Biomedical Engineering assignment

timer Asked: Sep 20th, 2017

Question Description

the assignment is attached. It is a biomedical Engineering assignment. You need to have a knowledge in differential equations, chemistry, and biology to solve this assignment.

1. (20 pts) The standard Michaelis-Menten model of enzyme kinetics is based on the following reaction: k1 k2 E + S↔ ES → E + P k−1 where E is free enzyme, S is the substrate being reacted (reactant), ES is the enzyme-substrate bound state, and P is the reacted product. k1 is the reaction rate constant for the forward direction of the first step, k-1 is the reaction rate constant for the reverse direction of the first step, and k2 is reaction rate constant for the the forward direction of the second step. It is assumed that the second step is irreversible and the slow step of the process. ! !" ). Assume steady state for the reaction a. (5 pts) Provide the rate of change equation for [ES] ( !! (assume the rate of formation = the rate of breakdown for ES) with this equation to solve for [ES] using algebra (you shouldn’t have a differential equation after the steady state assumption). !! b. (3 pts) Write an equation for v, the velocity of the reaction (= , the rate of production of P). !" This will be the equation you eventually will solve to get the full M-M equation. c. (7 pts) Derive the Michaelis-Menten equation from these equations. This should all be algebra. State all assumptions. Give the formula for KM in terms of all reaction rate constants. d. (5 pts) Provide substrate concentrations (as a percentage or multiple of KM) for which: (i) the reaction velocity has less than 1% error with the approximate equation 𝑣 = 𝑣!"# ; and (ii) the reaction velocity has less than 1% error with the approximate equation 𝑣 = 𝑣!"# ! !! . Show the math that proves the error is within the required ranges (i.e. plug in your answers to the M-M equation and show that it is within 1% of the approximate version of the equation). 2. (25 pts) Your research team has just found a new potentially useful enzyme for the digestion of amyloid β plaques in the brains of Alzheimer’s disease. Before you can begin clinical trials of the enzyme, you need to measure its enzymatic characteristic parameters, vmax and KM. You have measured the reaction rate velocity at various substrate concentrations, given in the table below. [S] (mM) 1.25 2.5 5 10 15 25 40 50 60 70 a. Reaction velocity, v (mM/min) 0.13 0.201 0.376 0.683 1.030 1.254 1.565 1.609 1.712 1.807 [S] (mM) 85 100 115 130 155 175 195 250 300 350 Reaction velocity, v (mM/min) 1.759 1.920 1.899 1.963 1.964 2.080 2.096 2.010 2.138 2.149 (5 pts) Find the approximate Michaelis constant using the graphical approach (KM = [S] at v = ½ vmax). State how you assumed a value for 𝑣!"# and the assumption you use to determine the substrate concentration that corresponds to the Michaelis constant (𝐾! ). b. (7.5 pts) A more exact method for determining the Michaelis constant and the maximum velocity of the reaction is to linearize the Michaelis-Menten equation by (properly) inverting both sides of the equation. Show this inversion and the final equation in a linear form (y = m·x +b). Show all of your intermediate steps. Note that y and x will not be the same as the M-M equation. c. (12.5 pts) Using your linearized M-M equation, determine vmax and KM (easiest using trendline in Excel). 3. (10 pts) A patient with a blood volume of 35L is given an I.V. injection of 200mg of drug (MW=30,000 Daltons) at time t=0. The drug is metabolized and/or excreted with a rate constant of kE = 0.15/hr. Use the single compartment model for drug concentration to plot for the blood concentration (in M) vs. number of hours post-injection. 4. (20 pts) A human bladder can hold 15.5 ounces of urine from the kidney. A patient has a chronic bladder leak due to insufficient closure on the urethra, with a leak rate constant, kl. The patient has a constant input rate from the kidneys, R. Solve a single compartment model of the patient’s bladder: a. (5 pts) First, write a change equation for the liquid volume ( !" !" = 𝑖𝑛𝑝𝑢𝑡 − 𝑜𝑢𝑡𝑝𝑢𝑡 + 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 − 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛) and put it into standard differential equation form dx(t)/dt + Ax(t)=B. b. (10 pts) Now, you should find the integration factor and solve the differential equation. Solve for the integration constant by using an initial volume within the bladder of V0. c. (5 pts) Given kl = 0.015/min, V0 = 0.65 ounces, and R = 0.275 ounce/minute, determine the amount of time until the patient has a full bladder and will need to relieve themselves (i.e. when there is 15.5 ounces in the bladder). 5. (25 pts) Pharmacokinetics modeling: You have invented a device that allows drug delivery through a patient’s skin using localized permeation enhancement. You want to demonstrate the advantages your system has over oral drug delivery. Your device consists of a reservoir of drug in a saline bag attached to the patient’s arm using an adhesive, can be refilled continuously to maintain a constant concentration and volume, and transport is a slow absorption through the skin. a. (17.5 pts) Derive a single compartment pharmacokinetics model assuming a constant concentration of drug coming through the skin into the patient’s arm (single compartment is the plasma). Define all your variables and state any assumptions. b. (7.5 pts) Plot the concentration in the body as a function of time for up to 7.5 hours. Assume a bag volume of 0.275L, a body (plasma) volume of 14.0L, a constant concentration of 2.5M in the bag, an absorption rate constant of 0.225 1/hour, and a plasma elimination rate constant of 0.95 1/hour. The initial drug concentration in the body is 0.001 mM.

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