Difference between revisions of "20.109(S21):M3D3"
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'''Semilog binding curves''' <br> | '''Semilog binding curves''' <br> | ||
− | By converting ligand concentrations to logspace, the dissociation constant is readily determined from the inflection point of the sigmoidal curve. The three curves each represent different ligand species. The middle curve has a <math>K_d</math> close to 10 nM, while the right-hand curve has a higher <math>K_d</math> and therefore lower affinity between ligand and receptor (vice-versa for the left-hand curve). | + | By converting ligand concentrations to logspace, the dissociation constant is readily determined from the inflection point of the sigmoidal curve. The three curves each represent different ligand species. The middle curve has a <math>K_d</math> close to 10 nM, while the right-hand curve has a higher <math>K_d</math> and therefore lower affinity between ligand and receptor (vice-versa for the left-hand curve). |
[[File:Sp21 M3D3 binding curves.png|thumb|600px|center|'''Binding curves are used to determine <math>K_d</math>.''' Titration data can be visualized using linear, or simple, plots (A) or semilog plots (B). In both representations, every ''L'' value is associated with a particular equilbrium value of ''y'', while the curve as a whole gives information on the global equilibrium constant <math>K_d</math>.]] | [[File:Sp21 M3D3 binding curves.png|thumb|600px|center|'''Binding curves are used to determine <math>K_d</math>.''' Titration data can be visualized using linear, or simple, plots (A) or semilog plots (B). In both representations, every ''L'' value is associated with a particular equilbrium value of ''y'', while the curve as a whole gives information on the global equilibrium constant <math>K_d</math>.]] | ||
− | + | Because inverse pericam has multiple binding sites the IPC-calcium binding is more complicated than the example above. The <math>K_d</math> reported in the IPC reference article by Nagai ''et. al.'' is called an ‘apparent <math>K_d</math>’ because it reflects the overall avidity of multiple calcium binding sites, not their individual affinities for calcium. Normally, calmodulin has a low affinity (N-terminus) and a high affinity (C-terminus) pair of calcium binding sites. However, the E104Q mutant, which is the version of CaM used in inverse pericam, displays low affinity binding at both termini. Moreover, the Hill coefficient, which quantifies cooperativity of binding in the case of multiple sites, is reported to be 1.0 for inverse pericam. This indicates that inverse pericam behaves as if it were binding only a single calcium ion per molecule. Thus, wild-type IPC is well-described by a single apparent <math>K_d</math>. | |
− | + | ||
For any given mutant, things may be more complicated. Keep in mind that we are not directly measuring calcium binding, but instead are indirectly inferring it based on fluorescence (for both mutant and wild-type IPC). A change in fluorescence requires the participation not only of calcium, but also of M13. In addition to the four separate calcium binding sites in calmodulin, the M13 binding site influences apparent affinity and apparent cooperativity. In short, be careful about how you describe the meanings of our binding parameters in your reports. | For any given mutant, things may be more complicated. Keep in mind that we are not directly measuring calcium binding, but instead are indirectly inferring it based on fluorescence (for both mutant and wild-type IPC). A change in fluorescence requires the participation not only of calcium, but also of M13. In addition to the four separate calcium binding sites in calmodulin, the M13 binding site influences apparent affinity and apparent cooperativity. In short, be careful about how you describe the meanings of our binding parameters in your reports. | ||
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==Protocols== | ==Protocols== |
Revision as of 19:37, 22 April 2021
Contents
Introduction
To test the effects of the mutations in the Variant IPC proteins, we will revisit several concepts that were discussed in Mod1. Specifically, a titration curve is used to calculate $ K_d $ of IPC binding to calcium! As a review, consider the simple case of a receptor-ligand pair that are exclusive to each other, and in which the receptor is monovalent. The ligand (L) and receptor (R) form a complex (C), which can be written
$ R + L \rightleftharpoons\ ^{k_f}_{k_r} C $
At equilibrium, the rates of the forward reaction (rate constant = $ k_f $) and reverse reaction (rate constant = $ k_r $) must be equivalent. Solving this equivalence yields an equilibrium dissociation constant $ K_d $, which may be defined either as $ k_r/k_f $, or as $ [R][L]/[C] $, where brackets indicate the molar concentration of a species. Meanwhile, the fraction of receptors that are bound to ligand at equilibrium, often called y or θ, is $ C/R_{TOT} $, where $ R_{TOT} $ indicates total (both bound and unbound) receptors. Note that the position of the equilibrium (i.e., y) depends on the starting concentrations of the reactants; however, $ K_d $ is always the same value. The total number of receptors $ R_{TOT} $= [C] (ligand-bound receptors) + [R] (unbound receptors). Thus,
$ \qquad y = {[C] \over R_{TOT}} \qquad = \qquad {[C] \over [C] + [R]} \qquad = \qquad {[L] \over [L] + [K_d]} \qquad $
where the right-hand equation was derived by algebraic substitution. If the ligand concentration is in excess of the concentration of the receptor, [L] may be approximated as a constant, L, for any given equilibrium. Let’s explore the implications of this result:
- What happens when L << $ K_d $?
