# Use of machine learning algorithms to classify binary protein sequences as highly-designable or poorly-designable

- Myron Peto
^{2}Email author, - Andrzej Kloczkowski
^{1, 2}, - Vasant Honavar
^{3}and - Robert L Jernigan
^{1, 2}

**9**:487

https://doi.org/10.1186/1471-2105-9-487

© Peto et al; licensee BioMed Central Ltd. 2008

**Received: **05 April 2008

**Accepted: **18 November 2008

**Published: **18 November 2008

## Abstract

### Background

By using a standard Support Vector Machine (SVM) with a Sequential Minimal Optimization (SMO) method of training, Naïve Bayes and other machine learning algorithms we are able to distinguish between two classes of protein sequences: those folding to highly-designable conformations, or those folding to poorly- or non-designable conformations.

### Results

First, we generate all possible compact lattice conformations for the specified shape (a hexagon or a triangle) on the 2D triangular lattice. Then we generate all possible binary hydrophobic/polar (H/P) sequences and by using a specified energy function, thread them through all of these compact conformations. If for a given sequence the lowest energy is obtained for a particular lattice conformation we assume that this sequence folds to that conformation. Highly-designable conformations have many H/P sequences folding to them, while poorly-designable conformations have few or no H/P sequences. We classify sequences as folding to either highly – or poorly-designable conformations. We have randomly selected subsets of the sequences belonging to highly-designable and poorly-designable conformations and used them to train several different standard machine learning algorithms.

### Conclusion

By using these machine learning algorithms with ten-fold cross-validation we are able to classify the two classes of sequences with high accuracy – in some cases exceeding 95%.

## Keywords

## Background

Elucidating the relationship between protein sequence and protein structure remains one of the most challenging unsolved problems in computational structural biology. One closely related specific problem is protein designability, that is, why real are proteins not random sequences of amino acids but rather exhibit regular patterns that encode protein structures within the limited number of folds. Reduced (coarse-grained) models of proteins enjoy considerable interest and applicability for studies in designability. In coarse-grained models of proteins a detailed atomistic description of the structure is replaced by a much simpler view where each amino acid is represented by a single point. Additionally, theoretical models of proteins frequently replace the 20-letter amino acid alphabet with a reduced alphabet, up to the limit of a much simpler binary hydrophobic/polar (H/P) representation and furthermore significantly restrict the conformational space by imposing lattices restrictions on the continuous space [1–23]. Through the use of complete enumerations of H/P sequences and compact lattice conformations it has been found that most protein sequences fold to a relatively small number of so called "highly-designable" conformations, while the remaining conformations have few, or no, sequences that fold to them [24–33]. In the present work we use a standard H/P alphabet and a 2D triangular lattice and apply machine learning algorithms to study protein designability for such a reduced model.

Much of the past work on protein designability has focused on searching for the most significant features of designable protein structures, for both lattice models and for real proteins, and relating them to energetic stability and evolution. Recently, it has been shown that proteins selected for thermal stability tend to be more highly designable, owing to their increased energetic stability [34–37]. There is contrary evidence suggesting that designable proteins are unfolded more easily, due to their greater flexibility [38]. Various studies have shown that designable conformations embedded on various lattices exhibit important traits of real proteins, such as symmetrical shapes and secondary structure elements [24–33]. In addition, recent studies suggest that designable lattice structures tend to have more peptide bonds between the protein core and its surface, which can increase protein flexibility [17, 38].

Those significant traits of designable conformations, found in previous works, suggested the use of machine learning algorithms to discriminate between sequences folding to highly- and poorly-designable structures. Symmetrical shapes, secondary structure elements, and extraordinary surface-core bonds can possibly appear as definitive patterns in the protein sequences; it has been our intention to exploit such features in this study to classify sequences folding to conformations of differing designability.

