International Journal of Biological Macromolecules 78 (2015) 257–265
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International Journal of Biological Macromolecules journal homepage: www.elsevier.com/locate/ijbiomac
A simplified electrostatic model for hydrolase catalysis Pedro de Alcantara Pessoa Filho a,b,∗ , John M. Prausnitz a a b
Department of Chemical and Biomolecular Engineering, University of California, Berkeley, CA, USA Department of Chemical Engineering, University of Sao Paulo, Sao Paulo, SP, Brazil
a r t i c l e
i n f o
Article history: Received 1 December 2014 Received in revised form 31 March 2015 Accepted 1 April 2015 Available online 13 April 2015 Keywords: Enzyme Hydrolase Model
a b s t r a c t Toward the development of an electrostatic model for enzyme catalysis, the active site of the enzyme is represented by a cavity whose surface (and beyond) is populated by electric charges as determined by pH and the enzyme’s structure. The electric field in the cavity is obtained from electrostatics and a suitable computer program. The key chemical bond in the substrate, at its ends, has partial charges with opposite signs determined from published force−field parameters. The electric field attracts one end of the bond and repels the other, causing bond tension. If that tension exceeds the attractive force between the atoms, the bond breaks; the enzyme is then a successful catalyst. To illustrate this very simple model, based on numerous assumptions, some results are presented for three hydrolases: hen-egg white lysozyme, bovine trypsin and bovine ribonuclease. Attention is given to the effect of pH. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Enzyme catalysis constitutes a well-established field of research in biochemistry. Pertinent literature reviews on this subject can be found in textbooks such as those by Frey and Hegeman [1], Bisswanger [2] and Purich [3], among others. In this work, we consider a fundamental question: For a specific substrate reaction at a fixed temperature and pH, will a specific enzyme provide catalysis or not? Toward an effort to answer this question, we initiate here a simple model based on electrostatics. To introduce the basic idea, consider two parallel metallic plates separated by a short distance. One plate bears a positive electric charge, while the other bears a negative electric charge, as indicated in Fig. 1. Into the gap between the plates, we put a dumbbell whose two ends are separated by a small distance compared with the height of the gap. The two ends of the dumbbell are charged; one end is positive and the other end is negative. The two charged plates generate an electric field perpendicular to the surfaces of the plates. This field attracts one end of the dumbbell while repelling the other, generating a tensile force in the rod that connects the two ends. If the tensile force is sufficiently strong, the rod breaks.
∗ Corresponding author at: Caixa Postal 61548, 05424-970, Sao Paulo, SP, Brazil. Tel.: +55 11 3091 1106; fax: +55 11 3091 2284. E-mail address: [email protected] (P.d.A. Pessoa Filho). http://dx.doi.org/10.1016/j.ijbiomac.2015.04.010 0141-8130/© 2015 Elsevier B.V. All rights reserved.
We use this electromechanical picture as a basis for a simple model of enzyme catalysis. The two charged plates are replaced by the surface of a spherical cavity that represents the immediate region of an enzyme’s active site. The charge on this surface is not uniform; it is determined by the positions of charges on the enzyme at a fixed pH. The dumbbell is replaced by the pertinent chemical bond in the substrate, inside the cavity. The ends of the dumbbell bear partial charges of opposite sign. When the electric field in the cavity exceeds the chemical bond strength, the bond is broken, giving a catalyzed reaction. Before presenting the essential details of our simple electrostatic model, we note that application of electrostatic physics to catalysis is not a new idea, as indicated by numerous previous publications, e.g. Warshel [4], Warshel et al. [5], Dao-Pin et al. [6]. The importance of the electric field in enzyme catalysis is evident in the THEMATICS method for identifying the active site [7–9]. Considering an enzyme of known structure but unknown active site, THEMATICS searches for unusual theoretically calculated titration curves, and associates the distortion in titration curves to catalytic activity. Electrostatic interactions also enhance the diffusion of substrate molecules to the enzyme’s active site [10,11] and the steering of substrate molecules inside the active site [12]. Fried et al. [13] showed recently that large electric fields found in the active pocket of ketosteorid isomerase are responsible for the unusually high rate constants of the reaction catalyzed by this enzyme. However, to our best knowledge, no previous publication has tried to answer the question of concern here: for a given situation (enzyme, substrate, temperature, pH), can a relationship between the electric
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Nomenclature Latin letters Ek electric field at position k Ex x-component of the electric field Ejk electric field generated by group j at position k electrostatic force acting on atom m Fm Fmn electrostatic force acting on atom n due to atom m N number of atoms in the enzyme molecule charge on atom i qi rjk vector from position (of atom/group) j to position (of atom/group) k Umn electrostatic energy between atoms m and n Greek letters ı cut-off distance in Eq. (7) ε0 vacuum permittivity dielectric constant ε k electric potential at position k function defined by Eq. (7)
does not interact with the active site. Instead, we imply that the effect of the charges of the substrate molecule on the electric field is the same, regardless of is environment. The effect of changes in the spatial arrangement in the substrate molecule is considered negligible for the calculation of the electric field. In all calculations, we set ε = 1.0 as discussed in Appendix. The electric potential k generated by the enzyme molecule at position k is obtained from: 1 qj 4εε0 |rjk | N
k =
(2)
j=1
where j is an atom in the enzyme. The summation is over all N atoms of the enzyme. While Ek is a vector, k is scalar. We now consider the bond formed by atoms m and n in the substrate in the absence of an external electric field due to the enzyme. Atoms m and n bear partial charges qm and qn , respectively. The electrostatic force Fmn acting on atom n due to atom m is a vector. This force is calculated from: Fmn =
1 qm qn rmn 4εε0 |rmn |3
(3)
We refer to the absolute value of force Fmn as the bond force. The energy Umn of bond mn in the substrate is calculated as the minimum work necessary to separate the charges on m and n from the bond length to infinity. In the absence of an external electric field, the bond energy is calculated in a molar basis as:
∞ Umn = NAV Fig. 1. Illustrative two-dimensional representation of bond cleavage in the active site. The two charged plates generate electric field E. (a) dumbbell representing the substrate’s pertinent bond in the active site. The negatively charged atom is red, while the positively charged atom is blue. The electric field attracts the blue atom and repels the red atom, causing a tensile force in the (dumbbell) bond. (b) Atoms after cleavage. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
field generated in the enzyme’s active site and the electric dipole of the bond in the substrate molecule be established? Based on simplifying assumptions, our model aims to provide an answer to the question. 2. Calculation of the Electric Field and the Electric Potential To calculate the electric field and electric potential inside the active pocket, the influence of all enzyme atoms is taken into account: each atom bears a full or partial charge that influences the electric field and the electric potential. The electric field Ek is a vector. At position k of the active pocket, the electric field is calculated using the basic equation of electrostatics: 1 qj rjk 3 4εε0 r N
Ek =
j=1
(1)
jk
where ε is the relative permittivity of the medium, ε0 is the permittivity of vacuum, qj is the charge on atom j, and rjk is the vector from the position of atom j in the enzyme to position k anywhere in the active pocket. The summation is carried out over all N atoms in the enzyme. In calculating Ek we do not consider full or partial charges on the substrate molecule. These charges are present even when the substrate is outside the active cavity. The electric field calculated through Eq. (1) is in effect an “excess” field caused by the enzyme’s active site. i.e. it is the net effect of the enzyme’s active site on the atoms that constitute the bond. By disregarding the influence of the substrate molecule, we do not suggest that the substrate molecule
Fmn · drmn = |r mn |
NAV qm qn 4εε0 |rmn |
(4)
where NAV is the Avogadro number. We now consider that bond mn in the substrate is placed in the active cavity of the enzyme. In the presence of the electric field due to the enzyme, the resulting force on atom m of bond mn is: Fm = qm Em + Fmn
(5)
A similar equation gives the force on atom n. The bond energy for bond mn placed in the cavity is the minimum work to separate qm and qn from the mn bond length to infinity, but now we must include the influence of the electric potential generated by the enzyme. The bond energy for bond mn placed in the active cavity is: Umn =
NAV qm qn + NAV (qm m + qn n ) 4εε0 |rmn |
(6)
where m and n are calculated at the positions of atoms m and n, respectively. The influence of the electric potential on the bond energy depends on the bond’s orientation. A specific enzyme may break a specific bond in the substrate, where the terminal atoms in the bond bear partial charges. Two methods are used here to estimate these partial charges. If partial charges for the ends of bond mn are given in one or more published force-field parameter sets, these partial charges are used, provided that the calculated bond energy (Eq. (4)) is close to the average bond energy of similar bonds. Otherwise, partial charges are estimated from Eq. (4) to be in general agreement with the average bond energy of similar bonds. In this case, we use partial charges of the same magnitude but opposite sign. Table 1 summarizes the partial charges used in our calculations. 3. Enzyme Structure and Ionization States To illustrate the simple model proposed here, three bondbreaking enzymes (hydrolases) were considered: lysozyme, trypsin
P.d.A. Pessoa Filho, J.M. Prausnitz / International Journal of Biological Macromolecules 78 (2015) 257–265 Table 1 Partial charges in substrate bonds. Enzyme
Atom 1
Atom 2
q1 /ea
q2 /e
Estimated from
Lysozyme Tryspin
C C
O N
Ribonuclease
P
O
0.60 0.61 0.51 0.55 0.96
−0.60 −0.42 −0.47 −0.40 −0.66
Bond-energy data AMBER CHARMM PARSE PARSE
a
e is the elementary charge; 1 e = 1.602·10−19 C.
and ribonuclease. To calculate the electric field (Eq. (1)) and the electric potential (Eq. (2)), we need the positions of all enzyme atoms. The enzymes’ crystal structure was used in the calculation of the electric field and the electric potential. Enzyme structure was obtained from the Protein Data-Bank (PDB). Table 2 presents the PDB code for all enzymes; nomenclature in the subsequent discussion follows the corresponding coordinate file for each enzyme. However, these coordinate files do not provide the positions of hydrogen atoms and the partial charges. Therefore, hydrogen atoms and partial charges were obtained using the PDB2PQR program [14,15] and applying the parameters of force-fields AMBER 94 [16,17], CHARMM 27 [18] and PARSE [19]. Ionization as a function of pH was calculated using the PROPKA method [20,21]. The medium permittivity was considered to be that of vacuum (i.e. ε = 1.0). All calculations were carried out through ad hoc developed programs written in FORTRAN 90, verified by comparison with calculations using the DELPHI package [22–24].
