Computational Studies of Water-Exchange Rates around Aqueous

Jun 27, 2014 - The water-exchange mechanisms occurring around aqueous divalent Mg2+ and Be2+ ions were studied using molecular dynamics simulations an...
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Computational Studies of Water-Exchange Rates around Aqueous Mg2+ and Be2+ Liem X. Dang Physical Sciences Division, Pacific Northwest National Laboratory, Richland, Washington 93352, United States ABSTRACT: The water-exchange mechanisms occurring around aqueous divalent Mg2+ and Be2+ ions were studied using molecular dynamics simulations and rate theory methods. Properties associated with the waterexchange process, such as ion−water potentials of mean force, time-dependent transmission coefficients, and rate constants, were examined using transitionrate theory and the reactive flux method, which includes the role of solvent friction. The effects of pressure on water-exchange rates and activation volumes also were studied. The simulated activation volume values and mechanism were different for Mg2+ and Be2+ because of the nature of their solvation shells. We found the agreement with experiments was improved when solvent effects were taken into account.

I. INTRODUCTION The kinetics and thermodynamics of water exchange around aqueous ions have been subjects of considerable theoretical and experimental interest because of their importance in many chemical and biological processes.1−10 Studies have shown that the mechanisms governing the exchange process can be characterized in terms of activation volume (ΔV‡). For example, the exchange process is an associative mechanism when the value of ΔV‡ is negative and a dissociative mechanism when the value of ΔV‡ is positive.9,10 Recently, we reported on two studies of ΔV‡ for aqueous Li+ and aqueous halides (Cl−, Br−, and I−).11,12 In both of these studies, the inclusion of solvent significantly affects the magnitude of the ΔV‡. For example, the computed ΔV‡ for the water-exchange process around aqueous alkali halides is negative using transition-state theory (TST) approximations, and it is positive when rates were corrected using the reactive flux (RF) method.13 We extend our molecular dynamics (MD) simulation studies performed on extended empirical models in which solvent molecules are treated explicitly using polarizable potential models. We present our MD and rate-theory results on waterexchange mechanisms around aqueous Mg2+ and Be2+. These results include ion−water potentials of mean force (PMF) and the corresponding rate constants obtained using TST and the RF method, including treatments of the dynamic response of the solvent.13 Pressure effects on these aqueous systems are of particular interest because of their relevance in determining ΔV‡ aqueous chemistry. The main reason we are studying Mg2+ and Be2+ systems is the availability of experimental data on activation energies, rate constants, and ΔV‡.9,10 This information will help us improve our approach and potential models. In addition, divalent ions such as Mg2+ and Ca2+ ions are ubiquitous in nature and also are well-known for their crucial roles in many biochemical © 2014 American Chemical Society

functions. For example, they can catalyze or enhance the reaction rate in RNA enzymatic reactions and the hydrolysis of adenosine triphosphate.14 Beryllium has no biological role, and its high toxicity probably results from the ability of Be2+ to displace Mg2+ from Mg-activated enzymes because of its stronger coordination.15 Given the ubiquitous use of Mg2+ in biology for structural stabilization and catalysis and its presence in a wide range of environmental substances, there have been few reports that focused on providing a fundamental understanding of the solvation structure of magnesium and beryllium in aqueous solutions. Ikeda and coworkers studied solvation structures of the divalent metal cations Mg2+ and Ca2+ in ambient water by applying a Car−Parrinello-based constrained MD method.16 That work provided detailed information about the first hydration shell as well as the activation free-energy profile of the exchange process. In another study on Mg2+solvation, Callahan et al. reported the PMF for the Mg2+−H2O complex and estimated the residence time for water in the first solvation shell of Mg2+ using TST to be 9.2 μs.17 Our work differs from early contributions in two main ways: we have made corrections to the TST and have characterized the influence of these changes on the rate results and ΔV‡. The remainder of our paper is organized as follows. In Section II, the potential models and simulation methods are described. Results and discussion are presented in Section III, and our conclusions are presented in Section IV. Special Issue: John C. Hemminger Festschrift Received: April 2, 2014 Revised: June 26, 2014 Published: June 27, 2014 29028

