- Research article
- Open Access
Protein secondary structure prediction for a single-sequence using hidden semi-Markov models
- Zafer Aydin^{1},
- Yucel Altunbasak^{1} and
- Mark Borodovsky^{2}Email author
https://doi.org/10.1186/1471-2105-7-178
© Aydin et al; licensee BioMed Central Ltd. 2006
- Received: 16 April 2005
- Accepted: 30 March 2006
- Published: 30 March 2006
Abstract
Background
The accuracy of protein secondary structure prediction has been improving steadily towards the 88% estimated theoretical limit. There are two types of prediction algorithms: Single-sequence prediction algorithms imply that information about other (homologous) proteins is not available, while algorithms of the second type imply that information about homologous proteins is available, and use it intensively. The single-sequence algorithms could make an important contribution to studies of proteins with no detected homologs, however the accuracy of protein secondary structure prediction from a single-sequence is not as high as when the additional evolutionary information is present.
Results
In this paper, we further refine and extend the hidden semi-Markov model (HSMM) initially considered in the BSPSS algorithm. We introduce an improved residue dependency model by considering the patterns of statistically significant amino acid correlation at structural segment borders. We also derive models that specialize on different sections of the dependency structure and incorporate them into HSMM. In addition, we implement an iterative training method to refine estimates of HSMM parameters. The three-state-per-residue accuracy and other accuracy measures of the new method, IPSSP, are shown to be comparable or better than ones for BSPSS as well as for PSIPRED, tested under the single-sequence condition.
Conclusions
We have shown that new dependency models and training methods bring further improvements to single-sequence protein secondary structure prediction. The results are obtained under cross-validation conditions using a dataset with no pair of sequences having significant sequence similarity. As new sequences are added to the database it is possible to augment the dependency structure and obtain even higher accuracy. Current and future advances should contribute to the improvement of function prediction for orphan proteins inscrutable to current similarity search methods.
Keywords
- Hide State
- Internal Position
- Amino Acid Pair
- Proximal Position
- Conversion Rule
Background
Accurate prediction of the regular elements of protein 3D structure is important for precise prediction of the whole 3D structure. A protein secondary structure prediction algorithm assigns to each amino acid a structural state from a 3-letter alphabet {H, E, L} representing the α-helix, β-strand and loop, respectively. Prediction of function via sequence similarity search for new proteins (function annotation transfer) should be facilitated by a more accurate prediction of secondary structure since structure is more conserved than sequence.
Algorithms of protein secondary structure prediction frequently employ neural networks [1–7], support vector machines [8–13] and hidden Markov models [14–16]. Parameters of the algorithm have to be defined by machine learning, therefore algorithm development and assessment usually contains four steps. The first one is a statistical analysis to identify the most informative correlations and patterns. The second one is the creation of a model that represents dependencies between structure and sequence elements. In the third step, the model parameters are derived from a training set. Finally, in the fourth step, the algorithm prediction accuracy is assessed on test samples (sets) with known structure.
There are two types of protein secondary structure prediction algorithms. A single-sequence algorithm does not use information about other (homologous) proteins. The algorithm should be suitable for a sequence with no similarity to any other protein sequence. Algorithms of another type are explicitly using sequences of homologous proteins, which often have similar structures. The prediction accuracy of such an algorithm should be higher than one of a single-sequence algorithm due to incorporation of additional evolutionary information from multiple alignments [17].
The estimated theoretical limit of the accuracy of secondary structure assignment from experimentally determined 3D structure is 88% [18]. The accuracy (see formal accuracy definition below) of the best current single-sequence prediction methods is below 70% [19]. BSPSS [14], SIMPA [20], SOPM [21], and GOR V [22] are examples of single-sequence prediction algorithms. Among the current best methods that use evolutionary information (multiple alignments, PSI-BLAST profiles), one can mention PSIPRED [1], Porter [23], SSpro [24], APSSP2 [2], SVMpsi [9], PHDpsi [25], JPRED2 [4] and PROF [26]. For instance, the prediction accuracy of Porter was shown to be as high as 80.4% [27]. The joint utilization of methods that specialize on single-sequence prediction and methods using homology information will definitely improve the prediction performance.
Single-sequence algorithms for protein secondary structure prediction are important because a significant percentage of the proteins identified in genome sequencing projects have no detectable sequence similarity to any known protein [28, 29]. Particularly in sequenced prokaryotic genomes, about a third of the protein coding genes are annotated as encoding hypothetical proteins lacking similarity to any protein with a known function [30]. Also, out of the 25,000 genes believed to be present in the human genome, no more than 40–60% can be assigned a functional role based on similarity to known proteins [31, 32]. For a larger picture, the Pfam database allows one to get information on the distribution of proteins with known functional domains in three domains of life (Table 1).
Number of proteins with known functional domains.
# Proteins | # Proteins with Pfam hit | (%) | |
---|---|---|---|
Bacteria | 623,037 | 450,962 | 72.38 |
Archaea | 50,406 | 33,259 | 65.98 |
Eukaryota | 284,392 | 187,472 | 65.92 |
Total | 957,835 | 671,693 | 70.13 |
Statistics of hypothetical proteins and orphan proteins observed in the recently sequenced genomes (year 2004).
# Proteins | (%) hypothetical proteins | (%) orphans in hypotheticals | |
---|---|---|---|
Sulfolobus islandicus (Archaea) | 197 | 65.98 | 57.69 |
Bacillus claussi (Bacteria) | 4121 | 31.64 | 18.66 |
Gallus gallus (Eukaryota) | 29,172 | 11.84 | 32.4 |
The matrix of transition probabilities, P(T_{ j }| T_{j-1}), used in the hidden semi-Markov model. Rows represent T_{j-1}values.
P(T_{ j }| T_{j-1}) | H | E | L |
---|---|---|---|
H | - | 0.031 | 0.969 |
E | 0.029 | - | 0.971 |
L | 0.314 | 0.686 | - |
Correlations at the amino acid level as characterized by the χ^{2} measure (PDB_SELECT set).
