8230
J. Phys. Chem. 1996, 100, 8230-8239
Ab Initio Studies on Organophosphorus Compounds. 5. Interactions of Dianionic Bisphosphonate Compounds with Magnesium and Calcium Jari P. Ra1 sa1 nen,† Esko Pohjala,‡ Hannu Nikander,§ and Tapani A. Pakkanen*,† Department of Chemistry, UniVersity of Joensuu, P.O. Box 111, FIN-80101, Joensuu, Finland, Leiras Oy, P.O. Box 33, FIN-33721, Tampere, Finland, and Leiras Oy, P.O. Box 415, FIN-20101, Turku, Finland ReceiVed: September 26, 1995; In Final Form: January 8, 1996X
Bisphosphonates are characterized by a P-C-P backbone angle and two phosphonic acid groups bonded to the same carbon atom. Two clodronate-type compounds, dianionic methylenebisphosphonate (MBP) and (dichloromethylene)bisphosphonate (Cl2MBP), were selected for study. Molecular geometries and energetics were calculated, and their interactions with cationic magnesium and calcium ions were investigated in the gas phase by ab initio molecular orbital methods. Energetically different metal-binding sites were identified. The metal-binding models were extended through the addition of six explicit water molecules to the complexes. Intermolecular hydrogen bonds were formed in the water sphere. The explicit models of MBP-Ca-water and MBP-Mg-water complexes were completed by adding a water medium produced by the self-consistent reaction field (SCRF) method. The 3-21G(*) basis set was used in all the ab initio MO calculations, and relaxed molecular geometries are reported for the studied complexes. Model reactions were used to study the energetics of the various complexes.
Introduction The bisphosphonates are synthetic analogues of pyrophosphate. Various types of bisphosphonate species are applied in the treatment of several bone diseases.5-16 For example, clodronate and its related species, etidronate and pamidronate, are potential inhibitors of bone resorption and are used successfully in the treatment of hypercalcemia and hypercalciuria.6-16 Diseases in which bone resorption take place are closely associated with the balance of calcium in the human body. As the fifth most abundant element in the body after carbon, hydrogen, oxygen, and nitrogen, calcium plays an essential role.17 The bones and tooth enamel contain 95% of body calcium, and the body fluids and soft tissues the remaining 5%.17 The intracellular calcium has an especially important role as a regulator of cell activities, including enzymic activities, blood clotting, and fertilization.17 Earlier studies have shown that bisphosphonates can inhibit osteoclast-mediated bone resorption and osteoclastic activation.6,7 Various hypotheses have been advanced to explain how bisphosphonates act against the osteoclasts,10,11 but the precise stimulation mechanism remains unclear.6,7 A direct effect on the metabolism of osteoclasts has been proposed in accounting for the action of intracellular clodronate against osteolytic bone diseases.10 It has also been suggested that the action of clodronate in osteolytic bone diseases is dependent on the interactions between cationic metal ions and dianionic bisphosphonate molecules.10 Water-soluble clodronate is one of the most thoroughly studied bisphosphonate compounds.10 Unlike pyrophosphates, the bisphosphonates, including clodronate, are completely resistant to enzymatic hydrolysis.6,16 Departing from most earlier studies on bisphosphonates, which have been pharmacological or clinical in nature, we investigated MBP and Cl2MBP complexes by ab initio molecular orbital methods. Although there are many theoretical ab initio MO studies on biomolecular phosphorus compounds and †
University of Joensuu. Leiras Oy, Tampere. § Leiras Oy, Turku. X Abstract published in AdVance ACS Abstracts, April 15, 1996. ‡
S0022-3654(95)02858-9 CCC: $12.00
their metal complexes,18,21-35 only a few studies deal with the bisphosphonates.4,18-20 Ab initio MO methods have also been applied in studies on hydrogen malonate, magnesium-malonate, and calcium-malonate species.36-41 Recently, the molecular geometry study was made on pyrophosphate-metal complexes,32 and the molecular geometries of water-calcium and water-magnesium clusters were claculated.42-46 In paper four of this series,1-4 molecular geometries and energies of dianionic MBP (a) and Cl2MBP (b) molecules and their complexes with up to six explicit water molecules were calculated by ab initio MO methods. In the present study, MBP-Ca (a) and Cl2MBP-Ca (b) complexes and the corresponding magnesium (c and d) complexes were calculated in the gas phase. The primary structures of the dianionic bisphosphonate a and b molecules are as shown below.
Magnesium and calcium ions were selected for the study because competitive binding properties of calcium and magnesium cations have been demonstrated in studies on osteocalcin protein.47 There is evidence to suggest that elevated intracellular Ca2+ inhibits osteoclast function by promoting cell-matrix dissociation and that Mg2+ ions play a critical role in osteoclastbone interactions.48 Interaction distances Ca‚‚‚O and Ca‚‚‚OH2 in [Ca‚Cl2MBP]pentahydrate species, earlier evaluated by crystallographic methods, were used as a reference for the theoretical studies.49 Computational Methods Ab initio Hartree-Fock SCF molecular orbital calculations on bisphosphonate-metal and bisphosphonate-metal-water complexes were performed with Gaussian 90 and 92 programs50 on SGI 4D/35, SGI Indigo R4000/Elan, and IBM RS6000 computers. The basis set for calcium was taken from Dobbs © 1996 American Chemical Society
Ab Initio Studies on Organophosphorus Compounds
J. Phys. Chem., Vol. 100, No. 20, 1996 8231
Figure 1. (A) Crystal structure of CaCl2MBP‚5H2O. (B) Single-point calculation model for calcium bonding. (C) Calcium-bonding sites in the structures used for the optimization studies. (D) Different conformations in which all bond lengths, bond angles, and torsional angles were fully optimized.
