Improved Model of Hydrated Calcium Ion for Molecular Dynamics Simulations Using Classical Biomolecular Force Fields

Abstract

Calcium ions (Ca²⁺) play key roles in various fundamental biological processes such as cell signaling and brain function. Molecular dynamics (MD) simulations have been used to study such interactions, however, the accuracy of the Ca²⁺ models provided by the standard MD force fields has not been rigorously tested. Here, we assess the performance of the Ca²⁺ models from the most popular classical force fields AMBER and CHARMM by computing the osmotic pressure of model compounds and the free energy of DNA–DNA interactions. In the simulations performed using the two standard models, Ca²⁺ ions are seen to form artificial clusters with chloride, acetate, and phosphate species; the osmotic pressure of CaAc₂ and CaCl₂ solutions is a small fraction of the experimental values for both force fields. Using the standard parameterization of Ca²⁺ ions in the simulations of Ca²⁺-mediated DNA–DNA interactions leads to qualitatively wrong outcomes: both AMBER and CHARMM simulations suggest strong inter-DNA attraction whereas, in experiment, DNA molecules repel one another. The artificial attraction of Ca²⁺ to DNA phosphate is strong enough to affect the direction of the electric field-driven translocation of DNA through a solid-state nanopore. To address these shortcomings of the standard Ca²⁺ model, we introduce a custom model of a hydrated Ca²⁺ ion and show that using our model brings the results of the above MD simulations in quantitative agreement with experiment. Our improved model of Ca²⁺ can be readily applied to MD simulations of various biomolecular systems, including nucleic acids, proteins and lipid bilayer membranes.

Introduction

The calcium ion (Ca²⁺) is an important secondary messenger in eukaryotic cell signaling that affects various vital functions such as development, homeostasis, neural activity, and immunity.^1–6 In the nucleus of eukaryotic cells, Ca²⁺ is known to play an important role in gene regulation^7,8 and in the case of neurons, brain function and memory.⁹ In prokaryotic cells, Ca²⁺ concentration is precisely regulated because of its broad effects on development and proliferation.¹⁰ In eukaryotic organisms, calcium is a critical component of the muscle contraction cycle and failure in calcium signaling can cause cardiovascular diseases such as hypertension and heart failure.² At the molecular level, Ca²⁺ ions interact electrostatically with acidic patches of proteins (e.g., EF hand motif),¹¹ anionic lipids (e.g., phosphatidyli-nositol),^12–14 and nucleic acids.¹⁵ Because of their +2e electrical charge, divalent cations such as Ca²⁺ and Mg²⁺ coordinate nearby electronegative atoms, e.g., water, acetate (Ac⁻), or phosphate oxygen atoms, more strongly than monovalent cations do.¹⁶

Ca²⁺ ions can bind to biomolecules in two distinct binding modes. Direct binding occurs in an environment that is relatively devoid of water molecules, such as in the binding pockets of proteins, RNA or chelators, where multiple negatively charged chemical groups, such as acetate and phosphate, form direct contacts with a Ca²⁺ ion. Direct binding tightly embeds a Ca²⁺ ion to a specific site and is known to play a role in the structural stability of RNA and proteins.^17,18 Structural biology techniques such as X-ray crystallography can elucidate the location of such directly bound, structural, Ca²⁺ ions, providing reliable initial molecular configurations for atomistic molecular dynamics (MD) simulations.

In an aqueous environment, solvated Ca²⁺ ions interact non-specifically with negatively charged species (e.g., phosphate groups of DNA and lipid) through bonds mediated by water molecules, an interaction that we refer to hereafter as indirect binding.¹⁹ Experimental characterization of the diffuse ionic atmosphere formed by solvated Ca²⁺ ions around an oppositely charged biomolecule is difficult because of the transient nature of the interactions. Bai et al. counted the numbers (not positions) of individual cationic species in the ionic atmosphere of a DNA duplex, revealing that Ca²⁺ ions bind to DNA at least an order of magnitude stronger than monovalent cations do.²⁰ For example, a DNA duplex submerged in an electrolyte solution containing either 1 mM Ca²⁺ or 20 mM Na⁺ will be surrounded by the same amount of ions. Due to their high affinity to DNA, Ca²⁺ ions can considerably reduce the electrostatic repulsion between DNA molecules. For example, the internal pressure of a DNA array is 20 bar at 250 mM NaCl but only 3 bar at 25 mM CaCl₂ at the same average nearest-neighbor distance (~30 Å) between the DNA molecules.¹⁵ The high affinity of solvated Ca²⁺ to phosphate groups can also affect the structure and mechanical properties of lipid bilayer membranes. For example, increasing concentrations of Ca²⁺ make lipid membranes more rigid by promoting lipid tail ordering.^21,22

Complementing experimental studies, the MD method has been used to elucidate the biological role of Ca²⁺ and other divalent ions.^23–30 In the framework of non-polarizable force fields such as AMBER and CHARMM, the models of divalent cations were developed to match the experimental solvation free energies of individual cations.^23,24 Recent simulations of the osmotic pressure of model solutions have shown that such non-polarizable models of divalent cations may not be accurate enough to properly describe the interactions of the cations with anions.^29,31,32 For example, we have previously shown that the attractive interaction of a Mg²⁺ ion with either a chloride (Cl⁻), acetate, or phosphate anion is significantly overestimated in both CHARMM and AMBER force fields.²⁹ Similarly, Saxena and Garcia found their custom multi-site models of Mg²⁺ and Ca²⁺ ions to overestimate the attraction of the ions to chloride when the latter was described using the standard CHRAMM force field.^31,32

Here, we examine the ability of popular MD force fields AMBER and CHARMM to describe the interaction of solvated Ca²⁺ ions with several representative anionic species: chloride, acetate, and phosphate. We find that the binding affinity of Ca²⁺ to all three anionic species is overestimated in both AMBER and CHARMM, causing artificial aggregation and clustering of the solutes. Following that, we describe an improved model of hydrated Ca²⁺ ions parametrized to reproduce experimental osmotic pressure data. Finally, we show that using our model of Ca²⁺ dramatically improves the realism of the MD simulations of DNA– DNA interactions and of Ca²⁺-mediated electrophoretic motion of DNA through a solid-state nanopore.

