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PROFESSOR: So, let me start
here with the temperature

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dependence of k.
 

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And this turns out to be
extremely important.

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And it's due to
Arrhenius in 1889.

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And Mr. Arrhenius is
famous for many reasons.

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Not just for his rate law.
 

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So I learned recently, turns
out that he was also one of the

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first people to calculate the
potential effect of carbon

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dioxide in the atmosphere.
 

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And and its potential
effect on global warming.

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This was the Industrial
Revolution and people were

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starting to use fossil fuels
at increasing amounts.

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And back then, already, some
people started to get worried.

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And so he did the calculation.
 

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It was a very crude
calculation.

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And he extrapolated, assuming
that the rate to fossil fuel

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use would keep going, he
got his calculation

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pretty much right.
 

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What he didn't get right was
the amount of fossil fuel

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that people would be using.
 

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And so when he did his
calculation, which is again a

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very crude calculation, he got
that we would get in trouble at

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about 2,000 years from the
time of his calculation.

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So he said, no problem, 2
years, we've got a lot of time.

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And ever since then people have
redone these calculations and

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more and more sophisticated.
 

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But his crude calculation
was good enough.

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And as people redo the
calculations, the time that

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they say we're in trouble to
the time of the calculation

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gets closer and
closer together.

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And the reason for that isn't
that their calculations are

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wrong, it's that the rate at
which we put out carbon dioxide

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just keeps getting faster
and faster and faster.

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So it's interesting to go back
to these calculations through

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the last century and
a half and see that.

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Anyway, so Mr. Arrhenius
predicted global warming.

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And he wrote his rate
equation, k is equal to e

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to the minus Ea over RT.
 

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Famous Arrhenius rate equation.
 

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If you plot it as log of k is
equal to log A minus Ea over

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RT, you find that it looks
like a straight line.

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With an intercept at log A,
where A is the pre-factor

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here and Ea is going to
be the activation energy.

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And the slope is Ea,
minus Ea over RT.

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R is the gas constant.
 

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T is the temperature.
 

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And if you put in some typical
numbers, activation energies

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typically are on the order of a
few tens to hundreds of

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kilojoules per mole.
 

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Let's say 50 to 300
kilojoules per mole.

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And typical activation factors,
typical pre-exponential

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factors, are depending if it's
a first order or second order.

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Let's say, first order 10 to
the 12th, 10 to the 15th, per

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second, so the A carries the
units for the rate here.

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So if you have A, so this is
for first order, if you have a

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second order, then the units
of A will be different.

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And typically they're going to
be on the order of 10 to the

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11th or so, 10 to the 10th, 10
to the 12th, one over molar

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per second, for second order.
 

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And it's interesting to do sort
of a back of the envelope

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example to figure out how
important this rate is.

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And one of the interesting
things to look at is, suppose

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that I'm at some temperature
and I want to know how do

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I change the temperature.
 

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How much do I need to
change the temperature

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to double the rate.
 

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And let's say my temperature's
at, say, T1 is on the order of

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300 degrees Kelvin,
room temperature.

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Room temperature, and let's
take a typical Ea on the order

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of 100 kilojoules per mole.
 

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Room temperature, or body
temperature are roughly

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the same, right?
 

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These things are.
 

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And so the question is, what
does T2 need to be to double

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the rate connected with an
activation energy which is

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fairly typical of 100
kilojoules per mole.

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So we want to know, what is,
if I have k2 over k1 is equal

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to two, what's the new T2.
 

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Alright so A e to the minus Ea
over R T2 divided by A e to the

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minus Ea over R T1, we
want that equal to two,

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the A's cancel out.
 

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So log k2 over k1, which is
equal to log two, is equal

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to Ea over R, one over
T1 minus one over T2.

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We know what this is, we know
what this is, we solve for T2

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and we find that T2 is
305 degrees kelvin.

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It's a pretty small change
to double the rate.

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So it's pretty important
that your body temperature

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doesn't change very much.
 

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If you have a fever and
your temperature goes up,

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rates start going up.
 

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In your body, you could cause a
lot of problems if the rates of

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some reactions go up
by a factor of two.

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So, very sensitive
to temperature.

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Rates are very sensitive
to temperature.

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What are these
things physically?

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Ea and and the
pre-exponent factor.

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Let's take a look at that.
 

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So physically what's going on
is, you have your two, let's

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say it's a bimolecular
reaction, A plus B goes to C.

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So mechanistically what we
think is happening is that

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the molecules come together
and collide, right?

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You have A and B getting
together and colliding.

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And the hypothesis is
that when they collide

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they form a complex.
 

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They sort of bind
together momentarily.

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And form this complex that has
all the kinetic energy, or part

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of the kinetic energy of this
collision as part of it.

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00:07:44 --> 00:07:50
So A and B are bound together
in some highly excited complex.

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Which then falls apart to give
the product by rearranging

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its atoms around.
 

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And if you plot the energy of
this process as a function of

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the reaction, we call this the
reaction coordinate, then

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we're going to start at some
delta G, or some energy.

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For the reactants.
 

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We're going to end up with
some energy for the products.

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And it could be
higher or lower.

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Let me make them higher here,
just because I don't have

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enough room on the board.
 

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Products.
 

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So if this is an endothermic
reaction, that's the energy for

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C, this is the energy for A
plus B, and then along the way

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00:08:59 --> 00:09:04
we have this complex that
captures some of the

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00:09:04 --> 00:09:06
energy of the collision.
 

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Up here somewhere.
 

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00:09:10 --> 00:09:17
And this is the energy, then,
at A B star, which we call

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00:09:17 --> 00:09:25
the activated complex.
 

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So for the reaction to happen,
then, we have to go over the

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barrier and then back
to the product.

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And this distance from the
energy of the reactants to the

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00:09:38 --> 00:09:41
top of the barrier, that's Ea.
 

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That's the energy
of activation.

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00:09:42 --> 00:09:45
How much extra energy you
have to put in there to

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00:09:45 --> 00:09:47
go over the barrier.
 

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00:09:47 --> 00:09:49
So it's very clear that as you
increase the temperature and

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00:09:49 --> 00:09:53
you increase the amount of
energy that the reactants

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00:09:53 --> 00:09:58
thermally have, or in terms of
their kinetic energy, as you

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raise the temperature you get
more and more kinetic energy,

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00:10:00 --> 00:10:02
you're going up higher and
higher in the Boltzmann

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distribution.
 

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00:10:03 --> 00:10:07
And the number of reactants
that can make it over the

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00:10:07 --> 00:10:10
barrier clearly goes
up exponentially.

