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

Karsten RippeKirchoff-Institut für PhysikMolecular Biophysics GroupIm Neuenheimer Feld 227Tel: 54-9270e-mail: Karsten.Rippe@kip.uni-heidelberg.de

• 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

The secret of life

“The secret of life is molecular recognition; theability of one molecule to "recognize" another throughweak bonding interactions.”

Linus Pauling at the 25th anniversary of the Institute ofMolecular 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 IItranscription initiation: astructural 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

+Pfree

!"DP K

1=Dfree #Pfree

DP

binding of the first proteins with the dissociation constant K1

Dfree, concentration free DNA; Pfree, concentration free protein

binding constant KB=

1

dissociation constant KD

What is the meaning of the dissociation constant forbinding of a single ligand to its site?

2. KD gives the concentration of ligand that saturates 50% of thesites (when the total sit concentration ismuch lower than KD)

3. Almost all binding sites are saturated if the ligandconcentration is 10 x KD

1. KD is a concentration and has units of mol per liter

4. The dissociatin constant KD is related to Gibbs free energy∆G by the relation ∆G = - R T ln(KD)

KD values in biological systems

Allosteric activators of enzymes e. g. NAD have KD 0.1 µM to 0.1 mM

Site specific binding to DNA KD 1 nM to 1 pM

Movovalent ions binding to proteins or DNA have KD 0.1 mM to 10 mM

Trypsin inhibitor to pancreatic trypsin protease KD 0.01 pM

Antibody-antigen interaction have KD 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 constanttemperature 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

• 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• for ∆G > 0 the input of energy is required to drive the reaction

!G =!H "T!SG !H "T S

• At constant pressure P and constant temperature T the system is described by the Gibbsfree energy:

the problem

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

The chemical potential µ of a substance is the partial molar Gibbs free energy

G i=!G

!ni

= µi

G = ni !µii=1

n

"

µi= µ

i

0 + RT lnCi

for an ideal solution it is:

µi= µ

i

0 for Ci=1 mol / l

Ci is the concentration in mol per liter

is the chemical potential of a substance at 1 mol/lµi

0

Changes of the Gibbs free energy ∆G of an reaction

!G =G(final state) "G(initial state)

from µi= µ

i

0 + RT lnCi

it follows:

aA +bB+...

!" gG+hH...

!G = gµG+ hµ

H+... " aµ

A"bµ

B" ...

!G = gµG0 + hµ

H0 +... " aµ

A0 "bµ

B0 " ... +RT ln

[G]g[H]h ...

[A]a[B]b ...

!G =!G0 +RT ln[G]g[H]h ...

[A]a[B]b ...

∆G of an reaction in equilibrium

0 =!G0 +RT ln[G]g[H]h ...

[A]a[B]b ...

"

# $

%

& ' Eq

!G0 = "RT ln[G]g[H]h ...

[A]a[B]b ...

#

$ %

&

' ( Eq

= "RT lnK

K =[G]g[H]h ...

[A]a[B]b ...

!

" #

$

% & Eq

= exp'(G0

RT

!

" #

$

% &

aA +bB+...

!" gG+hH...

Titration of a macromolecule D with n binding sitesfor the ligand P which is added to the solution

! =[boundligand P]

[macromolecule D]

!X

!Xmax ="

n =# (fraction saturation)

free ligand Pfree (M)

degr

ee o

f bin

ding

νn n binding sites

occupied

∆Xmax

∆X

0

Schematic view of gel electrophoresisto analyze protein-DNA complexes

“Gel shift”: electorphoretic mobility shift assay (“EMSA”)for DNA-binding proteins

Free DNA probe*

*Protein-DNA complex

1. Prepare labeled DNA probe2. Bind protein3. 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 DNAFrom (a) to (j) the concentrationof lac repressor is increased.

Complexes with

Free DNA

Measuring binding constants for lambda repressor on a gel

Principle of filter-binding assay

A macromolecule is dialyzed against a solution of ligand. Upon reaching equilibrium,the ligand concentration is measured inside and outside the dialysis chamber. The excessligand inside the chamber corresponds to bound ligand.

