Interaction between molecules



binding

Minimizing the total energy  binding configuration

+

n

interactio

AB

total

B

total

A

E

E

E

E







Attractive interaction between
red and green monomers

The binding – qualitative analysis

•

Binding is reversible, and the reaction can be

characterized by an equilibrium constant and
associated free energy.

RL

L

R

k

k

1

1







A

D

K

RL

L

R

K

1

]

[

]

][

[





•

The binding affinity is

measured by the dissociation
constant for

the binding

equilibrium

.

•

A reaction proceeds spontaneously only if there is

a decrease in free energy.

S

T

H

G











The change in bond

energy between

reactants and products.

The change in the

randomness.

A rough estimate of the contribution of the

various factors to ΔG

o

for binding of a ~ 30

residue proteins (T = 300

o

K).

Energetics of binding

The specificity – the interfaces must be

precisely complimentary.

Steric

complimentarity.

The

water

molecules making van der Waals contacts
need to be replaced by contacts with the
other subunit when the bond is made.

Hydrogen

bond

complimentarity.

Any

hydrogen bond donor on one
subunit must find a hydrogen
bond

acceptor

on

the

opposite.

Binding

of

methyl--

mannoside by concanavalin
A (the concerted hydrogen
bonding).

Ionic complimentarity.

All

possible salt bridges must
form.

0.5 s

10

-6

M

50 s

10

-8

M

500 s

10

-9

M

t

E

[L]

The lifetime of the empty receptor is the reciprocal

of the association rate.

]

[

1

1

L

k

t

E



Lifetime of the empty receptor

– depends on

the concentration of ligand, it is limited by diffusion
of ligand, and does not depend on the bond energy.

k

1

= 2x10

6

M

-1

s

-1

is the generic, diffusion limited

rate constant for protein-protein interaction.

Recepor – ligand interaction

RL

L

R

k

k

1

1







The lifetime of the complex t

C

= 1/k

-1

t

C

is

independent

of

the

concentration of free logand, it
depends on the bond energy (K

D

is

determined by k

-1

).

1

1

1

1

k

K

k

t

D

C







The dissociation constant, K

D

, is a measure of

how well a ligand binds to the macromolecule.

It is equivalent to the ligand concentration

required to saturate exactly half of the binding

sites on the macromolecule.

Ligands with low K

D

values bind

tightly

Ligands with high K

D

values bind weakly

Examples of (K

D

; t

C

):



The complex of trypsin-trypsin inhibitor (10

-13

; 58 days),



Actin (10

-6

; 0.5 s).

Experimental determination of K

D

1.

Equilibrium Dialysis

– a direct measurement of

the partitioning of a ligand between the bound and
free states.

It shpuld be known that

the signal change is
directly proportional to
degree of saturation.

They rely on a signal change that

accompanies complex formation.

2.

Spectroscopic Measurements

(fluorescence,

CD, Abs, etc.)

Basic Treatment of Binding Data

ML

L

M





The extent of binding

, given at any ligand

concentration.

]

[

]

[

]

[

]

[

]

[

ML

M

ML

M

L

Y

total

bound







]

[

]

[

L

K

L

Y

D





[M] is the macromolecule, [L] the ligand, and [ML] the
complex between the two. Y is the fraction of saturated
binding sites.

]

[

]

][

[

ML

L

M

K

D



D

A

K

K

1



The goal is to obtain a value for K

A

.

The

energetics

of

binding

– the relationship

between K

A

and G

o

.

A

K

RT

G

ln

0







The application of equilibrium dialysis

experiment to determine the

concentrations of M, L, and ML at

binding equilibrium.

Starting concentrations:

Right cell: [ML] = 0; [L] = 12;
[M] = 0.

Left cell: [ML] = 0; [L] = 0; [M] = 4.

The protein (M) is

present only in the

left cell of the

dialysis chamber.

The small

molecule (L) is

present only in

the right cell.

The cells are separated by a
semipermeable membrane,
through which only the ligand
can pass.

However, because the protein can bind the

ligand, the concentration of total ligand will be
higher in the left cell.

When equilibrium is reached, the concentration of

free ligand will be the same in both cells.

Equilibrium concentrations:

Right cell: [ML] = 0; [L] = 5;
[M] = 0.

Left cell: [ML] = 2; [L] = 5; [M] = 2.

Features of this binding equilibrium:

[L] = 5 in both cells[L]

total

= 7 in the left cell

[L]

total

- [L] = 2 = [ML] in the left cell

Now we can calculate
the K

D

:

5

2

5

2

]

[

]

][

[









ML

L

M

K

D

A complete binding curve of Y as a function of

ligand concentraton.

