Direct measurement of the dielectric polarization properties of DNA

Significance

The strength of DNA–DNA and DNA–ligand electrostatic interactions crucially depends on the electric polarizability of DNA, represented by its dielectric constant. This has remained unknown owing to the lack of experimental techniques able to measure it. Here, we experimentally determined the dielectric constant of double-stranded DNA in a native condensed state inside a single bacteriophage as well as the dielectric constants of the protein shell and tail that compose the viral capsid using scanning force microscopy. We supported the experimental data by theoretically determining the DNA dielectric constant using atomistic simulations. Both approaches yield a dielectric constant of DNA around 8, sensibly higher than commonly assumed, thus revealing a DNA intrinsic property essential for realistic computational description of DNA.

Abstract

The electric polarizability of DNA, represented by the dielectric constant, is a key intrinsic property that modulates DNA interaction with effector proteins. Surprisingly, it has so far remained unknown owing to the lack of experimental tools able to access it. Here, we experimentally resolved it by detecting the ultraweak polarization forces of DNA inside single T7 bacteriophages particles using electrostatic force microscopy. In contrast to the common assumption of low-polarizable behavior like proteins (ε_r ∼ 2–4), we found that the DNA dielectric constant is ∼8, considerably higher than the value of ∼3 found for capsid proteins. State-of-the-art molecular dynamic simulations confirm the experimental findings, which result in sensibly decreased DNA interaction free energy than normally predicted by Poisson–Boltzmann methods. Our findings reveal a property at the basis of DNA structure and functions that is needed for realistic theoretical descriptions, and illustrate the synergetic power of scanning probe microscopy and theoretical computation techniques.

Electrostatic forces have long been recognized to inherently influence the DNA structure and interactions (1, 2) including DNA bending and folding (3, 4), DNA packaging (5–7), and DNA–ligand recognition (8–10), owing to the high charge density of the DNA molecule. In particular, the crucial role of coulombic forces in the binding affinity of molecules to a specific DNA sequence, such as clinically important drugs into minor grooves (9) and protein complexes into major grooves (11), has been well established (12). However, although detailed knowledge of DNA–DNA and DNA–ligand interactions can nowadays be obtained by high-resolution structural techniques, energetic analysis remains experimentally challenging because it requires quantification of the different contributions to such interactions, in particular the electrostatic energy term.

To study the DNA electrostatics, theoretical methods are normally used. All of them require to precisely know the DNA polarization properties in addition to the detailed molecular structure and charge distribution (13–15). In the commonly used mean-field approximation, such as the Poisson–Boltzmann theory, this means including explicitly or implicitly the dielectric constant of DNA, ε_DNA, a measure of the screening of coulombic forces between charges due to the presence of the DNA molecule. However, despite its crucial impact, the dielectric constant of DNA has remained unknown so far. In the absence of a precise knowledge of ε_DNA, theoretical models typically assume DNA to be a low-polarizable medium with ε_DNA ∼ 2–4 like dry proteins surrounded by a highly screening solvent with the dielectric constant of the bulk water ∼80 (13–17). However, this is a simplified picture that lacks experimental validation and introduces a major source of uncertainty, which can give completely biased results in theoretical methods.

Direct experimental evaluation of ε_DNA has been practically impossible because of its inherently microscopic nature that requires high-resolution techniques together with the complex structure and chemical composition of DNA and its interaction with the solvent. In particular, standard dielectric characterization tools such as impedance spectroscopy (18, 19) and dielectrophoresis (20, 21) only yield average values of DNA polarizability in bulk solution that include major structural contributions and DNA–solvent interfacial effects. Microscopic measurements with dielectric sensitive fluorescence dyes (22, 23) reflect the DNA microenvironment rather than the DNA polarizability. Measured dielectric constants associated to DNA in the literature are therefore relatively high values in the range of 20–60 (20–23), which account for large solvent artifacts.

Recently, we demonstrated that the dielectric constant of a single bacteriophage T7 can be measured using electrostatic force microscopy (EFM) (24). Bacteriophage T7 is an ideal system for probing the dielectric constant of DNA in a native state because the DNA is confined inside the capsid at such high density that it shows a nearly crystalline concentration. Furthermore, our approach makes it possible to measure it in dry conditions, avoiding major distortions caused by the solvent. However, dissecting the dielectric contribution of DNA confined into the bacteriophage has proved challenging because the dielectric properties of the protein capsid are also unknown (25–29).

