A normalized root-mean-spuare distance for comparing protein three-dimensional structures

Abstract

The degree of similarity of two protein three-dimensional structures is usually measured with the root-mean-square distance between equivalent atom pairs. Such a similarity measure depends on the dimension of the proteins, that is, on the number of equivalent atom pairs. The present communication presents a simple procedure to make the root-mean-square distances between pairs of three-dimensional structures independent of their dimensions. This normalization may be useful in evolutionary and fold classification studies as well as in simple comparisons between different structural models.

Quantitative comparison of three-dimensional structures is a fundamental task in structural biology (Carugo and Eisenhaber 1997; Peters-Libeu and Adman 1997), especially in such fields as domain fold classification and structural evolution studies (Domingues et al. 2000; Yang and Honig 2000). A very popular quantity used to express the structural similarity is the root-mean-square distance (rmsd) calculated between equivalent atoms in two structures, defined as

1

where d is the distance between each of the n pairs of equivalent atoms in two optimally superposed structures. The rmsd is 0 for identical structures, and its value increases as the two structures become more different. Rmsd values are considered as reliable indicators of variability when applied to very similar proteins, like alternative conformations of the same protein. On the other hand, rmsd data calculated for structure pairs of different sizes cannot be directly compared, because the rmsd value obviously depends on the number of atoms included in the structural alignment. Clearly, an rmsd value of, say, 3 Å has a different significance for proteins of 500 residues than for those of 50 residues; accordingly, the structural variability of fold types cannot be easily compared in quantitative terms (Irving et al. 2001). In other words, rmsd is a good indicator for structural identity, but less so for structural divergence.

The present communication aims to define a normalized, size-independent rmsd formula that could help to overcome this problem. In order to derive a formula between rmsd and protein dimension, one would need a database of structural alignments, in which all other parameters, such as secondary structure content and amino acid composition of the protein, are either constant (which is not possible) or are evenly distributed with respect to protein chain length. Such experimental data are presently not available. For example, the FSSP database (Holm and Sander 1996) contains a reasonably high number of structural alignments (about 23,000), but 80% of these have small rmsd values (0–2 Å), which reflects the fact that the percentage of sequence identity is very high (more than 90% residue identity in 60% of the alignments).

We therefore decided to create a large artificial set of rmsd values via extensive self-comparison of 180 nonhomologous (maximal identity 25%) protein structures, selected from the protein data bank (Berman et al. 2000) using the PDB_SELECT (Hobohm and Sander 1994) algorithm. These proteins were selected so as to represent the largest possible variability of amino acid content, sequence length as well as secondary structure content (Table 1). Each structure was compared, using the algorithm of Kabsch (1976, 1978), with 400,000 of its randomized variants created through random shuffling of the C_α equivalencies. All C_α atoms were included in superposing each structure with all its variants. Overall, we obtained 400,000 rmsd observations in each of the 180 randomization experiments, which corresponds to a database of 72 million structural alignments. As expected, the distribution of rmsd values thus obtained depends on the size of the protein. The rmsd values are not evenly distributed, rather, the histograms are biased toward the high rmsd values (Fig. 1a ▶). Moreover, there are characteristic differences between proteins of different length, illustrated by, for example, the different rmsd limits of the 2000 smallest rmsd values in the two experiments, as shown by the shaded areas in Figure 1a ▶.

Table 1.