!

" =[X]

in# [X]

out

M

- direct measurement of binding

-non-specific binding will obscure results, work at moderate ionic strength (≥50 to avoid the Donnan Effect (electrostatic interactions between themacromolecule and a charged ligand.

- needs relatively large amounts of material

Binding measurments by equilibrium dialysis

Analysis of binding of RNAP·σ54 to a promoter DNA sequenceby measurements of fluorescence anisotropy

Rho

Rho

+RNAP·σ54

promoter DNA

RNAP·σ54-DNA-Komplex

Kd

free DNA with a fluorophorewith high rotational diffusion

-> low fluorescence anisotropy rmin

RNAP-DNA complexwith low rotational diffusion

-> high fluorescence anisotropyrmax

How to measure binding of a protein to DNA?One possibility is to use fluorescence anisotropy

z

y

x sample

I⊥

verticalexcitation

filter/mono-chromator

polarisatorIII

filter/monochromator

polarisator

measuredfluorescence

emission intensity

r =III! I"

III+ 2I"

Definition of fluorescenceanisotropy r

The anisotropy r reflectsthe rotational diffusion of afluorescent species

Measurements of fluorescence anisotropy tomonitor binding of RNAP·σ54 to different promoters

Vogel, S., Schulz A. & Rippe, K.

0

0.2

0.4

0.6

0.8

1

0.01 0.1 11 0 100 1000

nifH nifLglnAp2

RNAP σ54 (nM)

200 mM K-Acetate

Ptot = KD

θ = 0.5

Titration of a macromolecule D with n binding sitesfor the ligand P which is added to the solution

! =[boundligand P]

[macromolecule D]

!X

!Xmax ="

n =# (fraction saturation)

free ligand Pfree (M)

degr

ee o

f bin

ding

νn n binding sites

occupied

∆Xmax

∆X

0

Example: binding of a protein P to a DNA-fragment D with one or two binding sites

Dfree

+Pfree

!"DP K

1=Dfree #Pfree

DP

binding of the first proteins withthe dissociation constant K1

Dfree, concentration free DNA; Pfree, concentration free protein;DP, complex with one protein; DP2, complex with two proteins;

binding constant KB=

1

dissociation constant KD

DP+Pfree

!"DP

2 K

2=DP#Pfree

DP2

binding of the second proteins withthe dissociation constant K2

D+2Pfree

!"DP

2 K

2

* =Dfree #Pfree

2

DP2

K 2

* =K1#K

2alternative expression

Definition of the degree of binding ν

ν for n binding sites (Adair equation)

! =

i"1

Ki

"Dfrei "Pfreii

i=1

n

#1

Ki

"Dfrei"Pfrei

i

i=0

n

#=

i"1

Ki

"Pfreii

i=1

n

#1

Ki

"Pfrei

i

i=0

n

#mit K

0=1

!1=

DP

Dfree +DP!2=

DP+2"DP2

Dfree +DP+DP2! =

[boundligand P]

[macromolecule D]

degree of binding ν ν for one binding site ν for two binding sites

Binding to a single binding site: Deriving an expressionfor the degree of binding ν or the fraction saturation θ

KD=Dfree !Pfree

DP

Dfree

+Pfree

!"DP

!1=" =

1

KD

#Pfree

1+1

KD

#Pfree

$ !1=" =

Pfree

KD+Pfree

from the Adair equation we obtain:

!1=Dtot+P

tot+K

D" D

tot+P

tot+K

D( )2"4#Dtot #Ptot2#Dtot

Pfree

=Ptot!"

1#D

tot

Often the concentration Pfree can not be determined butthe total concentration of added protein Ptot is known.

Stoichiometric titration to determinethe number of binding sites

To a solution of DNA strands with a single binding site small amounts of protein P areadded. Since the binding affinity of the protein is high (low KD value as compared to the totalDNA concentration) practically every protein binds as long as there are free binding sites onthe DNA. This is termed “stoichiometric binding” or a “stoichiometric titration”.

