH (Å)	b(× 10−9 J/m)	λD (Å)	c
Osmotic pressure (25, 45)	1 M Na+	1.7 ± 0.9	2.9 ± 0.2	0	3.08	1.2 ± 0.1
	0.5 M Na+	1.4 ± 0.5	2.88	3 ± 2.6	4.36	1.3 ± 0.2
	0.1 M Na+	1.1 ± 0.3	2.88	0.41 ± 0.03	9.74	0.8 ± 0.06
	25 mM Mg2+	0.4 ± 0.05	2.88	0.004 ± 0.001	11.1	1 ± 0.1
Packaging force (21)	0.1 M Na+,1 mM Mg2+	4.8 ± 0.5	2.88	0.19 ± 0.02	9.47	1 ± 0.1
	50 mM Na+, 5 mM Mg2+	3.7 ± 0.3	2.88	0.08 ± 0.02	11.92	0.6 ± 0.1

Some experiments indicate that DNA ejection from the capsid is suppressed by the external osmotic pressures (19, 31, 32, 54, 55), but the strong force in the capsid generated by confined genome is capable of powering the entire genome to eject from the viral capsid. Ignoring the friction and inertia, we take the internal force in Eq. 4 as the driven force during the ejection, and the net force resulted by the internal driven force and the inhibited force can be expressed as

$$F_{\text{net}}=F_{\text{int}}\left(d,L_{\text{in}}\right)+F_{\text{ex}}\left(\Omega \right),$$ (5)

where the inhibited force is caused by the external pressure and expressed as

$$F_{\text{ex}}=-\mathrm{\pi }{R}_0^2\Omega \text{,}$$ (6)

where R₀ is the radius of a single DNA chain and R₀ = 1 nm for dsDNA (17, 36, 56); Ω is the external osmotic pressure. The ejection process is stalled when the net force F_net equals zero. In fact, considering the DNA chain as a semiflexible polymer chain through a cylindrical tube of radius R′ with an average speed v (57), there should be friction between DNA and its surrounding liquid due to viscous drag. The total friction coefficient is expressed as (57)

$$\begin{array}{c}{\eta }_{\text{total}}=\frac{2\text{π}\mu L_{\text{in}}}{\ln \left[\left(d-R_{\text{0}}\right)/R_{\perp }\left(d-R_{\text{0}}\right)\right]}+\frac{2\text{π}\mu L_{\text{tail}}}{\ln \left[R_{\text{tail}}/R_{\perp }\left(R_{\text{tail}}\right)\right]}\\ +\frac{3\text{π}{\mu }_{\text{sol}}{L}_{\text{out}}}{\ln \left[L_{\text{out}}/2R_{\text{0}}+\exp \left(4\sqrt{L_{\text{out}}}/\sqrt{3\text{π}{\xi }_{\text{p}}}\right)\right]},\end{array}$$ (7)

where the first term on the right of Eq. 7 represents the part inside the capsid or phage head, the second term represents the part in the tail tube (for viruses without tails, this term disappears), and the last term is the part in the external solution. For simplicity, we take the viscosity coefficient as μ_sol ≈ μ in a simple buffer solution. R_⊥(R′) means the effective transverse displacement of the polymer chain.

During the packaging or ejection, the DNA chain moves slowly in solution; according to viscous fluid mechanics, the viscous frictional force F_vis can be expressed as

$$F_{\text{vis}}={\eta }_{\text{total}}v,$$ (8)

where v is the packaging or ejection speed. Neglecting the kinetic energy due to the tiny inertial force, the total work needed due to the packing energetics and viscous dissipation is obtained as

$$E_{\text{pack}}={\int }_0^L{F}_{\text{pack}}\text{d}x+{\int }_0^L{F}_{\text{vis}}\text{d}x,$$ (9)

where the two terms on the right side of Eq. 9 represent the works done by the packaging force and the viscous frictional force, respectively. Note that both the packaging force F_pack and viscous frictional force F_vis are discrete. Therefore, the numerical trapezoid formula will be used in the following calculations.