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contains amino acids HIS-12 (neutral) and HIS-119 (positively charged) [29,30]. The cleavage of the P O bond occurs after the abstraction of a proton from the oxygen atom by HIS-12. The ionization states of other amino acids were obtained at pH 6.0. 4. Examples To calculate the effect of the electric field on the pertinent substrate bond, we must specify the partial charges at the ends of the bond and the putative position it occupies in the active-site cavity. Firstly, we consider hen egg-white lysozyme that may break the substrate’s glycoside bond between a carbon atom and an oxygen atom. The average length of a C O bond is 0.143 nm, and its average bond energy is 350 kJ mol−1 [31]. From Eq. (4), and assuming |qm | = |qn |, the partial charges qm and qn can be estimated; they are −0.6e for the oxygen atom and 0.6e for the carbon atom. During the cleavage step, the C O bond lies approximately between the carboxyl groups of amino acids GLU-35 and ASP-52 in lysozyme. Fig. 2a shows some of the main amino-acids in the active pocket of lysozyme. Table 3 presents the electric field calculated for the mid-point between carbon CG of ASP-52 and carbon CD of GLU-35 at pH 5.0. For these enzyme calculations, we used partial charges obtained from three different force-field parameter
3.1. Ionization states and mechanisms For the examples given in the next section, we consider the ionization and catalytic mechanism for three enzymes. Lysozyme is a glycoside hydrolase. It hydrolyzes the glycosidic bond between the oxygen atom of N-acetylmuramic acid and the carbon atom of N-acetylglucosamine in peptidoglycans. The active site of hen egg-white lysozyme comprises two ionizable groups: GLU-35 and ASP-52. The enzyme is active when GLU-35 is neutral and ASP-52 is negatively charged [25,26]. The active dyad of human lysozyme contains the same amino acids in similar positions, with ASP-53 instead of ASP-52. The oxygen atom of N-acetylmuramic acid acts as proton-acceptor for the GLU-35 protonated carboxylic acid group. Subsequent steps restore the GLU-35 group to the protonated state by abstracting a proton of a water molecule. The ionization states of other amino acids were obtained at pH 5.0. Trypsin is a protease. It hydrolyzes the peptide bond at the carboxyl side of some amino-acids, chiefly lysine and arginine. The active tryad of bovine and porcine trypsin contains amino acids HIS57 (neutral), ASP-102 (negatively charged) and SER-195 (neutral) [27,28]. The ionization states of other amino acids were obtained at pH 8.0. Calculations were also conducted for bovine chymotrypsin, another protease with the same active tryad. Ribonuclease hydrolyzes the P O bond of cytidine and uridine residues of ribonucleic acid. The active dyad of ribonuclease Table 2 Protein Data Bank (PDB) codes for the structure of enzymes. Enzyme
Organism
PDB code
Lysozyme
Gallus gallus Homo sapiens Bos taurus Sus scofa Bos taurus Bos taurus
1lys 1lz1 2ptna 1s81b 3a1r 2cga
Trypsin Ribonuclease Chymotrypsinogen a b
Considering ALA-221. Considering PHE-184, LYS-188 and ALA-221.
Fig. 2. (a) Partial structure of the hen egg-white lysozyme active site, indicating GLU-35 and ASP-52; (b) placement of bond C O inside the active site of lysozyme. Gray spheres: carbon atoms; blue spheres: nitrogen atoms, and red spheres: oxygen atoms. Sphere radii do not correlate with atomic radii. Other atoms in enzyme and substrate omitted for clarity.
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Table 3 Results for the electric field (V nm−1 ) and electric potential (V) for the active pocket of enzymes. Enzyme
pH
Force field
Hen lyzozyme (1lys)a
5.0
AMBER
8.0
CHARMM PARSE AMBER
6.0
Bovine trypsin (2ptn)b
Bovine ribonuclease (3a1r)c
Ex
|E|
Ey
Ez
−2.30
−0.36
−10.0 −11.7 −13.9
−1.87 −3.92 6.18
−0.87 1.50 8.52
10.2 12.4 17.4
6.37 6.26 2.16
CHARMM PARSE AMBER
−13.6 −16.1 12.0
6.52 5.99 1.96
10.4 7.09 12.1
18.4 18.6 17.2
1.87 1.20 11.4
CHARMM PARSE
13.0 14.3
3.77 9.01
12.9 19.8
18.7 26.0
11.5 12.5
−8.96
9.26
6.53
a The x-axis is defined by carbon CG of ASP-52 (origin) and carbon CD of GLU-35 (positive direction); the y-axis is approximately the axis defined by the by the origin and carbon C of ILE-58 (negative direction); the z-axis is defined in such a way that the basis is right handed. b The x-axis is defined by nitrogen NE2 of HIS-57 (origin) and oxygen OG of SER195 (positive direction); the y-axis is approximately the axis defined by the origin and nitrogen N in TYR-172 (positive direction); the z-axis is defined in such a way that the basis is right handed. c The x-axis is defined by nitrogen ND1 of HIS-119 (origin) and nitrogen NE2 of HIS-12 (positive direction); the y-axis is approximately the axis defined by the origin and nitrogen N of ASP 121 (negative direction); the z-axis is defined in such a way that the basis is right handed.
sets. The results show that the electric field is oriented in the direction connecting these atoms in the enzyme molecule, and that the calculated electric field is almost independent of the force-field parameter set used to obtain the partial charges for all atoms in the enzyme. To overcome the electrostatic attraction between the carbon atom and the oxygen atom in the substrate (calculated using Eq. (3) and partial charges presented in Table 1) requires a force of 4.1 nN (nano-Newton, i.e. 4.1 × 10−9 N). If the C O bond is placed in the active pocket, midway between the CG carbon of ASP-52 and the CD carbon of GLU-35, with the oxygen atom closer to the ASP-52 residue (Fig. 2b), the estimated force is 2.3−3.0 nN, depending on the choice of force-field parameter set. Because this force is in the bond direction, it is a tensile force. The energy of the C O bond in the active pocket, calculated from Eq. (6), gives 255 kJ mol−1 (calculations with PARSE parameters) or 280 kJ mol−1 (calculations with AMBER parameters). These results show that the tensile force due to the electric field acting on the bond atoms has the same magnitude as the attractive force between these atoms. Moreover, the energy of the bond inside the active site is significantly lower than the bond energy in the absence of the electric field (350 kJ mol−1 ). Because the energy barrier for bond cleavage is lower, the bond breaks. In summary, bond breaking is highly favored in the active site, as observed. Secondly, we consider trypsin-catalyzed proteolysis. Trypsin breaks the peptide bond between a carbon atom and a nitrogen atom. The average length of a C N bond is 0.147 nm, and its average bond energy is 308 kJ mol−1 [31]. However, in a peptide bond, the C N bond is known to have a length of 0.133 nm and an energy lower (in magnitude) than the average energy of other C N bonds. Estimating the partial charges from force-field parameter sets, the calculated bond energy ranges from 230 (PARSE) to 266 kJ mol−1 (AMBER). During the bond cleavage by bovine trypsin, the peptide bond in the substrate is located approximately parallel to the HIS-57 and SER-195 groups in the enzyme, as shown in Fig. 3a. These amino acids are so close that the peptide bond of the substrate
Fig. 3. (a) Partial structure of the bovine trypsin active site, indicating HIS-57 and SER-195; (b) placement of the peptide bond C N inside the active site of trypsin. Gray spheres: carbon atoms; blue spheres: nitrogen atoms, and red spheres: oxygen atoms. Sphere radii do not correlate with atomic radii. Other atoms in enzyme and substrate omitted for clarity.
cannot occupy the space between them. The electric field and electric potential were calculated for a point equidistant from NE2 in HIS-57 and OG in SER-195, but at a distance 0.2 nm from the line defined by these atoms, toward outside the cavity. Results are presented in Table 3. Because the electric field is oriented in the direction of the C N bond in the substrate, it is a tensile force. Calculated bond forces for the peptide bond range from 2.9 to 3.1 nN, depending on the choice of partial charges as shown in Table 1. If the peptide bond is placed at a distance 0.2 nm of the straight line between NE2 in HIS-57 and OG in SER-195, as shown in Fig. 3b, the calculated force acting on the peptide-bond atoms due to the electric field is 2.0 (AMBER and CHARMM) to 2.4 nN (PARSE). This force is obtained using the force component acting in the same direction as that of the bond. Therefore, the force that attracts the atoms in the peptide bond and the tensile force arising due to the electric field in the enzyme’s active pocket have the same order of magnitude. Moreover, the calculated bond energy for the peptide bond inside the active pocket lies between 110 (PARSE) and 157 kJ mol−1 (CHARMM). These energies are lower than that calculated for the peptide bond outside the active pocket (230−266 kJ mol−1 ). Hence, the C−N bond experiences a tensile force of the same magnitude as the bond force, and its energy is significantly lower inside the active site. We conclude that peptide bond breaking is favored in the active site. The third example concerning ribonuclease is counter-intuitive. This enzyme can break the P O bond in RNA. The P O bond energy can be estimated from force-field parameter sets. The P O bond distance from crystallographic data for RNA is 0.148 nm. From these data, the bond energy is 846 (AMBER), 1090 (CHARMM) and 598 kJ mol−1 (PARSE). This latter energy is compatible with the
P.d.A. Pessoa Filho, J.M. Prausnitz / International Journal of Biological Macromolecules 78 (2015) 257–265
expected energy for a P O bond. Therefore, we use the partial charges from PARSE: −0.66e for the oxygen atom and 0.96e for the phosphorus atom. The bond force (Eq. (3)) calculated with these partial charges is 6.7 nN. Table 3 gives the calculated electric field for the active pocket of pancreatic bovine ribonuclease. The electric fields were obtained for the mid-point between the nitrogen atom NE2 in HIS-12 and the nitrogen atom ND1 in HIS-119, shown in Fig. 4a. To calculate the force acting upon the atoms in the P O bond of the substrate, the position of this bond was considered perpendicular to the line joining these nitrogen atoms in enzyme, with the oxygen atom in the P O bond occupying the mid-point of this line and the phosphorus atom pointing outside the cavity, as shown in Fig. 4b. Using PARSE force-field parameters, the force generated by the electric field is 1.2 nN, significantly lower than the bond force, 6.7 nN. Because we know that ribonuclease can break P O bonds, this calculated result appears to be counter to our proposed hypothesis. However, in this case, due to the orientation of the bond in the active pocket, the pertinent force is not a tensile force but a shear force. The shear force acting on the oxygen atom is 2.8 nN (PARSE), while the phosphorus atom is not affected by a shear force. The calculated bond energy is lowered to 301 kJ mol−1 . Therefore, the same pattern observed for lysozyme and trypsin is also observed for ribonuclease: the resulting forces break the bond and the bond energy is lowered in the active site. We conclude that the enzyme provides enhanced breaking of the P O bond.
