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Journal of Biomolecular Structure and Dynamics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tbsd20
Molecular Modeling of Proteins: A Strategy for Energy Minimization by Molecular Mechanics in the AMBER Force Field a
R. Manjunatha Kini & Herbert J. Evans
a
a
Department of Biochemistry and Molecular Biophysics , Medical College of Virginia Virginia Commonwealth University , Richmond , Virginia , 23298-0614 Published online: 21 May 2012.
To cite this article: R. Manjunatha Kini & Herbert J. Evans (1991) Molecular Modeling of Proteins: A Strategy for Energy Minimization by Molecular Mechanics in the AMBER Force Field, Journal of Biomolecular Structure and Dynamics, 9:3, 475-488, DOI: 10.1080/07391102.1991.10507930 To link to this article: http://dx.doi.org/10.1080/07391102.1991.10507930
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or
indirectly in connection with, in relation to or arising out of the use of the Content.
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This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Journal of Biomolecular Structure & Dynamics, /SSN 0739-1102 Volume 9, Issue Number 3 (1991), "'Adenine Press (1991).
Molecular Modeling of Proteins: A Strategy for Energy Minimization by Molecular Mechanics in the AMBER Force Field
Downloaded by [Rutgers University] at 22:02 09 April 2015
R. Manjunatha Kini and Herbert J. Evans Department of Biochemistry and Molecular Biophysics Medical College of Virginia Virginia Commonwealth University Richmond, Virginia 23298-0614 Abstract Energy minimization is an important step in molecular modeling of proteins. In this study, we sought to develop a minimization strategy which would give the best final structures with the shortest computer time in the AMBER force field. In the all-atom model, we performed energy minimization of the melittin (mostly a-helical) and cardiotoxin (mostly ~-sheet and ~-turns) crystal s"tructures by both constrained and unconstrained pathways. In the constrained path, which has been recommended in the energy minimization of proteins, hydrogens were relaxed first, followed by the side chains of amino acid residues, and finally the whole molecule. Despite the logic of this approach, however, the structures minimized by the unconstrained path fit the experimental structures better than those minimized by constrained paths. Moreover, the unconstrained path saved considerable computer time. We also compared the effects of the steepest descents and conjugate gradients algorithms in energy minimization. Previously, steepest descents has been used in the initial stages of minimization and conjugate gradients in the final stages of minimization. We therefore studied the effect on the final structure of performing an initial minimization by steepest descents. The structures minimized by conjugate gradients alone resembled the structures minimized initially by the steepest descents and subsequently by the conjugate gradients algorithms. Thus an initial minimization using steepest descents is wasteful and unnecessary, especially when starting from the crystal structure. Based on these results, we propose the use of an unconstrained path and conjugate gradients for energy minimization of proteins. This procedure results in low energy structures closer to the experimental structures, and saves about 70-800/0 of computer time. This procedure was applied in building models of lysozyme mutants. The crystal structure of native T 4 lysozyme was mutated to three different mutants and the structures were minimized. The minimized structures closely fit the crystal structures of the respective mutants ( < 0.3 Arootmean-square, RMS, deviation in the position of all heavy atoms). These results confirm the efficiency of the proposed minimization strategy in modeling closely related homo logs. To determine the reliability of the united atom approximation, we also performed all of the above minimizations with united atom models. This approximation gave structures with similar but slightly higher RMS deviations than the all-atom model, but gave further savings of 60-70% in computer time. However, we feel further investigation is essential to determine the reliability of this approximation. Finally, to determine the limitation of the procedure, we built the melittin molecule interactively in an a-helical conformation and this model showed an RMS deviation greater than 2.8 A when compared to the melittin crystal structure. This
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Kini and Evans
model was minimized by various strategies. None of the minimized structures converged towards the crystal structure. Thus, although the proposed method seems to give valid structures starting from closely related crystal structures, it cannot predict the native structure when the starting structure is far from the native structure. From these results, we recommend the use of the proposed strategy of minimizing by an unconstrained path using the conjugate gradients algorithm, but only for modeling of closely related structural homologs of proteins.
