127

Biochimica et BiophysicaActa, 1077(1991) 127-130

© 1991 ElsevierSciencePublishersB.V.0167-4838/91/$03.50 ADONIS 016748389100136K BBAPRO30282

BBA Report

The effect of the conformer state and the model size chosen on the force field of the polypeptide backbone A n d r f i s Balfizs, P & e r R a j e z y a n d G ~ t b o r H o r v h t h Deparlment of Atomic Physics, E6tv6s l. University. Budapest (Hungary)

(Received3 October 1990)

Key words: Polypeptide;Harmonicforce field; Conformerstate; Modelsize We have attempted to check the validity of our previously calculated ab initio force field for the polypeptide hackboee, for the variance with conformer state and model size chosen. For the previous problem we applied our usual ab in_'qio Hartre~Fock S Q M method, also used at the original development of the force field, while for the biter we utilized the semiemplrical M N D O CO-gradient method, it is shown that the in-plane force field is reasonably stable for change el conformational state, while the out. of-plane force field can be assigned good average guess values. The effect of the model size is shown to be moderate ulmn introducing a dipeptlde into a polymer chain.

In recent years, we have made a considerable effort to develop a general valence force field for the polypeptide backbone, and suggested it be used in conformational energy-minimizing/protein dynamics programs as input data. While such attempt are not rare in the literature, we claimed that our force field is the best available guess. Our claim is based on the fact that we used an ab initio method (Pulay's ab imtio gradient (force) method [1]), a medium-sized sp basis (4-21) [2] and SQM (scaled quantum mechanical) technics [3]. According to this latter, Frm~ =

Ct/2FC~CI/2

that is, the directly calculated harmonic force constant matrix is corrected by "scaling coefficients' determined on smaller, related systems. To obtain these coefficients, we optimized them on the experimental vibrational spectra of formamide, acetamide, trans N-methylformamide [4] and checked them on the spectra of Nmethylacetamide and c/s N-methylformamide [5]. Thus, we proceeded onto the problem of the dipeptide approximation of the poiypeptide chain. We have calculated the full force field of N-acetylN'-methylalaninamide 0teretoforth AMAA) and we suggest use of the force constants thus obtained in wider scope (empirical energy calculating) programs [6].

Correspondence:A. BalD.s.Departmentof AtomicPhysics.E0tv0sL. University.H-1088 BudapesLPuskinu. 5-7. Hangar,/.

Our ab initio calculations were performed by Pulay's TEXAS ab initio gradient program [7]. While our approach is presumably more advanced than other related calculations, our claim is justified only if the relevance of the force field is checked against several conditions. Notably, if it is reasonably stable for: (1) the introduction of different side-chains; (2) different conformational (rotamer) states; and (3) the model being large enough for simulating the polypeptide chain. Point (1) was discussed briefly previously [8]. As to point (2), a general treatment was given recently to the problem by Ha and Giinthard [9]. They predicted by the isometric-group approach an up to ~ 5% variance of the force field by rotamer state. However, we have to see if this qualitative treatment is true for our present problem. Concerning point (3), we have to incorporate the dipeptide model into an infinite periodic polypeptide chain (which can be treated by the Hartree-Fock crystal orbital scheme) and see the shift of the force field (or, as we actually did, to take the polymer and gradually weaken the interaction between neighboring elementary cells). For this purpose, we used a modified version of Yamaguchi's M N D O CO-gradient program [10]. We had to use a semiempirical method here, because of the size of the problem. The original calculation of the full force field has been carried out at the C ~ conformation, supposedly the global minimum (see Ref. 6 and references cited therein). The additional conformers introduced were

128 similarly treated, namely, we optimized the full geometry in internal coordinates, until the residual forces along the individual internal coordinates were not larger than 10 -4 m d y n in order of magnitude. Afterwards, systeomtic 'empirical' corrections were applied to the ¢quifibrium geometries adopted from ReL 11 and through this to the force constant via the relation AF,~ = Fc~Aqt. Here, F,~ is the cubic diagonal anharmonlcity constarq, and Zig is the difference in internal coordinates. This correction, of course, is i m p o r t a n t only for the stretching type diagonals. Also, we corrected similarly for unidkectiunal numer/cal second derivatives (used to save computer time) and applied the SQM procedure b y the scaling factors of ReL 4, also adopted in Ref. 6 (for the C ~ conformer). By conformer state, we m e a n the different ~b, ~, values, and the conformers thus chosen were those termed "C5' and "i82". While the previous is a c o m m o n n a m e of a fully e x t e n d e d / 3 conformation, 182 is n a m e d b y Scarsdale et aL [12] for one lying in a 'corridor" of low energy connecting the ~8 region with the ap region. The expected effect is of a purely "geometric" nature [8]. W e have chosen 15-15 diagonal force constants a n d these are compared in Tab!e I. It can ~ seen that the "in-plane" constants are very similar, the average deviation being ---6~ from the C ~ values. "Out-of-plana" force constants are differenL Their values exhibit such large scatter, that the problem of total unreliability emerges. It must b e noted, however, that there is a very substantial contribution (66 and 89~) to the gross M% deviation from the ,b dihedral angle. This latter shows

