J. Mol. Biol. (1996) 256, 623–644 Residue–Residue Potentials with a Favorable Contact Pair Term and an Unfavorable High Packing Density Term, for Simulation and Threading Sanzo Miyazawa 1 and Robert L. Jernigan 2 * Attractive inter-residue contact energies for proteins have been re-evaluated 1 Faculty of Technology Gunma University, Kiryu with the same assumptions and approximations used originally by us in 1985, but with a significantly larger set of protein crystal structures. An Gunma 376, Japan additional repulsive packing energy term, operative at higher densities to 2 Room B-116, Building 12B prevent overpacking, has also been estimated for all 20 amino acids as a Laboratory of Mathematical function of the number of contacting residues, based on their observed Biology, DBS, National distributions. The two terms of opposite sign are intended to be used Cancer Institute, National together to provide an estimate of the overall energies of inter-residue Institutes of Health, Bethesda interactions in simplified proteins without atomic details. To overcome the MD 20892-5677, USA problem of how to utilize the many homologous proteins in the Protein Data Bank, a new scheme has been devised to assign different weights to each protein, based on similarities among amino acid sequences. A total of 1168 protein structures containing 1661 subunit sequences are actually used here. After the sequence weights have been applied, these correspond to an effective number of residue–residue contacts of 113,914, or about six times more than were used in the old analysis. Remarkably, the new attractive contact energies are nearly identical to the old ones, except for those with Leu and the rarer amino acids Trp and Met. The largest change found for Leu is surprising. The estimates of hydrophobicity from the contact energies for non-polar side-chains agree well with the experimental values. In an application of these contact energies, the sequences of 88 structurally distinct proteins in the Protein Data Bank are threaded at all possible positions without gaps into 189 different folds of proteins whose sequences differ from each other by at least 35% sequence identity. The native structures for 73 of 88 proteins, excluding 15 exceptional proteins such as membrane proteins, are all demonstrated to have the lowest alignment energies. 7 1996 Academic Press Limited Keywords: hydrophobicity; contact energy; residue packing energy; sequence threading; sequence sampling weights *Corresponding author Introduction The understanding of protein folding is a long-standing goal in structural biology. Although a large number of native structures of proteins are already known and more are being elucidated rapidly, usually only relatively small fluctuations near native structures have been examined in detail with molecular dynamics simulations that employ potentials with full atomic representations of proteins. Much less is known about the denatured state and the full breadth of the protein folding process. Simulating the entire protein folding process, which occurs on a time scale ranging from milliseconds to seconds, would require enormous computational power because of the high dimen- sional space of protein conformations and the complexity of the energy surface. The present-day capabilities of computers usually limit the time scale of molecular dynamics simulations to nanoseconds. Consequently, simplified models are required for the study of the protein folding process. Simplifica- tions can be made to both geometry and potential functions. Here, we address the design of simplified but realistic free energy potentials that include solvent effects. Previously we evaluated empirically a set of such Abbreviations used: PDB, Protein Data Bank; s.d., standard deviation. 0022–2836/96/080623–22 $12.00/0 7 1996 Academic Press Limited
Transcript
J. Mol. Biol. (1996) 256, 623–644
Residue–Residue Potentials with a Favorable ContactPair Term and an Unfavorable High Packing DensityTerm, for Simulation and Threading
Sanzo Miyazawa 1 and Robert L. Jernigan 2*
Attractive inter-residue contact energies for proteins have been re-evaluated1Faculty of TechnologyGunma University, Kiryu with the same assumptions and approximations used originally by us in
1985, but with a significantly larger set of protein crystal structures. AnGunma 376, Japanadditional repulsive packing energy term, operative at higher densities to2Room B-116, Building 12B prevent overpacking, has also been estimated for all 20 amino acids as a
Laboratory of Mathematical function of the number of contacting residues, based on their observedBiology, DBS, National distributions. The two terms of opposite sign are intended to be usedCancer Institute, National together to provide an estimate of the overall energies of inter-residueInstitutes of Health, Bethesda interactions in simplified proteins without atomic details. To overcome theMD 20892-5677, USA problem of how to utilize the many homologous proteins in the Protein
Data Bank, a new scheme has been devised to assign different weights toeach protein, based on similarities among amino acid sequences. A total of1168 protein structures containing 1661 subunit sequences are actually usedhere. After the sequence weights have been applied, these correspond toan effective number of residue–residue contacts of 113,914, or about sixtimes more than were used in the old analysis. Remarkably, the newattractive contact energies are nearly identical to the old ones, except forthose with Leu and the rarer amino acids Trp and Met. The largest changefound for Leu is surprising. The estimates of hydrophobicity from thecontact energies for non-polar side-chains agree well with the experimentalvalues. In an application of these contact energies, the sequences of 88structurally distinct proteins in the Protein Data Bank are threaded at allpossible positions without gaps into 189 different folds of proteins whosesequences differ from each other by at least 35% sequence identity. Thenative structures for 73 of 88 proteins, excluding 15 exceptional proteinssuch as membrane proteins, are all demonstrated to have the lowestalignment energies.
The understanding of protein folding is along-standing goal in structural biology. Although alarge number of native structures of proteins arealready known and more are being elucidatedrapidly, usually only relatively small fluctuationsnear native structures have been examined in detailwith molecular dynamics simulations that employpotentials with full atomic representations ofproteins. Much less is known about the denaturedstate and the full breadth of the protein folding
process. Simulating the entire protein foldingprocess, which occurs on a time scale ranging frommilliseconds to seconds, would require enormouscomputational power because of the high dimen-sional space of protein conformations and thecomplexity of the energy surface. The present-daycapabilities of computers usually limit the time scaleof molecular dynamics simulations to nanoseconds.Consequently, simplified models are required forthe study of the protein folding process. Simplifica-tions can be made to both geometry and potentialfunctions. Here, we address the design of simplifiedbut realistic free energy potentials that includesolvent effects.
Previously we evaluated empirically a set of suchAbbreviations used: PDB, Protein Data Bank; s.d.,
effective inter-residue contact energies for all pairsof the 20 amino acids, under the basic assumptionthat the average characteristics of residue–residuecontacts observed in a large number of crystalstructures of globular proteins represent the actualintrinsic inter-residue interactions. For this purpose,the Bethe approximation (quasi-chemical approxi-mation) with an approximate treatment of theeffects of chain connectivity was employed with thenumber of residue–residue close contacts observedin protein crystal structures (Miyazawa & Jernigan,1985). This empirical energy function includedsolvent effects, and provided an estimate of thelong-range component of conformational energieswithout atomic details. A comparison of thesecontact energies with the Nozaki–Tanford transferenergies (Nozaki & Tanford, 1971) showed a highcorrelation, although on average the contact energiesyielded about twice the energy gain indicated bythe Nozaki–Tanford transfer energies (Miyazawa& Jernigan, 1985). However, when these contactenergies are applied together with a simpleassumption about the compactness of the denaturedstate to estimate the unfolding Gibbs free energychanges for single amino acid mutants of thetryptophan synthase a subunit (Yutani et al., 1987)and bacteriophage T4 lysozyme (Matsumura et al.,1988), they yield estimates that exhibit a strongcorrelation with the observed values, especially forhydrophobic amino acids. Also, the calculatedenergy values had the same magnitudes as theobserved values for both proteins. This method canalso explain the wide range of unfolding Gibbs freeenergy changes for single amino acid replacementsat various residue positions of staphylococcalnuclease (Miyazawa & Jernigan, 1994). These factsall indicate that the inter-residue contact energiesproperly reflect actual inter-residue interactions,including hydrophobic effects originating in solventeffects.
It is generally thought that the native confor-mations of proteins correspond to the structures oflowest free energy. Thus, successful potentialfunctions, such as those based on native structures,ought to yield the lowest free energy for the nativeconformations. It was demonstrated that classicalsemi-empirical potentials such as CHARMM(Brooks et al., 1983) cannot always identifynon-native folds of proteins (Novotny et al., 1984).Our contact energies were demonstrated to dis-criminate successfully between native-like andincorrectly folded conformations in a lattice study offive small proteins (Covell & Jernigan, 1990). Sippl(1990) evaluated the potentials of mean force as afunction of distance for two-body interactionsbetween amino acids in protein structures from theradial distribution of amino acids in known proteinnative structures. The potentials of mean force forthe interactions between Cb atoms of all amino acidpairs were used to calculate the conformationalenergies of amino acid sequences in a number ofdifferent folds, and it was found that theconformational energy of the native state is the
Table 1. Summary of protein structures used in thepresent analysisNumber of protein structuresa 1168Number of protein subunit structures 1661Number of protein familiesb 424Effective number of proteins (Siwi) 251
a Structures whose resolutions were higher than 2.5 A andwhich were determined by X-ray analyses and are larger than 50residues. See text for details.
b A set of proteins with less than 95% sequence identitybetween any pair.
lowest among the alternatives (Hendlich et al., 1990;Sippl & Weitckus, 1992; Jones et al., 1992). Pairwisecontact potentials depending on inter-residuedistance were also estimated by Bryant & Lawrence(1993).
It should be noted here that a two-bodyresidue–residue potential of mean force based onthe radial distribution of residues will manifestpeaks and valleys as a function of distance, even forhard spheres, which are effects of close residuepacking. However, these would not be present inactual interaction potentials. That is, such apotential of mean force reflects not only the actualinter-residue interactions, but also includes theaverage effects of other residues upon the targetresidue pair, including those interposed betweenthe target pair and especially the significant effectsof residue packing in protein structures. There willbe an over-counting if the sum of the potential istaken over all residue pairs. Thus, if the residue–residue potential in a protein is approximated bysuch a potential of mean force, the sum of thepotential over all residue pairs is unlikely to yieldthe correct value for the total residue–residueinteraction energy. In addition, even though theseeffective potentials have the important character-istics of low values for the native folds of proteins,they are unlikely to succeed in representing theactual potential surface far from the nativeconformation. Therefore, such potentials of meanforce may not be appropriate for application in astudy of a wide range of conformations, from thedenatured state to the native conformation.
On the other hand, a direct account of therequirement that the native state be lowest in energywas taken by Crippen (1991), and Maiorov &Crippen (1992), who tried to fit empirically afeasible set of parameters, which corresponded tocontact energies between amino acid groupsseparated at certain ranges of distance, in such away that the total contact energies of nativeconformations were lower than other alternatives.As with all of these potentials, it is unknown howwell this potential represents the actual potentialsurface far from the native conformation.
Luthy et al. (1992) developed an empirical methodto evaluate the correctness of protein models.Pseudo-potentials have been devised to find outwhich amino acid sequences fold appropriately intoa known three-dimensional structure (Bowie et al.,
Tabl
e2.
