- Research article
- Open Access
A Hidden Markov Model method, capable of predicting and discriminating β-barrel outer membrane proteins
© Bagos et al; licensee BioMed Central Ltd. 2004
Received: 21 November 2003
Accepted: 15 March 2004
Published: 15 March 2004
Integral membrane proteins constitute about 20–30% of all proteins in the fully sequenced genomes. They come in two structural classes, the α-helical and the β-barrel membrane proteins, demonstrating different physicochemical characteristics, structure and localization. While transmembrane segment prediction for the α-helical integral membrane proteins appears to be an easy task nowadays, the same is much more difficult for the β-barrel membrane proteins. We developed a method, based on a Hidden Markov Model, capable of predicting the transmembrane β-strands of the outer membrane proteins of gram-negative bacteria, and discriminating those from water-soluble proteins in large datasets. The model is trained in a discriminative manner, aiming at maximizing the probability of correct predictions rather than the likelihood of the sequences.
The training has been performed on a non-redundant database of 14 outer membrane proteins with structures known at atomic resolution; it has been tested with a jacknife procedure, yielding a per residue accuracy of 84.2% and a correlation coefficient of 0.72, whereas for the self-consistency test the per residue accuracy was 88.1% and the correlation coefficient 0.824. The total number of correctly predicted topologies is 10 out of 14 in the self-consistency test, and 9 out of 14 in the jacknife. Furthermore, the model is capable of discriminating outer membrane from water-soluble proteins in large-scale applications, with a success rate of 88.8% and 89.2% for the correct classification of outer membrane and water-soluble proteins respectively, the highest rates obtained in the literature. That test has been performed independently on a set of known outer membrane proteins with low sequence identity with each other and also with the proteins of the training set.
Based on the above, we developed a strategy, that enabled us to screen the entire proteome of E. coli for outer membrane proteins. The results were satisfactory, thus the method presented here appears to be suitable for screening entire proteomes for the discovery of novel outer membrane proteins. A web interface available for non-commercial users is located at: http://bioinformatics.biol.uoa.gr/PRED-TMBB, and it is the only freely available HMM-based predictor for β-barrel outer membrane protein topology.
Integral membrane proteins are divided into two distinct structural classes, the α-helical membrane proteins and the β-barrel membrane proteins. The former class is the more abundant and well studied, since proteins of that type are located mostly in the cell membranes of both prokaryotic and eukaryotic organisms, performing a variety of biologically important functions. Proteins of that class have their membrane spanning regions forming α-helices, which consist mainly of hydrophobic residues . A variety of algorithms and computational techniques have been proposed for the prediction of the transmembrane segments of α-helical membrane proteins, with high accuracy and precision. The members of the latter class (β-barrel membrane proteins) are located in the outer membrane of gram-negative bacteria, and presumably in the outer membrane of chloroplasts and mitochondria. The members of that class are having their membrane spanning segments formed by antiparallel β-strands, creating a channel in a form of a barrel that spans the outer membrane . The first known members of that class were found to be the bacterial trimeric porins, forming water-filled channels that mediate the passive transport of ions and small molecules through the outer membrane . During the last few years, more β-barrel proteins were found in the bacterial outer membrane, and a number of structures have been solved in atomic resolution . These proteins perform a wide variety of functions such as active ion transport, passive nutrient uptake, membrane anchoring, adhesion, and catalytic activity. A large number of pathogens are actually bacteria belonging to the gram negative bacteria class. Considering additionally the important biological functions in which outer membrane proteins are involved in, it is not a surprise that those proteins attract an increased medical interest. This is confirmed by the continuously increasing number of completely sequenced genomes of gram-negative bacteria deposited in the public databases. On the other hand, the extensive study of the structure of transmembrane β-barrel proteins, could reveal special aspects of the process of protein folding, and give us useful insights on protein structure and function. For the reasons mentioned above, there is clearly a need to develop computational tools for predicting the membrane spanning strands of those proteins, and also discriminating them from water-soluble proteins when searching entire genomes.