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II. SIMULATIONS AND METHODS Polarizable force-field parameters for Mg2+ (α = 0.094 Å3, σ = 2.022 Å, ε = 0.1 kcal/mol, and q = 2e) and Be2+ (α = 0.008 Å3, σ = 1.202 Å, ε = 0.1 kcal/mol, and q = 2e) were parametrized to reproduce experimental hydration enthalpies of aqueous ions.18 In this particular study, we employed the RPOL water model developed by Dang.19 It is a revised version of the model published by Caldwell et al.20 It has three interaction sites and the same rigid tetrahedral geometry as simple point charge (SPC) models. There are fixed-charge and point polarizabilities attached to each atomic site, along with a Lennard-Jones (LJ) interaction centered on the oxygen. Each ion in the aqueous solution simulations is represented similarly by a fixed-charge plus point polarizability and a LJ term. Ion−water cross terms are defined using the usual sum rules (e.g., Bertholet−Lorentz mixing rules). In Figure 1, we show computed ion−water radial

F (r ) =

1 ⟨ ru⃗ •(FA⃗ − FB⃗ )⟩ 2

(1)

In this expression, FA and FB are the forces acting on the solutes. The term ru⃗ , which is a unit vector along the AB direction, is defined by eq 2 ru⃗ = rAB ⃗ /|rA − rB|

(2)

and the PMF, W(r), is calculated by eq 3 W (r ) = −

∫r

o

rs

⟨F(r )⟩ dr

(3)

We evaluated PMFs along the center-of-mass separation between the ion and water, with the center of mass separation between the ion and water incremented by 0.1 Å. At each center of mass separation, the average F(r) was determined from a 2 ns simulation time, preceded by a 500 ps equilibration period. The uncertainty for the PMF was ±0.05 kcal/mol, as estimated by determining the force averaged (the corresponding PMF) over four equally spaced time frames during production. In principle, one can extract the PMF from the pair correlation function (i.e., ln g(r)). However, in some cases, we and others have found that for systems such as the Li+− H2O, Mg2+−H2O, and Be2+−H2O the first minimum of the computed g(r) was not that well-defined (i.e., the statistical uncertainty of the radial distribution function in the region between the first and second hydration shells makes it difficult to obtain a reliable PMF).1−3 We found the approach of computing the full mean force and the corresponding PMF provides a smoother description of the transition state of the free-energy profile. This finding, in turn, is important in calculating the rate constants and the related kinetics properties. The systems investigated included Mg2+/Be2+ and 550 water molecules. All simulations were performed in an NVT ensemble at 300 K, with periodic boundary conditions applied in all three directions with a time step of 2 fs. In this study, we used the Langevin dynamics approach (option ntt = 3 and gamma_ln = 5.0 ps−1 in Amber 9) to monitor the temperature in the NVT ensemble. In addition, for the calculations of timedependent transmission coefficients, we used the NVE ensemble (option ntt = 0) to carry out MD simulation; the computed final average temperature of the system is ∼298 K. We monitored our simulations closely, and we did not notice any unusual behavior of the system. To estimate the ΔV‡, we performed the system studies at 0, 1, and 2 kbar, which correspond to cubic box lengths of 25.34, 25.05, and 24.80 Å, respectively. The Ewald summation technique was used to handle long-range electrostatic interactions,23 and the SHAKE algorithm was used to fix the internal water geometry.24

Figure 1. (a) Computed RDFs for the aqueous Mg2+ at 300 K and 1 atm. (b) Computed RDFs for the aqueous Be2+at 300 K and 1 atm.

