binding_kinetics.doc

8/14/2019 Binding_Kinetics.doc

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Peter Fajer Biophysical Methods in Biology Lecture 10

Ligand Binding

Consider ligand Abinding to macromolecule P:

A P AP+ [1]

Equilibrium constant is defined as:

[ ] [ ]

[ ]K

P A

PAd

= [2]

(note that Kdis often referred wrongly as binding constant: Kb= 1/Kd)

define fractional occupancy (moles of ligandmole of macromolecule) r:

[ ]

[ ]

[ ]

[ ] [ ]r

A

P

PA

P PA

bound

total

= =+

[!]

substitute [P][A]/Kd for [PA]:

[ ][ ]

rA

K Ad

=

+

["]

this hyperbolic dependence is called binding isothermor (Langmuir isotherm)#

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binding isotherm

$

$#2

$#"

$#%

$#&

1

1#2

$ ' 1$ 1' 2$ 2'

[A]

r

d1

d'

d'

d1$

*ote that although the isotherm has the concentration for free ligand [A]as well as bound ligand([+]boundin r) you ha,e to measure only one (bound or free) because conser'ation o( mass the totalligand [A]total= [A]+[PA](and total macromolecule [P]total= [P]+[PA])#

-easure the concentrations of free or bound ligand by ,ariety of methods: chromatography.equilibrium dialysis. ultrafiltration. spectroscopy#

Multiple binding sites

-acromolecule can ha,e more than one site for binding the ligand# /hese sites can be independent:binding of one ligand does not influence binding of the ne0t. or cooperative(binding of one affectsbinding of another)#

Independent

A P A P A P+ + +1 2 ....... [']

identical sites: +1+2 or

different sites: +1+2

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Identical sites

macromolecule

site 1

site 2

all sites are independent and identical so that total number of sites is n[P]totalrather than [P]total:

[ ]

[ ]AK

Annrr

d

siteglesitesall+

==_sin_ [%]

Scatchard plot

plot the ratio of r/[A],ersus r:

[ ]

r

A

n

K

r

Kd d

= [&]

or identical binding this results in a linear plot# 3e,iation from linearity implies non4identical sites#

r

r/[A]

Dierent sites

to be ,ery general: each class of sites with different Kdican ha,e niidentical sites

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macromolecule

site class 1

site class 2

/he binding isotherm is a sum of the single binding isotherms. each corresponding to different class ofsites:

[ ]

[ ]

[ ]

[ ]r r r

n A

K A

n A

K Ad d

= + + =

+

+

+

+1 2

1

1

2

2

..... ........, ,

[5]

!ooperati"it#

6inding to one site on a molecule modulates binding of another#

or strong cooperati,ity. binding of one ligand triggers binding to n sites:

PAPnAn

+ [7]

[ ] [ ][ ]PA

APK

n

n

d = [1$]

[ ]

[ ]

[ ]

[ ] [ ][ ]

[ ] nd

n

n

n

total

bound

AK

An

PAP

PAn

P

Ar

+

=

+

== [11]

or measure the ratio of filled sites$ %$=r/n&to the unfilled sites (1'$):

[ ]

d

n

K

A

rn

r

Y

Y=

=

1[12]

*ill+s e,uation)

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(ill)s Plot

lot log%$/%1'$& "* log[A]. the slope is n8 an intercept is 1/Kd#

no cooperati"it# n=1

negati"e cooperati"it# n=+*,

positi"e cooperati"it# n=-

log [+]

log[r(n4r)]

Kinetics

/ime course of reaction. 9inetics gi,e information about the rates rather than equilibria# f courserates and equilibria are related Keg= .+1/.'1. howe,er the ratio of rates tells us nothing about the ,alueof each rate i#e# is it fast or is it slow#

/wo sorts of 9inetics are usually considered:

stead#'state: in which the forward flu0 bac9ward flu0 resulting in no change of concentration8

transient .inetics: non4equilibrium 9inetics when concentrations are changing#

steady-state (Michaelis-Menten)

E S X E Pk

k

k

k

+ +

1

1

2

2

[1!]

for a reaction with an intermediate the rate of intermediate;s production is equal to that of itsdisappearance:

[ ]( ) [ ] [ ] [ ] [ ] [ ] = + =

d X

dt k k X k E S k E P 1 2 1 2 0

[1"]

/he rates of substrate disappearance and the appearance of the product are identical:

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[ ] [ ][ ][ ] [ ] = =

d S

dt

d P

dt k E S k X

1 1[1']

rom mass conser,ation we ha,e:

[ ] [ ] [ ]

[ ] [ ] [ ]

E E X

S S P

o

o

= +

= +

[1%]

+t the beginning of the reaction:

[ ]P = 0 [15]

thus the initial ,elocity "can be sol,ed since we ha,e fi,e equations with " un9nowns (concentrationsof E. e that initial ,elocity is a ma0imum ,elocity reduced by the partial en>ymeoccupancy. i#e#:

v rV=max [1&]

then substitute "for rin the equation for Langmuirisotherm#)

t0rno"er

kV

Ecat

o

= max

[17]

non'stead# state


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E S Xk

k

+ 1

1

1[2$]

the rate equations are:

[ ] [ ] [ ][ ] [ ] [ ] = = =

d E

dt

d S

dt

d X

dt k E S k X

1

1 1 1[21]

with two mass con,ersation relationships:

[ ] [ ] [ ]

[ ] [ ] [ ]

E E X

S S X

o

o

= +

= +

1

1

[22]

/he differential (rate) equation can be sol,ed but it is a mess since the equation is non4linear (the ratedepends on product of [E] and [


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the rate of en>yme disappearance is:

[ ][ ] [ ] [ ] =

d E

dt k E S k X 1 1 1

[25]

the rate of product appearance is:

[ ][ ] [ ] =

d X

dt k X k X

2

2 2 2 1[2&]

+t this point you are ad,ised to gi,e up# @hat you ha,e is the set of simultaneous/ nonlineardi((erential e,uationswhich e,en your grandma can;t sol,e#

6ill Aates to the rescue8 integrate the set using a C and any mathematical pac9age with n0mericalintegrationroutines e#g# -athcad. -athematica. -atlab#

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