BIOPOLYMERS
VOL. 15, 1061-1080 (1976)
Calculation of the Molecular Parameters of the aHelix-Coil and @-Structure-CoilTransitions T. M. BIRSHTEIN and A. M. SKVORTSOV, Institute of Macromolecular Compounds, Academy of Sciences of the U.S.S.R., Leningrad, U.S.S.R., and V. I. ALEXANYAN, Institute of Cytology, Academy of Science of the U.S.S.R., Leningrad, U.S.S.R.
Synopsis Monte-Carlo calculations of geometric and thermodynamic characteristics of the ahelix and the &structure of polypeptides have been carried out. T o describe a hydrogen bond both the Lippincott-Schroeder and Morse potentials were used. The internal rotation angles (o and $ in the a-helix have been shown to fluctuate in the range of f 7 " . The distribution functions on angles p and $ and on hydrogen bond lengths and angles in the a-helix have been computed and compared with those in myoglobin and lysozyme. Thermodynamic characteristics of the a-helix calculated in different approximations with the two forms of the hydrogen bond potentials have also been compared. The data obtained are close to the experimental values for polypeptides in neutral solution. Some geometric and thermodynamic characteristics of the regular parallel and antiparallel and irregular antiparallel p-structure have been found. In the b-structure the interAn increase in the cross and longinal rotation angles vary within the interval f15-20'. tudinal dimensions of the &structure only slightly influence both the geometric and thermodynamic characteristics.
INTRODUCTION Polypeptide chains in solution can form two types of secondary structures: a-helix and @-structure. Phenomenological theories of the a helix-coil, intramolecular @-structure-coil, and a @-coilt r a n ~ i t i o n s l - ~ provide all characteristics of the transitions in terms of parameters s = e-AFfRT and u = e-AF1nlt/RT, where ,IF = AHs - T - A S , = F o r d - Fcollis the difference of the free energies of a monomer in ordered and coiled regions, and AFlnlt = A H , - T-AS, characterizes the boundary effects in ordered regions (for the a-helix AF,,,t = 2 ( F b o u n d - Fc0dwhere F b o u n d is the free energy of a monomer a t the boundary of the structure'). In the case of the &structure there is also a parameter which is determined by the free energy of the @-structurebend. T h e direct numerical calculation of these parameters starting from the structure of the molecules is possible if the potentials of nonbonded interactions are known. T h e present paper is an attempt t o calculate the thermodynamic parameters of both coiled and ordered states. For the a-helix similar computations were carried out by GO, GO and Scher-
+
1061 01976 by John Wiley & Sons, Inc
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN
1062
TABLE I Thermodynamic Characteristics of the Poly( L-alanine) Monomer Unita ______~
__
~
Eo dip.
__
ECOd
-
scoil
-
E~p dip.
RT
RT ( p = 120". $ = 130")
$ = 300 ? 20')
Fcoil -
~~
RT (@ = 5 0
f
Conditions
RT
R
Electrostatic interactions are taken into account Electrostatic interactions are not taken into account
2.0
9.6
-7.6
2.7
1.7
3.0
10.1
-7.1
1.3
1.0
-
-
lo",
_______. ~
a The absolute minimum of the energy on the map was taken to be zero; the
R T value was thought to be equal to 0.6 kcal/mole and the entropy was calculated on the assumption that each state occupies an area I" X I" on the map.
aga5,6accepting the assumption that the potential curve near the minimum has a parabolic form. Within the scope of a classical approach we employed different procedures for the calculation. Like Go et al.,596!8 we ignored the dependence of the macromolecular kinetic energy on conforrnati~n.~~~ Experimental values of the AF and A F i n i t , are generally related to the polypeptides in organic solvents or in water. Therefore it is necessary to take into account the interactions of the macromolecule with the solvent. We assumed however that in inert solvent such an effect may be ignored and the calculation of the thermodynamic characteristics of the individual chain would be sufficient. Moreover, we considered only the properties of the main chain, regardless of the influence of the side chains. So the results should be compared with the data for the polypeptides with short, nonpolar side chains. The simplest part of the problem is a calculation of the thermodynamic and geometric characteristics of the polypeptide in the coiled state. In this case, conformations of neighboring monomers are independent and the chain can be considered as a linear set of the noninteracted units (dipeptide approximation). The partition function for a residue may be approximated as the sum of Boltzmann factors, taken a t equal intervals of cp and $. That is 2 = L: Z [exp -E(cp,$)/RT], and S = R In Z (E)/T. The value of entropy'dzpends of course on the interterval chosen for cp and $, that is, the entropy is subject inevitably to an additive constant, and hence only the differences between free energies and entropies have significance. The values of the energy E c o i l I R T , entropy S c o i l / R , and free energy F c o i l / R T of a monomer in the coiled state, which have been derived from the conformational energy maps of the L-alanine dipeptide, are given in Table I. The maps also allow the contribution of the intrachain interactions of the neighbor atoms to the conformational energy of the a- an &structures to be determined. These values E n d i p and Epdip. (i.e., dipeptide energy in the conformations close to a-helix or 0-structure) are also given in Table I.
+
MOLECULAR PARAMETERS OF TRANSITIONS
1063
T h e difficulty in calculating the thermodynamic and geometric characteristics of the ordered states is connected with the necessity of taking into account not only the interactions of the neighboring atoms but also those of the atoms brought together, due to the chain twisting (a-helix) or of the atoms of the adjacent chain (@-structure). T h e main problem is to find the structure entropies. T h e energy can be derived from the analysis of the regular a - and 0-structures. (In such structures all pairs of the rotation angles cp and $ are supposed to be equal, and the entropy is equal to zero.) This energy is the sum of the hydrogen bond energy and the conformational one Ereg.
= EHreg.
+ Econf.reg.
(1)
Econf.reg. is the sum of the energies of the short- and long-range interactions. The former is Endip. for the a-helix and Eodjp. plus the energy of the adjacent chains interaction for the 0-structure. T h e latter, as has been shown,lOJ1are determined to a great extent by the electrostatic interactions of the nonadjacent peptide groups. There is apparently certain screening of the polar groups far removed from one another; therefore we assumed that the conformational energy of the a-helix is Ehel.conf. = Econf.reg. = Endip. and in the case of the &structure, only interactions of the given monomer with the nearest ones were taken into account.'O The routine procedure of minimization of the value Ereg.provides the internal rotation angles for the regular c ~ - h e l i x (cp ~ ~=~125", J ~ $ = 130°), and for the parallel (cp = 53", $ = 313") and antiparallel (cp = 37", $ = 319") o-structures.lo I t has been found that both in the regular ahelix12 and antiparallel 0-structurelo hydrogen bonds are slightly distorted and the bond energies are close t o the optimal value (approximately 80-90% of the E Hoptjm.). On the other hand, in the regular parallel 0-structure, the hydrogen bonds are always distorted; the energy E H reg. is about half of E H and has no fixed minimum. (The minimum is spread over the region cp X $ = 10" X 10°.lo) Thus, the antiparallel 0-structure is more rigid than the parallel one. Hence, only the antiparallel 0-structure and a-helix will be considered. METHOD OF CALCULATION A N D HYDROGEN B O N D POTENTIALS T h e Monte-Carlo t e c h n i q ~ e 'was ~ employed t o obtain the average statistical values. We generated a lot of conformations picked a t random in the vicinity of the regular structures, determined their energies, and averaged over them the parameters we were interested in.14 T h e angles cp and $ were picked a t random over 1" from the intervals cp = 128" f 16", I,$ = 128" f 16" for the a-helix and cp = 38" f 20°, I,$ = 320" f 20" for the antiparallel &structure. This procedure also gave the probabilities for angles cp and $ (i.e., the distribution functions for angles cp and $).
1064
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN 16
2.0
3.0
4.0
P i
E”/RT Fig. 1. Lippincott-Schroeder potential of the hydrogen bond r is the bond length; 6 is the bond angle; ro = 1.7 8, is the equilibrium bond length. Since at r < 1.6 8, the Lippincott-Schroeder potential is unreal, E H in this region is assumed to be infinite.
The distance between two identical chains of the 0-structure was taken to be equal to 9.3 A. Nonbonded and electrostatic energies of all atoms except the atoms 0 . . . H-N forming a hydrogen bond were assumed to be constant. This assumption undoubtedly is true for the individual chain because there is a broad flat area on the dipeptide energy map in the regions of the angles cp and II/ corresponding to the antiparallel 0-structure and a - h e l i ~ . Therefore ~ the internal rotation is restricted mainly by the interchain and intrachain hydrogen bonds. In our calculations two hydrogen bond potentials were used: the LippincottSchroeder potential15 (L-S) and that of Morse.16 In the first case the energy is a function both of the H . . . 0 bond length r , and of the ONH bond angle 0 (Fig. 1). The two potentials computed with the same equilibrium distance ro = 1.7 A are presented in Figure 2. The Morse potential is somewhat
ri
Fig. 2. Lippincott-Schroeder’ and Morse2potentials of the hydrogen bond.