Results

Standard parameterization of the CHARMM and AMBER force fields overestimates attraction of Ca²⁺ ions to biological anions

To determine how accurate an all-atom MD simulation is in describing the interactions of Ca²⁺–Cl⁻ and Ca²⁺–Ac⁻ solute pairs, we simulated solutions of the binary mixtures (CaCl₂ or CaAc₂), measured the dependence of the osmotic pressure on the solute concentration, and compared the result with experimental osmotic pressure data. Following the protocols described in the previous studies,^29,33–36 the osmotic pressure was simulated using a two-compartment setup, Fig. 1A,B. Briefly, the simulation box was divided into two compartments (“solute” and “water”) by means of two half-harmonic potential walls (dashed lines in Fig. 1A,B), that acted on the heavy atoms of solutes only. In equilibrium, the solute compartment is pressurized due to osmosis. The osmotic pressure is determined by measuring forces exerted by the solutes on the walls. The Materials and Methods section provides a more detailed description of the simulation procedures.

Figure 1.

Standard molecular force fields overestimate association of Ca²⁺ with anionic species. (A,B) Representative configuration of solutes in MD simulations of 2 m CaAc₂ (A) and 3 m CaCl₂ (B) solutions performed using the standard CHARMM force field.³⁷ Only a 1 nm wide slice through the system is shown for clarity. In each panel, the blue semi-transparent surface illustrates the dimensions of a unit simulation cell; dashed lines indicate the presence of ideal semi-permeable membranes that confine solutes to one of the two compartments; the other compartment contains pure water. (C,D) Comparison of experimental (black square³⁸ or black line³⁹) and simulated (red or blue) osmotic pressure values for the CaAc₂ (C) and CaCl₂ (D) solutions as a function of the solutes’ concentration. The data were obtained using the standard AMBER (blue) and CHARMM (red) parameter sets. The osmotic pressure of an ideal 2:1 solution (osmotic coefficient = 1) is shown using dotted lines. The standard error of 1 ns block averages of 2 ps sampled osmotic pressure data are smaller than the symbols.

Our osmotic pressure simulations of the CaAc₂ solution using the standard MD force fields (both CHARMM and AMBER) showed considerable deviation between the simulation outcomes and experiment.³⁸ Extensive cluster formation was observed in the simulations of the CaAc₂ solution using both force fields, Fig. 1A. Such excessive cluster formation results in significant underestimation of the osmotic pressure at any solute concentration below the solubility limit (~2.2 M at room temperature). At ~2 m, the osmotic pressure was an order of magnitude smaller than the experimental value³⁸ for both force fields, Fig. 1C. Although the only direct experimental measurement of the osmotic pressure is available for a fixed ~2 m concentration of CaAc₂, at lower concentrations the osmotic pressure should be larger than that of an ideal 2:1 solution (dotted line in Fig. 1C).^39,40 At ~0.5 m, the simulated osmotic pressure was one fifth of the ideal value, Fig. 1C. Given the fact that Ac⁻ is an essential chemical group of proteins and lipid membranes, the profound inaccuracy of the standard force field in the description of Ca²⁺–Ac⁻ interaction warrant urgent improvement of the force field model.

In the case of a CaCl₂ solution, our MD simulations revealed inaccuracies of the standard CHARMM and AMBER force fields comparable to those of the CaAc₂ solutions. Ca²⁺ and Cl⁻ ions were seen to form direct pairs, leading to clustering, Fig. 1B, and significant underestimation of the osmotic pressure, Fig. 1D. At ~0.8 m, the simulated osmotic pressure was ~50–70% of the experimental value, depending on the force field used, Fig. 1D. The discrepancy between the simulated and experimental osmotic pressure increased with the solute concentration, Fig. 1D. At ~3 m, the simulated osmotic pressure was ~20–30% of the experimental value.

To evaluate the interactions of Ca²⁺ with phosphates, we computed the interaction free energy and the effective force between two parallel double stranded DNA (dsDNA) molecules. Experimentally, the interaction free energy of an array of aligned dsDNA molecules was quantitatively characterized at a 25 mM CaCl₂ solution using an osmotic pressure setup.¹⁵ We have previously shown that the interaction free energy of a pair of parallel dsDNA molecules is directly related to the experimentally measured osmotic pressure of a DNA array⁴¹ and can be used as a target for improvement of the MD force field.³⁴

To compute the interaction free energy of a pair of parallel dsDNA molecules, we constructed a system containing two effectively infinite dsDNA molecules submerged in a 25 mM solution of CaCl₂, Fig. 2A. The interaction free energy, ΔG, as a function of the inter-axial distance, d, was computed using the conventional umbrella sampling and weighted histogram analysis methods,⁴² see Materials and Methods for details. In our simulations using the standard force fields (CHARMM or AMBER), we observed direct pair formation between Ca²⁺ and phosphate oxygen atoms. The inset of Fig. 2A demonstrates a representative configuration of Ca²⁺ ions forming ionic “bridges” between two dsDNA molecules at d = 24 Å. Such direct binding of Ca²⁺ ions induces strong Ca²⁺-mediated inter-DNA (or inter-phosphate) attraction, indicated by a sharp decrease of the DNA–DNA free energy at d <~ 28 Å, which was observed for both force field models, Fig. 2B. Thus, the simulated pairwise free energy suggests strong attraction between two dsDNA molecules, which is in qualitative disagreement with experimental measurements that show that DNA–DNA interaction is repulsive.¹⁵ Quantitatively, the experimental ΔG at d = 25 Å was measured to be ~0.4 kcal/mol per DNA turn, whereas the simulation suggests a much greater and opposite sign value: −6 kcal/mol per turn, Fig. 2B. In both experiment and simulation, ΔG is set to zero at large DNA–DNA separations.