and Hehre,51 where the 3-21G* standard basis was extended by adding a set of single d-type Gaussian functions. All molecular geometries reported for the complexes were optimized in the gas phase with the 3-21G(*) basis set. Correlation corrections were made with calculations by the MP2/3-21G(*)// 3-21G(*) method.50 The basis set superposition error (BSSE) was determined for the bisphosphonate-metal ion pairs. The optimized molecular geomtries agree with the experimental values and with results calculated in the higher basis set levels.24,51-53 Recently, the 3-21G(*) basis set was successfully used in interaction studies on calcium-malonate and magnesium-malonate complexes.40 Calcium- and magnesium-binding sites and the effects of explicit water molecules around the bisphosphonate-metal complexes were studied. The presence of water molecules had a stabilizing effect on the ionic interactions between the MBPmetal cation pairs. To further extend the water medium around complexes, a self-consistent reaction field (SCRF) model implemented in Gaussian 90 was used in calculations.50 The SCRF approach ( ) 78.5454) was based on the code of Rivail and Rinaldi.55-59 The charge distribution of the solute was described with multipole expansion up to l ) 6. Van der Waals surfaces were used to determine an ellipsoidal cavity around the complexes. The multipole expansions (l ) 6) that were used exhibited serious convergence problems, probably caused by underestimation of the cavity surfaces of the compounds. These problems were solved by adding 0.5 Å to all X, Y, and Z dimensions of the ellipsoidal cavities. With the expanded cavities, multipole expansions with reasonably converged series were
obtained. The cavity expansion used in this study was in accordance with the method discussed in the review by Tomasi and Persico.60 Mulliken charges were calculated for the bisphosphonates. Calculations and Results Complexes of Bisphosphonate and One Calcium Ion. The molecular backbone of bisphosphonate needed for ab initio MO calculations was taken from the crystal structure of CaCl2MBP‚ 5H2O (Figure 1).49 In a rigid conformational analysis, the calcium ion was guided by different values of the P-O‚‚‚Ca bond angle and C-P-O‚‚‚Ca torsional angle, the latter of which was turned around the P-O axis in steps of 30° (Figure 1). Various calcium-binding sites were obtained by single-point calculations and these calcium-bonded conformations were used as an initial guess as in the further full optimization studies, where all bond lengths, bond angles, and torsional angles were relaxed. Stable conformations a1, a1′, and a2 were obtained for MBP-Ca complexes, and the corresponding conformations b1 and b2 for Cl2MBP-Ca complexes, respectively (Table 1). The global minimum energy structures were the a1 and b1 conformations (Figure 1 and 2). The calcium-binding site evaluated in the structure of the CaCl2MBP‚5H2O crystal49 (Figure 1) was not found in our studies. By way of comparison, earlier calculated lithium and sodium pyrophosphate (LiH3P2O7 and NaH3P2O7) complexes have local minimum energy conformations like those calculated for the a1 and a1′ complexes.32 The monophosphonic acid groups in the bisphosphonates had full freedom to turn around the C-P bond axis. In calculated
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Figure 2. Calculated structures of bisphosphonate complexes a(1-4) and b(1-4), with Ca‚‚‚O bonding lengths and P-C-P angles indicated. Molecular geometries of the Mg complexes c(1-2) and d(1-2) are similar to those calculated for Ca conformations a(1-2) and b(1-2).
TABLE 1: Selected Bond Lengths (pm) of Structures Calculated for Bisphosphonate-Calcium Complexesa a1 a5 a8 a13 b1 b5 b13
TABLE 2: Selected Parameters (pm, deg) for Bisphosphonate-Calcium Interactionsa
P-O1H
P-O5H
P-O6
P-O8
P-O9
P-O11
P-C
158.2 156.0 158.4 159.0 156.8 155.0 157.7
160.5 156.0 158.9 162.8 158.9 155.0 160.9
152.2 151.9 151.3 151.3 151.0 150.6 150.1
146.7 151.9 149.8 152.0 145.6 150.6 150.1
152.1 151.9 151.1 151.6 150.6 150.6 150.2
153.2 151.9 151.6 146.3 151.8 150.6 145.5
181.4 180.6 181.0 180.6 187.0 185.4 185.9
a
a1 is MBP-Ca, a5 is Ca-MBP-Ca, a8 is [MPB-Ca]‚(H2O)1, a13 is [MBP-Ca]‚(H2O)6. In the series b, the complex structures are the same as in the series a, but the bisphosphonate is Cl2MBP.
conformations a(1-3) and b(1-2), the bridging calcium was tridentately or tetradentately bonded to the bisphosphonates. Corresponding rotations around the O-P axis have recently been reported for pyrophosphates (P-O-P backbone).32 In the a1 complex, the bisphosphonate acted as a tridentate ligand, and the calcium was tightly bound in a bridging position (Figures 1 and 2) at interaction distances O‚‚‚Ca of 214 and 231 pm (Table 2). P-C-P backbone angles of 116.3° and 111.7° were calculated for the a1 and a1′ complexes, respec-
a1 a5 a8 a9 a10 a11 a12 a13 b1 b5 b13
Ca-O6
Ca-O8
Ca-O9
Ca-O11
231.0 218.3 241.4 244.6 227.4 228.3
231.4 218.3*
213.9 218.3* 235.7 237.3
218.3 223.9 233.4
225.2 238.1 230.6
233.8 219.5
219.6 231.5
238.7 236.9 242.8 219.7 238.6
237.9 219.5
P-C-P 116.3 111.1 111.5 111.7 112.4 112.4 113.3 115.1 115.1 111.5 113.7
a Index * means the binding distance values for the second calcium. See labels of oxygen atoms from Figure 1. a1 is MBP-Ca, a5 is CaMBP-Ca, and a8-a13 are [MBP-Ca]‚(H2O)n, where n ) 1-6. In the b series bisphosphonate is Cl2MBP.
