Is the G-Quadruplex an Effective Nanoconductor ... - ACS Publications

Jan 3, 2014 - ... of Southern California, Los Angeles, California 90089, United States. ‡ ... Jarzynski's equality, we find that the G-quadruplex se...
2 downloads 0 Views 4MB Size
Download PDF
Article pubs.acs.org/JPCB

Is the G‑Quadruplex an Effective Nanoconductor for Ions? Van A. Ngo,*,† Rosa Di Felice,†,‡ and Stephan Haas† †

Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, United States Center S3, CNR Institute of Nanoscience, Via Campi 213/A, 41125 Modena, Italy

‡

S Supporting Information *

ABSTRACT: We use a stepwise pulling protocol in molecular dynamics simulations to identify how a G-quadruplex selects and conducts Na+, K+, and NH4+ ions. By estimating the minimum free-energy changes of the ions along the central channel via Jarzynski’s equality, we find that the G-quadruplex selectively binds the ionic species in the following order: K+ > Na+ > NH4+. This order implies that K+ optimally fits the channel. However, the features of the free-energy profiles indicate that the channel conducts Na+ best. These findings are in fair agreement with experiments on Gquadruplexes and reveal a profoundly different behavior from the prototype potassiumion channel KcsA, which selects and conducts the same ionic species. We further show that the channel can also conduct a single file of water molecules and deform to leak water molecules. We propose a range for the conductance of the G-quadruplex.

■

INTRODUCTION Nucleic acid sequences that appear in single-stranded fashion in the chromosomal telomeres, whose length and folding control aging processes and affect genome instabilities,1 are able to form quadruplexes of guanines (Gs), or G-quadruplexes (GQs), or G4-DNA. Evidence for the presence of GQ motifs in the human telomeric region in vivo was gained only very recently,2 but this form of nucleic acid has been known and studied for several years. In particular, GQs can be synthesized in vitro, and structural NMR characterization has demonstrated different folding patterns: monomolecular, bimolecular, and tetramolecular.3,4 Reviews5−7 on the structure and stability of GQs disclose their utmost importance in various contexts, though their role is still an object of investigation in the medical, biological, biochemical, and chemical physics communities. In 1994, the first high-resolution X-ray crystal structure of a parallel-stranded GQ was published (PDB entry 244D, resolution 1.2 Å),8 which was further refined in 1997 (PDB entry 352D, resolution 0.95 Å).9 These works gave an unprecedented precise knowledge of how guanines arrange and form quadruple helices. The unit component of a GQ is the G-quartet, which is a very strong stacking assembly made of four coplanar guanine bases sustained by eight hydrogen bonds. GQs from telomeric sequences are stabilized by monovalent cations such as Na+, K+, or NH4+, which are coordinated by the carbonyl O6 atoms lining up a central pore or channel. Because of the existence of such a central channel and stable structures, a lively debated issue is whether GQs can be used to select and conduct ions.10 Hud and colleagues11 argued that because the hydration free energies of K+ and Na+ are different, the pore of a GQ is able to distinguish between ions. Davis and coworkers12 successfully synthesized a unimolecular GQ, which folds into a conformation to conduct Na+ along its symmetry axis across phospholipid bilayer membranes. © 2014 American Chemical Society

In potassium-selective KcsA channels, ion selectivity presumably results not only from the channel characteristics but also from inherent properties of K+.13−15 This selectivity occurs at the filter region where four identical subunits have the TVGYG amino acid sequence. The crystal structure16 of this filter region reveals that the amino acids have negatively charged carbonyl oxygen atoms, which line up the channel in a way that is similar to that in the GQ pore. These amino acids allow conducting potassium ions sandwiched by water molecules, while they inhibit flow of sodium ions. The favorable conduction of K+ goes together with the evidence found from the crystal structure that the ionic size of K+ fits the KcsA pore more favorably than Na+. For GQs, do the guanine bases selectively bind and conduct ions? If so, how does it happen? So far, it has been rather well-established that GQs bind K+ more strongly than Na+; however, understanding the mobility of different ions inside GQs is still elusive. Deng and Braunlin17 showed by experiments using 23Na NMR that the selectivity of a GQ d(G4T4G4) favors K+ over Na+. The freeenergy change measured for the conversion of [d(G3T4G3)]2 bonded by two Na+ to [d(G3T4G3)]2 bonded by two K+ is −1.7 ± 0.2 kcal/mol.11 This free-energy change suggests that the fit of K+ to the GQ pore is better than that of Na+, also indicating the preference for K+. The lifetime of bound Na+ in GQs, ∼10−250 μs,17 is much shorter than that of NH4+, 250 ms,18 which would be similar to that of K+ because of the close ionic sizes. This indirect evidence suggests that Na+ moves along the GQ channels much faster and more easily than the other ions. Therefore, the concepts of ion selectivity and speed of ion transport in GQs seem to be in conflict, at odds with the behavior of KcsA. Understanding the origin of this discrepancy Received: August 12, 2013 Revised: December 2, 2013 Published: January 3, 2014 864

dx.doi.org/10.1021/jp408071h | J. Phys. Chem. B 2014, 118, 864−872

The Journal of Physical Chemistry B

Article

KcsA,16,25 in which ion motion is accompanied with water. We present the results of such simulations for d[G9]4 channels that contain K+, Na+, or NH4+ ions in the pore, in explicit water solution (Figure 1). While this system is not related to any