- →Then y ~ $ L/K_d $, and the binding fraction increases in a first-order fashion, directly proportional to L.
- What happens when L >> $ K_d $?
- →In this case y ~1, so the binding fraction becomes approximately constant, and the receptors are saturated.
- What happens when L = $ K_d $?
- →Then y = 0.5, and the fraction of receptors that are bound to ligand is 50%. When y = 0.5, the concentration of free calcium (our [L]) is equal to $ K_d $.
The figures below demonstrate how to read $ K_d $ from binding curves:
Simple binding curve
The binding fraction y at first increases linearly as the starting ligand concentration is increased, then asymptotically approaches full saturation (y=1). The dissociation constant $ K_d $ is equal to the ligand concentration [L] for which y = 1/2.
Semilog binding curves
By converting ligand concentrations to logspace, the dissociation constant is readily determined from the inflection point of the sigmoidal curve. The three curves each represent different ligand species. The middle curve has a $ K_d $ close to 10 nM, while the right-hand curve has a higher $ K_d $ and therefore lower affinity between ligand and receptor (vice-versa for the left-hand curve).
![](/w/images/thumb/b/b5/Sp21_M3D3_binding_curves.png/600px-Sp21_M3D3_binding_curves.png)
Because inverse pericam has multiple binding sites the IPC-calcium binding is more complicated than the example above. The $ K_d $ reported in the IPC reference article by Nagai et. al. is called an ‘apparent $ K_d $’ because it reflects the overall avidity of multiple calcium binding sites, not their individual affinities for calcium. Normally, calmodulin has a low affinity (N-terminus) and a high affinity (C-terminus) pair of calcium binding sites. However, the E104Q mutant, which is the version of CaM used in inverse pericam, displays low affinity binding at both termini. Moreover, the Hill coefficient, which quantifies cooperativity of binding in the case of multiple sites, is reported to be 1.0 for inverse pericam. This indicates that inverse pericam behaves as if it were binding only a single calcium ion per molecule. Thus, wild-type IPC is well-described by a single apparent $ K_d $.
For any given mutant, things may be more complicated. Keep in mind that we are not directly measuring calcium binding, but instead are indirectly inferring it based on fluorescence (for both mutant and wild-type IPC). A change in fluorescence requires the participation not only of calcium, but also of M13. In addition to the four separate calcium binding sites in calmodulin, the M13 binding site influences apparent affinity and apparent cooperativity. In short, be careful about how you describe the meanings of our binding parameters in your reports.
Protocols
Part 1: Participate in Comm Lab workshop
Our communication instructors, Dr. Prerna Bhargava and Dr. Sean Clarke, will join us today for a discussion on preparing a Research proposal presentation.
Part 2: Prepare samples for titration curve
- Obtain a black 96-well plate and review the plate map schematic at right: the top two rows will be loaded with WT IPC, the next two rows will be loaded with your X#Z Variant IPC, and the final row will be loaded with BSA.
- BSA will be used to measure the background signal.
- A black plate is used to minimize "cross-talk" (i.e., light leakage) between samples in adjacent wells.
- Aliquot the WT IPC protein into the appropriate wells of the plate.
- Add 30 μL of the purified protein to each well in rows A and B.
- Add 30 μL of the purified protein to each well in rows C and D.
- Aliquot 30 μL of 0.1% BSA to each of the wells in row E.
- The calcium solutions are at the front bench in reservoirs that will be shared across the class.
- Carefully carry your plate to the front bench to add these solutions.
- From shared reservoir #1 (calcium concentration = 0 nM), use the multi-channel pipet to add 30 μL to the top five rows in the first column of the plate. Discard the pipet tips.