Helix | Strand | Loop | ||||
---|---|---|---|---|---|---|
Separation | χ ^{2} | # of pairs | χ ^{2} | # of pairs | χ ^{2} | # of pairs |
1 | 1854.34 | 118,324 | 2579.85 | 60,423 | 9600.85 | 154,404 |
2 | 7008.83 | 103,853 | 1832.78 | 44,121 | 5774.58 | 124,249 |
3 | 2454.03 | 89,414 | 1116.65 | 30,909 | 4828.13 | 100,325 |
4 | 5095.27 | 77,302 | 535.02 | 20,336 | 2276.21 | 80,930 |
5 | 2052.68 | 67,036 | 461.70 | 12,584 | 1298.16 | 66,109 |
6 | 1295.46 | 57,602 | 398.44 | 7361 | 950.66 | 54,993 |
7 | 2196.94 | 49,017 | 392.93 | 4196 | 895.42 | 46,391 |
8 | 627.00 | 41,350 | 355.81 | 2292 | 761.48 | 39,611 |
KL distance between distributions of amino acids in proximal and internal positions (PDB_SELECT set).
KL-dis | N1 | N2 | N3 | N4 | C4 | C3 | C2 | C1 |
---|---|---|---|---|---|---|---|---|
α-Helix | 0.402 | 0.194 | 0.100 | 0.053 | 0.018 | 0.018 | 0.020 | 0.036 |
β-Strand | 0.047 | 0.025 | 0.019 | - | - | 0.021 | 0.039 | 0.074 |
Loop | 0.045 | 0.019 | 0.008 | 0.003 | 0.004 | 0.008 | 0.026 | 0.028 |
Position specific correlations as characterized by the χ^{2} measure in α-helix proximal positions (PDB_SELECT set).
χ ^{2} | i - 5 | i - 4 | i - 3 | i - 2 | i - 1 | i + 1 | i + 2 | i + 3 | i + 4 | i + 5 |
---|---|---|---|---|---|---|---|---|---|---|
N1 | 380.33 | 491.35 | 416.40 | 524.46 | 708.76 | 770.43 | 982.18 | 875.59 | 1132.54 | 487.41 |
N2 | 410.29 | 409.30 | 637.47 | 1029.24 | 770.43 | 805.38 | 993.92 | 619.44 | 872.68 | 594.32 |
N3 | 421.77 | 591.87 | 2000.33 | 731.30 | 661.04 | 702.25 | 844.97 | 697.18 | 694.83 | 652.90 |
N4 | 538.17 | 482.28 | 552.22 | 649.11 | 614.21 | 470.58 | 827.26 | 465.17 | 1055.63 | 468.99 |
C4 | 604.79 | 830.18 | 696.89 | 1082.25 | 481.05 | 463.31 | 933.17 | 578.54 | 657.20 | 527.83 |
C3 | 628.98 | 963.03 | 632.99 | 1181.51 | 497.00 | 527.86 | 903.81 | 485.95 | 443.62 | 370.97 |
C2 | 549.77 | 1261.04 | 624.90 | 1270.42 | 603.44 | 717.04 | 591.22 | 507.21 | 397.44 | 378.79 |
C1 | 563.37 | 1213.12 | 714.48 | 1300.46 | 810.35 | 1266.49 | 631.45 | 482.92 | 476.19 | 454.61 |
Positional dependencies within structural segments for the models $\mathcal{M}$1, $\mathcal{M}$2, and $\mathcal{M}$3. ${h}_{j}^{3}$ ∈ {hydrophobic, neutral, hydrohilic} indicates the hydrophobicity class of the amino acid R_{ j }, where hydrophobic = {A, M, C, F, L, V, I}, neutral = {P, Y, W, S, T, G}, hydrophilic = {R, K, N, D, Q, E, H}. ${h}_{j}^{5}$ is a 5 letter alphabet with groups defined as {P, G}, {E, K, R, Q}, {D, S, N, T, H, C}, {I, V, W, Y, F}, {A, L, M}.
$\mathcal{M}$1 | $\mathcal{M}$2 | $\mathcal{M}$3 | ||
---|---|---|---|---|
H | Int | ${h}_{i-2}^{5},{h}_{i-3}^{3},{h}_{i-4}^{5},{h}_{i-7}^{3}$ | ${h}_{i-2}^{3},{h}_{i-3}^{3},{h}_{i-4}^{3},{h}_{i+2}^{3},{h}_{i+4}^{3}$ | ${h}_{i+2}^{5},{h}_{i+3}^{5},{h}_{i+4}^{5}$ |
N1 | ${h}_{i-1}^{5},{h}_{i-2}^{5}$ | ${h}_{i-1}^{5},{h}_{i+2}^{5}$ | ${h}_{i+2}^{5},{h}_{i+4}^{5}$ | |
N2 | ${h}_{i-1}^{3},{h}_{i-2}^{3},{h}_{i-3}^{3}$ | ${h}_{i-2}^{3},{h}_{i+2}^{3},{h}_{i+4}^{3}$ | ${h}_{i+1}^{3},{h}_{i+2}^{3},{h}_{i+4}^{3}$ | |
N3 | ${h}_{i-1}^{3},{h}_{i-2}^{3},{h}_{i-3}^{3}$ | ${h}_{i-2}^{3},{h}_{i-3}^{3},{h}_{i+2}^{3}$ | ${h}_{i+1}^{3},{h}_{i+2}^{3},{h}_{i+4}^{3}$ | |
N4 | ${h}_{i-1}^{3},{h}_{i-2}^{3},{h}_{i-3}^{3}$ | ${h}_{i-1}^{3},{h}_{i+2}^{3},{h}_{i+4}^{3}$ | ${h}_{i+1}^{3},{h}_{i+2}^{3},{h}_{i+4}^{3}$ | |
C1 | ${h}_{i-1}^{3},{h}_{i-2}^{3},{h}_{i-4}^{3}$ | ${h}_{i-2}^{3},{h}_{i-4}^{3},{h}_{i+1}^{3}$ | ${h}_{i+1}^{3},{h}_{i+2}^{3},{h}_{i+3}^{3}$ | |
C2 | ${h}_{i-2}^{3},{h}_{i-3}^{3},{h}_{i-4}^{3}$ | ${h}_{i-2}^{3},{h}_{i-4}^{3},{h}_{i+1}^{3}$ | ${h}_{i+1}^{3},{h}_{i+2}^{3},{h}_{i+3}^{3}$ | |
E | Int | ${h}_{i-1}^{5},{h}_{i-2}^{5},{h}_{i-3}^{3}$ | ${h}_{i-1}^{3},{h}_{i-2}^{3},{h}_{i+1}^{3},{h}_{i+2}^{3}$ | ${h}_{i+1}^{5},{h}_{i+2}^{5},{h}_{i+3}^{3}$ |
N1 | ${h}_{i-1}^{5},{h}_{i-2}^{5}$ | ${h}_{i-1}^{5},{h}_{i+1}^{5}$ | ${h}_{i+1}^{5},{h}_{i+2}^{5}$ | |
