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Deoxymyoglobin studied by the conformational normal mode analysis
J. Mol. Biol. (1990) 216, 95-109
D e o x y m y o g l o b i n Studied by the Conformational N o r m a l M o d e Analysis I. Dynamics of Globin and the Heme-Globin Interaction Yasunobu Senol"
Department of Physics, Faculty of Science Nagoya University, Nagoya 464, Japan and Nobuhiro G 6
Department of Chemistry, Faculty of Science Kyoto University, Kyoto 606, Japan (Received 2 January 1990; accepted 30 May 1990) Dynamic properties of deoxymyoglobin are studied theoretically by the analysis of conformational fluctuations. Root-mean-square atomic fluctuations and distance fluctuations between different segments reveal the mechanical construction of the molecule. Eight a-helices behave as relatively rigid bodies and corner regions are more flexible, showing larger fluctuations. More particularly, corner regions EF and GH are specific in that flanking a-helices extend their rigidity up to a point in the corner region and the two rigid segments are connected flexibly at that point. The FG corner is exceptional. A segment from the F helix to the beginning of the G helix, in which the FG corner is included, becomes relatively rigid by means of strong interactions with the heme group. The whole myoglobin molecule is divided into two large units of motion, one extending from the B to the E helix, and the other from the F to the H helix. These two units are connected covalently by the EF corner. However, dynamic interactions between these two units take place mainly through contacts between helices B and G and not through the E F corner. From correlation coefficients between fluctuational motions of residues and the heme group, 55 residues are identified as having strong dynamic interactions with the heme moiety. Among them, 18 residues in the three segments, one consisting of residues from the C helix to the CD corner, a second consisting of the E helix, and a third from the F helix to the beginning of the G helix, are in close contact with the heme group. Twenty-two of the 55 residues are within four residues of the 18 residues in their sequential residue number and are more than 3 A away from the heme group. The other 15 residues are located further in the sequential residue number and are all found in helices A and H. They are more than 6 A away from the heme group. By the use of correlation coefficients of fluctuations between residues, it is found that dynamic interaction with the heme group is transmitted to the A helix and the beginning of the H helix in the direction Leu(E15) -* (Val(A11) and Trp(A12)}. The transmission to the C-terminal end of the H helix is mediated by a long segment, from the end of the EF corner to the beginning of the G helix, that lies on the heme group and has close contacts over a wide range. The method for identifying the path of transmission of dynamic interaction by correlation coefficients of fluctuations will be useful also for studies of other functional proteins. globin have been widely applied to studies of other proteins. Three-dimensional structural data of highresolution X-ray crystallography have been accumulated for both the deoxy form and the liganded form of hemoglobin; by comparison of these, mechanism of co-operative ligand binding has been formulated (Perutz, 1965, 1970; Muirhead et at.,
1. I n t r o d u c t i o n
Study of the structure-function relationship of hemoglobin is most advanced in protein chemistry. The concepts and methods developed with hemot Author to whom all correspondence should be addressed. 0022-2836/90/200095-15 $03.00/0
95
© 1990 Academic Press Limited
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Y. Seno and N . G5
1967; Anderson, 1973; Fermi, 1975; Baldwin & Chothia, 1979; Baldwin, 1980; Shaanan, 1983; Fermi et al., 1984). The structural change of hemoglobin upon ligand binding is regarded as comprising three levels: the geometrical change, of the heme group upon ligand binding, the tertiary structural change in each subunit, and the quaternary structural change in the tetramer. For complete understanding of the mechanism of cooperative ligand binding it is necessary to characterize each of the three levels and the specific interactions between them, and to identify the lowenergy path of the structural change in the conformational space of the whole tetrameric molecule. In view of the rapid progress in recent computational chemistry, a theoretical study based on the conformational energy function is expected to be a powerful tool for approaching these problems. In pioneering work on an a-subunit of deoxyhemoglobin, by the use of energy minimization with atomic Cartesian co-ordinates as independent variables, Gelin et al. (1983) studied a wide range of important problems: the geometry of the heme group, the interaction between the heme group and the globin molecule, and the tertiary structural change upon ligand binding (Gelin & Karplus, 1977; Gelin et al., 1983). Though their work is very stimulating and suggestive, some difficulties remain in the method of their calculation and in the interpretation of results. Firstly, as to the energy minimization, the steepest descent method of 200 cycles seems to be too simple and insufficient to treat a large molecule such as one subunit of hemoglobin. From the magnitude of residual gradients, their energy minimization was probably terminated at a point far from a true minimum. Secondly, because the X-ray structure of an a-subunit isolated from deoxyhemoglobin is taken as the starting conformation of energy minimization, the absence of quaternary constraint may be serious when the energy-minimized structure is interpreted as being in the deoxy form of hemoglobin. Meanwhile, vibrational normal mode analysis, a method for studying dynamic properties of protein molecules within the harmonic approximation of a conformational energy function, has been developed (G5 et al., 1983; Brooks & Karplus, 1983; Levitt et al., 1985; Nishikawa & GS, 1987). By the use of dihedral angles as independent variables of the eonformational energy function, energy minimization can be performed relatively easily exactly to the point where the second derivative matrix is positive definite and the first derivatives virtually vanish. The analytical nature of the normal mode analysis allows us easily to compute conformational fluctuations and their correlations in thermal equilibrium. In the thorough analysis of the dynamic properties of bovine pancreatic trypsin inhibitor (BPTIt) as a typical example of a small globular protein (Nishikawa & GS, 1987), normal modes in t Abbreviations used: BPTI, bovine pancreatic trypsin inhibitor; r.m.s., root-mean-square.
different frequency ranges have been characterized by the degree of localization and spatial continuity of atomic displacement vectors. The importance of low-frequency normal modes, particularly, is notable because of their dominant contribution to the magnitude and site dependence of atomic fluctuations. From the analysis of space correlation functions of atomic fluctuations, the protein molecule is found to be made up of relatively rigid elements held together by very flexible parts. Here and in the accompanying paper, the method of normal mode analysis is applied to deoxymyoglobin. Myoglobin is also an oxygen-binding hemoprotein, and its structure is very similar to that of one subunit of hemoglobin (Takano, 1977a,b; Phillips, 1980). It is easier to carry out conformational energy analysis on myoglobin, due to its molecular size, than on hemoglobin. Though the intersubunit interactions existing in tetrameric hemoglobin cannot be treated, problems concerning the tertiary structure can be investigated in myoglobin. For the calculation of deoxymyoglobin, which consists of one polypeptide chain and the heme group, as a hetero (i.e. non-peptide) molecule, we developed a new F O R T R A N computer program adapted for vector processing. This program is based on the formulations developed at first to rapidly compute a conformational energy function and its first and second derivatives (Noguti & G5, 1983b; Abe et al., 1984; Wako & GS, 1987), and then extended to treat a system composed of two or more molecules (Braun et al., 1984; Higo et al., 1985). This paper involves two sets of calculations. Firstly, after the complete energy minimization, starting from the X-ray conformation, conformational fluctuations and their correlations are examined on the basis of normal mode analysis. From site dependence of root-mean-square (r.m.s.) fluctuations of atomic displacements and from distance fluctuations between different segments, the mechanical construction of the molecule can be elucidated. Secondly, we inquire into the nature of dynamic interactions between the heme group and the globin molecule with the aid of correlation coefficients of fluctuations. Important residues in the heme-globin interaction are identified as having large correlation coefficients. Then from the correlation coefficients of fluctuations between the selected residues, we can trace the path along which the effect of the geometrical change at the heme-globin linkage is transmitted to residues distant from the heme. On the basis of these analyses of the dynamic properties, the conformational change upon ligand binding is calculated in the accompanying paper (Seno & G5, 1990) within the range of harmonic approximation of the energy function. The method is essentially the same as that used in the calculation of the conformational change of BPTI induced by a chemical modification (Yoshioki et al., 1983). Forces acting upon the linkage between the heme group and the globin molecule are guessed by repro-
Dynamics of Myoglobin Conformation ducing, within the harmonic approximation, the local geometrical change actually observed at the linkage. Overall conformational change of the whole molecule b y these forces can then be calculated easily. The energetics of the conformational change is developed within the harmonic approximation, and the numerical results are discussed. Atomic displacements are also examined in relation to the mechanical construction and the nature of heme-globin interactions. In the a c c o m p a n y i n g paper, the dominating roles of low frequency normal modes in the conformational change of the molecule are also examined by decomposition of the calculated conformational change to contributions from individual vibrational normal modes.
97
t2"
\
/
N
2. Methods (a) X-ray co-ordinates
C"
Atomic co-ordinates determined by X-ray crystallography are necessary for the initial step of our calculation. As described below, atomic co-ordinates of deoxymyoglobin are used in regularization of the globin molecule and determination of some parameters in the treatment of the heme group. Those of oxymyoglobin are also referenced in the analysis of con|brmational change upon ligand binding. Atomic co-ordinates of deoxymyoglobin and oxymyoglobin are taken from 1MBD and 1MBO, respectively (Protein Data Bank; Bernstein et at., 1977), both of which were deposited by Phillips (1980).
Figure 1. Heine skeleton and the imidazole ring of the proximal His93. The origin of the X YZ-system is the atom N"2. The Z-axis is taken as bisecting the angle C~I-N~2-C62 and in the direction of the heine, the X-axis is in the plane of the imidazole ring and in the direction of C~2. The origin of the ~t/~-system is the Fe atom. The ~-axis is perpendicular to the mean plane of the heine skeleton, and the ~-axis lies in the plane formed by the ~-axis and the bond Fe-N c.
(b) Conformational space and energy function
Higo el al., 1985) is applied here in the following way. In order to express position and orientation of the heine skeleton relative to the globin molecule, 2 co-ordinate systems are introduced: one is fixed to the imidazole ring of the proximal His93 (X YZ-system) and the other to the heme skeleton (~t/~-system; Fig. 1). Position and orientation of the heine skeleton with respect to the globin moiety are expressed by 3 components of a translational vector t and 3 components of a rotational vector r that, together, move the X YZ-system to the ~t/~-system. These 6 components, defined with respect to the X YZ-system, are included in the set of independent variables of the energy function and referred to as external variables. Thus, the total conformational space consists of 925 dihedral angles, 0~, i -- l, 2 . . . . . 925, and 6 external variables, e~, i -- 1, 2 . . . . . 6. These independent variables, put together, are denoted by qt, i = l, 2 . . . . . 931.
(i) Conformational space In the present calculation, the conformational energy function is constructed basically according to ECEPP (Momany et al., 1975; N~methy et al., 1983). In the scheme of ECEPP, hydrogen atoms are explicitly taken into account. Then, the conformation of the globin molecule is expressed by 911 dihedral angles around all rotatable chemical bonds, and atomic co-ordinates of the globin are generated by the use of the standard residue data of ECEPP. Expression of the motion of the heine group needs some consideration. It is well known from X-ray crystallography that the structure of the heme skeleton changes upon ligand binding from a domed to a more planar form. In the scheme of the present calculation in the dihedral angle space, however, inclusion of the out-of-plane motion of the heme skeleton is not straightforward, although not impossible. In the present works we are primarily interested in the "soft" dynamic properties of the molecule, i.e. such collective motions occurring in the timerange of picoseconds or longer and involving at least a few residues. The out-of-plane motion of the heme skeleton is much harder than these soft motions and has no significant interactions with soft variables. With this consideration, the heme skeleton is treated as rigid and its geometry is fixed to the X-ray structure of deoxymyoglobin in this series of papers. The 14 dihedral angles in the 8 side-chains of the heine group, which are free to move, are treated as independent variables. Bond angles and bond lengths in these side-chains are fixed to those of the X-ray structure. If we assume a rigid heme skeleton, deoxymyoglobin becomes a molecular system composed of 2 chains. The theory for treating a 2-chain system (Braun et al., 1984;
(ii} Energy terms and enerlly parameters The ECEPP data of torsional and interatomic energy parameters are used as they are for the globin part. As for the heine group, each of the rotatable bonds in the 8 sidechains has a chemical bond of the same type in the ECEPP residue data, and also heme atoms other than Fe are of the same atom types in the residue data. Therefore, except for those for the Fe atom, the ECEPP data are used. Non-bonded interactions involving the Fe atoms are tentatively neglected because there are no available parameters. This neglect is not expected to give rise to any serious artificial results when conclusions are drawn in the following sections, because the Fe atom is, in a sense, separated from atoms belonging to globin by the surrounding heme atoms. As parameters of electrostatic interaction, the atomic charges in the heme skeleton are set to values obtained by an ab ini~io molecular orbital calculation on a model compound of deoxyheme (Obara &
Y. Seno and N . G5
98
Kashiwagi,, 1982) and provided by H. Kashiwagi (personal communication). Because the model compound has no side-chains, atomic charges in the side-chains are determined according to our CNDO/2 calculation on the whole heme group with a point charge 2e instead of the F e atom (where e is the electronic charge unit). In order to conserve the total charge of the heme group, the differences in electron distribution in the heme skeleton between the 2 types of calculation are distributed proportionally to the atomic electron distributions in the side-chains. Though the heme group is permitted to move freely in globin by the 6 external variables introduced above, its translational and rotational motions relative to the globin are restricted to a large extent by the co-ordination bond between the Fe atom and N ~2 of the imidazole ring of the proximal His93(F8). This bond is exceptional in the sense t h a t the bond lengths and bond angles associated with it are changeable. Together with a torsional energy around the bond, a force field is taken into account, with a stretching energy term of F e - N ~2, bending terms of C~1-N~2-Fe, C~2-N~2-Fe and 4 angles N~2-Fe-Np, and with an energy term for deformation from planarity about the N ~2 atom (where Np represents 4 pyrrole nitrogen atoms of the heme group, and C~1 and C~2 are the atoms in the imidazole ring of His93(F8). Values of torsional energy parameters, force constants of the stretching and bending terms and of the equilibrium bond length F e - N ~2 are chosen to be the same as those which were used in the energy refinement of the X-ray diffraction study of carbonmonoxyhemoglobin (Baldwin, 1980). Equilibrium values of the 2 bond angles around the N r2 atom are taken to be the same as the angle between the bond N~2-C~I and the Z-axis in Fig. ], and an equilibrium value of the angles N~2-Fe-Np is determined by averaging the 4 corresponding angles in the X-ray geometry of the heme group. An energy term of non-planarity about the N ~2 atom is quadratic in the angle of deviation of the bond N~2-Fe from the plane through 3 atoms (yl, N~2 and Ca2. The value of the force constant for the bending of C~I-N~2-Fe (C62-N~2-Fe) is tentatively assigned to this term of nonplanarity. The above energy terms of torsion, stretching, bending and non-planarity associated with the co-ordination bond can be expressed by external variables only.