Here, to experimentally determine the dielectric properties of DNA and bridge the gap between experimental data and computational analysis, we have achieved precise measurement of the dielectric constants of the main capsid components of the bacteriophage T7, and therefore we obtained the first (to our knowledge) experimental value of the dielectric constant of DNA inside the bacteriophage. We found that ε_DNA ∼ 8, a value considerably high compared with the common assumption and in particular to the value of ∼3 that we obtained for the shell and tail proteins of the virus in line with expected values for dry proteins (ε_protein ∼ 2–5) (25–31). We supported the experimental findings with computational analysis and calculated the dielectric constant of DNA by means of atomistic molecular dynamic (MD) simulations in explicit solvent. The theoretical dielectric constant of ∼8 that we obtained perfectly matches the experimental data. By solving the Poisson–Boltzmann equation (PBE) for DNA–ligand systems, we illustrate the importance of using current dielectric constant in mean-field approaches to avoid overestimating the affinity of DNA for ligands.

Results and Discussion

Sample Preparation and Characterization.

The T7 particle consists of double-stranded DNA (dsDNA) condensed inside a 2.5-nm-thick protein shell of ∼60 nm in diameter that assembled with a tail builds the capsid (Fig. 1A). The DNA is highly packed, forming concentric layers with center-to-center distance of 2.7 nm as shown by cryo-electron microscopy (cryo-EM) (Fig. 1 B and C, arrows) (32, 33). The capsid presents a tail complex that protrudes from one vertex of the shell. It has a conical shape of ∼30-nm length and ∼18-nm maximum width (32, 34) (Fig. 1 C and E), and it drives the ejection of the DNA via a small channel (2–4 nm in diameter) after the recognition of the host cell receptors.

Fig. 1.

Electron microscopy (EM) micrographs of the T7 bacteriophage and its components. (A) Simplified drawings of the DNA-containing virus and the capsid components, the shell and the tail. (B) Cryo-EM images of DNA-containing (B, Upper) and empty viruses (B, Lower). (C) A plane of the 3D reconstruction of T7 bacteriophage (32, 33). The arrow in B and C shows DNA filaments. (D) Micrograph of negatively stained samples of the empty shells treated with 4 M guanidinium chloride (GuHCl) and (E) of tail complexes. The diameter of ∼70–80 nm of the shells indicates that the shells collapsed in a double layer but did not open up into a single layer. (Scale bar: D, 50 nm; E, 30 nm.)

The dielectric constant of the whole T7 particle (Fig. 1A), ε_virus ∼ 6.3, which we measured in ref. 24 after desiccation [relative humidity (RH) < 1%], is an effective value (Fig. 1A) that accounts for the contribution of the condensed DNA, the proteins of the shell, and any proteins of the core–tail complex that may have been retained. To dissect the contribution of the DNA, here we separately measured the dielectric constants of the DNA-free shell, ε_shell, and of the tail complex, ε_tail.

We isolated the protein shell by treating T7 particles first with EDTA to separate the tails and eject the DNA in vitro (Figs. S1 and S2), and then with 4 M guanidinium chloride (GuHCl) to force their collapse, thus avoiding the formation of a water meniscus inside (35). In transmission electron microscopy (TEM) images (Fig. 1D), the shells appeared squashed but preserved a rather spheroidal shape. Their average diameter of ∼70–80 nm, significantly larger than the ∼60 nm of the shells before GuHCl treatment (Fig. S2B), together with their staining characteristics, clearly indicate that GuHCl-treated shells must have completely flattened into a double layer, but did not open further up into a single layer. Atomic force microscopy (AFM) topography images of the shells adsorbed on a graphite substrate in dry air (RH < 1%) for dielectric characterization (Fig. 2A and Fig. S3 A and B) confirmed the TEM findings. They showed shells with a circular shape of average diameter of 86 ± 5 nm with no visible pinholes and an almost flat profile between 5 and 10 nm with some corrugations but measuring ∼5 nm in thickness in grooves and flat regions, thus confirming the total collapse of the capsid into a double layer.

