2160
J. Phys. Chem. B 2009, 113, 2160–2169
Mediating Role of Multivalent Cations in DNA Electrostatics: An Epsilon-Modified Poisson-Boltzmann Study of B-DNA-B-DNA Interactions in Mixture of NaCl and MgCl2 Solutions Sergei Gavryushov† Engelhardt Institute of Molecular Biology, 32 VaViloVa St., Moscow, Russia ReceiVed: October 19, 2008; ReVised Manuscript ReceiVed: December 14, 2008
Potentials of mean force acting between two ions in SPC/E water have been determined via molecular dynamics simulations using the spherical cavity approach (J. Phys. Chem. B 2006, 110, 10878). The potentials were obtained for Me2+-Me+ pairs, where Me2+ means cations Mg2+ and Ca2+ and Me+ denotes monovalent ions Li+, Na+, and K+. The hard-core interaction distance for effective Me2+-Me+ potentials appears to be of about 5 Å that looks like a sum of the effective radii of a Me2+ ion (3 Å) and of an alkali metal ion Me+ (about 2 Å). These ion-ion interaction parameters were used in the epsilon-Modified Poisson-Boltzmann (ε-MPB) calculations (J. Phys. Chem. B 2007, 111, 5264) of ionic distributions around DNA generalized for the arbitrary mixture of different ion species. Ionic distributions around an all-atom geometry model of B-DNA in solution of a mixture of NaCl and MgCl2 were obtained. It was found that even a small fraction of ions Mg2+ led to sharp condensation of Mg2+ near the phosphate groups of DNA due to polarization deficiency of cluster [Mg(H2O)6]2+ in an external field. The ε-MPB calculations of the B-DNA-B-DNA interaction energies suggest that adding 1 mM of Mg2+ to 50 mM solution of NaCl notably affects the force acting between the two macromolecules. Being compared to Poisson-Boltzmann results and to MPB calculations for the primitive model of ions, the ε-MPB results also indicate an important contribution of dielectric saturation effects to the mediating role of divalent cations in the DNA-DNA interaction energies. Introduction For many years there has been much interest in the theoretical study of DNA electrostatics. Although molecular dynamics simulations of DNA with water and mobile ions have been reported,1-10 all-atom simulations require enormous computational resources. A reasonable alternative to speed up the calculations is approaches based on the implicit solvent approximation. This implies a nondiscrete continuum solvent, although its properties can, in general, be described as spatially inhomogeneous and dependent on the presence of macromolecules. In many cases the macromolecule conformation can be regarded as fixed (e.g., at calculation of the free energy of noncovalent electrostatic interaction between well separated molecules), which allows us to apply such computationally efficient implicit-solvent methods as Monte Carlo (MC) simulations11-17 and the Poisson-Boltzmann (PB)18 or modified Poisson-Boltzmann (MPB)19,20 calculations. The simplest and widely used implicit solvent description of hydrated ions is the primitive model (PM) treating them as hard spheres immersed into dielectric continuum. The further development of the implicit-solvent approach should include (i) a more realistic description of the short-range part of the potential of mean force (PMF) acting on ions and (ii) inclusion of polarization effects due to low permittivity of the macromolecule’s interior and dielectric saturation of the ion hydration shell. The detailed effective short-range ion-atom and ion-ion potentials can be extracted from the all-atom MD simulations and then used in the MC simulations for large systems.21 On the other hand, inclusion of the polarization properties of ions into MC procedure can only be done at great computational † E-mail:
[email protected].