tively. The distance between calcium atoms was found to be only 97 pm when the a1 and a1′ conformations were compared by superposing P2, C, and P4 atoms. In the a1 species there were no internal hydrogen bonds, but in a1′ an internal hydrogen
Ab Initio Studies on Organophosphorus Compounds bond (O‚‚‚H distance 197.2 pm) connected the monophosphonic acid ends together. In the a2 complex, four oxygen atoms of different monophosphonic acid groups were symmetrically bonded to the metal cation, and Ca‚‚‚O distances in this tetradentate binding ranged from 239.6 to 239.8 pm (Figure 1). Correspondingly, in b2 complexes with chlorine substituents, the Ca‚‚‚O distances ranged from 239.3 to 241.6 pm. The effects of chlorine substituents on the Ca‚‚‚O distances were small. The energies of the different conformations a1′, a2, and b2 were 3.6, 5.9, and 5.7 kJ mol-1 higher than the energies of global minimum structures a1 and b1. Ab initio calculations with the DZP basis set, made for several neutral and ionic pyrophosphate complexes with K, Na, and Li catins, have given P-O-P angles from 117.7° to 130.2°.32 Experimentally determined values reported for the P-C-P angle range from 113.6° to 117.2°.49,61,62 The P-C-P angles of our a1, a2, b1, and b2 complexes all fell within this range, while the P-C-P angle of the a1′ complex decreased to 111.7° by an internal hydrogen bond (Table 2). Average bond lengths and other angles were nearly the same in the a(1-2) and b(1-2) structures, indicating that the chlorine substituents had little effect on the optimized bond parameters of the bisphosphonate. In a next set of studies, two binding sites, 1 and 2 (Figure 1), were used in the initial guess and after full optimization complexes a3 and a4 were obtained, respectively. In the conformation a3, the monophosphonic acid group with calcium was turned about the C-P axis and a calcium bridge was formed between the two monophosphonic acid ends (Figures 1 and 2). The a3 complex was 11.3 kJ mol-1 higher in energy than the global minimum structure a1 (Figures 1 and 2). In the a4 conformation, the O-P-O part of the monophosphonic acid group provided the bidentate bonding for the calcium. The average O‚‚‚Ca distance was 218 pm and the bidentate bonding remained unchanged despite full relaxation of the molecular geometry. (Figures 1 and 2). Energetically, the a4 structure was 236 kJ mol-1 higher in energy than the global minimum structure a1 due to the close contact between two negative charges in the same monophosphonic acid group in the a4 complex. Proton-transferring effects were also found in the b3 and b4 complexes with chlorine substituents. As in a4, one monophosphonic acid end of the MBP was bound bidentately to the metal (Figure 2), and two negative charges were concentrated on the phosphorus group so bound. Relative energies for b3 and b4 were higher by 182 and 198 kJ mol-1 than for the global minimum b1, where charges were distributed over the entire molecular backbone. As in a4, there were internal hydrogen bonds in the b3 and b4 conformations, with lengths 164.4 and 154.4 pm, respectively. The average calculated O‚‚‚Ca distances were increased by 4 pm in the b3 and b4 complexes relative to the a4 complex. The pyrophosphate salt structures with Li+, Na+, and K+ ions have internal hydrogen bonds of 201, 190, and 185 pm, respectively.32 In the NaH2PO4 and KH2PO4 complexes, bidentate chelation with bonding distances of 225 and 260 pm was calculated.32 Complexes of Bisphosphonate and One Magnesium Ion. The magnesium ions were added to the various binding sites of the bisphosphonates, and molecular backbone relaxation was allowed. All bond lengths, bond angles, and torsional angles of the bisphosphonate-magnesium complexes were fully optimized. Different conformations were calculated for magnesium binding in the MBP (c) and Cl2MBP (d) complexes. The binding sites of magnesium in the bisphosphonates c(1-2) and d(1-2) were similar to those calculated for the a-
J. Phys. Chem., Vol. 100, No. 20, 1996 8233 TABLE 3: Selected Bond Lengths (pm) of Structures Calculated for Bisphosphonate-Magnesium Complexesa c1 c5 c8 c13 d1 d5 d13
P-O1H
P-O5H
P-O6
P-O8
P-O9
P-O11
P-C
156.9 153.8 157.4 158.9 155.6 152.9 158.0
159.7 153.8 158.3 163.2 158.1 152.9 161.2
153.2 152.8 152.2 150.5 152.2 151.7 148.5
146.2 152.8 149.1 151.7 145.3 151.4 150.8
153.2 152.8 151.9 152.3 151.7 151.4 151.4
154.5 152.8 152.6 146.2 153.0 151.7 145.5
181.4 180.4 181.2 180.4 186.7 185.1 186.3
a c1 is MBP-Mg, c5 is Mg-MBP-Mg, c8 is [MBP-Mg]‚(H O) , 2 1 c13 is [MBP-Mg]‚(H2O)6. In the series d, the complex structures are the same as in the series c, but the bisphosphonate is Cl2MBP.
TABLE 4: Selected Parameters (pm, deg) for Bisphosphonate-Magnesium Interactionsa c1 c5 c8 c9 c10 c11 c12 c13 d1 d5 d13
Mg-O6
Mg-O8
197.3 184.0 205.5
184.0*
194.1 200.3
194.0 200.4 195.0
198.5 184.2
186.2* 196.2
Mg-O9
Mg-O11
P-C-P
197.2 184.0* 204.2 192.3
182.3 184.0 191.1 192.3
202.5 199.6 197.5 186.2* 197.8
205.8
114.2 110.2 109.0 110.3 112.3 111.9 111.4 113.6 113.6 110.3 112.2
183.0 184.2
a Index * means the binding distance values for Mg2. See labels of oxygen atoms from Figure 1. c1 is MBP-Mg, c5 is Mg-MBP-Mg, and c8-c13 are [MBP-Mg]‚(H2O)n, where n ) 1-6. In the d series bisphosphonate is Cl2MBP.
