Lecture notes. Written from the material of MIT 8.591J 2004, licensed CC BY-NC-SA 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.
17. Michaelis-Menten Kinetics and Cooperativity¶
What this covers¶
This chapter answers two related questions. First, why does an enzyme-catalysed reaction saturate — why does its rate rise linearly with substrate at low concentration and then level off, rather than keep climbing — and what does the resulting Michaelis–Menten law mean physically. Second, when a protein has several binding sites for the same ligand, when does the binding curve stay a simple hyperbola and when does it turn sigmoidal (the signature of cooperativity), and how many genuinely different ways are there to produce that sigmoid. Both questions are answered the same way: write mass-action kinetics for a small reaction scheme, then solve it either in a steady state (Michaelis–Menten) or in full equilibrium (binding and cooperativity). The chapter assumes only that a reaction rate is proportional to the product of the reactant concentrations, and comfort setting time derivatives to zero to find a steady or equilibrium state.
The enzyme–substrate reaction and its kinetics¶
Enzymes are proteins that bind a substrate — another protein, a nucleic acid, or a small molecule — and catalyse its conversion to a product without being consumed themselves. Three examples the lecture uses to show the same scheme recurring: hexokinase converting glucose into glucose-6-phosphate; RNA polymerase binding a promoter and transcribing it into mRNA; and the phosphate CheZ converting the flagellar-motor regulator CheY into its phosphorylated form CheY-P. All three fit the same two-step scheme: reversible binding of substrate and enzyme, followed by an irreversible conversion to product,
Note \(k_1\) and \(k_{-1}\) carry different units (\(1/(\mathrm{M\,s})\) and \(1/\mathrm{s}\)), because the first step depends on the product of two concentrations and the second on one. Mass action — reaction rate proportional to the product of the concentrations of the reactants — turns this scheme into four coupled ODEs:
with initial conditions \([\mathrm{S}]_{t=0}=S_o\), \([\mathrm{E}]_{t=0}=E_o\), \([\mathrm{ES}]_{t=0}=0\), \([\mathrm{P}]_{t=0}=0\). The turnover rate \(v\), by definition, is how fast product accumulates, and it is directly proportional to how much enzyme is currently tied up as complex.
Because the enzyme is a catalyst, it is neither created nor destroyed — only shuttled between free and bound forms — so the total enzyme concentration \(E_o=[\mathrm{E}]+[\mathrm{ES}]\) is a conserved quantity. Using it to eliminate \([\mathrm{E}]\) collapses the system to three equations:
This system can be integrated exactly, and the lecture does so numerically (feeding \(k_1=10^3\,\mathrm{M^{-1}s^{-1}}\), \(k_{-1}=1\,\mathrm{s^{-1}}\), \(k_2=0.05\,\mathrm{s^{-1}}\), \(E_o=0.5\times10^{-3}\,\mathrm{M}\), \(S_o=10^{-3}\,\mathrm{M}\) into an ODE solver) to see what the concentrations actually do over time. The regime plotted is \(k_1[S_o]\approx k_{-1}\gg k_2\) — binding and unbinding equilibrate on a timescale much faster than turnover into product — which is also the biologically relevant regime, since it is what makes an enzyme useful as a catalyst rather than a one-shot reagent. Plotted on a log time axis, \([\mathrm{ES}]\) and \([\mathrm{E}]\) shoot to a plateau almost immediately and then stay nearly flat while \([\mathrm{S}]\) slowly drains and \([\mathrm{P}]\) slowly fills — two clearly separated timescales.
There is a true equilibrium at \(t\to\infty\): \([\mathrm{S}]=[\mathrm{ES}]=0\), \([\mathrm{E}]=E_o\), \([\mathrm{P}]=S_o\) — everything has been converted. That end state is uninteresting; the useful regime is the long, flat middle stretch where \([\mathrm{ES}]\) and \([\mathrm{E}]\) have already settled down but \([\mathrm{S}]\) has barely moved. This is the quasi-equilibrium or pseudo-steady state: after a short initial transient, complex is formed exactly as fast as it breaks apart (in either direction), so \([\mathrm{ES}]\) holds nearly constant even while \([\mathrm{S}]\) and \([\mathrm{P}]\) are still slowly evolving.