Figure 2.

MD simulation of Ca²⁺-mediated DNA–DNA interaction free energy using the standard CHARMM and AMBER force fields. (A) Representative configuration of DNA and ions in umbrella sampling simulations of the DNA–DNA potential of mean force. The DNA molecules are shown in gray. A blue semitransparent surface depicts the volume occupied by the solvent in a unit simulation cell. Colored spheres indicate the locations of calcium (blue), phosphate oxygen (red) and chloride (green). This particular image shows the DNA system at an inter-DNA distance of 24 Å; the bulk concentration of CaCl₂ is 25 mM. The inset shows a pair of phosphate groups stabilized by multiple Ca²⁺ ions directly bound to the phosphates forming an ion bridge between the two DNA molecules. (B) The interaction free energy of two parallel dsDNA molecules versus the DNA–DNA distance at 25 mM CaCl₂ computed using the standard parameterization of the CHARMM (red) and AMBER (blue) force fields. The experimental dependence is shown as a black line.¹⁵

Improved model of a Ca²⁺ heptahydrate complex

To improve the parameterization of hydrated Ca²⁺ ions, we consider a Ca²⁺ heptahydrate complex, Ca²⁺(H₂O)₇, that consists of one Ca²⁺ ion described using the CHARMM parameter set²⁴ and seven TIP3P water molecules⁴³ with modified dipole moments, Fig. 3. In doing so, we limit the applicability of our model to systems where Ca²⁺ ions can be expected to remain fully hydrated. In the case of CaCl₂, X-ray absorption fine structure (XAFS) spectroscopy,^44,45 large-angle X-ray scattering (LAXS),²⁵ and neutron diffraction⁴⁶ have shown that Ca²⁺ and Cl⁻ ions do not form direct contacts at room temperature even at CaCl₂ concentrations as high as 6 m. For Ac⁻, formation of a direct contact pair with Ca²⁺ is also unlikely because the osmotic pressures of CaAc₂ and CaCl₂ are comparable: 240 and 250 bar at 2.2 m of CaAc₂ and CaCl₂, respectively.^38,39 Experimental DNA array pressure measurements in 25 mM MgCl₂ and CaCl₂ solutions are quantitatively similar.¹⁵ Previously, we have shown that simulated pressure of a DNA array in MgCl₂ can be matched with experiment only in the absence of direct contacts between Mg²⁺ ions and DNA phosphates.²⁹ Thus, it is reasonable to expect that the number of direct contacts between Ca²⁺ and DNA phosphates is negligible as well.

Figure 3.

Reparameterization of a hydrated calcium ion using a heptahydrate model. In each heptahydrate complex, Ca(H₂O)₇, seven water oxygens are harmonically restrained to a calcium atom. To account for Ca²⁺-induced polarization of water molecules,^47,48 the dipole moment of each of the seven water molecules in the heptahydrate complex is increased by 0.5 debye by adjusting the partial charges of water oxygen and water hydrogen atoms. Following that, the LJ σ parameters describing specific interactions between water oxygens of Ca²⁺ (H₂O)₇ and the target anionic species (phosphate oxygen, acetate oxygen, and Cl⁻) are refined to match the experimental osmotic pressure data.

Although the exact number of water molecules coordinating a Ca²⁺ ion is not known, quantum calculations⁴⁹ and Car-Parrinello MD simulations^28,48 suggest a range of 7–8 and 6–8, respectively, whereas the coordination number of Ca²⁺ in the standard CHARMM force field is 7.²⁴ Because we base our model on the CHARMM force field model of Ca²⁺, using seven water molecules to form a hydrated Ca²⁺ ion complex comes as a natural choice. Furthermore, independent simulation studies based on the AMOEBA polarizable model^27,47 and the Car-Parrinello model,⁵⁰ as well as EXAFS⁴⁵ and neutron diffraction⁴⁶ experiments, also predict the hydration number of a Ca²⁺ ion to be 7.

To distinguish seven water molecules forming the Ca²⁺(H₂O)₇ complex from bulk water, we prevent exchange of water molecules between the Ca²⁺(H₂O)₇ complex and bulk water. Following the strategy applied to model Mg²⁺-hexahydrate,²⁹ we use a half-harmonic restraining potential between Ca²⁺ and water oxygens with a 3.0 Å equilibrium distance and a 50,000 kJ/(mol·nm²) force constant to ensure that all seven water molecules remain in the complex with Ca²⁺. Because the equilibrium distance between Ca²⁺ and water oxygen is ~2.5 Å, the potential exerts force on the restraining oxygen only when the atom attempts to escape from the complex. Because such restraints neither altered the equilibrium Ca–O distance nor the vibration dynamics of water, the effects of using such a restraining potential on the outcome of an MD simulation is negligible under assumption that the ions remain fully hydrated.