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00:10:10 --> 00:10:14
As the temperature goes up.
 

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So this is Ea for the
forward reaction.

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Clearly there's going to be an
equivalent activation energy

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00:10:22 --> 00:10:23
for the backward reaction.
 

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The two are related.
 

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The difference between the
forward and backward reactions,

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so E backwards minus Ea forward
gives you the delta G for

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the reaction itself.
 

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So, Ea backwards minus Ea
forwards gives you delta

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00:10:44 --> 00:10:49
E for the reaction.
 

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00:10:49 --> 00:10:55
Where E could be enthalpy or
it could be free energy.

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So that's the physical
origin of this Ea, this

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activation energy.
 

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00:11:00 --> 00:11:05
Now, A, the pre-exponential
factor, that's the rate of

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00:11:05 --> 00:11:11
attempt for the molecules to
try to go over the barrier.

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So it has units that
make sense for that.

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And this A is different
than this molecule here.

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Rate of attempt.
 

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How many times per unit time,
or how many times per second,

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00:11:37 --> 00:11:40
do these two molecules
try to collide?

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They never collide, they
never try, they'll

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never make it over.
 

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So it's one over second,
or one over mole second.

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So this is the rate of attempt.
 

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Then e to the minus Ea over RT
is the probability of success,

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of making it over the barrier.
 

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So the rate of attempt times
the probability of success

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gives you the rate per unit
time of making it

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over the barrier.
 

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Any questions on the dissection
of this Arrhenius rate law?

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Alright, so here's some
examples, a few examples

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that you know of.
 

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Let's look at OH minus
plus methyl bromide.

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Displacement reaction going
to this OH minus attacks

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the carbon right here.
 

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To form an activated complex
H, H, H, with the OH

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00:13:11 --> 00:13:12
coming in like this.
 

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Let me go like this.
 

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OK, there's the
intermediate right here.

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The activated complex
in your reaction.

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Which then falls apart.
 

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It can fall apart two ways.
 

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The OH can be spit out again,
forming back to the reactants,

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or the bromine can be spit
out, forming the product.

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Which in this case
here is methanol.

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H, H, H, OH, plus Br minus.
 

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Typical example.
 

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So, given this picture here,
which you're probably already

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somewhat familiar with, we can
then move on to talk about

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00:14:02 --> 00:14:04
how to affect this barrier.
 

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How to change the rate.
 

201
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Oh, I should say a
couple more things.

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One more thing, too.
 

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00:14:10 --> 00:14:15
Now, the rate of, so clearly
the Ea's are related.

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Because the difference of
the two Ea's is the delta

205
00:14:18 --> 00:14:21
E for the reaction.
 

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00:14:21 --> 00:14:25
But the pre-exponential
factors, these A's, for the

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00:14:25 --> 00:14:27
forward reaction and the
backward reactions aren't

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necessarily related.
 

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00:14:29 --> 00:14:31
You can imagine that in one
case you have, first of all

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00:14:31 --> 00:14:34
they don't even have to
have the same units.

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00:14:34 --> 00:14:37
You could have a bimolecular
reaction one way.

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00:14:37 --> 00:14:42
And a unimolecular
reaction the other way.

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00:14:42 --> 00:14:45
So you can't really say
anything about the forward

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00:14:45 --> 00:14:49
versus the backward
rate this way.

215
00:14:49 --> 00:14:54
All you know is
about the energies.

216
00:14:54 --> 00:15:01
OK, catalysis.
 

217
00:15:01 --> 00:15:05
So now we have this barrier
we have to overcome.

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00:15:05 --> 00:15:14
And, so suppose that you
have two motivators.

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00:15:14 --> 00:15:17
Suppose you have an equilibrium
case which is slow.

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00:15:17 --> 00:15:19
And a reaction which is
slow in both directions,

221
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k1, k minus one.
 

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00:15:23 --> 00:15:25
And you don't want to
wait for this to happen.

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00:15:25 --> 00:15:28
And this could be in a
biological environment.

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00:15:28 --> 00:15:30
Or it could be an industrial
process, like the

225
00:15:30 --> 00:15:33
Haber process.
 

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00:15:33 --> 00:15:34
You don't want to wait.
 

227
00:15:34 --> 00:15:36
And so one of the ways that you
can speed it up, you know is

228
00:15:36 --> 00:15:38
by changing the temperature.
 

229
00:15:38 --> 00:15:41
You change the temperature,
the rate goes up.

230
00:15:41 --> 00:15:43
You change it by five
degrees, the rate goes

231
00:15:43 --> 00:15:44
up by a factor of two.
 

232
00:15:44 --> 00:15:47
You can make a huge change by
changing the temperature.

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00:15:47 --> 00:15:56
But, so if you raise
T, speeds up.

234
00:15:56 --> 00:16:02
But, K equilibrium
also changes.

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00:16:02 --> 00:16:04
And we saw that when we
did the Haber process.

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We raised the temperature.
 

237
00:16:08 --> 00:16:11
The rate speeds up, but
the equilibrium switches

238
00:16:11 --> 00:16:12
to the reactants.
 

239
00:16:12 --> 00:16:15
That's no good.
 

240
00:16:15 --> 00:16:18
So changing the temperature is
not always the best thing to do

241
00:16:18 --> 00:16:21
if you want to change the rate.
 

242
00:16:21 --> 00:16:30
Instead, what you can do is use
a catalyst. if you find one.

243
00:16:30 --> 00:16:38
A molecule that reacts with
your reactant to form a

244
00:16:38 --> 00:16:42
product, that's B, spitting
back that molecule C again

245
00:16:42 --> 00:16:45
without using it up.
 

246
00:16:45 --> 00:16:49
And now with the activation
energy, we can understand

247
00:16:49 --> 00:16:50
what the catalyst does.
 

248
00:16:50 --> 00:16:55
What the catalyst does is
speeds up those rates, k2 and

249
00:16:55 --> 00:17:02
k minus one, by changing
the activation energy.

250
00:17:02 --> 00:17:04
By making the E smaller.
 

251
00:17:04 --> 00:17:10
So now I have a catalyst, and I
can make this energy smaller.

252
00:17:10 --> 00:17:21
So this would be A plus
A C activated, getting

253
00:17:21 --> 00:17:27
ready to spit out B.
 

254
00:17:27 --> 00:17:33
So in my example here where I
have A plus B goes to products,

255
00:17:33 --> 00:17:37
so I would have A, some
combination of A, B, and C

256
00:17:37 --> 00:17:42
together, to give
out the products.

