A theory of protein molecule self-organization

.I. Mol. Biol. (1976) 103, 15-24

A Theory of Protein Molecule Self-organization IV+. Helical and Irregular Local Structures of Unfolded Protein Chains A. V. FINKELSTEIN

Institute

AND 0. B. PTITSYN

of Protein Research, Academy of Sciences of the U.S.S.R., Poustchino, Moscow Region, U.S.S.R.

(Received 28 Augud

1975, and in revised form 12 Janzrary

1976)

A theory of formation of a-helices and irregular local structures in unfolded protein chains is presented. It haa been shown that practically all the native a-helices pre-exist in the unfolded chain as fluctuating embryos though not all the a-helices pre-existing in the unfolded chain are contained in native protoins. It, has also been shown that the overwhelming majority of the p-bends and other irregular local structures do not pm-exist in the unfolded chain. From this a conclusion ha;r been drawn that a-helices can initiate self-organization of the tertiary structure serving EM blocks for its assembling while t.he irregular local st rurt~nms cannot, initiate protein folding.

1. Introduction The t,he & (cf.

process of protein chain folding into the native globule presumably starts with formation of fluctuating local secondary structures (like a-helices, p-bends and helical fragements) stabilized by local interactions in an unfolded protein chain Lewis et al., 1971; Ptitsyn et al., 1972; Anfinsen, 1972,1973; Ptitsyn, 1973: Nagano, 1974). The formation time of these structures must be very short, being. seconds for a-helices growth (Cummings & Eyring, 1976: e.g. as small as ~10-~ Zana, 1975). In any case, it is much shorter than the time of p-structure formation or the time of any protein folding which needs at least more than 10e2 seconds (Anfinsen, 1972,1973). This allows us to distinguish the first stage of the protein molecule self-organization at which an equilibrium between different local secondary structures has been already established, while the formation of the fi-structure and elements of t.he tertiary structure has not yet begun. Therefore, quasi-equilibrium at this stage can be considered within the framework of statistical mechanics. Such n consideration seems very important as it must indicate which local secondary structures can initiate the tertiary structure formation and thus lay the foundations for considering the subsequent stages of protein self-organization. Scheraga and his collaborators (Lewis et al., 1970; Lewis & Scheraga, 1971a.b) have proposed the first theory of the helix formation in an unfolded protein chain. They have divided amino acid residues into “helix-forming”, “helix-breaking” and and have ascribed to them helix-coil equilibrium constants equal “helix-indifferent”, to 1.05, 0,385 and 1.00, respectively. Small s values for helix-breaking residues t Paper III

of this series is: Fink&t&

(1976d). I5

16

A. V. FINKELSTEIN

AND

0.

B. PTITSYN

adopted in these papers led to very low helical state probability in unfolded chains (of the order of several per cent) which contradicts experimental values of the degree of helicity, for example, of myoglobin fragments (Epand & Scheraga, 1968; Singhal & Atassi, 1970; Atassi & Singhal, 1970). In our previous papers (Ptitsyn et al., 1973 ; Denesyuk et al., 1974) we ascribed to amino acid residues s constants based on experimental data on synthetic polypeptides. As a result, we obtained more realistic absolute values of the degree of helicity, confirming in particular experimental data on myoglobin fragments. However, both these theories do not take into account a number of essential factors, the most important among which are the competition between helical regions and other types of the secondary structure and the specific behaviour of various residues at N- and C-termini of helical regions (specific “initiation” and “termination” of helices). The account of these factors requires theoretical evaluation of capabilities of different residues to build-in into various positions at the ends of helical regions, as well as their capabilities to build-in into other local secondary structures, since experimental estimations of the corresponding parameters are practically absent. It was also necessary to develop a mathematical apparatus to account for the competition between u-helices and different irregular local structures of the chain. Solutions of these two problems are given in papers I-III of this series (Finkelstein & Ptitsyn, 1976; Finkelstein et al., 1976; Finkelstein, 1976d), the results of which are digested in the following section. The present paper studies the probabilities of the formation of different local secondary structures in unfolded protein chains.