compared to the errors in determination of atomic coordinates by X-ray crystallography. Initial values of the 6 external variables of the heme-globin linkage are easily calculated from the regularized conformation of the globin molecule best-fitted to the X-ray structure. Initial values of dihedral angles in the side-chains of the heme group can be equated to values in the X-ray conformation. (ii) Energy minimization The energy minimization, starting from the regularized conformation, is performed by a Newton method, i.e. a restricted step method with line search (Fletcher, 1980; Gill et al., 1981). The Newton method has become very efficient due to the recently developed algorithm used to calculate rapidly the 1st and 2nd derivatives of the conformational energy function in their analytical forms (Noguti & GS, 1983b; Abe el al., 1984). With the full use of the algorithm, a new F O R T R A N program adapted for vector processing has been developed to study large protein molecules (Wako &.G5, 1987). F o r the present study of deoxymyoglobin, this program has been extended to treat a biopolymer system consisting of 2 chains on the basis of the formalism mentioned above (Braun et al., 1984; Higo el al., 1985). Required central processor unit time of 1 evaluation of the conformational energy function of deoxymyoglobin and all of its 1st and 2nd derivatives is 16"87 s by the vector processor HITAC S-820. One evaluation of the energy function only takes 0.57 s. After 152 cycles of iteration of the Newton method, a minimum point is reached where all eigenvalues of the second derivative matrix are positive, and the absolute value of the residual gradient of energy is less than 2"0 x l0 -9 (kcal/mol per rad), 4-6 x l0 -12 (kcal/mol per A) and l ' 6 x l0 -11 (kcal]mol per rad) for a dihedral angle (i cal = 4"184 J), a component of the translational vector t and a component of the rotational vector r, respectively. The required central processor unit time for this energy minimization is 54'5 min. The r.m.s, atomic shift in the minimization process is l'001 A and 1"214 A for mainchain atoms and all heavy-atoms, respectively. This movement is the same in magnitude as is the case for energy minimization of other smaller proteins with dihedral angles as independent variables (Levitt et al., 1985; Nishikawa & GS, 1987).
(e) Regularization and energy minimization (i) Regularization The normal mode analysis is carried out a t the minimum point of the eonformational energy function. In order to use the X-ray conformaton as the starting point of energy minimization, we must first determine the set of dihedral angles of the globin molecule t h a t generates the conformation nearest to t h a t of the X-ray structure. This process is referred to as regularization of Cartesian coordinates of atoms with bond lengths and bond angles fixed to the standard values of ECEPP. The regularization is implemented by a F O R T R A N computer program of distance geometry t h a t has also been used to determine protein solution conformations from p r o t o n - p r o t o n distance constraints (Braun & G5, 1985). F o r the purpose of preserving the exact position of the side-chain of the proximal His93(F8), a larger number of distance constraints between the imidazole ring and other parts of the globin molecule than our usual regularization process are included here. r.m.s, deviation of the regularized conformation from the X-ray structure is 0"182 A (l A = (~l rim)for all heavy-atoms, and 0"124 A for mainchain atoms. These deviations are not so large when
(d) Vibrational normal modes The method of protein normal mode analysis has been described in detail {Levitt et al., 1985; Nishikawa & GS, 1987). Here, we summarize only a mathematical framework to fix notations in order. The conformational energy function is expanded at the energy minimum point in terms of qi (i = I, 2 . . . . . 931), which hereinafter denotes infinitesimal changes of independent variables from corresponding values at the minimum point. By retaining terms up to 2nd-order, we have:
E = ½Y~,E/,,qm,
(l)
where subscripts i and j in summations go from I to 931. The kinetic energy is expressed as:
T = ½~.,~lH~1(dq,/dt)(dqj/dt).
(2)
Matrices, whose elements are F 0 and Hi j, are referred to as F - m a t r i x and H-matrix, respectively. The F - m a t r i x is evaluated a t the energy minimum point by the use of the above program to calculate the 2nd derivatives of the energy function. The H - m a t r i x is also evaluated in an
Dynamics of Myoglobin Conformation analytical form (Noguti & GS, 1983a), where overall translational and rotational motion of the whole molecule is separated from internal motions and the contributions of external motions are excluded from the kinetic energy. By a linear transformation of qi to new variables Q~ (n = l, 2 . . . . . 931): q~ = ~., VI,Q,,
(3)
the conformational energy and the kinetic energy given by eqns (l) and (2), respectively, can be transformed to simple sums: E = .~V , ~ . (.0202 ....
(4)
T = ~,.(dQ~/dt) 2.
(5)
The variables Q. are referred to as normal mode variables, and they satisfy independently the equations of motion of harmonic oscillation with angular frequencies ran:
d2Q,/dt 2 + co,2Q~= o.
(6)
The transformation matrix whose elements are Vi, in eqn (3) is the matrix of eigenvectors. The inverse transformation is written as:
Q, = ~ ( V- t),iqi,
(7)
where V -1 is the inverse matrix of V. By this equation, any vector with components qt in the tangential vector space at the energy minimum point can be decomposed by eigenvectors of normal modes with Q~ as decomposition coefficients. (e) Thermal fluctuations I n the method described in the previous works (Levitt etal., 1985; Nishikawa & G5, 1987), thermal fluctuations are calculated for each normal mode, and total fluctuations are given by sums of contributions from all normal modes. Because we are confined with total fluctuations in thermal equilibrium later in this work, somewhat more straightforward expressions are briefly given to the quantities calculated here. (i) Fluctuation of independent variables An averaged value of a product of fluctuations qi and ql in thermal equilibrium is given by:
(8)
where k s is the Boltzmann constant, T is the absolute temperature, which is fixed to 300 K throughout this work, and F - t denotes the inverse matrix of F. I n the ease of the same suffix (i = j), this equation expresses the squared amplitude of fluctuation in % and in other cases the correlation of fluctuations between q~ and ql" Next, as a measure of strength of dynamic interaction, we use the correlation coefficients of fluctuations of independent variables:
P(qi, qj) = (q~qj)/{((qi)2)((qi)2)} t/2.
(9)
Using this quantity, r.m.s, correlation coefficients Pkt between the kth and t h e / t h residues are defined by: ( P u ) 2 = ( /1
N ~~N ~ ,)~.,~.qP(q,,qi),2
(10)
where N~ and N~ denote the number of dihedral angles qi in the kth residue and qi in the lth. residue, respectively, and summations are taken over all dihedral angles in the 2 residues. Because the translational and rotational motion of the heme group relative to the globin molecule are expressed by
99
external variables e~ (i = 1, 2 . . . . . 6), correlation coefficients p(e l, Oj) from eqn (9) indicate the strength of dynamic interaction between the motion of the heme group expressed by changes of ei and the conformational change of the globin molecule caused by a change in dihedral angle 0i. F o r each of 6 external variables, r.m.s. correlation coefficients Pk(ei) between the external motion of the heme group and the conformational change of the kth residue are defined by: {~k(S,)}2 = (l/N~)~,q{p(e,, Oj)}2,
(l l)
where the summation extends over all dihedral angles in the kth residue. (ii) Fluctuations of atomic co-ordinates Infinitesimal displacements of a-components (a -- 1, 2, 3, or x, y, z) of an atom a, Ax,~, are derived from changes in independent variables qi (i = 1, 2, . . . , 931): Ax.~-- ~K,o.,q,,
(12)
where, in addition to the H-matrix, the K-matrix, whose elements are Kao.~, is calculated analytically so as to exclude external motions of the whole molecule (Noguti & G5, 1983b). By the use of eqns (8) and (12), correlations of fluctuations of atomic co-ordinates are calculated as: (AxaaAXb~) = ~.i~tKaa.iKbrj(q~i) =
k.TZ,ZjKa..,Kb.j(F-'), j.
(13)
From this we calculate r.m.s, fluctuations of atomic displacements in a segment, distance fluctuations, and a type of measure of dynamic interaction other than Pu from eqn (10) between a pair of segments. Nishikawa & G5 (1987) discussed the distance fluctuation, which is defined as follows. Change of distance between a pair of atoms, a and b, is given by: Arab ~- (Ara-Arb)- (r.--rb)/]r.--rbL
(14)
where r, and r b denote position vectors of atoms a and b in the energy-minimized conformation, and Ar, and Ar b are their infinitesimal changes whose components are Ax,~ and AXb, (a ----1, 2, 3), respectively. Distance change Adkl between the kth segment and the /th is defined as the average over pairs of atoms:
adkl -~ ( l / NkN,) ~a~b Arab,
(15)
where atoms a and b belong to the kth and the /th segments, respectively, and N k and N~ are numbers of atoms considered in 2 segments. Here, a segment is taken to be the main-chain of a residue, and, in thermal equilibrium, distance fluctuation Dal between main-chain parts of the kth and t h e / t h residues is defined by: (Dkl)2 = ((Adkl)2).
(16)
Another correlation coefficient ~ between a pair of residues k and l is defined as follows, on the basis of fluctuations of Cartesian co-ordinates of atoms:
(~1)2 = (1tN:N~ )ZaZb × { Z ~ L ( a x a ~ X b t )2/(Ara2) (arb2) },
(17)
where N~, and N~ are numbers of all atoms in the kth and /th residues, respectively, and a, r = l, 2, 3. This quantity takes values from 0 to 1. Whereas the correlation coefficient of eqn (10) focuses on correlation of the internal conformational changes of a pair of residues, eqn (17) gives also correlation of the external (i.e. translational and rotational rigid body) motions between a pair of residues.
100
Y. Seno and N. G5
Similarly we define:
fl
B
C
0
E
F
0
H
{hI(s,)}~ = (l/Nl)~,~.,,(,~z, oe,>~/{< ~ ) (e~ >,
(18) where the 1st summation is taken over all atoms in the kth residue, and the 2nd summation is over components a = l, 2 and 3. 3. Results and I:)iscussion
(a) Dynamic properties of the globin molecule (i) r.m.s, atomic fluctuations It has been shown in previous papers that a protein molecule is mechanically composed of relatively rigid and flexible parts !Levitt et al., 1985; Nishikawa & GS, 1987). In myoglobin, which is a protein with a large helix content, we again see this clearly, firstly by the calculation of atomic fluctuations. The calculated r.m.s, fluctuation (averaged over atoms in each amino acid residue) is plotted against residue number in Figure 2(a) for mainchain atoms, and Figure 2(b) for all heavy-atoms. The broken lines are plots of the experimental fluctuations obtained from temperature factors of an X-ray diffraction analysis (Phillips, 1980). In both cases ((a) and (b)), calculated and observed r.m.s. fluctuations exhibit almost the same patterns of variation with residue number, although the calculated values are smaller than those observed. In the case of main-chain atoms (Fig. 2(a)), both calculated and observed r.m.s, fluctuations have a tendency to become small at the central parts of the eight helices {from A to H) and to take large values at terminal sides of helices and at the corner regions connecting them (except for the case of the observed value at the C helix). In the calculated plot, a strong peak is observed in each corner region except for in FG, i.e. at AlaI9(ABI), LysS0(CD8), Ala84(EF7) and Alal21(GH3). Although a small peak is observed at Lys96(FG1), calculated fluctuations at the FG corner are smaller than at other corners. The segment from the F helix to the beginning of the G helix, including the FG corner, is located near the heme group and has strong contacts with it. The small r.m.s, fluctuations of main-chain atoms in this region reflect stabilization of conformation by these interactions. In the plot of the r.m.s, fluctuations of all heavyatoms (Fig. 2(b)), sharp peaks appear periodically every three or four residues. These peaks come from large fluctuations of side-chain atoms of hydrophilic residues located at the surface of the globin molecule. The large contribution to the r.m.s, fluctuations from hydrophilic residues at the surface is confirmed by calculations of solvent accessibility, which is defined here as the ratio of area of exposure of an amino acid residue X in globin to that of the residue in a tripeptide Gly-X-Gly with fully extended conformation (Shrake & Rupley, 1973). In Figure 2(e), the calculated solvent accessibilities are plotted. Peaks and troughs of the r.m.s, fluctuations coincide well with those of solvent accessibility, except for a few points.