MOLECULAR PARAMETERS OF TRANSITIONS
1065
I
-L -ri
-i -i
i,
4 ir iz
I)
A’P
Fig. 3. Total distribution function n(p) in the nine-monomer segment of the a-helix, calculated with the Morse potential (-) and with the Lippincott-Schroeder potential (-O-); A p = 128’ - p.
deeper and steeper than that of Lippincott-Schroeder, which should lead to a decrease in the internal rotation freedom in the a-helix and pstructure.
GEOMETRIC CHARACTERISTICS OF THE (Y-HELIX The distribution function on the angles p averaged over the all monomer conformations in the nine-monomer helical segment is shown in Figure 3. The standard deviation 6p = [ ( Ap2) - ( A P ) ~ ] , ” for ~ the L-S potential is equal t o 7.2’, which suggests the possibility of significant torsional oscilations within the a-helix. The same results were obtained for the distribution function on angles $. The average values of the internal rotation angles were found to be (p) = 126.8’, ($) = 131.3’, i.e., they are very close to their standard values in the a-helix cp = 123’ and $ = 132’. Thus, in spite of the considerable internal mobility, the a-helix can be treated as “ideal.” T h e use of the steeper Morse potential func-
0.I
0 f.0
f. 5
2.0
2.5.
Fig. 4. Distribution function on the hydrogen bond length in the a-helix. LippincottSchroeder potential (1);without any potential (2). Q ( r ) calculated with the Morse potential does not differ markedly from curve (1).
1066
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN
10
PO
Fig. 5. Distribution function on the hydrogen bond angle in the a-helix: with use of the Lippincott-Schroeder potential (1); without any potential (2).
tion results in a slight narrowing of the distribution functions Q(q)and N $ ) (Fig. 3). The distribution function on the hydrogen bond lengths Q ( r )is given in Figure 4. The sharp drop of Q ( r )a t r < 1.6 %, is connected with the unrestricted growth of the L-S potential which we assumed in our calculations. It is seen that Y3 of the hydrogen bonds are of lengths r = 1.6-1.9 A which correspond to the minimum of EH(r,B). The function R(r) obtained with the hydrogen bond potential being omitted is also presented in Figure 4. In this case all angles cp and $ from the given in) n($)have the form of a recterval are of equal probability and n ( ~and tangular impulse in the interval cp,$ = 128O f 16'. As can be seen, the use of the hydrogen bond potential considerably enriches Q ( r )in the region close to ro mainly a t the expense of the states with the small r. A t the same time even without potential, 40% of the bond are of length r = 1.75 f 0.15 A, that is, this value of r is more favorable from an entropy point of view than others. Hence there are a number of ways to form the lengths r = 1.75 f 0.15 %, in the a-helix a t the variation of three pairs of angles cp and $. The distribution functions Q ( 0 ) calculated both with the potential 'y3
'y3
rsb.
iro" (a)
ffOO
f30"
fP0'
(b)
Fig. 6. Maps of lengths (a) and angles (b) of the hydrogen bond of the three-monomer segment of the a-helix. rp1 = = 123O; $1 = $2 = 132O; a and $3 are varied.
MOLECULAR PARAMETERS OF TRANSITIONS
1067
0.I 0.05
-16
-8
0
(d)
b
16
4 y4 (el
Fig. 7. Distribution functions in the different monomers of the a-helix ( A @ = 128O pz). The Morse potential.
and without it are given in Figure 5. T h e distinction between these functions is greater than t h a t for the functions Q ( r ) . As a result of the strong dependence of the L-S potential on the angle B the maximum of the Q(0) is in the region 5-10', though the value 0 = 15' is more favorable from an entropy point of view. Figures 4 and 5 show that in spite of considerable variation of the angles cp and $, the hydrogen bond parameters in the irregular a-helix are more or less close to their optimal values. This is apparently the result of the fitting of the angles cp and $ in the three-monomer segment. T h e hydrogen bond lengths and angles of the three monomer segment are shown in Figure 6. The conformations of the first two monomers of the segment were fixed (at the standard values of angles cp and $), while two angles in the third monomer were varied. The figures show t h a t there is a wide interval of the angles cp and $ which corresponds t o the almost linear hydrogen bond with the length close t o ideal. I t is obvious t h a t simultaneous alteration of the angles in three successive monomers can only expand the region of possible variation of the angles cp and $. T h e distribution functions on angles cp and $, averaged over the conformations of each monomer taken separately, are more detailed characteristics of the internal rotation (Figs. 7a-e). These functions show the change of internal freedom while removal from the end of the helix takes place. T h e maximum of the distribution function is displaced t o the right until it reaches its extreme value a t cp 3 124' (Figs. 7d, 7e); simultaneously, the width of the distribution function decreases from 20' for the terminal monomer to 14' for middle ones, thus pointing t o the significant differences between the monomer conformations. However,
qi+i
-0.02 0.12 -0.14 -0.08 -0.09 -0.03 -0.04 -0.02 -0.04 -0.01
i
14 13 12 11 10 9 8 7 6 5
-0.24 0.15 -0.04 0.03 0.05 0.04 0.03 0.01 0.02 0.06
Cpilpi- 1
-0.22 -0.33 -0.38 -0.37 -0.35 -0.38 -0.30 -0.32 -0.31 -0.28
qiGi-1
-
-
-0.11
0.37 -
-
-0.16 -0.13 -0.18 -0.14 -0.14 -0.13 -0.09
c~id~i-2
0.51 0.49 0.54 0.49 0.47 0.40 0.43
qi~i.2
-0.10 -0.08 -0.12
-0.10
-0.04 -0.03 -0.07 -0.10 -0.08 -0.08
$iPi-1
-0.002 -0.12 -0.13 -0.02 -0.05 -0.02 -0.08 -0.12 -0.05 -0.10
$i$i-1
-
-
-0.16
-0.08 -0.16 -0.14 -0.18 -0.23 -0.13 -0.13
Giqi-2
-
-0.004 -
0.02 0.03 0.0004 -0.02 0.001 -0.05 -0.006
$i*i-2
TABLE I1 The Correlation Coefficients of the I4 Monomer Segment of the a-Helix
0.05 0.02 -0.04 -0.08 -0.06 0.02 0.03
-0.14 -0.16 -0.13 -0.19 -0.09 -0.18 -0.19 -0.13 -0.19
-
-0.03
cPi+i, ~i-2$i-2
-0.0001
cPi$i> qi-l+i-1
?2
F 2
M
gu
,C
g
4
p 0
2 CJI
"8
c3
3:
5Cn3
W
MOLECULAR PARAMETERS OF TRANSITIONS
1069
A 'p7 f6
-
!6
- I6
-8
0
(a)
-
-8
0
8
(b)
* 'p9
Fig. 8. Bivariate d i h b u t i o n function on the internal rotation angles. (a) Q(a,w); (b) Corresponding correlation coefficients are K = +0.4 and K = -0.38. The figures indicate the probabilities of realization of the given pair of rotation angles. Q(~&X).
middle and terminal monomer entropies being the logarithmic function of the number of states differ ~ 1 i g h t l y . l ~ The energy of the hydrogen bonds between i and i 3 monomer in the a-helix is determined by the six angles pi, # i , p i + l , $;+I ~ ; + 2 ,#i+2. I t is obvious that the angles cannot change independently. T h e degree of the correlation between different angles can be estimated by calculating the correlation coefficient
+
T h e value of K is within the limits -1 < K < 1, while the values K = fl correspond t o a linear dependence between the pair of parameters under consideration and the sign defines a slope of the plot. In the ab= ($;) ( $ j ) and the correlation coeffisence of the correlation ($i$;) cient is equal to zero. We consider the a-helical segment t o consist of fourteen monomers. T h e correlation coefficient was obtained for different angles belonging either to the same monomers or to different ones, but not far removed from one another. As can be seen from the results given in Table 11, the correlation coefficient changes within a wide interval from -0.4 t o +0.54. T h e value of K for any chosen pair of internal rotation angles does not depend on the indexes of the monomers dealt with. T h e only exception is when one or both angles is in the first or second monomer. T h e maximum correlation is between angles pi; $;-I and pi, p i - 2 in the different monomers; near angles pi, $; and pi, pi-1 as well as ones farther do not show a strong correlation. T h e binary distribution functions Q ( ( o s , ~ and ) Q((ps,$s) for strong correlated an-
1070
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN
gles are pictured in Figures 8a and 8b. Both functions possess the obvious asymmetry that accounts for the value and sign of the correlation coefficient. Besides the correlations between single angles those of the pairs of angles vi$i and cpi-l$i-l as well as cpi$i and cpi-24i-2 were calculated. It was found that the former are weak enough -K N f0.15 and the latter are about zero. The variation of the internal rotation angles in the a-helix seems to be the main cause of its flexibility. The helical segments considered are too short to estimate the persistence length of the a-helix from their dimensions. Nevertheless, there are analytic expressions connecting the persistence length and the angles of internal rotation for the independent oscillation of neighbor monomers.17 For this approximation the obtained value of the persistence length is equal to 200 A, which is several times less then the experimental one. Being taken into account the correlations between the internal rotation angles lead to the greater values of the persistence length.
