J. Phys. Chem. 1994, 98, 9422-9430
9422
A Molecular Dynamics Study of Valinomycin and the Potassium-Valinomycin Complex Timothy R. Forester,*Jgs William Smith,t98 and Julian H. R. Clarkeb3s Daresbury Laboratory, Warrington WA4 4AD, U.K., and Department of Chemistry, UMIST, Manchester M60 IQD, U.K. Received: April 20, 1994; In Final Form: July 5, 1994@
An adapted AMBER force field for valinomycin is presented and used to study valinomycin and its potassium complex in a series of molecular dynamics simulations. The force field faithfully models the crystallographic structures of both valinomycin and the potassium complex. Isolated molecule simulations at 100 K and temperatures between 293 and 370 K are reported for both valinomycin and the potassium complex. In these simulations a stochastic thermal bath is used as a first approximation to including solvent interactions. Simulations of uncomplexed valinomycin yield the experimental dipole moment of the molecule in carbon tetrachloride. Structural and dynamical properties of the systems studied are presented and discussed.
1. Introduction Valinomycin, (L-Val-D-Hyl-D-Val-L-Lac)3, is a fascinating biopolymer of considerable biological and biotechnological importance. A cyclododecapsipeptide, it is a natural “cation carrier” for transport of potassium across cell membranes. This capability is a consequence of its unique chemical structure, and there are significant changes in the stereochemistry of valinomycin upon complexation with potassium ions. In this article we describe the development of an adapted AMBER force field which correctly reproduces most of the features of the crystal structures of both the uncomplexed and the chelated molecule. As a prelude to more complete calculations in an explicit solvent we also describe some long simulations (up to 1 ns) of the isolated molecule in a stochastic thermal bath at ambient temperatures in order to explore some of the structural fluctuations. In a future article we will describe studies of the mechanism of the complexation reaction. The structural and dynamical properties of valinomycin are ideal candidates for investigation by computer simulation. Molecular dynamics (MD) calculations are capable of providing detailed information on both structural and time-dependent properties of molecular systems at finite temperatures.’ As a simple ionophore, valinomycin has been the subject of numerous energy minimization studies;*however, there have been surprisingly few MD studies of this important system. The most significant studies have been by Eisenman and co-workers who have studied the relative free energies of alkali metal complexes in vacuo (100 K)384 and in methan01.~ These authors were primarily concerned with structures close to the crystal structure of Neupert-Laves et aL6 In addition, Shobana and Vishveshwara have published a 40 ps simulation of uncomplexed valinomycin in vacuo at 100 K.’ In this paper we describe our own investigations of valinomycin and the potassium-valinomycin system. We begin with the development of a new set of partial charges for valinomycin from ab initio calculations. Our work differs from previous studies in that the force field we proposed is validated by simulation of the periodic crystals of both uncomplexed valinomycin and the potassium-valinomycin complex. Isolated
* Address correspondence to this author.
’ Daresbury Laboratory.
* UMIST.
8 Email address: T. R. Forester,
[email protected];W. Smith,
[email protected] J. H. R. Clarke,
[email protected]. @Abstractpublished in Advance ACS Abstracts, August 15, 1994.
I
I
8 8
CH3 CH
cH3 CH3
I CH3
3
I
I
I
8 8 8
CH3 CH3 CH3CH3 CH3 CH3
Figure 1. A schematic diagram of valinomycin. See text for definition of torsional angles.
