Biochemistry II: Binding of Ligands to a Macromolecule (Or the Secret of Life Itself...)

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: [email protected] 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 #Pfree Dfree + Pfree "DP K1 = DP

binding of the ﬁrst proteins with the dissociation constant K1

Dfree, concentration free DNA; Pfree, 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. KD is a concentration and has units of mol per liter

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

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

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

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

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

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 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 !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 • 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

n !G G = = µ G = ni !µi i !n i " i i=1

0 for an ideal solution it is: µi = µi + RT lnCi

Ci is the concentration in mol per liter 0 µ i is the chemical potential of a substance at 1 mol/l

0 µi = µi for Ci =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 µG + hµH +... " aµA "b µB "...

0 from µi = µi + RT lnCi it follows: [G]g[H]h ... !G = g µ0 + hµ0 +... " aµ 0 "b µ0 "... + RT ln G H A B [A]a[B]b ... [G]g[H]h ... !G = !G0 + RT ln [A]a[B]b ... ∆G of an reaction in equilibrium

! aA +bB+... " gG+hH...

" [G]g[H]h ...% 0 = !G0 + RT ln $ [A]a[B]b ... ' # & Eq

[G]g[H]h ... G0 = RT ln = RT ln K [A]a[B]b ... Eq

[G]g[H]h ... G0 K = = exp [A]a[B]b ... RT Eq Titration of a macromolecule D with n binding sites for the ligand P which is added to the solution

n n binding sites occupied ν

g in d in b

o ∆ Xmax e e r g

e ∆X d

0 free ligand Pfree (M)

X [boundligand P] = = (fraction saturation) ! = Xmax n [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

Advantage: sensitive, fmol DNA 1. Prepare labeled DNA probe 2. Bind protein Disadvantage: requires stable complex; 3. Native gel electrophoresis 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.

[X] #[X] " = in 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 the macromolecule and a charged ligand. - needs relatively large amounts of material Analysis of binding of RNAP·σ54 to a promoter DNA sequence by measurements of ﬂuorescence anisotropy

RNAP·σ54 + free DNA with a ﬂuorophore Rho with high rotational diffusion promoter DNA -> low ﬂuorescence anisotropy rmin

RNAP-DNA complex

Rho with low rotational diffusion RNAP·σ54-DNA-Komplex -> high ﬂuorescence anisotropy rmax How to measure binding of a protein to DNA? One possibility is to use ﬂuorescence anisotropy

vertical excitation ﬁlter/mono- III ! I" chromator polarisator r = x sample III + 2I"

polarisator Definition of fluorescence III anisotropy r I⊥ filter/monochromator

y The anisotropy r reflects measured fluorescence the rotational diffusion of a emission intensity fluorescent species Measurements of fluorescence anisotropy to monitor binding of RNAP·σ54 to different promoters

1 200 mM K-Acetate

0.8 glnAp2 nifH nifL 0.6 θ = 0.5

0.4

0.2 Ptot = KD 0

0.01 0.1 11 0 100 1000 54 RNAP σ (nM)

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 n binding sites occupied ν

g in d in b

o ∆ Xmax e e r g

e ∆X d

0 free ligand Pfree (M)

!X " [boundligand P] = =# (fraction saturation) ! = !Xmax n [macromolecule D] Example: binding of a protein P to a DNA- fragment D with one or two binding sites

D #P D + P ! DP K = free free binding of the ﬁrst proteins with free free " 1 the dissociation constant K DP 1

Dfree, concentration free DNA; Pfree, concentration free protein;

DP, complex with one protein; DP2, complex with two proteins;

DP#P DP+ P ! DP K = free binding of the second proteins with free " 2 2 the dissociation constant K DP2 2

2 D #P ! * free free * alternative expression D+2Pfree " DP2 K 2 = K 2 = K1# K2 DP2 1 binding constant KB = dissociation constant KD Deﬁnition of the degree of binding ν

[boundligand P] DP DP+2"DP2 ! = !1 = !2 = [macromolecule D] Dfree +DP Dfree +DP+DP2

