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2 calculations of the relative stability of poly (l -proline I) and poly (l -proline II)
CNDO/2 calculations of the relative stability of poly (L-proline I) and poly (L-proline II) Masaru Ohsaku Department of Chemistry, Faculty of Science, H iroshima University, H igashisenda-machi, H iroshima 730, Japan and Akira I m a m u r a Department of Chemistry, Shiga University of Medical Science, Setatsukinowa-cho, Otsu, Shiga 520-21, Japan (Received 11 September 1979; revised 3 December 1979) The CNDO/2 method using the tight binding approximation Jbr polymers was applied to poly(L-proline I) and poly(L-proline II). The calculations were also carried out for poly(L-alanines) and model molecules which have the same hackhone geometries as those of poly(L-prolines). The results obtained show that both Jorms of poly(Lproline I) and poly(L-proline 11) have nearly the same energy in agreement with experimental results. From the analysis of the total energy, it wasfound that the intrasegment energy of poly(L-proline II) was lower than that of poly(L-proline I) while the intersegment energy of poly(L-proline 1) was lower than that of poly(L-proline 11). This result can be considered to correspond well with the experimental.fact that poly( L-proline 11) is more stable in good or polar solvents and poly(L-proline I) in poor or non-polar solvents. The analysis of the total energy of poly(Lproline) leads us to the conclusion that the ot and fl carbons play an important role in determining the relative stability between poly(L-proline I) and poly(L-proline II) and the Y carbon does not have a marked effect on the electronic structures of" the polymers in question. This conclusion was also confirmed by comparison of" the electronic structures ofpol)~L-prolines) with those of poly(L-alanines) and the model compounds concerned.
One centre term:
Introduction Poly(L-proline) exists as two conformational isomers, I and II, the structures of which have been elucidated by Xray analysis 1-4. For simplicity we will hereafter use Pro-I and Pro-II for poly(L-proline I) and poly(L-proline II), respectively. These two forms exhibit markedly different optical rotation s. In Pro-I or Pro-lI there are no hydrogen atoms present to form hydrogen bonds and the rigidity and unique shape of the pyrrolidine ring make it very difficult for this residue to fit into the normal s-helix conformation. Therefore, it is expected that when Lproline is the major constituent in a high polymer, a unique type of ordered structure will result 6. This is the case for collagen. On the other hand when L-proline is the minor constituent in a polymer, a natural distortion or degtruction of the normal s-helical structure can be expected. In the present work, in order to obtain information on the structural role of L-proline sequences in a protein or on its behaviour in solution, we have carried out C N D O / 2 calculations 7 for this species using the tight-binding approximation s .
onA
EA = Y. P , £ - ~ / , + A , ) - ( Z 4 - ~,z., '*'' o.o.j + /
onA onA
½ZZ [P.P,,,,-~(P.,) ,
t
lj
Intrasegment t w o centre term: onA onB
(o,o)_ 2flaB o ~ ~' E4.. ~ - p, , ~.~(o,o) ,, _ -
l onA o n B
~lllt]
I
rp
-}- L
~, .(o,o)
4 4--BB/4, B
p
B
-- Z BB~4[4,B
(3)
--[~4B
o n 4 onB
Z
~' p+j ~(o.j) ~
--t,t
~,t,
onA o n B
Z
The total energy of a polymer can be expressed as follows9:
l
+ ~rp
o
.,~o.j~ o
2L~4.41BB/4,B
Z E A + E E E'°'°' A,B + E E E E(°'/' ,,
0141 8130/80/060347 08502.00 ©1980 IPC Business Press
~ ~,(o,o) D 7 ,,(o,o)
-- --AA~B/A.
Intersegment two centre term:
M e t h o d o f calculation
A B<4
14,B
II
+ ( Z ~Z~e2/R 4R)]
4,B
A
fl
½Z Z'"
ElO,j)_ o
Etotal =
(2)
] , 4..4
J
4
B
(1)
alll
i ~.B
tl
7 .,(o.j~
-- ~44~B[4,
B
--
p
HB
Z .,qo.j~ 4/
4,B
+ (Z 4ZBe2/R 4a)]
Int. J. Biol. Macromol., 1980, Vol 2, December
(4)
347
Relative stability of poly(L-prolines): M. Ohsaku and A. Irnamura
02
02
C1
C'1
/
\
N3"---'------~q 4~
H14~CA/ /
\\
~
/
~
~5"~C5
/\
H12
-
HT/
=z/\
C4~H 8 /
._.__..-H9
C15
C1
/
N3
H8
H18
C19
/
N3
C4,..~
H20//H14~c7.//
/"\ ~'~H8 ~/H9
\
~Hg
Hll
/\
H1Q
a
H16
H12
b
Hll
c
Figure I Schematic structures and atom numberings of: (a) poly(L-proline); [b) polylL-alanine); lc) model molecule. With model molecules in relation to the methyl groups, the atomic position N3 C4 C 15 H 16 was assumed to be trans and N3 C 1 C 19 H20 to be <'i.~
Table I
Atom coordinates of Pro-I and Pro-II Pro-I
Atom a CI 02 N3 C4 C5 C6 C7 H8 H9 H10 HI1 H12 HI3 H14
x 1.46772 2.09614 1.97621 1.31209 2.41165 3.56292 3.41775 0.84822 2.69295 2.09606 4.48391 3.62234 4.11885 3.50031
Pro-ll
y -0.84179 - 1.19737 0.00000 0.47925 0.98977 1.26772 0.54135 -0.29034 0,21376 1,92000 0,95565 2,33928 -0,29324 1,26410
:
x
-0.84047 - 1.88856 0.00000 1.22072 2.11746 1.22827 -0.06010 1.65952 2.82933 2.58986 1.72071 1.03771 -0.06352 -0.87184
-0.00195 1.14683 -1.04627 -0.89322 -2.27636 - 3.21795 -2.52354 -0.30964 - 2.42519 -2.42486 -4.04883 -3.60872 -2.87104 - 2.65916
y -0.28927 0.14103 0.00000 0.89572 1.39446 0.42509 -0.37917 1.66448 2.37198 1.40554 0.96323 -0.23634 -0.05408 - 1.43458
: -0.75373 -0.57277 0.00000 1,15549 1,46909 0,86035 -0.17293 0.89387 1.01040 2.54887 0.40412 1.63359 -1.15357 0.06333
Atom numberings are shown in Fioure la
where A or B denotes atom, t or t~ atomic orbital, j the n u m b e r of segments, and 0 the central segment. The other notations are the same as those usually used. In equations (3) and (4), the first term of the right hand side is the core resonance term, the second the exchange term, and the third the electrostatic term respectively. Numerical calculations were made according to the previous paper 9. Schematic structures and a t o m numberings are shown in Fi.qure 1. The geometries of Pro-I and Pro-II were referred to those reported 14. At the start the c o m m o n structural parameters both for Pro-I and Pro-If were used for the bond lengths r ( C l N3), r(N3 C4), r(C4C'I), and r(C1 O 2 ) , and the geometry of a ring part C5 ~ C 7 and H 9 ~ H 1 4 , and then the ring part was adequately corrected for each ring to form a cyclic ring exactly. The bond angles (CIN3C4), (N3C4C'1), and (O2C1N3) were transferred from the data of X-ray analysis 1.4. The torsional angles
348
Int. J. Biol. Macromol., 1980, Vol 2, December
Ala-I or Ala-II and Model-1 or Model-II for poly(Lalanine I) or poly(L-alanine II) and model molecules, respectively. Here, the backbone of Ala-I and M o d e m is the same as that of Pro-I, and the b a c k b o n e of Ala-II and Model-II is the same as that of Pro-II. The coordinates estimated are summarized in Table 2. Non-listed atomic coordinates are available from Table 1.
Results and discussion Pro-I and Pro-ll The total energies calculated are shown in Table 3. Up to the present time, some empirical and q u a n t u m mechanical conformational analyses of L-prolyl residues, poly(L-prolines), and their model molecules have been reported~l 25. The P r o - l ~ P r o - I I cooperative order ~ o r d e r transition has been treated by Schwarz 26", Applequist26L and Ganser et al.2% Mattice and Mandelkern 27 have proposed the r a n d o m coil model for Pro-II in water, trifluoroethanol, acetic acid, propionic acid, and benzyl alcohol. The statistical treatment of the cooperative transition between two ordered conformations of Pro-I and Pro-lI and of the r a n d o m coiled
R e l a t i v e s t a b i l i t y o f poly(L-prolines): M . O h s a k u a n d A. I m a m u r a
Table 2
Atom coordinates of Ala-I, Ala-II, Model-I, and Model-II
Atom a'b
x
H6 H7
3.25898 2.91166
C15 H16 H17 H18 C19 H20 H21 H22
0.35871 -0.09080 -0.42485 0.91258 0.06018 -0.34549 0.10499 -0.58171
3'
z
Ala-I 1.19434 0.35130 Model-I 1.65352 1.94796 1.35422 2.49495 -1.39558 -0.98397 -2.48207 -1.11524
x
1.46302 -0.03900
-2.96881 -2.00871
0.99042 1.93876 0.29433 0.57410 -0.65017 0.27400 -0.57512 -1.48532
-0.24481 -0.13944 0.73848 -0.87153 -0.29047 -1.34644 0.31528 -0.04579
)' Ala-ll 0.68158 -0.24703 Model-I1 0.15411 0.83276 -0.20855 -0.69021 -1.23117 -1.50137 -2.13176 -0.73140
--
1.02142 -0.11266 2.32616 3.17258 2.02664 2.61326 -1.92440 - 1.92160 - 1.82394 -2.86166
Atom numberings are shown in Figures lh and lc b Non-listed atomic coordinates are available from Table I Table 3
Total energy (eV) of Pro-I and Pro-II
Energy Total Total intrasegment Total one centre Total two centre Total intersegment Total two centre 0 0 0 0 " See footnote **
1" 2 3 4
Pro-I
Pro-II
- 1937.03 - 1908.44 - 1553.41 - 355.03 - 28.60 - 14.23 - 0.02 -0.03 -0.02
- 1937.03 - 1908.77 - 1553.39 - 355.39 - 28.25 - 14.11 - 0.02 0.00 0.00
(Energy, see equations 1 4.)