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Introduction Models of proteins help in planning and interpretation ofbiochemical and mutagenesis experiments, in probing structure-function relationships and functional properties, determination of mechanisms of catalysis, and in designing specific ligands and inhibitors. Most of these models are based on either X-ray diffraction data or nuclear magnetic resonance spectra. Despite recent progress (1-3), the number of proteins whose structure is solved completely is a small fraction (less than 1%) of the number of proteins whose amino acid sequences are known (4).1t is evident from the large number of protein sequences extant that proteins, although unique in their sequences, share significant sequence homology and can be grouped in families. Solving crystal structures of all the homologs in a protein family would be monotonous and unproductive. Thus model building based on comparable structures and by simulations of protein folding with computational methods is becoming more popular (5-8). After building such models, the structures are subjected to energy minimization to obtain the low energy, and hence stable, conformations of the molecules. The final structure, however, depends on several parameters chosen during the minimization. Calculation of potential energy forms the basis of these computations. Therefore the reliability of the mathematical expressions, which describe the molecular energies and interactions between the atoms as a function of their coordinates, plays an important role in energy minimization. Although much progress has been made in derivation of valid energy functions and parameters to simulate various macromolecular systems, this work is still incomplete. At present, several force fields are available to calculate the potential energy of proteins and perform energy minimizations (9-13). After defining the force field, target functions which include external restraining terms to bias the minimization are constructed. By these biasing functions a variety of minimization strategies can be developed. Such strategies and thus the target functions significantly determine the resulting minimized structure. Finally, during the minimization calculations, the conformation of the molecule is adjusted to lower the energy value of the target function by different algorithms. We have been working on structure-function relationships of toxic proteins from snake venoms (14-18). Several of these toxic proteins share structural homology with nontoxic mammalian proteins, making the structure-function relationships complicated and elusive. Venom phospholipases provide good examples of such subtleties (19). Despite sharing homology in primary, secondary and tertiary structures with nontoxic mammalian pancreatic phospholipases (20-22), the venom enzymes exhibit a wide variety of pharmacological effects (19). The protein models of these homologs should contribute to our understanding of the structure-function relationships. We plan to build models ofhomologs by mutating the crystal structure from the protein data bank and using energy minimization methods. For these reasons we examined
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Energy Minimization in the AMBER Force Field
477
various minimization strategies to determine the optimal conditions for energy minimization. Earlier, Hall and Pavitt showed that the AMBER (Assisted Model Building with Energy Refinement) force field is superior to other alternative force fields (23). Whitlow and Teeter (24) have examined the effects of the form of electrostatic potential energy, i. e., constant and distance-dependent, the value of the dielectric constant, the size of the van der Waals radii in the united atom model, the non bonding cutoff, the inclusion ofHC, and the addition of a crystal environment and of explicit water in the AMBER force field. In a recent paper, Dauber-Osguthorpe et al. described various minimization techniques applied in the consistent valence force field (13). The all-atom minimized structures fit the experimental structures much better than did the united atom minimized structures. Our goal was to develop a procedure which can routinely be used to build models of homologous proteins starting from the experimental structure of a closely related protein. In this study, we investigated the validity of various minimization strategies in the AMBER force field (9,10). We compared the minimization of proteins by both the all-atom method and the united atom approximation, and addressed the reliability of the united atom approximation. We also examined the contribution of two different algorithms, steepest descents and conjugate gradients, to the energy minimizations and indicate the limitations of molecular mechanics in building a protein model.
Materials and Methods Theoretical Considerations
X-ray crystallographic structures of proteins describe the average position of various heavy atoms in space in a given molecule. In nature molecules tend to be in their most stable, low energy conformations. Therefore it is logical to assume that the crystal structure of a protein is the average of all low energy conformations. Energy minimization of these structures should result in low energy structures, but not too distorted from the observed crystal structures. We therefore used various methods to minimize the crystal structures of proteins in order to determine the validity of these techniques. The minimization strategy resulting in a low energy conformation and closest to the crystal structure was considered the superior method. Choice of Proteins
For our initial studies we chose two small polypeptides representing different conformational classes. Melittin is a 26-residue hemolytic peptide isolated from Apis mellifera bee venom. Structurally melittin forms an a helix with a kink in the middle due to the presence of a Pro residue (25). Cardiotoxin V!1isolated from Naja mossambica mossambica venom is a 60-residue peptide. This polypeptide consists of ~-sheet and ~-turns (26). By energy minimization of these two proteins, we were able to develop certain minimization strategies. In the later stages, we have applied these strategies to the energy minimization of a larger protein, lysozyme. Lysozyme from T 4 bacteriophage has all classes of secondary conformations (10 a-helices, 4 ~-sheets, 4 ~ turns and random coils) (27). Furthermore, crystal structures of several mutants of this lysozyme are readily available. Thus T4 lysozyme and its mutants provide good opportunities to determine the validity of the minimization method in modeling
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Get the Crystal Structure From Brookhaven Data Bank
~
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Add Hydrogens to the structure
~ Calculate the Electrostatic Charges
~ Relax the Hydrogens
[g}
~Relax the
Hydrogens
Relax the Side Chains
/
Minimize the Whole Structure Figure 1: Schematic representation of various minimization strategies.
structures of homologs. To determine the limitations of energy minimization by molecular mechanics, we examined the effectiveness of this standardized method in predicting the structure of a simple a helical molecule, melittin. Crystal Structures
The crystal structures of the following proteins were from the Brookhaven protein data bank: melittin (lMLT), T 4 lysozyme (3LZM), Ala 82 Pro mutant (1L24), Pro 86 Ser mutant (1L32) and Lys 124 Gly mutant (1L22). The structural data for cardiotoxin V! 1were supplied by Dr. Bernard Rees. Since the structural water molecules are not defined in these crystal structures, we removed all water molecules to maintain uniformity during energy minimizations. In the case of melittin, only monomer A was used for minimization calculations.
Energy Minimization in the AMBER Force Field
479
Graphics and Software
Graphics modeling was performed interactively on an Evans & Sutherland PS390 terminal, using the Graphics program SYBYL (version 5.32, Tripos Associates). All the energy minimization calculations were performed on a VAX 6420 or VAX 8650 running VMS version 5.4.
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Minimization Strategy
The crystal structures generally do not include hydrogen coordinates. There is an option available with respect to addition of hydrogens: one can include all hydrogens or only polar hydrogens. In the latter case, the nonpolar hydrogens are merged with the parent carbon. To compensate for the lack of hydrogens the van der Waals radii of heavy atoms were increased. This united atom approximation reduces the number of atoms significantly and hence saves computation time. To determine the adequacy of this approximation, we minimized the protein structures by both of these options separately. The hydrogens and lone pairs are reviewed and added by SYBYL software based on the above option. The electrostatic charges on various atoms are defined by the AMBER force field. In the all-atom method, by using external restraining terms to bias the minimization, three different procedures were designed to minimize the protein molecule (Figure 1). Procedure Cis a typical approach for relaxing a protein system (28). In this procedure, all the heavy atoms are constrained and only hydrogens are relaxed in a first step. In the second step, all the backbone atoms are constrained and the side chains are relaxed. Finally, after removing all the constraints, the whole molecule is relaxed (Figure I). These constraints are designed to remove the close contacts that may have been overlooked during the addition of hydrogens. In procedures A and B, some of the initial steps have been eliminated. All other parameters for minimization were the default parameters in SYBYL. All the minimizations were carried out in the AMBER force field (9,10). Structures were refined until the final RMS energy gradient was less than 0.1 Kcal/mol. At each stage the minimization was complete, unless otherwise mentioned. In the case of the united atom approximations, energy minimization was carried out to relax the whole protein molecule directly by each algorithm separately. To determine the role of different algorithms in energy minimizations, two separate sets of minimizations were performed, using the steepest descents and the conjugate gradients algorithms. The use of the steepest descents algorithm has been recommended for the initial stages of minimization and the conjugate gradients for the final stages of convergence (28). We also minimized the structures by this recommended method. Comparison of Structures
The protein structures obtained after energy minimization were compared with the experimental structures. This was achieved by the superposition of the minimized structure onto the crystal structure using a least-squares fit of all the heavy atoms or a-carbons. The root-mean-square (RMS) deviations of all the heavy atoms or acarbons were used as the estimate of fit.
