A Molecular Dynamics Study of Thermodynamic and Structural Aspects of the Hydration of Cavities in Proteins REBECCA C. WADE,' MICHAEL H. MAZOR,' J. ANDREW McCAMMON,' and FLORANTE A. QUIOCHO' 'Department ot Chemistry, University of Houston, Houston, Texas 77204-5641, and 'Howard Hughes Medical Institute, Departments of Biochemistry and of Physiology and Molecular Biophysics, Baylor ( ollvge of Medicine, Houston, Texas 77030
SYNOPSIS
The structure and activity of a protein molecule are strongly influenced by the extent of hydration of its cavities. This is, in turn, related to the free energy change on transfer of a water molecule from bulk solvent into a cavity. Such free energy changes have been calculated for two cavities in a sulfate-binding protein. One of these cavities contains a crystallographically observed water molecule while the other does not. Thermodynamic integration and perturbation methods were used to calculate free energies of hydration for each of the cavities from molecular dynamics simulations of two separate events: the removal of a water molecule from pure water, and the introduction of a water molecule into each protein cavity. From the simulations for the pure water system, the excess chemical potential of water was computed to be -6.4 k 0.4 kcallmol, in accord with experiment and with other recent theoretical calculations. For the protein cavity containing an experimentally observed water molecule, the free energy change on hydrating it with one water molecule was calculated as -10.0 +- 1.3 kcallmol, indicating the high probability that this cavity is occupied by a water molecule. By contrast, for the cavity in which no water molecules were experimentally observed, the free energy change on hydrating it with one water molecule was calculated as 0.2 f 1.5 kcallmol, indicating its low occupancy by water. The agreement of these results with experiment suggests that thermodynamic simulation methods may become useful for the prediction and analysis of internal hydration in proteins.
INTRODUCTION Water is essential to the properties of biomolecules.'.' It influences their structure, function, and dynamics..' Its importance may be illustrated by the loss of enzymatic activity on d e h y d r a t i ~ n Solvent .~ interactions, including hydrophobic interactions, are one of the dominant factors in guiding the folding of a protein polypeptide chain and maintaining its ~tability' . ~Water molecules that interact with proteins may be classified into three categories: buried internal, ordered surface, and d i s ~ r d e r e d . "Internal ~,~ water molecules are of particular importance and are the subject of this paper. They fill cavities in proteins and their number generally increases with
Biopolymers. Vol 31. 919-931 (1991) 0 1991 John Wiley & Sons, Inc.
CCC 0006-3525/91/080919-13$04.00
protein size.' They may frequently be considered as a n integral part of a protein because of their structural importance. For instance, they may stabilize sharp internal turn^,^ coordinate metal ions, or mediate salt bridges or hydrogen bonds between internal hydrophilic moieties of protein residues."' Water molecules play a n important role in intermolecular interactions such as those between a drug and its receptor, a n enzyme and its substrate, or DNA and a binding protein. Internal water molecules may be displaced from a protein on binding of a ligand," or a ligand may bind via internal water molecules to a target protein.'*-l3 Ordered water molecules on protein surfaces may also be displaced by the approach of a ligand, e.g., in antibody-antigen complexes, l 4 or they may be internalized in cavities formed a t the ligand-protein interface, e.g., in repressor-operator complexes.'5~'6Water molecules a t molecular interfaces influence the thermodynamics 919
920
WADE ET AL.