2. Thermodynamic

Parameters and Method of Calculation

In paper I of this series we have estimated thermodynamic parameters controlling building-in of the natural residues into different positions of local secondary structures of the unfolded protein chain (a-helices, p-bends, 3,, helices and their combinations). The estimation is based on stereochemical analysis of local interactions in protein molecules (Finkelstein, 1976aJ). The obtained equilibrium constants between all the positions in local structures and com$etely un$xed state (the coil without any local structures) are presented in Table VI of paper I. The deduced (Finkelstein, 1976d) approximate equilibrium constants s between the a-helix and a real coil (i.e. a coil with p-bends and other irregular local structures) are presented in Table 1. These constants refer to residues included as “guests” in poly( Gln) and practically also to residues in homopolymers. They are in good accordance with the available data on corresponding synthetic polypeptides (see paper II). The non-specific (i.e. independent of the amino acid sequence) initiation constant of the u-helix is U’ = exp(2AFJkT) = 8.10m4, where AF, = -2.2 kcal/mol is the energy of the hydrogen bond in water chosen so as to provide the experimental value m 10m3 (Platzer et al., 1972). Besides this, the initiation constant is influenced Of uAla by hydrogen bonds, electrostatic and steric interactions of side groups with the Nand C-butts of the helix, and also by the preference of the transition of the C-t,erminus of the a-helix into helix 31,, which follows from our stereochemical analysis. These effects depend on the amino acid residues occupying the first three fixed positions at both the termini of the helix (positions 1, 2, 3) and the first unfixed positions beyond both termini (positions 0). For poly(Ala) these effects almost compensate each other (uAla = 1.lO - “). At every replacement of alanine by other residues in

STRUCTURE

OF UNFOLDED TABLE

Helix-coil

PROTEIN

Ii

CHAINS

1

equilibrium wntstnhs

Residues

8

Ala

(14Wt

L3U

Glu, Lys, Met Phe, Trp, Tyr Arg, Ile His V&l Cys, Gln Am, Asp Ser, Thr G’Y Pro

1.2, 1.1, 1.1 1.0, (& 0.95 0.8s 0.7, 0.5, O§

t Taken from the experiment of Sugiyama & Node (1970) and Platzor et aZ. (1972). $ Taken from the experiment of Alter et al. (1973). 5 Proline can build-in only into the first turn of a-helioes (see Table 2).

the above terminal positions, the constant cAla must be additionally multiplied by the corresponding factor a,. Table 2 presents approximate values of those of the factors mentioned which differ considerably from unity. It was shown in paper II that these factors are in accord with the amino acid residue distribution at the globular protein helix termini. A description of different irregular structures demands a great number of thermodynamic parameters (see Table VI in paper I). Table 3 gives, as an example, approximate equilibrium constants between different positions of the most probable b-bends and the real coil. The initiation constant of p-bends (as well as of other irregular TABLE 2

Terminal factors ui for helical regionsi N-terminal positions

Residues

Asp Glu Ser Asn, Thr GUY His ‘$9 LYS Ile, Val

t All factors 2

3

0

1, 2, 3

0

3.2 1.6 1.7 1.6

1.7 1.7

0.6 0.6

0.6 0.4 1.6

0.6 0.6

0.6 0.6

1.4 2.0 1.8

0.4 1.7 1.2 6.5

0.7 1.4 1.4 0.7

1.7 I.7

0

0

0

0

1.3 (1st and 2nd positions) 0.6 0.5

6.6 1st position:

Pro

C-terminal positions 1 2

2nd end 3rd positions

6 6 after Gly 0 after Pro 1 after others

not shown in the Table are near to unity.

18

A. V. FINKELSTEIN

AND

Residues

Am, Asp, His, SW, Thr

1 m 1.5

G’Y 110, Val PI.0 others

1.5 -1

constants for various

0.7

PTITSYN

of /3-be,&

positions

p-bends I + III 2 3

4

1

2

2.5-s

ml

5yl

ml.7

1.2

-1

*l

-1

ml

0.8 0.3 eO.8

Wl

B.