Equilibrium dialysis done at several starting

concentrations of ligand.

Obtaining K

D

directly from the

binding curve.

Plot Y as a function of [L]. Interpolate to find the
ligand concentration that gives Y=0.5, this is the
K

D

.

Linearize the binding equation.



Obtaining K

D

from the binding curve is quick, but

really only uses the data from ligand concentrations
that give Y = 0.5, the other data points at low and
high ligand concentration do not contribute to the
analysis.

D

D

K

Y

K

L

Y

1

1

]

[









The binding curve can be transformed into a

straight line, to give the Scatchard equation:

]

[

]

][

[

ML

L

M

K

D



]

[

]

[

]

[

]

[

]

[

ML

M

ML

M

L

Y

total

bound







]

[

]

[

L

K

L

Y

D





A plot of Y/[L] versus Y
will give a straight line
with a slope of -1/K

D

.

Non-cooperative binding to multiple
sites

(n-sites).

The

fractional

saturation, Y, is replaced by , the total

amount of ligand bound/macromolecule
(varies from 0 to n).

 can be easily converted to Y by

simply dividing it by n (the number of
sites):

n

v

Y 

The binding equation is:

]

[

]

[

L

K

L

n

v

d





The Scatchard Plot

d

d

K

v

K

n

L

v





]

[

The number of binding sites, n, can be obtained
from the x-intercept.

If two different ligands interact this way it is

heterotropic

.

Cooperativity of binding

Binding at one ligand binding site can affect binding
at another site on a protein.

If two identical ligands interact this way, the

interaction is

homotropic

.



The second ligand binding has higher affinity

–

positive cooperativity.



The second ligand

binding has lower affinity
–

negative

cooperativity

.

]

[

]

[

]

[

ML

L

M





]

][

[

]

[

1

L

M

K

ML

A



]

[

]

[

]

[

2

ML

L

ML





2

2

1

2

2

]

][

[

]

][

[

]

[

L

M

K

K

L

ML

K

ML

A

A

A





2

2

2

2

1

2

2

1

2

2

2

2

]

[

1

]

[

]

][

[

]

[

]

][

[

]

[

]

[

]

[

2

2

1

]

[

]

[

]

[

]

[

2

]

[

2

1

L

K

L

K

L

M

K

K

M

L

M

K

K

ML

M

ML

ML

ML

M

ML

ML

Y

A

A

A

A

























The Hill equation:

]

log[

log

1

log

L

n

K

Y

Y

h





















n

h

- Hill coefficient.

Cooperative binding

two binding sites

Hill Plot

1.

Define

Y

Y







1

or equivalently

n

v

n

v







1

2.

Plot

log Θ versus

log[L]

iv.

n

h

= n (number of sites) for infinitely

positive cooperativity.

3.

Slope at log Θ =0 [Y=0.5] is the Hill coefficient n

h

i.

n

h

= 1 for non-cooperative binding

ii.

n

h

< 1 for negative cooperativity

iii.

n

h

> 1 for positive cooperativity

4.

Intercept at log Θ =0 [Y=0.5]

gives logK

D

Ave

, or the average

dissociation constant.

At log Θ =0

]

[

]

[

1

]

log[

log

]

log[

log

1

]

log[

log

0

1

L

K

L

K

L

K

L

K

n

L

n

K

Ave

D

n

n

























The Hill plot for oxygen

binding to normal adult

hemoglobin.

The Hill coefficient, n

h

=

3. Cooperativity is positive

and fairly high.

The average affinity
constant is approximately
10

5

(K

D

=10

-5

M)

Its thermodinamical stability

results from molecular non-

covalent interactions within the

phospholipid bilayer and water

phase.

Biological membrane

Life’s Border

Functions of the cell

Functions of the cell

membrane

membrane

Isolation,
compartmentalization,
protection

Regulate
transport

Diffusio
n

Active transport
Vesicular
transport

Signaling, signal
processing

Receptor –ligand
interaction

(hormons, growth factors, neurotransmitters,
etc.)

Selective signal recognition and
transduction by transmembrane
receptors

Electric activity

Membrane
potencial

Allow cell to cell interaction,

Allow cell to cell interaction,

cell recognition

cell recognition

Immune recognition,
synchronization of cellular
activities

Biochemical
activity

Provide stable site for the
binding and catalysis of
enzymes

Compartment separation for
chemiosmosis

ATP sysnthesis in mitochondria and
chloroplasts

Functions of the cell

Functions of the cell

membrane

membrane

Cell shape and
motility

cytoskeleton,
cilia and flagella

asymmetric functions



asymmetric structures

Membrane

asymmetry

Carbohycrate asymmetry

Found only on outside surface of cell

(glycocalyx).