is a fundamental question that has potential impact for the design of molecular machines. From the experimental point of view, it is problematic to attain a direct comparison between different ionic species in the same GQ. Therefore, simulations assume a particularly relevant role for addressing this problem. However, simulations also have their difficulties. The characteristic times of the involved phenomena, namely ions entering the GQ channel from solution and moving through the pore, are too long for standard molecular dynamics (MD) techniques, and enhanced techniques must be invoked. As a matter of fact, the sole simulation of the problem was published in 2012.19 Akhshi et al. investigated ion mobility in a parallel-stranded d[TG4T]4 Gquadruplex by computing the potential of mean force (PMF) profiles for Na+, K+, and NH4+ ions along the central pore.19 We summarize their findings here for reference, and we comment in the Final Remarks how we add relevant knowledge. 1. The analysis of energy profiles gives the following information:19 • the energy barriers between any two subsequent binding sites along the core of the GQ are about 4−5 kcal/mol for Na+ and 13−15 kcal/mol for K+ and NH4+ ⇒ Na+ ions move faster than the other species; this is rationalized in terms of atomic sizes because the small Na+ ion does not need to perturb the internal size of the channel during its axial motion, at odds with K+ and NH4+; • the leakage of internal ions from the sides is blocked by very high energy barriers (>50 kcal/mol); this finding of large energy barriers for sideways escape of ions from the channel into the solution is a clear proof of experimental indications of long binding times of NH4+.20 • energy barriers for leakage through the edges are also large, ∼20 kcal/mol for K+ and NH4+ and ∼14 kcal/mol for Na+; thus, Na+ motion from the channel into the solution is less energetically costly, which is in line with experimental measurements of residence times.21,22 2. The analysis of the motion of water molecules gives the following information:19 • all ionic species are dehydrated inside the pore and fully hydrated in solution; • occasionally, water molecules enter the channel and approach the first coordination shell of the ions. These findings shed light on important aspects of ion motion along the GQ axis. However, other important issues related to ion stability and motion in GQs are not tackled.8,12,23 No comments are given on different residence sites for the different ionic species. No explicit comments are given on energy barriers for entering the channel from the solution, though their data in Figure 1 suggests that K+ and NH4+ experience an energy barrier of approximately 4−5 kcal/mol, while Na+ does not experience an entrance energy barrier. Furthermore, the energy profiles are compared to those of KcsA ion channels, but conductivity values are not estimated. To shed direct light on the selectivity and conduction properties of GQs, solve uncertain points from the previous simulation,19 and unveil whether this system can be exploited as a natural or artificial ion channel, we have carried out a computational study using biased molecular dynamics according to an efficient protocol developed by one of us.24 We simulate a condition typical of natural ion channels, e.g.,

Figure 1. Snapshot of the G-quadruplex in the presence of K+ ions after 10 ns equilibration. The quadruplex is visualized in a ribbon mode that highlights the backbone; water molecules (O, red; H, white) and potassium ions (green) are visualized as ball-and-stick models. A planar position is defined as the crossing point of a Gquartet plane and the z-axis, while a cage-like position (S1 to S8) is the center of any two successive G-quartets along the z-axis. The threedimensional structure is rendered by VMD.33

telomeric sequence, d[G9]4 is related to long G4-DNA wires that can be produced by enzymatic synthesis, can exist in the absence of inner cations, and have been proposed for nanoelectronics applications.26−29 Note that their existence in the absence of stabilizing cations was proved by both experiments26,28 and simulations.23 Furthermore, though no X-ray or NMR structure could be resolved so far, d[G9]4 exhibits a stacking motif that conforms to the three-dimensional structure of tetramolecular GQs.23,30 We compute the freeenergy profiles using Jarzynski’s equality (JE)24,31,32 to examine how ions of the different species accommodate and move in the d[G9]4 channel. Our results show that the GQ is selective for K+ ions, but can conduct Na+ with almost no energy barrier within its channel, whereas the guanine bases are disturbed distinguishably as ions enter the channel. We also observe in the trajectories that the ion movement along the axis of the helix is accompanied by spontaneous leaking of water molecules from the channel. Finally, we estimate the conductance via the equivalent relation between work computed in the pulling protocol and heat generated by a current.

■

METHOD We use the average structure of a G-quadruplex (GQ) from a 5 ns free MD trajectory at room temperature and standard pressure.23 This structure contains nine G-quartets organized in four parallel strands, having a total length of 30.6 Å that is the same as that of the GQ synthesized to be a Na+ transporter.12 Specifically, we start from the equilibrated 9-plane GQ filled with 8 K+ ions, remove the inner ions, solvate the structure with TIP3P water molecules, and randomly neutralize it with potassium ions in the solution at the concentration of 0.57 M. These simulation strategies for solvation and neutralization are standard for the vast majority of biological systems that have been addressed by MD.23,34 We remark, as we mentioned in the Introduction, that long G-quadruplexes can exist in the 865

dx.doi.org/10.1021/jp408071h | J. Phys. Chem. B 2014, 118, 864−872

The Journal of Physical Chemistry B

Article

absence of internal cations,26−29 at odds with the situation in short G-quadruplexes.19 We choose to simulate a condition in which the ions of different species pass through a channel that is not lined with ions because this is similar to the conduction state in natural KcsA ion-channels.16,25 Keeping all 8 K+ ions at the core during the pulling simulations would not be a typical conduction state because water molecules also should have access to the conduction path and play essential roles in ion channels, at least in the systems that are nowadays understood.16,25 The system has 9238 atoms in a supercell of 44 × 44 × 48 Å3. Its symmetry axis is along the z-direction centered at x0 = y0 = 0. We apply harmonic restraints to the phosphate groups with a spring constant of 2.0 kcal mol−1 Å−2. These restraints allow us to preserve the equilibrated structure as required for a durable channel. They are necessary to impede the translation of the whole GQ, which would be pulled along with the ions by the application of an external force. Similar restraints have been used for wire-like channels also within more conventional methodologies to compute free-energy profiles.35 The restrained condition limits only the motion of the external boundary of the channel, yet enabling fluctuations of the bases. Therefore, the bases maintain degrees of freedom to respond to the passage of ions, in a way that may depend on the ionic species. It is a common practice in studying physical and chemical reactions to focus the investigation on the reaction center, with the use of structural models that mimic the reaction center, while the surroundings are approximated or neglected (e.g., in QM/MM simulation approaches). Of course all the quantitative aspects may be affected, but the overall behavior of the reaction can be grasped. This is the same principle that we are adopting in this study, focusing our attention on the inner core that hosts the ionic motion. We do not claim that we attain precision on all the computed quantities, but definitely this is satisfactory to grasp the physicochemical principles. We use the parm99 AMBER force field36 and other simulation parameters as assessed before.23 The parm99 force field reproduces canonical DNA structures in simulations of double-stranded DNA on the time scale below ∼10 ns37 and has given good results on G-quadruplexes on the time scale 5− 20 ns.23 Because our simulations at each pulling step are of 5 ns, we are confident that the chosen computational setup is appropriate for a correct description of the structure at hand. For the energetics, parm99 gives accurate values of H-bonding and stacking energies against experimental data and quantum chemistry calculations.38,39 More precise force field parameters are nowadays available to include ion polarization effects37,40,41 and to avoid incorrect transitions, but these are not critical in our problem (see discussion in the Supporting Information). Confident in this setup, we equilibrate the system at T = 300 K and P = 1 atm. We first minimize the energy of the system using conjugate gradients for 5000 steps, after which we perform a MD run for 0.5 ns with a time step of 1 fs at T = 200 K using Langevin dynamics with a damping constant equal to 1 ps−1. Then the system is further equilibrated for 10 ns at T = 300 K and P = 1 atm using Langevin piston dynamics with a time step of 1 fs, which produces isothermal−isobaric ensembles.42 To prepare the systems with Na+ and NH4+ ions, we simply mutate potassium in the 10 ns equilibration system to sodium and ammonium and then apply the same multistep minimization−equilibration procedure. Finally, each equilibrated quadruplex is subjected to ion pulling: an ion just