According to the results of MD simulations employing a polarizable AMOEBA force field or the Car-Parrinello scheme,^47,48 the dipole moment of water molecules forming the first solvation shell of a divalent cation is ~0.5 debye larger than that of a bulk water molecule. To account for this polarization effect in the framework of a non-polarizable force field, we increased the dipole moments of water molecules in each Ca²⁺(H₂O)₇ complex by 0.5 debye by changing the charge of the oxygen atom from −0.834e to − 1.012e and the charge of each hydrogen atom from 0.417e to 0.506e. A similar approach was used in our previous parameterization of the Mg²⁺ hexahydrate complex,²⁹ where increasing the dipole moment of the first solvation shell water molecules was found to be essential to match the osmotic pressure data.

Atom pair-specific Lennard-Jones parameters describing interactions of Ca²⁺(H₂O)₇ with anionic species

With all Ca²⁺ ions replaced by our Ca²⁺(H₂O)₇ complexes, we recomputed the osmotic pressure of a 2 m CaAc₂ solution using both standard CHARMM and AMBER force fields. The replacement improved the agreement between simulation and experiment, increasing the osmotic pressure from 10–20 to 50 bar for both force fields, Fig. S1A,B. To further refine the interaction between Ca²⁺(H₂O)₇ and Ac⁻, we gradually increased the Lennard-Jones (LJ) σ parameter for the Ca²⁺(H₂O)₇ oxygen and Ac⁻ oxygen pairs, Fig. 3, which is similar to the approach we previously used to refine parameters of Mg²⁺.²⁹ As σ increased, the simulated osmotic pressure monotonically increased as well, approaching the experimental value (~250 bar)³⁸ at σ = 0.07 Å for both force fields, Fig. S1A,B.

Next, we repeated the same procedure for a 3 m CaCl₂ solution, Fig. S2. In the case of CaCl₂, simple replacement of Ca²⁺ ions with Ca²⁺(H₂O)₇ dramatically improved agreement between simulated and experimental osmotic pressure values, Fig. S2A,B. Full agreement between simulation and experiment was observed when the LJ σ parameter for the Ca²⁺(H₂O)₇ oxygen and Cl⁻ pairs was increased by 0.01 Å for both force fields, Fig. S2A,B.

The dramatic improvements of the all-atom model in describing the osmotic pressure of CaAc₂ and CaCl₂ solutions result from prevention of direct ion pair formation. In contrast to molecular configurations obtained using the standard parameterization, Fig. 2A,B, the use of our Ca²⁺(H₂O)₇ model results in homogeneous distribution of the solutes within the solute compartment, Fig. 4A,B. Recomputing the osmotic pressure of the two solutions for a full range of solute concentration reveals excellent agreement between the simulations carried out using our Ca²⁺(H₂O)₇ model and experiment, Fig. 4C,D.

Figure 4.

MD simulations of CaAc₂ and CaCl₂ solutions using the refined force field models. (A,B) Representative configuration of solutes in MD simulations of 2 m CaAc₂ (A) and 3 m CaCl₂ (B) solutions performed using our improved parameterization of Ca²⁺ ions. In each panel, the blue semi-transparent surface illustrates the dimensions of a unit simulation cell; dashed lines indicate the presence of ideal semi-permeable membranes that confine the solutes to one of the two compartments; the other compartment contains pure water. Only a 1 nm-wide slice of the system is shown for clarity. (C,D) Comparison of experimental (black square³⁸ or black line³⁹) and simulated (red or blue) osmotic pressure values for the CaAc₂ (C) and CaCl₂ (D) solutions as a function of the solute concentration. The data were obtained using our calcium heptahydrate parameterization of Ca²⁺ ions in AMBER (blue) and CHARMM (red) force fields. The osmotic pressure of an ideal 2:1 solution (osmotic coefficient = 1) is shown using dotted lines. The standard error of 1 ns block averages of 2 ps sampled osmotic pressure data are smaller than the symbols.

Parameterization of the hydrated Ca²⁺ ion interaction with DNA

Finally, we recomputed the free energy, ΔG, of two parallel dsDNA molecules versus the inter-DNA distance after having all Ca²⁺ ions replaced by our Ca²⁺(H₂O)₇ complexes. Because acetate and phosphate oxygen atoms are chemically similar (note that both CHARMM and AMBER force fields use identical LJ parameters for both oxygen types), it is reasonable to assume that using parameters optimized for a CaAc₂ solution to describe interactions between oxygens of Ca²⁺(H₂O)₇ and DNA phosphate will improve the overall agreement between simulated and experimental free energy of DNA–DNA interactions. Indeed, ΔG computed using the LJ σ correction for CaAc₂ was in a much better agreement with experiment than the original ΔG curve, Fig. 5A,B. However, the resulting dependence of ΔG on the inter-DNA distance was too repulsive in comparison to the experimental one, indicating that the nonbonded interactions between Ca²⁺ and phosphate and between Ca²⁺ and acetate are different. To refine the interaction between Ca²⁺(H₂O)₇ and DNA phosphate groups, we further adjusted the LJ σ parameter for the Ca²⁺(H₂O)₇ water oxygen and phosphate oxygen pairs. Increasing the LJ σ parameters by only 0.02 and 0.04 Å with respect to the standard value for the CHARMM and AMBER force fields, respectively, yielded the best agreement of the ΔG curve with experimental data for both force fields, Fig. 5A,B.

Figure 5.