ν or

θ

0

0.2

0.4

0.6

0.8

1

0 1·10 -10

Ptot (M)

equivalence point1 protein per DNA

2·10 -10

Dtot = 10-10 (M)

KD = 10-14 (M)

KD = 10-13 (M)

KD = 10-12 (M)

!

n ="

for n=1

! ="

Binding to a single binding site. Titration of DNA with aprotein for the determination of the dissociation constant KD

!1=

Pfree

Pfree +KD

"Ptot

Ptot +KD

if Pfree

"Ptot

d. h. 10#Dtot$ K

D

Ptot (M)

ν or

θ

0

0.2

0.4

0.6

0.8

1

0 2 10

-9 4 10

-9 6 10

-9 8 10

-9 1 10

-8

Ptot = KD

ν or θ = 0.5

Dtot = 10-10 (M)

KD = 10-9 (M)

KD = 10-9 (M)

!1=

Pfree

Pfree +KD

!1=

Ptot

Ptot +KD

Increasing complexity of binding

all binding sites areequivalent and independent

cooperativity

heterogeneity

all binding sites areequivalent and not independent

cooperativityheterogeneity

all binding sites arenot equivalent and not independent

all binding sites areindependent but not equivalent

simple

difficult

verydifficult

Binding to n identical binding sites

!1=

Pfree

Pfree +KDbinding to a single binding site

!n=

n"Pfree

kD+Pfree

binding to n independent andidentical binding sites

D+n!Pfree

"#DP

n K

n=Dfree !Pfree

n

DPn

$n=

n!Pfree

n

Kn +Pfree

n

strong cooperative binding to n identical binding sites

!n=

n"Pfree#H

K#H +P

free

#H

approximation for cooperative binding ton identical binding sites, αH HiIl coefficient

Difference between microscopic andmacroscopic dissociation constant

1 2

1 2 1 2

1 2

kD kD

kD kD

Dfree

DP

DP2

K1

K2

=kD 2

2!kD=1

4

K1=kD

2

K2=2 !k

D

microscopic binding macroscopic binding

2 possibilities forthe formation of DP

2 possibilities for thedissociation of DP2

Cooperativity: the binding of multiple ligandsto a macromolecule is not independent

ν 2

P free (M)

0

0.5

1

1.5

2

0 2 10

-9 4 10

-9 6 10

-9 8 10

-9 1 10

-8

Adair equation: !2=

K2"Pfree +2"Pfree2

K1"K2 +K2 "Pfree +Pfree2

independent bindingmicroscopic binding constantkD = 10-9 (M)macroscopic binding constantsK1 = 5·10-10 (M); K2 = 2·10-9 (M)

cooperative bindingmicroscopic binding constantkD = 10-9 (M)macroscopic binding constantsK1 = 5·10-10 (M); K2 = 2·10-10 (M)

Logarithmic representation of a binding curve

P free (M)

0

0.5

1

1.5

2

10-11 10-10 10-9 10-8 10-7

ν 2

• Determine dissociation constants over a ligand concentration of at least three orders ofmagnitudes• Logarithmic representation since the chemical potential µ is proportional to the logarithmof the concentration.

independent bindingmicroscopic binding constantkD = 10-9 (M)macroscopic binding constantsK1 = 5·10-10 (M); K2 = 2·10-9 (M)

cooperative bindingmicroscopic binding constantkD = 10-9 (M)macroscopic binding constantsK1 = 5·10-10 (M); K2 = 2·10-10 (M)

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

Visualisation of binding data - Scatchard plotν/

P fre

i

ν

0

1 10

9

2 10

9

3 10

9

0 0.5

1 1.5

2

intercept = n/kD

slope = - 1/kD

intercept =n

!n=

n"Pfree

kD+Pfree#

!n

Pfree

=n

kD

$!n

kD

independent bindingmicroscopic binding constantkD = 10-9 (M)macroscopic binding constantsK1 = 5·10-10 (M); K2 = 2·10-9 (M)

cooperative bindingmicroscopic binding constantkD = 10-9 (M)macroscopic binding constantsK1 = 5·10-10 (M); K2 = 2·10-10 (M)

Binding to n identical binding sites

!1=

Pfree

Pfree +KDbinding to a single binding site

!n=

n"Pfree

kD+Pfree

binding to n independent andidentical binding sites

D+n!Pfree

"#DP

n K

n=Dfree !Pfree

n

DPn

$n=

n!Pfree

n

Kn +Pfree

n

!