Fig. 2.

Dielectric constant measurement of the shells and of the tails. (A and C) Measured topography and (B and D) dielectric images at constant height of the T7 shell and its tail (scan height z = 23.5 and 21 ± 0.5 nm) and (E) profiles taken in air. The shell is collapsed, showing some corrugations and measuring ∼5 nm in thickness in flat regions. The tail is a little higher but much smaller in width. (F) Matching the maximum dielectric signals (symbols) with calculations using a homogeneous oblate spheroid model gives the dielectric constants ε_shell = 3.20 ± 0.15 (cyan) and ε_tail = 3.15 ± 0.18 (blue). Despite the different size of the tip (radius R = 10.5 and 4.25 nm) and the different structure of the particles, we obtained the same dielectric constant for the shell and the tail, which reflects their protein composition. The error bar in F is smaller than the symbol. Two almost overlapped color-coded regions (cyan and blue) indicate the SDs of the two particles, as obtained in Fig. 4, by repeating the experiment on different particles. (Scale bars: A and B, 45 nm; C and D, 30 nm.)

We purified the tail complex by using a recombinant approach (Methods). The T7 tail is composed of several proteins, namely, connector (gp8), tubular tail proteins (gp11 and gp12), and the fiber (gp17) (34). Here, we used a purified recombinant tail complex composed by proteins gp8, gp11, and gp12 (Fig. S1B). TEM images (Fig. 1E) show the tails have expected shapes and sizes in agreement with those found in viral particles. AFM topography images of the tails adsorbed on a graphite substrate in dry air (Fig. 2C and Fig. S3C) confirmed the TEM results. They show that the tails are clearly smaller in lateral width than the shells with a slightly heterogeneous elongated shape as better appreciated in the AFM phase images (Fig. S3D).

Dielectric Measurement of the T7 Shell and Tail Complex.

We measured the dielectric constants of the T7 capsid components by quantitative EFM and applied the same procedure demonstrated in ref. 24.

Fig. 2B shows the dielectric image—the capacitance gradient, dC/dz—of a representative shell measured at constant height from the substrate. The tip radius R = 10.5 ± 0.25 nm was precisely calibrated in situ (Methods). Given the not perfectly flat shape of the shells (Fig. 2E, dash line), typically showing a slightly higher region near to the center, we numerically modeled them as an oblate spheroid—and not as a flat disk—of dielectric constant ε_shell and maximum height corresponding to the maximum measured height (Fig. S4), thus using 2D axisymmetric calculations instead of complex 3D models. By matching the dielectric signals with finite-element calculations, we found ε_shell = 3.2 ± 0.15 (Fig. 2F). We repeated the experiment at four different scan heights and found an average value on the set of measurements of ε_shell = 3.22 ± 0.02. The small error thus obtained proves the consistency of our model; otherwise, the obtained dielectric constant is likely to diverge with the scan height.

To further assess the robustness of the model against geometrical contributions, we repeated the experiment on different shells and with different tips. Fig. 3 shows the tip radius R, the height h and width w—the main geometrical parameters of the system—of different shells (cyan symbols) as a function of the obtained dielectric constant. We found an average value of ε_shell = 3.47 ± 0.48, with an error that remained small. We therefore conclude that the obtained dielectric constant, ε_shell ∼ 3.5, is essentially independent of the detailed shape of the shell (the corrugations and diameter) and of tip used. It captures the dielectric response of the protein gp10 assembled to build the shell. This value is remarkably close to the theoretical estimations for the interior of proteins, which is predicted to be a low-polarizable medium with ε_r ∼ 2–3 (28, 29). It is also within the range of experimental values reported for dry protein powders in bulk (ε_r ∼ 2−5) (30, 31), reflecting the similar dry environment.

Fig. 3.