480
Kini and Evans Table I Effect of Minimization Strategy on Energy Minimization by the All-Atom Method
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Final Energy Number of Evaluation (Kcal/mol)" Iterations Count
A. Melittin Conjugate gradients Procedure A Procedure B Procedure C Steepest descents Procedure A Procedure B Procedure C B. Cardiotoxin Conjugate gradients Procedure A Procedure B Procedure C Steepest descents Procedure A Procedure B Procedure C
(A)h
CPUTimec (Hrs:min)
Root Mean Square
52.5 82.0 236.7
202 314 351
1178 2040 2206
0.653 (0.344) 0.890 (0.502) 0.766 (0.376)
2:14 3:31 2:36
114.4 216.4 204.9
828 582 937
4973 3440 5738
0.229 (0.169) 0.361 (0.174) 0.3 72 (0.17 5)
21:55 6:23 7:49
610.3 604.6 684.9
102 309 630
837 2556 4913
0.308 (0.207) 0.590 (0.417) 0.648 (0.427)
4:11 12:11 23:26
648.9 1534.0 1425.4
2448 622 2203
20748 5246 18672
0.227 (0.152) 0.690 (0.417) 0.716 (0.450)
112:54 23:54 63:40
• The energy zero is arbitrary. The potential energy of one protein cannot be compared with that of the other. However, the relative energies of the same molecule in different conformations can be compared. The initial potential energies for the melittin and cardiotoxin structures were 1011.4 and 8508.9 Kcal/ mol, respectively. h RMS deviations represent the average deviation in the position of all heavy atoms. The numbers in parentheses represent the RMS deviations of all a-carbon atoms. RMS deviations are between the initial crystal structure and the final minimized structure. c Energy minimization calculations were performed on either a VAX 6420 or a VAX8650 computer. CPU times for the calculations on these computers vary by ± 10%.
Results and Discussion In several cases, planning of experiments and interpretation of results are dependent on the accuracy of the protein models. Due to inherent problems, it is difficult to obtain experimental structures for all the proteins and their homologs. With the advent of better computers and computational methods, model building based on structural homology is both feasible and encouraging (5-7). Although there are several inherent drawbacks, computational methods provide us with fairly accurate protein models. In such model building, the energy minimization procedure is of critical importance, as it contributes significantly to the final model and hence to the conclusions based on the model. We investigated several minimization strategies and various algorithms used in minimization. Role of Constraints in Energy Minimizations
During minimization, constraints can be used to define the target functions, which in turn bias the minimization. By using different constraints a variety of minimization
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Energy Minimization in the AMBER Force Field
481
strategies can be developed. We have studied the role of three different sets of constraints in energy minimization (Figure 1). In procedure C, the hydrogens added to the crystal structure are allowed to find a low energy conformation in the first step. In two subsequent steps, the side chains of amino acid residues and the whole molecule are relaxed (Figure 1). This strategy for energy minimization appears to be most logical. In procedures A and B, some of the initial steps have been eliminated. Table I shows the results of energy minimization of melittin and cardiotoxin using these three procedures in both the steepest descents and conjugate gradients algorithms. Surprisingly, procedure C, the most commonly used approach for minimization (28), not only takes a large amount of computer time, but also results in models with high RMS deviations from the crystal structure (Table 1). This may be due to the change in the position of the atoms to reach the minimized structure in the initial stages. Procedure A, which minimizes the whole molecule without any of the initial stages, results in structures with comparatively low RMS deviations from the starting crystal structures in shorter computational times (Table I). Based on the final minimized structures and the faster convergence, we conclude that procedure A is the best of the three minimization strategies described here. In all of the subsequent studies, only procedure A has been used for energy minimization. Whitlow and Teeter also studied the effects of constrained and unconstrained paths, similar to procedures C and A, respectively, of minimization in building models of homologous proteins (29). In the case of crambin, both paths resulted in comparable structures (RMS deviations of0.84 Aand 0.88 A. respectively). But the unconstrained path used 40% fewer iterations to reach convergence (29). However, those studies were made using the united atom approximation. Role ofAlgorithms in Energy Minimization
The conformation of the molecule is adjusted to lower the energy value of the target function. There are several algorithms which choose the directions for line search during minimization. Thus the final minimized structure depends on the algorithm used in minimization. We investigated the role of the steepest descents and conjugate gradients algorithms both separately and in combination in energy minimization. (Other algorithms were not used in our minimizations since they require large amounts of computer memory). In all of the minimization paths (Figure 1), the steepest descents algorithm results in structures with higher energy, but lower RMS deviations compared to the structures minimized using conjugate gradients (Table I). This may be due to the differential ability of these two algorithms. Although steepest descents is a robust algorithm, it sometimes retraces the same positions of the atoms and hence is a slowest converging algorithm (28). This algorithm has been recommended for only the initial stages of minimization when the structures are far from the low energy conformations. Subsequently, energy minimization is carried out using the conjugate gradients algorithm (28). To determine the effect of initial minimization using the steepest descents algorithm, we minimized the structures, which were initially minimized either for 100 iterations or to convergence with the steepest descents algorithm, using conjugate gradients. In general, the total potential energy and the RMS deviations of these structures from the crystal structures are comparable to those of the structures which are minimized using only