In contrast to the α-helical membrane proteins, whose membrane spanning segments can be identified by statistical methods, neural networks, or Hidden Markov Models with high accuracy, this task is more difficult in the case of the β-barrel membrane proteins of the outer membrane. This is due to the lack of a clear pattern in their membrane spanning strands, such as the stretch of 15–30 consecutive hydrophobic residues or the Positive Inside rule, which occur in the α-helical proteins. Furthermore, discrimination of transmembrane strands from other β-strands, forming β-barrel structures in water-soluble proteins, is even more difficult. The reason for that is the fact that water-soluble proteins that form β-barrel structures, share (up to a certain degree) common features with the transmembrane strands of the bacterial outer membrane proteins, such as amphipathicity.
A few approaches have been made, in the direction of predicting the transmembrane strands of outer membrane proteins and/or identifying those proteins when searching large data sets; they are based on study of the physicochemical properties of the β-strands, such as hydrophobicity and amphipathicity , statistical analyses based on the amino acid composition of the known structures , or machine learning techniques like neural network predictors [5, 6], and Hidden Markov Models [4, 7, 8]. Recently, a method based on a sequence profile-based HMM , requiring as input evolutionary information derived from multiple alignments, achieved the highest accuracy.
The non-redundant data set of outer membrane proteins used in this study.
Number of β-strands
Results and discussion
Overall measures of accuracy, obtained in the Self-consistency and in the Jacknife testing.
Type of test
The result of the jacknife test concerning the correctly predicted TM strands was 204 out of 214 (95.3%), with 12 over-predicted strands. The overall number of correctly predicted topologies was 9 out of 14. When counting the predicted misplaced strands, the number of correctly predicted topologies raises to 10 out of 14. Once again when comparing against the manual annotation, the second TM strand of 1I78, is predicted slightly misplaced.
The per-residue accuracies and correlation coefficients (see Materials and methods) for both the self-consistency and jacknife tests are listed also in Table 2, with respect to either the PDB annotation or the manual annotation used for training. Apparently, the significantly lower percentages reported in the case of comparison with the PDB annotation, is a clear consequence of the fact that the strands extend in some cases far beyond the lipid bilayer. These strands could not have been predicted as transmembrane along their entire length, and our model predicts only the part of the strand that it is inserted into the membrane. 1I78 is a perfect example of such a case since all of its strands are clearly extending far beyond the membrane by 8 or more residues. In addition to the self-consistency and jacknife test, we performed another independent test. We divided the training set in two datasets of seven proteins each and used the one for training and the other for testing. This procedure was repeated 5 times, choosing randomly 7 different proteins each time, and the results concerning the per-residue accuracy and the correlation coefficient were in the range 0.78 – 0.80 and 0.57 – 0.71 respectively. We also tested the performance of the model on the Neisserial Surface Protein A (Nspa) , the Outer Membrane Enzyme Pagp from E. coli , and the Outer Membrane Cobalamin Transporter (Btub) from E. coli  The structures of these proteins have been very recently solved, they have not been included in the training set, and they do not show any significant homology with any protein of the training set. For NspA, and BtuB the model correctly locates all the transmembrane strands and the proteins' full topologies, whereas for Pagp we get two over-predicted strands. For the three proteins the per-residue accuracy is 90.9% and the correlation coefficient is 0.78.