Pro-II have been reported by T a n a k a a n d Scheraga 28. At the present time it seems that there are n o completely settled models to explain the molecular structure of poly(L-proline) in solution. P u l l m a n et al. have carried out the P C I L O calculations for a model molecule of the Lprolyl residue in its h o m o p o l y m e r s 23. F r o m the calculations they have shown that in these c o m p o u n d s the stable c o n f o r m a t i o n s are limited, both for cis a n d t r a n s polymers (only to qJ centred a r o u n d 150 °) a n d that the t r a n s form should be a b o u t 1.4 kcal m o l - 1 more stable t h a n the cis form. In the present work, the total energy is nearly equal both in Pro-I a n d Pro-II. This means that both forms can coexist energetically, a n d this finding is in complete agreement with the experimental result that I a n d II forms coexist in the solid state i n d e p e n d e n t l y a n d in solution d e p e n d i n g u p o n the kinds or the c o m p o s i t i o n of the solvents 5. It should be pointed out that the structure of the polymer in solution may n o t be an ordered one but a mixed one. However, as is pointed out by T a n a k a a n d Scheraga 28, there are no marked differences between the all t r a n s Pro-II a n d the mixed structure of 5'Yocis a n d 95'y,, trans conformations. C o n s e q u e n t l y , it may be expected that the present result gives i n f o r m a t i o n qualitatively on the relative stability of Pro-1 a n d Pro-II in solution. The total intrasegment energy of Pro-II is smaller t h a n that of Pro-I, whereas the total intersegment energy is smaller in P r o - I I t h a n that in Pro-I. That is, the total i n t r a s e g m e n t energy favours the trans form P r o - l I , while the reverse is true for the total intersegment energy. After b a l a n c i n g these two terms, it is found that the total energy is nearly equal in Pro-I and Pro-I1. A brief statistical a n d t h e r m o d y n a m i c a l t r e a t m e n t of the free energy of this
polymer is discussed in the Appendix. By c o m p a r i n g the total i n t r a s e g m e n t one centre and two centre energies, it is found that the former contributes nearly equally to Pro-I a n d Pro-II. The energy of the r e m a i n i n g two centre term of P r o - l I is lower by 0.36 eV (8.3 kcal) per segment t h a n that of Pro-I. We will discuss in more detail the total i n t r a s e g m e n t and total intersegment energies below. As for an L-prolyl residue, the t r a n s form is indeed more stable t h a n the cis form j u d g i n g from the intrasegment energy. If a single residue of L-proline is isolated in a polypeptide chain, the residue may favour the t r a n s c o n f o r m a t i o n . Shimmel and Flory have also noted after e x a m i n a t i o n of the structure of polypeptide copolymers c o n t a i n i n g L-proline in low proportions, that the polymer residue is in its t r a n s c o n f o r m a t i o n as in Pro-II 2°. This is also confirmed by X-ray crystallography. For" example, from the crystal structure analyses of oligomers such as L-leucyl-L-prolyl-glycine29 a n d acetyl-L-prolineN - m e t h y l - a m i d e 3° it was concluded that the peptide groups should have the t r a n s c o n f o r m a t i o n . However, in the case of h o m o p o l y m e r s such as the case treated here,* Pro-I gets much more energy from the intersegment interactions t h a n Pro-lI. The Pro-I form is more condensely packed than Pro-II. Therefore the interactions between segments are stronger in Pro-I t h a n that in ProII. This is very interesting in c o n n e c t i o n with the conform a t i o n a l change of poly(L-proline) d e p e n d i n g u p o n the properties or c o m p o s i t i o n s of solvents in solutions*. In n o n - p o l a r (poor) solvents the more packed form Pro-I * In this work, we have taken into consideration up to nine monomer units in each case. In the calculation with less than seven units, the trans form should be more stable than the cis form. Tanaka and Scheraga have also concluded that with less than seven units the Pro-ll is energetically more stable, and the Pro-I becomesthe stable one for more than or equal to seven segments.28b The present result, therefore, corresponds well with Tanaka and Scheraga's work, although the calculation is entirely different. 5 The method of calculation used in the present article is very difficult to apply to the random coiled state. Tanaka and Scheraga have explained without contradiction the relation of the results between the random coiled Pro-ll and the Pro-I~Pro-II interconversion as a cooperative processTM. According to the calculations by Tanaka and Scheraga, the 5% cis (Pro-l) and 95% trans (Pro-Ill state explains better the experimental intrinsic viscositydata 27, and also the trans form ProI1 lies close to the experimental range. This means that, for example, the completely ordered trans structure Pro-ll and the one consisting of 5% cis and 95% trans give nearly the same physical properties. Thus, the data obtained from the ordered structures in the present article can be used to explain the experimental information in solution.
Int. J. Biol. Macromol., 1980, Vol 2, December
349
Relative stability of poly(L-prolines): M. Ohsaku and A. lmamura Table 4
Differencein two centre interaction energy (eV) in the central segment between Pro-I and Pro-II ~
Resonance term O2 h N3 C4 C5 C6 C7 Exchange term 02 N3 C4 C5 C6 C7 Electrostatic term 02 N3 C4 C5 C6 C7 Total 02 N3 C4 C5 C6 C7
CIb
02
0.10 - 0.09 0.25
0.06 -0.I I
-0.19
0.07
N3
C4
C5
C6
Total
Symmetric -0.49 0.03 - 0.02 0.47
0.41 - 0.02 0.01
- 0.04 0.03
0.14
0.57
0.01
0.09
0.03 -0.01
-0.10
-0.31
- 0.02 0.02
0.05
0.36
0.01 Symmetric
0.01
-0.01
-0.06 0.05 -0.01
0.01 0.02 - 0.08
0.01
0.05
0.05
0.03
Symmetric -0.11
0.03
-0.05
0.13 - 0.06 0.18 0.01
0.07 -0.08
-0.15
0.04
-0.12
-0.01
Symmetric -0.52 0.02 - 0.01 0.41
0.36 - 0.02 -0.01
° Digitsshowtheenergydifference:Pro-I Pr•-••.Energyterms:abs••uteva•ues•essthan•.••eVaren•tcata••gued.(Seetextandequati•n•3)f•rm•re details.t h Atom numberings are shown in Fi~lure la
should be the stable form, while in polar (good) solvents the less closely packed form Pro-lI becomes dominant. In other words, in non-polar solvents the polymer may interact with solvent molecules repulsively, and in the polar solvents the polymer interacts with solvents attractively. Actually gelatin has the Pro-I form in non-aqueous acid-diluent systems 5s. In conclusion, in poor solvents the segment segment interactions are more dominant than the polymer solvent interactions to stabilize the cis form, Pro-l. In polar solvents, the polymer solvent interactions are more dominant than the segment segment interactions to stabilize the trans form, Pro-lI. It should be mentioned that we can discuss only the relative stabilities of the two conformational isomers qualitatively since the solvent polymer interaction energies are not calculated at all. Next, we will analyse the contribution of each atom or bond to the difference in the relative stability of the two conformational isomers. From the two centre term in the central segment Pro-ll gets more energy than Pro-l. Partitioning this energy into three components, resonance, exchange, and electrostatic, we can easily recognize that the resonance and exchange terms stabilize Pro-II, 0.57 eV (13.1 kcal) and 0.09 eV (2.1 kcal) respectively, relative to Pro-I as shown in Table 4. On the other hand the electrostatic term : stabilizes Pro-I 0.31 eV (7.1 kcal) more than Pro-ll. A large difference (Pro-I Pro-ll) appeared in the terms in relation to the L-prolyl skeleton. According to the resonance terms, the elements (C4, C l),
350
Int. J. Biol. Macromol., 1980, Vol 2, December
(C7, N3), and (C5, C4) stabilize the Pro-II form more than the Pro-I form, on the other hand for the elements (C7, C l) and (C4, N3) the reverse is true. Among these, we may suppose that the elements (C4, C1) and (C7, C1), and (C4, N3) and (C7, N3) are cancelling with each other from the geometrical consideration of the bonds (C4-C 1), (C 7-C 1), (C4 N3), and (C7 N3). As a result an element (C5, C4) still remains. This element largely contributes to the stabilization of Pro-II. The term (C7, C6) is also postulated to stabilize the Pro-II form. There appeared no extremely large difference in the exchange term. As for the electrostatic term, the elements (C4, C 1) and (C7, C1), (C4, 02) and (C7, O2), and (C4, N3) and (C7, N3) are fully or partly cancelled with each other. The elements (C5, C4) and (C7, C6) contribute to stabilize the Pro-I form. From the total term in Table 4, we can see that the difference in stability between Pro-I and Pro-II is due mainly to the contribution of the element (C5, C4). From the discussions mentioned above, we can deduce that the intrasegment energy of poly(L-proline) is governed by the C7, N3, C4, and C5 part of the pyrrolidine ring in addition to the C1 part. Moreover, it should be stressed again that the (C5, C4) interaction can be considered to determine the The electrostatic effect on the P r o - l ~ P r o - l l transition have been discussed by Holzwarth and Backman, (Biochemistry 1969, 8, 883). The electrostatic energies obtained by the present calculation and by Holzwarth et al. correspond well with each other. However, as is shown in Table 4, we can readily recognize that the conformational stability is not governed only by the electrostatic energy.