/~"~%i!;"k]l~?liiil
[i
;~,.ll;li:~ :iJt %~,,'1,=.'~::~i/!
-~. ~0. 0
20
40
60 80 100 Residue number
120
140
Figure 2. r.m.s, fluctuations of atoms in each residue plotted against residue number in (a) and (b). Continuous line, calculated r.m.s, fluctuations at 300 K; broken line, experimental Lm.s. fluctuations derived from the observed temperature factors. Locations of 8 helices, A to H, are indicated. (a) Main-chain atoms; (b) all heavy atoms. In (c) solvent aecessibilities are calculated for the energy minimum structure.
(ii) Correlation of fluctuations A three-dimensional picture of the dynamic structure of a protein molecule can be obtained from fluctuations of distances between different segments. Small distance fluctuations within a segment of a molecule mean that the segment behaves as a relatively rigid body in the molecule. I f they are small between a pair of segments, two segments show a tendency to move together as if they were one body. Pairs of residues with small distance fluctuations in the sense defined by equation (16) "are shown in Figure 3. In this Figure, triangular structures of dense shading corresponding to a-helices appear along the diagonal line. This means that a-helices are fluctuating as relatively rigid bodies in the molecule. Among the eight helices, B, F, G and H are seen to be more rigid than others from completeness of the corresponding triangles, and helices A, C and E seem to be more flexible. Detailed structures of shaded triangles can be interpreted in relation to the calculated r.m.s, fluctuations shown in Figure 2(a). In each of the two corner regions (EF and GH), there is a strong peak of r.m.s, fluctuation centered at AIa84(EF7) and AIal21(GH3) (mentioned previously). In this case, two triangles corresponding to a-helices extend from both sides up to the residue with the large fluctuation. This means the existence of a flexible point in the corner region as a joint between two rigid segments. There are no corner regions between adjacent helices B and C,
Dynamics of MyoglobinConformation
101
E i £ ; ; ; ;;, :: ~ . ~ l b : ~ a
50
-
-::
~iEiiii
C
'0 m
g~ . . .
_..
4--
I00 = _--
_--
7-
_
_..
-
- -
-
-~
.
.
.
.
.
Tt
I 50
I00
Residue
I
I 150
number
Figure 3. A map of small fluctuations of distances between residues. Distance fluctuations are averaged over mainchain atoms. (*)Less than 0"25A, (+) from 0"25A to 0"35A, ( - ) from 0'35A to 0"40A, where A (=0"476 A) is the maximum value of the distance fluctuation. The porphyrin ring of the heme group is treated as the 154th residue, and the helical regions are shown by the names of helices along the diagonal line.
and between D and E. In these cases, two triangles for two consecutive helices merge into one relatively rigid body. In inner parts of the map there are several regions of small distance fluctuations. Among them the following four wide regions are conspicuous: one is between segment B-C (from the B to the C helices) and segment D - E (region I), a second is between segment B - C and the G helix (region II), a third is between the G helix and the H helix (region III), and a fourth is between the H helix and segment F - F G (region IV). In each of these four regions a pair of helices are in close contact in a wide range. This is seen in the map shown in Figure 4, where minimum atomic distances between residues are plotted. Three lines of close contact extend perpen-
diculady to the diagonal line (antiparallel arrangement of helices), at upper-left, lower-left and lowerright, to pass through the above-mentioned regions I, I I and III, respectively. A line of contact between the H helix and the segment F - F G (parallel arrangement) appears at lower-center in Figure 4, and this line is located at region IV shown in Figure 3. In this way we can see t h a t two rigid segments or helices tend to move together as ff as one body when t h e y are in close contact in a parallel or antiparaUel arrangement. Contact interactions do not necessarily m e a n a region of small distance fluctuations. The A helix has two lines of antiparallel contact interactions with helix E and helix H (Fig. 4). However, corresponding regions of small distance fluctuations are
102
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incomplete. This may be due to large flexibility of the A helix or its external motion relative to the other helices. Further investigation of the internal and external motions of a-helices in the globin molecule is now in progress. From Figure 3 we see that a long segment from the B helix to the E helix, except for the internal CD corner, forms one relatively rigid unit because of the presence of region I. Another long segment from the F helix to the H helix, except for the GH corner, also forms a relatively rigid unit because of the presence of regions I I I and IV. Thus, aside from the flexible terminal sides NA, A and HC, the globin molecule appears to be composed of two large relatively rigid units. These two units are connected covalently by the E F corner. However, these two units are held together relatively rigidly mainly by region II and not by the E F corner.
In hemoglobin, the alfll(a2fl2) interface is formed by parts of the helices B, G and H of the a-subunit packing against helices B, 13 and H of the fl-subunit. Essentially no structural differences are observed between deoxyhemoglobin and liganded hemoglobin in this interface (Baldwin & Chothia, 1979). It is interesting that the contact region between helices B and G plays an important role in holding the two relatively rigid units together within myoglobin, and it is also involved in the intersubunit interaction in hemoglobin. In Figure 3 we see that the neighborhood of the proximal His93(F8) residue has small distance fluctuations between a large portion of the globin molecule, as is shown by the horizontally and vertically shaded regions. This is because motion of this segment is mediated to other parts of globin by the heme group, which interacts with a large portion of
Dynamics of Myoglobin Conformation the globin molecule. The FG corner (96 to 99) also interacts with the heme group and therefore has small distance fluctuations from it. This corner is more rigid, as seen in Figure 2(a), than others because of the interaction with the heine group. The interactions, with the heine group of the segment from the center of the F helix to the G helix, contribute to making the above-mentioned unit from the F helix to the H helix relatively rigid. (iii) Justification of the harmonic approximation The results so far discussed are based on the normal mode analysis, which in turn is based on the harmonic approximation of the conformational energy function within the range of thermal fluctuation. In Figure 2 we see that the residue number dependence of the calculated r.m.s, atomic fluctuation reproduces the experimental results quite well, at least qualitatively. This result may suggest that the approximation is justified. However, experimental studies such as multipleflash photolysis with carbonmonoxymyoglobin and MSssbauer spectroscopy from low to room temperature indicate that this protein at room temperature can go through many local minimum energy conformations (Austin et al., 1975; Parak et al., 1982; Frauenfelder et al., 1988). In other words the energy function within the range of thermal fluctuation is so highly anharmonic as to have many local minima. Such a highly anharmonic nature of the energy function of proteins has also been shown computationally by molecular dynamics simulation of myoglobin by Elber & Karplus (1987) and Monte Carlo simulation of BPTI by Noguti & G5 (1989a,b,c,d,e). Therefore, proteins appear to have a peculiar property in that they behave as if they were a good harmonic system, even though their energy fluctuation is very highly anharmonic. In order to solve this apparent paradox, we are now carrying out normal mode analyses at a number of local minimum energy conformations. Tentative results indicate that individual normal mode at one energy minimum point is generally quite different from any modes at other points. However, the subspace spanned by a set of normal modes with very low frequencies is very stable as a whole, i.e. does not depend sensitively on a particular minimum at which normal mode analysis is carried out. This fact implies that the multiple minima shown to exist both experimentally and theoretically have such a common subspace spanned by the low frequency normal modes. If so, calculated r.m.s, fluctuations would be almost the same for different local minima, because the thermal fluctuation is dominated by the low frequency normal modes. The above paradox may be solved in this way. In other words the use of normal mode analysis can be justified by a rather complex mechanism, even when the energy function is highly anharmonic. The above justification, however, is not yet complete. With this situation in mind, at this stage, we would justify our approach by the result shown
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in Figure 2, i.e. the results obtained by this treatment agree well with the experimental data, even though we are aware that the reason for this agreement is quite complex. (b) Dynamic structure of the heme-globin interaction One of the important problems concerning the tertiary structure of oxygen-binding hemoprotein is the dynamic interaction between the heme group and globin; e.g. which parts of the globin molecule are most strongly affected by the geometrical change at the linkage between the heine group and the proximal histidine residue, and by what mechanism is the effect transmitted to other parts of the globin molecule? Because the dynamic interaction should be reflected in conformation fluctuations, we will use the r.m.s, correlation coefficients ~k(si) and ~(ei) given by equations (11) and (18), respectively, as measures of strength of the dynamic interaction between the kth residue and the heme group. The quantity ~k(e~) is relevant to local conformational changes within the residue but not to translational and rotational motions of the entire residue. Correlations to the latter rigid body motions can be detected by ~(8i). As mentioned, the motion of the heme group is expressed by six external variables 8i (i = 1, 2 . . . . . 6), which represent three components of the translational vector (i = 1, 2, 3) and three
104
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components of the rotational vector (i-- 4, 5, 6) with respect to the imidazole ring of His93(F8). For each of these six components of external motion of the heme group, dependence of P~(Si) and ~5~(~) on residue k is shown in Figure 5. In the same way as for the dynamic heme-globin interaction, dynamic interactions between residues within the giobin molecule are examined using two types of r.m.s. correlation coefficients, Pkl and ~5~1 given by equations (10) and (17), respectively. They are mapped in Figure 6(a) and (b), respectively. The above r.m.s, correlation coefficients have generally strong correlation with minimum distances between segments. This is seen by comparison of Figure 6 with Figure 4, and by the fact that Pk(~i) and P~(~i) in Figure 5 take large values around residues that are in close contact with the
heme group (the bottom row of Fig. 4). They offer us, however, information about mediation of dynamic interactions. We should pay special attention to points or regions shown in Figures 5 and 6 where pairs of segments have exceptionally large r.m.s, correlation coefficients, even for long distances. In such cases there must be some mechanism to mediate interactions between them. In the following, important residues in the dynamic heme-globin interactions are identified by using r.m.s, correlation coefficients pk(e~) and P~(~i) and the minimum distances from the heme group. Then, dynamic interactions between the selected residues are examined using Pu and P~t. First of all, 18 residues with minimum distances less than 3 A from the heme group, which are shown by the symbol (*) in the bottom row of Figure 4, are
Dynamics of MyoglobinConformation
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identified as nearest-neighbor residues and are shown underlined in Table 1 (see below). Among the 18 nearest-neighbor residues, four residues are found in the segment from the C helix to the CD corner, five residues (including the distal His64(E7)) in the E helix, eight residues (including the proximal His93(F8)) in the segment from the F helix to the beginning of the G helix, and one residue (Phe138) in the H helix. When the 153 residues are arranged in descending order of magnitude of the r.m.s, correlation coefficients ~(e~) and ~(e~) for each of the six external variables ei, these 18 residues are usually found towards the top in the 12 lists, six lists of Pk(~i) and six lists of ~(8f), showing strong dynamic interactions with the heme group. Except for six residues (67, 68, 71, 72, 107 and 138) 12 residues are
included in the top 20 members in at least one of the six lists of Pk(e~), and in the same way, 15 residues are included in the six lists of P~(ei), except for 64, 68 and 138. From the exceptional residues, we can see that the segment in the E helix from the distal His64(E7) has relatively weak dynamic interactions with the heme group. Other residues have minimum distances larger than 3"0 A from the heme group, but may still have strong dynamic interactions with it. Besides the above-mentioned 18 nearest-neighbor residues, 37 residues are found in the top 20 members of at least one of the 12 lists mentioned above. These 37 residues are also selected as the residues having strong dynamic interactions with the heine group. In total, 55 (---18+37) residues are selected as
Y. Seno and N. G5
106
i m p o r t a n t in dynamic heme-globin interactions. No residues in the B helix or in the C-terminal side of the G helix are included in these 55 residues. Next, these 55 residues are classified into clusters according to their sequence proximity in such a way t h a t each residue in one cluster is not separated from at least one member residue in the same cluster by more than four in residue sequence number. This sequential classification is based on the fact t h a t strong dynamic interactions, measured by ~ : and P~l, are generally observed for pairs of residues with sequential distances less than or equal to 4, which corresponds to one helical turn. By this means, the 55 residues are classified into the six clusters shown in Table 1. The r.m.s, correlation coefficients Pkt and P~l within the 55 residues are shown in Figure 7, and spatial arrangements of residues and the heme group are illustrated in Figure 8. Because this classification is based mainly on sequential residue number, some of the clusters m a y be located spatially close to each other. In fact, clusters 1 and 5, and 4 and 6, are relatively close in space. A part of cluster 2 and another part of cluster 3 are close, and a similar situation occurs between clusters 2 and 4. No nearest-neighbor residues are found in clusters 1 and 5. Only one, Phe138, is found in cluster 6. In fact all member residues in these clusters, except for Phel38, are spatially a p a r t from the heine group by more than 6 A in minimum distance. Their strong interactions with the heme group should be mediated by some member residues belonging to other clusters. The mediation of dynamic interactions can be traced by the r.m.s, correlation coefficients Pkt and P~l, and by the minimum distances between residues. The mediation to cluster 1 is traced as follows. The nearest-neighbor residue Leu72(El5) in cluster 3 has a strong dynamic interaction with
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Membert Leu2 Vail3 Pro37 Asp44 His64 Glu83 Pro88 His93 Lys98 Tyrl03 Ala127 Glu136 Lys145
Gly5 Trpl4 Thr39 Arg45 Thr67 Ala84 Leu89 Ala94 Ile99 Leul04 Metl31 Phe138 Tyr146
Location:~
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Lys42 Phe43 C-CD His48 Ala71 Leu72 E Leu86 Lys87 EF-G Gln91 Set92 Lys96 His97 Ilel01 Lysl02
H I1e142 Ala143 Ala144 H-HC Tyr149 Tyrl51
t Nearest-neighbor residues that have close contact interactions with the heine group are underlined. :~Locations of member residues with respect to the secondary structures.
