doing a final pass for typos, etc. now
This is it, folks! Moment of truth. Time to find out how the proteins that you worked so hard to make, purify, and test really behave. Although you should be able to produce reasonable titration curves by following the example of Nagai, the introduction/review of binding constants below may help contextualize your analysis.
Let’s start by considering 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 reaction can be written
At equilibrium, the rates of the forward reaction (rate constant = kf) and reverse reaction (rate constant = kr) must be equivalent. Solving this equivalence yields an equilibrium dissociation constant KD, which may be defined either as kr / kf, 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 / RTOT, where RTOT 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, KD is always the same value. The total number of receptors RTOT= [C] (ligand-bound receptors) + [R] (unbound receptors). Thus,
where the right-hand equation was derived by algebraic substitution. If the ligand concentration is in excess of that 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 << KD?
- →Then y ~ L / KD, and the binding fraction increases in a first-order fashion, directly proportional to L.
- What happens when L >> KD?
- →In this case y ~1, so the binding fraction becomes approximately constant, and the receptors are saturated.
- What happens when L = KD?
- →Then y = 0.5, and the fraction of receptors that are bound to ligand is 50%. This is why you can read KD directly off of the plots in Nagai’s paper (compare Figure 3 and Table 1). When y = 0.5, the concentration of free calcium (our [L]) is equal to KD. This is a great rule of thumb to know.
The figures at below demonstrate how to read KD from binding curves. You will find semilog plots right particularly useful today, but the linear plot (left) can be a helpful visualization as well. Keep in mind that 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 KD.
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 KD
is equal to the ligand concentration [L]
for which y
Semilog Binding Curves
By converting ligand concentrations to logspace, the dissociation constants are readily determined from the sigmoidal curves' inflection points. The three curves each represent different ligand species. The middle curve has a KD
close to 10 nM, while the right-hand curve has a higher KD
and therefore lower affinity between ligand and receptor (vice-versa for the left-hand curve).
Of course, inverse pericam has multiple binding sites, and thus IPC-calcium binding is actually more complicated than in the example above. The KD reported by Nagai is called an ‘apparent KD’ 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 KD.
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). Said change in fluorescence requires the participation not only of calcium, but also of M13. So besides 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.
Returning to the big picture: when you write your research article, be sure to consider how changes in both binding affinity and cooperativity (and even potentially raw fluorescence differences) can affect the practical utility of a sensor.
Begin by applying the practice analysis from Day 3, Part 5 of this module to your real data. Recall that here you plot titration curves in Excel and make a first crude estimate of KD values. Next, you will use MATLAB to get improved estimates of KDs and also assess cooperativity. The MATLAB code is now up to date.
- Download these three files: S12_Fit_Main, Fit_SingleKD, and Fit_KDn. If using a computer in 16-336, the files will be available in the Downloads folder under your username. Move them to the username/Documents/MATLAB folder on your PC.
- 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 S12_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 functioname. 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 S12_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 S12_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 use in the model?
- What KD 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 S12_Fit_Main program. Part 2 also fits the data to a model with a single, ‘apparent’ value of KD, 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 KD compare for the two models? How does the value of n compare to the implicitly assumed value in Part 1?
- Do you see changes in binding affinity and/or cooperativity between the wild-type, E67K/T79P/M124S, 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 S12_Fit_Main program. Don’t worry too much about the coding details, but 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 KD and Hill coefficient (n) do you get for your three proteins? How do the KD’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 KD and/or n, and intercept and KD 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 KD's (or if you are interesting in exploring some sample data that is), download and run Fit_TwoKD and Fit_TwoKD_Func.
For next time
- Results outline
- Extra credit, OR REQUIRED THIS YEAR?: Prepare a figure and caption for your SDS-PAGE, along with a paragraph of text for the associated results section. Look up the expected molecular weight using the IPC sequence document and this or a similar website. Be sure to add ~ 3 KDa for the size of the N-terminus of pRSET (His tag, etc). If you see two strong bands, what do you think the second one is?