Results and Discussion

Packaging

Bacteriophage phi29 is taken as a sample for the study of the packaging process. The related parameters are given as follows: the Boltzmann constant k_B = 1.38 × 10⁻²³ J/K; the temperature T = 298 K; and the persistence length of dsDNA chain ξ_p = P₀ + (4λ_D²λ_Β)⁻¹, where P₀ is the nonelectrostatic contribution to the persistence length, P₀ = 50 nm, and l_B is the Bjerrum length (7.14 Å in water at 298 K for dsDNA) (58). The geometrical shape of phage phi29 is simplified as a cylinder. The total dsDNA chain length L = 6500 nm (21, 59), the outer radius of DNA spool R_out = 17 nm, the height of DNA spool z = 45 nm, and the length of phi29 phage tail L_tail = 40 nm (59). Equation 4 and the related fitting parameters in Table 1 are used to obtain the packaging forces in different ionic solutions. The variation of the packaging force versus the length of DNA packaged into the bacteriophage phi29 capsid is shown in Fig. 2. The three solid curves represent the packaging forces predicted in 0.1, 0.5, and 1 M Na⁺ solutions, respectively. The solid square and triangle represent the packaging forces measured in 100 mM Na⁺ plus 1 mM Mg²⁺ and 50 mM Na⁺ plus 5 mM Mg²⁺ mixed solutions, respectively (21); the two dashed curves are the corresponding fitting curves.

Figure 2.

Packaging force versus packaged DNA length under different ionic conditions for bacteriophage phi29.

As shown in Fig. 2, the three curves of the packaging forces predicted by using the Na⁺-related parameters (25) exhibit the same trend as the experimental results in mixed solutions (21): they all rise monotonically with the increasing packaged DNA length. This is because the interchain spacing decreases as the packaged length increases, which results in the growth of the internal energy barrier. In addition, the comparison of three solid curves indicates that the packaging force in monovalent cationic solutions with lower concentration is greater than that with higher concentration. A similar ionic effect was also observed in the experiments of multivalent cationic effect on the mechanical properties of the phage phi29 (34, 35). Nevertheless, some deviations between the predicted solid curves and the experimental scatter points exist, which were also found in the predictions of Fuller et al. (21) by using the model of Purohit et al. (27) and bulk DNA parameters (25). Theses deviations mainly come from different solution conditions. In the packaging force experiment, the divalent Mg²⁺ ions were mixed. Mg²⁺ has greater potential to screen the electrostatic interaction of DNA than Na⁺, so the monovalent cationic effect is often ignored in physiological solutions containing varied ions (13, 19, 60). Besides, the DNA strands constrained in the capsid are also different from the bulk ensemble in the osmotic pressure experiments (21, 45, 61). To alleviate this deficiency, we supplement some fitting parameters for the two mixed solutions into Table 1.

Once the viscous friction coefficient during the packaging is obtained via Eq. 7, the packaging speed measured by Fuller et al. (21) is substituted into Eq. 8 to obtain the viscous frictional force F_vis. Then Eq. 9 is applied to obtain the total work due to the packing energetics and viscous dissipation during the packaging. As shown in Fig. 3, the work done by the motor approximately is one-third of the chemical energy from ATP hydrolysis (per ATP consumed for ∼0.68 nm movement); this result is consistent with the previous experiments (51, 62), and the remaining 70% of energy is responsible for chemical cycles and heat releasing due to enthalpy change during DNA translocation (10, 62). After mixing the divalent Mg²⁺ into the solution, the total work becomes smaller than that in the monovalent cationic solution because of the reduction of the packaging force and velocity. This phenomenon suggests that the total work in cationic solutions with higher concentration will be less than that with lower concentration. During the packaging, the energy due to the viscous dissipation is several orders of magnitudes less than the work done by the packaging force; this indicates the appropriateness of the quasistatic assumption in the study of the packaging process (21, 27, 36).

Figure 3.

The energy and work during the viral packaging process under different ionic conditions for bacteriophage phi29.

Ejection

We attempt to study microscopic interactions for DNA ejection from the capsid to the external environment. HSV-1 in a monovalent cationic solution and phi29 phage in a divalent cationic solution are taken as calculation samples. The capsid of HSV-1 is an icosahedron (63), but we model it as a sphere for simplification (28). The total length of the HSV-1 DNA chain is 51,500 nm (64), and the outer radius of the DNA spool R_out = 43 nm (64). The parameters of phi29 phage are the same as the previous section, and the prescribed external osmotic pressure is 3.5 atm, the same as the previous ejection experiment (19).

Fig. 4 shows the percentage of HSV-1 DNA ejection length versus the external osmotic pressure. As shown in Fig. 4, the percentages of DNA length ejected from HSV-1 capsid all decrease with the increasing external pressure under three different ionic conditions, and the predicted curve for 1 M Na⁺ solution agrees well with the experiments of Bauer et al. (54). When the resistant force caused by the external pressure is greater than the ejection-driven force caused by the rest of DNA in the capsid, the DNA ejection terminates. In addition, in a monovalent cationic solution with higher ionic concentration, viral DNA ejection can be inhibited by a lower external pressure, which reflects the ionic influence on DNA ejection.