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

n n 1 i 1 i i" "Dfrei "Pfrei i" "Pfrei # Ki # Ki i=1 i=1 ! = n = n mit K0 =1 1 i 1 i "Dfrei "Pfrei "Pfrei #Ki # Ki i=0 i=0 ν for n binding sites (Adair equation) Binding to a single binding site: Deriving an expression for the degree of binding ν or the fraction saturation θ

D !P D + P ! DP K = free free free free " D DP from the Adair equation we obtain: 1 #P K free P ! =" = D $ ! =" = free 1 1 1 KD +Pfree 1+ #Pfree KD

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

Pfree = Ptot !"1# Dtot

2 Dtot +Ptot +KD " (Dtot +Ptot +KD) "4#Dtot #Ptot !1 = 2#Dtot Stoichiometric titration to determine the number of binding sites

1 -14 ! equivalence point KD = 10 (M) =" 0.8 1 protein per DNA n

θ -10

0.6 D = 10 (M)

r tot o for n =1 -13 ν KD = 10 (M) 0.4

! =" 0.2 -12 KD = 10 (M) 0 0 1·10 -10 2·10 -10

Ptot (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 KD 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 KD

P 0. ! = free 8 K = 10-9 (M) 1 D Pfree +KD

θ 0.

r 6

o ν or θ = 0.5

ν 0. 4 Ptot K = 10-9 (M) !1 = -10 D P K 0. Ptot = KD Dtot = 10 (M) tot + D 2

0 0 2 1 -9 4 1 -9 6 1 -9 8 1 -9 1 1 -8 0 0 0 0 0 Ptot (M)

Pfree Ptot !1 = " if Pfree " Ptot d. h. 10# Dtot $ KD Pfree +KD Ptot +KD Increasing complexity of binding

all binding sites are simple equivalent and independent

cooperativit heterogeneity y all binding sites are all binding sites are difﬁcult equivalent and not independent independent but not equivalent

heterogeneity cooperativity

all binding sites are very not equivalent and not independent difﬁcult Binding to n identical binding sites

Pfree !1 = binding to a single binding site Pfree +KD

n"Pfree binding to n independent and !n = kD +Pfree identical binding sites

n n " Dfree !Pfree n!Pfree D+n! Pfree # DPn Kn = $n = n DPn Kn +Pfree

strong cooperative binding to n identical binding sites

n"P#H ! = free approximation for cooperative binding to n #H #H n identical binding sites, α HiIl coefﬁcient K +Pfree H Difference between microscopic and macroscopic dissociation constant

microscopic binding macroscopic binding

1 2 Dfree

2 possibilities for kD kD kD the formation of DP K = 1 2 1 2 1 2 DP

2 possibilities for the dissociation of DP K = 2 !k kD kD 2 2 D

1 2 DP2 K k 2 1 1 = D = K2 2!kD 4 Cooperativity: the binding of multiple ligands to a macromolecule is not independent

2 independent binding microscopic binding constant k = 10-9 (M) 1. D 5 macroscopic binding constants K = 5·10-10 (M); K = 2·10-9 (M)

2 1 2 1 ν cooperative binding 0. microscopic binding constant 5 -9 kD = 10 (M)

0 macroscopic binding constants -9 -9 -9 -9 -8 -10 -10 0 2 1 4 1 6 1 8 1 1 1 K1 = 5·10 (M); K2 = 2·10 (M) 0 0 0 0 0 P free (M)

2 K2"Pfree +2"Pfree Adair equation: !2 = 2 K1"K2 +K2 "Pfree +Pfree Logarithmic representation of a binding curve

2 independent binding microscopic binding constant -9 1.5 kD = 10 (M) macroscopic binding constants -10 -9 2

K = 5·10 (M); K = 2·10 (M) 1 1 2 ν

cooperative binding 0.5 microscopic binding constant -9 kD = 10 (M) 0 macroscopic binding constants -11 -10 -9 -8 -7 10 10 10 10 10 -10 -10 K1 = 5·10 (M); K2 = 2·10 (M)

