A zero one programming model for RNA structures with arc length 4

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Philomena McGee
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1 Iraia Joural of Matheatical Cheistry, Vol. 3, No.2, Septeber 22, pp IJMC A zero oe prograig odel for RNA structures with arc legth 4 G. H. SHIRDEL AND N. KAHKESHANI Departet of Matheatics, Faculty of Basic Scieces, Uiversity of Qo, Qo, Ira Received August, 22) ABSTRACT I this paper, we cosider RNA structures with arc-legth 4. First, we represet these structures as atri odels ad zero-oe liearprograig probles. The, we obtai a optial solutio for this probleusig a iplicit eueratio ethod. The optial solutio correspods toa RNA structure with the aiu uber of hydroge bods. Keywords: RNA structure,zero-oe liear prograig proble, additive algorith.. INTRODUCTION A proble i atheatical biology is eueratio of RNA structures. The RNAhas a iportat role withi cells ad also, its fuctios deped o the structure of therna olecules. Hece, uderstadig of its helical cofiguratio is iportat. TheRNA olecule is a sequece of four ucleotides A, C, G ad U which plays a iportatrole i Biological reactios. These ucleotides are coected to each other via hydrogebods. The foratio of these bods stabilizes the olecule by lowerig its free eergy[2]. The RNA structures ca be displayed i various ways such as tree, liear ecodigof tree, coarse graied represetatio, hoeoorphically irreducible tree ad diagra[3, 4]. I this paper, we represet aother two odels for RNA structures. Accordigto [4], the diagra represetatio is defied as followig: Let G V, E ) be a directed graph such that G G V G,, ad E i, i j. [ ] G VG ad E G are called the sets of vertices ad arcs, respectively. Each directed graphca be displayed as a diagra i which the vertices,, are placed o a horizotallie ad the arcs i,, where i j, ca be displayed above the lie. Because of liearorderig of the vertices, the directio of the arcs is oitted. The vertices adarcs show the ucleotides ad

First, we represet these structures as atri odels ad zero-oe liearprograig probles. The, we obtai a optial solutio for this probleusig a iplicit eueratio ethod.

2 86 G. H. SHIRDEL AND N. KAHKESHANI hydroge bods, respectively. We attribute two paraeters to the diagras: the iiu arc-legth,, the iiu stac-legth,. I diagra represetatio, the legth of a arc i, is j i ad a stac of legth is a sequece of the parallel arcs lie i,, i, j ),, i ), j ))), see Figure. We deote the uber of RNA structures with 4 ad over [] by S ). 4, Figure. RNA Structure with 4, 2. The reaider of this paper is orgaized as follows. I sectios 2 ad 3, we represet RNA structures with arc-legth 4 ad stac-legth as atri odels ad zero-oe liear prograig probles, respectively. Also, we epress the results foreueratio of RNA structures. I sectio 4, we use the additive algorith for solvig liear prograig probles with biary variables oly []. The geeral otio of the additivealgorith is based o testig a few uber of possible solutios, 2 i which is theuber of variables), of a proble istead of all solutios. I other words, through thisethod, soe of the possible solutios of the proble are left ueaied. Also,the zero-oe liear prograig probles ca be solved usig each of the geeraliteger prograig techiques. Fially, coclusios ad future wors are discussed isectio MATRIX MODELS Each RNA structure over [], G, correspods to a atri by M G ) [ ] such that if i, EG if i, EG atri. We display this Theore. Suppose G is a RNA structure with arc-legth 4 over []. The we have A A2 M G ) ) A2 A22 where A 4, A 2 4 4, A 22 4 ad A2 is the upper triagular atri..

We deote the uber of RNA structures with 4 ad over [] by S ). 4, 2 3 4 5 6 7 8 9 2 Figure. RNA Structure with 4, 2. The reaider of this paper is orgaized as follows.

3 A zero oe prograig odel for RNA structures 87 Proof. For all i,, where i j, i, E G. Therefore, M G ) is a upper triagular atri. Sice G is a directed graph with arc legth 4, we have i, E G ad for all i,, where j i 3. Therefore, three diagoals above thepricipal diagoal are zero. The, the atri M G ) is of the for ). We ow write the upper triagular atri A 2 as follows: Theore 2. S 4, ) is equal to the uber of situatios i which the etries abovead o the pricipal diagoal i A2 ca be equal to such that for each i 4 ad 5 j, at ost oe of the etries be. i i,, i, j j,, j, i,, i, j,, j Proof. LetGis a RNA structure with arc-legth 4 over []. The degree of each verte G is at ost. Therefore, each row ad colu A 2 is or e, where 4. Suppose that there eist i 4 ad 5 j so that. The i, E G. Sice the degree of each verteg is at ost,we have, i), i, h) E G for each i ad h i, where h j. Siilarly, we have j, h),, for each h j ad j, where i. This copletes our arguet. 3. ZERO-ONE LINEAR PROGRAMMING PROBLEMS Here, we preset a zero-oe liear prograig proble for displayig of RNA structures with arc-legth 4 ad stac-legth.sice each row of the atri A 2 is or e, where 4, we have the followig costraits: E G

We ow write the upper triagular atri A 2 as follows: 5 6 2 Theore 2.