In past studies of protein designability amino acid sequences were threaded onto all possible compact conformations for a given shape, and for each case the total energy of the structure was computed based on a specified energy function. If, for a given amino acid sequence, there is a conformation having a total energy lower than all other conformations, it was assumed that the sequence would fold to that specific structure. If many different sequences fold to the same conformation it was assumed that such a structure has high *designablility*. There were also conformations with few or even no sequences folding to them, *i.e*. having poor designability. Additionally many sequences do not fold uniquely, having similar lowest energies for different structures. We may however expect that such a degeneracy effect would rapidly diminish if a simple 2-letter (H/P) amino acid alphabet were replaced with a more complex one. Previous studies that examined the property of protein designability were mostly focused on the conformations within regular lattice shapes in 2D and 3D, such as a 6 × 6 square or a 3 × 3 × 3 cube. Results of these studies imply the existence of only a few highly designable conformations among a much larger number of less or non-designable structures. The results obtained for lattice proteins also suggest that, as for real proteins, designable conformations tend to exhibit structural symmetries. These findings show that a simple lattice model can demonstrate important traits that are mirrored in real proteins.

^{19}(≅ 5.2 × 10

^{5}) and 2

^{21}(≅ 2.1 × 10

^{6}) different H/P sequences for each shape. (Our model has no sequence symmetry because of the difference between the C and the N terminals). Because of relatively small numbers of possible H/P sequences and the numbers of all possible compact (no voids allowed) self-avoiding walks unrelated by shape symmetries for the hexagon (20,843) and the triangle (22,104), we are able to enumerate them completely and perform complete designability computations. Similarly, as in previous studies, we find that certain distinct conformations have many sequences folding to those structures, while other have few or no sequences folding to them.

After finding highly- and poorly-designable structures we then compare the sequences that fold to these two classes of conformations and test whether we could classify them by using standard machine learning algorithms. We used the Waikato Environment for Knowledge Analysis (WEKA) software [39, 40] available at http://weka.sourceforge.net as a platform for our classification computations, testing several different algorithms such as Support Vector Machine [41], Naïve Bayes [42] and a Decision Tree [43]. We first trained those statistical learning algorithms on a randomly chosen subset of our data (training set) and then checked the prediction accuracy on a test set. We have performed ten-fold cross-validation experiments to eliminate possible biases. By using a Support Vector Machine with a Sequential Minimal Optimization method of training we are able to obtain highly accurate predictions, often with an accuracy exceeding 90%, depending upon how the binary sequence was represented to the learning algorithm. We are quite optimistic that our approach can also be successfully applied to real proteins to distinguish protein-like sequences folding to distinct native structures from random and non-protein-like sequences that carry no significant structural signal.

## Methods

The complete enumeration of all possible compact conformations for each shape was performed by using a backtracking algorithm generating walks on a tree that checks for all accessible nodes for the next step of the walk. If none of the nodes is available then the algorithm backtracks to the first node offering a different path. Each of nodes must be visited once and only once, with unoccupied voids and chain overlaps not allowed. For longer chains this algorithm suffers from significant attrition and is less efficient than the alternative attrition-free transfer matrix approach developed by us previously [13–15]. However for the relatively short chains containing 19 or 21 nodes studied here a backtracking algorithm is simpler to use. The energy functions that we use for calculating the total energy of each fold obtained by threading of a sequence through a conformation are based only on non-bonded nearest-neighbor contacts. Two neighbors can either be both hydrophobic (with interaction energy E_{HH}), one hydrophobic and one polar (E_{HP} = E_{PH}), or both polar (E_{PP}). We use a standard energy function, used in references [24, 38], that sets E_{HH} = -2.3, E_{HP} = E_{PH} = -1.0 and E_{PP} = 0 in dimensionless energy units. The energy function was derived from real-protein interaction data of amino acids, based on the frequency of non-bonded contacts in protein structures, summarized in the Miyazawa-Jernigan matrix of contact potentials. This function satisfies two significant physical requirements: (i) E_{HH} < E_{HP} < E_{PP} and (ii) 2E_{HP} > E_{PP} + E_{HH}. The first requirement minimizes the number hydrophobic residues on protein surface, and the second condition allows for the segregation of different amino acid types. This potential will preferentially yield overall a hydrophobic core and a polar exterior.

Because we were interested in a complete enumeration of both the sequence space and the conformational space of our model, we restricted ourselves to the HP binary alphabet. A complete enumeration of the sequence space using the full alphabet of 20 amino acids would not be computationally feasible, as the size of space grows as 20^{n}, where *n* is the length of the protein chain. We would need to sample of the sequence space that gives us less insight than the full enumeration. Previous studies have shown, however that the reduced 2-letter HP alphabet model reflects most of the aspects of real protein folding. However, it remains to be seen whether improvements in classification between sequences folding to poorly- and highly-designable conformations could be achieved by using an expanded amino acid alphabet.