P free (M)

• 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. Y = θ (degree of binding) L: free ligand Visualisation of binding data - Scatchard plot

independent binding microscopic binding constant -9 9 k = 10 (M) 3 1 D 0 macroscopic binding constants intercept = n/kD -10 -9

i K1 = 5·10 (M); K2 = 2·10 (M)

e 9 r f 2 1

slope = - 1/kD P 0

ν cooperative binding

9 microscopic binding constant 1 1 intercept = k = 10-9 (M) 0 n D macroscopic binding constants 0 K = 5·10-10 (M); K = 2·10-10 (M) 0 0. 1 1. 2 1 2 5 ν 5

n"Pfree !n n ! n !n = # = $ kD +Pfree Pfree kD kD

Binding to n identical binding sites

Pfree !1 = binding to a single binding site Pfree +KD

n"Pfree binding to n independent and !n = kD +Pfree identical binding sites

n n " Dfree !Pfree n!Pfree D+n! Pfree # DPn Kn = $n = n DPn Kn +Pfree

Pn or divided by n free " = n Kn +Pfree

! All or none binding (very high cooperativity)

n Pfree or " = n Kn +Pfree

! Binding to n identical binding sites

Pfree !1 = binding to a single binding site Pfree +KD

n"Pfree binding to n independent and !n = kD +Pfree identical binding sites

n Pn free strong cooperative binding "n = n Kn +Pfree to n identical binding sites, n with Kn = (kd) n"P#H ! ! = free approximation for cooperative binding to n K#H P#H + free n identical binding sites, αH HiIl coefﬁcient P#H "= free #H #H K +Pfree

! Hill coefﬁcient and Hill plot

#H approximation for cooperative binding to Lfree " = n identical binding sites, α HiIl coefﬁcient K#H +L#H H free Lfree is free ligand

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

αH > 1, the system shows positive cooperativity αH = n, the cooperativity is inﬁnite αH = 1, the system is non-cooperative αH < 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

#H αH HiIl coefﬁcient Lfree " = Lfree is free ligand #H #H K +Lfree K average microscopic binding constan

rearrange the terms to get !

L"H # free = K"H 1$#

which yields the Hill equation !

$ " ' * log& ) =* log L # log K H %1 #" ( H free

! Visualisation of binding data - Hill plot

#H 2 2 n"Pfree K "P +2"P ! % K "P +2"P ! = ! = 2 free free # 2 = = 2 free free n #H #H 2 2 K +Pfree K1"K2 +K2 "Pfree +Pfree 2$!2 1$% 2"K1"K2 +K2 "Pfree

2 & $ 2 ) & K2 ) ! ] Pfree " # log( + = log(Pfree )%log( + ) 1. ' 2%$ 2 * ' 2 * θ strong 5 – bindin

1 1 ( g slope α H =1.5 limit θ/

[ 0. g 5 lo # 2

% ( 0 P ! K " log' * =+ ,log Pfree $+ ,log K = free H ( ) H ( ) ] & 2$#2 ) )

ν - –

2 0.5 ( - weak

ν/ bindin

[ 1 % #2 ( g g K / 2·K - 2 1 Pfree ! 0 " log' * =log(Pfree )$ log(2K1)

lo limit 1.5 2 & 2$# 2 ) - 21 -11 1 -10 1 -9 1 -8 0 0 0 0 log (P free)

Binding of dioxygen to hemoglobin The Monod-Wyman-Changeau (MWC) model for cooperative binding

T T T 0 1 2 T3 T4 +P +P T conformation + P +P P P P P P P P (all binding sites P P P are weak) kT kT kT kT

L0 L1 L2 L3 L4

+P +P +P P +P P P P P P P R conformation (all binding sites P P P are strong) kR kR kR kR R0 R1 R2 R3 R4

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

β-conformation α-conformation α-conformation (induced by (facilitated binding) ligand binding)

+P +P P P P

k k1 2 +P k’2

P P

• 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.

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)