Important in cellular recognition

processes

carbohydrate

Types of asymmetry

lipids

proteins

salt

composition

Glycolipids



Tend to self-associate (lipid
rafts)



Surface-associated sugers



Exclusively on noncytosolic side

The diversity of carbohydrate chains is

enormous, providing each individual with a

unique cellular “fingerprint”.



Cell recognition and
adhesion

Functions



Protection of cell surface



Electrical effects



Lectins, or carbohydrate-binding proteins, are also

involved in this layer

Lipid asymmetry

Asymmetry

of

interactions



A second protein called “scramblase” becomes

active and transfers phospholipids nonspecifically in
both directions resulting in exposure of PS on the
outside of the cell.



Phosphatidylserine
is restricted to the
cytoplasmic leaflet
in a healthy cell.

An example of a function for the asymmetry



The asymmetry is maintained by a “phospholipid

translocator” that transports PS to the inner leaflet.
This is inactivated in dying cells.



Macrophages detect the PS and destroy the dead cell.

Protein asymmetry

Active sites of

enzymes are located
depending

on

location

of

substrates.

Transport proteins

(especially

active

transporters) work in
only one direction.

Receptor proteins

bind ligand on only
one

side

of

the

membrane.

Adhesion proteins

only

on

the

extracellular side of
the membrane.

Ion

Intracellul

ar

Concentra

tion

(mM)

Extracellu

lar

Concentra

tion

(mM)

Catio

ns

Na

+

K

+

Mg

+

2

Ca

+2

5 - 15

140
0.5

1 x10

-5

145

5

1 - 2

1 - 2

Anion

s

Cl

-

P

i

5 – 15

40

110

2

Glucose

1

5.6

Mammalian Cell Intracellular and

Extracellular Ion Concentrations

Lipid
microdomain
s in plasma
membrane

Caveola



Rich in sphingolipids and

cholesterol



Longer, saturated chains of sphingolipids cause membrane

thickening



Such lipid arrangement recruits particular proteins and

facilitates their transport or function as a group

Membrane

electrastatics

Charges separated by moving ions all the way
across the membrane from the aqueous medium
on one side to the aqueous medium on the other.

Transmembr

ane

potential

Because water has a high
dielectric constant (80)
charge separation across
water will tend to create a
relatively small electric
field and small electrical
potential differences.

Charge separation across

organic phases ( = 1.89)

will create large electric

fields.

0



S

q

E 

Membranes have planar
symmetry

A

charge

smoothly

distributed on a plane will
create an electric field
perpendicular to the plane.

The electric field will be

proportional to the charge density

on the plane (q

s

).

Membrane potential

Two parallel

planes of

opposite charges

0



S

q

E 



















x

x

s

s

x

q

dx

q

Edx

0

0

0

0







•

The electrical potential difference between the

plates

x

A

q

C

C

0











The capacitance of

biological membrane is

about

1 µF/cm

2

(35 Å thick

with a dielectric constant

of 4).

The electric field associated with the membrane

potential acts on dipolar groups in membrane
proteins and may regulate the activities of these
proteins.

The total difference in electrical potential

(

membrane potential

) is the force that drives ions

across the membrane.

The membrane potential partly determines

the energy stored in ion concentration
gradients

.

Voltage-dependent

channels

open

and

close in response to
changes

in

the

membrane potential.

Fixed charges bound to the membrane surface will
be neutralized by counterions present in the
adjacent aqueous solution.

Because these counterions will tend to diffuse away
from the membrane surface, there will be a charge
separation.

Surface potential

Origin of surface charge

Ionisation of surface groups (ionisation of

carboxyl and amino groups to give COO

-

and NH

3

+

ions)

Ion adsorption

 Cations more hydrated than anions, have
greater tendency to reside in bulk.
 Anions have greater tendency to be
specifically adsorbed.

Any

ion

whose

adsorption at surface is
influenced

by

forces

other

than

simple

electrostatic

potential

can be regarded as
specifically adsorbed.

The charge separation

depends on:

 a dynamic tension with diffusion
pushing the counterions away from
the surface

 electrical attraction pulling
them toward the surface

Generally membranes have a negative charge (10 -
20% anionic lipids, charge from gangliosides and
proteins)

Membrane surface electrostatic potential

A plane of fixed charges
with a surface charge
density σ

s

.