cost.22 Moreover, the ion-image interactions due to the dielectric interface between the macromolecule and water cannot be described in MC procedure except for the simplest geometries of this interface. The MPB theory incorporates dielectric properties in a more natural way than any other theory of the electrical double layer. For example, it correctly describes the image interactions of ions within the framework of the PM.20,23,24 This theory is a higher level closure of the Kirkwood hierarchy of ion distribution functions than the zero level in the mean-field PB approximation23 widely used in studies of noncovalent electrostatic interactions of macromolecules. In this sense, the set of MPB equations is an extension of the PB equation. The theory describes the interionic correlation and ion volume exclusion effects. For the PM electrolyte, the MPB calculations showed good predictions of the MC results for electrical double layers with divalent cations where the PB equation fails.23 It is important that the MPB theory can incorporate such a solvent dielectric saturation effect as the polarization deficiency of the hydration shell of a cation in an external field. The shell polarization, being lower in comparison with that of the equal volume of pure water, affects the long-range electrostatics of biological macromolecules because of decrease of the mean permittivity of electrolyte solution. The polarization deficiency of hydrated cations such as Na+ or Ca2+ in the external field was studied via all-atom simulations with SPC/E model of water.25 The dielectric approximation of this effect was called the primitive polarization model (PPM) describing hydrated ions as spheres of permittivity lower than that of water. The effect is especially pronounced for multivalent cations that possess highly saturated and stiff inner shells of hydration.
10.1021/jp809245a CCC: $40.75 2009 American Chemical Society Published on Web 01/28/2009
Multivalent Cations in DNA Electrostatics The development of the MPB theory for the PPM ions was named the ε-MPB theory.26 Recently published ε-MPB results27 indicate that the solvent dielectric saturation effects are important for DNA-DNA interaction in a solution of CaCl2. The peculiar polarization of the cation hydration shell in the external field leads to sharp condensation of ions Ca2+ onto [PO2]- groups of B-DNA. It results in repulsion between two B-DNA double helices in a solution of CaCl2 in accordance with experimental data,28 whereas MC12,29 and MPB27 calculations for the 2:1 PM electrolyte always lead to an attractive force between the two DNA macromolecules. In the case of DNA-DNA interaction in a solution of NaCl, these polarization effects are weak and the electrostatic forces are well approximated by the PB equation.27 The derivation of the ε-MPB equations26 is based on the PPM approximating the short-range ion-ion interactions and polarization of the ion hydration shell in the external field. The PPM parameters were fitted by means of all-atom MD simulations for small systems.25,26,30,31 An important assumption of the theory is that ions always retain their hydration clusters such as [Mg(H2O)6]2+ or [Ca(H2O)6]2+.32 For Mg2+ and DNA, it was confirmed in all-atom MD simulations: magnesium ions were always localized around DNA atoms in a hydrated form.7 As follows from the effective Ca2+-Cl- or Ca2+-Ca2+ potentials in SPC/E water,30 the hydration shell of ion Ca2+ is also stiff enough, so this assumption looks reasonable for both biologically important ions Ca2+ and Mg2+. As to monovalent ions, the solvent dielectric saturation effects are weak for DNA in a solution of NaCl27 and one can disregard the distortion of the hydration shell of ion Na+. In the present state, the ε-MPB theory describes neither the site-specific binding of ions accompanied by destruction of their hydration shells nor the properties of the Stern region around the macromolecule’s atoms and the related surface energy effects. In respect to the Stern region and its dielectric properties, accuracy of the model is not higher than at the standard PB calculations. This is acceptable because the present ε-MPB study is pointed to explore the polarization effects on the long-range electrostatic interactions. In the present work, the ε-MPB theory is extended to the mixture of electrolytes. It should be noted that such phenomena as DNA condensation are induced by addition of multivalent electrolytes to a solution at physiological conditions (0.1 M of NaCl).33 On the other hand, in theoretical studies of the electrical double layers, systems with mixtures of electrolytes have been scarcely addressed.34-37 The main objective of the present research is evaluation of an impact of the peculiar polarization of the [Mg(H2O)6]2+ cluster on ionic distributions around DNA in the mixture of NaCl and MgCl2. It can improve our understanding of the role of alkaline earth metal cations in biological processes. The effect of solvent dielectric saturation on long-range electrostatic forces can be extracted by means of a comparison of the ε-MPB calculated energies of DNA-DNA interactions with the same energies obtained from MPB results for nonpolarizable ions and from the PB calculations. The paper is organized as follows. The next section includes a description of some simulations of the ion-ion PMFs needed to approximate the model for ε-MPB calculations. In section “Theory”, the ε-MPB theory for a mixture of ions is discussed. After that, results related to DNA-DNA interactions are described.