257
00:17:42 --> 00:17:45
In this case, the difference
in energies doesn't change.

258
00:17:45 --> 00:17:51
All that you're changing is
that the hump, in both ways.

259
00:17:51 --> 00:17:53
Equilibrium constant
doesn't change.

260
00:17:53 --> 00:17:58
Just the rate changes through
the Arrhenius rate law.

261
00:17:58 --> 00:18:00
And this is extremely powerful.
 

262
00:18:00 --> 00:18:04
Especially in biology.
 

263
00:18:04 --> 00:18:10
So let me give you
some examples here.

264
00:18:10 --> 00:18:16
This is sort of a typical
example of increasing rates.

265
00:18:16 --> 00:18:18
Let's say you have the
reaction, hydrogen peroxide,

266
00:18:18 --> 00:18:28
goes to water plus oxygen.
 

267
00:18:28 --> 00:18:31
If you take a little bit of
hydrogen peroxide and you

268
00:18:31 --> 00:18:33
put it on your skin,
it starts to bubble.

269
00:18:33 --> 00:18:37
But if you let it sit on the
bottle nothing happens to it.

270
00:18:37 --> 00:18:40
Put it on your hair,
your hair turns white.

271
00:18:40 --> 00:18:43
Blond.
 

272
00:18:43 --> 00:18:45
But again, if you just
let it sit in the bottle

273
00:18:45 --> 00:18:47
very little happens.
 

274
00:18:47 --> 00:18:51
Well, it happens but very,
very slowly over time.

275
00:18:51 --> 00:18:56
So if you look at the rate of
this reaction here, if the

276
00:18:56 --> 00:19:06
rate, moles per second, with no
catalyst at all, the rate is

277
00:19:06 --> 00:19:10
10 to the minus 8
molar per second.

278
00:19:10 --> 00:19:12
Which is very slow.
 

279
00:19:12 --> 00:19:21
And the activation energy
in kilojoules per mole,

280
00:19:21 --> 00:19:25
in this case here is 71
kilojoules per mole.

281
00:19:25 --> 00:19:27
Now we can start
adding catalysts.

282
00:19:27 --> 00:19:29
We can start adding
inorganic catalysts

283
00:19:29 --> 00:19:31
like hydrogen bromide.
 

284
00:19:31 --> 00:19:35
Increase the rate at
10 to the minus 4.

285
00:19:35 --> 00:19:40
So this creates a complex with
the hydrogen peroxide, which

286
00:19:40 --> 00:19:45
lowers the barrier to 50
kilojoules per mole, a

287
00:19:45 --> 00:19:46
small amount of lowering.
 

288
00:19:46 --> 00:19:51
But because the energy is in
the exponent there, it makes

289
00:19:51 --> 00:19:52
a big change in the rate.
 

290
00:19:52 --> 00:19:53
Yes.
 

291
00:19:53 --> 00:20:01
STUDENT: [INAUDIBLE]
 

292
00:20:01 --> 00:20:02
PROFESSOR: The rate is
independent of the

293
00:20:02 --> 00:20:04
catalyst concentration.
 

294
00:20:04 --> 00:20:05
STUDENT: [INAUDIBLE]
 

295
00:20:05 --> 00:20:09
PROFESSOR: The rate
would have to change.

296
00:20:09 --> 00:20:19
You're right.
 

297
00:20:19 --> 00:20:24
That's a good question and I'm
not prepared to answer it.

298
00:20:24 --> 00:20:27
I'm going to have to
think about this.

299
00:20:27 --> 00:20:31
Let's pretend now that we are
at per unit concentration

300
00:20:31 --> 00:20:33
of the catalyst.
 

301
00:20:33 --> 00:20:37
And then, and I'll
look into it.

302
00:20:37 --> 00:20:40
OK, so this is what happens
with an inorganic catalyst.

303
00:20:40 --> 00:20:46
And instead if you use a
generic biological catalyst, an

304
00:20:46 --> 00:20:50
enzyme catalase, it's a sort
ubiquitous enzyme which is why

305
00:20:50 --> 00:20:54
you have it on your skin, et
cetera, then this rate

306
00:20:54 --> 00:20:57
become 10 to the 7th.
 

307
00:20:57 --> 00:21:08
And the activation energy drops
to eight kilojoules per mole.

308
00:21:08 --> 00:21:11
So your question really has
to do with the units of A.

309
00:21:11 --> 00:21:17
Of the pre-factor right there.
 

310
00:21:17 --> 00:21:21
OK, and so there are all
these examples of reactions.

311
00:21:21 --> 00:21:24
That are very important
biologically.

312
00:21:24 --> 00:21:28
Where with an inorganic
catalyst, an organic catalyst,

313
00:21:28 --> 00:21:31
you can change the rate by
maybe a few orders

314
00:21:31 --> 00:21:32
of magnitude.
 

315
00:21:32 --> 00:21:35
But as soon as you put in an
enzyme, the rate changes by

316
00:21:35 --> 00:21:37
ten orders of magnitude.
 

317
00:21:37 --> 00:21:40
Or in this case eight
plus seven, fifteen

318
00:21:40 --> 00:21:42
orders of magnitude.
 

319
00:21:42 --> 00:21:46
Humongous change in the rate.
 

320
00:21:46 --> 00:21:50
So you've probably done some
enzyme catalysis before.

321
00:21:50 --> 00:22:04
But it's probably a good idea
to quickly do it again.

322
00:22:04 --> 00:22:07
Because it's just so important.
 

323
00:22:07 --> 00:22:10
And it ties together our
approximations that

324
00:22:10 --> 00:22:15
we've learned about.
 

325
00:22:15 --> 00:22:20
So enzyme catalysis, so enzymes
can be either, could be

326
00:22:20 --> 00:22:22
heterogeneous catalysis, can
be homogeneous catalysis.

327
00:22:22 --> 00:22:25
Enzymes are ubiquitous in
the biological environment.

328
00:22:25 --> 00:22:31
They serve to regulate
the cellular activity in

329
00:22:31 --> 00:22:35
very complicated ways.
 

330
00:22:35 --> 00:22:40
The cell will up-regulate or
down-regulate the concentration

331
00:22:40 --> 00:22:45
of enzymes as it needs to
make more or less products.

332
00:22:45 --> 00:22:52
And there's whole cascades of
events that happen in this way.

333
00:22:52 --> 00:22:57
And they're in very
small concentrations.

334
00:22:57 --> 00:22:59
But they play an extremely
important role.

