Close encounters with DNA

Abstract

Over the past ten years, the all-atom molecular dynamics method has grown in the scale of both systems and processes amenable to it and in its ability to make quantitative predictions about the behavior of experimental systems. The field of computational DNA research is no exception, witnessing a dramatic increase in the size of systems simulated with atomic resolution, the duration of individual simulations and the realism of the simulation outcomes. In this topical review, we describe the hallmark physical properties of DNA from the perspective of all-atom simulations. We demonstrate the amazing ability of such simulations to reveal the microscopic physical origins of experimentally observed phenomena. We also discuss the frustrating limitations associated with imperfections of present atomic force fields and inadequate sampling. The review is focused on the following four physical properties of DNA: effective electric charge, response to an external mechanical force, interaction with other DNA molecules and behavior in an external electric field.

Figure 1

Chemical model of DNA. DNA is a polymer composed of nucleotides, each having a negatively charged phosphate, a deoxyribose sugar ring and one of the four nucleobases: adenine, thymine, guanine, cytosine. Two single DNA strands form a double helix held together through non-covalent interactions. In addition to the four types of DNA nucleotides, chemical modification of DNA occurs frequently and includes methylation and hydroxymethylation.

Figure 2

Nucleic acid systems that represent the range of scales amenable to various computational and theoretical methods. From left to right, generally increasing in number of nucleotides: (a) Base-stacking interactions can be studied through quantum mechanics calculations, and reveal asymmetric van der Waals radii [47]. Adapted with permission from Ref. 47. The all-atom MD method with explicit or implicit solvent provides a balance between computational speed and descriptive detail. This method can be used to study systems ranging from a few nucleotides to (b) several turns of DNA [48] to (c) hundreds nucleotides as in the nucleosome [49]. (d) A large multi-layer, curved DNA origami construct described by MD simulation with explicit solvent [50]. Coarse-grained simulations can describe DNA with a wide range of detail, from near atomic (e.g. to describe (e) a plectoneme [51]) to many nucleotides per site (e.g. to describe (f) the packaging of a virus [52]). The two dominant polymer models of DNA are (g) the wormlike chain (WLC) model [9], and the freely jointed chain (FJC) model [7, 8]. The WLC model describes DNA under tension better than FJC, but FJC is convenient for very coarse descriptions of DNA, (h) such as at the level of chromatin [53]. Panels adapted with permission from (a) Ref. 47, (b) Ref. 48, (d) Ref. 50, (e) Ref. 51, (f) Ref. 52, and (h) Ref. 53.

Figure 3

Intra-DNA and DNA-ion interactions. (a) Schematic illustration of the chemical structure of DNA. Seven torsion angles are required to describe the conformation of a nucleotide unit. (b) DNA-cation interactions. The phosphate groups of the DNA backbone strongly interact with cations because of electrostatic attraction. The accuracy of these opposite-charge interactions is essential for simulations of DNA–ion and ion-mediated DNA–DNA interactions [60]. Magnesium is shown in a Mg(H₂O)₆ complex to emphasize its pervasive ability to coordinate water molecules. See Section 4.2.2 for a detailed discussion of Mg²⁺ parameterization. Bulk water and DNA hydrogen atoms are omitted for clarity. (c) Custom NBFIX corrections improve accuracy of DNA array simulations. The panels show representative configurations of 64 dsDNA molecules (white circles) confined in a cylindrical volume (not shown). The color indicates the local density of cations. Left. The artificially strong attractions between cations and DNA phosphate results in erroneous clustering of DNA. Right. Using custom parameters to describe cation–DNA phosphate interactions recovers hexagonal packing of DNA helices [60], in agreement with experiments [74].

Figure 4

Typical distribution of monovalent ions around duplex DNA. (A) Density of Na⁺ (solid) and Cl⁻ (dashed) ions as a function of distance from the axis of a DNA molecule corresponding to bulk ion concentrations of 170 (blue) and 320 (green) mM. (B) Charge of ions within the specified distance from the DNA axis, normalized by the DNA charge. Colors are as in (A). The Manning radius is indicated for each ion concentration as a vertical dashed line. The ion distributions were taken from trajectories reported in Ref 153. The background shows the DNA molecule (cyan) surrounded by ions (green) superimposed from several snapshots of an MD trajectory. The molecular image is arranged to scale with the data so that the axis of the DNA corresponds to the zero mark of the graph’s x-axis.

Figure 5

Competitive binding of Mg²⁺ and Na⁺ to dsDNA. (a) A representative configuration of ions near a 24-bp duplex. The two DNA strands are shown in black and gray; Na⁺ and Cl⁻ ions are shown as yellow and green spheres. The first solvation shell of each Mg²⁺ (pink) ion is explicitly shown. (b) Volumetric representations of DNA (gray), Mg²⁺ (blue), and Na⁺ (green) obtained by averaging a ~100 ns trajectory. The density maps are shown as isosurfaces of 0.5 and 0.02 Åatom number density for DNA and ions, respectively. (c) Ion concentration as a function of the radial distance from the DNA axis. Data in panels a-c correspond to bulk ion concentrations of ~5-mM Mg²⁺ , ~40-mM Na⁺, and ~50-mM Cl⁻. (d) Simulated excess number of Mg²⁺ (red), Na⁺ (blue), and Cl⁻ (green) ions as a function of Na⁺ concentration at the background of 5 mM Mg²⁺. For comparison, experimental data [129] are shown in black. Reprinted with permission from Ref. 128. Copyright 2012 American Chemical Society.

