Analysis of Protein Three-Dimension Structure Using Amino Acids Depths Shiyi Shen, 1 Gang Hu, 1,3 and Jack A. Tuszynski 2 The issue of amino acid depth in proteins gives important insights to our understanding of proteinÕs three-dimensional structure. There has already been much research done in mathematical and statistical sciences regarding the general definitions, properties and algorithms describing the particle depth of spatially extended systems. We constructed a method of calculating the amino acids depths and applied it to a set of 527 protein structures. We propose the introduction of amino acid depth tendency factors for three-dimensional structures of proteins. The depth tendency factors relate not only to the hydrophobicity indices but also to the electrostatic charge. We found a relationship between the protein size and the number of residues using the distance between the deepest residue and surface residues. We made a prediction regarding the number of residues on the surface of a protein, the deepest amino acid, and the average depth, all of which are fitted well to a linear functional relationship with the length of the protein. Finally, we have predicted the depths of multiple peptides in proteinÕs three-dimension structure. KEY WORDS: Statistics depth; depth tendency factor; protein depth indexes; protein size. 1. INTRODUCTION The solvent-accessible surface area (ASA) (Lee and Richards, 1971) has been commonly used in the analysis of protein structure, stability and protein– protein interactions (Serrano et al., 1992; Jackson et al., 1993; Jones and Thornton, 1996). It has also proved to be very useful in the analysis of atoms located on the protein surface. ASA is defined as the surface area generated by rolling a probe over the proteinÕs surface. However, it cannot effectively measure the depths of atoms buried in the protein interior. Based on ASA, various calculations have been carried out that give the measurements of some properties of the proteinÕs interior (Pedersen et al., 1991; Chakravarty and Varadarajan, 1999; Pintar et al., 2003). These methods estimate the depths of residues and atoms in the proteinÕs inte- rior by calculating their distance to the proteinÕs surface. In this article, we introduce a novel statisti- cal method to measure amino acid depth. The sta- tistical depth does not depend on ASA but is related to a number of chemical and physical char- acteristics developed for proteins. The concept of an extended objectÕs depth origi- nated in statistics where it is still extensively used to- day. In mathematics and statistics, one can find a number of definitions, characteristics, and algorithms for the particle depth in a general spatially-extended system (Rousseeuw and Struyf, 1998; Liu et al., 1999; Zuo and Serfling, 2000). Statistical depth functions have become increasingly pursued as a useful tool in nonparametric inference for multivariate data. (Zuo and Serfling, 2000). Tukey proposed a ‘‘halfspace’’ depth to define the multivariate analogues of univari- 1 School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China. 2 Cross Cancer Institute, 11560 University Avenue, Edmonton, AB, Canada T6G 1Z2. 3 To whom correspondence should be addressed. e-mail: huggs@ nankai.edu.cn Abbreviations: ASA, solvent-accessible surface area; PDB, protein data bank. 183 1572-3887/07/0400-0183/0 Ó 2007 Springer Science+Business Media, LLC The Protein Journal, Vol. 26, No. 3, April 2007 (Ó 2007) DOI: 10.1007/s10930-006-9060-1