A DNA Coding Scheme for Searching Stable Solutions

Transcription

1 A DNA odng Scheme for Searchng Stable Solutons Intaek Km, HeSong Lan, and Hwan Il Kang 2 Department of ommuncaton Eng., Myongj Unversty, , Yongn, South Korea kt@mju.ac.kr, hslan@hotmal.net 2 Department of Informaton Eng., Myongj Unversty, , Yongn, South Korea hwan@mju.ac.kr Abstract. Ths paper presents a novel method for searchng stable solutons usng a DNA codng scheme. Often there s more than one soluton that satsfes the system requrements. These solutons can be vewed as extremes n the multmodal functon. All extremes are not the same n that some of them are senstve to nose or perturbaton. Ths paper addresses the method that selects a soluton that meets the system requrements n terms of output performance and s tolerant to the perspectve nose or perturbaton. A new method called a gradent DNA odng s proposed to acheve such objectves. A numercal example s presented and comparng DNA codng wth genetc algorthm s also gven. Keywords: DNA codng, genetc algorthms, gradent DNA codng. Introducton The genetc algorthm (A) has been wdely used n many optmzaton problems. The A offers an effcent way to search a global soluton n the multmodal functon [][2]. The multmodal functon may have several solutons, but some of them are very senstve to small perturbatons of ther parameter values. They may be not good solutons n certan stuatons. In many optmzaton tasks, there s a need to fnd solutons whose performance wll not change much due to small varaton of the parameter values. In ths paper, we defne a stable soluton as one whose varaton results n a small amount of change n output performance that satsfes the system requrement. We propose a new codng method for searchng stable solutons. It s based on the bologcal DNA and a mechansm of artfcal DNA [4][5]. A gradent nformaton s utlzed to fnd solutons and t s named a gradent DNA codng method. In addton, the comparson between methods usng the smple DNA codng and the gradent DNA codng s presented. In the next secton, the DNA codng method s descrbed. The proposed algorthm s gven n secton 3 and t s followed by smulaton to show the effectveness of the proposed method. The fnal secton s followed for the concluson and future work. V.S. Sunderam et al. (Eds.): IS 2005, LNS 354, pp , Sprnger-Verlag Berln Hedelberg 2005

2 594 I. Km, H. Lan, and H.I. Kang 2 DNA odng Method The bologcal DNA conssts of nucleotdes whch have four bases, Adenne(A), uanne(), ytosne() and Thymne(T)[4][5][6]. A messenger RNA (mrna) s frst syntheszed from the DNA. In the synthess of RNA, each base s translated nto the complementary base and the unused parts are cut out. Ths operaton s a splcng. After ths splcng the mrna s completed. Three successve bases called codons are allocated sequentally n the mrna. These codons are the codes for amno acds. 64 knds of codons correspond to 20 knds of amno acds as shown n Table. The detals of translaton nto amno acd from codons are omtted here. Ths allocaton of amno acd makes protens, and protens make up cells. A Fgure shows an example of the DNA chromosome and ts translaton mechansm. A gene begns wth the start codon AT, and closes wth end codons TA, TAA or TA [6]. The Fgure ndcates that ene conssts of eght condons:, T,, T and they are translated nto amno acds: Arg, Arg,, Ser, respectvely. Each amno acd has a number from 0 to 9 as shown n Table 2. The sum of the acds value represents gene s value and t s plugged n as a parameter value. For example, ene2 conssts of acds: Arg, ly, Phe, Leu, Ala, Ser and ly and ts value becomes by addng the value of each acds. Table. RNA(DNA) odon and amno acd T A T T TTT TT TAT TT Phe Tyr Ser ys A TT T TA T TTA TA TAA TA Stop A Stop TT T TA T Trp TT T AT T T Leu Hs T A Pro Arg TA A AA A A ln T T A ATT AT AAT AT T Asn Ser AT Ile A AA A Thr ATA AA AAA AA A Lys Arg AT Met A AA A TT T AT T T Asp T A T Val Ala ly TA A AA A A lu T A

