# MATHEMATICAL SIMULATION OF MASS SPECTRUM

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1 MATHEMATICA SIMUATION OF MASS SPECTUM.Beánek, J.Knížek, Z. Pulpán 3, M. Hubálek 4, V. Novák Univesity of South Bohemia, Ceske Budejovice, Chales Univesity, Hadec Kalove, 3 Univesity of Hadec Kalove, Hadec Kalove, 4 Univesity of Defense, Bno Abstact The contibution pesents shotly simulation of mass spectum. This was necessay fo debugging and testing of the mathematical algoithms fo the pocessing of data fom mass spectoscopy Intoduction Mass specta epesent vey valuable infomation souce which can be used at solving eseach and diagnostics poblems in biology and expeimental medicine. Many studies deal (fo example [], [], [3], [4] with the poblem of statistic evaluation of mass specta fo the puposes of biology eseach. ong line of woks uses these statistical pocedues to the evaluation of mass specta within the scope of biology eseach. Mass spectum can be descibed as a dependence of elative ion pecentage intensity on its effective mass. elative ion pecentage intensity is the intensity of ion elated to the intensity of maximum ion in a given spectum. It is denoted in liteatue by the chaacte I [%]. Effective mass is the atio of ion mass and its chage m / z []. This atio is chaacteistic fo evey paticle. Mass spectum can seve to pefom an exact identification of a given paticle unde the condition of sufficient esolution of instument and its pecise calibation. Data Mass spectomety geneates (pimay data: single shots of mass spectal values y( x t fo evey x t value, t =,, T, of abscise vecto, i.e. fo x = ( x,, x T, whee T is 4 the numbe of abscises (o measued places, T. The data stuctue of an mass spectum can be mathematically witten in the fom: y x y x T T. ( The less bief notation, that could be also used, is yt y( xt, whee t =,, T. The spectum y( x is nomalized so that the maximum of the highest peak gets the value ymax ( x = %. The value of independent vaiable x moves appoximately within the inteval ;.. = l ; u. The eal -shot mass spectum is displayed on the fig. 5.

2 Voyage Spec #=>AdvBC(3,.5,.=>NF.7[BP = 573.3, 46846] E+4 % Intensity Mass (m/z Figue : mass spectum (Voyage - DE ST, Voyage instument contol panel vesion 5., Data exploe vesion Mathematical simulation of mass spectum et us suppose that the andom sample is an expeimental epesentation of a cetain andom quantity Y ( x which epesents consideed spectum. Agument x gathes only the known values x { x,, x T }. The andom quantity Y ( x can be expessed in the fom Y ( x = y ( x + e( x, x { x,, x T }, ( whee e( x consists of esidual eo afte the pocessing of ough specta, of andom distubances which epesent e.g. biological vaiability, and of laboatoy eo of expeiments e.g. in molecula biology etc. y( x is the exact value. 3. One peak simulation A peak is defined by its coodinates x and y of peak maximum, i.e. by the numbes x and y, and by the atio of height and width of peak at half its height. The modified functional dependence of nomal distibution pobability density function in the fom ( x µ K f ( x = e, x l; u (3 can be used fo peak simulation. To achieve the coect shape of the peak whose ight side tail is always visibly highe then the left side tail, the peak is modeled with the help of nomal distibution pobability function so that the left side peak is expessed by the equation (3, and fo the ight side the equation (3 is modified so that is multiplied by the empiical coefficient ζ. Hence the left peak side: ( x µ K f ( x = e, x ( l; µ, (4

3 ight peak side: ( x µ K ( ζ f ( x = e, x ( µ ; u. (5 ζ Futhe it has to hold that the ation of peak height and width in its half height is constant: y /( z z =, (6 whee z < z ae elated peak x -coodinates in the half of its height. Thus, it applies f ( x = z = f ( x = z = y /, wheeas y = f,max ( x = f,max( x. 3.. Deivation of computational fomulas fo z and z It can be easily deived fom the elation (3 (o (4 and (5 that the coodinate x of peak maximum is identical to the position paamete µ : x µ, thus f ( x = y = K / a f ( x = y = K /( ζ. (7 E.g. fo the left peak side it is valid Afte the aangement: ( z µ y K K = f ( z = e. (8 ( z µ = e. (9 Afte aangement of (9 we eceive the quadatic equation z µ z + µ ln =, ( and afte its solving the esult has fom z = z ( = µ ln 4. Analogously, fo the ight peak side z = z( ζ = µ + ζ ln 4, whee ζ is the coefficient which enables to modify the shape of the ight peak tail, ζ 5. Explicit elations fo the calculation of constants K o K can be then expessed by the help of elations (7 (poviding is known, see below: K = y and K = y ζ. 3.. The calculation of paamete To calculate paamete it is necessay to solve tanscendental equation (e.g. by the method egula falsi