In order to classify the sequences folding into highly- and poorly-designable structures we use the WEKA machine learning workbench [39, 40] and several classification algorithms, including Support Vector Machine (SVM), Decision Tree, and Naïve Bayes. As input to the statistical learning algorithms we use two different representations of the binary amino acid sequence. Because all sequences for a given shape have the same length (21 residues for the triangle and 19 for the hexagon) it is possible simply to use the binary sequence itself as input. The input vector is thus x= (*x*_{1}, *x*_{2},..., *x*_{
n
}) with elements *x*_{
i
}(1 ≤ *i* ≤ *n*) defined as members of the set *x* ∈ {0,1}, corresponding to either a hydrophobic or polar amino acid. In addition, we also tried using as input a percentage count of different tripeptides from the set {HHH, HHP, HPH, PHH, PPH, PHP, HPP, PPP}. The input vector is then x= (*x*_{1}, *x*_{2}, *x*_{3}, *x*_{4}, *x*_{5}, *x*_{6}, *x*_{7}, *x*_{8}) with *x*_{
i
}(1 ≤ *i* ≤ 8) corresponding to the percentage of each *i*^{th} tripeptide in the sequence. Encoding a sequence in this manner allows us to compare sequences of unequal lengths. The resulting classifiers classify a target sequence as either folding to a conformation of high designability or of low designability.

*False Positives (FP)*constitute the sequences that fold to conformations of low designability but are incorrectly labeled as folding to conformations of high designability,

*True Positives (TP)*are sequences that are correctly labeled as folding to conformations of high designability,

*False Negatives (FN)*are sequences that are incorrectly labeled as folding to conformations of low designability, and

*True Negatives (TN)*are sequences that are correctly labeled as folding to conformations of low designability. We can define sensitivity and specificity as statistical measures of the performance of the binary classification test, namely:

## Results

^{19}and 2

^{21}(524,288 and 2,097,152) for the binary H/P case; combined with the 20,843 and 22,104 conformations for each shape, respectively. We then count the number of different sequences folding to a given conformation with energy lower than all other conformations for a given shape and store the counts. These results are shown in Figure 2a for the hexagon, and Figure 2b for the triangle, where the logarithm of the number of conformations log

*N*

_{conf}having

*N*

_{s}sequences folding to them is plotted against

*N*

_{s}. These two graphs express qualitatively the same ideas reported in earlier studies [17, 24, 28, 29, 33, 38]. There are many conformations with relatively few (or no) sequences folding to them and a rather smaller number of conformations that have many sequences folding to these structures. The latter conformations are named designable conformations and the former are called poorly designable conformations. We used the top 10% and bottom 10% of conformations for the two respective groups.

_{S}). Similarly as observed in previous studies [24, 26, 27, 38] we find a marked tendency for the energy gap to increase for more designable conformations. This trend seems weaker for larger

*N*

_{s}, which may be a result of having too few conformations to obtain a reliable average. For the hexagonal shape there are fewer than 40 conformations with more than 38 sequences folding to them; whereas there are more than 20,000 conformations with fewer sequences folding to them.

In addition to the general results presented above, we apply machine learning algorithms to distinguish between sequences folding to highly designable and poorly designable conformations. In our first attempt we define two subsets from the set of all possible sequences: those folding to the bottom 10% of designable conformations and those folding to the top 10% of designable conformations. As there were 54 sequences folding to the most designable conformation for the triangular shape, this would mean, for example, that conformations having 49 sequences folding to them (i.e. within the 10% range from the most designable structure) would be also included in the "highly-designable" set of conformations; and sequences folding to those conformations would be classified as highly-designable sequences. The efficiency of the application of statistical machine learning methods such as SVM depends considerably on the representativeness of the learning sample set used for the training purposes that should include both positive data and the negative ones. We have used both types of these well balanced data for training the models. In order to have a balanced dataset since the number of sequences in both subsets differs greatly, and to reduce the computational cost, we utilize a random sample of sequences from each group. We could not compare sequences corresponding to different shapes, since the triangle has 21 residues while the hexagon has 19.