molecule simulations have also been performed at ambient temperature as well as at 100 K. A significant aspect of these simulations is their length. Such precautions were deemed necessary since it is recognized that insufficient statistical sampling occurs in similar systems (e.g., crown ethers8s9)on a subnanosecond time scale. A schematic diagram of valinomycin is shown in Figure 1. The molecule consists of three identical segments linked to form a single ring. Around the ring there are six amide groups and six ester groups arranged alternately. Attached to the ring are nine isopropyl and three methyl side groups. The potassium complex (Figure 2a) crystallizes into an approximately spherical structure in which the molecule folds into a “tennis ball seam” arrangement around the cation6 Other alkali cations form similar complexes. The cation sits at the center of the structure while the side groups reside on the outside producing a hydrophobic exterior which facilitates the passage of the complex through the interior of a lipid membrane. The complex is also stabilized by the presence of six hydrogen bonds, each one involving an amide proton and the carbonyl of an adjacent amide group. When the cation is absent, valinomycin can be crystallized from apolar and medium polar solvents into a “twisted bracelet” conformer (Figure 2b) with six internal hydrogen bonds.l0 However, when crystallized from a polar solvent such as DMSO, valinomycin crystallizes into an open “dish-shaped” conformation with three of the amide hydrogens acting as hydrogen bond donors to the solvent rather than to an adjacent amide carbonyl.” In this structure, the ester carbonyls are not involved in internal hydrogen bonds. Spectroscopic measurements indicate that the same “dish-shaped” conformation predominates in polar solu-
0022-3654/94/2098-9422$04SO~O 0 1994 American Chemical Society
Valinomycin and Potassium-Valinomycin Complex
J. Phys. Chem., Vol. 98, No. 38, 1994 9423
t
-s
N-1 N
3[4.[iti’;
Figure 2. Crystal structures of valinomycin. (a) The potassium
complex from ref 6. The potassium is in the centre of the structure chelated by the six ester carbonyl groups. (b) The uncomplexed molecule from ref 10. tions such as DMS0,12 although the data have also been interpreted in terms of a “propeller” conformation with threefold ~ y m m e t r y . ’ ~In J ~any event complexation with potassium has not been observed in such solvent^.'^ Spectroscopic measurements of uncomplexed valinomycin in vivo (in the lipid membrane) and in apolar solution are very similar, implying the same conformers predominate in both media and that apolar solvents provide a reasonable but simple model for the lipid membrane environment. In chloroform and lipid bilayers, FTIR spectral5 are consistent with a conformation in which all six amide carbonyl groups are internally hydrogen bonded while the ester groups are not. The line widths in the ‘HNMR spectra in a phospholipid bilayer16 are broad (20-40 Hz) indicative of considerable molecular motion but consistent with a “prebinding” structure with the same hydrogen bonding pattern as the alkali complex. Spectra for the apolar solvent and the lipid bilayer conformations of the alkali complex are consistent with the “tennis ball seam” conformation found in the solid state. Despite some conjecture in the literature, little is actually known about the mechanism of the formation of the cation complex and the actual conformers involved in the complexation process. In the following section we describe the derivation of the valinomycin force field and the computational techniques we have employed in our simulations. In section 3 we present the results of the simulations and relate these to the experimental data. In section 4 we provide our conclusions.
2. Computational Procedures 2.1. Derivation of the Force Field. Our force field for the valinomycin molecule is based on the all-atom version of the AMBER force field,17though with some significant adaptations. The general potential function is of the form
-
A
B
---
+
(3+-]
(l)
Hydrogen bonding is accounted for by the 12-10 potential while other short-range nonbonded interactions are treated with the Lennard-Jones 12-6 potential. The final term of eq 1 accounts for the electrostatic interactions. Parameters for the bond angle, dihedral angle, and van der Waals terms were taken directly from the AMBER force field; however, rigid bond constraints were used for all formally bonded atom pairs so that the first term in the general potential is redundant. Adaptations to the AMBER force field were required for the following reasons. First, Lennard-Jones parameters for unsolvated potassium were required in place of the implicitly solvated ion values found in the standard parameter set. We used values of B = 3.13 A, E = 1.356 kJ mol-’ for potassium and applied the standard Lorentz-Berthelot mixing rules’ to obtain the parameters for interactions between potassium and other atoms. When used with the SPC model of water18 the potassium parameters we used gave good agreement with the height and positions of the first peaks in the pair distribution functions g K , O ( r ) and gK.H(T) reported by Bounds1g in his simulation of potassium in water. (Note that in valinomycin three types of oxygen are distinguishable and for the rest of this paper we mark them with the following notation: “ 0 ’ is the amide carbonyl oxygen; “Oe)’is the ester carbonyl oxygen and; “Os” is the bridging oxygen of the ester linkage.) Second, the standard AMBER force field did not provide entries for ester groups and therefore an extension for the bond angle and dihedral terms involving the bridging oxygen (OS) was required. The additional bond angle and dihedral terms were taken from the ether parametrizations of Cruzeiro-Hansson et aLZ0and Pedersen et aLZ1 These nonelectrostatic parameters are summarized in Table 1.
