代写BioSci 746: Assignment 2 (Reversible Molecular Interactions)调试Haskell程序

日期：2024-09-12 11:05

BioSci 746: Assignment 2 (Reversible Molecular Interactions)

Question 1 (60%).  The binding of ATP to a protein was studied in the presence and absence of Magnesium using intrinsic tryptophan fluorescence. The protein was titrated with ATP and fluorescence data collected under equilibrium conditions using the appropriate excitation and emission wavelengths. The observed fluorescence data were expressed relative to the protein fluorescence in the absence of any ligand. The following tables give the data as a function of the total concentrations of protein [P_T] and Ligand [L_T]

ATP binding with no Magnesium

[PT] μM	[LT] μM	Frelative
1	0	1
1	10.038	0.985
1	20.382	0.964
1	30.137	0.942
1	75.403	0.913
1	100.157	0.89
1	150.343	0.877
1	199.825	0.871

ATP binding with Magnesium

[PT] μM	[LT] μM	Frelative
1	2.064	0.988
1	5.315	0.978
1	10.237	0.96
1	20.211	0.931
1	30.324	0.912
1	40.309	0.899
1	60.321	0.879
1	80.513	0.874
1	100.216	0.868
1	150.558	0.86

Part A. Obviously this experiment has been conducted with a considerable excess of ligand over protein. Also - from simple inspection of the data  - we can see that the equilibrium dissociation constant appears to be in the “micromolar” range.  Assuming that the equilibrium dissociation constant is  ~ 50 x 10^-6 M (50 μM) calculate the percentage error that results when you approximate the free ligand concentration [L] with the total ligand concentration [L_T]. Consider the situation at both the beginning of the titrations ( take [P_T] = 1.0 μM  and [L_T] = 2 μM) and the end of the titrations ( take [P_T] = 1.0 μM  and [L_T] = 200 μM). Based on this, decide if the approximation is justified and can be carried forward into part B.

Part B. Assuming a simple 1:1 binding scheme, use the expression for the fluorescence signal developed in class to fit these two experimental data sets, using non-linear least squares, hence determining the equilibrium constant governing binding of ATP, as well as the Fluorescence enhancement factor F’_PL associated with binding.  The Microsoft Excel Solver add-in can be used for this purpose (Kemmer and Keller, Nature Protocols (2010) 5, 267-281). However any other software can be employed so long as you document your procedure.  

Your report should begin with a brief description of the procedure used to find the minimum sum-of-squares residuals (the model fitted and the software employed).

Then for each data set report:

I) Your best estimates for K_D and  F’_PL obtained from the model fitting process, and the corresponding sum-of-squares residuals calculated for all data

II) A plot showing the experimental data and the best fit model. On the Y axis report F_relative . On the X axis report either the total Ligand concentration [L_T] or the molar ratio of the reactants [L_T]/[P_T]. By convention use symbols (like filled or hollow circles) to represent the experimental data, which is measured at discrete points. Use a solid line to represent the model, since the predictions of the model are continuous. That's the standard approach in the literature, and one that makes it easy to visualize the overall characteristics of the model, and its agreement with the data.

III) A plot of the residuals - that is the difference between the model and experiment for each data point (Note - do not plot the squared residuals here). Comment on whether the residuals appear random, or indicate systematic departures between data and model.

IV) Some assessment of how well K_D and F’_PL  are determined by the data, as discussed in Kemmer and Keller (2010). How reliable are the parameter estimates ?

Conclude by addressing the following pair of biological questions:

1. The authors of the study claim that ATP binding is modestly but significantly strengthened in the presence of Magnesium. Is that conclusion supported by the data ?

2. Is there any evidence that Magnesium influences the intrinsic tryptophan fluorescence of the protein ?

Question 2 (10%). Consider a protein that self-associates according to the following monomer-trimer scheme

Give the equations that would be combined to derive an expression for the trimer concentration [A₃] in terms of total amount of protein present [A_T], and the equilibrium dissociation constant (K_D) governing the reaction.  Then, show the first step in substituting one equation into the other so that an expression for [A₃] could be derived.  You do not need to rearrange this into the final cubic equation or find the analytical solutions of that equation.

Similarly, give the equations that would be combined to derive an expression for the monomer concentration [A] in terms of total amount of protein present [A_T], and the equilibrium association constant (K_A) governing the reaction. Then, show the first step in substituting one equation into the other so that an expression for [A] could be derived. Again, you do not need to rearrange this into the final cubic equation or find the analytical solutions of that equation.

Hint: See the supplementary material at the bottom of the Module 2 page for the approach to deriving equations for a protein that self-associates according to a monomer-dimer scheme.

Question 3 (30%) Consider the formation of a ternary complex ABC. B and C can’t bind to one another in the absence of A. Through experiment, the following dissociation constants have been determined at 25 °C

Draw two thermodynamic cycles representing formation of the ternary complex, showing the standard Gibbs energies of binding associated with each step.

For the first of these cycles,  represent each step with a single equilibrium constant and and associated Gibbs Energy term. What is the coupling Gibbs energy (ΔΔG)? Is binding of B and C to A positively cooperative, negatively cooperative, or independent?

For the second of these cycles, eliminate the redundancy in the parameters, and describe the binding cycle using two equilibrium constants and an interaction parameter (and associated Gibbs Energy terms). In this formulation, what is the magnitude and sign of the interaction term, when expressed as a Gibbs Energy?