Use of  tetrapeptide signals for protein secondary-structure prediction

Abstract

This paper develops a novel sequence-based method, tetra-peptide-based increment of diversity with quadratic discriminant analysis (TPIDQD for short), for protein secondary-structure prediction. The proposed TPIDQD method is based on tetra-peptide signals and is used to predict the structure of the central residue of a sequence fragment. The three-state overall per-residue accuracy (Q ₃) is about 80% in the threefold cross-validated test for 21-residue fragments in the CB513 dataset. The accuracy can be further improved by  taking long-range sequence information (fragments of more than 21 residues) into account in prediction. The results show the tetra-peptide signals can indeed reflect some relationship between an amino acid’s sequence and its secondary structure, indicating the importance of  tetra-peptide signals as the protein folding code in the protein structure prediction.

Appendices

Appendix 1

Table 4

Table 4 Probability parameters in defining tetra-peptide structural words in CB513 dataset

Appendix 2: the deduction of quadratic discriminant analysis in two-group case by using Bayesian theorem

For a sequence X to be classified between group ω ₁ and group ω ₁, assuming ω ₁ is positive set and ω ₂ is negative set, the discriminant function is defined by

$$ \xi = \ln p(\omega _{1} |x) - \ln p(\omega _{2} |x). $$

(10)

According to Bayes’ theorem,

$$ p(\omega _{l} |x) = p(\omega _{l} )p(x|\omega _{l} )/p(x)\quad (l = 1,2) $$

(11)

where p(ω _l ) is the probability a priori of set l (l = 1, 2), inserting Eq. 11 into Eq. 10, we obtain

$$ \xi = \ln \frac{{p(\omega _{1} )}} {{p(\omega _{2} )}} + \ln \frac{{p(x|\omega _{1} )}} {{p(x|\omega _{2} )}}. $$

(12)

Assume normal distribution of feature variables (M-dimensional vector) in two sets

$$ p(x|\omega _{l} ) = \frac{1} {{Z_{l} }}\exp \left( - \frac{1} {2}(x - \mu _{l} )^{T} \sum ^{{ - 1}}_{l} (x - \mu _{l} )\right) $$

(13)

$$ Z_{l} = (2\pi )^{{M/2}} {\left| {\Sigma _{l} } \right|}^{{1/2}} \quad (l = 1,2) $$

where μ _l (M-dimensional vector) and $\Sigma _{l}$ (M × M matrix) are the mean and covariant of feature variables over positive and negative sets respectively. Inserting Eq. 13 into Eq. 12, we obtain

$$ \xi = \ln \frac{{p(\omega _{1} )}} {{p(\omega _{2} )}} - \frac{1} {2}((x - \mu _{1} )^{T} \sum ^{{ - 1}}_{1} (x - \mu _{1} ) - (x - \mu _{2} )^{T} \sum ^{{ - 1}}_{2} (x - \mu _{2} )) - \frac{1} {2}\ln \frac{{{\left| {\Sigma _{1} } \right|}}} {{{\left| {\Sigma _{2} } \right|}}} $$

(14)

$$ {\text{Set}}\quad \delta _{l} = (x - \mu _{l} )\;\sum ^{{ - 1}}_{l} \;(x - \mu _{l} )\quad (l = 1,\;2) $$

(15)

$$ {\text{So}}\quad \xi _{{ij}} = \ln \frac{{p_{i} }} {{p_{j} }} - \frac{{\delta _{i} - \delta _{j} }} {2} - \frac{1} {2}\ln \frac{{{\left| {\Sigma _{i} } \right|}}} {{{\left| {\Sigma _{j} } \right|}}}. $$

(16)

This result can easily be generalized to more than two groups as shown in text.