20.109(S20):Analyze titration curves (Day5)

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20.109(S20): Laboratory Fundamentals of Biological Engineering

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Spring 2020 schedule        FYI        Assignments        Homework        Class data        Communication
       1. Screening ligand binding        2. Measuring gene expression        3. Engineering antibodies              


Introduction

The antigen, lysozyme (L), and antibody (Ab) form a complex (C), which can be written

$ L + Ab \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 $ [Ab][L]/[C] $, where brackets indicate the molar concentration of a species. Meanwhile, the fraction of antibody that are bound to antigen at equilibrium, often called y or θ, is $ C/Ab_{TOT} $, where $ Ab_{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 antibody $ Ab_{TOT} $= [C] (L-bound Ab) + [Ab] (unbound Ab). Thus,

$ \qquad y = {[C] \over Ab_{TOT}} \qquad = \qquad {[C] \over [C] + [Ab]} \qquad = \qquad {[L] \over [L] + [K_d]} \qquad $

where the right-hand equation was derived by algebraic substitution. If the antigen concentration is in excess of the concentration of the antibody, [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 antibodies are saturated.
  • What happens when L = $ K_d $?
→Then y = 0.5, and the fraction of antibodies that are bound to ligand is 50%. This is why you can read $ K_d $ directly off of the plots. When y = 0.5, the concentration of lysozyme (our [L]) is equal to $ K_d $. This is a great rule of thumb to know.

The figures below demonstrate how to read $ K_d $ 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 $ K_d $.

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 antigen 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 antibody clones. 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).


Protocols

Today, we will be analyzing the flow cytometry data that we collected last class in order to assess the binding of your scFv to lysozyme as compared to the control scFv Ab14989. On M3D3, you compared the sequences to the control scFv DNA from Ab31375 which has a Kd of 6 nM for binding to lysozyme. To assess differences in Kd between our data and the control, we have elected to use control scFv Ab14989 which has a Kd of 650 nM for binding to lysozyme.

Part 1: Analyze flow cytometry scatter plots using FlowJo and generate MFI table

Download FlowJo on to your computer and activate free 30-day trial

  1. Go to https://www.flowjo.com/solutions/flowjo/free-trial and click “Download”. Complete the software download instructions to install FlowJo v10.6.2 for your system (Mac or Windows).
  2. Once the software installation has finished, open the program and locate your computer’s hardware ID number in the license pop-up window. Complete the form using your email and ID number at the above link. If the license pop-up window does not automatically appear with your ID number, instructions to find the number through FlowJo are listed here: https://docs.flowjo.com/flowjo/faq/general-faq/locating-hwa/.
  3. Use the serial number sent to your email to activate your 30-day free trial.

Analyze scFv Flow Cytometry Titration Samples in FlowJo

  1. Download the following zipfile of flow cytometry samples (The file is also linked to the Class Data page). Unzip and save the files on your computer.
  2. When you have activated your trial, a blank workspace should have opened up in FlowJo. If not, open the FlowJo_v10.6.2 program and create a new workspace by clicking the “new” button.
  3. Add all of the sample files to your workspace (FlowJo Tab --> Add samples)
  4. Click on the first sample to open FSC-A vs. SSC-A plot. Change the axis to log scale by pressing the large [T] buttons by the axis labels. Use the oval gating tool at the top to select yeast cells (and exclude extraneous particles). Label this population as “yeast”. Your plot should look like Figure 1 below:

    Protocol Figure 1

  5. Close the graph window and go back to the main workspace page and click on now gated “yeast” population below the first sample. This will open up a new graph with only the gated cells.
  6. We will now change our axis on the yeast scatterplot to analyze the data based on the fluorescence markers used in the secondary staining. Change the x-axis of the plot from “FSC-A::FSC-A” to “FL1-A::FL1-A” and the y-axis of the plot from “SSC-A::SSC-A” to “FL4-A::FL4-A”.
    • The FL1-A channel and x-axis will measure the fluorescence intensity of the AF488 marker on our cells. Thus, dots farther to the right on the x axis represent cells that displayed more scFv on their surface.
    • The FL4-A channel and y-axis will measure the fluorescence intensity of the AF647 marker on our cells. Thus, dots higher on the y axis represent cells with scFvs that bound lysozyme.
  7. We will now set two gates: yeast that displayed scFvs and yeast that did not display scFv. Despite the induction with galactose media not all yeast cells will display scFv on their surface (ex. Dead cells, budding cells). We want to analyze only the cells that displayed scFvs on their surface (right population) but we can use the non-displaying cell population (left population) to subtract the background fluorescence signal.
    • Use the rectangle gating tool at the top to make a “displaying yeast” population and a “non-displaying yeast” population. Your plot should look like Figure 2 below:

      Protocol Figure 2

  8. We will now add statistics to our graph to obtain the Median Fluorescence Intensity of the y-axis AF647 binding signal. Under the “Edit” tab, choose “Add a Statistic”. A pop-up menu will appear.
    • Choose the “displaying yeast” as your population drop-down option.
    • Choose “Median” in the statistic options menu on the left.
    • Choose “FL4-A::FL4-A” as the statistic options menu on the left and select “Add” at the bottom.
    • Repeat the above steps but now add the same statistic to the “non-displaying yeast” population
  9. Close the graph and return to the workspace screen. Your first sample should now have the following sub-populations listed below:

    Protocol Figure 3

  10. We now need to copy all of the sub-populations and statistics below the first sample to the other samples in our workspace. Select all of the subpopulations and statistics (select top, hold shift key, select bottom) and copy them (via ctrl+c, or copy in edit tab). Select the remaining samples and paste the subpopulations and statistics. All of the samples should now have the same five sub-populations and statistics.
  11. At this point, we need to check our control samples to ensure our gates are set correctly. The below image shows the three controls for Clone5, one sample with no staining, one sample with only AF488 staining, and one sample with only AF647 staining. The few errant dots in the AF647 control shows that yeast cells can sometimes be sticky to AF647 even without displaying scFv. We want to make sure that our "displaying yeast" gate excludes cells that do not display scFvs but may be slightly sticky to AF488. Fortunately, as you can see in the first two controls, we have set out gate to include minimal sticky cells.

    Protocol Figure 4

  12. Next, we will export the statistics in a table. Under the “FlowJo” tab, select “Table Editor”
    • Under the “Edit” tab in the Table Editor pop up window, select “Add Column”.
    • Add the two statistics to your table that you just added to your samples in your workspace using the same populations (displaying and then non-displaying), statistics (Median), and parameter (FL4-A::FL4-A). Your window should look like the following:

      Protocol Figure 5

    • Return to the “Table Editor” tab and “Create Table” by pressing the gear symbol.
    • Save the table as an excel file for further analysis.

Part 2: Plot titration binding curves in excel

  1. Open your new table in excel. Add a column with the lysozyme concentration of each sample (ignore the control samples). The sample file names and their corresponding concentrations are listed below:
  • File Name scFv Name Lysozyme Concentration [nM]
    A01 C2 1 Ab14989 10,000
    A02 C2 2 Ab14989 3,160
    A03 C2 3 Ab14989 1,000
    A04 C2 4 Ab14989 316
    A05 C2 5 Ab14989 100
    A06 C2 6 Ab14989 31.6
    A07 C2 7 Ab14989 10
    A08 C2 no stain Ab14989 No Lysozyme, Control
    A09 C2 AF488 only Ab14989 No Lysozyme, Control
    C01 H2 1 Clone 2 316
    C03 H2 3 Clone 2 31.6
    C04 H2 4 Clone 2 10
    C05 H2 5 Clone 2 3.16
    C06 H2 6 Clone 2 1
    C07 H2 7 Clone 2 0.316
    C08 H2 8 Clone 2 0.1
  • File Name scFv Name Lysozyme Concentration [nM]
    G02 H5_2 Clone 5 100
    G03 H5_3 Clone 5 31.6
    G04 H5_4 Clone 5 10
    G05 H5_5 Clone 5 3.16
    G06 H5_6 Clone 5 1
    G07 H5_7 Clone 5 0.316
    G08 H5_8 Clone 5 0.1
    G09 H5_9 no stain Clone 5 No Lysozyme, Control
    G010 H5_10 488 only Clone 5 No Lysozyme, Control
    G011 H5_11 647 only Clone 5 No Lysozyme, Control
  1. Create a new column of MFI data for each sample subtracting the background signal (Displaying MFI – Non-displaying MFI).
  2. Create another new column of MFI data for each sample dividing by the sample with the highest fluorescence signal for a given clone. In other words, look through the column of data that you just made and select the cell with the highest value for Clone 2, Clone 5, and Ab14989. Then, divide each data point from the (Displaying MFI – Non-displaying MFI) column by their respective highest value (i.e. divide Clone 2 samples by the highest Clone 2 data point, Clone 5 by highest Clone 5, etc.) This will normalize the data to fit approximately on a normalized MFI y-axis from 0 to 1. Thus, this normalized axis will represent no binding (Normalized MFI = 0) to fully saturated binding (Normalized MFI = 0).
  3. Plot the normalized MFI data (y-axis) for the control scFv and the two clones pulled out of the sort vs. the concentration (x-axis) on a semilog plot (logarithmic scaling on the x-axis). Your plot should look like the figure below. Do not be concerned if the exact values are different as a result of slight gating differences.

    Protocol Figure 6

  4. Save your plot or take a screenshot of your excel window for your lab report.

Estimate the Kd of each of the clones from your graph. Which scFv has the highest Kd? Which scFv has the lowest Kd? Rank the scFvs in order from weakest to strongest lysozyme binders in your lab notebook

Reagents list

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