Secondary Structure Prediction Based on
Statistical Mechanics
Yukio Kobayashi 1
koba@t.soka.ac.jp

12Nobuhiko Sait^o 2nsaito@mn.waseda.ac.jp

Department of Information Systems Science, Faculty of Engineering, Soka University
1-236 Tangi-cho, Hachioji-shi, Tokyo 192-8577, Japan
Department of Applied Physics, Waseda University, Shinjuku, Tokyo 169-8555, Japan

1 Introduction
Recent genome science has been developing from two sides: one is biological evolution due to mutationof DNA sequence and the other is biological function of protein molecules formed according to aminoacid sequences. We have devoted ourselves to elucidate the mechanism of protein folding, and todevelop the method for protein structure prediction [1, 2]. Our method has successfully yielded thenative-like tertiary structures of various proteins, by using secondary structure information. Secondarystructure prediction with a high precision, however, is essential to proteins of unknown structures. Ourmethod for three-state ( -helix, -strand and coil) prediction have reached the prediction accuracy of
only 68%, which accuracy may not be sucient for packing secondary structures into correct tertiarystructure. Some methods [3] for secondary structure prediction exceed a little the accuracy achievedby our method. Contrary to the other methods, we can analyze the reasons for the poor accuracy andthus are attempting to improve the present accuracy. In this article, we describe the way to improve
the prediction accuracy presented in Genome Informatics Workshop IV (GIW '93) [4].

2 Formulation
We formulated the statistical mechanical method for secondary structure prediction as follows: (1)The ith and the (i + k)th residues can interact with each other if and only if all of the ith to the(i + k)th residues exist in the same state ( -helix or -strand). If even one residue in between is in adi erent state, the ith and the (i + k)th residues do not interact with each other. (2) A residue in coildoes not interact with any other residues. (3) At least one residue in coil exists between neighboring
secondary structures. (4) The distance of interaction k is taken up to 4 for -helix and up to 2 for-starnd. This is justi ed by considering the hydrogen bonds between the ith and the (i +4)th residues
in -helix and those between the ith and the (i + 2)th residues in -strand. In GIW'93 k was taken
up to 4 also for -strand.
We can obtain the probabilities P ( )i , P ( )i and P (c)i for the ith residue by calculating thepartition function with recurrence relations [5]. Here, we need the statistical weights of 1620(=20 + 20 2 20 2 4) residue pairs for -helix and, in addition, 820(= 20 + 20 2 20 2 2) residue pairs for-starnd. In contrast to GIW'93 the weights for k = 3 and 4 in -strands were set at 1. To determine
these weights, we optimized the objective function.

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Table 1: Result of Prediction on 13 Proteins for Accuracy Estimation.
PDB-ID No. of Residues Accuracy (Previous) [4] Accuracy (Present)
1ACX
108
58.33362.037
1CTF
68
58.82452.9411LH1
15385.62184.967.
1UBQ
76
69.73777.632
2ALP
198
60.60651.0102CDV
10773.83277.5702CI2
6586.15458.462
2WRP
104
53.84656.731
4RHV
255
62.23865.8823CLN
143
81.88462.9374FXN
138
61.31489.8555LYZ
12956.58964.341
7RSA
124
74.19472.581
Total
1668
67.68667.626

3 Results
We increased the number of proteins for optimization from 65 in [4] to 80 and additionally corrected theupper limit of the distance of the interactions from 4 to 2 between residues in -strand as described inSection 2. We optimized the weights of 2440 (= 1620 + 820) pairs by referring to 80 proteins which do
not have sequence homology among them, and then estimated the prediction accuracy for 13 proteins
not included in, and not homologous with, the above 80 proteins. The accuracies were 67.6% for threestates, 65.4% for -helix and 45.9% for -starnd.

4 Discussion
Now, we consider the reasons why the present accuracies are poor: (1) 265 pairs among 2440 ones
are not found in the secondary structures of the 80 proteins for optimization. The weights of themissing pairs cannot be optimized properly. (2) The weights of 2175(= 2440 0 265) pairs have notyet optimized suciently. The accuracies of prediction for the 80 proteins for optimization are only
78.3% in the present case. (3) An important problem is that the secondary structures, especially thosearound the edges of the structures are more or less deformed at the nal stage of folding. Consequently,
the secondary structure prediction cannot yield the correct results. In this connection we refer theproposal of Rost, Sander, and Schneider [6]. They showed that considerable variation in the positionand length of secondary structure segments can be accommodated within the same three-dimensionalstructure. Thus, the goal of the precision accuracy can be reduced to some extent, and a new measureof segment overlap is introduced to compromise between permissiveness and precision.
The total number of the residues predicted correctly in the 80 referring proteins is increasing
through the optimization, especially from 1527 residues to 1709 ones in -strands. Thus we have ahope to improve the prediction accuracy by the present method.

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Acknowledgements
This work has been supported by the Grants-in-Aid for Scienti c Research No.10680644 from theMinistry of Education, Science, Sports and Culture of Japan.

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[6] Rost, B., Sander, C., and Schneider, R., Rede ning the goals of protein secondary structure
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