## Starting points

Starting points for biochemists:
 A protein assay An enzyme kinetic plot Visual Index Study at Cardiff

## Tuesday, 26 May 2015

### Plotting and fitting enzymology data...

Professor Rob Beynon put together this example using some enzymology data. The script creates a plot, fits the data and calculates Vmax and Km for the enzyme.

Wikipedia has some useful information if you want to know more about enzyme kinetics. The plots are named after the scientists that described them: Eadie-Hostee

Here's the plot:

Here is the script:

# As a biochemist, one of the first data types we play with is enzymology data.
# Here is an example of some enzyme simple kinetic data
# Eight values for substrate concentration with corresponding enzyme velocities.
# The goals are to plot the data, in different formats, and also, find the best fit values of Km and Vmax
# all of this is complete in base R, so no packages should be needed

# Entered as vectors
S <- c(0,1,2,5,8,12,30,50)
v <- c(0,11.1,25.4,44.8,54.5,58.2,72.0,60.1)

# simple plot, rather ugly
plot (S,v)

#  A better plot: Michaelis Menten hyperbola
plot (S,v,
xlab="Subtrate (mM)",
ylab="Velocity (nmol/s)",
main="Michaelis-Menten",
pch=17, col="red")

# Tranform the data to plot a Lineweaver-Burk plot (1/v as a function of 1/S)
# This plot is not recommended becuase it distorts the error structure of the data
plot (1/S,1/v,
xlab="Subtrate (mM)",
ylab="Velocity (nmol/s)",
main="Lineweaver-Burk",
pch=17, col="blue", cex=1.5)

# Tranform the data to plot a Eadie-Hostee plot (v as a function of v/S)
plot (v,v/S,
xlab="velocity/Subtrate(nmol.s-1/mM)",
ylab="Velocity (nmol.s-1)",
pch=17, col="blue", cex=1.5)

# We can also build a simple data frame from (S,v) data
kinData <- data.frame(S,v)

# simple plot of the dataframe
plot (kinData)

# Store some colours as variables for later
c1 <- "red"
c2 <- "blue"

# More complex plot, adding axis labels, changing symbol and colour
# Simple Michaelis-Menten plot
plot (kinData,
xlab="Subtrate (mM)",
ylab="Velocity (nmol/s)",
pch=17, col=c1, cex=2)

# And this shows how simple it is to plot a Lineweaver-Burk plot
# (1/v as a function of 1/S)
plot (1/kinData,
xlab="1/Subtrate (mM)",
ylab="1/Velocity (nmol/s)",
pch=17, col=c1, cex=2)

# Next step - how do we get the values of Km and Vmax?

# This is the theoretical formula
# "velocity = Vmax times S divided by (Km plus S)", stored in MMcurve
MMcurve<-formula(v~Vmax*S/(Km+S))

# nls is non-linear least squares optimiser.
# If you're not sure why a Michael-Menten equation is non-linear, ask
# (Hint: it is not because it is a curve when plotted!)

# start sets the initial guess - do not have to be very good
# should be possible to make reasonable guesses
# from a quick inspection - this is why we visualise the data first
bestfit <- nls(MMcurve, kinData, start=list(Vmax=50,Km=2))

# Build a theoretical line defining the best fit curve
# First, make a finely detailed set of points between 0 and 50, at 0.1 intervals
# These will be the substrate concentrations that are used to calculate
# the predicted velocities
SconcRange <- seq(0,50,0.1)

# Then, calculate the predicted velocities using the predict function
theorLine <- predict(bestfit,list(S=SconcRange))

# Best fit values of Km and Vmax obtained by coef function, stored in bestFitVals
bestFitVals <- coef(bestfit)

# Now plot the data, the best fit line, and put the best fit coefficients in the plot
plot (kinData,
xlab="Subtrate (mM)",
ylab="Velocity (nmol/s)",
title(main="Fitted MM data"),
pch=17, col=c2, cex=2)

# Draw the line
# lines() function adds to the existing plot as does points()
# here, we want a line of best fit, not points, so we use lines()
lines(SconcRange,theorLine,col=c1)

# Add text with the calculated values
# This is a fudge, there must be better ways, also adding errors on parameter values.

text(30,30, "Vmax=")
text(35,30,round(bestFitVals[1],2))
text(30,27, "Km=")
text(35,27,round(bestFitVals[2],2))

# END OF SCRIPT