Lecture 7
Enzyme Kinetics: the
Michaelis-Menten System
Reading: The material in this lecture appears in Sections 6.1–6.3 of
James D. Murray (2002), Mathematical Biology I: An introduction, 3rd
edition, Springer Interdisciplinary Applied Mathematics Series, Vol. 17,
ISBN 0-387-95223-2.
Available online at http://bit.ly/OnlineMurrayVol1.
I also drew inspiration from
Desmond J. Higham (2008), Modeling and simulating chemical reactions,
SIAM Review, 50:347–368. DOI 10.1137/060666457
and Chapter 2 of
Edda Klipp, Wolfram Liebermeister, Christoph Wierling, Axel Kowald,
Hans Lehrach, and Ralf Herwig (2009), Systems Biology: A Textbook,
Wiley-Blackwell, Weinheim, ISBN 978-3-527-31874-2.
7.1
The Michaelis-Menten system
This lecture is a lightning introduction to the next main theme of the course, the
study of systems of biochemical reactions. It is meant to be slightly too quick and
sketchy for comfort—important observations will be plucked out thin air—but my
aim is to provoke the reader’s curiosity and make her wish for the more systematic
approach that will follow in subsequent lectures. Despite the brisk pace, I hope that
you’ll notice that the ODEs that appear today look very similar to those we have
been studying in the lectures on population dynamics. The vocabulary is similar
too: we’ll use the term chemical species to refer to di↵erent kinds of molecules or
chemically distinct forms of the same molecule.
7.1
Our main example will be the Michaelis-Menten system, which was originally
proposed around a century ago1 and is the standard model of enzyme catalysis, a
process through which one kind of molecule, an enzyme, facilitates the conversion of
a second type of molecule, the enzyme’s substrate, into a third type of molecule, the
reaction’s product. Although the enzyme participates in this process, it is neither
created nor destroyed—it acts as a catalyst. Enzymes can accelerate the rates of
reactions very substantially, causing them to occur 106 –1012 times faster than they
would in the absence of a catalyst.
The following concise representation of the chemistry
k
k2
1
S + E )* C !
P +E
k
1
involves three distinct reactions
Complex formation:
S+E
Dissociation:
Product formation:
C
C
k1
k
!
1
!
!
k2
C
S+E
P +E
The first of these is a reversible step (“reversible” because we also include a reaction
that undoes the process of complex formation) in which one molecule each of the
enzyme E and substrate S combine to form the so-called activated complex C (also
called complex for short). This complex can either dissociate—fall back apart into
its constituent E and S molecules—or the action of the enzyme can transform the
substrate molecule into a molecule of product P , liberating the enzyme to form a
new complex. The product formation step is taken to be irreversible and so one
expects that all the substrate will eventually be converted to product.
Chemical reactions such as these are usually modelled with the law of mass action, which says that the rate at which a reaction proceeds (measured in reaction
events per unit volume per unit time) is proportional to the product of the concentrations of the reactants. So, for example, the rate of the complex-formation
reaction
k1
S+E !
C
is k1 [S][E], where the square brackets indicate concentrations (measured in particles
per unit volume) and constants of proportionality such as k1 are called rate constants.
Rate laws of this form are referred to as mass-action kinetics.
Many of the chemical reactions in living things (and, indeed, elsewhere) involve
such huge numbers of particles that chemists have a special counting word (not so
much a unit like “seconds” or “meters” as a term like “a hundred” or “a thousand”
used to denote quantity), the mole, to refer to the vast quantities2 of molecules in
play. Concentrations are thus often reported in moles/liter. Later in the term we
will revisit this issue and discuss how to model chemistry in settings such as the
1
Leonor Michaelis and Maud Leonora Menten (1913), Die kinetic der invertinwirkung, Biochemische Zeitschrift, 49:333-369.
2
A mole is an Avogadro number, L ⇡ 6.02 ⇥ 1023 , of atoms or molecules of a substance. It
is currently defined as the number of atoms in a 12 gram sample of pure 12C, the carbon isotope
whose nucleus has six neutrons and six protons.
7.2
nucleus of a cell, where some important chemical species—notably genes, of which
there are often only one or two copies in a given cell—exist in such low numbers
that one cannot justify the use of mass-action kinetics.