THERMODYNAMIC CHARACTERISTICS OF THE a-HELIX As distinct from the random coil state, the presence of the hydrogen bonds in the a-helix leads to the correlation of the conformations of the monomer units. Hence, the different microstates of the N-monomer helical segment are determined by the set of 2N internal rotation angles cp&. The energy of each microstate is the sum of the energies of ( N 2) hydrogen bonds and the conformational energy of the a-helix. The partition function of the system may be found by the use of the average value (eEIRT) = K-1
K C eEiIRT i=l
(3)
where Ei is the energy of the i-th conformation, K is the number of conformations over which the averaging is carried out. Indeed in the canonical ensemble
where ro = J’; w(E)dE is the number of all possible chain conformations with a given N , and hence Z ( N ) = I’o/(eEIRT) (in our case the number of microstates of each monomer is g = 322 and r o = g N = 322N). For the method described above, the slow convergence of the results, and hence a low accuracy of calculations are inherent. Indeed, the main contribution to ( eE/RT) is due to the less favorable conformations (conformations with high positive energy). The accidental appearance of several unfavorable conformations failing to affect the average statisti-
MOLECULAR PARAMETERS OF TRANSITIONS
I: '
1071
\,
8.0 0.2
0.4 '/N
Fig. 9. Dependence of the monomer free energy in the a-helix on 1/N (the conformational energy of the helix is not taken into account). The Lippincott-Schroeder potential.
cal characteristics considerably changes the computed value of the free energy. Therefore the method is unapplicable to the systems with the areas unfavorable energetically in which the system occasionally appears. T h e more energetically heterogenous the system (for example, there is a series of the narrow energy wells), the greater the error in calculation of the free energy. In our case the hydrogen bond potentials have a single minimum, and hence the system is energetically homogeneous. Nevertheless, to check on the calculation we also found the free energy of the system using a statistical ensemble of a variable number of monomers.14 In the a-helix the extreme values of a monomer free energy found by both methods differ only slightly. Since we presumed that the difference in energies E; of microstates is determined only by the hydrogen bond energies, the value F(N)IRT = -In Z ( N ) = -In I'o In ( e E I R T )which is related t o the free energy of the a-helix as follows
+
may be found. On the other hand, and thus
L-x 0
Fhel.(N)
= ( N - 2)Fhel.
. 5
+ 2Fbound.
N
10
Fig. 10. Dependence of the average hydrogen bond energy (Lippincott-Schroeder potential) on the number of monomers in the helical segment.
1072
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN
0
4.0
5.0
-RT En
Fig. 11. Distribution function on the hydrogen bond energies in the nine-monomer helical segment (seven hydrogen bonds).
Therefore F(N) (see Eq. (5)) may be presented as
- Ehel.conf. + -1 ( A F i n i t . - 2 W (7) RT N RT The dependence of F(N)IN R T on 1/N according to Eq. (7) is presented in Figure 9. As can be seen the value F(N)/N R T depends linF(N) -NRT-
Fhel.
early on 1/N, thus allowing the values of two parameters (Fhel. = -11.4 and ( A F i n ; t . - 2AF)/RT N 9.4 to be determined. Supposing that Ehel,conf. = Econf,reg. = E a d i p . and using the data in Table I, we obtain F h e l . / R T = -8.7, AFIRT = -1.15 and AFini,/RT = 7.3. Hence the parameters of the helix-coil transition are s N 3 and CJ N 7.10-4. Figure 10 shows the dependence of the average hydrogen bond energy (EH)= (E)/(N- 2) on the number of monomers in the helical segment (the distribution function ~ ( E His) presented in Figure 11). The extreme value ( E H ) / R T = -4.8 is about 85% of the minimum energy for the L-S potential. Combining the plots in Figures 9 and 10 one can obtain the dependence of the average entropy per monomer on the number of monomers in the helical segment (see Fig. 12). A t N = 1 or N = 2 the entropy is S h e l . / R = In g = 6.93. It is also possible to estimate the helical state entropy in the presence of the hydrogen bonds (N 2 3) using the distribution functions Q(v) and Q($) and assuming that there is no correlation between angles cp and #. According to the Boltzmann formula it is EheI.conf.)/RT
where the summation extends over all angles of internal rotation. The value S h e l . / R = 6.75 obtained in such a way is shown in Figure 12 by a dotted line and is just the upper value of the S h e l . because the presence of the correlations between the angles of different monomers may only decrease the entropy of the system. If the free energy of the monomer in the helix Fhel. is known, it is also possible to obtain the value of the conformational free energy Fhel.conf. =
-6.6 -8.5 -7.5
-5.1 -4.1 -0.9
A
-6.4 -5.4 -2.3
B
-4.1 -3.9 -3.1
Ref. 5
A -4.0 -3.3 -3.3
B
Present worka
R
AS,
-3.7 -3.0 -3.0
~
-2.5 -4.6 -4.4
Ref. 5
a A represents data obtained with the Lippincott-Schroeder
Gly Ala(R) Ala(L)
Polypeptide Ref. 5
Present worka
RT
AHS
~
-2.4 -2.1 1.0
B 6.5 9.3 9.0
Ref. 5
AH,
-0.5 1.3 7.8
A -0.6 1.4 7.8
B
Present worka
RT
-7.5 -5.1 -5.7
Ref. 5
4 . 0
-7.5 -6.0
A
-8.0 -6.6 -6.6
B
Present worka
R
As, ~
14.1 14.4 14.7
Ref. 5
7.0 7.3 13.8
A
7.4 8.0 14.4
B
Present worka
RT
AFinit.
potential; B represents data obtained with the Morse potential.
-1.3 -1.1 2.1
A
Present worka
RT
AF
TABLE I11 Thermodynamic Characteristics of Different Polypeptides
1074
BIRSHTEIN, SKVORTSOV, A N D ALEXANYAN
5
10
Fig. 12. Dependence of the monomer entropy on the number of monomers in the helical segment.
(Ehel.conf. - TShe1.J = (Fh,l. - E H ) which is equal to -4 R T or -2.4 kcal/mole. Fbound. = (AFiniJ2 Fcoil) is about the same value; therefore Fhel,conf, is the same a t the edges and in the middle of the helix. It arises from the weak dependence of She1. on the number of hydrogen bonds (Fig. 12). All thermodynamic characteristics of the poly(L-alanine) a-helix (left and right) obtained with the L-S and Morse potentials are presented in Table 111. The data pertaining to the a-helix of the glycine, which can be easily obtained from the dipeptide conformational energy maps, are also included into the table. As can be seen, the thermodynamic characteristics of the helical state are stable regarding the choice of the hydrogen bond potential. The data in Table I11 also allow the results obtained in different approximations and by different methods to be compared. GO et al.5.6 have calculated the thermodynamic parameters of the a-helix, assuming that the oscillations of the angles cp and rc/ are small (harmonic approximation). In this approximation ( ( E ) - Emin.)/RT = 1 (Emin. is the minimum energy of the regular conformation). Our calculations show that ( ( E ) - Emin.)/RT is equal to 0.8 for the L-S potential and 0.6 for the Morse potential. The entropy differences A S , and A S , quoted by GO et al.5 and in the present work are similar, so both the harmonic approximation and the assumption about the independence of the conformational energy on the angles cp and rc/ proved to be correct. It is to be noted that the difference of the right and left a-helix entropies for the poly(L-alanine) quoted by GOet al.5 is equal to 0.3 but we supposed it to be zero. The distinction characterizes the degree of accuracy of these calculations. In contrast, the energy differences A H , obtained by various methods differ considerably in values. The distinction is mainly associated with the fact that we used the dipeptide approximation while GO et aL5 took into account long-range interactions, which are determined to a great extent by electrostatic interactions of the neighbor peptide groups. The value A H , / R T * 10 for the right helix of the poly(L-alanine) obtained5 at t = 4 leads to a high value AF;,;,/RT N 15, which differs sharply from the experimental values 5-71° (calculation at e = 1 gives unreal values A H J R T = 20 and AFinit./RT = 40). A t the same time our data A F i , i , / R T N 8 and A H J R T N 0 are in exact accor-
+
MOLECULAR PARAMETERS OF TRANSITIONS
1075
0.10
0.05
-
0
20
AY
Fig. 13. Distribution function of angle q in t h e helical segments of myoglohin
dance with experimental results. This apparently means t h a t a t real conditions there is a strong screening of the polar groups removed far from one another. Therefore, use of the general value t = 3-4 excessively increases the contributions of distant groups and electrostatic potential should not be taken into account a t the distances more than 6 A.6
ac
1
0
II 1P--0
10
10
..
rA
2.5
2.0
1.5
30
8"
(b) Fig. 14. Distribution function of lengths ( a ) and angles ( h ) of the hydrogen bond in myoglobin (1) and lysozyme (2).