(1-2) and b(1-2) calcium complexes (Figure 2). Even the interatomic torsional angles HO-P‚‚‚P-OH through the molecular backbone were similar. The global minimum energy structures were c1 and d1 complexes (Table 3). In the c(12), c1′, and d(1-2) structures, magnesium was bonded between the monophosphonic acid ends. Internal hydrogen bonds were not formed in the c(1-2) and d(1-2) complexes, but a weak internal hydrogen bond length 214 pm was formed in the c1′ complex, which was 8.7 kJ mol-1 higher in energy than the c1 complex. The P-C-P angles in the c1, c1′, and d1 complexes were 114.4°, 111.1°, and 113.6°, respectively (Table 4). The angles in c1 and d1 correlated well with the experimental values ranging from 113.6° to 117.2°.49,61,62 In the c1′ complex, the internal hydrogen bond causes slight closing of the P-C-P angle. The P-C-P angles of the magnesium complexes were closed relative to those calculated for the calcium complexes. Being sterically smaller, the Mg2+ ion can approach closer to the dianionic bisphosphonate molecule than the Ca2+ ion. In c(1-2) and d(1-2), the Mg‚‚‚O distances were in the range 182.3-210.6 pm, whereas in the corresponding calcium complexes a(1-2) and b(1-2) the average Ca‚‚‚O distances were 35 pm longer (Tables 2 and 4). By way of comparison, M‚‚‚O distances of 178.8 and 209.6 pm, respectively, have been calculated for the malonate-magnesium and malonate-calcium complexes.40 In studies on pyrophosphate salts, O‚‚‚Li and O‚‚‚Na distances 177.8-233.8 pm have been calculated.32 The calculated Mg‚‚‚O distances agree well with the reference values for several phosphorus compounds.32,40 Complexes of Bisphosphonate and Two Calcium Ions. Two metal ions were added to the different binding sites around the bisphosphonate backbone, and all bond lengths, bond angles, and torsional angles of complexes were fully optimized. As a result of molecular geometry optimizations, three different conformations a5, a6, and a7 were obtained for the a species,
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Figure 3. Calculated structures of bisphosphonate complexes a(5-7) and b(5-7), with Ca‚‚‚O bonding lengths and P-C-P angles indicated. Molecular geometries of the Mg complexes c(5-7) and d(5-7) are similar to those calculated for Ca conformations a(5-7) and b(5-7).
and related conformations b5, b6, and b7 were obtained for the b species (Table 1 and Figure 3). The global minimum energy structures were those of the a5 and b5 complexes (Figure 3). The binding sites of the two calcium atoms corresponded to the site determined for one calcium binding to Cl2MBP in the CaCl2MBP‚5H2O crystal structure.49 Theoretical calculations have predicted similar binding sites for two-cation bonding in pyrophosphate species with Li+, Na+, and K+.32 An average Ca‚‚‚O distance of 236.2 pm has been reported in the CaCl2MBP‚5H2O crystal structure (Figure 1),49 and Ca‚‚‚O distances of 235.2 and 242.0 pm for calcium (1-hydroxyethylidene)bisphosphonate dihydrate (CaH2HEBP‚2H2O) crystal.61 Ca‚‚‚O distances in the a5 and b5 complexes were 218.3 and 219.6 pm, respectively (Figure 3 and Table 2). These distances are about 17 pm shorter than those in the CaCl2MBP‚5H2O crystal structure.49 In the a(6-7) and b(6-7) complexes, as in the tridentate bonding of the a1 and b1 species, one calcium ion was bound between the monophosphonic acid ends of the bisphosphonate. The Ca1‚‚‚O distances of this calcium ranged from 225.6 to 254.3 pm. The second calcium was bonded to just one monophosphonic acid end, which was acting like a bidentate ligand (Figure 3). These bidentate interaction Ca2‚‚‚O distances were in the range 219.2-259.1 pm, and the Ca2 was in the O-P-O plane. In the a6, a7, b6, and b7 complexes there were internal hydrogen bonds between the monophosphonic acid ends. The energies of the a6 and a7 compounds were 152 and 212 kJ mol-1 higher level than the energy of the global minimum a5. Correspondingly, the energies of the chlorine species b6 and b7 were 162 and 233 kJ mol-1 higher than the energy of
b5. In the a(5-7) and b(5-7) complexes, the P-C-P angles ranged from 106.8° to 111.5° (Table 2). These values are smaller than the experimental values.49,61 A P-O-P angle of 123.7° has been calculated for dilithium pyrophosphate (Li2H2P2O7).32 The similar calcium binding and molecular backbone geometries for the MBP a(5-7) and Cl2MBP b(57) complexes suggest that the structural changes due to the chlorine atoms are minor. According to our results for species with two calcium ions, MBP and Cl2MBP molecules are able to bind many calcium ions at the same time. Intriguing results were found for Ca‚‚‚Ca distances in the optimized two-calcium structures a(5-7) and b(5-7) (Figure 3). The Ca‚‚‚Ca distances were 542.8 and 571.9 pm in the global minimum energy complexes a5 and b5, respectively. The Ca‚‚‚Ca distances of a6, a7, b6, and b7, in turn, ranged more widely from 473 to 621 pm. We compared these calculated distances with those determined in the hydroxyapatite (HA) crystal, which is widely accepted as presenting a prototype of bone structure.63 The HA crystal includes eight- and nine-coordinated calcium.63 These Ca1 and Ca2 centers have five and six calcium atom neighbors, respectively, within a radius of 621.3 pm. In the xy plane (001) of the HA crystal, for example, there is a Ca‚‚‚Ca1 distance of 544 pm, which is almost the same as the value we found in a5. The wider ramification of our study on bisphosphonate complexes with two calcium atoms is that bisphosphonates may have the ability to be bonded to the bone matrix through many calcium atoms at the same time and, correspondingly, to many calcium atoms in body fluids. Complexes of Bisphosphonate and Two Magnesium Ions. Three different stable conformations c(5-7) and d(5-7) were
Ab Initio Studies on Organophosphorus Compounds
J. Phys. Chem., Vol. 100, No. 20, 1996 8235
Figure 4. Calculated structures of bisphosphonate complexes a(8-13) and b(13), with Ca‚‚‚O bonding lengths and P-C-P angles indicated. Molecular geometries of the Mg complexes c(8-13) and d(13) are almost similar to those calculated for Ca conformations a(8-13) and b(13).
