Enzyme kinetic modelling and analytical solution of nonlinear rate equation in the transformation of D-methionine into L-methionine in batch reactor using the new homotopy perturbation method

Sivasamy, Pavithra; Ganapathy, Jansi Rani Palaniyandi; Thinakaran, Iswarya; Lakshmanan, Rajendran

doi:10.21577/0100-4042.20160155

Pavithra Sivasamy

Department of Mathematics, Sethu Institute of Technology, Pulloor, Kariapatti - 626115, Virudunagar District - Tamilnadu, India

Jansi Rani Palaniyandi Ganapathy

Department of Mathematics, Sethu Institute of Technology, Pulloor, Kariapatti - 626115, Virudunagar District - Tamilnadu, India

Iswarya Thinakaran

Department of Mathematics, Sethu Institute of Technology, Pulloor, Kariapatti - 626115, Virudunagar District - Tamilnadu, India

Rajendran Lakshmanan ^**e-mail: dr.rajendran.l@gmail.com

Department of Mathematics, Sethu Institute of Technology, Pulloor, Kariapatti - 626115, Virudunagar District - Tamilnadu, India

*e-mail: dr.rajendran.l@gmail.com

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Figure 1
Kinetic scheme of D-AAO, L-PheDH, FDH and Coupled system of D-methionine bioconversion into L-methionine¹1 Findrik, Z.; Vasić-Rački, Ð.; Biotechnol. Bioeng. 2007, 98, 956.

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Figure 2
D-AAO kinetics (a) Comparison between the theoretical results ( and ) and simulation results. (b) Comparison between theory () and simulations results. (c) Comparison between theory () and simulation results. (d)-(e) Comparison of theoretical results with experimental results¹1 Findrik, Z.; Vasić-Rački, Ð.; Biotechnol. Bioeng. 2007, 98, 956. for the reaction rate r₁. The numerical value of the kinetic parameters used for the above figure is given in

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Figure 3
L-pHeDH kinetics (a) Comparison between the theoretical results ( and ) and simulation results. (b) Comparison between the theoretical results (, and ) and simulation results. Concentration of (c) L-methionine (), (d) 2-oxo-4-methylthiobutyric acid (), (e)Ammonium (), (f) Nicotinamide adenine dinucleotide (), (g) Nicotinamide adenine dinucleotide hydrogen () versus time t. The numerical value of the kinetic parameters used for the above figure is given in . Key to the graph: (___) represents the (-) and (....) represents the numerical results

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Figure 4
Comparison of theoretical results with the experimental results¹1 Findrik, Z.; Vasić-Rački, Ð.; Biotechnol. Bioeng. 2007, 98, 956. for the reaction rate r₂ and r₃.The numerical value of the kinetic parameters used for the above figure is given in

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Figure 5
FDH kinetics (a) Comparison between the theoretical results ( and ) and simulation results. (b) Concentration of Nicotinamide adenine dinucleotide (), (c) Nicotinamide adenine dinucleotide hydrogen () versus time t. (d-f) Comparison of theoretical results with the experimental results ¹1 Findrik, Z.; Vasić-Rački, Ð.; Biotechnol. Bioeng. 2007, 98, 956. for the reaction rate r₄.The numerical value of the kinetic parameters used for the above figure is given in

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Figure 6
(a) Variation of the concentration of 2-oxo-4-methylthiobutyric acid (c_2-oxo) with time t at different γ_D-AAO using . (b) Variation of the concentration of ammonium (c_NH4+) with time t at different γ_D-AAO using

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Figure 7
Estimation of kinetic parameters K_m^D-met and K_i^2-oxo using .The numerical value of the kinetic parameters used for the above figure is given in