Accuracy of three different machine learning prediction algorithms – J48 Decision Tree, Naïve Bayes and SVM with SMO training – using binary H/P sequences.^{a}

J48 | Naïve Bayes | SMO | |
---|---|---|---|

a) Sequences folding to the top 10% and the bottom 10% of designable conformations for the hexagon | 96.8% correct | 95.8% correct | 98.3% correct |

AUC .97 | AUC 0.99 | AUC 0.98 | |

Sens: 1.0 | Sens: 1.0 | Sens: 0.997 | |

Spec: 0.94 | Spec: 0.92 | Spec: 0.97 | |

b) Sequences folding to the top 10% and the bottom 10% of designable conformations for the triangle | 92.7% correct | 82.4% correct | 95.0% correct |

AUC 0.93 | AUC 0.92 | AUC 0.95 | |

Sens: 0.93 | Sens: 0.76 | Sens: 0.92 | |

Spec: 0.92 | Spec: 0.86 | Spec: 0.97 |

We repeat the above analysis using a different representation of the binary sequences with the sequence being represented by the percent composition of the different tripeptides; for a binary alphabet, there are 8 triplets, HHH, HHP, HPH, PHH, HPP, PHP, PPH, and PPP. Using the frequency of occurrences of such short segments gives us the advantage of being able to compare sequences of varying lengths across different shapes, allowing us to examine whether the designability traits encoded within the binary sequences are a general feature independent of the specific protein shape.

Accuracy of three different machine learning prediction algorithms (J48 Decision Tree, Naïve Bayes and SVM with SMO training) using the frequencies of all possible short tripeptide binary segments.^{a}

J48 | Naïve Bayes | SMO | |
---|---|---|---|

a) Sequences folding to the top 10% and the bottom 10% of designable conformations for the hexagon | 89.7% correct | 78.8% correct | 91.0% correct |

AUC 0.95 | AUC 0.92 | AUC 0.91 | |

Sens: 0.91 | Sens: 0.85 | Sens: 0.84 | |

Spec: 0.90 | Spec: 0.77 | Spec: 0.91 | |

b) Sequences folding to the top 10% and the bottom 10% of designable conformations for the triangle | 67.8% correct | 56.7% correct | 57.8% correct |

AUC 0.69 | AUC 0.61 | AUC 0.58 | |

Sens: 0.68 | Sens: 0.58 | Sens: 0.64 | |

Spec: 0.68 | Spec: 0.57 | Spec: 0.57 |

From the J48 decision tree results we are able to identify the tripeptide sequences containing the most information. For the hexagon shape the two most defining tripeptides are HHH and PPP; for the triangle shape the two most defining tripeptides are PPH and HHH. This means that the percentage of HHH and PPH sequences often was used by the classifier for determining whether sequences were highly- or poorly-designable for conformations in the triangle shape, and likewise PPH and HHP for the hexagon shape. This could be related to the number of interior/exterior peptide bonds, since more interior/exterior bonds would lead to more boundaries between H and P in the triplets (P residues are more often found on the surface and H residues more often in the interior).

^{19}(or 2

^{21}) possible binary sequences and performed machine learning predictions for all these sets. Tables 3 and 4 show the results for these cases.

Accuracy of machine learning predictions classifying sequences folding to the most designable conformations among random binary sequences for a) hexagonal and b) triangular shapes.^{a}

J48 | Naïve Bayes | SMO | |
---|---|---|---|

a) Sequences folding to the top 10% of designable structures vs. random binary sequences of length 19 for the hexagon | 97.2% correct | 94.2% correct | 97.3% correct |

AUC 0.97 | AUC 0.98 | AUC 0.98 | |

Sens: 1.0 | Sens: 1.0 | Sens: 0.997 | |

Spec: 0.94 | Spec: 0.89 | Spec: 0.95 | |

b) Sequences folding to the top 10% of designable structures vs. random binary sequences of length 21 for the triangle | 90.3% correct | 84.4% correct | 95.2% correct |

AUC 0.91 | AUC 0.92 | AUC 0.95 | |

Sens: 0.93 | Sens: 0.92 | Sens: 0.97 | |

Spec: 0.90 | Spec: 0.82 | Spec: 0.94 |

Accuracy of machine learning predictions classifying sequences folding to the least designable conformations among random binary sequences for a) hexagonal and b) triangular shapes.^{a}

