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Comparison of experimentally determined protein structures by solution of Bloch equations
Biochimica et Biophysica Acta 953 (1988) 61-69
61
Elsevier BBA 33073
Comparison of experimentally determined protein structures by solution of Bloch equations Marcela Madrid and Oleg Jardetzky Stanford Magnetic Resonance Laboratory, Stanford University, Stanford, CA (U.S.A.)
(Received 10 August 1987)
Key words: Protein structure prediction; NMR; Nuclear Overhauser enhancement spectroscopy; Bloch equation
Nuclear Overhauser enhancement (NOESY) spectra were theoretically generated by solving the generalized BIoch equations with the appropriate initial conditions. The input to the equations were the coordinates of the protons of two similar crystal structures of basic pancreatic trypsin inhibitor. The two NOESY spectra obtained were compared to published experimental spectra of the protein in solution. It was found that the two crystal structures of basic pancreatic trypsin inhibitor give different theoretical spectra. The solution of the Bloch equations is very sensitive to small variations in the distance between protons (approx. 0.2 A), and to differences in the surrounding configurations. The method allows a detailed comparison of the crystal and solution structures of proteins. The structure of the trypsin inhibitor in solution was found to be similar to either one or the other crystal forms in different regions of the molecule.
Introduction To an ever increasing extent, information on the solution structure of proteins is being derived from high-resolution nuclear magnetic resonance ( N M R ) data. However, important questions remain concerning the limits within which the structural interpretation of such data is valid. The most abundant source of structural information from N M R is the set of interatomic distances estimated from the intensities of the crosspeaks in a two-dimensional nuclear Overhauser enhancement (NOESY) experiment [1,2]. For rigid molecules at short mixing times, the intensity of a N O E S Y crosspeak is inversely proportional to the distance between the two nuclei to the sixth power [3]. For flexible molecules and longer mixing times, this simple relation does not apply. We have reCorrespondence: O. Jardetzky, Stanford Magnetic Resonance Laboratory, Stanford University, Stanford CA 94305-5055, U.S.A.
cently shown that even for mixing times as short as 50 ms, spin diffusion can play an important role in magnetization transfer and, hence, significantly affect the distance estimate [4,5]. A simple calibration of N O E S Y crosspeak intensities in terms of distances can lead to erroneous conclusions, and the solution of the generalized Bloch equations, starting from a known structure, may be required to approach an accurate interpretation of a N O E S Y spectrum [6-8]. This paper reports calculations defining the sensitivity of the N M R spectrum to variations in structure and the magnitude of errors which can be expected when a N O E S Y spectrum has to be interpreted without prior knowledge of the structure.
Methods The time evolution of a system of N spins is described by its density matrix [9]. This treatment is computationally prohibitive for systems of the size of proteins and can be approximated by the
0167-4838/88/$03.50 © 1988 Elsevier Science Publishers B.V. (Biomedical Division)
62
generalized Bloch equations [5,10]:
The intensity of a NOESY crosspeak is defined as
N
dm,/dt=-pimi- E~jmj j=l
(1)
FEll¸
I = ~
m;
where rn i are the deviations of the longitudinal magnetization from their thermal equilibrium values. The spin-lattice relaxation rate p, and crossrelaxation rate oo are calculated considering dipole-dipole interactions between unlike spins I and S [3] and: + ~J,~(~)
k (2) O,s = v]v2shZI( I +
1) ( - a~jo ( ~1 - Ws) + ~J2s( w, + Ws) }
(3) where ~01, ~os are the Larmor frequencies of the spins. The spectral density functions J~k are: 24r 6 J,*= ~ff-rik(l + o~2.r 2 )
J°(~):Jl(w):J2(~o)
= 6:1:4
(4)
The fluctuations in the interaction arise from the rotations of the molecule due to its Brownian motion. The correlation time for tumbling is given by the Stokes-Einstein relation: ~v r=~
(5)
where V = hydrated molecular volume, ~ = solvent viscosity and k is the Boltzmann constant * * We would like to thank Dr. D. McCain for suggesting that we place a stronger emphasis on the fact that the spectral density function ~k used in the calculation is at best a crude approximation to the unknown real function, and that errors in ~k could cause major errors in the calculated crosspeak intensities. This is indeed a fundamental point that deserves to be stressed, and we have discussed its implications at greater length in our previous publications (Refs. 2, 4, 5 and 7). From data in Table I he also calculated the average experimental crosspeak intensity to be 1.79% with the average calculated intensities of 2.55% for crystal form I and 2.60% for form II, and suggested that the calculated and experimental intensities could be brought into agreement if Eqn. 4 included an order parameter approximately equal to
(6)
where m ° is the thermal equilibrium value of the spin. If the coordinates of the atoms in the crystal structure are known, the time evolution of the spins can be simulated, taking into account spin diffusion, by integration of the generalized Bloch equations [10]. For a certain set of initial conditions of the spins, the solution of the Bloch equations gives the theoretical intensities of the NOESY crosspeaks [7]. The calculated crosspeaks of the protein in the solution state are compared to the experimental ones. Differences between theoretical and experimental spectra imply static or dynamic differences between the coordinates of the atoms in the crystal and solution states. The Bloch equations therefore provide a method to compare in detail the solution and crystal structure of proteins, provided that the crystal coordinates are known. The objective of the present study was to analyze the sensitivity of the method. We have addressed the question of whether two slightly different structures give appreciably different NOESY crosspeaks, and whether these differences can be experimentally observed. The method was tested on basic pancreatic trypsin inhibitor, a small globular protein of 58 residues. The protein grows in two different crystal forms, and the coordinates for both forms have been previously obtained by X-ray or neutron diffraction techniques [11,12]. The overall conformations of basic pancreatic trypsin inhibitor in both crystal forms are very similar, the rms deviation for main-chain atoms is 0.4 ,~, but larger
0.7. We find this calculation to be an excellent illustration of the extent to which internal motion attenuates magnetization transfer on the average and are including it here with his permission. For individual pairs of atoms, the attentuation could be smaller or greater, and the resulting uncertainty precludes the possibility of interpreting a single observed N O E in terms of a precise interatomic distance. The magnitude of the potential errors is discussed more fully in Ref. 5. S 2 =
63 deviations exist in several regions of the chain [12]. Two different N O E S Y spectra were theoretically generated by solving the Bloch equations from the coordinates of the protons of each of the crystal forms. All pathways of magnetization transfer were considered by °solving the equations for all protons within a 6 A distance from each other, for each molecular structure. The obtained spectra were compared to each other. In order to determine whether either one of the crystal forms or both of them satisfy the experimental data, the calculated spectra were compared to published N O E S Y experiments of basic pancreatic trypsin inhibitor in H 2 0 and 2H20 solutions [13]. To further establish the sensitivity of the Bloch equations to small variations in the distance between protons, the coordinates of one of the crystal forms were randomly varied from 0.3 to 0.8 A, and new spectra were calculated. For the basic pancreatic trypsin inhibitor molecule, Eqn. 1 represents a set of approx. 380 coupled equations. They were solved in a C R A Y Supercomputer, using Euler's integration method, with a step size of 2 ms. The input to the program were the Cartesian coordinates of the protons and the spectral frequency ~0 = 500 MHz. The correlation time was estimated from the molecular weight of the trypsin inhibitor, Mr 6500. The interactions between spins further than 6 apart were neglected to simplify the computational task. This approximation is amply justified by the fact that only protons at distances d < 3 ~, gave crosspeak intensities I > 1%. Each methyl group was considered as a single nucleus of spin I = 3/2, located at the average position of the three protons. Basic pancreatic trypsin inhibitor was chosen for this work because of the existence of known crystal structures, an almost complete set of N M R assignments, and extensive N O E S Y data [13].
been investigated by X-ray and neutron diffraction to 1 ,~ resolution for the X-ray and 1.8 resolution for the neutron data [12]. The major differences between the two crystal forms are (from Ref. 12): (1) Main-chain deviations of over 0.5 ,~ for C, atoms: 4 residues near the N terminus, between Lys-15 and Ile-19, Lys-26, Leu-29, between Arg-39 and Lys-41, between Ser-47 and Asp-50, 3 residues at the C terminus. (2) Side-chain deviations of over 1.0 ,~ are observed for Asp-3, Lys-15, Arg-17, Ile-19, Tyr-21, Lys-26, Leu-29, Arg-39, Lys-41, Arg-42, Glu-49, Asp-50, and both termini. (3) Side-chains reoriented in form II: Arg-17, Tyr21, Leu-29, Arg-39, Arg-42 and Glu-49. We have solved Eqn. 1 using as input the Cartesian coordinates of the protons from crystal forms I and II. The coordinates of the protons of the crystal form I were calculated from the heavy atoms positions determined by X-ray diffraction using the standard valences, angles and bond lengths for amino acids. To avoid van der Waals violations, the minimum allowed distance between protons was set at 1.8 .~. Only intensities I > 1% were considered. The parameters used in all the calculations were spectral frequency ~00 = 500
Results and Discussion
Fig. 1. Some of the NOESY crosspeaks (+) calculated by solving Eqn. 1 for crystal form I versus the interproton distance. The solid line is the NOESY intensity calculated from a two-spin approximation. The dispersion of the crosses from the solid line is a consequence of spin diffusion. The parameters used are to = 500 MHz and ~-=1.3 ns. Essentially the same plot was obtained using the NOESY intensities calculated for crystal form II.