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Symbol Meaning r 1 Enzyme reaction rate of D-AAO catalyzed D-methionine oxidation (mmol dm-3 min-1 mg-1) r 2 Enzyme reaction rate of L-PheDH catalyzed 2-oxo-4- methylthiobutyric acid reductive amination (mmol dm-3 min-1 mg-1) r 3 Enzyme reaction rate of L-PheDH catalyzed L-methionine oxidation (mmol dm-3 min-1 mg-1) r 4 Enzyme reaction rate of FDH catalyzed NAD+ reduction (mmol dm-3 min-1 mg-1) Vm1 Maximum enzyme reaction rate (mmol dm-3 min-1 mg-1) Vm2 Maximum enzyme reaction rate (mmol dm-3 min-1 mg-1) Vm3 Maximum enzyme reaction rate(mmol dm-3 min-1 mg-1) Vm4 Maximum enzyme reaction rate (mmol dm-3 min-1 mg-1) cD–met Concentration of D-methionine (mmol dm-3) c 2–oxo Concentration of 2-oxo-4-methylthiobutyric acid (mmol dm-3) cL–met Concentration of L-methionine (mmol dm-3) cNH 4 + Concentration of ammonium (mmol dm-3) cNAD+ Concentration of nicotinamide adenine dinucleotide coenzyme (mmol dm-3) cNADH Concentration of nicotinamide adenine dinucleotide hydrogen (mmol dm-3) cF Concentration of formate (mmol dm-3) KmD-met Michaelis-Menten constant of D-methionine (mmol dm-3) Ki2-oxo Product inhibition constant of 2-oxo-4-methylthiobutyric acid (mmol dm-3) KiNAD Product inhibition constant of Nicotinamide adenine dinucleotide hydrogen (mmol dm-3) Km2-oxo Michaelis-Menten constant of 2-oxo-4-methylthiobutyric acid (mmol dm-3) KmNADH Michaelis-Menten constant of Nicotinamide adenine dinucleotide hydrogen (mmol dm-3) KmNH4+ Michaelis-Menten constant of Ammonium (mmol dm-3) KmL-met Michaelis-Menten constant of L-methionine (mmol dm-3) KmNAD+ Michaelis-Menten constant of Nicotinamide adenine dinucleotide coenzyme (mmol dm-3) KmF Michaelis-Menten constant of Formate (mmol dm-3) a 1 to a 4 Enzyme reaction rate (mmol dm-3 min-1) K 1 to K 10 Michaelis-Menten constant (min-1)

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Table 1
Comparison of analytical result with simulation result for the concentration of c_D-met using

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Table 2
Comparison of analytical result with Simulation result for the concentration of c_2-oxo using

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Table 3
Experimental Values of Parameter Use in This Work and Findrik et al.¹1 Findrik, Z.; Vasić-Rački, Ð.; Biotechnol. Bioeng. 2007, 98, 956.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8a)

(8b)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

Symbol	Meaning
r ₁	Enzyme reaction rate of D-AAO catalyzed D-methionine oxidation (mmol dm^-3 min^-1 mg^-1)
r ₂	Enzyme reaction rate of L-PheDH catalyzed 2-oxo-4- methylthiobutyric acid reductive amination (mmol dm^-3 min^-1 mg^-1)
r ₃	Enzyme reaction rate of L-PheDH catalyzed L-methionine oxidation (mmol dm^-3 min^-1 mg^-1)
r ₄	Enzyme reaction rate of FDH catalyzed NAD⁺ reduction (mmol dm^-3 min^-1 mg^-1)
V_m₁	Maximum enzyme reaction rate (mmol dm^-3 min^-1 mg^-1)
Vm₂	Maximum enzyme reaction rate (mmol dm^-3 min^-1 mg^-1)
Vm₃	Maximum enzyme reaction rate(mmol dm^-3 min^-1 mg^-1)
Vm₄	Maximum enzyme reaction rate (mmol dm^-3 min^-1 mg^-1)
c_D–met	Concentration of D-methionine (mmol dm^-3)
c _2–oxo	Concentration of 2-oxo-4-methylthiobutyric acid (mmol dm^-3)
c_L–met	Concentration of L-methionine (mmol dm^-3)
c_NH ₄ ⁺	Concentration of ammonium (mmol dm^-3)
c_NAD⁺	Concentration of nicotinamide adenine dinucleotide coenzyme (mmol dm^-3)
c_NADH	Concentration of nicotinamide adenine dinucleotide hydrogen (mmol dm^-3)
c_F	Concentration of formate (mmol dm^-3)
K_m^D-met	Michaelis-Menten constant of D-methionine (mmol dm^-3)
K_i^2-oxo	Product inhibition constant of 2-oxo-4-methylthiobutyric acid (mmol dm^-3)
K_i^NAD	Product inhibition constant of Nicotinamide adenine dinucleotide hydrogen (mmol dm^-3)
K_m^2-oxo	Michaelis-Menten constant of 2-oxo-4-methylthiobutyric acid (mmol dm^-3)
K_m^NADH	Michaelis-Menten constant of Nicotinamide adenine dinucleotide hydrogen (mmol dm^-3)
K_m^NH4+	Michaelis-Menten constant of Ammonium (mmol dm^-3)
K_m^L-met	Michaelis-Menten constant of L-methionine (mmol dm^-3)
K_m^NAD+	Michaelis-Menten constant of Nicotinamide adenine dinucleotide coenzyme (mmol dm^-3)
K_m^F	Michaelis-Menten constant of Formate (mmol dm^-3)
a ₁ to a ₄	Enzyme reaction rate (mmol dm^-3 min^-1)
K ₁ to K ₁₀	Michaelis-Menten constant (min^-1)