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Cluster number (a)
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Table 1 Classification of the 55 amino acid residues that have strong dynamic interactions with the heme group
3
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Figure 7. Maps of the r.m.s, correlation coefficients Pkt and P~l between the 55 residues that have strong dynamic interactions with the heme group. They are clustered and ordered as given in Table 1. (a) Map of ~kl: (*) larger than 0-08, (+) from 0"06 to 0"08, ( - ) from 0"04 to 0"06. (b) Map of ~t: (*) larger than 0"2, (+) from 0-15 to 0"20, ( - ) from 0-10 to 0"15.
Trpl4(A12) in cluster 1 due to a close contact interaction. The other three residues of cluster 3, Thr67(E10), Va168(Ell) and Ala71(E14), have moderately strong dynamic interactions with T r p l 4 and its neighboring residue V a i l 3 ( A l l ) in cluster 1. Within cluster l, residues 13 and 14, respectively, have a strong and a moderately strong dynamic interaction with Leu9(A7). This Leu residue transmits further to the residues located at the N-terminal side. Thus, the dynamic interactions between cluster 1 and the heme group are mediated by
Dynamics of Myoglobin Conformation
107
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40
41
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members of cluster 3, which are a part of the E helix. As for cluster 5, each of two member residues Ala127(H4) and Metl3l(HS) have strong and/or moderately strong dynamic interactions with residues Leu9 and Vall3 in cluster 1. These interactions are due to close contacts. Thus, the dynamic interactions between cluster 5 and the heme group are mediated again first by the same part of the E helix and then by a part of cluster 1. As for cluster 6, strong dynamic interactions are observed with cluster 4. Three residues of cluster 6, Ala143(H20), Ala144 and Lys145, have strong or moderately strong dynamic interactions with most
of the six residues from Glu83(EF6) to Pro88(F3) in cluster 4. These strong interactions are due a to widerange of contact interactions between the clusters. Tyr146 has a strong interaction with Prol00(G]) and moderately strong interactions with Ala94(F9), Ile99(FG4) and Ilel01 (G2). We see that Tyrl46(H23) is in close contact with these four residues in cluster 4. Thus, the strong dynamic interactions between the heme group a n d the C-terminal side of cluster 6 are mediated by cluster 4, which is a long segment composed mainly of the F helix, the FG corner and the beginning of the G helix. From the strength of the dynamic inter-
108
Y. Seno and N. G5
~c)
Figure 8. Stereo drawings, drawn by NAMOD (Beppu, 1989), showing spatial arrangements of residues and the heme moiety in 6 clusters. The a-carbon atoms are distinguished from other atoms by heavy circles. (a) Clusters 1, 3 and 5; (b) cluster 2; (c) clusters 4 and 6.
actions between residues, Lysl45(H22) and Tyr146(H23) are seen to play a central role in the mediation. So far we have discussed pathways of mediation of dynamic interactions between the heme group and various parts of the globin molecule. The latter has been divided into clusters. However, the clusters are not completely separate entities dynamically, because sometimes strong dynamic interactions are observed between them. Between clusters 2 and 3, strong dynamic interactions are observed between residues Arg45(CD3) and Phe46(CD4) in cluster 2 and the distal His64(E7) residue in cluster 3. Between clusters 2 and 4, Lys42(C7) in cluster 2 has moderately strong dynamic interactions with two residues in cluster 4, Ile99(FG4) and Tyrl03(G4). These strong or moderately strong dynamic interactions are again due to the close contact interactions, between clusters, mentioned above. 4. Conclusion The conformational normal mode analysis has been applied to deoxymyoglobin to calculate: (1) distance fluctuations between various parts of the protein, (2) correlation coefficients of fluctuations of internal variables in each residue and the berne group, and (3) correlation coefficients of fluctuations of Cartesian co-ordinates of atoms in each residue and the heine group. From (1), mechanical construction of this molecule has been studied, u-Helices were found to be more rigid than other parts of the molecule. Two larger structures, each consisting of a few ~-helices, i.e. one from the B helix to the E helix and the other
from the F helix to the H helix, were found to behave as dynamic units. These two units are held together by the interactions between helices B and G. From the analyses based on the quantities (2) and (3), we obtained a picture of the dynamic heme-globin interaction: 55 residues were identified as having strong dynamic interactions with the heme group. These residues were classified into six clusters. Three clusters (2, 3 and 4) surround the heme group and have strong dynamic interactions with each other as well as with the heme group. Clusters 3 and 4 mediate the dynamic interactions to the A helix and the C'terminal side of the H helix, respectively. The method of application of normal mode analysis to study the dynamic properties of the myoglobin molecule may also be useful in functionally related dynamic studies of other protein molecules. We are grateful to Dr H. Kashiwagi for providing calculated atomic charges of the heme groups. Computations reported in this paper were done at the Computer Centers of Nagoya University and of the Institute of Molecular Science. This work was supported in Kyoto University by grants to N.G. from the Ministry of Education, Science and Culture, Japan, and from the Science and Technology Agency, Japan. References Abe, H., Braun, W., Noguti, T. & G5, N. (1984). Comput. Chem. 8, 239-247. Anderson, L. (1973). J. Mol. Biol. 79, 495-506. Austin, R. H., Beeson, K. W., Eisenstein, L. & Frauenfelder, H. (1975). Biochemistry, 14, 5355-5373. Baldwin, J. M. (1980). J. Mol. Biol. 136, 103-128.
Dynamics of M yoglobin Conformation
Baldwin, J. M. & Chothia, C. (1979). J. Mol. Biol. 129, 175-220. Beppu, Y. (1989). Comput. Chem. 13, 101. Bernstein, F. C., Koetzle, T. F., Williams, G . J . B . , Meyer, E. F., Brice, M. D., Rodgers, J. R., Kennard, 0., Shimanouchi, T. & Tasumi, M. (1977). J. Mol. Biol. 122, 535-542. Braun, W. & G5, N. (1985). J. Mol. Biol. 186, 611-626. Braun, W., Yoshioki, S. & G5, N. (1984). J. Phys. Soc. Japan, 53, 3269-3275. Brooks, B. & Karplus, M. (1983). Proc. Nat. Acad. Sci., U.S.A. 80, 6571-6575. Elber, R. & Karplus, M. (1987). Science, 235, 318-321. Fermi, G. (1975). J. Mol. Biol. 97, 237-256. Fermi, G., Perutz, M. F., Shaanan, B. & Fourme, R. (1984). J. Mol. Biol. 175, 159-174. Fletcher, R. (1980). Practical Methods of Optimization, vol. l, John Wiley and Sons, New York and Toronto. Frauenfelder, H., Parak, F, & Yang, R. D. (1988). Annu. Rev. Biophys. Biophys. Chem. 17, 451-479. Gelin, B. R. & Karplus, M. (1977). Proc. Nat. Acad. Sei., U.S.A. 74, 801-805. Gelin, B. R., Lee, A. W. & Karplus, M. (1983). J. Mol. Biol. 171,489-559. Gill, P. E., Murray, W. & Wright, M. H. (1981). Practical Optimization, Academic Press, London, New York, Toronto, Sydney and San Francisco. G5, N., Noguti, T. & Nishikawa, T. (1983). Proc. Nat. Acad. Sci., U.S.A. 80, 3696-3700. Higo, J., Seno, Y. & G5, N. (1985). J. Phys. Soc. Japan, 54, 4053-4058. Levitt, M., Sander, C. & Stern, P. S. (1984). J. Mol. Biol. 181, 432-447.
109
Momany, F. A., McGuire, R. F., Burgess, A.W. & Seheraga, H. A. (1975). J. Phys. Chem. 79, 2361-2381. Muirhead, H., Cox, J. M., Mazzarella, L. & Perutz, M. F. (1967). J. Mol. Biol. 28, 117-156. N~methy, G., Pottle, M. S. & Scheraga, H.A. (1983). J. Phys. Chem. 87, 1883-1887. Nishikawa, T. & GS, N. (1987). Proteins, 2, 308-329. Noguti, T. & GS, N. (1983a). J. Phys. Soc. Japan, 52, 3283-3288. Noguti, T. & G5, N. (1983b). J. Phys. Soc. Japan, 52, 3685-3690. Noguti, T. & G5, N. (1989a). Proteins, 5, 97-103. Noguti, T. & GS, N. (1989b). Proteins, 5, 104-112. Noguti, T. & GS, N. (1989e). Proteins, 5, 113-124. Noguti, T. & GS, N. (1989d). Proteins, 5, 125-131. Noguti, T. & G5, N. (1989e). Proteins, 5, 132-138. Obara, S. & Kashiwagi, H. (1982). J. Chem. Phys. 77, 3155-3165. Parak, F., Knapp, E. W. & Kucheida, D. (1982). J. Mol. Biol. 161, 177-194. Perutz, M. F. (1965). J. Mol. Biol. 13, 646-668. Perutz, M. F. (1970). Nature (London), 228, 726-739. Phillips, S. E. (1980). J. Mol. Biol. 142, 531-554. Seno, Y. & G5, N. (1990). J. Mol. Biol. 216, lll-126. Shaanan, B. (1983). J. Mol. Biol. 171, 31-59. Shrake, A. & Rupley, J. A. (1973). J. Mol. Biol. 79, 351-371. Takano, T. (1977a). J. Mol. Biol. 110, 537-568. Takano, T. (1977b). J. Mol. Biol. 110, 569-584. Wako, H. & GS, N. (1987). J. Comp. Chem. 8, 625-635. Yoshioki, S., Abe, H., Noguti, T., G5, N. & Nagayama, K. (1983). J. Mol. Biol. 170, 1031-1036.