Furthermore, we developed a protocol that allowed us to apply the newly developed method to search the complete proteome of E. coli  for β-barrel membrane proteins. The protocol consists of three steps: In the first step we perform a search using the PRED-CLASS algorithm , with the aim of identifying and removing α-helical membrane proteins. According to the PRED-CLASS prediction, 1157 proteins out of 5361 were classified as transmembrane and hence they were removed. In the second step, the remaining proteins were filtered with the SignalP program, for finding the secreted proteins, since it is apparent that the majority, if not all, of the outer membrane proteins posses a signal peptide sequence necessary for the translocation to the outer membrane. In order not to discard potential candidates for being identified as outer membrane proteins, we applied both versions of SignalP, the Neural Network  and the Hidden Markov Model , and if at least one of them indicated the presence of a signal peptide, the protein was not discarded. Then, the signal peptide predicted by the Neural Network algorithm of SignalP was removed, since this algorithm is more accurate than the Hidden Markov algorithm in locating the correct splicing site from all the candidate proteins. This procedure resulted to 978 proteins. In the third and final step, the remaining 978 protein sequences were submitted to our HMM predictor having set the discrimination score at the pre-specified threshold mentioned above. In total, 236 proteins scored below the threshold for the outer membrane proteins (after excluding fragments and sequences shorter than 60 residues) accounting for 4.4% of the complete proteome. Among the top scoring proteins, 42 are well known outer membrane proteins according to the existing annotation, including many fimbrial proteins, usher proteins and transporters, and 87 were proteins whose annotation was "putative" but suggested that their localization was to the outer membrane. The number of proteins whose annotation suggested that they were misclassified, including a lot of periplasmic proteins and enzymes, was 34, whereas the "putative" misclassified proteins were 23. Finally, the remaining 50 proteins were hypothetical proteins or proteins with completely unknown function. Apparently, the 57 over-predictions probably are resulting from the fact that outer membrane proteins are only a small fraction compared to the whole proteome. Since it is believed that outer membrane proteins constitute around 2–4% of the complete proteomes, it is natural that even a method with 99% of correct predictions, will result in a large number of false positives. Nevertheless, given the constrains mentioned above, this method clearly offers a useful tool for the automatic annotation of entire proteomes, since the false positives could be easily removed considering other sequence characteristics.
Comparing our method with the best method proposed so far for the prediction of transmembrane β-barrel proteins by Martelli et al. , as well as with the HMM method proposed by Liu et al. , the following points should be mentioned. The HMM-profile based method by Martelli et al. , uses as input the evolutionary information included in multiple alignments. The method proposed here, uses as input only the amino acid sequence of the protein, hence it is computationally simpler. Even though our method does not outperform the profile based HMM method by Martelli et al.  in the per residue accuracy, when it comes to the number of correctly predicted transmembrane strands and overall topologies, the two methods are practically equivalent. The same argument holds for the case of the discriminative power of the two methods, since the percentage of correctly classified β-barrel proteins was 84%, and percentage of correctly classified water soluble proteins was 90% as reported in , showing that better results can be obtained even without the use of evolutionary information. Concerning the method proposed by Liu et al, which uses as input single sequences, the results about strand localization and overall topology assignment are also comparable with our method, but no discrimination could be performed between outer membrane proteins and soluble ones in their method, thus requiring a separate method for the discrimination purposes. Furthermore in Liu et al , no overall measures of accuracy were reported.
Both methods mentioned above use HMMs with architectures quite similar to the model shown here, with minor differences, and this is not a surprise. For the sake of argument, the two most successful methods for the prediction of transmembrane segments of α-helical membrane proteins use a similar architecture; that architecture reflects the most obvious way to map the biological features of transmembrane proteins to the mathematical formalism of the Hidden Markov Model. Finally, the methodology that we used for the training and the decoding is completely different from those used by Martelli et al. , and Liu et al. . Our model was trained according to the Conditional Maximum Likelihood criterion, which differs significantly from the Maximum Likelihood training scheme, performed with the Baum-Welch algorithm, by the two methods mentioned. For the decoding, Martelli et al use the so-called posterior decoding method, with the aid of a dynamic programming algorithm, whereas Liu et al, rely on the traditional Viterbi algorithm. Even when in our tests the N-best decoding does not outperform significantly the Viterbi decoding (data not shown), when it comes to newly discovered proteins, the option to perform decoding with the best method available is a clear advantage.