obtained as a result of studies where all bond lengths, bond angles, and torsional angles were fully optimized (Figure 3, Tables 3 and 4). Magnesium binding was similar to that found for the corresponding bisphosphonate-calcium systems a(5-
7) and b(5-7). The global minimum of energy structures were c5 and d5 species, and Mg‚‚‚O distances there were 184 and 186 pm, respectively (Table 4). The P-C-P angles of the c(5-7) and d(5-7) complexes ranged from 105.3° to 110.3°.
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TABLE 5: Metal-Water Bonding Distances (pm) of Bisphosphonate-Metal-Water Complexesa
a
M‚‚‚O(W1)
M‚‚‚O(W2)
M‚‚‚O(W3)
M‚‚‚O(W4)
M‚‚‚O(W5)
M‚‚‚O(W6)
a8 a9 a10 a11 a12 a13 b13
229.8 230.8 251.3 231.2 233.2 312.7 314.2
231.3 228.9 249.2 241.5 319.9 320.5
232.0 234.5 235.6 259.3 257.6
235.2 304.8 235.8 235.5
232.8 235.5 235.0
240.8 239.9
c8 c9 c10 c11 c12 c13 d13
198.3 195.5 205.7 205.8 206.8 348.6 361.8
195.5 204.9 205.9 204.4 314.5 315.6
197.8 207.8 207.7 326.9 341.1
207.7 322.9 198.3 198.4
203.8 205.1 202.9
204.0 203.3
a8-a13 are [MBP-Ca]‚(H2O)n, where n ) 1-6, and b13 is [Cl2MBP-Ca]‚(H2O)6. In the c and d groups the metal is magnesium.
Figure 5. Optimized complex b13 presented as stereo pair.
Both the Mg‚‚‚O distances and the P-C-P angles were smaller than those of the corresponding calcium complexes. The Mg‚‚‚Mg distances ranged from 385.9 to 539.2 pm. Complexes of Bisphosphonate, Calcium, and One to Six Explicit Water Molecules. To extend the interaction models, one to six explicit water molecules were added to the MBPmetal complexes. Since the full coordination sphere of calcium is reported to include six to nine water molecules, the complex of MBP, calcium, and six water molecules is large enough to present the first coordination shell of calcium ion reliably.42 The complex a2 was used as an initial guess of the complex structure, and water molecules were bonded to the calcium ion, one at the time, at the farthest possible distance from bisphosphonate. All bond lengths, bond angles, and torsional angles of complexes were optimized. The complexes calculated were a(8-13) and b13 (Figure 4, Tables 1 and 2). The monophosphonic acid groups in the bisphosphonate-calcium-water complexes were turned about the P-C axis. In addition, the explicit water molecules around the calcium were reoriented during geometry relaxation, and strong hydrogen bonds were formed. Internal hydrogen bonds between the monophosphonic acid ends were not found. Strong interactions between ionic species made the complexes very stable. As seen in all complexes, four water molecules could be arranged in very close contact to the calcium ion (Table 5). Steric hindrance forced the fifth and sixth water molecules to positions further from the calcium ion, to Ca‚‚‚O distances of over 300 pm, but even then, these two water molecules were tightly bonded to the complex via Ca‚‚‚O and hydrogen bond interactions. In a8 and a9, calcium was bonded to three oxygen atoms of MBP at distances less than 245 pm (Table 2). In the remaining compounds, a(10-13) and b13, calcium was bonded to two oxygen atoms of the bisphosphonate at distances less than 245 pm. In all a(8-13) and b13 complexes, the nearest
oxygen neighbors around the calcium centers were within a radius of 224-245 pm. By way of comparison, in the crystal structure of a high-temperature phase of the R-calcium diphosphate, Ca‚‚‚O distances in eight-coordinate binding sites range from 225.0 to 280.7 pm.64 In the crystal structure of the lower temperature phase of β-calcium diphosphate, Ca‚‚‚O distances in seven-, eight-, and nine-coordinate sites range from 231.8 to 292.7 pm.64 Krauss and Stevens42 have calculated that the radius of the first solvation shell of the Ca(H2O)6 and Ca(H2O)9 complexes is 371 and 436 pm, respectively. Nine or ten water molecules form the first coordination sphere of calcium in aqueous media.42 In the six-water complexes a13 and b13, there was no space left in the first coordination shell of the calcium ions for other molecules (Figure 4 and Table 2). The Ca‚‚‚O distances are presented in Tables 2 and 5. In the a(8-13) and b13 complexes with explicit water molecules, the distances correlated well with the average values of Ca‚‚‚O, 236.2 pm, and Ca‚‚‚O(water), 242.2 pm, determined for the CaCl2MBP‚5H2O crystal structure.49 The molecular dynamic results of Charifson et al.22 predict that, in dilauroylphosphatidylserine, Ca‚‚‚O distances in bidentate metal binding are 230 and 231 pm, respectively. In the [Ca(H2O)n]2+ (n ) 5, 6, 7) complex the Ca‚‚‚O distances are from 240 to 252 pm.22,65 The closest Ca‚‚‚O(water) distances in the molecular structures of a(8-13) and b13 bisphosphonate complexes range from 228.9 to 259.3 pm (Table 5) and agree well with the experimentally and theoretically obtained results.22,49,65 However, although the formation of complexes was dominated by ionic interactions, there were also hydrogen bonds ranging from 147 to 271 pm, to connect the water and MBP molecules together. In addition, hydrogenbonded water-water interactions were found in the a(1013) and b13 complexes, with bond