Measured dielectric constants of the protein shell (gp10) and tail (gp8, gp11, gp12) for a set of different particles. We show the measured geometric parameters for each experiment—tip radius, maximum height, and width of the particles—of flattened shells (filled cyan) and tails (filled blue) as a function of the dielectric constant ε_r obtained assuming a uniform ellipsoidal model. For comparison, the data of mature viruses (unfilled magenta) are reported here from ref. 24, for which we obtained ε_virus = 6.25 ± 0.37. The average ε_shell = 3.47 ± 0.48 (shell) and ε_tail = 3.38 ± 0.38 (tail) obtained here nicely agree, showing that the dielectric constant of the protein capsid is ∼3.5 in dry conditions independently of geometrical variations of the tip and of the particles. Each point is the average of four measurements taken at different scan heights and the error bar is the SD (not shown when smaller than the symbol); the color-coded regions indicate the obtained SD of the particles.

Fig. 2D shows the dielectric image of a representative tail complex. Because the shape is only slightly elongated and with a clearly higher region at the center, we applied the 2D symmetric spheroidal model also for the tails. By fitting the measurement (Fig. 2F), we found a dielectric constant of ε_tail = 3.15 ± 0.18; the average value of repeated measurements at different scan heights was 2.9 ± 0.15. This value remarkably agrees with the value obtained for the shell reported in the same figure, despite the different size of the two particles (compare width and height in Fig. 2E) and the fact that they were obtained in different experiments, that is, over different graphite substrates using different tips: radius R = 4.25 nm (tail) and R = 10.5 nm (shell). By repeating the experiment over a set of different tails of slightly different size and shape (average h = 8.4 nm, w = 40 nm), we found that the dielectric constant of the tail is on average ε_tail = 3.38 ± 0.38 (Fig. 3, blue symbols). Again, the small uncertainty obtained proves that the measured value of ε_tail is independent of geometrical parameters.

Remarkably, the obtained dielectric constant for the tail proteins (gp8, gp11, gp12) is essentially the same as that obtained for the gp10 shell protein. Despite the different amino acid composition and structural features, we found that the protein complexes that compose the bacteriophage have similar dielectric properties, supporting the theoretical predictions of a low dielectric constant for the interior of proteins independently of the amino acid sequence (28, 29), which presumably reflects a similar composition of the protein cores and their analogy with polyamides (ε_r ∼ 3–4) (36).

Measured Dielectric Constant of Condensed DNA.

Having precisely quantified the dielectric contributions of the protein capsid, we can determine the dielectric constant of the DNA confined inside from the effective dielectric constant of the whole virus, ε_virus = 6.25 ± 0.37, previously measured; for clarity, the corresponding data were reported here in Fig. 3 from ref. 24. This value, clearly higher than the dielectric constant of the shell, ε_shell ∼ 3.47, and of the tail, ε_tail ∼ 3.38, that we found here (Fig. 3), indicates a larger dielectric contribution from the condensed DNA.

To precisely quantify ε_DNA, we performed finite-element calculations with a core–shell oblate spheroid model with shell thickness of 2.5 nm and dielectric constant, ε_shell ∼ 3.47 ± 0.48, and unknown core dielectric constant, ε_DNA (Fig. S4B). We assumed the core to be fully occupied by DNA because upon desiccation the internal phage proteins were presumably ejected together with a fraction of condensed DNA (24, 35). If, on the other hand, a small amount of internal phage proteins had remained inside, the fact that they have been found to give the same dielectric response as the shell proteins and occupy a small fraction of the whole viral volume would make our model a reasonable approximation. Fig. 4 shows the calculated effective dielectric constant of the core–shell model of the virus, ε_virus, with known dielectric constant of the shell measured here, ε_shell ∼ 3.47, as a function of the unknown dielectric constant of the core, ε_DNA. Continuous lines correspond to the average values, whereas the dashed lines correspond to the average values ± SD. By matching the calculations with the measured dielectric constant of the virus, ε_virus ∼ 6.25 ± 0.37, here we obtained that ε_DNA = 8.5 ± 1.4. We then conclude that the dielectric constant of the DNA condensed inside the bacteriophage is ∼8.5, noticeably higher than the low values (ε_r ∼ 2−4) typically assumed in theoretical calculations. This value was obtained in perfectly dry conditions that ensured minimum free water content, which is also reflected in the low dielectric constants that we measured for the protein complexes. We can reasonably rule out a significant contribution from water molecules that might have remained tightly bound to DNA (37). Under such dry conditions, tightly bound water would occupy an extremely small fraction of volume (37), and in addition it would show a significantly reduced dielectric response with respect to bulk water (38, 39). This suggests that the higher dielectric constant found here for DNA reflects the dielectric behavior of the DNA molecule and not of water or other molecules composing the virus.