When our method is compared against the methods developed by Zhai and Saier  and Wimley , we observe that none of the above methods controlled for the number of false positives and false negatives, since they were not validated statistically. They both report the finding of a number of predicted outer membrane proteins, for which the genome annotation suggested localization to the outer membrane. The fact that we report 236 predicted outer membrane proteins in E. coli proteome, compared to 118 in  and 200 in , reflects the fact that we chose to retain the threshold obtained from cross-validation. Clearly, in real life applications using our method we could lower the threshold and obtain fewer predictions (<200), with the cost of loosing 5–10 outer membrane proteins.
Materials and methods
The Hidden Markov Model
Hidden Markov Models have been extensively used for pattern recognition problems, with the most known example found in the speech recognition methodology . Hidden Markov Models have been used in bioinformatics during the last few years for generating probabilistic profiles for protein families , the prediction of transmembrane helices in proteins [18, 19], the prediction of signal peptides and their cleavage sites , the prediction of genes  and recently for the prediction of the transmembrane β-strands [7, 8]. An excellent introduction of those applications in molecular biology is the book of Durbin et al  whose notation will follow hereafter.
The Hidden Markov Model is a probabilistic model consisting of several states, connected by means of the transition probabilities, thus forming a markov process. If we consider an aminoacid sequence of a protein with length L, denoted by:
x = x1, x2,..., x L ,
with a labeling (corresponding to transmembrane, intracellular and extracellular regions):
y = y1, y2,..., y L
then, the transition probability for jumping from a state k to a state l is defined as:
α kl = P(π i = l|πi-1= k)
Where π is the "path" in the particular position of the amino acid sequence (i.e. the sequence of states, as opposed to the sequence of symbols). Each state k is associated with a distribution of emission probabilities, meaning the probabilities that any particular symbol could be emitted by the current state. Assuming an alphabet Σ, consisting of the symbols corresponding to the 20 amino acids, the probability that a particular amino-acid b is emitted from state k is defined as:
e k (b) = P(x i = b|π i = k)
The term 'hidden' is justified by the fact that when one observes the emitted symbols he cannot see the underlying states, thus the true state process is hidden from the observer. The total probability of the observation sequence given the model, P(x|θ), is computed using the efficient Forward algorithm , whereas the joint probability of the sequence and the labeling denoted by P(x,y|θ), by its trivial modification proposed by Krogh .
Training and decoding algorithms
Traditionally, the parameters of a Hidden Markov Model are optimized according to the Maximum Likelihood criterion ,
A widely used algorithm for this task is the efficient Baum-Welch algorithm (also known as Forward-Backward) [16, 23], which is a special case of the Expectation-Maximization (EM) algorithm, proposed for Maximum Likelihood (ML) estimation for incomplete data . The algorithm, updates iteratively the model parameters (emission and transition probabilities), with the use of their expectations, computed with the use of the Forward and Backward algorithms. Convergence to at least a local maximum of the likelihood is guaranteed. The main disadvantage of ML training is that it is not discriminative. In this work, we used Conditional Maximum Likelihood (CML) training for labeled data, as proposed by Krogh . The Conditional Maximum Likelihood criterion is:
where L is the length of the sequence. We studied the proportion of correctly classified proteins as a function of the discrimination score used as the threshold. We defined the optimal threshold as the value that maximizes that function. Proteins with score values below the threshold should be declared as beta-barrel membrane proteins. All algorithms and tools used throughout this work have been implemented by the authors, using the Java programming language by Sun Microsystems.