J48 | Naïve Bayes | SMO | |
---|---|---|---|

a) Sequences folding to the bottom 10% of designable structures vs. random binary sequences of length 19 for the hexagon | 57.5% correct | 55.6% correct | 57.9% correct |

AUC 0.58 | AUC 0.59 | AUC 0.58 | |

Sens: 0.62 | Sens: 0.55 | Sens: 0.61 | |

Spec: 0.56 | Spec: 0.55 | Spec: 0.57 | |

b) Sequences folding to the bottom 10% of designable structures vs. random binary sequences of length 21 for the triangle | 50.1% correct | 52.3% correct | 56.0% correct |

AUC 0.50 | AUC 0.53 | AUC 0.56 | |

Sens: 0.54 | Sens: 0.67 | Sens: 0.59 | |

Spec: 0.53 | Spec: 0.54 | Spec: 0.58 |

For each class there were approximately 300 sequences, chosen to allow a sufficient number to train the classifier but limited by the extent of computations. We test using a larger set of sequences, on the order of 1000, and observe qualitatively the same results as for the smaller set. (The random sequences are generated using standard C++ tools.) In all cases we are careful to ensure that we use two similar sized sets of sequences for our classification tests, as imbalances between the sizes of two classes can artificially enhance the performance of machine learning algorithms.

The general result is that we are quite successful in classifying sequences that fold to highly designable structures among random sequences but are far less successful in classifying sequences folding to poorly- and non-designable structures among randomly chosen sequences. This observation is true of all machine learning algorithms and for both shapes studied.

Finally, in order to further elucidate whether binary sequences carry the shape information in their designability patterns, we attempt to classify both sequences folding to highly designable and poorly designable conformations of the hexagonal shape and the triangular shape. We have also tried machine learning methods to distinguish sequences folding to highly designable conformations folding in the hexagonal shape from poorly-designable sequences folding in the triangular shape as well as highly-designable sequences folding in the triangular shape from poorly-designable sequences folding in the hexagonal shape. Again, because we were classifying binary sequences of unequal lengths, we use the vector of percentages of all tripeptides as the input to our classifiers.

Accuracy of machine learning predictions.^{a}

J48 | Naïve Bayes | SMO | |
---|---|---|---|

a) Sequences folding to the top 10% of designable structures vs. sequences folding to the bottom 10% of designable structures for both shapes | 69.5% correct | 65.0% correct | 65.6% correct |

AUC 0.73 | AUC 0.69 | AUC 0.67 | |

Sens: 0.67 | Sens: 0.66 | Sens: 0.71 | |

Spec: 0.71 | Spec: 0.65 | Spec: 0.64 | |

b) Sequences folding to the top 10% of designable structures of hexagonal shape vs. sequences folding to the bottom 10% of designable structures in the triangular shape | 98.1% correct | 84.9% correct | 87.0% correct |

AUC 0.99 | AUC 0.92 | AUC 0.87 | |

Sens: 0.98 | Sens: 0.82 | Sens: 0.84 | |

Spec: 0.98 | Spec: 0.90 | Spec: 0.92 | |

c) Sequences folding to the top 10% of designable structures of triangular shape vs. sequences folding to the bottom 10% of designable structures in the hexagonal shape | 98.0% correct | 65.8% correct | 64.3% correct |

AUC 0.99 | AUC 0.70 | AUC 0.63 | |

Sens: 0.98 | Sens: 0.64 | Sens: 0.75 | |

Spec: 0.98 | Spec: 0.72 | Spec: 0.66 |

## Discussion

The protein structural designability results obtained in the present paper for two regular shapes on the 2D triangular lattice are not qualitatively different from results obtained in numerous earlier studies [17, 24, 28, 29, 33, 38]. We found that designable conformations having many sequences folding to them are relatively rare among a large number of conformations that have few or no sequences folding to them with the lowest energy. We have also found that the average energy gap between the ground state and next lowest energy state increases with increasing designability of structures; similarly as observed earlier by [24, 26].

The most interesting results obtained in our present study relate to our ability to successfully classify sequences folding to highly- and poorly-designable conformations using several standard freely available machine learning algorithms. For both of the shapes studied (the hexagon and the triangle) we are able to classify successfully the sequences using their full binary representation, which we may ascribe to the fact that there are relatively few highly designable conformations, and sequences folding to them probably share similar patterns in the distribution of hydrophobic and polar residues along the protein sequence.