TABLE 1: Additional Parameters for the All-Atom AMBER Force Field of Valinomycin ks = 70” eo = 123.00 0-c-os CT-C-Os ke = 70’ eo = 117.00 ke = 70t eo = 109.50 C-CT-Os ke = 100t eo = 116.40 C-Os-CT x-c-os-x V,/2 = 4.6b y = 180”,n = 2 kcal/mol/rad2. kcaymol. The third modification, and the most important, was the production of a set of appropriate partial charges for the molecule. It is important that this was done in a manner consistent with the derivation of the standard AMBER charges. Thus we conducted SCF calculations at the ST03G level in accordance with the method of Kollman et al.17 The quantum chemistry package GAMESSZ2was used for these calculations. Atomic partial charges were derived which best represented the electrostatic potential on a number of potential energy surfaces, generated from 1.4, 1.6, 1.8, and 2.0 times the van der Waals atomic radii. This “potential surface derived partial charges”
Forester et al.
9424 J. Phys. Chem., Vol. 98, No. 38, 1994 TABLE 2: Partial Charges in Valinomycin
AMBER type
atom
de
Amide N H C 0 C, H
N H C 0 CT HC
0,
os
C 0, CLl H
C 0 CT HC
C H CWe) HW)
CT HC CT HC
C H
CT HC
-0.401 0.21 1 0.477 -0.405 -0.073 0.058
Ester -0.455 0.692 -0.416 0.272 -0.009
Isopropyl -0.0183 0.024 -0.139 0.0473
Methyl -0.1495 0.058
(PSDPC) procedure was first checked on glycine and toluidine fragments and showed excellent agreement with the standard AMBER charges. Similar calculations on linear polyglycine fragments with adjacent carbonyl groups arranged in the anti conformation were then performed. Two calculations were performed on each fragment: one on the standard peptide (fragment “N’));and another on a substituted peptide in which the central amide linkage had been replaced by an ester linkage (fragment “0’).It was reasoned that with a sufficient number of units, fragment “0” would produce a reasonable model of the ester linkage in valinomycin. We found it necessary to use a tetrapeptide fragment in order to achieve acceptable agreement in the PSDPC’s for the terminal atoms of the two fragment forms. Using the tetrapeptide fragment as the reference, partial charges for the valinomycin molecule were assigned as follows. The ester group atoms, and atoms in alkyl groups either side of the ester group, were assigned charges from the central portion of fragment “ 0 ’ . These were the atoms whose PSDPC were the most influenced by the ester linkage. The amide group atoms were assigned charges from the central amide group of fragment “N’. This assignment left a net charge of 0.0245 e to be taken by each alkyl residue (the isopropyl and methyl side groups). In the standard AMBER force field each alkyl residue carries at net charge of 0.016 e; thus, to assign charges for the residues in valinomycin the standard AMBER charges were scaled by a factor of 1.53. The complete partial charge assignments for valinomycin are given in Table 2. It transpires that the partial charges on the carbonyl oxygens that we have derived are quite close to those originally derived by Eisenmann et as those that best reproduce the ion-selectivity pattern for isolated valinomycin. While it is difficult to make simplisitic comparisons between force fields (Eisenman et al. used the GROMOS and MOLARIS force fields) it is gratifying that some correspondence exists with out partial charge set. We stress that the ion-selectivity sequence for valinomycin was not used in parametrizing our partial charges nor is it the subject of this study. The charges we have derived do not take account of polarization effects arising from the presence of a cation. This may be particularly important when the ion-valinomycin complex is embedded in polar solvents. For example, Aqvist et al. found that a larger carbonyl dipole moment was required to reproduce the correct ~
1
.