The quasi-steady-state approximation and the Michaelis–Menten law¶
Setting \(\mathrm{d}[\mathrm{ES}]/\mathrm{d}t=\mathrm{d}[\mathrm{E}]/\mathrm{d}t=0\) in the reduced system gives the complex concentration and turnover rate directly in terms of the instantaneous substrate level:
If substrate is in large excess over enzyme (\(S_o\gg E_o\)), this pseudo-steady state is reached before any appreciable amount of substrate has been consumed, so \([\mathrm{S}]\approx S_o\) throughout the approach to it. Substituting gives the Michaelis–Menten equation for the initial turnover rate as a function of the initial substrate concentration:
\(K_m\), the Michaelis constant, has units of concentration and sets the scale of the curve: it is the substrate concentration at which the rate is exactly half its maximum, \(v(K_m)=\tfrac12 v_{\max}\). A small \(K_m\) means strong affinity — the enzyme reaches half-saturation at low substrate — and a large \(K_m\) means weak affinity. \(v_{\max}=k_2E_o\) is the rate once every enzyme molecule is permanently tied up as complex, so it scales linearly with how much enzyme is present and with how fast the catalytic step itself runs.
Multiple binding sites and Adair's equation¶
The Michaelis–Menten picture has one substrate per enzyme. Many proteins carry several binding sites for the same ligand — haemoglobin's four oxygen sites are the standard example — and here the question shifts from kinetics (how fast does turnover happen) to equilibrium binding (how much ligand is bound at a given free ligand concentration). Write \(\mathrm{P}_j\) for the protein bound to \(j\) of its \(n\) sites, so ligand binds and unbinds one at a time,
For the first step, mass action gives \(\mathrm{d}[\mathrm{P}_0]/\mathrm{d}t = -k_{+1}[\mathrm{P}_o][\mathrm{S}] + k_{-1}[\mathrm{P}_1]\), and at steady state this defines an association constant \(K_a=k_{+1}/k_{-1}=[\mathrm{P}_1]/([\mathrm{P}_o][\mathrm{S}])\) (its reciprocal, \(K_d\), is the dissociation constant). More generally each of the \(n\) steps has its own stepwise association constant
The individual \([\mathrm{P}_j]\) are hard to measure directly; what an experiment actually reports is the average number of ligands bound per protein,
Substituting the \(K_j\)'s expresses \(r\) purely in terms of the association constants and the free ligand concentration — Adair's equation:
Since \(0<r<n\), it is common to normalise to the saturation function \(Y=r/n\in(0,1)\), the fraction of all sites occupied. Adair's equation is the exact, general statement for \(n\) sites; the rest of the chapter is about what happens to it under different assumptions about how the \(K_j\) relate to one another.
Identical, independent binding sites¶
Suppose the \(n\) sites are chemically identical and bind completely independently — occupying one site has no effect on any other. Then a single pair of rate constants \(k_+,k_-\) describes binding and unbinding at any site, and the stepwise constants \(K_j\) differ from each other only through combinatorics: going from \(\mathrm{P}_0\) to \(\mathrm{P}_1\), there are \(n\) empty sites the incoming ligand could land on, but only one way to leave \(\mathrm{P}_1\) back to \(\mathrm{P}_0\); going from \(\mathrm{P}_1\) to \(\mathrm{P}_2\) there are \((n-1)\) empty sites left but \(2\) ways to lose one of the two occupied ones. Writing the intrinsic constant as \(K\equiv k_+/k_-\), steady state at each step gives \(K_1=nK\), \(K_2=(n-1)K/2\), and in general
Feeding this into Adair's equation and summing the resulting binomial series (the lecture points to Bisswanger 2002, p. 11–16 for the algebra) collapses the whole sum to one hyperbola:
This has exactly the shape of the Michaelis–Menten curve, but it is a fundamentally different object: it is an equilibrium relation between bound ligand and free ligand concentration, not a steady-state kinetic rate. The same result falls out with no combinatorics at all if the sites are treated as an undifferentiated pool rather than as clustered on one protein — which is legitimate precisely because they are independent. Let \([\mathrm{F}]\) and \([\mathrm{B}]\) be the concentrations of free and bound sites; the association constant for this one-site-at-a-time equilibrium is \(K=[\mathrm{B}]/([\mathrm{F}][\mathrm{S}])\), and with \(n[\mathrm{P}]=[\mathrm{F}]+[\mathrm{B}]\) as the total site count, \(r=[\mathrm{B}]/[\mathrm{P}]\) reduces to the same hyperbola. The two derivations agreeing is itself informative: independence is what let the many-site problem collapse back to a one-site problem.