J. Phys. Chem. B, Vol. 113, No. 7, 2009 2161 TABLE 1: Lennard-Jones Parameters σ and ε for Ionsa SPCb ion
σ
SPC/Ec ε
RPOLd
σ
ε
2.584 3.332
0.100 0.100
+
Li Na+ K+ Mg2+ Ca2+
1.398 2.361
σ
ε
1.506
0.165
0.875 0.45
a
Units: σ in Å and ε in kcal/mol. b Reference 45. Åqvist provides ion-water oxygen LJ parameters, which here have been converted to ion LJ parameters using the Lorentz-Berthelot mixing rules and LJ parameters of the SPC water model. c Reference 46. d Reference 47.
Effective Potentials of Interaction between Alkali and Alkaline Earth Metal Ions in SPC/E Water To apply a theory of the electrical double layer to the mixture of electrolytes, one has to explore the effective PMFs acting between two hydrated ions. For alkali and alkaline earth metal ions, most of such potentials have been reported.21,30 For example, in ref 30 the Cl--Cl- and all Me+-Cl-, Me+-Me+, Me2+-Cl-, and Me2+-Me2+ potentials were obtained through MD simulations with SPC/E model of water (here Me+ denotes Li+, Na+, and K+ and Me2+ denotes Mg2+, Ca2+, Sr2+, and Ba2+). Those effective potentials were used in implicit-solvent MC simulations to obtain the radial distribution functions of ions at different ionic concentrations and finally to approximate the hard-core repulsion of hydrated ions.30,31 Those approximations together with results of studies of the ion hydration shell polarization25 were used for the parametrization of the ε-MPB equations.26,27 In ref 30 the spherical cavity model was used in MD simulations to calculate the effective ion-ion potentials in SPC/E water. The obtained PMFs were in a good agreement with previously published results.21,38,39 Calculated PMFs for Me2+-Me2+ interactions suggested that the effective diameters of hydrated ions Mg2+, Ca2+, Sr2+, and Ba2+ were about 6 Å. In the present work the spherical cavity model is applied to similar calculations of effective Me2+-Me+ potentials in SPC/E water. Here Me2+ denotes ions Mg2+ and Ca2+, whereas Me+ is cations Li+, Na+, and K+. Such potentials will complete the set of interionic PMFs needed for the present study of DNA in a solution of electrolyte including monovalent ions and such divalent cations as Ca2+ or Mg2+. The latter two play a regulatory role in many biological processes involving polyanions. In the MD simulations, two ions and 500 water molecules are enclosed in a spherical cavity of radius RRF ) 18.1 Å surrounded by a dielectric medium with permittivity εRF representing water. Details of this method of the PMF evaluation are given in ref 30. The rigid SPC/E water model40 is used. Water molecules interact with the surrounding via the shortrange water-water potential centered at the boundary RRF. Inside the spherical cavity, all interactions are evaluated explicitly. The Lorentz-Berthelot mixing rules are applied. The leading reaction field of the dielectric medium acting on water molecules and ions is represented by image charges41 using the dielectric constant of the SPC/E water model, εSPC/E ) 71.25,30,42-44 The Lennard-Jones (LJ) parameters (σ and ε) of ions45-47 are given in Table 1. The choice of the LJ potentials follows from implicit-water MC simulations of the mean activity coefficients of electrolytes with the effective ion-ion potentials.30 In ref 30 the best agreement with experimental activities of NaCl and KCl was obtained for LJ parameters from refs 46 and 47 and
2162 J. Phys. Chem. B, Vol. 113, No. 7, 2009
Gavryushov nating from the molecular nature of the solvent. At any r, the effective ion-ion potentials uij(r) are constructed as30
uij(r) )
Figure 1. Short-range terms of effective alkaline earth ion-alkali ion potentials from MD simulations using SPC/E water with Mg2+ (thick curves), Ca2+ (thin curves), Li+ (solid curves), Na+ (long-dashed curves), and K+ (short-dashed curves). The LJ parameters are given in Table 1. tMD ) 1.8 ns (Ca2+-K+) and 0.9 ns (other potentials). The uSR(r) curves were vertically adjusted to agree with upol(r) at r ) 10 Å. See text for further details.