Time t (min)	Concentration of c_D-met (mmol dm^-3)		% of deviation between simulation result and Equation 5
Time t (min)	Simulation result	Analytical result Equation 5	% of deviation between simulation result and Equation 5
	9.992700	10	0.07
1	9.402076	9.401787	0.003077
10	5.396787	5.396398	0.007204
20	2.910123	2.912111	-0.06832
30	1.572126	1.571491	0.040373
40	0.849807	0.848039	0.208076
50	0.457956	0.457636	0.070032
60	0.245974	0.246958	-0.40035

Time t (min)	Concentration of c_2-oxo (mmol dm^-3)		% of deviation between simulation result and Equation 6
Time t (min)	Simulation result	Analytical result Equation 6	% of deviation between simulation result and Equation 6
0	0.100200	0.100000	0.20
1	0.697964	0.698213	-0.03572
10	4.703253	4.703602	-0.00742
20	7.189917	7.187889	0.02821
30	8.527914	8.528509	-0.00697
40	9.250233	9.251961	-0.01868
50	9.642084	9.642364	-0.00291
60	9.854066	9.853042	0.010399

\frac{d c_{D - m e t}}{d t} = r_{1} (t) = \frac{V_{m 1} . c_{D - m e t} (t) γ_{D - A A O}}{K_{m}^{D - m e t} + \frac{K_{m}^{D - m e t}}{K_{i 2}^{2 - o x o}} [(c_{D - m e t})_{0} + (c_{2 - o x o})_{0} - (c_{D - m e t})_{0} e x p [- (K_{0} t]] + c_{D - m e t} (t)}

\frac{d c_{2 - o x o}}{d t} = r_{1} (t) = \frac{V_{m 1} . (c_{D - m e t})_{0} (c_{2 - o x o})_{0} + (c_{D - m e t})_{0} \exp [- (K_{0} t)] γ_{D - A A O}}{K_{m}^{D - m e t} + \frac{K_{m}^{D - m e t}}{K_{i 2}^{2 - o x o}} c_{2 - o x o} (t) + (c_{D - m e t})_{0} (c_{2 - o x o})_{0} + (c_{D - m e t})_{0} \exp [- (K_{0} t)]}

r_{2} (t) = \frac{V_{m 2} . c_{2 - o x o} (t) . c_{N A D H} (t) . c_{N H_{4}^{+}} (t) . γ_{L - p h e D H}}{(K_{m}^{2 - o x o} + c_{2 - o x o} (t)) . (K_{m}^{N A D H} + c_{N A D H} (t)) . (K_{m}^{N H_{4}^{+}} + c_{N H_{4}^{+}} (t))}