Figure 6

MD simulations of the solvent force on dsDNA. (a) Simulation set-up. A fragment of dsDNA (green and purple) is surrounded by an aqueous solution of K⁺ (gold) and Cl⁻ (cyan) ions. Bonds connecting atoms in each urea molecule are shown as lines; water molecules are not shown. (b,c) Schematic representation of the lateral (panel b) and vertical (panel c) mechanical forces (represented by arrows) on each nucleotide in 0.1 M KCl. (d) Cartesian components of the force acting on each nucleotide of the DNA at 0.1 M KCl. (e) Same as in panel d, but in the presence of 4 M urea. In panels d and e, nucleotides that belong to the one strand of dsDNA are numbered from 1 to 20 and to the other strand from 21 to 40; the force along the z-axis is shown using dashed lines.

Figure 7

SMD simulation of ssDNA stretching. (a) Setup of the simulations. A single DNA strand (vdW spheres) is placed in electrolyte solution (not shown). One end of the DNA is fixed in space; the other end is coupled through a harmonic potential of the spring constant k to a template particle. Moving the template with a constant velocity V stretches the DNA. The difference between the coordinates of the template and the end of the DNA reports the applied force. (b) The force-extension dependence of a a poly(dA)₂₀ fragment simulated using SMD. The inset shows three representative conformations of ssDNA corresponding to different amounts of stretching (L = L₀, 1.5L₀ and 1.9L₀), respectively. L₀ denotes the extension of a single DNA strand in a DNA double helix.

Figure 8

Stretching DNA using anisotropic pressure control. (a) Simulation set-up. A fragment of a DNA helix (orange and blue) is placed in a rectangular box of water (semi-transparent surface) and surrounded by potassium (tan) and chloride (cyan) ions. The strands of DNA are linked to themselves across the periodic boundary. The lateral pressures (P_xx,P_yy) are maintained at 1 bar, while the longitudinal (along the DNA helical axis) pressure (P_zz) is set to a negative value to stretch DNA. (b) A DNA overstretching processs, simulated using anisotropic pressure control. (Top) Densities of the bulk water and electrolyte away from DNA versus simulation time. (Bottom) The length of the DNA fragment versus simulation time. (c) Simulated force-extension dependences of torsionally constrained and nicked dsDNA molecules. The figures are adapted from Luan and Aksimentiev, Ref. 207, with permission. Copyright 2008 by the American Physical Society.

Figure 9

MD simulation of twist-stretching coupling. (A) Definition of a base pair geometry. The orientation of a base pair is characterized using a plane passing through atoms N9, N1, and C6, and a vector, $\overrightarrow{\mathit{v}}$, connecting atoms N9 and N1. (B) Scatter plot of instantaneous versus instantaneous rise from a 50 ns simulation of dsDNA. The red line shows a linear fit to the scatter plot, indicating slight (R² =0.12) anticorrelation of twist and rise.

Figure 10

SMD simulation of nucleosomal DNA unwrapping. (a–e) Representative conformations of a partially unwrapped nucleosome at several instances of the SMD trajectory. To realize the SMD protocol, a harmonic spring potential was applied between the centers of mass of the terminal fragments (10 bp) of DNA. The equilibrium length of the spring was increased at a rate of v to unwrap the nucleosome. The inset in panel e illustrates local melting of DNA produced by the excessive SMD force (~100 pN). (f) SMD force as a function of the simulation time. Note that the pulling rate was reduced by a factor of 5 at t ~ 18 ns to facilitate structural relaxation.

Figure 11

Schematic representing the experimental protocol used to produce plectonemic supercoils [48]. DNA is tethered to a stationary surface and a magnetic bead. A constant tension is applied to the DNA as it is twisted. At some critical number of twists, the DNA buckles to form a plectoneme (red curve). MD simulations quantified the interactions between the two halves of the plectoneme by averaging the force needed to restrain two effectively infinite DNA fragments about various separations. A typical simulation system is shown below the drawing of the experimental setup. Water is shown a transparent molecular surface. Ions (green and cyan) and DNA atoms (grey, red backbone) are shown as vdW spheres. A small spring is drawn between the DNA, representing the method used to obtain the force of the interaction.