{
uijSR(r) + uijCoulomb(r;εw), uijpol(r)
+
uijCoulomb(r;εw),
r e rmax r > rmax
(2)
where εw is the dielectric constant of water (εw ) 78.5), uSR ij (r) is extracted from eq 1 and upol ij (r) represents a repulsive contribution arising from a polarization deficiency of the ion hydration shell in an external field. The term u0ij(rmax) is set equal Coulomb (rmax;εSPC/E), making uij(r) continuous at to upol ij (rmax) + uij r ) rmax. The term upol ij (r) is connected to experimental coefficients δi of the linear decrease of the electrolyte solution permittivity with growth of ionic concentrations ci:25,44,50-52
εsolution(c) ) εw -
∑ δici
(3)
i
they are applied in the present study. For solutions of chlorides of alkaline earth metals, the accuracy of the obtained effective ion-ion potentials was found not to be sufficient to resolve the predicted activity coefficients for different cations. As the LJ approximations for Ca2+ and Mg2+ from ref 45 were used in ref 30, they are also chosen in the present work just for comparison between the Me2+-Me2+ and Me2+-Me+ potentials. It should be noted that the LJ parameters of ions were originally fitted for different water models, viz. the SPC, SPC/ E, or RPOL models (Table 1), but here the SPC/E water model will be used. Even though simulations of hydrated ions apparently require a polarizable water model, the choice of the SPC/E model was dictated by the amount of computations for the spherical cavity model. One more reason could be successful predictions of the mean electrolyte activity coefficients by MC simulations with the effective ion-ion potentials obtained for the rigid SPC/E water.30 A comparison between the Na+-ClPMFs in RPOL water and in SPC/E water48 showed that the difference in the two potentials was not high and it was lower than the change of the Na+-Cl- PMF caused by the choice of the LJ approximation for Na+.30 The mean force acting on an ion was evaluated for fixed positions of the ions. The simulation times were tMD ) 0.9 or 1.8 ns, in addition to preceding equilibration. The desired temperature was 298 K, and the averaged temperatures obtained were between 299 and 300 K. The ion-ion separations were taken between 4 Å and rmax ) 10 Å with a step of h ) 0.1 Å, implying 61 simulations for each mean-force curve. The PMF at ion-ion separation r, u0(r), was integrated as described in ref 30. As follows from independent simulations starting from different initial configurations of water molecules at each r, the accuracy of the effective ion-ion potentials is estimated within 0.1 kT over the entire range. All MD simulations were carried out employing the MOLSIM package.49 The effective ion-ion potential (at infinite dilution) u0ij(r) obtained through MD simulations for a pair of ions i and j can be divided into the two terms:30
uij0 (r) ) uijSR(r) +
qiqj , 4πε0εSPC/Er
r e rmax
(1)
where the second term is the long-range Coulomb potential (r;εSPC/E) assuming a homogeneous permittivity εSPC/E of uCoulomb ij the system and uSR ij (r) is the remaining short-range part origi-
and
uijpol(r) ) RiEj2(r) + RjEi2(r)
(4)
where
Ri )
1 ε0 δ 1000NA 2 i
(5)