The two crystal structures of basic pancreatic trypsin inhibitor, forms I and II, are similar. The three-dimensional structure of crystal form I has been determined by X-ray diffraction techniques to 1.5 ,~ resolution [11]. The crystal form II has
~+~ 2~
~
2~
2~
~
~
32~
3,2
~
64
MHz, temperature T = 68 ° C and tumbling correlation time ~-= 1.3 ns. The spectra were anlyzed at a mixing time t = 100 ms. The calculated NOESY spectra were compared to those measured at the aforementioned co0 and T by Wagner and Wi~thrich [13]. The frequency regions analyzed were those published with enough resolution as to allow a determination of crosspeak intensities: (3.6 to 5.4 p p m ) x (6.6 to 10.0 ppm), (6.6 to 8.7 ppm) x (6.6 to 8.7 ppm), (3.2) to 5.4 p p m ) x (6.6 to 9.2 ppm), (0 to 4.2 p p m ) x (0 to 5 ppm), and (2.4 to 3.7 ppm) x (6.5 to 7.5 ppm).
Comparison between the calculated spectra of the two crystal forms The theoretical NOESY spectra obtained for each of the two crystal forms consists of approx. 800 crosspeaks of intensities bigger than 1%, and are available from the authors upon request. Only protons at distances d < 3.3 .~ contribute to the crosspeaks at 100 ms mixing time. Most of the crosspeaks have similar intensities for both crystal forms; however, approx. 11% of them differ significantly in their intensities. Differences in the distance between protons for the two crystalforms involve practically all the residues along the basic pancreatic trypsin inhibitor chain. The distance differences relative to crystal form I range from 0.45% to 198% and give NOESY intensities that differ from 10.5% to 100%, respectively. Another factor contributing to the intensity differences between the two spectra is spin diffusion; the intensities depend not only on the distance between the two connected protons,
TABLE I T H E O R E T I C A L L Y P R E D I C T E D NOESY CROSSPEAKS T H A T W E R E N O T E X P E R I M E N T A L L Y OBSERVED Connectivity
HN HA HN HN HN HA
A s p 3 - H N Phe4 A r g 2 0 - H D Arg20 A r g 3 9 - H A Cys38 A r g 3 9 - H B Cys38 A r g 3 9 - H A Cys 38 A r g 4 2 - H G Arg42
but also on the surrounding configurations. As a consequence, pairs of protons with the same interproton distance might have different crosspeak intensities. Such is the case, to cite an example, of the protons HA Argl and 1HD Pro2; the distance between them is practically the same in the two crystal forms (2.22 and 2.23 ,~), but the corresponding crosspeak intensity is 10% larger in the spectra corresponding to crystal form II. The configurations surrounding the two protons include, in the crystal form I: HA Argl-2HD Pro2 separated by a distance d I = 2.01 A, and 1HD Pro22HD Pro2 at d 2 = 1.8 ,~ from each other. In the crystal form II the distance d 2 is the same, but d 1 = 1.8 A. The decrease in dl increases the interaction HA Argl-2HD Pro2, and 2HD Proc2 becomes a more effective pathway of magnetization transfer to 1 H D Pro2. The NOESY crosspeaks calculated from crystal form I have been plotted against the interproton distance in Fig. 1. The atoms that represent methyl groups have been excluded from the figure for the purpose of intensity comparisons, since their larger crosspeak intensities are due to their 3-times larger spin, as explained in Methods. Essentially, the same graph was obtained from the calculated intensities of crystal form II. The error, E, in the theoretical data, resulting from the integration of the Bloch equations by the method of Euler with a stepsize of 2 ms is E < 1%, which is of the size of the data symbols in the figure for the largest intensities shown, and smaller than the symbols otherwise. The 10-20% dispersion observed in the intensity for a given distance is a consequence of spin diffusion, the effect is more significant if one of the connected protons is very close (d--- 1.8 A) to a third proton. The solid line in Fig. 1 represents the NOESY intensity that would be obtained if only two spins were present. It was calculated solving Eqn. 1 for two spins:
Crosspeak intensity (%) form I
form II
1.1 1.4 4.7 1.8 4.2 4.0
1.4 2.0 2.7 1.7 2.7 1.1
m(t) = -1/2 e -(p+°)'+1/2 e -(p-o)'
(7)
It is seen that for d < 3 ,~ the intensity calculated from the generalized Bloch equations is smaller than the two-spin predictions. The discrepancy between the exact solution and the two-spin approximation corresponds to differences in inter-
65
T A B L E II T H E O R E T I C A L L Y C A L C U L A T E D NOESY C R O S S P E A K INTENSITIES
Theoretically calculated NOESY crosspeak intensities for crystal forms I and II, distances between the connected protons and estimated experimental intensities. The errors in the experimental intensities are qualitative estimates. The data are consistent with a solution structure of basic pancreatic trypsin inhibitor in which different regions of the molecule are similar to either crystal forms I or II. In addition to the N O E s reported in the table, a substantial n u m b e r of N O E s was calculated which would not be unequivocally indentified in the experimental spectrum because of line overlap. These additional calculated values are available to any interested readers from the authors. Connectivity
Theoretical
Experi-
intensity (%)
H N Phe4-HD2 Phe4 H A Phe4-HG Glu7 1 HB Phe4-HD2 Phe4 H N Cys5-HB Cys5 HB Cys5-HA Cys5 H A G l u 7 - H D Pro8 H A Pro8-HD Pro9 H A Pro9-HN T y r l 0 HA Pro9-HB Pro9 HA T y r l 0 - H D 1 T y r l 0 HA T h r l 1-HE1 Phe33 1 HB T h r l l - H N T h r l l HG1 T h r l 1-HA Thrl 1 HA Cysl4-HB C y s l 4 HA A r g l 7 - H G Val34 H D Arg20-HA Arg20 HD1 Tyr21-2HB Tyr21 HE1 Phe22-HA Asn24 H A Tyr23-HD2 Tyr23 H A Asn24-HN Ala25 H D 2 Asn24-HE2 Gin31 H N Lys26-HN Ala27 H A GIy28-HN Gly28 H A Leu29-HB Leu29 H N Cys30-HA Leu29 H A Gln31-1HB Gln31 HE2 Phe33-HA Asn44 H A Tyr35-HN Gly36 HB Tyr35-HA Tyr35 H D 2 Tyr35-HE2 Tyr35 HE1 Tyr35-HA Ala40 HA Cys38-HN Arg39 H A Cys38-1HB Cys38 H A Cys38-2HB Cys38 H A Lys41-HG Lys41 H A Arg42-HB Arg42 H A Arg42-HG Arg42 H N Asn43-HB Asn43 HB Phe45-HN Ser47 HB Ser47-HN Gin49 H A AIa48-HB Ala48 H A GIu49-HG Ghi49 H A Met52-HB Met 52 H N Arg53-HNThr54 H N Thr54-HB Thr54 H N Thr54-HG1 Thr54 H N AIa58-HA Ala58 a n.s., not
seen.