~
3
~
ion-selectivity sequence when the alkali complex was placed in methan01.~ However, it is not clear that such ad hoc adjustments of the partial charges are consistent with the overall parametrization of the force field and they are certainly inappropriate for the uncomplexed molecule. 2.2. Molecular Dynamics Simulations. All the molecular dynamics simulations reported in this paper were carried out using the parallel macromolecular simulation package DL-POLY25-26 developed at Daresbury Laboratory and were run on an INTEL iPSC/860 parallel computer and Hewlett Packard Series 9000 Model 735 workstations at Daresbury Laboratory. The equations of motion were integrated using the Verlet leapfrog scheme coupled to a NosC-Hoover thermostat2’ with a thermostating rate of 10 ps-’. The rigid bond dynamics were handled with a parallel version of the SHAKE algorithm28 known as RD-SHAKE26with a relative tolerance in dz of The MD time step was 2.5 fs except in the simulation of the potassium-valinomycin crystal where a time step of 1.0 fs was used. In the solid-state simulations electrostatic terms were handled using the Ewald sum, coverged within a relative error of in conjunction with a multiple time step algorithm.29 In the nonperiodic isolated molecule simulations a cutoff of 20-30 8, was applied to the nonbonded atom-atom potential energy functions. This was sufficient to ensure that all atom pairs were included within the nonbonded cutoff sphere. These simulations also employed a stochastic thermal bath as a first approximation to including solvent interactions. In a future article we hope to report the effect of explicit solvent. In the stochastic thermal bath modelz3 the motion of valinomycin corresponds to free propagation of the equations of motion in between instantaneous collisions which occur with an average frequency v and whose incidence corresponds to a Poisson process. Each collision consists of replacing the velocities of a subset of atoms by new ones sampled from a Boltzmann distribution. The stochastic thermal bath is a particular version of Andersen’s constant temperature m e t h ~ d ~ Oand , ~ lyields static properties corresponding to a canonical distribution at constant tem~erature?~It has the advantage over Brownian dynamics’ in that the diffusion coefficients of valinomycin and potassium in the lipid membrane are not required by the algorithm. In valinomycin the sp3 hybridized carbons reside primarily on the exterior of the molecule and were thus the subset of atoms subject to the stochastic collisions. The collisions where implemented at a mean rate of v = 1 ps-’. 2.3. Structural Properties. We have used several measures of local structure to characterize the simulations and to facilitate comparison with X-ray data. The first set of measures are the standard partial radial distribution functions (PRDFs), g,p(r), which describe interatomic separations. The second are the dihedral angles around the ring. The ring has 12 unique types of dihedral angle which we label as a1, ,&, y1, 61, 61, 51, a2, ,&, y2, 8 2 , €2, and 52. The central bond for each dihedral is indicated in Figure 1. Thus al, for example, refers to the N-C(i-Pr)-C(O,)-Os dihedral angle with y2 refers to the C(O,)-Os-C(Me)-C(O) dihedral angle. The dihedral potential term is symmetrical about 0” and 180” so we have reduced histograms of the distributions to be in the range [0”,180”].This, in effect, removes stereochemical information from the distributions. We have also measured some aspects of the global structure of the molecule. The first was the asphericity order parameter, A3, of Rudnick and G a ~ p a r i ~ ~
. I . Phys. Chem., Vol. 98, No. 38, 1994 9425
Valinomycin and Potassium-Valinomycin Complex
TABLE 3: Potassium Complex Crystal Properties (293 K) g ( r ) Fist Peak Position (A)
X-ray
MD
i=l
where (...) denotes an ensemble average and the i l i are the three principal moments of inertia of the valinomycin molecule. The principal moments of inertia are obtained by diagonalization of the inertia tensor T, the components of which are given by
K-0,
K-0
K-Os
C-K
N-0
0-H
2.76 2.80
4.80 4.81
4.81 4.90
3.88 3.95
2.91 2.88
2.03 2.00
al
a2
Mean Dihedral Angles (den) B1 BZ y1 y2 61 8 2 el
€2
C1
52
X-rav 132 131 175 176 82 75 6 18 177 178 58 58 MD' 130 126 174 172 71 81 28 15 171 170 67 60 Rms Deviations from Crystal Structure Positions (A)
,\-
i= 1
where a and p (x, y, or z), m, is the mass of the ith particle, rr is the a component of the position vector of particle i, rzamis the a component of the center of mass of valinomycin, and iris 168 - the number of atomic sites in valinomycin. A3 is 0 for a perfectly spherical object, and 1.0 for an infinitely thin rod. The second measure of global structure used was the mean squared radius of gyration, defined as I
