Biochemistry II: Binding of ligands to a macromolecule (or the secret ...

Biochemistry II: Binding of ligands to a macromolecule
(or the secret of life itself...)
• http://www.kip.uni-heidelberg.de/chromcon/publications/pdf-files/Rippe_Futura_97.pdf
• Principles of Physical Biochemistry, van Holde, Johnson & Ho, 1998.
• Slides available at
• http://www.kip.uni-heidelberg.de/chromcon/teaching/index_teaching.html
Karsten Rippe
Kirchoff-Institut für Physik
Molecular Biophysics Group
Im Neuenheimer Feld 227
Tel: 54-9270
e-mail: Karsten.Rippe@kip.uni-heidelberg.de

The secret of life
“The secret of life is molecular recognition; the
ability of one molecule to "recognize" another through
weak bonding interactions.”
Linus Pauling at the 25th anniversary of the Institute of
Molecular Biology at the University of Oregon

Binding of dioxygen to hemoglobin
(air)

Binding of Glycerol-Phosphate to triose phosphate isomerase
(energy)

Complex of the HIV protease the inhibitor SD146
(drugs)

Antibody HyHEL-10 in complex with Hen Egg White Lyzoyme
(immune system)

X-ray crystal structure of the TBP-promoter DNA complex
(transcription starts here...)
Nikolov, D. B. & Burley, S. K.
(1997). RNA polymerase II
transcription initiation: a
structural view. PNAS 94, 15-
22.

Structure of the tryptophan repressor with DNA
(transcription regulation)

Binding of ligands to a macromolecule
• General description of ligand binding
– the esssentials
– thermodynamics
– Adair equation
• Simple equilibrium binding
– stoichiometric titration
– equilibrium binding/dissociation constant
• Complex equilibrium binding
– cooperativity
– Scatchard plot and Hill Plot
– MWC and KNF model for cooperative binding

The mass equation law for binding of a protein P to its DNA D
Dfree + P !
free"DP
K1 = Dfree #Pfree DP
binding of the first proteins with the dissociation constant K 1
D free, concentration free DNA; P free, concentration free protein
1
binding constant KB =
dissociation constant KD

What is the meaning of the dissociation constant for
binding of a single ligand to its site?
1. K D is a concentration and has units of mol per liter
2. K D gives the concentration of ligand that saturates 50% of the
sites (when the total sit concentration ismuch lower than K D)
3. Almost all binding sites are saturated if the ligand
concentration is 10 x K D
4. The dissociatin constant K D is related to Gibbs free energy
∆G by the relation ∆G = - R T ln(K D)

K D values in biological systems
Movovalent ions binding to proteins or DNA have K D 0.1 mM to 10 mM
Allosteric activators of enzymes e. g. NAD have K D 0.1 µM to 0.1 mM
Site specific binding to DNA K D 1 nM to 1 pM
Trypsin inhibitor to pancreatic trypsin protease K D 0.01 pM
Antibody-antigen interaction have K D 0.1 mM to 0.0001 pM

What is ∆G? The thermodynamics of a system
• Biological systems can be usually described as having constant pressure P and constant
temperature T
– the system is free to exchange heat with the surrounding to remain at a constant temperature
– it can expand or contract in volume to remain at atmospheric pressure

Some fundamentals of solution thermodynamics
• At constant pressure P and constant temperature T the system is described by the Gibbs
free energy:
G ! H "T S
• H is the enthalpy or heat content of the system, S is the entropy of the system
• a reaction occurs spontaneously only if ∆G < 0
• at equilibrium ∆G = 0
!G = !H "T !S
• for ∆G > 0 the input of energy is required to drive the reaction

the problem
In general we can not assume that the total free energy G of a
solution consisting of N different components is simply the sum of the
free energys of the single components.

The chemical potential µ of a substance is
the partial molar Gibbs free energy
G i = !G
!n i
for an ideal solution it is:
n
= µ G = " n i i!µ i
i=1
µ i = µ i 0 for Ci =1 mol / l
0 µ i = µ i + RT lnCi
C i is the concentration in mol per liter
0
µ i
is the chemical potential of a substance at 1 mol/l

Changes of the Gibbs free energy ∆G of an reaction
aA +bB+...
!
" gG+hH...
!G = G(final state) "G(initial state)
!G = g µ G + hµ H +... " aµ A "b µ B "...
from µ i = µ i 0 + RT lnCi it follows:
!G = g µ G 0 + hµH 0 +... " aµA 0 "b µB 0 "... + RT ln [G] g [H] h ...
[A] a [B] b ...
!G = !G 0 + RT ln [G]g [H] h ...
[A] a [B] b ...