distance (,~)
mental
form I
form II
form I
form II
1.1 2.4 3.1 2.6 2.7 2.7 1.9 3.9 3.1 1.0 1.2 < 1.0 1.0 3.2 1.2 4.2 2.2 < 1.0 1.4 4.4 9.9 1.6 4.3 3.4 2.4 3.1 1.0 4.2 2.1 3.2 1.0 3.9 3.1 3.8 2.5 2.7 4.1 1.9 1.7 < 1.0 7.2 3.2 2.6 2.0 <1.0 <1.0 2.0
2.1 1.9 1.2 5.3 1.4 1.7 2.5 13.3 1.4 2.3 2.4 4.1 3.2 1.8 2.8 1.6 1.4 2.5 3.4 2.8 < 1.0 2.7 2.5 <1.0 4.8 2.0 2.1 2.4 6.2 2.7 2.4 2.7 2.6 2.5 1.6 1.6 2.0 3.0 2.4 1.7 3.6 1.4 3.8 1.2 2.5 3.8 < 1.0
2.84 2.41 2.33 2.39 2.38 2.43 2.59 2.27 2.33 2.89 2.77 3.47 2.93 2.33 2.77 2.18 2.46 5.84 2.70 2.22 1.80 2.61 2.19 2.31 2.45 2.33 2.84 2.20 2.35 2.36 2.79 2.22 2.30 2.24 2.41 2.39 2.21 2.57 2.55 2.92 2.31 2.34 2.42 2.55 3.47 3.57 2.49
2.53 2.51 2.72 2.11 2.72 2.70 2.44 1.80 2.68 2.51 2.46 2.24 2.35 2.59 2.38 2.60 2.66 2.46 2.32 2.40 3.04 or 4.17 2.40 2.42 3.00 2.18 2.54 2.53 2.43 2.03 2.44 2.46 2.42 2.45 2.47 2.59 2.62 2.53 2.37 2.42 2.55 2.62 2.71 2.22 2.76 2.43 2.28 2.89
2.3 _+0.5 1.5 + 0.5 1.1 +0.2 4 _+1 1 +0.2 1.3 + 0.4 1.5 _+0.4 3.5 + 0.9 1 +0.2 2.5 + 0.8 1.2 + 0.3 1.5 + 0.5 1.3 + 0.3 1 +0.4 2 +1 n.s. a 3 +0.5 2.5 +0.5 1.5 + 0.4 4 + 1 3.5 + 1.5 2 +0.8 2 + 0.5 1 +0.3 4.5 _+ 1 1.7 + 0.5 3 + 1 2.3 + 1 1.5 + 0.2 2.5 + 0.5 2.0 + 0.4 n.s. 1.5 + 0.5 1.0 + 0.3 1.0 + 0.5 1 + 0.6 n.s. 2 +0.3 0.8 + 0.2 0.9 + 0.2 2 +1 1.3 -1-0.3 1.5 + 0.5 1.7 + 0.4 3 +0.5 3 +0.5 2 +0.3
66 proton distances that are smaller than 0.25 A. These results are inherent in the small correlation time and mixing time used in the calculations; bigger disagreements are expected for larger values of ~- or t (Lane, A.N., personal communication).
Comparison with the experimental data The intensities of the experimental crosspeaks were estimated from the size of each outermost ring. It was found that, for isolated crosspeaks, intensity differences as small as 20-50%, depending on the particular crosspeak, were easily discernible. The following two factors, however, preclude in some cases a precise intensity estimate: (1) protons that have very close resonance frequencies produce crosspeaks that are too close to the diagonal of the NOESY spectra; (2) pairs of protons that produce crosspeaks in approximately the same frequency region result in a superposition of crosspeaks. In the present study, the comparison between experimental and theoretical spectra was further limited by the lack of N M R resonance assignments for the residues Arg-1, Pro-2 and Pro-3, and some side-chain atoms of the basic pancreatic trypsin inhibitor molecule. The presence of overlapping crosspeaks and lack of these resonance assignments prevent us from determining whether there are connectivities in the experimental data that have not been theoretically predicted. We can however, discuss the opposite case. There are some connectivities that have been theoretically predicted but are not experimentally observed. Their calculated intensities for the two crystal forms are detailed in Table I. The absence of calculated crosspeaks implies that the atoms are further away from each other in solution than in the crystal state. A single-time, single-temperature NOESY experiment does not provide enough information to decide whether the atoms are farther apart because of a static deformation or dynamical fluctuations. The protons that occupy similar positions in the two crystal structures produce crosspeaks of close intensities for the two crystal forms. These crosspeaks compose the majority of the calculated spectra, and they satisfy the experimental data, with the few exceptions listed in Table I.
Table II details the theoretical calculated NOESY crosspeak intensities and the distance between the connected protons for the two crystal forms. Two possible distances are given when it was not possible to decide between two equivalent protons, for example, between two HB protons. Only those crosspeaks that differ significantly in intensity between the two crystal forms and that could be compared with experimental results are listed. There are approx. 100 additional calculated crosspeaks that have different intensities for each of the crystal forms, but for them an estimate of the experimental intensity was not possible because of the aforementioned limitations. In the last column of Table II we show the estimated experimental intensities with the purpose of analyzing whether they agree with those calculated for crystal forms I or II. The errors in the intensities shown are qualitative estimates. The information that can be extracted about the distance between protons in the solution state and those measured in the two crystal states is given in Table III. Since there are very few disagreements between theory and experiment, the data do not support the hypothesis of a solution structure exchanging between several significantly different conformers. Instead, the solution structure of basic pancreatic trypsin inhibitor appears to be very close to that of the crystal states. There are in principle two possibilities that should be distinguished. (1) The solution state might be a mixture, with different populations, of molecules of crystal forms I and II. As a special case, the population of one of the crystal forms can be zero, and the solution structure will be equal to just one of the crystal forms. The molecules can be exchanging between the two configurations at a slow or fast rate compared to the N M R time scale. A slow exchanging rate would give crosspeaks corresponding to both crystal forms. A fast exchange rate would give a NOESY spectrum corresponding to a weighted average of the distances between the two crystal forms. (2) The molecule of basic pancreatic trypsin inhibitor might have clusters of atoms whose configuration is closer to that of crystal form I, and other regions closer to crystal form II. Slow exchange between the two crystal forms is
67 TABLE III Distance between protons Argl Pro2 HN-HD2 Phe4 HA Phe4-HG Glu7 HB-HD2 Phe4 HN-HB Cys5 HB-HA Cys5 HA Glu7-HD Pro8 HA Pro8-HD Pro9 HA Pro9-HN Tyrl0 HA-HB Pro9 HA-HD1 Tyrl0 HA Thrll-HE1 Phe33 1 HB-HN Thrll HG1-HA Thrll HA-HB Cysl4 HA Argl7-HG Va134 HD1-2HB Tyr21 HE1 Phe22-HA Asn24 HA-HD2 Tyr23 HA Asn24-HN Ala25 HD2 Asn24-HE2 GLn31 HN Lys26-HN Ala27 HA-HN Gly28 HA-HB Leu29 HN Cys30HA Leu29 HA-1HB Gln31 HE2 Phe33-HA Asn44 HA Tyr35-HN Gly36 HB-HA Tyr35 HD2-HE2 Tyr35 HE1 Tyr35-HA Ala40 HA-1HB Cys38 HA-2HB Cys38 HA-HG Lys41 HA-HB Arg42 HN-HB Asn43 HB Phe45-HN Ser47 HB Ser47-HN Glu49 HA-HG Glu49 HA-HB Met52 HN-HB Thr54 HN-HA Ala58
Appear to be closer to crystal a
a form II form II form II form If form II form II form I form I form II form II form I form I form I form II form If b form II form I form I _c a form II form II form II form II form II form II form I form II form II _c - c form II form II form I b form I form II b
form II form lI
No comparison available. b Experimental intensity is larger than predicted, but closer 10 form II. c No agreement with experiment. d Experimental intensity is within the predictions for both crystal forms. e Experimental intensity smaller than predicted. a
n o t consistent with our results, because, in this case, we w o u l d expect to see m o r e e x p e r i m e n t a l crosspeaks that c a n n o t be p r e d i c t e d f r o m our calculations. The results are consistent with a solution structure where several regions of the molecule are closer to either one or the o t h e r crystal form. If r a p i d exchange occurs, then the weighted average in some regions of the molecule is closer to structure I a n d in o t h e r regions is closer to structure II.