Relative stability ol'poly(L-prolines): M. Ohsaku and A. Imamura Table 5
Difference in two centre interaction energy (eV) in the (~1 segments between Pro-I and Pro-IV 1C1
Resonance term °Clb'" o02 °N3 °C4 °C5 °H8 Exchange term oC4 °C5 °H8
IO2
1N3
IC4
IC7
- 0.01 0.02 - 0.02
0.06
- 0.06
- 0.02
0.04
0.01
-0.01
Total
0.01 0.01 0.04 - 0.20 0.02
-0.14
-0.01 0.01
Electrostatic term °C 1 002 °N3 °C4 °C5 °H8
- 0.03 0.03 -0.01
Total °Cl 002 °N3 °C4 °C5 °H8
-0.01 0.09 0.04 - 0.02 -0.19 0.01
- 0.02 0.09
0.01 - 0.05 -0.01 0.02
0.01 - 0.05 -0.01 0.02
-0.01 0.01
-0.01 0.02
0.01
-0.01
0.01
-0.01
0.01 - 0.04
-0.01 0.01
0.02
- 0.01 0.02 -0.02
0.08 0.01 -0.01
0.01 - 0.04
-0.01
0.03
- 0.08 0.03
-0.12
" See footnote a of Table 4. ISee text and equation 14) for more details} b See footnotett ' See footnote h of Table 4 relative stability of the central segment of Pro-I and P r o : II. We will now discuss the intersegment interactions. Table 5 shows a part of the two centre interactions between the central and the first nearest neighbour segments** in the form of the energy difference between Pro-I and Pro-II. As for the total term, a large difference ( P r o - I - P r o - I I ) appeared in the following elements: (°O2, 1C1), (°N3, 1C1), (°C5, tC1), (o02, 102), (o02, 1N3), (°C4, 1C4), and °C4, IC7)**. A m o n g them, the element (°C5, ~C1) has the largest contribution of all to the difference in the 0 1 intersegment energy. This element stabilizes Pro-I 0.19 eV (4.4 kcal) more than Pro-lI. Therefore, it was found that from the intersegment energy the interaction energy between the fl carbon of the central segment and the carbonyl carbon of the nearest neighbour segment is the d o m i n a n t term in determining the relative stability in poly(L-prolines). This stabilization originates from the element (°C5, 1C1) of the resonance term.
Hydrogen bonding Sasisekharan has studied the arrangement of poly(Lproline II) 4 and concluded that for the polymer chain there is a minimum of short contacts, the only one which occurs being a C 6 - H . . . O 2 hydrogen b o n d between different polymer chains. In the present work, we have ** For simplicity, (~1 (segments) means the central and the first nearest neighbour segments, (~2, 0-3 .... refers to the central and the second, third .... nearest neighbour segments. tt For example, (°02, ~C1) refers to the interaction term between the 02 oxygen atom in the central segment and the C1 carbon atom in the first nearest neighbour segment.
calculated the energy for only one polymer chain and therefore do not discuss this problem. There are no short contacts of H . . . O below 0.25 nm both in the Pro-I and P r o - I I forms in the present case.
Alanines having Pro-I and Pro-ll skeletons The C N D O / 2 calculations were also carried out for alanines assuming the poly(L-proline) skeletons in order to confirm the characteristic properties in stabilization in poly(L-proline). The calculated results are shown in Table 6. F r o m the total energy calculated it is deduced that Ala-I is more stable than Ala-lI by 0.07 eV (1.6 kcal). The intrasegment energy is smaller in Ala-II than in Ala-I, while the intersegment energy is in the reverse order as was already shown in Pro-I and Pro-II. We will discuss in more detail the results of the energy partitioning using Tables 7 and 8. Ala-II is more stable 0.40 eV (9.2 kcal) in the two centre term in the central segment than Ala-1. O n the other hand Ala-I is more stable by 0.18 eV (4.1 kcal) in the two centre term in the (~ l segments than Ala-ll. In the two centre term in the central segment, the difference (Ala-I Ala-II) mainly appeared in the following elements: (C4, Cl), (C4, N3), (C5, C4), and (HT, N3). The element (C4, N3) stabilizes the Ala-I form, and the other elements stabilize the Ala-ll form. The tendency of the energy contribution for the two centre term of the central segment in poly(L-alanine) strongly resembles the tendency shown in poly(L-proline). In the O-1 segments, fairly large difference (Ala-I Ala-lD appeared in the following four elements, (°02, 1el), (°C4, 1C4), (°C4, 1H7), and (°C5, tCl). A m o n g these elements, the resonance element of (°C5, 1C 1) contributes largely to
Int. J. Biol. Macromol., 1980, Vol 2, December
351
Relative stability of poly(L-prolines): M. Ohsaku and A. lmamura the stabilization of Ala-I. This is also seen in the case of P r o d and Pro-II. F r o m the calculations of Pro-I, Pro-It, and Ala-I, and Ala-I1, it was found that there was a very strong resemblance of the energy c o n t r i b u t i o n between poly(t.proline) and poly(L-alanine). It may be seen from these calculations that the c a r b o n a t o m of the fl position plays an i m p o r t a n t role in the c o n f o r m a t i o n a l stability of poly(L-proline). As a result, the methylene g r o u p of the 7 position does not c o n t r i b u t e so much to the energy difference between Pro-I and Pro-II. In other words, the tendency of the energies of the i n t r a s e g m e n t and the intersegment is governed mostly by the main chain d i m e n s i o n s including C5 and C7 a t o m s in poly(t.prolines). Actually, M a d i s o n has proposed that the pyrrolidine ring is generally somewhat flexible 3~. From the Table
6
Total energy (eV) of Ala-I and Ala-ll
Energy Total Total intrasegment Total one centre Total two centre Total intersegment Total two centre () 0 0 0 " See footnote**.
Table
7
1" 2 3 4
Ala-I
Ala-ll
- 1503.28 - 1474.62 1242.61 - 232.00 -28.66 - 14.27 - 0.02 - 0.02 - 0.02
- 1503.21 - 1475.00 - 1242.60 - 232.40 -28.21 - 14.09 - 0.01 - 0.02 - 0.02
(Energy, see equations 1 4)
Model molecules We now take up the model molecules which have structures with two methyl groups attached to the Lprolyl residue as is shown in Fiqure lc. The C N D O / 2 calculations were also carried out for those molecules. The total energies o b t a i n e d are - 2 4 4 8 . 5 5 and - 2 4 4 8 . 6 8 eV for Model-I a n d Model-II, respectively, and the difference is 0.13 eV (3.0 kcal mol 1). The order of the total energy between Model-I and Model-II is in good agreement with that of the intrasegment energy of Pro-I and Pro-ll. Scheraga et al. have also treated the E H T and C N D O / 2 calculations on a model of poly(L-proline) such as acetylu-prolineamide 22. They have o b t a i n e d the energy difference AE,, .......... =5.3 and 1.2 kcal mo1-1 by E H T and C N D O / 2 method, respectively. O u r values agree well with these previously reported values. Scheraga et al. have also made a n o t h e r a p p r o a c h to u n d e r s t a n d the cooperative transition between Pro-I and P r o - l l ash. Energy difference AE,~......... between Pro-I and P r o - l I u n d e r v a c u u m at room temperature varies from - 3.0 to 3.6 kcal mol t with increasing n u m b e r , n, of the considered segment from 4 to 10. In the case n = 9, the difference AE is
Difference in two centre interaction energy (eV) in the central segment between Ala-1 and Ala-ll" C 1h
Resonance term O2 I' N3 C4 C5 H6 H7 Exchange term 02 N3 C4 C5 H7 Electrostatic term 02 N3 C4 C5 H6 H7 Total 02 N3 C4 C5 H6 H7
c a r b o n - 13 l o n g i t u d i n a l relaxation time, it is indicated that the fl and ,, c a r b o n s are more mobile than the z~ and 6 c a r b o n s in n u m e r o u s c o m p o u n d s which have the pyrr o l i d i n e r i n g 32. Therefore it seems that the present result closely relates to those n.m.r, studies, although the direct c o m p a r i s o n between them is very difficult.
0.09 -0.08 0.25
02
0.08 -0.11
N3
0.08
0.01 -0.01
-0.06
0.01
0.02
0.04
0.02 0.03 - 0.07
0.04
0.01 0.04
-0.01 - 0.06
0.03 -0.01 -0.01
0.01 0.01
C5
H6
0.01 0.01
0.01
Total
Symmetric
-
-0.50 0.03 -0.01 0.21
-0.10
C4
0.42 -0.01 - 0.01
0.41
Symmetric 0.05
0.11 - 0.05 0.18 0.01 0.01 - 0.05
0.10
Symmetric
0.09 -0.08 -0.01 0.04
-0.10 -0.12
- 0.01
Symmetric -0.53 0.03 -0.02 0.24
0.37 -0.01 - 0.02
0.01 0.01
0.01
0.40
" Digitsshow the energydifference:Ala-! Ala-1I. Energy terms:absolute valueslessthan 0.01 eV are not catalogued. [Seetext and equation (3) for more details.) h Atom numberings are shown in Fixture Ib
352
Int. J. Biol. Macromol., 1980, Vol 2, December
Relative stability o f poly(L-prolines): M. Ohsaku and A. I m a m u r a Table 8
Difference in two centre interaction energy (eV) in the (~1 segments between Ala-I and Ala-II" I C l b."
Resonance term o c i b.c °N3 °C4 °C5 oH 8
0.01 0.04 -0.01 - 0.21 0.02
Exchange term °C4 °C5
-0.01
Electrostatic term °C1 002 °N3 °C4 °C5 °H8 Total °Cl 002 °N3 °C4 °C 5 °H8
-0.03 0.08
Io2
IN3
IC4
~H7
-0.01
-0.01 0.02 - 0.01
0.06
-0.07
- 0.02
0.01
0.01
-0.01
Total
-0.22
-0.01
- 0.04
- 0.02 0.03 -0.01
0.02
-0.02 0.09 0.04 - 0.03 - 0.19 0.01
0.01 - 0.04 -0.01 0.01
0.01 - 0.05 -0.01
-0.01 0.01 0.01
-0.01 0.03 0.01 - 0.01
0.01
-0.01
0.01 - 0.05
-0.01 0.01
- 0.01 0.02 -0.01
0.08
-0.01 0.03 0.01 - 0.09
-0.01
0.05
-0.18
" See footnote a of Table 7. {See text and equation (4) for more details.) See footnote H ' See footnote h of Table 7 e s t i m a t e d as 2.0 kcal m o l - ~ 2Sb. T h e difference e s t i m a t e d in the p r e s e n t w o r k , w h i c h has also t a k e n i n t o c o n s i d e r a t i o n 9 units, a g r e e s v e r y well w i t h the value, a l t h o u g h the g e o m e t r i e s used in the p r e s e n t article are a little different f r o m t h o s e of S c h e r a g a et al. 28b.
