Numerical Solution for Fuzzy Enzyme Kinetic Equations by the Runge Kutta Method

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1 Mahemaical and Compuaional Applicaions Aricle Numerical Soluion for Fuzzy Enzyme Kineic Equaions by he Runge Kua Mehod Yousef Barazandeh and Bahman Ghazanfari * ID Deparmen of Mahemaics, Loresan Universiy, Khorramabad , Iran; barazandeh.yu@fs.lu.ac.ir * Correspondence: ghazanfari.ba@lu.ac.ir Received: 8 February 28; Acceped: 4 March 28; Published: 8 March 28 Absrac: One of he mos imporan biochemical reacions is caalyzed by enzymes. A numerical mehod o solve nonlinear equaions of enzyme kineics, known as he Michaelis and Menen equaions, ogeher wih fuzzy iniial values is inroduced. The numerical mehod is based on he fourh order Runge Kua mehod, which is generalized for a fuzzy sysem of differenial equaions. The convergence and sabiliy of he mehod are also presened. The capabiliy of he mehod in fuzzy enzyme kineics is demonsraed by some numerical examples. Keywords: enzyme kineic equaions; sysem of fuzzy differenial equaions; fuzzy fourh order Runge Kua mehod. Inroducion Enzymes found in naure have been used since ancien imes in he producion of food producs. Enzymes are currenly used in several differen indusrial producs and processes. In a world wih a rapidly-increasing populaion and approaching he exhausion of many naural resources, enzyme echnology offers a grea poenial for many indusries o help mee he challenges hey will face in years o come []. Enzymes are among he caalyss for chemical reacions. Mos enzymes are proeins, alhough a few are RNAs. Enzymes help subsrae molecules become producs. Enzymes can muliply he speed of a reacion up o a million imes. Wihou enzymes, many of he reacions become slow or impossible [2 5]. Enzyme kineics deals wih he sudy of reacions ha are performed by enzymes. In 93, Michaelis and Menen offered a simple model for enzyme reacion, as seen below [3,5,6]. Suppose E is he enzyme and S is he subsrae, and heir combinaion is C. C irreversibly becomes an enzyme and he produc P, ha is: E + S C E + P Using he conceps of mass acion and he mass conservaion law, we ge o he sysem of differenial equaions. For simpliciy, we will use he same symbols C, S, E and P o refer o boh he chemicals C, S, E and P and heir concenraions. de = η ES + (η + η 2 )C, ds = η ES + η C, dc = η ES (η + η 2 )C, dp = η 2 C, () Mah. Compu. Appl. 28, 23, 6; doi:.339/mca236

2 Mah. Compu. Appl. 28, 23, 6 2 of 5 where η, η and η 2 are he raes of he reacion consans; ogeher wih iniial condiions S() = S, E() = E, C() = C and P() =. These iniial condiions are physically suppored in he sense ha, if no reacion akes place, we canno have a produc. From Equaion (), we have: d(e + C) = = E + C = E = E = E C Hence, Equaion () can be wrien as follows: dc = f (, C, S) = λs() (θ + η S())C(), ds = f 2 (, C, S) = λs() + (η S() + η )C(), C() = C, S() = S (2) where θ = η + η 2 and λ = η E. Various numerical mehods o solve Equaion () have been offered. Sen in [7] applied he Adomian decomposiion mehod; Khader in [8] applied he Picard Pade echnique; and Baiha and Baiha in [9] applied he differenial ransformaion mehod. In he presen sudy, he numerical soluions of enzyme kineic Equaion (2) are sudied by inroducing a new fuzzy enzyme kineic models in a way ha he iniial condiions are fuzzy numbers. The concep of fuzzy mahemaics was firs inroduced by Zadeh []. Then, many researchers developed i in many differen areas, including fuzzy differenial equaions. The heoreical framework of he fuzzy iniial value problem (FIVP) has been an acive research field over he las few years. The concep of he fuzzy derivaive was firs inroduced by Chang and Zadeh in []. This was followed up by Dubois and Prade in [2], who defined and used he exension principle. A comprehensive approach o FIVPs has been given in he work of Seikkala [3] and Kaleva [4,5], especially in is generalized form given by Buckly and Feuring [6]. Their work is imporan as i overcomes he exisence of muliple definiions of he derivaive of fuzzy funcions; see [2,3,7 9]. Furhermore, [6] compares he various soluions o he FIVP ha one may obain using he differen derivaives. Nieo [2] considered he Cauchy problem for coninuous fuzzy differenial equaions (FDEs). The generalized fuzzy differeniabiliy was suggesed more recenly by Sefanini and Bede [2]. Fuzzy differenial equaions can be used as a model o describe he phenomenon ha includes uncerainy parameers. The resuls of [3] on a cerain caegory of fuzzy differenial equaions (FDEs) have inspired several auhors who have applied numerical mehods for he soluion of hese equaions. The mos imporan conribuion o hese numerical mehods is he Euler mehod provided by Ma, Friedman and Kandel in [8]. Abbasbandy and Viranloo in [22,23] developed fourh order Runge Kua mehods for a Cauchy problem wih a fuzzy iniial value and fuzzy differenial equaions of order n. Furhermore, in [24], he auhors applied Runge Kua mehods o a more general caegory of problems, and hey proved convergence for s-sage Runge Kua mehods. The basics of various approaches of FDEs can be found in [25]. The organizaion of he paper is as follows: In Secion 2, some basic resuls from he fourh order Runge Kua mehod and he fuzzy numbers are presened. In Secion 3, he sysem of fuzzy iniial value problems is discussed. In Secion 4, he numerical soluion of he fuzzy enzyme kineic equaions hrough he fourh order Runge Kua mehod is illusraed. The convergence and sabiliy of he mehod are presened in Secion 5. Some numerical cases are inroduced by an example in Secion 6. Finally, he conclusion is given in Secion Preinaries Given he iniial value problem (IVP): { y = f (, y), y(a) = y a. a b (3)