Figure 12

Interaction free energy of two parallel DNA helices. (a) Simulation setup contains two parallel DNA helices (G₂₀ ·C₂₀), which are effectively infinite under periodic boundary condition. The DNA helices are shown using a cartoon representation; Na⁺, Cl⁻, and spermine ions are shown as vdW spheres. Water is not shown for clarity. (b) The free energy of two DNA helices versus the DNA–DNA distance. The simulations were performed for the following three onic conditions: [Na⁺] = 200 mM (red); [Na⁺] = 200 mM, [Mg²⁺] = 20 mM (black); [Na⁺] = 200 mM, [spermine] < 1 mM (blue). Consistent with experimental data [74, 176], attraction is observed only in the presence of polyamine. Data are taken from Yoo and Aksimentiev (unpublished).

Figure 13

End-to-end association of duplex DNA fragments. (A) Simulation system containing a solution of 458 DNA fragments near the density of the experimentally determined isotropic–nematic phase transition [273]. Most DNA fragments are shown in grey. Those fragments that formed the ten largest end-to-end chains at the end of the simulation are shown in color. The simulation used periodic boundary conditions, lasted 260 ns and included about 1,500,000 atoms. Water in the simulation unit cell is shown as a semitransparent molecular surface. (B) The ten largest end-to-end chains at the end of the simulation. The kinetic association and dissociation rates enabled estimation of the standard binding free energy of the end-to-end interaction. (C) End-to-end attraction in different DNA systems. The effective binding free energy (black) and the fraction of bound DNA ends (red) are plotted against the reference concentration of DNA ends. Images illustrate four DNA systems in which the end-to-end attraction may or may not play a role. From top left to bottom right: blunt-ended DNA circles (orange); repair of DNA during non-homologous end joining [276] (purple); small-angle x-ray scattering experiments [130] (blue); DNA aggregation into liquid crystal phases [273] (green). Figures adapted with permission from Ref. 75.

Figure 14

MD simulation of the effective force on DNA in a nanopore. (a) Simulation setup. Two strands of DNA are colored in purple and blue; K⁺ (tan) and Cl⁻ (cyan) ions are shown as spheres; water (green) is shown as a semitransparent surface. A mechanical tether force is applied to the DNA via a harmonic potential (a virtual spring). One end of the spring is fixed in space whereas the other end is attached to the center of mass of the DNA fragment. A uniform external electric field E is applied to the whole system. The DNA molecule is made effectively infinite by connecting the backbone to its image across the periodic boundary. (b) The restraining force versus the simulation time for several values of the applied electric field. The color of the lines corresponds to the color of the symbols in panel c. (c) The average effective force acting on DNA versus the nominal force of the electrostatic field QE, where Q is the nominal electrical charge of DNA. Figures were adapted from Ref. 316 with permission. Copyright 2008 by the American Physical Society.

Figure 15

Electroosmotic screening of the DNA charge. (a) A stationary DNA fragment is subject to a restraining force Fteth and an electrophoretic force Felec. (b) The DNA fragment is displaced through the nanopore with a constant velocity by a mechanical pulling force F_mech. (c) The DNA fragment is displaced through the nanopore by an electrophoretic force F_elec. (d) Water velocity as a function of radial position, calculated from all-atom MD simulations corresponding to the setups shown in panels a–c. The DNA surface is located at approximately 11 Å; the surface of the nanochannel is located at ~30 Å. A superposition (blue stars) of the flow profiles observed in mechanical pulling (cyan diamonds; panel b) and electrophoresis (black triangles; panel c) simulations reproduces the flow profile observed in the simulations of the effective electrophoretic force (red circles; panel a). Positive values of the water velocity corresponds to the upward direction in panels a–c. The data were taken from Ref. 316.

Figure 16

The effective force acting on a DNA fragment in a solid-state nanochannel of surface charge density σ. The force was measured using a setup similar to that shown in Figure 14a. The data were taken from Ref. 318.

Figure 17

The type of monovalent ions affect the translocation velocity of DNA. (a) A snapshot from an MD simulations illustrating binding of Li⁺ ions to dsDNA. DNA is shown in gray, lithium is shown in yellow, and water is shown in red and white. Only those water molecules involved in the lithium-DNA bonds are shown. (b) A toy model describing ion binding to DNA. Red and black lines schematically represent 1D potentials describing affinity of two different cations to DNA. The depth of the free-energy minima are the same and so is the instantaneous number of ions bound to DNA. The barrier for hopping between the adjacent sites along DNA varies with the ion size and so is the strength of the bond. (c) The barrier height (the strength of the cation-DNA bond) affects the effective charge of DNA. The effect of water flow is neglected in this model. Adapted with permission from Ref. [319]. Copyright 2012 American Chemical Society.

Figure 18

Reversal of the electrophoretic motion of DNA in a multivalent electrolyte. (a) Setup of MD simulations. The two strands of a DNA duplex are shown using a molecular surface representation, spermidine molecules directly bound to the grove of DNA are shown using vdW spheres, spermidine molecules dissolved in solution are shown using a ball-and-stick model. Subject to electric field, the DNA molecule moves through the nanochannel (gray molecular surface). (b) Center-of-mass displacement of the DNA duplex versus simulation time at three different concentrations of spermidine counterions. The data were taken from Ref. 182.