N2 | ${h}_{i-1}^{5},{h}_{i-2}^{5}$ | ${h}_{i-1}^{5},{h}_{i+2}^{5}$ | ${h}_{i+1}^{5},{h}_{i+2}^{5}$ | |
C1 | ${h}_{i-1}^{3},{h}_{i-3}^{3},{h}_{i-4}^{3}$ | ${h}_{i-1}^{3},{h}_{i-3}^{3},{h}_{i+1}^{3}$ | ${h}_{i+1}^{3},{h}_{i+2}^{3},{h}_{i+3}^{3}$ | |
C2 | ${h}_{i-1}^{5},{h}_{i-2}^{5}$ | ${h}_{i-1}^{5},{h}_{i+1}^{5}$ | ${h}_{i+1}^{5},{h}_{i+2}^{5}$ | |
L | Int | ${h}_{i-1}^{5},{h}_{i-2}^{5},{h}_{i-3}^{3},{h}_{i-4}^{3}$ | ${h}_{i-1}^{5},{h}_{i-2}^{3},{h}_{i+1}^{5},{h}_{i+2}^{3}$ | ${h}_{i+1}^{5},{h}_{i+2}^{5},{h}_{i+3}^{3},{h}_{i+4}^{3}$ |
N1 | ${h}_{i-1}^{5},{h}_{i-2}^{5},{h}_{i-3}^{3}$ | ${h}_{i-1}^{5},{h}_{i-2}^{5},{h}_{i+1}^{3}$ | ${h}_{i+1}^{5},{h}_{i+2}^{5},{h}_{i+3}^{3}$ | |
N2 | ${h}_{i-1}^{5},{h}_{i-2}^{3},{h}_{i-4}^{3}$ | ${h}_{i-1}^{5},{h}_{i-2}^{3},{h}_{i+1}^{3}$ | ${h}_{i+1}^{5},{h}_{i+2}^{3},{h}_{i+4}^{3}$ | |
N3 | ${h}_{i-1}^{5},{h}_{i-2}^{3},{h}_{i-3}^{3}$ | ${h}_{i-1}^{5},{h}_{i-2}^{3},{h}_{i+1}^{3}$ | ${h}_{i+1}^{5},{h}_{i+2}^{3},{h}_{i+3}^{3}$ | |
N4 | ${h}_{i-1}^{5},{h}_{i-2}^{3},{h}_{i-3}^{3}$ | ${h}_{i-1}^{5},{h}_{i-2}^{3},{h}_{i+1}^{3}$ | ${h}_{i+1}^{5},{h}_{i+2}^{3},{h}_{i+3}^{3}$ | |
C1 | ${h}_{i-1}^{5},{h}_{i-2}^{5},{h}_{i-3}^{3}$ | ${h}_{i-1}^{5},{h}_{i-2}^{5},{h}_{i+1}^{3}$ | ${h}_{i+1}^{5},{h}_{i+2}^{5},{h}_{i+3}^{3}$ | |
C2 | ${h}_{i-1}^{5},{h}_{i-2}^{5}$ | ${h}_{i-1}^{5},{h}_{i+3}^{5}$ | ${h}_{i+1}^{5},{h}_{i+2}^{5}$ | |
C3 | ${h}_{i-1}^{3},{h}_{i-2}^{3},{h}_{i-3}^{3}$ | ${h}_{i-1}^{3},{h}_{i+1}^{3},{h}_{i+2}^{3}$ | ${h}_{i+1}^{3},{h}_{i+2}^{3},{h}_{i+3}^{3}$ | |
C4 | ${h}_{i-1}^{3},{h}_{i-2}^{3},{h}_{i-3}^{3}$ | ${h}_{i-2}^{3},{h}_{i-3}^{3},{h}_{i+1}^{3}$ | ${h}_{i+1}^{3},{h}_{i+2}^{3},{h}_{i+3}^{3}$ |
For an orphan protein, any method of secondary structure prediction performs as a single-sequence method. Developing better methods of protein secondary structure prediction from single-sequence has a definite merit as it helps improving the functional annotation of orphan proteins. In this work, we describe a new algorithm for protein secondary structure prediction, which develops further the model suggested by Schmidler et al. [14]. We consider the protein secondary structure prediction as a problem of maximization of a posteriori probability of a structure, given primary sequence, as defined by a hidden semi-Markov model (HSMM). To determine the architecture of this HSMM, we performed a statistical analysis identifying the most informative correlations between sequence and structure variables. We specifically considered correlations at proximal positions of structural segments and dependencies to upstream and downstream residues. Finally, we proceeded with an iterative estimation of the HSMM parameters.
Results and discussion
We first compared the performances of BSPSS [14] and IPSSP in strict jacknife conditions. In our computations, we used the EVA set of "sequence-unique" proteins derived from the PDB database (see the Methods section). We removed sequences shorter than 30 amino acids and arrived to a set of 2720 proteins. The performances of IPSSP and BSPSS were evaluated by a leave-one-out cross validation experiment (jacknife procedure) on this reduced set.
Then we evaluated and compared the performances of BSPSS, IPSSP and PSIPRED on the set of 81 CASP6 targets (see the Methods section) that are available in the PDB. This evaluation is at the "single-sequence condition" implying no additional evolutionary information is available. We used the software "PSIPRED_single", version 2.0, which uses a set of fixed weight matrices in the neural network and does not employ PSI-BLAST profiles. This program was downloaded from the PSIPRED server [33] with the available training data (see the Methods section). We used the same training set to estimate the parameters of BSPSS and IPSSP.