3 Mah. Compu. Appl. 28, 23, 6 3 of 5 The fourh order Runge Kua mehod [26] is as follows: k = f ( i, y i ), k 2 = f ( i + h 2, y i+ h 2 k ), k 3 = f ( i + h 2, y i+ h 2 k 2), k 4 = f ( i + h, y i+ hk 3 ), y i+ = y i + h 6 (k + 2k 2 + 2k 3 + k 4 ), (4) where a =... N = b, h = (b a) N = i+ i and i =,,..., N. In his par, some definiions and he necessary fuzzy mahemaics (see [7]) are reviewed. The se of all real numbers is denoed by R, and he se of all fuzzy numbers on R is denoed by R F. A fuzzy number is a mapping ω : R [, ] wih he following properies: (a) ω is upper semi-coninuous on R. (b) ω is fuzzy convex, i.e., for λ and any x, y R, ω(λx + ( λ)y) min{ω(x), ω(y)}. (c) ω is normal, i.e., x R for which ω(x ) =. (d) [ω] = cl{x R : ω(x) > } is he suppor of ω, and is closure is compac. For r (, ], define [ω] r = {x R : ω(x) r} = [ω(r), ω(r)]. Then from (a) (d), i follows ha for all r [, ], he r-level se [ω] r is a closed bounded inerval. Le I be a real inerval; a mapping ω : I R F is called a fuzzy process, and is r-level se is denoed by: ω(; r) = [ω(; r), ω(; r)], I, r [, ]. We can define a riangular fuzzy number ω by hree real numbers v v 2 v 3 where he graph of ω(x), he membership funcion of he fuzzy number ω = (v, v 2, v 3 ), is a riangle wih he base on he inerval v, v 3 and he verex a x = v 2. A riangular fuzzy number ω = (v, v 2, v 3 ) can be wrien as α-cu for r [, ] as ω(r) = v + (v 2 v )r and ω(r) = v + (v 2 v )r. Assume v, w R F and s is a posiive scalar, hen for r (, ], we have: [v + w] r = [v(r) + w(r), v(r) + w(r)] [v w] r = [v(r) w(r), v(r) w(r)] [v.w] r = [min {v(r).w(r), v(r).w(r), v(r).w(r), v(r).w(r)}, max {v(r).w(r), v(r).w(r), v(r).w(r), v(r).w(r)}, [sv] r = s[v] r. The Hausdorff disance beween fuzzy numbers is given by: D[ω, ω 2 ] = sup max{ ω (r) ω 2 (r), ω (r) ω 2 (r) } r The meric space (R F, D) is complee, separable and locally compac. Definiion. A funcion f : R R F is called a fuzzy funcion. If for arbirary fixed R and ε >, δ > such ha: < δ D( f (), f ( )) < ε exiss, f is said o be coninuous. Definiion 2. (see [27]) Le u, v R F. If here exiss w R F such ha u = v + w, hen w is called he Hukuhara-difference or H-difference of u and v, and i is denoed by u v:

4 Mah. Compu. Appl. 28, 23, 6 4 of 5 Definiion 3. (see [27]) Le f : (a, b) R F and x (a, b). We say ha f is srongly generalized differeniable a x, if here exiss an elemen f (x ) R F such ha (i) for all h > sufficienly small, f (x + h) f (x ), f (x ) f (x h), and he is (in he meric D): f (x + h) f (x ) f (x = ) f (x h) = f (x ) h h h h or (ii) for all h > sufficienly small, f (x ) f (x + h), f (x h) f (x ), and he is (in he meric D): f (x ) f (x + h) f (x = h) f (x ) = f (x ) h h h h or (iii) for all h > sufficienly small, f (x + h) f (x ), f (x h) f (x ), and he is (in he meric D): f (x + h) f (x ) f (x = h) f (x ) = f (x ) h h h h or (iv) for all h > sufficienly small, f (x ) f (x + h), f (x ) f (x h), and he is (in he meric D): f (x ) f (x + h) f (x = ) f (x h) = f (x ). h h h h If f is differeniable wih (i) of Definiion 3, hen f is (I)-differeniable, and is differeniable wih (ii) of Definiion 3, hen f is (II)-differeniable. Theorem. (see [28]) Le f : (a, b) R F and f (; r) = [ f (; r), f (; r)] for any r [, ]. Then: If f is (I)-differeniable, hen f (; r) and f (; r) are differeniable funcions and: [ f (; r)] = [ f (; r), f (; r)]. If f is (II)-differeniable, hen f (; r) and f (; r) are differeniable funcions and: 3. A Sysem of Fuzzy Iniial Value Problems [ f (; r)] = [ f (; r), f (; r)]. Consider he following sysem of fuzzy iniial value problems [23]: y = f (, y,..., y n ),. y n = f (, y,..., y n ), y () = y [],..., y n() = y [] n (5) where f i ( i n) are coninuous mappings from R + R n ino R and y [] i are fuzzy numbers in R F wih r-level inervals: [y [] i ] r = [y i [] (r), y i [] (r)], for i =,..., n and r (, ]. (6)

5 Mah. Compu. Appl. 28, 23, 6 5 of 5 Suppose y is a (I)-differeniable, hen y = (y,..., y n ) is a fuzzy soluion of (5) on he inerval I, if: y i (; r) = min{ f i(, u,..., u n ); u j [y j (; r), y j (; r)]} = f i (, y(; r)), y i (; r) = max{ f i(, u,..., u n ); u j [y j (; r), y j (; r)]} = f i (, y(; r)), y i (; r) = y i [] (r), y i (; r) = y i [] (r), i =, 2,..., n. Therefore, for fixed r, we have [ a sysem of ] iniial value problem in R 2n. If i is solved, we have only o verify ha he inervals, y j (; r), y j (; r) define a fuzzy number y i () R F. Now, le: { y [] (r) = (y [] (r),..., y n [] (r)), y [] (r) = (y [] (r),..., y n [] (r)) regarding he above-menioned indicaor, we can wrie sysem: { y () = F(, y()), y() = y [] R F n. (7) Wih assumpion y(, r) = [y(; r), y(; r)] and y (; r) = [y (; r), y (; r)], where: y(; r) = (y (; r),..., y n (; r)), y(; r) = (y (; r),..., y n (; r)), y (; r) = (y (; r),..., y n (; r)), y (; r) = (y (; r),..., y n (; r)) (8) and wih assumpion F(, y(; r)) = [F(, y(; r)), F(, y(; r))], where: { F(, y(; r)) = ( f (, y(; r)),..., f n (, y(; r))), F(, y(; r)) = ( f (, y(; r)),..., f n (, y(; r))). or: Thus, y() is a fuzzy soluion of (5) on he inerval I for all r (, ], if: y (, r) = F(, y(; r)) y (, r) = F(, y(; r)) y(, r) = y [] (r), y(; r) = y [] (r), { y (; r) = F(, y(; r)) y(; r) = y [] (r) R n F. In he same way, if y is (II)-differeniable, hen we can wrie: y (; r) = F(, y(; r)) y (; r) = F(, y(; r)) y(; r) = y [] (r), y(; r) = y [] (r). In his paper, we suppose ha y is (I)-differeniable: (9) () ()