outside the 5′ end is dragged into the channel to span its axis via a stepwise protocol.24 We find that in each of the three equilibrated systems there is one ion capping at each edge of the channel (see Figure 1), either slightly outside (5′) or slightly inside (3′). The ion at the 3′ end moves into the channel during the equilibration phase simply by thermal motion and attraction of the channel. At the 5′ end, where we note somewhat larger distortions, in line with previous findings, the ions can be internal or external depending on the species.23 During the 10 ns equilibration, we also find that there are about seven to eight water molecules stably located within the channels at the centers of sites S1 to S8 (see Figure S2 in Supporting Information); thus, it is sufficient to use such well-equilibrated configurations for the following pulling simulations. Note that water inside GQs was indeed experimentally observed.22 The stepwise pulling protocol requires relaxation times for all discretized pulling steps and allows us to estimate free-energy profiles. In each pulling simulation, one ion at different subsequent positions along the axis of the GQ is subjected to a harmonic potential 0.5k[(x − x0)2 + (y − y0)2 + (z − z0 − λ)2], where x, y, and z are the ion coordinates and k is 0.6 kcal/mol ∼ kBT; λ is a control parameter that scans the direction of motion. The same reference coordinates x0 = y0 = z0 = 0 Å are used for potassium, sodium, and ammonium ions. Every τ = 5 ns, we instantaneously increase λ by 1.0 Å, starting from λ = −17 Å, so that the external force is increased in a modest manner to pull an ion along the positive z-axis. The relaxation time, τ = 5 ns, is ten times larger than the minimum tested value.24 We perform s = 25 discretized steps of pulling NH4+ and Na+ or K+ ions to pass the middle of the longitudinal channel. Every 50 fs during the interval τ = 5 ns at each value of λ, we collect z to construct the distribution gi(z) and compute average positions ⟨z⟩i = ∑zgi(z)z, where i from 1 to s is the index for the i-th step of increasing λ. From the gi(z), we compute free-energy changes ΔF(λ1,λ) along the z-axis, which directly indicate amounts of work including entropic effects.24,31,32 Because the work applied to an ionic charge q is equivalent to the heat emitted by a resistance R = G−1, we propose a simple expression of conductance G = ⟨I 2(t j − ti + (j − i)τ )⟩/⟨W ⟩

(1)

where I = q[z(tj) − z(ti)]/[(tj − ti + (j − i)τ)(λj − λi)] and z(ti) is the ion position along the axis of motion at time ti ∈ [0:5] ns, with j > i being the index of the pulling step. Here, the brackets denote the average taken over all possible trajectories between ith and jth pulling steps and ⟨W⟩ is the averaged value of work over such trajectories. The heat of ionic flow is I2(tj − ti+ (j − i)τ) = [q2/(λj − λi)2][z(tj) − z(ti)]2/(tj − ti + (j − i)τ), whose average is equal to a diffusion coefficient43 Dij multiplied by the term in the first square bracket. Because work for a pair having j > i + 1 is more biased than that for a pair with j = i + 1,24 we compute the conductance Gi+1 for a pair of successive steps i and (i+1), in which the work ⟨Wi+1⟩ is averaged from U(z(ti),λi + Δλ) − U(z(ti),λi) over all possible values of z(ti). Note that Gi+1 can largely fluctuate along the channel due to the fact that ⟨Wi+1⟩ and Dij are not the same for all pairs of successive steps. We compute a non-negative total conductance as follows 1 Gtotal 866

s−1

=

∑ i=1

1 = Gi

s−1

∑ i = 1, j = i + 1

Wtotal ⟨Wi ⟩ ∼ Dij [Dij]max

(2)

dx.doi.org/10.1021/jp408071h | J. Phys. Chem. B 2014, 118, 864−872

The Journal of Physical Chemistry B

Article

Because the total work Wtotal = ∑iWi to enforce ions through the channel and the maxima of Dij are always positive, this total conductance can be positive for a sufficiently large number of pulling steps (∼15−30 steps). The first equality in eq 2 is known for a total resistance of multiple resistors in series. The steered MD method adopted in this work, in conjunction with Jarzynski’s equality, produces a free-energy profile of stretching deca-alanine in excellent agreement with steered MD simulations evaluating the potential of mean force (PMF).24,44,45 It offers an efficient way to both estimate freeenergy profiles and sample reaction pathways. Importantly, it outdoes PMF for the characterization of thermal fluctuations and estimation of conductance.

agreement with the absence of an entrance barrier found for Na+ ions by Akhshi et al.19 The order of these averaged positions for ions close to the entrance, ⟨zNa⟩ > ⟨zK⟩ > ⟨zNH4⟩, is consistent with that of the ionic sizes rNa < rK < rNH4, and it reflects the fact that the smaller the ion size is, the more easily the ion enters the channel. The distinguishable peaks of the histograms (e.g., around z = −12 Å) are consistent with the observations (see Figures 2 and 3) that K+ and NH4+ ions are preferably accommodated at the cages between G-quartets,17,18,23 while Na+ ions can be metastable both between and within the planes of G-quartets.8,30

■

RESULTS AND DISCUSSION We remark that the three equilibrated systems contain one ion in the GQ core at the 3′ ter (S8 in Figure 1): Na+ in a planar site and K+ and NH4+ in a cage site. These are indeed binding sites for the ions, and consequently they are not abandoned by the unrestrained ions. The pulling protocol is applied only to the ion that is located at the mouth of the channel at the 5′ ter at the end of the equilibration procedure. At the 3′ ter each of the three simulated GQs is clogged. The structural portion around the 5′ ter is, however, rather far from the obstructed portion of the pore and can be retained as representative of a range open to free motion. Thus, the results in this part of the systems can be generalized to interpret absolute and relative ion mobility. Binding Sites. To show how the ions stably bind to the Gquadruplex channel, we plot cumulative distributions of zcoordinates and the averaged positions at each value of λ during the pulling (see Figure 2) for each of the three investigated Figure 3. Three-dimensional structures of the investigated systems at different stages of the simulations, illustrating the axial motion of Na+ (orange, panels a−e), K+ (green, panels f−j) and NH4+ (blue and white, panels k−o) ions in the GQ channel. The snapshots are at 5 ns of each pulling step.