Non-identical, independent binding sites¶
If instead the \(n\) sites fall into families — \(n_1\) sites with constant \(K_1\), \(n_2\) with \(K_2\), and so on — independence still lets each family be treated on its own, and \(r\) is just their sum:
At low ligand concentration the high-affinity family fills first; only at higher \([\mathrm{S}]\) does the low-affinity family start to fill. The binding curve is still a sum of hyperbolas — never sigmoidal — because nothing here couples one site's occupancy to another's.
Two interacting sites: cooperativity¶
Sigmoidal binding needs sites that talk to each other: occupying one site changes the affinity of the others. Restrict to the simplest case, \(n=2\), with both sites identical. There are three states — \(\mathrm{P}_0\), \(\mathrm{P}_1\), \(\mathrm{P}_2\) — and now two intrinsic constants: \(K=k_+/k_-\) for the empty-to-singly-bound transition, and \(K^*=k_+^*/k_-^*\) for the singly-bound-to-doubly-bound transition, allowed to differ because the second binding event happens on a protein already occupied at the other site. The combinatorial factors from the identical-independent case still apply (\(K_1=2K\), \(K_2=\tfrac12K^*\)), so Adair's equation for \(n=2\) gives
If \(K=K^*\) — the second site is exactly as easy to bind as the first — this reduces to the ordinary hyperbola \(\tilde Y=K[\mathrm{S}]/(1+K[\mathrm{S}])\), the \(n=2\) independent case. The deviation from that hyperbola is what cooperativity is:
Positive cooperativity is defined as \(Y>\tilde Y\), equivalently \(K^*>K\): binding the first ligand makes the second site more attractive. Negative cooperativity is \(Y<\tilde Y\), i.e. \(K^*<K\) — the first binding event makes the second harder.
Two ways to define cooperativity: affinity shift versus sigmoidicity¶
The affinity-shift definition above (\(K^*\) vs \(K\)) is not the only one in use. A second, older definition calls a curve cooperative when it is genuinely sigmoidal — S-shaped — which happens exactly when the second derivative of \(Y\) with respect to \([\mathrm{S}]\) changes sign. Writing \(x=K[\mathrm{S}]\) and \(\beta=K^*/K\) as dimensionless variables,
Working through when the numerator of the second derivative can change sign shows it needs \(\beta>2\), not merely \(\beta>1\). So the two definitions genuinely disagree: by the affinity-shift definition any \(K^*>K\) (\(\beta>1\)) counts as cooperative, but the curve is only visibly sigmoidal once \(\beta>2\). The chapter — and the rest of the course — sticks with the first, affinity-shift definition, but it is worth knowing the second exists and sets a stricter bar.
The Hill limit and the Hill number¶
A useful limiting case is when the singly-bound state is so unstable that it is essentially never observed — binding is effectively all-or-nothing, \(\mathrm{P}_0+2\mathrm{S}\leftrightarrow \mathrm{P}_2\) directly. Then
Hill's device for reading off this kind of relation from data is the Hill plot: plot \(\ln[Y/(1-Y)]\) against \(\ln[\mathrm{S}]\). Its slope is the Hill number \(n_H\), which for the all-or-nothing limit above is exactly \(2\) — matching the number of sites, which is exactly why the Hill number is often quoted as an estimate of site count. That estimate is only as good as the all-or-nothing assumption behind it. Allowing the intermediate, singly-bound state back in (the full two-site cooperative model above) gives instead
which only approaches \(2\) in the double limit of large \(\beta\) and small \(x\), and sits below \(2\) everywhere else — reaching \(1\) (no cooperativity) when \(\beta=1\). So a measured Hill number between \(1\) and the true site count is completely normal for a real, intermediate-allowing cooperative protein; it is a lower bound on cooperativity, not a literal site count, unless the all-or-nothing assumption actually holds.