6 Mah. Compu. Appl. 28, 23, 6 6 of 5 Theorem 2. [23] If f i (, u,..., u n ) for i =, 2,..., n are coninuous funcions of and saisfy he Lipschiz condiion in u = (u,..., u n ) in he region D = {(, u) I = [, ], < u i < for i =, 2,..., n} wih consan L i, hen he iniial value problem (5) has a unique fuzzy soluion in each case. 4. Fuzzy Runge Kua Mehod of Order Four for he Fuzzy Kineic Enzyme Reacion Consider Equaion (2) in which he iniial values are fuzzy, i.e., { dc = f (, C, S) = λs() (θ + η S())C(), ds = f 2 (, C, S) = λs() + (η S() + η )C(), (2) and: C(; r) = [C(; r), C(; r)], S(; r) = [S(; r), S(; r)]. (3) Suppose ha he exac soluion C(; r) = [C(, r), C(, r)] and S(; r) = [S(, r), S(, r)] of (2) wih ogeher (3) is approximaed by c(; r) = [c(, r), c(, r)] and s(; r) = [s(, r), s(, r)]. By he fourh order Runge Kua mehod, hen for i =, 2, 3, 4 and j =, 2, we have: k ij (, c(; r),s(; r)) = l ij (, c(; r),s(; r)) = min max(j){ f ( + I(i) h, u, v) 2 u [p i (, c(; r), s(; r)), p i 2 (, c(; r), s(; r))], v [q i (, c(; r), s(; r)), q i 2 (, c(; r), s(; r))]} min max(j){ f 2 ( + I(i) h, u, v) 2 u [p i (, c(; r), s(; r)), p i 2 (, c(; r), s(; r))], v [q i (, c(; r), s(; r)), q i 2 (, c(; r), s(; r))]} where min max() = min, min max(2) = max and I(i) = 2i 3 5i i 24, such ha I() =, I(2) = I(3) = 6 and I(4) = 2, and also: wih c = c, c 2 = c, s = s and s 2 = s. And, p ij (, c(; r), s(; r)) = c j (; r) + I(i) h 2 k i j(, c(; r), s(; r)) q ij (, c(; r), s(; r)) = s j (; r) + I(i) h 2 l i j(, c(; r), s(; r)), ψ [(, c(; r), s(; r))] = k (, c(; r), s(; r)) + 2k 2 (, c(; r), s(; r))+ 2k 3 (, c(; r), s(; r)) + k 4 (, c(; r), s(; r)) ψ 2 [(, c(; r), s(; r))] = k 2 (, c(; r), s(; r)) + 2k 22 (, c(; r), s(; r))+ 2k 32 (, c(; r), s(; r)) + k 42 (, c(; r), s(; r)) ψ 3 [(, c(; r), s(; r))] = l (, c(; r), s(; r)) + 2l 2 (, c(; r), s(; r))+ 2l 3 (, c(; r), s(; r)) + l 4 (, c(; r), s(; r)) ψ 4 [(, c(; r), s(; r))] = l 2 (, c(; r), s(; r)) + 2l 22 (, c(; r), s(; r))+ 2l 32 (, c(; r), s(; r)) + l 42 (, c(; r), s(; r)).

7 Mah. Compu. Appl. 28, 23, 6 7 of 5 Finally, we now have: c( n+ ; r) = c( n ; r) + h 6 ψ [(, c(; r), s(; r))] c( n+ ; r) = c( n ; r) + h 6 ψ 2[(, c(; r), s(; r))] s( n+ ; r) = s( n ; r) + h 6 ψ 3[(, c(; r), s(; r))] (4) s( n+ ; r) = s( n ; r) + h 6 ψ 4[(, c(; r), s(; r))]. 5. Convergence and Sabiliy Consider he sysem of differenial equaions: { y () = F(, y()), y() = y [] R n. (5) Definiion 4. A one-sep mehod for approximaing he soluion of (5) can be wrien in he form: w( m+ ) = w( m ) + hφ( m, w( m ), h). (6) Theorem 3. [29] If Φ(, y, h) saisfies a Lipschiz condiion in y, hen he mehod given by (6) is sable. Lemma. [8] If a sequence of numbers {W n } n=n n=, {V n} n=n n= saisfies: W n+ W n + A max{ W n, V n } + B for some given posiive consan A and B, we suppose U n = W n + V n, n N hen U n Ā n U + B Ān Ā, where Ā = + 2A and B = 2B. Now, suppose v(; r) = (v (; r),..., v n (; r)) is a fuzzy approximae soluion of sysem () by following he one-sep mehod: v( m+ ; r) = v( m ; r) + hφ( m, v( m ; r), h), h > (7) where [Φ] r = [Φ, Φ], [v] r = [v, v]. We can wrie is componens as follows: v i ( m+ ; r) = v i ( m ; r) + hφ i ( m, v( m ; r), h) (8) v i ( m+ ; r) = v i ( m ; r) + hφ i ( m, v( m ; r), h). (9) Theorem 4. For fixed r [, ], in Relaion (), if F(, y) saisfies a Lipschiz condiion in y, hen he mehod given by (7) is sable. Theorem 5. Consider he one-sep mehod (7) where v( ; r) = y( ; r) and Φ, Φ is uniformly coninuous in, y, h for l, h h o and all y, and if i saisfies he Lipschiz condiion in he region D = {(, x, z, h) l, < x i z i, < x i +, h h o, i =, 2,..., n} and Φ(, y(; r), ) = F(, y(; r)), hen he approximae mehod (7) is convergen o he exac soluion of (). Proof. We define runcaion errors, T i (m; r), T i (m; r), of he mehod by: y i ( m+ ; r) = y i ( m ; r) + hφ i ( m, y( m ; r), h) + ht i (m; r) (2)