∆G of an reaction in equilibrium
aA +bB+...
0 = !G0 + RT ln [G]g [H] h ...
[A] a [B] b " %
$
# ... &
' Eq
!G0 = "RT ln [G]g [H] h ...
[A] a [B] b # &
%
$ ... ( = "RT ln K
' Eq
K = [G]g [H] h ...
[A] a [B] b ! $
#
" ... %
!
" gG+hH...
& Eq
= exp '(G0 ! $
#
" RT &
%

Titration of a macromolecule D with n binding sites
for the ligand P which is added to the solution
n
degree of binding ν
!X
!X max
0
= "
n
∆X
free ligand P free (M)
=# (fraction saturation)
! =
n binding sites
occupied
∆X max
[boundligand P]
[macromolecule D]

Schematic view of gel electrophoresis
to analyze protein-DNA complexes

*
*
“Gel shift”: electorphoretic mobility shift assay (“EMSA”)
for DNA-binding proteins
Protein-DNA complex
Free DNA probe
1. Prepare labeled DNA probe
2. Bind protein
3. Native gel electrophoresis
Advantage: sensitive, fmol DNA
Disadvantage: requires stable complex;
little “structural” information about which
protein is binding

EMSA of Lac repressor binding to operator DNA
From (a) to (j) the concentration
of lac repressor is increased.
Complexes with
Free DNA

Measuring binding constants for lambda repressor on a gel

Principle of filter-binding assay

Binding measurments by equilibrium dialysis
A macromolecule is dialyzed against a solution of ligand. Upon reaching equilibrium,
the ligand concentration is measured inside and outside the dialysis chamber. The excess
ligand inside the chamber corresponds to bound ligand.
- direct measurement of binding
!
" = [X] in #[X] out
M
-non-specific binding will obscure results, work at moderate ionic strength (≥
50 to avoid the Donnan Effect (electrostatic interactions between the
macromolecule and a charged ligand.
- needs relatively large amounts of material

Analysis of binding of RNAP·σ 54 to a promoter DNA sequence
by measurements of fluorescence anisotropy
RNAP·σ 54
+
promoter DNA
K d
Rho
Rho
RNAP·σ 54 -DNA-Komplex
free DNA with a fluorophore
with high rotational diffusion
-> low fluorescence anisotropy r min
RNAP-DNA complex
with low rotational diffusion
-> high fluorescence anisotropy
r max

How to measure binding of a protein to DNA?
One possibility is to use fluorescence anisotropy
x
filter/monochromator
y
I⊥
measured
fluorescence
emission intensity
polarisator
III
vertical
excitation
polarisator
filter/monochromator
z
sample
r = I II ! I "
I II + 2I "
Definition of fluorescence
anisotropy r
The anisotropy r reflects
the rotational diffusion of a
fluorescent species

Measurements of fluorescence anisotropy to
monitor binding of RNAP·σ 54 to different promoters
1
0.8
0.6
0.4
0.2
0
200 mM K-Acetate
θ = 0.5
glnAp2
nifH nifL
0.01 0.1 11 0 100 1000
RNAP σ 54 (nM)
P tot = K D
Vogel, S., Schulz A. & Rippe, K.

Titration of a macromolecule D with n binding sites
for the ligand P which is added to the solution
n
degree of binding ν
!X
!X max
0
= "
n
∆X
free ligand P free (M)
=# (fraction saturation)
! =
n binding sites
occupied
∆X max
[boundligand P]
[macromolecule D]

Example: binding of a protein P to a DNAfragment
D with one or two binding sites
Dfree + P !
free"DP
K1 = Dfree #Pfree DP
D free, concentration free DNA; P free, concentration free protein;
DP, complex with one protein; DP 2, complex with two proteins;
DP+ P free
!
"DP 2 K 2 = DP#P free
DP 2
D+2P !
* D
free"
DP2 K 2 = free#Pfree DP2 binding constant K B =
2
1
dissociation constant K D
binding of the first proteins with
the dissociation constant K 1
binding of the second proteins with
the dissociation constant K 2
K 2 * = K1 # K 2
alternative expression

[boundligand P]
! =
[macromolecule D]
! =
Definition of the degree of binding ν
n
#
i=1
n
#
i=0
i" 1 i
"D
K
frei"Pfrei i
1
K i
! 1 =
i
"Dfrei "Pfrei DP
D free +DP
=
n
#
i=1
n
#
i=0
i" 1 i
"Pfrei Ki ν for n binding sites (Adair equation)
1
K i
i
"Pfrei ! 2 = DP+2"DP 2
D free +DP+DP 2
degree of binding ν ν for one binding site ν for two binding sites
mit K 0 =1

Binding to a single binding site: Deriving an expression
for the degree of binding ν or the fraction saturation θ
Dfree + P !
free"DP
! 1 =" =
1
K D
#P free
1+ 1
#Pfree KD K D = D free!P free
DP
from the Adair equation we obtain:
$ ! 1 =" =
P free
K D +P free
Often the concentration P free can not be determined but
the total concentration of added protein P tot is known.
P free = P tot !" 1 # D tot
! 1 = Dtot +Ptot +K D " Dtot +Ptot +K ( D)
2 "4#Dtot #Ptot 2#Dtot

