Diffusion
Thermal fluctuations
Low Reynold’s number
The radius of a water
molecule is about 0.1
nm.
Protein radius is in
the range 2 - 10 nm.
Fluid can be
considered as a
continuum
Diffusion
Thermal fluctuations
Low Reynold’s number
The radius of a water
molecule is about 0.1
nm.
Protein radius is in
the range 2 - 10 nm.
Fluid can be
considered as a
continuum
The distribution of molecular
speeds with temperature and
molar mass
Actual velocity Maxwell’s distribution
RT
Mv
e
v
RT
M
v
f
2
2
2
/
3
2
2
4
)
(
Molecular speed
2
1
2
3
2
1
2
3
M
kT
v
v
M
kT
For T = 300 K
500 Da (ATP) – v = 70 m/s
50 000 Da (protein) – v = 7 m/s
6.25 GDa (200 nm diameter vesicle) –
v = 600 μm/s
An experimental apparatus to
measure the distribution of
molecular speeds
The mean-square dosplacement in
a one-dimensional random walk.
Dt
x
N
2
2
Dt
z
y
x
r
N
N
N
N
6
2
2
2
2
3D diffusion.
No net movement
occurs.
0
x
The distribution is symmetrical.
Diffusion
Net solute movement is from higher to lower
concentration,
even
though
individual
particles move completely randomly.
The non-uniform distribution provides an
energy gradient to drive the overall process of
net movement from high to low concentration.
In thermodynamic terms, we're watching the
increase in entropy within a small, isolated
system without an input of energy.
Diffusion of selected molecules
D
10
-5
for most small molecules in water
Force
dx
potential
d
J
)
(
•
More probability seems to be „pushing” the
particles.
•
The random walk results with deterministic flow of particles.
•
But only when number of particles is large.
Flux of particles due to diffusion
dx
dc
D
Adt
dm
j
Fick’s first low
D – Diffusion
constant (m
2
/s)
Flux into volume A
l
l
J
Aldt
JAdt
t
c
Flux out of volume A
l
l
J
Aldt
Adt
J
t
c
'
'
'
Flux through volume A
l
l
J
J
t
c
'
2
2
'
'
x
c
Dl
l
x
c
c
x
D
x
c
D
x
c
D
x
c
D
J
J
Fick’s 2
nd
law
2
2
x
c
D
t
c
x
c
D
J
t
z
y
x
dz
c
d
dy
c
d
dx
c
d
D
dt
z
y
x
dc
2
2
2
2
2
2
,
,
)
,
,
(
Solution to diffusion equation
Dt
x
t
t
x
c
4
exp
)
,
(
2
2
/
1
Conditions:
At t = 0 all N
0
particles at x = 0
2
/
1
2
2
/
1
2
4
exp
D
dx
Dt
x
t
cdx
N
using
0
2
/
1
2
2
2
exp
r
dx
x
r
therefore
2
/
1
2 D
N
Dt
x
Dt
N
t
x
c
4
exp
2
)
,
(
2
2
/
1
In terms of particles per unit area
Strength of
source
Diffusion in space and time
The traveling wave
0
0
j
dt
dc
Inflection point
the curvature
changes sign
Migration down gradients
Thermal conductiity
dz
dT
energy
J
)
(
dz
dv
momentum
the
of
component
x
J
x
)
(
Viscosity
A charged molecules in an electric field
dz
dV
J
1
)
charge
(
Diffusive transport in biology
dx
dC
D
J
x
A concentration penalty
– diffusive
transport requires a concentration gradient.
The time penalty
– diffusive transport
time scales as the square of the distance or
<X
2
> = 4Dt
No
directional
specificity
The size limit
As a cell gets bigger there
will come a time when its
surface area is insufficient
to meet the demands of the
cell's volume and the cell
stops growing or it will
divide.
Forces acting on a particle due to the
solvent:
(i)
Stochastic thermal (Langevin) force
:
Averageing over a large number of particles
The Langevin approach –
dissipative force
changes direction and magnitude
averages to zero over time
0
)
(
t
(ii)
a viscous drag force that always slows
the motions.
v
f
friction (damping) coeff.
viscosity
Stokes law
R
6
The thermal forces
nN
f
5
.
4
The gravitational force
f
nN
F
g
14
10
Newton’s law for the protein motion in a one-
dimensional domain of length L, x(t):
L
t
x
t
f
v
dt
dv
m
v
dt
dx
B
)
(
0
)
(
,
)
(
)
(
2
)
(
2
2
2
2
2
2
t
xf
dt
x
d
mv
dt
x
d
m
B
The average over a large number of proteins
)
(
2
2
2
2
2
2
2
t
xf
dt
x
d
mv
dt
x
d
m
B
The mass, m, of a typical protein is about 10
-21
kg
Integrating twice between t = (0, t) with
x(0) = 0:
)
1
(
2
),
1
(
2
2
2
t
B
t
B
e
t
T
k
x
e
T
k
dt
x
d
where
=
m/
.
T
k
E
B
2
1
The random impulses from
the water molecules are
uncorrelated with position.
0
)
(
)
(
t
f
t
x
B
The protein behaves as a ballistic particle moving
with a velocity
v = (k
B
T/m)
1/2
. For a protein with m
= 10
-21
kg,
v = 2 m/s.
In a fuid the protein moves at this velocity only
for a time
~ m/ = 10
-13
sec
– shorter than any
motion of interest in a molecular motor.
During this time the protein travels a distance
v
· ~ 0.01 nm
before it collides with another
molecule.
For short times,
t <<
,
the exponential can be
expand to second
order:
)
(
2
2
t
t
m
T
k
x
B
)
1
(
2
2
t
B
e
t
T
k
x
)
(
2
2
t
t
T
k
x
B
When t >> , the exponential term disappears and:
For protein typically D ~ 10
-11
m
2
/sec.
Because
<x
2
> = 2Dt
(Einstein
relation – 1905):
T
k
D
B
Friction is
quantitatively related
to diffusion
)
1
(
2
2
t
B
e
t
T
k
x
External forces acting on
macromolecules
)
(
)
,
(
t
f
t
t
x
dt
dx
B
)
(
)
,
(
t
f
t
t
x
dt
dx
B
Langevin equation
The inertial term
is neglected.
)
(
)
,
(
t
f
t
x
F
dt
dx
B
Forces acting on proteins can be characterized by
a potential
x
t
x
t
x
F
)
,
(
)
,
(
It quantifies the relative importance of
friction and inertia
The Reynolds Number
The transition to turbulent flow in a pipe
occurs for R ~ 1000
Low Reynolds Number = Laminar
Flow
m
va
term
friction
term
inertial
R
a – radius of a particle
v – particle
velocity
m
– medium
density
When the Reynolds
number ‘R’ is small the
viscous forces dominate.
Size spectrum of living organisms and
the biological and physical properties
associated with the scale.
10
-1
10
-1
1 min
10
-6
100 μm
Protozoan
10
3
5
1 day
1
1 cm
Shrimp
10
8
1000
10
3
years
10
9
10 m
Whale
10
-5
10
-3
1 msec
10
-12
1 μm
Bacterium
Reynol
ds
number
Swimmi
ng speed
[cm/s]
Diffusi
on time
Mass
[g]
Leng
th
10
5
20
1 week
10
2
10 cm
Herring
10
7
100
1 year
10
6
1 m
Tuna
Without external force
such object will stop after
about 0.1 Å.
It takes it about 0.6 μsec
to stop.
Inertia plays no
role whatsoever !!
R ~ 10
–5
The objects which are the order of a μm in size is
moving in water with a typical speed of 30 μm/s.
(kg/m•sec at
20
o
C)
Water
10
-3
Olive oil
0.084
Glycerine
1.34
Glucose
10
13
There are two solutions to the
problem of swimming at low
Reynolds number.
If the animal tries
to swim by a
reciprocal motion,
it can’t go
anywhere.
Fast or slow, it
exactly retraces its
trajectory and it’s
back where it
started.
In addition the viscous friction
coefficient can be anizotropic.
II
The flexible oar
The other possibility
E.
coli
Rotary motion is
periodic but not
reciprocal.
The helix can translate and it
can rotate.
D
C
B
A
P
The
propulsio
n matrix
At low Reynolds number everything is linear.
B
Av
F
Force
D
Cv
N
Torgue
v – velocity, Ω –angular velocity
A „corkscrew”
(flagella)
It is a rigid, helical object.
At low Reynolds
number you can't shake
off your environment.
If you move, you take it
along; it only gradually
falls behind.
A covalent bond
is formed between the two non-
metals which share a pair of valance electrons so
that each obtains a filled valence shell.
A non-polar covalent bond
–
the electrons are shared equally
between the two atoms.
A polar covalent bond
– the
electrons are shared unequally
between the two atoms.
Bio-polymers are held together
by covalent bonds between
subunits.
Ionic Bonds
The valence electrons interact and the metal
transfers its valence electrons to the non-metal.
Delocaltzed bonding
– a resonance hybrid
between alternate structures e.g., benzene
Bond
energy
The macromolecular structures have extremely
many degrees of conformational freedom.
Intra- and Inter-molecular
interactions is what biology
is all about
Every residue of a macromolecule, and every
bit of their surfaces, are interacting with their
surroundings.
These interactions are
what set the stability criteria
and the dynamic constraints
of the biological structures.
All that allow for dynamic interactions
(
Interactions form, break, re-form constantly)
steric
repulsion
Noncovalent interactions
hydrogen bonds
Noncovalent Interactions
are Weak
They are ~ 10-100 times weaker than covalent bonds
(< 10 kJ/mol).
Noncovalent interactions
essentially hold together the entire
organism.
electrostatics
van der Waals (repulsive and attractive)
Interaction
Energy [kJ/mol]
Ion-ion
56 (ε = 8, r = 3Å)
Ion-dipol
–8 to +8 (μ = 2 debye)
Dipole-dipole
–2 to +2
Ion-induce dipole
0.24 (α = 10
-24
cm
3
)
Dispersion, and stacking
of aromatic ring system
0 to 40
H-bond
– 5 ± 2.5
Hydrophobic
– 5± 2 per CH
2
Hydration force
– 0.4 to 0.4 per residue
Thermal energy
3.7
Weak interactions
There are two types of
interactions
)))
(
cos(
1
(
2
1
2
1
4
0
2
0
,
,
,
,
,
,
2
0
,
,
,
6
,
,
12
,
,
,
0
n
k
k
b
r
k
r
B
r
A
r
q
q
V
dihedrals
angels
k
j
i
k
j
i
k
j
i
bonds
j
i
j
i
b
j
i
j
i
j
i
j
i
j
i
j
i
j
i
j
i
j
i
noncovalent
covalent
Structure of macromolecules and aggregates
Thermal motion
- biomolecules are stable
enough to make things work, and yet allow
the systems to play around in order to allow
the evolution
B
A
o
AB
G
E
a
A
B
T
k
G
B
A
B
o
B
A
e
A
B
K
]
[
]
[
At equilibrium
0
3 k T
3 5 k T
3
k T
3 0 0 k T
E =
How long it takes for
Brownian motion to
overcome an energy
barrier ?
Brownian death
pico-seconds
Biology
seconds
Too strong
10
67
years
E
a
kT
kT
E
a
e
t
t
0
t
0
~ 10
-12
seconds
Hairpin unfolding
accurs at 14.5 pN
(a)
Force-extension curves of an
RNA hairpin. Scretching and
relaxing
curves
are
superimposed.
DNA folding as a two-state system
Effect of mechnical forcen on the
rate of RNA folding. Length
versus time traces of the RNA
hairpin
at
various
constant
forces.
E
Control in Biology is accomplished by reducing
energy barriers.
Statistics
Unlikely events
will occur over
short time
periods.
Probable
Improbable
Disorder is Favorable
Disorder is Favorable
Entropic
Forces
Pressure
Tension
To create order
To create order
‘work’ must be
‘work’ must be
done
done
The entropic forces can create a
situation where two molecules will
interact strongly, although there is
not a direct “force” between them.
Electrostatics
Charge-charge interaction
r
q
q
r
U
B
A
0
4
)
(
r
r
r
q
q
r
F
B
A
2
0
4
)
(
Electric field of point charges
r
q
r
V
A
0
4
)
(
r
r
r
q
r
E
A
2
0
4
)
(
E
q
r
F
B
)
(
)
(r
V
q
Energy
B
k
dz
r
V
j
dy
r
V
i
dx
r
V
r
V
E
)
(
)
(
)
(
)
(
Interaction between two charge
distributions
j
i
j
i
AB
ij
B
j
A
i
AB
ij
d
q
q
U
U
,
,
0
4
A
i
B
j
AB
ij
r
r
R
d
B
j
A
j
A
j
B
j
A
j
B
j
AB
ij
r
r
r
R
r
R
r
r
R
d
2
2
2
)
(
)
(
2
2
2
R
r
and
R
r
B
i
A
i
2
2
2
2
2
2
2
2
/
1
2
2
2
2
3
2
3
1
3
1
1
2
2
2
)
(
)
(
1
1
R
r
r
R
R
r
r
R
R
R
r
R
r
R
r
r
R
r
R
R
r
R
R
r
r
r
R
r
R
r
r
R
d
B
j
B
i
A
j
A
i
B
i
A
i
B
j
A
i
B
i
A
i
B
j
A
j
A
j
B
j
A
j
B
j
AB
ij
j
i
j
i
AB
ij
B
j
A
i
AB
ij
d
q
q
U
U
,
,
0
4
charge-quadrupole
3
0
2
2
3
0
2
2
8
3
8
3
R
r
r
R
q
q
R
r
r
R
q
q
j
B
j
B
j
B
j
i
A
i
i
A
i
A
i
A
i
j
B
j
dipole-dipole
3
0
4
3
R
r
q
R
r
q
R
r
q
r
q
j
B
j
B
j
i
A
i
A
i
i
j
B
j
B
j
A
i
A
i
charge-charge
R
q
q
i
j
B
j
A
i
0
4
charge-dipole
2
0
2
0
4
4
R
r
q
R
q
R
r
q
R
q
i
j
B
j
B
j
A
i
j
i
A
i
A
i
B
j
i
i
i
r
q
p
Electric dipole
moment
r
q
p
p
p = 3.8 Debya
E
p
U
Interaction energy of a dipole with a field
1 Debye = 0.2 electron-Angstroms
= 3.3x10
–23
Cm
Electric field of a
charge
r
r
r
q
r
E
2
0
4
)
(
Interaction energy between a charge and a dipole
p
r
r
r
q
E
p
U
2
0
4
Charge-dipole interaction
The energy of a dipole p in an electric
field
- the angle between the dipole and the
electric field.
cos
E
p
E
p
U
p
i
A
i
A
i
A
r
q
p
j
B
j
B
j
B
r
q
p
r
R
A
B
E
p
U
Electric field of a dipole
3
0
4
3
r
r
r
p
r
r
p
E
A
A
A
3
0
4
3
R
r
q
R
r
q
R
r
q
r
q
U
j
B
j
B
j
i
A
i
A
i
i
j
B
j
B
j
A
i
A
i
Dipole-dipole interaction
3
0
4
3
r
p
r
r
p
r
r
p
p
U
B
A
B
A
A
E
p
0
– polarizability of the molecule
Inducede
dipole
Interaction energy between the induced dipole and the
inducing field
2
0
0
0
0
2
1
E
E
d
E
E
d
p
U
E
E
Induced dipoles: dependent
upon
polarizability
of
molecule, how easily electrons
can shuffle around to react to
a charge
Electric field of a
charge
r
r
r
q
r
E
A
2
0
4
)
(
Interaction energy between a charge and its induced dipole
4
0
2
2
2
2
0
32
2
1
r
q
E
U
Charge-induced dipole interaction
Electric field of a dipole
3
0
4
3
r
r
r
p
r
r
p
E
Dipole-induced dipole interaction
Interaction energy between a dipole and its induced dipole
6
0
2
2
2
32
3
r
p
r
r
p
U
Induced dipole-induced dipole interaction
Interaction between two harmonic oscillators with
natural frequency
1
and
2
and polarizabilities
1
and
2
.
6
2
1
2
1
2
1
)
(
2
3
r
v
v
v
hv
U
Summary
System
Potential
dependen
ce on
distance
Energy
[kJ/mol]
ion-ion
r
–1
250
ion-dipol
r
–2
15
dipol –
dipol
r
–3
2
London
r
–6
1
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0
2
4
6
8
10
r
-1
r
-2
r
-3
r
-6
Water Screens
Electrostatic
Forces
r
r
q
q
F
3
2
1
0
4
1
~ 1-10 for organic substances
= 80 for pure water
Summary