Mathematical Modeling of Metabolic Processes in a Living Organism in Relation to Nutrition

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1 Mathematical Modeling of Metabolic Pocee in a Living Oganim in Relation to Nutition Dimitova N., Makov S. Depatment Biomathematic Intitute of Mathematic and Infomatic Bulgaian Academy of Science 8 Acad. G. Bonchev St., 1113 Sofia, Bulgaia nelid@bio.ba.bg, makov@bio.ba.bg Summay: The pape i devoted to the mathematical modeling of metabolim and nutition baed on enzyme-kinetic eaction decibed by Michaeli-Menten equation. Popoed ae imple model of metabolic pocee in a living ytem which explain cetain effect elated to vaiou egime of nutition and fating. Keywod: Metabolic Pocee, Enzyme-kinetic Reaction, Michaeli-Menten Equation. 1. INTRODUCTION AND MOTIVATION The food ubtance enteing a living oganim gadually undego biochemical change duing digetion and metabolic pocee. It i chaacteitic fo biochemical pocee that enzyme play an impotant ole a catalit. Biochemical pocee involving enzyme can be effectively decibed mathematically by ytem of nonlinea diffeential equation. The baic model of enzyme kinetic i popoed by Michaeli and Menten [3]. A ubtate S convet into a poduct P in the peence of an enzyme E. Theeby S and E bind into an enzymeubtate complex C, which then diolve into P and E. Schematically, k1 k2 S + E C P + E (1) k 1 whee k 1, k -1 and k 2 ae coefficient of popotionality. The fit poce the binding of S and E into the complex C i eveible, the poduction of P i ieveible. Accoding to the Ma Action Law, the ate of a eaction i popotional to the poduct of the 1

2 concentation of the eactant. The coeponding ytem of odinay diffeential equation i, ee e.g. [6]: d / dt = k e + k c 1 1 ( ) ( ) dc / dt = k e k + k c de / dt = k e + k + k c dp / dt = k c (2) wheein = [S], e = [E], c = [C], p = [P] ae the concentation involved and the initial condition ae ( ) ( ) ( ) ( ) 0 =, c 0 = c, e 0 = e, p 0 = 0. (3) The equation fo p i uncoupled, o we hall futhe concentate on the ytem of the fit thee coupled equation. Uing the following nondimenionalization, cf. [6], = ( ) = ( ) ( ) = ( ) ( τ ) = ( ) / 0, λ = 1 / ( 1 0 ), = ( + ) / ( ), ε = / τ k e t, u τ t /, υ τ c t / e, w e t e k k K k k k e (4) we obtain the ytem u = uw + λυ ευ = uw Kυ ε w = uw + Kυ (5) with initial condition ( ) υ ( ) w( ) u 0 = 1, 0 = 0, 0 = 1. (6) 7 2 Typically we have ε 10,10, which make ytem (5) tiff. Namely, the ubtate vaiable (u) change nea 0 much lowe than the enzyme (υ) and complex vaiable (w) change. We can exclude one of the vaiable υ o w fom ytem (5), educing thu the 2

3 numbe of equation fom 3 to 2. Howeve, even the educed ytem cannot be olved in a cloed fom, ee [6]. Theefoe we need to make ue of numeical method, ee Fig. 1 and 2. Remak. A well-known, the olution fo in (2) can be appoximated by the olution σ of the imple DE dσ / dt = k e σ / k + σ. Howeve, we hall not be able to ue ( ) uch an appoximation, a we hall need to upply (2) by additional nonlinea tem. In an oganim the enzyme themelve ae a poduct of biochemical eaction. Thi obevation lie at the bai of the popoed model. The concentation of enzyme in the oganim change: namely the concentation diminihe becaue of a natual wah-out of enzyme and inceae due to a epoduction of enzyme. The coeponding model ae conideed in Section 3. In Section 4 the eult of numeical expeiment with the popoed model ae peented. 2. ASSUMPTIONS OF THE MODEL In the peent wok we popoe a global mathematical model of the metabolic pocee in a living oganim unde the following aumption, cf. [2]: 1. All ubtance enteing the oganim (food, wate, oxygen etc.) ae conideed a ubtate, involved in ubequent pocee catalized by the enzyme peent in the oganim. 2. Theoetically all enzyme-catalytic eaction can be decibed mathematically uing Michaeli-Menten equation involving pecific paamete. Howeve, the mathematical deciption of even a mall numbe of eaction lead to a complex mathematical ytem of nonlinea diffeential equation, which cannot be olved analytically and whoe numeical tudy i tediou. In ode to keep the mathematical model a imple a poible, we unify the biochemical eaction in lage goup unde cetain chaacteitic popetie. Fo intance, in the model popoed no ubtantial ditinction i made between catabolic and anabolic pocee and between digetic and metabolic pocee all thee pocee ae conideed fom the point of view of enzyme kinetic. 3