that the r o l a i i o n a r o u n d the Ca-N b o n d is much more hindered in the C ~ conformation than at the Cs, even more a t B2; in the latter, there is practically n o rotational barrier. Separating ~, the average deviation is 25%. ~ is still rather high, so we must state that all we can d o is to give values averaged over the three coformer state, as the best guesses. According to this: ~,N~H~, 0.152; ¢ C , - N , , 0.075; %bJ2-H2, 0.108; and ¢C2~r2, 0.082. I t is, however, encouraging that the deviation from the C~q conformer is similar in direction and magnitude, even in details, a n d that our conformers cover ~ 6 0 ~ of the "allowed region" of the ,b-6 map. W e c a n add that the intramolecular H - b o n d has its effects on the y N t H ] d i a g o a a l value of the C~ conformer (as this is its highest value, higher b y = 2 0 ~ than at the C ~ confor~a~). Anyway, it can perhaps he stated that our "in-plane" force field is reasnably stable for ~, ~, rotation, whereas for the "out-of-plane" force field our guessed average values are the best estinmtes. A s it is c o m m o n l y accepted that in the case of infinite periodic polymers the elements of the Fock mah-ix begin t o stabilize after t a k i n g a n elementary cell of = 8 - 1 0 ~t, length d u r i n g the crystal orbital approach, the question arises if the dipeptide model is relevant or not concerning its size, that is, h o w deeply is the dipoptide model perturbed being built into a long polymer cha;n F o r technical reasons, we proceeded as follows. W e have calculated the factor group (unit cell-) vibration of

TABLE I The ~

of son~

d/agona/forceconstantsof A M A A

in ~ffere~ conformational.o.alesn

Average of AFa for the in-plane coordinate& 5.8~[ and average of dFu for the out-of-plalle coordinates. 33.99~. Units are consistent with the energy measuxedin aJ=10-1sJ: stretching type coordinates in A; bending tapes in radial. C~ -84.5 ° 62.2° 11.089 6.815 6.986 1.982 0.567 11.625 7.091 6.736 1.784 0.641

¢,

vC~-Ot(2, 2) gCI- NI(3, 3) ~,Nn- HI(5, 5)

BNIO~CI(6,6 ) 8Nt - Hn(9,9) vC~-O2(16,16) pC2- N2(17,17) uN2-//2(19,19)

BNzDzC~(20.20) 8N2//2(23, 23) TN n//1(30,3 0 ) 'rCt - N~(31, 31) ¥NzHz(37,37) "rC2- N2(38, 38) ¢{ ~C~ - N z C ' C ~ X 4 5 , n

45)

CS -165.7 ° 1673 ° 12.329 7.276 6.800 1.777 0.559 12.236 7.387 6.964 1.775 0.568

0.146

0.176

0.095 0.138 0.102

0.073 0.093 0.67

0.494

0.171

Deviation (%)

11.2 6.8 -- 2.7 - 10.4

- 1.4 5.3 4.2 3.4 -0.9

172 134.2° 38.1° 12.705 6.815 6.819 1.815 0.574 12.455 7335 7.060

Deviation (~)

14.6 0.0 -- 2 . 4

-925 1.2 7.1 3.4 4.8

1.756

- 1.9

- 11.4 20.2

0.552 0.134

- 22.7 -33.6 - 23.4 - 65.5

0.057 0.092 0.078 0.056

- 13.9 - 8.5 --40.1

-33.6 - 23.6 - 88.7

For the numbering and definitions of internal coordinates see Table I of ReL 6. Atoms belonging to the half of the molecule in the C a ~ N direction is marked by "1' subscripL whereas those belonging to the half in the C" -* C" direction by "2'. v-stretching~&bending, ,/-out-of-plane bending, c-torsion.

129 TABLE 11 Fhe faclor group Omit cell-) vibrations of polyglycine 1 at different translational vector lengths (cm- JJ 0) Parent

(2l +1A

(3) +2A

(1) Parent

3615 3562 3518 3464 3460 3386 2292 2093 2074 1796 1683 1534 1500 1472 1423 1392 1302 1244 1241

3618 3569 3460 3430 3385 3368 2083 2062 1762 1656 1494 1453 1420 1409 1390 1317 1272 1217 1173

3619 3569 3460 3414 3384 3_a64 2072 2023 1752 1619 1480 1430 1422 1406 1389 1298 1232 1182 1160