Ei is the electric field intensity emancipating from ion i at separation r, NA is the Avogadro number, and δi is measured in M-1. The values δi are 22 M-1 for Mg2+ (ref 50) and 21 M-1 for Ca2+ (ref 51). In the case of monovalent cations, these factors are determined as 12 M-1 for Li+ (ref 51), 10 M-1 for Na+ (ref 52), and 8 M-1 for K+ (ref 50). In all cases, values δi were evaluated from experimental data for solutions of chlorides of these metals50-52 assuming that this factor for Cl- is between 2 and 3 M-1.25,50 The spline-interpolated curves of uSR ij (r) from eq 2 are shown in Figure 1 for all Me2+-Me+ interactions, where Me2+ is Mg2+ and Ca2+ cations and Me+ denotes Li+, Na+, and K+ cations. The curves look similar to Me2+-Me2+ or Li+-Li+ potentials;30 i.e., the magnitude of the solvent-driven oscillation is low (about 0.5 kT) and there is a water molecule between the two ions. The energy of removal of this molecule is about 3 kT, which can be roughly estimated from the visible inflection of the curves between r ) 4 Å and r ) 4.5 Å (not shown). It is much less than such energy for Me2+-Me2+ interactions reaching tens of kT.30 The only exception in Figure 1 is the Ca2+-K+ potential, where the water molecule between ions appears to be easily removed at less than about 5 Å separation between the ion centers. We can assume that it happens due to larger sizes of both cations of calcium and potassium than of other ions studied. Except for this pair, the effective sizes of ions are found in a surprisingly good agreement with the effective sizes estimated from interactions between identical cations. In ref 30 the “sizes” of ions at Me2+-Me2+ or Me+-Me+ interactions were defined SR (r) exceeds 2 kT. These as ion-ion separation r where uMeMe values are collected in Table 2. They were independently verified at calculations of the mean activities of electrolytes based on the effective interionic potentials.31 In Table 3, sums of the
Multivalent Cations in DNA Electrostatics
J. Phys. Chem. B, Vol. 113, No. 7, 2009 2163
TABLE 2: Estimated Hard-Core Diameters of Hydrated Ionsa Me
dMeMe (Å)
+
3.8 (*) 3.2 3.6 5.9 (*) 6.0 (*)
Li Na+ K+ Mg2+ Ca2+ a
dii denotes the separation at which uijSR(r) attain 2 kT for interactions of like-charge ions. The asterisk denotes a solvent-separated ion pair. The data are taken from ref 30.
TABLE 3: Comparison of Hard-Core Distances dij between Ions (Figure 1) and Values (dii + djj)/2 from Table 2 j ) Li+ (dii + djj)/2
Me i ) Mg i ) Ca2+
2+
a
a
4.9 4.9
j ) Na+
j ) K+
dij
(dii + djj)/2
dij
(dii + djj)/2
dij
4.8 4.9
4.6 4.6
4.8 4.8
4.8 4.8
4.7 4.2
All data are in Angstroms.
Figure 2. Illustration of the model of ion-macromolecule interaction used in the ε-MPB calculations. The permittivity of macromolecule’s interior εm extends to the layer of thickness of ∆ε outside the van der Waals radius of each atom of the macromolecule. The short-range repulsion between hydrated ions and macromolecule’s atoms is approximated by the exclusion layer of thickness ∆HC. The cluster of water molecules around the ion is approximated by a dielectric sphere of low permittivity εi (∼10) and radius aiε (∼4 Å). The distance of the hard core repulsion of hydrated ions i and j is aHC ij . The water permittivity εW depends on the mean electric field E according to the Booth theory. See text for further details.