8 Mah. Compu. Appl. 28, 23, 6 8 of 5 y i ( m+ ; r) = y i ( m ; r)h + Φ i ( m, y( m ; r), h) + ht i (m; r). (2) Assume δ i (m; r) = y i ( m ; r) v i ( m ; r), δ i (m; r) = y i ( m ; r) v i ( m ; r). By subracing (8) from (2), we obain: δ i (m + ; r) = δ i (m; r) + h[φ i ( m, y( m ; r), h) Φ i ( m, v( m ; r), h)] + ht i (m; r) (22) Denoe: δ i (m + ; r) δ i (m; r) + hl { n j= ( δ j (m; r) + δ j (m; r) )} + ht i (m; r). (23) m ax{ δ j(m; r) } = p δ i (m; r), j n max{p, p } = p m ax{ δ j(m; r) } = p δ i (m; r) j n and so: δ i (m + ; r) δ i (m; r) + 2nhL p max{ δ i (m; r), δ i (m; r) } + ht i (m; r). (24) Similarly, here is p 2, such ha: By seing: and similarly: δ i (m + ; r) δ i (m; r) + 2nhL 2 p 2 max{ δ i (m; r), δ i (m; r) } + ht i (m; r). (25) max{l, L 2 } = L, max{p, p 2 } = p, max{t i (m; r), T i (m; r)} = Ω i (m; r) δ i (m + ; r) δ i (m; r) + 2nhLp max{ δ i (m; r), δ i (m; r) } + hω i (m; r) (26) δ i (m + ; r) δ i (m; r) + 2nhLp max{ δ i (m; r), δ i (m; r) } + hω i (m; r). (27) If we suppose i (m; r) = δ i (m; r) + δ i (m; r) and use Lemma (): i (m; r) ( + 4nhLp) m i (; r) + 2hΩ i (m; r) ( + 4nhLp)m. (28) 4nhLp Since + 4nhLp exp(4nhlp): i (m; r) exp(4mnhlp) i (; r) + Ω i (m; r) exp(4mnhlp). (29) 2nLp Now, we wan o check wha happens o (29) if h. By assumpions of he heorem, we rewrie Relaion (2) as follows: T i (m; r) =[ y i( m+ ; r) y i ( m ; r) f h i ( m, y( m ; r))] + [Φ i ( m, y( m ; r), ) Φ i ( m, y( m ; r), h)]. (3) By he mean value heorem, he firs bracke is y i (ξ; r) y i ( m ; r), where ξ [ m, m+ ]. Since y is uniformly coninuous, hence for any ɛ >, here is h (ɛ), so ha: y i (ξ; r) y i ( m ; r) 2 ɛ f or h < h (ɛ)

