Modelling the dynamics of an antigenic peptide using NMR relaxation data

Biochimie (1992) 74, 815-824

815

© Soci6t6 fran~aise de biochimie et biologie mol6culaire / Elsevier, Paris

Modelling the dynamics of an antigenic peptide using NMR relaxation data B Kieffer, P Koehl, JF Lef'evre* IBMC du CNRS, 15, rue Descartes, 67084 Strasbourg Cedex, France

(Received 2 June 1992; accepted 2 July 1992)

Summary P The internal dynamics of a cyclic peptide which was designed to mimic an antigenic loop of the haemagglutinin, is studied through heteronuclear relaxation along the 13Ca-ilia vectors and through homonuclear relaxation along the IH,~-IHNand IHl~IHa, vectors. Order parameters are extracted from the longitudinal and cross-relaxation data. Molecular dynamics simulations are performed and the order parameters are calculated in different ways from the trajectories. The simulation, which is performed in vacuo, gives smaller order parameters (vector motions of larger amplitude) than the experimental results. However, the general features of the experimental order parameters are reproduced by the molecular dynamics simulation. The flexibility of the molecule can then be investigated from the results of the molecular dynamics. It shows that the mobility observed through the order parameters is due to motions in flanking regions, remote from the observed vectors. NMR / peptide / dynamics Introduction Several examples have demonstrated that the internal molecular dynamics play a major role in the molecular recognition process and the activity of proteins [1, 2]. Interaction between antigens and antibodies, for instance, may use the plasticity of the molecular structure to ensure a good conformational fit between the partners [3-5]. The dynamics are also of particular interest when peptides are studied in solution. The conformational space available to these molecules is usually observed to be large. The analysis of the internal motions leads to define this conformational space. Relaxation processes in nuclear magnetic resonance spectroscopy are sensitive to motions and have been widely used to analyze them. Here we study the dynamics of an antigenic peptide which has been designed in order to mimic the structure of an antigenic loop from the haemagglutinin of the influenza virus [6]. The sequence of the peptide (called D-loop) is given in figure 1. The peptide is cyclised through a peptide bond between t,~ side chain carboxylic group of Asp l0 and the terminal amide of the peptide. We

have measured the homonuclear and the heteronuclea~ relaxation parameters along various vectors in the molecule. The data were analyzed by making use of molecular dynamics calculations, which allow to define the conformational space sampled by internal motion. Theoretical background

In this section we describe briefly the theoretical background of the interpretation of relaxation parameters to get insight into the internal dynamics. The approach which is presented in the following can be applied to any vectors joining two nuclei that are in dipolar interaction. For the present study, we have analyzed the relaxation along vectors pointing in various directions: the ~3C~-IH,~ and the ~H,~-~HN vectors which are informative of motions in the backbone of the peptide,, and. the IH.~-IH'~. vectors which are sensitive to moUons m the sldechams. Here, the formalism is applied to the dynamics of the ~3C,~-~H~ vectors. The relaxation mechanisms

CYS-LYS-ARG-GLY-PRO-GLY-SER-ASP-PHE-ASP-TYR *Correspondence and reprints Abbreviations: DSS, 3-(trimethylsilyl)-l-propanesulfonic acid; rid, free induction decay; HSQC, heteronuclear single quantum correlation; MD, molecular dynamics; NMR, nuclear magnetic resonance; nOe, nuclear Overhauser enhancement.

I Fig 1. Primary structure of the synthetic peptide D-loop. The side chain of the aspartic acid (solid line) is used as a linker.