Na+ also moves along the channel more easily than the other ions; for instance, at λ = −8 Å, Na+ is at ⟨z⟩ ∼ −8 Å, farther from the entrance than the other ions (see Figure 2b). Figure 3 shows snapshots of the ions moving in the progression S1 → S2 → S3 → S4 at the end of each pulling step characterized by a value of λ. The series of Figure 3a−o shows that at the same cage-like positions, λNa ≤ λK ≤ λNH4. Because the external force f due to the harmonic potential is equal to −k(z − λ), at the same z it requires greater forces to pull larger ions to upper cage-like positions: f Na ≤ f K ≤ f NH4. When the initial Na+ ion moves in the GQ core from 5′ to 3′, it is accompanied by other ions and water molecules. The movement of the Na+ ions implies the conduction of multiple ions similar to the “knock-on” mechanism14 in which the channel attracts extra ions in support of the conduction of the first ion. In the case of potassium ions, most simulation stages are characterized by a conduction configuration of −K+−H2O− K+−H2O− in which all cage positions (namely, positions between two consecutive G-quartets) are occupied by either ions or water molecules (see Figure 3h). Even though exceptional situations occur, in which one or more cage positions are empty (see Figure 3j), this conduction behavior resembles that observed in potassium-selective channels.25

Figure 2. (a) Histograms of ion positions along the z-direction in all pulling steps. The histograms are the combination of all separated normalized gi(z) for s = 25 pulling steps for each ion. (b) Averaged position ⟨z⟩i = ∑zgi(z)z of ions along the z-direction.

systems. The plot in Figure 2a reveals the most probable values of the z-coordinate across the whole pulling protocol in the three cases, while the plot in Figure 2b gives the average at each value of the control parameter λ. We note that the simulation of the system with Na+ does not give a probability peak for z ∼ −15 Å below the 5′ entrance of the channel, as the other two simulations do. In fact, inspecting the average positions, for λ = −17 Å (namely, for the initial step that corresponds to an ion at the entrance of the channel in 5′) we find that ⟨zNa⟩ = −12.7, ⟨zK⟩ = −14.5, and ⟨zNH4⟩ = −15.3 Å. These results about the position distributions together indicate that while potassium and ammonium ions reside for a finite time outside the channel, sodium ions enter directly. This evidence does not imply that the GQ prefers Na+ to the other ionic species; it is in qualitative 867

dx.doi.org/10.1021/jp408071h | J. Phys. Chem. B 2014, 118, 864−872

The Journal of Physical Chemistry B

Article

Note that a spatial arrangement of oxygen atoms (−0.51|e| in CHARMM force fields) 46 in potassium-selective KcsA channels,16 which are decisive to select ions, is similar to the structure of oxygen atoms lining the GQ channel. We show below, on the basis of further analysis of free-energy profiles, that the GQ channel has a higher affinity (selectivity) to K+ over Na+ ions because of the optimal fit to the cages of Gquartets. However, it allows the easiest conduction of Na+. Water Molecules in the Channel. We examine if and how water molecules accompany the motion of ions in the channel. Before the pulling protocol is applied to each system, seven water molecules fill the channel in the equilibrated structure of the GQ in the presence of Na+ and K+ ions, whereas eight water molecules fill the channel in the presence of NH4+ ions. During the simulations the internal water molecules form two opposite zigzag chains of hydrogen bonds with variable lengths (denoted by arrows in panels a, h, and p of Figure 4). These two opposite zigzag chains compete with each other to (1) prevent ions from moving farther into the channel, (2) exert forces on the upper ions, and (3) knock water molecules out of the channel from the channel sides. This competition is a reason for the ejection of water molecules as the ion motion proceeds (see Figure 5). Water molecules are also able to move around the pulled ions to lower positions as shown in panels f, g, n, and t in Figure 4. Another interesting behavior that emerges from our simulations is how water molecules escape from the channel as Na+ and K+ move along the symmetry axis. Figure 5a−c shows a water molecule leaking from the channel after 50 ps when K+ is located at a metastable cage-like position (⟨zK⟩ ≈ −8.6 Å) and faces a water complex in a triangle configuration. They also show some bases tilted by about 20 degrees: this deformation allows a water molecule to interfere with the Hbond pattern of the guanines, eventually loosening such pattern to find a way out. Figure 5d shows that Na+ located at a metastable position with ⟨zNa⟩ ≈ −5 Å also faces a triangle of water molecules. The zigzag water chain in this configuration is under strong compression because of the movement of the pulled ion. This compression causes the top ion (see Figure 4d) to move up, toward the 3′ ter, and release the stress on the water chain. Then this top ion, because of the strong attraction of the channel, returns to its original metastable position and continues to compress the water chain. At the same time the compression causes the bases, especially those around the water triangle, to be disturbed significantly. One base is tilted by 29°, and a water molecule slides into the region-bond pattern of that base, eventually abandoning the channel. These processes occur at 0.17 ns. It then takes 1.52 ns for the second water molecule to escape the channel while Na+ is still at the same stable position. Fluctuations of Bases around Ions. In Figure 6 we examine the response of the channel to the ions by plotting the probability density of the root-mean-square deviation (RMSD) of the first 16 guanines, with respect to the equilibrated structure,23 for pulling with λ = −11 to −7 Å. Clear differences emerge in how the channel responds to Na+, K+, and NH4+. The RMSD in the presence of NH4+ ions is peaked at smaller values than in the presence of the other ionic species, independent of the value of λ at least in the early stages considered here. A reason for this might be that eight water molecules in the channel stabilize the overall base motif better than seven water molecules. Furthermore, the ionic size of NH4+ limits the movement of bases more than the other smaller ions; it is interesting to note that despite the small size

Figure 4. Positions of water molecules and ions along the axis of motion at various stages of the dynamical simulations, pruned from three-dimensional snapshots of the entire systems. (a−g) Subsystem extracted from the simulation with Na+ ions (orange spheres). (h−n) Subsystem extracted from the simulation with K+ ions (green spheres). (o−t) Subsystem extracted from the simulation with NH4+ ions (blue and white spheres). The arrows indicate directions of hydrogen bonds starting from oxygen and ending at hydrogen atoms. The snapshots are at 5 ns of each pulling step.