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4.1. Influence of enzyme structure and pH Our calculations assume invariance of the primary structure of the enzyme. However, for many cases, the same enzyme produced by different organisms catalyzes the same reaction. Thus, despite different origins and differences in amino-acid sequence, it appears that the same enzyme generates similar electric fields and potentials. To assess the effect of enzyme structure, calculations similar to those above were carried out for human lysozyme and porcine trypsin. Table 4 gives results for the electric field and electric potential for human lysozyme. For 129 amino-acid groups of hen lysozyme and 130 amino-acid groups of human lysozyme, human and hen egg-white lysozymes share 79 amino acids in corresponding positions. Despite differences in the amino acids, the electric field and the electric potential calculated for human lysozyme are similar to those for hen egg-white lysozyme. Table 4 also shows results for the electric field for porcine trypsin. The variation between porcine and bovine trypsin is larger than that between the two lysozymes, even though both trypsins share a higher proportion of amino-acid groups (183 out of 223) than the lysozymes. Unlike lysozyme, for trypsin partial charges of neutral amino acids, instead of ionized ones, have a predominant effect on the electric field, as discussed in the Appendix. Chymotrypsin is a serine protease with the same catalytic tryad as that for trypsin. Both enzymes cleave the same bond. Side chains in the substrate determine whether a specific peptide bond can be broken. We expect that, for the active cavity, the electric field and electric potential for chymotrypsin is similar to those for trypsin. Table 4 shows the calculated electric field for the same position in chymotrypsinogen A, the inactive form of chymotrypsin. The catalytic tryad in chymotrypsin and that in chymotrypsinogen A have the same amino acids and the same spatial orientation, but chymotrypsinogen A is inactive due to differences in primary/tertiary structure. As expected, the electric field generated in the active pocket of chymotrypsinogen A is similar to that of trypsin, as both enzymes cleave the same peptide bond. The pH affects the overall charge of the protein molecule and the charges of the ionizable groups in the active site. There is extensive experimental evidence that the ionization state of the ionizable groups in the active pocket determines the presence or absence
Table 4 Results for the electric field (V nm−1 ) and electric potential (V) for the active pocket of variant forms of enzymes.
Fig. 4. (a) Partial structure of bovine ribonuclease active site, indicating HIS-12 and HIS-119; (b) placement of the bond P O inside the active site of trypsin. Gray spheres: carbon atoms; blue spheres: nitrogen atoms; red spheres: oxygen atoms, and orange sphere: phosphorus atom. Sphere radii do not correlate with atomic radii. Other atoms in enzyme and substrate omitted for clarity.
|E|
pH
Forcefield
Ex
Ey
Human lyzozyme (1lys)a
5.0
AMBER CHARMM PARSE
−11.6 −12.6 −14.1
1.31 2.00 1.27
2.91 2.02 5.15
12.1 12.9 15.1
5.99 5.69 5.36
Porcine trypsin (1s81)b
8.0
AMBER CHARMM PARSE
5.79 6.74 5.45
−9.08 −10.5 −8.51
11.9 13.5 12.0
−1.42 −1.73 −2.46
Bovine chymotryspinogen (2cga)c
8.0
AMBER CHARMM PARSE
5.10 3.79 7.45
7.29 6.22 7.94
17.2 16.1 21.7
1.12 0.99 0.84
−4.97 −5.16 −6.51 −14.8 −14.3 −18.8
Ez
Enzyme
a The x-axis is defined by carbon CG of ASP-52 (origin) and carbon CD of GLU-35 (positive direction); the y-axis is approximately the axis defined by the by the origin and carbon C of ILE-58 (negative direction); the z-axis is defined in such a way that the basis is right handed. b The x-axis is defined by nitrogen NE2 of HIS-57 (origin) and oxygen OG of SER195 (positive direction); the y-axis is approximately the axis defined by the origin and nitrogen N in TYR-172 (positive direction); the z-axis is defined in such a way that the basis is right handed. c The x-axis is defined by nitrogen ND1 of HIS-119 (origin) and nitrogen NE2 of HIS-12 (positive direction); the y-axis is approximately the axis defined by the origin and nitrogen N of ASP 121 (negative direction); the z-axis is defined in such a way that the basis is right handed.
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of catalytic activity [3]. The equilibrium among the different ionization states of the ionizable groups in the active site results in the usual bell-shape curve for the relative activity as a function of pH. Therefore, provided that the groups in the active site remain in the active configuration (i.e. the configuration that assures the presence of proton-donor and proton-acceptor groups for the reaction pathway), according to our model the electric field should be insensitive to changes in the overall ionization state of the entire enzyme. To illustrate the effect of pH, we calculated the electric field for lysozyme at pH 2.0−9.0, the pH range where lysozyme is reported to be enzymatically active. Table 5 presents the electric field as a function of pH for the mid-point between carbon CG of ASP-52 and carbon CD of GLU-35. Results are based on CHARMM force-field parameters; results for other force-field parameters are comparable. Despite the change in the overall charge of the enzyme that drops by approximately half in this pH range, the electric field remains almost constant. The opposite result is obtained when the ionization state of the ionizable amino-acids in the active pocket changes. On the one hand, if both ASP-52 and GLU-35 are neutral, the absolute value of the electric field decreases from 10.2 to 2.3 V nm−1 , using the CHARMM force-field parameters. This reduction is primarily related to a reduction in the x-component of the electric field; this component is in the direction of the C O bond of the substrate during the cleavage step. This x-component decreases from −10.0 to −2.0 V nm−1 . There is no significant decrease in the calculated C O bond energy: the bond energy decreases from 350 to 331 kJ mol−1 . On the other hand, if both ASP-52 and GLU-35 are ionized, the x-component of the electric field changes sign, from −10.0 to 4.0 V nm−1 , and the effect of the electric field is to stabilize the C O bond of the substrate: the calculated bond energy increases from 350 to 404 kJ mol−1 . Therefore, bond cleavage is not favored in these ionization states. 4.2. Comparison with other methods The most common procedure to calculate protein electrostatics is to use the two-dielectric assumption: the protein molecule is considered a low-dielectric medium, with ε between 2 and 10 (the so-called “internal” dielectric constant), surrounded by an aqueous solvent, with ε equal to 78 or 80 and a certain ionic strength [32]. This internal dielectric constant accounts for the dipoles in the protein molecule and approximately for atomic polarizability [32]. The Poisson-Boltzmann equation is then solved numerically. The electric fields presented in Table 3 disagree with those presented by Dao-Pin et al. [6] for the electric field calculated in the vicinity of GLU-35 of hen egg-white lysozyme. Considering no solvent charges or partial charges in the enzyme, these authors arrived at electric fields between 0.16 and 0.77 V nm−1 . The main Table 5 Results for the electric field (V nm−1 ) for the active pocket of hen egg-white lysozyme (1lys). Calculations based on CHARMM force-field parameters. pH
Overall chargea
Ex
Ey
Ez
|E|
2 3 4 5 6−9b
17 16 12 10 9
−10.4 −11.1 −11.2 −10.0 −9.87
0.05 −0.19 −0.67 −1.87 −2.18
−1.17 −0.97 −0.71 −0.87 −0.72
10.4 11.2 11.3 10.2 10.1
a The overall charge is calculated considering that the ionized groups involved in the catalytic mechanism are in the active state, even if the most common form is different. b There is no difference between the ionized states calculated in this pH range; the only change that would occur is the ionization of ASP-52.
source of this disagreement is the use of a different solvent permittivity, along with the use of the Poisson-Boltzmann equation, by Dao-Pin et al. [6]. If we consider the two-dielectric model, we obtain results similar to those by Dao-Pin et al. [6]. For the midpoint between CG carbon of ASP-52 and the CD carbon of GLU-35 of hen egg-white lysozyme, the electric field calculated using the DELPHI package is 0.22 V nm−1 ; in this calculation, the ionic strength is 0.1 mol·kg−1 , the internal dielectric constant is 4 and the solvent dielectric constant is 78. Besides the solvent permittivity, the internal permittivity influences the calculated electric field. When ε for the internal medium changes from ε = 1 to ε = 4, the electric field calculated with the DELPHI package for the lysozyme molecule, at the mid-point between GLU-35 and ASP-52, decreases from 12.0 to 7.19 V nm−1 ; when ε = 10, we obtain 4.53 V nm−1 . As pointed out by Li et al. [32], there is no general solution for this question. The two dielectric assumption, while adequate for calculating pKA changes and solvation energies, is not suitable for the calculation of the electric field inside the enzyme’s active pocket [33,34]. The solvent cannot be assumed to be a continuum at the atomistic scale corresponding to our calculations, and implicit solvent assumptions may be unreasonable [33]. Moreover, the presence of the substrate inside the active pocket results in at least partial exclusion of solvent molecules. Therefore, for the very small space in the active cavity, the dielectric constant has no meaning. A more comprehensive and accurate description of the energy change should include contributions from electron exchange, delocalization (i.e. related to deformations in charge distribution) and correlation [37], obtained in the framework of first-principle calculations. Examples of first-principle calculations include chorismate mutase [38] and protein kinase [39], among others. These works present a detailed description of enzyme catalysis concerning energy calculations, and can be used in a predictive way. However, they also show that the electrostatic contribution is predominant to the activation energy lowering, and hence to the catalytic activity, which is the basis of the model presented here. 5. Conclusions This paper investigates the relationship between the electric field and the electric potential generated in the active pocket of an enzyme, and the force and energy of the pertinent chemical bond in the substrate. Illustrative calculations are presented for three hydrolases: hen egg-white lysozyme, bovine trypsin, and bovine ribonuclease. For lysozyme and trypsin, the tensile forces resulting from the electric field in the active pocket have the same magnitude as the electrostatic forces that bind the atoms in the substrate’s pertinent bond. The electric potential inside the active pocket lowers the calculated substrate bond energy significantly for all enzymes. Provided that the enzyme’s ionized residues in the active pocket remain in the active ionization state, the calculated electric field is not sensitive to the overall protein charge and to the organism from which the enzyme originated. The simple model presented here is crude because it is based on a variety of simplifying assumptions. Nevertheless, we believe that it reflects the essential features of enzymatic catalysis. The model cannot be used blindly. To obtain useful predictions, it is necessary to use insightful judgment when interpreting calculated results. Acknowledgments For helpful discussions, the authors are grateful to Clayton Radke, Jack Kirsch and Judith Klinman (University of California, Berkeley, USA) and Luis Fernando Mercier Franco (University of Sao Paulo, Brazil). P.A.P.F. is grateful to the financial support of Sao Paulo
P.d.A. Pessoa Filho, J.M. Prausnitz / International Journal of Biological Macromolecules 78 (2015) 257–265
263
Research Foundation (FAPESP), through grant 2012/23860-2, and CNPq. Appendix A. Methodological questions The equations used in the calculations are simple, but their application to a particular system requires choices. Moreover, some simplifications introduced in the calculations need justification. First, we must decide how protein charges are assigned when calculating the electric field and the electric potential. The crystallographic structure can be assumed to define the positions of carbon, oxygen, and nitrogen atoms. However, the partial charges of each atom are necessary. Different force-field parameter sets can be used to estimate the partial charges of each atom in the enzyme. Three used here are: AMBER 94, CHARMM 27, and PARSE. Results presented in the text show weak dependence of the electric field and electric potential on the choice of force-field sets. This weak dependence can be assessed by the auxiliary function
(ı) =
j
< Ejk · Ek >
< Ek · Ek >
with |rjk | ≤ ı
(7)
where vector Ejk is the electric field generated by all atoms of amino acid j at position k, Ek is the electric field at position k resulting from the entire enzyme, and ı is the cut-off distance; for a given value of ı, we consider only those amino-acids whose distance to j is lower than ı. For calculating function , the position of each amino acid was considered to be the position of its ˛-carbon. For hen egg-white lysozyme, results are presented in Fig. 5, where some ionizable groups are indicated. Fig. 5 shows that the electric field results not only from charged amino acids in the active site; other ionized amino-acids also influence the electric field, but their importance fades as distance j-k increases beyond 2.0 nm. The influence of each amino acid depends on its charge, on its distance to the active site and on the orientation toward the active site. Fig. 6 presents function for bovine trypsin. In this case, the amino-acids mostly responsible for the calculated electric field are those in the catalytic tryad (HIS-57, SER-195, and ASPS-102). These examples show that care must be exercised when choosing a cut-off radius.