335
00:22:59 --> 00:23:01
Because they're also
extremely specific.

336
00:23:01 --> 00:23:06
So you can have an enzyme that
will only act on one part

337
00:23:06 --> 00:23:09
of the biological cycle.
 

338
00:23:09 --> 00:23:11
And affect it by 10 orders
of magnitude or 15

339
00:23:11 --> 00:23:12
orders of magnitude.
 

340
00:23:12 --> 00:23:16
But not affect any other
protein that's around.

341
00:23:16 --> 00:23:20
And that's amazing.
 

342
00:23:20 --> 00:23:29
And it turns out that a lot of
the diseases of old age like,

343
00:23:29 --> 00:23:34
what I'm about to face, or am
facing already, you, not yet,

344
00:23:34 --> 00:23:44
but, are the result of some of
these enzyme up-regulation,

345
00:23:44 --> 00:23:46
down-regulation,
beginning to break down.

346
00:23:46 --> 00:23:52
So we have all these reactions
going on that need to

347
00:23:52 --> 00:23:53
be essentially perfect.
 

348
00:23:53 --> 00:23:58
When you have DNA replication
or protein folding,

349
00:23:58 --> 00:24:01
or things like this.
 

350
00:24:01 --> 00:24:02
And errors are made.
 

351
00:24:02 --> 00:24:06
Errors are made all the
time in these processes.

352
00:24:06 --> 00:24:09
And errors cause diseases.
 

353
00:24:09 --> 00:24:15
And so, even as a
baby, your biological

354
00:24:15 --> 00:24:17
processes make errors.
 

355
00:24:17 --> 00:24:20
But you have these processes,
these enzymes especially based

356
00:24:20 --> 00:24:25
on enzymes, that can go in
there and sort of fix things.

357
00:24:25 --> 00:24:29
Fix things and make sure that
you don't end up getting

358
00:24:29 --> 00:24:33
Alzheimer's at age six months.
 

359
00:24:33 --> 00:24:40
But as we get older, for some
reason, these repair processes

360
00:24:40 --> 00:24:41
lose their bearings.
 

361
00:24:41 --> 00:24:43
Just like we lose our bearings.
 

362
00:24:43 --> 00:24:48
And and they can't
repair things any more.

363
00:24:48 --> 00:24:50
They don't do it very well.
 

364
00:24:50 --> 00:24:54
And that causes all
sorts of diseases.

365
00:24:54 --> 00:24:57
Cancer is probably one of the
diseases, due to the lack of

366
00:24:57 --> 00:24:59
being able to repair problems.
 

367
00:24:59 --> 00:25:00
Alzheimer's.
 

368
00:25:00 --> 00:25:02
All sorts of dementias.
 

369
00:25:02 --> 00:25:07
MS. Just name a chronic disease
and it's probably due to a

370
00:25:07 --> 00:25:10
problem with up-regulation
or down-regulation of some

371
00:25:10 --> 00:25:17
proteins, some enzymes, that
are due to a repair process.

372
00:25:17 --> 00:25:25
So anyway, these enzymes
are big proteins.

373
00:25:25 --> 00:25:29
10 to the 4th, 10 to the 6th
molecular weight proteins.

374
00:25:29 --> 00:25:30
On that order or so.
 

375
00:25:30 --> 00:25:35
They tend to be fairly
large in size.

376
00:25:35 --> 00:25:36
On the order of nanometers.
 

377
00:25:36 --> 00:25:39
Let's say ten nanometers
to 100 nanometers.

378
00:25:39 --> 00:25:41
That could be, that's a
little bit on the big side.

379
00:25:41 --> 00:25:44
Probably closer to
ten nanometers.

380
00:25:44 --> 00:25:46
Ten nanometers is kind
of big, for any sort of

381
00:25:46 --> 00:25:51
biological molecule.
 

382
00:25:51 --> 00:25:54
And they always end
with their name ase.

383
00:25:54 --> 00:26:00
Like catalase, uriase, rnase.
 

384
00:26:00 --> 00:26:04
Esterase, clips ester bonds.
 

385
00:26:04 --> 00:26:06
Your liver is full
of esterases.

386
00:26:06 --> 00:26:09
Because it likes to break
things down into smaller and

387
00:26:09 --> 00:26:12
smaller pieces and lots of
ester bonds and things that

388
00:26:12 --> 00:26:14
are not very biologically
interesting.

389
00:26:14 --> 00:26:22
And that's one way of the
liver breaking things down.

390
00:26:22 --> 00:26:29
So the way it works is that you
have your reactants, which in

391
00:26:29 --> 00:26:39
the biological language are
called your substrate,

392
00:26:39 --> 00:26:44
come in, into an enzyme.
 

393
00:26:44 --> 00:26:50
Gets bound up in a
pocket of, there's the

394
00:26:50 --> 00:26:52
substrate, reactant.
 

395
00:26:52 --> 00:26:55
There's the enzyme here.
 

396
00:26:55 --> 00:26:58
Forms a complex.
 

397
00:26:58 --> 00:27:06
And then product gets spit out.
 

398
00:27:06 --> 00:27:08
And the product floats off.
 

399
00:27:08 --> 00:27:09
And does its thing.
 

400
00:27:09 --> 00:27:12
It probably gets bound to
another enzyme, which

401
00:27:12 --> 00:27:13
makes another product.
 

402
00:27:13 --> 00:27:15
Et cetera, and the
cascade goes on.

403
00:27:15 --> 00:27:18
The signaling cascade goes on.
 

404
00:27:18 --> 00:27:22
And this enzyme is in very
small concentration.

405
00:27:22 --> 00:27:25
The product goes away, so
that's going to stay as a very

406
00:27:25 --> 00:27:32
small concentration as well.
 

407
00:27:32 --> 00:27:44
So let's observe
experimentally, then.

408
00:27:44 --> 00:27:50
Experimentally, what's seen
is that the substrate

409
00:27:50 --> 00:27:55
makes products in the
presence of the enzyme.

410
00:27:55 --> 00:28:03
With a rate dP/dt, that's, at t
equals zero this is the initial

411
00:28:03 --> 00:28:06
rate is proportional to the
concentration of the enzyme.

412
00:28:06 --> 00:28:10
This is initial rate.
 

413
00:28:10 --> 00:28:14
And in the language of
enzymatic kinetics, this is

414
00:28:14 --> 00:28:20
called the velocity. dP/dt is
also called the velocity.

415
00:28:20 --> 00:28:23
Velocity, moles per unit time.
 