Figure 4.

The percentage of DNA length ejected from HSV-1 versus the external osmotic pressure.

The divalent ionic influence on DNA ejection from phi29 phage is more directly shown in Fig. 5. If the external pressure remains unchanged, the ejection length fraction will become less with the increasing concentration of MgCl₂. Because of different phages in different solutions, the difference between the theoretical curve and the experimental scatter (19) is obvious, but their descending tendencies are almost the same. In vivo, the external osmotic pressure caused by cytoplasm or nucleus is almost constant. Therefore, increasing the concentration of cations in solutions may block the viral DNA ejection into the host cell to stall the infectious process. In addition, the percentage of ejection length tends to be stable as the MgCl₂ concentration increases; this implies that some microinteraction contributions to the ejection-driven force are insensitive to the ionic variation. Therefore, it is necessary to distinguish the microscopic interactions for DNA ejection from virus.

Figure 5.

The percentage of DNA length ejected from phi29 phage or λ phage versus MgCl₂ concentration.

Fig. 6 shows the changes of microscopic interaction contributions to the mechanical properties of HSV-1 DNA during its ejection. As shown in Fig. 6 A, the entropic contribution to ejection force or free energy rises up gradually with the decrease of remaining DNA length until reaching its peak. This implies an increase of the disorder level. During this stage, the entropic contribution to free energy or ejection force is not negligible. This result supports the previous conclusion obtained in MD simulations that DNA configurational entropy is a dominant contribution to free energy (40). After less than 20% length of DNA remains in capsid, the interchain spacing is so large that the interactions between DNA chains become very weak. Therefore, the contribution of fluctuation-enhanced interaction, namely entropic effect, decreases abruptly in the final stage of ejection.

Figure 6.

The contribution percentage of (A) configurational entropy, (B) electrostatic repulsion, (C) hydration repulsion, and (D) elastic bending to ejection-driven force and free energy versus percentage of DNA length remaining in the capsid for HSV-1.

As shown in Fig. 6 B, the electrostatic repulsion contribution to free energy or ejection-driven force also rises at the first stage, then falls down. Compared with the curves of entropic contribution, the peak range of the electrostatic repulsion contribution is wider (from 60 to 40% of remaining DNA length). This range just corresponds to the interchain spacing 30–35 Å, which is dominated by the electrostatic repulsion (30). While the DNA continues to eject out, the electrostatic contribution decreases as the chain spacing increases in the capsid because of the Coulomb effect (65).

The hydration repulsion plays a predominant role with a shorter spacing of <30 Å (28, 43), and its contribution as shown in Fig. 6 C declines after the start of ejection because the DNA interchain spacing exceeds the predominant range of hydration repulsion. Different from the above three kinds of interaction contributions, the bending contribution to free energy in Fig. 6 D rises up when the DNA density reduces, whereas its contribution to ejection force first decreases slightly, then rises up abruptly in the last 20% of the remaining DNA length.

The comparison of four graphs in Fig. 6 suggests the nonmonotonicity in entropic and electrostatic repulsion and bending contributions to ejection force and the monotonicity in hydration repulsion contribution to ejection force. The competition between interaction and bending elasticity leads to the equilibrium of the DNA state in the capsid (13); the nonmonotonicity or monotonicity of microscopic interactions actually reflects this competition during DNA ejection. Similarly, the nonmonotonic variations of ejection fractions and interchain spacing versus the concentration of divalent cations were observed in previous DNA ejection experiments (19). We also use the same method to analyze the microscopic interaction contributions of phi29 phage DNA in 0.1 M NaCl; the related results are shown in Fig. 7 and given in Appendix A. Note that the regular packaging with a hexagonal arrangement of DNA cannot exist when only 20% or less of the genome is inside the capsid (66). However, the interaction model of Parsegian et al. is still applicable for DNA in several phases regardless of the hexagonal arrangement (44, 45), and the effective range of interchain spacing even exceeds 100 Å (45); such a large range suffices for the level of the interchain spacing corresponding to the entire ejection process.

Figure 7.

The contribution percentage of (A) configurational entropy, (B) electrostatic repulsion, (C) hydration repulsion, and (D) elastic bending to ejection-driven force and free energy versus percentage of DNA length remaining in the capsid for phi29 phage.