1076
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN
COMPARISON WITH X-RAY ANALYSIS DATA It is possible to obtain the internal rotation angles cp and J/ in the helical segments and to calculate the hydrogen bond parameters on the basis of the X-ray analysis of protein crystals. The detailed conformations of the protein monomers are obviously determined not only by interactions along the chain, but also by the globular packing of the neighboring regions of the chain. Nevertheless, the comparison of the experimental data with the results of calculations for the individual helix can be useful for the right choice of the effective hydrogen bond potential for proteins. The distribution function on the angles cp and J/ in the myoglobin helical segments is given in Figure 13. It is interesting to note that this function almost coincides with that on angle cp which we have calculated for the individual chain (Fig. 3, L-S potential). The distribution functions on the hydrogen bond lengths and angles in the myoglobin and lysozyme are shown in Figure 14. As can be seen from Figure 14a the distribution function Q ( r )for the myoglobin has a maximum near r = 1.7-1.9 A, but the distribution function n(6) (Fig. 14b) has a maximum a t the 0 = 10-15' (see for comparison Figures 4 and 5). The distribution functions Q ( r )and Q(6) (dotted line in Figure 14) are broader than that for myoglobin; this could be caused by the small number of the helical monomers in lysozyme. The results obtained show that the effective hydrogen bond potential in the helical segments of globular proteins is sure to give a smooth growth of the energy, while r increases from its optimal value and a weak angular dependence, i.e., the potential is more gentle than those of L-S and Morse.
GEOMETRIC AND THERMODYNAMIC CHARACTERISTICS OF THE ANTIPARALLEL STRUCTURE
8-
A t first we shall consider "hairpin" structures consisting of two neighboring chains with one, two, three, and five monomers per chain. The results obtained show that the thermodynamic and geometric parameTABLE IV Geometric and Thermodynamic Parameters of the &Structure of Different Types
-
37"
319"
10"
38"
319"
8"
/VV\J
37"
320"
7"
1.84
4.2
7.2
11.4
1.8
4.7
6.5
11.2
1.73
4.9
6.5
11.4
MOLECULAR PARAMETERS OF TRANSITIONS
1077
0.15 0.05
0.fO 0.05
0
0.fO 0.05
0
E/R T
Fig. 15. Distribution function of the internal rotation angle q (a), hydrogen bond length (b), angle (c), and energy (d) for the &structure (two chains with three monomers per each chain).
ters of the “hairpin”-&structure do not depend on its length. T h e average geometric parameters of the “hairpin” of three monomers, as well as certain thermodynamic characteristics, are given in Table IV (first line). T h e distribution function on the angle p of the three-monomer system is shown in Figure 15a (the distribution on the angle $ is of the same shape). T h e plot shows considerable freedom of the internal rotation in such a system (about f15’), and a n increase in the “hairpin” size fails to give a diminution of the allowed interval of the angles. However, in spite of the great deviation of the angles 9 and $ from their standard values, the hydrogen bonds are only slightly distorted. In Figures 15b and 15c the distribution functions on the lengths and angles of the hydrogen bonds are shown. The two distributions turn out to be narrow and therefore the average energy of the hydrogen bonds can reach 75% of the minimum energy. The distribution on the hydrogen bond energies is presented in Figure 15d. The entropy per monomer is equal to 7.2. In the case of the free rotation around the bonds N-C and C-C’ in the intervals f 2 0 ” the entropy is SolR = 2 In 40 = 7.4. The close values of S and So also point out the great freedom of the internal rotation in the p-structure.
1078
aro
BIRSHTEIN, SKVORTSOV, A N D ALEXANYAN
0.90
’
0.05
AfO
D
20
40
0.20-
OdO
-
Fig. 16. The same as in Fig. 15. One of the chains in @-structureis fixed.
Besides the “hairpin” structure, the three-chain system was also calculated. An increase in the cross dimensions slightly influences both the geometric and thermodynamic characteristics. Therefore, the segments of the antiparallel P-structure, irrespective of its cross and longitudinal dimensions, possess considerable freedom of internal rotation. The internal rotation angles change within the interval f15-20°, that is, a t least two times more than the allowed interval for the a-helix. In our calculations the parameters sp and a p of the 0-structure-coil transition were also obtained. It was found that the free energy of a monomer in the middle of every chain of the 0-structure is about that at the edge of the @-structureand hence ag N 1. Thus, as distinguished from the a-helix, where the initiation of a helical region is strongly hindered, in the &structure both its initiation and continuation (increase in longitudinal dimensions) are thermodynamically equivalent. A t the same time the equilibrium constant sp of the 0-structure-coil transition (per monomer) is roughly equal to that of the a-helix-coil transition. I t is interesting to note the restriction of freedom of the internal rotation when one chain in @-structureis fixed. Such a situation may occur
MOLECULAR PARAMETERS OF TRANSITIONS
1079
in the sorption of the polypeptide chain on a flat surface. For this purpose the two chain &structure with the same pairs of angles p and $ in one of the chains was calculated. T h e result proved t h a t the average values of parameters of such a system (see Table IV) are also independent of the number of monomers in the chain. T h e distribution functions on the angle p, on lengths, angles, and energies of the hydrogen bonds are shown in Figures 16a-16d. The figure shows that despite the almost 50% decrease in the allowed interval of angles, the latter remains wide enough. There are no essential changes in energy (its decrease is about 10%). In the system of three chains the maximum restriction will occur if the variations of the internal rotation angles take place only in the middle chain. In such a system (average values see in Table IV) all distribution functions narrow but slightly. Thus the entropy and energy of the middle chain change slightly while the neighboring chains are fixed. T h e monomer entropy remains close t o SIR N 6.5, and the average energy of hydrogen bond is ( - E I R T ) = 4.7-4.9, i.e., about 3 kcallmole. We found that the entropy of the a-helix is Shel.lR N 6.7 and hence, it is close to t h a t for the 0-structure, while the hydrogen bond energies of those conformations are roughly equal. Since the conformational energy of the monomer in the 0-structure is approximately the same as that in the a-helix, the free energies of these states are close to each other. On the other hand, the poly(L-alanine) in solution is observed either in the helical or in the coiled state and does not form any antiparallel pstructure. The origin of the discrepancy may be thought of as follows. Folding of the polypeptide chain leads to the rise of unfavorable bends. This fact was not taken into account and therefore we could not estimate the free energy of the 0-structure bend. Besides, the finite dimensions of the chains slightly influence the energy of helical state (in any one-dimensional system about 1/N of a whole energy is lost a t the boundary), whereas in the 0-structure the loss of the energy decreases as 1/N1/2with growth of the system, i.e., much more slowly.
References 1. Birshtein, T. M. & Ptitsyn, 0. B. (1966) Conformations of Macromolecules, Interscience, New York. 2. Birshtein, T. M., Eljaschevitch, A. M. & Skvortsov, A. M. (1971) Mol. B i d , 5 , 7890. 3 . Adonts, V. G., Birshtein, T. M. & Skvortsov, A. M. (1973) Repts. Akad. Sci. Arm. SSR, 3,168-173. 4. Adonts, V. G., Birshtein, T. M., Eljaschevitch, A. M. & Skvortsov, A. M. (1976) Riopolymers, 15,0000-0000. 5. GO, M., GO, N. & Scheraga, H. A. (1970) J . Chem. Phys., 52,2060-2079. 6. Go, M., GO, N. & Scheraga, H. A. (1971) J . Chem. Phys., 54,4489-4503. 7. Volkenstein, M. V. (1955) Structure a n d Physical Properties of Molecules, Acad. of Sci. USSR, Moscow. 8. GO, N., GO, M. & Scheraga, H. A. (1968) Proc. Natl. Acad. Sci. ( U . S . ) ,59, 10301037.
1080 9. York. 10. 11. 12.
BIRSHTEIN, SKVORTSOV, A N D ALEXANYAN Flory, P. I. (1969) Statistical Mechanics of Chain Molecules, Interscience, New
Alexanyan, V. I. & Skvortsov, A. M. (1974) Mol. Biol.,8,182-192. Brant, D. A. (1968) Makromolecules, 1,291-298. Skvortsov, A. M., Birshtein, T. M. & Zalensky, A. 0. (1971) Mol. Biol.,5.69-77. 13. Hamersley, I. M. & Handscomb, D. C. (1964) Monte-Carlo Methods, John Wiley, New York. 14. Skvortsov, A. M., Birshtein, T. M. & Zalensky, A. 0. (1971) Mol. Biol.,5.390-398. 15. Schroeder, R. & Lippincott, E. R. (1957) J. Phys. Chem., 61,921-932. 16. Popov, E. M., Dashevsky, V. G., Lipkind, G. M. & Arkhipova, C. F. (1968) Mol. Biol., 2, 612-621. 17. Birshtein, T. M. (1974) Vysokomol. Soedin., 16A. 54-62. 18. Ptitsyn, 0. B. (1969) in The Modern Problems of Peptides and Proteins Chemistry, 95, Nauka, Moskow.
Received July 9,1975 Accepted November 4,1975
VOL. 15, 1061-1080 (1976)
Calculation of the Molecular Parameters of the aHelix-Coil and @-Structure-CoilTransitions T. M. BIRSHTEIN and A. M. SKVORTSOV, Institute of Macromolecular Compounds, Academy of Sciences of the U.S.S.R., Leningrad, U.S.S.R., and V. I. ALEXANYAN, Institute of Cytology, Academy of Science of the U.S.S.R., Leningrad, U.S.S.R.
Synopsis Monte-Carlo calculations of geometric and thermodynamic characteristics of the ahelix and the &structure of polypeptides have been carried out. T o describe a hydrogen bond both the Lippincott-Schroeder and Morse potentials were used. The internal rotation angles (o and $ in the a-helix have been shown to fluctuate in the range of f 7 " . The distribution functions on angles p and $ and on hydrogen bond lengths and angles in the a-helix have been computed and compared with those in myoglobin and lysozyme. Thermodynamic characteristics of the a-helix calculated in different approximations with the two forms of the hydrogen bond potentials have also been compared. The data obtained are close to the experimental values for polypeptides in neutral solution. Some geometric and thermodynamic characteristics of the regular parallel and antiparallel and irregular antiparallel p-structure have been found. In the b-structure the interAn increase in the cross and longinal rotation angles vary within the interval f15-20'. tudinal dimensions of the &structure only slightly influence both the geometric and thermodynamic characteristics.
INTRODUCTION Polypeptide chains in solution can form two types of secondary structures: a-helix and @-structure. Phenomenological theories of the a helix-coil, intramolecular @-structure-coil, and a @-coilt r a n ~ i t i o n s l - ~ provide all characteristics of the transitions in terms of parameters s = e-AFfRT and u = e-AF1nlt/RT, where ,IF = AHs - T - A S , = F o r d - Fcollis the difference of the free energies of a monomer in ordered and coiled regions, and AFlnlt = A H , - T-AS, characterizes the boundary effects in ordered regions (for the a-helix AF,,,t = 2 ( F b o u n d - Fc0dwhere F b o u n d is the free energy of a monomer a t the boundary of the structure'). In the case of the &structure there is also a parameter which is determined by the free energy of the @-structurebend. T h e direct numerical calculation of these parameters starting from the structure of the molecules is possible if the potentials of nonbonded interactions are known. T h e present paper is an attempt t o calculate the thermodynamic parameters of both coiled and ordered states. For the a-helix similar computations were carried out by GO, GO and Scher-
+
1061 01976 by John Wiley & Sons, Inc
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN
1062
TABLE I Thermodynamic Characteristics of the Poly( L-alanine) Monomer Unita ______~
__
~
Eo dip.
__
ECOd
-
scoil
-
E~p dip.
RT
RT ( p = 120". $ = 130")
$ = 300 ? 20')
Fcoil -
~~
RT (@ = 5 0
f
Conditions
RT
R
Electrostatic interactions are taken into account Electrostatic interactions are not taken into account
2.0
9.6
-7.6
2.7
1.7
3.0
10.1
-7.1
1.3
1.0
-
-
lo",
_______. ~
a The absolute minimum of the energy on the map was taken to be zero; the
R T value was thought to be equal to 0.6 kcal/mole and the entropy was calculated on the assumption that each state occupies an area I" X I" on the map.
aga5,6accepting the assumption that the potential curve near the minimum has a parabolic form. Within the scope of a classical approach we employed different procedures for the calculation. Like Go et al.,596!8 we ignored the dependence of the macromolecular kinetic energy on conforrnati~n.~~~ Experimental values of the AF and A F i n i t , are generally related to the polypeptides in organic solvents or in water. Therefore it is necessary to take into account the interactions of the macromolecule with the solvent. We assumed however that in inert solvent such an effect may be ignored and the calculation of the thermodynamic characteristics of the individual chain would be sufficient. Moreover, we considered only the properties of the main chain, regardless of the influence of the side chains. So the results should be compared with the data for the polypeptides with short, nonpolar side chains. The simplest part of the problem is a calculation of the thermodynamic and geometric characteristics of the polypeptide in the coiled state. In this case, conformations of neighboring monomers are independent and the chain can be considered as a linear set of the noninteracted units (dipeptide approximation). The partition function for a residue may be approximated as the sum of Boltzmann factors, taken a t equal intervals of cp and $. That is 2 = L: Z [exp -E(cp,$)/RT], and S = R In Z (E)/T. The value of entropy'dzpends of course on the interterval chosen for cp and $, that is, the entropy is subject inevitably to an additive constant, and hence only the differences between free energies and entropies have significance. The values of the energy E c o i l I R T , entropy S c o i l / R , and free energy F c o i l / R T of a monomer in the coiled state, which have been derived from the conformational energy maps of the L-alanine dipeptide, are given in Table I. The maps also allow the contribution of the intrachain interactions of the neighbor atoms to the conformational energy of the a- an &structures to be determined. These values E n d i p and Epdip. (i.e., dipeptide energy in the conformations close to a-helix or 0-structure) are also given in Table I.
+
MOLECULAR PARAMETERS OF TRANSITIONS
1063
T h e difficulty in calculating the thermodynamic and geometric characteristics of the ordered states is connected with the necessity of taking into account not only the interactions of the neighboring atoms but also those of the atoms brought together, due to the chain twisting (a-helix) or of the atoms of the adjacent chain (@-structure). T h e main problem is to find the structure entropies. T h e energy can be derived from the analysis of the regular a - and 0-structures. (In such structures all pairs of the rotation angles cp and $ are supposed to be equal, and the entropy is equal to zero.) This energy is the sum of the hydrogen bond energy and the conformational one Ereg.
= EHreg.
+ Econf.reg.
(1)
Econf.reg. is the sum of the energies of the short- and long-range interactions. The former is Endip. for the a-helix and Eodjp. plus the energy of the adjacent chains interaction for the 0-structure. T h e latter, as has been shown,lOJ1are determined to a great extent by the electrostatic interactions of the nonadjacent peptide groups. There is apparently certain screening of the polar groups far removed from one another; therefore we assumed that the conformational energy of the a-helix is Ehel.conf. = Econf.reg. = Endip. and in the case of the &structure, only interactions of the given monomer with the nearest ones were taken into account.'O The routine procedure of minimization of the value Ereg.provides the internal rotation angles for the regular c ~ - h e l i x (cp ~ ~=~125", J ~ $ = 130°), and for the parallel (cp = 53", $ = 313") and antiparallel (cp = 37", $ = 319") o-structures.lo I t has been found that both in the regular ahelix12 and antiparallel 0-structurelo hydrogen bonds are slightly distorted and the bond energies are close t o the optimal value (approximately 80-90% of the E Hoptjm.). On the other hand, in the regular parallel 0-structure, the hydrogen bonds are always distorted; the energy E H reg. is about half of E H and has no fixed minimum. (The minimum is spread over the region cp X $ = 10" X 10°.lo) Thus, the antiparallel 0-structure is more rigid than the parallel one. Hence, only the antiparallel 0-structure and a-helix will be considered. METHOD OF CALCULATION A N D HYDROGEN B O N D POTENTIALS T h e Monte-Carlo t e c h n i q ~ e 'was ~ employed t o obtain the average statistical values. We generated a lot of conformations picked a t random in the vicinity of the regular structures, determined their energies, and averaged over them the parameters we were interested in.14 T h e angles cp and $ were picked a t random over 1" from the intervals cp = 128" f 16", I,$ = 128" f 16" for the a-helix and cp = 38" f 20°, I,$ = 320" f 20" for the antiparallel &structure. This procedure also gave the probabilities for angles cp and $ (i.e., the distribution functions for angles cp and $).
1064
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN 16
2.0
3.0
4.0
P i
E”/RT Fig. 1. Lippincott-Schroeder potential of the hydrogen bond r is the bond length; 6 is the bond angle; ro = 1.7 8, is the equilibrium bond length. Since at r < 1.6 8, the Lippincott-Schroeder potential is unreal, E H in this region is assumed to be infinite.
The distance between two identical chains of the 0-structure was taken to be equal to 9.3 A. Nonbonded and electrostatic energies of all atoms except the atoms 0 . . . H-N forming a hydrogen bond were assumed to be constant. This assumption undoubtedly is true for the individual chain because there is a broad flat area on the dipeptide energy map in the regions of the angles cp and II/ corresponding to the antiparallel 0-structure and a - h e l i ~ . Therefore ~ the internal rotation is restricted mainly by the interchain and intrachain hydrogen bonds. In our calculations two hydrogen bond potentials were used: the LippincottSchroeder potential15 (L-S) and that of Morse.16 In the first case the energy is a function both of the H . . . 0 bond length r , and of the ONH bond angle 0 (Fig. 1). The two potentials computed with the same equilibrium distance ro = 1.7 A are presented in Figure 2. The Morse potential is somewhat
ri
Fig. 2. Lippincott-Schroeder’ and Morse2potentials of the hydrogen bond.