r_{3} (t) = \frac{V_{m 3} . c_{L - m e t} (t) . c_{N A D^{+}} (t) . γ_{L - P h e D H}}{(K_{m}^{N A D^{+}} . (1 + c_{N A D H} (t) / K_{m}^{N A D^{+}}) + c_{N A D^{+}} (t)) . (K_{m}^{L - m e t} + c_{L - m e t} (t))}

c_{N A D^{+}} = (c_{N A D^{+}})_{0}, c_{N A D H} = (c_{N A D H})_{0}, c_{2 - o x o} = (c_{2 - o x o})_{0}, c_{L - m e t} = (c_{L - m e t})_{0}, c_{N H_{4}^{+}} = (c_{N H_{4}^{+}})_{0}, a t t = 0

r_{4} (t) = \frac{V_{m 4} . c_{F} (t) . c_{N A D^{+}} (t) . γ_{F D H}}{(K_{m 2}^{N A D^{+}} . (1 + c_{N A D H} (t) / K_{i 3}^{N A D H}) + c_{N A D^{+}} (t)) . (K_{m}^{F} + c_{F} (t))}

r_{1} (t) = \frac{V_{m 1} . c_{D - m e t} (t) . γ_{D - A A O}}{(K_{m}^{D - m e t} . (1 + c_{2 - o x o} (t) / K_{i 2}^{2 - o x o} + c_{N A D H} (t) / K_{i}^{N A D H}) + c_{D - m e t} (t))}

\frac{1}{r_{1}} = [\frac{K_{m}^{D - m e t} - K_{i}^{2 - o x o}}{V_{m 1} γ_{D - A A O} K_{i}^{2 - o x o}}] - [\frac{K_{m}^{D - m e t} (K_{i}^{2 - o x o} + c_{i})}{V_{m 1} γ_{D - A A O} K_{i}^{2 - o x o}}] \frac{1}{c_{D - m e t}}

b (1) = \frac{V_{m 1} γ_{D - A A O} K_{i 2}^{2 - o x o}}{α_{2}}, b (2) = \frac{V_{m 1} γ_{D - A A O} K_{i 2}^{2 - o x o} α_{1}}{α_{2}^{2}} α_{1} = K_{m}^{D - m e t} (K_{i 2}^{2 - o x o} + c_{i}), α_{2} = K_{i 2}^{2 - o x o} - K_{m}^{D - m e t} and c_{i} = (c_{D - m e t})_{0} + (c_{2 - o x o})_{0}

[1] *e-mail: dr.rajendran.l@gmail.com

Parameter	Value	Parameter	Value
D-AAO kinetics		FDH kinetics
V_m1	37.93 (U mg^-1) or 63.2167 (mmol dm^-3 mg^-1 min^-1)	V_m4	0.63 (U mg^-1) or 1.05 (mmol dm^-3 mg^-1 min^-1)
K_m^D-met	0.23 (mmol dm^-3)	K_m^F	12.80 (mmol dm^-3)
K_i^2-oxo	1.26 (mmol dm^-3)	K_m2^NAD+	0.039 (mmol dm^-3)
K_i^NADH	0.022 (mmol dm^-3)	K_i3^NADH	0.289 (mmol dm^-3)
γ _D-AAO	0.01 (mg)	γ _FDH	0.85 (mg)
L-pheDH kinetics
V_m2	59.211 (U mg^-1) or 98.65 (mmol dm^-3 mg^-1 min^-1)	V_m3	0.42 (U mg^-1) or 0.7 (mmol dm^-3 mg^-1 min^-1)
K _m^2-oxo	1.06 (mmol dm^-3)	K_m^L-met	4.02 (mmol dm^-3)
K_m^NADH	0.05 (mmol dm^-3)	K_m^NAD+	0.44 (mmol dm^-3)
K_m^NH₄+	76.99 (mmol dm^-3)	N_i2^NADH	0.0027 (mmol dm^-3)
γ _L-pheDH	10 (mg)

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Enzyme kinetic modelling and analytical solution of nonlinear rate equation in the transformation of D-methionine into L-methionine in batch reactor using the new homotopy perturbation method