Conclusions T h e c o n f o r m a t i o n a l stability o f P r o - I a n d P r o - I I in the solid state or in s o l u t i o n is a c c o u n t e d for very well by the C N D O / 2 m e t h o d u s i n g the t i g h t - b i n d i n g a p p r o x i m a t i o n . T h e c o n t r i b u t i o n s of the i n t r a s e g m e n t a n d the i n t e r s e g m e n t t e r m s to the e n e r g y difference b e t w e e n P r o - I a n d P r o - I I are n e a r l y e q u a l in a b s o l u t e m a g n i t u d e but h a v e different signs. T h e difference in the i n t r a s e g m e n t e n e r g y of poly(Lp r o l i n e ) is g o v e r n e d by the i n t e r a c t i o n s b e t w e e n the ~ a n d t h e / 3 c a r b o n s of the p y r r o l i d i n e ring. T h e difference in the i n t e r s e g m e n t e n e r g y of poly(Lp r o l i n e I) a n d p o l y ( L - p r o l i n e II) is g o v e r n e d by t h e i n t e r a c t i o n s b e t w e e n the c e n t r a l s e g m e n t [3 c a r b o n a n d the nearest neighbour carbonyl carbon. F r o m the c a l c u l a t i o n of the c o n t r i b u t i o n of the t o r s i o n a l m o t i o n a r o u n d the C , - C' b o n d to the free energy, it was f o u n d t h a t this e n e r g y is sufficient to c h a n g e the conformational stability of poly(L-proline) (see Appendix).
Acknowledgement T h i s w o r k was s u p p o r t e d in p a r t by a G r a n t - i n - A i d for Scientific R e s e a r c h f r o m the M i n i s t r y of E d u c a t i o n , for w h i c h t h e a u t h o r s express t h e i r g r a t i t u d e . T h e a u t h o r s w o u l d like to a c k n o w l e d g e the g u i d a n c e a n d e n c o u r a g e -
m e n t of P r o f e s s o r H i r o m u M u r a t a . T h e c o m p u t a t i o n was c a r r i e d o u t o n a F A C O M M 1 9 0 of the D a t a P r o c e s s i n g C e n t e r , K y o t o U n i v e r s i t y . T h a n k s are a l s o d u e to P r o f e s s o r H. A. S c h e r a g a for s e n d i n g us the rel:;rints of R e f e r e n c e 28 a n d for p r o v i d i n g useful i n f o r m a t i o n in notes.
References 1 2 3 4 5
6 7
Traub, W. and Shmueli, U. Nature ILondon) 1963, 198, 1165 Traub, W. and Shmueli, U. in ~Aspects of Protein Structure', (Eds. W. Traub and U. Schmueli), Academic Press, New York, NY, 1963, pp. 81 92 Cowan, P. W. and McGavin, S. Nature (London) 1955, 176, 501 Sasisekharan, V. Acta Crystallo~tr. 1959, 12, 897 (a) Downie, A. R. and Randall, A. A. Trans. Farady Soc. 1959, 55, 2132; (b) Steinberg, i. Z., Harrington, W. F., Berger, A., Sela, M. and Katchalski, E. J. Am. Chem. Soc. 1960, 82, 5263; (c) Gornick, F., Mandelkern, L., Diorio, A. F. and Roberts, D. E. ibid. 1964, 86, 2549; (d) Engel, J. Biopolymers 1966, 4, 945: (e) Engel, J. in "Conformation of Biopolymers', (Ed. G. N. Ramachandran), Academic Press, New York, 1967, Vol 2, pp. 483 497:11) Veis, A., Kaufman, E. and Chao, C. C. W. in 'Conformation of Biopolymers', (Ed. G. N. Ramachandran) Academic Press, New York, 1967, Vol. 2, pp. 499-512; [g) Carver, J. P. and Blout, E. R. in "Treatise on Collagen', (Ed. G. N. Ramachandran) Academic Press, New York, 1967, Vol. 1, pp. 441 526: see also (hi Strassmair, H., Engel, J. and Knof, S. Biopolymers 1971, 10, 1759; 6) Knof, S., Strassmair, Engel, J., Rothe, M. and Steffen, K. D. ibid. 1972, I !, 731 : (j) the preference of proline peptides for certain conformational states in solution is adequately reviewed, see Scheraga, H. A. Chem. Rev. 1971, 71, 195; and Deber, C. M., Madison, V. and Blout, E. R. Ace. Chem. Res. 1976, 9, 106 Mandelkern, L. in 'Poly
Int. J. Biol. M a c r o m o l . , 1980, Vol 2, D e c e m b e r
353
Relatit:e stability qf poly(L-prolines): M. Ohsaku and A. lmamura 8 9 10
1
2 3
14 15 16 17 18 19 20 21 22 23
24 25 26 27 28
29 30 31 32
For example. Slater, J. C. in 'Quantum Theory of Molecules and Solids', McGraw-Hill, New York, 1965 lmamura, A. and Fujita, H. J. Chem. Phys. 1974, 61, 115 (at Ramachandran, G. N. and Sasisekharan, V. Adr. Protein Chem. 1968, 23, 283: (b) according to IUPAC Commission on nomenclature [Biochemistry, 197(I, 9, 3471 ] DeSantis, P.,Giglio, E.,Liquori, A . M . a n d R i p a m o n t i , A. Nuture (London) 1965, 206, 456 Damiani, A., De Santis, P. and Pizzi, A. ibid. 1970, 226, 542 Balasubramanian, R. Lakshminarayanan, A. V., Sabesan, M. N., Tegoni, G., Venkatesan, K. and Ramachandran, G. N. Int. J. Protein Res. 1971.3, 25 Ramachandran, G. N., Lakshminarayanan, A V., Balasubramanian, R. and Tcgoni, G. Bioehim. Biophys. Aeta 197[1, 221, 165 Tonnelli, A. g. J. Am. Chem. Soc. 197(I, 92, 6187 Englcrt, A.. Furnc'monl, J. and Ldonis, J. Mueromolecules 1971,4, 768 :l-onnelli, A. g. J. Am. Chem. Soc. 1972, 94, 346 Scheraga, H. A. Adr. Phys. Org. Chem. 1968, 6, 103 Schimmel, P. R. and FIory, P. J. Proc. Natl. Aead. Sei. USA t967, 58, 52 Schimmel, P. R. and Flory, P. J. J. Mol. Biol. 1968, 34, 105 Flory, P. J. in 'Statistical Mechanics of Chain Molecules" John Wiley & Sons, New York, 1969 Yam J. F., Momany, F. A., Hoffmann, R. and Scheraga, H. A. J. Phys. Chem. 1970, 74, 420 (at Maigret, B., Perahia, D. and Pullman, B. J. Theor. Biol. 1970, 29, 275: (b) Pullman, B. and Pullman, A. in 'Advances in Protein Chemistry', (Eds. C. B. Afinsen, J. T. Edsail and F. M. Richards) Academic Press, New York, 1974, pp. 347 526 Maigrel, B., Pullman, B. and Caillet, J. Bioehem. Biophys. Res. Commun. 1970, 40, 808 Nishikawa, K. and Ooi, T. Bull. Inst. Chem. Res. 1972, 50, 94 (a) Schwarz, G. Biopolymers 1968, 6, 873: (b) Applequist, J. ibid. 1968, 6, 117: (c) Ganser, V., Engel, J., Winklmair, D. and Krause, G. ibid. 1970, 9, 329 Mattice, W. L. and Mandelkern, L. J. Am. Chem. Soc. 1971, 93, 1769 (at Tanaka, S. and Scheraga, H. A. Maeromoleeules 1975, 8, 494: (b) Tanaka, S. and Scheraga, H. A. ind. 1975, 8, 504: (c) Tanaka, S. and Scheraga, H. A. ibid. 1975, 8, 516:(d) Tanaka, S. and Scheraga, H. A. ind. 1975, 8, 623 Leung, Y. C. and Marsh, R. E. Aeta Crystallogr. 1958, II, 17 Matsuzaki, T. and litaka, Y. Aeta Crystallogr. (B] 1971, 27, 507 Madison, V. Biopolymers 1977, 16, 2671 (al Torchia, D. A. and Lyerla, J. R. Jr. Biopolymers 1974, 13, 97;(b) Deslauriers, R., Smith, I. C. P. and Walter, R. J. Biol. Chem. 1974, 249, 7006: (el Fossel, E. T., Easwaran, K. R. K. and Blout, E. R. Biopolymers 1975, 14, 927: (d)Somorjai, R. L. and Deslauriers, R. J. Am. Chem. Soc. 1976, 98, 6460
/
/
/ -1936 50
\
/
\ \
L
/
I \
I \
o -1937.0C L_ : ~....
loo
I
~ " - ~ . . . . . . . . . ~ \ , , , . ~ . / ----0" ~ ' ' "
12o
14o 16o q~(aeg)
18o
- 20
Figure AI
Total energy as a function of t~ in Pro-I ( • ) and Pro-lI (o). The following angles were fixed during the calculations: Pro-l;
quadratic equation. The minimum values of the curvatures are nearly equal to each other for Pro-I and Pro-II as is described in the preceding section. However, their shapes are very different from each other. That is, in Pro-I the energetically allowed region is narrow, but in Pro-It it is wide. Moreover, it should be pointed out that the observed ~ vaJues by X-ray analyses described above are well reproduced by the present C N D O / 2 calculations (Pro-I, 0 = 157.5, Pro-II, 0 = 142.9). Now, the partition function was proportional to the equation: rU3
Z,= f exp[-(ax2+bx+c)/kT]dx
(A1)
-- rr,'3
Appendix
and the free energy F can be expressed as
Cm!fbrmational stability and the contribution of the torsional motion to the.Jree ener.qy
F= - k Tln Z r
In order to investigate the relative stability of the two structures of poly(L-proline) statistically and thermodynamically, we have evaluated the contribution of the torsional motion about the C, C' bond to the free energy. As is mentioned in the preceding section, the structure of this polymer can be described by the three internal rotation angles: u), ¢p, and 0. Among them, to can be selected to be 0' or 180 under the planarity of the amide group, q) may only take values within a very narrow region, because the monomer is a ring. From this we have assumed the angles of the fixed values as follows, for Pro-I o r P r o - l l : ~ o = 0 or 180, a n d t p = - 8 3 ' o r - 7 7 . 2 " ' ; a n d is varied around the observed equilibrium values ~''*, 158' and 145.9' for Pro-I and Pro-II, respectively. Using these internal rotation angles, the total energies are calculated. The results are shown in Figure A1. Here, we have assumed that the curvature can be described by a certain
The integration of equation (A1) between - n / 3 and n/3 centred at the energy minimum with regard to X was carried out by using Simpson's formula. The values calculated for Pro-I and Pro-II are -1937.09 eV and -1937.13 eV respectively, and the difference between both forms is estimated to be 0.04 eV (0.92 kcal). The present result shows that the torsional motion around the C, C' bond may change the relative stability of the polymer in the Pro-I and Pro-II forms. In other words, with regard to Pro-It, the fluctuation of the structure about the most stable geometry should be fairly large, while in the case of Pro-I, the fluctuation is expected to be relatively small. The free energy difference between Pro-I and Pro-t1 which includes the contribution of the torsional motion is, however, not large enough to deny the coexistence of these two forms in the solid state or in solution.