Sensitivity of the theory to small variations in distance T h e question arises as to how m u c h should the p o s i t i o n of an a t o m in the crystal differ from that in the solution to observe an a p p r e c i a b l e difference b e t w e e n the c a l c u l a t e d a n d m e a s u r e d spectra. To illustrate the sensitivity of the m e t h o d , the c o o r d i n a t e s o f the a t o m s of crystal f o r m II were r a n d o m l y altered, a n d new spectra were c a l c u l a t e d using these c o o r d i n a t e s as input. Some of the results were p l o t t e d to simulate e x p e r i m e n tal spectra. Fig. 2a (region b e l o w the d i a g o n a l ) shows the N O E S Y crosspeaks c o r r e s p o n d i n g to H B Cys5, as c a l c u l a t e d from the crystal c o o r d i nates. Fig. 2b (region a b o v e the d i a g o n a l ) shows the c a l c u l a t e d intensities after c h a n g i n g the x coo r d i n a t e of HB Cys5 b y 0.3 .~. T h e s a m e c o m p a r i s o n is m a d e in Fig. 3 between the H A Ile19 crosspeaks of the crystal (Fig. 3a) a n d the c a l c u l a t e d crosspeaks after m o d i f y i n g the x c o o r d i n a t e of H A Ile19 b y 0.8 ,~. If the c h a n g e in c o o r d i n a t e s h a d no effect on the spectra, Figs. 2 a n d 3 w o u l d b e s y m m e t r i c with respect of the diagonal. Instead, the lack of s y m m e t r y shows that differences in the a t o m s ' c o o r d i n a t e s as small as 0.3 ~, p r o d u c e discernible differences in the spectra. In s o m e cases, the intensities of the c r o s s p e a k s are m a r k e d l y different, in others, there a p p e a r new N O E S Y crosspeaks. N O E S Y cross p e a k s are observed, at 100 ms, between 1 HB Cys5 a n d the a t o m s 2 H B Cys5 a n d H N Cys 5. A c h a n g e of 0.3 ,~ in the x c o o r d i n a t e of 1 HB Cys5 m o v e s the p r o t o n closer to H N Cys5 a n d farther a w a y from 2 H B Cys5 b y a p p r o x . 0.25 A, dou~ b l i n g the 1 of one of the crosspeaks a n d r e d u c i n g the other to one half of its initial I. In the case of H A Asn43, a change of 0.8 ~, in the x c o o r d i n a t e increases the intensities of crosspeaks that h a d
68
b
I 5
10
r~l
('~11(ppm)
[]
"HNCys5
a t lO ~O (.D2(ppm) "r rn
Fig. 2. Effect of changing the x coordinate of HB Cys5 by 0.3 ,~. The NOESY connectivities below the diagonal (a) were calculated from the coordinates of crystal form II, and the upper region (b) corresponds to the changed coordinates. The sensitivity of the theory to a small change in the coordinates of the protons is reflected in the lack of symmetry with respect to the diagonal of the spectra.
b
- ~
~D/
10
(~I(ppm)
-5
5
C02(ppm)
Fig. 3. Effect of changing the x coordinate of HA lle19 by 0.8 ,~ on its NOESY connectivities. The spectra below the diagonal (A) were calculated from the coordinates of crystal form II, and the upper region (b) corresponds to the changed coordinate.
I < 1% in the spectra calculated from the crystal coordinates: HA Asn43-HD Phe4 and HA Asn43H N Asn43. The crosspeak HA Asn43-HE1 Phe45 had I = 1.6% in the original spectra; this crosspeak disappears after modifying the coordinate. Changes in I are also observed for HA Asn43-HB Asn43 and HA Asn43-HN Asn44 (Fig. 3). Approximately the same effects detailed here were observed for all the changed coordinates in the molecule. The exact effect of a distance variation in the NOESY intensity depends on the surroundings of the atom and on the mixing time. Conclusions The NOESY spectra generated by solving the generalized Bloch equations are very sensitive to small (of the order of 0.2 ,~) differences in the distance between protons. Because the equations take into account spin diffusion, the solution is also sensitive to the surrounding configuration of protons. The two spectra obtained from slightly different crystal structures of basic pancreatic trypsin inhibitor reflect the differences in interproton distances existing between the two crystals. Approx. 11% of the crosspeaks calculated have substantial intensity differences that could be experimentally discernible, under ideal experimental conditions. A difference in the interproton distance of 0.2 ,~ between the two crystal structures produces a relative change in the crosspeak intensity of approx. 63%. The exact intensity depends, however, on the geometry of the surrounding protons. Depending on the extent of spin diffusion, there might be a 10-20% variation in the crosspeak intensity for the same interproton distance. If the interproton distance is calculated from the observed crosspeak intensity by means of the two spin approximation, the obtained distance would be larger than the actual. The difference in the distance estimate, at 100 ms mixing time and ~-= 1.3 ns, is = 0.1-0.25 ,~. The comparison between theoretical and experimental spectra gives some detailed information on the similarities or discrepancies between the solution and crystal states of basic pancreatic trypsin inhibitor. For those regions of the molecule where the two crystal structures are very
69 similar, a single spectrum was obtained as a result of the calculations, and very good agreement was found with the experimental data. For those regions of the molecule where there exist substantial differences between the crystal forms, the experimental crosspeak intensities were, in general, close to the calculated intensities of either one or the other of the crystal forms. These results are consistent with a solution structure of basic pancreatic trypsin inhibitor in which different regions of the molecule are alternatively similar to the crystal form I or to the crystal form II. Six connectivities predicted by the two crystal spectra were not found in the experimental data. These were the only discrepancies found in the frequency region of the spectra studied; they indicate that the position of these atoms in the solution state differ from those measured in the crystals. The atoms are all located near the surface of the basic pancreatic trypsin inhibitor, and are therefore more susceptible to static or dynamic distortions because they experience weaker packing forces. Measuring the spectra as a function of temperature could help distinguish between the static and dynamic possibilities, since dynamic fluctuations are expected to have a much stronger temperature dependence than static distortions. Some of the information obtained in the theoretical simulation of NOESY spectra are unfortunately masked in the experimental data because of unresolved resonances or overlapping crosspeaks. In these cases, it was found that an experimental NOESY spectrum at a single mixing time is not enough to discriminate between two close structures, or to decide whether the experimental data present connectivities that were not predicted by the theory. Additional information must be obtained by following the time evolution of those spins that produce the unresolved connectivities and comparing it with theoretical simulations of one-dimensional NOEs.
Acknowledgements The authors are gateful to Dr. Andrew N. Lane for the preliminary stages of the computer simulations and for many helpful discussions; to James R. Williamson for his H Y D R O program, and to Drs. Olivier Lichtarge and Robert L. Domenick for interesting remarks. This work was supported by N I H grant number RR02300.
References 1 Wiithrich, K. (1986) NMR of Proteins and Nucleic Acids, John Wiley& Sons, New York. 2 Jardetzky, O. and Roberts, G.C.K. (1981) NMR in Molecular Biology,Academic Press, New York. 3 Abragam, A. (1961) The Principles of Nuclear Magnetism, Clarendon, Oxford. 4 Jardetzky, O., Lane, A., Lefrvre, J.F., Lichtarge, O., Hayes-Roth, B., Altman, R. and Buchanan, B. (1986) in Proceedings XXIII Congress Ampere on Magnetic Resonance (Maraviglia, B., De Luca, F. and Campanella, R., eds.), pp. 64-69, Rome, Italy. 5 Jardetzky, O. and Lane, A. (1988) in Proceedings of the International School of Physics, July 8-18, 1986, Enrico Fermi, Varenna, Italy, in press. 6 Gariepy, J., Lane, A., Frayman, F., Wilbur, D., Robien, W., Schoolnik, G.K. and Jardetzky, O. (1986) Biochemistry25, 7854-7866. 7 Lef~vre, J.F., Lane, A.N. and Jardetzky, O. (1987) Biochemistry 26, 5076-5090. 8 Duncan, B., Buchanan, B.G., Hayes-Roth, B., Lichtarge, O., Altman, R., Brinkley, J., Hewett, M., Cornelius, C. and Jardetzky, O. (1986) Bull. Mag. Res. 8, 111-119. 9 Bothner-By,A. and Noggle, J.H. (1979) J. Am. Chem. Soc. 101, 5152. 10 Kalk, A. and Berendsen, H.J.C. (1976) J. Mag. Res. 24, 343. 11 Deisenhofer, J. and Steigemann, W. (1975) Acta Crystallogr., Sect. B, 31,238. 12 Wlodawer,A., Walter, J., Buber, R. and Sjolin, L. (1984) J. Mol. Biol. 182, 295-315. 13 Wagner, G. and Wiithrich, K. (1982) J. Mol. Biol. 155, 347-366.