Edited by P. E. Wright
D e o x y m y o g l o b i n Studied by the Conformational N o r m a l M o d e Analysis I. Dynamics of Globin and the Heme-Globin Interaction Yasunobu Senol"
Department of Physics, Faculty of Science Nagoya University, Nagoya 464, Japan and Nobuhiro G 6
Department of Chemistry, Faculty of Science Kyoto University, Kyoto 606, Japan (Received 2 January 1990; accepted 30 May 1990) Dynamic properties of deoxymyoglobin are studied theoretically by the analysis of conformational fluctuations. Root-mean-square atomic fluctuations and distance fluctuations between different segments reveal the mechanical construction of the molecule. Eight a-helices behave as relatively rigid bodies and corner regions are more flexible, showing larger fluctuations. More particularly, corner regions EF and GH are specific in that flanking a-helices extend their rigidity up to a point in the corner region and the two rigid segments are connected flexibly at that point. The FG corner is exceptional. A segment from the F helix to the beginning of the G helix, in which the FG corner is included, becomes relatively rigid by means of strong interactions with the heme group. The whole myoglobin molecule is divided into two large units of motion, one extending from the B to the E helix, and the other from the F to the H helix. These two units are connected covalently by the EF corner. However, dynamic interactions between these two units take place mainly through contacts between helices B and G and not through the E F corner. From correlation coefficients between fluctuational motions of residues and the heme group, 55 residues are identified as having strong dynamic interactions with the heme moiety. Among them, 18 residues in the three segments, one consisting of residues from the C helix to the CD corner, a second consisting of the E helix, and a third from the F helix to the beginning of the G helix, are in close contact with the heme group. Twenty-two of the 55 residues are within four residues of the 18 residues in their sequential residue number and are more than 3 A away from the heme group. The other 15 residues are located further in the sequential residue number and are all found in helices A and H. They are more than 6 A away from the heme group. By the use of correlation coefficients of fluctuations between residues, it is found that dynamic interaction with the heme group is transmitted to the A helix and the beginning of the H helix in the direction Leu(E15) -* (Val(A11) and Trp(A12)}. The transmission to the C-terminal end of the H helix is mediated by a long segment, from the end of the EF corner to the beginning of the G helix, that lies on the heme group and has close contacts over a wide range. The method for identifying the path of transmission of dynamic interaction by correlation coefficients of fluctuations will be useful also for studies of other functional proteins. globin have been widely applied to studies of other proteins. Three-dimensional structural data of highresolution X-ray crystallography have been accumulated for both the deoxy form and the liganded form of hemoglobin; by comparison of these, mechanism of co-operative ligand binding has been formulated (Perutz, 1965, 1970; Muirhead et at.,
1. I n t r o d u c t i o n
Study of the structure-function relationship of hemoglobin is most advanced in protein chemistry. The concepts and methods developed with hemot Author to whom all correspondence should be addressed. 0022-2836/90/200095-15 $03.00/0
95
© 1990 Academic Press Limited
96
Y. Seno and N . G5
1967; Anderson, 1973; Fermi, 1975; Baldwin & Chothia, 1979; Baldwin, 1980; Shaanan, 1983; Fermi et al., 1984). The structural change of hemoglobin upon ligand binding is regarded as comprising three levels: the geometrical change, of the heme group upon ligand binding, the tertiary structural change in each subunit, and the quaternary structural change in the tetramer. For complete understanding of the mechanism of cooperative ligand binding it is necessary to characterize each of the three levels and the specific interactions between them, and to identify the lowenergy path of the structural change in the conformational space of the whole tetrameric molecule. In view of the rapid progress in recent computational chemistry, a theoretical study based on the conformational energy function is expected to be a powerful tool for approaching these problems. In pioneering work on an a-subunit of deoxyhemoglobin, by the use of energy minimization with atomic Cartesian co-ordinates as independent variables, Gelin et al. (1983) studied a wide range of important problems: the geometry of the heme group, the interaction between the heme group and the globin molecule, and the tertiary structural change upon ligand binding (Gelin & Karplus, 1977; Gelin et al., 1983). Though their work is very stimulating and suggestive, some difficulties remain in the method of their calculation and in the interpretation of results. Firstly, as to the energy minimization, the steepest descent method of 200 cycles seems to be too simple and insufficient to treat a large molecule such as one subunit of hemoglobin. From the magnitude of residual gradients, their energy minimization was probably terminated at a point far from a true minimum. Secondly, because the X-ray structure of an a-subunit isolated from deoxyhemoglobin is taken as the starting conformation of energy minimization, the absence of quaternary constraint may be serious when the energy-minimized structure is interpreted as being in the deoxy form of hemoglobin. Meanwhile, vibrational normal mode analysis, a method for studying dynamic properties of protein molecules within the harmonic approximation of a conformational energy function, has been developed (G5 et al., 1983; Brooks & Karplus, 1983; Levitt et al., 1985; Nishikawa & GS, 1987). By the use of dihedral angles as independent variables of the eonformational energy function, energy minimization can be performed relatively easily exactly to the point where the second derivative matrix is positive definite and the first derivatives virtually vanish. The analytical nature of the normal mode analysis allows us easily to compute conformational fluctuations and their correlations in thermal equilibrium. In the thorough analysis of the dynamic properties of bovine pancreatic trypsin inhibitor (BPTIt) as a typical example of a small globular protein (Nishikawa & GS, 1987), normal modes in t Abbreviations used: BPTI, bovine pancreatic trypsin inhibitor; r.m.s., root-mean-square.
different frequency ranges have been characterized by the degree of localization and spatial continuity of atomic displacement vectors. The importance of low-frequency normal modes, particularly, is notable because of their dominant contribution to the magnitude and site dependence of atomic fluctuations. From the analysis of space correlation functions of atomic fluctuations, the protein molecule is found to be made up of relatively rigid elements held together by very flexible parts. Here and in the accompanying paper, the method of normal mode analysis is applied to deoxymyoglobin. Myoglobin is also an oxygen-binding hemoprotein, and its structure is very similar to that of one subunit of hemoglobin (Takano, 1977a,b; Phillips, 1980). It is easier to carry out conformational energy analysis on myoglobin, due to its molecular size, than on hemoglobin. Though the intersubunit interactions existing in tetrameric hemoglobin cannot be treated, problems concerning the tertiary structure can be investigated in myoglobin. For the calculation of deoxymyoglobin, which consists of one polypeptide chain and the heme group, as a hetero (i.e. non-peptide) molecule, we developed a new F O R T R A N computer program adapted for vector processing. This program is based on the formulations developed at first to rapidly compute a conformational energy function and its first and second derivatives (Noguti & G5, 1983b; Abe et al., 1984; Wako & GS, 1987), and then extended to treat a system composed of two or more molecules (Braun et al., 1984; Higo et al., 1985). This paper involves two sets of calculations. Firstly, after the complete energy minimization, starting from the X-ray conformation, conformational fluctuations and their correlations are examined on the basis of normal mode analysis. From site dependence of root-mean-square (r.m.s.) fluctuations of atomic displacements and from distance fluctuations between different segments, the mechanical construction of the molecule can be elucidated. Secondly, we inquire into the nature of dynamic interactions between the heme group and the globin molecule with the aid of correlation coefficients of fluctuations. Important residues in the heme-globin interaction are identified as having large correlation coefficients. Then from the correlation coefficients of fluctuations between the selected residues, we can trace the path along which the effect of the geometrical change at the heme-globin linkage is transmitted to residues distant from the heme. On the basis of these analyses of the dynamic properties, the conformational change upon ligand binding is calculated in the accompanying paper (Seno & G5, 1990) within the range of harmonic approximation of the energy function. The method is essentially the same as that used in the calculation of the conformational change of BPTI induced by a chemical modification (Yoshioki et al., 1983). Forces acting upon the linkage between the heme group and the globin molecule are guessed by repro-
Dynamics of Myoglobin Conformation ducing, within the harmonic approximation, the local geometrical change actually observed at the linkage. Overall conformational change of the whole molecule b y these forces can then be calculated easily. The energetics of the conformational change is developed within the harmonic approximation, and the numerical results are discussed. Atomic displacements are also examined in relation to the mechanical construction and the nature of heme-globin interactions. In the a c c o m p a n y i n g paper, the dominating roles of low frequency normal modes in the conformational change of the molecule are also examined by decomposition of the calculated conformational change to contributions from individual vibrational normal modes.
97
t2"
\
/
N
2. Methods (a) X-ray co-ordinates
C"
Atomic co-ordinates determined by X-ray crystallography are necessary for the initial step of our calculation. As described below, atomic co-ordinates of deoxymyoglobin are used in regularization of the globin molecule and determination of some parameters in the treatment of the heme group. Those of oxymyoglobin are also referenced in the analysis of con|brmational change upon ligand binding. Atomic co-ordinates of deoxymyoglobin and oxymyoglobin are taken from 1MBD and 1MBO, respectively (Protein Data Bank; Bernstein et at., 1977), both of which were deposited by Phillips (1980).
Figure 1. Heine skeleton and the imidazole ring of the proximal His93. The origin of the X YZ-system is the atom N"2. The Z-axis is taken as bisecting the angle C~I-N~2-C62 and in the direction of the heine, the X-axis is in the plane of the imidazole ring and in the direction of C~2. The origin of the ~t/~-system is the Fe atom. The ~-axis is perpendicular to the mean plane of the heine skeleton, and the ~-axis lies in the plane formed by the ~-axis and the bond Fe-N c.
(b) Conformational space and energy function
Higo el al., 1985) is applied here in the following way. In order to express position and orientation of the heine skeleton relative to the globin molecule, 2 co-ordinate systems are introduced: one is fixed to the imidazole ring of the proximal His93 (X YZ-system) and the other to the heme skeleton (~t/~-system; Fig. 1). Position and orientation of the heine skeleton with respect to the globin moiety are expressed by 3 components of a translational vector t and 3 components of a rotational vector r that, together, move the X YZ-system to the ~t/~-system. These 6 components, defined with respect to the X YZ-system, are included in the set of independent variables of the energy function and referred to as external variables. Thus, the total conformational space consists of 925 dihedral angles, 0~, i -- l, 2 . . . . . 925, and 6 external variables, e~, i -- 1, 2 . . . . . 6. These independent variables, put together, are denoted by qt, i = l, 2 . . . . . 931.
(i) Conformational space In the present calculation, the conformational energy function is constructed basically according to ECEPP (Momany et al., 1975; N~methy et al., 1983). In the scheme of ECEPP, hydrogen atoms are explicitly taken into account. Then, the conformation of the globin molecule is expressed by 911 dihedral angles around all rotatable chemical bonds, and atomic co-ordinates of the globin are generated by the use of the standard residue data of ECEPP. Expression of the motion of the heine group needs some consideration. It is well known from X-ray crystallography that the structure of the heme skeleton changes upon ligand binding from a domed to a more planar form. In the scheme of the present calculation in the dihedral angle space, however, inclusion of the out-of-plane motion of the heme skeleton is not straightforward, although not impossible. In the present works we are primarily interested in the "soft" dynamic properties of the molecule, i.e. such collective motions occurring in the timerange of picoseconds or longer and involving at least a few residues. The out-of-plane motion of the heme skeleton is much harder than these soft motions and has no significant interactions with soft variables. With this consideration, the heme skeleton is treated as rigid and its geometry is fixed to the X-ray structure of deoxymyoglobin in this series of papers. The 14 dihedral angles in the 8 side-chains of the heine group, which are free to move, are treated as independent variables. Bond angles and bond lengths in these side-chains are fixed to those of the X-ray structure. If we assume a rigid heme skeleton, deoxymyoglobin becomes a molecular system composed of 2 chains. The theory for treating a 2-chain system (Braun et al., 1984;
(ii} Energy terms and enerlly parameters The ECEPP data of torsional and interatomic energy parameters are used as they are for the globin part. As for the heine group, each of the rotatable bonds in the 8 sidechains has a chemical bond of the same type in the ECEPP residue data, and also heme atoms other than Fe are of the same atom types in the residue data. Therefore, except for those for the Fe atom, the ECEPP data are used. Non-bonded interactions involving the Fe atoms are tentatively neglected because there are no available parameters. This neglect is not expected to give rise to any serious artificial results when conclusions are drawn in the following sections, because the Fe atom is, in a sense, separated from atoms belonging to globin by the surrounding heme atoms. As parameters of electrostatic interaction, the atomic charges in the heme skeleton are set to values obtained by an ab ini~io molecular orbital calculation on a model compound of deoxyheme (Obara &
Y. Seno and N . G5
98
Kashiwagi,, 1982) and provided by H. Kashiwagi (personal communication). Because the model compound has no side-chains, atomic charges in the side-chains are determined according to our CNDO/2 calculation on the whole heme group with a point charge 2e instead of the F e atom (where e is the electronic charge unit). In order to conserve the total charge of the heme group, the differences in electron distribution in the heme skeleton between the 2 types of calculation are distributed proportionally to the atomic electron distributions in the side-chains. Though the heme group is permitted to move freely in globin by the 6 external variables introduced above, its translational and rotational motions relative to the globin are restricted to a large extent by the co-ordination bond between the Fe atom and N ~2 of the imidazole ring of the proximal His93(F8). This bond is exceptional in the sense t h a t the bond lengths and bond angles associated with it are changeable. Together with a torsional energy around the bond, a force field is taken into account, with a stretching energy term of F e - N ~2, bending terms of C~1-N~2-Fe, C~2-N~2-Fe and 4 angles N~2-Fe-Np, and with an energy term for deformation from planarity about the N ~2 atom (where Np represents 4 pyrrole nitrogen atoms of the heme group, and C~1 and C~2 are the atoms in the imidazole ring of His93(F8). Values of torsional energy parameters, force constants of the stretching and bending terms and of the equilibrium bond length F e - N ~2 are chosen to be the same as those which were used in the energy refinement of the X-ray diffraction study of carbonmonoxyhemoglobin (Baldwin, 1980). Equilibrium values of the 2 bond angles around the N r2 atom are taken to be the same as the angle between the bond N~2-C~I and the Z-axis in Fig. ], and an equilibrium value of the angles N~2-Fe-Np is determined by averaging the 4 corresponding angles in the X-ray geometry of the heme group. An energy term of non-planarity about the N ~2 atom is quadratic in the angle of deviation of the bond N~2-Fe from the plane through 3 atoms (yl, N~2 and Ca2. The value of the force constant for the bending of C~I-N~2-Fe (C62-N~2-Fe) is tentatively assigned to this term of nonplanarity. The above energy terms of torsion, stretching, bending and non-planarity associated with the co-ordination bond can be expressed by external variables only.