Space charge

density ρ

v

(x)

C

∞

- bulk ion concentration at ∞, Z -

ion valence



charges are smeared out on the

surface



ions in solution are “point”

charges



image effects - ignored



the dielectric constant is

constant

Assumptions of Gouy-Chapman theory:

The Gouy-Chapman theory

(1910-1913)

The ion distributions as a
function of distance from
the membrane surface.

Mean field

approximati

on

Each ion is moving under the influence of an
electric potential created by the average charge
density of the others <ρ

q

>.

Electroneutrality – the total space charge be

equal and opposite to the sum of the fixed
charge q

s

and the capacitative charge q

c

.













0

)

( dx

x

q

q

q

q

v

s

c

s

The surface charge

The space charge at a point x is





i

i

i

v

x

C

z

F

x

q

)

(

)

(

For each ion species i, z

i

is the valence and

C

i

is the concentration. F is the Faraday

constant.

q

c

 10

-7

C/cm

2

(a 100 mV membrane potential

when membrane capacitance is 1 µF/cm

2

).

S

c

q

q 

The ion concentrations depends on the electrical
potential according to the Boltzmann distribution:











 



RT

x

F

z

C

x

C

i

i

i

)

(

exp

)

(

0



C

io

is the concentration of ion species i far from the

membrane (x = ).

Hence













 



i

i

i

i

v

RT

x

F

z

C

z

F

x

q

)

(

exp

)

(

0



From the Poisson's equation:

2

2

0

)

(

dx

d

x

q

v









If the single plane of
charge is assumed to
have a thickness dx:

0

2

2





S

q

dx

dE

dx

d









dx

dx

d

dx

x

q

q

v

s















0

0

2

2

0

)

(





If () = 0

dx

d

q

s

0

0



















 





i

i

i

i

RT

F

z

C

z

F

dx

d







exp

0

0

2

2

Multiplying both sides by 2d/dx and integrating



















 















i

i

i

const

RT

F

z

C

RT

dx

d







exp

2

0

0

2

At x = ,  = 0 and d/dx = 0, so



















i

i

C

RT

const

0

0

2





The general form of the Gouy-Chapman

equation relating the surface charge to the

surface potential.





2

0

0

2

1

exp

2

s

i

i

i

q

RT

F

z

C

RT

dx

d



























 























If all of the ions in the aqueous phase are univalent,
C

io

= C

o

for both anions and cations.



















RT

F

RTC

q

s

2

sinh

8

0

2

/

1

0

0





The sinh can be presented as a power series and
higher order terms dropped (assume



o

is small),

then

2

/

1

2

0

0

0

2















o

s

RT

F

C

q

























RT

F

RTC

q

s

2

8

0

2

/

1

0

0





A constant  with units of

m

-1

is customarily defined

as

















i

i

i

z

C

RT

F

2

0

0

2

2





Increasing salt in the solution shrinks the diffuse layer

For the aqueous solution (ε = 80) the

Debye length for ultrapure water is 190

nm and for 1 mM KCl 9.7 nm.

Debye length for

0.1 M NaCl, L

D

= 0.96

nm,

0.01 M NaCl L

D

= 3.04

nm

0.01 M MgCl

2

L

D

= 1.75

nm

1/is a measure of the thickness of

the diffuse double layer.

The surface charge density =
0.0158 C/m

2

. Univalent

electrolyte,  = 80 and T = 20°C.

The decay of

potential

from a

surface

Univalent electrolyte at
10 mM concentration,
 = 80 and T = 20°C.

Surface

potential as a

function of the
surface charge

The surface potential may act to repel or attract

other surfaces.

Surface charge has a number of effects

A negative surface charge attracts cations to

the membrane surface

 enhanceing the binding of cations to the membrane surface.

 increaseing the conductance of the membrane to cations

The surface potential has a greater affect on the

distributions of divalent ions than on distributions
of monovalent ions (high Ca

2+

concentrations at

the membrane surface).

The surface potential changes the pH near the

surface of the membrane.

Because the ester linkages between fatty acids and
the glycerol backbones of the membrane lipids are
dipolar in character, alignment of these dipoles
creates a charge separation which gives rise to the
dipole potential.

Dipole potential

Origin of a dipole potential Ψ

d

Primery determinant sn

2

-carbonyls

Little effect of sn

1

-carbonyls

–

P – N

+

dipole

P = O bonds of phosphate groups

Dipoles of hydration water which

are ordered and oriented at the

membrane surface.

As a result, lipid membranes exhibit a

substantial (up to six orders of

magnitude) difference in the penetration

rates between positively and negatively

charge hydrophobic ions.

The dipole potential

is a major factor in

determining the ionic

permeability of the

lipid bilayer.