816 which govern the return of the ~3C populations to their equilibrium after an excitation are dominated by the dipolar interactions between the carbon and proton nuclei. The phenomenological description of the ~3C relaxation involves the use of the longitudinal and the cross-relaxation rate constants (Pc and riCH respectively) which are expressed by:

assumes that the overall molecular tumbling and intemal motions are not correlated. This assumption is reasonable since these dynamics involve very different time scales. The correlation function could then be written as the product of the correlation function of the tumbling (Co(t)) and of internal motions (C,(t)):

P c - - - [ J ( O ) H - - ~ c ) + 3J(00c)+6J(cth +¢-Oc)] (1)

If the overall tumbling of the molecule is isotropic, Co (t) is a simple mono-exponential curve with a correlation time Xo The simplest way to model the correlation function of internal motions is also a monoexponential with a value at infinite time equal to the order parameter $2 and a value of one at the origin. The internal motion is characterized by a unique correlation time xi which is defined by the integral of its correlation function. The order parameter is a measure of the space sampled by the vector during its internal motion. $2 is equal to one when the vector is rigidly attached to the molecule, and to zero when it moves isotropically within the molecule. It can be evaluated using the geometric, time independent equation [8]:

IX

GcH --

~

[6 J((% + Oc)-J(tow-~c)]

(2)

~.~.//o~ -

,where T are the gyromagnetic factors, and 10 h the Planck constant. In the case of proton-carbon interaction, cx = 3.56 109 s-2 A 6. rcn is the constant distance between the carbon and the proton. When this distance is fluctuating, one should consider the time (t) averaged value: 1 -

(6)

C(t) = Co(t).C~(t)

$2

= IIPeq(['~l)P2(COS(OI2))Peq(~"~2)d~md~2

(7)

I =<

~

~

(3)

r(0)-~.r(t)3

The information about the dynamics of the C-H vector is contained in the spectral density function ,1(o)) which is the Fourier transform of the correlation function C(t) characterizing the motion of the vector joining the two spins. The equation of the correlation function tbr a vector in motion is: I C(t) = ~ < P, (llLv(O)'IILI~(I))'> 5

(4)

where P2(x) = 1/2 (3x2-1) is the second Legendre polynomial and ULF(t') a normalized vector along the C-H vector with a time-dependent orientation. In the case of an isotropic motion, with one correlation time (zc), the correlation function is modelled with a monoexponential decay and leads to the classical equation of the spectral density function: 't'c

J(co) =

(5) 1 + 0#.'~c2

In the presence of internal motion, which is very likely in macromolecules, the correlation function is more complicated involving more than one time component [ 1]. Several attempts have been made to model its behavior [7-10], the most simple being the one proposed by Lipari and Szabo [8]. This approach

where On is the angle between the two normalized vectors um and t,2, ~i are the polar angles of ui and Peq(fi) the normalized distribution function of u. This description of internal motions leads to a modified equation of the spectral density function: % J((o) = S 2 •

% + (I-$2).

I + ¢OLZc2

(8)

1 + 0.~.%2

The first term of this equation corresponds to the spectral density function of the whole molecule, weighed by the order parameter $2. The second term in (eqn 8) contains a correlation time % which is a combination of the overall correlation t i m e xc and a p s e u d o internal correlation t i m e x~: 1

1

1

(9) When the modelled internal correlation time xi is small compared to I;o the second term of the spectral density function (eqn 8) becomes negligible and this approach results in a scaling of the overall tumbling spectral density function by the order parameter. In this case, the ratio OcH/Pc becomes independent of $2, and is only a function of the overall correlation time %. Ttle analysis of the ratio ocdpc for 13C~-IH~ vectors along a peptide backbone could then be used to estimate a value of the overall correlation time %. This method has been shown to be suitable for corm-

817 lation t i m e s in the r a n g e o f 1O0 ps to 1 ns ( K o e h l e t al, in p r e p a r a t i o n ) . A b o v e this r a n g e , the d e t e r m i n a t i o n o f Xc d e m a n d s quite a g o o d a c c u r a c y on the m e a s u r e m e n t o f r e l a x a t i o n rate c o n s t a n t s .