The kinds of metabolic processes described by the Michaelis-Menten system do,
however, involve sufficiently many molecules that mass-action kinetics are a good
approximation and so we’ll write down ODEs that describe the time evolution of
the concentrations of the various chemical species. If we use lowercase symbols to
denote the various concentrations
s = [S], e = [E], c = [C] and p = [P ]
then the relevant system of ODEs is
de
dt
ds
dt
dc
dt
dp
dt
=
k1 es + (k
=
k1 es + k 1 c
= k1 es
(k
1
1
+ k2 )c
+ k2 )c
= k2 c
(7.1)
The righthand sides of these equations have a positive term for every reaction that
creates particles of the given species and negative terms for those that consume the
species. Thus the equation for ds/dt has two terms,
ds
=
dt
k es
+
k 1c
| {z1 }
|{z}
complex formation dissociation
We’ll work primarily with the initial value problem in which
e(0) = e0 , s(0) = s0 and c(0) = p(0) = 0.
(7.2)
where e0 and s0 are positive constants. That is, we’ll think about the situation just
after we’ve added some enzyme to a container full of substrate.
7.2
Simplifying the Michaelis-Menten system
One can simplify the system (7.1) by noting that product formation is such a uncomplicated business that given c(t), the time course of the concentration of the
complex, one can obtain p(t) via straightforward integration:
Z T
p(T ) = p(0) +
k2 c(t) dt.
0
Also, as enzyme molecules can be free or locked up in a complex with substrate, but
are neither created nor destroyed, one has that
de dc
+
= [ k1 es + (k
dt dt
1
+ k2 )c] + [k1 es
7.3
(k
1
+ k2 )c] = 0
or equivalently,
e(0) + c(0) = e(t) + c(t)
so
e(t) = e0
c(t),
where the last equation holds only for the initial conditions in Eqn. (7.2).
One can use this result to eliminate e(t) from the ODE for the substrate concentrations as follows:
ds
=
dt
=
=
=
k1 es + k 1 c
k1 (e0 c)s + k 1 c
k1 e0 s + k1 sc + k 1 c
k1 e0 s + (k1 s + k 1 )c.
Similar calculations also yield
dc
= k1 es (k 1 + k2 )c
dt
= k1 (e0 c)s (k 1 + k2 )c
= k1 e0 s k1 sc (k 1 + k2 )c
= k1 e0 s (k1 s + k 1 + k2 )c
and together these observations mean that one can reduce the Michaelis-Menten
system to the following system
ds
= k1 e0 s + (k1 s + k 1 )c
dt
dc
= k1 e0 s (k1 s + k 1 + k2 )c
dt
(7.3)
which begins to look very similar to a two-species population model.
7.3
The quasi-steady-state approximation
The standard account of (7.3) investigates the initial value problem that corresponds
to Eqn. (7.2) with the added assumption that there is much, much more substrate
than enzyme, so
e(0) = e0 , s(0) = s0
e0 and c(0) = p(0) = 0.
(7.4)
This means that the processes of complex formation and breakdown (whether via
dissociation or product formation) are in approximate equilibrium and so
dc
⇡ 0.
dt
This is called the quasi-steady-state approximation (QSSA) or in the more mathematical literature, the quasi-steady-state hypothesis.
7.4
Under the QSSA we can set the expression for dc/dt for zero and solve to obtain
c as a function of s:
dc
= 0 = k1 e0 s (k1 s + k 1 + k2 )c
dt
k1 e0 s = (k1 s + k 1 + k2 )c
k1 e 0 s
c=
.
k1 s + k 1 + k2
which is then rewritten in the form
c(t) =
e0 s(t)
s(t) + Km
where
Km =
k
+ k2
k1
1
(7.5)
(7.6)
is called the Michaelis constant.
Putting the expression from Eqn. (7.5) into the ODE for ds/dt yields
ds
=
dt
k1 e0 s + (k1 s + k 1 )c
✓
◆
e0 s
= k1 e0 s + (k1 s + k 1 )
s + Km
✓
◆
k1 s + k 1
= e0 s
k1
s + Km
✓
◆
k1 s + k 1
= e0 s
k1
s + (k 1 + k2 )/k1
✓ 2
◆
k1 s + k1 k 1
= e0 s
k1
k 1 s + k 1 + k2
✓ 2
◆
k1 s + k1 k 1
k1 (k1 s + k 1 + k2 )
= e0 s
k s + k 1 + k2
k1 s + k 1 + k 2
✓ 1
◆
k1 k2
= e0 s
k s + k 1 + k2
✓ 1
◆
k2
= e0 s
s + (k 1 + k2 )/k1
✓
◆
k2 e 0 s
=
s + Km
The last line is usually rewritten as
ds
=
dt
k2 e 0 s
=
s + Km
sVmax
s + Km
(7.7)
where Km is as above and
Vmax = k2 e0
is called the maximum reaction velocity.
