Peter FajerBiophysical Methods in BiologyLecture 10
Ligand Binding
Consider ligand A binding to macromolecule P:
[1]
Equilibrium constant is defined as:
[2]
(note that Kd is often referred wrongly as binding constant: Kb = 1/Kd)
define fractional occupancy (moles of ligand/mole of macromolecule) r:
[3]
substitute [P][A]/Kd for [PA]:
[4]
this hyperbolic dependence is called binding isotherm or (Langmuir isotherm).
Note that although the isotherm has the concentration for free ligand [A] as well as bound ligand ([A]bound in r ) you have to measure only one (bound or free) because conservation of mass the total ligand [A]total = [A]+[PA] (and total macromolecule [P]total = [P]+[PA]).
Measure the concentrations of free or bound ligand by variety of methods: chromatography, equilibrium dialysis, ultrafiltration, spectroscopy.
Multiple binding sites
Macromolecule can have more than one site for binding the ligand. These sites can be independent: binding of one ligand does not influence binding of the next, or cooperative(binding of one affects binding of another).
Independent
[5]
- identical sites: A1P=A2P or
- different sites: A1PA2P
Identical sites
all sites are independent and identical so that total number of sites is n[P]total rather than [P]total :
[6]
Scatchard plot
plot the ratio of r/[A] versus r:
[8]
For identical binding this results in a linear plot. Deviation from linearity implies non-identical sites.
Different sites
to be very general: each class of sites with different Kdi can have ni identical sites
The binding isotherm is a sum of the single binding isotherms, each corresponding to different class of sites:
[7]
Cooperativity
Binding to one site on a molecule modulates binding of another.
For strong cooperativity, binding of one ligand triggers binding to n sites:
[9]
[10]
[11]
or measure the ratio of filled sites Y (Y=r/n) to the unfilled sites (1-Y):
[12]
(Hill’s equation)
Hill’s Plot
Plot log(Y/(1-Y) v. log[A], the slope is n; an intercept is 1/Kd.
Kinetics
Time course of reaction, kinetics give information about the rates rather than equilibria. Of course rates and equilibria are related Keg = k+1/k-1, however the ratio of rates tells us nothing about the value of each rate i.e. is it fast or is it slow.
Two sorts of kinetics are usually considered:
- steady-state: in which the forward flux = backward flux resulting in no change of concentration;
- transient kinetics: non-equilibrium kinetics when concentrations are changing.
steady-state (Michaelis-Menten)
[13]
for a reaction with an intermediate X the rate of intermediate’s production is equal to that of its disappearance:
The rates of substrate disappearance and the appearance of the product are identical:
[15]
From mass conservation we have:
[16]
At the beginning of the reaction:
[17]
thus the initial velocity v can be solved since we have five equations with 4 unknowns (concentrations of E, S, X, P)
[18]
(dirty shortcut: realize that initial velocity is a maximum velocity reduced by the partial enzyme occupancy, i.e.:
[18]
then substitute v for r in the equation for Langmuir isotherm.)
turnover
[19]
non-steady state
Steady-state approximation is not always applicable, for general reaction:
[20]
the rate equations are:
[21]
with two mass conversation relationships:
[22]
The differential (rate) equation can be solved but it is a mess since the equation is non-linear (the rate depends on product of [E] and [S]), better approximate that So > Eo so that concentration of substrate never changes (pseudo-first order approximation):
[23]
This equation is a linear in X1 and it can be integrated to give the concentration of X1 as function of time:
[24]
Presence of two intermediates complicates the affairs considerably:
[25]
conservation of mass gives:
[26]
the rate of enzyme disappearance is:
[27]
the rate of product appearance is:
[28]
At this point you are advised to give up. What you have is the set of simultaneous, non-linear differential equations which even your grandma can’t solve.
Bill Gates to the rescue; integrate the set using a PC and any mathematical package with numerical integration routines e.g. Mathcad, Mathematica, Matlab.
Binding_Kinetics.doc3/4/04 11:08 AM1