difference between NH4+ and K+ (0.1 Å, smaller than the 0.38 Å difference between K+ and Na+), the channel distinguishes between the two ionic species, which points to a possible role of physicochemical effects. When the NH4+ ion is at S1 (λ = −11 and λ = −7 Å), the RMSD is larger than when it is still external at the mouth of the channel (λ = −17 Å). The situation is opposite for Na+ and K+ ionic species in the sense that the RMSD decreases when an ion enters the GQ channel relative to the values when it is at the mouth of the tube. The RMSD curves for GQ in the presence of Na+ are less broad at λ = −11 Å, and smaller at λ = −7 Å than those in the presence of K+. This implies that Na+ can maintain the initial canonical structure of the bases somewhat better than K+, which may be attributed to the in-plane favorable location of Na+ ions. Cavallari and co-workers23 reported that the total RMSD computed on guanine heavy atoms for GQs in which the 868

dx.doi.org/10.1021/jp408071h | J. Phys. Chem. B 2014, 118, 864−872

The Journal of Physical Chemistry B

Article

Figure 5. Three-dimensional structures of portions of the simulated systems at selected snapshots in a ball-and-stick representation, to visualize the escape of water molecules from the GQ core. Water molecules that abandon the channel are represented in black. Na+ and K+ ions are at λ = −6 and −4 Å, respectively. Na+ and K+ ions are represented as orange and green spheres, respectively. Carbon, oxygen, nitrogen, and hydrogen atoms are shown as cyan, red, blue and white spheres, respectively.

Figure 7. Free-energy profiles for the motion of different ionic species in the GQ channel d[G9]4 as a function of the parameter λ that scans the axis of the quadruplex.

The minima of the free-energy profiles are ΔFKmin = −7.78 kcal/mol at λ = −11 Å, ΔFNamin = −7.49 kcal/mol at λ = −8 Å, and ΔFNH4min = −5.71 kcal/mol at λ = −11 Å, in the order ΔFKmin < ΔFNamin < ΔFNH4min. Although the difference ΔFKmin − ΔFNamin = −0.3 kcal/mol is very small, the discrimination between the two atomic species is significant, as supported by the order of magnitude of the energy gain found experimentally for changing d[(G3T4G3]2 bound to Na+ to d[(G3T4G3]2 bound to K+, of ∼0.8 kcal mol−1 ion−1.11 The order of minima in Figure 7 reveals that the GQ binds to K+ slightly stronger than to Na+, and more sensibly stronger (by 2 kcal/mol) than to NH4+, whose ionic size is marginally larger than K+. These minima are not related to the facility of transporting ions but to the stabilization effects of different ionic species, which agrees with the analysis presented in Figure 2 and with previous indications18,47 that K+ fits to GQs better than Na+ fits to GQs. These values are consistent with the minima that we find for the attraction between the selectivity filter of KcsA and K+ or Na+ ions.48 The attraction is attributed to the negatively charged carbonyl oxygen atoms in the filter, which line the KcsA channel in the same pattern as the O6 atoms of guanine in the G-quadruplex. However, the binding positions of K+ and Na+ ions at the energy minima are different from those in KcsA. While Na+ at the minimum binds in the KcsA filter at a position more external than that of K+, in the GQ channel Na+ penetrates farther and faster than K+. This

Figure 6. Normalized histogram of RMSD evaluated over the four nearest-neighbor G-quartets to a given ion (16 guanine bases), with respect to the equilibrated structure.23 The RMSD is evaluated on the heavy atoms of the bases, excluding the backbone. Because we do not allow shrinking of the channel, the fluctuations do not represent the overall stability of the GQ due to different ions. The RMSDs are collected over 5000 frames during 5 ns for each λ.

channel is filled with K+ and Na+ ions, relative to the equilibrated structure, is 0.6 Å and 0.9 Å, respectively, which is in agreement with the fact that K+ is a better stabilizer; therefore, a K+-filled pore is less flexible. This does not contradict our present RMSD analysis because we are focusing here on the reaction of the guanines surrounding a certain ion to the passage of the ion. Indeed, we also find that the binding of K+ to a cage-like position is stronger than that of Na+ to a planar position, meaning that K+ has a more powerful stabilization effect. Free-Energy Profiles. Figure 7 illustrates the free-energy profiles of the three ionic species. These free-energy profiles directly indicate energy or work required to induce transitions along reaction pathways. The curves in Figure 7 thus express the free-energy changes of the ions binding to the GQ channel with respect to their energies in the solution. These values include entropic effects encompassed by the formalism of Jarzynski’s equality.24,31,32 A comparison with potential of mean force profiles19 is given in the Supporting Information. 869

dx.doi.org/10.1021/jp408071h | J. Phys. Chem. B 2014, 118, 864−872

The Journal of Physical Chemistry B

Article

from eq 2 G = 1/∑G−1 i is physically meaningful because it is always positive in our simulations, reflecting the fact that the total work due to external forces is positive. The total conductance computed from our steered MD trajectories is 7.7, 4.0, and 2.6 pS for conducting Na+, K+, and NH4+, respectively, in the d[G9]4 quadruplex, approximately in the range of 1−10 pS. This means that the G-quadruplex conducts Na+ better than K+ and NH4+, which was already stated from the free-energy profiles of Figure 7 and from the previous simulation of a shorter quadruplex.19 These values compare unfavorably by about 1 order of magnitude with the estimated conductance of 97−150 pS50 for the KcsA potassium channel. The same order of magnitude distinguishes the GQs for conducting different ionic species.