Fig. 6. Comparison of function for bovine trypsin calculated with different forcefield parameters. AMBER: continuous line; CHARMM: dotted line; PARSE: dashed line. Calculations with ionization state at pH 8.0. For this enzyme, the electric field is generated mainly by the amino acids belonging to the active tryad (HIS 57, ASP 102 and SER 195).
An alternate method to calculate the electric field is to consider only the integer charges of ionized amino-acids in the enzyme molecule. In this work, two methods were applied: the “discrete” and the “semi-discrete”. The “discrete” method considers only integer charges in oxygen and nitrogen atoms, and the “semi-discrete” method considers that each oxygen atom in an ionized carboxylic acid group bears a −0.5 charge due to resonance. Table 6 shows the electric fields calculated by both methods. For hen egg-white lysozyme and bovine ribonuclease, the discrete and semi-discrete methods yield results similar to those obtained using force-field partial charges (Table 3). However, for bovine trypsin, the discrete and semi-discrete methods give calculated fields significantly different from those previously calculated (Table 3). The main reason for this difference is that the electric field for trypsin is calculated close to non-ionized amino acids, but close to atoms (NE2 in HIS-57 and OG in SER-195) with large partial charges. Secondly, we must justify the suitability of using the electrostatic equations as presented to calculate the electric field and the electric potential. To verify the results, for comparison, similar calculations were made using the DELPHI package, extensively used to calculate electrostatic properties of proteins. Results are presented in Table 7 for hen egg-white lysozyme. These calculations were made with internal and external dielectric constants equal to 1.0 and with zero ionic strength. The results in Table 7 are similar to those in Tables 3 and 6: both procedures yield similar results. As DELPHI also considers only electrostatic interactions, the small differences observed can be ascribed to the numerical method used
Table 6 Results for the electric field (V nm−1 ) and electric potential (V) calculated using the discrete and semi-discrete methods.
Fig. 5. Comparison of function for hen-egg lysozyme calculated with different force-field parameters. AMBER: continuous line; CHARMM: dotted line; PARSE: dashed line. Calculations with ionization state at pH 5.0. For this enzyme, the electric field is generated by ionized groups in the active site (GLU 35 and ASP 52) and outside the active site.
Enzyme
Approach
Ex
Ey
Ez
|E|
Lysozyme/1lys
Discrete Semi-discrete Discrete Semi-discrete Discrete Semi-discrete
−11.4 −10.4 −0.05 −0.20 9.74 10.26
3.24 1.01 1.41 1.61 −5.24 −5.48
−0.95 −1.57 1.63 2.01 5.19 5.35
11.9 10.5 2.15 2.58 12.21 12.80
5.77 6.04 2.54 2.49 11.7 11.8
Trypsin/2ptn Ribonuclease/3a1r
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Table 7 Results for the electric field (V nm−1 ) and electric potential (V) for the active pocket of hen egg-white lysozyme (1lys)a . Calculations based on DELPHI package with ionization states at pH 5.0. Force-field
Ex
Ey
Ez
|E|
AMBER CHARMM PARSE Discrete Semi-discrete
−8.10 −9.56 −11.9 −11.5 −10.5
−3.32 −3.15 −4.08 3.06 0.98
−0.63 −1.29 1.24 −0.95 −1.52
8.78 10.1 12.6 12.0 10.6
6.30 6.10 6.02 5.58 5.88
a
Axes defined as in Table 3.
in DELPHI calculations. These differences rise as the number of charges increase. Thirdly, we must ask whether the crystallographic structure of the enzyme is adequate to calculate the electric field. The crystallographic structure is not exactly the same as that of a protein in solution; comprehensive studies on this subject were conducted by Sikic et al. [35] and Andrec et al. [36]. We must ask whether differences in the positions of the atoms in solution, compared with the fixed positions in the crystal, influence the calculated electric field. To assess qualitatively this difference, we calculated a molecular-dynamic simulation of hen-egg lysozyme at zero ionic strength, with the protein molecule at infinite dilution (a single 1lys molecule with ionization state at pH 5.0 in 23878 SPC water molecules, simulated using the OPLS force field) and with charge calculated at pH 5.0. 9 negative ions (Cl− ) were added to the water bath to counterbalance the protein positive net charge. The electric field was calculated using Eq. (1) at the mid-point (updated at each time step) between carbon CG of ASP-52 and carbon CD of GLU-35. The time-average of the absolute value of the electric field (|Ek |) is 11.3 ± 4.7 V nm−1 , and the projection at the direction given by carbon CG of ASP-52 and carbon CD of GLU-35 (updated at each time step) is 8.9 ± 3.5 V nm−1 . These results are similar to those presented in Table 3 for the electric field magnitude (|E|) and for the x component of the electric field (Ex ), calculated using the crystal structure. This comparison shows that changes in spatial arrangement (as calculated by molecular dynamic simulation) influence the calculated electric field, but its time-average value is approximately the same as that calculated using the crystal structure. Fourth, we must consider the relation between partial charges in the substrate molecule and the bond energy. Eqs. (3) and (4) are based on the hypothesis that only electrostatic interactions are considered, and that van der Waals forces between atoms in a bond are neglected. These equations also imply a separation of partial charges that corresponds neither to a heterolytic bond cleavage (which would result in an ionic pair) nor to a homolytic bond cleavage (which would result in the formation of free radicals). However, the separation of partial charges following the bond cleavage does not imply that isolated fractional charges are created. For example, in the peptide bond cleavage, the two separated amino-acid groups have overall integer charges, as calculated by the force-field parameter sets–even though the carbon and the nitrogen atoms bear fractional charges. The use of force-field parameters to estimate bond energies and bond forces in the substrate molecule is beyond the hypotheses that lead to these partial charges. When using force fields in molecular dynamic simulations, the so-called non-bonded terms (i.e. Coulomb and van der Waals forces) are used to calculate the interaction only between atoms that are not bonded to each other. However, the bond energies calculated with partial charges are checked against experimental bond energies, and only those partial charges that result in a good agreement with experiment are used in subsequent calculations. Furthermore, by considering that a chemical bond has a purely electrostatic nature, partial charges of the atoms belonging to that bond are somehow overestimated. As the
bond force and bond energy are proportional to the product of the partial charges, and the influence of the electric field is proportional to each individual partial charge, the calculated results constitute a worst-case scenario. Fried et al. [13] showed that the electric field in the enzyme’s active site is the main factor that reduces the activation barrier for reaching the transition state. In their work, the influence of the electric field on the dipole of the bond in the substrate molecule is considered in both the free molecule and in the transition state. Our model somehow involves a much simplified version of their conclusions, in which only the influence of the electric field on the bond dipole of the free substrate molecule is taken into account. A similar question arises from the influence of the electric field on the electron cloud of the substrate bond. The presence of the electric field shifts this electron cloud toward a different configuration, thus leading to different partial charges in the bond, and to a different bond energy in the substrate molecule. However, the assignment of charges to points, at the atomic scale, can be considered a rough approximation. The extensive literature on protein electrostatic calculations is ultimately based on this assignment. The literature shows that this approximation can lead to reliable results. Fifth, we must consider the ionized state of the amino acids belonging to the enzyme’s active dyad/tryad and that of the substrate molecule during the cleavage step. In a hydrolysis reaction, the bond-breaking step occurs with the substrate molecule and the active site in different ionization states, after the abstraction of a proton by either the substrate or the enzyme. For a lysozymecatalyzed reaction, prior to or at the bond-breaking step, a proton is abstracted from the GLU-35 amino-acid by the substrate molecule. For trypsin-catalyzed proteolysis, the bond-breaking step occurs after a proton is transferred from HIS-57 to SER-195, with a subsequent abstraction of this proton by the nitrogen atom of the peptide bond of the substrate molecule. For a ribonuclease-catalyzed reaction, the mechanism involves the transfer of a proton from HIS-119 to the oxygen atom of the P O bond of RNA. The effect of the electric field is not the only driving-force in these reactions; other aspects must be accounted for. Otherwise, the sole presence of charged residues in the protein surface, in whose closest vicinity large electric fields can be encountered, would catalyze a reaction. Because chymotrypsinogen and trypsin generate similar electric fields, it illustrates that the electric field is a necessary, but not sufficient, aspect to be considered. Chymotrypsinogen is the inactive form of chymotryspin, but the absence of enzymatic activity is related to conformational differences, not necessarily in the active site.