To reduce eight secondary structure states used in the DSSP notation to three, it is possible to use different conversion rules. Here we considered the following three rules: (i) H, G and I to H; E, B to E; all other states to L, (ii) H, G to H; E, B to E; all other states to L, (iii) H to H; E to E; all other states to L. The first rule is also known as the 'EHL' mapping [34, 35], the second rule is the one used in PSIPRED [1] and earlier outlined by Rost and Sander [36], while, finally, the third rule is the common 'CK' mapping, which is the one used in BSPSS and other methods [17, 37, 38]. We also analyzed the effect of making further adjustments after applying either of the three conversion rules. We used the adjustments proposed by Frishman and Argos [39] that lead to a secondary structure sequence with the minimum β-strand length of 3 and the minimum α-helix length of 5. In our simulations, we used D = 50 for the maximum allowed segment length. This value is sufficiently large to cover almost all observed uniform secondary structure segments (see the Bayesian formulation section). For the IPSSP method, we performed 2 iterations and used a percentage threshold value of 35% in the dataset reduction step (see the Iterative model training section).
Performance measures
We have compared the performances of the methods in terms of four measures: the Sensitivity, Specificity, Matthew's correlation coefficient and Segment Overlap score. We use the three-state-per-residue accuracy (Q_{3}), defined in Eq. 1 as the overall sensitivity measure:
${Q}_{3}\left(\%\right)=\frac{{N}_{c}}{N}\times 100.\text{}\left(1\right)$
Here, N_{ c }is the total number of residues with correctly predicted secondary structure, N is the total number of observed amino acids. The same measure can be used for each type of secondary structure, Q_{ α }, Q_{ β }and Q_{ L }(Eq. 2):
${Q}_{i}\left(\%\right)=\frac{{N}_{c}^{i}}{{N}^{i}}\times 100,\text{}\left(2\right)$
Prediction sensitivity measures, Q.(%) evaluated on the EVA set under the single-sequence condition.
Sensitivity | Q_{3}(%) | Q_{ α }(%) | Q_{ β }(%) | Q_{ L }(%) |
---|---|---|---|---|
BSPSS | 68.400 | 63.203 | 36.737 | 82.167 |
IPSSP | 70.300 | 65.934 | 45.445 | 81.280 |
Segment border sensitivity values, Q._{-sb}(%), evaluated on the EVA set under the single-sequence condition.
Sensitivity | Q_{3_sb}(%) | Q_{α _sb}(%) | Q_{β _sb}(%) | Q_{L _sb}(%) |
---|---|---|---|---|
BSPSS | 62.207 | 52.634 | 24.215 | 81.903 |
IPSSP | 63.883 | 55.669 | 32.754 | 80.303 |
The specificity measure SP_{ i }is defined for individual types of secondary structure as follows:
$S{P}_{i}\left(\%\right)=\frac{{N}_{c}^{i}}{{N}_{p}^{i}},\text{}\left(3\right)$
Prediction specificity measures, SP.(%), evaluated on the EVA set under the single-sequence condition.
Specificity | SP._{ α }(%) | SP._{ β }(%) | SP_{ L }(%) |
---|---|---|---|
BSPSS | 68.636 | 59.728 | 69.832 |
IPSSP | 72.132 | 59.203 | 72.002 |
The Matthew's correlation coefficient [40] is a single parameter characterizing the extent of a match between the observed and predicted secondary structure. Matthew's correlation is defined for each type of secondary structure as follows:
$MCC=\frac{TP*TN-FP*FN}{{\left[\left(TN+FN\right)\left(TN+FP\right)\left(TP+FN\right)\left(TP+FP\right)\right]}^{1/2}}\text{}\left(4\right)$
Matthew's correlation coefficient values, C., evaluated on the EVA set under the single-sequence condition.
MCC | C _{ α } | C _{ β } | C _{ L } |
---|---|---|---|
BSPSS | 0.5195 | 0.3849 | 0.4468 |
IPSSP | 0.5638 | 0.4312 | 0.4764 |
In terms of the Segment Overlap scores, IPSSP performs uniformly better than BSPSS [see Additional file 1]. We also assessed the reliability of predictions (prediction confidence) produced by the methods BSPSS and IPSSP [see Additional file 2]. The results lead us to the conclusion that IPSSP is better than BSPSS in terms of the reliability measures.
Prediction sensitivity measures, Q.(%), analyzed with respect to three conversion rules and length adjustments, evaluated on the EVA set under the single-sequence condition.
Sensitivity | Q_{3}(%) | Q_{ α }(%) | Q_{ β }(%) | Q_{ L }(%) |
---|---|---|---|---|
BSPSS Rule 1 | 65.177 | 65.655 | 38.844 | 76.644 |
BSPSS Rule 2 | 65.175 | 65.640 | 38.814 | 76.658 |
BSPSS Rule 3 | 67.218 | 64.048 | 38.071 | 80.491 |
BSPSS Rule 1 + Length adj | 68.060 | 63.775 | 37.022 | 81.378 |
BSPSS Rule 2 + Length adj | 68.078 | 63.793 | 37.017 | 81.399 |
BSPSS Rule 3 + Length adj | 68.400 | 63.203 | 36.737 | 82.167 |
IPSSP Rule 1 | 67.415 | 68.115 | 46.386 | 76.340 |
IPSSP Rule 2 | 67.421 | 68.089 | 46.395 | 76.363 |
IPSSP Rule 3 | 69.096 | 66.559 | 45.319 | 79.893 |
IPSSP Rule 1 + Length adj | 70.027 | 66.557 | 45.588 | 80.577 |
IPSSP Rule 2 + Length adj | 70.036 | 66.554 | 45.559 | 80.602 |
IPSSP Rule 3 + Length adj | 70.300 | 65.934 | 45.445 | 81.280 |
Prediction sensitivity measures, Q.(%), evaluated on the CASP6 targets.