lengths from 171 to 234 pm. The total number of hydrogen bonds increased as more water molecules were attached to the complexes, and the a13 complex included altogether nine hydrogen bonds (Figure 4). Eight hydrogen bonds were found in b13. By way of comparison, six water molecules, with maximum bond length of 210 pm, have been found hydrogen bonded to the dianionic pentacoordinate oxyphosphorane backbone.31 In studies on the crystal structures of bisphosphonate species, P-O double-bond distances were determined to range from 149.6 to 151.1 pm and P-O(H) single bonds of 157.6 pm were found.49,66 The P-O bond lengths 149.7 and 152.9 pm have been reported for β-calcium diphosphate crystal structure.64 The average P-O and P-O(H) bond lengths 150.9 and 158.1 pm (Tables 1 and 3) calculated for the a, b, c, and d type
Ab Initio Studies on Organophosphorus Compounds bisphosphonate-metal complexes of our study compare well with these experimental values. The P-C-P angles of several crystallized bisphosphonate compounds range from 113.6° to 117.2°.49,61,62 The P-C-P backbone angles 111.5-115.2° calculated for our a(8-13) and b13 complexes agree well those determined for several crystal structures of bisphosphonates. As seen from these values (Table 2), the P-C-P angle opens as more water molecules are added to the complex. Comparison of the a and b type bisphosphonates suggested that the steric and electronic effects of chlorine substituents in the bisphosphonate backbone close the P-C-P angles of molecular structures throughout the calculated series. All Ca2+ complexes were stable in the gas phase because of the strong ionic interactions between the dianionic bisphosphonate and metal ions. Water molecules approaching the bisphosphonate-calcium complexes seemed to prefer bonding as near as possible to the Ca2+ ions. The Ca‚‚‚OH2 interactions were supported by hydrogen bonds linking the water molecules to the bisphosphonate backbone. In both a13 and b13 complexes, one water molecule acted as a bridge between the monophosphonic acid ends of the bisphosphonate. Corresponding water bridges between the monophosphonic acid groups were found in the MBP-[water]n complexes (n ) 1-6) of our previous study.4 Oxygen atoms of the bisphosphonate molecules occupied two or three coordination sites of calcium, and water molecules occupied all the remaining sites. In our previous study,4 the double-donor double-acceptor interactions were calculated for hydrogen-bonded bisphosphonate-[water]6 complexes. The water molecules cover gradually the whole bisphosphonate-calcium complex. Complex of Bisphosphonate, Magnesium, and One to Six Explicit Water Molecules. The bisphosphonate-magnesium complexes were similarly studied with one to six explicit water molecules added. All bond lengths, bond angles, and torsional angles of complexes were optimized. As a result, the optimized magnesium complexes c(8-13) and d13 were obtained (Tables 3 and 4). The molecular backbones and metal-binding sites were almost the same as in the calcium case. The magnesium ions were bonded to the dianionic MBP at distances of 191.1308.8 pm (Table 4). The explicit water molecules were bonded to the free coordination sites of the magnesium ion at distances ranging from 195.5 to 315.6 pm (Table 5). By way of comparison, Mg‚‚‚O distances from 170.0 to 214.5 pm were calculated with the SCF/6-31G* basis set for magnesium-water complexes.45,46,67 Mg‚‚‚O distances in the [Mg(H2O)6]2+ complex determined by neutron diffraction studies ranged from 205.8 to 207.1 pm, and the oxygen atoms were in the second coordination sphere when the distance from the Mg2+ ion was increased to 430 pm.45 The P-C-P angles in Mg2+ species were reduced by an average of 1.3° relative to the corresponding calcium complexes (Tables 2 and 4). The effects of molecular geometry due to chlorine substituents in the MBP (complex d13) were small. Optimized molecular geometries of the magnesium and calcium complexes were similar (Figure 4). Calculated with the 3-21G(*) basis set, Mulliken charges of bisphosphonate backbones in Ca2+ and Mg2+ complexes a1 and c1 were -1.08e and -0.77e. In complexes a8 and c8, where one water molecule was bonded to the metal ion, the corresponding charges were -0.99e and -0.75e. With an increasing number of water molecules in the complexes, the charges of the bisphosphonate backbone increased to the maximum values of -1.11e and -1.11e in the a13 and c13 complexes, respectively. The total charges of the bisphosphonate molecules were almost the same in the Mg2+ and Ca2+ complexes after
J. Phys. Chem., Vol. 100, No. 20, 1996 8237 TABLE 6: Mulliken Charges (e) for Selected Atomsa a1 a8 a9 a10 a11 a12 a13 b1 b13
Ca
P2
P4
1.08 0.89 0.77 0.74 0.63 0.57 0.47 1.11 0.48
1.65 1.64 1.64 1.66 1.64 1.65 1.67 1.73 1.75
1.60 1.64 1.65 1.65 1.66 1.64 1.62 1.67 1.67
c1 c8 c9 c10 c11 c12 c13 d1 d13
Mg
P2
P4
0.77 0.59 0.69 0.62 0.48 0.52 0.47 0.78 0.53
1.74 1.72 1.65 1.63 1.62 1.61 1.64 1.83 1.70
1.59 1.63 1.65 1.63 1.62 1.63 1.60 1.67 1.65
a a1 is MBP-Ca and a8-a13 are [MBP-Ca]‚(H2O)n, where n ) 1-6. b1 is Cl2MBP-Ca and b13 is [Cl2MBP-Ca]‚(H2O)6.