(4) where M is molecular mass. 2.4. Bonding Correlations. The persistence of individual (intramolecular) hydrogen bonds can be monitored through the autocorrelation of a function H(t) in which H(t) = 1 if a given hydrogen bond was present at time t and had been in continuous existence since t = 0; and is 0 otherwise.33 The autocorrelation function
therefore describes the mean decay of the hydrogen bonds with increasing time and the integral corerlation time, z = &'C(t) dt, gives the mean persistence time of the bonds. Whereas the PRDFs give a static average of interatomic separations, the "bonding" functions give ensemble averages of the dynamical properties of hydrogen bonds. We have used a simple geometric criterion to define the hydrogen bonds: the H-O distance had to be less than 3.0 8, for a bond to be classified as present. The persistence time was inevitably sensitive to the cutoff criterion used; however, the value we have used is in keeping with intermolecular hydrogen bonding studies in other biopolymer systems34and was large enough to exclude small fluctuations in local configurations resetting the values of H(t). The coordination of specific carbonyl oxygens to the cation was monitored through an analogous set of functions. The cutoff distance for the K - 0 bonding function was set near the minimum between the first two peaks in the PRDF gK.O.(T) viz. at 3.5 A. 3. Results and Discussion
3.1. The Solid-state Potassium Complex. The crystalline potassium complex was simulated at 293 K in the canonical (NVT) ensemble. The system was allowed to equilibrate for 5 ps and statistical data collected over the following 10 ps of
CT
C
N
Os
0,
0
K
I
0.17
0.15
0.21
0.23
0.20
0.26
0.33
0.22
simulation. As in all the simulations reported here, all covalent bonds were kept of fixed length by use of the SHAKE algorithm. To our knowledge this is the first reported simulation of the crystalline state of this molecule yet this is a key contact between simulation and experiment and provides an appropriate test of the force field. Comparison of simulation results with X-ray data is made in Table 3. The initial configuration for potassium-valinomycin complex was taken from the crystallographic data of Neupert-Laves et aL6 The unit cell is orthorhombic with dimensions a = 13.342 A, b = 24.648 A, and c = 46.961 A. Two unit cells, stacked in the a direction, were used as the basic MD cell. This was to allow use of a reasonable (11 8,) short-range cutoff. The resultant MD cell contained 16 [potassium-valinomycin]+ complexes, 8 Is- ions and 8 13- ions. The rms deviation of atomic sites from the X-ray structure is 0.19 8,. The largest deviations are associated with the potassium (0.33 A) moving around the valinomycin cage (Table 3). In keeping with the small rms deviations in position the simulation also reproduces most of the key interatomic distances to within the 2 0 confidence limit of the X-ray data (0.04 8,). The only exception is the C-K distance where the difference between the experimental and simulation mean separations is 0.07 8,. This reflects a slightly more linear K-0,-C angle in the simulation. Some angle straightening is anticipated in a partial charge model as anisotropy in the electron density of the carbonyl oxygens (due to lone pairs) is not accounted for. The simulation also provides excellent agreement with X-ray estimates of the dihedral angles. The mean dihedral angles from the simulation are compared with the mean angle from the X-ray structure in Table 3. In most cases the values agree to within the 2 0 confidence limit of the experimental results (7"). The agreement tends to be worst for those angles that showed the largest variance in the X-ray values: for y 2 the spread in the X-ray values was lo", for 81 and 8 2 it was 13". Other angles had a spread of 7" ( y l ) or less. Throughout the simulation the global structural parameters A3 and s2 remained very close to the average values quoted in Table 4, indicating that the global structure is quite stable. Moreover, for most of the PRDFs there was little difference in the functions collected during the equilibration stage and the production stage of the simulation. The PRDFs of prime interest are shown in Figure 3. shows a first In the starting configuration the PRDF peak at 4.8 8, corresponding to 0 atoms in the second coordination sphere of the potassium. (The function gK.O,(r) shows a very strong first peak at 2.8 8, corresponding to six ester carbonyls in the first coordination sphere of the cation). After 2 ps of equilibration the formation of an additional peak at 3.0 8,in gK.O(r) was observed. This corresponds to an amide
Forester et al.