III. RESULTS AND DISCUSSION III.A. Mg2+−H2O. We begin this section by presenting the results for Mg2+−H2O and Be2+−H2O results that will be followed. The computed PMF for the Mg2+−water pair at 300 K 0 bar is shown in Figure 2. Closely examining the PMF, we observed that there is a barrier height to dissociation of ∼9.75 kcal/mol with the transition distance at 3.075 Å. This value is closely comparable to the estimated activation energy of 9.5 kcal/mol at 300 K for water exchange in Mg2+(H2O)6 obtained from 17O nuclear magnetic resonance (NMR) spectroscopy experiments.25 In addition, our barrier height also is very similar to the simulation study reported by Callahan et al.17 using

distribution functions (RDFs) for Mg2+−H2O and Be2+−H2O; the computed coordination numbers 4 and 6 for Be2+ and Mg2+, respectively, are in good agreement with results from other simulations and with X-ray diffraction data.21 We used a modified version of the Amber 9 software package to perform all MD simulations.22 The mathematical expressions used to calculate the ion−water mean force as an average over the different solvent configurations are presented. 29029

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Figure 3. Computed PMF for the Mg2+−H2O system as the function of pressure at 300 K.

Figure 2. Computed PMF for the Mg2+−H2O system and its components at 300 K.

∗

k

similar approaches and potential models. The entropic and enthalpic contributions to the free energy also were computed using the finite difference method, which is based on the following relationship.26 ⎛ ∂ΔAij ⎞ = −ΔSij ⎜ ⎟ ⎝ ∂T ⎠V , N

(6)

⎛ ∂ ln(k) ⎞ ⎟ ΔV ‡ = −RT ⎜ ⎝ ∂P ⎠T

(4)

(7)

where ΔV‡ is the activation volume, T is the temperature, R is the gas constant, k is the rate constant, and P is the pressure. An approximate solution to the above equation is given by ⎛ kp ⎞ P ln⎜ ⎟ = −ΔV ‡ RT ⎝ k0 ⎠

ΔAij (T + ΔT ) − ΔA(T − ΔT ) 2ΔT

=

k bT (r ∗)2 e−βW (r ) ∗ 2πμ ∫ r r 2e−βW (r) dr 0

where r*is defined as the position of the barrier top, μ is the ion−water reduced mass, kb is the Boltzmann constant, and T is the temperature. Pressure dependence of the rate constant at constant temperature is given by following relationship10

The term ΔAij is the Helmholtz free-energy difference between the states i and j, and ΔSij is the entropy difference between the states. Using the above equation and performing the free-energy simulations at different temperatures, one can compute the difference in entropy using the following relationship −ΔSij =

TST

(5)

(8)

We use eq 8 to compute the activation volumes where kp and k0 are the rate constants at pressures P and 0. Using the computed PMFs and the transition-state distances, we computed the rate constant, kTST, for the exchange process using eq 6; the results for kTST are 1.84 ± 0.02 × 10−6/ps, 1.68 ± 0.02 × 10−6/ps, and 1.57 ± 0.02 × 10−6/ps at pressures of 0, 1, and 2 kbar, respectively. Thus, kTST decreases with increasing pressure, and this trend would be expected by examining at the PMF barrier heights. Our computed kTST under ambient conditions is about three times larger than that determined from the results obtained from the 17O NMR experiments of 5.3 × 10−7/ps and 6.7 × 10−7/ps.27 The computational results are summarized in Table 1. Using the computed kTST, a positive ΔV‡ value of 1.9 ± 0.3 cm3/mol was obtained using eq 7. This result showed that an exchange process with an increasing pressure and an increasing free energy at the barrier would lead to a positive activation

We chose ΔT = 20 K and performed two more sets of freeenergy calculations at 280 and 320 K. The obtained free-energy profiles at ΔT and −ΔT are substituted into eq 5 to get the entropic contribution along the separation at 300 K. Also in Figure 2, along with the PMFs, we show the entropic and enthalpic components. The results show that the enthalpic contribution is significant and added entropy only partially compensates it. Our estimated values for enthalpic and entropic components at the transition state are 10.5 and 0.7 kcal/mol; these values can be compared with experimental data of 11.7 and 2.2 kcal/mol, respectively, which was reported by Bleuzen et al.25 The different between two approaches can be attributed to the sensitivity in the finite differences method. Figure 3 shows the computed PMFs obtained at 300 K for the three pressures, normalized to the contact Mg2+−water pair free-energy minimum at 0 bar. As expected, the shapes of the computed PMFs are very similar because of strong Mg2+−water interactions, and the changes are rather small. We observe two effects: (1) increasing the pressure destabilizes the contact ion− water pair and (2) the free-energy barrier for escaping the first hydration shell increases from 9.75 ± 0.05 kcal/mol at 0 bar to 9.83 ± 0.05 kcal/mol at 2 kbar. These small changes imposed no change in the transition-state distances. For a given PMF, the rate constant for the exchange process can be computed using TST as follows