416
00:28:23 --> 00:28:27
And this would be called, then,
the initial velocity of v

417
00:28:27 --> 00:28:29
initial. v initial's
proportional to the

418
00:28:29 --> 00:28:33
enzyme concentration.
 

419
00:28:33 --> 00:28:37
And what else is seen?
 

420
00:28:37 --> 00:28:42
For fixed, if I fix my
concentration of enzyme, I look

421
00:28:42 --> 00:28:46
at the velocity over time,
let's say minus dS/dt, which is

422
00:28:46 --> 00:28:57
dP/dt, I find that that's
proportional to S.

423
00:28:57 --> 00:29:00
For small S.
 

424
00:29:00 --> 00:29:03
Small concentration
of substrate.

425
00:29:03 --> 00:29:13
And then at large
concentration, it's a constant.

426
00:29:13 --> 00:29:22
So it's not a straight line.
 

427
00:29:22 --> 00:29:28
In fact, I don't want to
do it on this board here.

428
00:29:28 --> 00:29:41
I'll do it on this board here.
 

429
00:29:41 --> 00:29:44
Plot the concentration of
substrate on this axis, and

430
00:29:44 --> 00:29:48
I plot the velocity, or
the rate, of the reaction

431
00:29:48 --> 00:29:49
on that axis here.
 

432
00:29:49 --> 00:29:53
What I find is that
it's a constant.

433
00:29:53 --> 00:29:55
So it's going to saturate,
the rate is going to

434
00:29:55 --> 00:29:59
saturate to some number.
 

435
00:29:59 --> 00:30:03
And it's going to start
linear with substrate

436
00:30:03 --> 00:30:04
at the beginning.
 

437
00:30:04 --> 00:30:07
So it's going to be a straight
line at the beginning.

438
00:30:07 --> 00:30:09
And eventually it
will saturate.

439
00:30:09 --> 00:30:09
That sort of slope.
 

440
00:30:09 --> 00:30:14
And the saturation point,
that's the maximum velocity, or

441
00:30:14 --> 00:30:16
maximum rate that it can have.
 

442
00:30:16 --> 00:30:20
So we call this v max.
 

443
00:30:20 --> 00:30:34
And this part here, the
velocity is proportional to S.

444
00:30:34 --> 00:30:39
And somehow we have to
find, explain this.

445
00:30:39 --> 00:30:43
Using what we know
from kinetics.

446
00:30:43 --> 00:30:45
So we have to come up with
a mechanism, and then

447
00:30:45 --> 00:30:48
solve the mechanism.
 

448
00:30:48 --> 00:30:53
And make sure that it
reproduces the data.

449
00:30:53 --> 00:30:54
So we're not the first ones
to do this, obviously.

450
00:30:54 --> 00:31:02
Michaelis and Menton did
this many years ago.

451
00:31:02 --> 00:31:04
For this mechanism.
 

452
00:31:04 --> 00:31:08
And the idea is, you have
the enzyme plus a substrate

453
00:31:08 --> 00:31:15
react, k1, k minus one,
to form a complex.

454
00:31:15 --> 00:31:16
Enzyme substrate complex.
 

455
00:31:16 --> 00:31:20
But unlike what we've drawn
before in terms of this hump,

456
00:31:20 --> 00:31:23
where there's an activated
complex which is not stable, in

457
00:31:23 --> 00:31:28
this case here this enzyme
substrate combination lasts a

458
00:31:28 --> 00:31:36
long enough time that it's
basically a stable complex.

459
00:31:36 --> 00:31:41
So we're going to write this
as a real intermediate that

460
00:31:41 --> 00:31:44
lasts long enough for you
to be able to fish it out.

461
00:31:44 --> 00:31:47
And characterize it.
 

462
00:31:47 --> 00:31:51
And then eventually, that,
k2, k minus two, goes

463
00:31:51 --> 00:31:54
to product plus enzyme.
 

464
00:31:54 --> 00:31:58
So the enzyme is a
catalyst that forms a

465
00:31:58 --> 00:31:59
long-lived complex.
 

466
00:31:59 --> 00:32:04
And if you were to draw this,
then, in our energy diagram,

467
00:32:04 --> 00:32:10
where we have the reaction
coordinate here, you start out

468
00:32:10 --> 00:32:16
with your enzyme
plus substrate.

469
00:32:16 --> 00:32:23
Go up, and you form your
intermediate up here.

470
00:32:23 --> 00:32:23
ES.
 

471
00:32:23 --> 00:32:26
And I put a little dimple in
there, because it's stable

472
00:32:26 --> 00:32:29
enough that it's not on
the top of the hump, but

473
00:32:29 --> 00:32:32
it lives long enough.
 

474
00:32:32 --> 00:32:38
And it comes back down
to form the product.

475
00:32:38 --> 00:32:39
Plus the enzyme.
 

476
00:32:39 --> 00:32:42
Without the enzyme in
there, this hump would

477
00:32:42 --> 00:32:44
be way, way up there.
 

478
00:32:44 --> 00:32:47
Would be maybe a
factor of ten higher.

479
00:32:47 --> 00:32:51
So this really lowers it a lot.
 

480
00:32:51 --> 00:32:54
Now we can start to
solve this mechanism.

481
00:32:54 --> 00:32:56
And I forgot one arrow.
 

482
00:32:56 --> 00:33:00
There which is the
arrow going back.

483
00:33:00 --> 00:33:01
Which you usually don't
see, but we might as

484
00:33:01 --> 00:33:03
well keep it there.
 

485
00:33:03 --> 00:33:15
For the sake of completeness.
 

486
00:33:15 --> 00:33:16
So what do we know?
 

487
00:33:16 --> 00:33:22
We know that this intermediate
concentration is very small.

488
00:33:22 --> 00:33:24
The enzyme concentration
itself is very small.

489
00:33:24 --> 00:33:26
Intermediate is very small.
 

490
00:33:26 --> 00:33:29
And it doesn't
change very much.

491
00:33:29 --> 00:33:36
Not changing much.
 

492
00:33:36 --> 00:33:38
So that means that we
need to use a steady

493
00:33:38 --> 00:33:39
state approximation.
 

494
00:33:39 --> 00:33:46
So let's write down the rate
for this intermediate,

495
00:33:46 --> 00:33:49
d[ES]/dt.
 

496
00:33:49 --> 00:33:53
It gets formed through
the forward process.

497
00:33:53 --> 00:33:57
I'm going to put my brackets
back in because E and ES would

498
00:33:57 --> 00:33:59
look the same otherwise.
 

499
00:33:59 --> 00:34:02
Gets destroyed.
 