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Contents lists available at ScienceDirect
International Journal of Biological Macromolecules journal homepage: www.elsevier.com/locate/ijbiomac
A simplified electrostatic model for hydrolase catalysis Pedro de Alcantara Pessoa Filho a,b,∗ , John M. Prausnitz a a b
Department of Chemical and Biomolecular Engineering, University of California, Berkeley, CA, USA Department of Chemical Engineering, University of Sao Paulo, Sao Paulo, SP, Brazil
a r t i c l e
i n f o
Article history: Received 1 December 2014 Received in revised form 31 March 2015 Accepted 1 April 2015 Available online 13 April 2015 Keywords: Enzyme Hydrolase Model
a b s t r a c t Toward the development of an electrostatic model for enzyme catalysis, the active site of the enzyme is represented by a cavity whose surface (and beyond) is populated by electric charges as determined by pH and the enzyme’s structure. The electric field in the cavity is obtained from electrostatics and a suitable computer program. The key chemical bond in the substrate, at its ends, has partial charges with opposite signs determined from published force−field parameters. The electric field attracts one end of the bond and repels the other, causing bond tension. If that tension exceeds the attractive force between the atoms, the bond breaks; the enzyme is then a successful catalyst. To illustrate this very simple model, based on numerous assumptions, some results are presented for three hydrolases: hen-egg white lysozyme, bovine trypsin and bovine ribonuclease. Attention is given to the effect of pH. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Enzyme catalysis constitutes a well-established field of research in biochemistry. Pertinent literature reviews on this subject can be found in textbooks such as those by Frey and Hegeman [1], Bisswanger [2] and Purich [3], among others. In this work, we consider a fundamental question: For a specific substrate reaction at a fixed temperature and pH, will a specific enzyme provide catalysis or not? Toward an effort to answer this question, we initiate here a simple model based on electrostatics. To introduce the basic idea, consider two parallel metallic plates separated by a short distance. One plate bears a positive electric charge, while the other bears a negative electric charge, as indicated in Fig. 1. Into the gap between the plates, we put a dumbbell whose two ends are separated by a small distance compared with the height of the gap. The two ends of the dumbbell are charged; one end is positive and the other end is negative. The two charged plates generate an electric field perpendicular to the surfaces of the plates. This field attracts one end of the dumbbell while repelling the other, generating a tensile force in the rod that connects the two ends. If the tensile force is sufficiently strong, the rod breaks.
∗ Corresponding author at: Caixa Postal 61548, 05424-970, Sao Paulo, SP, Brazil. Tel.: +55 11 3091 1106; fax: +55 11 3091 2284. E-mail address: [email protected] (P.d.A. Pessoa Filho). http://dx.doi.org/10.1016/j.ijbiomac.2015.04.010 0141-8130/© 2015 Elsevier B.V. All rights reserved.
We use this electromechanical picture as a basis for a simple model of enzyme catalysis. The two charged plates are replaced by the surface of a spherical cavity that represents the immediate region of an enzyme’s active site. The charge on this surface is not uniform; it is determined by the positions of charges on the enzyme at a fixed pH. The dumbbell is replaced by the pertinent chemical bond in the substrate, inside the cavity. The ends of the dumbbell bear partial charges of opposite sign. When the electric field in the cavity exceeds the chemical bond strength, the bond is broken, giving a catalyzed reaction. Before presenting the essential details of our simple electrostatic model, we note that application of electrostatic physics to catalysis is not a new idea, as indicated by numerous previous publications, e.g. Warshel [4], Warshel et al. [5], Dao-Pin et al. [6]. The importance of the electric field in enzyme catalysis is evident in the THEMATICS method for identifying the active site [7–9]. Considering an enzyme of known structure but unknown active site, THEMATICS searches for unusual theoretically calculated titration curves, and associates the distortion in titration curves to catalytic activity. Electrostatic interactions also enhance the diffusion of substrate molecules to the enzyme’s active site [10,11] and the steering of substrate molecules inside the active site [12]. Fried et al. [13] showed recently that large electric fields found in the active pocket of ketosteorid isomerase are responsible for the unusually high rate constants of the reaction catalyzed by this enzyme. However, to our best knowledge, no previous publication has tried to answer the question of concern here: for a given situation (enzyme, substrate, temperature, pH), can a relationship between the electric
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Nomenclature Latin letters Ek electric field at position k Ex x-component of the electric field Ejk electric field generated by group j at position k electrostatic force acting on atom m Fm Fmn electrostatic force acting on atom n due to atom m N number of atoms in the enzyme molecule charge on atom i qi rjk vector from position (of atom/group) j to position (of atom/group) k Umn electrostatic energy between atoms m and n Greek letters ı cut-off distance in Eq. (7) ε0 vacuum permittivity dielectric constant ε k electric potential at position k function defined by Eq. (7)
does not interact with the active site. Instead, we imply that the effect of the charges of the substrate molecule on the electric field is the same, regardless of is environment. The effect of changes in the spatial arrangement in the substrate molecule is considered negligible for the calculation of the electric field. In all calculations, we set ε = 1.0 as discussed in Appendix. The electric potential k generated by the enzyme molecule at position k is obtained from: 1 qj 4εε0 |rjk | N
k =
(2)
j=1
where j is an atom in the enzyme. The summation is over all N atoms of the enzyme. While Ek is a vector, k is scalar. We now consider the bond formed by atoms m and n in the substrate in the absence of an external electric field due to the enzyme. Atoms m and n bear partial charges qm and qn , respectively. The electrostatic force Fmn acting on atom n due to atom m is a vector. This force is calculated from: Fmn =
1 qm qn rmn 4εε0 |rmn |3
(3)
We refer to the absolute value of force Fmn as the bond force. The energy Umn of bond mn in the substrate is calculated as the minimum work necessary to separate the charges on m and n from the bond length to infinity. In the absence of an external electric field, the bond energy is calculated in a molar basis as:
∞ Umn = NAV Fig. 1. Illustrative two-dimensional representation of bond cleavage in the active site. The two charged plates generate electric field E. (a) dumbbell representing the substrate’s pertinent bond in the active site. The negatively charged atom is red, while the positively charged atom is blue. The electric field attracts the blue atom and repels the red atom, causing a tensile force in the (dumbbell) bond. (b) Atoms after cleavage. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
field generated in the enzyme’s active site and the electric dipole of the bond in the substrate molecule be established? Based on simplifying assumptions, our model aims to provide an answer to the question. 2. Calculation of the Electric Field and the Electric Potential To calculate the electric field and electric potential inside the active pocket, the influence of all enzyme atoms is taken into account: each atom bears a full or partial charge that influences the electric field and the electric potential. The electric field Ek is a vector. At position k of the active pocket, the electric field is calculated using the basic equation of electrostatics: 1 qj rjk 3 4εε0 r N
Ek =
j=1
(1)
jk
where ε is the relative permittivity of the medium, ε0 is the permittivity of vacuum, qj is the charge on atom j, and rjk is the vector from the position of atom j in the enzyme to position k anywhere in the active pocket. The summation is carried out over all N atoms in the enzyme. In calculating Ek we do not consider full or partial charges on the substrate molecule. These charges are present even when the substrate is outside the active cavity. The electric field calculated through Eq. (1) is in effect an “excess” field caused by the enzyme’s active site. i.e. it is the net effect of the enzyme’s active site on the atoms that constitute the bond. By disregarding the influence of the substrate molecule, we do not suggest that the substrate molecule
Fmn · drmn = |r mn |
NAV qm qn 4εε0 |rmn |
(4)
where NAV is the Avogadro number. We now consider that bond mn in the substrate is placed in the active cavity of the enzyme. In the presence of the electric field due to the enzyme, the resulting force on atom m of bond mn is: Fm = qm Em + Fmn
(5)
A similar equation gives the force on atom n. The bond energy for bond mn placed in the cavity is the minimum work to separate qm and qn from the mn bond length to infinity, but now we must include the influence of the electric potential generated by the enzyme. The bond energy for bond mn placed in the active cavity is: Umn =
NAV qm qn + NAV (qm m + qn n ) 4εε0 |rmn |
(6)
where m and n are calculated at the positions of atoms m and n, respectively. The influence of the electric potential on the bond energy depends on the bond’s orientation. A specific enzyme may break a specific bond in the substrate, where the terminal atoms in the bond bear partial charges. Two methods are used here to estimate these partial charges. If partial charges for the ends of bond mn are given in one or more published force-field parameter sets, these partial charges are used, provided that the calculated bond energy (Eq. (4)) is close to the average bond energy of similar bonds. Otherwise, partial charges are estimated from Eq. (4) to be in general agreement with the average bond energy of similar bonds. In this case, we use partial charges of the same magnitude but opposite sign. Table 1 summarizes the partial charges used in our calculations. 3. Enzyme Structure and Ionization States To illustrate the simple model proposed here, three bondbreaking enzymes (hydrolases) were considered: lysozyme, trypsin
P.d.A. Pessoa Filho, J.M. Prausnitz / International Journal of Biological Macromolecules 78 (2015) 257–265 Table 1 Partial charges in substrate bonds. Enzyme
Atom 1
Atom 2
q1 /ea
q2 /e
Estimated from
Lysozyme Tryspin
C C
O N
Ribonuclease
P
O
0.60 0.61 0.51 0.55 0.96
−0.60 −0.42 −0.47 −0.40 −0.66
Bond-energy data AMBER CHARMM PARSE PARSE
a
e is the elementary charge; 1 e = 1.602·10−19 C.
and ribonuclease. To calculate the electric field (Eq. (1)) and the electric potential (Eq. (2)), we need the positions of all enzyme atoms. The enzymes’ crystal structure was used in the calculation of the electric field and the electric potential. Enzyme structure was obtained from the Protein Data-Bank (PDB). Table 2 presents the PDB code for all enzymes; nomenclature in the subsequent discussion follows the corresponding coordinate file for each enzyme. However, these coordinate files do not provide the positions of hydrogen atoms and the partial charges. Therefore, hydrogen atoms and partial charges were obtained using the PDB2PQR program [14,15] and applying the parameters of force-fields AMBER 94 [16,17], CHARMM 27 [18] and PARSE [19]. Ionization as a function of pH was calculated using the PROPKA method [20,21]. The medium permittivity was considered to be that of vacuum (i.e. ε = 1.0). All calculations were carried out through ad hoc developed programs written in FORTRAN 90, verified by comparison with calculations using the DELPHI package [22–24].