4 3. The ole of enzyme in biochemical pocee i twofold. Fom one ide, enzyme ae catalyt of thee pocee needed fo the poduction of cetain poduct. On the othe ide, enzyme ae themelve poduct of the metabolim. A a conequence, the biochemical pocee can be conditionally ubdivided into two lage goup. In the fit one we claify catabolic enzyme-catalytic eaction which ae not diectly involved in the poduction of enzyme; typically hee belong eaction, pataking in digetion and lowe metabolic cycle. In the econd goup we claify anabolic eaction eponible fo the poduction of new enzyme. 4. Fo implicity we can aume that catabolic eaction take place mainly in the extacellula pat of the oganim and that thei main pupoe i the beakdown of the nutient ubtance up to amino acid. On the othe ide anabolic eaction occu mainly in the citoplam of the cell leading to the ynthei of amino acid up to potein. We can conide the extacellula and the intacellula pat of the oganim a two epaate compatment aiving thu to a twocompatmental model. 5. In an oganim, the concentation of enzyme (both in bound and fee fom) undegoe change. One eaon i the outflow of enzyme with the excement of the oganim. We thu intoduce a wah-out function γe in the equation fo e a follow: d / dt = k e + k c 1 1 ( ) ( ) dc / dt = k e k + k c de / dt = k e + k + k c γ e dp / dt = k c (7) whee the initial condition ae again (3) and γ 0 i a wah-out contant. Uing fomulae (4) togethe with δ γ / ( k ) = we obtain the ytem 1 0 u = uw + λυ ευ = uw Kυ ε w = uw + Kυ δ w (8) 4

5 with initial condition (6). Fig. 3 and 4 viualize the numeical olution to (7), ep. (8). 3. MODELS WITH TWO TYPES OF SUBSTRATES In what follow we mathematically decibe the imultaneou tanfomation of two diffeent type of nutient ubtate S and R. We aume that the enzyme E tand fo the et of all enzyme neceay fo the tanfomation of S and R and that E i patially epoduced fom the ubtate S, R in the ene that cetain component of S and R ae ued fo the poduction of new enzyme. The nutient (ubtate) ae diffeentiated a follow: ubtate S do not diectly contibute to the fomation of potein, wheea ubtate R ae eaily conveted to potein and effectively contibute to the epoduction of enzyme needed fo the biochemical activity in the oganim. Amino acid belong to goup R. The coeponding enzyme-kinetic can be chematically decibed in two poible way. Vaiant 1. Hee it i aumed that the nutient S and R ae patially tanfomed into enzyme accoding to the following cheme k1 k2 k 1 k3 k4 k 3 ( 1 α ) S + E C Q + α E + E 1 ( 1 β ) R + E RE Q + β E + E 2 (9) whee 0 α < β 1. In paticula, if α = 0, β = 1, (9) obtain the fom k1 k2 S + E SE Q + E k 1 k3 k4 R + E RE E + E = 2E k 3 (10) 5

6 We hall aume that nutient both of type S and R ae enteing the oganim, which will be eflected by mean of function U U t U = U t. = ( ), ( ) The cheme (9) lead to the following ytem of diffeential equation: d / dt = k e + k c + U 1 1 d / dt = k e + k c + U 3 3 ( ) ( ) ( α ) ( 1 β ) dc / dt = k e k + k c dc / dt = k e k + k c ( ) ( ) de / dt = k e + k + 1+ k c k e + k + + k c γ e (11) wheein = [S], = [R], e = [E], c = [C], c = [RE] ae the concentation of the coeponding ubtance in (9) and the U U t U = U t peent the (ate of) intoduction function = ( ), ( ) of nutient in the oganim. The atuation of the enzyme E in the left-hand ide of the equation fo de/dt in (11) i limited again by the wah-out function γe. The initial condition ae ( ) ( ) ( ) ( ) ( ) 0 =, 0 =, c 0 = c 0 = 0, e 0 = e Numeical olution to (11) ae given on Fig. 5 and 6. Vaiant 2. Hee we aume that the nutient S i patially tanfomed into nutient R and then R i patially tanfomed into enzyme. k1 k2 k 1 k3 k4 k 3 ( 1 α ) S + E C Q + α R + E 1 ( 1 β ) R + E RE Q + β E + E 2 (12) Thi cheme lead to the following ytem of ODE': 6