Protein structures examined in the present work

idcode	n	h	e	t	o	idcode	n	h	e	t	o	idcode	n	h	e	t	o
1a7kA	358	31	25	25	19	1agnA	373	30	26	21	23	1ak0_	264	64	4	20	12
1amx_	150	8	54	21	17	1aojA	60	5	45	22	28	1ap8_	213	21	23	26	29
1auxA	292	29	31	20	20	1avgI	142	8	41	27	25	1axn_	323	79	0	15	6
1b12A	239	8	46	20	26	1b6bA	168	26	30	21	23	1b6rA	349	34	29	19	19
1b87A	181	27	30	28	15	1b9lA	119	32	39	17	12	1b9xA	340	7	46	29	18
1bq2_	323	32	28	21	19	1bhu_	102	4	25	35	36	1boeA	46	0	24	43	33
1bor_	56	0	4	55	41	1bp3A	186	60	0	22	18	1bqv_	110	46	0	23	31
1bu2A	229	60	0	27	14	1buyA	166	57	1	20	22	1bxwA	172	0	60	12	28
1by1A	209	64	0	25	11	1bynA	128	5	48	18	29	1cl7M	142	84	0	7	9
1c20A	128	65	3	15	17	1ceuA	51	51	0	25	24	1cflD	368	7	49	21	24
1ckv_	141	16	21	36	27	1cn3A	283	7	41	25	27	1d0mA	312	49	9	22	20
1dldA	220	58	0	25	17	1d2hA	252	32	31	19	18	1de9A	276	27	26	22	25
1dgvA	183	46	2	31	20	1dtjB	62	44	32	6	18	1dujA	187	28	29	18	25
1eus_	358	3	44	28	25	1evtD	192	9	52	13	26	1ewiA	114	11	25	31	33
1gcf_	109	0	44	20	36	1gnhA	206	9	44	21	26	1gsa_	314	35	29	18	18
1hcd__	118	3	49	30	19	1hoe_	74	0	49	26	26	1hsm_	79	62	0	22	16
1ihfA	96	42	29	9	20	1iyu_	79	0	47	25	28	1jlyA	299	7	45	30	18
1ksr_	100	0	43	36	21	1lbd_	238	66	3	20	12	1liaA	164	76	0	15	9
1mtyB	384	64	1	19	16	1nfdA	203	5	43	22	30	1oczB	227	30	25	22	23
1pgs_	311	5	47	22	26	1pho_	330	2	56	25	16	1pslA	304	72	0	14	14
1pyaA	81	27	27	22	23	1qhkA	47	32	28	21	19	1qklA	127	20	14	32	33
1qleC	273	68	1	16	16	1qmcA	52	6	56	21	17	1qqvA	67	42	3	30	25
1qrjB	199	57	1	22	20	1qslA	402	34	29	19	18	1qsoA	149	34	30	19	17
1qstA	160	35	29	22	14	1qu0C	183	3	51	22	24	1qu5A	182	10	22	35	33
1r63_	63	67	0	16	17	1rgs_	264	34	27	17	21	1rip_	81	0	11	28	60
1stu_	68	38	31	12	19	1svpA	160	6	46	26	23	1tbaA	67	21	6	39	34
1tig_	88	35	35	16	14	1tiv_	86	0	0	50	50	1tnm_	91	0	46	27	26
1upuA	224	39	27	15	18	1xikA	340	70	4	13	14	1xrc_	378	32	25	26	17
2af8_	86	50	0	21	29	2cgpA	200	40	30	19	12	2def_	146	22	20	25	33
2ezl_	99	59	0	19	22	2jhbA	143	26	25	24	24	2myo_	118	47	0	34	19
2pcbA	294	49	7	22	21	2pcfB	250	8	40	23	28	2pii_	112	26	29	15	29
2qwc_	385	3	45	25	26	2tbd_	134	28	27	21	24	2tmvP	154	45	5	26	25
2yfpA	224	9	53	24	14	3cla_	213	30	29	22	20	3csmA	252	64	0	17	19
3sil_	378	6	47	25	22	7prcC	332	42	4	30	24	8prn_	289	4	55	24	17
8rucI	123	22	24	27	28	1a3k_	137	2	58	18	21	1a79A	171	34	31	20	15
1a7m_	180	58	0	22	19	1a8p_	257	26	34	23	18	1aep_	153	80	0	14	7
1aru_	336	43	7	24	26	1avwB	171	2	42	20	36	1awj_	77	0	5	27	68
1b3kA	373	31	31	21	17	1b65A	363	31	26	18	24	1bec_	238	5	49	20	26
1beg_	97	61	4	14	21	1bmy_	107	48	0	27	25	1bu9A	168	48	1	27	24
1bw6A	56	55	0	23	21	1bx9A	210	56	10	23	11	1cd3_	294	63	2	20	16
1cby_	227	30	30	16	24	1cczA	171	4	51	22	24	1cjkA	189	38	30	18	15
1cmyA	141	74	0	15	11	1cpzA	68	28	29	28	15	1d4uA	111	23	11	38	29
1d8lA	140	39	29	19	14	1dipA	77	40	0	31	29	1dj7B	73	4	44	27	25
1dkdA	146	34	29	17	19	1dztA	183	10	44	21	25	1eioA	127	9	58	21	11
1ej3A	187	61	4	18	17	1eqfA	267	64	0	21	15	1exg_	110	3	56	22	19
1gdoA	238	25	34	21	20	1ghj_	79	0	42	25	33	1hhhB	100	0	49	21	30
1irl_	133	52	3	22	23	1lkfA	292	3	59	20	18	1mrj_	247	40	25	21	13
1mut_	129	12	23	28	37	1nflA	259	58	0	29	13	1nfa_	178	2	24	26	49
1otfA	59	42	32	10	15	1ounB	121	28	49	12	11	1p32A	182	36	26	17	20
1pdnC	123	47	5	24	24	1pex_	192	7	42	33	18	1qgiA	259	58	6	21	15
1qhgA	163	56	10	20	15	1ghlA	203	37	27	19	16	1qj8A	148	0	82	14	5
1qkfA	73	22	21	27	30	1qovM	302	64	4	14	18	1qpvA	133	30	30	23	17
1qu8A	46	0	0	50	50	1r2aA	46	63	0	11	26	1rof_	60	12	10	40	38
1rypL	212	39	33	16	12	1sxl_	97	18	13	30	39	1tif_	76	36	30	17	17
1tiiD	98	26	39	21	14	1tuc_	61	5	44	21	30	1u2fA	90	16	18	38	29
1vcaA	199	5	57	20	19	2abd_	86	60	0	19	21	2atcB	152	6	5	41	48
2aviA	121	2	51	22	24	2ayh_	214	6	52	19	23	2bby_	69	51	7	10	32
2bidA	197	57	0	21	22	2nlrA	222	6	50	23	21	2nmbA	147	20	20	31	29
2shl_	48	0	52	27	21	2trxA	108	33	28	27	12	3ncmA	92	4	51	27	17
3stdA	162	30	48	14	9	5daaA	277	27	31	23	19	6gsvA	217	49	9	28	14