Num
ber
ofco
ntac
ts:
upp
ertr
iang
lea
for
rand
omm
ixin
gan
dlo
wer
tria
ngle
bfo
rac
tual
coun
tsin
the
prot
ein
sam
ple
Ni
SLV
cC
ysM
etPh
eIl
eL
euVa
lTr
pTy
rA
laG
lyT
hrSe
rA
snG
lnA
spG
luH
isA
rgL
ysPr
o
314,
460
1089
528
1723
3429
7813
5970
299
1780
10,2
1991
6048
9467
9828
8817
6048
5441
0073
024
9251
2322
73C
' ii24
,518
21,1
2441
,637
57,5
3689
,945
76,8
8015
,672
40,7
7299
,089
95,3
0570
,556
80,4
5952
,629
41,6
5270
,989
64,3
2424
,806
47,8
8872
,052
47,8
60SL
Vc
SLV
c12
,785
,07
4,01
0,16
921
2811
7824
9636
2453
8845
2110
3326
7044
1043
9530
3234
6921
8416
3024
7620
6011
9119
6521
6020
65C
ysC
ys10
,318
12,3
7350
0394
629
6141
6165
8150
4196
424
1650
2340
1530
2030
4619
7416
0026
3824
1012
2120
8921
9519
40M
etM
et10
,354
7172
1247
1125
3566
8302
13,3
3510
,531
2170
5294
10,1
7885
1665
9066
2542
7734
0556
5348
4425
9341
4045
0841
09Ph
ePh
e21
,163
6577
2573
4546
4778
6620
18,6
2214
,980
2820
7004
14,5
0812
,158
9153
9075
5808
4644
7478
6746
3424
5795
6186
5678
Ile
Ile
29,0
6912
,219
3830
5188
10,1
8384
3216
,070
23,5
0245
4010
,854
23,1
4618
,339
14,1
0214
,095
8989
7655
11,5
8410
,788
5514
9456
9895
8882
Leu
Leu
44,4
8313
,346
5395
8218
17,2
2826
,396
21,4
3210
,497
3612
8844
19,2
2915
,857
11,7
8611
,964
7356
5923
9312
8345
4422
7147
7867
7256
Val
Val
38,2
0726
,678
4768
5343
12,7
4619
,830
32,6
2314
,091
525
1998
3503
3064
2393
2452
1541
1236
1880
1640
940
1524
1499
1545
Trp
Trp
8160
5866
1168
1393
2498
3101
4810
3723
519
2752
8598
7554
5875
5917
3874
3001
4811
4030
2221
3662
3759
3654
Tyr
Tyr
20,3
6521
,398
2045
2765
5549
6770
10,8
0677
7719
7524
9110
,471
15,6
4411
,298
11,4
4672
1257
9492
2281
0342
8970
7776
1269
23A
laA
la47
,643
97,6
1343
0650
1410
,516
16,4
6524
,837
21,7
3334
5379
4211
,789
7237
9785
9962
6133
4860
7736
6570
3640
5864
6138
5981
Gly
Gly
46,3
8112
0,77
341
6833
7665
2390
7113
,533
13,5
8429
4672
3517
,079
10,8
8140
4175
5146
7937
5758
9950
7827
3445
0947
2445
11T
hrT
hr32
,587
82,7
9524
7525
9552
8179
5811
,387
9906
1708
5102
11,3
2311
,312
4387
4266
4743
3747
5854
4957
2767
4479
4692
4639
Ser
Ser
36,7
1011
1,60
930
1222
0555
6764
4310
,084
9059
1889
5537
11,4
4712
,019
9465
5458
1766
2396
3791
3232
1784
2853
3126
2873
Asn
Asn
24,2
3676
,966
1549
1410
2940
3331
5751
4792
1381
3792
6465
7319
5790
5920
2605
1148
2989
2619
1375
2404
2420
2306
Gln
Gln
19,1
9958
,151
1375
1490
3056
3772
6091
4541
1057
3350
5037
4918
4412
3957
3368
1253
2721
4301
2255
3683
3992
3648
Asp
Asp
32,7
4910
7,79
015
8413
8932
1743
1857
2946
0614
4451
0578
3686
3273
6283
2760
7535
5029
5121
5420
0434
5936
7832
29G
luG
lu30
,640
98,6
4611
8918
2731
1444
9867
0652
4115
2146
0159
3352
4861
7263
4345
5030
9640
0118
8476
218
2018
6818
17H
isH
is11
,907
24,9
0812
0912
7325
8125
2541
3332
1510
9124
4136
0034
2530
7130
7320
5313
3236
0328
0012
2417
6629
4628
89A
rgA
rg22
,741
62,3
9411
2014
6329
5640
6566
5348
6715
8844
3352
0556
5647
2047
9033
2530
6289
8185
7018
8315
5819
7229
13L
ysL
ys32
,058
124,
460
1200
1456
3098
4199
6531
5437
1367
4678
6019
6054
4940
5117
4260
3304
9501
10,2
3415
5418
0313
5416
24Pr
oPr
o24
,594
63,9
5819
8319
1639
3046
0076
3265
1822
5346
5363
6868
2748
0648
4330
7727
7934
8329
8920
9730
3626
6518
24
Ni
SLV
cC
ysM
etPh
eIl
eL
euVa
lTr
pTy
rA
laG
lyT
hrSe
rA
snG
lnA
spG
luH
isA
rgL
ysPr
o
Tota
lsar
e:N
r=
54,3
56.2
,N
rr=
113,
913.
5,2N
r0=
113,
569.
1.a
Scal
ing
fact
ors
are
C' ii
×10
,C
' i0×
20,
Cii
×10
,an
dC
ij×
20.
bSc
alin
gfa
ctor
sar
eN
i×
10,
Nii
×10
,N
ij×
20.
cSL
V,ef
fect
ive
solv
ent
mol
ecul
es.
Tabl
e3.
Con
tact
ener
gies
inR
Tun
its;
e ijfo
rup
per
half
and
dia
gona
lan
de' ij
for
low
erha
lfC
ysM
etPh
eIl
eL
euVa
lTr
pTy
rA
laG
lyT
hrSe
rA
snG
lnA
spG
luH
isA
rgL
ysPr
o
Cys
−5.4
4−4
.99
−5.8
0−5
.50
−5.8
3−4
.96
−4.9
5−4
.16
−3.5
7−3
.16
−3.1
1−2
.86
−2.5
9−2
.85
−2.4
1−2
.27
−3.6
0−2
.57
−1.9
5−3
.07
Cys
Met
0.46
−5.4
6−6
.56
−6.0
2−6
.41
−5.3
2−5
.55
−4.9
1−3
.94
−3.3
9−3
.51
−3.0
3−2
.95
−3.3
0−2
.57
−2.8
9−3
.98
−3.1
2−2
.48
−3.4
5M
etPh
e0.
54−0
.20
−7.2
6−6
.84
−7.2
8−6
.29
−6.1
6−5
.66
−4.8
1−4
.13
−4.2
8−4
.02
−3.7
5−4
.10
−3.4
8−3
.56
−4.7
7−3
.98
−3.3
6−4
.25
Phe
Ile
0.49
−0.0
10.
06−6
.54
−7.0
4−6
.05
−5.7
8−5
.25
−4.5
8−3
.78
−4.0
3−3
.52
−3.2
4−3
.67
−3.1
7−3
.27
−4.1
4−3
.63
−3.0
1−3
.76
Ile
Leu
0.57
0.01
0.03
−0.0
8−7
.37
−6.4
8−6
.14
−5.6
7−4
.91
−4.1
6−4
.34
−3.9
2−3
.74
−4.0
4−3
.40
−3.5
9−4
.54
−4.0
3−3
.37
−4.2
0L
euVa
l0.
520.
180.
10−0
.01
−0.0
4−5
.52
−5.1
8−4
.62
−4.0
4−3
.38
−3.4
6−3
.05
−2.8
3−3
.07
−2.4
8−2
.67
−3.5
8−3
.07
−2.4
9−3
.32
Val
Trp
0.30
−0.2
90.
000.
020.
080.
11−5
.06
−4.6
6−3
.82
−3.4
2−3
.22
−2.9
9−3
.07
−3.1
1−2
.84
−2.9
9−3
.98
−3.4
1−2
.69
−3.7
3Tr
pTy
r0.
64−0
.10
0.05
0.11
0.10
0.23
−0.0
4−4
.17
−3.3
6−3
.01
−3.0
1−2
.78
−2.7
6−2
.97
−2.7
6−2
.79
−3.5
2−3
.16
−2.6
0−3
.19
Tyr
Ala
0.51
0.15
0.17
0.05
0.13
0.08
0.07
0.09
−2.7
2−2
.31
−2.3
2−2
.01
−1.8
4−1
.89
−1.7
0−1
.51
−2.4
1−1
.83
−1.3
1−2
.03
Ala
Gly
0.68
0.46
0.62
0.62
0.65
0.51
0.24
0.20
0.18
−2.2
4−2
.08
−1.8
2−1
.74
−1.6
6−1
.59
−1.2
2−2
.15
−1.7
2−1
.15
−1.8
7G
lyT
hr0.
670.
280.
410.
300.
400.
360.
370.
130.
100.
10−2
.12
−1.9
6−1
.88
−1.9
0−1
.80
−1.7
4−2
.42
−1.9
0−1
.31
−1.9
0T
hrSe
r0.
690.
530.
440.
590.
600.
550.
380.
140.
180.
14−0
.06
−1.6
7−1
.58
−1.4
9−1
.63
−1.4
8−2
.11
−1.6
2−1
.05
−1.5
7Se
rA
sn0.
970.
620.
720.
870.
790.
770.
300.
170.
360.
220.
020.
10−1
.68
−1.7
1−1
.68
−1.5
1−2
.08
−1.6
4−1
.21
−1.5
3A
snG
ln0.
640.
200.
300.
370.
420.
460.
19−0
.12
0.24
0.24
−0.0
80.
11−0
.10
−1.5
4−1
.46
−1.4
2−1
.98
−1.8
0−1
.29
−1.7
3G
lnA
sp0.
910.
770.
750.
710.
890.
890.
30−0
.07
0.26
0.13
−0.1
4−0
.19
−0.2
4−0
.09
−1.2
1−1
.02
−2.3
2−2
.29
−1.6
8−1
.33
Asp
Glu
0.91
0.30
0.52
0.46
0.55
0.55
0.00
−0.2
50.
300.
36−0
.22
−0.1
9−0
.21
−0.1
90.
05−0
.91
−2.1
5−2
.27
−1.8
0−1
.26
Glu
His
0.65
0.28
0.39
0.66
0.67
0.70
0.08
0.09
0.47
0.50
0.16
0.26
0.29
0.31
−0.1
9−0
.16
−3.0
5−2
.16
−1.3
5−2
.25
His
Arg
0.93
0.38
0.42
0.41
0.43
0.47
−0.1
1−0
.30
0.30
0.18
−0.0
7−0
.01
−0.0
2−0
.26
−0.9
1−1
.04
0.14
−1.5
5−0
.59
−1.7
0A
rgL
ys0.