7.5
(7.8)
Vmax/2
0
dp/dt
Vmax
Michaelis-Menten: rate of product formation
0
2Km
4K m
6Km
s
Figure 7.1: The rate of product formation as a function of substrate concentration
in the Michaelis-Menten system ubder the QSSA: half-maximal production occurs
when s = Km .
To see why, first note that the full Michaelis-Menten system has another conserved quantity:
ds dc dp
+
+
= [k1 es + k 1 c] + [k1 es
dt dt
dt
(k
1
+ k2 )c] + k2 c = 0.
Under the QSSA—where we assume dc/dt = 0—this implies that
dp
=
dt
ds
sVmax
=
.
dt
s + Km
(7.9)
The ratio at right is a montone increasing function of s with the property that
✓
◆
sVmax
lim
= Vmax .
s!1
s + Km
and so Vmax is the maximal rate of product formation. Further, Km is the substrate
concentration for which the rate of product formation is half-maximal. Figure 7.1,
which is reminiscent of the plot (Fig. 6.1) of the predation term in the spruce budworm model, illustrates these features.
The quantities Km and Vmax are the standard biochemical parameters used to
characterise an enzyme and considerable ingenuity has been devoted to finding ways
to estimate them from experimental data. If this interests you, have a look at
Chapter 2 of the book by Klipp et al.. This chapter also includes more elaborate
analogs of Eqn. (7.9) that apply to more complicated enzymatic processes.
7.6
7.4
Afterword: on the QSSA
I’d like to conclude this lecture by saying a bit about the validity of the QSSA. One
can integrate the ODE in Eqn. (7.7) to obtain the following implicit formula for
s(t), the time course of the substrate concentration:
✓
◆
s(t)
s(t) + Km ln
= s0 + k2 e0 t.
(7.10)
s0
One might then hope to solve the reduced Michaelis-Menten system (7.3) by substituting the solution for s(t) implied by the formula above into the QSSA condition
Eqn. (7.5)—which gives c(t) as a function of s(t)—to obtain a complete solution.
Unfortunately this approach doesn’t quite work. Although the implicit solution
above gives the correct result when t = 0, if we then try to use the QSSA condition
to infer c(0) we end up with
c(0) =
e0 s(0)
e 0 s0
=
6= 0,
s(0) + Km
s 0 + Km
which contradicts the initial data in Eqn. (7.4). The issue is that the QSSA does
not apply immediately: it’s not initially true that dc/dt ⇡ 0. Instead there is a brief
transient phase during which complex formation is very rapid.
Figure 7.2 illustrates both the issue and its resolution: if c(0) = 0 the relationship
implied by the QSSA fails to hold at t = 0, but rapidly becomes valid and then
provides a good approximation until most of the substrate has been converted to
product. Murray’s Sections 6.2 and 6.3 review an approach pioneered by the great
Lee Segel3 , that uses the tools of asymptotic analysis to establish that if
✓
◆
e0
k2
"=
⌧ 1
(7.11)
s 0 + Km k 2 + k 1 + s 0 k 1
then the QSSA applies after a time tc with
tc ⇡
1
.
k1 (s0 + Km )
(7.12)
The attraction of Segel’s approach is its generality: it applies even when e0 ⇡ s0 ,
provided that Km
e0 .
3
Lee A. Segel and Marshall Slemrod (1989), The quasi-steady-state assumption: A case study
in perturbation, SIAM Review, 31(3): 446–477. DOI: 10.1137/1031091.
7.7
0.6
0.4
0.0
0.2
c(t)/e0
0.8
Numerical solution vs. QSSA
0
50
100
150
200
t
Figure 7.2: The ratio c(t)/e0 , where c(t) is the concentration of complex and e0
is its maximum possible value. The green curve shows a numerical solution to the
full system of ODEs in Eqn. (7.1) while the orange dashed curve shows the QSSA
result. To draw it, I solved Eqn. (7.7) numerically to get an approximation s̃(t) for
the substrate concentration, then applied Eqn. (7.5) to get the QSSA approximation
c̃(t) = c(s̃(t)). The dashed vertical line shows tc from Eqn. (7.12), which is an
estimate for the duration of the initial transient phase during which c(t) increases
rapidly. The parameters are k1 = 106 , k 1 = 10 4 , k2 = 0.1, e0 = 5 ⇥ 10 8 and
s0 = 5.0 ⇥ 10 7 , which means Km ⇡ 10 7 and the small parameter in Eqn. (7.11)
is " ⇡ 0.014.
7.8