" =Pfree

n

Kn+P

free

nor divided by n

All or none binding (very high cooperativity)

!

" =Pfree

n

Kn+P

free

nor

Binding to n identical binding sites

!1=

Pfree

Pfree +KDbinding to a single binding site

!n=

n"Pfree

kD+Pfree

binding to n independent andidentical binding sites

strong cooperative bindingto n identical binding sites,

with Kn = (kd)n

!n=

n"Pfree#H

K#H +P

free

#H

approximation for cooperative binding ton identical binding sites, αH HiIl coefficient

!

"=Pfree

#H

K#H+P

free

#H

!

"n

=n P

free

n

Kn+P

free

n

Hill coefficient and Hill plot

approximation for cooperative binding ton identical binding sites, αH HiIl coefficientLfree is free ligand

!

" =Lfree

#H

K#H+L

free

#H

The HiIl αH coefficient characterizes the degree of cooperativitiy. It variesfrom 1 (non-cooperative vinding) to n (the total number of bound ligands)

αH > 1, the system shows positive cooperativityαH = n, the cooperativity is infiniteαH = 1, the system is non-cooperativeαH < 1, the system shows negative cooperativity

The Hill coefficient and the ‘average’ Kd can be obtained from a Hill plot,which is based on the transformation of the above equation

Hill coefficient and Hill plot

αH HiIl coefficientLfree is free ligandK average microscopic binding constan

!

" =Lfree

#H

K#H+L

free

#H

rearrange the terms to get

!

log"

1#"

$

% &

'

( ) =*H logLfree # logK

*H

!

Lfree

"H

K"H

=#

1$#

which yields the Hill equation

Visualisation of binding data - Hill plot

!2=

K2 "Pfree +2"Pfree2

K1"K2 +K2 "Pfree +Pfree2#

!2

2$!2=

%

1$%=

K2"Pfree +2"Pfree2

2"K1"K2 +K2 "Pfree

Pfree

! 0 " log#2

2$#2

%

& '

(

) * = log Pfree( )$ log 2K1( )

Pfree

!" # log$2

2%$2

&

' (

)

* + = log Pfree( )% log

K2

2

& ' (

) * +

Pfree

!K " log#2

2$#2

%

& '

(

) * =+H

,log Pfree( )$+H,log K( )

!n=

n"Pfree#H

K#H +P

free

#H

log[ν/

(2–ν

)]= lo

g[θ/

(1–θ

)]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

10

-11 10

-10 10

-9 10

-8

strongbindin

glimit

slope α H =1.5

log (P free)

weakbindin

glimit

2·K1K2/2

Binding of dioxygen to hemoglobin

The Monod-Wyman-Changeau (MWC)model for cooperative binding

+P

T0 T1 T2

R0 R1 R2

L0 L1 L2

kT

+P

kR

+P

kR

+P

kT

T conformation(all binding sites

are weak)

R conformation(all binding sites

are strong)

• in the absence of ligand P the the T conformation is favored• the ligand affinity to the R form is higher, i. e. the dissociation constant kR< kT.• all subunits are present in the same confomation• binding of each ligand changes the T<->R equilibrium towards the R-Form

P P P P PP

P PP P

+P

kT

+P

kT

T3 T4

P P P P P

P P

P P

P

L3 L4

+P

kR

+P

kRR3 R4

The Koshland-Nemethy-Filmer (KNF)model for cooperative binding

+P

k1

+P

k2

• Binding of ligand P induces a conformation change in the subunit to which it binds 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 k2 < k’2.• subunits can adopt a mixture of α−β confomations.

α-conformationβ-conformation(induced byligand binding)

+P

P

P

k’2

P PP

α-conformation(facilitated binding)

Summary

• Thermodynamic relation between ∆G und KD

• 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)