That there was a significant difference in the classification accuracy between the two shapes came as a surprise to us. The hexagonal shape, being more compact, resembles real protein structures more than the less compact triangular shape. There were a similar number of total conformations for each shape, even though the triangular shape had 21 vertices and the hexagonal shape had only 19. Perhaps the corners of the triangle placed restrictions on all of the conformations such that the differences between the poorly- and highly-designable conformations were less pronounced. This could lead to smaller differences between the sequences folding to each set and hence poorer classification accuracy.

Additionally, our further testing of sequences folding to the most designable structures among completely random sequences seems to suggest that the structural designability pattern is somehow encoded in the sequence. If the structural designability information is indeed encoded in the binary sequence we would expect to discern sequences folding to highly designable structures among random sequences much more effectively than sequences folding to poorly-designable structures. The results of our computations fully support these expectations. We are able to classify sequences folding to highly-designable structures among random sequences with an accuracy exceeding 90%; whereas for sequences folding to poorly- and non-designable structures our accuracy of prediction among random sequences was not much better than random. Our testing of sequences folding to designable conformations in different shapes suggests that the overall shape of the fold may also be encoded in the protein sequence.

The results presented here lend further support to the use of simple H/P lattice models developed for protein structural studies. Our success in classifying sequences folding to conformations in the triangular lattice, a lattice without the parity effects of the square or cubic lattice, offers evidence of the usefulness of such simple models. As mentioned earlier, an interesting next step would be to test our machine learning algorithms on sequences of real proteins which fold to higher or lower designable states. Recent work [35–37] finds that proteins of thermophilic organisms tend also to be more designable than proteins in mesothermic organisms. We are working on classifying these two sets of protein sequences using the same tools used in this study. It would be rather remarkable if a designability footprint exists for real protein sequences.

The real protein folding problem is, of course, significantly more complicated than folding on simple lattices with a reduced 2-letter HP alphabet. The present success in applying statistical machine learning algorithms to distinguish between highly-designable and poorly-designable sequences for lattice proteins suggest that similar approach can be applied to real proteins. Statistical machine learning algorithms are already extremely useful in bioinformatics for prediction of protein secondary structure from the amino acid sequence, prediction of protein classes, protein-protein, protein-RNA, protein-DNA, or protein-ligand binding sites, prediction of intrinsically disordered regions in proteins, prediction of phosphorylation and other post-translationally modified sites, and many other purposes. The main problem is a proper choice of training (positive and negative) sets for the learning process. It is a difficult endeavor, since sometimes a single mutation changes protein structure. We are currently working on this problem for real proteins and hope that our approach will help to a certain degree in protein folding studies.

## Declarations

### Acknowledgements

We gratefully acknowledge support from NIH Grants 1R01 GM-072014 and 1R01 GM081680, and NSF Grant CNS-05-515. We would like also to thank Cornelia Caragea, Carson Andorf, Yasser EL-Manzalawy, and Leelananda Sumudu for helping us with the paper.