9 Mah. Compu. Appl. 28, 23, 6 9 of 5 Φ i ( m, y( m ; r), ) Φ i ( m, y( m ; r), h) 2 ɛ f or h < h 2 (ɛ). If h (ɛ) = min{h (ɛ), h 2 (ɛ)}, hen: and similarly, here is h (ɛ) such ha: T i (m; r) < ɛ f or h < h (ɛ) T i (m; r) < ɛ f or h < h (ɛ). If h(ɛ) = min{h (ɛ), h (ɛ)}, hen we can say for any ɛ >, here is h(ɛ), so ha: Ω i (m; r) < ɛ f or h < h(ɛ). Now, in Relaion (29), if h, hen i (m; r), and so, δ i (m; r), δ i (m; r) and also δ(m; r), δ(m; r). Therefore, he menioned numerical mehod is convergen o he exac soluion. 6. Examples In his secion, given he examples wih differen values of parameers and fuzzy iniial values in Equaion (2), we plo he approximaion of C(), S(). Example. λ = ; η =.2; η =.; θ =.7 ogeher wih fuzzy iniial values: C(; r) = [ 3 ( + r), 3 ( r)], S(; r) = [ 3 (999 + r), 3 ( r)]. In Figures and 2, c() and s() for 4 wih h [] =.25 are ploed, respecively. In Figure 3, c() and s() are ploed for differen values of r. Example 2. λ = 4; η = ; η =.5; θ = 3 ogeher wih fuzzy iniial values: C(; r)[ 3 ( + r), 3 ( r)], S(; r) = [ 3 (999 + r), 3 ( r)]. In Figures 4 and 5, c() and s() for wih h [] =.25 are ploed, respecively. In Figure 6, c() and s() are ploed for differen values of r. Example 3. λ = 2; η = ; η = ; θ = 2 ogeher wih fuzzy iniial values: C(; r) = [ 3 ( + r), 3 ( r)], S(; r) = [9.9 +.r,..r]. In Figures 7 and 8, c() and s() for. wih h [] =.25 are ploed, respecively. In Figure 9, c() and s() are ploed for differen values of r.

10 Mah. Compu. Appl. 28, 23, 6 of 5 r c() Figure. The approximae soluions of C for differen values of r and. r s() Figure 2. The approximae soluions of S for differen values of r and.

11 Mah. Compu. Appl. 28, 23, 6 of c(),s() Figure 3. The approximae soluions of C and S for differen values of r =,.2,.4,...,. r c() Figure 4. The approximae soluions of C for differen values of r and.

12 Mah. Compu. Appl. 28, 23, 6 2 of r s().5 Figure 5. The approximae soluions of S for differen values of r and c(),s() Figure 6. The approximae soluions of C and S for differen values of r =,.2,.4,...,.

13 Mah. Compu. Appl. 28, 23, 6 3 of 5 r c() 4 Figure 7. The approximae soluions of C for differen values of r and. r s(). Figure 8. The approximae soluions of S for differen values of r and.

14 Mah. Compu. Appl. 28, 23, 6 4 of c(),s() Figure 9. The approximae soluions of C and S for differen values of r =,.2,.4,...,. In Figures, 2, 4, 5, 7 and 8, he approximae fuzzy values obained by he fuzzy fourh order Runge Kua mehod are presened for differen values of parameers, which shows us ha hey are all fuzzy numbers. In Figures 3, 6 and 9, he speed of reaching he equilibrium poin is shown by increasing he concenraion, which corresponds o he real-world enzyme reacions. 7. Conclusions In his paper, we inroduced he fuzzy fourh order Runge Kua mehod for approximae soluions of he fuzzy enzyme kineic reacion equaions and illusraed i by numerical examples. Since he Runge Kua mehod has a high convergence order, i can provide appropriae approximae soluions o solve fuzzy enzyme kineics equaions. The presened examples show ha he speed of reaching he equilibrium poin is shown by increasing he concenraion, which corresponds o he real-world enzyme reacions and confirms he suiabiliy of he menioned mehod. The algorihm presened here can easily be used for his ype of fuzzy sysem. Auhor Conribuions: Boh auhors conribued equally o his work. Conflics of Ineres: The auhors declare no conflic of ineres. References. Kirk, O.; Borcher, T.V.; Fuglasang, C.C. Indusrial enzyme applicaions. Curr. Opin. Bioechnol. 22, 3, Buxbaum, E. Fundamenals of Proein Srucure and Funcion, 2nd ed.; Springer Inernaional Publishing: New York, NY, USA, Keener, J.; Sneyd, J. Mahemaical Physiology; Springer: New York, NY, USA, Berg, J.M.; Tymoczko, J.L.; Sryer, L. Biochemisry, 5h ed.; W. H. Freeman: San Francisco, CA, USA, Chasnov, J.R. Mahemaical Biology Lecure noes for MATH 365; The Hong Kong Universiy of Science and Technology: Hongkong, China, Chapwanya, M.; Lubuma, J.M.; Mickens, R.E. From enzyme kineics o epidemiological models wih Michaelis Menen conac rae: Design of nonsandard finie difference schemes. Compu. Mah. Appl. 22, 64, 2 23.

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