means that the GQ is not only able to choose one species of ions for the optimal fit to its pore but also capable of choosing a different species of smaller ions for the easiest ion conduction. Figure 7 shows that all the ionic species do not experience energy barriers for entering the channel. This is consistent with the fact that during the initial 10 ns equilibration of the system ions get close to the edges and enter the channel (see Figures 1 and 3). This finding is in apparent discrepancy with the PMF profiles computed via the adaptive biasing force method for a [d(TG4T]4 G-quadruplex,19 which indicate that K+ and NH4+ ions encounter energy barriers of about 3−4 kcal/mol to enter the pore. We discuss this aspect in the final remarks. The opposite values of the free-energy minima in Figure 7 represent the free-energy barriers for the ions to move from the channel into the bulk water. These values are in the range of folding free-energy barriers of GQs, which were estimated between −3 and −15 kBT (2−9 kcal mol−1) from temperature dependence studies and Kramers’s theory.49 The computed value for NH4+ is half the experimentally measured free-energy barrier.20 The significant difference may be due to the fact that we impose restraints on phosphate groups to maintain the overall quadruplex symmetry, while experimental GQs are able to shrink for further energy reduction, thus increasing freeenergy barriers for ions to move out into the bulk water. Thus, if we take into account the shrinking effect and subtract the estimated “deformation” energy (∼2−9 kcal/mol)49 from the measured free-energy barrier, we find that our data is in fair agreement with experimental outcome. The broad free-energy minimum of Na+ indicates that Na+ is moving with a small energy cost from λ = −13 (or ⟨z⟩ = −12) to λ = −3 (or ⟨z⟩ = −4) Å. Thus, the GQ can conduct Na+ very well in this region of two G-quartet steps and also through the entire channel if one edge is not capped (see Figure 3). This is in qualitative agreement with the PMF-based finding that Na+ ions move faster than K+ and NH4+ ions within a [d(TG4T]4 Gquadruplex because of the small energy barriers between subsequent metastable binding sites.19 Note that the shape of each free-energy profile is affected by the presence of one capping ion at the 3′ end (Figure 1); because no restraints are applied to the capping ion during the simulation, it remains at its metastable binding site S8. Therefore, as we noted before, the channel is obstructed at one side, and consequently after some steps of 5′-to-3′ motion the internal ions start to feel the electrostatic repulsion of the capping ion and the free energy becomes positive. The free-energy profile for the Na+−GQ becomes positive at λ = 2 Å after the ion has gone through an axial distance of about 10 Å (or three G-quartet steps). The motion of the ammonium ion is the most limited: ΔF quickly becomes positive at λ = −7 Å after the ion has moved by only one G-quartet step (∼3.4 Å). K+ can go farther, by about 10 Å, but requires a large energy cost because the free-energy profile of K+ becomes positive at λ = −4 Å; thus, a K+ ion moves exothermically for a distance of only about 3.4 Å, as evident in Figure 3h−j, in which a free ion at S1 is unable to enter farther. Conductance of G-Quadruplex. We can finally estimate by eq 1 the conductance G for the Na+−GQ, K+−GQ, and NH4+−GQ systems, namely the efficiency of the G-quadruplex for transporting Na+, K+, and NH4+ ions through the core. First we note that the sequential values of Gi estimated from eq 1 fluctuate significantly from negative to positive values because of the signs and nonequilibrium values of work measured for all instantaneous pairs of steps. Such instantaneous conductance values are therefore trivial. Nonetheless, the total conductance

■

FINAL REMARKS We have presented the results of biased molecular dynamics simulations to disclose the binding and motion of K+, Na+, and NH4+ ions along the core of a d[G9]4 G-quadruplex in the presence of explicit water molecules in the surrounding solution and partially filling the G-quadruplex channel. We have shown results on (1) metastable binding sites for different ionic species, (2) free-energy barriers for ionic motion in the channel and for entrance into the channel, (3) hydration patterns, and (4) species-dependent ionic conductivity and performance as ion channels. We point out here the novelty and timeliness of our results and their relation to previous knowledge on the subject. First, we note that experimental investigations on ion mobility in GQs are essentially restricted to NH4+ ions and that there exists a single theoretical work on a quadruplex that is different from that considered by us. Our results are in agreement with experimental data. They are also qualitatively in agreement with the theoretical data, while they add compelling evidence on some aspects and reveal new features. Akhshi’s communication based on a PMF approach was focused on energy profiles and did not target the dynamics, except for hydration features. A critical summary of those results, pertaining to a steady-state ion motion, is traced in the Introduction. Our analysis here is more elaborate and brings new insights into the mechanism of ion motion through a G-quadruplex because we have simulated transitional states when ions enter the channel and have also characterized the binding locations and the values of conductance. Furthermore, we have investigated a different system, which allows us to extend our results to longer channels that may be more relevant for practical applications. (1) Na+ ions can be transported within the quadruplex axis with minimal cost, namely, with a vanishing energy cost for displacement between subsequent binding sites even though there is an energy cost of moving Na+ from the inside to the outside of the channel. There is, instead, a finite energy cost of about 4 kcal/mol for the motion of K+ from the minimum at λ = −9 Å (⟨z⟩ = −12 Å) to λ = −6 Å (⟨z⟩ = −8.5 Å), corresponding to an advancement by one G-quartet step into the channel. These findings are qualitatively in agreement with the simulation of the shorter quadruplex,19 but the energy profiles are different. Our simulation generating transitional states gives us access to the energy gain or cost of entering and exiting the pore and of moving through the pore, 870

dx.doi.org/10.1021/jp408071h | J. Phys. Chem. B 2014, 118, 864−872

The Journal of Physical Chemistry B

Article

European Science Foundation through COST Action MP0802 “Self-Assembled guanosine structures for molecular electronic devices”.

neglecting the energy barriers for hopping between successive metastable states, due to local maxima (see Figure S1C in Supporting Information). The relation between the computed energy profile and the ionic conductance is direct through eq 2. (2) We have shown that K+ ions optimally fit the GQ core at cage positions. Interplane positions are also metastable binding sites for NH4+ ions, while Na+ ions are preferentially bound at in-plane sites. This evidence is in line with previous experiments and simulations. It also further rationalizes the energetic behavior: a Na+ ion passes through an “empty” cage space for moving from one in-plane binding site to the next, while a K+ ion needs to squeeze in G-quartet plane for passing from one cage binding site to the next, which has a deformation energy cost. (3) We have described the interference between the motion of the ions and the motion of water molecules that penetrate the channel, with particular reference to triangular configurations of water molecules that perturb the G-quartet H-bonding pattern and thus find a way out. (4) Finally, we propose an order of 1−10 pS for the conductance of the synthetic nine-base G-quadruplex. In summary, we have reported on the second simulation of ion mobility through a G-quadruplex, finding qualitative agreement with the previous results19 that the GQ favors K+ ions for binding and favors the motion of Na+ ions. We extend this behavior to the longer GQ treated in this work, which is a step toward generalization for what concerns quadruplex length. We have calculated the conductance of this potential artificial ion channel and related it to that of the natural ion channel KcsA. Although the motion of Na+ through GQ is somewhat faster than the motion of K+, this unbalance is tiny relative to their noticeable selectivity ratios in the KcsA. Therefore, it is still doubtful whether the motion of Na+ ions through GQ can be a better basis for the implementation of an artificial ion channel.