1220 1160 1128 979 828 725 686 620 555 482 462 375 334 307 263 254 213 171 150

(2) +1A

(3) +2.~

1130 1064 968 776 682 625 590 551 485 381 359 327 320 295 244 226 205 189 179

1075 985 970 730 612 554 544 439 387 344 331 325 301 244 239 207 201 187 86

polyglycine I (a #-pleated sheet), also its harmonic force field, by our MNDO CO-grad/ent program in the neare~.t neighbor approximation. The conformational parameters adopted were 4 , = - 1 4 0 °, g,= 140 °, and "standard' polypeptide dimensions [13]. After,Woards, we lengthened the translational vector by + 1 A, + 1.5 ~ . + 2.5 A, + 3 A, and malyzed the change of the force field (and also of the frequencies). First of all, the SCF CO procedure varied from fast convergence ( + 1 A) to divergence ( + 3 A). Evaluating the results, we must keep in mind the widlcy known incorrect dissociational properties of the Har~ee-Fock

! !

e,

/

No

Ii i.

.

.

.

.

.

.

1,

/\ .

.

.

.

.

.

.

j

Fig. I. The unit cell and numhering of atoms of Imlyglycine I. Atoms which are invariant for introducing the monomer into the polymer are encircled.

method and our experience that the basic effects already fully appear in the case of + 1 A, and the situation only quantitatively evolves further at longer translational vectors. We give the factor group frequencies (without sealing) in Table II. It can be seen that the frequencies of the elongated polymer are shifted downwards as compared with the parent polymer, probably be,auseof the large (negative) anharmonicity effects. At + 1 A = 70 cm -~, while at + 2 A = 1 0 0 cm -~ is the shift as an average. This reflects about 20-25~ change in the force field. This effect, however, concerns only = 30~ of it according to our calculations, notably by the foilowing interactions (see Fig. 1): (a) the direct effect of bond dissociation: C4, N2, O5, Ns, C10 diagonal and off-diagonal (Cartesian) force constants referring to the same atomic nucleus; (b) unit cell-unit cell (neighbor) interactions: N2-C m, C4-C10, C4-Hn, Os-Clo (and transposed elements); and (c) small shifts inside the 'monomer' between the following atom pairs: N2-C4, N2-H3, G-O~, C4-H3, Ns-GoThese interactions, as can be seen by inspection of Fig. 1. arise in the immediate vicinity of the dissociated bonds. The monomer unit (which is somewhat smaller than the dipeptide model) possesses an internal domain which is exactly invariant for introducing it into the .polymer. According to our results, thus, the dipeptide model and the monomer of polyglycine I are in good correspondance, and terminating the molecule by two terminal methyl g~oups is certainly less drastic than the bond-lengthening (dissociating) we performed. That is, nearly half of the atoms of the monomer (6) do not suffer any perturbation by terminating the interaction with neighboring unit cells, but of course the new interactions which arise do reflect the wrong description of bond dissociation, the big. direct (sometimes indirect) change of neighbor interactions. Taking the view that the latter are exaggerated, our estimated - 2 0 ~ ,force field change is, as and extreme case, acceptable. In our view, the dipeptide model probably /s of a proper size for calculating harmonic potential constants. The authors are indebted to Professor James E. Boggs and Dr. Mauricio Alcolea for performing the HartrceFeck ab initio calculation. This part of the study was supported by the Robert A. Welch Foundation.

Refen~es 1 Pulasf, P. (1969) Mot. Fh3~. !7, 197-2'34. 2 Pulay, P., Fogarasi, G., Pang, F. and ~-v~gs, J.E. (1979) J. Amer. Chem. Soc. 101, 2~50-7.560. 3 (a) Fogarasi, G., Pulay, P. (1984) Annu. Re,/. Chem. Phys. 35, 191-213. 3 {b) Pulay, P., Fogarasi, G., Pongor, G., Boggs, .I.E. and Vargha, A. (1983) J. Amer. Chem. Soc. 105, 7037-7047. 4 Fogarasi, G. and BalD.s, A. (1985) J. Mol. Struct. 133, 105-123.

130 ~; [~l~s, A. (1987) J. MoL Str~,-t. 153. 103-120. 6 ~ ~ ( l ~ ) J. ~ i ~ ~ , 2755-2763. 7 Pulay, P. (1979) Theor. Ckirm hera 50, 299-312. 8 B a l m A. (1989) B/och/m. B/ophys_ Acta 998, 215-217. 9 Ha. T.-K- and G~athard. H.H. (1989) Chem. Phys. 134, 203-228. 10 Dewar. MJ-~. YamaguchL Y_ and Suck. S.H. 11979) 43, 145-156.

11 Fagara~ G.. Pulay. P_ T-oraL F. and Boggs. J.E (1979) J. Mol. StmcL 57, 259-270. 12 Scarsd~¢. J.N~ Van Ahenoy. Co Klira~Jti, VJ., Schiller, L. and Momany. F.A. 11983) J. Amer. Chem. Soc, 105, 3438-3.445. 13 Schulz. G . E and Sehirmer. ILH. (1979) priadples of Protein S t m c m m p. 18 Springer Vedag, New York.