Materials and methods Sample

The synthesis of the D-loop peptide has been previously described [11]. It was performed using the solid phase method [12]. The cyclisation was directly performed on the solid support prior to final cleavage of the peptides. The purity was checked by analytical HPLC and by consideration of the l-D NMR spectrum. 10 mg of the peptide was dissolved in 500 pl D20 buffered with a deuterated acetate buffer 20 mM (pH 4) (pH meter reading). The final concentration of the D-loop peptide was 15 raM. DSS (0.1 mM final concentration) was added as internal standard.

I3C-#H relaxation rate measurements The relaxation rate constants which are of interest, are the longitudinal relaxation rate of the carbon and the cross-relaxation rate constant between the 13C and its attached t H. THe first constant is obtained from an inversion recovery experiment on the t3C (fig 2a), with saturation of the proton during the carbon relaxation in order to insure a mono-exponentiai behavior of the relaxation curve [13]. The cross-relaxation rate constant is extracted from inversion recovery experiments on the proton, with various initial conditions. The method will be described elsewhere. Briefly, two inversion recovery experiments are performed: one by selectively inverting only the protons (attached to the t3C which are in natural abundance) and the other by inverting both the protons and their attached carbons (fig 2b, c). The relaxation curves of the protons are affected differently by the cross-relaxation between the nuclei. A local relaxation matrix is fitted iteratively in order to account for the measured relaxation curves. This matrix involves the carbon, its attached proton and a second proton which allows for magnetisation leakage of the first one due to its environment. As shown by the pulse sequences in figure 2, the read-out of the longitudinal magnetisation of either the 13C or the IH, are performed by a two-dimensional heteronuclear single quantum coherence spectrum (HSQC). All experiments were performed at 25°C, using a Bruker AMXS00 spectrometer, operating at a proton frequency of 500.13 MHz. Using sequences of figure 2, the purging spin lock pulses were set to a power of 20 kHz and a duratiun of 1 and 2 ms for the first and second pulse respectively. 16 relaxation delay time points were used to describe a relaxation curve. For 2-D experiments, 2048 points were acquired in the t2 domain for a sweep width of 5000 Hz, and 64 rids were collected in the t~ dimension. In the carbon dimension, the spectrum was folded and the sweep width was reduced to 4000 Hz. Apodisation with a decreasing exponential function was applied prior to Fourier transform, with a line broadening of 2 Hz in the t2 dimension and 4 Hz in tin. Free M D simulation

Energy minimisation and molecular dynamics simulations were performed in vacuo using GROMOS and its force field [14].

The charge of side chain atoms was set to zero in order to minimize side chain-backbone interactions. The starting coordinates of the D-loop (D174) belong to the conformational space, which is in agreement with homonuclear nOe and coupling constant data [15] and which have been defined using a distance geometry procedure [16]. This structure has been relaxed by 1400 steps of conjugate gradients energy minimisation without constraint before the simulation. For the molecular dynamics simulation, the initial velocities were taken from a Maxwellian distribution at 300 K. The system was then equilibrated during 60 ps and 740 ps was used for the analysis. The time step was set to 1 fs and the C-C bond lengths were constrained using the SHAKE algorithm. During the simulation a weak coupling to a thermal bath at 300 K was used with a time constant of 10 fs. The non-bounded pair list was updated every 10 steps (10 fs). The global rotational and translational motions were stopped by superimposing all time frames on the first one by a least-square fit. The coordinates of the molecule were saved every 0.2 ps. If we assume that the simulation is long enough to allow the sampling of the conformational space explored by internal motions, the correlational functions can be calculated using a single trajectory replacing the ensemble average in equation (4) by a time averaging. The calculations presented here were averaged over 160 ps (800 structures). The order parameter S 2 was extracted by considering the value of the correlation function at the plateau. If the molecular dynamics trajectory is long enough to allow the intra-molecular vectors to sample the conformational space induced by their internal motions, the order parameter S z can also be evaluated using equation (7). This calculation was performed using a Monte-Carlo method. 50 000 pairs of vectors were randomly chosen among the 4000 conformations of the trajectory. The analysis of the hydrogen bonding patterns during the dynamics was done assuming that a hydrogen bond occurs if the distance between the acceptor and the donor was less than