## Authors’ Affiliations

## References

- Chan HS, Dill KA: The effects of internal constraints on the configurations of chain molecules.
*J Chem Phys*1990, 92: 3118–3135. 10.1063/1.458605View ArticleGoogle Scholar - Chan HS, Dill KA: Origins of structure in globular proteins.
*Proc Natl Acad Sci USA*1990, 87: 6388–6392. 10.1073/pnas.87.16.6388PubMed CentralView ArticlePubMedGoogle Scholar - Chan HS, Dill KA: Compact polymers.
*Macromolecules*2003, 22: 4559. 10.1021/ma00202a031View ArticleGoogle Scholar - Covell DG, Jernigan RL: Conformations of Folded Proteins in Restricted Spaces.
*Biochemistry*1990, 29: 3287–3294. 10.1021/bi00465a020View ArticlePubMedGoogle Scholar - Crippen GM: Enumeration of cubic lattice walks by contact class.
*J Chem Phys*2000, 112: 11065–11068. 10.1063/1.481746View ArticleGoogle Scholar - des Cloizeaux J, Jannink G:
*Polymers in solution*. Oxford, New York: Oxford University Press; 1989.Google Scholar - Guttmann AJ, Enting IG: Solvability of some statistical mechanical systems.
*Physical Review Letters*1996, 76: 344–347. 10.1103/PhysRevLett.76.344View ArticlePubMedGoogle Scholar - Jensen I: Enumeration of compact self-avoiding walks.
*Comput Phys Communications*2003, 142: 109–113. 10.1016/S0010-4655(01)00340-XView ArticleGoogle Scholar - Madras N, Slade G:
*The self-avoiding walk.*Boston: Birkhauser; 1993.Google Scholar - Shakhnovich E, Gutin A: Enumeration of all Compact Conformations of Copolymers with Random Sequnce of Links.
*J Chem Phys*1990, 93: 5967–5971. 10.1063/1.459480View ArticleGoogle Scholar - Shakhnovich EI: Modeling protein folding: The beauty and power of simplicity.
*Fold Design*1996, 1: R50-R54. 10.1016/S1359-0278(96)00027-2View ArticleGoogle Scholar - Kloczkowski A, Jernigan RL: Computer generation and enumeration of compact self-avoiding walks within simple geometries on lattices.
*Comput Theoret Polymer Sci*1997, 7: 163–173. 10.1016/S1089-3156(97)00022-6View ArticleGoogle Scholar - Kloczkowski A, Jernigan RL: Efficient method to count and generate compact protein lattice conformations.
*Macromolecules*1997, 30: 6691–6694. 10.1021/ma970662hView ArticleGoogle Scholar - Kloczkowski A, Jernigan RL: Transfer matrix method for enumeration and generation of compact self-avoiding walks. II. Cubic lattice.
*J Chem Phys*1998, 109: 5147–5159. 10.1063/1.477129View ArticleGoogle Scholar - Kloczkowski A, Jernigan RL: Transfer matrix method for enumeration and generation of compact self-avoiding walks. 1. Square lattices.
*J Chem Phys*1998, 109: 5134–5146. 10.1063/1.477128View ArticleGoogle Scholar - Schmalz TG, Hite GE, Klein DJ: Compact self-avoiding circuits on two dimensional lattices.
*J Phys A*1984, 17: 445–453. 10.1088/0305-4470/17/2/029View ArticleGoogle Scholar - Cejtin C, Edler J, Gottlieb A, Helling R, Li H: Fast Tree Search for Enumeration of a Lattice Model of Protein Folding.
*J Chem Phys*2002, 116: 352–359. 10.1063/1.1423324View ArticleGoogle Scholar - Mansfield ML: Unbiased sampling of lattice Hamiltonian path ensembles.
*J Chem Phys*2006, 125: 154103. 10.1063/1.2357935View ArticlePubMedGoogle Scholar - Peto M, Sen TZ, Jernigan RL, Kloczkowski A: Generation and enumeration of compact conformations on the 2D triangular and 3D fcc lattices.
*J Chem Phys*2007, 127: 10. 10.1063/1.2751169View ArticleGoogle Scholar - Shakhnovich EI, Gutin AM: Engineering of stable and fast folding sequences of model proteins.
*Proc Natl Acad Sci USA*1993, 90: 7195–7199. 10.1073/pnas.90.15.7195PubMed CentralView ArticlePubMedGoogle Scholar - Shakhnovich EI: Proteins with selected sequences fold into unique native conformation.
*Phys Rev Letts*1994, 72: 3907–3910. 10.1103/PhysRevLett.72.3907View ArticleGoogle Scholar - Gutin AM, Abkevich VI, Shakhnovich EI: Evolution-like selection of fast-folding model proteins.
*Proc Natl Acad Sci USA*1995, 92: 1281–1286. 10.1073/pnas.92.5.1282View ArticleGoogle Scholar - Yue K, Dill KA: Inverse protein folding problem: designing polymer sequences.