Fig. 4.

Measured dielectric constant of the condensed DNA of bacteriophage T7. Effective dielectric constant of the core–shell model of the virus with measured dielectric constant of the shell, ε_shell = 3.47 ± 0.48 (black lines) as a function of the dielectric constant of the core, ε_DNA. By matching the calculations with the measured effective dielectric constant of the virus, ε_virus = 6.25 ± 0.37 (magenta lines), we obtained ε_DNA = 8.5 ± 1.4 for dsDNA (magenta color bar) condensed inside the virus. Core–shell model calculations were performed using the average sizes measured in the experiments of the viruses, namely, height h = 36 nm and full-width w = 54.8 nm, shell thickness of 2.5 nm, and average tip radius R = 7.8 and half-angle of 10°. The continuous lines are the average values, and the dashed lines are the average values ± SD.

Calculated DNA Dielectric Constant with Molecular Dynamic Simulations.

Direct comparison of the dielectric constant of DNA obtained here with previous observations is difficult owing to the lack of data on isolated DNA. As described above, only experimental data that indicate high dielectric constants ∼20–60 (20–23) have been reported based on electrokinetic and fluorescence experiments with DNA in solution, which include major contaminations from solvent screening. These values tend to reflect the DNA local environment that is likely to have a dielectric constant smaller than that of bulk water due to water structuring near the charged DNA surface. The only reported value that refers to an isolated DNA molecule is a theoretical prediction for triple-stranded DNA, estimated ∼16 with a high contribution coming from the sugar–phosphate backbone ∼21 (38).

To test our experimental findings, we theoretically determined the dielectric constant of DNA with atomistic molecular dynamic simulations. We followed Pettitt’s procedure (38) but considered a duplex DNA and simulated it in the microsecond timescale using our last-generation force-field: parmBSC0 (40). The selected sequence was the well-characterized Drew–Dickerson dodecamer, which we simulated in a box of TIP3P water molecules with minimum salt, periodic boundary conditions, and particle mesh Ewald correction for long-range electrostatic interactions (41, 42). The dielectric constant for the 12-mer B-DNA, obtained from our stable and fully converged trajectory (Fig. S5 A and B), was computed using 10 different 100-ns-long segments (replicas) taken from the last 2 μs of simulation (out of 4 μs of total simulation time). We found the dielectric constant for the naked DNA to be ε_DNA ∼ 8.3 ± 0.4. The obtained dielectric constant is again sensibly higher than normally assumed and in perfect agreement with the experimental observation. The high dielectric constant of DNA arises from the sugar–phosphate backbone that gives the largest contribution to the total dielectric response, as shown in Table 1.

Table 1.

Dielectric constants and volumes obtained from MD simulations of the Drew–Dickerson dodecamer

Component	Volume, nm3	Dielectric constant, εr
Simulation box	166.6 ± 0.9	83.6 ± 0.6
DNA	10.1 ± 0.2	8.3 ± 0.4
DNAsugar–phosphate	3.4 ± 0.2	16.4 ± 1.4
DNAsugar–base	7.6 ± 0.2	5.6 ± 0.6

The remarkable agreement between the experimental and predicted value of ε_DNA supports our conclusion that the dielectric constant measured in a single bacteriophage reflects the DNA molecule and particularly its composition. It suggests that shape effects due to DNA condensation inside the viral capsid as well as effects of water molecules that may have remained bound to DNA are negligible. In particular, concerning the potential influence of remaining waters, we note that we could not dissect the contribution of strongly bonded waters from that of the total simulation box (see Fig. S5C for a concise description of the simulated system), which shows a dielectric constant only slightly higher than the nominal value for bulk water (∼84; Table 1). However, the perfect matching of the experimental dielectric constant with that obtained theoretically for naked-DNA dodecamer suggests that residual water molecules in the experiment, if any, were negligible or dielectrically inactive (“ice-like” hypothesis) (43).