61
Elsevier BBA 33073
Comparison of experimentally determined protein structures by solution of Bloch equations Marcela Madrid and Oleg Jardetzky Stanford Magnetic Resonance Laboratory, Stanford University, Stanford, CA (U.S.A.)
(Received 10 August 1987)
Key words: Protein structure prediction; NMR; Nuclear Overhauser enhancement spectroscopy; Bloch equation
Nuclear Overhauser enhancement (NOESY) spectra were theoretically generated by solving the generalized BIoch equations with the appropriate initial conditions. The input to the equations were the coordinates of the protons of two similar crystal structures of basic pancreatic trypsin inhibitor. The two NOESY spectra obtained were compared to published experimental spectra of the protein in solution. It was found that the two crystal structures of basic pancreatic trypsin inhibitor give different theoretical spectra. The solution of the Bloch equations is very sensitive to small variations in the distance between protons (approx. 0.2 A), and to differences in the surrounding configurations. The method allows a detailed comparison of the crystal and solution structures of proteins. The structure of the trypsin inhibitor in solution was found to be similar to either one or the other crystal forms in different regions of the molecule.
Introduction To an ever increasing extent, information on the solution structure of proteins is being derived from high-resolution nuclear magnetic resonance ( N M R ) data. However, important questions remain concerning the limits within which the structural interpretation of such data is valid. The most abundant source of structural information from N M R is the set of interatomic distances estimated from the intensities of the crosspeaks in a two-dimensional nuclear Overhauser enhancement (NOESY) experiment [1,2]. For rigid molecules at short mixing times, the intensity of a N O E S Y crosspeak is inversely proportional to the distance between the two nuclei to the sixth power [3]. For flexible molecules and longer mixing times, this simple relation does not apply. We have reCorrespondence: O. Jardetzky, Stanford Magnetic Resonance Laboratory, Stanford University, Stanford CA 94305-5055, U.S.A.
cently shown that even for mixing times as short as 50 ms, spin diffusion can play an important role in magnetization transfer and, hence, significantly affect the distance estimate [4,5]. A simple calibration of N O E S Y crosspeak intensities in terms of distances can lead to erroneous conclusions, and the solution of the generalized Bloch equations, starting from a known structure, may be required to approach an accurate interpretation of a N O E S Y spectrum [6-8]. This paper reports calculations defining the sensitivity of the N M R spectrum to variations in structure and the magnitude of errors which can be expected when a N O E S Y spectrum has to be interpreted without prior knowledge of the structure.
Methods The time evolution of a system of N spins is described by its density matrix [9]. This treatment is computationally prohibitive for systems of the size of proteins and can be approximated by the
0167-4838/88/$03.50 © 1988 Elsevier Science Publishers B.V. (Biomedical Division)
62
generalized Bloch equations [5,10]:
The intensity of a NOESY crosspeak is defined as
N
dm,/dt=-pimi- E~jmj j=l
(1)
FEll¸
I = ~
m;
where rn i are the deviations of the longitudinal magnetization from their thermal equilibrium values. The spin-lattice relaxation rate p, and crossrelaxation rate oo are calculated considering dipole-dipole interactions between unlike spins I and S [3] and: + ~J,~(~)
k (2) O,s = v]v2shZI( I +
1) ( - a~jo ( ~1 - Ws) + ~J2s( w, + Ws) }
(3) where ~01, ~os are the Larmor frequencies of the spins. The spectral density functions J~k are: 24r 6 J,*= ~ff-rik(l + o~2.r 2 )
J°(~):Jl(w):J2(~o)
= 6:1:4
(4)
The fluctuations in the interaction arise from the rotations of the molecule due to its Brownian motion. The correlation time for tumbling is given by the Stokes-Einstein relation: ~v r=~
(5)
where V = hydrated molecular volume, ~ = solvent viscosity and k is the Boltzmann constant * * We would like to thank Dr. D. McCain for suggesting that we place a stronger emphasis on the fact that the spectral density function ~k used in the calculation is at best a crude approximation to the unknown real function, and that errors in ~k could cause major errors in the calculated crosspeak intensities. This is indeed a fundamental point that deserves to be stressed, and we have discussed its implications at greater length in our previous publications (Refs. 2, 4, 5 and 7). From data in Table I he also calculated the average experimental crosspeak intensity to be 1.79% with the average calculated intensities of 2.55% for crystal form I and 2.60% for form II, and suggested that the calculated and experimental intensities could be brought into agreement if Eqn. 4 included an order parameter approximately equal to
(6)
where m ° is the thermal equilibrium value of the spin. If the coordinates of the atoms in the crystal structure are known, the time evolution of the spins can be simulated, taking into account spin diffusion, by integration of the generalized Bloch equations [10]. For a certain set of initial conditions of the spins, the solution of the Bloch equations gives the theoretical intensities of the NOESY crosspeaks [7]. The calculated crosspeaks of the protein in the solution state are compared to the experimental ones. Differences between theoretical and experimental spectra imply static or dynamic differences between the coordinates of the atoms in the crystal and solution states. The Bloch equations therefore provide a method to compare in detail the solution and crystal structure of proteins, provided that the crystal coordinates are known. The objective of the present study was to analyze the sensitivity of the method. We have addressed the question of whether two slightly different structures give appreciably different NOESY crosspeaks, and whether these differences can be experimentally observed. The method was tested on basic pancreatic trypsin inhibitor, a small globular protein of 58 residues. The protein grows in two different crystal forms, and the coordinates for both forms have been previously obtained by X-ray or neutron diffraction techniques [11,12]. The overall conformations of basic pancreatic trypsin inhibitor in both crystal forms are very similar, the rms deviation for main-chain atoms is 0.4 ,~, but larger
0.7. We find this calculation to be an excellent illustration of the extent to which internal motion attenuates magnetization transfer on the average and are including it here with his permission. For individual pairs of atoms, the attentuation could be smaller or greater, and the resulting uncertainty precludes the possibility of interpreting a single observed N O E in terms of a precise interatomic distance. The magnitude of the potential errors is discussed more fully in Ref. 5. S 2 =
63 deviations exist in several regions of the chain [12]. Two different N O E S Y spectra were theoretically generated by solving the Bloch equations from the coordinates of the protons of each of the crystal forms. All pathways of magnetization transfer were considered by °solving the equations for all protons within a 6 A distance from each other, for each molecular structure. The obtained spectra were compared to each other. In order to determine whether either one of the crystal forms or both of them satisfy the experimental data, the calculated spectra were compared to published N O E S Y experiments of basic pancreatic trypsin inhibitor in H 2 0 and 2H20 solutions [13]. To further establish the sensitivity of the Bloch equations to small variations in the distance between protons, the coordinates of one of the crystal forms were randomly varied from 0.3 to 0.8 A, and new spectra were calculated. For the basic pancreatic trypsin inhibitor molecule, Eqn. 1 represents a set of approx. 380 coupled equations. They were solved in a C R A Y Supercomputer, using Euler's integration method, with a step size of 2 ms. The input to the program were the Cartesian coordinates of the protons and the spectral frequency ~0 = 500 MHz. The correlation time was estimated from the molecular weight of the trypsin inhibitor, Mr 6500. The interactions between spins further than 6 apart were neglected to simplify the computational task. This approximation is amply justified by the fact that only protons at distances d < 3 ~, gave crosspeak intensities I > 1%. Each methyl group was considered as a single nucleus of spin I = 3/2, located at the average position of the three protons. Basic pancreatic trypsin inhibitor was chosen for this work because of the existence of known crystal structures, an almost complete set of N M R assignments, and extensive N O E S Y data [13].
been investigated by X-ray and neutron diffraction to 1 ,~ resolution for the X-ray and 1.8 resolution for the neutron data [12]. The major differences between the two crystal forms are (from Ref. 12): (1) Main-chain deviations of over 0.5 ,~ for C, atoms: 4 residues near the N terminus, between Lys-15 and Ile-19, Lys-26, Leu-29, between Arg-39 and Lys-41, between Ser-47 and Asp-50, 3 residues at the C terminus. (2) Side-chain deviations of over 1.0 ,~ are observed for Asp-3, Lys-15, Arg-17, Ile-19, Tyr-21, Lys-26, Leu-29, Arg-39, Lys-41, Arg-42, Glu-49, Asp-50, and both termini. (3) Side-chains reoriented in form II: Arg-17, Tyr21, Leu-29, Arg-39, Arg-42 and Glu-49. We have solved Eqn. 1 using as input the Cartesian coordinates of the protons from crystal forms I and II. The coordinates of the protons of the crystal form I were calculated from the heavy atoms positions determined by X-ray diffraction using the standard valences, angles and bond lengths for amino acids. To avoid van der Waals violations, the minimum allowed distance between protons was set at 1.8 .~. Only intensities I > 1% were considered. The parameters used in all the calculations were spectral frequency ~00 = 500
Results and Discussion
Fig. 1. Some of the NOESY crosspeaks (+) calculated by solving Eqn. 1 for crystal form I versus the interproton distance. The solid line is the NOESY intensity calculated from a two-spin approximation. The dispersion of the crosses from the solid line is a consequence of spin diffusion. The parameters used are to = 500 MHz and ~-=1.3 ns. Essentially the same plot was obtained using the NOESY intensities calculated for crystal form II.