830.
310.
330.
320.
370.
33−0
.10
−0.4
60.
110.
03−0
.19
−0.1
5−0
.30
−0.4
6−1
.01
−1.2
80.
230.
24−0
.12
−0.9
7L
ysPr
o0.
530.
160.
250.
390.
350.
31−0
.33
−0.2
30.
200.
130.
040.
140.
18−0
.08
0.14
0.07
0.15
−0.0
5−0
.04
−1.7
5Pr
o
e rr−
2.55
e ir−3
.57
−3.9
2−4
.76
−4.4
2−4
.81
−3.8
9−3
.81
−3.4
1−2
.57
−2.1
9−2
.29
−1.9
8−1
.92
−2.0
0−1
.84
−1.7
9−2
.56
−2.1
1−1
.52
−2.0
9e r
−3.
60e i
−4.2
9−4
.73
−5.5
7−5
.29
−5.7
1−4
.72
−4.4
1−3
.87
−3.1
7−2
.53
−2.6
3−2
.27
−2.1
4−2
.35
−2.0
2−2
.07
−2.9
4−2
.43
−1.8
2−2
.53
f r−
3.60
f i−5
.58
−6.1
4−7
.39
−7.0
9−7
.88
−6.1
5−5
.34
−4.6
0−3
.24
−2.2
2−2
.48
−1.9
2−1
.74
−1.9
3−1
.54
−1.4
9−2
.91
−2.0
7−1
.17
−1.9
7N
ir/
Ni
2.09
62.
723
2.72
22.
780
2.81
12.
893
2.72
82.
537
2.49
32.
143
1.84
01.
973
1.77
11.
699
1.72
01.
598
1.50
82.
075
1.78
71.
343
1.62
9q i
7.16
26.
281
6.64
66.
137
5.87
06.
042
6.08
76.
155
5.79
36.
037
6.33
46.
284
6.48
66.
582
6.57
46.
469
6.48
76.
235
6.24
16.
318
6.56
95.
858
Residue–Residue Potentials 627
Figure 1. A comparison of the coordination number, qi ,between the previous analysis (1985) and this work foreach type of amino acid. Slv denotes solvent. The ordinateindicates the values previously reported by Miyazawa &Jernigan (1985) and the abscissa the values obtained inthis work. A continuous line shows the regression linethat is y = −0.19 + 1.03x; the correlation coefficient is 0.98.
would be developed. In the Bethe approximation,the effects of interactions are taken into account toestimate the average numbers of contacts. Thus, theBethe approximation is the lowest order approxi-mation to be able to provide an estimate of a set ofcontact energies between species from a given set ofthe average numbers of contacts between them.
Therefore, if residue–residue contacts in proteinstructures can be reliably represented to be thesame as those in mixtures of unconnected aminoacids and solvent molecules, the Bethe approxi-mation will give us a reasonable estimate of actualinteraction energies between amino acids. Ofcourse, it must be examined whether or not thisbasic approximation is appropriate to describe thecontacts in real protein structures. Also, a limitation,both of this method and of methods using apotential of mean force to evaluate inter-residueinteractions, lies in the fact that the effects of specificamino acid sequences on the statistical distributionof inter-residue distances are completely neglected,although the characteristics of the protein being achain are included as a mean field.
Here, the effective inter-residue contact energiesfor all amino acid pairs are re-evaluated using thesame method as before, but with significantly moreprotein structures than before. In the original work,only 42 globular proteins were used to calculatecontact frequencies between amino acids. Since1985, many additional protein structures have beenreported, and now more than 1000 proteinstructures are available. However, one complicationarises from the fact that there are many homologousproteins in the Protein Data Bank (PDB; Bernsteinet al., 1977). For example, the structures of manysingle amino acid mutants of T4 lysozyme areincluded in the PDB. Therefore, to use all of thisstructural data, an unbiased sampling of proteinstructures from the PDB is required in thecalculation of the contact frequencies. Here, asampling weight for each protein has been devisedbased on a sequence homology matrix giving theextent of sequence identity of all pairs of sequences.
In the estimation of the attractive contact energies,the Bethe lattice is used for conformational space.However, an additional repulsive potential based onmany-body residue packing is needed to properlyestimate the long range energies of proteinconformations. Here, the repulsive energies thatresult from tight packing of residues are evaluatedas a function of the numbers of contacting residuesbased on their distributions in known proteinstructures.
Results
Sample weighting
According to the procedure described inMethods, 1168 protein structures, whose structureshave been analyzed by X-ray and whose resolutionhas been reported to be better than 2.5 A, arechosen, and then each of the 1661 sequences in
1991; Nishikawa & Matsuo, 1993). The pseudo-potential devised by Nishikawa & Matsuo (1993)was composed of four terms, side-chain packing,hydration, hydrogen bonding and local confor-mational potentials, and was empirically derivedfrom the statistical features observed in 101 knownprotein structures. A slightly modified form of theSippl potential was used to take account of the effectof side-chain packing in proteins. All other termswere also evaluated as potentials of mean force.This function was also demonstrated to be anappropriate measure of the compatibility betweensequences and structures of proteins. They share anumber of common characteristics with empiricalenergy potentials, but they are not designed to beused explicitly as an empirical energy function. Inthe case of Nishikawa & Matsuo (1993), each of thefour terms are summed with weights in the totalenergy.
On the other hand, the present two-body contactenergies are estimated with the Bethe approxi-mation with the basic assumption of regardingresidue–residue contacts in protein structures to bethe same as those in mixtures of unconnected aminoacids and solvent molecules. The Bethe approxi-mation is a well-known second-order approximationto the mean field approximation used to describea system consisting of a mixture of multiplemolecular species interacting with each other (Hill,1960). Both approximations are usually used tocalculate a partition function for such a systemfrom a given set of interaction energies betweenmolecules. In the mean field approximation, con-tacts between species would be approximated to berandom, and a partition function of the system
Residue–Residue Potentials628
A
B
C
Figure 2(A–C)
those structures is sampled with a weight deter-mined as described below on the basis of thesequence identity matrix. Sequences are aligned ina conventional pairwise manner with a globalalignment method (Needleman & Wunsch, 1970),using the log odds matrix for 250 PAM (Dayhoffet al., 1978) as a scoring matrix for amino acidsimilarity. Penalty for a gap (of k residues) is takento be 12 + 4(k − 1) with the cut-off value of 48, butno penalty is used for terminal gaps. Sequenceidentities are then calculated for the aligned pairs ofsequences. In the original work, the numbers ofcontacts in protein structures were counted incomplete assembly. In the present calculation, onlythe coordinates of subunits explicitly given in thePDB files are used. As listed in Table 1, the effectivenumber of proteins is 251. The effective number ofresidues is 54,356, and the total effective number ofcontacts is 113,914 in the present analysis; bycontrast, these values were 9040 and 18,192 in theoriginal analysis, respectively. The total effectivenumber of contacts is 6.3 times the previous data.Thus, on average, the standard deviations in thenumbers of contacts could be expected to bereduced by a factor of 1/2.5.
Estimates of contact energies
The effective numbers of contacts observed in theprotein structures are listed in the lower triangle ofTable 2. The entries in Table 2 have been multipliedby ten for diagonal elements and by 20 foroff-diagonal elements. The numbers in the uppertriangle of Table 2 correspond to the correctionfactors, C and C ', for the estimation of contactenergies; see equations (29), (30), (36) and (37) inthis paper, and also equations (10) to (15) ofMiyazawa & Jernigan (1985). The coordinationnumber for each residue type has been estimated inthe same way with equation (33) of Miyazawa &Jernigan (1985) from the volume of each type ofresidue at the center and the average volume of itssurrounding residues. These newly estimatedcoordination numbers are listed in the last row ofTable 3 and are very similar to the previous
Figure 2. A, A comparison of the average contactenergy, ei , in RT units for each type of amino acidbetween the previous analysis (1985) and this work. Theordinate values are those reported by Miyazawa &Jernigan (1985) and the abscissa the values obtained in thepresent analysis. The continuous line is the regressionline, y = 0.06 + 0.92x; the correlation coefficient is 0.98.B, A comparison of the contact energy, eij , in RT units ofeach type of contact between the previous analysis (1985)and this work. A continuous line shows the regressionline that is y = 0.03 + 0.93x; the correlation coefficient is0.97. C, A comparison of the energy, e'ij , in RT units, whichis the energy difference accompanying the formation of acontact pair i–j from contact pairs i–i and j–j, between theprevious analysis (1985) and this work. The regressionline is y = −0.02 + 0.96x, and the correlation coefficient is0.88.
Residue–Residue Potentials 629
estimates (1985) as shown in Figure 1; the regressionline is y = −0.19 + 1.03x and the correlation coeffi-cient is 0.98.
Effective contact energies are estimated fromthese numbers and are listed in dimensionlessunits, units of RT, in Table 3. Figure 2 shows acomparison between the new contact energies andthose from the previous analysis. The averagecontact energies, ei , for each type of residue arecompared in Figure 2A, and each of the contactenergies, eij , is compared in Figure 2B. The ordinatevalues are those reported by Miyazawa & Jernigan(1985) and the abscissa values those obtained in thepresent work. Correlations between both estimatesare high; the correlation coefficient is 0.98 for ei
and 0.97 for eij . The difference between these twoestimates tends to be larger for hydrophobic aminoacids than for hydrophilic ones. On average, thepresent estimates of contact energies are slightlymore negative than the previous estimates, perhapsreflecting large structures present in the new data;the regression line is y = 0.06 + 0.92x for ei , andy = 0.03 + 0.93x for eij . The present estimate of themean contact energy, er , is equal to −3.6, which ismore negative than the old estimate, −3.2.
Also, the differences between individual newand old contact energies tend to be large forinfrequent amino acids such as Trp and Met. Thismight be expected. However, there is one largedifference for a common amino acid, Leu. Thecontact energy of any pair with Leu is significantlymore negative in the present analysis. All of the topten contact pairs with large differences involve pairswith Leu. The difference in the contact energy forthe Leu–Leu pair is the largest among them, and thepresent estimate, −7.37, is more negative than theprevious one, −5.79. If the coordination number forLeu were estimated to be smaller in the presentwork than previously, then the contact energy forany pair with Leu would be estimated to be morenegative. However, the two estimates of thecoordination number of Leu are almost identical(see Table 2 and Figure 1). The comparison of thedistribution of the number of contacts for Leuclearly shows that the present distribution is shiftedtoward more contacts than the previous one.Therefore, there are actually more contacts with Leuin the present data than before, but the reason isunclear.
