Binding and Kinetics

1

Lecture 8

Binding and kinetics

Antoine van Oijen

BCMP201 Spring 2008

Donald T. HayneBiological Thermodynamics

James Goodrich, Jennifer KugelBinding and Kinetics for Molecular Biologists

2

Goals

- Quantitative measurements of biological binding reactions

- Affinities- Cooperativity in binding- Kinetics

Practical use!!!

Assays: how much is bound?

• Assays that separate complexes from a solution- Filter-binding (or cell-binding)- Gel-filtration chromatography- Electrophoretic mobility shift assays (EMSAs/ gel-shift)

• Assays that detect complexes in solution- Fluorescence (quenching, anisotropy, FRET)- Protection assays (Rnase, Dnase footprinting)

• Assays in which a biomolecule is bound- Affinity resins- Surface plasmon resonance

(More details later in the semester)

Protein-protein, protein-DNA, protein-ligand, …

3

Bimolecular interactions

A + B ABkon

koff

Binding is not all-or-nothing:

Portion of A and B will be bound, portion will be free

X Ykon

koff

!

d[Y]

dt= [X ] " k

on# [Y] " k

off= 0

Equilibrium

Reaction is in equilibrium when concentrations do not change:

(unimolecular reaction)

(mass action law)

4

Equilibrium

Reaction is in equilibrium when concentrations do not change:

!

d[AB]

dt= [A] " [B] " k

on# [AB] " k

off= 0 (mass action law)

A + B ABkon

koff

Equilibrium is reached when:

!

[A] " [B] " kon

= [AB] " koff

Equilibrium is still dynamic!!!

Binding (bimolecular reaction):

Equilibrium dissociation constant KD

Rearrange to define equilibrium dissociation constant KD:

!

KD

=k

off

kon

=[A] " [B]

[AB]

When [A]=Keq, 50% of B is bound to A

Equilibrium is reached when:

!

[A] " [B] " kon

= [AB] " koff

5

Units

Units:

!

KD

=[A] " [B]

[AB]

!

{M} ={M} " {M}

{M}

(Conversely, equilibrium binding constant, KB, is defined as:

!

KB

=[AB]

[A] " [B]

!

{M"1} ={M}

{M} # {M})

!

KD

=k

off

kon

=[A] " [B]

[AB]

koff: {s-1}

kon: {M-1·s-1 }

Rate constants:

Where does this KD come from?

From Lecture 5:

6

How to measure KD ?

!

KD

=[A] " [B]

[AB]Measure [A], [B], and [AB]?

Introducing [A]Total=[A]+[AB]:

!

[AB]

[A]Total

=[B]

KD

+ [B]

D

Figure from: G

oodrich, Kugel

Experimental considerations

• [A] constant; titrate B• Measure fraction bound

!

[AB]

[A]Total

=[B]free

Keq + [B]free

If [A]Total << KD, then [B]≈[B]+[AB]

No need to measure [B],Just take [B]Total!

Figure from: G

oodrich, Kugel

7

Logarithmic versus linear display

As a corollary: Choose your titrations logarithmically!

1, 3, 10, 30, 100, 300 nM, or2, 4, 8, 16, 30, 60, 180, 360 nM, instead of50, 100, 150, 200, 250, 300 nM

Figure from: G

oodrich, Kugel

Example: Repressor binding to DNA

DNA + R DNA-Rkon

koff

In E. coli, how much repressor is bound non-specifically to DNA and how much is free?

[non-operator DNA] ≈ 106 / 1 µm3 ≈ 10 mM (107 bp/genome; 10 bp/site; volume. E.coli 1 µm3)

!

F =[R]

[R] + [R "DNA]=

[R][DNA

non]

[R "DNA]

[R][DNA

non]

[R "DNA]+ [R "DNA]

[DNAnon

]

[R "DNA]

=K

D

KD

+ [DNAnon

]=

10#4M

10#4M +10

-2M

= 0.01

Hardly any free repressor; almost all bound to nonspecific DNA!

