Chaos game representation of functional protein sequences, and simulation and multifractal analysis of induced measures

Yu Jun-Wu^{a}, Vo Anh^{b}, Xiao Qian-Jun^{c}, Shi Long^{c}, Yu Zu-Guo^{d}

^{a} Department of Mathematics and Computational Science, Hunan
University of Science and Technology, Xiangtan 411201, China; ^{b} School of Mathematical Sciences, Queensland University of
Technology, GPO Box 2434, Brisbane, Q 4001, Australia; ^{c} School of Mathematics and Computational Science, Xiangtan
University, Xiangtan 411105, China; ^{d} School of Mathematics and Computational Science, Xiangtan
University, Xiangtan 411105, China;School of Mathematical Sciences, Queensland University of
Technology, GPO Box 2434, Brisbane, Q 4001, Australia

Abstract Investigating the biological function of proteins is a key
aspect of protein studies. Bioinformatic methods become important
for studying the biological function of proteins. In this paper, we
first give the chaos game representation (CGR) of randomly-linked
functional protein sequences, then propose the use of the recurrent
iterated function systems (RIFS) in fractal theory to simulate the
measure based on their chaos game representations. This method helps
to extract some features of functional protein sequences, and
furthermore the biological functions of these proteins. Then
multifractal analysis of the measures based on the CGRs of
randomly-linked functional protein sequences are performed. We find
that the CGRs have clear fractal patterns. The numerical results
show that the RIFS can simulate the measure based on the CGR very
well. The relative standard error and the estimated probability
matrix in the RIFS do not depend on the order to link the functional
protein sequences. The estimated probability matrices in the RIFS
with different biological functions are evidently different. Hence
the estimated probability matrices in the RIFS can be used to
characterise the difference among linked functional protein
sequences with different biological functions. From the values of
the D_q curves, one sees that these
functional protein sequences are not completely random.
The D_q of all linked functional proteins studied are
multifractal-like and sufficiently smooth for the C_q (analogous
to specific heat) curves to be meaningful. Furthermore, the D_q
curves of the measure \mu based on their CGRs for different orders
to link the functional protein sequences are almost identical if
q\geq 0. Finally, the C_q curves of all linked functional
proteins resemble a classical phase transition at a critical point.

Fund:Project partially supported by the
National Natural Science Foundation of China (Grant No.~30570426),
the Chinese Program for New Century Excellent Talents in University
(Grant No.~NCET-08-06867), Fok Ying Tung Education Foundation (Grant
No.~101004), and

Yu Zu-Guo,Xiao Qian-Jun,Shi Long et al . Chaos game representation of functional protein sequences, and simulation and multifractal analysis of induced measures[J]. Chin. Phys. B, 2010, 19(6):068701.

Venter J C, Adams M D, Myers E W, et al . 2001 Science 291 1304

[2]

Pandey A and Mann M 2000 Nature 405 837

[3]

Jeffrey H J 1990 Nucleic Acids Research 18 2163

[4]

Goldman N 1993 Nucleic Acids Research 21 2487

[5]

Deschavanne P J, Giron A, Vilain J, Fagot G and Fertil B 1999 Mol. Biol. Evol. 16 1391

[6]

Almeida J S, Carrico J A, Maretzek A, Noble P A and Fletcher M 2001 Bioinformatics 17 429

[7]

Joseph J and Sasikumar R 2006 BMC Bioinformatics 7 243(1-10)

[8]

Gao J and Xu Z Y 2009 Chin. Phys. B 18 370

[9]

Gao J, Jiang L L and Xu Z Y 2009 Chin. Phys. B 18 4571

[10]

Fiser A, Tusnady G E and Simon I 1994 J. Mol. Graphics 12 302

[11]

Basu S, Pan A, Dutta C and Das J 1998 J. Mol. Graphics and Modelling 15 279

[12]

Yu Z G, Anh V V and Lau K S 2004 J. Theor. Biol. 226 341

[13]

Yu Z G, Anh V V and Lau K S 2004 Physica A 337 171

[14]

Dill K A 1985 Biochemistry 24 1501

[15]

Wang J and Wang W 2000 Phys. Rev. E 61 6981

[16]

Brown T A 1998 Genetics 3rd ed. (London: Chapman &Hall)

[17]

Huang Y Z and Xiao Y 2003 Chaos, Solitons and Fractals 17 895

[18]

Huang Y Z, Li M F and Xiao Y 2007 Chaos, Solitons and Fractals 34 782

[19]

Feng J, Liu J H and Zhang H G 2008 Acta Phys. Sin. 57 6868 (in Chinese)

[20]

Chen Y P, Fu P P, Shi M H, Wu J F and Zhang C B 2009 Acta Phys. Sin. 58 7050 (in Chinese)

[21]

Yu Z G and Anh V V 2001 Chaos, Solitons and Fractals 12(10) 1827

[22]

Yu Z G and Wang B 2001 Chaos, Solitons and Fractals 12 519

[23]

Yu Z G, Anh V V, Gong Z M and Long S C 2002 Chin. Phys. 11 1313

[24]

Barnsley M F and Demko S 1985 Proc. R. Soc. London Ser. A 399 243

[25]

Falconer K 1997 Techniques in Fractal Geometry (London: John Wiley &Sons)

[26]

Vrscay E R 1991 Fractal Geometry and Analysis ed. Belair J and Dubuc S (Dordrecht: Kluwer) pp.~405--468

[27]

Anh V V, Lau K S and Yu Z G 2002 Phys. Rev. E 66 031910

[28]

Yu Z G, Anh V V and Lau K S 2001 Phys. Rev. E 64 031903

[29]

Yu Z G, Anh V V and Lau K S 2003 Int. J. Mod. Phys. B 17 4367

[30]

Yu Z G, Anh V V and Lau K S 2003 J. Xiangtan Univ. (Natural Science Edition) 25(3) 131

[31]

Wanliss J A, Anh V V, Yu Z G and Watson S 2005 J. Geophys. Res. 110 A 08214

[32]

Anh V V, Yu Z G, Wanliss J A and Watson S M 2005 Nonlin. Processes Geophys. 12 799

[33]

Yu Z G, Anh V V, Wanliss J A and Watson S M 2007 Chaos, Solitons and Fractals 31 736

[34]

Hentschel H G E and Procaccia I 1983 Physica D 8 435

[35]

Gutierrez J M, Iglesias A and Rodriguez M A 1998 Chaos and Noise in Biology and Medicine ed. Barbi M and Chillemi S (Singapore: World Scientific) pp.~315--319

[36]

Gutierrez J M, Rodriguez M A and Abramson G 2001 Physica A 300 271

[37]

Yu Z G, Anh V V, Lau K S and Zhou L Q 2006 Phys. Rev. E 63 031920

[38]

Yang J Y, Yu Z G and Anh V V 2009 Chaos, Solitons and Fractals 40 607

[39]

Barnley M F, Elton J H and Hardin D P 1989 Constr. Approx. B 5 3

[40]

Halsy T, Jensen M, Kadanoff L, Procaccia I and Schraiman B 1986 Phys. Rev. A 33 1141

[41]

Tel T, Fulop A and Vicsek T 1989 Physica A 159 155