MOLECULAR PARAMETERS OF TRANSITIONS
1065
I
-L -ri
-i -i
i,
4 ir iz
I)
A’P
Fig. 3. Total distribution function n(p) in the nine-monomer segment of the a-helix, calculated with the Morse potential (-) and with the Lippincott-Schroeder potential (-O-); A p = 128’ - p.
deeper and steeper than that of Lippincott-Schroeder, which should lead to a decrease in the internal rotation freedom in the a-helix and pstructure.
GEOMETRIC CHARACTERISTICS OF THE (Y-HELIX The distribution function on the angles p averaged over the all monomer conformations in the nine-monomer helical segment is shown in Figure 3. The standard deviation 6p = [ ( Ap2) - ( A P ) ~ ] , ” for ~ the L-S potential is equal t o 7.2’, which suggests the possibility of significant torsional oscilations within the a-helix. The same results were obtained for the distribution function on angles $. The average values of the internal rotation angles were found to be (p) = 126.8’, ($) = 131.3’, i.e., they are very close to their standard values in the a-helix cp = 123’ and $ = 132’. Thus, in spite of the considerable internal mobility, the a-helix can be treated as “ideal.” T h e use of the steeper Morse potential func-
0.I
0 f.0
f. 5
2.0
2.5.
Fig. 4. Distribution function on the hydrogen bond length in the a-helix. LippincottSchroeder potential (1);without any potential (2). Q ( r ) calculated with the Morse potential does not differ markedly from curve (1).
1066
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN
10
PO
Fig. 5. Distribution function on the hydrogen bond angle in the a-helix: with use of the Lippincott-Schroeder potential (1); without any potential (2).
tion results in a slight narrowing of the distribution functions Q(q)and N $ ) (Fig. 3). The distribution function on the hydrogen bond lengths Q ( r )is given in Figure 4. The sharp drop of Q ( r )a t r < 1.6 %, is connected with the unrestricted growth of the L-S potential which we assumed in our calculations. It is seen that Y3 of the hydrogen bonds are of lengths r = 1.6-1.9 A which correspond to the minimum of EH(r,B). The function R(r) obtained with the hydrogen bond potential being omitted is also presented in Figure 4. In this case all angles cp and $ from the given in) n($)have the form of a recterval are of equal probability and n ( ~and tangular impulse in the interval cp,$ = 128O f 16'. As can be seen, the use of the hydrogen bond potential considerably enriches Q ( r )in the region close to ro mainly a t the expense of the states with the small r. A t the same time even without potential, 40% of the bond are of length r = 1.75 f 0.15 A, that is, this value of r is more favorable from an entropy point of view than others. Hence there are a number of ways to form the lengths r = 1.75 f 0.15 %, in the a-helix a t the variation of three pairs of angles cp and $. The distribution functions Q ( 0 ) calculated both with the potential 'y3
'y3
rsb.
iro" (a)
ffOO
f30"
fP0'
(b)
Fig. 6. Maps of lengths (a) and angles (b) of the hydrogen bond of the three-monomer segment of the a-helix. rp1 = = 123O; $1 = $2 = 132O; a and $3 are varied.
MOLECULAR PARAMETERS OF TRANSITIONS
1067
0.I 0.05
-16
-8
0
(d)
b
16
4 y4 (el
Fig. 7. Distribution functions in the different monomers of the a-helix ( A @ = 128O pz). The Morse potential.
and without it are given in Figure 5. T h e distinction between these functions is greater than t h a t for the functions Q ( r ) . As a result of the strong dependence of the L-S potential on the angle B the maximum of the Q(0) is in the region 5-10', though the value 0 = 15' is more favorable from an entropy point of view. Figures 4 and 5 show that in spite of considerable variation of the angles cp and $, the hydrogen bond parameters in the irregular a-helix are more or less close to their optimal values. This is apparently the result of the fitting of the angles cp and $ in the three-monomer segment. T h e hydrogen bond lengths and angles of the three monomer segment are shown in Figure 6. The conformations of the first two monomers of the segment were fixed (at the standard values of angles cp and $), while two angles in the third monomer were varied. The figures show t h a t there is a wide interval of the angles cp and $ which corresponds t o the almost linear hydrogen bond with the length close t o ideal. I t is obvious t h a t simultaneous alteration of the angles in three successive monomers can only expand the region of possible variation of the angles cp and $. T h e distribution functions on angles cp and $, averaged over the conformations of each monomer taken separately, are more detailed characteristics of the internal rotation (Figs. 7a-e). These functions show the change of internal freedom while removal from the end of the helix takes place. T h e maximum of the distribution function is displaced t o the right until it reaches its extreme value a t cp 3 124' (Figs. 7d, 7e); simultaneously, the width of the distribution function decreases from 20' for the terminal monomer to 14' for middle ones, thus pointing t o the significant differences between the monomer conformations. However,
qi+i
-0.02 0.12 -0.14 -0.08 -0.09 -0.03 -0.04 -0.02 -0.04 -0.01
i
14 13 12 11 10 9 8 7 6 5
-0.24 0.15 -0.04 0.03 0.05 0.04 0.03 0.01 0.02 0.06
Cpilpi- 1
-0.22 -0.33 -0.38 -0.37 -0.35 -0.38 -0.30 -0.32 -0.31 -0.28
qiGi-1
-
-
-0.11
0.37 -
-
-0.16 -0.13 -0.18 -0.14 -0.14 -0.13 -0.09
c~id~i-2
0.51 0.49 0.54 0.49 0.47 0.40 0.43
qi~i.2
-0.10 -0.08 -0.12
-0.10
-0.04 -0.03 -0.07 -0.10 -0.08 -0.08
$iPi-1
-0.002 -0.12 -0.13 -0.02 -0.05 -0.02 -0.08 -0.12 -0.05 -0.10
$i$i-1
-
-
-0.16
-0.08 -0.16 -0.14 -0.18 -0.23 -0.13 -0.13
Giqi-2
-
-0.004 -
0.02 0.03 0.0004 -0.02 0.001 -0.05 -0.006
$i*i-2
TABLE I1 The Correlation Coefficients of the I4 Monomer Segment of the a-Helix
0.05 0.02 -0.04 -0.08 -0.06 0.02 0.03
-0.14 -0.16 -0.13 -0.19 -0.09 -0.18 -0.19 -0.13 -0.19
-
-0.03
cPi+i, ~i-2$i-2
-0.0001
cPi$i> qi-l+i-1
?2
F 2
M
gu
,C
g
4
p 0
2 CJI
"8
c3
3:
5Cn3
W
MOLECULAR PARAMETERS OF TRANSITIONS
1069
A 'p7 f6
-
!6
- I6
-8
0
(a)
-
-8
0
8
(b)
* 'p9
Fig. 8. Bivariate d i h b u t i o n function on the internal rotation angles. (a) Q(a,w); (b) Corresponding correlation coefficients are K = +0.4 and K = -0.38. The figures indicate the probabilities of realization of the given pair of rotation angles. Q(~&X).
middle and terminal monomer entropies being the logarithmic function of the number of states differ ~ 1 i g h t l y . l ~ The energy of the hydrogen bonds between i and i 3 monomer in the a-helix is determined by the six angles pi, # i , p i + l , $;+I ~ ; + 2 ,#i+2. I t is obvious that the angles cannot change independently. T h e degree of the correlation between different angles can be estimated by calculating the correlation coefficient
+
T h e value of K is within the limits -1 < K < 1, while the values K = fl correspond t o a linear dependence between the pair of parameters under consideration and the sign defines a slope of the plot. In the ab= ($;) ( $ j ) and the correlation coeffisence of the correlation ($i$;) cient is equal to zero. We consider the a-helical segment t o consist of fourteen monomers. T h e correlation coefficient was obtained for different angles belonging either to the same monomers or to different ones, but not far removed from one another. As can be seen from the results given in Table 11, the correlation coefficient changes within a wide interval from -0.4 t o +0.54. T h e value of K for any chosen pair of internal rotation angles does not depend on the indexes of the monomers dealt with. T h e only exception is when one or both angles is in the first or second monomer. T h e maximum correlation is between angles pi; $;-I and pi, p i - 2 in the different monomers; near angles pi, $; and pi, pi-1 as well as ones farther do not show a strong correlation. T h e binary distribution functions Q ( ( o s , ~ and ) Q((ps,$s) for strong correlated an-
1070
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN
gles are pictured in Figures 8a and 8b. Both functions possess the obvious asymmetry that accounts for the value and sign of the correlation coefficient. Besides the correlations between single angles those of the pairs of angles vi$i and cpi-l$i-l as well as cpi$i and cpi-24i-2 were calculated. It was found that the former are weak enough -K N f0.15 and the latter are about zero. The variation of the internal rotation angles in the a-helix seems to be the main cause of its flexibility. The helical segments considered are too short to estimate the persistence length of the a-helix from their dimensions. Nevertheless, there are analytic expressions connecting the persistence length and the angles of internal rotation for the independent oscillation of neighbor monomers.17 For this approximation the obtained value of the persistence length is equal to 200 A, which is several times less then the experimental one. Being taken into account the correlations between the internal rotation angles lead to the greater values of the persistence length.