Simulations of DNA–DNA potential of mean force using our refinement of the CHARMM and AMBER force fields. (A,B) The interaction free energy of two parallel dsDNA molecules versus the DNA–DNA distance at 25 mM CaCl₂ computed using our refined parameterization of the CHARMM (A) and AMBER (B) force fields. The experimental dependence is shown as a black line.¹⁵ The red lines show the results obtained using the same adjustments of the σ parameter for the Ca²⁺ (H₂O)₇ oxygen-DNA phosphate oxygen pair as for the Ca²⁺ (H₂O)₇ oxygen-Ac⁻ oxygen pair (Δσ = 0.07 Å for both force fields). Blue lines show the results obtained using the σ parameters adjusted to reproduce experimental DNA–DNA interaction free energy (Δσ = 0.02 and 0.04 Å for CHARMM and AMBER, respectively). (C) Representative configuration of DNA and ions in umbrella sampling simulations of the DNA–DNA PMF carried out using our refined version of the CHARMM force field. The DNA molecules are shown in gray, blue semi-transparent surface depicts the volume occupied by the solvent in a unit simulation cell. Colored spheres indicate the locations of calcium (blue), phosphate oxygen (red) and chloride (green). This particular image shows the DNA system at the inter-DNA distance of 24 Å; the bulk concentration of CaCl₂ is 25 mM.

Fig. 5C shows a typical configuration of the two DNA system at d = 24 Å simulated using our optimized Ca²⁺(H₂O)₇ model. Unlike the conformations obtained using the standard force fields, Fig. 2A, there are no direct contact between DNA and Ca²⁺ in the simulation carried out using our optimized parameters. Furthermore, Cl⁻ ions are almost completely excluded from the volume surrounding DNA, Fig. 2A. The excessive presence of Cl⁻ ions near DNA in the simulations performed using the standard force fields was caused by direct binding of Cl⁻ ions to Ca²⁺ ions bound to the phosphate groups of DNA.

The effect of Ca²⁺ models on MD simulations of DNA translocation through a solid-state nanopore

Optical detection of DNA translocation through a solid-state nanopore^51,52 allows for massive parallelization of the DNA translocation measurements, which is required to make nanopore sequencing of DNA practical. In one optical detection method, DNA and calcium sensitive dye are placed on the one side of a membrane containing a nanopore, whereas the solution on the other side of the membrane contains calcium ions. An electric field is applied to drive calcium ions from one side of the membrane to the other through a nanopore. When Ca²⁺ ions reach the other side, they combine with the calcium sensitive dye and emit a fluorescent signal. However, the presence of DNA in the nanopore reduces the flow of calcium ions, which in turn reduces the fluorescence of the dyes. Thus, a measurement of the fluorescence intensity can report on the presence of DNA in the nanopore.⁵²

To investigate the effect of Ca²⁺ parameterization on the outcome of simulations designed to reproduce optical detection of DNA translocation, we built two all-atom models containing the same nanopore in a Si₃N₄ membrane and a 36 bp fragment of DNA threaded through it, Fig. 6A. The initial conformation of the DNA molecule along with any bound Ca²⁺ ions were obtained from separate equilibration simulations of DNA in a bulk electrolyte solution lasting hundreds of nanoseconds, see Materials and Methods for details. The systems were then solvated with a hexagonal prism volume of electrolyte solution containing 0.4 M KCl and 65 mM CaCl₂, typical conditions realized in experiment.⁵² The only difference between the two systems was in the description of the hydrated Ca²⁺ ions: standard CHARMM parameterization was used for one system whereas the other system was described using our custom Ca²⁺(H₂O)₇ model and the appropriate NBFIX corrections, Table 1. After short energy minimization, both systems were equilibrated for 2 ns allowing them to obtain an equilibrium volume. Each system was then simulated under applied electric field of different magnitudes, corresponding to a transmembrane bias of 1V, 200 mV and 100 mV.

Figure 6.

Simulations of electric field-driven nanopore transport of DNA in the presence of Ca²⁺ ions. (A) The microscopic configuration of the nanopore system at the beginning of the DNA translocation simulations. DNA is shown using tan spheres; calcium ions are shown as blue spheres. The volume occupied by water is indicated by a semitransparent surface. The Si₃N₄ membrane is shown cut away, with surface atoms shown as black spheres and the interior atoms shown as grey spheres. In addition to calcium, the solution contains 0.4 M of KCl (not shown). The arrow illustrates the direction of applied electric field producing a positive transmembrane bias. (B–D) The z coordinate of the DNA’s center of mass (CoM) during MD simulations at a transmembrane bias of 1 V (B), 200 mV (C), and 100 mV (D). Data from simulations performed using the standard parameterization of the CHARMM force field are shown in blue; orange lines illustrate the results of simulations performed using our parameterization of Ca²⁺ ions for the CHARMM force field. The z coordinate is defined in panel A. (E–G) The number of calcium ions directly bound to DNA (black) and the number of chloride ions directly bound to calcium ions that are already bound to DNA (blue) during the simulations of DNA translocation performed at a transmembrane bias of 1 V (E), 200 mV (F), and 100 mV (G). These data characterize the simulations carried out using the standard CHARMM force field. No Cl⁻-Ca²⁺-DNA binding (defined as a Cl⁻ ion remaining within 0.4 nm of a Ca²⁺ ion for more than 30 ps) was observed in MD simulations employing our parameterization of Ca²⁺ ions.

Table 1.

Reparametrization of Lennard-Jones interactions between Ca²⁺(H₂O)₇ water oxygen and anionic species (Cl⁻, acetate oxygen, and phosphate oxygen). Listed are pair-specific adjustments to the LJ σ parameter, expressed as the difference (Δσ, Å) between the optimal value of this parameter for the specific ion pair and the standard value for that pair. The partial charges of the oxygen and hydrogen atoms within a Ca²⁺(H₂O)₇ complex are −1.012e and 0.506e, respectively.