A)

±y

±y

,,n' l'J' H' nlnn "°n''° nnnn .... , B)

±y

,,n" ll" R..... 1'1];,11 ,Nn nn

n,nn = ~y

I

I

li

c)

,,n"rl"n ....... i,r17 plrl,n,I] I] Nn ,, I]nrl Fig 2. Pulse sequences used to measure the inversion recovery of (a) carbon, (b) proton attached to a carbon, by inverting the proton alone, (c) proton attached to a carbon, by inverting the proton and the carbon. Unless indicated, all phases are x. The acquisition is made in TPPI mode. For observation of CH, 1; is set to 1/4 JCH.

818

Results Asp

Order parameters and correlation times

I0

The n3C,~carbon assignments have been obtained from a H S Q C proton-carbon correlation spectrum (fig 3), using the H~ proton assignments determined previously [ 15]. Except for C y s l and A s p l 0 (for which the IH~-13Ccx correlation cross-peaks are broader than for the other aminoacids), the fitting procedure of the local heteronuclear relaxation matrices gave good results according to the Khi2 values which measure the differences between the calculated (using the refined relaxation parameters) and the experimental values of the relaxation curves (table I). The order parameters for the IH~-laC(x vectors of C y s l and A s p l 0 shown in figure 7b are probably wrong values, since they have been calculated assuming, as for the other residues, that the correlation time for the internal motions was very small compared to those of the overall tumbling

_ Asp 8'

~

AP9 "3

L,vs 2

Q~

,

~

P/"le 9 /

T~,r" 1 ,'

o.

o~

~ ~

Set" 7

Pr'o

5

CI (t)

4.7'

4.5

4.5

4.4

DI

4.3

4.2

4.1

(ppm)

0.s ii. ,...,. . . . . . . . .

Fig 3. nH~,-t.~C,, region of the HSQC spectrum of the D-loop.

%,,

o

2.2 A and the acceptor-hydrogen-donor angle was greater than 160°. All calculations have been performed on a IBM RS 6000 workstation.

~

gt~

,.l*

0

C)

~,i, .~, !%

,

•..t,

,

IiX)

50

..

,

l

150

200

Time(ps)

Fig 4. Correlation function calculated from MD trajectory, for the tH~-t3C~ vector of Phe9 (up) and Pro5 (down).

Table I. Relaxation data measured along the C~-H~,vector of the D-loop peptide using the local relaxation matrix approach. ¢~Hcis the cross-relaxation rate constant and Pc is the longitudinal relaxation rate of the carbon. The reported Khi 2 values are criteria for the fit between the experimental and the calculated relaxation curves. The order parameters for the C,~-H~, the HN-Ha and the Ha-Ha' vectors, were extracted from the relaxation measurements (experimental S2) and from molecular dynamics simulation (modelled S2). Experimental S 2 Residue

~nc( S-t )

p d s-t )