9426 J. Phys. Chem., Vol. 98, No. 38, 1994
TABLE 4: Summary of Potassium-Valinomycin Studies T(K) 293" 100 293 310 330 350 370 a
time (PSI
(9) (Az)
(A3)
0.045 0.053 0.031 0.016 0.062 0.049 0.024
10 500 1000 1000 1000 500 500
nHO
THO (PSI
5.9 4.6 0.8 0.4 0.8 1.1 0.4
27.9 27.7 21.9 27.5 28.6 28.5 28.0
nCHO,b
THO, (PS)
2.8 (0.1) 1.3 (0.0) 1.9(1.0) 1.6 (0.9) 1.4 (0.7) 1.8 (0.7) 1.5 (0.5)
C C
1.15 0.95 0.82 0.98 0.24
Solid state. Values in parentheses estimated from radial distribution functions.
80
60
40
20
0
2
3
4
5
6
7
10
5
0
1
2
3
4
5
r 1 Angstrom
6
7
8 r /Angstrom
Figure 3. Partial radial distribution functions (PRDFs) g H O ( r ) , gH oe(r), go,o,(r), g K O ( r ) , and gKO,(r). In each results from the simulations are labeled as (a) crystalline potassium complex, 293 K; (b) isolated potassium complex, 310 K; (c) crystalline valinomycin, 293 K and; d isolated uncomplexed valinomycin, 3 10 K. Note that the PRDFs have been offset from zero to facilitate viewing and that gKo(r) for a has been multiplied by five.
carbonyl entering the first coordination sphere of the cation making the total coordination number 7 rather than 6. This occurred in only 4 of the 16 valinomycin molecules in the MD cell and was accomplished without substantial changes in the global structure of the complex. In the X-ray structure each amide oxygen sits adjacent to three ester carbonyls (the mean 0-0, distance is 3.4 A) but on the other side of the plane containing the three 0, atoms from the cation. The site has approximately threefold symmetry with respect to the K - 0 vector. The consequence is that the amide carbonyl can move into the first coordination sphere of the cation with only a minor change in 0, and potassium positions. The potassium moves away from the centre of the (distorted) 0, octahedron toward the incoming 0 in the process. That potassium has a preference for more than 6 oxygen ligands in its first coordination shell is well documented: In the potassium-nonactin complex the coordination number is 8;37 MD simulations of potassium in water19$35$36 predict the
c c
0.69 0.65 1.03 0.33 0.35 T
nKO
0.4 1.6 3.0 3.9 2.5 2.6 3.4
TKO (PS) C
C
4.0 3.6 2.9 2.1 1.7
nTKQ
5.9 6.0 4.0 3.2 4.1 3.9 3.4
TKO, (PS) C C
6.2 7.3 5.0 4.1 1.9
nTa
6.3 7.6 7.0 6.6 6.5 6.5 6.8
too long for reliable estimate from simulation.