three of more water molecules were added. In the Cl2MBP complexes b13 and d13, the charges of the bisphosphonate molecular backbones were -1.17e and -1.19e, respectively (Table 6). The SCRF model predicted similar but slightly more negative values for the charges of bisphosphonate molecules. Models Reactions for the Complexations. Our first step in studying model reactions was to optimize [Ca(H2O)n]2+ and [Mg(H2O)n]2+ complexes (n ) 1-6) for use as reactants with the bisphosphonate species. The Ca‚‚‚M distances 205 and 237 pm were calculated for the freely optimized [Mg(H2O)6]2+ and [Ca(H2O)6]2+ complexes with the 3-21G(*) basis set. Model reactions and corresponding energies calculated with the 3-21G(*) basis set and MP2/3-21G(*)//3-21G(*) and SCRF methods are presented in Table 7. The basis set superposition error (BSSE), at the 3-21G basis set level, was 18% in the H2PO4-‚Na+ complex and below 5% in the H2PO4- complexes with Li+ and Mg2+.68 The BSSEs ranging from 15 to 28% for the malonate-calcium and formate-calcium complexes were calculated with the minimum basis set.69,70 We calculated the BSSE corrections in the counterpoise (CP) method ranging from 3 to 9% for phosphonate-calcium and phosphinate-calcium species.3 The BSSE corrections were calculated only for the bisphosphonatecalcium and bisphosphonate-magnesium pairs, because the methods used and the results for the multimolecular BSSE corrections are not unambiguous even in the linear HF systems.71 In the a1, b1, c1, and d1 species, the amount of BSSE correction was about 8% for the formation energies calculated at the 3-21G(*) basis set level (Table 7). The formation energies calculated for the ion pairs were very high due to ionic interactions. The chlorine and non-chlorine systems have almost the same formation energies. The highly exothermic reaction energies decreased when the SCRF model was included to the study. Values of -212 and -231 kJ mol-1 were found in SCRF calculations when [MBP-[H2O]6]2- and [M(H2O)6]2+ complexes (M ) Ca2+ and Mg2+) reacted and the metal ion was bonded to the bisphosphonate. In addition, reaction paths were modeled for the metal exchange processes. According to the results, magnesium and calcium bonding in bisphosphonate complexes is competing, with slight magnesium preference. In SCRF calculations, Ca2+ was replaced by Mg2+ ion in both MBP-metal-[H2O]6 and Cl2MBP-metal-[H2O]6 complexes. The model reactions for the formation of the studied MBPCa-[water]n complexes (n ) 1-6) are presented in Table 7. Dianionic MBP and dicationic Ca2+ and Mg2+ produced bisphosphonate-metal complexes with energies of -2163 and -2629, kJ mol-1 respectively. These exothermic energies calculated with the 3-21G(*) basis set are high relative to the values -366 and -344 kJ mol-1 obtained in SCRF calculations (Table 7). The SCRF model reaction energies for the formation of the bisphosphonate-metal-water complexes with six explicit
8238 J. Phys. Chem., Vol. 100, No. 20, 1996
Ra¨sa¨nen et al.
TABLE 7: Model Reactions for the Compounds Studieda [Ca(H2O)6]2+
reactants [H2O]6 [MBP‚(H2O)6]2[H2O]6 [MBP‚(H2O)6]2[CaMBP‚(H2O)6] [CaCl2MBP‚(H2O)6] [CaMBP‚(H2O)5] [CaMBP‚(H2O)5] [MBP]2[Cl2MBP]2+ [CaMBP] [CaCl2MBP] [MBP]2[Cl2MBP]2[MgMBP] [MgCl2MBP]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a
+ + + + + + + + + + + + + + + +
Ca2+ [Ca(H2O)6]2+ Mg2+ [Mg(H2O)6]2+ [Mg(H2O)6]2+ [Mg(H2O)6]2+ [Mg(H2O)5]2+ [Mg(H2O)6]2+ Ca2+ Ca2+ [H2O]6 [H2O]6 Mg2+ Mg2+ [H2O]6 [H2O]6
f f f f f f f f f f f f f f f f
[Ca(H2O)6]2+ [CaMBP‚(H2O)6] [Mg(H2O)6]2+ [MgMBP‚(H2O)6] [MgMBP‚(H2O)6] [MgCl2MBP‚(H2O)6] [MgMBP‚(H2O)5] [MgMBP‚(H2O)5] [CaMBP] [CaCl2MBP] [CaMBP‚(H2O)6] [CaCl2MBP‚(H2O)6] [MgMBP] [MgCl2MBP] [MgMBP‚(H2O)6] [MgCl2MBP‚(H2O)6]
+ [H2O]6 + [H2O]6 + + + + +
[H2O]6 [Ca(H2O)6]2+ [Ca(H2O)6]2+ [Ca(H2O)5]2+ [Ca(H2O)6]2+
3-21G(*)
MP2
SCRF
-1068.2 -1388.1 -1476.0 -1418.6 -30.5 -39.1 -59.5 -53.6 -2162.7 -2055.9 -543.1 -542.1 -2628.5 -2504.2 -515.7 -540.6
-761.1 -1480.3 -1510.0 -1488.0 -7.7 -19.9 -63.0 -46.6 -2021.9 -1882.1 -465.9 -495.6 -2689.2 -2565.5 -555.2 -581.0
-321.5 -212.1 -405.7 -230.8 -18.7 -29.5 -66.7 -51.4 -365.5 -276.9 -319.9 -335.2 -344.2 -252.5 -444.0 -473.4
BSSE
-178.7 -161.6 -178.3 -164.9
Energies, EF, in kJ mol-1.