Each entry is identified by its four-letter identification code, followed by the chain identifier. The following features are indicated for each entry: the number of residues (n) and the percentages of residues in helical (h), extended (e), turn (t), and other (o) backbone conformation. The secondary structures, as assigned by DSSP, were simplified as follows: helical if 310-α or π-helix (G, H, and I respectively in DSSP), extended if β-bulge or strand (B and E), turn if bend or reverse turn (S and T), and others in the remaining cases.

Fig. 1.

(a) Typical distributions of rmsd values after 400,000 random superpositions for proteins of different sizes (PDB codes indicated in parentheses); the percentage of observations in each range of 0.4 Å is reported. (b) A typical rmsd-versus-chain-length plot; the upper limits of the smallest 2000 observations (examples indicated by perpendicular lines in a) are plotted for each of the 180 experiments. (c) The dependence of the rmsd-versus-chain-length plots as a function of the different number of smallest observations (indicated in parentheses); the lines were determined by fitting a logarithmic equation of the form y = a + b ln(x) to the data (0.95 < r < 1.00); ◯ is a reference value, corresponding to 100 residues, chosen to normalize the curves. (d) Dividing the rmsd values by the corresponding reference value (indicated with ◯ in c) causes the curves in the previous figure to collapse into one single curve.

In order to check the effect of the uneven distribution of rmsd values, we prepared separate rmsd-versus-chain-length plots for different subsets of the database, selected to represent different rmsd ranges without changing the other parameters (secondary structure content, amino acid composition, etc.). This was achieved by first ordering the structural alignments in growing order of the rmsd values in each of the 180 data sets and then selecting the first n smallest rmsd values from each data set. This procedure guarantees that the data sets will be equal with respect to all parameters; only the range of the rmsd values will be different: that is, gradually increasing the number n of observations in the data sets means not only an increase of the data size but also an inclusion of higher rmsd values. The data of each subset could be fitted with a logarithmic function with correlation coefficients higher than 0.95 (an example is shown in Fig. 1b ▶). The fitted curves are different as higher rmsd data are included in the calculation, which results in the series of curves shown in Figure 1c ▶. This observation therefore confirms that the uneven distribution of rmsd values would bias the parameters obtained by simple curve fitting. Interestingly, dividing the rmsd values with a reference value, chosen here as the value of the fitted rmsd curve at 100 residues, rmsd₁₀₀ (Fig. 1c ▶), makes the curves collapse into one single logarithmic curve (Fig. 1d ▶) that is described by the following equation:

2

where N is the number of amino acid residues. This curve is accordingly independent of both the number n of observations included in the calculation and the magnitude of rmsd values; a statistical bias is therefore not likely. Given that −1.3 ≅ 1 − ln(10), the equation can be rearranged to give

3

It is interesting to note that the value 100, the residue number corresponding to the chosen reference value, rmsd₁₀₀, appears in the equation. We repeated the normalization procedure on the entire data set with residue numbers of 50, 75, 150, and 200, respectively, and in fact found that a generalized equation is valid with correlation coefficients 0.96–0.99:

4

where L is the number of residues chosen as a reference. In other words, the relative root-mean-square distance rmsd/rmsd_L is a simple function of the relative dimension N/L. Equation 3 can be simply rearranged to give a formula for a normalized rmsd value:

5

The chain length of 100 residues was primarily chosen because this is the mean number of amino acids per domain (Xu and Nussinov 1998). rmsd₁₀₀ is therefore an rmsd value that would be observed for a pair of structures of 100 residues exhibiting the same degree of similarity as the structures actually compared. In other words, the rmsd₁₀₀ value can be considered as a normalized, size-independent indicator of structural variability. For example, suppose that the C_α atoms of two pairs of protein structures, 50 and 200 residues long, respectively, can be superposed to give a final rmsd value of 1.0 Å. For the first pair of sequences sharing N = 50 equivalent residues, the corresponding rmsd₁₀₀ value will be 1.524 Å The second pair of structures (N = 200) is considerably more similar to each other (rmsd₁₀₀ = 0.741 Å) despite the fact that the crude rmsd values are the same. In other words, the normalized rmsd₁₀₀ qualitatively reflects the intuitive view that larger structures have a higher probability to differ one from the other. Because the data were derived from proteins with more than 40 residues we suggest that the rmsd₁₀₀ formula should be applied to alignments that include more than 40 residues. On the other hand, it follows from the mathematical form of the equation that the formula can be applied only for structural alignments with more than 14 residues; for smaller N values the ratio in equation 2 would be negative.

We think that the normalized rmsd can be useful in estimating the quality of an NMR ensemble of models, in applying multivariate statistical techniques to structural bioinformatic problems, as well as in comparing limited sets of protein three-dimensional structures.