Non-identical, interacting sites¶
The most general two-site case drops both simplifications at once: the two sites can differ and interact. There are still four states, but now the two singly-bound states — call them \(\mathrm{P}_1\) (site 1 occupied) and \(\mathrm{P}_1'\) (site 2 occupied) — are distinct, each reachable from \(\mathrm{P}_0\) by binding at one particular site, and each able to proceed to the fully-bound state \(\mathrm{P}_2\) by binding at the other. Four intrinsic constants \(K_1,K_2,K_3,K_4\) describe the four edges of this diagram:
Because this is a true thermodynamic equilibrium, there can be no net circulation around the loop — the two routes from \(\mathrm{P}_0\) to \(\mathrm{P}_2\) must be consistent, which forces \(K_1K_3=K_2K_4\): only three of the four constants are free. Writing the saturation function out and using that constraint to eliminate \(K_4\),
which, as detailed balance guarantees, does not depend on \(K_4\) at all. Defining effective constants
puts \(Y\) back into exactly the same universal shape as the identical-sites case, \(Y=x'(1+\beta'x')/(1+2x'+\beta'x'^2)\). Two limits check that this is the right generalisation: if \(K_1=K_2\) the two sites are equally easy to bind first, and \(x'=x\), \(\beta'=\beta\) recovers the identical-interacting-sites result exactly; if instead \(K_1=K_3\) and \(K_2=K_4\) — each site's affinity is unaffected by whether the other is occupied — the whole thing collapses back to the sum of two independent hyperbolas, \(Y=K_1[\mathrm{S}]/(1+K_1[\mathrm{S}])+K_2[\mathrm{S}]/(1+K_2[\mathrm{S}])\) divided appropriately, and in that case
with equality only when \(K_1=K_2\). This is the sting in the tail: \(\beta'<1\) — which reads as negative cooperativity under the affinity-shift definition — is exactly what two genuinely independent but non-identical sites also produce. A binding curve that looks like mild negative cooperativity cannot, from its shape alone, distinguish "two sites that don't talk to each other but have different affinities" from "two sites that do talk to each other and the second binding event is discouraged." Telling those apart needs more than the equilibrium binding curve.
Sources¶
- MIT OCW 8.591J Systems Biology (Fall 2004), lecture 2 handout, reconstructed transcript,
section I "Michaelis-Menten kinetics" — the enzyme–substrate scheme, the reduced ODEs, the
quasi-steady-state derivation and the Michaelis–Menten law (equations I.1–I.6 in the source), plus
the MATLAB simulation described there
(
docs/computational-biology/mit-ocw/8591j-2004/recordings/l2-syllabus-transcript/01-i-michaelis-menten-kinetics.md). - Same lecture, section II "Equilibrium binding and cooperativity" — multiple binding sites,
Adair's equation, identical/non-identical and independent/interacting sites, the two definitions
of cooperativity, the Hill limit and Hill number, and the non-identical interacting-sites diagram
(equations II.1–II.31 in the source)
(
docs/computational-biology/mit-ocw/8591j-2004/recordings/l2-syllabus-transcript/02-ii-equilibrium-binding-and-cooperativity.md). - Both files are model reconstructions of a PDF with no text layer; their own banner flags every equation in them as unverified, so this chapter should be read as a guide back to the original 8.591J lecture 2 handout, not as a citable derivation in its own right.
- The lecture points to, but does not itself reproduce, the combinatorial derivation of the identical-independent-sites result: H. Bisswanger, Enzyme Kinetics (Wiley-VCH, 2002), pp. 11–16. It also names, without drawing on directly, D. Fell, Understanding the Control of Metabolism (Portland Press, 1997); J. D. Murray, Mathematical Biology (Springer-Verlag, 1989); and L. A. Segel, Biological Kinetics (Cambridge University Press, 1991), as further reading on enzyme kinetics and cooperativity.
- No slides, written notes or exercise sets were supplied for this lecture, so there is no Exercises section.
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