of intermolecular interactions as well as having structural and functional importance. A variety of experimental techniques may be used to study protein hydration, but only a few provide information a t a molecular level on the structural and energetic properties of individual water molecules hydrating proteins. The principal technique supplying this information is x-ray d i f f r a ~ t i o n . ' ~ - ' ~ At a resolution of better than 2 8, (18, = 0.1 n m ) , it is possible to observe the oxygen atoms of ordered water molecules as discrete peaks in electron density maps. Hydrogen atoms cannot be detected and so the orientation of the water molecules cannot be determined except by inference from hydrogenbonding patterns. Neutron diffraction enables hydrogen atoms t o be located directly, and hence gives the orientations of ordered water molecules.20However, it has only applied to a few proteins. Recently, it has also become possible to locate individual internal water molecules in proteins by nmr.2' In view of the restricted information available from experimental methods, a widely applicable and quantitative theoretical method of determining the structural and energetic properties of individual internal water molecules would be of value. This might be used in, for example, the design of ligands, such as drugs; the alteration of proteins by site-directed mutagenesis; the study of protein-protein interfaces and protein folding; and the location of water molecules in protein structures solved by x-ray crystallography for which the resolution is insufficient for their detection in the electron density map. Protein hydration has been studied by many theoretical techniques 22 ranging from graphical approaches 2'3,24 to ab initio quantum mechanical calculations." The solvent-accessible area of a protein may be found by rolling a sphere of radius 1.4 8, over the protein and mapping the locus of its center.2,'3By summing over atomic groups, the solventaccessible area may be used in conjunction with atomic solvation parameters (derived from experimental free energies of transfer of amino acids) to determine solvation free energies on protein folding and binding.26Techniques that enable the positions of individual ordered water molecules to be determined include quantum mechanics methods such as the "supermolecule" approach in which tightly bound water molecules are located by ab initio or semiempirical methods and included in the calculation of the properties of small solute molecule^.^^^^^ For large molecules, a full-scale quantum mechanical treatment is not feasible. An alternative method to determine preferred solvent locations and orientations around large molecules is to apply a n electro-
static approximation to the results from ab initio calculations for small molecules by assuming additivity.'3s28Another approach is t o use a n empirical molecular mechanics energy function based on experimental observations of water molecule locations to determine solvation sites on biological molecules.z9-:" Water-biomolecule interactions may also be studied by computer simulation using Monte Carlo " and molecular dynamics method^.^"."^ These simulation methods can incorporate large numbers of water molecules and can yield free energies. Here, we propose the use of statistical thermodynamic perturbation and integration techniques in order to determine the energetic and structural properties of internal water molecules in proteins. These techniques have previously been used to compute the excess chemical potential of water, 35-38 and the relative free energies of hydration of small molecules and ions."845 T h e extent of hydration of protein cavities is determined by the excess chemical potential of water. This is constant throughout a system a t equilibrium and is given by the free energy change associated with hydrating any given position in bulk solvent with one water molecule. Cavities in a protein will tend t o be occupied by solvent if the corresponding free energy of hydration is less than that of the bulk solvent. In this work, this free energy has been computed for two cavities in a protein. In the crystal structure of the protein, one of these cavities contains a water molecule while the other does not. These two cavities, therefore, provide a test of the ability of thermodynamic simulation techniques to determine the degree to which protein cavities are hydrated. T h e computed water occupancies of these cavities were found to be consistent with those observed experimentally. Preliminary results of this work have been reported elsewhere; 46 here, we give a complete account of the calculations and a full analysis and discussion of the results.
THEORY T h e free energy change on hydration of a protein cavity determines its occupancy and the associated thermodynamic equilibrium constant K e g ,which is given by
Keg= exp(-AGo/RT)
(1)
where AGO is the standard Gibbs free energy, R is the molar gas constant, and T is the absolute temperature.
THERMODYNAMIC AND STRUCTURAL ASPECTS OF HYDRATION
Free energy is a thermodynamic state function and therefore the free energy difference between two states is independent of the path traveled between them as long as the transition between the two states is made reversibly. This means that the free energy of hydration of a protein cavity with one water molecule may be given by the sum of the free energy changes for two processes: ( 1)the removal of a water molecule W from pure water HPObulk,
and ( 2 ) the introduction of a water molecule W into a cavity of a protein P ,
P-+P+
w.