Atom type	σstandard	Δσ	σNBFIX
CHARMM Cl−	3.61	0.01	3.62
CHARMM acetate(O)	3.09	0.07	3.16
CHARMM phosphate(O)	3.09	0.02	3.11
AMBER Cl−	3.81	0.01	3.82
AMBER acetate(O)	3.06	0.07	3.13
AMBER phosphate(O)	3.06	0.04	3.10

Fig. 6B – D characterizes the displacement of the DNA strand in response to applied electric field. At 1 V, the choice of a calcium model determines the direction of DNA translocation: the standard CHARMM model predicts DNA translocation in the direction of the applied electric field, whereas our improved model indicates translocation in the direction opposite to the applied electric field, the latter being in agreement with experiment.⁵² At lower biases, a more complex dependence is observed in the simulations performed using the standard CHARMM force field. At 200 mV, the DNA moves initially in the direction of the applied field but then reverses the translocation direction; at 100 mV, the DNA very gradually moves along the direction of the applied electric field. In contrast, reducing the bias in the simulation carried out using our improved model of Ca²⁺ is seen to only reduce the overall speed of DNA translocation in proportion to the bias magnitude, without altering its direction.

The counterintuitive behavior observed in the CHARMM-based simulations is explained by overcharging of the DNA molecule. The equilibration simulations performed in a solution containing 65 mM of CaCl₂ and 400 mM of KCl resulted in 79 Ca²⁺ ions bound to a DNA molecule that contained only 70 phosphate groups, Figure S5. Furthermore, the Ca²⁺ binding to DNA did not show signs of saturation, and we expect that if the equilibration simulations were continued, many more Ca²⁺ would bind to the DNA, up to a maximum of nearly 140 Ca²⁺ ions (two Ca²⁺ per phosphate group). Without taking into account the secondary binding of Ca²⁺ to Cl⁻ ions, the DNA molecule and bound Ca²⁺ ions together had a charge of +88e at the end of the equilibration simulation. When the equilibrated DNA molecule was transferred to the nanopore system along with the bound Ca²⁺ ions, the secondary bound Cl⁻ ions (i.e., Cl⁻ ions bound to Ca²⁺ ions bound to DNA) were not transferred and hence the DNA-Ca²⁺ construct was positively charged at the beginning of the nanopore simulations. In the applied field simulations, the complex initially moves in the direction of the applied field. As the simulations progress, the number of bound calcium does not change, but the number of secondary bound Cl⁻ ions increases, Fig. 6E–G. At 1 V, the DNA escapes the nanopore before significant number of Cl⁻ ions binds to DNA. At 200 mV, the DNA translocation is considerably slower, so the DNA charge does become negative and the DNA reverses the direction of its motion. At 100 mV, the DNA charge is close to zero, leading to a small displacement in the applied electric field.

All of the above indicates that MD simulation of electric field-driven motion of DNA in the presence of Ca²⁺ done using the standard parameterization of the CHARMM force field leads to considerable artifacts, including outcomes that are in qualitative disagreement with experiment. In contrast, our model predicts DNA translocation behavior consistent with experimental observation, making computational studies of such systems and processes possible.

Discussion

Because Ca²⁺ ions play a crucial role in cell signaling that occurs in the vicinity of cell membranes, the interaction of Ca²⁺ with lipid bilayers has been an important subject of MD simulations.⁵³ Several groups used MD simulations to investigate the effects of Ca²⁺ on the structure and organization of lipid bilayers, reporting Ca²⁺-induced formation of clusters in anionic phosphatidylserine (PS)^13,54 and neutral phosphatidylcholine (PC)^55,56 membranes. The force that drove such lipid clustering was the direct interaction of Ca²⁺ ions with phosphate (PC and PS) and acetate (PS) groups of multiple lipid molecules, which is similar to the artificial ion bridging observed in our simulations of two DNA molecules using the standard CHARMM force field, Fig. 2A. Experimentally, lipid head groups are known to be highly hydrated.⁵⁷ Therefore, it is plausible that our Ca²⁺(H₂O)₇ model of hydrated Ca²⁺ ions may provide a more realistic description of Ca²⁺ ions interaction with lipid bilayers than the standard parameterization of the CHARMM force field.

A potential drawback of our Ca²⁺(H₂O)₇ model is that it does not allow a direct contact pair formation between Ca²⁺ and anionic species. As discussed in detail in the one of the previous sections (see “Improved model of a Ca²⁺-heptahydrate complex”), a number of experimental studies suggest that Ca²⁺ is unlikely to form a direct contact pair with acetate, phosphate, or Cl⁻ in an aqueous environment.^{15,25,38–40,45,46} One can, however, imagine a situation where ability to simulate direct binding would be desirable, for example, in the simulations of a chelator function or allosteric change upon Ca²⁺ binding.⁵⁸ For such simulations, the MD code can be modified to allow for probabilistic exchange of water from the first solvation shell of a Ca²⁺ ion with an atom of a protein or a chelator. Development of such an algorithm is one of our future goals. An alternative approach is to use the multi-site model that allows for direct contact pair formation.³² We should, however, point out that the latter model does not take the first solvation shell polarization into account nor has it been validated yet for simulations of systems containing acetate or phosphate groups.

