A Mathematical Model for Enzyme Kinetics: The Runge-Kutta Method Justin Saliba Advisor: Muhammed Usman. Ph.D Abstract In this work we study a basic mathematical model for enzyme-substrate reaction. We summarize the enzyme kinetics model from J. D. Murray's Mathematical Biology book. We simplify the equations via nondimensionalization. Then we use the numerical solver called the Runge-Kutta methods to solve the system of differential equations describing the reaction. Introduction An enzyme is a protein molecule that catalyzes a biochemical reaction [1]. Enzymes can act as activators or inhibitors and regulate many biological processes. A useful enzyme in your body is Pepsin. This enzyme is found in your digestive track and helps metabolize your food. Enzyme Kinetics is the study of how an enzyme effects the speed at which a reaction occurs. My goal is to study a mathematical model of enzyme-substrate reactions in order to better understand their kinetics. Using the law of mass Action, Using the Runge-Kutta Method to Solve for the Reduced System: K1= [SE]/[S][E] K_1=[S][E]/[SE] K2=[P][E]/[SE] The following graphs of numerical solutions to system (1) were generated using Matlab. The numerical solutions were found using ode45. you can derive the following equations: s0 = 5, t=10, e0 = 1, k_1 = 1, k1 = 2 and k2 = 3 Let s=[S], e=[E],c=[SE], and p=[P] where [ ] denotes the concentration. From [1]: S’=-k1*e*s + k-1*c E’=k1*e*s + (k-1+k2)*c C’=k1*e*s – (k-1+k2)*c P’=k2*c S(0)=s0, e(0)=e0, c(0)=0, p(0)=0 Michaelis and Menten’s Enzyme Model S=Substrate; E=Enzyme; ES=Substrate-enzyme complex P=Product k1,k-1,k2 are all rate constants This is a basic enzyme reaction model that shows that the substrate-enzyme complex can decompose back into enzyme and substrate or may form one molecule of the product and one molecule of the enzyme [1,2]. C’+E’=0 Thus: c(t)+e(t)=e0 The following system of equations can now be reduced: s0=3, t=5 e0=1, k_1 = 1, k1=2 and k2 =3 S’= -k1*e0*s + (k1*s + k-1)c C’= k1*e0*s – (k1s + k-1 + k2)c (1) S(0) = s0, c(0)= 0 The Runge-Kutta Method Yn+1=Yn +1/6(K11+2*K12+2*K13+K14) K11 = h*f(Xn,Yn) K12 = h*f(Xn + 1/2h, Yn + 1/2K11) K13 = h*f (Xn + 1/2h, Yn + 1/2K12) K14 = h*f(Xn + h,Yn + k13) Conclusion: After studying a system of differential equations for enzyme-substrate reactions, I have found that the solutions can be approximated using the Runge-Kutta numerical method. Finding out the time or concentration for certain substrate-enzyme mechanisms to reach equilibrium can be experimentally obtained, but that will lose you both time and money. By understanding Michaelis and Menten’s Enzyme Model, and by using the Law of Mass Action, you can obtain differential equations that can be solved numerically with the Runge-Kutta method to approximate rates, concentrations, and equilibrium constants in enzyme kinetics. References: [1]. Muniz-Alicea, Roberto. "A Mathematical Model for Enzyme Kinetics: Multiple Timescales Analysis." Dynamics at the Horsetooth Focused Issue: Asymptotics and Pertubations 2A (2010) [2]. J. D. Murray, Mathematical Biology.