The model architecture
The model that we used is cyclic, consisting of 61 states, (Figure 4). The architecture has been chosen so that it could fit as much as possible to the limitations imposed by the known structures. The model consists of three "sub-models" corresponding to the three desired labels to predict, the TM (transmembrane) strand sub-model and the inner and outer loops sub-models respectively. The TM strand model incorporates states to model the special architecture of the transmembrane strands. Thus, there are states that correspond to the core of the strand and the aromatic belt located at the lipid bilayer interface. Furthermore, other states correspond to the amino acid residues facing the bilayer (the external side of the barrel) and the residues facing the barrel interior. All states are connected with the appropriate transition probabilities in order to be consistent with the known structures (i.e. to ensure appropriate length distributions and to model the alternating pattern of hydrophobic-non hydrophobic residues, corresponding to the external-internal residues of the barrel). The minimum allowed length for a transmembrane strand is 7 residues, whereas the maximum is 17.
The inner and outer loops are modeled with a "ladder" architecture, whereas at the top of the outer loop there is a self transitioning state corresponding to residues too distant from the membrane; these cannot be modeled as loops, hence that state is named "globular". The "inner" loop sub-model has no corresponding "globular" state, reflecting the fact that inner loops are significantly shorter than the outer ones, since none of the known structures possesses an inner loop longer than twelve residues. In order to capture the fact that all known structures are having their N-terminal tail falling into the periplasmic space (the "inside" with respect to the outer membrane) we allowed the begin state of the model to be followed only by states belonging to the inner loop or to TM strands directing to the external side of the outer membrane. Finally, we allowed a self-transitioning absorbing state to follow the inner loop states, in order to correctly model sequences that have a long C-terminus falling in the periplasmic space. States expected to have the same emission probabilities are tied together.
Training and testing sets
The training set that we used has been compiled with consideration of the SCOP classification . In particular, we selected all PDB codes from SCOP that belong to the fold "Transmembrane beta-barrels", and obtained the corresponding structures from the Protein Data Bank (PDB) . For variants of the same protein, we kept the structure solved at the highest resolution, and we removed multiple chains, keeping only one chain for each structure. The sequences of the remaining structures have been submitted to a redundancy check, removing chains with a sequence identity above some threshold. We considered two sequences as being homologues, if they demonstrated an identity above 30% in a pairwise alignment, in a length longer than 80 residues. For the pairwise local alignment we used BlastP  with default parameters, and the homologous sequences were removed implementing Algorithm 2 from Hobom et al . The remaining 14 outer membrane proteins constitute our training set (Table 1). The structures of TolC  and alpha-hemolysin , were not included in the training set for the following reasons: TolC is a mixed beta-barrel and alpha-helical protein which spans both the outer membrane and the periplasmic space of gram negative bacteria. Three TolC protomers assemble to form a continuous, solvent accessible conduit, a "channel-tunnel" over 140 Å long. Each monomer of the trimer contributes 4 β-strands to the 12 strand β-barrel. Alpha-hemolysin secreted from S. aureus is active as a transmembrane heptamer, where the transmembrane domain is a 14-strand antiparallel β-barrel, in which two strands are contributed by each monomer. Both structures are not included in the fold "transmembrane beta-barrel" of the SCOP database. In summary, the set includes proteins being monomeric, dimeric or trimeric, with a number of TM β-strands ranging from 8 to 22, and is representative of the known functions of outer membrane proteins. As an independent test set of outer membrane proteins, we chose the dataset used in the validation of the PSORT-B algorithm , consisting of 377 proteins. This set was also checked for redundancy with the same criteria mentioned above, and the closest homologues were removed along with the proteins showing similarity to at least one protein from the training set, leaving us with 119 outer membrane proteins. To test the discriminative power of the model we used an additional dataset of globular proteins, with 3-dimensional structures deposited in PDB . This set was compiled using the PAPIA  server, with the sequence similarity threshold set to 25%, and excluding membrane proteins, proteins with a length lower than 80 residues, and proteins with at least one unidentifiable residue in the sequence; finally we came up with 1100 sequences of such globular proteins.