*Proc Natl Acad Sci USA*1992, 89: 4163–4167. 10.1073/pnas.89.9.4163PubMed CentralView ArticlePubMedGoogle Scholar - Li H, Helling R, Tang C, Wingreen N: Emergence of Preferred Structures in a Simple Model of Protein Folding.
*Science*1996, 273: 666–669. 10.1126/science.273.5275.666View ArticlePubMedGoogle Scholar - Li H, Tang C, Wingreen NS: Nature of driving force for protein folding: A result from analyzing the statistical potential.
*Phys Rev Letts*1997, 4: 765–768. 10.1103/PhysRevLett.79.765View ArticleGoogle Scholar - Li H, Tang C, Wingreen N: Designability of protein structures: a lattice-model study using the Miyazawa-Jernigan matrix.
*PROTEINS: Struct, Funct Genetics*2002, 49: 403–412. 10.1002/prot.10239View ArticleGoogle Scholar - Wingreen N, Li H, Tang C: Designability and thermal stability of protein structures.
*Polymer*2004, 45: 699–705. 10.1016/j.polymer.2003.10.062View ArticleGoogle Scholar - Shahrezaei V, Ejtehadi MR: Geometry selects highly designable structures.
*J Chem Phys*2000, 113: 6437–6442. 10.1063/1.1308514View ArticleGoogle Scholar - Shahrezaei V, Hamedani N, Ejtehadi MR: Protein ground state candidates in a simple model: An enumeration study.
*Phys Rev E*1999, 60: 4629–4636. 10.1103/PhysRevE.60.4629View ArticleGoogle Scholar - Ejtehadi MR, Hamedani N, Shahrezaei V: Geometrically reduced number of protein ground state candidates.
*Phys Rev Letts*1999, 82: 4723–4726. 10.1103/PhysRevLett.82.4723View ArticleGoogle Scholar - Ejtehadi MR, Hamedani N, Seyed-Allaei H,
*et al*.: Highly designable protein structures and inter-monomer interactions.*J Phys A Math General*1998, 31: 6141–6155. 10.1088/0305-4470/31/29/006View ArticleGoogle Scholar - Ejtehadi MR, Hamedani N, Seyed-Allaei H, et al.: Stability of preferable structures for a hydrophobic-polar model of protein folding. Phys Rev E 57(3):3298–3301. 10.1103/PhysRevE.57.3298Google Scholar
- Peto M, Kloczkowski A, Jernigan RL: Shape-dependent designability studies of lattice proteins.
*J Phys Condensed Matter*2007, 19: 11. 10.1088/0953-8984/19/28/285220View ArticleGoogle Scholar - Shakhnovich B, Deeds E, Delisi C, Shakhnovich EI: Protein structure and evolutionary history determine sequence space topology.
*Genome Res*2005, 15: 385–392. 10.1101/gr.3133605PubMed CentralView ArticlePubMedGoogle Scholar - England JL, Shakhnovich B, Shahknovich EI: Natural selection of more designable folds: A mechanism for thermophilic adaptation.
*Proc Natl Acad Sci USA*2003, 100: 8727–8731. 10.1073/pnas.1530713100PubMed CentralView ArticlePubMedGoogle Scholar - Berezovsky IN, Shahknovich EI: Physics and evolution of thermophilic adaptation.
*Proc Natl Acad Sci USA*2005, 102: 12742–12747. 10.1073/pnas.0503890102PubMed CentralView ArticlePubMedGoogle Scholar - Berezovsky IN, Zeldovich KB, Shahknovich EI: Positive and Negative Design in Stability and Thermal Adaptation of Natural Proteins.
*PLoS Comput Biol*2007, 3(3):e52. 10.1371/journal.pcbi.0030052PubMed CentralView ArticlePubMedGoogle Scholar - Dias CL, Grant M: Designable Structures Are Easy to Unfold.
*Phys Rev E Stat Nonlin Soft Matter Phys*2006, 74(4 Pt 1):042902.View ArticlePubMedGoogle Scholar - Weka 3 – Data Mining with Open Source Machine Learning Software, The University of Waikato, New Zealand[http://weka.sourceforge.net]
- Witten IH, Frank E:
*"Data Mining: Practical machine learning tools and techniques".*2nd edition. Morgan Kaufmann, San Francisco; 2005.Google Scholar - Vapnik VN Statistical Learning Theory Wiley Press, NY; 1998.Google Scholar
- Mitchell T Machine learning New York, USA: McGraw Hill, NY; 1997.Google Scholar
- Quinlan JR: The effect of noise on concept learning. In
*Machine learning: An artificial intelligence approach*.*Volume 2*. Edited by: Michalski RS, Carbonell JG, Mitchell TM. Morgan Kaufman, San Francisco; 1986.Google Scholar

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