Impact of the DNA Dielectric Constant in DNA–Binder and DNA–Cation Interaction.

The higher dielectric constant found here compared with low values traditionally assumed (13–17) has profound effects in the predictions of continuum calculations. In particular, it can drastically change the prediction of DNA–ligand interaction energies derived from Poisson–Boltzmann calculations (8–12). We qualitatively illustrate it here by solving the linear PBE for DNA–ligand binding systems and for different dielectric constants, that is, for the commonly used values ε_DNA = 1−4 and for the value ε_DNA ∼ 8 found here.

Two protein–DNA complexes, the high mobility group (HMG) domain and the repressor of the phage 434 protein (Fig. 5A, Upper), and the two most abundant physiological monovalent cations (Na⁺, K⁺) (Fig. 5A, Lower) were used to estimate the interaction free energy using PBE. Fig. 5A gives the calculated changes of the electrostatic free energy upon binding, which is favorable to the binding for all of the dielectric constants considered, both for major groove (repressor 434) and minor groove (HMG domain and monovalent cations) binders. Although at relative large distances the interaction energy cost is essentially insensitive to the choice of the dielectric constant, it turns extremely sensitive to ε_DNA at small distances, strongly increasing the magnitude of the binding with decreasing ε_DNA. For ε_DNA ∼ 8, the binding strengths is almost one-half of the predicted for low dielectric constants (ε_DNA ∼ 2–4), thus resulting into a strongly decreased contribution to the binding energy, which is normally overestimated in linear PBE calculations (44). Similar conclusions on the sensitivity of computed DNA–drug binding energies to the dielectric constant have been previously reached using linear PBE, nonlinear PBE, and experimental binding affinities (9).

Fig. 5.

Impact of changing the DNA internal dielectric constant in DNA–protein and DNA–cation interaction. (A) Calculated single-point interaction free energy (electrostatics and van der Waals) of DNA major and minor groove binders, obtained by solving the linear PBE using different dielectric constants for DNA (ε_DNA = 1, 2, 4, and 8): repressor of the phage 434, a major groove binder; HMG domain of the human SRY factor interacting with the minor groove of DNA; DNA–sodium and DNA–potassium interaction in the minor groove of the Drew–Dickerson dodecamer. The initial structure was taken from a representative snapshot during the MD simulation (42b). (B) Molecular structure of Drew–Dickerson dodecamer and calculated molecular interaction potential using sodium (Na⁺) for a constant energy threshold (−29 kJ/mol). The gray mesh corresponds to a low internal dielectric constant (ε_DNA = 1), and the red mesh to the value obtained here (ε_DNA = 8).

To further analyze the impact of higher internal dielectric constant of DNA, we used sodium as a probe to compute the molecular interaction potential for different values of ε_DNA. A representative snapshot during the MD simulation of the Drew–Dickerson dodecamer was chosen and a constant energy threshold was used to compare the 3D maps for ε_DNA = 1 and ε_DNA = 8 (Fig. 5B). We paid special attention to the interaction of sodium or potassium in the minor groove of the AATT central tetranucleotide. Cations were already found to interact in the minor groove of the AATT sequence of the Drew–Dickerson dodecamer in long-microsecond MD simulations (41), as confirmed by high-resolution X-ray diffraction techniques (45). The PBE-based molecular interaction potential enabled us to roughly estimate the impact of the DNA internal dielectric constant on the ability of DNA to recognize cations (or protein residues) in a region where they are expected to be found (1, 10, 46). As shown in Fig. 5B, the DNA–cation interacting surface decreases ∼60% with increasing ε_DNA from 1 to 8, resulting in fewer molecular contacts available for DNA binding. The dramatic narrowing of ion localization, in turns, indicates an increased specificity of DNA–protein interaction, with fewer nucleobase and amino acid residues that are likely to contribute to the binding than predicted previously. The higher specificity can also be extended to the interaction of DNA with small ligands such as drugs and molecular probes. These observations can profoundly impact DNA–ligand binding affinities and sites, and indicate that current dielectric constant must be redefined to account for the realistic prediction of the electrostatic component of the DNA–ligand interaction with computational modeling.