■

■

(1) O’Sullivan, R. J.; Karlseder, J. Telomeres: Protecting Chromosomes Against Genome Instability. Nat. Rev. Mol. Cell Biol. 2010, 11, 171−181. (2) Biffi, G.; Tannahill, D.; McCafferty, J.; Balasubramanian, S. Quantitative Visualization of DNA G-Guadruplex Structures in Human Cells. Nat. Chem. 2013, 5, 182−186. (3) Karsisiotis, A. I.; Hessari, N. M.; Novellino, E.; Spada, G. P.; Randazzo, A.; Webba da Silva, M. Topological Characterization of Nucleic Acid G-Quadruplexes by UV Absorption and Circular Dichroism. Angew. Chem., Int. Ed. 2011, 50, 10645−10648. (4) Hu, L.; Lim, K. W.; Bouaziz, S.; Phan, A. T. Giardia Telomeric Sequence d(TAGGG)4 Forms Two Intramolecular G-Quadruplexes in K+ Solution: Effect of Loop Length and Sequence on the Folding Topology. J. Am. Chem. Soc. 2009, 131, 16824−16831. (5) Maizels, N. Dynamic Roles for G4 DNA in the Biology of Eukaryotic Cells. Nat. Struct. Mol. Biol. 2006, 13, 1055−1059. (6) Davis, J. T. G-Quartets 40 Years Later: From 5′-GMP to Molecular Biology and Supramolecular Chemistry. Angew. Chem., Int. Ed. 2004, 43, 668−698. (7) Alberti, P.; Bourdoncle, A.; Sacca, B.; Lacroix, L.; Mergny, J.-L. DNA Nanomachines and Nanostructures Involving Quadruplexes. Org. Biomol. Chem. 2006, 4, 3383−3391. (8) Laughlan, G.; Murchie, A. I. H.; Norman, D. G.; Moore, M. H.; Moody, P. C. E.; Lilley, D. M. J.; Luisi, B. The High-Resolution Crystal Structure of a Parallel-Stranded Guanine Tetraplex. Science 1994, 265, 520−524. (9) Phillips, K.; Dauter, Z.; Murchie, A. I. H.; Lilley, D. M. J.; Luisi, B. The Crystal Structure of a Parallel-Stranded Guanine Tetraplex at 0.95 Å Resolution. J. Mol. Biol. 1997, 273, 171−182. (10) Forman, S. L.; Fettinger, J. C.; Pieraccini, S.; Gottarelli, G.; Davis, J. T. Toward Artificial Ion Channels: A Lipophilic GQuadruplex. J. Am. Chem. Soc. 2000, 122, 4060−4067. (11) Hud, N. V.; Smith, F. W.; Anet, F. A. L.; Feigon, J. The Selectivity for K+ Versus Na+ in DNA Quadruplexes Is Dominated by Relative Free Energies of Hydration: A Thermodynamic Analysis by 1 H NMR†. Biochemistry 1996, 35, 15383−15390. (12) Kaucher, M. S.; Harrell, W. A.; Davis, J. T. A Unimolecular GQuadruplex That Functions as a Synthetic Transmembrane Na+ Transporter. J. Am. Chem. Soc. 2005, 128, 38−39. (13) Roux, B. Ion Conduction and Selectivity in K+ Channels. Annu. Rev. Biophys. Biomol. Struct. 2005, 34, 153−171. (14) Hodgkin, A. L.; Keynes, R. D. The Potassium Permeability of a Giant Nerve Fibre. J. Physiol. 1955, 128, 61−88. (15) Bezanilla, F.; Armstrong, C. M. Negative Conductance Caused by Entry of Sodium and Cesium Ions into the Potassium Channels of Squid Axons. J. Gen. Physiol. 1972, 60, 588−608. (16) Zhou, Y.; Morais-Cabral, J. H.; Kaufman, A.; MacKinnon, R. Chemistry of Ion Coordination and Hydration Revealed by a K+ Channel-Fab Complex at 2.0 Å Resolution. Nature 2001, 414, 43−48. (17) Deng, H.; Braunlin, W. H. Kinetics of Sodium Ion Binding to DNA Quadruplexes. J. Mol. Biol. 1996, 255, 476−483. (18) Hud, N. V.; Schultze, P.; Sklenár,̌ V.; Feigon, J. Binding Sites and Dynamics of Ammonium Ions in a Telomere Repeat DNA Quadruplex. J. Mol. Biol. 1999, 285, 233−243. (19) Akhshi, P.; Mosey, N. J.; Wu, G. Free-Energy Landscapes of Ion Movement through a G-Quadruplex DNA Channel. Angew. Chem., Int. Ed. 2012, 51, 2850−2854. (20) Šket, P.; Plavec, J. Not All G-Quadruplexes Exhibit IonChannel-like Properties: NMR Study of Ammonium Ion (Non)movement Within the d(G3T4G4)2 Quadruplex. J. Am. Chem. Soc. 2007, 129, 8794−8800.

ASSOCIATED CONTENT

S Supporting Information *

Explanation of our methodology and the differences with potential of mean force approaches, including Figure S1 (Section S1, “Step-Wise Pulling Protocol to Analyze and Control Non-Equilibrium Dynamics”); three-dimensional structures that we include as separate files (Section S2); stable and saturated positions of water molecules in the GQ channel (Section S3); and further discussion of the choice of force fields (Section 4). This material is available free of charge via the Internet at http://pubs.acs.org.

■

REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank R. Farley for his comments. We acknowledge financial support by the Department of Energy grant DE-FG0205ER46240. Work by R.D.F. was funded by the Italian Institute of Technology through project MOPROSURF, by the CRMO Foundation through project “DNA-NANO” and by the 871