354
Int. J. Biol. Macromol., 1980, Vol 2, December
(A2)
One centre term:
Introduction Poly(L-proline) exists as two conformational isomers, I and II, the structures of which have been elucidated by Xray analysis 1-4. For simplicity we will hereafter use Pro-I and Pro-II for poly(L-proline I) and poly(L-proline II), respectively. These two forms exhibit markedly different optical rotation s. In Pro-I or Pro-lI there are no hydrogen atoms present to form hydrogen bonds and the rigidity and unique shape of the pyrrolidine ring make it very difficult for this residue to fit into the normal s-helix conformation. Therefore, it is expected that when Lproline is the major constituent in a high polymer, a unique type of ordered structure will result 6. This is the case for collagen. On the other hand when L-proline is the minor constituent in a polymer, a natural distortion or degtruction of the normal s-helical structure can be expected. In the present work, in order to obtain information on the structural role of L-proline sequences in a protein or on its behaviour in solution, we have carried out C N D O / 2 calculations 7 for this species using the tight-binding approximation s .
onA
EA = Y. P , £ - ~ / , + A , ) - ( Z 4 - ~,z., '*'' o.o.j + /
onA onA
½ZZ [P.P,,,,-~(P.,) ,
t
lj
Intrasegment t w o centre term: onA onB
(o,o)_ 2flaB o ~ ~' E4.. ~ - p, , ~.~(o,o) ,, _ -
l onA o n B
~lllt]
I
rp
-}- L
~, .(o,o)
4 4--BB/4, B
p
B
-- Z BB~4[4,B
(3)
--[~4B
o n 4 onB
Z
~' p+j ~(o.j) ~
--t,t
~,t,
onA o n B
Z
The total energy of a polymer can be expressed as follows9:
l
+ ~rp
o
.,~o.j~ o
2L~4.41BB/4,B
Z E A + E E E'°'°' A,B + E E E E(°'/' ,,
0141 8130/80/060347 08502.00 ©1980 IPC Business Press
~ ~,(o,o) D 7 ,,(o,o)
-- --AA~B/A.
Intersegment two centre term:
M e t h o d o f calculation
A B<4
14,B
II
+ ( Z ~Z~e2/R 4R)]
4,B
A
fl
½Z Z'"
ElO,j)_ o
Etotal =
(2)
] , 4..4
J
4
B
(1)
alll
i ~.B
tl
7 .,(o.j~
-- ~44~B[4,
B
--
p
HB
Z .,qo.j~ 4/
4,B
+ (Z 4ZBe2/R 4a)]
Int. J. Biol. Macromol., 1980, Vol 2, December
(4)
347
Relative stability of poly(L-prolines): M. Ohsaku and A. Irnamura
02
02
C1
C'1
/
\
N3"---'------~q 4~
H14~CA/ /
\\
~
/
~
~5"~C5
/\
H12
-
HT/
=z/\
C4~H 8 /
._.__..-H9
C15
C1
/
N3
H8
H18
C19
/
N3
C4,..~
H20//H14~c7.//
/"\ ~'~H8 ~/H9
\
~Hg
Hll
/\
H1Q
a
H16
H12
b
Hll
c
Figure I Schematic structures and atom numberings of: (a) poly(L-proline); [b) polylL-alanine); lc) model molecule. With model molecules in relation to the methyl groups, the atomic position N3 C4 C 15 H 16 was assumed to be trans and N3 C 1 C 19 H20 to be <'i.~
Table I
Atom coordinates of Pro-I and Pro-II Pro-I
Atom a CI 02 N3 C4 C5 C6 C7 H8 H9 H10 HI1 H12 HI3 H14
x 1.46772 2.09614 1.97621 1.31209 2.41165 3.56292 3.41775 0.84822 2.69295 2.09606 4.48391 3.62234 4.11885 3.50031
Pro-ll
y -0.84179 - 1.19737 0.00000 0.47925 0.98977 1.26772 0.54135 -0.29034 0,21376 1,92000 0,95565 2,33928 -0,29324 1,26410
:
x
-0.84047 - 1.88856 0.00000 1.22072 2.11746 1.22827 -0.06010 1.65952 2.82933 2.58986 1.72071 1.03771 -0.06352 -0.87184
-0.00195 1.14683 -1.04627 -0.89322 -2.27636 - 3.21795 -2.52354 -0.30964 - 2.42519 -2.42486 -4.04883 -3.60872 -2.87104 - 2.65916
y -0.28927 0.14103 0.00000 0.89572 1.39446 0.42509 -0.37917 1.66448 2.37198 1.40554 0.96323 -0.23634 -0.05408 - 1.43458
: -0.75373 -0.57277 0.00000 1,15549 1,46909 0,86035 -0.17293 0.89387 1.01040 2.54887 0.40412 1.63359 -1.15357 0.06333
Atom numberings are shown in Fioure la
where A or B denotes atom, t or t~ atomic orbital, j the n u m b e r of segments, and 0 the central segment. The other notations are the same as those usually used. In equations (3) and (4), the first term of the right hand side is the core resonance term, the second the exchange term, and the third the electrostatic term respectively. Numerical calculations were made according to the previous paper 9. Schematic structures and a t o m numberings are shown in Fi.qure 1. The geometries of Pro-I and Pro-II were referred to those reported 14. At the start the c o m m o n structural parameters both for Pro-I and Pro-If were used for the bond lengths r ( C l N3), r(N3 C4), r(C4C'I), and r(C1 O 2 ) , and the geometry of a ring part C5 ~ C 7 and H 9 ~ H 1 4 , and then the ring part was adequately corrected for each ring to form a cyclic ring exactly. The bond angles (CIN3C4), (N3C4C'1), and (O2C1N3) were transferred from the data of X-ray analysis 1.4. The torsional angles
348
Int. J. Biol. Macromol., 1980, Vol 2, December
Ala-I or Ala-II and Model-1 or Model-II for poly(Lalanine I) or poly(L-alanine II) and model molecules, respectively. Here, the backbone of Ala-I and M o d e m is the same as that of Pro-I, and the b a c k b o n e of Ala-II and Model-II is the same as that of Pro-II. The coordinates estimated are summarized in Table 2. Non-listed atomic coordinates are available from Table 1.
Results and discussion Pro-I and Pro-ll The total energies calculated are shown in Table 3. Up to the present time, some empirical and q u a n t u m mechanical conformational analyses of L-prolyl residues, poly(L-prolines), and their model molecules have been reported~l 25. The P r o - l ~ P r o - I I cooperative order ~ o r d e r transition has been treated by Schwarz 26", Applequist26L and Ganser et al.2% Mattice and Mandelkern 27 have proposed the r a n d o m coil model for Pro-II in water, trifluoroethanol, acetic acid, propionic acid, and benzyl alcohol. The statistical treatment of the cooperative transition between two ordered conformations of Pro-I and Pro-lI and of the r a n d o m coiled
R e l a t i v e s t a b i l i t y o f poly(L-prolines): M . O h s a k u a n d A. I m a m u r a
Table 2
Atom coordinates of Ala-I, Ala-II, Model-I, and Model-II
Atom a'b
x
H6 H7
3.25898 2.91166
C15 H16 H17 H18 C19 H20 H21 H22
0.35871 -0.09080 -0.42485 0.91258 0.06018 -0.34549 0.10499 -0.58171
3'
z
Ala-I 1.19434 0.35130 Model-I 1.65352 1.94796 1.35422 2.49495 -1.39558 -0.98397 -2.48207 -1.11524
x
1.46302 -0.03900
-2.96881 -2.00871
0.99042 1.93876 0.29433 0.57410 -0.65017 0.27400 -0.57512 -1.48532
-0.24481 -0.13944 0.73848 -0.87153 -0.29047 -1.34644 0.31528 -0.04579
)' Ala-ll 0.68158 -0.24703 Model-I1 0.15411 0.83276 -0.20855 -0.69021 -1.23117 -1.50137 -2.13176 -0.73140
--
1.02142 -0.11266 2.32616 3.17258 2.02664 2.61326 -1.92440 - 1.92160 - 1.82394 -2.86166
Atom numberings are shown in Figures lh and lc b Non-listed atomic coordinates are available from Table I Table 3
Total energy (eV) of Pro-I and Pro-II
Energy Total Total intrasegment Total one centre Total two centre Total intersegment Total two centre 0 0 0 0 " See footnote **
1" 2 3 4
Pro-I
Pro-II
- 1937.03 - 1908.44 - 1553.41 - 355.03 - 28.60 - 14.23 - 0.02 -0.03 -0.02
- 1937.03 - 1908.77 - 1553.39 - 355.39 - 28.25 - 14.11 - 0.02 0.00 0.00
(Energy, see equations 1 4.)