3

TABLE

Equilibrium

0.

ml.5

4

0

0.8

2.5.-s

0 al

0.6 e 1.6

p-bends II 3 R30.3 (0 for Thr)

4 -1.0-1.5

I.2

al

0

*l

Wl

8

0

ml

e 1.7

.?z0.3

0 -1

structures) is oirr = exp(flF,/kT), which at d F, = -2.2 kcal/mol (see above) gives oirr m 0.03. The distribution of residues in p-bends of globular proteins on the whole agrees with the data of Table 3 (see paper II). The Tables show that in the unfolded protein chain the helices must be localized in the regions enriched by hydrophobic and long hydrophilic side groups and depleted of short hydrophilic side groups and glycine. Proline can be included only in the N-terminal turn of the helix, initiating its formation. The helix is initiated (and to some extent terminated) also by short hydrophilic side groups which can form additional hydrogen bonds at the helix termini. An important role is played as well by the interaction of the charged groups with the backbone partial charges at the helix termini, which leads to a’n asymmetric distribution of the charged groups near these termini. Localization of /3-bends in the unfolded chain must be determined mainly by the great tendency of prolines to build-in into position 2 (the first fixed position) of p-bends I-III, and also by the inclination of glycines to build-in into position 3 of p-bends II, averting the other fixed positions of the bends considered. Moreover, note should be taken of the initiating role of short hydrophilic side groups as well as that of glycines for /I-bends I and III. To calculate the probability of different local secondary structures, it is necessary to use parameter values referred to the completely unfixed state of the coil. The whole set of these values is presented in Tables V and VI of paper I. The mathematical apparatus for these calculations and principles of program development are given in paper III. The difference of this apparatus from the usual matrix method of the Ising model (Zimm C%Bragg, 1959) lies in the following: with the use of the matrix method of the Ising model the state of each residue is described in terms of conformations of all the residues of a segment, the length of which, m, is determined by the maximally remote residues the interactions of which arc taken into consideration (e.g. for a-helices, ‘m = 3). The apparatus, proposed in paper 111, describes the state of each residue in terms of conformations of only those residues with which it practically interacts in the given local secondary structure. This approach permits us to eliminate the application of the matrix apparatus of the co-operative system to a consideration of physically non-co-operative conformations of the chain. As a result, the rank of the matrices without limitations as to generality of the consideration decreases from

STRUCTURE

OF

UNFOLDED

PROTEI1;

I !I

CHAINS

II,“’ to w/t m (where n is the number of states of the amino acid residue considered) i.e. from ~10~ to ~50. The mathematical apparatus used, as well as the usual matrix apparatus of the Ising model, is precise, i.e. the accuracy of the results obtained is limited only by the class of the structures taken for consideration and the accuracy of thv applied thermod~ynamic parameters. The calculations were doncx on a M220M. , 1,cu 1at)‘1011\\‘i\s ahout IO t,o computer (USSR) at a speed of ~20,000 s-l. Tl w t’lrne 0 f ca I5 minutes for every 100 residues of the chain. The appropriate set of thermodynamic parameters is presented in paper I (Tables V and YT).

3. Helical Regions Figures 1 t)o 4 represent the calculated probability of the helical state for some globular proteins. They also represent the probabilities of localization of helical Nand C-ends, i.e. the first and the last residues with a fixed helical conformation (including the 3,, helical conformation at the C-end of helical regions). The Figures also represent the probabilities of the most probable irregular local secondary structures (isolated helices 31,. p-bends and their combinations). Analogous calculations were

0

50 w

-

I-

mm

100 m

‘ii4

la)

(b) ) of the N. (-------I. (a) Probabilities of the helical state (-a****~*--* ) and localizationa ) termini of helical regions in the ribonuclease unfolded chain. and c‘- ( The most probable positions of helical regions in the unfolded chain. The dotted line represents t,he alternative localization of the structures. The native ribonuclease S secondary structure is pictured under the Figure according t’o t,he X-ray data (Wyokoff et’ nl., 1970) (m, CChelices; m, p-structure). ) and the helical (b) Probabilities of the most likely irregular secondary structure ( ) of the ribonuclease unfolded chain. n, the localization of p-bends in the native state (.-NMribonuclease 8 (Lewis et al., 1973). FIG.

20

A. V. FINKELSTEIN

AND

0.

B. PTITSYN

20

IO

0 0

50 ---

100

153

(I----

FIG. 2. Secondary structure of myoglobin. Probability of the most powerful irregular structure is marked only for the regions in which it surpasses the helical stclte probability. Designations in Figures 2 to 4 the same as in Fig. 1. X-ray data from Watson (1969).