TABLE 8: Total Energies of the Selected Compounds Calculated with the 3-21G(*) Basis Set and the Total Solvation Energies (Esolv) Calculated by Subtracting the SCF Energies from the SCRF Energies 3-21G(*)
a1 a5 a13 b1 b5 b13 c1 c5 c13 d1 d5 d13
-1838.722 618 -2511.869 567 -2292.608 662 -2752.339 434 -3425.482 449 -3206.225 064 -1363.789 116 -1561.914 443 -1817.664 706 -2277.399 276 -2475.515 984 -2731.284 367
MP2 E/au
SCRF
Esolv/ kJ mol-1
-1839.659 910 -2512.799 530 -2294.273 315 -2753.539 342 -3426.650 594 -3206.225 064 -1364.803 151 -1562.920 008 -1819.450 574 -2278.688 709 -2476.799 168 -2733.345 935
-1838.823 789 -2512.476 734 -2292.634 728 -2752.431 136 -3425.961 069 -3206.247 928 -1363.827 405 -1562.228 106 -1817.685 646 -2277.433 574 -2475.803 555 -2731.302 983
-266 -1594 -68 -241 -1257 -60 -101 -824 -55 -90 -755 -49
water molecules were -320 and -444 kJ mol-1 as obtained for Ca2+ and Mg2+ species, respectively. Formation energies for Ca2+ and Mg2+ species were virtually converged when at least four water molecules were included in the complexes. Explicit water molecules from one to four contributed almost the same energy when bound to the complexes, whereas the energies contributions of the fifth and the sixth water molecules were somewhat less. The results indicate that an explicit first water sphere with SCRF solvent sphere should be used in the energy studies of reactions of bisphosphonate complexes. Total energies are reported in Table 8. Conclusions Complexation between dianionic MBP and Cl2MBP species and dicationic calcium and magnesium ions was studied by ab initio MO methods. The effects of aqueous solution were modeled. All bond lengths, bond angles, and torsional angles of the studied species were optimized. The molecular structures calculated for the bisphosphonates agreed well with the theoretical and experimental values reported for related compounds. In the minimum energy structures of the MBP-metal and Cl2MBP-metal complexes, the bisphosphonate molecules acted as tridentate ligands when bound to a single metal. Tridentate metal bonding was replaced by bidentate metal bonding in MBP-metal-water and Cl2MBP-metal-water complexes when more than three explicit water molecules were included in the complex. However, the ionic interactions were so strong that water molecules were unable to break the bisphosphonatemetal ion pair. Bidentate bonding was also preferred in the complexation with two metal ions. The calculated Ca‚‚‚Ca
distances in the Ca-MBP-Ca complexes were similar to Ca‚‚‚Ca distances in the crystal structure of the hydroxyapatite. The Mg‚‚‚Mg distances were shorter than those calculated for bi-calcium species. The present study demonstrated that more than one calcium may be bonded to the same bisphosphonate molecule. This observation may explain the tight bonding of the bisphosphonates to the bone structure. Oxygen atoms of the phosphorus groups prefer to form strong hydrogen bonds with hydrogens of water molecules. Calcium and magnesium ions were bonded to similar sites. The smaller ionic size of magnesium allowed closer contacts for Mg‚‚‚O interactions, and the Ca‚‚‚O distances were longer, respectively. The bisphosphonate-magnesium complexes were more compact and the P-C-P angles more closed relative to the corresponding bisphosphonate-calcium complexes. Calculated global minimum energy structures and the other calculated conformations give useful background information on the precipitation reactions of bisphosphonates and the various bonding possibilities of bisphosphonate molecules to bone matrix. The calculated MBP- and Cl2MBP-metal-water complexes can be regarded as prototypes for bisphosphonate-calcium-water and bisphosphonate-magnesium-water complexations in human body fluids. Furthermore, the molecular geometries obtained by ab initio MO methods were in good agreement with experimental findings, and the theoretical calculations of the bisphosphonates accordingly can be considered a reliable source of information on the molecular scale. References and Notes (1) Pera¨kyla¨, M.; Pakkanen, T. A.; Bjo¨rkroth, J.-P.; Pohjala, E. J. Chem. Soc., Perkin Trans. 2 1992, 1167. (2) Ra¨sa¨nen, J. P.; Pera¨kyla¨, M.; Pohjala, E.; Pakkanen, T. A. J. Chem. Soc., Perkin Trans. 2 1994, 1055. (3) Ra¨sa¨nen, J. P.; Pohjala, E.; Pakkanen, T. A. J. Chem. Soc., Perkin Trans. 2 1994, 2485. (4) Ra¨sa¨nen, J. P.; Pohjala, E.; Pakkanen, T. A. J. Chem. Soc., Perkin Trans. 2 1996, 39. (5) Mundy, G. R. Bone 1991, 12, 1. (6) Kanis, J. A.; McCloskey, E. V.; Taube, T.; O’Rourke, N. Bone 1991, 12, 13. (7) Bonjour, J.-P.; Rizzoli, R. Bone 1991, 12, 19. (8) Delmas, P. D. Bone 1991, 12, 31. (9) Burki, F. Bone 1991, 12, 35. (10) Van Rooijen, N. Calcif. Tissue Int. 1993, 52, 407. (11) Papapoulos, S. E.; Landman, J. O.; Bijvoet, O. L. M.; Lo¨wik, C. W. G. M.; Valkema R.; Pauwels, E. K. J.; Vermeij, P. Bone 1992, 13, 41. (12) Powles, T. Bone 1991, 12, 43. (13) Biermann, W. A.; Cantor, R. I.; Fellin, F. M.; Jakobowski, J.; Hopkins, L.; Newbold, R. C., III. Bone 1991, 12, 37. (14) Hannuniemi, R.; Lauren, L.; Puolijoki, H. Med. Actual. 1991, 27, 375. (15) Kanis, J. A.; McCloskey, E. V. Calcium Metab. Prog. Basic Clin. Pharmacol. 1990, 4, 89.
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