As shown in equation (27), the estimation ofcontact energies, eij , requires the estimation of n00,which is not so accurate. On the other hand, relativedifferences between contact energies do not dependon such a parameter; see equation (28). As alreadystated, the absolute values of the contact energiesmight be expected to be less reliable than theirrelative differences. The required scaling factor forabsolute energy specification could be adjusted bymaking err consistent with experimental estimates ofthe average attractive energy between residues.
Figure 2C shows the comparison of the values ofe'ij between the two estimates. The estimation of e'ij ,which is the energy difference accompanying the
Figure 3. Transfer free energies of amino acids relativeto glycine between octanol and water corrected by Sharpet al. (1991) and the corresponding values of −0.6qiei /2 inthis work. −RTqiei /2 corresponds to the average contactenergy gain of an i type residue completely surroundedby other residues in the protein crystal structures.RT = 0.6 kcal/mol has been employed to transform thedimensionless contact energies into kcal/mol units. Themodified transfer free energies of 20 N-acetyl amino acidamides between octanol and water are taken fromTable III of Sharp et al. (1991); these values include volumecorrections for solute–solvent size differences. The line ofunit slope is shown by a dotted line. Filled circles showthe values for non-polar side-chains (Ala, Val, Ile, Leu,Phe) and open circles for other amino acids; Gly is locatedat the origin. The regression line, not shown here, isy = −0.14 + 0.96x, and the correlation coefficient is 0.98 fornon-polar side-chains. Although polar amino acids aregiven in this Figure, the comparison is strictly meaningfulonly for non-polar residues.
formation of the two contact pairs i–j from thecontact pairs i–i and j–j, does not require estimationof n00. Therefore, their estimates might be expectedto be more accurate than the absolute values of thecontact energies, eij . The regression line in Figure 2Calmost passes through the origin with unit slope; itis y = −0.02 + 0.96x. Even though the correlationbetween the two estimates is clearly not as good asfor the contact energies eij , its value is still quitegood at 0.88.
The important qualitative characteristics of thecontact energies observed in the previous analysishold in the present results; the values of e'ij clearlyshow: (1) a relatively large energy loss accompany-ing the formation of Cys–X contacts from Cys–Cysand X–X contacts, probably reflecting the loss ofdisulfide bonds; (2) the large favorable electrostaticinteractions between positively charged (Lys, Arg)and negatively charged (Glu, Asp) amino acids; and(3) the segregation of hydrophobic and hydrophilicresidues.
Residue–Residue Potentials630
A C
B D
Figure 4. The frequency distribution of the number of contacts for each type of amino acid. The distribution for eachtype of amino acid is shown by order of increasing values of eir , which measures the hydrophobicity of the amino acid,in A to D. Total in C shows the frequency distribution of the number of contacts, irrespective of amino acid type.
Comparison with experimentaltransfer energies
Figure 3 shows the comparison of the contactenergies with the transfer free energies, taken fromTable III of Sharp et al. (1991), for amino acidsrelative to glycine between octanol and water. Theabscissae show values of −0.6qiei /2 that correspondto the average contact energy gain of an i type ofresidue completely surrounded by other residues inprotein crystal structures. For comparison, a valueof RT = 0.6 kcal/mol has been employed in thisFigure to express the dimensionless contact energiesin kcal/mol units. Sharp et al. (1991) re-evaluatedthe transfer free energies of 20 amino acids betweenoctanol and water taken from Fauchere & Pliska(1983) by including volume corrections for solute–solvent size differences. The filled circles show thevalues for non-polar side-chains, Ala, Val, Ile, Leuand Phe, and the open circles for other amino acids;Gly is located at the origin. The plots for the mosthydrophobic side-chains are located nearly directlyon the dotted line of the unit slope, i.e. the twovalues are nearly identical. Their coincidencestrongly supports the present estimates of long-range interactions between side-chains. Although
polar amino acids are also plotted in this Figure, thecomparison is really meaningful only for non-polarresidues, because organic solvents cannot representcircumstances surrounding polar residues in nativeprotein structures; ei for polar residues includesnot only hydrophobic energies but also the averageof other interaction energies with surroundingresidues and water in proteins, such as hydrogenbonds and electrostatic energies. Thus, the extent ofagreement is, overall, better than might be expected.
Distributions of the number ofresidue contacts
The distribution of the number of contacts foreach of the 20 types of amino acids in proteincrystals is shown in order of increasing values of eir ,which measure the hydrophobicity of amino acids,in Figure 4A to 4D. Amino acids whose values ofeir are similar will show similar frequency distri-butions, although the coordination numbers mustbe taken into account. For example, the distributionfor Cys is shifted somewhat toward more contactsthan that for Trp, even though ecys,r for Cys is morepositive than that for Trp, because the coordinationnumber for Cys is significantly larger than that for
Residue–Residue Potentials 631
Trp. The distributions for non-polar amino acidshave single peaks around nc = 6 and those for themost polar amino acids near nc = 2. These numbersof contacts are typical for residues buried com-pletely inside of proteins or for residues exposed onthe surfaces of proteins. This indicates that thesystem consists nearly of two phases, buriedresidues and surface residues. The distribution forSer clearly shows the presence of a shoulder nearnc = 6, as well as a single peak at nc = 2. For suchresidue types that are ambivalent in character, thedistributions can readily be decomposed into twopeaks with maxima near 2 and 6.
Table 4 shows only the high density portion of thedistribution of the number of contacts for each ofthe 20 types of amino acid in protein crystals. Eachnumber in the table is an effective number becauseit is the sum of the numbers of contacts weightedby the sampling weight of equation (50) to removeany sampling bias from homologous proteinsequences included in the protein sample. Repul-sive packing energies have been estimated directlyfrom the values in this table by using equation (43).
Average residue–residue energies
Replacing eipj in equation (19) by the averagecontact energy eip of the ip type of amino acid, theaverage contact energy for the pth residue isrepresented by:
�Ecp�01
2 eipncp (1)
and depends linearly on the number of contactswith the pth residue, nc
p. Figure 5 shows thedependence of the long range interaction energy onthe number of contacts for each type of amino acid.The ordinate corresponds to the sum of the averageattractive contact energy (equation (1)) and therepulsive packing energy (equation (43)). In therepulsive region shown in this Figure, the repulsivepacking energy, the second term in equation (43),does not depend so much on the type of aminoacid. This is expected, because repulsive packingenergies should reflect only packing density and notdepend strongly on the type of amino acid at thecenter; likewise, the coordination number of theamino acid does not depend much on the type ofamino acid.
Total energies of individual proteins
As described in Methods, the long-range energydefined by equation (13) has been calculated for aset of 189 representatives of protein structures in thePDB, which differ from each other by at least 35%sequence identity, but do not include too manyunknown atomic coordinates, which were selectedby Orengo et al. (1993).
The numbers of residues in contact with eachresidue along a chain are calculated by countingresidues within 6.5 A of each residue according toequation (14): the contact energies between these
residue pairs are then evaluated from Table 3 andsummed up according to equation (19). Alsorepulsive energies are estimated for the residueaccording to equation (40). The hard core repulsiondefined by equation (41) is not included here withehc = 0, because known protein structures wereusually refined to remove such close contacts, andsuch close contacts are easily removed by structurerefinement. The repulsive packing energy is esti-mated according to equation (43), if the number ofcontacts at the pth residue is above the thresholdvalue qip . Then, those contact energies and repulsiveenergies are summed over all residues in a proteinto estimate the total long-range energy.
As discussed in the previous work, the totalcontact energies are expected to consist of twoterms, a term that is proportional to the chain lengthand another term that is proportional to the surfacearea of the proteins, that is, the two-thirds power ofthe chain length for proteins whose shapes aresimilar to each other:
sp
Ecp = s
i($0)
sj($0)
eijnij (2)
= ernrr (3)
2ev si($0)
12 qini − esnr0 (4)
where
ev = 0 si($0)
eiqiNi1>0 si($0)
qiNi1 (5)
= −3.26
es = 0 si($0)
eiNi01/Nr0 (6)
= −2.57
In Figure 6A, the long-range energies per residuecalculated with equation (13) are plotted indimensionless units against the inverse of theone-third power of their chain lengths. Monomericproteins, whose shapes could be similar to eachother, are shown by filled circles, and the regressionline for them is shown by a continuous line that isgiven by Elong/nr = −8.5 + 10.1n−1/3
r . The correlationcoefficient is 0.67. The value of the intersect of theregression line is more positive than expected fromequation (4), probably because of repulsive packingenergies included in the total long-range energies ofthe proteins.
Alignment energy for residues within aprotein structure
In the present contact energies, any contact has afavorable contribution to protein stability, even if itis between polar residues, because contact energiesbetween residues are all negative. Therefore,
Tabl
e4.
The
high
dens
ity
por
tion
inth
ed
istr
ibut
ion
ofth
enu
mbe
rof
cont
acts
for
each
amin
oac
idnc
Tota
lC
ysM
etPh
eIl
eL
euVa
lTr
pTy
rA
laG
lyT
hrSe
rA
snG
lnA
spG
luH
isA
rgL
ysPr
o
580
57.8
212.
116
7.2
415.
452
7.5
720.
465
4.2
185.
642
5.0
754.
267
7.2
485.
147
0.1
333.
524
3.9
364.
230
6.2
204.
633
8.7
251.
832
0.8
673
72.4
242.
120
9.7
461.
461
9.4
947.
380
4.8
172.
740
2.6
780.
649
9.2
366.
837
8.5
201.
117
2.1
204.
520
4.6
152.
021
9.3
125.
420
8.4
753
34.9
178.
719
4.4
368.
854
9.7
921.
268
6.4
95.5
258.
049
4.4
288.
023
5.1
237.
911
9.9
90.8
143.
910
8.4
95.4
98.5
53.1
116.
88
2747
.786
.410
7.9
213.
031
8.6
553.
639
2.8
53.3
121.
221
3.0
115.
811
6.1
88.5
47.8
52.0
70.0
40.7
55.4
33.8
17.3
50.4
988
3.3
20.3
42.7
69.2
110.
020
5.8
124.
714
.936
.457
.835
.945
.227
.36.
513
.716
.212
.414
.211
.44.
913
.810
226.
94.
56.
123
.728
.859
.930
.02.
911
.811
.616
.09.
57.
11.
91.
14.
90.
72.
90.
61.
71.
111
28.4
0.5
0.0
2.5
4.9
5.6
4.7
0.0
0.8
2.8
1.9
1.8
0.1
0.0
0.0
0.1
0.0
2.0
0.0
0.0
0.8
120.
80.
00.
00.
20.
00.
00.
20.
00.
00.
20.
20.
00.
00.
00.
00.
00.
00.
00.
00.
00.
013
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
q i6.
286.
656.
145.
876.
046.
096.
165.
796.
046.
336.
286.
496.
586.
576.
476.
496.
246.
246.
326.
575.