40

s 0

30

20

IO

0 0

50 mmFIG. 3. Secondary

100 -

structure

-m of rtdenylate

-mkinase. X-ray

150 -ma

194 B

tlnta from Schulz et al. (1974).

made for many other proteins, including 35 proteins with a known three-dimensional structure. The E’ignres show directly one important general feature concerning unfolded protein chains: the probability of the helical structure 8, in unfolded protein chains is, as a rule, considerably greater than the probability of any other fixed local structure. The probability of the helical state of a region with the fixed &mini in the unfolded chain is very small (equal to ~%ls,, i.e. -10v3 - 10e2). However a fluctuation of helical termini gives rise to a considerable gain of helix entropy. This increases the

STRUCTURE

0

OF UNFOLDED

PROTEIN

n FIG. 4. Secondary

.5i structure

n

mm

n

n

of a-chymotrypsin.

200

150

100

0

X-ray

21

CHAINS

I-

I

data from Birktoft

mm & Mow

I##-

245

(1972).

probability of helical state for central parts of a helical region by one or two orders of magnitude. As a result, the average degree of helicity of unfolded protein chains turns out to be considerably great (10% for ribonuclease, 167$ for adenylate kinase and even 3196 for myoglobin). Probabilities of the helical state in some regions can even exceed 50% (the helical state of these regions being more probable than the totality of all other states). The probability of the helical state of unfolded fragments of sperm whale myoglobin calculated by our theory is 10 to 30% which is close to experimental data (Epand 6 Scheraga, 1968; Singhal & Atassi, 1970; Atassi & Singhal, 1970). Localization of fluctuating helical regions in the unfolded chain is of course determined primarily by localization of prolines which can only enter the first turn of the a-helix: for the majority of sufficiently long regions of the chain “from proline to proline” the helical state is the most probable one. However in a number of cases the probability of helices proves to be lower than that of an irregular secondary structure in regions of concentration of other helix-breaking residues (see Table 1). As a result, fluctuating u-helices are formed in the regions of the unfolded protein chain enriched by hydrophobic and long hydrophilic groups. Their N-termini are mainly localized directly after short hydrophilic groups and near prolines or negatively charged groups, and their C-termini near positively charged groups (see Table 2). It is seen from Figures 1 to 4 that all or almost all helical regions in native globules of these protein have correspondi ig maxima or manifest shoulders of the helical state probability in the unfolded chain rising against the background of irregular structures. This is also true for all the other proteins considered. In the 35 proteins with a known three-dimensional structure that we considered there are about 160 native helical regions; almost 150 of them correspond to the pronounced maxima or shoulders of the helical state probability in the unfolded chain. This means that the overwhelming h e1ices are outlined by local interactions majority of the “native” already at the unfolded chain stage. At the same time the opposite is not true: in the unfolded chain there are a great number of pronounced maxima and shoulders which does not correspond to the x-helical structure of native globules. In the 35 considered proteins there are about 140 such “excessive” maxima or shoulders; most of them (more than 100) correspond to the regions of the chain having the /&structure in the native globules (see, in particular. Fig. 4 for a-chymotrypsin). It should be remembered that the theory does

22

A.

V.

FINKELSTETN

AND

0.

B.

PTITRYK

not consider the formation of the B-structure which needs much more time than the formation of local secondary structures. Figures 1 to 4 show also that the N- and C-termini of native helical regions in globular proteins are localized near the probability maxima of these termini in an unfolded chain. However, one can see that local interactions in an unfolded chain often outline several plausible localizations for one helical terminus. In conclusion we note that the proposed theory of local secondary structure of unfolded protein chains can be applied to the first stage of self-organization not only of globular but also of other proteins. For fibrillar proteins it predicts the secondary structure of the protein chain by itself (i.e. in the absence of intermolecular interactions); e.g. it predicts a very high probability of the a-helical structure for the unfolded C-terminus fragment of tropomyosin (Ptitsyn et al., 1975) and the practical absence of a-helical structures for the high-sulphur fraction of keratin, and, certainly, for collagen.