compared to the errors in determination of atomic coordinates by X-ray crystallography. Initial values of the 6 external variables of the heme-globin linkage are easily calculated from the regularized conformation of the globin molecule best-fitted to the X-ray structure. Initial values of dihedral angles in the side-chains of the heme group can be equated to values in the X-ray conformation. (ii) Energy minimization The energy minimization, starting from the regularized conformation, is performed by a Newton method, i.e. a restricted step method with line search (Fletcher, 1980; Gill et al., 1981). The Newton method has become very efficient due to the recently developed algorithm used to calculate rapidly the 1st and 2nd derivatives of the conformational energy function in their analytical forms (Noguti & GS, 1983b; Abe el al., 1984). With the full use of the algorithm, a new F O R T R A N program adapted for vector processing has been developed to study large protein molecules (Wako &.G5, 1987). F o r the present study of deoxymyoglobin, this program has been extended to treat a biopolymer system consisting of 2 chains on the basis of the formalism mentioned above (Braun et al., 1984; Higo el al., 1985). Required central processor unit time of 1 evaluation of the conformational energy function of deoxymyoglobin and all of its 1st and 2nd derivatives is 16"87 s by the vector processor HITAC S-820. One evaluation of the energy function only takes 0.57 s. After 152 cycles of iteration of the Newton method, a minimum point is reached where all eigenvalues of the second derivative matrix are positive, and the absolute value of the residual gradient of energy is less than 2"0 x l0 -9 (kcal/mol per rad), 4-6 x l0 -12 (kcal/mol per A) and l ' 6 x l0 -11 (kcal]mol per rad) for a dihedral angle (i cal = 4"184 J), a component of the translational vector t and a component of the rotational vector r, respectively. The required central processor unit time for this energy minimization is 54'5 min. The r.m.s, atomic shift in the minimization process is l'001 A and 1"214 A for mainchain atoms and all heavy-atoms, respectively. This movement is the same in magnitude as is the case for energy minimization of other smaller proteins with dihedral angles as independent variables (Levitt et al., 1985; Nishikawa & GS, 1987).
(e) Regularization and energy minimization (i) Regularization The normal mode analysis is carried out a t the minimum point of the eonformational energy function. In order to use the X-ray conformaton as the starting point of energy minimization, we must first determine the set of dihedral angles of the globin molecule t h a t generates the conformation nearest to t h a t of the X-ray structure. This process is referred to as regularization of Cartesian coordinates of atoms with bond lengths and bond angles fixed to the standard values of ECEPP. The regularization is implemented by a F O R T R A N computer program of distance geometry t h a t has also been used to determine protein solution conformations from p r o t o n - p r o t o n distance constraints (Braun & G5, 1985). F o r the purpose of preserving the exact position of the side-chain of the proximal His93(F8), a larger number of distance constraints between the imidazole ring and other parts of the globin molecule than our usual regularization process are included here. r.m.s, deviation of the regularized conformation from the X-ray structure is 0"182 A (l A = (~l rim)for all heavy-atoms, and 0"124 A for mainchain atoms. These deviations are not so large when
(d) Vibrational normal modes The method of protein normal mode analysis has been described in detail {Levitt et al., 1985; Nishikawa & GS, 1987). Here, we summarize only a mathematical framework to fix notations in order. The conformational energy function is expanded at the energy minimum point in terms of qi (i = I, 2 . . . . . 931), which hereinafter denotes infinitesimal changes of independent variables from corresponding values at the minimum point. By retaining terms up to 2nd-order, we have:
E = ½Y~,E/,,qm,
(l)
where subscripts i and j in summations go from I to 931. The kinetic energy is expressed as:
T = ½~.,~lH~1(dq,/dt)(dqj/dt).
(2)
Matrices, whose elements are F 0 and Hi j, are referred to as F - m a t r i x and H-matrix, respectively. The F - m a t r i x is evaluated a t the energy minimum point by the use of the above program to calculate the 2nd derivatives of the energy function. The H - m a t r i x is also evaluated in an
Dynamics of Myoglobin Conformation analytical form (Noguti & GS, 1983a), where overall translational and rotational motion of the whole molecule is separated from internal motions and the contributions of external motions are excluded from the kinetic energy. By a linear transformation of qi to new variables Q~ (n = l, 2 . . . . . 931): q~ = ~., VI,Q,,
(3)
the conformational energy and the kinetic energy given by eqns (l) and (2), respectively, can be transformed to simple sums: E = .~V , ~ . (.0202 ....
(4)
T = ~,.(dQ~/dt) 2.
(5)
The variables Q. are referred to as normal mode variables, and they satisfy independently the equations of motion of harmonic oscillation with angular frequencies ran:
d2Q,/dt 2 + co,2Q~= o.
(6)
The transformation matrix whose elements are Vi, in eqn (3) is the matrix of eigenvectors. The inverse transformation is written as:
Q, = ~ ( V- t),iqi,
(7)
where V -1 is the inverse matrix of V. By this equation, any vector with components qt in the tangential vector space at the energy minimum point can be decomposed by eigenvectors of normal modes with Q~ as decomposition coefficients. (e) Thermal fluctuations I n the method described in the previous works (Levitt etal., 1985; Nishikawa & G5, 1987), thermal fluctuations are calculated for each normal mode, and total fluctuations are given by sums of contributions from all normal modes. Because we are confined with total fluctuations in thermal equilibrium later in this work, somewhat more straightforward expressions are briefly given to the quantities calculated here. (i) Fluctuation of independent variables An averaged value of a product of fluctuations qi and ql in thermal equilibrium is given by:
(8)
where k s is the Boltzmann constant, T is the absolute temperature, which is fixed to 300 K throughout this work, and F - t denotes the inverse matrix of F. I n the ease of the same suffix (i = j), this equation expresses the squared amplitude of fluctuation in % and in other cases the correlation of fluctuations between q~ and ql" Next, as a measure of strength of dynamic interaction, we use the correlation coefficients of fluctuations of independent variables:
P(qi, qj) = (q~qj)/{((qi)2)((qi)2)} t/2.
(9)
Using this quantity, r.m.s, correlation coefficients Pkt between the kth and t h e / t h residues are defined by: ( P u ) 2 = ( /1
N ~~N ~ ,)~.,~.qP(q,,qi),2
(10)
where N~ and N~ denote the number of dihedral angles qi in the kth residue and qi in the lth. residue, respectively, and summations are taken over all dihedral angles in the 2 residues. Because the translational and rotational motion of the heme group relative to the globin molecule are expressed by
99
external variables e~ (i = 1, 2 . . . . . 6), correlation coefficients p(e l, Oj) from eqn (9) indicate the strength of dynamic interaction between the motion of the heme group expressed by changes of ei and the conformational change of the globin molecule caused by a change in dihedral angle 0i. F o r each of 6 external variables, r.m.s. correlation coefficients Pk(ei) between the external motion of the heme group and the conformational change of the kth residue are defined by: {~k(S,)}2 = (l/N~)~,q{p(e,, Oj)}2,
(l l)
where the summation extends over all dihedral angles in the kth residue. (ii) Fluctuations of atomic co-ordinates Infinitesimal displacements of a-components (a -- 1, 2, 3, or x, y, z) of an atom a, Ax,~, are derived from changes in independent variables qi (i = 1, 2, . . . , 931): Ax.~-- ~K,o.,q,,
(12)
where, in addition to the H-matrix, the K-matrix, whose elements are Kao.~, is calculated analytically so as to exclude external motions of the whole molecule (Noguti & G5, 1983b). By the use of eqns (8) and (12), correlations of fluctuations of atomic co-ordinates are calculated as: (AxaaAXb~) = ~.i~tKaa.iKbrj(q~i) =
k.TZ,ZjKa..,Kb.j(F-'), j.
(13)
From this we calculate r.m.s, fluctuations of atomic displacements in a segment, distance fluctuations, and a type of measure of dynamic interaction other than Pu from eqn (10) between a pair of segments. Nishikawa & G5 (1987) discussed the distance fluctuation, which is defined as follows. Change of distance between a pair of atoms, a and b, is given by: Arab ~- (Ara-Arb)- (r.--rb)/]r.--rbL
(14)
where r, and r b denote position vectors of atoms a and b in the energy-minimized conformation, and Ar, and Ar b are their infinitesimal changes whose components are Ax,~ and AXb, (a ----1, 2, 3), respectively. Distance change Adkl between the kth segment and the /th is defined as the average over pairs of atoms:
adkl -~ ( l / NkN,) ~a~b Arab,
(15)
where atoms a and b belong to the kth and the /th segments, respectively, and N k and N~ are numbers of atoms considered in 2 segments. Here, a segment is taken to be the main-chain of a residue, and, in thermal equilibrium, distance fluctuation Dal between main-chain parts of the kth and t h e / t h residues is defined by: (Dkl)2 = ((Adkl)2).
(16)
Another correlation coefficient ~ between a pair of residues k and l is defined as follows, on the basis of fluctuations of Cartesian co-ordinates of atoms:
(~1)2 = (1tN:N~ )ZaZb × { Z ~ L ( a x a ~ X b t )2/(Ara2) (arb2) },
(17)
where N~, and N~ are numbers of all atoms in the kth and /th residues, respectively, and a, r = l, 2, 3. This quantity takes values from 0 to 1. Whereas the correlation coefficient of eqn (10) focuses on correlation of the internal conformational changes of a pair of residues, eqn (17) gives also correlation of the external (i.e. translational and rotational rigid body) motions between a pair of residues.
100
Y. Seno and N. G5
Similarly we define:
fl
B
C
0
E
F
0
H
{hI(s,)}~ = (l/Nl)~,~.,,(,~z, oe,>~/{< ~ ) (e~ >,
(18) where the 1st summation is taken over all atoms in the kth residue, and the 2nd summation is over components a = l, 2 and 3. 3. Results and I:)iscussion
(a) Dynamic properties of the globin molecule (i) r.m.s, atomic fluctuations It has been shown in previous papers that a protein molecule is mechanically composed of relatively rigid and flexible parts !Levitt et al., 1985; Nishikawa & GS, 1987). In myoglobin, which is a protein with a large helix content, we again see this clearly, firstly by the calculation of atomic fluctuations. The calculated r.m.s, fluctuation (averaged over atoms in each amino acid residue) is plotted against residue number in Figure 2(a) for mainchain atoms, and Figure 2(b) for all heavy-atoms. The broken lines are plots of the experimental fluctuations obtained from temperature factors of an X-ray diffraction analysis (Phillips, 1980). In both cases ((a) and (b)), calculated and observed r.m.s. fluctuations exhibit almost the same patterns of variation with residue number, although the calculated values are smaller than those observed. In the case of main-chain atoms (Fig. 2(a)), both calculated and observed r.m.s, fluctuations have a tendency to become small at the central parts of the eight helices {from A to H) and to take large values at terminal sides of helices and at the corner regions connecting them (except for the case of the observed value at the C helix). In the calculated plot, a strong peak is observed in each corner region except for in FG, i.e. at AlaI9(ABI), LysS0(CD8), Ala84(EF7) and Alal21(GH3). Although a small peak is observed at Lys96(FG1), calculated fluctuations at the FG corner are smaller than at other corners. The segment from the F helix to the beginning of the G helix, including the FG corner, is located near the heme group and has strong contacts with it. The small r.m.s, fluctuations of main-chain atoms in this region reflect stabilization of conformation by these interactions. In the plot of the r.m.s, fluctuations of all heavyatoms (Fig. 2(b)), sharp peaks appear periodically every three or four residues. These peaks come from large fluctuations of side-chain atoms of hydrophilic residues located at the surface of the globin molecule. The large contribution to the r.m.s, fluctuations from hydrophilic residues at the surface is confirmed by calculations of solvent accessibility, which is defined here as the ratio of area of exposure of an amino acid residue X in globin to that of the residue in a tripeptide Gly-X-Gly with fully extended conformation (Shrake & Rupley, 1973). In Figure 2(e), the calculated solvent accessibilities are plotted. Peaks and troughs of the r.m.s, fluctuations coincide well with those of solvent accessibility, except for a few points.