Pro-II have been reported by T a n a k a a n d Scheraga 28. At the present time it seems that there are n o completely settled models to explain the molecular structure of poly(L-proline) in solution. P u l l m a n et al. have carried out the P C I L O calculations for a model molecule of the Lprolyl residue in its h o m o p o l y m e r s 23. F r o m the calculations they have shown that in these c o m p o u n d s the stable c o n f o r m a t i o n s are limited, both for cis a n d t r a n s polymers (only to qJ centred a r o u n d 150 °) a n d that the t r a n s form should be a b o u t 1.4 kcal m o l - 1 more stable t h a n the cis form. In the present work, the total energy is nearly equal both in Pro-I a n d Pro-II. This means that both forms can coexist energetically, a n d this finding is in complete agreement with the experimental result that I a n d II forms coexist in the solid state i n d e p e n d e n t l y a n d in solution d e p e n d i n g u p o n the kinds or the c o m p o s i t i o n of the solvents 5. It should be pointed out that the structure of the polymer in solution may n o t be an ordered one but a mixed one. However, as is pointed out by T a n a k a a n d Scheraga 28, there are no marked differences between the all t r a n s Pro-II a n d the mixed structure of 5'Yocis a n d 95'y,, trans conformations. C o n s e q u e n t l y , it may be expected that the present result gives i n f o r m a t i o n qualitatively on the relative stability of Pro-1 a n d Pro-II in solution. The total intrasegment energy of Pro-II is smaller t h a n that of Pro-I, whereas the total intersegment energy is smaller in P r o - I I t h a n that in Pro-I. That is, the total i n t r a s e g m e n t energy favours the trans form P r o - l I , while the reverse is true for the total intersegment energy. After b a l a n c i n g these two terms, it is found that the total energy is nearly equal in Pro-I and Pro-I1. A brief statistical a n d t h e r m o d y n a m i c a l t r e a t m e n t of the free energy of this
polymer is discussed in the Appendix. By c o m p a r i n g the total i n t r a s e g m e n t one centre and two centre energies, it is found that the former contributes nearly equally to Pro-I a n d Pro-II. The energy of the r e m a i n i n g two centre term of P r o - l I is lower by 0.36 eV (8.3 kcal) per segment t h a n that of Pro-I. We will discuss in more detail the total i n t r a s e g m e n t and total intersegment energies below. As for an L-prolyl residue, the t r a n s form is indeed more stable t h a n the cis form j u d g i n g from the intrasegment energy. If a single residue of L-proline is isolated in a polypeptide chain, the residue may favour the t r a n s c o n f o r m a t i o n . Shimmel and Flory have also noted after e x a m i n a t i o n of the structure of polypeptide copolymers c o n t a i n i n g L-proline in low proportions, that the polymer residue is in its t r a n s c o n f o r m a t i o n as in Pro-II 2°. This is also confirmed by X-ray crystallography. For" example, from the crystal structure analyses of oligomers such as L-leucyl-L-prolyl-glycine29 a n d acetyl-L-prolineN - m e t h y l - a m i d e 3° it was concluded that the peptide groups should have the t r a n s c o n f o r m a t i o n . However, in the case of h o m o p o l y m e r s such as the case treated here,* Pro-I gets much more energy from the intersegment interactions t h a n Pro-lI. The Pro-I form is more condensely packed than Pro-II. Therefore the interactions between segments are stronger in Pro-I t h a n that in ProII. This is very interesting in c o n n e c t i o n with the conform a t i o n a l change of poly(L-proline) d e p e n d i n g u p o n the properties or c o m p o s i t i o n s of solvents in solutions*. In n o n - p o l a r (poor) solvents the more packed form Pro-I * In this work, we have taken into consideration up to nine monomer units in each case. In the calculation with less than seven units, the trans form should be more stable than the cis form. Tanaka and Scheraga have also concluded that with less than seven units the Pro-ll is energetically more stable, and the Pro-I becomesthe stable one for more than or equal to seven segments.28b The present result, therefore, corresponds well with Tanaka and Scheraga's work, although the calculation is entirely different. 5 The method of calculation used in the present article is very difficult to apply to the random coiled state. Tanaka and Scheraga have explained without contradiction the relation of the results between the random coiled Pro-ll and the Pro-I~Pro-II interconversion as a cooperative processTM. According to the calculations by Tanaka and Scheraga, the 5% cis (Pro-l) and 95% trans (Pro-Ill state explains better the experimental intrinsic viscositydata 27, and also the trans form ProI1 lies close to the experimental range. This means that, for example, the completely ordered trans structure Pro-ll and the one consisting of 5% cis and 95% trans give nearly the same physical properties. Thus, the data obtained from the ordered structures in the present article can be used to explain the experimental information in solution.
Int. J. Biol. Macromol., 1980, Vol 2, December
349
Relative stability of poly(L-prolines): M. Ohsaku and A. lmamura Table 4
Differencein two centre interaction energy (eV) in the central segment between Pro-I and Pro-II ~
Resonance term O2 h N3 C4 C5 C6 C7 Exchange term 02 N3 C4 C5 C6 C7 Electrostatic term 02 N3 C4 C5 C6 C7 Total 02 N3 C4 C5 C6 C7
CIb
02
0.10 - 0.09 0.25
0.06 -0.I I
-0.19
0.07
N3
C4
C5
C6
Total
Symmetric -0.49 0.03 - 0.02 0.47
0.41 - 0.02 0.01
- 0.04 0.03
0.14
0.57
0.01
0.09
0.03 -0.01
-0.10
-0.31
- 0.02 0.02
0.05
0.36
0.01 Symmetric
0.01
-0.01
-0.06 0.05 -0.01
0.01 0.02 - 0.08
0.01
0.05
0.05
0.03
Symmetric -0.11
0.03
-0.05
0.13 - 0.06 0.18 0.01
0.07 -0.08
-0.15
0.04
-0.12
-0.01
Symmetric -0.52 0.02 - 0.01 0.41
0.36 - 0.02 -0.01
° Digitsshowtheenergydifference:Pro-I Pr•-••.Energyterms:abs••uteva•ues•essthan•.••eVaren•tcata••gued.(Seetextandequati•n•3)f•rm•re details.t h Atom numberings are shown in Fi~lure la
should be the stable form, while in polar (good) solvents the less closely packed form Pro-lI becomes dominant. In other words, in non-polar solvents the polymer may interact with solvent molecules repulsively, and in the polar solvents the polymer interacts with solvents attractively. Actually gelatin has the Pro-I form in non-aqueous acid-diluent systems 5s. In conclusion, in poor solvents the segment segment interactions are more dominant than the polymer solvent interactions to stabilize the cis form, Pro-l. In polar solvents, the polymer solvent interactions are more dominant than the segment segment interactions to stabilize the trans form, Pro-lI. It should be mentioned that we can discuss only the relative stabilities of the two conformational isomers qualitatively since the solvent polymer interaction energies are not calculated at all. Next, we will analyse the contribution of each atom or bond to the difference in the relative stability of the two conformational isomers. From the two centre term in the central segment Pro-ll gets more energy than Pro-l. Partitioning this energy into three components, resonance, exchange, and electrostatic, we can easily recognize that the resonance and exchange terms stabilize Pro-II, 0.57 eV (13.1 kcal) and 0.09 eV (2.1 kcal) respectively, relative to Pro-I as shown in Table 4. On the other hand the electrostatic term : stabilizes Pro-I 0.31 eV (7.1 kcal) more than Pro-ll. A large difference (Pro-I Pro-ll) appeared in the terms in relation to the L-prolyl skeleton. According to the resonance terms, the elements (C4, C l),
350
Int. J. Biol. Macromol., 1980, Vol 2, December
(C7, N3), and (C5, C4) stabilize the Pro-II form more than the Pro-I form, on the other hand for the elements (C7, C l) and (C4, N3) the reverse is true. Among these, we may suppose that the elements (C4, C1) and (C7, C1), and (C4, N3) and (C7, N3) are cancelling with each other from the geometrical consideration of the bonds (C4-C 1), (C 7-C 1), (C4 N3), and (C7 N3). As a result an element (C5, C4) still remains. This element largely contributes to the stabilization of Pro-II. The term (C7, C6) is also postulated to stabilize the Pro-II form. There appeared no extremely large difference in the exchange term. As for the electrostatic term, the elements (C4, C 1) and (C7, C1), (C4, 02) and (C7, O2), and (C4, N3) and (C7, N3) are fully or partly cancelled with each other. The elements (C5, C4) and (C7, C6) contribute to stabilize the Pro-I form. From the total term in Table 4, we can see that the difference in stability between Pro-I and Pro-II is due mainly to the contribution of the element (C5, C4). From the discussions mentioned above, we can deduce that the intrasegment energy of poly(L-proline) is governed by the C7, N3, C4, and C5 part of the pyrrolidine ring in addition to the C1 part. Moreover, it should be stressed again that the (C5, C4) interaction can be considered to determine the The electrostatic effect on the P r o - l ~ P r o - l l transition have been discussed by Holzwarth and Backman, (Biochemistry 1969, 8, 883). The electrostatic energies obtained by the present calculation and by Holzwarth et al. correspond well with each other. However, as is shown in Table 4, we can readily recognize that the conformational stability is not governed only by the electrostatic energy.
Relative stability ol'poly(L-prolines): M. Ohsaku and A. Imamura Table 5
Difference in two centre interaction energy (eV) in the (~1 segments between Pro-I and Pro-IV 1C1
Resonance term °Clb'" o02 °N3 °C4 °C5 °H8 Exchange term oC4 °C5 °H8
IO2
1N3
IC4
IC7
- 0.01 0.02 - 0.02
0.06
- 0.06
- 0.02
0.04
0.01
-0.01
Total
0.01 0.01 0.04 - 0.20 0.02
-0.14
-0.01 0.01
Electrostatic term °C 1 002 °N3 °C4 °C5 °H8
- 0.03 0.03 -0.01
Total °Cl 002 °N3 °C4 °C5 °H8
-0.01 0.09 0.04 - 0.02 -0.19 0.01
- 0.02 0.09
0.01 - 0.05 -0.01 0.02
0.01 - 0.05 -0.01 0.02
-0.01 0.01
-0.01 0.02
0.01
-0.01
0.01
-0.01
0.01 - 0.04
-0.01 0.01
0.02
- 0.01 0.02 -0.02
0.08 0.01 -0.01
0.01 - 0.04
-0.01
0.03
- 0.08 0.03
-0.12
" See footnote a of Table 4. ISee text and equation 14) for more details} b See footnotett ' See footnote h of Table 4 relative stability of the central segment of Pro-I and P r o : II. We will now discuss the intersegment interactions. Table 5 shows a part of the two centre interactions between the central and the first nearest neighbour segments** in the form of the energy difference between Pro-I and Pro-II. As for the total term, a large difference ( P r o - I - P r o - I I ) appeared in the following elements: (°O2, 1C1), (°N3, 1C1), (°C5, tC1), (o02, 102), (o02, 1N3), (°C4, 1C4), and °C4, IC7)**. A m o n g them, the element (°C5, ~C1) has the largest contribution of all to the difference in the 0 1 intersegment energy. This element stabilizes Pro-I 0.19 eV (4.4 kcal) more than Pro-lI. Therefore, it was found that from the intersegment energy the interaction energy between the fl carbon of the central segment and the carbonyl carbon of the nearest neighbour segment is the d o m i n a n t term in determining the relative stability in poly(L-prolines). This stabilization originates from the element (°C5, 1C1) of the resonance term.