500
00:34:02 --> 00:34:03
Through the backward ways.
 

501
00:34:03 --> 00:34:05
And I'm going to use a
steady state approximation.

502
00:34:05 --> 00:34:07
So I'm already going to start
adding steady state here every

503
00:34:07 --> 00:34:09
time I see an intermediate.
 

504
00:34:09 --> 00:34:15
Get it destroyed
to make products.

505
00:34:15 --> 00:34:17
It gets recreated through
the backwards reaction

506
00:34:17 --> 00:34:24
from the products.
 

507
00:34:24 --> 00:34:25
And I'm going to set that
equal to zero for the

508
00:34:25 --> 00:34:28
steady state approximation.
 

509
00:34:28 --> 00:34:33
Now, we don't really
want to have this E

510
00:34:33 --> 00:34:34
floating around here.
 

511
00:34:34 --> 00:34:37
Because this is something that
is very hard to measure.

512
00:34:37 --> 00:34:40
It's much easier to measure
the initial concentration

513
00:34:40 --> 00:34:41
of the enzyme.
 

514
00:34:41 --> 00:34:44
What you put in there,
instead of the amount

515
00:34:44 --> 00:34:46
that's not being bound up.
 

516
00:34:46 --> 00:34:51
So we're going to solve,
instead, in terms of [E]

517
00:34:51 --> 00:34:55
is equal to [E]0 minus [ES],
where this is the initial

518
00:34:55 --> 00:34:58
concentration and this
is the amount that's

519
00:34:58 --> 00:34:59
binding substrate.
 

520
00:34:59 --> 00:35:03
And this is the amount of
free enzyme here then.

521
00:35:03 --> 00:35:08
And when you do that, and you
plug in here, and you plug in

522
00:35:08 --> 00:35:14
here, you get your result.
 

523
00:35:14 --> 00:35:20
Which is that [E]
 

524
00:35:20 --> 00:35:22
steady state, [ES]
 

525
00:35:22 --> 00:35:37
steady state, is this
ratio. k1 times [S]

526
00:35:37 --> 00:35:45
plus k minus two times the
product divided by k minus one

527
00:35:45 --> 00:35:51
plus k2 plus k1 times [S]
 

528
00:35:51 --> 00:35:57
plus k minus two times the
product, times proportional

529
00:35:57 --> 00:36:05
to the initial
concentration of enzyme.

530
00:36:05 --> 00:36:10
And then you can take your
steady state approximation for

531
00:36:10 --> 00:36:30
the intermediate and plug it
back into your rate equation.

532
00:36:30 --> 00:36:35
So the velocity, we defined
as the rate which is minus

533
00:36:35 --> 00:36:45
d[S]/dt, which is k1 times [E]
 

534
00:36:45 --> 00:36:48
times [S], this is the
destruction of the substrate

535
00:36:48 --> 00:36:52
minus k minus one, times [ES]
 

536
00:36:52 --> 00:36:57
steady state, the
backwards process.

537
00:36:57 --> 00:36:59
So we put in,
instead of this [E]

538
00:36:59 --> 00:37:00
we put in [E]
 

539
00:37:00 --> 00:37:06
is equal to [E]0 minus [ES]
 

540
00:37:06 --> 00:37:10
steady state.
 

541
00:37:10 --> 00:37:12
And then we put in for [ES]
 

542
00:37:12 --> 00:37:15
steady state, we put in
what we found here.

543
00:37:15 --> 00:37:19
We turn the crank
on the algebra.

544
00:37:19 --> 00:37:27
And we find that the velocity,
then, is k1 k2 times the

545
00:37:27 --> 00:37:29
substrate concentration.
 

546
00:37:29 --> 00:37:34
Minus k minus one times
k minus two times the

547
00:37:34 --> 00:37:36
product concentration.
 

548
00:37:36 --> 00:37:44
The whole thing times the
initial enzyme concentration.

549
00:37:44 --> 00:37:48
And then k minus one
plus k2 on the bottom.

550
00:37:48 --> 00:37:51
Plus k1 [S]
 

551
00:37:51 --> 00:37:57
plus k minus two
times the product.

552
00:37:57 --> 00:38:07
So, as I mentioned, this
product here usually

553
00:38:07 --> 00:38:08
just floats away.
 

554
00:38:08 --> 00:38:13
And so locally, where you're
doing the, and then it

555
00:38:13 --> 00:38:17
gets used by the next
step in the cycle.

556
00:38:17 --> 00:38:20
So this is pretty much zero.
 

557
00:38:20 --> 00:38:22
We can pretty much ignore this.
 

558
00:38:22 --> 00:38:25
We can ignore this
guy here too.

559
00:38:25 --> 00:38:27
Because it's not, you don't
have any steady state

560
00:38:27 --> 00:38:28
or an equilibrium.
 

561
00:38:28 --> 00:38:31
You have a steady state but
not an equilibrium situation.

562
00:38:31 --> 00:38:40
And so in that case here, you
can rewrite this then as k1 k2

563
00:38:40 --> 00:38:49
times the substrate times [E]0
divided by k minus one plus k2

564
00:38:49 --> 00:38:53
plus k1 times the substrate.
 

565
00:38:53 --> 00:39:00
And now we can look at these
experimental observations

566
00:39:00 --> 00:39:05
and see whether they
match our mechanism.

567
00:39:05 --> 00:39:12
If we can understand something.
 

568
00:39:12 --> 00:39:15
So let's look at
the initial rate.

569
00:39:15 --> 00:39:19
Initial rate is supposed to be
proportional to the substrate.

570
00:39:19 --> 00:39:22
Initial rate is supposed to be
proportional to the substrate.

571
00:39:22 --> 00:39:34
So the initial rate,
k1 k2 plus k1, if at

572
00:39:34 --> 00:39:43
early times, plus one.
 

573
00:39:43 --> 00:39:56
So the initial rate is going
to be, somehow I've got

574
00:39:56 --> 00:40:08
the product missing here.
 

575
00:40:08 --> 00:40:09
No, I got it backwards here.
 

576
00:40:09 --> 00:40:11
This is not the initial rate.
 

577
00:40:11 --> 00:40:15
This is the rate where [S]
 

578
00:40:15 --> 00:40:19
is small.
 

579
00:40:19 --> 00:40:23
The initial rate is when you
don't have any products made.

580
00:40:23 --> 00:40:25
Or when the product
concentration is very low.

581
00:40:25 --> 00:40:28
And because we're making the
assumption that the product

582
00:40:28 --> 00:40:31
concentration is basically
equal to zero here,

583
00:40:31 --> 00:40:33
this is basically the
initial rate here.