/~"~%i!;"k]l~?liiil
[i
;~,.ll;li:~ :iJt %~,,'1,=.'~::~i/!
-~. ~0. 0
20
40
60 80 100 Residue number
120
140
Figure 2. r.m.s, fluctuations of atoms in each residue plotted against residue number in (a) and (b). Continuous line, calculated r.m.s, fluctuations at 300 K; broken line, experimental Lm.s. fluctuations derived from the observed temperature factors. Locations of 8 helices, A to H, are indicated. (a) Main-chain atoms; (b) all heavy atoms. In (c) solvent aecessibilities are calculated for the energy minimum structure.
(ii) Correlation of fluctuations A three-dimensional picture of the dynamic structure of a protein molecule can be obtained from fluctuations of distances between different segments. Small distance fluctuations within a segment of a molecule mean that the segment behaves as a relatively rigid body in the molecule. I f they are small between a pair of segments, two segments show a tendency to move together as if they were one body. Pairs of residues with small distance fluctuations in the sense defined by equation (16) "are shown in Figure 3. In this Figure, triangular structures of dense shading corresponding to a-helices appear along the diagonal line. This means that a-helices are fluctuating as relatively rigid bodies in the molecule. Among the eight helices, B, F, G and H are seen to be more rigid than others from completeness of the corresponding triangles, and helices A, C and E seem to be more flexible. Detailed structures of shaded triangles can be interpreted in relation to the calculated r.m.s, fluctuations shown in Figure 2(a). In each of the two corner regions (EF and GH), there is a strong peak of r.m.s, fluctuation centered at AIa84(EF7) and AIal21(GH3) (mentioned previously). In this case, two triangles corresponding to a-helices extend from both sides up to the residue with the large fluctuation. This means the existence of a flexible point in the corner region as a joint between two rigid segments. There are no corner regions between adjacent helices B and C,
Dynamics of MyoglobinConformation
101
E i £ ; ; ; ;;, :: ~ . ~ l b : ~ a
50
-
-::
~iEiiii
C
'0 m
g~ . . .
_..
4--
I00 = _--
_--
7-
_
_..
-
- -
-
-~
.
.
.
.
.
Tt
I 50
I00
Residue
I
I 150
number
Figure 3. A map of small fluctuations of distances between residues. Distance fluctuations are averaged over mainchain atoms. (*)Less than 0"25A, (+) from 0"25A to 0"35A, ( - ) from 0'35A to 0"40A, where A (=0"476 A) is the maximum value of the distance fluctuation. The porphyrin ring of the heme group is treated as the 154th residue, and the helical regions are shown by the names of helices along the diagonal line.
and between D and E. In these cases, two triangles for two consecutive helices merge into one relatively rigid body. In inner parts of the map there are several regions of small distance fluctuations. Among them the following four wide regions are conspicuous: one is between segment B-C (from the B to the C helices) and segment D - E (region I), a second is between segment B - C and the G helix (region II), a third is between the G helix and the H helix (region III), and a fourth is between the H helix and segment F - F G (region IV). In each of these four regions a pair of helices are in close contact in a wide range. This is seen in the map shown in Figure 4, where minimum atomic distances between residues are plotted. Three lines of close contact extend perpen-
diculady to the diagonal line (antiparallel arrangement of helices), at upper-left, lower-left and lowerright, to pass through the above-mentioned regions I, I I and III, respectively. A line of contact between the H helix and the segment F - F G (parallel arrangement) appears at lower-center in Figure 4, and this line is located at region IV shown in Figure 3. In this way we can see t h a t two rigid segments or helices tend to move together as ff as one body when t h e y are in close contact in a parallel or antiparaUel arrangement. Contact interactions do not necessarily m e a n a region of small distance fluctuations. The A helix has two lines of antiparallel contact interactions with helix E and helix H (Fig. 4). However, corresponding regions of small distance fluctuations are
102
Y. Seno and LV. G6
,.-iL \
e
Ull im
50--
M
M
It-U ~ M
M IM
=E@~,
~5 if'~ M
L~MN
\
E
MM
II
c
M
•
M
I
100-•~-
I
im
M
MIII
-
M
|,,,
-
M
IIm¥ M
~N
114
w
le
M
l
_*I! M
4M
M
MW
MM
H1L,~
|. 150-÷
I
I
I
M MMM
I
w m
I 5O
m
1
t
-
I
I
l IO0
I
I
I
I
1 150
Residue number
Figure 4. Minimum atomic distance between residues (including H atoms). (*) Less than 3 A, ( + ) from 3 A to 3-5 A, ( - ) from 3-5 A to 4"0 A. The heme group, including side-chains, is treated as the 154th residue.
incomplete. This may be due to large flexibility of the A helix or its external motion relative to the other helices. Further investigation of the internal and external motions of a-helices in the globin molecule is now in progress. From Figure 3 we see that a long segment from the B helix to the E helix, except for the internal CD corner, forms one relatively rigid unit because of the presence of region I. Another long segment from the F helix to the H helix, except for the GH corner, also forms a relatively rigid unit because of the presence of regions I I I and IV. Thus, aside from the flexible terminal sides NA, A and HC, the globin molecule appears to be composed of two large relatively rigid units. These two units are connected covalently by the E F corner. However, these two units are held together relatively rigidly mainly by region II and not by the E F corner.
In hemoglobin, the alfll(a2fl2) interface is formed by parts of the helices B, G and H of the a-subunit packing against helices B, 13 and H of the fl-subunit. Essentially no structural differences are observed between deoxyhemoglobin and liganded hemoglobin in this interface (Baldwin & Chothia, 1979). It is interesting that the contact region between helices B and G plays an important role in holding the two relatively rigid units together within myoglobin, and it is also involved in the intersubunit interaction in hemoglobin. In Figure 3 we see that the neighborhood of the proximal His93(F8) residue has small distance fluctuations between a large portion of the globin molecule, as is shown by the horizontally and vertically shaded regions. This is because motion of this segment is mediated to other parts of globin by the heme group, which interacts with a large portion of
Dynamics of Myoglobin Conformation the globin molecule. The FG corner (96 to 99) also interacts with the heme group and therefore has small distance fluctuations from it. This corner is more rigid, as seen in Figure 2(a), than others because of the interaction with the heine group. The interactions, with the heine group of the segment from the center of the F helix to the G helix, contribute to making the above-mentioned unit from the F helix to the H helix relatively rigid. (iii) Justification of the harmonic approximation The results so far discussed are based on the normal mode analysis, which in turn is based on the harmonic approximation of the conformational energy function within the range of thermal fluctuation. In Figure 2 we see that the residue number dependence of the calculated r.m.s, atomic fluctuation reproduces the experimental results quite well, at least qualitatively. This result may suggest that the approximation is justified. However, experimental studies such as multipleflash photolysis with carbonmonoxymyoglobin and MSssbauer spectroscopy from low to room temperature indicate that this protein at room temperature can go through many local minimum energy conformations (Austin et al., 1975; Parak et al., 1982; Frauenfelder et al., 1988). In other words the energy function within the range of thermal fluctuation is so highly anharmonic as to have many local minima. Such a highly anharmonic nature of the energy function of proteins has also been shown computationally by molecular dynamics simulation of myoglobin by Elber & Karplus (1987) and Monte Carlo simulation of BPTI by Noguti & G5 (1989a,b,c,d,e). Therefore, proteins appear to have a peculiar property in that they behave as if they were a good harmonic system, even though their energy fluctuation is very highly anharmonic. In order to solve this apparent paradox, we are now carrying out normal mode analyses at a number of local minimum energy conformations. Tentative results indicate that individual normal mode at one energy minimum point is generally quite different from any modes at other points. However, the subspace spanned by a set of normal modes with very low frequencies is very stable as a whole, i.e. does not depend sensitively on a particular minimum at which normal mode analysis is carried out. This fact implies that the multiple minima shown to exist both experimentally and theoretically have such a common subspace spanned by the low frequency normal modes. If so, calculated r.m.s, fluctuations would be almost the same for different local minima, because the thermal fluctuation is dominated by the low frequency normal modes. The above paradox may be solved in this way. In other words the use of normal mode analysis can be justified by a rather complex mechanism, even when the energy function is highly anharmonic. The above justification, however, is not yet complete. With this situation in mind, at this stage, we would justify our approach by the result shown
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Figure 5. For each of the 6 external variables e~ (i = 1,2 . . . . . 6), the r.m.s, correlation coefficients Pk{ei) defined by eqn (11) and ~,(~) defined by eqn (18) are plotted against residue number k, using a continuous line and a broken line, respectively. (a} For the translation component e I -= tx, (b) for e2 - t~, (c) for es =- t=, (d) for the rotational component e4 ----rx, (e) for e~ -- r~, (f) for ~6 ~ Tz"
in Figure 2, i.e. the results obtained by this treatment agree well with the experimental data, even though we are aware that the reason for this agreement is quite complex. (b) Dynamic structure of the heme-globin interaction One of the important problems concerning the tertiary structure of oxygen-binding hemoprotein is the dynamic interaction between the heme group and globin; e.g. which parts of the globin molecule are most strongly affected by the geometrical change at the linkage between the heine group and the proximal histidine residue, and by what mechanism is the effect transmitted to other parts of the globin molecule? Because the dynamic interaction should be reflected in conformation fluctuations, we will use the r.m.s, correlation coefficients ~k(si) and ~(ei) given by equations (11) and (18), respectively, as measures of strength of the dynamic interaction between the kth residue and the heme group. The quantity ~k(e~) is relevant to local conformational changes within the residue but not to translational and rotational motions of the entire residue. Correlations to the latter rigid body motions can be detected by ~(8i). As mentioned, the motion of the heme group is expressed by six external variables 8i (i = 1, 2 . . . . . 6), which represent three components of the translational vector (i = 1, 2, 3) and three
104
Y. Seno and N. G5
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components of the rotational vector (i-- 4, 5, 6) with respect to the imidazole ring of His93(F8). For each of these six components of external motion of the heme group, dependence of P~(Si) and ~5~(~) on residue k is shown in Figure 5. In the same way as for the dynamic heme-globin interaction, dynamic interactions between residues within the giobin molecule are examined using two types of r.m.s. correlation coefficients, Pkl and ~5~1 given by equations (10) and (17), respectively. They are mapped in Figure 6(a) and (b), respectively. The above r.m.s, correlation coefficients have generally strong correlation with minimum distances between segments. This is seen by comparison of Figure 6 with Figure 4, and by the fact that Pk(~i) and P~(~i) in Figure 5 take large values around residues that are in close contact with the
heme group (the bottom row of Fig. 4). They offer us, however, information about mediation of dynamic interactions. We should pay special attention to points or regions shown in Figures 5 and 6 where pairs of segments have exceptionally large r.m.s, correlation coefficients, even for long distances. In such cases there must be some mechanism to mediate interactions between them. In the following, important residues in the dynamic heme-globin interactions are identified by using r.m.s, correlation coefficients pk(e~) and P~(~i) and the minimum distances from the heme group. Then, dynamic interactions between the selected residues are examined using Pu and P~t. First of all, 18 residues with minimum distances less than 3 A from the heme group, which are shown by the symbol (*) in the bottom row of Figure 4, are
Dynamics of MyoglobinConformation
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105
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Figure 6. Maps of the r.m.s, correlation coefficients (a) Pkt and (b) P~t defined by eqns (10) and (17), respectively, between residues. In (a), (*) larger than 0"05, (+) from 0"025 to 0"05, ( - ) from 0"01 to 0"025. In (b), (*) larger than 0"25, (+) from 0"15 to 0-25, ( - ) from 0-1 to 0"15.