dx.doi.org/10.1021/jp408071h | J. Phys. Chem. B 2014, 118, 864−872

The Journal of Physical Chemistry B

Article

(21) Wong, A.; Ida, R.; Wu, G. Direct NMR Detection of the “Invisible” Alkali Metal Cations Tightly Bound to G-quadruplex Structures. Biochem. Biophys. Res. Commun. 2005, 337, 363−366. (22) Snoussi, K.; Halle, B. Internal Sodium Ions and Water Molecules in Guanine Quadruplexes: Magnetic Relaxation Dispersion Studies of [d(G3T4G3)]2 and [d(G4T4G4)]2. Biochemistry (Mosc.) 2008, 47, 12219−12229. (23) Cavallari, M.; Calzolari, A.; Garbesi, A.; Di Felice, R. Stability and Migration of Metal Ions in G4-Wires by Molecular Dynamics Simulations. J. Phys. Chem. B 2006, 110, 26337−26348. (24) Ngo, V. A. Parallel-Pulling Protocol for Free-Energy Evaluation. Phys. Rev. E 2012, 85, 036702. (25) Jensen, M. Ø.; Borhani, D. W.; Lindorff-Larsen, K.; Maragakis, P.; Jogini, V.; Eastwood, M. P.; Dror, R. O.; Shaw, D. E. Principles of Conduction and Hydrophobic Gating in K+ Channels. Proc. Natl. Acad. Sci., U.S.A. 2010, 107, 5833−5838. (26) Kotlyar, A. B.; Borovok, N.; Molotsky, T.; Cohen, H.; Shapir, E.; Porath, D. Long, Monomolecular Guanine-Based Nanowires. Adv. Mater. 2005, 17, 1901−1905. (27) Cohen, H.; Sapir, T.; Borovok, N.; Molotsky, T.; Di Felice, R.; Kotlyar, A. B.; Porath, D. Polarizability of G4-DNA Observed by Electrostatic Force Microscopy Measurements. Nano Lett. 2007, 7, 981−986. (28) Borovok, N.; Iram, N.; Zikich, D.; Ghabboun, J.; Livshits, G. I.; Porath, D.; Kotlyar, A. B. Assembling of G-strands into Novel Tetramolecular Parallel G4-DNA Nanostructures Using Avidin−biotin Recognition. Nucleic Acids Res. 2008, 36, 5050−5060. (29) Shapir, E.; Sagiv, L.; Molotsky, T.; Kotlyar, A. B.; Felice, R. D.; Porath, D. Electronic Structure of G4-DNA by Scanning Tunneling Spectroscopy. J. Phys. Chem. C 2010, 114, 22079−22084. (30) Calzolari, A.; Di Felice, R.; Molinari, E.; Garbesi, A. G-quartet Biomolecular Nanowires. Appl. Phys. Lett. 2002, 80, 3331−3333. (31) Ngo, V. A.; Haas, S. Demonstration of Jarzynski’s Equality in Open Quantum Systems Using a Stepwise Pulling Protocol. Phys. Rev. E 2012, 86, 031127. (32) Ngo, V.; Haas, S. Non-equilibrium Approach to Thermodynamics Using Jarzynski Equality and Diagonal Entropy. arXiv:1303.2141 2013. (33) Humphrey, W.; Dalke, A.; Schulten, K. VMD - Visual Molecular Dynamics. J. Mol. Graphics 1996, 14, 33−38. (34) Schwede, T.; Peitsch, M. Computational Structural Biology: Methods and Applications; World Scientific: Singapore, 2008. (35) Kato, M.; Warshel, A. Through the Channel and Around the Channel: Validating and Comparing Microscopic Approaches for the Evaluation of Free Energy Profiles for Ion Penetration through Ion Channels. J. Phys. Chem. B 2005, 109, 19516−19522. (36) Cornell, W. D.; Cieplak, P.; Bayly, C. I.; Gould, I. R.; Merz, K. M.; Ferguson, D. M.; Spellmeyer, D. C.; Fox, T.; Caldwell, J. W.; Kollman, P. A. A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules. J. Am. Chem. Soc. 1995, 117, 5179−5197. (37) Pérez, A.; Marchán, I.; Svozil, D.; Sponer, J.; Cheatham, T. E., III; Laughton, C. A.; Orozco, M. Refinement of the AMBER Force Field for Nucleic Acids: Improving the Description of α/γ Conformers. Biophys. J. 2007, 92, 3817−3829. (38) Šponer, J.; Jurečka, P.; Hobza, P. Accurate Interaction Energies of Hydrogen-Bonded Nucleic Acid Base Pairs. J. Am. Chem. Soc. 2004, 126, 10142−10151. (39) Šponer, J.; Jurečka, P.; Marchan, I.; Luque, F. J.; Orozco, M.; Hobza, P. Nature of Base Stacking: Reference Quantum-Chemical Stacking Energies in Ten Unique B-DNA Base-Pair Steps. Chem. Eur. J. 2006, 12, 2854−2865. (40) Cheatham, T. E.; Young, M. A. Molecular Dynamics Simulation of Nucleic Acids: Successes, Limitations, and Promise. Biopolymers 2000, 56, 232−256. (41) Joung, I. S.; Cheatham, T. E. Determination of Alkali and Halide Monovalent Ion Parameters for Use in Explicitly Solvated Biomolecular Simulations. J. Phys. Chem. B 2008, 112, 9020−9041.

(42) Feller, S. E.; Zhang, Y.; Pastor, R. W.; Brooks, B. R. Constant Pressure Molecular Dynamics Simulation: The Langevin Piston Method. J. Chem. Phys. 1995, 103, 4613−4621. (43) Ermak, D. L.; McCammon, J. A. Brownian Dynamics with Hydrodynamic Interactions. J. Chem. Phys. 1978, 69, 1352−1360. (44) Park, S.; Khalili-Araghi, F.; Tajkhorshid, E.; Schulten, K. Free Energy Calculation from Steered Molecular Dynamics Simulations Using Jarzynski’s Equality. J. Chem. Phys. 2003, 119, 3559−3566. (45) Park, S.; Schulten, K. Calculating Potentials of Mean Force from Steered Molecular Dynamics Simulations. J. Chem. Phys. 2004, 120, 5946−5961. (46) Mackerell, A. D. Empirical Force Fields for Biological Macromolecules: Overview and Issues. J. Comput. Chem. 2004, 25, 1584−1604. (47) Ross, W. S.; Hardin, C. C. Ion-Induced Stabilization of the GDNA Quadruplex: Free Energy Perturbation Studies. J. Am. Chem. Soc. 1994, 116, 6070−6080. (48) Ngo, V. A.; Stefanovski, D.; Haas, S.; Farley, R. NonEquilibrium Dynamics Determine Ion Selectivity In The KcsA Channel. PLoS One, submitted for publication. (49) Ying, L.; Green, J. J.; Li, H.; Klenerman, D.; Balasubramanian, S. Studies on the Structure and Dynamics of the Human Telomeric G Quadruplex by Single-molecule Fluorescence Resonance Energy Transfer. Proc. Natl. Acad. Sci., U.S.A. 2003, 100, 14629−14634. (50) LeMasurier, M.; Heginbotham, L.; Miller, C. KcsA: It’s a Potassium Channel. J. Gen. Physiol. 2001, 118, 303−314.

872

dx.doi.org/10.1021/jp408071h | J. Phys. Chem. B 2014, 118, 864−872