86
Residue–Residue Potentials 633
A
B
C
Figure 5. The dependence of the average long-rangeenergy on the number of contacts for each residue type.A, Ranges of packing density as reflected in nc
p values.B, Values for the more polar residue types. C, The lesspolar residue types in order of decreasing values of eir ,which measures the hydrophobicity of the amino acid.The ordinates show the sum of the average contact energyand the repulsive packing energy, in dimensionless RTunits; see equation (1) for the definition of the averagecontact energy and equation (40) for the repulsive packingenergy. The hard core repulsion energy is not includedhere; ehc = 0 in equation (41).
non-representative proteins whose shapes differsignificantly from a globular form, or smallproteins like inhibitors, are often stabilized bydisulfide bonds whose effects on protein stabilityare not counted here. Other effects not accountedfor here are effects of the backbone and details ofdenatured state entropies. Also, as already noted,the estimate of err , which is defined by equation(34) and reflects the overall compactness ofproteins, is less reliable. Consequently, an assess-ment of protein stability based on the totalnumber of contacts in a protein could lead to anincorrect result. Thus, in order to measure thestability of any protein structure for a givensequence, it might be better to use the energythat does not include the homogeneous energyfor protein collapse but consists only of theremaining energy for aligning residues with thecontacts assumed to be present within the proteinstructure. The alignment energy of residueswithin a protein structure is therefore definedhere to be the total contact energy minus theaverage collapse energy:
Ecp (eij − err ) = Ec
p (eij ) − Ecp (err ) (7)
where Ecp (eij − err ) is a function of (eij − err ) that is
calculated by replacing eij with (eij − err ) in equation(18). eij − err here is named the alignment energy forcontacts between residues of type i and j, andrepresents the relative preference of the i–j contactscompared to the average attraction.
It should also be noted that Ec(eij − err ) does notdepend on nr0 but has only a linear dependence onthe chain length; because es is almost equal to err , thesecond term in equation (4) is negligible. The valueof err is estimated to be −2.55 in RT units as givenin Table 3.
It is clear from equation (7) that Ecp (eij − err ) will
be positive unless hydrophobic residues are buriedinside proteins and hydrophilic ones are exposedto solvent on the surfaces of proteins. In otherwords, the alignment energy, Ec
p (eij − err ), is consist-ent with the polar-out and non-polar-in ruleobserved in protein structures. Here it should benoted that equation (7) can be applied to proteinsin water, but should not be used in a hydrophobicenvironment.
The total alignment energy in the form ofequation (7) is appropriate for the considerationof the stabilities of protein conformations fora given sequence, but it is still inappropriatefor the stabilities of a given fold for differentprotein sequences. In the latter case, a simplecomparison of the energies of a given foldamong protein sequences would be meaningless,because the ensemble of protein conformationscould depend on the protein sequence. Thestability of a specific conformation for a proteinis determined in relation to the whole ensembleof protein conformations. Therefore, unless thewhole ensemble of protein conformations isknown, reference energies, each of which reflectsto some extent the partition function for a
conformations with the most contacts tend to bethe most stable in the assessment of the totalcontact energy. However, a rigorous treatment ofbinding requires an estimate of the entropy lossas well as the binding energy. On the other hand,
Residue–Residue Potentials634
A B
Figure 6. A, The dependence of the long-range energy per residue on the inverse of the one-third power of chainlength. B, The long-range energy per residue with a collapse energy subtracted to remove the protein size dependence.See equation (13) for the definition of the long-range energy, and equation (12) for DElong(eij − err ). The long-range energiesinclude the contact energies and the repulsive packing energies, but not the hard core repulsion energies, i.e. ehc = 0in equation (41). All energies here are given in RT units. The representative protein structures used here are 189 proteinstructures that differ from each other by at least 35% sequence identity and are those selected by Orengo et al. (1993)(see their Table 1). Proteins with many unknown atomic coordinates are not included. The filled circles show the valuesfor monomeric proteins determined by X-ray, not including membrane proteins, metal binding proteins, DNA bindingproteins and inhibitors, and multimeric proteins not given in their complete assembly in the coordinate files. The opencircles are proteins whose structures are given in an at least partial, if not full, assembly of subunits. A continuous lineshows the regression line for the monomeric proteins. In A, the regression line is Elong(eij )/nr = −8.5 + 10.1n − 1/3
r , and thecorrelation coefficient is 0.67. In B, a collapse energy has been removed, so that the regression line is almost flat,DElong(eij − err )/nr = 0.30 − 0.01n − 1/3
r , and the correlation coefficient is 0.001. The entry names and sequence identifiers ofthe PDB files used in preparing this Figure are:
sequence, are needed in order to measure thestabilities of a given fold for protein sequences.Here, we choose the total energy expected fora typical protein with the given amino acid
composition and chain length to be the referenceenergy of the native structure for the given proteinsequence, in order to compare the alignmentenergies of the native folds among a wide range of
Residue–Residue Potentials 635
protein sequences; see Miyazawa & Jernigan (1994)for a reference state that is appropriate to measurestability changes due to limited amino acid changesin a protein. That is, the following difference inenergy is considered:
DElong 0 Elong
− (Elong of a typical native structurefor a given sequence) (8)
0Elong − sp
fip
· (the average number of contacts perresidue in a typical native structure) (9)
where the latter sum is the sum of the averagecontact energies per residue of residue type ip overall positions in the protein. fi as a function of eij isestimated by averaging over all proteins; that is:
fi (eij )0eiNir
Nrr
Nr
Ni(10)
The values of fi (eij ) are given in Table 3. The averagenumber of contacts per residue in a typical nativestructure for a given sequence is estimated asfollows, ignoring the chain length dependence.
(the average number of contacts per residue in atypical native structure):
0Nrr /Nr = 2.096 (11)
Because the second term of equation (9) does notdepend on protein conformation but only on theamino acid composition and the length of theprotein, it is a scaling factor and does not have anyeffect in a comparison of different conformations forthe same protein. However, to a certain extent, itsuse will allow us to discuss how compatible a givenprotein sequence is with a certain structure, as wellas how stable a particular conformation is for acertain sequence.
Lastly, a linear dependence on chain length isremoved by comparison on a ‘‘per residue’’ basis,and the following quantity, which is appropriate forassessing the stability of one protein structureamong other folds, is obtained:
DElong(eij − err )/nr
0[Elong(eij − err ) − sp
fip (eij − err )
· (Nrr /Nr )]/nr (12)
where DElong(eij − err ) and fip (eij − err ) are calculatedby substituting eij with (eij − err ) in equations (9) and(10). DElong includes both attractive and repulsiveterms; the latter will be important principally forpoor quality structures.
In Figure 6B, the estimates of DElong(eij − err )/nr forthe protein representatives are plotted against n−1/3
r .As expected, overall there is no correlation betweenthe two quantities for monomeric proteins; themean for monomeric proteins is slightly morepositive than zero, because the repulsive packing
energy is included in DElong(eij − err )/nr . Therefore,this alignment energy has removed the dependenceon the size of the protein. Membrane proteins,which are shown as crosses, tend to have muchhigher values of DElong(eij − err )/nr than the mean formonomeric proteins. This is to be expected, becausemembrane proteins are not stable in water; inmembrane proteins, portions exposed and embed-ded in the membrane are highly hydrophobic, andthe surface is hydrophobic rather than hydrophilic,resulting in relatively high values of DElong(eij − err ).The same type of exception can be found for themetal binding proteins and the DNA bindingproteins, in which metals and DNA have beentreated here only as holes filled with water in thecalculation of the total contact energies. Also, themultimeric cases, shown as open circles, tend to belocated above the continuous line, probably becausesome coordinates of the inter-molecular neighborsare incompletely given in the partially assembledstructures. On the other hand, the high values ofenergies for proteins whose structures weredetermined by NMR may indicate the relativelypoorer resolution of these structures.
Discrimination for the native structures amongother folds
The threading of sequences into other folds(Hendlich et al., 1990; Jones et al., 1992) or moregenerally the inverse folding (Bowie et al., 1991) isa good method to evaluate how well a given energyscale can discriminate the native structures as thelowest energy conformations among other folds. Amore extensive study of inverse folding, in whichdeletions and additions are included, will bereported in a subsequent paper. Here, a result forsimple threading of protein sequences without gapsinto other folds is reported as a demonstration of thediscrimination power of our alignment energy.
A total of 88 proteins determined to a resolutionbetter than 2.5 A by X-ray analyses, which arestructurally dissimilar to each other with valuessmaller than 80 on the scale of Orengo et al. (1993)for structure similarity, were threaded into each ofthe 189 representatives of protein structures, whichdiffer from each other by at least 35% sequenceidentity and were selected by Orengo et al. (1993)(see their Table 1). The 88 proteins are a subset ofthe 189 protein representatives whose entry namesare listed in the caption of Figure 6. Coordinate fileswith too many unknown atomic coordinates areexcluded from these data sets. Proteins classifiedwithin the multidomain group by Orengo et al.(1993) are also excluded from the set of sequencesto be threaded.
DElong(eij − err ) is calculated for protein sequencesthreaded at all possible positions in all other proteinstructures, and their means and standard deviationsare also calculated; no gaps in either the sequencesor the structures are allowed. The positions of thenative energies in the distributions of all threadingsare then measured in units of standard deviation
Residue–Residue Potentials636
(s.d.) where negative values indicate the nativeenergies are below the mean. Table 5 lists values perresidue, DElong(eij − err )/nr , for the protein sequencesthreaded in their own native structures, as wellas the ranks and the positions of the nativeenergies in units of s.d. in the distributions of allthreadings; proteins are sorted by the increasingorder of the values, in units of standard deviation,of DElong(eij − err )/nr . Favorable cases for proteinswith more negative values than −5 s.d. are listed inTable 5A and the unfavorable higher ones inTable 5B.
For most proteins, the native structures have s.d.values of DElong(eij − err )/nr significantly large inmagnitude and are ranked at the lowest energy.However, there are some proteins for which thenative structures are not best or significantly betterthan all others. Proteins with values worse (higher)than −5 s.d., listed in Table 5B, are always membraneproteins or proteins, whose coordinates are given inisolated forms without their counterparts, such assmall inhibitors, multimeric proteins and proteinsbinding metallic ions or other molecules. Therelative proportions of binding regions on theirsurfaces may be significantly large for these smallproteins.
Figure 7 shows the correlation between thevalues of DElong(eij − err )/nr in RT units and in s.d.units. It should be noted that their values in s.d.units do not depend on the second term of equation(12), but their absolute values in RT units do. Sincethe native energies in s.d. units depend not only onthe native energies but also on the means andstandard deviations of the energy distributions ofthreadings, a good correlation is not expected,especially in the low energy region. However, a factthat insignificant native folds have relatively highenergies indicates that the energy function,DElong(eij − err )/nr , may not only be used as anenergy function to evaluate the stabilities of proteinstructures for a given sequence, but also as a scoringfunction to assess the compatibilities of proteinsequences to a given structure. A more detaileddiscussion is planned for a subsequent paper.