It is important to note that our modifications to the dipole moments of the first salvation shell water molecules appear to be an efficient approach for implementation of polarization effects^59–61 within the framework of a non-polarizable model. Because such modifications affect Coulombic interactions of Ca²⁺(H₂O)₇ or Mg²⁺(H₂O)₆ with all other chemical groups, using hydrated models of multivalent ions may improve the description of multivalent ion interactions with a range of polar groups. We stress, however, that applications of our Ca²⁺(H₂O)₇ model to systems not explicitly considered in the present work requires further validation of the model.

Materials and Methods

General MD methods

All MD simulations of model compound mixtures and inter-DNA free energy calculations were done using the Gromacs 4.5.5 package. The simulations employed a 2 fs time step for integration⁶² and an 8–10 Å switching scheme to evaluate van der Waals forces. Electrostatic forces were computed using particle-Mesh Ewald (PME) summation⁶³ with a 12 Å cutoff for the real-space Coulomb interaction and a 1.2 Å Fourier-space grid spacing. Temperature was kept constant at 298 K using the Nosé-Hoover scheme;^64,65 pressure was kept constant at 1 bar using the Parrinello-Rahman scheme applied semi-isotropically.⁶⁶ In the simulations of the osmotic pressure, the xy area of the simulation systems (the area of the semipermeable membranes) was kept constant but the systems were allowed to change their dimensions along the z axis to achieve the target pressure. In the free energy simulations, the systems’ dimensions along the DNA axis (z axis) was kept constant but the systems were allowed to change their dimensions along the x and y axes. SETTLE⁶⁷ and LINCS⁶⁸ algorithms were used to constrain covalent bonds to hydrogen atoms in water and non-water molecules, respectively.

All DNA translocation simulations were carried out using NAMD2,⁶⁹ with a 2 fs time step and periodic boundary conditions. Multiple timestepping was employed, with local interactions being calculated every time step, and full interactions calculated every three steps. The van der Waals forces were smoothly cut off from 10 to 12 Angstroms. The particle mesh Ewald method was used to evaluate long-range electrostatic interactions⁶³ over 1.0 Å-resolution grids. SETTLE⁶⁷ and RATTLE⁷⁰ algorithms were applied to covalent bonds involving hydrogen atoms in water and DNA, respectively. A Nose-Hoover Langevin⁷¹ piston was used for pressure control in the NPT simulations with a period of 400 fs and a damping time scale of 200 fs. Temperature in bulk solution simulations was controlled by a Langevin thermostat with a damping constant of 0.5 ps⁻¹. Temperature in nanopore translocation simulations was controlled by a Langevin thermostat acting on the membrane atoms with a damping constant of 1.0 ps⁻¹; all simulations under applied electric field were carried out in a constant temperature, volume and number of particles (NVT) ensemble.

Force fields

In this work, we considered both CHARMM and AMBER force fields.

The CHARMM parameters for DNA and Ac⁻ molecules were taken from the standard distribution of the CHARMM27 force field,⁷² which includes ion parameters for Ca^{2+ 24} and Cl⁻.⁷³ Although a newer version of DNA force field (CHARMM36) is now available,⁷⁴ the difference between the two force fields is in updated bonded parameters and therefore our updated parameterization of calcium-DNA interactions equally applies to both CHARMM27 and CHARMM36 parameters sets as the two have identical nonbonded parameters.

The AMBER parameters for DNA and Ac⁻ molecules were taken from the standard distribution featuring the AMBER99 parameter set^75,76 for biomolecules. The parameters for Cl⁻ were taken from the standard ion model developed by the Cheatham group for the AMBER force field.⁷⁷ Because parameters describing divalent ions in the standard AMBER force field²³ are known to be outdated,⁷⁷ we chose the Ca²⁺ model from the standard CHARMM force field²⁴ to be our baseline parameterization of Ca²⁺ for AMBER, which is a reasonable approximation because both AMBER and CHARMM parameters employ the same water model, TIP3P. In all simulations, we used the standard TIP3P model for water.⁴³

The standard Lennard-Jones (LJ) parameters (σ and ∊) for a pair of atoms were determined using the Lorentz-Berthelot mixing rule, which is the standard method for both CHARMM and AMBER force fields. Corrections to the LJ σ parameter of a specific atom pair were introduced using the nonbond_params entry in the Gromacs parameter file or NBFIX entry in the CHARMM parameter file. The final set of corrections is summarized in Table 1.

Protocol of osmotic pressure simulations

Each simulation system contained two compartments separated by two virtual semipermeable membranes aligned with the xy plane. One compartment contained an electrolyte solution while the other contained pure water. We modeled the semipermeable membranes using a half-harmonic planar potential that applied to solute molecules only, making it invisible to water:

$$F_i^{\text{memb}}=\{\begin{array}{ccc}-k(z_i-D/2)\hfill & \text{for}\hfill & z_i>D/2\hfill \\ 0\hfill & \text{for}\hfill & \left|z_i\right|\le D/2\hfill \\ -k(z_i+D/2)\hfill & \text{for}\hfill & z_i<-D/2\hfill \end{array}$$ (1)

where z_i is the z coordinate of ion i, D is the width of the electrolyte compartment, and the force constant k = 4000 kJ/(mol·nm²). Such half-harmonic potentials were applied using the mdrun program of Gromacs 4.5.5 package.⁶² During the simulations, we recorded the instantaneous force applied to both membranes by the solutes. The instantaneous pressure on the membranes was obtained by dividing the instantaneous total force on the membranes by the total area of the membranes.