Conclusions

The study of electrostatic interactions associated to the structure and functions of DNA relies on computational methods such as PB equations. It is well established, however, that theoretical predictions thus obtained are sensitive to the dielectric discontinuity between DNA and the surrounding solution, and therefore to the choice of the dielectric constant of DNA, ε_DNA, which has so far remained unknown. Here, we have successfully determined ε_DNA both experimentally and theoretically.

Our direct measurements based on a scanning probe microscopy technique have provided the intrinsic dielectric constants of the main proteins that form the bacteriophage T7. Despite their rather different oligomeric composition and topology, they have similar dielectric constants around 3, reflecting a similar core composition as predicted by previous theoretical calculations (28, 29). Based on this value, we have provided the first (to our knowledge) experimental determination of the dielectric constant of DNA in a native condensed state inside the viral capsid. We found it to be ∼8, clearly larger than assumed in previous mean-field theoretical calculations (8–17), but remarkably in agreement with the theoretical value that we obtained here by extended state-of-the-art atomistic MD simulations with explicit counterions and solvent.

Our experiments and simulations thus shed light on an inherent property of DNA and will allow a more realistic prediction of the electrostatic binding energies and electrostatic potentials of DNA. The fact that the dielectric constant of DNA is sensibly larger than previously assumed is expected to dramatically influence our estimates of DNA–ligand interactions, particularly DNA–protein recognition (8–11), as well as our understanding of how DNA is packed (5–7). We showed it for the case of DNA–ligand systems, in which the electrostatic free binding energy contribution has been largely overestimated by assuming low dielectric constants. With ε_DNA ∼ 8 found here, it decreases by one-half with respect to previous estimations, resulting into a sensibly lowered affinity to the binding, narrowing by about 60% the molecular contact regions and leading to a marked increase of specificity of DNA–ligand interactions.

Methods

Viral Particle Preparation.

Mature T7 virions and empty capsids were prepared as described in ref. 24. The flattened capsids were obtained after treatment of the empty capsids with 4 M ClGu for 15 min at 37 °C. The recombinant tails were produced as described before (34). Briefly, the genes of proteins gp8, gp11, and gp12 were cloned and expressed into pRSET plasmid. The recombinant complexes were purified using a TALON cobalt metal affinity resin followed by a size exclusion chromatography.

Dielectric Imaging and Tip Radius Calibration.

AFM and dynamic-EFM images were taken with a commercial AFM operated in dry-air conditions (1% RH) and room temperature as described in ref. 24. Briefly, the particles were deposited on HOPG (Scientec) and gently dried out using nitrogen gas. For each tip and particle, before and after taking the dielectric image, we measured an approach capacitance-gradient calibration curve on the substrate to determine the tip radius R and checked that it remained constant within ±0.25 nm during the dielectric image. We used highly doped silicon tips (PPP-CONTR; Nanosensors; spring constant, ∼0.1 N/m; nominal radius, <7 nm; half-angle, 10°). All EFM measurements were taken with an applied ac voltage of amplitude, v_ac, of 4 V and angular frequency, ω = 2π⋅1 kHz. Data analysis was performed using WSxM software and custom software in Matlab (The Mathworks).

Dielectric Quantification and Data Analysis.

Two-dimensional finite-element calculations (COMSOL 3.4) were used to fit both tip calibration curves and dielectric images. Calibration curves were fitted over the 0–50 nm using the radius R as the only fitting parameter and fixed half-angle at nominal 10°. Dielectric images were fitted using the dielectric constant as the only fitting parameter by matching the maximum dielectric signal measured at the center of the nanoobject with the calculated value using the calibrated radius. The probe–particle system was modeled as described in ref. 24. The width w of each particle was obtained from topography using the phenomenological deconvolution formula w(h) = h + FW_measured − FW_sphere(h), where FW_measured is the full width measured in the topography affected by tip shape and lateral scanning effects specific to our setup. FW_sphere(h) is the full width of a spherical nanoparticle of same height as the object that follows the phenomenological formula FW_sphere(h) = 1.6h + 29 ± 2 nm obtained by analyzing nanoparticles of different heights measured in ref. 24. For each viral particle, we repeated the dielectric image four times at four different heights. Fig. 2 gives the dielectric constant obtained at a single height and the error is due to the detection noise of 0.1 zF/nm. In Fig. 3, the dielectric constants and error bars of each particle are the average and SD of the four dielectric constants obtained at different scan heights.