khO

at~ "/Pc

Cysl Lys2 Arg3 Gly4 Pro5 Gly6 Set7 Asp8 Phe9 Asp 10 Tyrl I

!.12 0.57 0.44

2.2 3.5 3

378 95 92

0.47

3

47

0.56 0.56 0.87 ! .38 0.34

3.7 2.8 3.9 3.5 3.4

61 i 18 !42 58 ! 67

S~

,

(',42 9.13 0.12

I. ! 2 0.57 0.44

0. i 3

0.5 1 0.8 0.7

0.47

0.12 0.16 0.18 0.33 0.08

0.56 0.56 0.87 ! .38 0.34

0.6 0.8 0.7 0.9 0.8

Modelled S2 ,

,

0.57 0.86 0.86

0.41 0.56 0.69

0.47

0.13

0.42 0,57 0,72 0,61

0.18 0.45 0.66 0.54 0.4

,

0.59 0.59 0.72 0.22

0.21 0.13 0.09 0. l I

0.33 0.29 0.30 0.69 0.54 0.48

0.03 0.34 0.32 0.54 0.34

819 of the molecule. The relaxation parameters for the two glycines were not calculated as the delays in the relaxation experiments (fig 2) were not set to optimize the observation of CH2 groups. Within the limit of errors, the o/p ratios appear very similar from one ~H~-~3C~vector to another, giving an average value of 0.13. The corresponding correlation time for the overall tumbling of the molecule is equal to 850 + 350 ps. The order parameters for the ~H~~3C~ vectors calculated according to equation (8) are given in table I, together with the order parameters previously obtained for the ~H~-~HN vectors using a relaxation matrix approach [15].

Correlation functions obtained by MD From the MD trajectories, we have calculated the correlation functions for various vectors of fixed length. The effect of two different types of dynamic behaviors on the correlation function is shown in figure 4. The first 200 ps are represented for the I Ha-13C a vectors of the proline and the phenylalanine residues. For both vectors, the rapid initial decay which occurs in few picoseconds is followed by a plateau region. The value of the plateau obtained for the phenylalanine residue (C(t) = 0.8) indicates a spatial restriction of the internal motions of this vector and allows a direct estimation of the S 2 value. On the contrary, the value observed for the proline (C(t) = 0.2) shows that the IH~,-13C~ vector of this residue is affected by large amplitude motions. The comparison between S 2 values measured by the correlation functions and the Monte-Carlo calculations is shown in figure 5. For the majority of the vectors, both values are in agreement, the $2 value obtained by the correlation function being generally larger than that calculated by the Monte-Carlo method. This is due to the fact that contributions at

I

slower rares could arise and affect the latter value. This phenomenon affects in particular the $2 value of the Asp8 residue indicating that a rare event (on a time scale of several hundred picoseconds) may have occurred in this residue.

Dihedral angles fluctuation Dihedral angle fluctuations are the major source of internal motions. These fluctuations could result from high frequency oscillations within a single potential well or from transitions between different minima. The analysis of backbone dihedral angle values during the simulation provides some insight on the transitions which are responsible for the peptide dynamics. The time course of the main chain phi and psi dihedral angle values is shown in figure 6, illustrating three different types of behaviors. For the glycine 4, we observed rapid transitions between two phi angle values (tp = 120 and tp = -120). This behavior leads to a high standard deviation for this angle (SD = 66°). The psi angle of this residue exhibits fluctuations which are not correlated with the phi angle fluctuations. For Asp8, a concerted transition of the phi and psi angles occurs at 450 ps. This rare event could result in a bad evaluation of the order parameter evaluated by a Monte-Carlo calculation due to an incorrect averaging effect of this slow motion. This event which is observed only for the Asp8 residue is responsible for the high discrepancy between the S 2 value evaluated from the correlation function and the Monte-Carlo calculation for this residue. The phi and psi dihedral angles of the Gly6 residue are oscillating within a single potential minimum ( = 85 ° and <~> = 67 °) leading to small standard deviations (SD(¥) = 15° and SD(~) = 20°). The analysis of the time course of all residues indicates that no concerted transition occurs between dihedral angles of different residues. Such kinds of motion could have resulted in a bad estimation of the peptide dynamics from the S 2 values analysis.

09 0.8

Discussion

o.?

Time range of the internal dynamics in the linker and the peptide backbone

0.6

o.s 04

0.3 0.2

Ol o Cysl

Lys2

A.-83

Pro5

Set7

Asp8

Phe9

AsplO

Ty~Jl

Fig 5. Comparison of the order Parameters of the

Residue

IHa-13Ca

vectors calculated from MD simulation, by the Monte-Carlo method (gray) and by the correlation functions (white).