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Kini and Evans Table II Role of Algorithms in Energy Minimization of Melittin and Cardiotoxin Final Energy Number of Evaluation (Kcal/mol)" Iterations Count
(A)h
CPUTimec (Hrs:min)
Root Mean Square
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A. Melittin
All-Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradientsd Steepest descents 100/ conjugate gradients• United Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradientsd Steepest descents 100/ conjugate gradients•
52.5 114.4 49.0
202 828 979
1178 4973 5854
0.653 (0.344) 0.229 (0.169) 0.630 (0.320)
2:14 9:15 10:56
50.4
302
1781
0.654 (0.337)
3.06
-81.9 -45.3 -82.0
187 704 806
1101 4050 4652
0.736 (0.429) 0.400 (0.225) 0.753 (0.454)
0:34 2:44 3:08
-62.2
248
626
0.552 (0.334)
0:32
610.3 648.9 502.5
102 2448 2560
837 20748 21643
0.308 (0.207) 0.227 (0.152) 0.436 (0.293)
4:11 112:54 117:30
436.0
372
2947
0.568 (0.371)
18:53
155.8 359.2 138.0
102 541 621
841 4616 5278
0.387 (0.243) 0.153 (0.098) 0.437 (0.284)
2:12 8:04 9:37
135.8
212
1746
0.388 (0.245)
3:38
B. Cardiotoxin
All-Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradientsd Steepest descents 100/ conjugate gradients' United Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradientsd Steepest descents 100/ conjugate gradients•
• The energy zero is arbitrary. The potential energy of one protein cannot be compared with that of the other. The energies in the all-atom method cannot be compared with the united atom model. However, the relative energies of the same molecule in different conformations can be compared. The initial potential energies for the melittin and cardiotoxin structures were 1011.4 and 8508.9 Kcal/ mol, respectively, in the all-atom model, and 473.4 and 7766.4 KcaVmol, respectively, in the united atom model. h RMS deviations represent the average deviation in the position of all heavy atoms. The numbers in parentheses represent the RMS deviations of all a-carbon atoms. RMS deviations are between the initial crystal structure and the final minimized structure. cEnergy minimization calculations were performed on either a VAX 6420 or a VAX 8650 computer. CPU times for the calculations on these computers vary by ± 10%. d The structures initially minimized using the steepest descents algorithm to convergence were further minimized by conjugate gradients. • The structures initially minimized for 100 iterations using the steepest descents algorithm were further minimized by conjugate gradients.
483
Energy Minimization in the AMBER Force Field Table III Modeling ofT4 Lysozyme Mutants
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Final Energy Number of Evaluation (Kcal/mol)" Iterations Count A Ala82 Pro Crystal Structures All-Atom Method Conjugate gradients Steepest descents 100/ conjugate gradientsd United Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradients< Steepest descents 100/ conjugate gradientsd B. Pro 86 Ser Crystal Structures All-Atom Method Conjugate gradients Steepest descents 100/ conjugate gradientsd United Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradients< Steepest descents 100/ conjugate gradientsd C. Lys 124 Gly Crystal Structures All-Atom Method Conjugate gradients Steepest descents 100/ conjugate gradientsd United Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradients< Steepest descents 100/ conjugate gradientsd
(A)h
CPUTimec (Hrs:min)
0.078 (0.057)
None
Root Mean Square
-814.9 -928.6
102 262
819 2127
0.226 (0.156) 0.291 (0.191)
14:24 38:40
-1754.5 -1378.3 -1750.0
162 571 683
1270 4877 5756
0.349 (0.244) 0.145 (0.106) 0.355 (0.229)
7:46 28:10 33:57
-1736.7
232
1894
0.330 (0.229)
15:18
0.153 (0.122)
None
-822.3 -870.0
102 202
823 823
0.272 (0.197) 0.289 (0.206)
15:02 30:45
-1737.4 -1370.8 -1477.4
172 571 583
1358 4877 5006
0.376 (0.273) 0.209 (0.160) 0.229 (0.174)
8:30 28:28 29:29
-1719.6
202
1687
0.358 (0.256)
13:26
0.172 (0.145)
None
-818.5 -802.9
102 182
825 1507
0.269 (0.20 1) 0.255 (0.194)
15:13 27:03
-1775.0 -1393.5 -1733.6
182 571 703
1415 4877 5916
0.374 (0.256) 0.206 (0.166) 0.350 (0.241)
8:45 28:30 34:35
-1707.4
212
1746
0.336 (0.230)
10:33
" The energy zero is arbitrary. The potential energy of one protein cannot be compared with that of the other. The energies in the all-atom method cannot be compared with the united atom model. However, the relative energies of the same molecule in different conformations can be compared. Following were the initial potential energies for the lysozyme mutant structures in the all-atom and united atom models, respectively: Ala 82 Pro, 6876.2 and 4201.2 Kcal/mol; Pro 86 Ser, 6836.2 and 4182.6 Kcal/mol; and Lys 124 Gly, 6824.2 and 4152.1 Kcal/mol. h RMS deviations represent the average deviation in the position of all heavy atoms. The numbers in parentheses represent the RMS deviations of all a-carbon atoms. RMS deviations are between the initial crystal structure and the final minimized structure.