259
contains amino acids HIS-12 (neutral) and HIS-119 (positively charged) [29,30]. The cleavage of the P O bond occurs after the abstraction of a proton from the oxygen atom by HIS-12. The ionization states of other amino acids were obtained at pH 6.0. 4. Examples To calculate the effect of the electric field on the pertinent substrate bond, we must specify the partial charges at the ends of the bond and the putative position it occupies in the active-site cavity. Firstly, we consider hen egg-white lysozyme that may break the substrate’s glycoside bond between a carbon atom and an oxygen atom. The average length of a C O bond is 0.143 nm, and its average bond energy is 350 kJ mol−1 [31]. From Eq. (4), and assuming |qm | = |qn |, the partial charges qm and qn can be estimated; they are −0.6e for the oxygen atom and 0.6e for the carbon atom. During the cleavage step, the C O bond lies approximately between the carboxyl groups of amino acids GLU-35 and ASP-52 in lysozyme. Fig. 2a shows some of the main amino-acids in the active pocket of lysozyme. Table 3 presents the electric field calculated for the mid-point between carbon CG of ASP-52 and carbon CD of GLU-35 at pH 5.0. For these enzyme calculations, we used partial charges obtained from three different force-field parameter
3.1. Ionization states and mechanisms For the examples given in the next section, we consider the ionization and catalytic mechanism for three enzymes. Lysozyme is a glycoside hydrolase. It hydrolyzes the glycosidic bond between the oxygen atom of N-acetylmuramic acid and the carbon atom of N-acetylglucosamine in peptidoglycans. The active site of hen egg-white lysozyme comprises two ionizable groups: GLU-35 and ASP-52. The enzyme is active when GLU-35 is neutral and ASP-52 is negatively charged [25,26]. The active dyad of human lysozyme contains the same amino acids in similar positions, with ASP-53 instead of ASP-52. The oxygen atom of N-acetylmuramic acid acts as proton-acceptor for the GLU-35 protonated carboxylic acid group. Subsequent steps restore the GLU-35 group to the protonated state by abstracting a proton of a water molecule. The ionization states of other amino acids were obtained at pH 5.0. Trypsin is a protease. It hydrolyzes the peptide bond at the carboxyl side of some amino-acids, chiefly lysine and arginine. The active tryad of bovine and porcine trypsin contains amino acids HIS57 (neutral), ASP-102 (negatively charged) and SER-195 (neutral) [27,28]. The ionization states of other amino acids were obtained at pH 8.0. Calculations were also conducted for bovine chymotrypsin, another protease with the same active tryad. Ribonuclease hydrolyzes the P O bond of cytidine and uridine residues of ribonucleic acid. The active dyad of ribonuclease Table 2 Protein Data Bank (PDB) codes for the structure of enzymes. Enzyme
Organism
PDB code
Lysozyme
Gallus gallus Homo sapiens Bos taurus Sus scofa Bos taurus Bos taurus
1lys 1lz1 2ptna 1s81b 3a1r 2cga
Trypsin Ribonuclease Chymotrypsinogen a b
Considering ALA-221. Considering PHE-184, LYS-188 and ALA-221.
Fig. 2. (a) Partial structure of the hen egg-white lysozyme active site, indicating GLU-35 and ASP-52; (b) placement of bond C O inside the active site of lysozyme. Gray spheres: carbon atoms; blue spheres: nitrogen atoms, and red spheres: oxygen atoms. Sphere radii do not correlate with atomic radii. Other atoms in enzyme and substrate omitted for clarity.
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Table 3 Results for the electric field (V nm−1 ) and electric potential (V) for the active pocket of enzymes. Enzyme
pH
Force field
Hen lyzozyme (1lys)a
5.0
AMBER
8.0
CHARMM PARSE AMBER
6.0
Bovine trypsin (2ptn)b
Bovine ribonuclease (3a1r)c
Ex
|E|
Ey
Ez
−2.30
−0.36
−10.0 −11.7 −13.9
−1.87 −3.92 6.18
−0.87 1.50 8.52
10.2 12.4 17.4
6.37 6.26 2.16
CHARMM PARSE AMBER
−13.6 −16.1 12.0
6.52 5.99 1.96
10.4 7.09 12.1
18.4 18.6 17.2
1.87 1.20 11.4
CHARMM PARSE
13.0 14.3
3.77 9.01
12.9 19.8
18.7 26.0
11.5 12.5
−8.96
9.26
6.53
a The x-axis is defined by carbon CG of ASP-52 (origin) and carbon CD of GLU-35 (positive direction); the y-axis is approximately the axis defined by the by the origin and carbon C of ILE-58 (negative direction); the z-axis is defined in such a way that the basis is right handed. b The x-axis is defined by nitrogen NE2 of HIS-57 (origin) and oxygen OG of SER195 (positive direction); the y-axis is approximately the axis defined by the origin and nitrogen N in TYR-172 (positive direction); the z-axis is defined in such a way that the basis is right handed. c The x-axis is defined by nitrogen ND1 of HIS-119 (origin) and nitrogen NE2 of HIS-12 (positive direction); the y-axis is approximately the axis defined by the origin and nitrogen N of ASP 121 (negative direction); the z-axis is defined in such a way that the basis is right handed.
sets. The results show that the electric field is oriented in the direction connecting these atoms in the enzyme molecule, and that the calculated electric field is almost independent of the force-field parameter set used to obtain the partial charges for all atoms in the enzyme. To overcome the electrostatic attraction between the carbon atom and the oxygen atom in the substrate (calculated using Eq. (3) and partial charges presented in Table 1) requires a force of 4.1 nN (nano-Newton, i.e. 4.1 × 10−9 N). If the C O bond is placed in the active pocket, midway between the CG carbon of ASP-52 and the CD carbon of GLU-35, with the oxygen atom closer to the ASP-52 residue (Fig. 2b), the estimated force is 2.3−3.0 nN, depending on the choice of force-field parameter set. Because this force is in the bond direction, it is a tensile force. The energy of the C O bond in the active pocket, calculated from Eq. (6), gives 255 kJ mol−1 (calculations with PARSE parameters) or 280 kJ mol−1 (calculations with AMBER parameters). These results show that the tensile force due to the electric field acting on the bond atoms has the same magnitude as the attractive force between these atoms. Moreover, the energy of the bond inside the active site is significantly lower than the bond energy in the absence of the electric field (350 kJ mol−1 ). Because the energy barrier for bond cleavage is lower, the bond breaks. In summary, bond breaking is highly favored in the active site, as observed. Secondly, we consider trypsin-catalyzed proteolysis. Trypsin breaks the peptide bond between a carbon atom and a nitrogen atom. The average length of a C N bond is 0.147 nm, and its average bond energy is 308 kJ mol−1 [31]. However, in a peptide bond, the C N bond is known to have a length of 0.133 nm and an energy lower (in magnitude) than the average energy of other C N bonds. Estimating the partial charges from force-field parameter sets, the calculated bond energy ranges from 230 (PARSE) to 266 kJ mol−1 (AMBER). During the bond cleavage by bovine trypsin, the peptide bond in the substrate is located approximately parallel to the HIS-57 and SER-195 groups in the enzyme, as shown in Fig. 3a. These amino acids are so close that the peptide bond of the substrate
Fig. 3. (a) Partial structure of the bovine trypsin active site, indicating HIS-57 and SER-195; (b) placement of the peptide bond C N inside the active site of trypsin. Gray spheres: carbon atoms; blue spheres: nitrogen atoms, and red spheres: oxygen atoms. Sphere radii do not correlate with atomic radii. Other atoms in enzyme and substrate omitted for clarity.
cannot occupy the space between them. The electric field and electric potential were calculated for a point equidistant from NE2 in HIS-57 and OG in SER-195, but at a distance 0.2 nm from the line defined by these atoms, toward outside the cavity. Results are presented in Table 3. Because the electric field is oriented in the direction of the C N bond in the substrate, it is a tensile force. Calculated bond forces for the peptide bond range from 2.9 to 3.1 nN, depending on the choice of partial charges as shown in Table 1. If the peptide bond is placed at a distance 0.2 nm of the straight line between NE2 in HIS-57 and OG in SER-195, as shown in Fig. 3b, the calculated force acting on the peptide-bond atoms due to the electric field is 2.0 (AMBER and CHARMM) to 2.4 nN (PARSE). This force is obtained using the force component acting in the same direction as that of the bond. Therefore, the force that attracts the atoms in the peptide bond and the tensile force arising due to the electric field in the enzyme’s active pocket have the same order of magnitude. Moreover, the calculated bond energy for the peptide bond inside the active pocket lies between 110 (PARSE) and 157 kJ mol−1 (CHARMM). These energies are lower than that calculated for the peptide bond outside the active pocket (230−266 kJ mol−1 ). Hence, the C−N bond experiences a tensile force of the same magnitude as the bond force, and its energy is significantly lower inside the active site. We conclude that peptide bond breaking is favored in the active site. The third example concerning ribonuclease is counter-intuitive. This enzyme can break the P O bond in RNA. The P O bond energy can be estimated from force-field parameter sets. The P O bond distance from crystallographic data for RNA is 0.148 nm. From these data, the bond energy is 846 (AMBER), 1090 (CHARMM) and 598 kJ mol−1 (PARSE). This latter energy is compatible with the
P.d.A. Pessoa Filho, J.M. Prausnitz / International Journal of Biological Macromolecules 78 (2015) 257–265
expected energy for a P O bond. Therefore, we use the partial charges from PARSE: −0.66e for the oxygen atom and 0.96e for the phosphorus atom. The bond force (Eq. (3)) calculated with these partial charges is 6.7 nN. Table 3 gives the calculated electric field for the active pocket of pancreatic bovine ribonuclease. The electric fields were obtained for the mid-point between the nitrogen atom NE2 in HIS-12 and the nitrogen atom ND1 in HIS-119, shown in Fig. 4a. To calculate the force acting upon the atoms in the P O bond of the substrate, the position of this bond was considered perpendicular to the line joining these nitrogen atoms in enzyme, with the oxygen atom in the P O bond occupying the mid-point of this line and the phosphorus atom pointing outside the cavity, as shown in Fig. 4b. Using PARSE force-field parameters, the force generated by the electric field is 1.2 nN, significantly lower than the bond force, 6.7 nN. Because we know that ribonuclease can break P O bonds, this calculated result appears to be counter to our proposed hypothesis. However, in this case, due to the orientation of the bond in the active pocket, the pertinent force is not a tensile force but a shear force. The shear force acting on the oxygen atom is 2.8 nN (PARSE), while the phosphorus atom is not affected by a shear force. The calculated bond energy is lowered to 301 kJ mol−1 . Therefore, the same pattern observed for lysozyme and trypsin is also observed for ribonuclease: the resulting forces break the bond and the bond energy is lowered in the active site. We conclude that the enzyme provides enhanced breaking of the P O bond.