For a given condition (solute concentration, adjustments to interaction parameters), we performed at least a 10 ns equilibration simulation, followed by a production run of at least 100 ns. The osmotic pressure of each system was computed by averaging the instantaneous pressure. Statistical uncertainty in determination of the osmotic pressure was characterized as the standard error of 1 ns block averages of instantaneous pressure. Further details can be found in the description of our previous study.²⁹

Simulation protocol for computing DNA–DNA interaction free energy

We used the conventional umbrella sampling and weighted histogram analysis method⁴² to compute the interaction free energy between a pair of dsDNA molecules (dG₁₀-dC₁₀). The DNA helices were parallel to the z axis and covalently bound to themselves across the periodic boundary of the system. The simulation box was a water-filled hexagonal prism, which was ~ 9 nm on a side within the x − y plane and 3.4 nm along the z axis. Specific ionic conditions were obtained by replacing randomly chosen water molecules with ions. Each system was equilibrated for at least 10 ns at a specific ionic condition.

Umbrella sampling simulations were performed on the equilibrated systems. The force constant of the harmonic umbrella potentials was 1, 000 kJ/(mol·nm²). The reaction coordinate was defined as the distance between the centers of mass of the DNA molecules projected onto the xy plane; the range of ξ in the umbrella sampling was 24 to 40 Å with 1 Å spacing. Per each window, we performed at least 10 ns of equilibration simulation followed by a production run of at least 100 ns. Thus, ~20-µs-long sampling was required to compute each individual free energy curve reported in this study. The weighted histogram analysis method was used for the reconstruction of free energy from the umbrella sampling simulations.⁴² To estimate the error, we divided data in each window into four non-overlapping subsets, computed four free energy curves using the subsets, and calculated standard deviation of those four free energy curves.

Simulation of DNA translocation through a solid-state nanopore

To simulate transport of DNA through a solid-state nanopore, a (ACTG)₉ fragment of dsDNA was built using the 3D-DART server.⁷⁸ The DNA molecule was submerged in a cubic volume (14 nm on each side) of solution containing 0.4 M KCl and 65 mM CaCl₂, which is a typical experimental solution used in optical detection of DNA translocation.⁵² The system was equilibrated by restraining all DNA heavy atoms to their original positions with a spring constant of 1 kcal/(mol nm²) for 1 ns, and then restraining only phosphorous atoms using the same spring constant for 1 ns, followed by a free equilibration simulation of 10 ns. To decrease the computational cost of subsequent equilibration runs, the DNA molecules along with all bound Ca²⁺ ions were placed in a rectangular volume ( 6 nm×6 nm×18 nm) filled with 0.4 M KCl and 65 mM CaCl₂ aqueous solution. The system was equilibrated for 1 ns applying harmonic restraints of 1 kcal/(mol nm²) to all DNA heavy atoms and bound Ca²⁺ ions. Finally, the system was equilibrated for 490 ns under 1 kcal/(mol nm²) harmonic restraints applied to each phosphorous atom of the DNA. During the equilibration, calcium ions were observed to permanently bind to DNA, Figure S5, which lowered their concentration in solution. To keep the calcium concentration in solution constant at 65 mM, randomly chosen water molecules were replaced with calcium and chloride ions during the 490 ns equilibration run; six such replacements occurred during the equilibration run.

The equilibrated DNA molecule along with all permanently bound Ca²⁺ ions (defined as being within 0.3 nm of any phosphate oxygen atoms) were placed inside a 3.5 nm diameter nanopore in a 7 nm thick silicon nitride membrane, Fig. 6A. The nanopore was built following a previously described procedure;⁷⁹ the DNA axis was aligned with the nanopore axis to avoid direct contacts between DNA and the nanopore surface. Following that, the system was solvated to form a hexagonal prism volume 9 nm across and 19 nm in length. Potassium, calcium and chloride ions were added to bring the bulk concentration of ions to 0.4 M KCl and 65 mM CaCl₂. The system was equilibrated having the heavy atoms of DNA restrained for the first 2 ns of equilibration; during the first nanosecond, bound calcium ions were also restrained. Following that, the restraints on DNA were released and the system was simulated at constant volume conditions under external electric field producing a transmembrane bias of either 1.0 V, 200 mV or 100 mV.⁸⁰

A second nanopore system was built and simulated following the exact same procedures as above except that all Ca²⁺ ions were replaced by calcium heptahydrate complexes, Ca²⁺(H₂O)₇, and that NBFIX corrections, Table 1, were used to describe the interactions between Ca²⁺(H₂O)₇ and Cl⁻ and DNA phosphates, Fig. 3. The initial equilibration of DNA in bulk electrolyte containing 0.4 M KCl, and 65 mM Ca²⁺(H₂O)₇ lasted ~100 ns, no long lasting binding of Ca²⁺(H₂O)₇ to DNA (or Cl⁻) was observed during the equilibration run or simulations of DNA translocation through the nanopore driven by an applied electric field.

In all simulations of the nanopore systems, surface atoms of the silicon nitride membrane were restrained using harmonic potentials of 10 kcal/(mol·Å²) spring constants; the interior atoms were restrained using 1 kcal/(mol·Å²) harmonic potentials to give the membrane a relative bulk permittivity of 7.5.⁸¹ The interactions of silicon nitride atoms with the rest of the system were described by a custom force field.⁸⁰ All other interactions were governed by the CHARMM36 force field; previously described NBFIX corrections were applied to describe interactions of potassium ions with chloride ions and DNA phosphates.²⁹ A grid-based potential⁸² was applied to prevent calcium ions from sticking to the surface of Si₃N₄.⁸¹ The DNA was constrained to remain aligned along nanopore axis via a half-harmonic potential acting on the DNA phosphorous atoms; the onset of the half-harmonic potential began at 1.5 nm from the axis of the nanopore, and the spring constant was 1000 pN/nm.