Hydrogen bonding Sasisekharan has studied the arrangement of poly(Lproline II) 4 and concluded that for the polymer chain there is a minimum of short contacts, the only one which occurs being a C 6 - H . . . O 2 hydrogen b o n d between different polymer chains. In the present work, we have ** For simplicity, (~1 (segments) means the central and the first nearest neighbour segments, (~2, 0-3 .... refers to the central and the second, third .... nearest neighbour segments. tt For example, (°02, ~C1) refers to the interaction term between the 02 oxygen atom in the central segment and the C1 carbon atom in the first nearest neighbour segment.
calculated the energy for only one polymer chain and therefore do not discuss this problem. There are no short contacts of H . . . O below 0.25 nm both in the Pro-I and P r o - I I forms in the present case.
Alanines having Pro-I and Pro-ll skeletons The C N D O / 2 calculations were also carried out for alanines assuming the poly(L-proline) skeletons in order to confirm the characteristic properties in stabilization in poly(L-proline). The calculated results are shown in Table 6. F r o m the total energy calculated it is deduced that Ala-I is more stable than Ala-lI by 0.07 eV (1.6 kcal). The intrasegment energy is smaller in Ala-II than in Ala-I, while the intersegment energy is in the reverse order as was already shown in Pro-I and Pro-II. We will discuss in more detail the results of the energy partitioning using Tables 7 and 8. Ala-II is more stable 0.40 eV (9.2 kcal) in the two centre term in the central segment than Ala-1. O n the other hand Ala-I is more stable by 0.18 eV (4.1 kcal) in the two centre term in the (~ l segments than Ala-ll. In the two centre term in the central segment, the difference (Ala-I Ala-II) mainly appeared in the following elements: (C4, Cl), (C4, N3), (C5, C4), and (HT, N3). The element (C4, N3) stabilizes the Ala-I form, and the other elements stabilize the Ala-ll form. The tendency of the energy contribution for the two centre term of the central segment in poly(L-alanine) strongly resembles the tendency shown in poly(L-proline). In the O-1 segments, fairly large difference (Ala-I Ala-lD appeared in the following four elements, (°02, 1el), (°C4, 1C4), (°C4, 1H7), and (°C5, tCl). A m o n g these elements, the resonance element of (°C5, 1C 1) contributes largely to
Int. J. Biol. Macromol., 1980, Vol 2, December
351
Relative stability of poly(L-prolines): M. Ohsaku and A. lmamura the stabilization of Ala-I. This is also seen in the case of P r o d and Pro-II. F r o m the calculations of Pro-I, Pro-It, and Ala-I, and Ala-I1, it was found that there was a very strong resemblance of the energy c o n t r i b u t i o n between poly(t.proline) and poly(L-alanine). It may be seen from these calculations that the c a r b o n a t o m of the fl position plays an i m p o r t a n t role in the c o n f o r m a t i o n a l stability of poly(L-proline). As a result, the methylene g r o u p of the 7 position does not c o n t r i b u t e so much to the energy difference between Pro-I and Pro-II. In other words, the tendency of the energies of the i n t r a s e g m e n t and the intersegment is governed mostly by the main chain d i m e n s i o n s including C5 and C7 a t o m s in poly(t.prolines). Actually, M a d i s o n has proposed that the pyrrolidine ring is generally somewhat flexible 3~. From the Table
6
Total energy (eV) of Ala-I and Ala-ll
Energy Total Total intrasegment Total one centre Total two centre Total intersegment Total two centre () 0 0 0 " See footnote**.
Table
7
1" 2 3 4
Ala-I
Ala-ll
- 1503.28 - 1474.62 1242.61 - 232.00 -28.66 - 14.27 - 0.02 - 0.02 - 0.02
- 1503.21 - 1475.00 - 1242.60 - 232.40 -28.21 - 14.09 - 0.01 - 0.02 - 0.02
(Energy, see equations 1 4)
Model molecules We now take up the model molecules which have structures with two methyl groups attached to the Lprolyl residue as is shown in Fiqure lc. The C N D O / 2 calculations were also carried out for those molecules. The total energies o b t a i n e d are - 2 4 4 8 . 5 5 and - 2 4 4 8 . 6 8 eV for Model-I a n d Model-II, respectively, and the difference is 0.13 eV (3.0 kcal mol 1). The order of the total energy between Model-I and Model-II is in good agreement with that of the intrasegment energy of Pro-I and Pro-ll. Scheraga et al. have also treated the E H T and C N D O / 2 calculations on a model of poly(L-proline) such as acetylu-prolineamide 22. They have o b t a i n e d the energy difference AE,, .......... =5.3 and 1.2 kcal mo1-1 by E H T and C N D O / 2 method, respectively. O u r values agree well with these previously reported values. Scheraga et al. have also made a n o t h e r a p p r o a c h to u n d e r s t a n d the cooperative transition between Pro-I and P r o - l l ash. Energy difference AE,~......... between Pro-I and P r o - l I u n d e r v a c u u m at room temperature varies from - 3.0 to 3.6 kcal mol t with increasing n u m b e r , n, of the considered segment from 4 to 10. In the case n = 9, the difference AE is
Difference in two centre interaction energy (eV) in the central segment between Ala-1 and Ala-ll" C 1h
Resonance term O2 I' N3 C4 C5 H6 H7 Exchange term 02 N3 C4 C5 H7 Electrostatic term 02 N3 C4 C5 H6 H7 Total 02 N3 C4 C5 H6 H7
c a r b o n - 13 l o n g i t u d i n a l relaxation time, it is indicated that the fl and ,, c a r b o n s are more mobile than the z~ and 6 c a r b o n s in n u m e r o u s c o m p o u n d s which have the pyrr o l i d i n e r i n g 32. Therefore it seems that the present result closely relates to those n.m.r, studies, although the direct c o m p a r i s o n between them is very difficult.
0.09 -0.08 0.25
02
0.08 -0.11
N3
0.08
0.01 -0.01
-0.06
0.01
0.02
0.04
0.02 0.03 - 0.07
0.04
0.01 0.04
-0.01 - 0.06
0.03 -0.01 -0.01
0.01 0.01
C5
H6
0.01 0.01
0.01
Total
Symmetric
-
-0.50 0.03 -0.01 0.21
-0.10
C4
0.42 -0.01 - 0.01
0.41
Symmetric 0.05
0.11 - 0.05 0.18 0.01 0.01 - 0.05
0.10
Symmetric
0.09 -0.08 -0.01 0.04
-0.10 -0.12
- 0.01
Symmetric -0.53 0.03 -0.02 0.24
0.37 -0.01 - 0.02
0.01 0.01
0.01
0.40
" Digitsshow the energydifference:Ala-! Ala-1I. Energy terms:absolute valueslessthan 0.01 eV are not catalogued. [Seetext and equation (3) for more details.) h Atom numberings are shown in Fixture Ib
352
Int. J. Biol. Macromol., 1980, Vol 2, December
Relative stability o f poly(L-prolines): M. Ohsaku and A. I m a m u r a Table 8
Difference in two centre interaction energy (eV) in the (~1 segments between Ala-I and Ala-II" I C l b."
Resonance term o c i b.c °N3 °C4 °C5 oH 8
0.01 0.04 -0.01 - 0.21 0.02
Exchange term °C4 °C5
-0.01
Electrostatic term °C1 002 °N3 °C4 °C5 °H8 Total °Cl 002 °N3 °C4 °C 5 °H8
-0.03 0.08
Io2
IN3
IC4
~H7
-0.01
-0.01 0.02 - 0.01
0.06
-0.07
- 0.02
0.01
0.01
-0.01
Total
-0.22
-0.01
- 0.04
- 0.02 0.03 -0.01
0.02
-0.02 0.09 0.04 - 0.03 - 0.19 0.01
0.01 - 0.04 -0.01 0.01
0.01 - 0.05 -0.01
-0.01 0.01 0.01
-0.01 0.03 0.01 - 0.01
0.01
-0.01
0.01 - 0.05
-0.01 0.01
- 0.01 0.02 -0.01
0.08
-0.01 0.03 0.01 - 0.09
-0.01
0.05
-0.18
" See footnote a of Table 7. {See text and equation (4) for more details.) See footnote H ' See footnote h of Table 7 e s t i m a t e d as 2.0 kcal m o l - ~ 2Sb. T h e difference e s t i m a t e d in the p r e s e n t w o r k , w h i c h has also t a k e n i n t o c o n s i d e r a t i o n 9 units, a g r e e s v e r y well w i t h the value, a l t h o u g h the g e o m e t r i e s used in the p r e s e n t article are a little different f r o m t h o s e of S c h e r a g a et al. 28b.
Conclusions T h e c o n f o r m a t i o n a l stability o f P r o - I a n d P r o - I I in the solid state or in s o l u t i o n is a c c o u n t e d for very well by the C N D O / 2 m e t h o d u s i n g the t i g h t - b i n d i n g a p p r o x i m a t i o n . T h e c o n t r i b u t i o n s of the i n t r a s e g m e n t a n d the i n t e r s e g m e n t t e r m s to the e n e r g y difference b e t w e e n P r o - I a n d P r o - I I are n e a r l y e q u a l in a b s o l u t e m a g n i t u d e but h a v e different signs. T h e difference in the i n t r a s e g m e n t e n e r g y of poly(Lp r o l i n e ) is g o v e r n e d by the i n t e r a c t i o n s b e t w e e n the ~ a n d t h e / 3 c a r b o n s of the p y r r o l i d i n e ring. T h e difference in the i n t e r s e g m e n t e n e r g y of poly(Lp r o l i n e I) a n d p o l y ( L - p r o l i n e II) is g o v e r n e d by t h e i n t e r a c t i o n s b e t w e e n the c e n t r a l s e g m e n t [3 c a r b o n a n d the nearest neighbour carbonyl carbon. F r o m the c a l c u l a t i o n of the c o n t r i b u t i o n of the t o r s i o n a l m o t i o n a r o u n d the C , - C' b o n d to the free energy, it was f o u n d t h a t this e n e r g y is sufficient to c h a n g e the conformational stability of poly(L-proline) (see Appendix).