3 A DNA odng Scheme for Searchng Stable Solutons 595 Table 2. The value for each amno acd Phe 0.25 Pro 0.50 Hs 0.75 lu.00 Leu.25 Thr.50 ln.75 ys 2.00 Ile 3.25 Ala 2.50 Asn 2.75 Trp 3.00 Met 3.25 Tyr 3.50 Lys 3.75 Arg 4.00 Ser 4.25 Val 4.50 Asp 4.75 ly 5.00 Fg.. The example of genes overlappng and chromosome translaton mechansm 3 Algorthm In ths paper, the proposed algorthm can be summarzed n 7 steps as follows: Step : Intalzaton Determne the populaton sze, the chromosome length, and probabltes of crossover and mutaton. The smple A (genetc algorthm) s ntalzed wth the bnary number (0, ), but the DNA codng s ntalzed wth four bases: A(adenne), T(thymne), (guanne) and (cytosne). Step 2: Addton of the nose The nose s generated and added to parameters n the system. If a gven populaton sze s n, the parameter can be expressed as X = ( x, x2, Lxn) and the nose s gven by = ( δ, δ 2, Lδ n ) where δ k has a random varable from the nterval [0, 0.02]. Ether X + or X s consdered as a nose-added parameter. Ths nose vector wll be used n next step. Step 3: alculaton of the average value of gradents In ths procedure, the gradents S( x, x + δ ) and S( x, x δ ) wll be obtaned, so s the average value of absolute gradent, that s, ( S( x, x + δ ) + S( x, x δ ) ) / 2. Ths value wll be used n calculatng the ftness functon. Step 4: alculaton of the ftness The above average value wll be taken nto account n ths step. A ene wth a larger gradent value has a smaller ftness value so that t can be degenerated n the next

4 596 I. Km, H. Lan, and H.I. Kang generaton. Ftness values also nclude the object functon s value as well as one obtaned from the gradent nformaton. The optmal ftness functon requres two condtons: the hgh object functon value and the low average absolute gradent value. Step 5: Selecton The Roulette Wheel selecton method s used. It s better than Tournament selecton method to avod the object functon convergng nto local optmums. Step 6: rossover and mutaton The Eltst strategy s adopted as a crossover operator. It saves the best gene n every chromosome so that the best soluton wll never degenerate. The mutaton s performed randomly by the gven mutaton probablty. Step 7: go to step 2 untl the requred condtons are satsfed. 4 Smulaton In order to demonstrate the capablty of the proposed algorthm, we appled the algorthm to a -D functon that s the multmodal functon. In smulaton, the parameters were kept constant wth the mutaton probablty p m = 0. 02, the crossover probablty p c = 0. 6, the populaton sze N=60, the maxmum number of generaton 50. we performed 50 smulatons for each experment wth randomly ntalzng the populaton. ) Functon y : onsder a functon y [Fg. 2(a)], whch has fve unequal peaks n t he range 0 x and s a varant of the functon used n [3]. It s defned as Y e e x ln 2( ) 0.8 ( x) = x ln 2( ) 0.8 sn(5πx) sn (5πx) 0.4 < x 0.6 otherwse. As shown n Fg. 2(a) the global optmum s located at x = 0. wth the functon value. 0. There are four sharp peaks. The thrd peak s broad compared to others and s located at x = wth functon value Fg.2 (b), (c) shows a typcal dstrbuton of the ndvduals n the after 50 generatons for the smple DNA codng method and the gradent DNA codng method. Fg.2 (d) shows a convergence process of the mean value of parameter x n the populaton wth trals. The smple DNA codng method converged at x = 0., the center of the hghest peak. The gradent DNA codng method converged to the stable peak ( x = ) zone. Indeed, we can observe from Fg.2 (d) that t approached the broad peak.

5 A DNA odng Scheme for Searchng Stable Solutons 597 (a) (b) (c) (d) Fg. 2. (a) The orgnal functon y (b) A typcal dstrbuton of the ndvduals n functon y after 50 generaton for the smple DNA codng method (c) A typcal dstrbuton of the ndvduals n functon y after 50 generaton for the gradent DNA codng method (d) The varaton of mean (over the populaton) value of x wth functon evaluatons 5 onclusons Ths paper proposed a gradent DNA codng method, whch extends the applcaton of A s to domans that requre detecton of stable solutons. The gradent DNA codng method was found that ths approach can be effectve when we want to detect more than one stable solutons on dfferent peaks. The future work wll focus on analyzng the behavor of gradent DNA codng method on more complcated problems where many peaks nteract, evaluatng the gradent DNA codng method on real-world problems. Acknowledgment Ths work was supported by grant No. (R ) from the Basc Research Program of the Korea Scence & Engneerng Foundaton.

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