584
00:40:33 --> 00:40:36
So by taking the product equal
to zero here, we're also

585
00:40:36 --> 00:40:41
saying that this is the same
thing as the initial rate.

586
00:40:41 --> 00:40:44
So this is what it
looks like here..

587
00:40:44 --> 00:40:45
The initial rate here.
 

588
00:40:45 --> 00:40:54
And this is, this, you rewrite
as v initial is equal

589
00:40:54 --> 00:40:57
to k2 times [S]
 

590
00:40:57 --> 00:40:58
times [E]0.
 

591
00:40:58 --> 00:41:02
You divide by k1 up and down.
 

592
00:41:02 --> 00:41:07
And you have this k minus
one plus k2 over k1

593
00:41:07 --> 00:41:09
and then plus [S]
 

594
00:41:09 --> 00:41:11
sitting down there.
 

595
00:41:11 --> 00:41:18
And you define this ratio of
rates, k minus one plus k2 over

596
00:41:18 --> 00:41:25
k1 as the KM, the Michaelis
constant, by definition.

597
00:41:25 --> 00:41:26
And this is an
interesting ratio.

598
00:41:26 --> 00:41:30
This is the rate, k minus one
is the rate of destroying the

599
00:41:30 --> 00:41:34
enzyme substrate complex by
going back to the reactants.

600
00:41:34 --> 00:41:37
k2 is the destruction of the
complex by going

601
00:41:37 --> 00:41:38
to the product.
 

602
00:41:38 --> 00:41:40
And k1's the creation
of the complex.

603
00:41:40 --> 00:41:43
So this is the rate of
destruction of the complex

604
00:41:43 --> 00:41:47
divided by the rate of
creation of the complex.

605
00:41:47 --> 00:41:50
So if the rate of destruction
of the complex is much faster

606
00:41:50 --> 00:41:56
than the rate of creation,
meaning that this is a large

607
00:41:56 --> 00:41:59
number, then you're not going
to pile up any complex.

608
00:41:59 --> 00:42:02
It's going to be destroyed
as soon as you create it.

609
00:42:02 --> 00:42:11
So if KM is large, then the
concentration, [ES], is

610
00:42:11 --> 00:42:13
going to be very small.
 

611
00:42:13 --> 00:42:15
Compared to [E]0.
 

612
00:42:15 --> 00:42:19
But if the rate of destruction
of the complex is small

613
00:42:19 --> 00:42:22
compared to the rate of
creation, you create complexes,

614
00:42:22 --> 00:42:25
you create complexes but you
don't keep up in terms of

615
00:42:25 --> 00:42:27
destroying them, in terms of
making products, or going

616
00:42:27 --> 00:42:29
back to reactants.
 

617
00:42:29 --> 00:42:33
And so you end up
saturating your enzyme.

618
00:42:33 --> 00:42:38
Every enzyme ends up having
substrate bound to it.

619
00:42:38 --> 00:42:44
So when KM is very
small, then [ES]

620
00:42:44 --> 00:42:51
goes to saturation.
 

621
00:42:51 --> 00:42:56
Basically, when KM is very
small, then you're limited by

622
00:42:56 --> 00:43:00
the rate, the second rate
in the process, of the

623
00:43:00 --> 00:43:01
enzyme falling apart.
 

624
00:43:01 --> 00:43:03
To form the product.
 

625
00:43:03 --> 00:43:06
You have to wait
until that happens.

626
00:43:06 --> 00:43:09
Because then the product
just floats away.

627
00:43:09 --> 00:43:14
And that becomes your
rate limiting step.

628
00:43:14 --> 00:43:20
Another way that you also will
see this written is as this,

629
00:43:20 --> 00:43:28
then, is equal to, we'll define
k cat is equal to k2. k cat

630
00:43:28 --> 00:43:35
times the enzyme concentration
times the substrate, [E]0,

631
00:43:35 --> 00:43:46
divided by KM plus but the
substrate concentration.

632
00:43:46 --> 00:43:48
So let's look at a few
limiting cases then.

633
00:43:48 --> 00:43:51
First limiting case
is, suppose, that [S]

634
00:43:51 --> 00:43:59
is large.
 

635
00:43:59 --> 00:44:00
Let's take [S]
 

636
00:44:00 --> 00:44:03
to be much larger than KM.
 

637
00:44:03 --> 00:44:07
Because you look at the
denominator, and you see it's

638
00:44:07 --> 00:44:10
this ratio of rates but
there's this concentration

639
00:44:10 --> 00:44:12
that's important here.
 

640
00:44:12 --> 00:44:14
So in one case KM is going to
dominate, and in the other

641
00:44:14 --> 00:44:17
limiting case the substrate
concentration is

642
00:44:17 --> 00:44:17
going to dominate.
 

643
00:44:17 --> 00:44:20
So let's say that the substrate
concentration dominates.

644
00:44:20 --> 00:44:22
Meaning K sub M is small.
 

645
00:44:22 --> 00:44:23
In which case we already saw.
 

646
00:44:23 --> 00:44:27
If K sub M is small then
you reach saturation.

647
00:44:27 --> 00:44:33
And in the equation, then, the
velocity, KM is small, [S]

648
00:44:33 --> 00:44:38
is large, the [S]'s cancel out
and the velocity is equal

649
00:44:38 --> 00:44:43
to k cat times the initial
substrate concentration.

650
00:44:43 --> 00:44:44
Both of these are constants.
 

651
00:44:44 --> 00:44:46
The velocity is constant.
 

652
00:44:46 --> 00:44:50
And the rate is constant,
it's that limit up here.

653
00:44:50 --> 00:44:55
That's experimentally seen.
 

654
00:44:55 --> 00:44:58
And the rate is depending
on the initial substrate

655
00:44:58 --> 00:45:01
concentration.
 

656
00:45:01 --> 00:45:02
Initial enzyme concentration.
 

657
00:45:02 --> 00:45:04
All the enzymes have a
substrate in there.

658
00:45:04 --> 00:45:07
The more enzymes you have to
begin with, the more

659
00:45:07 --> 00:45:08
intermediates you're going to
have, the faster you

660
00:45:08 --> 00:45:11
going to make products.
 

661
00:45:11 --> 00:45:13
And then it's going to depend
on this on the rate, k cat,

662
00:45:13 --> 00:45:17
which is just k2, which is the
rate of formation of products.

663
00:45:17 --> 00:45:24
That becomes the rate
limiting step here.

