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BENG 449 Problem Set 7

# BENG 449 Problem Set 7 - BENG 449 Biomedical Data Analysis...

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BENG449BiomedicalDataAnalysis Unit7 Spring2008 Homework for Unit #7, Due Date: Friday April 4th o 1. If a ligand binds to a single receptor, the bound concentration B at equilibrium with total concentration B max and free ligand concentration L is: B = B max L L + K d Suppose the ligand binds to 2 binding sites, i.e., 2 different receptors B max (1) and B max (2) with dissociation equilibrium constants K d (1) and K d (2) , respectively. o a) What is the equation for the total bound ligand in this case. o b) Assume B max (1) and B max (2) are equal and K d (1) = 1 nM. Plot B vs. log L for 5 K d (2) values over the range from 1 to 100 nM. How different must the 2 K d values be to clearly see the presence of 2 binding sites. o 2. For enzyme catalyze reactions, the Michaelis-Menten rate equation for substrate C is: dC dt = .notdef.g0001 V max C K m + C o a) Use a numerical ODE solver in Matlab, to determine C(t) assuming that C(t=0)=C 0 =10, with V max = 1, K m = 5, and t ranges from 0 to 1000. o b) Integrate the above equation analytically to obtain an implicit solution, i.e., an expression in C and t, although not in the form C=… Use the fact that C(t=0)=C 0 . o c) With Matlab, check the accuracy of your implicit formula with the numerical solution for C(t), i.e,. calculate the left hand side and the right hand sides of your equation in b and compare. o 3. Using the receptor binding programs rec_occ.m or rec_occ2.m as a guide, perform 100 noisy simulations of data with 21 free ligand concentrations over the range 1 to 200, evenly spaced logarithmically (To evenly space values logarithmically from a to b, take log(a) and log(b), generate evenly spaced log values from log(a) to log(b) and exponentiate), B max = 25, K d = 10, and .notdef.g0005 = 1. Fit the data with the nonlinear model and the linear Scatchard reformulation. Present the sample mean and sample standard deviation of the two estimates. Compare the estimates from the linear and nonlinear methods. Discuss. o 4. Write a Matlab program to test the F-test for model comparison. Generate random data from one exponential (t=1,….,40, .notdef.g0002 1 exp(- .notdef.g0002 2 t), .notdef.g0002 1 = 20, .notdef.g0002 2 =.1, .notdef.g0005 = 2). Fit the data to one exponential (using exponential.m with 2 parameters) or one exponential plus a constant (3 parameters). Calculate the F statistic. Repeat the experiment 100 times. What fraction of the experiments did you consider to show a statistically significant difference at p < .05? o 5. Dunn, problem 7.2. Solve using ode45. o 6. (bonus) In problem 3, Calculate the predicted standard errors for the nonlinear and linear Scatchard fitting of B max and K d . You will have to use the formulas for functions of parameter estimates for the Scatchard fit. Do they agree with the sample standard deviations? If not, why not.

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4/3/08 5:34 PM MATLAB Command Window 1 of 2 >> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Alex Lemon % % BENG 449 -- Biomedial Data Analysis % % Problem Set 7, Question 1 % % Due April 4th, 2008 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Suppose that a ligand, L, binds to two different binding sites, B1 and % B2, with dissociation constants Kd1 and Kd2 respectively, and total % concentrations Bmax1 and Bmax2 respectively. Then the total bound ligand
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BENG 449 Problem Set 7 - BENG 449 Biomedical Data Analysis...

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