4 88 G. H. SHIRDEL AND N. KAHKESHANI ) ) 5) ) 5) 6) ) 6) 2) 6) 7) 3) 7) b a Siilarly, sice each colu of atri 2 A is or e, where 4, we have the followig costraits: We also ow that the degree of each verte is at ost. Therefore, we have thefollowig costraits: ) ) 2 ) 5) ) 2 ) 2) 6) 2) 2 2) 3) 7) 3) 2 3) 8) d c

5 A zero oe prograig odel for RNA structures 89 e) ) 5 6 6) 6) 6 7 7) 72) ) 83) 8 6) 2 6) ) 6) 6) 2) 6) ) 5) 2 5) 9) 5) 5) ) 5) 2 8) 6) Based o the followig lea, soe of the costraits are etra ad they ca beoitted. Lea. Let A b, Let two costraits S, where A is a atri with ra ad b ad i belog to the set of costraits A b. The the costrait bi is etra. j b i b R. Proof. Suppose that S be the feasible regio after deletig the costrait bi. Let the costrait bi is't etra. The S S. Let,, ) S ad,, ) S. Therefore, bi ad b. j i Itroducig the slac variables y ad z, we have the followig costraits i stadard for: Therefore, z bi ad j y bi. j y z. 3) O the other had, j y z,

The the costrait bi is etra. j b i b R. Proof. Suppose that S be the feasible regio after deletig the costrait bi. Let the costrait bi is't etra. The S S.

6 9 G. H. SHIRDEL AND N. KAHKESHANI Sice,, j, y ad z. But, this is a cotradictio. The S S. Accordig to Lea, the costraits b) ad c) are etra. Therefore, we ca delete the. Now, we defie the Proble A as follows: Proble A: Ma i, E s. t. ji4 i,2,3,4 j4 i j4 i, ji i j4 j j 3, 2,, 5,6,, 4 i, E where i, j i 4 E. The uber of variables ad costraits of the Proble A has bee preseted itable. Theore 3. S 4, ) is equal to the uber of feasible solutios of the Proble A. Also, theuber of optial solutios of the Proble A is equal to the uber of RNA structures with arc legth 4 ad aiu uber of arcs over []. Proof. The Proble A is writte o the basis of the atri odel ). Therefore, Theore 2 guaratees that S 4, ) is equal to the uber of feasible solutios. Sicethe objective fuctio is equal to the su of the variables ad the Proble A is theaiizatio proble, the aog the feasible solutios, the optial solutio belogs to the oe i which the aiu uber of variables would be equal to. So, the optial solutio has aiu uber of arcs over []. 4. USING ADDITIVE ALGORITHM FOR SOLVING THE PROBLEM A There are differet ethods for solvig a zero-oe liear prograig proble such as the additive algorith. For usig additive algorith, a proble ust possess three followig coditios:

The uber of variables ad costraits of the Proble A has bee preseted itable. Theore 3. S 4, ) is equal to the uber of feasible solutios of the Proble A.

7 A zero oe prograig odel for RNA structures 9. Its objective fuctio should be i the for of iiizatio. 2. The coefficiets of the objective fuctio should be oegative. 3. All the costraits ust be of the type. Table.The Nuber of Variables ad Costraits of the Proble A The uber of variables The uber of costraits Bysettig, the Proble A is coverted ito the followig proble: Proble B: Mi s. t. i, E j4 i j4 i ji4, i 4 5 j ji i j4 8 j i,2,3,4 j i, E 3, 2,, 5,6,, 4 Now, we ca apply the additive algorith for solvig the Proble B. I Table 2, we list the optial solutios for 6,,. Eaple. For = 7, the probles A is as follows: Ma s. t. z , 6 6 6, 7 7 7,,, Usig of additive algorith, the optial solutio of the Proble A is equal to

6 7 8 9 2 The uber of variables 3 6 5 2 28 8 6 The uber of costraits 2 4 6 9 Bysettig, th

8 92 G. H. SHIRDEL AND N. KAHKESHANI,, z * This solutio is correspodig to the RNA structure which is show i Figure Figure 2. Optial Structure for 7. The atri odel of this optial structure is as follows: Table2. The Optial Solutios of the Proble A, for 6,,. Optial solutio Optial solutio value 6, , ) ) , 28, ) 5) 38, 28 4, 4,, ) 5

2 3 4 5 6 7 Figure 2. Optial Structure for 7. The atri odel of this optial structure is as follows: Table2.

9 A zero oe prograig odel for RNA structures CONCLUSION Poolsap et al. i [5] represeted a iteger prograig proble for the RNA structures. Their odel is coplicated with ay variables. But, i here, we represeted aother prograig proble for the RNAstructures such that the uber of variables is lesstha the uber of variables i [5]. The optial RNA structure obtaied by solvig this proble is of arc-legth 4 ad has the aiu uber of hydroge bods. Iother words, foratio of these bods stabilizes the structure by lowerig its free eergy over []. If the eueratio of the feasible solutios of the zero-oe liear prograig proble is possible, the we are able to euerate the RNA structures with arc-legth 4 over [].also, i this case, the uber of optial solutios will be equal to the uber of optial RNA structures. Therefore, a future wor ca be the eueratio of the RNA structures usig atri ad liear prograig odels. REFERENCES. E. Balas, A additive algorith for solvig liear progras with zero-oe variables, Operatios Research, Vol. 3, No ) R. T. Batey, R. P. Rabo, J. A. Douda, Tertiary otifs i RNA structure ad foldig, Agew. Che. It. Ed ) I. L. Hofacer, P. schuster, P. F. Stadler, Cobiatorics of RNA secodary structures, Discrete Appl. Math ) E. Y. Ji, C. M. Reidys, Cobiatorial desig of pseudo ot RNA, Adv. Appl. Math ) U. Poolsap, Y. Kato, T. Autsu, Predictio of RNA secodary structure with pseudo ots usig iteger prograig, MBC Bioiforatics. Suppl. I):S38 29).

The optial RNA structure obtaied by solvig this proble is of arc-legth 4 ad has the aiu uber of hydroge bods.

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree