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
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