TABLE 4: ε-MPB Parameters of Ionsa
effective hard-core radii dMeMe/2 are compared to the effective hard-core distances dij between ions obtained from the present simulations. The same 2kT threshold of uSR ij (r) is applied. The accuracy of values in Table 3 is estimated as (0.1 Å. As seen from Table 3, the sums (dii + djj)/2 well agree with dij for all ions except for interaction between Ca2+ and K+. The agreement is especially good for the lithium cation. It is expected, because both Me2+-Me2+ and Li+-Li+ interactions are similar to the Me2+-Li+ interaction as retaining a water molecule between the two ions. In the case of Mg2+, the agreement is also acceptable for Na+ and K+ ions according to the accuracy of all values despite the different natures of the short-range Me+-Me+ repulsion. The same is observed for Ca2+-Na+ interaction and only the Ca2+-K+ potential does not obey the rule. Theory Model. The model of interactions between PPM ions and a macromolecule is described in refs 26 and 27. Here we only discuss its features arising from an arbitrary mixture of ions of different species. The parameters of the model are illustrated in Figure 2. As in the previous ε-MPB study,27 we will focus on evaluation of the polarization effects originating from the fact that the polarizability of the cluster [Mg(H2O)6]2+ in an external field is lower than that of equal volume of pure water. As the study is primarily focused on this long-range electrostatic effect, we will be most interested in evaluation of differences in the ionic distributions and free energies between ε-MPB results and MPB/PB results for nonpolarizable ions. Therefore, we can use a crude approximation of the Stern region of ionic exclusion around a macromolecule, as this description coincides for ε-MPB, MPB, and PB calculations. The zone of ionic exclusion is parametrized by values ∆ε and ∆HC (Figure 2), and we neglect the dependencies of these parameters on the ion species, macromolecule’s atoms, and geometry of the macromolecule/water interface. For the same reason, we also neglect a possible destruction of the hydration cluster of ions inside the ∆HC layer that might result in the site-specific short-range binding of ions. The parameters ∆ε and εi were fitted by means of comparisons of the electrostatic energies of the PPM ions near a spherical macroion with the ion-macroion PMFs obtained from all-atom MD simulations.26 The same PMFs lead to evaluation of the
aijHC/Å ion
Na+
Ca2+
Mg2+
aiε/Å
εi
∆HC/Å
∆ε/Å
δi (M-1)
Na+ Ca2+ Mg2+
4.5
5 6
5 6 6
3.8 4.0 4.1
25 7 7
2.5 2.5 2.5
0.8 0.8 0.5b
10 21 22
a
Data from ref 27 and MD simulations described in the present work. b Value of 0.8 Å was used in the present MPB calculations.
value ∆HC that appears to be almost independent of the applied electric field and cation species.26,27 All parameters of the model are collected in Table 4. Values aεi and εi fit both the shortrange repulsion of the ion near the macroion26 and experimental values δi from eq 3.25,26 Parameters aHC ij can be extracted from ion-ion RDFs obtained from MD and MC simulations of HC follows from MD ions.30,31 The value of 5 Å for aNaMg simulations of the present study (Figure 1). Quantities aHC ij approximate the hard-core repulsion of hydrated ions as sums of ion radii. It should be noted that they always represent the solvent-separated repulsion if a divalent cation is involved. Table 4 contains the PPM parameters of cations. Because the present ε-MPB study deals with DNA that is surrounded mainly by cations, the approximating parameters of Cl- are of minor importance.27 For us it is important that δCl , δNa;25 i.e., we can neglect polarization properties of hydrated anions. The choice of aHC MeCl does not affect results. This value was arbitrarily set to 4.5 Å for all metal ions. ε-MPB Equations. The Loeb closure of the Kirkwood hierarchy gives the following distribution function of PPM ion i:26
ln gi(r1) ≈
∑ ( 34 π(aikHC)3n0k - ∫|r-r |min aikHC or inside ∇(εi(r1, r) ∇φiδ(r1, r)) ) -qiδ3(r-r1)/ε0, |r - r1 |a 1
{
∇(εk(r′, r)∇φk(r′, r)) ) -qkδ (r - r′)/ε0, ∇(ε(r)∇φ0k (r′, r))
ε j
) -qkδ (r - r’)/ε0, and
Dk(rk, r) )
∫inside dφF(r′, r) + ∑ ∫|r-r |>a 1
HC ik
(1 -
∑ ∫|r-r|aiHC
-qiδ3(r - r1)/ε0, outside, |r - r1 | aHC ˜ i(r1,r) - ψ )ψ i :
sik(r1, r) φi(r1, r) +
2
k
ik
k
1
k
∑ (1 - s (r , r)) n (r)q ik
1
k
k
k
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