Journal of Biomolecular Structure and Dynamics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tbsd20
Molecular Modeling of Proteins: A Strategy for Energy Minimization by Molecular Mechanics in the AMBER Force Field a
R. Manjunatha Kini & Herbert J. Evans
a
a
Department of Biochemistry and Molecular Biophysics , Medical College of Virginia Virginia Commonwealth University , Richmond , Virginia , 23298-0614 Published online: 21 May 2012.
To cite this article: R. Manjunatha Kini & Herbert J. Evans (1991) Molecular Modeling of Proteins: A Strategy for Energy Minimization by Molecular Mechanics in the AMBER Force Field, Journal of Biomolecular Structure and Dynamics, 9:3, 475-488, DOI: 10.1080/07391102.1991.10507930 To link to this article: http://dx.doi.org/10.1080/07391102.1991.10507930
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or
indirectly in connection with, in relation to or arising out of the use of the Content.
Downloaded by [Rutgers University] at 22:02 09 April 2015
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Journal of Biomolecular Structure & Dynamics, /SSN 0739-1102 Volume 9, Issue Number 3 (1991), "'Adenine Press (1991).
Molecular Modeling of Proteins: A Strategy for Energy Minimization by Molecular Mechanics in the AMBER Force Field
Downloaded by [Rutgers University] at 22:02 09 April 2015
R. Manjunatha Kini and Herbert J. Evans Department of Biochemistry and Molecular Biophysics Medical College of Virginia Virginia Commonwealth University Richmond, Virginia 23298-0614 Abstract Energy minimization is an important step in molecular modeling of proteins. In this study, we sought to develop a minimization strategy which would give the best final structures with the shortest computer time in the AMBER force field. In the all-atom model, we performed energy minimization of the melittin (mostly a-helical) and cardiotoxin (mostly ~-sheet and ~-turns) crystal s"tructures by both constrained and unconstrained pathways. In the constrained path, which has been recommended in the energy minimization of proteins, hydrogens were relaxed first, followed by the side chains of amino acid residues, and finally the whole molecule. Despite the logic of this approach, however, the structures minimized by the unconstrained path fit the experimental structures better than those minimized by constrained paths. Moreover, the unconstrained path saved considerable computer time. We also compared the effects of the steepest descents and conjugate gradients algorithms in energy minimization. Previously, steepest descents has been used in the initial stages of minimization and conjugate gradients in the final stages of minimization. We therefore studied the effect on the final structure of performing an initial minimization by steepest descents. The structures minimized by conjugate gradients alone resembled the structures minimized initially by the steepest descents and subsequently by the conjugate gradients algorithms. Thus an initial minimization using steepest descents is wasteful and unnecessary, especially when starting from the crystal structure. Based on these results, we propose the use of an unconstrained path and conjugate gradients for energy minimization of proteins. This procedure results in low energy structures closer to the experimental structures, and saves about 70-800/0 of computer time. This procedure was applied in building models of lysozyme mutants. The crystal structure of native T 4 lysozyme was mutated to three different mutants and the structures were minimized. The minimized structures closely fit the crystal structures of the respective mutants ( < 0.3 Arootmean-square, RMS, deviation in the position of all heavy atoms). These results confirm the efficiency of the proposed minimization strategy in modeling closely related homo logs. To determine the reliability of the united atom approximation, we also performed all of the above minimizations with united atom models. This approximation gave structures with similar but slightly higher RMS deviations than the all-atom model, but gave further savings of 60-70% in computer time. However, we feel further investigation is essential to determine the reliability of this approximation. Finally, to determine the limitation of the procedure, we built the melittin molecule interactively in an a-helical conformation and this model showed an RMS deviation greater than 2.8 A when compared to the melittin crystal structure. This
475
476
Kini and Evans
model was minimized by various strategies. None of the minimized structures converged towards the crystal structure. Thus, although the proposed method seems to give valid structures starting from closely related crystal structures, it cannot predict the native structure when the starting structure is far from the native structure. From these results, we recommend the use of the proposed strategy of minimizing by an unconstrained path using the conjugate gradients algorithm, but only for modeling of closely related structural homologs of proteins.