For comparison, we also calculate the energy caused by the respective interactions by using the model of Podogornik et al. and their parameters for bulk DNA in 0.1 M NaCl solution (43, 46, 47). The relevant results are marked with black solid circles in Fig. 6. In the calculation, we take the same DNA interchain spacing as this work. However, the calculation of bending elastic energy in the model of Podogornik et al. needs to measure the value of osmotic pressure, but this is difficult for virus DNA in vivo (33). Although the respective percentages of electrostatic and hydration interactions in the model of Podogornik et al. have the similar trends as our predictions, there is an obvious quantitative difference. This discrepancy reflects the difference between bulk DNA and viral DNA. Nevertheless, our model did not reflect a discontinuous change between line hexatic-cholesteric phase transitions in the capsid like the model of Podogornik et al. (43), but it has its own applicability for dealing with the mechanical problems of virus DNA in vivo. Furthermore, similar to the theoretical studies of the packaging and ejection processes by Purohit et al. (29) and Evilevitch et al. (19), respectively, we assume the inverse spool as the optimal geometry throughout the dynamic process, so the relevant results based on our model need to be validated by more delicate experiments or by MD simulations in the future.

Conclusions

In this work, based on the DNA free-energy formula of Parsegian et al. (44, 45) and the DNA geometric model of Purohit et al. (27, 28, 29), an updated continuum elastic model of confined DNA in viral capsid is presented. The updated model can be used to identify the respective contributions of microscopic interactions such as electrostatic repulsion, hydration repulsion, configurational entropy, and elastic bending. During the packaging and ejection of viral DNA, these contributions of microscopic interactions play critical roles in mechanical properties such as packaging force, ejection-driven force, and ejection length. The influences of ionic kind and concentration on the internal free energy, packaging force, and ejection-driven force are also investigated symmetrically.

Our predictions quantitatively or qualitatively agree well with the relevant packaging and ejection experimental data of viral DNA (19, 21, 51, 54, 62). The configurational entropic term in this continuum model may reflect the DNA disordered level during the packaging and the ejection of DNA in the form of statistical mechanics. The change of DNA disordered level has been revealed as important in the previous cryo-EM experiment (66), osmotic experiment (43), and MD simulations (40, 67). During DNA ejection, the percentages of diverse microscopic interaction contributions to ejection-driven force and free energy versus the percentage length of DNA remaining in the capsid are nonmonotonic or monotonic. These variations actually reflect the competition between interchain interaction and bending elasticity (13). Similar nonmonotonicity of ejection fractions and interchain spacing versus the concentration of divalent cations was observed in the previous experiments (19).

We also use our model to study the ionic effect of viral DNA’s mechanical properties and obtain conclusions similar to previous experimental and theoretical results (21, 28): higher concentrations of monovalent or divalent cations will weaken the mechanical properties of viral DNA because they can counteract more negative charges of DNA chains so as to screen the electrostatic repulsion dramatically. Increasing the concentration of cations or mixing higher-valent cations may be a potential method to prevent viral infection and to regulate gene delivery. Recently, it was reported that multivalent cations such as Co³⁺ or spermidine can result in attraction interaction between DNA chains (13, 19, 33, 34, 35, 68); this phenomenon is an important topic to be studied in the future.

Author Contributions

N.-H.Z. designed the research. C.-Y.Z. contributed the calculation. All authors analyzed the data and wrote the manuscript.

Editor: Gijs Wuite.

APPENDIX A: Microscopic Interaction Contributions to Phi29 Phage DNA Ejection

Fig. 7 shows the respective microscopic interaction contributions to the mechanical properties versus remaining DNA length in the capsid during phi29 phage DNA ejection. As observed in Figs. 6 and 7, the variation tendencies of electrostatic and hydration contributions to the ejection-driven force and free energy for phi29 phage are almost consistent with that for HSV-1. As for as the entropic contribution, both curves increase to their maximums firstly, then drop abruptly in the final stage. However, there are obvious differences in their growth rates and convexity-concavity properties. The foremost difference between them lies in the bending contribution to ejection-driven force. The bending contribution to ejection-driven force for HSV-1 is nonmonotonic, whereas that for phi29 phage monotonically increases with the decrease of the remaining DNA length. We ascribe this diversity to different ionic conditions as well as different packaging efficiency (29). Therefore, a viral system in vivo has its own diversity and uniqueness.