Acknowledgement T h i s w o r k was s u p p o r t e d in p a r t by a G r a n t - i n - A i d for Scientific R e s e a r c h f r o m the M i n i s t r y of E d u c a t i o n , for w h i c h t h e a u t h o r s express t h e i r g r a t i t u d e . T h e a u t h o r s w o u l d like to a c k n o w l e d g e the g u i d a n c e a n d e n c o u r a g e -
m e n t of P r o f e s s o r H i r o m u M u r a t a . T h e c o m p u t a t i o n was c a r r i e d o u t o n a F A C O M M 1 9 0 of the D a t a P r o c e s s i n g C e n t e r , K y o t o U n i v e r s i t y . T h a n k s are a l s o d u e to P r o f e s s o r H. A. S c h e r a g a for s e n d i n g us the rel:;rints of R e f e r e n c e 28 a n d for p r o v i d i n g useful i n f o r m a t i o n in notes.
References 1 2 3 4 5
6 7
Traub, W. and Shmueli, U. Nature ILondon) 1963, 198, 1165 Traub, W. and Shmueli, U. in ~Aspects of Protein Structure', (Eds. W. Traub and U. Schmueli), Academic Press, New York, NY, 1963, pp. 81 92 Cowan, P. W. and McGavin, S. Nature (London) 1955, 176, 501 Sasisekharan, V. Acta Crystallo~tr. 1959, 12, 897 (a) Downie, A. R. and Randall, A. A. Trans. Farady Soc. 1959, 55, 2132; (b) Steinberg, i. Z., Harrington, W. F., Berger, A., Sela, M. and Katchalski, E. J. Am. Chem. Soc. 1960, 82, 5263; (c) Gornick, F., Mandelkern, L., Diorio, A. F. and Roberts, D. E. ibid. 1964, 86, 2549; (d) Engel, J. Biopolymers 1966, 4, 945: (e) Engel, J. in "Conformation of Biopolymers', (Ed. G. N. Ramachandran), Academic Press, New York, 1967, Vol 2, pp. 483 497:11) Veis, A., Kaufman, E. and Chao, C. C. W. in 'Conformation of Biopolymers', (Ed. G. N. Ramachandran) Academic Press, New York, 1967, Vol. 2, pp. 499-512; [g) Carver, J. P. and Blout, E. R. in "Treatise on Collagen', (Ed. G. N. Ramachandran) Academic Press, New York, 1967, Vol. 1, pp. 441 526: see also (hi Strassmair, H., Engel, J. and Knof, S. Biopolymers 1971, 10, 1759; 6) Knof, S., Strassmair, Engel, J., Rothe, M. and Steffen, K. D. ibid. 1972, I !, 731 : (j) the preference of proline peptides for certain conformational states in solution is adequately reviewed, see Scheraga, H. A. Chem. Rev. 1971, 71, 195; and Deber, C. M., Madison, V. and Blout, E. R. Ace. Chem. Res. 1976, 9, 106 Mandelkern, L. in 'Poly
Int. J. Biol. M a c r o m o l . , 1980, Vol 2, D e c e m b e r
353
Relatit:e stability qf poly(L-prolines): M. Ohsaku and A. lmamura 8 9 10
1
2 3
14 15 16 17 18 19 20 21 22 23
24 25 26 27 28
29 30 31 32
For example. Slater, J. C. in 'Quantum Theory of Molecules and Solids', McGraw-Hill, New York, 1965 lmamura, A. and Fujita, H. J. Chem. Phys. 1974, 61, 115 (at Ramachandran, G. N. and Sasisekharan, V. Adr. Protein Chem. 1968, 23, 283: (b) according to IUPAC Commission on nomenclature [Biochemistry, 197(I, 9, 3471 ] DeSantis, P.,Giglio, E.,Liquori, A . M . a n d R i p a m o n t i , A. Nuture (London) 1965, 206, 456 Damiani, A., De Santis, P. and Pizzi, A. ibid. 1970, 226, 542 Balasubramanian, R. Lakshminarayanan, A. V., Sabesan, M. N., Tegoni, G., Venkatesan, K. and Ramachandran, G. N. Int. J. Protein Res. 1971.3, 25 Ramachandran, G. N., Lakshminarayanan, A V., Balasubramanian, R. and Tcgoni, G. Bioehim. Biophys. Aeta 197[1, 221, 165 Tonnelli, A. g. J. Am. Chem. Soc. 197(I, 92, 6187 Englcrt, A.. Furnc'monl, J. and Ldonis, J. Mueromolecules 1971,4, 768 :l-onnelli, A. g. J. Am. Chem. Soc. 1972, 94, 346 Scheraga, H. A. Adr. Phys. Org. Chem. 1968, 6, 103 Schimmel, P. R. and FIory, P. J. Proc. Natl. Aead. Sei. USA t967, 58, 52 Schimmel, P. R. and Flory, P. J. J. Mol. Biol. 1968, 34, 105 Flory, P. J. in 'Statistical Mechanics of Chain Molecules" John Wiley & Sons, New York, 1969 Yam J. F., Momany, F. A., Hoffmann, R. and Scheraga, H. A. J. Phys. Chem. 1970, 74, 420 (at Maigret, B., Perahia, D. and Pullman, B. J. Theor. Biol. 1970, 29, 275: (b) Pullman, B. and Pullman, A. in 'Advances in Protein Chemistry', (Eds. C. B. Afinsen, J. T. Edsail and F. M. Richards) Academic Press, New York, 1974, pp. 347 526 Maigrel, B., Pullman, B. and Caillet, J. Bioehem. Biophys. Res. Commun. 1970, 40, 808 Nishikawa, K. and Ooi, T. Bull. Inst. Chem. Res. 1972, 50, 94 (a) Schwarz, G. Biopolymers 1968, 6, 873: (b) Applequist, J. ibid. 1968, 6, 117: (c) Ganser, V., Engel, J., Winklmair, D. and Krause, G. ibid. 1970, 9, 329 Mattice, W. L. and Mandelkern, L. J. Am. Chem. Soc. 1971, 93, 1769 (at Tanaka, S. and Scheraga, H. A. Maeromoleeules 1975, 8, 494: (b) Tanaka, S. and Scheraga, H. A. ind. 1975, 8, 504: (c) Tanaka, S. and Scheraga, H. A. ibid. 1975, 8, 516:(d) Tanaka, S. and Scheraga, H. A. ind. 1975, 8, 623 Leung, Y. C. and Marsh, R. E. Aeta Crystallogr. 1958, II, 17 Matsuzaki, T. and litaka, Y. Aeta Crystallogr. (B] 1971, 27, 507 Madison, V. Biopolymers 1977, 16, 2671 (al Torchia, D. A. and Lyerla, J. R. Jr. Biopolymers 1974, 13, 97;(b) Deslauriers, R., Smith, I. C. P. and Walter, R. J. Biol. Chem. 1974, 249, 7006: (el Fossel, E. T., Easwaran, K. R. K. and Blout, E. R. Biopolymers 1975, 14, 927: (d)Somorjai, R. L. and Deslauriers, R. J. Am. Chem. Soc. 1976, 98, 6460
/
/
/ -1936 50
\
/
\ \
L
/
I \
I \
o -1937.0C L_ : ~....
loo
I
~ " - ~ . . . . . . . . . ~ \ , , , . ~ . / ----0" ~ ' ' "
12o
14o 16o q~(aeg)
18o
- 20
Figure AI
Total energy as a function of t~ in Pro-I ( • ) and Pro-lI (o). The following angles were fixed during the calculations: Pro-l;
quadratic equation. The minimum values of the curvatures are nearly equal to each other for Pro-I and Pro-II as is described in the preceding section. However, their shapes are very different from each other. That is, in Pro-I the energetically allowed region is narrow, but in Pro-It it is wide. Moreover, it should be pointed out that the observed ~ vaJues by X-ray analyses described above are well reproduced by the present C N D O / 2 calculations (Pro-I, 0 = 157.5, Pro-II, 0 = 142.9). Now, the partition function was proportional to the equation: rU3
Z,= f exp[-(ax2+bx+c)/kT]dx
(A1)
-- rr,'3
Appendix
and the free energy F can be expressed as
Cm!fbrmational stability and the contribution of the torsional motion to the.Jree ener.qy
F= - k Tln Z r
In order to investigate the relative stability of the two structures of poly(L-proline) statistically and thermodynamically, we have evaluated the contribution of the torsional motion about the C, C' bond to the free energy. As is mentioned in the preceding section, the structure of this polymer can be described by the three internal rotation angles: u), ¢p, and 0. Among them, to can be selected to be 0' or 180 under the planarity of the amide group, q) may only take values within a very narrow region, because the monomer is a ring. From this we have assumed the angles of the fixed values as follows, for Pro-I o r P r o - l l : ~ o = 0 or 180, a n d t p = - 8 3 ' o r - 7 7 . 2 " ' ; a n d is varied around the observed equilibrium values ~''*, 158' and 145.9' for Pro-I and Pro-II, respectively. Using these internal rotation angles, the total energies are calculated. The results are shown in Figure A1. Here, we have assumed that the curvature can be described by a certain
The integration of equation (A1) between - n / 3 and n/3 centred at the energy minimum with regard to X was carried out by using Simpson's formula. The values calculated for Pro-I and Pro-II are -1937.09 eV and -1937.13 eV respectively, and the difference between both forms is estimated to be 0.04 eV (0.92 kcal). The present result shows that the torsional motion around the C, C' bond may change the relative stability of the polymer in the Pro-I and Pro-II forms. In other words, with regard to Pro-It, the fluctuation of the structure about the most stable geometry should be fairly large, while in the case of Pro-I, the fluctuation is expected to be relatively small. The free energy difference between Pro-I and Pro-t1 which includes the contribution of the torsional motion is, however, not large enough to deny the coexistence of these two forms in the solid state or in solution.
354
Int. J. Biol. Macromol., 1980, Vol 2, December
(A2)