The two crystal structures of basic pancreatic trypsin inhibitor, forms I and II, are similar. The three-dimensional structure of crystal form I has been determined by X-ray diffraction techniques to 1.5 ,~ resolution [11]. The crystal form II has
~+~ 2~
~
2~
2~
~
~
32~
3,2
~
64
MHz, temperature T = 68 ° C and tumbling correlation time ~-= 1.3 ns. The spectra were anlyzed at a mixing time t = 100 ms. The calculated NOESY spectra were compared to those measured at the aforementioned co0 and T by Wagner and Wi~thrich [13]. The frequency regions analyzed were those published with enough resolution as to allow a determination of crosspeak intensities: (3.6 to 5.4 p p m ) x (6.6 to 10.0 ppm), (6.6 to 8.7 ppm) x (6.6 to 8.7 ppm), (3.2) to 5.4 p p m ) x (6.6 to 9.2 ppm), (0 to 4.2 p p m ) x (0 to 5 ppm), and (2.4 to 3.7 ppm) x (6.5 to 7.5 ppm).
Comparison between the calculated spectra of the two crystal forms The theoretical NOESY spectra obtained for each of the two crystal forms consists of approx. 800 crosspeaks of intensities bigger than 1%, and are available from the authors upon request. Only protons at distances d < 3.3 .~ contribute to the crosspeaks at 100 ms mixing time. Most of the crosspeaks have similar intensities for both crystal forms; however, approx. 11% of them differ significantly in their intensities. Differences in the distance between protons for the two crystalforms involve practically all the residues along the basic pancreatic trypsin inhibitor chain. The distance differences relative to crystal form I range from 0.45% to 198% and give NOESY intensities that differ from 10.5% to 100%, respectively. Another factor contributing to the intensity differences between the two spectra is spin diffusion; the intensities depend not only on the distance between the two connected protons,
TABLE I T H E O R E T I C A L L Y P R E D I C T E D NOESY CROSSPEAKS T H A T W E R E N O T E X P E R I M E N T A L L Y OBSERVED Connectivity
HN HA HN HN HN HA
A s p 3 - H N Phe4 A r g 2 0 - H D Arg20 A r g 3 9 - H A Cys38 A r g 3 9 - H B Cys38 A r g 3 9 - H A Cys 38 A r g 4 2 - H G Arg42
but also on the surrounding configurations. As a consequence, pairs of protons with the same interproton distance might have different crosspeak intensities. Such is the case, to cite an example, of the protons HA Argl and 1HD Pro2; the distance between them is practically the same in the two crystal forms (2.22 and 2.23 ,~), but the corresponding crosspeak intensity is 10% larger in the spectra corresponding to crystal form II. The configurations surrounding the two protons include, in the crystal form I: HA Argl-2HD Pro2 separated by a distance d I = 2.01 A, and 1HD Pro22HD Pro2 at d 2 = 1.8 ,~ from each other. In the crystal form II the distance d 2 is the same, but d 1 = 1.8 A. The decrease in dl increases the interaction HA Argl-2HD Pro2, and 2HD Proc2 becomes a more effective pathway of magnetization transfer to 1 H D Pro2. The NOESY crosspeaks calculated from crystal form I have been plotted against the interproton distance in Fig. 1. The atoms that represent methyl groups have been excluded from the figure for the purpose of intensity comparisons, since their larger crosspeak intensities are due to their 3-times larger spin, as explained in Methods. Essentially, the same graph was obtained from the calculated intensities of crystal form II. The error, E, in the theoretical data, resulting from the integration of the Bloch equations by the method of Euler with a stepsize of 2 ms is E < 1%, which is of the size of the data symbols in the figure for the largest intensities shown, and smaller than the symbols otherwise. The 10-20% dispersion observed in the intensity for a given distance is a consequence of spin diffusion, the effect is more significant if one of the connected protons is very close (d--- 1.8 A) to a third proton. The solid line in Fig. 1 represents the NOESY intensity that would be obtained if only two spins were present. It was calculated solving Eqn. 1 for two spins:
Crosspeak intensity (%) form I
form II
1.1 1.4 4.7 1.8 4.2 4.0
1.4 2.0 2.7 1.7 2.7 1.1
m(t) = -1/2 e -(p+°)'+1/2 e -(p-o)'
(7)
It is seen that for d < 3 ,~ the intensity calculated from the generalized Bloch equations is smaller than the two-spin predictions. The discrepancy between the exact solution and the two-spin approximation corresponds to differences in inter-
65
T A B L E II T H E O R E T I C A L L Y C A L C U L A T E D NOESY C R O S S P E A K INTENSITIES
Theoretically calculated NOESY crosspeak intensities for crystal forms I and II, distances between the connected protons and estimated experimental intensities. The errors in the experimental intensities are qualitative estimates. The data are consistent with a solution structure of basic pancreatic trypsin inhibitor in which different regions of the molecule are similar to either crystal forms I or II. In addition to the N O E s reported in the table, a substantial n u m b e r of N O E s was calculated which would not be unequivocally indentified in the experimental spectrum because of line overlap. These additional calculated values are available to any interested readers from the authors. Connectivity
Theoretical
Experi-
intensity (%)
H N Phe4-HD2 Phe4 H A Phe4-HG Glu7 1 HB Phe4-HD2 Phe4 H N Cys5-HB Cys5 HB Cys5-HA Cys5 H A G l u 7 - H D Pro8 H A Pro8-HD Pro9 H A Pro9-HN T y r l 0 HA Pro9-HB Pro9 HA T y r l 0 - H D 1 T y r l 0 HA T h r l 1-HE1 Phe33 1 HB T h r l l - H N T h r l l HG1 T h r l 1-HA Thrl 1 HA Cysl4-HB C y s l 4 HA A r g l 7 - H G Val34 H D Arg20-HA Arg20 HD1 Tyr21-2HB Tyr21 HE1 Phe22-HA Asn24 H A Tyr23-HD2 Tyr23 H A Asn24-HN Ala25 H D 2 Asn24-HE2 Gin31 H N Lys26-HN Ala27 H A GIy28-HN Gly28 H A Leu29-HB Leu29 H N Cys30-HA Leu29 H A Gln31-1HB Gln31 HE2 Phe33-HA Asn44 H A Tyr35-HN Gly36 HB Tyr35-HA Tyr35 H D 2 Tyr35-HE2 Tyr35 HE1 Tyr35-HA Ala40 HA Cys38-HN Arg39 H A Cys38-1HB Cys38 H A Cys38-2HB Cys38 H A Lys41-HG Lys41 H A Arg42-HB Arg42 H A Arg42-HG Arg42 H N Asn43-HB Asn43 HB Phe45-HN Ser47 HB Ser47-HN Gin49 H A AIa48-HB Ala48 H A GIu49-HG Ghi49 H A Met52-HB Met 52 H N Arg53-HNThr54 H N Thr54-HB Thr54 H N Thr54-HG1 Thr54 H N AIa58-HA Ala58 a n.s., not
seen.
distance (,~)
mental
form I
form II
form I
form II
1.1 2.4 3.1 2.6 2.7 2.7 1.9 3.9 3.1 1.0 1.2 < 1.0 1.0 3.2 1.2 4.2 2.2 < 1.0 1.4 4.4 9.9 1.6 4.3 3.4 2.4 3.1 1.0 4.2 2.1 3.2 1.0 3.9 3.1 3.8 2.5 2.7 4.1 1.9 1.7 < 1.0 7.2 3.2 2.6 2.0 <1.0 <1.0 2.0
2.1 1.9 1.2 5.3 1.4 1.7 2.5 13.3 1.4 2.3 2.4 4.1 3.2 1.8 2.8 1.6 1.4 2.5 3.4 2.8 < 1.0 2.7 2.5 <1.0 4.8 2.0 2.1 2.4 6.2 2.7 2.4 2.7 2.6 2.5 1.6 1.6 2.0 3.0 2.4 1.7 3.6 1.4 3.8 1.2 2.5 3.8 < 1.0
2.84 2.41 2.33 2.39 2.38 2.43 2.59 2.27 2.33 2.89 2.77 3.47 2.93 2.33 2.77 2.18 2.46 5.84 2.70 2.22 1.80 2.61 2.19 2.31 2.45 2.33 2.84 2.20 2.35 2.36 2.79 2.22 2.30 2.24 2.41 2.39 2.21 2.57 2.55 2.92 2.31 2.34 2.42 2.55 3.47 3.57 2.49
2.53 2.51 2.72 2.11 2.72 2.70 2.44 1.80 2.68 2.51 2.46 2.24 2.35 2.59 2.38 2.60 2.66 2.46 2.32 2.40 3.04 or 4.17 2.40 2.42 3.00 2.18 2.54 2.53 2.43 2.03 2.44 2.46 2.42 2.45 2.47 2.59 2.62 2.53 2.37 2.42 2.55 2.62 2.71 2.22 2.76 2.43 2.28 2.89
2.3 _+0.5 1.5 + 0.5 1.1 +0.2 4 _+1 1 +0.2 1.3 + 0.4 1.5 _+0.4 3.5 + 0.9 1 +0.2 2.5 + 0.8 1.2 + 0.3 1.5 + 0.5 1.3 + 0.3 1 +0.4 2 +1 n.s. a 3 +0.5 2.5 +0.5 1.5 + 0.4 4 + 1 3.5 + 1.5 2 +0.8 2 + 0.5 1 +0.3 4.5 _+ 1 1.7 + 0.5 3 + 1 2.3 + 1 1.5 + 0.2 2.5 + 0.5 2.0 + 0.4 n.s. 1.5 + 0.5 1.0 + 0.3 1.0 + 0.5 1 + 0.6 n.s. 2 +0.3 0.8 + 0.2 0.9 + 0.2 2 +1 1.3 -1-0.3 1.5 + 0.5 1.7 + 0.4 3 +0.5 3 +0.5 2 +0.3
66 proton distances that are smaller than 0.25 A. These results are inherent in the small correlation time and mixing time used in the calculations; bigger disagreements are expected for larger values of ~- or t (Lane, A.N., personal communication).
Comparison with the experimental data The intensities of the experimental crosspeaks were estimated from the size of each outermost ring. It was found that, for isolated crosspeaks, intensity differences as small as 20-50%, depending on the particular crosspeak, were easily discernible. The following two factors, however, preclude in some cases a precise intensity estimate: (1) protons that have very close resonance frequencies produce crosspeaks that are too close to the diagonal of the NOESY spectra; (2) pairs of protons that produce crosspeaks in approximately the same frequency region result in a superposition of crosspeaks. In the present study, the comparison between experimental and theoretical spectra was further limited by the lack of N M R resonance assignments for the residues Arg-1, Pro-2 and Pro-3, and some side-chain atoms of the basic pancreatic trypsin inhibitor molecule. The presence of overlapping crosspeaks and lack of these resonance assignments prevent us from determining whether there are connectivities in the experimental data that have not been theoretically predicted. We can however, discuss the opposite case. There are some connectivities that have been theoretically predicted but are not experimentally observed. Their calculated intensities for the two crystal forms are detailed in Table I. The absence of calculated crosspeaks implies that the atoms are further away from each other in solution than in the crystal state. A single-time, single-temperature NOESY experiment does not provide enough information to decide whether the atoms are farther apart because of a static deformation or dynamical fluctuations. The protons that occupy similar positions in the two crystal structures produce crosspeaks of close intensities for the two crystal forms. These crosspeaks compose the majority of the calculated spectra, and they satisfy the experimental data, with the few exceptions listed in Table I.
Table II details the theoretical calculated NOESY crosspeak intensities and the distance between the connected protons for the two crystal forms. Two possible distances are given when it was not possible to decide between two equivalent protons, for example, between two HB protons. Only those crosspeaks that differ significantly in intensity between the two crystal forms and that could be compared with experimental results are listed. There are approx. 100 additional calculated crosspeaks that have different intensities for each of the crystal forms, but for them an estimate of the experimental intensity was not possible because of the aforementioned limitations. In the last column of Table II we show the estimated experimental intensities with the purpose of analyzing whether they agree with those calculated for crystal forms I or II. The errors in the intensities shown are qualitative estimates. The information that can be extracted about the distance between protons in the solution state and those measured in the two crystal states is given in Table III. Since there are very few disagreements between theory and experiment, the data do not support the hypothesis of a solution structure exchanging between several significantly different conformers. Instead, the solution structure of basic pancreatic trypsin inhibitor appears to be very close to that of the crystal states. There are in principle two possibilities that should be distinguished. (1) The solution state might be a mixture, with different populations, of molecules of crystal forms I and II. As a special case, the population of one of the crystal forms can be zero, and the solution structure will be equal to just one of the crystal forms. The molecules can be exchanging between the two configurations at a slow or fast rate compared to the N M R time scale. A slow exchanging rate would give crosspeaks corresponding to both crystal forms. A fast exchange rate would give a NOESY spectrum corresponding to a weighted average of the distances between the two crystal forms. (2) The molecule of basic pancreatic trypsin inhibitor might have clusters of atoms whose configuration is closer to that of crystal form I, and other regions closer to crystal form II. Slow exchange between the two crystal forms is
67 TABLE III Distance between protons Argl Pro2 HN-HD2 Phe4 HA Phe4-HG Glu7 HB-HD2 Phe4 HN-HB Cys5 HB-HA Cys5 HA Glu7-HD Pro8 HA Pro8-HD Pro9 HA Pro9-HN Tyrl0 HA-HB Pro9 HA-HD1 Tyrl0 HA Thrll-HE1 Phe33 1 HB-HN Thrll HG1-HA Thrll HA-HB Cysl4 HA Argl7-HG Va134 HD1-2HB Tyr21 HE1 Phe22-HA Asn24 HA-HD2 Tyr23 HA Asn24-HN Ala25 HD2 Asn24-HE2 GLn31 HN Lys26-HN Ala27 HA-HN Gly28 HA-HB Leu29 HN Cys30HA Leu29 HA-1HB Gln31 HE2 Phe33-HA Asn44 HA Tyr35-HN Gly36 HB-HA Tyr35 HD2-HE2 Tyr35 HE1 Tyr35-HA Ala40 HA-1HB Cys38 HA-2HB Cys38 HA-HG Lys41 HA-HB Arg42 HN-HB Asn43 HB Phe45-HN Ser47 HB Ser47-HN Glu49 HA-HG Glu49 HA-HB Met52 HN-HB Thr54 HN-HA Ala58
Appear to be closer to crystal a
a form II form II form II form If form II form II form I form I form II form II form I form I form I form II form If b form II form I form I _c a form II form II form II form II form II form II form I form II form II _c - c form II form II form I b form I form II b
form II form lI
No comparison available. b Experimental intensity is larger than predicted, but closer 10 form II. c No agreement with experiment. d Experimental intensity is within the predictions for both crystal forms. e Experimental intensity smaller than predicted. a
n o t consistent with our results, because, in this case, we w o u l d expect to see m o r e e x p e r i m e n t a l crosspeaks that c a n n o t be p r e d i c t e d f r o m our calculations. The results are consistent with a solution structure where several regions of the molecule are closer to either one or the o t h e r crystal form. If r a p i d exchange occurs, then the weighted average in some regions of the molecule is closer to structure I a n d in o t h e r regions is closer to structure II.