664
00:45:24 --> 00:45:29
The other special case is if
substrate concentration is

665
00:45:29 --> 00:45:33
very small compared to KM.
 

666
00:45:33 --> 00:45:38
And in that case here, and the
other thing that we're going to

667
00:45:38 --> 00:45:44
do is, we're going to, because
we now understand this as this

668
00:45:44 --> 00:45:49
maximum rate up there, we're
going to call this v max. k cat

669
00:45:49 --> 00:45:52
times [E]0, we're going
to call it v max.

670
00:45:52 --> 00:45:53
And so when [S]
 

671
00:45:53 --> 00:46:01
is very small, v, then,
is equal to v max, which

672
00:46:01 --> 00:46:03
is k cat times [E]0.
 

673
00:46:03 --> 00:46:04
This is v max here now.
 

674
00:46:04 --> 00:46:12
So we can rewrite this as v max
times substrate divided by KM

675
00:46:12 --> 00:46:14
plus the substrate
concentration.

676
00:46:14 --> 00:46:15
Another way of writing it.
 

677
00:46:15 --> 00:46:19
Capital K.
 

678
00:46:19 --> 00:46:23
v max times the substrate
concentration.

679
00:46:23 --> 00:46:25
And we have KM plus [S].
 

680
00:46:25 --> 00:46:25
But [S]
 

681
00:46:25 --> 00:46:27
is very small.
 

682
00:46:27 --> 00:46:28
So we drop it.
 

683
00:46:28 --> 00:46:33
And this is then proportional
to the substrate concentration.

684
00:46:33 --> 00:46:36
And then that's small substrate
concentration compared to KM,

685
00:46:36 --> 00:46:37
we're sitting right here.
 

686
00:46:37 --> 00:46:40
Where it's linear, where
the rate is linear.

687
00:46:40 --> 00:46:46
And there's a third place on
the graph which is interesting.

688
00:46:46 --> 00:46:52
Which is when the substrate
concentration is equal to KM.

689
00:46:52 --> 00:46:54
In that case there,
you plug [S]

690
00:46:54 --> 00:46:58
equal to KM, and you end up
with the velocity then is equal

691
00:46:58 --> 00:47:04
to v max divided by two.
 

692
00:47:04 --> 00:47:05
So when [S]
 

693
00:47:05 --> 00:47:10
is equal to KM, you
are halfway up.

694
00:47:10 --> 00:47:18
There's v max over two and
there's KM sitting here.

695
00:47:18 --> 00:47:18
When [S]
 

696
00:47:18 --> 00:47:24
is equal to KM, you're
at v max over two.

697
00:47:24 --> 00:47:27
And so enzymes, then, are
labeled by their KM's.

698
00:47:27 --> 00:47:31
Because then it becomes very
important to know how strongly

699
00:47:31 --> 00:47:34
they bind the substrate.
 

700
00:47:34 --> 00:47:37
Sometimes you want the enzyme
to bind it very strongly.

701
00:47:37 --> 00:47:40
Sometimes you don't, you
want it to be fleeting.

702
00:47:40 --> 00:47:45
Depends on the role
that the enzyme plays.

703
00:47:45 --> 00:47:52
Now there's a way to plot this
that extracts out these

704
00:47:52 --> 00:47:54
important numbers.
k cat and KM.

705
00:47:54 --> 00:47:57
 


706
00:47:57 --> 00:48:00
And that's the
Lineweaver-Burk plot..

707
00:48:00 --> 00:48:06
Lineweaver-Burk plot..
 

708
00:48:06 --> 00:48:08
And I just looked up this
morning to see if Mr.

709
00:48:08 --> 00:48:09
Lineweaver was still alive.
 

710
00:48:09 --> 00:48:12
And as far as I can
tell he's still alive.

711
00:48:12 --> 00:48:17
He was 97, in 2003.
 

712
00:48:17 --> 00:48:22
So as of 2007 he
was still alive.

713
00:48:22 --> 00:48:25
He's getting up there.
 

714
00:48:25 --> 00:48:27
One of the most cited papers
that you have in your notes is

715
00:48:27 --> 00:48:32
the in Jack's, was the paper
that showed how to go from this

716
00:48:32 --> 00:48:39
curved line to a straight line
by plotting one over v versus

717
00:48:39 --> 00:48:42
[S], instead of v versus [S].
 

718
00:48:42 --> 00:48:51
So if you take your equation
and massage it, one over v KM

719
00:48:51 --> 00:48:57
over v max times [S], plus one
over v max, we haven't done

720
00:48:57 --> 00:49:00
anything except rewrite the
equation in terms of one

721
00:49:00 --> 00:49:05
of v versus one over [S].
 

722
00:49:05 --> 00:49:08
So it becomes linear.
 

723
00:49:08 --> 00:49:14
In one over [S], and there's
one over v sitting here.

724
00:49:14 --> 00:49:18
And you get a straight line.
 

725
00:49:18 --> 00:49:23
With an intercept here
that's one over v max.

726
00:49:23 --> 00:49:29
And if you keep going,
extrapolate out, you get this

727
00:49:29 --> 00:49:40
point here to be minus 1
over KM and the slope

728
00:49:40 --> 00:49:45
is KM over v max.
 

729
00:49:45 --> 00:49:48
And v max was equal to k
cat times [E]0, so you

730
00:49:48 --> 00:49:53
get k cat out of this.
 

731
00:49:53 --> 00:49:57
So it turned out to be
a very useful plot.

732
00:49:57 --> 00:49:59
It's very easy to plot a
straight line, especially

733
00:49:59 --> 00:50:00
before computers.
 

734
00:50:00 --> 00:50:02
In the age of computers.
 

735
00:50:02 --> 00:50:06
And the referees, there were
six referees that got this

736
00:50:06 --> 00:50:10
paper and pretty much turned it
down because they didn't think

737
00:50:10 --> 00:50:11
there was any new
chemistry in it.

738
00:50:11 --> 00:50:12
Which is true, there's
no new chemistry.

739
00:50:12 --> 00:50:15
It's just a way of
rewriting the plot.

740
00:50:15 --> 00:50:18
But it was very
important nevertheless.

741
00:50:18 --> 00:50:24
OK, any questions on catalysis?
 

742
00:50:24 --> 00:50:27
Enzymes?
 

743
00:50:27 --> 00:50:30
Arrhenius?
 

744
00:50:30 --> 00:50:34
Alright, next time we'll
oscillating reactions

745
00:50:34 --> 00:50:37
and recap the course.
 

746
00:50:37 --> 00:50:37