Table 1. Computed Activation Barriers and Rate Coefficients As a Function of Pressure at 300 K of Aqueous Mg2+ pressure (bar) 0 1000 2000 ΔV‡ 29030

barrier height (kcal/mol) 9.75 9.80 9.83

kTST (ps−1) −6

1.84 × 10 1.68 × 10−6 1.57 × 10−6 1.9 cm3/mol

κRF 3.56 × 10−2 2.34 × 10−2 2.17 × 10−2 6.0 cm3/mol

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both the barrier height and the transmission coefficient contribute to the activation volume. III.B. Be2+−H2O. The computed PMF for the Be2+−water pair at 300 K and 0 bar is shown in Figure 5. A barrier height to

volume (i.e., a dissociative mechanism). It is not unexpected that this value significantly underestimates the 17O NMR study of 6.7 ± 0.7 cm3/mol on the same Mg2+(H2O)6 system. It is well-established that TST can overestimate the values of rate constants because of the assumption that, once the reactive species reach the transition state, they will directly end up on the products side without recrossing to the reactants side. The RF method and GH theory are well known, and popular corrections to the TST that have been reported in the literature.1,13 In the RF method, the transmission coefficient, κRF, is extracted from the plateau value of the time-dependent transmission coefficient, as calculated using the following equation k(t ) =

⟨r(0) ̇ θ[r(t ) − r *]⟩c ⟨r(0) ⟩c ̇ θ[r(0)] ̇

(9)

where θ(x) is the Heaviside function, which is 1 if x is larger than 0 and 0 otherwise, and ṙ(0) is the initial ion−water velocity along the reaction coordinate. The subscript c means the initial configurations have been generated in the constrained reaction coordinate ensemble. To compute the time dependence transmission coefficient, we generated a series of starting configurations by running a simulation in which the distance between the Mg2+ and a selected water molecule was constrained to be the transition-state distance. A 10 ns simulation was carried out, and a configuration was collected every 4 ps to obtain 2500 configurations. Then, each configuration was run both backward and forward for 2 ps with a NVE ensemble, and the value of κRF was determined by averaging k(t) over the last 0.5 ps of the rate constants. In Figure 4, we present the computed time-dependent k(t) at three different pressures. From the data, we observed that the

Figure 5. Computed PMF for the Be2+−H2O system at 300 K.

dissociation of ∼16.5 kcal/mol at a transition distance of 1.67 Å was observed. The dissociation value is comparable to the activation energy of 13.6 kcal/mol at 300 K for water exchange in Be2+(H2O)4 estimated from results from the 17O NMR experiments. 28 Furthermore, this simulated value of Be2+(H2O)4 is significantly greater than the corresponding barrier height for the Mg2+−water system; this result would be expected because the beryllium ion is significantly smaller and its hydration energy is much larger than that of Mg2+. We carried out additional PMF calculations at 300 K and pressures of 1 kbar and 2 kbar to determine ΔV‡. The shapes of the computed PMFs are very similar because of the strong Be2+− water interactions, and the changes are rather small. The freeenergy barrier for escaping the first hydration shell decreases from 16.5 ± 0.05 kcal/mol at 0 bar to 16.2 ± 0.05 kcal/mol at 2 kbar. These small changes imposed no change in the transitionstate distances. Using the computed PMFs and the transitionstate distances, we calculated the rate constant, kTST, for the exchange process; the results are 3.30 ± 0.02 × 10−11/ps, 4.23 ± 0.02 × 10−11/ps, and 4.96 ± 0.02 × 10−11/ps at 0, 1, and 2 kbar, respectively. Thus, kTST increases with increasing pressure, and this trend would be expected by examining the PMF barrier heights. A negative ΔV‡ value of −5.1 ± 0.3 cm3/mol was obtained using eq 7 as compared with the experimental value of −13.6 cm3/mol on the same Be2+(H2O)4 system.28 This result showed that an exchange process with an increasing pressure and a decreasing free energy at the barrier will lead to a negative activation volume (i.e., an associative mechanism). These results are presented in Table 2. In Figure 6, we present the computed time-dependent k(t) at three different pressures, and we observed that the rate constants decrease as pressure increases. The transmission

Figure 4. Pressure dependence of the transmission coefficients, k(t), of Mg2+−H2O system.