7 d / dt = k e + k c + U 1 1 d / dt = k e + k c + αk c + U ( ) ( ) ( ) ( 1 β ) dc / dt = k e k + k c dc / dt = k e k + k c de / dt = k e + k + k c ( ) k e + k + + k c γ e (13) The meaning of the function U = U ( t), U U ( t) =, γe a well a the initial condition ae ame in (11). Numeical olution to (13) ae viualized on Fig. 7 and Fig COMMENTS ON THE NUMERICAL EXPERIMENTS Fo the numeical olution we ue an Eule method and a unifom meh which i malle in the bounday laye, e.g. the tepize i h = fo the fit 150 point tating fom 0, and then it become lage, h = It ha been hown [1, 5], that by mean of uch a imple meh one can achieve ame unifom eo (accuacy) a moe ophiticated mehe can poduce. Fig. 1 peent the olution to (2) with the following value fo the paamete and initial data: k = 5, k = 1, k = 4, = 10 / 3, e = 1. (14) Fig. 2 peent the olution to (5) coeponding to the above data tanfomed by (4). Fig. 3 and 4 viualize the olution to (7) and (8) epectively uing data (14) with γ = 0.5. The wah-out effect of the paamete γ (with epect to the enzyme) i clealy obeved. To viualize the olution to the next two model (11) and (13) we ue the following value fo the paamete and the initial data: 7

8 k = 5, k = 1, k = 4, k = 5, k = 1, k = 4, α = 0.2, β = 0.5, 3 4 γ = 0.3, = 10 / 3, = 2, e = (15) Fig. 5 peent gaphically the olution of (11) within U t = U t =. The next Fig. 6 viualize the output to (11) with ( ) ( ) 0 ( ) = { } ( ) = { } U t 3, 4 t 5; 0, othewie ; U t 2, 2 t 3; 0, othewie. (16) Fig. 7 and 8 peent the olution to (13) uing data (15); on Fig. 7 we have U ( t) U ( t) 0 U t fom = =, Fig. 8 ue U ( t ) and ( ) (16). A poible pupoe of any nutition egime (diet) could be to keep the enzyme concentation above a cetain limit. Thu the U t play the ole of contol vaiable. ubtate intake function ( ) We may fomulate vaiou optimization o contol poblem like keeping e above a cetain level, minimizing the quantity of food (the U t ). integal of ( ) Fig. 1. Solution to (2) uing data (14) 8

9 Fig. 2. Solution to (5) uing data (14) Fig. 3. Solution to (7) uing data (14) and γ = 0.5 Fig. 4. Solution to (8) uing data (14) and γ = 0.5 9

10 Fig. 5. Solution to (11) uing data (15) within α = 0.2, β = 0.5, γ = 0.3 and U ( t) = U ( t) = 0 Fig. 6. Solution to (11) uing data (15) within α = 0.2, U t fom (16) β = 0.5, γ = 0.3 and ( ) U t, ( ) Fig. 7. Solution to (13) uing data (15) within α = 0.2, β = 0.5, γ = 0.3 and U ( t) = U ( t) = 0 10

11 Fig. 8. Solution to (13) uing data (15) within α = 0.2, U t fom (16) 5. CONCLUSION β = 0.5, γ = 0.3 and ( ) U t, ( ) We peent and numeically tudy two enzyme-kinetic model with the pupoe to model baic metabolic activity of an oganim. It i obeved that diffeent type of ubtate contibute diffeently to the (e)poduction of enzyme. Thee i a tong feedback expeed in a timulating effect on the concentation of enzyme when the ubtate exhibit a etoation quality (α cloe to 1), o in an inhibiting effect wheneve the ubtate doe not poe uch qualitie (α cloe to 0). The effectivene of the above mentioned feedback i checked in the popoed model by the numeical imulation of vaiou type of diet (that i vaiou egime of nutition and fating). It i well-known that well-expeed ymptomatic phenomena can be obeved unde vaiou type of diet, uch a a low etoation of the metabolic activity of the oganim afte a polonged fating, a poibility fo poioning when conuming cetain type of food afte fating, etc. The inability of the metabolic ytem to poce the nutient ubtate can be intepeted a poioning of the oganim due to the lack of uitable enzyme needed to catabolize the incoming nutient. The numeical expeiment with the popoed model ugget that they can be ued fo checking vaiou hypothee elated to dieting, fo an altenative model ee [4]. We hope that on the bae of the above model moe ophiticated model involving cetain pecific metabolic cicle can be developed. 11

12 REFERENCES 1. Doolan E. P., J. J. H. Mille, W. H. A. Schilde, Unifom Numeical Method fo Poblem with Initial and Bounday Laye, Boole Pe, Makov S., P. Zlateva, M. Candev, Mathematical Model of Bioconveion Pocee in Living Oganim, Poc. IFIP Conge, Paga, 1995, Michaeli L., M. Menten, Die Kinetik de Invetinwikung, Biochem., 1913, 49, Micken R. E., D. N. Bewley, M. L. Ruell, A Model of Dieting, SIAM Rev., 1998, 40(3), Mille, J. J. H., E. O'Riodan, G. I. Shihkin, Fitted Numeical Method fo Singula Petubation Poblem, Wold Scientific, Muay J., Mathematical Biology, Spinge,