- Work from reservoirs #2 to #12 (lowest calcium concentration to highest calcium concentration), and from the left-hand to the right-hand columns on your plate.
- Be sure to use fresh pipet tips each time! If you do contaminate a solution, let the Instructor know so the solution can be replaced. Honesty about a mistake is far preferred here to affecting every downstream experiment.
- When you are done, alert the Instructor and you will be taken in small groups to measure the fluorescence values for your samples.
- Samples will be measured using a platereader.
- Settings included an excitation wavelength of 485 nm and an emission wavelength at 515 nm.
In your laboratory notebook, complete the following:
- Recall your analysis of the SDS-PAGE and microBCA completed on M3D1. Were the Variant IPC and WT IPC concentrations consistent / similar across each of the purified protein samples?
- Given that the same volume is added to each the titrations completed with each of Variant IPC and WT IPC (30 µL of each purified protein sample was added), might this impact the results from the titration curve experiment? How?
- What change(s) could be made to the protocol to ensure the amount of Variant IPT or WT IPC was used in each titration curve experiment? Be specific on how you might address this issue.
Part 3: Analyze titration curve data
You will analyze your calcium titration assay data in two steps. First, you will get a rough sense for how the mutations changed (or didn't) binding activity compared to WT IPC by plotting the averages of the two replicate values for each Variant IPC, in both raw and processed form. Second, you will take the average processed values and use MATLAB code that will more precisely determine the affinity and cooperativity for the Variant IPC and WT IPC with respect to calcium.
The spreadsheet with the data that will be used for the analysis is linked [[ |here]].
Plot titration curve in Excel to determine first estimate of Kd
Today you will analyze the fluorescence data that you got last time. Begin by analyzing the wild-type protein as a check on your work (your curve should resemble Nagai's Figure 3L), and then move on to your mutant samples. If you are not familiar with manipulations in Excel, use the Help menu or ask the teaching faculty for assistance.
- Open an Excel file for your data analysis. Begin by making a column of the free calcium concentrations present in your twelve test solutions. Assuming a 1:1 dilution of protein with calcium, the final concentrations are: 0 nM, 8.5 nM, 19 nM, 32.5 nM, 50 nM, 75 nM, 112.5 nM, 175 nM, 301 nM, 675 nM, 1.505 μM, 19.5 μM. Be sure to convert all concentrations to the same units.
- Now open the text file containing your raw data as a tab-delimited file in Excel (you can download the file from the M1D7 Discussion page). Convert the row-wise data to column-wise data (using Paste Special → Transpose), and transfer each column to your analysis file. Add column headers to indicate which protein is which, and analyze each replicate separately for now. Also include a column of your control samples that did not contain protein.
- Begin by calculating the average of your blank samples, and bold this number for easy reference. It is the background fluorescence present in the calcium solutions and should be quite low. If necessary, subtract this background value from each of your raw data values. It may help to have a 6-column series called “RAW”, and another called “SUBTRACTED.”
- Next you should normalize your data. The maximum and minimum fluorescence values for a given titration series should be defined as 100% and 0% fluorescence, respectively, and every other fluorescence value should be expressed as a percentage in between. Think about how to mathematically express these conditions.
- First calculate the percent fluorescence for both replicates. Then make a new column and calculate the average percentage as well.
- Alternatively, average your data first, and then normalize the average data. How do the "average then normalize" and the "normalize then average" curves compare? Which one will you include in your Protein engineering summary?
- If one data point seems really off from the other replicate and from the expected trend, you might consider it an outlier and delete it, especially if you have good reason to believe that there was a reason (error in pipetting, air bubble in that well) for the anomaly. Otherwise, you might be losing valuable information, and/or misleading anyone who tries to interpret your data.
- For each protein, plot this normalized data versus calcium concentration. Save these plots in case you want to include them in your report.
- You might plot the two replicates as points and their average value as a dashed line (see front page of this module).
- Note down the approximate inflection points of the curves, which should occur at half-saturation: these indicate the approximate values of the apparent $ K_d $ for each sample.
Use MATLAB modeling to estimate Kd
- Download these three files: F15_Fit_Main, Fit_SingleKD, and Fit_KDn. Move them to the username/Documents/MATLAB folder on your computer.
- Double-click on the MATLAB icon to start up this software.