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4.1. Influence of enzyme structure and pH Our calculations assume invariance of the primary structure of the enzyme. However, for many cases, the same enzyme produced by different organisms catalyzes the same reaction. Thus, despite different origins and differences in amino-acid sequence, it appears that the same enzyme generates similar electric fields and potentials. To assess the effect of enzyme structure, calculations similar to those above were carried out for human lysozyme and porcine trypsin. Table 4 gives results for the electric field and electric potential for human lysozyme. For 129 amino-acid groups of hen lysozyme and 130 amino-acid groups of human lysozyme, human and hen egg-white lysozymes share 79 amino acids in corresponding positions. Despite differences in the amino acids, the electric field and the electric potential calculated for human lysozyme are similar to those for hen egg-white lysozyme. Table 4 also shows results for the electric field for porcine trypsin. The variation between porcine and bovine trypsin is larger than that between the two lysozymes, even though both trypsins share a higher proportion of amino-acid groups (183 out of 223) than the lysozymes. Unlike lysozyme, for trypsin partial charges of neutral amino acids, instead of ionized ones, have a predominant effect on the electric field, as discussed in the Appendix. Chymotrypsin is a serine protease with the same catalytic tryad as that for trypsin. Both enzymes cleave the same bond. Side chains in the substrate determine whether a specific peptide bond can be broken. We expect that, for the active cavity, the electric field and electric potential for chymotrypsin is similar to those for trypsin. Table 4 shows the calculated electric field for the same position in chymotrypsinogen A, the inactive form of chymotrypsin. The catalytic tryad in chymotrypsin and that in chymotrypsinogen A have the same amino acids and the same spatial orientation, but chymotrypsinogen A is inactive due to differences in primary/tertiary structure. As expected, the electric field generated in the active pocket of chymotrypsinogen A is similar to that of trypsin, as both enzymes cleave the same peptide bond. The pH affects the overall charge of the protein molecule and the charges of the ionizable groups in the active site. There is extensive experimental evidence that the ionization state of the ionizable groups in the active pocket determines the presence or absence
Table 4 Results for the electric field (V nm−1 ) and electric potential (V) for the active pocket of variant forms of enzymes.
Fig. 4. (a) Partial structure of bovine ribonuclease active site, indicating HIS-12 and HIS-119; (b) placement of the bond P O inside the active site of trypsin. Gray spheres: carbon atoms; blue spheres: nitrogen atoms; red spheres: oxygen atoms, and orange sphere: phosphorus atom. Sphere radii do not correlate with atomic radii. Other atoms in enzyme and substrate omitted for clarity.
|E|
pH
Forcefield
Ex
Ey
Human lyzozyme (1lys)a
5.0
AMBER CHARMM PARSE
−11.6 −12.6 −14.1
1.31 2.00 1.27
2.91 2.02 5.15
12.1 12.9 15.1
5.99 5.69 5.36
Porcine trypsin (1s81)b
8.0
AMBER CHARMM PARSE
5.79 6.74 5.45
−9.08 −10.5 −8.51
11.9 13.5 12.0
−1.42 −1.73 −2.46
Bovine chymotryspinogen (2cga)c
8.0
AMBER CHARMM PARSE
5.10 3.79 7.45
7.29 6.22 7.94
17.2 16.1 21.7
1.12 0.99 0.84
−4.97 −5.16 −6.51 −14.8 −14.3 −18.8
Ez
Enzyme
a The x-axis is defined by carbon CG of ASP-52 (origin) and carbon CD of GLU-35 (positive direction); the y-axis is approximately the axis defined by the by the origin and carbon C of ILE-58 (negative direction); the z-axis is defined in such a way that the basis is right handed. b The x-axis is defined by nitrogen NE2 of HIS-57 (origin) and oxygen OG of SER195 (positive direction); the y-axis is approximately the axis defined by the origin and nitrogen N in TYR-172 (positive direction); the z-axis is defined in such a way that the basis is right handed. c The x-axis is defined by nitrogen ND1 of HIS-119 (origin) and nitrogen NE2 of HIS-12 (positive direction); the y-axis is approximately the axis defined by the origin and nitrogen N of ASP 121 (negative direction); the z-axis is defined in such a way that the basis is right handed.
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of catalytic activity [3]. The equilibrium among the different ionization states of the ionizable groups in the active site results in the usual bell-shape curve for the relative activity as a function of pH. Therefore, provided that the groups in the active site remain in the active configuration (i.e. the configuration that assures the presence of proton-donor and proton-acceptor groups for the reaction pathway), according to our model the electric field should be insensitive to changes in the overall ionization state of the entire enzyme. To illustrate the effect of pH, we calculated the electric field for lysozyme at pH 2.0−9.0, the pH range where lysozyme is reported to be enzymatically active. Table 5 presents the electric field as a function of pH for the mid-point between carbon CG of ASP-52 and carbon CD of GLU-35. Results are based on CHARMM force-field parameters; results for other force-field parameters are comparable. Despite the change in the overall charge of the enzyme that drops by approximately half in this pH range, the electric field remains almost constant. The opposite result is obtained when the ionization state of the ionizable amino-acids in the active pocket changes. On the one hand, if both ASP-52 and GLU-35 are neutral, the absolute value of the electric field decreases from 10.2 to 2.3 V nm−1 , using the CHARMM force-field parameters. This reduction is primarily related to a reduction in the x-component of the electric field; this component is in the direction of the C O bond of the substrate during the cleavage step. This x-component decreases from −10.0 to −2.0 V nm−1 . There is no significant decrease in the calculated C O bond energy: the bond energy decreases from 350 to 331 kJ mol−1 . On the other hand, if both ASP-52 and GLU-35 are ionized, the x-component of the electric field changes sign, from −10.0 to 4.0 V nm−1 , and the effect of the electric field is to stabilize the C O bond of the substrate: the calculated bond energy increases from 350 to 404 kJ mol−1 . Therefore, bond cleavage is not favored in these ionization states. 4.2. Comparison with other methods The most common procedure to calculate protein electrostatics is to use the two-dielectric assumption: the protein molecule is considered a low-dielectric medium, with ε between 2 and 10 (the so-called “internal” dielectric constant), surrounded by an aqueous solvent, with ε equal to 78 or 80 and a certain ionic strength [32]. This internal dielectric constant accounts for the dipoles in the protein molecule and approximately for atomic polarizability [32]. The Poisson-Boltzmann equation is then solved numerically. The electric fields presented in Table 3 disagree with those presented by Dao-Pin et al. [6] for the electric field calculated in the vicinity of GLU-35 of hen egg-white lysozyme. Considering no solvent charges or partial charges in the enzyme, these authors arrived at electric fields between 0.16 and 0.77 V nm−1 . The main Table 5 Results for the electric field (V nm−1 ) for the active pocket of hen egg-white lysozyme (1lys). Calculations based on CHARMM force-field parameters. pH
Overall chargea
Ex
Ey
Ez
|E|
2 3 4 5 6−9b
17 16 12 10 9
−10.4 −11.1 −11.2 −10.0 −9.87
0.05 −0.19 −0.67 −1.87 −2.18
−1.17 −0.97 −0.71 −0.87 −0.72
10.4 11.2 11.3 10.2 10.1
a The overall charge is calculated considering that the ionized groups involved in the catalytic mechanism are in the active state, even if the most common form is different. b There is no difference between the ionized states calculated in this pH range; the only change that would occur is the ionization of ASP-52.
source of this disagreement is the use of a different solvent permittivity, along with the use of the Poisson-Boltzmann equation, by Dao-Pin et al. [6]. If we consider the two-dielectric model, we obtain results similar to those by Dao-Pin et al. [6]. For the midpoint between CG carbon of ASP-52 and the CD carbon of GLU-35 of hen egg-white lysozyme, the electric field calculated using the DELPHI package is 0.22 V nm−1 ; in this calculation, the ionic strength is 0.1 mol·kg−1 , the internal dielectric constant is 4 and the solvent dielectric constant is 78. Besides the solvent permittivity, the internal permittivity influences the calculated electric field. When ε for the internal medium changes from ε = 1 to ε = 4, the electric field calculated with the DELPHI package for the lysozyme molecule, at the mid-point between GLU-35 and ASP-52, decreases from 12.0 to 7.19 V nm−1 ; when ε = 10, we obtain 4.53 V nm−1 . As pointed out by Li et al. [32], there is no general solution for this question. The two dielectric assumption, while adequate for calculating pKA changes and solvation energies, is not suitable for the calculation of the electric field inside the enzyme’s active pocket [33,34]. The solvent cannot be assumed to be a continuum at the atomistic scale corresponding to our calculations, and implicit solvent assumptions may be unreasonable [33]. Moreover, the presence of the substrate inside the active pocket results in at least partial exclusion of solvent molecules. Therefore, for the very small space in the active cavity, the dielectric constant has no meaning. A more comprehensive and accurate description of the energy change should include contributions from electron exchange, delocalization (i.e. related to deformations in charge distribution) and correlation [37], obtained in the framework of first-principle calculations. Examples of first-principle calculations include chorismate mutase [38] and protein kinase [39], among others. These works present a detailed description of enzyme catalysis concerning energy calculations, and can be used in a predictive way. However, they also show that the electrostatic contribution is predominant to the activation energy lowering, and hence to the catalytic activity, which is the basis of the model presented here. 5. Conclusions This paper investigates the relationship between the electric field and the electric potential generated in the active pocket of an enzyme, and the force and energy of the pertinent chemical bond in the substrate. Illustrative calculations are presented for three hydrolases: hen egg-white lysozyme, bovine trypsin, and bovine ribonuclease. For lysozyme and trypsin, the tensile forces resulting from the electric field in the active pocket have the same magnitude as the electrostatic forces that bind the atoms in the substrate’s pertinent bond. The electric potential inside the active pocket lowers the calculated substrate bond energy significantly for all enzymes. Provided that the enzyme’s ionized residues in the active pocket remain in the active ionization state, the calculated electric field is not sensitive to the overall protein charge and to the organism from which the enzyme originated. The simple model presented here is crude because it is based on a variety of simplifying assumptions. Nevertheless, we believe that it reflects the essential features of enzymatic catalysis. The model cannot be used blindly. To obtain useful predictions, it is necessary to use insightful judgment when interpreting calculated results. Acknowledgments For helpful discussions, the authors are grateful to Clayton Radke, Jack Kirsch and Judith Klinman (University of California, Berkeley, USA) and Luis Fernando Mercier Franco (University of Sao Paulo, Brazil). P.A.P.F. is grateful to the financial support of Sao Paulo
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Research Foundation (FAPESP), through grant 2012/23860-2, and CNPq. Appendix A. Methodological questions The equations used in the calculations are simple, but their application to a particular system requires choices. Moreover, some simplifications introduced in the calculations need justification. First, we must decide how protein charges are assigned when calculating the electric field and the electric potential. The crystallographic structure can be assumed to define the positions of carbon, oxygen, and nitrogen atoms. However, the partial charges of each atom are necessary. Different force-field parameter sets can be used to estimate the partial charges of each atom in the enzyme. Three used here are: AMBER 94, CHARMM 27, and PARSE. Results presented in the text show weak dependence of the electric field and electric potential on the choice of force-field sets. This weak dependence can be assessed by the auxiliary function
(ı) =
j
< Ejk · Ek >
< Ek · Ek >
with |rjk | ≤ ı
(7)
where vector Ejk is the electric field generated by all atoms of amino acid j at position k, Ek is the electric field at position k resulting from the entire enzyme, and ı is the cut-off distance; for a given value of ı, we consider only those amino-acids whose distance to j is lower than ı. For calculating function , the position of each amino acid was considered to be the position of its ˛-carbon. For hen egg-white lysozyme, results are presented in Fig. 5, where some ionizable groups are indicated. Fig. 5 shows that the electric field results not only from charged amino acids in the active site; other ionized amino-acids also influence the electric field, but their importance fades as distance j-k increases beyond 2.0 nm. The influence of each amino acid depends on its charge, on its distance to the active site and on the orientation toward the active site. Fig. 6 presents function for bovine trypsin. In this case, the amino-acids mostly responsible for the calculated electric field are those in the catalytic tryad (HIS-57, SER-195, and ASPS-102). These examples show that care must be exercised when choosing a cut-off radius.