Sensitivity | Q_{3}(%) | Q_{ α }(%) | Q_{ β }(%) | Q_{ L }(%) |
---|---|---|---|---|
BSPSS | 66.541 | 75.177 | 41.743 | 72.696 |
IPSSP | 67.899 | 74.984 | 46.087 | 73.755 |
PSIPRED | 67.680 | 76.066 | 52.032 | 69.028 |
Matthew's correlation coefficient values, evaluated on the CASP6 targets.
MCC | C_{ α }(%) | C_{ β }(%) | C_{ L }(%) |
---|---|---|---|
BSPSS | 0.5403 | 0.4354 | 0.4457 |
IPSSP | 0.5657 | 0.4486 | 0.4696 |
PSIPRED | 0.5465 | 0.4801 | 0.4646 |
Improvements over the BSPSS method
Performances of the BSPSS, IPSSP with dependency models, $\mathcal{M}$_{1}, $\mathcal{M}$_{2}, $\mathcal{M}$_{3}, and IPSSP with the combined model, $\mathcal{M}$_{ C }(obtained using an averaging filter), evaluated on the EVA set under the single-sequence condition.
Sensitivity | Q_{3}(%) | Q_{ α }(%) | Q_{ β }(%) | Q_{ L }(%) |
---|---|---|---|---|
BSPSS | 65.175 | 65.640 | 38.814 | 76.658 |
IPSSP-$\mathcal{M}$_{1} | 65.968 | 66.199 | 45.387 | 75.043 |
IPSSP-$\mathcal{M}$_{2} | 66.003 | 66.606 | 46.952 | 74.108 |
IPSSP-$\mathcal{M}$_{3} | 66.315 | 67.012 | 45.005 | 75.364 |
IPSSP-$\mathcal{M}$_{ C } | 67.421 | 68.089 | 46.395 | 76.363 |
Comparison of prediction sensitivity measures, Q.(%), for BSPSS, IPSSP and PSSP (the method that does not use iterative training), evaluated on the EVA set under the single-sequence condition.
Sensitivity | Q_{3}(%) | Q_{ α }(%) | Q_{ β }(%) | Q_{ L }(%) |
---|---|---|---|---|
BSPSS | 65.175 | 65.640 | 38.814 | 76.658 |
IPSSP | 67.421 | 68.089 | 46.395 | 76.363 |
PSSP | 66.840 | 66.945 | 44.566 | 76.761 |
Single-sequence vs. sequence-unique condition
We would like to emphasize that throughout the paper, we use the term single-sequence prediction in its strict meaning, i.e. the prediction method does not exploit information about any protein sequence similar to the sequence in question as for a true single-sequence such information does not exist. The "single-sequence" concept should be distinguished from the concept of the "sequence-unique" category. The "sequence-unique" condition requires the absence of significant similarity between proteins in the test and in the training set. However, this condition leaves an opportunity to use the sequence profile information that typically improves the prediction accuracy by several percentage points in comparison with the single-sequence condition, in which such profiles are not available. Indeed, methods such as APSSP2 [2] and SVMpsi [9] achieved values around 78% in the "sequence-unique" category of CASP [41] and CAFASP [42] experiments. Similarly, the SSPAL method [43] was cited [14] to have 71% accuracy in terms of Q_{3}(%) again in the "sequence-unique" category. Single-sequence condition, as defined, is more stringent. This condition is common for "orphan" proteins, which have no detectable homologs. Improvement of structural prediction under the single-sequence condition should contribute to the improvement of function prediction for orphan proteins, which are not easy targets for functional characterization.
Conclusions
We have shown that new dependency models and training methods bring further improvements to single-sequence protein secondary structure prediction. The results are obtained under cross-validation conditions using a dataset with no pair of sequences having significant sequence similarity. As new sequences are added to the database it is possible to augment the dependency structure and obtain even higher accuracy.
Typically protein secondary structure prediction methods suffer from low accuracy in predicting β-strands, in which non-local correlations have a significant role. In this work, we did not specifically address this problem, but showed that improvements are possible when higher order dependency models are used and significant correlations outside the segments are considered. To achieve substantial improvements in the prediction accuracy, it is necessary to develop models that incorporate long-range interactions in β-sheets. The advances in secondary structure prediction should contribute to the improvement of function prediction for orphan proteins inscrutable to current similarity search methods.
Methods
The representative set of 2482 amino acid sequences (PDB_SELECT) for preliminary statistical analysis were obtained from [44]. The procedure used to generate the PDB_SELECT list was described earlier [45]. In this set, the percentage of identity between any pair of sequences is less than 25%. The 3324 "sequence-unique" proteins were downloaded from the EVA server ftp site, as of 2004_05_09, and the copy of the data were placed at [46]. The proteins in this set, which was used in leave-one-out cross validation experiments were selected to satisfy the condition that percentage of identity between any pair of sequences should not exceed the length dependent threshold S (for instance, for sequences longer than 450 amino acids, S = 19.5) [47]. CASP6 targets were downloaded from [48], and the PDB definitions were used for the amino acid sequences and secondary structure assignments. PSIPRED training data was downloaded from [49].
Bayesian formulation
The linear sequence that defines a secondary structure of a protein can be described by a pair of vectors (S, T), where S denotes the structural segment end (border) positions and, T determines the structural state of each segment (α-helix, β-strand or loop). For instance, for the secondary structure shown in Fig. 2, S = (4, 9, 12, 16, 21, 28, 33) and T = (L, E, L, E, L, H, L).
$P\left(S,T|R\right)=\frac{P\left(R|S,T\right)P\left(S,T\right)}{P\left(R\right)}\text{}\left(5\right)$
where P(R | S, T) denotes the likelihood and P(S, T) is the a priori probability. Since P(R) is a constant with respect to (S, T), maximizing P(S, T | R) is equivalent to maximizing P(R | S, T)P(S, T). To proceed further, we need models for each of these probabilistic terms. We model the distribution of a priori probability P(S, T) as follows:
$P\left(S,T\right)={\displaystyle \prod _{j=1}^{m}P\left({T}_{j}|{T}_{j-1}\right)P\left({S}_{j}|{S}_{j-1},Tj\right).}\text{}\left(6\right)$
Here, m denotes the total number of uniform secondary structure segments. P(T_{ j }| T_{j-1}) is the probability of transition from segment with secondary structure type T_{j-1}to a segment with secondary structure type T_{ j }. Table 3 shows the transition probabilities P(T_{ j }| T_{j-1}), estimated from a representative set of 2482 "unrelated" proteins (see the Methods section). The third term, P(S_{ j }| S_{j-1}, T_{ j }), reflects the length distribution of the uniform secondary structure segments. It is assumed that
P(S_{ j }| S_{j-1}, T_{ j }) = P(S_{ j }- S_{j-1}| T_{ j }), (7)
where S_{ j }- S_{j-1}is equal to the segment length (Fig. 2). The segment length distributions for different types of secondary structure have been determined earlier [50].