4 y z ( ζ z ( = U ( fo the unknown vaiable. The value of z( and zu ( ζ can be calculated by means of elations fom the last section. The iteation pocess is to be epeated so long until the value does not change in two successive iteations with an accuacy of equied numbe of significant figues d :, ( d whee is the ode numbe of the final iteation. The solution should always find itself in 3 the inteval ;, whee = and =, fo x µ l; u, 3 y %;% and. It is necessay to solve by means of egula falsi the modified tanscendental equation with egad to big ode diffeence between and : U y = + ζ z ( ( U log z (3 fo unknown = log and coespondent bounds = 3 and =. The esult of these calculations is demonstated in the pictue. Figue : One peak simulation 3. Spectum simulation Ideal (un-noised by andom distubances dependence of m-peaks simulated spectum can be witten in the explicit elation j= m y( x = f ( x, x, y, µ,, K, K, ζ, (4 j=, j, j j j, j U, j whee f (. ae functions of paticula peaks fom 3.. X -coodinates of maximums x, j, j =,, m, of all peaks, i.e. of poteins (and thei fagments, ae in all n shots of specta

5 identical. They ae as though natual constants. Thei values ae geneated with the help of geneatos of andom numbe of ectangula distibution in the inteval of all measued ange 5 of x specta coodinates, i.e. in the inteval ;.. = l ; u. The heights of all peaks y, j =,, m ae geneated with the help of geneato of andom numbe of ectangula, j distibution in the inteval (%;%. The heights of all peaks y, j =,, m, epesent in the famewok of one n-shots mass, j spectum as though constants. But at simulation of data (epesented by moe n-shots mass specta these heights of all peaks y, j =,, m, ae similaly andom quantities noised by, j andom distubances which epesent e.g. biological vaiability, laboatoy eo of expeiments e.g. in molecula biology etc. Ideal spectum calculated accoding the descibed steps in vaious scales is demonstated in the pictues and 3. Figue 3: Ideal spectum simulation in the selected inteval Figue 4: Ideal spectum simulation in the whole inteval 4 Conclusion The pupose of ou futue wok is to design the new methodology of mathematicalstatistical and fuzzy-logical identification and decision making in the domain of potein biomakes fom mass specta. The descibed simulation of mass spectum is necessay fo debugging and testing of the mathematical algoithms fo the pocessing of data fom mass spectoscopy. Fo these puposes the Matlab envionment is vey pope tool.

6 efeences [] J. Knížek, Z. Pulpan, M. Hubalek,. Beanek, P. Pokony. Stochastic model of mass spectum amdom distibution and its simulation, to be published in Jounal of Mass Specktomety [] G. Ball, S. Mian, F. Holding,.O. Allibone, J. owe, S. Ali, G. i, S. McCadle, I.O. Ellis, C. Cease,.C. ees An integated appoach utilizing atificial neual netwoks and SEDI mass spectomety fo the classification of human tumous and apid identification of potential biomakes. Bioinfomatics., Ma; 8(3: [3] E.P. Diamandis. Mass Spectomety as a Diagnostic and a Cance Biomake Discovey Too., Molecula and Cellula Poteomics, 4, 3:7-4. [4]. Tibshiani, T. Hastie, B. Naasimhan, S. Soltys, G. Shi, A. Koong, Q.T. e. Sample classification fom potein mass spectomety by peak pobability contasts. Bioinfomatics - Bioinfomatics Advance Access, Oxfod Univesity Pess, 4, -. Autho Contact infomation Depatment of Compute Science, Faculty of Education in Ceske Budejovice, Univesity of South Bohemia, Jeonymova, 37 5 Ceske Budejovice, Czech epublic, Autho Contact infomation Depatment of Medical Biophysics, Faculty of Medicine in Hadec Kalove, Chales Univesity in Pague, Simkova 87, 5 38 Hadec Kalove, Czech epublic Autho3 Contact infomation Depatment of Mathematics, Faculty of Education, Univesity of Hadec Kalove, okitanskeho 6, 5 3 Hadec Kalove, Czech epublic Autho4 Contact infomation Institute of Molecula Pathology, Faculty of Militay Health Sciences, Univesity of Defense in Bno, Tebeska 575, 5 Hadec Kalove, Czech epublic Autho5 Contact infomation Depatment of Compute Science, Faculty of Education in Ceske Budejovice, Univesity of South Bohemia, Jeonymova, 37 5 Ceske Budejovice, Czech epublic

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