Sensitivity of the theory to small variations in distance T h e question arises as to how m u c h should the p o s i t i o n of an a t o m in the crystal differ from that in the solution to observe an a p p r e c i a b l e difference b e t w e e n the c a l c u l a t e d a n d m e a s u r e d spectra. To illustrate the sensitivity of the m e t h o d , the c o o r d i n a t e s o f the a t o m s of crystal f o r m II were r a n d o m l y altered, a n d new spectra were c a l c u l a t e d using these c o o r d i n a t e s as input. Some of the results were p l o t t e d to simulate e x p e r i m e n tal spectra. Fig. 2a (region b e l o w the d i a g o n a l ) shows the N O E S Y crosspeaks c o r r e s p o n d i n g to H B Cys5, as c a l c u l a t e d from the crystal c o o r d i nates. Fig. 2b (region a b o v e the d i a g o n a l ) shows the c a l c u l a t e d intensities after c h a n g i n g the x coo r d i n a t e of HB Cys5 b y 0.3 .~. T h e s a m e c o m p a r i s o n is m a d e in Fig. 3 between the H A Ile19 crosspeaks of the crystal (Fig. 3a) a n d the c a l c u l a t e d crosspeaks after m o d i f y i n g the x c o o r d i n a t e of H A Ile19 b y 0.8 ,~. If the c h a n g e in c o o r d i n a t e s h a d no effect on the spectra, Figs. 2 a n d 3 w o u l d b e s y m m e t r i c with respect of the diagonal. Instead, the lack of s y m m e t r y shows that differences in the a t o m s ' c o o r d i n a t e s as small as 0.3 ~, p r o d u c e discernible differences in the spectra. In s o m e cases, the intensities of the c r o s s p e a k s are m a r k e d l y different, in others, there a p p e a r new N O E S Y crosspeaks. N O E S Y cross p e a k s are observed, at 100 ms, between 1 HB Cys5 a n d the a t o m s 2 H B Cys5 a n d H N Cys 5. A c h a n g e of 0.3 ,~ in the x c o o r d i n a t e of 1 HB Cys5 m o v e s the p r o t o n closer to H N Cys5 a n d farther a w a y from 2 H B Cys5 b y a p p r o x . 0.25 A, dou~ b l i n g the 1 of one of the crosspeaks a n d r e d u c i n g the other to one half of its initial I. In the case of H A Asn43, a change of 0.8 ~, in the x c o o r d i n a t e increases the intensities of crosspeaks that h a d
68
b
I 5
10
r~l
('~11(ppm)
[]
"HNCys5
a t lO ~O (.D2(ppm) "r rn
Fig. 2. Effect of changing the x coordinate of HB Cys5 by 0.3 ,~. The NOESY connectivities below the diagonal (a) were calculated from the coordinates of crystal form II, and the upper region (b) corresponds to the changed coordinates. The sensitivity of the theory to a small change in the coordinates of the protons is reflected in the lack of symmetry with respect to the diagonal of the spectra.
b
- ~
~D/
10
(~I(ppm)
-5
5
C02(ppm)
Fig. 3. Effect of changing the x coordinate of HA lle19 by 0.8 ,~ on its NOESY connectivities. The spectra below the diagonal (A) were calculated from the coordinates of crystal form II, and the upper region (b) corresponds to the changed coordinate.
I < 1% in the spectra calculated from the crystal coordinates: HA Asn43-HD Phe4 and HA Asn43H N Asn43. The crosspeak HA Asn43-HE1 Phe45 had I = 1.6% in the original spectra; this crosspeak disappears after modifying the coordinate. Changes in I are also observed for HA Asn43-HB Asn43 and HA Asn43-HN Asn44 (Fig. 3). Approximately the same effects detailed here were observed for all the changed coordinates in the molecule. The exact effect of a distance variation in the NOESY intensity depends on the surroundings of the atom and on the mixing time. Conclusions The NOESY spectra generated by solving the generalized Bloch equations are very sensitive to small (of the order of 0.2 ,~) differences in the distance between protons. Because the equations take into account spin diffusion, the solution is also sensitive to the surrounding configuration of protons. The two spectra obtained from slightly different crystal structures of basic pancreatic trypsin inhibitor reflect the differences in interproton distances existing between the two crystals. Approx. 11% of the crosspeaks calculated have substantial intensity differences that could be experimentally discernible, under ideal experimental conditions. A difference in the interproton distance of 0.2 ,~ between the two crystal structures produces a relative change in the crosspeak intensity of approx. 63%. The exact intensity depends, however, on the geometry of the surrounding protons. Depending on the extent of spin diffusion, there might be a 10-20% variation in the crosspeak intensity for the same interproton distance. If the interproton distance is calculated from the observed crosspeak intensity by means of the two spin approximation, the obtained distance would be larger than the actual. The difference in the distance estimate, at 100 ms mixing time and ~-= 1.3 ns, is = 0.1-0.25 ,~. The comparison between theoretical and experimental spectra gives some detailed information on the similarities or discrepancies between the solution and crystal states of basic pancreatic trypsin inhibitor. For those regions of the molecule where the two crystal structures are very
69 similar, a single spectrum was obtained as a result of the calculations, and very good agreement was found with the experimental data. For those regions of the molecule where there exist substantial differences between the crystal forms, the experimental crosspeak intensities were, in general, close to the calculated intensities of either one or the other of the crystal forms. These results are consistent with a solution structure of basic pancreatic trypsin inhibitor in which different regions of the molecule are alternatively similar to the crystal form I or to the crystal form II. Six connectivities predicted by the two crystal spectra were not found in the experimental data. These were the only discrepancies found in the frequency region of the spectra studied; they indicate that the position of these atoms in the solution state differ from those measured in the crystals. The atoms are all located near the surface of the basic pancreatic trypsin inhibitor, and are therefore more susceptible to static or dynamic distortions because they experience weaker packing forces. Measuring the spectra as a function of temperature could help distinguish between the static and dynamic possibilities, since dynamic fluctuations are expected to have a much stronger temperature dependence than static distortions. Some of the information obtained in the theoretical simulation of NOESY spectra are unfortunately masked in the experimental data because of unresolved resonances or overlapping crosspeaks. In these cases, it was found that an experimental NOESY spectrum at a single mixing time is not enough to discriminate between two close structures, or to decide whether the experimental data present connectivities that were not predicted by the theory. Additional information must be obtained by following the time evolution of those spins that produce the unresolved connectivities and comparing it with theoretical simulations of one-dimensional NOEs.
Acknowledgements The authors are gateful to Dr. Andrew N. Lane for the preliminary stages of the computer simulations and for many helpful discussions; to James R. Williamson for his H Y D R O program, and to Drs. Olivier Lichtarge and Robert L. Domenick for interesting remarks. This work was supported by N I H grant number RR02300.
References 1 Wiithrich, K. (1986) NMR of Proteins and Nucleic Acids, John Wiley& Sons, New York. 2 Jardetzky, O. and Roberts, G.C.K. (1981) NMR in Molecular Biology,Academic Press, New York. 3 Abragam, A. (1961) The Principles of Nuclear Magnetism, Clarendon, Oxford. 4 Jardetzky, O., Lane, A., Lefrvre, J.F., Lichtarge, O., Hayes-Roth, B., Altman, R. and Buchanan, B. (1986) in Proceedings XXIII Congress Ampere on Magnetic Resonance (Maraviglia, B., De Luca, F. and Campanella, R., eds.), pp. 64-69, Rome, Italy. 5 Jardetzky, O. and Lane, A. (1988) in Proceedings of the International School of Physics, July 8-18, 1986, Enrico Fermi, Varenna, Italy, in press. 6 Gariepy, J., Lane, A., Frayman, F., Wilbur, D., Robien, W., Schoolnik, G.K. and Jardetzky, O. (1986) Biochemistry25, 7854-7866. 7 Lef~vre, J.F., Lane, A.N. and Jardetzky, O. (1987) Biochemistry 26, 5076-5090. 8 Duncan, B., Buchanan, B.G., Hayes-Roth, B., Lichtarge, O., Altman, R., Brinkley, J., Hewett, M., Cornelius, C. and Jardetzky, O. (1986) Bull. Mag. Res. 8, 111-119. 9 Bothner-By,A. and Noggle, J.H. (1979) J. Am. Chem. Soc. 101, 5152. 10 Kalk, A. and Berendsen, H.J.C. (1976) J. Mag. Res. 24, 343. 11 Deisenhofer, J. and Steigemann, W. (1975) Acta Crystallogr., Sect. B, 31,238. 12 Wlodawer,A., Walter, J., Buber, R. and Sjolin, L. (1984) J. Mol. Biol. 182, 295-315. 13 Wagner, G. and Wiithrich, K. (1982) J. Mol. Biol. 155, 347-366.