TITLE: Fitting enzyme kinetics data with Solver
AUTHORS: Nithesh Chandrasekharan, Chris Berndsen

[DESCRIPTION]
<div class = "text-blocks"><div class = "text-block"><span>Reaction kinetics are a fundamental component of the biochemical characterization of a biomolecule. The V</span><span style = "vertical-align:sub;">max</span><span> and the K</span><span style = "vertical-align:sub;">M</span><span> values derived from experiments describe how fast a biomolecule catalyzes the reaction and the approximate affinity between biomolecule and substrate. For almost a century, these properties have been used to describe the mechanism and functions of enzyme and other biomolecules. Fitting data to calculate kinetic values like the V</span><span style = "vertical-align:sub;">max</span><span> and the K</span><span style = "vertical-align:sub;">M</span><span> values is a fundamental skill in biochemistry. </span></div><div class = "text-block"><span>This protocol describes fitting data determined at several substrate concentrations to calculate V</span><span style = "vertical-align:sub;">max</span><span> and K</span><span style = "vertical-align:sub;">M</span><span> values using Excel/Google Sheets and the macro with these programs Solver. This protocol is based on the work of Kemmer and Keller. </span></div><div class = "text-block">This protocol is written for students in Biochemistry at James Madison University, however the approach is general enough for any enzyme kinetics data.</div></div>

[STEPS]
?. [Calculating intial rates]
Calculate the initial rate from the slope of the product formed vs. time plot.
It is important to calculate the rate when the plot of product formed vs. time is linear with time. This ensures that the enzyme is in the steady-state.  Typically this is in the early part of the reaction before more than 10% of the substrate has been converted to product (see figure). The rate is the slope of the line.In the steady-state, the concentration of enzyme bound complexes is not changing with time and the enzyme is catalyzing the reaction at the fastest possible rate at that concentration of substrate.
?. [Calculating intial rates]
Repeat for each concentration of substrate.
?. [Calculating intial rates]
Subtract the rate of the background reaction from each rate.
Background rate is typically from a reaction performed in the absence of enzyme which shows the rate of the reaction in the absence of catalysis.
?. [Calculating intial rates]
Record the rate at each concentration of substrate in the table below. Be sure to modify the units to match the units from the experiment. Add rows to the table as needed.  AB1Concentration of substrate (uM)Observed Rate (uM/sec)234567891011121314
AB1Concentration of substrate (uM)Observed Rate (uM/sec)234567891011121314
?. [Plotting data to estimate kinetic values]
Plot rate (y-axis) vs Substrate concentration (x-axis) and estimate the Vmax and KM values using the plot. Plot the data as a scatter plot (no lines)
The Vmax estimate is the y-value where the graph levels out. The KM is the x-value at 1/2 Vmax.
?. [Plotting data to estimate kinetic values]
Record your estimates with units of Vmax and KM in the table below.  ABC1Vmax2Km
ABC1Vmax2Km
?. [Fitting data to calculate Vmax and Km]
To draw a trendline, you'll need to calculate theoretical values for rates using the estimated Vmax and KM values from 2.1 and the Michaelis-Menten Equation. This is your V Calculated (V calc). This will also help you calculate the correct Vmax and Km values later on.
?. [Fitting data to calculate Vmax and Km]
In Microsoft Excel or Google Sheets, set up a table with headers as shown below:  ABCDEFGH1Concentration of substrateObserved rateTheoretical rate (V Calc)Squared residualVariablesEstimatesUnits2Vmax3Km4SSR
ABCDEFGH1Concentration of substrateObserved rateTheoretical rate (V Calc)Squared residualVariablesEstimatesUnits2Vmax3Km4SSR
?. [Fitting data to calculate Vmax and Km]
Input your data table from step 1.3 into columns A and B.