Discussion
A basic assumption underlying the presentestimation of contact energies is that, for a largeenough sample, the effects of specific amino acidsequences will average out, and then the numbersof residue–residue contacts observed in a largenumber of protein crystals will represent the actualintrinsic interresidue interactions. As already notedin the original paper, this assumption is compatiblewith the ‘‘principle of structural consistency’’originally proposed by Go (1983) and also called the‘‘principle of minimal frustration’’ in the energylandscape view of proteins by Bryngelson &Wolynes (1987), because the present assumption isequivalent to the assumption that on average theintrinsic contact interactions are those consistentwith the high stability of native structures. This
Table 5. Positions of native folds in the energydistributions of threadingsA. Proteins with favorable native threadings
DElong (eij − err ) In unitsPDB name Length Rank Threadings /nr
a Alignment energies per residue in RT units; see equation (12) fordefinition.
b The position of the native energy in the distribution of allthreadings in units of s.d., where negative values are for nativeenergies below the mean.
c Photosynthetic reaction center; four protoporphyrin IX are bound.d High potential iron sulfur protein; four Fe and four S are bound.e Bence-Jones protein.f Rubredoxin; small Fe binding protein.
Residue–Residue Potentials 637
Table 5—continued
B. Proteins with less favorable native threadingsDElong (eij − err ) In units
PDB name Length Rank Threadings /nra of s.d.b Comment
2OVO 56 1 28,983 0.56 −4.9 Ovomucoid third domain; 3 S–S bonds2STV 184 1 11,413 0.72 −4.5 Coat protein of S.T. virus; multimeric2WRP-R 104 1 20,812 0.60 −4.5 Trp repressor; DNA binding1SN3 65 5 27,410 0.66 −4.1 Scorpion neurotoxin; membrane binding protein1TPK-A 88 11 23,674 0.69 −4.1 Tissue plasminogen activator, Kringle-2 domain3EBX 62 15 27,925 0.78 −3.8 Erabutoxin B; inhibitor to acetylcholine receptor1UTG 70 15 26,567 0.89 −3.7 Uteroglobin; progesterone binding protein1PI2 61 26 27,752 0.78 −3.7 Bowman–Birk proteinase inhibitor; no enzyme5PTI 58 44 28,625 0.75 −3.6 Trypsin inhibitor; no enzyme2POR 301 3 4778 1.05 −3.3 Porin; integral membrane protein2CDV 107 50 20,812 1.01 −2.8 Cytochrome c3; small protein with 4 hemes1CRN 46 >100 30,832 1.01 −2.6 Crambin1HOE 74 >100 25,905 1.06 −2.2 a-Amylase inhibitor without enzyme1PRC-L 273 >100 6205 0.88 −2.0 Photosynthetic reaction center; membrane protein1CY3 118 >100 19,333 1.28 −1.4 Cytochrome c3; small protein with 4 hemes
a Alignment energies per residue in RT units; see equation (12) for definition.b The position of the native energy in the distribution of all threadings in units of s.d., where negative values are for native energies
below the mean.
assumption is also equivalent to the assumptionthat the distribution of the numbers of contacts inprotein structures is a ‘‘self-averaging property’’ inthe terms of Bryngelson et al. (1995), which meansit is, in the present case, the property ofheteropolymers rather than of the detailed aminoacid order of protein sequences. Gutin et al. (1992)indicated that the Boltzmann-like statistics observedin protein structures are a general property ofthe stable structures of heteropolymer chains, andthat the ‘‘temperature’’ in these statistics is not theusual temperature of the medium but a ‘‘selectivetemperature’’, at which the native structure is‘‘frozen out’’ from an exponentially large set ofother structures; see also Sali et al. (1994) for theestimation of the selective temperature or criticaltemperature.
We have not stated explicitly what temperatureshould be taken for the conversion of the contactenergies from RT units to kcal/mol in our originalpaper, but a melting temperature is implied, whichis just low enough for the native structure to bemarginally stable and high enough for the energylandscape of protein to show minimum frustrationand for the ‘‘self-averaging property’’ of contacts tobe satisfied. In the analyses given in our papers(Miyazawa & Jernigan, 1994, 1985), however, roomtemperature was used to translate the RT units intokcal/mol. The reason is that the contact energiesestimated here are free energies which depend ontemperature. In general, the conformational energyof a protein is implicitly not an energy but rather afree energy, because it is usually a potential of meanforce that is obtained by integrating over solventcoordinates and therefore includes energetic andentropic solvent effects. Therefore, as long as themelting temperature is not too far away from roomtemperature, it is preferable to use the experimentalroom temperature for unit conversions. The useof average melting temperature or room tempera-ture for the conversion differs significantly from
the claim of Gutin et al. (1992) that a criticaltemperature estimated to be a factor of 1.5 higherthan room temperature should be used for theconversion, but does not contradict the claim of Saliet al. (1994) that the critical temperature could alsobe less than room temperature, because rapidfolding into the stable native structure occurredslightly above the critical temperature and thefolding temperature could be equated with roomtemperature.
The hydrophobic effect is a dominant force instabilizing the native structure of a globular protein.However, there is a lack of agreement as to itsprecise magnitude. The hydrophobicity of a smallmolecule is usually measured by the transferenergies, for example, of amino acids from anon-polar solvent to water. However, it is notimmediately evident that the free energy changeaccompanying a process in which residues areburied in the folding process of protein is the sameas that accompanying the transfer process of aminoacids from water to a nonaqueous solvent, becauseboth processes are obviously different (Lee, 1993).In our previous work on contact energies, wepointed out that the estimates of the contactenergies of amino acid side-chains were almosttwice as large in magnitude as the usual estimatesof hydrophobicities from the transfer energies ofNozaki & Tanford (1971). In order to measuredirectly the contribution of the hydrophobic effectto the stability of proteins, many experimentsmeasuring the stability change caused by aminoacid replacements have been performed (Yutaniet al., 1984; Matsumura et al., 1988; Shortle et al.,1990). These experiments showed that the unfoldingfree energy changes upon single amino acidreplacements could be significantly larger inmagnitude than those expected from the transferfree energies of amino acids (Yutani et al., 1984;Shortle et al., 1990), but were also highly variable(Shortle et al., 1990).
Residue–Residue Potentials638
Figure 7. The correlation between the values ofDElong(eij − err )/nr of the native structures of 88 proteins inRT units and those in standard deviation units from itsdistribution of threaded structures. A total of 88 proteinsdetermined to a resolution better than 2.5 A by X-rayanalyses, which are structurally dissimilar to each otherwith values smaller than 80 on the scale of Orengo et al.(1993) for structure similarity, are threaded into each ofthe 189 representatives of protein structures, which differfrom each other by at least 35% sequence identity andwere those selected by Orengo et al. (1993) (see theirTable 1). DElong(eij − err )/nr is then calculated according toequation (12) for all threadings, with no gaps allowed, andits mean and standard deviation are calculated. The 189protein representatives are the same ones used for Figure6. A total of 88 proteins are a subset of these 189 proteinrepresentatives and are listed in Table 5. Coordinate fileswith too many unknown atomic coordinates are excludedfrom these data sets. The correlation coefficient is 0.75.
not fully account for changes in volume entropy, andthey provided new estimates by including a volumecorrection term. Their overall estimates were almosttwice as large as the previous ones and similar inmagnitude to the estimates in this work of thecontact energies for the transfer energy of side-chains from the outside to the interior of a protein.Pace (1992) also reported that the estimate ofhydrophobicity with the volume correction couldexplain the values of unfolding free energy changesobserved for 72 aliphatic side-chain mutants fromfour proteins in which a larger side-chain wasreplaced by a smaller side-chain.
As shown in Figure 3, the estimates by Sharp et al.(1991) of the transfer free energies between octanoland water for non-polar side-chains are almostidentical with the present estimates of equivalentquantities for hydrophobic side-chains. Here, itshould be noted that the contributions of amino acidside-chains to the contact energy are calculated tobe equal to the difference of the contact energiesbetween Gly and other amino acids and theseestimates are expected to be more reliable than theabsolute values of the contact energies of aminoacids.
Miyazawa & Jernigan (1994) showed, based onthe previous estimates of contact energies, that thecontact energy changes of protein structures due tosingle amino acid replacements were large enoughto account for the observed values of unfoldingfree energy changes. Also, the large variation ofunfolding free energy changes among residuepositions for single amino acid replacements,observed by Shortle et al. (1990), was accounted forif the free energy changes in the denatured statewere taken into account. The free energy change inthe denatured state of a protein due to single aminoacid replacements had not been considered andhad usually been ignored. The large variation ofunfolding free energy changes may be due to thedifferent environments surrounding each residue inboth the native structure and the denatured state.Miyazawa & Jernigan (1994) indicated that staphylo-coccal nuclease, under the experimental conditionsof Shortle et al. (1990), was not fully expanded in thedenatured state and also that the compact de-natured state might have a substantially native-liketopology.
In the original work to evaluate the contactenergies, the number of protein structures usedwas only 42, including 30 monomeric proteins(Miyazawa & Jernigan, 1985). These proteins werechosen with the criteria that their chain lengthswere longer than 100 residues and that few atomicpositions were missing. Also, in cases wherecoordinates are available for several closely homolo-gous proteins, only one representative was used; theminimum difference between amino acid sequencesfor homologous proteins was 50%. The numbers ofcontacts were calculated for the complete assembly.Now, more than 1000 protein structures areavailable. Here, we use only the coordinates ofsubunits as given in the PDB files, without
The effect of a cavity created by such amino acidreplacements was considered to be one of the factorscausing this discrepancy (Shortle et al., 1990;Nicholls, 1991). Eriksson et al. (1992) reported thatthe unfolding free energy changes showed corre-lations with the increases in the sizes of cavitiesobserved in the protein structures of mutant T4lysozymes, and that the value of the unfolding freeenergy change extrapolated to zero cavity sizecoincided with the value expected from the transferfree energy of an amino acid. Lee (1993) estimatedthe free energy changes for cavities formed byreplacing a bulky side-chain with a smallerside-chain, assuming the protein structure toremain completely rigid. He pointed out that mostexperimental values of unfolding free energychanges for such replacements fell into a rangebetween the maximum expected for the full cavitysize and the minimum expected for no cavity,because rearrangement of the protein occurs to fillthe cavity.
On the other hand, Sharp et al. (1991) argued thatthe previous estimates of the hydrophobic effectderived from analyses of solute partition data did
Residue–Residue Potentials 639
calculating any additional subunit intersections.That is, if a PDB file is given in a monomericstate, the numbers of contacts have been countedin the monomeric state, even though matrices togenerate multiple symmetric subunits are givenin the PDB file. The ratio of molecular surfacecontacts to the total number of contacts
6 Nro
(Nrr + Nro)is about 0.33.