On the HSQC spectrum (fig 3) the IHa-13Ca c o r r e l a t i o n cross-peaks of Cysl and Aspl0 exhibit broader lines than the other aminoacids. This might reflect peculiar dynamics at the level of these two aminoacids which are at both ends of the linker closing the loop. This part of the molecule is different from the neighboring peptide backbone. The linker, which utilizes the side chain of Aspl0 is analogous to an o~

820

A

B

Time (ps)

!i i,i , lR il?i

C

Time (ps)

so

Phi

Phi .*t

~mc(~)

.~Pml~'~ln"rm mlmln "m""v'" ~ Irq~lm''rnr'r''''~'~ °

°

. . . . . . . . . . . .

..

.

.

.Ioo

o

i~

i~o

IGo

~

~

Ioo

Psi

Psi

'~t

..

..

..

:

:

:

:

:

:

:

:

Fig 6. Time course of the phi and psi dihedral angle fluctuations during the MD trajectory. A. Gly4. B. Asp8. C. Gly6. amino 13 acid. Similarly to 13 aminoacids, this type of residues is known to produce unstable polymers, due to a high conformational entropy related to the flexibility of the backbone [17]. It is possible that several conformations are available to the linker, which interconverts in a time range that broadens the proton and carbon resonances of the involved aminoacids. The relaxation curves for these two residues are consequently noisy, and the values for the relaxation rate constants are not accurate. The dynamics of the linker are likely to be complicated. The broadening of the resonance lines indicated above is induced by motions in the millisecond to second time range. Measurements of the relaxation rates in the rotating frame (Tip, data not shown) using various spin lock fields 1181 did not show any motion in the microsecond time range. However, motions occurring at a rate comparable to the overall tumbling of the peptide might also be present. This would explain why the a/p ratios (table I) are so different from those of the other residues. The constant value of the a/p ratios for the rest of the IHct-13C a vectors shows that the complicated dynamics of the linker do not influence the motions in the other part of the molecule. In this region, internal motions appear to be characterized by a correlation time "r~ which is much smaller than that of the !umbling. "r~ must be in the picosecond range. This is m agreement with the molecular dynamics calculation where it can be observed that the correlation functions for the internal motion reach their plateau values after few tens of picoseconds (fig 4). Of course, a longer correlation time could not be observed in these calculations as the trajectory was run tbr only 740 ps. However, experimental results tell us that they do not occur.

Comparison between order parameters measured from relaxation data and from molecular dynamics calculation along the backbone For several residues, the order parameter which characterizes the motion of the backbone depends on whether it is measured on the nH,~-n3C~or the IH~-nHN vectors. This is observed with Lys2, Arg3, Ser7 and Tyrl I for which the difference between the two order parameters is larger than the expected error. This reflects the anisotropy of the internal motion, as the two vectors point in different directions with respect to the backbone, and experience differently the rotations of the dihedral angles of the peptide chain. Grossly, the behavior of the order parameters for the nH~-13C~ or the ~H~-nHN vectors along the sequence is quite well reproduced by the molecular dynamics calculation. The vectors around the proline move with large amplitude motions. On the contrary, the regions of Arg3 and Phe9 are less mobile. It is interesting to note that the temperature coefficient of the Phe9 amide proton was found to be somewhat smaller than that of the other amide protons [15], suggesting that this amide, thus protected against the solvent, might be engaged in an hydrogen bond. This can also be observed in the molecular dynamics calculation where the Phe9 NH lies 17% of the time within the hydrogen bonding distance of the carbonyl group of Lys2. This hydrogen bond would contribute to the smaller mobility of the vectors in the two connected regions of Lys2 and Phe9. While the features of the motions are well described by the calculation when one part of the molecule is compared with another, it is, however, necessary to scale down the experimental S 2, in order to make them comparable in magnitude with the calcu-

