JOHN: I'm going to demonstrate here
how to use polymath to carry out nonlinear regression
to determine the kinetic parameters
in the Michaelis-Menten kinetics expression
and two parameters are maximum rate
and Km where S is the substrate concentration,
that's our independent variable,
v is the velocity or the rate of reaction
or dependent variable.
I'm going to you use the data from this reference,
and I'm using the same format,
so S is in micromols
but obviously, this would be for some volume
and just to be consistent with the reference,
I'll use it this way
and so what I'm gonna determine is both the parameters
and the confidence limits.
So let's look at what the results are
and then we'll go to the Polymath program.
So here are the parameters.
Notice that the confidence limits are a small fraction,
particularly for Vmax of the value,
r squared value is large
variances is relatively small
and if we look at a plot,
which I'll show you here,
I'm plotting the rate versus S substrate concentration
and the data, the data are just the green squares
to fit is the blue.
Now this is rather than drawing a curve for the equation,
this is just an Excel connecting the dots.
So that's why it's not a smooth curve
but you see the fit is really very good
for nonlinear regression
and we show another screencast if we were to use
the line weaver per linear regression approach.
We wouldn't get a very good fit to the data.
So let's go and look at the Polymath program.
So here's a Polymath program.
What I've already done is,
I went in the program and selected the regression
and now we select nonlinear regression.
I'm gonna copy the data
so it's data from a spreadsheet,
I copied into the program,
and now I label the column.
So I go to CO1,
that's the substrate concentration,
and then CO2, the second column is the rate.
So let's call it v, since that's the notation
we use in Michaelis-Menten.
Now, I'm gonna do a regression
so to do nonlinear regression on the right,
I select a nonlinear regression, I have to input the model.
So v is equal to Vmax,
that's my first parameter multiplied by S,
and then divided by Km plus S.
So you'll notice that
Polymath shows immediately the two parameters,
because in not only regression,
they have to estimate the value.
So let's try one and 400
since I have some information
about what I expect the answer to be
and if the guess is too far off,
the regression is not going to work.
I can get output of the residuals a report.
Let's just look at those two for now.
I've already showed you the graph of the fit.
So if I select this purple arrow,
it regresses gives us the parameters
544 for Km 0.3767 for the max.
R squared is pretty large
and if we look at the residuals,
so the residuals, of course are the difference
between the experimental values and the values
from the model and distribution looks random.
We can calculate the run number
which we've described in other screencast
but if I, let me bring up those results.
So the run number is just the residuals,
which is the last column here,
the Delta difference in the rates measured and calculated
and every time there's a sign change that's a new run.
So this is one, the first data point
and it's negative for two value
that's two plus again, is three, negative,
that's four and plus again, it's five,
we expect approximately six for a random distribution,
this number 10 number of data points
divided by two plus one.
So this indicates that this is a reasonable fit
that there's not some pattern
and so this demonstrates how quickly
we can do nonlinear regression with polymath
to get kinetic parameters.