Dipole potential

modifies the electric

field inside the

membrane,

producing a virtual

positve charge in the

apolar bilayer center.

++++++

The dipole potential

0





D





D is the surface
dipole density in
C/m.

A dipole moment of phospholipid =

1.5 Debyes

- 5 x 10

-30

C/m

Surface area per lipid =

60 A

2

A surface dipole density =

8.34 x 10

-12

C/m

A dipole potential =

1000/ mV

The dipole

potential is not a

significant

component of the

membrane

potential because

the dipoles on

opposite surfaces

of the membrane

are oriented in

opposite directions

and tend to cancel

each other.

 Modulates membrane enxymes
activities.

Dipole potential functions:

 Effects the membrane permeability for
lipophilic ions and drugs.

 Modulates the binding of peptides and
biologically active molecules to cell plasma
membranes.

 Directs the insertion and filding of
peptides in the membranes.

Examples of potential profiles



Negative

surface

charge on both sides.



A large surface

charge on one side.



Surface charge

on one side, dipole
potential

and

transmembrane
potential.

Classes of ligands interacting with the bilayer

Nonpolar solutes

– the bilayer is a 2D

fluid, i.e. “solvent” for small nonpolar
molecules

e.g. benzene - localizes in the bilayer
interior

Amphipatic molecules

– with polar and nonpolar

moieties

- adhesion, insertion,

- at high concentration: disruption of the

membrane

e.g. anesthetics, drugs,
tranquilizers, antibiotics, bile
salts, fatty acids, fluorescent
probes

Benzene

The binding constant K
reflects the standard free
energy change upon binding.





















]

][

[

]

[

ln

0

M

L

LM

RT

G

G

The Binding Constant

L + M



LM

L is the free ligand, M are empty binding sites on the
membrane, and LM represents ligand bound to the
membrane.

At
equilibrium:

K

RT

G

ln

0





0



G

]

[

]

][

[

LM

M

L

K 

L

m

– the concentration of bound ligand ([LM])

L

max

– the total number of binding sites on the

membrane ([M]+[LM]).

m

m

L

L

L

L

K

)

](

[

max





The amount of bound ligand L

m

)

]

([

]

[

max

K

L

L

L

L

m





The Scatchard equation

A plot of L

m

/[L] vs. L

m

is linear and

may be used to define the
parameters L

max

and K.

K

L

K

L

L

L

m

m





max

]

[

The Effect of Surface Charge



[L] is the concentration of ligand at the surface of

the membrane.



This is related to [L]



, the

concentration far from the
membrane, by the Boltzmann
equation:











 





RT

F

z

L

L

i

0

exp

]

[

]

[





If the surface potential is -18 mV and the ligand is

a univalent anion, [L] will be 50% of [L]



.

Specific and non-specific binding

Specifically binds to a receptor or other membrane-bound protein

Non-specifically binds to the membrane (because

ligand is hydrophilic or amphiphilic)

Total binding to the membrane will be the sum of
the specific (sp) and nonspecific (ns) contributions:

ns

ns

sp

sp

ns

m

sp

m

m

K

L

L

L

K

L

L

L

L

L

L













]

[

]

[

]

[

]

[

max

max

The affinity of the non-specific binding

sites is very weak compared to the affinity

of the specific sites (K

ns

>> K

sp

).

ns

ns

sp

sp

m

K

L

L

K

L

L

L

L

]

[

]

[

]

[

max

max







To measure specific binding ligand concentrations
should range about K

sp

so [L] will be very small

relative to the non-specific binding constant

[L] <<

K

ns

),

Non-specific binding

will be directly

proportional to the

ligand concentration

over this range.

Specific binding can be calculated by subtracting

non-specific binding from total binding.

Under these conditions

ns

ns

sp

m

K

L

L

L

L

]

[

max

max





Because

L

max

sp

<< L

max

ns

,

the measured binding L

m

will all be non-specific.

This gives the slope of the non-specific binding line

(L

max

ns

/ K

ns

).

The

non-specific

binding

can

be

measured by adding a
high concentration of
ligand (

[L] >> K

sp

).

K

obs

should be measured at very low

ligand concentrations (one ligand

molecule per 100 lipid molecules) in

order to avoid non-ideal behavior.

Partition

The

mole-fraction

partition

coefficient

K

obs

determined in equilibrium dialysis experiments

]

/[

]

[

]

/[

]

[

W

P

L

P

K

water

mem

obs



[P]

mem

and [P]

water

are the bulk molar concentrations of

ligand atributable to ligand in the membrane and
water phases, respectively

[L] and [W] are the molar concentration of water and lipid