- The main window that opens is called the command window: here is where you run programs (or directly input commands) and view outputs. You can also see and access the command history, workspace, and current directory windows, but you likely won’t need to today.
- In the command window, type more on; this command allows you to scroll through multi-page output (using the spacebar), such as help files.
- In addition to the command area, MATLAB comes with an editor. Click File → Open and select the program F15_Fit_Main. It has the .m extension and thus is executable by MATLAB. Read the introductory comments (the beginning of a comment is indicated by a % sign), and then input your fluorescence data.
- Read through the program, and as you encounter unfamiliar terms, return to the workspace and type help functionname. Feel free to ask questions of the teaching faculty as well.
- You might read about such built-in functions as logspace and nlinfit.
- You will also want to open and read Fit_SingleKD – a user-defined function called by F15_Fit_Main – in the MATLAB editor.
- If you type help function you will learn the syntax for a function header.
- Note that a dot preceeding an operator (such as A ./ B or A .* B) is a way of telling MATLAB to perform element-by-element rather than matrix algebra.
- Also note that when a line of code is not followed by a semi-colon, the value(s) resulting from the operation will be displayed in the command window.
- Once you more-or-less follow Part 1 of the program, type F15_Fit_Main in the workspace, hit return to run the program, and consider the following questions:
- Why must the fluorescence data be transformed (from S to Y) prior to using in the model?
- What $ K_d $ values are output in the command window, and how do they compare to the values you estimated from your Excel plots?
- Figure 1 should display your wild type and mutant data points and model curves. How do they look in comparison to the curves you plotted in Excel?
- Figure 2 should display the residuals (difference between data and model) for your three proteins. If the absolute values are low, this indicates good agreement between the model and the data numerically. Whether or not this is the case, another thing to look for is whether the residuals are evenly and randomly distributed about the zero-line. If there is a pattern to the errors, likely there is a systematic difference between the data and the model, and thus the model does not reflect the actual binding process well. What are the residuals like for each of your modeled proteins?
- Now move on to Part 2 of the F15_Fit_Main program. Part 2 also fits the data to a model with a single, ‘apparent’ value of $ K_d $, but it allows for multiple binding sites and tests for cooperativity among them. The parameter used to measure cooperativity is called the Hill coefficient. A Hill coefficient of 1 indicates independent binding sites, while greater or lesser values reflect positive or negative cooperativity, respectively. Let the following questions guide you as you proceed:
- Visually, which model appears to fit your wild-type data better (Fig. 3 vs. Fig. 1)? Your mutant data?
- Do the respective residuals support your qualitative assessment (Fig. 4 vs. Fig. 2)?
- Numerically, how do the values of $ K_d $ compare for the two models? How does the value of n compare to the implicitly assumed value of 1 in Part 1?
- Do you see changes in binding affinity and/or cooperativity between the wild-type and X#Z samples? Do they match your a priori predictions?
- Don't forget to save any figures you want to use in your report! If the legends are covering up your data, you can simply move them over with your mouse.
- Finally, you can skim Part 3 of the F15_Fit_Main program. Make sure you update the range of the linear transition region for each IPC sample, but beyond this, don’t worry too much about the coding details; rather do read through the comments.
- Look at Part 1 of Figure 5: are the binding curves asymptotic, sigmoidal, or other? What does this shape indicate? You can use the zoom button to get a closer look at part of the plot, or the axis command present in the code. (Don't worry too much about this question if it is unclear.)
- Now look in the command window. What values of $ K_d $ and Hill coefficient (n) do you get for your three proteins? How do the $ K_d $’s from Part 3 compare to the ones from Parts 1 and 2? Don’t be discouraged if your wild-type values do not exactly match Nagai’s work, or if there is variation between Parts 1, 2, and 3.
- Comparing the model and data points by eye (Part 2 of Figure 5), do you think it is a good model for any of your proteins? If so, which ones? What experimental limitations might prevent Hill analysis from working well, especially for some mutants?
- Why should only the transition region be analyzed in a Hill plot?
- What is the relationship between slope and $ K_d $ and/or n, and intercept and $ K_d $ and/or n?
- If your mutant proteins are not well-described by any of the models so far, what kind of model(s) (qualitatively speaking) do you think might be useful?
- Optional: If your data might be well-described by a model with two $ K_d $'s (or if you are interesting in exploring some sample data that is), download and run Fit_TwoKD and Fit_TwoKD_Func.
Next day: Design new IPC variant