TUTOR: Here I'm going to demonstrate
that we should use nonlinear regression
if we wanna get the parameters
in the Michaelis-Menten kinetics expression rather
than something like the Lineweaver-Burke linear regression.
And so first let's look at what the equation is
for the Michaelis-Menten kinetics
which shows up often in bio enzyme kinetics.
So, V is used for the rate
and Vmax is one of the parameters,
S is the substrate concentration,
Km the other parameter and S shows up again.
So, remember this is the rate
and typically we vary the substrate concentration.
So this would be the independent variable,
and we measure the rate,
so this would be the dependent variable.
There are other screen casts
that describe how this rate expression arises,
what we're interested here is comparing
if we carry out nonlinear regression,
and then we carry out Linear regression,
how these differ and fit in the data.
So both types regression assume that the data scatter
is gonna be in a Gaussian distribution
around the dependent variable,
around the rate in this case,
it's reasonably good assumption for nonlinear regression.
We're varying the substrate concentration
and in measuring the rate.
Now, if instead of using nonlinear regression,
we linearize and let me write down the expression
I won't derive it but it follows fairly straight forward.
So over the plot one over the rate, one over the velocity
versus one over substrate concentration,
we expect a straight line,
and then this will be the intercept
and this would be the slope
and so this is a way to fit the data.
However, what we're doing is compressing points
that are the high substrate concentrations
and we're putting more emphasis on points
at the low substrate concentration,
and these are ones that are often the least accurate.
Remember that in regression we're assuming the scatter
about the measured values is the same,
when we take the inverse of the rate, that's no longer true.
So let's look at values that we get
when you use some data from a reference.
So let me pull up that data.
So what I've done, I've taken data,
show it in the table from reference one,
this reference is at the end of this screen cast.
Done non-linear regression and notice
that the confidence limits are relatively small
which is what we look for to have a good fit,
and then I compared it
to the Lineweaver-Burke linear regression.
A couple of things to note, first the values
for the Vmax and Km are really quite different.
Notice how large the 95% confidence limits are for the a0,
the Vmax it's almost as large as the actual value.
So, this is really a red flag saying this
is not a very good fit.
Here I've plotted the rate
versus the substrate concentration.
We're just showing the micromol,
obviously it's micromols per volume.
The green squares are the actual data
that we're trying to fit.
The blue line then is the nonlinear regression.
See, it's pretty good fit with lineweaver-Burke fit,
the linear regression, the orange line and data and points
and notice it's not a very good fit,
it's really quite a bit off from the data.
So it's clearly an indication this
is not a good approach to use.
We also can look at some
of the other approaches that are used.
So [indistinct] choosing couple other models
in linear regression but a different arrangement
of the equation.
You noticed I've repeated here
the nonlinear regression value so you have a comparison.
Notice that the value for Km is quite a bit different.
Contrast the fit using the Hanes-Woolf linear regression,
it is relatively close.
So this, of the ones I show here, this is...
This model is clearly much better,
excuse me, this linearization I should say
is clearly much better but in general nonlinear regression
is still the best approach to use.
And I've listed a couple of references that discuss this
in more detail but the final takeaway is that
we should not be using the Lineweaver-Burke linearization
to determine Michaelis-Menten kinetic parameters.
We should instead be using nonlinear regression.