Generally, subunit–subunit interfaces and en-zyme–substrate interfaces are not the major por-tion of a molecular surface. Therefore, ignoringthe molecular binding in some proteins shouldnot affect the estimated values of contact energiesso much.
There are many homologous proteins in the PDB.Any statistical analysis of sequence–structurerelations of proteins requires an unbiased samplingof proteins. One way is to choose representative setsof proteins that are sufficiently dissimilar to eachother. Usually, selections of protein representativesare based on sequence dissimilarity (Hobohm et al.,1992) or structure dissimilarity (Orengo et al., 1993;Fischer et al., 1996). The alternative method usedhere is to assign a different sampling weight to eachprotein according to the extent of redundancy(sequence identity), to remove sampling bias. Thishas not been done previously, but we show thatsuch a sampling weight for each protein can bebased on a similarity matrix between proteins in thedata set. Here, each element of the similarity matrixhas been evaluated as the ratio of identical residuesbetween aligned protein sequences, and then thesequence identity matrix has been diagonalized.The present data set contains 1661 proteinsequences. To reduce computational time for thecalculation of the 1661 by 1661 sequence identitiesamong sequences and for diagonalization of theresulting sequence identity matrix, the matrix sizewas reduced by regarding highly similar sequencesas identical, i.e. if they are more than 95% identical.If sequence identities lower than a certain valueought to be ignored, then matrix elements ofsequence identity, equation (44), whose values fallbelow a threshold value, could be replaced by zero;30% identity, where sequence pairs may beevolutionally unrelated, may be a good choice forsuch a threshold. For statistical analyses similar tothe present one, this method will certainly be betterif homologous proteins are equally good structures,because all available data are used and then thestatistical error becomes lower for a larger numberof samples.
Remarkably, all characteristics of the contactenergies found in the original analysis hold for thepresent results, for which the total effective numberof contacts is 6.3 times more than in the originaldata. The new values differ from the originalestimates of the contact energies to a significantextent only for the infrequent amino acids Trp andMet and the one other exception, Leu. The reason
why the contacts with Leu are more frequentlyobserved is not clear.
The contact energies are attractive energies only,and volume exclusion between residues needs tobe taken into account, unless a lattice model witha coordination number small enough to preventoverpacking is used. In a lattice model, volumeexclusion between residues can be satisfiedreadily by assuming that a lattice site cannot beoccupied by multiple residues. When a latticemodel is not used, a repulsive potential is neededfor evaluation of the conformational packingenergy of a protein. This has been one motivationfor the present work. The van der Waals’ surfacesof side-chains are highly anisotropic, makingspherically symmetric repulsive potentials inap-propriate to represent effective two-body repulsiveinteractions between two residues. The presentmany-body potential has been formulated todepend instead on packing density to representthe repulsive interactions around a central residuein a protein and to treat packing density effectsmore realistically than would be possible with aspherically symmetric two-body potential. How-ever, such a repulsive potential is not in effect forsmall numbers of neighboring residues. In ad-dition, there is a hard core, which cannot bepenetrated, for any side-chain with any confor-mation. However, a preliminary investigationshowed that there is relatively little residue typedependence for these hard cores, and that theminimum size of such a hard core may correspondclosely to the van der Waals’ size of a methylgroup. Therefore, the repulsive potential betweenresidues here is represented as the sum of twokinds of potentials, a hard core potential that is atwo-body potential and a repulsive packingpotential that is a many-body potential dependingon the number of residues immediately surround-ing a residue. The repulsive packing potentials for20 types of amino acids have been estimated as afunction of the number of contacts from the highdensity portion of the distributions observed inknown protein structures. This tail portion of thedistribution, which is defined as the region of highcoordination numbers of amino acids, shouldreflect the effective repulsive packing energybetween residues.
We have shown how this new potential forinteractions among residues can be used to calculatethe total energies of a set of proteins and how theseenergies behave roughly as expected. Also, calcu-lations of threading sequences into other folds havedemonstrated that the native structures have thelowest alignment energies of residues among allother folds for most of protein representatives inthe PDB, the exceptions being proteins such asmembrane proteins and others having boundligands. The application to threading here may notbe the most demanding test of the potentials,because it only requires discrimination amongrelatively few conformations. However, the resultsare excellent and give, at least, a taste of the level
Residue–Residue Potentials640
of success achievable with it. This new potentialis appropriate for application to a broad rangeof protein conformation simulations and inversefolding calculations.
Methods
Long-range inter-residue interaction energy
Long-range interaction energies are simplified forinter-residue interactions at the residue level withoutany atomic details of side-chains. They are approxi-mated to consist of two terms, a short-range attractiveterm that becomes effective only when two residues are inclose proximity, and a repulsive term that results from theoverlap of residues at high packing densities. In the caseof a lattice model, volume exclusion between residues isimplicitly included in the model, because a site cannot beoccupied by more than one residue. However, ifconformational space is defined continuously rather thandiscretely, a repulsive energy potential is essential toaccount for volume exclusion among residues. In thefollowing, the long range inter-residue energy of a proteinis represented as a sum of these two terms over allresidues in a protein:
Elong = sp
(Ecp + Er
p ) (13)
where p is a residue sequence index in a protein.
Contact energy
Residues are represented by single points at thecenters of their side-chain atom positions; the positionsof Ca atoms are used for glycine residues. Residueswhose centers are closer than Rc are defined to bein contact. The limiting value Rc = 6.5 A for contactswas chosen on the basis of the occurrence of the first peakin the radial distribution of residues in the interiorof proteins (Miyazawa & Jernigan, 1985).
Intra-residue interactions and nearest-neighbor inter-actions also lead to contact formation among nearestneighbor residues, and therefore in the present evaluationof the long range interactions, these nearest-neighborcontacts are explicitly excluded.
Thus, a contact between the pth and qth residues isdefined using:
Dcpq060 if =p − q =E1
H(Rc − dpq ) if =p − q = > 1(14)
Rc06.5 A (15)
where H is the Heaviside function defined as:
H(x)061 if xe0
0 if x < 0(16)
and dpq is the distance between the pth and qthresidues. Maiorov & Crippen (1992) pointed out that,in the use of discrete functions for the definitions ofcontacts, slight changes in interatomic distances canproduce significantly different lists of contacts. Sinceresidue–residue contacts are defined here by usingthe distance between their side-chain centers, theyare relatively insensitive to small variations in inter-atomic distances. However, a sigmoidal function with
a transition width of about 1 A might have beenused to advantage, instead of the Heaviside function,in equation (16) to account for uncertainties inthe boundary region. The number of residues of type jin contact with the pth residue whose amino acid is ip
type is:
ncipj = s
q($p)
djqjDcpq (17)
where d is the Kronecker d function.The contact energy for the pth residue, Ec
p , in which theenergy of a conformation with no residue–residuecontacts is defined to be zero, is represented as:
Ecp (eij ) = 1
2 sq($p)
eip jq Dcpq (18)
= 12 s
j($0)
eip jncipj (19)
where eij is the energy difference accompanying theformation of contacts between i and j types of amino acidsfrom those amino acids exposed to solvent, and is definedas follows (Miyazawa & Jernigan, 1985):
eij0Eij + E00 − Ei0 − Ej0 (20)
where 0 means effective solvent molecules, and E00 andEi0 are the absolute interaction energies between a pairof solvents and between an i type of residue andan effective solvent, respectively. Here, it should benoted that a lattice model is used to take account ofinteractions among solvent molecules and amino acidsin a protein. Residues in a protein are assumed tooccupy lattice sites or cells as a linear chain. Eachvacant cell is regarded as being occupied by an effectivesolvent molecule.
ncipj can be summed to produce the following equation:
nij = 12 s
p
sq($p)
DcipiDc
jqj = 12 s
p
ncipjdipi (21)
12 qini = s
j = 0
nij (22)
nij = nji (23)
where nii and 2nij are the total numbers of contactsbetween two residues of the same type, i, and betweeni and j types of amino acids; ni is the total number of itype amino acids in a protein and qi is the coordinationnumber for the i type of amino acid, that is, the averagenumber of residues that completely surround the i typeof amino acid. Let us define here the following quantitiesfor the typical residue r:
nir = nri0 sj($0)
nij (24)
nrr0 si($0)
nir (25)
nr0 si($0)
ni (26)
The summations above are taken over all 20 amino acidtypes.
Residue–Residue Potentials 641
How to estimate contact energies
The contact energies have been re-evaluated by usinga newer, much larger protein data set with the sameprocedure described by Miyazawa & Jernigan (1985), inwhich the following assumptions and approximationswere employed; the original notation is used here.
(1) For a large enough sample, the effects of specificamino acid sequences average out and the numbers ofnon-bonded residue–residue contacts observed in a largenumber of protein crystal structures will then representthe actual intrinsic differences of interactions amongresidues in proteins. Of course, nearest-neighbor contactsalong chains are significantly affected by the amino acidsequences of proteins. Therefore, the contacts betweennearest-neighbor residues are explicitly excluded in thecounting of contacts. The effect of a protein being a chainmay remain and might affect somewhat the observednumber of residue–residue contacts. Therefore, thecontact energies estimated here should be regarded aseffective inter-residue energies.
(2) By taking account of the effects of the chainconnectivity as imposing a limit to the size of the system,i.e. the total number of lattice sites, the system is thenregarded as an equilibrium mixture of unconnectedresidues and effective solvent molecules.
(3) The Bethe approximation (quasi-chemical approxi-mation), which gives an exact solution for the Bethelattice, and in which contact pair formation can beregarded as a chemical reaction, is used to estimate thecontact energies from the numbers of contacts observed inknown protein structures. In the Bethe approximation, eij
satisfies the following equilibrium equation:
exp(−eij ) = nijn00
ni0nj0(27)
where nij represents the statistical average of nij . Note thatall energies here are taken to be dimensionless, i.e. inunits of RT. Usually this equation is used to evaluate nij
from known eij , but it is used inversely here to estimatethe contact energies from the numbers of contactsobserved in known protein structures. However, theequation above includes n00, the number of contactsbetween effective solvent molecules, which is ratherdifficult to estimate accurately. However, the differencesbetween contact energies, such as e'ij defined in the nextparagraph, do not depend on n00 and so the estimates ofsuch relative quantities ought to be more reliable than theabsolute values of contact energies; e.g.:
exp(eij − ekl ) = nijnk0nl0
ni0nj0nkl(28)
does not include n00.(4) n00 is evaluated through an estimation of the
number of effective solvent molecules. The number ofeffective solvent molecules for each protein is chosen toyield the total number of residue–residue contacts equalto its expected value for the hypothetical case of hardsphere interactions with eij = 0 among residues andeffective solvent molecules, representing the effects ofchain connectivity. An effective solvent molecule is takento have the same volume (of water molecules) as anaverage residue.