THERMODYNAMIC CHARACTERISTICS OF THE a-HELIX As distinct from the random coil state, the presence of the hydrogen bonds in the a-helix leads to the correlation of the conformations of the monomer units. Hence, the different microstates of the N-monomer helical segment are determined by the set of 2N internal rotation angles cp&. The energy of each microstate is the sum of the energies of ( N 2) hydrogen bonds and the conformational energy of the a-helix. The partition function of the system may be found by the use of the average value (eEIRT) = K-1
K C eEiIRT i=l
(3)
where Ei is the energy of the i-th conformation, K is the number of conformations over which the averaging is carried out. Indeed in the canonical ensemble
where ro = J’; w(E)dE is the number of all possible chain conformations with a given N , and hence Z ( N ) = I’o/(eEIRT) (in our case the number of microstates of each monomer is g = 322 and r o = g N = 322N). For the method described above, the slow convergence of the results, and hence a low accuracy of calculations are inherent. Indeed, the main contribution to ( eE/RT) is due to the less favorable conformations (conformations with high positive energy). The accidental appearance of several unfavorable conformations failing to affect the average statisti-
MOLECULAR PARAMETERS OF TRANSITIONS
I: '
1071
\,
8.0 0.2
0.4 '/N
Fig. 9. Dependence of the monomer free energy in the a-helix on 1/N (the conformational energy of the helix is not taken into account). The Lippincott-Schroeder potential.
cal characteristics considerably changes the computed value of the free energy. Therefore the method is unapplicable to the systems with the areas unfavorable energetically in which the system occasionally appears. T h e more energetically heterogenous the system (for example, there is a series of the narrow energy wells), the greater the error in calculation of the free energy. In our case the hydrogen bond potentials have a single minimum, and hence the system is energetically homogeneous. Nevertheless, to check on the calculation we also found the free energy of the system using a statistical ensemble of a variable number of monomers.14 In the a-helix the extreme values of a monomer free energy found by both methods differ only slightly. Since we presumed that the difference in energies E; of microstates is determined only by the hydrogen bond energies, the value F(N)IRT = -In Z ( N ) = -In I'o In ( e E I R T )which is related t o the free energy of the a-helix as follows
+
may be found. On the other hand, and thus
L-x 0
Fhel.(N)
= ( N - 2)Fhel.
. 5
+ 2Fbound.
N
10
Fig. 10. Dependence of the average hydrogen bond energy (Lippincott-Schroeder potential) on the number of monomers in the helical segment.
1072
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN
0
4.0
5.0
-RT En
Fig. 11. Distribution function on the hydrogen bond energies in the nine-monomer helical segment (seven hydrogen bonds).
Therefore F(N) (see Eq. (5)) may be presented as
- Ehel.conf. + -1 ( A F i n i t . - 2 W (7) RT N RT The dependence of F(N)IN R T on 1/N according to Eq. (7) is presented in Figure 9. As can be seen the value F(N)/N R T depends linF(N) -NRT-
Fhel.
early on 1/N, thus allowing the values of two parameters (Fhel. = -11.4 and ( A F i n ; t . - 2AF)/RT N 9.4 to be determined. Supposing that Ehel,conf. = Econf,reg. = E a d i p . and using the data in Table I, we obtain F h e l . / R T = -8.7, AFIRT = -1.15 and AFini,/RT = 7.3. Hence the parameters of the helix-coil transition are s N 3 and CJ N 7.10-4. Figure 10 shows the dependence of the average hydrogen bond energy (EH)= (E)/(N- 2) on the number of monomers in the helical segment (the distribution function ~ ( E His) presented in Figure 11). The extreme value ( E H ) / R T = -4.8 is about 85% of the minimum energy for the L-S potential. Combining the plots in Figures 9 and 10 one can obtain the dependence of the average entropy per monomer on the number of monomers in the helical segment (see Fig. 12). A t N = 1 or N = 2 the entropy is S h e l . / R = In g = 6.93. It is also possible to estimate the helical state entropy in the presence of the hydrogen bonds (N 2 3) using the distribution functions Q(v) and Q($) and assuming that there is no correlation between angles cp and #. According to the Boltzmann formula it is EheI.conf.)/RT
where the summation extends over all angles of internal rotation. The value S h e l . / R = 6.75 obtained in such a way is shown in Figure 12 by a dotted line and is just the upper value of the S h e l . because the presence of the correlations between the angles of different monomers may only decrease the entropy of the system. If the free energy of the monomer in the helix Fhel. is known, it is also possible to obtain the value of the conformational free energy Fhel.conf. =
-6.6 -8.5 -7.5
-5.1 -4.1 -0.9
A
-6.4 -5.4 -2.3
B
-4.1 -3.9 -3.1
Ref. 5
A -4.0 -3.3 -3.3
B
Present worka
R
AS,
-3.7 -3.0 -3.0
~
-2.5 -4.6 -4.4
Ref. 5
a A represents data obtained with the Lippincott-Schroeder
Gly Ala(R) Ala(L)
Polypeptide Ref. 5
Present worka
RT
AHS
~
-2.4 -2.1 1.0
B 6.5 9.3 9.0
Ref. 5
AH,
-0.5 1.3 7.8
A -0.6 1.4 7.8
B
Present worka
RT
-7.5 -5.1 -5.7
Ref. 5
4 . 0
-7.5 -6.0
A
-8.0 -6.6 -6.6
B
Present worka
R
As, ~
14.1 14.4 14.7
Ref. 5
7.0 7.3 13.8
A
7.4 8.0 14.4
B
Present worka
RT
AFinit.
potential; B represents data obtained with the Morse potential.
-1.3 -1.1 2.1
A
Present worka
RT
AF
TABLE I11 Thermodynamic Characteristics of Different Polypeptides
1074
BIRSHTEIN, SKVORTSOV, A N D ALEXANYAN
5
10
Fig. 12. Dependence of the monomer entropy on the number of monomers in the helical segment.
(Ehel.conf. - TShe1.J = (Fh,l. - E H ) which is equal to -4 R T or -2.4 kcal/mole. Fbound. = (AFiniJ2 Fcoil) is about the same value; therefore Fhel,conf, is the same a t the edges and in the middle of the helix. It arises from the weak dependence of She1. on the number of hydrogen bonds (Fig. 12). All thermodynamic characteristics of the poly(L-alanine) a-helix (left and right) obtained with the L-S and Morse potentials are presented in Table 111. The data pertaining to the a-helix of the glycine, which can be easily obtained from the dipeptide conformational energy maps, are also included into the table. As can be seen, the thermodynamic characteristics of the helical state are stable regarding the choice of the hydrogen bond potential. The data in Table I11 also allow the results obtained in different approximations and by different methods to be compared. GO et al.5.6 have calculated the thermodynamic parameters of the a-helix, assuming that the oscillations of the angles cp and rc/ are small (harmonic approximation). In this approximation ( ( E ) - Emin.)/RT = 1 (Emin. is the minimum energy of the regular conformation). Our calculations show that ( ( E ) - Emin.)/RT is equal to 0.8 for the L-S potential and 0.6 for the Morse potential. The entropy differences A S , and A S , quoted by GO et al.5 and in the present work are similar, so both the harmonic approximation and the assumption about the independence of the conformational energy on the angles cp and rc/ proved to be correct. It is to be noted that the difference of the right and left a-helix entropies for the poly(L-alanine) quoted by GOet al.5 is equal to 0.3 but we supposed it to be zero. The distinction characterizes the degree of accuracy of these calculations. In contrast, the energy differences A H , obtained by various methods differ considerably in values. The distinction is mainly associated with the fact that we used the dipeptide approximation while GO et aL5 took into account long-range interactions, which are determined to a great extent by electrostatic interactions of the neighbor peptide groups. The value A H , / R T * 10 for the right helix of the poly(L-alanine) obtained5 at t = 4 leads to a high value AF;,;,/RT N 15, which differs sharply from the experimental values 5-71° (calculation at e = 1 gives unreal values A H J R T = 20 and AFinit./RT = 40). A t the same time our data A F i , i , / R T N 8 and A H J R T N 0 are in exact accor-
+
MOLECULAR PARAMETERS OF TRANSITIONS
1075
0.10
0.05
-
0
20
AY
Fig. 13. Distribution function of angle q in t h e helical segments of myoglohin
dance with experimental results. This apparently means t h a t a t real conditions there is a strong screening of the polar groups removed far from one another. Therefore, use of the general value t = 3-4 excessively increases the contributions of distant groups and electrostatic potential should not be taken into account a t the distances more than 6 A.6
ac
1
0
II 1P--0
10
10
..
rA
2.5
2.0
1.5
30
8"
(b) Fig. 14. Distribution function of lengths ( a ) and angles ( h ) of the hydrogen bond in myoglobin (1) and lysozyme (2).