4. Irregular Secondary Structures Figures 1 to 4 show that the probability of each irregular secondary structure as a rule is considerably smaller than helix probability and commonly is about 4 to 7%. All the well pronounced maxima of this probability shown in the Figures are connected with prolines, as the building-in of a proline residue into a secondary structure requires the fixing of only one and not two angles of internal rotation. The most probable irregular secondary structure is in almost all cases one type of p-bend. For all regions not containing glycines it is a p-bend of type I or III (Venkatachalam, 1968), but glycines are more likely to enter p-bends of type II or II’. The probability of p-bends not containing prolines increases in the regions containing short hydrophilic groups, but never exceeds loo/, . The probability of other irregular secondary structures (helices 3,, and structures stabilized only by side chain-backbone hydrogen bonds) not containing prolines does not exceed 3 to 5%. The probability of an irregular secondary structure with fixed termini is significantly greater than the probability of an x-helix with fixed ends. This is due to a comparatively low co-operativity of irregular structures. However, unlike a-helices, irregular secondary structures cannot be prolongated. Therefore their probability cannot increase owing to the fluctuations of their termini. As a result, the probability of irregular structures is much smaller than the probability of fluctuating a-helices. The localization of p-bends and other irregular structures (not containing prolines) is significantly less specific than the localization of a-helices. The comparison of /I-bend probability in an unfolded chain with localization of p-bends in globular proteins shows positive correlation (cf. Ptitsyn et al., 1975). But like the case of a-helices there are many p-bends outlined by local interactions which are not fixed in compact globules. Moreover, unlike the case of a-helices, in this case we cannot even say that all or almost all native p-bends are outlined by local interactions (see e.g. Fig. 1).

5. Conclusions A priori three types of protein secondary structures can be proposed as initiators for the self-organization of the tertiary structure. These are a-helices, irregular structures like &bends and p-structural hair-pins or sheets. The first two types of these structures need a short time for their formation and therefore can be formed at

STRUCTURE

OF UNFOLDED

PROTEII”

23

CHAIFB

the first stage of self-organization. These two structures were investigated in this paper. It has been shown that cc-helices do pre-exist in an unfolded chain, and therefore t,he tertiary structure can be assembled from their fluctuating embryos which serve as blocks. The approach of helical regions to each other can ensure a simuItaneous interaction of a great number of groups stabilizing the centres of crystallization of the tertiary structure which appear during this process. The theoretically predicted differences in the helical probabilities of various unfolded chain regions are large enough to ensure a collision of two helical regions with high helical probability of one to t,wo orders of magnitude greater than a collision of two helical regions with small helical probability. The long-range interactions forming a compact globule, as a rule, do not make new a-helices, but build-up the pre-existing helices which satisfy certain stereochemical conditions analyzed by Lim (1974). The B-bends and other irregular local structures as a rule do not pre-exist in an unfolded chain. Therefore they cannot serve as initiators for the tertiary structure. The small probability of b-bends, as well as their low specificity, does not permit them t,o direct the mutual approach to helices separated by these bends (an exception can be p-bends containing prolines and having a high probability in the unfolded chain). Presumably most of the p-bends are formed at later stages of self-organization in the regions where local interactions facilitate a bending of the chain. The /3Aructural hair-pins and sheets with their long formation time are apparent,ly formed at later stages of self-organization. Therefore, the questions of the mechanism of the formation of those structures and their probable role in the assembling of the tertiary st,ructure are not considered here. ‘I’hv interpretation of the data obtained is f’he result, of many Dr V. 1. Lim t,c) whom the authors are very grateful.