Fig. 6. Comparison of function for bovine trypsin calculated with different forcefield parameters. AMBER: continuous line; CHARMM: dotted line; PARSE: dashed line. Calculations with ionization state at pH 8.0. For this enzyme, the electric field is generated mainly by the amino acids belonging to the active tryad (HIS 57, ASP 102 and SER 195).
An alternate method to calculate the electric field is to consider only the integer charges of ionized amino-acids in the enzyme molecule. In this work, two methods were applied: the “discrete” and the “semi-discrete”. The “discrete” method considers only integer charges in oxygen and nitrogen atoms, and the “semi-discrete” method considers that each oxygen atom in an ionized carboxylic acid group bears a −0.5 charge due to resonance. Table 6 shows the electric fields calculated by both methods. For hen egg-white lysozyme and bovine ribonuclease, the discrete and semi-discrete methods yield results similar to those obtained using force-field partial charges (Table 3). However, for bovine trypsin, the discrete and semi-discrete methods give calculated fields significantly different from those previously calculated (Table 3). The main reason for this difference is that the electric field for trypsin is calculated close to non-ionized amino acids, but close to atoms (NE2 in HIS-57 and OG in SER-195) with large partial charges. Secondly, we must justify the suitability of using the electrostatic equations as presented to calculate the electric field and the electric potential. To verify the results, for comparison, similar calculations were made using the DELPHI package, extensively used to calculate electrostatic properties of proteins. Results are presented in Table 7 for hen egg-white lysozyme. These calculations were made with internal and external dielectric constants equal to 1.0 and with zero ionic strength. The results in Table 7 are similar to those in Tables 3 and 6: both procedures yield similar results. As DELPHI also considers only electrostatic interactions, the small differences observed can be ascribed to the numerical method used
Table 6 Results for the electric field (V nm−1 ) and electric potential (V) calculated using the discrete and semi-discrete methods.
Fig. 5. Comparison of function for hen-egg lysozyme calculated with different force-field parameters. AMBER: continuous line; CHARMM: dotted line; PARSE: dashed line. Calculations with ionization state at pH 5.0. For this enzyme, the electric field is generated by ionized groups in the active site (GLU 35 and ASP 52) and outside the active site.
Enzyme
Approach
Ex
Ey
Ez
|E|
Lysozyme/1lys
Discrete Semi-discrete Discrete Semi-discrete Discrete Semi-discrete
−11.4 −10.4 −0.05 −0.20 9.74 10.26
3.24 1.01 1.41 1.61 −5.24 −5.48
−0.95 −1.57 1.63 2.01 5.19 5.35
11.9 10.5 2.15 2.58 12.21 12.80
5.77 6.04 2.54 2.49 11.7 11.8
Trypsin/2ptn Ribonuclease/3a1r
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Table 7 Results for the electric field (V nm−1 ) and electric potential (V) for the active pocket of hen egg-white lysozyme (1lys)a . Calculations based on DELPHI package with ionization states at pH 5.0. Force-field
Ex
Ey
Ez
|E|
AMBER CHARMM PARSE Discrete Semi-discrete
−8.10 −9.56 −11.9 −11.5 −10.5
−3.32 −3.15 −4.08 3.06 0.98
−0.63 −1.29 1.24 −0.95 −1.52
8.78 10.1 12.6 12.0 10.6
6.30 6.10 6.02 5.58 5.88
a
Axes defined as in Table 3.
in DELPHI calculations. These differences rise as the number of charges increase. Thirdly, we must ask whether the crystallographic structure of the enzyme is adequate to calculate the electric field. The crystallographic structure is not exactly the same as that of a protein in solution; comprehensive studies on this subject were conducted by Sikic et al. [35] and Andrec et al. [36]. We must ask whether differences in the positions of the atoms in solution, compared with the fixed positions in the crystal, influence the calculated electric field. To assess qualitatively this difference, we calculated a molecular-dynamic simulation of hen-egg lysozyme at zero ionic strength, with the protein molecule at infinite dilution (a single 1lys molecule with ionization state at pH 5.0 in 23878 SPC water molecules, simulated using the OPLS force field) and with charge calculated at pH 5.0. 9 negative ions (Cl− ) were added to the water bath to counterbalance the protein positive net charge. The electric field was calculated using Eq. (1) at the mid-point (updated at each time step) between carbon CG of ASP-52 and carbon CD of GLU-35. The time-average of the absolute value of the electric field (|Ek |) is 11.3 ± 4.7 V nm−1 , and the projection at the direction given by carbon CG of ASP-52 and carbon CD of GLU-35 (updated at each time step) is 8.9 ± 3.5 V nm−1 . These results are similar to those presented in Table 3 for the electric field magnitude (|E|) and for the x component of the electric field (Ex ), calculated using the crystal structure. This comparison shows that changes in spatial arrangement (as calculated by molecular dynamic simulation) influence the calculated electric field, but its time-average value is approximately the same as that calculated using the crystal structure. Fourth, we must consider the relation between partial charges in the substrate molecule and the bond energy. Eqs. (3) and (4) are based on the hypothesis that only electrostatic interactions are considered, and that van der Waals forces between atoms in a bond are neglected. These equations also imply a separation of partial charges that corresponds neither to a heterolytic bond cleavage (which would result in an ionic pair) nor to a homolytic bond cleavage (which would result in the formation of free radicals). However, the separation of partial charges following the bond cleavage does not imply that isolated fractional charges are created. For example, in the peptide bond cleavage, the two separated amino-acid groups have overall integer charges, as calculated by the force-field parameter sets–even though the carbon and the nitrogen atoms bear fractional charges. The use of force-field parameters to estimate bond energies and bond forces in the substrate molecule is beyond the hypotheses that lead to these partial charges. When using force fields in molecular dynamic simulations, the so-called non-bonded terms (i.e. Coulomb and van der Waals forces) are used to calculate the interaction only between atoms that are not bonded to each other. However, the bond energies calculated with partial charges are checked against experimental bond energies, and only those partial charges that result in a good agreement with experiment are used in subsequent calculations. Furthermore, by considering that a chemical bond has a purely electrostatic nature, partial charges of the atoms belonging to that bond are somehow overestimated. As the
bond force and bond energy are proportional to the product of the partial charges, and the influence of the electric field is proportional to each individual partial charge, the calculated results constitute a worst-case scenario. Fried et al. [13] showed that the electric field in the enzyme’s active site is the main factor that reduces the activation barrier for reaching the transition state. In their work, the influence of the electric field on the dipole of the bond in the substrate molecule is considered in both the free molecule and in the transition state. Our model somehow involves a much simplified version of their conclusions, in which only the influence of the electric field on the bond dipole of the free substrate molecule is taken into account. A similar question arises from the influence of the electric field on the electron cloud of the substrate bond. The presence of the electric field shifts this electron cloud toward a different configuration, thus leading to different partial charges in the bond, and to a different bond energy in the substrate molecule. However, the assignment of charges to points, at the atomic scale, can be considered a rough approximation. The extensive literature on protein electrostatic calculations is ultimately based on this assignment. The literature shows that this approximation can lead to reliable results. Fifth, we must consider the ionized state of the amino acids belonging to the enzyme’s active dyad/tryad and that of the substrate molecule during the cleavage step. In a hydrolysis reaction, the bond-breaking step occurs with the substrate molecule and the active site in different ionization states, after the abstraction of a proton by either the substrate or the enzyme. For a lysozymecatalyzed reaction, prior to or at the bond-breaking step, a proton is abstracted from the GLU-35 amino-acid by the substrate molecule. For trypsin-catalyzed proteolysis, the bond-breaking step occurs after a proton is transferred from HIS-57 to SER-195, with a subsequent abstraction of this proton by the nitrogen atom of the peptide bond of the substrate molecule. For a ribonuclease-catalyzed reaction, the mechanism involves the transfer of a proton from HIS-119 to the oxygen atom of the P O bond of RNA. The effect of the electric field is not the only driving-force in these reactions; other aspects must be accounted for. Otherwise, the sole presence of charged residues in the protein surface, in whose closest vicinity large electric fields can be encountered, would catalyze a reaction. Because chymotrypsinogen and trypsin generate similar electric fields, it illustrates that the electric field is a necessary, but not sufficient, aspect to be considered. Chymotrypsinogen is the inactive form of chymotryspin, but the absence of enzymatic activity is related to conformational differences, not necessarily in the active site.
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