1076
BIRSHTEIN, SKVORTSOV, AND ALEXANYAN
COMPARISON WITH X-RAY ANALYSIS DATA It is possible to obtain the internal rotation angles cp and J/ in the helical segments and to calculate the hydrogen bond parameters on the basis of the X-ray analysis of protein crystals. The detailed conformations of the protein monomers are obviously determined not only by interactions along the chain, but also by the globular packing of the neighboring regions of the chain. Nevertheless, the comparison of the experimental data with the results of calculations for the individual helix can be useful for the right choice of the effective hydrogen bond potential for proteins. The distribution function on the angles cp and J/ in the myoglobin helical segments is given in Figure 13. It is interesting to note that this function almost coincides with that on angle cp which we have calculated for the individual chain (Fig. 3, L-S potential). The distribution functions on the hydrogen bond lengths and angles in the myoglobin and lysozyme are shown in Figure 14. As can be seen from Figure 14a the distribution function Q ( r )for the myoglobin has a maximum near r = 1.7-1.9 A, but the distribution function n(6) (Fig. 14b) has a maximum a t the 0 = 10-15' (see for comparison Figures 4 and 5). The distribution functions Q ( r )and Q(6) (dotted line in Figure 14) are broader than that for myoglobin; this could be caused by the small number of the helical monomers in lysozyme. The results obtained show that the effective hydrogen bond potential in the helical segments of globular proteins is sure to give a smooth growth of the energy, while r increases from its optimal value and a weak angular dependence, i.e., the potential is more gentle than those of L-S and Morse.
GEOMETRIC AND THERMODYNAMIC CHARACTERISTICS OF THE ANTIPARALLEL STRUCTURE
8-
A t first we shall consider "hairpin" structures consisting of two neighboring chains with one, two, three, and five monomers per chain. The results obtained show that the thermodynamic and geometric parameTABLE IV Geometric and Thermodynamic Parameters of the &Structure of Different Types
-
37"
319"
10"
38"
319"
8"
/VV\J
37"
320"
7"
1.84
4.2
7.2
11.4
1.8
4.7
6.5
11.2
1.73
4.9
6.5
11.4
MOLECULAR PARAMETERS OF TRANSITIONS
1077
0.15 0.05
0.fO 0.05
0
0.fO 0.05
0
E/R T
Fig. 15. Distribution function of the internal rotation angle q (a), hydrogen bond length (b), angle (c), and energy (d) for the &structure (two chains with three monomers per each chain).
ters of the “hairpin”-&structure do not depend on its length. T h e average geometric parameters of the “hairpin” of three monomers, as well as certain thermodynamic characteristics, are given in Table IV (first line). T h e distribution function on the angle p of the three-monomer system is shown in Figure 15a (the distribution on the angle $ is of the same shape). T h e plot shows considerable freedom of the internal rotation in such a system (about f15’), and a n increase in the “hairpin” size fails to give a diminution of the allowed interval of the angles. However, in spite of the great deviation of the angles 9 and $ from their standard values, the hydrogen bonds are only slightly distorted. In Figures 15b and 15c the distribution functions on the lengths and angles of the hydrogen bonds are shown. The two distributions turn out to be narrow and therefore the average energy of the hydrogen bonds can reach 75% of the minimum energy. The distribution on the hydrogen bond energies is presented in Figure 15d. The entropy per monomer is equal to 7.2. In the case of the free rotation around the bonds N-C and C-C’ in the intervals f 2 0 ” the entropy is SolR = 2 In 40 = 7.4. The close values of S and So also point out the great freedom of the internal rotation in the p-structure.
1078
aro
BIRSHTEIN, SKVORTSOV, A N D ALEXANYAN
0.90
’
0.05
AfO
D
20
40
0.20-
OdO
-
Fig. 16. The same as in Fig. 15. One of the chains in @-structureis fixed.
Besides the “hairpin” structure, the three-chain system was also calculated. An increase in the cross dimensions slightly influences both the geometric and thermodynamic characteristics. Therefore, the segments of the antiparallel P-structure, irrespective of its cross and longitudinal dimensions, possess considerable freedom of internal rotation. The internal rotation angles change within the interval f15-20°, that is, a t least two times more than the allowed interval for the a-helix. In our calculations the parameters sp and a p of the 0-structure-coil transition were also obtained. It was found that the free energy of a monomer in the middle of every chain of the 0-structure is about that at the edge of the @-structureand hence ag N 1. Thus, as distinguished from the a-helix, where the initiation of a helical region is strongly hindered, in the &structure both its initiation and continuation (increase in longitudinal dimensions) are thermodynamically equivalent. A t the same time the equilibrium constant sp of the 0-structure-coil transition (per monomer) is roughly equal to that of the a-helix-coil transition. I t is interesting to note the restriction of freedom of the internal rotation when one chain in @-structureis fixed. Such a situation may occur
MOLECULAR PARAMETERS OF TRANSITIONS
1079
in the sorption of the polypeptide chain on a flat surface. For this purpose the two chain &structure with the same pairs of angles p and $ in one of the chains was calculated. T h e result proved t h a t the average values of parameters of such a system (see Table IV) are also independent of the number of monomers in the chain. T h e distribution functions on the angle p, on lengths, angles, and energies of the hydrogen bonds are shown in Figures 16a-16d. The figure shows that despite the almost 50% decrease in the allowed interval of angles, the latter remains wide enough. There are no essential changes in energy (its decrease is about 10%). In the system of three chains the maximum restriction will occur if the variations of the internal rotation angles take place only in the middle chain. In such a system (average values see in Table IV) all distribution functions narrow but slightly. Thus the entropy and energy of the middle chain change slightly while the neighboring chains are fixed. T h e monomer entropy remains close t o SIR N 6.5, and the average energy of hydrogen bond is ( - E I R T ) = 4.7-4.9, i.e., about 3 kcallmole. We found that the entropy of the a-helix is Shel.lR N 6.7 and hence, it is close to t h a t for the 0-structure, while the hydrogen bond energies of those conformations are roughly equal. Since the conformational energy of the monomer in the 0-structure is approximately the same as that in the a-helix, the free energies of these states are close to each other. On the other hand, the poly(L-alanine) in solution is observed either in the helical or in the coiled state and does not form any antiparallel pstructure. The origin of the discrepancy may be thought of as follows. Folding of the polypeptide chain leads to the rise of unfavorable bends. This fact was not taken into account and therefore we could not estimate the free energy of the 0-structure bend. Besides, the finite dimensions of the chains slightly influence the energy of helical state (in any one-dimensional system about 1/N of a whole energy is lost a t the boundary), whereas in the 0-structure the loss of the energy decreases as 1/N1/2with growth of the system, i.e., much more slowly.
References 1. Birshtein, T. M. & Ptitsyn, 0. B. (1966) Conformations of Macromolecules, Interscience, New York. 2. Birshtein, T. M., Eljaschevitch, A. M. & Skvortsov, A. M. (1971) Mol. B i d , 5 , 7890. 3 . Adonts, V. G., Birshtein, T. M. & Skvortsov, A. M. (1973) Repts. Akad. Sci. Arm. SSR, 3,168-173. 4. Adonts, V. G., Birshtein, T. M., Eljaschevitch, A. M. & Skvortsov, A. M. (1976) Riopolymers, 15,0000-0000. 5. GO, M., GO, N. & Scheraga, H. A. (1970) J . Chem. Phys., 52,2060-2079. 6. Go, M., GO, N. & Scheraga, H. A. (1971) J . Chem. Phys., 54,4489-4503. 7. Volkenstein, M. V. (1955) Structure a n d Physical Properties of Molecules, Acad. of Sci. USSR, Moscow. 8. GO, N., GO, M. & Scheraga, H. A. (1968) Proc. Natl. Acad. Sci. ( U . S . ) ,59, 10301037.
1080 9. York. 10. 11. 12.
BIRSHTEIN, SKVORTSOV, A N D ALEXANYAN Flory, P. I. (1969) Statistical Mechanics of Chain Molecules, Interscience, New
Alexanyan, V. I. & Skvortsov, A. M. (1974) Mol. Biol.,8,182-192. Brant, D. A. (1968) Makromolecules, 1,291-298. Skvortsov, A. M., Birshtein, T. M. & Zalensky, A. 0. (1971) Mol. Biol.,5.69-77. 13. Hamersley, I. M. & Handscomb, D. C. (1964) Monte-Carlo Methods, John Wiley, New York. 14. Skvortsov, A. M., Birshtein, T. M. & Zalensky, A. 0. (1971) Mol. Biol.,5.390-398. 15. Schroeder, R. & Lippincott, E. R. (1957) J. Phys. Chem., 61,921-932. 16. Popov, E. M., Dashevsky, V. G., Lipkind, G. M. & Arkhipova, C. F. (1968) Mol. Biol., 2, 612-621. 17. Birshtein, T. M. (1974) Vysokomol. Soedin., 16A. 54-62. 18. Ptitsyn, 0. B. (1969) in The Modern Problems of Peptides and Proteins Chemistry, 95, Nauka, Moskow.
Received July 9,1975 Accepted November 4,1975