(5) The numbers of contacts, nij , are counted for eachprotein structure and the sums of nij over all proteinsamples, Nij , are calculated. The numbers of contacts witheffective solvent molecules, ni0, are estimated withequation (22); the coordination number for each aminoacid type is estimated from the volume of each type of
amino acid at the center and the average volume of itssurrounding residues. That is, incomplete coordinationspheres are completed with solvent molecules. Then, thecontact energies are estimated from Nij with a compo-sition correction, so that if all residues and effectivesolvent molecules were randomly mixed, the estimatedvalues of the contact energies would be zero:
e'ij = − 12 ln0N2
ijCiiCjj
NjjNjjC2ij1 for i, j$0 (29)
e'i0 = − 12 ln0 N2
i0
NiiN00
C'iiC'00
C'2i0 1 (30)
where:
e'ij0eij − (eii + ejj )/2 (31)
eij = e'ij + e'00 − e'i0 − e'j0 (32)
Cij and C'ii are correction factors and are equal to thenumber of contacts expected between residues of i and jtypes and the number of contacts expected betweenresidues of the same type i, respectively, when residuesand effective solvent molecules are randomly mixed; seeequations (10) to (15) of Miyazawa & Jernigan (1985) fortheir detailed definitions. e'ij is the energy differencedetailing the residue interaction specificity accompanyingthe formation of a contact pair i–j from contact pairs i–iand j–j.
According to the procedure described by Miyazawa &Jernigan (1985) and briefly summarized above, the contactenergies, eij , between i and j types of amino acids arere-evaluated with equations (29) and (30) above, orequations (10) to (15) of Miyazawa & Jernigan (1985).However, the further useful quantities for interactionswith the average residue r are defined as:
exp(−eir )0nirn00
ni0nr0(33)
exp(−err )0nrrn00
nr0nr0(34)
These are not estimated by the procedure described in theearlier paper, but directly from the data by the followingequations:
eir = e'ir − e'i0 − e'r0 (35)
e'ir = − 12 ln(Cii /Nii ) (36)
err = −2e'r0 = ln(C'00/N00) (37)
As stated by Miyazawa & Jernigan (1985), if eir is morenegative than err , then amino acids of the i type will tendto be buried inside a protein; otherwise, they will tend tobe exposed to solvent on the surface of the protein.
Also, the average contact energy of each type of aminoacid is estimated by:
ei = sj($0)
eijNij /Nir (38)
er = si($0)
eiNir /Nrr (39)
ei may be used to compare with other estimates ofhydrophobic energies. However, eir will be moreappropriate as a one-dimensional measure of hydropho-bicity. ei and eir correspond to the mean energy and the
Residue–Residue Potentials642
effective mean energy, respectively; refer to equation (35)of Miyazawa & Jernigan (1985).
Repulsive energy
Here, each residue has been represented by the pointat the center of its side-chain heavy atom positions.Generally, the van der Waals’ surface of a side-chain ishighly anisotropic, and its hard core would not bewell represented by a sphere. However, if a repulsivepotential between residues is averaged over allpossible side-chain conformations, then the anisotropiccharacter in the average repulsive potential will beweakened by the flexibility manifested in the side-chainconformations. Therefore, a symmetric repulsive potentialis appropriate at the present level of simplification.However, a repulsive force resulting from van der Waals’overlaps is a very short range force, and therefore evena soft core symmetric repulsive potential may beinappropriate. Here, the repulsive force of van der Waals’overlaps is approximated as the sum of two terms, ahard core repulsion between residues and a repulsivepacking potential, depending on the packing density ofresidues:
Erp = ehc
p + erp (40)
The hard core potential is taken, in turn, as a two-bodyinteraction potential given by:
ehcp 01
2 sq($p)
ehcH(rc − dpq ) (41)
ehc is the parameter for the positive energy of hard corerepulsion and will have a large positive value. Dependingon side-chain conformation, the side-chain centers of tworesidues can sometimes come as close as the van derWaals’ separation between two methyl groups. Therefore,the hard core radius, rc/2, is independent of residue typeand here is simply the van der Waals’ core of a methylgroup, with a value of 1.9 A.
The repulsive packing potential is a many-bodyinteraction potential dependent on nc
p , the number ofresidues in contact with the pth residue:
ncp0 s
j($0)
ncipj (42)
This packing density energy in RT units is estimated asfollows:
erp0H(nc
p − qip ) $0qip
ncp
− 11 Ecp − ln0N(ip , nc
p ) + eN(ip , qip ) + e1% (43)
N(i, nc) is the total number of residues of type i that aresurrounded by nc residues in the set of protein structures.The value of N(i, qi ) is obtained by interpolation. To avoidthe divergence of the logarithm function in equation (43),a small positive number, e, has been added to both thenumerator and the denominator, so that the sum of thecontact energy and repulsive energy takes on a positivevalue even at N(i, nc) = 0 for any amino acid; here, a valueof e = 10−6 is employed.
The first term of the repulsive packing potential inequation (43) represents the effect of exceeding thelimiting number of contacts, which is equal to thecoordination number qip , to offset the extra contactenergy; the actual value of Ec
p from equation (19) isused. The second term is estimated from the ratio of
the observed frequency of packing density to thatfor the limiting value for packing density. The repulsiveenergy is applied through the Heaviside function onlyif the number of surrounding residues exceeds athreshold value, qip . The distribution of the number ofcontacts for each amino acid is assumed in the Betheapproximation to be determined by the contact energies,except for the high density region, which should reflectthe repulsive energies among residues. It can be seenin Table 3 that the estimated coordination numbers forthe 20 types of amino acids here range only from 5.79 forTrp to 6.65 for Cys.
Protein structures used in the presentstatistical analysis
The proteins used here are all those in the Protein DataBank (PDB) (Bernstein et al., 1977) that satisfy thefollowing conditions.
(1) Proteins whose structures are determined by X-rayanalysis and whose resolution is equal to, or better than,2.5 A. All protein structures determined by NMR areexcluded.
(2) Protein subunits composed of 50 or more residues.(3) Membrane proteins are excluded, because, inappro-
priate to them, it is assumed that incomplete coordinationspheres are completed with water molecules. Thus, thenumber of inter-residue contacts compiled here are thoseobserved only in soluble proteins in order to derivecontact energies appropriate for soluble proteins. How-ever, some characteristics of neighbor pairs might bepreserved, except for the overall inversion of structurecompared to soluble proteins.
The number of protein subunit structures satisfying thesecriteria is 1661 (see Table 1).
In this data set, there are many proteins whosesequences are similar to one another. To obtainstatistically unbiased results, unbiased sampling isrequired. There are two ways to do this: either (i) useprotein representatives that are sufficiently dissimilar toeach other in their sequences; or (ii) use a differentstatistical weight for each protein related to its extent ofsimilarity to other sequences. So far, most statisticalanalyses have used a representative set of proteins.Usually, protein representatives are chosen by specifyingan upper limit for sequence identity (Hobohm et al., 1992)or structural similarity (Orengo et al., 1993; Fischer, et al.,1996, among them). However, it is not clear what valueis best as an upper limit of similarity in proteinrepresentatives. Also, in such a method, many goodstructures may be discarded. In this work, the secondapproach has been taken.
Sampling weight
A sequence identity matrix is defined here as:
Ismn0sequence identity between sequence m and n
0
2 × (number of identicalresidues in the alignment)
(length of sequence m)+ (length of sequence n)
(44)
The sequence identity is taken as the fraction of identicalresidues in the alignment of two sequences. The sequenceidentity matrix is a real symmetric, non-negative matrix.
Residue–Residue Potentials 643
Each element of Ismn is between 0 and 1, and the diagonal
elements are equal to one:
0EIsmn = Is
nmE1
Ismm = 1 (45)
The sequence identity matrix is similar to a correlationmatrix in the sense that the value of Is
mn represents thecorrelation between the amino acid sequences m and n.Let us define lm as the mth eigenvalue and Vm the mthcolumn eigenvector that is orthogonal to all others andnormalized to be equal to one. That is:
IsVm = lmVm (46)
VTm Vn = dmn (47)
VTm means the transpose of Vm. The total sum of the
eigenvalues is equal to the trace of the sequence identitymatrix, and so it is equal to the number of sequencesused:
sm
lm = Tr Is = sm
1 = Nprot (48)
where Nprot is the number of sequences used. Fromequation (45), it follows that a sequence identity matrixmust be positive semidefinite:
0EliENprot (49)
Let us consider some special cases. If a whole set ofsequences can be divided into groups where individualsequences from different groups are completely dissimi-lar, each group can be handled independently. In casethere is a sequence which is completely dissimilar to anyother sequence, at least one eigenvalue will be equal toone. If all sequences are the same, all other eigenvalues,except one, must be equal to zero; if Is
mn = 1 for any m andn, rank Is = 1, and therefore the number of non-zeroeigenvalues must be equal to one.
On the basis of these characteristics, the followingprocedure may be used to remove redundant informationthat comes from similarities among sequences. Thesampling weight, wn for the nth sequence is taken to be:
wn00sm (lm − (lm − 1) · H(lm − 1)) · VmVTm1nn
(50)
0 < wnE1 (51)
This definition of sampling weight satisfies all require-ments. If, and only if, a sequence has zero sequenceidentity with any other sequence, then the samplingweight for that sequence will be equal to one. If Nprot
sequences in the data set are all identical, then thestatistical weights for those sequences will be equal to1/Nprot. Generally, sampling weights take a value betweenone and 1/Nprot, and are approximately equal to theinverse of the number of similar sequences. The effectivenumber of proteins can be defined as the total sum of thesampling weights:
Neffectiveprot 0s
mwmENprot (52)
In a similar way, the effective number of residues isdefined.
Please note the use of sequence identity here. Althoughthis approach for weighting could be used equally wellwith any sequence similarity measure, the presentproblem dictates that only identities be considered;
otherwise part of the data for deriving contact pairs wouldhave been discarded.
The number of protein subunits chosen according tothe criteria described in the previous section is more than1600. To reduce computational time expediently for thecalculation of the sequence identity matrix and itsdiagonalization, proteins that have more than 0.95sequence identity are regarded as the same sequence andare removed from the sequence identity matrix. Then astatistical weight for each of the individual proteins istaken to be wm/m, where wm is the weight for that proteinfamily m, and m is the number of members in the family.Protein representatives for this purpose are chosen in thefollowing procedures. (1) If a protein is less similar than0.95 sequence identity to any protein representativealready chosen, this protein is regarded as a new protein.(2) The above procedure is iterated until all proteins areexamined. There are 424 protein representatives remain-ing with less than 0.95 sequence identity to any other inthe PDB, and the total effective number of proteins isfound with equation (52) to be 251.
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Edited by B. Honig
(Received 17 April 1995; accepted in revised form 17 November 1995)
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