!
n ="
for n =1
! ="
ν or θ
Stoichiometric titration to determine
the number of binding sites
1
0.8
0.6
0.4
0.2
equivalence point
1 protein per DNA
0
0 1·10 -10
P tot (M)
D tot = 10 -10 (M)
2·10 -10
K D = 10 -14 (M)
K D = 10 -13 (M)
K D = 10 -12 (M)
To a solution of DNA strands with a single binding site small amounts of protein P are
added. Since the binding affinity of the protein is high (low K D value as compared to the total
DNA concentration) practically every protein binds as long as there are free binding sites on
the DNA. This is termed “stoichiometric binding” or a “stoichiometric titration”.

Binding to a single binding site. Titration of DNA with a
protein for the determination of the dissociation constant K D
ν or θ
0.
8
0.
6
0.
4
0.
2
1
0
0 2 1
0
! 1 =
P tot = K D
-9
P free
P free +K D
ν or θ = 0.5
-9
4 1 6 1
0 0
Ptot (M)
"
D tot = 10 -10 (M)
-9
P tot
P tot +K D
8 1
0
-9
1 1
0
-8
K D = 10 -9 (M)
K D = 10 -9 (M)
! 1 =
if P free " P tot d. h. 10# D tot $ K D
P free
P free +K D
P tot
! = 1
Ptot +K D

cooperativit
y
Increasing complexity of binding
all binding sites are
equivalent and independent
all binding sites are
equivalent and not independent
heterogeneity
heterogeneity
all binding sites are
not equivalent and not independent
all binding sites are
independent but not equivalent
cooperativity
simple
difficult
very
difficult

Binding to n identical binding sites
! 1 =
P free
P free +K D
! n = n"P free
k D +P free
D+n! P "
free
# DPn Kn = D n
free!Pfree binding to a single binding site
binding to n independent and
identical binding sites
DP n
$ n = n!P n
free
n
Kn +Pfree strong cooperative binding to n identical binding sites
! n =
#
n"P H
free
K # H
#
+Pfree
H
approximation for cooperative binding to
n identical binding sites, α H HiIl coefficient

k D
Difference between microscopic and
macroscopic dissociation constant
1 2
1 2 1 2
k D
microscopic binding macroscopic binding
1 2
k D
k D
2 possibilities for
the formation of DP
2 possibilities for the
dissociation of DP 2
K1 K2 = k D 2
2!k D
= 1
4
D free
DP
DP 2
K 1 = k D
2
K 2 = 2 !k D

ν 2
1.
5
0.
5
2
1
0
Cooperativity: the binding of multiple ligands
to a macromolecule is not independent
0 2 1
0
-9
-9
4 1 6 1
0 0
P free (M)
-9
8 1
0
Adair equation: ! 2 =
-9
1 1
0
-8
independent binding
microscopic binding constant
k D = 10 -9 (M)
macroscopic binding constants
K 1 = 5·10 -10 (M); K 2 = 2·10 -9 (M)
cooperative binding
microscopic binding constant
k D = 10 -9 (M)
macroscopic binding constants
K 1 = 5·10 -10 (M); K 2 = 2·10 -10 (M)
2
K2"Pfree +2"Pfree K 1"K 2 +K 2"P free +P free
2

ν 2
2
1.5
1
0.5
0
10 -11
Logarithmic representation of a binding curve
10 -10
10 -9
P free (M)
10 -8
• Determine dissociation constants over a ligand concentration of at least three orders of
magnitudes
• Logarithmic representation since the chemical potential µ is proportional to the logarithm
of the concentration.
10 -7
independent binding
microscopic binding constant
k D = 10 -9 (M)
macroscopic binding constants
K 1 = 5·10 -10 (M); K 2 = 2·10 -9 (M)
cooperative binding
microscopic binding constant
k D = 10 -9 (M)
macroscopic binding constants
K 1 = 5·10 -10 (M); K 2 = 2·10 -10 (M)

Y = θ (degree of binding)
L: free ligand

ν/ P frei
3 1
0
2 1
0
1 1
0
9
9
9
0
Visualisation of binding data - Scatchard plot
intercept = n/k D
0 0.
5
! n = n"P free
k D +P free
slope = - 1/k D
ν
1 1.
5
# ! n
P free
intercept =
n
= n
k D
$ ! n
k D
2
independent binding
microscopic binding constant
k D = 10 -9 (M)
macroscopic binding constants
K 1 = 5·10 -10 (M); K 2 = 2·10 -9 (M)
cooperative binding
microscopic binding constant
k D = 10 -9 (M)
macroscopic binding constants
K 1 = 5·10 -10 (M); K 2 = 2·10 -10 (M)