(3)
These free energies may be calculated by using statistical thermodynamic integration and perturbation techniques. These methods have been used in liquid state theory for some time,47and have more recently been applied to biological s y ~ t e m s . ~They ' permit the simulation of nonphysical processes which may allow free energy differences to be computed faster and more easily than is possible by simulating physical processes. Helmholtz free energies may be obtained from simulations performed in the canonical ensemble in which the number of particles N , the volume V ,and the temperature T are constant, whereas Gibbs free energies may be obtained from simulations in the isobaric, isothermal ensemble. For the simulation of protein solutions and bulk water, the difference between the Helmholtz and Gibbs free energies may be assumed to be negligible.49,s0 In the canonical ensemble, the Helmholtz free enerk3 A is given by
A
=
-kTln Q
92 1
T h e free energy change on addition of one particle t o a given site in a system is equal to the excess chemical potential pe given by
where pigis the ideal-gas chemical potential.'" Thus, the free energy change for the first process described above, the removal of a water molecule from pure water, is the negative of the excess chemical potential of water. In thermodynamic perturbation theory, a parameter A is introduced into the Hamiltonian H ( p , q , A ) that characterizes the perturbation of the system between two states. In the canonical ensemble, the kinetic energy of the system is constant and the potential energy term U (q , A ) of the Hamiltonian is a function of A. Here, U ( q , A ) is given by the following equations:
U
=
( 1 - A ) U,,
u = (2
-
X)U,.,,
+ XU,,,
for 0
I A I 1, and
forl
SYNOPSIS
The structure and activity of a protein molecule are strongly influenced by the extent of hydration of its cavities. This is, in turn, related to the free energy change on transfer of a water molecule from bulk solvent into a cavity. Such free energy changes have been calculated for two cavities in a sulfate-binding protein. One of these cavities contains a crystallographically observed water molecule while the other does not. Thermodynamic integration and perturbation methods were used to calculate free energies of hydration for each of the cavities from molecular dynamics simulations of two separate events: the removal of a water molecule from pure water, and the introduction of a water molecule into each protein cavity. From the simulations for the pure water system, the excess chemical potential of water was computed to be -6.4 k 0.4 kcallmol, in accord with experiment and with other recent theoretical calculations. For the protein cavity containing an experimentally observed water molecule, the free energy change on hydrating it with one water molecule was calculated as -10.0 +- 1.3 kcallmol, indicating the high probability that this cavity is occupied by a water molecule. By contrast, for the cavity in which no water molecules were experimentally observed, the free energy change on hydrating it with one water molecule was calculated as 0.2 f 1.5 kcallmol, indicating its low occupancy by water. The agreement of these results with experiment suggests that thermodynamic simulation methods may become useful for the prediction and analysis of internal hydration in proteins.
INTRODUCTION Water is essential to the properties of biomolecules.'.' It influences their structure, function, and dynamics..' Its importance may be illustrated by the loss of enzymatic activity on d e h y d r a t i ~ n Solvent .~ interactions, including hydrophobic interactions, are one of the dominant factors in guiding the folding of a protein polypeptide chain and maintaining its ~tability' . ~Water molecules that interact with proteins may be classified into three categories: buried internal, ordered surface, and d i s ~ r d e r e d . "Internal ~,~ water molecules are of particular importance and are the subject of this paper. They fill cavities in proteins and their number generally increases with
Biopolymers. Vol 31. 919-931 (1991) 0 1991 John Wiley & Sons, Inc.
CCC 0006-3525/91/080919-13$04.00
protein size.' They may frequently be considered as a n integral part of a protein because of their structural importance. For instance, they may stabilize sharp internal turn^,^ coordinate metal ions, or mediate salt bridges or hydrogen bonds between internal hydrophilic moieties of protein residues."' Water molecules play a n important role in intermolecular interactions such as those between a drug and its receptor, a n enzyme and its substrate, or DNA and a binding protein. Internal water molecules may be displaced from a protein on binding of a ligand," or a ligand may bind via internal water molecules to a target protein.'*-l3 Ordered water molecules on protein surfaces may also be displaced by the approach of a ligand, e.g., in antibody-antigen complexes, l 4 or they may be internalized in cavities formed a t the ligand-protein interface, e.g., in repressor-operator complexes.'5~'6Water molecules a t molecular interfaces influence the thermodynamics 919
920
WADE ET AL.
of intermolecular interactions as well as having structural and functional importance. A variety of experimental techniques may be used to study protein hydration, but only a few provide information a t a molecular level on the structural and energetic properties of individual water molecules hydrating proteins. The principal technique supplying this information is x-ray d i f f r a ~ t i o n . ' ~ - ' ~ At a resolution of better than 2 8, (18, = 0.1 n m ) , it is possible to observe the oxygen atoms of ordered water molecules as discrete peaks in electron density maps. Hydrogen atoms cannot be detected and so the orientation of the water molecules cannot be determined except by inference from hydrogenbonding patterns. Neutron diffraction enables hydrogen atoms t o be located directly, and hence gives the orientations of ordered water molecules.20However, it has only applied to a few proteins. Recently, it has also become possible to locate individual internal water molecules in proteins by nmr.2' In view of the restricted information available from experimental methods, a widely