Table 2. Computed Activation Barriers and Rate Coefficients as a Function of Pressure at 300 K of Aqueous Be2+

rate constants decrease as pressure increases. The transmission coefficients, kRF, estimated as outlined above are 3.56 ± 0.05 × 10−2, 2.34 ± 0.05 × 10−2, and 2.17 ± 0.05 × 10−2. The simulated ΔV‡ using the computed kRFkTST is 6.0 ± 1.0 cm3/ mol, which is greater that the value extracted using kTST. This value also is in very good agreement with results from the 17O NMR study of 6.7 ± 0.7 cm3/mol on the same Mg2+(H2O)6 system. Thus, we conclude that the pressure dependence of

pressure (bar) 0 1000 2000 ΔV‡ 29031

barrier height (kcal/mol) 16.50 16.36 16.21

kTST (ps−1) −11

3.30 × 10 4.23 × 10−11 4.96 × 10−11 −5.1 cm3/mol

κRF 1.06 × 10−1 1.14 × 10−1 1.23 × 10−1 −7.0 cm3/mol

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Our future research efforts will focus on understanding the influence of nuclear quantum-mechanical effects on the properties of ions in aqueous solution such as the waterexchange rate and the corresponding transmission coefficients using ring polymer MD techniques.30

■ ■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS The Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences (BES) of the U.S. Department of Energy (DOE) funded this work. Battelle operates Pacific Northwest National Laboratory for DOE. The calculations were carried out using computer resources provided by BES.

■

Figure 6. Pressure dependence of the transmission coefficients, k(t), of Be2+−H2O system.

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coefficients, kRF, estimated as previously outlined are 1.06 ± 0.05 × 10−1, 1.14 ± 0.05 × 10−1, and 1.23 ± 0.05 × 10−1. The simulated ΔV‡ using the computed kRFkTST is ∼7.0 ± 1.0 cm3/ mol, which is a bit greater that the computed value using kTST, and agreement with results from the 17O NMR study is somewhat improved.28 These results are also summarized in Table 2.

IV. CONCLUSIONS In summary, to enhance our understanding of the mechanism of water exchange around aqueous Mg2+and Be2+, we carried out systematic studies using MD simulations and rate theory methods. Our work differs from early contributions in two main ways: we have made corrections to the TST and have characterized the influence of these changes on the rate results and ΔV‡. To complete this activity, we computed most of the properties associated with the water-exchange process, such as ion−water PMFs, time-dependent transmission coefficients, rate constants, and activation volumes. We found the agreement with experiments was improved when solvent effects were taken into account. The mechanisms of water exchange around aqueous divalent Mg2+ and Be2+ ions were found to be different because of the nature of their solvation shells. In this particular study, we employed the RF to calculate the transmission coefficient. One of the major approximations in this procedure is the convergence of the plateau of the k(t) in the last 0.5 ps. In our experience, convergence was achieved quite easily for ion−water systems such as the Li+−H2O, Mg2+−H2O, and Be2+−H2O. For systems with lower barrier exchanges such as halide−water systems, the convergence is quite difficult to attain, and averaging it over many runs is required. Moreover, the water exchange rates can be estimated directly from MD simulations using the residence time method developed by Impey and coworkers.29 However, this method recently has been called into questions because of its sensitivity to the t* tolerance time value (designed to account for barrier recrossing effects). In addition, in some cases, it can yield overestimated residence times.8 In our experience, the RF approach employed in the present study is as solid and reliable as other methods in computing the transmission coefficient available in literature. 29032

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