The Michaelis-Menten model describes the rate of an enzyme-mediated reaction: [E] + [9] kf [E]+[S][ES] kr...

  

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The Michaelis-Menten model describes the rate of an enzyme-mediated reaction: [E] + [9] kf [E]+[S][ES] kr Keat -> [E] + [P] where [E] is the enzyme concentration, [S] is the substrate concentration, [ES] is the complex concentration, [P] is the concentration of the product, ky is the forward rate constant, k, is the reverse rate constant, and keat is the catalytic rate constant. Constant Value Units k 0.001 S-1 kr 0.0001 S-1 kcat 0.1 S-1 [E]o 50 M Slo 100 M 0 M 0 0 M [ES] [P] Table 1: Rate constants and initial conditions for all reactions and species, respectively. Your tasks: a.) By hand, use the law of mass action to derive the four ODEs that describe the change in concentrations of [E], [S], [ES] and [P]. Use the constants and initial conditions in Table 1. b.) If possible, write the system of ODEs in matrix form and analytically solve the eigenvalue problem to obtain each concentration as a function of time. If it is not possible to analytically solve, show why by hand and then proceed to step c.) to solve numerically. c.) Use Euler's Method to obtain approximate concentration values for each species ([E], [S], [ES] and [P]) from 0 to 150 seconds. d.) Overlay the concentration values for all four species onto a single plot. Be sure to format your plot correctly including a legend and axes labels. See my plot template for help. e.) Use MATLAB's built-in ode45 function to solve the same problem for comparison. Plot results in the same exact way as you did for Euler's method, but in a different figure. Show that you obtain identical results. The Michaelis-Menten model describes the rate of an enzyme-mediated reaction: [E] + [9] kf [E]+[S][ES] kr Keat -> [E] + [P] where [E] is the enzyme concentration, [S] is the substrate concentration, [ES] is the complex concentration, [P] is the concentration of the product, ky is the forward rate constant, k, is the reverse rate constant, and keat is the catalytic rate constant. Constant Value Units k 0.001 S-1 kr 0.0001 S-1 kcat 0.1 S-1 [E]o 50 M Slo 100 M 0 M 0 0 M [ES] [P] Table 1: Rate constants and initial conditions for all reactions and species, respectively. Your tasks: a.) By hand, use the law of mass action to derive the four ODEs that describe the change in concentrations of [E], [S], [ES] and [P]. Use the constants and initial conditions in Table 1. b.) If possible, write the system of ODEs in matrix form and analytically solve the eigenvalue problem to obtain each concentration as a function of time. If it is not possible to analytically solve, show why by hand and then proceed to step c.) to solve numerically. c.) Use Euler's Method to obtain approximate concentration values for each species ([E], [S], [ES] and [P]) from 0 to 150 seconds. d.) Overlay the concentration values for all four species onto a single plot. Be sure to format your plot correctly including a legend and axes labels. See my plot template for help. e.) Use MATLAB's built-in ode45 function to solve the same problem for comparison. Plot results in the same exact way as you did for Euler's method, but in a different figure. Show that you obtain identical results.