821 lated ones. In figures 7a, b, the experimental order parameters have been multiplied by a factor of 0.7. This value gives the best agreement between the experimental and the calculated $2. This reflects the fact that the motion in the molecular dynamics trajectory which is calculated in vacuo is of a larger amplitude than it is in solution. The solvent seems to have a damping effect on the amplitude of the motions. In order to be rigorous, the calculation should have been done with solvent molecules. However, this was beyond our computing capabilities. In few points of the molecule, the calculated order parameters do not compare well with the experimental values. This is the case for the ~Ha-~HN vectors of Gly4, Ser7 and Asp8 and for the IHa-13C ~ vectors of Arg3, Pro5 and Ser7 (fig 7a, b). Two arguments can be proposed in order to explain these discrepancies: i) the multiplying factor which has been artificially used to scale down the experimental order parameters may not be the same for all residues. The solvent effect might damp differently the motion of different residues, depending on their exposure, the nature of their side chain, etc; ii) as observed in figure 6, some angles just oscillate around a mean value during the trajectory, while for Gly4, the angle values flip frequently between two mean values which differ by 60 ° . These large fluctuations, by increasing the space explored by the vectors, decrease drastically the value of the attached order parameters. In solution, these large oscillations may be damped and may occur with a much lower frequency so that they are not reflected in the experimental S 2. The calculation of the order parameters around Gly4 might be biased by these internal motions of large amplitude. Side chain motions are not properly modelled in vacuo

The experimental and calculated order parameters of the IHa-~Ha' vectors which reflect the motions of the side chains are greatly divergent (table I). The simulation in vacuo gives a large freedom of motion to the side chains and the simulated order parameters are smaller than the experimental ones by a factor of 4-6. This is clearly due to the absence of solvent and of charges on polar side chains during the molecular dynamics simulation. The calculation with solvent molecules seems mandatory in the case of the side chains. Analysis o f the flexibili~ of the backbone

Given some interpretation of the data, like the damping effect of the solvent, one can say that the general features of the dynamics of the molecule are well reproduced by the trajectory simulation. In other

words, bearing in mind that some fluctuations which are calculated in vacuo might be attenuated in solution, the results of the molecular dynamics can be used to interpret the experimental data. Figure 8 shows the mean values of the fluctuations of the phi and psi angles calculated from one trajectory. The other trajectories gave very similar results. In this trajectory, Gly4 was undergoing large conformational changes as shown in figure 8. This explains the large fluctuations observed for the phi and the psi angles of this residue. The interesting feature in the curves of figure 8 is that they seem not to correlate directly with the S2 values measured along the backbone. Grossly, S 2 values are small around the proline and large around the Lys2 and Phe9. It is precisely around these two residues that the angular fluctuations are larger. The mobility observed in the sequence around the proline appears to originate in the angular fluctuations occurring in the flanking regions. The order parameters attached to a vector are a measure of the mobility of this vector, but not of the flexibility of dihedral angles directly connected to it. Only a molecular dynamics analysis can give a picture of this flexibility. The IHa-tHs vectors can be used for probing motions

One question which might be raised is whether the order parameter of the ~H~-IHN vector which is extracted from the homonuclear relaxation between these protons, reflects only the reorientation factor or contains also the fluctuation in length of this vector. The averaged length of the ~Ha-IHN vector is deduced from the averaged scalar coupling constant measured between the alpha and the amide protons. This is legitimate only when the phi dihedral angle lies between -60 ° and -120 ° [19]. In this region the curve of proton-proton distance as a function of phi is monotonous. In order to test the influence of the fluctuation of the inter-proton distance on the estimate of the order parameter, the S 2 value was calculated from the molecular dynamics simulation in two ways: i) the calculation was conducted by the Monte-Carlo method, taking only the orientation factor into account (equation 7); ii) a slightly modified expression was calculated: S'~ = < r 3 . r ~ . S 2

(10)

where S2 is the generalized order p "meter which takes the fluctuation of distance into account: P,(cos(Oi2)) 82 = II Peq(['~l )

"

.3 .3

I "i,I 2

Peq(f~2) df~ld~2

( 11 )

L

t.-i ,,.