applicable and quantitative theoretical method of determining the structural and energetic properties of individual internal water molecules would be of value. This might be used in, for example, the design of ligands, such as drugs; the alteration of proteins by site-directed mutagenesis; the study of protein-protein interfaces and protein folding; and the location of water molecules in protein structures solved by x-ray crystallography for which the resolution is insufficient for their detection in the electron density map. Protein hydration has been studied by many theoretical techniques 22 ranging from graphical approaches 2'3,24 to ab initio quantum mechanical calculations." The solvent-accessible area of a protein may be found by rolling a sphere of radius 1.4 8, over the protein and mapping the locus of its center.2,'3By summing over atomic groups, the solventaccessible area may be used in conjunction with atomic solvation parameters (derived from experimental free energies of transfer of amino acids) to determine solvation free energies on protein folding and binding.26Techniques that enable the positions of individual ordered water molecules to be determined include quantum mechanics methods such as the "supermolecule" approach in which tightly bound water molecules are located by ab initio or semiempirical methods and included in the calculation of the properties of small solute molecule^.^^^^^ For large molecules, a full-scale quantum mechanical treatment is not feasible. An alternative method to determine preferred solvent locations and orientations around large molecules is to apply a n electro-
static approximation to the results from ab initio calculations for small molecules by assuming additivity.'3s28Another approach is t o use a n empirical molecular mechanics energy function based on experimental observations of water molecule locations to determine solvation sites on biological molecules.z9-:" Water-biomolecule interactions may also be studied by computer simulation using Monte Carlo " and molecular dynamics method^.^"."^ These simulation methods can incorporate large numbers of water molecules and can yield free energies. Here, we propose the use of statistical thermodynamic perturbation and integration techniques in order to determine the energetic and structural properties of internal water molecules in proteins. These techniques have previously been used to compute the excess chemical potential of water, 35-38 and the relative free energies of hydration of small molecules and ions."845 T h e extent of hydration of protein cavities is determined by the excess chemical potential of water. This is constant throughout a system a t equilibrium and is given by the free energy change associated with hydrating any given position in bulk solvent with one water molecule. Cavities in a protein will tend t o be occupied by solvent if the corresponding free energy of hydration is less than that of the bulk solvent. In this work, this free energy has been computed for two cavities in a protein. In the crystal structure of the protein, one of these cavities contains a water molecule while the other does not. These two cavities, therefore, provide a test of the ability of thermodynamic simulation techniques to determine the degree to which protein cavities are hydrated. T h e computed water occupancies of these cavities were found to be consistent with those observed experimentally. Preliminary results of this work have been reported elsewhere; 46 here, we give a complete account of the calculations and a full analysis and discussion of the results.
THEORY T h e free energy change on hydration of a protein cavity determines its occupancy and the associated thermodynamic equilibrium constant K e g ,which is given by
Keg= exp(-AGo/RT)
(1)
where AGO is the standard Gibbs free energy, R is the molar gas constant, and T is the absolute temperature.
THERMODYNAMIC AND STRUCTURAL ASPECTS OF HYDRATION
Free energy is a thermodynamic state function and therefore the free energy difference between two states is independent of the path traveled between them as long as the transition between the two states is made reversibly. This means that the free energy of hydration of a protein cavity with one water molecule may be given by the sum of the free energy changes for two processes: ( 1)the removal of a water molecule W from pure water HPObulk,
and ( 2 ) the introduction of a water molecule W into a cavity of a protein P ,
P-+P+
w.
(3)
These free energies may be calculated by using statistical thermodynamic integration and perturbation techniques. These methods have been used in liquid state theory for some time,47and have more recently been applied to biological s y ~ t e m s . ~They ' permit the simulation of nonphysical processes which may allow free energy differences to be computed faster and more easily than is possible by simulating physical processes. Helmholtz free energies may be obtained from simulations performed in the canonical ensemble in which the number of particles N , the volume V ,and the temperature T are constant, whereas Gibbs free energies may be obtained from simulations in the isobaric, isothermal ensemble. For the simulation of protein solutions and bulk water, the difference between the Helmholtz and Gibbs free energies may be assumed to be negligible.49,s0 In the canonical ensemble, the Helmholtz free enerk3 A is given by
A
=
-kTln Q
92 1
T h e free energy change on addition of one particle t o a given site in a system is equal to the excess chemical potential pe given by
where pigis the ideal-gas chemical potential.'" Thus, the free energy change for the first process described above, the removal of a water molecule from pure water, is the negative of the excess chemical potential of water. In thermodynamic perturbation theory, a parameter A is introduced into the Hamiltonian H ( p , q , A ) that characterizes the perturbation of the system between two states. In the canonical ensemble, the kinetic energy of the system is constant and the potential energy term U (q , A ) of the Hamiltonian is a function of A. Here, U ( q , A ) is given by the following equations:
U
=
( 1 - A ) U,,
u = (2
-
X)U,.,,
+ XU,,,
for 0
I A I 1, and
forl