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Introduction Models of proteins help in planning and interpretation ofbiochemical and mutagenesis experiments, in probing structure-function relationships and functional properties, determination of mechanisms of catalysis, and in designing specific ligands and inhibitors. Most of these models are based on either X-ray diffraction data or nuclear magnetic resonance spectra. Despite recent progress (1-3), the number of proteins whose structure is solved completely is a small fraction (less than 1%) of the number of proteins whose amino acid sequences are known (4).1t is evident from the large number of protein sequences extant that proteins, although unique in their sequences, share significant sequence homology and can be grouped in families. Solving crystal structures of all the homologs in a protein family would be monotonous and unproductive. Thus model building based on comparable structures and by simulations of protein folding with computational methods is becoming more popular (5-8). After building such models, the structures are subjected to energy minimization to obtain the low energy, and hence stable, conformations of the molecules. The final structure, however, depends on several parameters chosen during the minimization. Calculation of potential energy forms the basis of these computations. Therefore the reliability of the mathematical expressions, which describe the molecular energies and interactions between the atoms as a function of their coordinates, plays an important role in energy minimization. Although much progress has been made in derivation of valid energy functions and parameters to simulate various macromolecular systems, this work is still incomplete. At present, several force fields are available to calculate the potential energy of proteins and perform energy minimizations (9-13). After defining the force field, target functions which include external restraining terms to bias the minimization are constructed. By these biasing functions a variety of minimization strategies can be developed. Such strategies and thus the target functions significantly determine the resulting minimized structure. Finally, during the minimization calculations, the conformation of the molecule is adjusted to lower the energy value of the target function by different algorithms. We have been working on structure-function relationships of toxic proteins from snake venoms (14-18). Several of these toxic proteins share structural homology with nontoxic mammalian proteins, making the structure-function relationships complicated and elusive. Venom phospholipases provide good examples of such subtleties (19). Despite sharing homology in primary, secondary and tertiary structures with nontoxic mammalian pancreatic phospholipases (20-22), the venom enzymes exhibit a wide variety of pharmacological effects (19). The protein models of these homologs should contribute to our understanding of the structure-function relationships. We plan to build models ofhomologs by mutating the crystal structure from the protein data bank and using energy minimization methods. For these reasons we examined
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Energy Minimization in the AMBER Force Field
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various minimization strategies to determine the optimal conditions for energy minimization. Earlier, Hall and Pavitt showed that the AMBER (Assisted Model Building with Energy Refinement) force field is superior to other alternative force fields (23). Whitlow and Teeter (24) have examined the effects of the form of electrostatic potential energy, i. e., constant and distance-dependent, the value of the dielectric constant, the size of the van der Waals radii in the united atom model, the non bonding cutoff, the inclusion ofHC, and the addition of a crystal environment and of explicit water in the AMBER force field. In a recent paper, Dauber-Osguthorpe et al. described various minimization techniques applied in the consistent valence force field (13). The all-atom minimized structures fit the experimental structures much better than did the united atom minimized structures. Our goal was to develop a procedure which can routinely be used to build models of homologous proteins starting from the experimental structure of a closely related protein. In this study, we investigated the validity of various minimization strategies in the AMBER force field (9,10). We compared the minimization of proteins by both the all-atom method and the united atom approximation, and addressed the reliability of the united atom approximation. We also examined the contribution of two different algorithms, steepest descents and conjugate gradients, to the energy minimizations and indicate the limitations of molecular mechanics in building a protein model.
Materials and Methods Theoretical Considerations
X-ray crystallographic structures of proteins describe the average position of various heavy atoms in space in a given molecule. In nature molecules tend to be in their most stable, low energy conformations. Therefore it is logical to assume that the crystal structure of a protein is the average of all low energy conformations. Energy minimization of these structures should result in low energy structures, but not too distorted from the observed crystal structures. We therefore used various methods to minimize the crystal structures of proteins in order to determine the validity of these techniques. The minimization strategy resulting in a low energy conformation and closest to the crystal structure was considered the superior method. Choice of Proteins
For our initial studies we chose two small polypeptides representing different conformational classes. Melittin is a 26-residue hemolytic peptide isolated from Apis mellifera bee venom. Structurally melittin forms an a helix with a kink in the middle due to the presence of a Pro residue (25). Cardiotoxin V!1isolated from Naja mossambica mossambica venom is a 60-residue peptide. This polypeptide consists of ~-sheet and ~-turns (26). By energy minimization of these two proteins, we were able to develop certain minimization strategies. In the later stages, we have applied these strategies to the energy minimization of a larger protein, lysozyme. Lysozyme from T 4 bacteriophage has all classes of secondary conformations (10 a-helices, 4 ~-sheets, 4 ~ turns and random coils) (27). Furthermore, crystal structures of several mutants of this lysozyme are readily available. Thus T4 lysozyme and its mutants provide good opportunities to determine the validity of the minimization method in modeling
478
Kini and Evans
Get the Crystal Structure From Brookhaven Data Bank
~
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Add Hydrogens to the structure
~ Calculate the Electrostatic Charges
~ Relax the Hydrogens
[g}
~Relax the
Hydrogens
Relax the Side Chains
/
Minimize the Whole Structure Figure 1: Schematic representation of various minimization strategies.
structures of homologs. To determine the limitations of energy minimization by molecular mechanics, we examined the effectiveness of this standardized method in predicting the structure of a simple a helical molecule, melittin. Crystal Structures
The crystal structures of the following proteins were from the Brookhaven protein data bank: melittin (lMLT), T 4 lysozyme (3LZM), Ala 82 Pro mutant (1L24), Pro 86 Ser mutant (1L32) and Lys 124 Gly mutant (1L22). The structural data for cardiotoxin V! 1were supplied by Dr. Bernard Rees. Since the structural water molecules are not defined in these crystal structures, we removed all water molecules to maintain uniformity during energy minimizations. In the case of melittin, only monomer A was used for minimization calculations.
Energy Minimization in the AMBER Force Field
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Graphics and Software
Graphics modeling was performed interactively on an Evans & Sutherland PS390 terminal, using the Graphics program SYBYL (version 5.32, Tripos Associates). All the energy minimization calculations were performed on a VAX 6420 or VAX 8650 running VMS version 5.4.
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Minimization Strategy
The crystal structures generally do not include hydrogen coordinates. There is an option available with respect to addition of hydrogens: one can include all hydrogens or only polar hydrogens. In the latter case, the nonpolar hydrogens are merged with the parent carbon. To compensate for the lack of hydrogens the van der Waals radii of heavy atoms were increased. This united atom approximation reduces the number of atoms significantly and hence saves computation time. To determine the adequacy of this approximation, we minimized the protein structures by both of these options separately. The hydrogens and lone pairs are reviewed and added by SYBYL software based on the above option. The electrostatic charges on various atoms are defined by the AMBER force field. In the all-atom method, by using external restraining terms to bias the minimization, three different procedures were designed to minimize the protein molecule (Figure 1). Procedure Cis a typical approach for relaxing a protein system (28). In this procedure, all the heavy atoms are constrained and only hydrogens are relaxed in a first step. In the second step, all the backbone atoms are constrained and the side chains are relaxed. Finally, after removing all the constraints, the whole molecule is relaxed (Figure I). These constraints are designed to remove the close contacts that may have been overlooked during the addition of hydrogens. In procedures A and B, some of the initial steps have been eliminated. All other parameters for minimization were the default parameters in SYBYL. All the minimizations were carried out in the AMBER force field (9,10). Structures were refined until the final RMS energy gradient was less than 0.1 Kcal/mol. At each stage the minimization was complete, unless otherwise mentioned. In the case of the united atom approximations, energy minimization was carried out to relax the whole protein molecule directly by each algorithm separately. To determine the role of different algorithms in energy minimizations, two separate sets of minimizations were performed, using the steepest descents and the conjugate gradients algorithms. The use of the steepest descents algorithm has been recommended for the initial stages of minimization and the conjugate gradients for the final stages of convergence (28). We also minimized the structures by this recommended method. Comparison of Structures
The protein structures obtained after energy minimization were compared with the experimental structures. This was achieved by the superposition of the minimized structure onto the crystal structure using a least-squares fit of all the heavy atoms or a-carbons. The root-mean-square (RMS) deviations of all the heavy atoms or acarbons were used as the estimate of fit.