identified as nearest-neighbor residues and are shown underlined in Table 1 (see below). Among the 18 nearest-neighbor residues, four residues are found in the segment from the C helix to the CD corner, five residues (including the distal His64(E7)) in the E helix, eight residues (including the proximal His93(F8)) in the segment from the F helix to the beginning of the G helix, and one residue (Phe138) in the H helix. When the 153 residues are arranged in descending order of magnitude of the r.m.s, correlation coefficients ~(e~) and ~(e~) for each of the six external variables ei, these 18 residues are usually found towards the top in the 12 lists, six lists of Pk(~i) and six lists of ~(8f), showing strong dynamic interactions with the heme group. Except for six residues (67, 68, 71, 72, 107 and 138) 12 residues are
included in the top 20 members in at least one of the six lists of Pk(e~), and in the same way, 15 residues are included in the six lists of P~(ei), except for 64, 68 and 138. From the exceptional residues, we can see that the segment in the E helix from the distal His64(E7) has relatively weak dynamic interactions with the heme group. Other residues have minimum distances larger than 3"0 A from the heme group, but may still have strong dynamic interactions with it. Besides the above-mentioned 18 nearest-neighbor residues, 37 residues are found in the top 20 members of at least one of the 12 lists mentioned above. These 37 residues are also selected as the residues having strong dynamic interactions with the heine group. In total, 55 (---18+37) residues are selected as
Y. Seno and N. G5
106
i m p o r t a n t in dynamic heme-globin interactions. No residues in the B helix or in the C-terminal side of the G helix are included in these 55 residues. Next, these 55 residues are classified into clusters according to their sequence proximity in such a way t h a t each residue in one cluster is not separated from at least one member residue in the same cluster by more than four in residue sequence number. This sequential classification is based on the fact t h a t strong dynamic interactions, measured by ~ : and P~l, are generally observed for pairs of residues with sequential distances less than or equal to 4, which corresponds to one helical turn. By this means, the 55 residues are classified into the six clusters shown in Table 1. The r.m.s, correlation coefficients Pkt and P~l within the 55 residues are shown in Figure 7, and spatial arrangements of residues and the heme group are illustrated in Figure 8. Because this classification is based mainly on sequential residue number, some of the clusters m a y be located spatially close to each other. In fact, clusters 1 and 5, and 4 and 6, are relatively close in space. A part of cluster 2 and another part of cluster 3 are close, and a similar situation occurs between clusters 2 and 4. No nearest-neighbor residues are found in clusters 1 and 5. Only one, Phe138, is found in cluster 6. In fact all member residues in these clusters, except for Phel38, are spatially a p a r t from the heine group by more than 6 A in minimum distance. Their strong interactions with the heme group should be mediated by some member residues belonging to other clusters. The mediation of dynamic interactions can be traced by the r.m.s, correlation coefficients Pkt and P~l, and by the minimum distances between residues. The mediation to cluster 1 is traced as follows. The nearest-neighbor residue Leu72(El5) in cluster 3 has a strong dynamic interaction with
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2
2
--~
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6
-
Cluster 1 2 3 4
5 6
Membert Leu2 Vail3 Pro37 Asp44 His64 Glu83 Pro88 His93 Lys98 Tyrl03 Ala127 Glu136 Lys145
Gly5 Trpl4 Thr39 Arg45 Thr67 Ala84 Leu89 Ala94 Ile99 Leul04 Metl31 Phe138 Tyr146
Location:~
Glu6
Trp7 Leu9
A
Glu41 Phe46 Va168 Glu85 Alag0 Thr95 Prol00 Ilel07
Lys42 Phe43 C-CD His48 Ala71 Leu72 E Leu86 Lys87 EF-G Gln91 Set92 Lys96 His97 Ilel01 Lysl02
H I1e142 Ala143 Ala144 H-HC Tyr149 Tyrl51
t Nearest-neighbor residues that have close contact interactions with the heine group are underlined. :~Locations of member residues with respect to the secondary structures.
-
++~6"~
Cluster number (a)
2t
C
-
-
- -
--7- -5 : : : . '~ . -
~-
-
Table 1 Classification of the 55 amino acid residues that have strong dynamic interactions with the heme group
3
I
2
--:
-
5
-
3
4
5
6
Cluster number (b)
Figure 7. Maps of the r.m.s, correlation coefficients Pkt and P~l between the 55 residues that have strong dynamic interactions with the heme group. They are clustered and ordered as given in Table 1. (a) Map of ~kl: (*) larger than 0-08, (+) from 0"06 to 0"08, ( - ) from 0"04 to 0"06. (b) Map of ~t: (*) larger than 0"2, (+) from 0-15 to 0"20, ( - ) from 0-10 to 0"15.
Trpl4(A12) in cluster 1 due to a close contact interaction. The other three residues of cluster 3, Thr67(E10), Va168(Ell) and Ala71(E14), have moderately strong dynamic interactions with T r p l 4 and its neighboring residue V a i l 3 ( A l l ) in cluster 1. Within cluster l, residues 13 and 14, respectively, have a strong and a moderately strong dynamic interaction with Leu9(A7). This Leu residue transmits further to the residues located at the N-terminal side. Thus, the dynamic interactions between cluster 1 and the heme group are mediated by
Dynamics of Myoglobin Conformation
107
!S "71
72 :s
(o)
48
40
41
41
(b}
Fig. 8.
members of cluster 3, which are a part of the E helix. As for cluster 5, each of two member residues Ala127(H4) and Metl3l(HS) have strong and/or moderately strong dynamic interactions with residues Leu9 and Vall3 in cluster 1. These interactions are due to close contacts. Thus, the dynamic interactions between cluster 5 and the heme group are mediated again first by the same part of the E helix and then by a part of cluster 1. As for cluster 6, strong dynamic interactions are observed with cluster 4. Three residues of cluster 6, Ala143(H20), Ala144 and Lys145, have strong or moderately strong dynamic interactions with most
of the six residues from Glu83(EF6) to Pro88(F3) in cluster 4. These strong interactions are due a to widerange of contact interactions between the clusters. Tyr146 has a strong interaction with Prol00(G]) and moderately strong interactions with Ala94(F9), Ile99(FG4) and Ilel01 (G2). We see that Tyrl46(H23) is in close contact with these four residues in cluster 4. Thus, the strong dynamic interactions between the heme group a n d the C-terminal side of cluster 6 are mediated by cluster 4, which is a long segment composed mainly of the F helix, the FG corner and the beginning of the G helix. From the strength of the dynamic inter-
108
Y. Seno and N. G5
~c)
Figure 8. Stereo drawings, drawn by NAMOD (Beppu, 1989), showing spatial arrangements of residues and the heme moiety in 6 clusters. The a-carbon atoms are distinguished from other atoms by heavy circles. (a) Clusters 1, 3 and 5; (b) cluster 2; (c) clusters 4 and 6.
actions between residues, Lysl45(H22) and Tyr146(H23) are seen to play a central role in the mediation. So far we have discussed pathways of mediation of dynamic interactions between the heme group and various parts of the globin molecule. The latter has been divided into clusters. However, the clusters are not completely separate entities dynamically, because sometimes strong dynamic interactions are observed between them. Between clusters 2 and 3, strong dynamic interactions are observed between residues Arg45(CD3) and Phe46(CD4) in cluster 2 and the distal His64(E7) residue in cluster 3. Between clusters 2 and 4, Lys42(C7) in cluster 2 has moderately strong dynamic interactions with two residues in cluster 4, Ile99(FG4) and Tyrl03(G4). These strong or moderately strong dynamic interactions are again due to the close contact interactions, between clusters, mentioned above. 4. Conclusion The conformational normal mode analysis has been applied to deoxymyoglobin to calculate: (1) distance fluctuations between various parts of the protein, (2) correlation coefficients of fluctuations of internal variables in each residue and the berne group, and (3) correlation coefficients of fluctuations of Cartesian co-ordinates of atoms in each residue and the heine group. From (1), mechanical construction of this molecule has been studied, u-Helices were found to be more rigid than other parts of the molecule. Two larger structures, each consisting of a few ~-helices, i.e. one from the B helix to the E helix and the other
from the F helix to the H helix, were found to behave as dynamic units. These two units are held together by the interactions between helices B and G. From the analyses based on the quantities (2) and (3), we obtained a picture of the dynamic heme-globin interaction: 55 residues were identified as having strong dynamic interactions with the heme group. These residues were classified into six clusters. Three clusters (2, 3 and 4) surround the heme group and have strong dynamic interactions with each other as well as with the heme group. Clusters 3 and 4 mediate the dynamic interactions to the A helix and the C'terminal side of the H helix, respectively. The method of application of normal mode analysis to study the dynamic properties of the myoglobin molecule may also be useful in functionally related dynamic studies of other protein molecules. We are grateful to Dr H. Kashiwagi for providing calculated atomic charges of the heme groups. Computations reported in this paper were done at the Computer Centers of Nagoya University and of the Institute of Molecular Science. This work was supported in Kyoto University by grants to N.G. from the Ministry of Education, Science and Culture, Japan, and from the Science and Technology Agency, Japan. References Abe, H., Braun, W., Noguti, T. & G5, N. (1984). Comput. Chem. 8, 239-247. Anderson, L. (1973). J. Mol. Biol. 79, 495-506. Austin, R. H., Beeson, K. W., Eisenstein, L. & Frauenfelder, H. (1975). Biochemistry, 14, 5355-5373. Baldwin, J. M. (1980). J. Mol. Biol. 136, 103-128.
Dynamics of M yoglobin Conformation
Baldwin, J. M. & Chothia, C. (1979). J. Mol. Biol. 129, 175-220. Beppu, Y. (1989). Comput. Chem. 13, 101. Bernstein, F. C., Koetzle, T. F., Williams, G . J . B . , Meyer, E. F., Brice, M. D., Rodgers, J. R., Kennard, 0., Shimanouchi, T. & Tasumi, M. (1977). J. Mol. Biol. 122, 535-542. Braun, W. & G5, N. (1985). J. Mol. Biol. 186, 611-626. Braun, W., Yoshioki, S. & G5, N. (1984). J. Phys. Soc. Japan, 53, 3269-3275. Brooks, B. & Karplus, M. (1983). Proc. Nat. Acad. Sci., U.S.A. 80, 6571-6575. Elber, R. & Karplus, M. (1987). Science, 235, 318-321. Fermi, G. (1975). J. Mol. Biol. 97, 237-256. Fermi, G., Perutz, M. F., Shaanan, B. & Fourme, R. (1984). J. Mol. Biol. 175, 159-174. Fletcher, R. (1980). Practical Methods of Optimization, vol. l, John Wiley and Sons, New York and Toronto. Frauenfelder, H., Parak, F, & Yang, R. D. (1988). Annu. Rev. Biophys. Biophys. Chem. 17, 451-479. Gelin, B. R. & Karplus, M. (1977). Proc. Nat. Acad. Sei., U.S.A. 74, 801-805. Gelin, B. R., Lee, A. W. & Karplus, M. (1983). J. Mol. Biol. 171,489-559. Gill, P. E., Murray, W. & Wright, M. H. (1981). Practical Optimization, Academic Press, London, New York, Toronto, Sydney and San Francisco. G5, N., Noguti, T. & Nishikawa, T. (1983). Proc. Nat. Acad. Sci., U.S.A. 80, 3696-3700. Higo, J., Seno, Y. & G5, N. (1985). J. Phys. Soc. Japan, 54, 4053-4058. Levitt, M., Sander, C. & Stern, P. S. (1984). J. Mol. Biol. 181, 432-447.
109
Momany, F. A., McGuire, R. F., Burgess, A.W. & Seheraga, H. A. (1975). J. Phys. Chem. 79, 2361-2381. Muirhead, H., Cox, J. M., Mazzarella, L. & Perutz, M. F. (1967). J. Mol. Biol. 28, 117-156. N~methy, G., Pottle, M. S. & Scheraga, H.A. (1983). J. Phys. Chem. 87, 1883-1887. Nishikawa, T. & GS, N. (1987). Proteins, 2, 308-329. Noguti, T. & GS, N. (1983a). J. Phys. Soc. Japan, 52, 3283-3288. Noguti, T. & G5, N. (1983b). J. Phys. Soc. Japan, 52, 3685-3690. Noguti, T. & G5, N. (1989a). Proteins, 5, 97-103. Noguti, T. & GS, N. (1989b). Proteins, 5, 104-112. Noguti, T. & GS, N. (1989e). Proteins, 5, 113-124. Noguti, T. & GS, N. (1989d). Proteins, 5, 125-131. Noguti, T. & G5, N. (1989e). Proteins, 5, 132-138. Obara, S. & Kashiwagi, H. (1982). J. Chem. Phys. 77, 3155-3165. Parak, F., Knapp, E. W. & Kucheida, D. (1982). J. Mol. Biol. 161, 177-194. Perutz, M. F. (1965). J. Mol. Biol. 13, 646-668. Perutz, M. F. (1970). Nature (London), 228, 726-739. Phillips, S. E. (1980). J. Mol. Biol. 142, 531-554. Seno, Y. & G5, N. (1990). J. Mol. Biol. 216, lll-126. Shaanan, B. (1983). J. Mol. Biol. 171, 31-59. Shrake, A. & Rupley, J. A. (1973). J. Mol. Biol. 79, 351-371. Takano, T. (1977a). J. Mol. Biol. 110, 537-568. Takano, T. (1977b). J. Mol. Biol. 110, 569-584. Wako, H. & GS, N. (1987). J. Comp. Chem. 8, 625-635. Yoshioki, S., Abe, H., Noguti, T., G5, N. & Nagayama, K. (1983). J. Mol. Biol. 170, 1031-1036.
Edited by P. E. Wright