It is noteworthy that even in structures known at atomic resolution, the exact boundaries of the TM strands are not obvious, and in some situations the PDB annotations for the strands are clearly extending far beyond the membrane. Since our primary objective was to predict the TM segments of the strands rather than the entire β-strands, the model was trained to identify these particular segments. It is well known that discriminative training algorithms are very sensitive to data mislabeling, thus the training could not have been performed with labels based on the PDB annotation for the TM-strands. In , an automated method for re-labeling the data was proposed, but in our case since the training data set was limited we chose a manual approach. For the training purposes, the labels for the TM segments were set manually, by identifying the aromatic belts of the barrel  after inspection of the 3-dimensional structures of the proteins of the training set using molecular graphics. All residues contained between the two aromatic belts of each β-strand of the β-barrel were set to define a TM segment, including the residues of the belts. In cases where the aromatic belt residues of a β-strand are not clearly defined, neighbouring β-strands of the β-barrel helped in the "belt" identification. The resulting dataset used for training is shown in additional file 1 (second column, TM).
Measures of accuracy
To assess the accuracy of the predictions, we used several measures. For the transmembrane strand predictions we calculated the number of correctly predicted strands (True Positives, TP), the number of missed strands (False Negatives, FN) and the number of the over-predicted strands (False Positives, FP). We also calculated the total number of correctly predicted topologies, i.e. when both the strand localization and the loops topology have been predicted correctly. As measures of the accuracy per residue, we report here both the total fraction of the correctly predicted residues (Qβ) in a two-state model (transmembrane versus non-transmembrane), and the well known Matthews Correlation Coefficient (Cβ) . The comparisons have been performed against our manual annotation of the TM segments to show the efficiency of the model as well as against the PDB annotation for the transmembrane strands, for demonstration purposes. We feel that this had to be done in order to allow a fair comparison with other published methods [6–8], since in each one of the published methods the comparisons were performed against the PDB annotation.
The authors would like to thank the two anonymous referees for their constructive criticism.
- von Heijne G: Recent advances in the understanding of membrane protein assembly and function. Q Rev Biophys 1999, 32: 285–307. 10.1017/S0033583500003541View ArticlePubMedGoogle Scholar
- Schulz GE: The structure of bacterial outer membrane proteins. Biochim Biophys Acta 2002, 1565: 308–317. 10.1016/S0005-2736(02)00577-1View ArticlePubMedGoogle Scholar
- Zhai Y, Saier M. H., Jr.: The beta-barrel finder (BBF) program, allowing identification of outer membrane beta-barrel proteins encoded within prokaryotic genomes. Protein Sci 2002, 11: 2196–2207. 10.1110/ps.0209002PubMed CentralView ArticlePubMedGoogle Scholar
- Wimley WC: Toward genomic identification of beta-barrel membrane proteins: composition and architecture of known structures. Protein Sci 2002, 11: 301–312. 10.1110/ps.29402PubMed CentralView ArticlePubMedGoogle Scholar