480
Kini and Evans Table I Effect of Minimization Strategy on Energy Minimization by the All-Atom Method
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Final Energy Number of Evaluation (Kcal/mol)" Iterations Count
A. Melittin Conjugate gradients Procedure A Procedure B Procedure C Steepest descents Procedure A Procedure B Procedure C B. Cardiotoxin Conjugate gradients Procedure A Procedure B Procedure C Steepest descents Procedure A Procedure B Procedure C
(A)h
CPUTimec (Hrs:min)
Root Mean Square
52.5 82.0 236.7
202 314 351
1178 2040 2206
0.653 (0.344) 0.890 (0.502) 0.766 (0.376)
2:14 3:31 2:36
114.4 216.4 204.9
828 582 937
4973 3440 5738
0.229 (0.169) 0.361 (0.174) 0.3 72 (0.17 5)
21:55 6:23 7:49
610.3 604.6 684.9
102 309 630
837 2556 4913
0.308 (0.207) 0.590 (0.417) 0.648 (0.427)
4:11 12:11 23:26
648.9 1534.0 1425.4
2448 622 2203
20748 5246 18672
0.227 (0.152) 0.690 (0.417) 0.716 (0.450)
112:54 23:54 63:40
• The energy zero is arbitrary. The potential energy of one protein cannot be compared with that of the other. However, the relative energies of the same molecule in different conformations can be compared. The initial potential energies for the melittin and cardiotoxin structures were 1011.4 and 8508.9 Kcal/ mol, respectively. h RMS deviations represent the average deviation in the position of all heavy atoms. The numbers in parentheses represent the RMS deviations of all a-carbon atoms. RMS deviations are between the initial crystal structure and the final minimized structure. c Energy minimization calculations were performed on either a VAX 6420 or a VAX8650 computer. CPU times for the calculations on these computers vary by ± 10%.
Results and Discussion In several cases, planning of experiments and interpretation of results are dependent on the accuracy of the protein models. Due to inherent problems, it is difficult to obtain experimental structures for all the proteins and their homologs. With the advent of better computers and computational methods, model building based on structural homology is both feasible and encouraging (5-7). Although there are several inherent drawbacks, computational methods provide us with fairly accurate protein models. In such model building, the energy minimization procedure is of critical importance, as it contributes significantly to the final model and hence to the conclusions based on the model. We investigated several minimization strategies and various algorithms used in minimization. Role of Constraints in Energy Minimizations
During minimization, constraints can be used to define the target functions, which in turn bias the minimization. By using different constraints a variety of minimization
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Energy Minimization in the AMBER Force Field
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strategies can be developed. We have studied the role of three different sets of constraints in energy minimization (Figure 1). In procedure C, the hydrogens added to the crystal structure are allowed to find a low energy conformation in the first step. In two subsequent steps, the side chains of amino acid residues and the whole molecule are relaxed (Figure 1). This strategy for energy minimization appears to be most logical. In procedures A and B, some of the initial steps have been eliminated. Table I shows the results of energy minimization of melittin and cardiotoxin using these three procedures in both the steepest descents and conjugate gradients algorithms. Surprisingly, procedure C, the most commonly used approach for minimization (28), not only takes a large amount of computer time, but also results in models with high RMS deviations from the crystal structure (Table 1). This may be due to the change in the position of the atoms to reach the minimized structure in the initial stages. Procedure A, which minimizes the whole molecule without any of the initial stages, results in structures with comparatively low RMS deviations from the starting crystal structures in shorter computational times (Table I). Based on the final minimized structures and the faster convergence, we conclude that procedure A is the best of the three minimization strategies described here. In all of the subsequent studies, only procedure A has been used for energy minimization. Whitlow and Teeter also studied the effects of constrained and unconstrained paths, similar to procedures C and A, respectively, of minimization in building models of homologous proteins (29). In the case of crambin, both paths resulted in comparable structures (RMS deviations of0.84 Aand 0.88 A. respectively). But the unconstrained path used 40% fewer iterations to reach convergence (29). However, those studies were made using the united atom approximation. Role ofAlgorithms in Energy Minimization
The conformation of the molecule is adjusted to lower the energy value of the target function. There are several algorithms which choose the directions for line search during minimization. Thus the final minimized structure depends on the algorithm used in minimization. We investigated the role of the steepest descents and conjugate gradients algorithms both separately and in combination in energy minimization. (Other algorithms were not used in our minimizations since they require large amounts of computer memory). In all of the minimization paths (Figure 1), the steepest descents algorithm results in structures with higher energy, but lower RMS deviations compared to the structures minimized using conjugate gradients (Table I). This may be due to the differential ability of these two algorithms. Although steepest descents is a robust algorithm, it sometimes retraces the same positions of the atoms and hence is a slowest converging algorithm (28). This algorithm has been recommended for only the initial stages of minimization when the structures are far from the low energy conformations. Subsequently, energy minimization is carried out using the conjugate gradients algorithm (28). To determine the effect of initial minimization using the steepest descents algorithm, we minimized the structures, which were initially minimized either for 100 iterations or to convergence with the steepest descents algorithm, using conjugate gradients. In general, the total potential energy and the RMS deviations of these structures from the crystal structures are comparable to those of the structures which are minimized using only