KD≈10-10 M for operator DNA (specific binding)KD≈10-4 M for non-operator DNA (non-specific binding)

8

Protein

B BB

000 + BK

00B

Not Cooperative

00B + BK

0BB

Protein

B BB Cooperative

000 + BK

00B

00B + BτK

0BB

Non-cooperative versus cooperative

τ can be positive or negative (positive or negative cooperativity)

Cooperative binding

A + nB ABn

kon

koff

(perfect cooperativity)

Simplification:

!

KD

=[A] " [B]n

[ABn]

Rearrange (next Problem Set???):

!

logY

1"Y

#

$ %

&

' ( = n

H) log[B] " logK

D ,

where Y=[ABn]/[A]total

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Cooperative binding

!

logY

1"Y

#

$ %

&

' ( = n

H) log[B] " logK

D ,

where Y=[ABn]/[A]total

Figure from: G

oodrich, Kugel

Hemoglobin

10

Reaction kinetics

Equilibrium thermodynamics does not provide anyinformation on rates of chemical changes!

Energy profile for ageneric chemical reaction:

Gibbs free energy (ΔG0) determines ratio of reactants/products(thermodynamic properties), activation energy (ΔG++) determinesrates (kinetics)

(dynamite versus nitroglycerin)

Figure from: H

aynie, Biological Therm

odynamics

++

Rate of reaction

Reaction rate = a measure of how fast the concentration of reactants /products changes with time

Example: hydrolysis of ATP into ADP

ATP ADP + Pi

Reaction rate:

!

J = "d[ATP]

dt= +

d[ADP]

dt= +

d[Pi]

dt

Figure from: H


odynamics

11

Rate constant and order of reaction

!

J = k[A]n

Reaction rate/velocity is related to concentration of reactant:

n is order of reaction (often identical to stoichiometry)k is rate constant (don’t confuse with binding constant)

We saw that , so k will have:

Per second (s-1) as unit for 1st order reaction,Per molar per second (M-1s-1) as units for 2nd order reaction

!

J = "d[A]

dt

1st order reaction

1st order reaction A P

!

J = k[A]

!

J = "d[A]

dtCombining with gives:

!

"d[A]

dt= k[A]

!

1

[A]d[A] = "kdt

Integrate ( ):

!

ln[A] = ln[A]0 " kt

!

1

x" dx = ln x + C

!

[A]

[A]0

= e("kt)

Figure from: H


odynamics

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2nd order reaction

2nd order reaction 2A P

!

J = k[A]2

!

J = "d[A]

dtCombining with gives:

!

"d[A]

dt= k[A]2

!

1

[A]2d[A] = "kdt

Integrate ( ):

!

1

[A]=

1

[A]0

+ kt

!

1

x2

" dx = #1

x+ C

!

[A]

[A]0

=1

1+ kt

Figure from: H


odynamics

1st and 2nd order reactions

1st order:

!

[A]

[A]0

=1

1+ kt

!

[A]

[A]0

= e("kt)

2nd order:

Figure from: H


odynamics

13

Half-times and rate constants

Half time t1/2 is not the same as k-1 :

!

[A]

[A]0

= 0.50 = e("kt1/ 2 )

# " ln2 = "kt1/ 2 # t1/ 2 =ln2

k$

0.693

k

Temperature effects

Rates depend on temperature

Arrhenius:

!

k = Ae("#G

++ / RT)

A+B ABkon

koff

!

lnk = ln A "#G++

/ RT

14

Reversible reaction

A + B ABkon

koff

!

d

dt[A "B] = k

on[A][B] # k

off[A "B]

Formation(2nd order)

Dissociation(1st order)

Under equilibrium, equals zero:

!

d

dt[A "B]

!

[A][B]

[A "B]=

koff

kon

= KD

In terms of free energies:

!