coordination number to be in the range 7.6 f 0.3. In water potassium does not have a clearly defined first coordination ~ h e 1 1since ' ~ ~ gK.O,(r) ~~ has a sizable nonzero minimum between first and second peaks. We observe parallels with both these phenomena in the crystal structure of valinomycin and in the isolated complex (see below). We note that the first peak in the potassium-oxygen PRDFs in both water and in valinomycin is 2.8 A and so the presence in the valinomycin simulations of a poorly defined first coordination shell and a coordination number greater than 6 is not unexpected. In any event the deviation from the X-ray structure is not serious: the total coordination number of the potassium evaluated at 3.5 A was 6.3, and these results were obtained without artifically restraining atoms to the X-ray positions. These structural changes are subtle effects. They occur almost without disruption of the six hydrogen bonds to the amide oxygens (Table 4). The accompanying P D F , gH.O(r) (Figure 3), shows no anomalous features nor do the functions go.o(r),goe.oe(r),and go.q(r) which are virtually unchanged from the X-ray structure values. Commensurate with the slight disruption of the octahedron of ester carbonyls around the cation is a small peak in gH.O,(T) at 2.0 A (Figure 3). This is indicative of the oxygen of the ester carbonyl functioning as a hydrogen bond receptor. The incidence of such events is low (an average of 0.1 events per valinomycin molecule) but presages behavior seen in the isolated molecule simulations described below. The dihedral distributions for the ring are shown in Figure 4. The distributions for both the ester groups p2) and the amide groups (61, € 2 ) are peaked at 180" implying both fluctuate around planar geometry. The amide groups are the more flexible of the two. Other reasonably sharply peaked distributions are those for the a, y , and 5 torsions. Only for the y torsions is a difference in the peak maxima for the two sets ( y l and 72) apparent. The distribution for y2 is broader than that for y1 presumably a reflection of the smaller steric hindrance for the Me side group (y2) compared to the i-Pr group ( ~ 1 ) . 3.2. Uncomplexed Valinomycin. The uncomplexed valinomycin crystal was simulated at 293 K in the canonical (NVT) ensemble. The system was allowed to equilibrate for 5 ps and statistical data collected over the following 20 ps of simulation. The starting configuration was taken from the crystallographic data of Karle'O for the "twisted bracelet" conformer. Comparison of simulation results with the crystallographic data is made in Table 5. The simulation box contained four unit cells (two molecules per unit cell) and was monoclinic with cell vectors a = (23.144, 0,O) A, b = (0,20.694, 0) A, c = (-4.830,0, 28.468) A. Given that the X-ray data are not particularly well resolved (the R factor is 12.0%), excellent agreement was found between the experimental and calculated structural properties (Table 5). As in the potassium complex simulation the rms deviations in atomic position from the crystal structure are small (0.19 A) and most of the key interatomic separations are reproduced to within the limits of experimental yror. An exception is the 0,-0, separation which is 0.16 A larger in the simulations than in the
Valinomycin and Potassium-Valinomycin Complex
J. Phys. Chem., Vol. 98, No. 38, 1994 9427
TABLE 6: Summary of Uncomplexed Valinomycin Studies T(K) length (PS) 6 4 3 ) (9) (A2)nHo THO (PSI nH9.b THO. (PS)
b : isolaled K-vln
0.30
100 293" 293 310 330 350 370
0.25
,$
2
0.20
0 20
0 15
500
0.125 0.163 0.133 0.152 0.116 0.121 0.122
20 lo00 lo00 lo00 500 500
27.9 29.3 29.5 29.5 29.6 29.8 29.9
4.8 3.4 1.2 1.2 1.2 1.1 1.1
c
5.4 1.1 1.0
0.6 0.8 0.8
3.6(1.2) 3.6(1.2) 4.4(1.0) 4.1 (1.0) 3.7 (1.0) 4.0(1.0) 4.3 (1.0)
15.6 1.9 10.5 7.9 3.4 2.6 3.7
Solid state. Values in parentheses estimated from radial distribution functions. t too long for reliable estimate from simulation.
0.10
0.05
0.00 0.0
o,30
30.0
(L
60.0
900
000 1200 150.0 180.0 0
--.
a-.'
30
60
30
60
90
120
150
180
90 120 / degrees
150
180
a :crystalline K-vin
025 21 VI
5 0.20 .= 21 a 0.15 $
B 0.10
0.05
0.00
0
30
60
90
120
150
160
0
0 i degrees
t$
Figure 4. Dihedral angle distributions. The labels a, b, c, and d refer to the same simulations as in Figure 3. Definitions of the torsional angles are given in the text and illustrated in Figure 1. In each ase the solid line refers to al,PI, etc. and the dashed line to az,8 2 , etc. Distributions are normalized with respect to integration over $J (in degrees) and offset from zero to facilitate viewing.