482
Kini and Evans Table II Role of Algorithms in Energy Minimization of Melittin and Cardiotoxin Final Energy Number of Evaluation (Kcal/mol)" Iterations Count
(A)h
CPUTimec (Hrs:min)
Root Mean Square
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A. Melittin
All-Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradientsd Steepest descents 100/ conjugate gradients• United Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradientsd Steepest descents 100/ conjugate gradients•
52.5 114.4 49.0
202 828 979
1178 4973 5854
0.653 (0.344) 0.229 (0.169) 0.630 (0.320)
2:14 9:15 10:56
50.4
302
1781
0.654 (0.337)
3.06
-81.9 -45.3 -82.0
187 704 806
1101 4050 4652
0.736 (0.429) 0.400 (0.225) 0.753 (0.454)
0:34 2:44 3:08
-62.2
248
626
0.552 (0.334)
0:32
610.3 648.9 502.5
102 2448 2560
837 20748 21643
0.308 (0.207) 0.227 (0.152) 0.436 (0.293)
4:11 112:54 117:30
436.0
372
2947
0.568 (0.371)
18:53
155.8 359.2 138.0
102 541 621
841 4616 5278
0.387 (0.243) 0.153 (0.098) 0.437 (0.284)
2:12 8:04 9:37
135.8
212
1746
0.388 (0.245)
3:38
B. Cardiotoxin
All-Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradientsd Steepest descents 100/ conjugate gradients' United Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradientsd Steepest descents 100/ conjugate gradients•
• The energy zero is arbitrary. The potential energy of one protein cannot be compared with that of the other. The energies in the all-atom method cannot be compared with the united atom model. However, the relative energies of the same molecule in different conformations can be compared. The initial potential energies for the melittin and cardiotoxin structures were 1011.4 and 8508.9 Kcal/ mol, respectively, in the all-atom model, and 473.4 and 7766.4 KcaVmol, respectively, in the united atom model. h RMS deviations represent the average deviation in the position of all heavy atoms. The numbers in parentheses represent the RMS deviations of all a-carbon atoms. RMS deviations are between the initial crystal structure and the final minimized structure. cEnergy minimization calculations were performed on either a VAX 6420 or a VAX 8650 computer. CPU times for the calculations on these computers vary by ± 10%. d The structures initially minimized using the steepest descents algorithm to convergence were further minimized by conjugate gradients. • The structures initially minimized for 100 iterations using the steepest descents algorithm were further minimized by conjugate gradients.
483
Energy Minimization in the AMBER Force Field Table III Modeling ofT4 Lysozyme Mutants
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Final Energy Number of Evaluation (Kcal/mol)" Iterations Count A Ala82 Pro Crystal Structures All-Atom Method Conjugate gradients Steepest descents 100/ conjugate gradientsd United Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradients< Steepest descents 100/ conjugate gradientsd B. Pro 86 Ser Crystal Structures All-Atom Method Conjugate gradients Steepest descents 100/ conjugate gradientsd United Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradients< Steepest descents 100/ conjugate gradientsd C. Lys 124 Gly Crystal Structures All-Atom Method Conjugate gradients Steepest descents 100/ conjugate gradientsd United Atom Method Conjugate gradients Steepest descents Steepest descents/ conjugate gradients< Steepest descents 100/ conjugate gradientsd
(A)h
CPUTimec (Hrs:min)
0.078 (0.057)
None
Root Mean Square
-814.9 -928.6
102 262
819 2127
0.226 (0.156) 0.291 (0.191)
14:24 38:40
-1754.5 -1378.3 -1750.0
162 571 683
1270 4877 5756
0.349 (0.244) 0.145 (0.106) 0.355 (0.229)
7:46 28:10 33:57
-1736.7
232
1894
0.330 (0.229)
15:18
0.153 (0.122)
None
-822.3 -870.0
102 202
823 823
0.272 (0.197) 0.289 (0.206)
15:02 30:45
-1737.4 -1370.8 -1477.4
172 571 583
1358 4877 5006
0.376 (0.273) 0.209 (0.160) 0.229 (0.174)
8:30 28:28 29:29
-1719.6
202
1687
0.358 (0.256)
13:26
0.172 (0.145)
None
-818.5 -802.9
102 182
825 1507
0.269 (0.20 1) 0.255 (0.194)
15:13 27:03
-1775.0 -1393.5 -1733.6
182 571 703
1415 4877 5916
0.374 (0.256) 0.206 (0.166) 0.350 (0.241)
8:45 28:30 34:35
-1707.4
212
1746
0.336 (0.230)
10:33
" The energy zero is arbitrary. The potential energy of one protein cannot be compared with that of the other. The energies in the all-atom method cannot be compared with the united atom model. However, the relative energies of the same molecule in different conformations can be compared. Following were the initial potential energies for the lysozyme mutant structures in the all-atom and united atom models, respectively: Ala 82 Pro, 6876.2 and 4201.2 Kcal/mol; Pro 86 Ser, 6836.2 and 4182.6 Kcal/mol; and Lys 124 Gly, 6824.2 and 4152.1 Kcal/mol. h RMS deviations represent the average deviation in the position of all heavy atoms. The numbers in parentheses represent the RMS deviations of all a-carbon atoms. RMS deviations are between the initial crystal structure and the final minimized structure.