KD

=k

off

kon

=Ae

("#Goff

++ / RT)

Ae("#Gon

++ / RT)= e

("(#Goff

++"#Gon

++ ) / RT)= e

("#G0 / RT)

ΔG0= ΔGoff*-ΔGon

*

Figure from: H


odynamics

!

[A][B]

[A "B]=

koff

kon

= KD

Relation between KD, kon/off, and ΔG

A+B

AB

++++ ++

15

Rates of binding and dissociation

A+B ABkon

koff

Association rate for two objects with diffusion coefficients D1 and D2and diameter r1 and r2:

kdiff=4πNA(D1+D2)(r1+r2) (units: {mol-1}{cm2s-1}{cm} = {M-1s-1} )

For a small ligand and protein: kdiff ≈ 109 M-1s-1,for two proteins: kdiff ≈ 106 - 107 M-1s-1

This rate can be further slowed down if a conformational changeneeds to take place before binding

Example: Repressor binding to DNA

DNA + R DNA-Rkon

koff

!

d

dt[R "DNA] = k

on[R][DNA] # k

off[R "DNA]

Formation(2nd order)

Dissociation(1st order)

It takes 0.1 seconds to switch off gene expression in E.coli afterlactose depletion. What is kon?

!

d

dt[R "DNA] # k

on[R][DNA]

With ~10 repressors per E.coli and [DNA]≈10-9 M (1 operator sequence in 1 µm3 cell),kon needs to be at least 109 M-1s-1 (is actually measured to be 1010 M-1s-1)

How come this is much faster than diffusion limit???

16

1D sliding along DNA to speed up kon

BWH (Berg, Winter, von Hippel) model:Combine 3D diffusion (‘hopping’) with1D diffusion (‘sliding’). Scan short stretch of DNA by 1Dsearch, then jump to different area.

!

"L(# ) = D1D#

Length explored by one 1D sliding event:

(1D random walk)

Typical duration will be τ=1/knonsp.off:

!

"L = D1D / knonsp.off

Remember, repressors spend 99% of time on nonspecific DNA:

!

L(t) = tknonsp.off D1D / knonsp.off Total length explored L(t) is linear with time!

1D sliding: the numbers

D1D ≈ 10-9 cm2/s (limited by rotational drag)knonsp.off ≈ 10 s-1

L(τ) ≈ 100 nm (300 bp)100 kb of DNA is searched by singlerepressor in half a minute

Searching 100 kb with only 1D sliding would takeTtotal = L2

total/D1D ≈ 3 hours!

Now we understand why 99% of repressor is bound to nonspecific DNA:They’re actively involved in the search process.

17

Folding revisited: a riboswitch

• Folded RNA that binds small molecule (aptamer)• Plays role in regulation of gene expression

How does it fold?

?

Single-molecule probing of RNA folding

Liphardt et al., Science (2001)

18

Pulling at an RNA hairpin

Force-extension curve ofsingle RNA unfolding/folding


Along the reaction coordinate, anamount of energy equal to forcetimes displacement is added

Force tilts free-energy diagrams

ΔG = -F·Δx

ΔG = ΔG0 - F·Δx

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!

P(unfolded)

P(folded)= exp "

#G

kT

$

% &

'

( ) = exp "

#G0 "F#x

kT

$

% &

'

( )


Pulling at an RNA hairpin


Single-molecule kinetics:Direct observation of kopen and kclose

Pulling at an RNA hairpin: kinetics

20

Unfolding a riboswitch

Unfolding a riboswitch

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Take-home message

Equilibrium constant K is related to free energy difference ΔG0

between initial and final state, rates k are related to free energydifferences ΔG‡ between initial/final state and transition state

!

[A][B]

[A "B]=

koff

kon

= KD

!

KD

=k

off

kon

=Ae

("#Goff

++ / RT)

Ae("#Gon

++ / RT)= e

("(#Goff

++"#Gon

++ ) / RT)= e

("#G0 / RT)

A+B

AB

++++ ++ΔG0= ΔGoff

*-ΔGon*