D+n! P "
free
Binding to n identical binding sites
! 1 =
P free
P free +K D
! n = n"P free
k D +P free
# DP K = n n D n
free!Pfree DP n
or divided by n
!
binding to a single binding site
binding to n independent and
identical binding sites
$ = n n!P n
free
n
Kn +Pfree " = P n
free
n
Kn +Pfree

All or none binding (very high cooperativity)
or
!
" = P n
free
n
Kn +Pfree

!
!
Binding to n identical binding sites
! 1 =
P free
P free +K D
! n = n"P free
k D +P free
" = n n P n
free
n
Kn +Pfree ! n =
"=
# H n"Pfree K # H
#
+Pfree
H
#
P H
free
K # H #
+Pfree
H
binding to a single binding site
binding to n independent and
identical binding sites
strong cooperative binding
to n identical binding sites,
with K n = (k d ) n
approximation for cooperative binding to
n identical binding sites, α H HiIl coefficient

!
" =
#
L H
free
K # H #
+Lfree
H
Hill coefficient and Hill plot
approximation for cooperative binding to
n identical binding sites, α H HiIl coefficient
L free is free ligand
The HiIl α H coefficient characterizes the degree of cooperativitiy. It varies
from 1 (non-cooperative vinding) to n (the total number of bound ligands)
α H > 1, the system shows positive cooperativity
α H = n, the cooperativity is infinite
α H = 1, the system is non-cooperative
α H < 1, the system shows negative cooperativity
The Hill coefficient and the ‘average’ K d can be obtained from a Hill plot,
which is based on the transformation of the above equation

!
!
!
" =
Hill coefficient and Hill plot
#
L H
free
K # H #
+Lfree
H
"
L H
free
K " H
rearrange the terms to get
= #
1$#
which yields the Hill equation
log " $ '
& ) =* log L #log K H free
% 1#" (
* H
α H HiIl coefficient
L free is free ligand
K average microscopic binding constan

log[ν/ (2–ν)]= log[θ/ (1–θ)]
! n =
2
1.
5
1
0.
5
0
-
0.5
-
1
-
1.5
-
21
0
#
n"P H
free
K # H
#
+Pfree
H
-11
weak
bindin
g
limit
Visualisation of binding data - Hill plot
slope α H =1.5
1
0
K 2/
2
-10
! 2 =
1
0
log (P free)
2
K2"Pfree +2"Pfree K 1"K 2 +K 2"P free +P free
-9
2·K 1
strong
bindin
g
limit
1
0
-8
2 # ! 2
2$! 2
P ! " # log
&
free (
'
%
Pfree ! K " log'
&
%
Pfree ! 0 " log'
&
= %
1$% = K 2
2"Pfree +2"Pfree 2"K1"K 2 +K2"P free
$ 2
2%$ 2
# 2
2$# 2
# 2
2$# 2
)
+ = log( Pfree )%log
*
K & 2 )
( +
' 2 *
(
* =+ H ,log( Pfree )$+ H ,log K
)
(
* =log Pfree )
( )$ log 2K 1
( )
( )

Binding of dioxygen to hemoglobin

+
P
The Monod-Wyman-Changeau (MWC)
model for cooperative binding
T 0 T 1 T 2
k T
L 0 L 1 L 2
+P
k R
+P
P P P P P
k T
+P
k R
R 0 R 1 R 2
• in the absence of ligand P the the T conformation is favored
T conformation
(all binding sites
are weak)
R conformation
(all binding sites
are strong)
• the ligand affinity to the R form is higher, i. e. the dissociation constant k R< k T.
• all subunits are present in the same confomation
• binding of each ligand changes the TR equilibrium towards the R-Form
+P
k T
+P
P
T 3
+P
k T
T 4
P P
P P
P P P P P
P P
k R
L 3
P
R 3
+P
k R
L 4
P P
R 4

α-conformation
The Koshland-Nemethy-Filmer (KNF)
model for cooperative binding
+P
k 1
+P
• Binding of ligand P induces a conformation change in the subunit to which it binds
+P
from the α into the β-conformation (“induced fit”).
• The bound ligand P facilitates the binding of P to a nearby subunit
in the α-conformation (red), i. e. the dissociation constant k 2 < k’ 2.
• subunits can adopt a mixture of α−β confomations.
P
P
α-conformation
(facilitated binding)
P
k’ 2
k 2
P P
β-conformation
(induced by
ligand binding)

Summary
• Thermodynamic relation between ∆G und K D
• Stoichiometry of binding
• Determination of the dissociation constant for simple systems
• Adair equation for a general description of binding
• Binding to n binding sites
• Visualisation of binding curves by Scatchard and Hill plots
• Cooperativity of binding (MWC and KNF model)