- Diederichs K, Freigang J, Umhau S, Zeth K, Breed J: Prediction by a neural network of outer membrane beta-strand protein topology. Protein Sci 1998, 7: 2413–2420.PubMed CentralView ArticlePubMedGoogle Scholar
- Jacoboni I, Martelli PL, Fariselli P, De Pinto V, Casadio R: Prediction of the transmembrane regions of beta-barrel membrane proteins with a neural network-based predictor. Protein Sci 2001, 10: 779–787. 10.1110/ps.37201PubMed CentralView ArticlePubMedGoogle Scholar
- Liu Q, Zhu YS, Wang BH, Li YX: A HMM-based method to predict the transmembrane regions of beta-barrel membrane proteins. Comput Biol Chem 2003, 27: 69–76. 10.1016/S0097-8485(02)00051-7View ArticlePubMedGoogle Scholar
- Martelli PL, Fariselli P, Krogh A, Casadio R: A sequence-profile-based HMM for predicting and discriminating beta barrel membrane proteins. Bioinformatics 2002, 18 Suppl 1: S46–53.View ArticlePubMedGoogle Scholar
- Vandeputte-Rutten L, Bos MP, Tommassen J, Gros P: Crystal structure of Neisserial surface protein A (NspA), a conserved outer membrane protein with vaccine potential. J Biol Chem 2003, 278: 24825–24830. 10.1074/jbc.M302803200View ArticlePubMedGoogle Scholar
- Hwang PM, Choy WY, Lo EI, Chen L, Forman-Kay JD, Raetz CR, Prive GG, Bishop RE, Kay LE: Solution structure and dynamics of the outer membrane enzyme PagP by NMR. Proc Natl Acad Sci U S A 2002, 99: 13560–13565. 10.1073/pnas.212344499PubMed CentralView ArticlePubMedGoogle Scholar
- Chimento DP, Mohanty AK, Kadner RJ, Wiener MC: Substrate-induced transmembrane signaling in the cobalamin transporter BtuB. Nat Struct Biol 2003, 10: 394–401. 10.1038/nsb914View ArticlePubMedGoogle Scholar
- Hayashi T, Makino K, Ohnishi M, Kurokawa K, Ishii K, Yokoyama K, Han CG, Ohtsubo E, Nakayama K, Murata T, Tanaka M, Tobe T, Iida T, Takami H, Honda T, Sasakawa C, Ogasawara N, Yasunaga T, Kuhara S, Shiba T, Hattori M, Shinagawa H: Complete genome sequence of enterohemorrhagic Escherichia coli O157:H7 and genomic comparison with a laboratory strain K-12. DNA Res 2001, 8: 11–22.View ArticlePubMedGoogle Scholar
- Pasquier C, Promponas VJ, Hamodrakas SJ: PRED-CLASS: cascading neural networks for generalized protein classification and genome-wide applications. Proteins 2001, 44: 361–369. 10.1002/prot.1101View ArticlePubMedGoogle Scholar
- Nielsen H, Engelbrecht J, Brunak S, von Heijne G: Identification of prokaryotic and eukaryotic signal peptides and prediction of their cleavage sites. Protein Eng 1997, 10: 1–6. 10.1093/protein/10.1.1View ArticlePubMedGoogle Scholar
- Nielsen H, Krogh A: Prediction of signal peptides and signal anchors by a hidden Markov model. Proc Int Conf Intell Syst Mol Biol 1998, 6: 122–130.PubMedGoogle Scholar
- Rabiner LR: A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 1989, 77: 257–286. 10.1109/5.18626View ArticleGoogle Scholar
- Eddy SR: Profile hidden Markov models. Bioinformatics 1998, 14: 755–763. 10.1093/bioinformatics/14.9.755View ArticlePubMedGoogle Scholar
- Krogh A, Larsson B, von Heijne G, Sonnhammer EL: Predicting transmembrane protein topology with a hidden Markov model: application to complete genomes. J Mol Biol 2001, 305: 567–580. 10.1006/jmbi.2000.4315View ArticlePubMedGoogle Scholar
- Tusnady GE, Simon I: Principles governing amino acid composition of integral membrane proteins: application to topology prediction. J Mol Biol 1998, 283: 489–506. 10.1006/jmbi.1998.2107View ArticlePubMedGoogle Scholar
- Krogh A, Mian IS, Haussler D: A hidden Markov model that finds genes in E. coli DNA. Nucleic Acids Res 1994, 22: 4768–4778.PubMed CentralView ArticlePubMedGoogle Scholar