?. [Fitting data to calculate Vmax and Km]
Input your estimates of Vmax and KM from step 2.1 into the appropriate box in column G
?. [Fitting data to calculate Vmax and Km]
In column C, insert the following code to use the Michaelis-Menten equation to calculate the theoretical rate given the estimates for Vmax and KM.=($G$2 * A2)/($G$3 + A2)
This code uses the Michaelis-Menten equation   to calculate the rate based on the estimates. The $ locks those values in so that the equation can be "dragged" down for all concentrations of substrate without needing to re-write the equation.v = \frac{V_{max} \times [S]}{K_M + [S]}
?. [Fitting data to calculate Vmax and Km]
"Drag" the equation down until the theoretical rate has been calculated for all the concentrations of substrate.  After this step the table will appear similar to below:
?. [Fitting data to calculate Vmax and Km]
In Column D, insert the following code to calculate the squared residual value. =(B2-C2)^2
The squared residual will be used to calculate error and refine the estimates of Vmax and KM. It is the squared difference between the real rate and the calculated rate. For a perfect match the squared residual should be zero.
?. [Fitting data to calculate Vmax and Km]
"Drag" the equation down until the squared residual has been calculated for all the concentrations of substrate.
?. [Fitting data to calculate Vmax and Km]
In cell G4, insert the following code to calculate the sum of squared residuals. Replace DXX with D followed by the number of the last substrate concentration (e.g. D10 if there are 9 substrate concentrations).  The resulting number is the sum of squared residuals or SSR. =sum(D2:DXX)
?. [Fitting data to calculate Vmax and Km]
Use SOLVER to fit the data by changing the Vmax and KM estimates to minimize the sum of squared residuals.
?. [Fitting data to calculate Vmax and Km]
If Solver has not been used before, it may not have been installed as described below. If it has been used before skip to step 6.2.Note: For some users of Excel, add-ins are disabled, therefore Google Sheets has to be used. FOR EXCEL USERS:If Solver has not been installed, it can be added to Excel by going to File -> Options -> Add-ins -> Manage Excel Add-ins -> Solver Add-inFOR GOOGLE SHEETS USERS:If Solver has not been installed, it can be added to Sheets by going to Add-ons -> Get Add-ons, and searching for Solver.
?. [Fitting data to calculate Vmax and Km]
Load the Solver window. In Excel, Solver is found in the Data tab. In Sheets, it is found in the Add-ons menu.
?. [Fitting data to calculate Vmax and Km]
The Solver window (example from Excel shown below) has 4 parameters to set:Set Objective: $G$4To: MinBy Changing Variable Cells: $G$2, $G$3Select a Solving Method: GRG NonlinearThe Objective and Variable cells can also be selected using the button at the right of the window and using the cursor to click on the appropriate cells.
?. [Fitting data to calculate Vmax and Km]
Click Solve and wait for the fitting to finish (
Solver will subsitute in various values for Vmax and KM to reduce the SSR value as much as possible which means that the theoretical values match the experimental values closely. At the end, the values will be changed automatically in the spreadsheet
?. [Fitting data to calculate Vmax and Km]
In Sheets: Solver will run and change the values. In Excel: A window will pop-up asking to accept the values and display if there were errors. Click OK and the estimated  Vmax and KM will be changed to the optimized values.
?. [Fitting data to calculate Vmax and Km]
Record the values Vmax, KM, and SSR below. These Vmax and KM values are accurate compared to your original estimates.  AB1Vmax2Km3SSR
AB1Vmax2Km3SSR
?. [Visualizing the fit to the data]
Add a second set of data to the original plot of observed rate vs. concentration of substrate.
?. [Visualizing the fit to the data]
Plot the theoretical (V Calc) rates vs. concentration of substrate.
?. [Visualizing the fit to the data]
Click on the newly plotted V Calc data and edit the data set to show only the line. To show only the line, highlight the Vcalc data by left-clicking on the data and then right clicking to bring up the pop-up menu. Format the data series to add the line.The plot should look similar to that shown below:
?. [Visualizing the fit to the data]
Label the plot axes with appropriate labels and units.
?. [Visualizing the fit to the data]
Save your plot as a picture and upload the image.
?. [Where to start]
If you have calculated rate data already skip to step 2.3. If you have time course data and need to calculate the rate, start with step 2.