TABLE 5: Uncomplexed Valinomycin Crystal Properties (293 K) g ( r ) First Peak Position (A)
X-ray MD
2.91 2.88
2.03 2.00
~
a1
a2
2.25 2.22
2.34 2.36
Mean Dihedral Angles (ded BI BZ Y I y2 61 82
3.84 4.00
€1
4.66 4.68
€2
51
f2
X-ray 110 111 175 175 107 111 5 40 174 176 76 76 MD 112 105 172 171 98 112 15 29 171 173 72 80
Rms Deviations from Crystal Structure Positions (A) CT
C
N
0 9
0.17
0.20
0.21
0.20
0,
0
0.21
0.27
X-ray structure. Good agreement is also seen in the mean dihedral angles, although it is apparent that the corresponding distributions (Figure 4)are more diffuse than their counterparts in the crystalline potassium complex and in the case of the a, y , and 5 distributions are double peaked. The double peaks are inherent to the "twisted bracelet" conformation. For example, in each valinomycin molecule two of three a1 torsions are near 130" and the other near 67". Other results from the simulation are summarized in Table 6 and the key PRDFs shown in Figure 3. The asphericity parameter, A3, takes an average value close to 1/6 (Table 6)-the value for an infinitely thin uniform disk, although the global structure is more reminiscent of a squat cylinder. The mean squared radius of gyration, p,is slightly larger than in the
potassium complex (29.3 vs 27.9 Az) consistent with the nonspherical structure it adopts. Neither A3 nor p fluctuated greatly during the course of the simulation. The hydrogen bond network in the molecule remained close to the X-ray structure. The mean lifetime of H-O bonds was ~ H . O= 5.4 ps while the H-0, bonds were shorter lived ( Z H . ~ ,= 1.9 ps) and less frequent (Table 6 ) . The PRDFs (Figure 3) show a strong first peak in gH.O(r) at 2.0 8, corresponding to the H-0 hydrogen bonds. The function gH.O,(r) is more complex. It has a shoulder around 2.2 8, but the main f i s t peak is at 3.2 8,. The differences in the PRDFs for the two crystal structures manifest themselves in the respective dihedral distributions (Figure 4). The general pattern is one of much broader distributions (and hence more torsion freedom) for the uncomplexed valinomycin. As in the potassium complex the torsional distributions @I, /?z) and (€1, €2) are peaked at 180"; however, the /3 distributions (ester groups) are noticeably broader, indicating the influence cation coordination has on these distributions in the potassium complex. The key to the conformational flexibility of valinomycin is to be found in the a, y , 6, and torsions. A prime feature to note for the valinomycin crystal is the difference between the distributions for y1 and y2. In the potassium complex y~ and y2 are both reasonably sharp functions peaked around 70"-80". In the valinomycin crystal the y2 function in particular is much broader and the maximum shifted to higher 4. The differences between these two distributions will become more apparent when results from the isolated molecule simulations are considered below. For the moment we note that the broadeness of the y2 distribution indicates that there is considerable torsional freedom about the 0-C(Me) ester linkage. 3.3. Isolated Potassium-Valinomycin. We carried out simulations of the isolated potassium complex at 100 K with our model and where structural comparisons can be made (e.g. K-0, distances) the agreement with Eisenmann's et al. results3s4 was excellent. We have also carried out extensive simulations from 293 to 370 K in order to assess the stability and flexibility of the complex. Table 4 summarizes the results of these simulations. The measures of global structure, A3 and 9, indicate the complex is stable over the entire temperature range studied. Average values show little change from the 293 K solid-state values. There is a small mean expansion of the structure at 330 K and above (i.e., p increases) but it does not appear to be particularly significant. As in all the isolated molecule simulations A3 and p showed large fluctuations on the picosecond time scale indicative of a highly flexible structure. Although the potassium remains fully coordinated the manner in which valinomycin chelates the cation in the isolated molecule studies differ from that seen in the solid state. First, amide carbonyl groups are now able to coordinate to the potassium in significant amounts (Table 4). At 100 K this is in addition to full ester coordination. Between 293 and 370 K further amide coordination is at the expense of ester coordination. That is,