~

I

I

o

i

I

.o

!!~!!!!!!!!!~!!!~!!!!!!!!!~:f::!!!!~!!:!!~

..........

~~

~r

~[ "["

o

(-}

I

o

I

o

I

o

I

.o

.....

I

=_.............

_.__.7

-

I

iiiiiiiiiiii

I

o

I

.~

I

.o

I

Or}

i I

o 0

o 0 i I

o 0 i I

t ........................ i

(,'3

o~

~t

~H

nnnnnr

i i

.o 0 i w

o 0

I

I

l .................................

>t .............................

-...I

o

>'L

~ ' ~ - ~

(~

0

o

.

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.

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l i

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.o 0

.

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.

.

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00 t~ I',,}

823 Fig 7. Comparison between order parameters extracted from the relaxation data (white) and calculated from molecular dynamics calculation. The experimental order parameters have been scaled down by a factor of 0.7, in order to make them more alike to the calculated S2. a. JH,~-]HNvectors. b. IH,~-13C,,vectors. The values for Cysl and Asp8 (*~ are wrong values because of non-appropriate treatment of the relaxation data.

s

2

0.80 0.70 0.60 0.50

0.40 0.30 0.20 0.10

100.00

0.00

90.00 I 80.00

I

1HN IHA

2HN 2HA

I

t

|

I

I

I

3HN 4HN 4HN 6HN 6 t I N 7HN 3HA 4HA1 4HA2 6HAl 6ttA2 7HA

I

~ttN 8HA

i

t

I

9HN IOItN I IHN 91tA 10ttA IlHA

Vectors

70.00

Fig 9. Effect of the fluctuation of distance between ~H,, and ~HN on the determination of the order parameter of the corresponding vector. Comparison between the order parameter calculated by the Monte-Carlo method (~) according to equation (7), (tt) according to equation (10).

60.00 50.00 40.00 30.00 ~.00 I0.00 0.00 Ci

I

I

I

I

I

I

I

t

I

I

K2

R3

G4

P5

G6

$7

D8

F9

DI0

YII

Residue

Fig 8. Standard deviations of phi (i) and psi (::) angles between the structures of the MD trajectory. Figure 9 shows that both order parameters are very similar indicating that the fluctuations of the distances between the alpha and the amide protons have a negligible effect on $2.

Conclusions The conformation of a peptide or a protein is always rapidly fluctuating around one or several mean structures. This fast internal motion lies in the picosecond to nanosecond time range and corresponds to the exploration by the molecule of a potential well. This motion can be probed by laboratory frame relaxation measurements. The time scale of these fast internal motions can only be estimated, but their amplitudes can be measured through the order parameters. These order parameters can also be calculated from molecular dynamics trajectories. The comparison between measured and calculated order parameters allows an evaluation of the relevance of the simulated motions. If these parameters agree, very precise informations on the molecular flexibility can be obtained. This approach should lead to a better understanding of the behavior of biological active peptide in solution. Very often, it is hardly possible to find any evidence of a defined structure for a linear peptide in solution. The approach which was used here should

allow to determine whether the peptide explores a large or a restrained conformational space. Work is in progress to study few very active linear peptides of this kind. Another application of the throrough analysis of the dynamics of a molecule is to improve the determination of its structure by NMR. The structure determination procedure relies on the values of the interproton relaxation rates which contain the magnitude and the dynamics of the vector joining the interacting protons. Understanding the dynamics should lead to a better translation of the cross-relaxation rate constants into distances. It should be considered that in a procedure of refinement where the $2 are introduced in order to take into account the internal motions of the vectors, the dynamics simulation and the structure are mutually dependent. Such a structure refinement procedure has already been proposed and applied in a case of small DNA oligomers structure determination [20]. It might be easier to use normal mode analysis [21,221 rather than trajectory simulations which are time consuming. Moreover, an analytical expression of the order parameter as a function of normal modes variables exists [23]. This could greatly simplify the refinement procedure. We are now exploring these possibilities.

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