- Durbin R, Eddy S, Krogh A, Mithison G: Biological sequence analysis, probabilistic models of proteins and nucleic acids. Cambridge University Press 1998.View ArticleGoogle Scholar
- Krogh Anders.: Hidden Markov models for labelled sequences. Proceedings of the12th IAPR International Conference on Pattern Recognition 1994, 140–144.Google Scholar
- Baum L: An inequality and associated maximization technique in statistical estimation for probalistic functions of Markov processes. Inequalities 1972, 3: 1–8.Google Scholar
- Dempster AP, Laird NM, Rubin DB: Maximum likelihood from incomplete data via the EM algorithm. J Royal Stat Soc B 1977, 39: 1–38.Google Scholar
- Krogh A: Two methods for improving performance of an HMM and their application for gene finding. Proc Int Conf Intell Syst Mol Biol 1997, 5: 179–186.PubMedGoogle Scholar
- Krogh A, Riis SK: Hidden neural networks. Neural Comput 1999, 11: 541–563. 10.1162/089976699300016764View ArticlePubMedGoogle Scholar
- Schwartz R, Chow YL: The N-Best Algorithm: An Efficient and Exact Procedure for Finding the N Most Likely Sentence Hypotheses. Proc IEEE Int Conf Acoust, Speech, Sig Proc 1990, 1: 81–84.Google Scholar
- Lo Conte L, Brenner SE, Hubbard TJ, Chothia C, Murzin AG: SCOP database in 2002: refinements accommodate structural genomics. Nucleic Acids Res 2002, 30: 264–267. 10.1093/nar/30.1.264PubMed CentralView ArticlePubMedGoogle Scholar
- Berman HM, Battistuz T, Bhat TN, Bluhm WF, Bourne PE, Burkhardt K, Feng Z, Gilliland GL, Iype L, Jain S, Fagan P, Marvin J, Padilla D, Ravichandran V, Schneider B, Thanki N, Weissig H, Westbrook JD, Zardecki C: The Protein Data Bank. Acta Crystallogr D Biol Crystallogr 2002, 58: 899–907. 10.1107/S0907444902003451View ArticlePubMedGoogle Scholar
- Altschul SF, Madden TL, Schaffer AA, Zhang J, Zhang Z, Miller W, Lipman DJ: Gapped BLAST and PSI-BLAST: a new generation of protein database search programs. Nucleic Acids Res 1997, 25: 3389–3402. 10.1093/nar/25.17.3389PubMed CentralView ArticlePubMedGoogle Scholar
- Hobohm U, Scharf M, Schneider R, Sander C: Selection of representative protein data sets. Protein Sci 1992, 1: 409–417.PubMed CentralView ArticlePubMedGoogle Scholar
- Koronakis V, Sharff A, Koronakis E, Luisi B, Hughes C: Crystal structure of the bacterial membrane protein TolC central to multidrug efflux and protein export. Nature 2000, 405: 914–919. 10.1038/35016007View ArticlePubMedGoogle Scholar
- Gouaux E: alpha-Hemolysin from Staphylococcus aureus: an archetype of beta-barrel, channel-forming toxins. J Struct Biol 1998, 121: 110–122. 10.1006/jsbi.1998.3959View ArticlePubMedGoogle Scholar
- Gardy JL, Spencer C, Wang K, Ester M, Tusnady GE, Simon I, Hua S, deFays K, Lambert C, Nakai K, Brinkman FS: PSORT-B: Improving protein subcellular localization prediction for Gram-negative bacteria. Nucleic Acids Res 2003, 31: 3613–3617. 10.1093/nar/gkg602PubMed CentralView ArticlePubMedGoogle Scholar
- Noguchi T, Akiyama Y: PDB-REPRDB: a database of representative protein chains from the Protein Data Bank (PDB) in 2003. Nucleic Acids Res 2003, 31: 492–493. 10.1093/nar/gkg022PubMed CentralView ArticlePubMedGoogle Scholar
- Baldi P, Brunak S, Chauvin Y, Andersen CA, Nielsen H: Assessing the accuracy of prediction algorithms for classification: an overview. Bioinformatics 2000, 16: 412–424. 10.1093/bioinformatics/16.5.412View ArticlePubMedGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article: verbatim copying and redistribution of this article are permitted in all media for any purpose, provided this notice is preserved along with the article's original URL.