The likelihood term P(R | S, T) can be written as:
$\begin{array}{c}P\left(R|S,T\right)={\displaystyle \prod _{j=1}^{m}P\left({R}_{\left[{S}_{j-1}+1:{S}_{j}\right]}|S,T\right)\text{}\left(8\right)}\\ ={\displaystyle \prod _{j=1}^{m}P\left({R}_{\left[{S}_{j-1}+1:{S}_{j}\right]}|{S}_{j-1},{S}_{j},{T}_{j}\right)}\end{array}$
Here, R_{[p:q]}denotes the sequence of amino acid residues with position indices from p to q. The probability of observing a particular amino acid sequence in a segment adopting a particular type of secondary structure is P(${R}_{\left[{S}_{j-1}+1:{S}_{j}\right]}$ | S, T). This term is assumed to be equal to P(${R}_{\left[{S}_{j-1}+1:{S}_{j}\right]}$ | S_{j-1}, S_{ j }, T_{ j }), thus this probability depends only on the secondary structure type of a given segment, and not of adjacent segments. Note that, we ignore the non-local interactions observed in β-sheets. This simplification allows us to implement an efficient hidden semi-Markov model.
To elaborate on the segment likelihood terms in Eq. 8, we have to consider the correlation patterns within the segment with uniform secondary structure. These patterns reflect the secondary structure specific physico-chemical interactions. For instance, α-helices are strengthened by hydrogen bonding between amino acid pairs situated at specific distances. To correctly define the likelihood term we should also pay attention to proximal positions, typically the four initial and the four final positions of a secondary structure segment. In particular, α-helices include capping boxes, where the hydrogen bonding patterns and side-chain interactions are different from the internal positions [51, 52]. The observed distributions of amino acid frequencies in proximal (capping boxes) and internal positions of α-helix segments are depicted in Schmidler et al. [14], and show noticeably distinct patterns.
Presence of this inhomogeneity in the statistical model leads to the following expression for P(${R}_{\left[{S}_{j-1}+1:{S}_{j}\right]}$ | S_{ j }, S_{j-1}, T_{ j }):
$\begin{array}{c}P({R}_{\left[{S}_{j-1}+1:{S}_{j}\right]}|{S}_{j},{S}_{j-1},{T}_{j})={P}_{{N}_{1}}\left({R}_{{k}_{b}+1}\right){\displaystyle \prod _{i={k}_{b}+2}^{{l}_{N}+{k}_{b}}{P}_{{N}_{i}-{k}_{b}}\left({R}_{i}|{R}_{i-1},\dots ,{R}_{{k}_{b}+1}\right)\text{}\left(9\right)}\\ \times {\displaystyle \prod _{i={l}_{N+{k}_{b}+1}}^{{k}_{n}-{l}_{C}}{P}_{Int}\left({R}_{i}|{R}_{i-1},\dots ,{R}_{{k}_{b}+1}\right)}\\ \times {\displaystyle \prod _{i=-{l}_{C}+1}^{0}{P}_{{C}_{1-i}}\left({R}_{i+{k}_{n}}|{R}_{i+{k}_{n}-1},\dots ,{R}_{{k}_{b}+1}\right).}\end{array}$
Here, the first and third sub-products represent the probability of observing l_{ N }and l_{ C }specific amino acids at the segment's N-terminal and C-terminal, respectively. The second sub-product defines the observation probability of given amino acids in the segment's internal positions. Note that, k_{ b }and k_{ e }designate S_{j-1}+ 1 and S_{ j }, respectively. The probabilistic expression (9) is generic for α-helices, β-strands and loops. Formula (9) assumes that, the probabilistic model is fully dependent within a segment, i.e. observation of an amino acid at a given position depends on all previous amino acids within the segment. However, at this time, the Protein Data Bank (PDB, [53]) does not have a sufficient amount of experimental data to reliably estimate all the parameters of a fully dependent model. Therefore, it is important to reduce the dependency structure and keep only the most important correlations. In order to achieve this goal, we performed the statistical analysis described in the following section.
Correlation patterns of amino acids
Amino acids have distinct propensities for the adoption of secondary structure conformations [54]. These propensities are in the heart of many secondary structure prediction methods [51, 52, 55–62]. Our goal is to come up with a dependency pattern that is comprehensive enough to capture the essential correlations yet simple enough in terms of the number of model parameters to allow reliable parameter estimation from the available training data. Therefore, we performed a χ^{2}-test to identify the most significant correlations between amino acid pairs located in adjacent and non-adjacent positions for each type of secondary structure segments. The χ^{2}-test compared empirical distribution of amino acid pairs with the respective product of marginal distributions. Therefore, a 20 × 20 contingency table was computed for the frequencies of possible amino acid pairs observed in different structural states. In this test, the threshold was computed as 404.6 for a statistical significance level of 0.05.
We first considered the correlations between amino acid pairs at various separation distances (Table 4). In α-helix segments, a residue at position i is highly correlated with residues at positions i - 2, i - 3 and i - 4. Similarly, a β-strand residue had its highest correlations with residues at positions i - 1, i - 2, and a loop residue had its most significant correlation with a residue at position i - 1. The test statistics for the remaining pairs were above the threshold for statistical significance but these values were considerably lower than the ones listed above. The dependencies that were identified by the statistical analysis are in agreement with the well known physical nature of the secondary structure conformations.