Molecular Dynamic Calculations.

A previously extended (42) MD simulation of the Drew–Dickerson dodecamer (41) was used to compute the DNA dielectric constant. Ten segments of 100 ns, sampled every 1 ps, were extracted from the last two microseconds of the total 4-μs-long simulation. The total dipole moment and its fluctuations were computed following the protocol suggested by Pettitt and coworkers (38). The final dielectric constant of each component (n = water, ions, DNA, etc.) was computed considering that the dielectric constant of a system surrounded by an infinite dielectric medium is related to the dipole fluctuations as follows:

$$\epsilon =1+\frac{\langle {\overrightarrow{M_n}}^2\rangle -{\langle \overrightarrow{M_n}\rangle }^2}{3{\epsilon }_0{V}_n{k}_{B}T},$$

where M_n, V_n, and T are the total dipole moment of the component n, the volume of the component n, and the temperature of the system, respectively, and ε₀ and k_B are the electric permittivity of vacuum and the Boltzmann constant. Following the work of Pettitt and coworkers (38), to obtain the dipole moment fluctuations of the total system (or when combining groups, i.e., DNA base–sugar), we took into account the individual contribution of each component plus the corresponding cross-correlation terms. Concerning the calculation of the dielectric constant for the components with net charges (for which the dipole moment is ill defined), we chose the instantaneous center of mass of the DNA as the origin to eliminate the conductance component due to the DNA translational motion (38). As noted by Pettitt and coworkers, this choice is particularly meaningful in our case, being, physically speaking, equivalent to fixing or immobilizing the DNA, as in the virus capsid, and considering only the fluctuations. The convergence of all of the computed properties was carefully examined (Figs. S5 A and B and S6). The volume of each component was computed across the 10 100-ns-long segments using the double cubic lattice method (47). For the sake of completeness, and to ensure that our results are consistent with other state-of-the-art force field, the same system was simulated under identical conditions for only a half microsecond, but using CHARMM27—another popular force field for the simulation of nucleic acids (48, 49). A converged value of ε_DNA = 12.0 ± 1.4 was obtained (using three 100-ns-long segments), showing a good agreement between both force fields, as suggested by previous consensus study of the DNA structure and flexibility (50). Due to larger atomic charges (and hence, larger dipole moments), the ε_DNA value obtained with CHARMM27 is slightly higher compared with the perfect agreement found between the experiment and parmBSC0 (ε_DNA ∼ 8.5), but still in good accordance with the experimental results, thus fully supporting the main conclusions of this work.

PBE Calculations.

The linear PBE in its simplest form (without considering dielectric self-interaction), as implemented in CMIP (46), was used to compute free interaction energies between DNA and different ligands. Although numerical solutions of PBEs provide exact representation of the classical electrostatic solvation of a rigid molecule, with a well-defined solvent/solute interface and a nonpolarizable charge distribution, it is clear that the method neglects a variety of effects, which could be very significant with increasing the charge density of the molecule (for discussion, see ref. 51). This is particularly true when working with highly charged polyelectrolyte-like DNA molecules, and consequently one must be careful not to fall into too quantitative interpretations of the PBE results. Experimentally determined conformations were used as initial structures for the protein–DNA complexes, whereas a representative snapshot during the MD simulation was used to study the DNA–cation interaction. The HMG domain of the human SRY factor protein determined by means of solution NMR was used as an example of minor groove binder [Protein Data Bank (PDB) ID code 1J46]. The crystal structure of the repressor of phage 434 was used to illustrate the binding of protein through the major groove of DNA (PDB ID code 2OR1). Ten single-point calculations were performed increasing the separation between the ligands and DNA every 0.1 nm. The ionic strength and the reaction-field dielectric constant were set to 0.15 M and 79.8, respectively. The van der Waals radii, taken from the parmbsc0 force field (40), were scaled by a factor of 0.8. The molecular interaction potential was computed for a representative snapshot of the Drew–Dickerson dodecamer during the MD simulation, using sodium as a probe. The resulting 3D map was plotted using VMD (52).