helpful

discussions

with

REFERENCES Alter,

J. E., Andreatta, R. H., Taylor, G. T. & Scheraga, H. A. (1973). Macromolecules, 6, 564-570. Anfinsen, C. B. (1972). Biochem. J. 128, 737-749. 181, 223-230. Bnfinsen. C. B. (1973). Science, Atassi, M. Z. & Singhal, R. P. (1970). J. Biol. Chem. 245, 5122-5128. Birkt,oft, J. J. & Blow, D. M. (1972). J. Mol. Bid. 68, 187-240. Cummings, A. L. & Eyring, E. M. (1975). Biopolymers, 14, 2107-2114. Denesyuk, a. I., Ptitsyn, 0. B. & Finkelstein, A. V. (1974). Biofizizika (U.S.S.R.). 19. 549-560. Epand, H. M. & Scheraga, H. A. (1968). Biochemistry, 7, 2864.-2872. Pink~~lstvin. A. V. (1976a). ~Vol. BioZ. ((T.S.S.R.), 10, 3 (in thr press). Vinkvlst,cin, A. V. (1976b). ,l//oZ. BioZ. ((‘.S.S.R.) 10, 4 (in t,he press), b’inkt~lstcin, A. lT. ( 1976~). Biopolymers, in the press. Finkelstein, A. V. (1976d). Biopolymers, in the press. Finkelstcin, A. V. & Ptitsyn, 0. B. (1976). Biopolymers, in the press. Pinkclstt~in, A. \‘., Pt.itsyll, 0. B. 85 Kozitsyn, S. A. (1976). Biopolymer.v, in the press. Lewis, P. N. & Sclleraga, H. A. (1971~). ilrch. Biochem. Biophys. 144, 576-583. I,cwis, P. N. & Sclloraga, H. A. (197lb) Arch. Biochem. Biophys. 144, 584-588. ll(>wis, P. N., Gd, N., Gc, M., Kotelchuk, D. & Scheraga, H. A. (1970). Proc. Nat. Acad. Sci., U.S.A. 65, 810-815. Lewis, P. N., Momany, F. A. & Scheraga, H. A. (1971). Proc. Nut. Acad. Sci., U.S.A. 65, “293 ~2297. I,tswis, P. N., Momany, F. A. & Schnmga, H. A. (1973). Riochim. Riophys. .4&x, 303. 2 I 1~ “s!).

24

A. V. FINKELSTEIN

AND

0. B. PTITSYN

Lim, V. I. (1974). J. Mol. Biol. 88, 857-871. Maxfield, F. R. & Scheraga, H. A. (1975). Macromoleculee, 8, 491-493. Maxfield, F. R., Alter, J. E., Taylor, G. T. & Scheraga, H. A. (1975). Macromoleculee, 8, 479-491. Nagano, K. (1974). J. Mol. Biol. 84, 337-372. Platzer, K. E. B., Ananthanarayanan, V. S., Andreatta, R. H. & Scheraga, H. A. (1972). Macromoleculea, 4, 417-424. Ptitsyn, Q. B. (1973). Dokl. Akad. Nauk8.S.S.R. 210, 1213-1216. Ptitsyn, 0. B., Lim, V. I. & Finkelstein, A. V. (1972). Proc. VIllth FEBS Meeting, “Analysis and Simulation of Biochemical Systems” (Heiss, H. & Hemker, H. C., eds), vol. 25, pp. 421-431, North-Holland Publ. Co., Amsterdam. Ptitsyn, 0. B., Denesyuk, A. I., Finkelstein, A. V. & Lim, V. I. (1973). FEBS Letters, 34, 55-57. Ptitsyn, 0. B., Finkelstein, A. V. & Lim, V. I. (1975). Proc. IXth FEBS Meeting, vol. 31, pp. 145-160, Akademiai diado, Budapest. Schulz, G. E., Elzinga, M., Marx, F. & Schirmer, R. H. ( 1974). Nature (London), 260, 129-123. Singhal, R. P. & Atassi, M. Z. (1970). Biochemistry, 9, 4252-4259, Sugiyama, H. & Noda, H. (1970). Biopolymera, 9, 459-469. Venkataohalam, C. M. (1968). Biopolymers, 6, 1425-1430. Watson, H. C. (1969). Progress in St-ereochemistry (Aylett, B. S. & Harris, M. M., eds), vol. 4, pp. 299-333, Butterworth & Co. (Publishers) Ltd., London. Wyckoff, H. W., Tsernoglou, D., Hanson, A. W., Knox, J. R., Lee, B. & Richards, F. M. (1970). J. Bid. Chem. 245, 305-328. Zana., R. (1975). Biopolymera, 14, 2425-2428. Zimm, B. H. & Bragg, J. K. (1959). J. Chem. Phya. 31, 526-536. Note added in proof: The a constants for Glu, Lys, Arg, His and Asp shown in Table 1 refer to uncharged residues. At ionization of these residues, their a constants, according to a theoretical evaluation (A. V. Finkelstein, unpublished results), decrease by about 200/o, which is corroborated by experimenta, data for Glu (Maxfield et al., 1976) and for Lys (M. K. Dygert, G. T. Taylor, F. Cardinaux & H. A. Scheraga, unpublished work). Simultaneously electrostatic interactions appear between ionized groups leading, on the average, to stabilization of the helical state (Maxfield & Scheraga, 1975). In this paper we do not take into account both these effects as they are of the same order and at least partly compensate each other.