Next, we analyzed proximal positions and a representative set of internal positions. Frequency patterns in proximal positions deviate from the patterns observed in internal positions [51, 58]. For a better quantification, we computed the Kullback-Liebler (KL) distance between probability distributions of the proximal and internal positions (Table 5). The observation that the KL distance is significantly higher for positions closer to segment borders suggests that amino acids in proximal locations have significantly different distributions from those at internal regions.
Finally, we performed a χ^{2}-test for proximal positions to identify the correlations between amino acid pairs at various separation distances. As can be seen from the results for α-helix segments (Table 6), the general assumption of segment independence does not hold as statistically significant correlations were observed between residues situated on both sides of the segment borders. For instance, the second amino acid i in the α-helix N-terminal significantly correlates with the previous amino acid at position i - 2, which is outside the segment. This correlation can be caused by physical interactions between nearby residues [51]. Also, the strength of correlation for the i + (downstream) residues was different from the strength observed for i - (upstream) residues (Table 6). This fact indicates an asymmetry in correlation behavior for i+ and i- residues. The parameters of position specific correlations were also computed for β-strand and loop segments (data not shown).
A similar asymmetry in the correlation patterns was also observed in internal positions (data not shown). For instance, for α-helices, the i^{ th } residue in an internal position is highly correlated with the i - 2^{ th }, i - 3^{ th }, i - 4^{ th }, i + 2^{ th } and i + 4^{ th } residues. The parameters of correlation between i^{ th } and i - 2^{ th } residues is different from the parameters of correlation between i^{ th } and i + 2^{ th } residues.
In the next section, we will refine the probabilistic model needed to determine P(${R}_{\left[{S}_{j-1}+1:{S}_{j}\right]}$ | S_{j-1}, S_{ j }, T_{ j }) using the most significant correlations identified by the statistical analysis.
Reduced dependency model
Correlation analysis allows one to reduce the alphabet size in the likelihood expression (Eq. 9) by selecting only the most significant correlations. The dependence patterns revealed by the statistical analysis are shown in Table 7 divided into panels for α-helix (H), β-strand (E), and loop (L) structures. To reduce the dimension of the parameter space, we grouped the amino acids into three and five hydrophobicity classes. We used five classes only for those positions, which have significantly high correlation measures. In Table 7, ${h}_{i-1}^{3}$ stands for the dependency of an amino acid at position i to the hydrophobicity class of an amino acid at position i - 1, and the superscript 3 represents the total number of hydrophobicity classes.
To better characterize the features that define the secondary structure, we distinguished positions within a segment as well as segments with different lengths. We identified as proximal positions those in which the amino acid frequency distributions significantly deviate from ones in internal positions in terms of the KL distance (Table 5). Based on the available training data, we chose 6 proximal positions (N1-N4, C1-C2) for α-helices, 4 proximal positions (N1-N2, C1-C2) for β-strands, and 8 proximal positions (N1-N4, C1-C4) for loops. The remaining positions are defined as internal positions (Int). In addition to position specific dependencies, we derived separate patterns for segments with different lengths. Table 7 shows the dependence patterns for segments longer than L residues, where L is 5 for α-helices, and 3 for β-strands and loops. For shorter segments, we selected a representative set of patterns from Table 7 according to the available training data.
To fully utilize the dependency structure, we found it useful to derive three separate dependency models. The first model, $\mathcal{M}$1, uses only dependencies to upstream positions, (i-), the second model, $\mathcal{M}$2, includes dependencies to upstream (i-), and downstream (i+) positions simultaneously, and the third model, $\mathcal{M}$3, incorporates only downstream (i+) dependencies. For each dependency model ($\mathcal{M}$1-$\mathcal{M}$3), the probability of observing an amino acid at a given position is defined using the dependence patterns selected from Table 7. For instance, according to the model $\mathcal{M}$2, the conditional probability of observing an amino acid at position i = N 3 of an α-helix segment becomes PN3(R_{i}l ${h}_{i-2}^{3},{h}_{i-3}^{3},{h}_{i+2}^{3}$). By multiplying the conditional probabilities selected from Table 7 as formulated in Eq. 9, we obtain the propensity value for the observation of the amino acid segment under the specified model. In the case of $\mathcal{M}$1, and $\mathcal{M}$3, this product gives the segment likelihood expression, which is a properly normalized probability value P(${R}_{\left[{S}_{j-1}+1:{S}_{j}\right]}$ | S, T). Hence, $\mathcal{M}$1, and $\mathcal{M}$3 are probabilistic models. For the model $\mathcal{M}$2, we rather obtain a score Q(${R}_{\left[{S}_{j-1}+1:{S}_{j}\right]}$ | S, T) that represents the potential of a given amino acid segment to adopt a particular secondary structure conformation. This scoring system can be used to characterize amino acid segments in terms of their propensity to form structures of different types and when uniformly applied to compute segment potentials, allows to implement algorithms following the theory of hidden semi-Markov models. Implementing three different models enables to generate three predictions each specializing in a different section of the dependency structure. Those predictions can then be combined to get a final prediction sequence, as explained in the next section.
The hidden semi-Markov model and computational methods
Amino acid and DNA sequences have been successfully analyzed using hidden Markov models (HMM) as the character strings generated in "left-to-right" direction. For a comprehensive introduction to HMMs, see [63].
Having defined this HSMM, we can consider the protein secondary structure prediction problem as the problem of finding the sequence of hidden states with the highest a posteriori probability given the amino acid sequence. One efficient algorithm to solve this optimization problem is well known. Given an amino acid sequence R, the vector (S, T)* = arg max P(S, T | R) can be found using the Viterbi algorithm. Here lies a subtle difference between the result that can be delivered by the Viterbi algorithm and the result needed in the traditional statement of the protein secondary structure prediction problem.
The Viterbi path does not directly optimize the three-state-per residue accuracy (Q_{3}):
${Q}_{3}=\frac{\text{Total\#ofcorrectlypredictedstructuralstates}}{\text{Total\#ofobservedaminoacids}}.\text{}\left(10\right)$
Also, the Viterbi algorithm might generate many different segmentations, which might have significant probability mass but are not optimal [14]. As an alternative to the Viterbi algorithm, we can determine the sequence of structural states that are most likely to occur in each position. This approach will use forward and backward algorithms (posterior decoding) generalized for HSMM [63]. Although the prediction sequence obtained by forward and backward algorithms might not be a perfectly valid state sequence (i.e. it might not be realized given the parameters of HSMM), the prediction measure defined as the marginal posterior probability distribution (Eq. 14) correlates very strongly with the prediction accuracy (Q_{3}) [14]. The performance of the Viterbi and forward-backward algorithms are compared in Schmidler et al. [14]. Here, forward and backward variables are defined as follows (n is the total number of amino acids in a sequence):
$\begin{array}{c}{\alpha}^{\theta}\left(j,t\right)={P}^{\theta}\left({R}_{\left[1:j\right]},S=j,T=t\right)\\ ={\displaystyle \sum _{v=1}^{j-1}{\displaystyle \sum _{l\in SS}{\alpha}^{\theta}\left(v,l\right){P}^{\theta}\left({R}_{\left[v+1:j\right]}|{S}_{prev}=v,S=j,T=t\right)}}\\ \times {P}^{\theta}\left(S=j|T=t,{S}_{prev}=v\right){P}^{\theta}\left(T=t|{T}_{prev}=l\right)\\ {\alpha}^{\theta}\left(1,t\right)={P}^{\theta}\left({R}_{\left[1\right]}|T=t,S=1\right){P}^{\theta}\left(S=1|T=t\right){P}^{\theta}\left(T=t\right)\\ j=1,\dots ,n\end{array}\text{}\left(11\right)$
$\begin{array}{c}{\beta}^{\theta}\left(j,t\right)={P}^{\theta}\left({R}_{\left[j+1:n\right]}|S=j,T=t\right)\\ ={\displaystyle \sum _{v=j+1}^{n}{\displaystyle \sum _{l\in SS}{\beta}^{\theta}\left(v,l\right){P}^{\theta}\left({R}_{\left[j+1:v\right]}|{S}_{prev}=v,S=j,{T}_{next}=l\right)}}\\ \times {P}^{\theta}\left({S}_{next}=v|S=j,{T}_{next}=l\right){P}^{\theta}\left({T}_{next}=l|T=t\right)\\ {\beta}^{\theta}\left(n,t\right)=1\\ j=n,\dots ,1\end{array}\text{}\left(12\right)$
The forward variable α^{ θ } (j, t) is the joint probability of observing the amino acid sequence up to position j, and a secondary structure segment that ends at position j with type t. Here, θ represents the statistical dependency model. Similarly, the backward variable β^{ θ } (j, t) defines the conditional probability of observing the amino acid sequence in positions j + 1 to n, and a secondary structure segment that ends at position j with type t. Then, the a posteriori probability for a hidden state in position i to be either an α-helix, β-strand or loop is computed via all possible segmentations that include position i (Eq. 13). The hidden state at position i is identified as the state with maximum a posteriori probability. Finally, the whole predicted sequence of hidden states is defined by Eq.14.
$\begin{array}{c}{P}^{\theta}\left({T}_{{R}_{i}}|R\right)={\displaystyle \sum _{j=1}^{i-1}{\displaystyle \sum _{k=i}^{n}{\displaystyle \sum _{l\in SS}{\alpha}^{\theta}\left(j,l\right)}{\beta}^{\theta}\left(k,t\right)}{P}^{\theta}\left(T=t|{T}_{prev}=l\right)}\\ \times {P}^{\theta}\left(S=k|{S}_{prev}=j,T=t\right)\\ \times {p}^{\theta}\left({R}_{\left[j+1:k\right]}|{S}_{prev}=j,S=k,T=t\right)/{P}^{\theta}\left(R\right)\end{array}\text{}\left(13\right)$
${\left(S,T\right)}^{*}=\mathrm{arg}\underset{(S,T)}{\mathrm{max}}{\left\{{P}^{\theta}\left({T}_{{R}_{i}}|R\right)\right\}}_{i=1}^{n}\text{}\left(14\right)$
The computational complexity of this algorithm is O(n^{3}). If the maximum size of a segment is limited by a value D, the first summation in Eq. 11 starts at (j - D), and the first summation in Eq. 12 ends at (j + D) reducing the computational cost to O(nD^{2}).
Note that, forward and backward variables are computed by multiplying probabilities, which are less than 1, and as the sequence gets longer, these variables approximate to zero after a certain position. Hence it is necessary to introduce a scaling procedure to prevent numerical underflow. The scaling for a "classic" HMM is described in [63]. This procedure can easily be generalized for an HSMM, where the scaling coefficients are introduced at every D positions.
This completes the derivation of the algorithm for a single model. Since we are utilizing three dependency models, θ = $\mathcal{M}$1, $\mathcal{M}$2, $\mathcal{M}$3, it becomes necessary to combine the outputs of the three models with an appropriate function. In our simulations, we implemented averaging and maximum functions to perform this task and observed that the averaging function gives a better performance. The final prediction sequence is then computed as:
$\begin{array}{ccc}{P}^{C}({T}_{{R}_{i}}|R)& =& ({P}^{\mathcal{M}1}({T}_{{R}_{i}}|R)+{P}^{\mathcal{M}2}({T}_{{R}_{i}}|R)+{P}^{\mathcal{M}3}({T}_{{R}_{i}}|R))/3\\ {(S,T)}^{*}& =& \mathrm{arg}\underset{(S,T)}{\mathrm{max}}{\left\{{P}^{C}({T}_{{R}_{i}}|R)\right\}}_{i=1}^{n}\end{array}\text{}\left(15\right)$
Iterative model training
Although affinity in structural composition does not guarantee structural similarity, using this measure allows us to reduce the training set to proteins that belong to more closely related SCOP families [67]. Thus, for example, a prediction of a structure from an all-α class is likely to be followed by a training using proteins having high α-helix content. In our simulations, we observed that after several iterations (no more than 3) the predicted secondary structure sequence did not change indicating the algorithm convergence.
Declarations
Acknowledgements
Yucel Altunbasak and Zafer Aydin were supported by grant CCR-0105654 from the NSF-SPS and Mark Borodovsky was supported in part by grant HG00783 from the NIH.
Authors’ Affiliations
References
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