Brownian motion
• Robert Brown, botanist, 1827: pollen particles suspended in water
perform zigzag random motion: ”Brownian motion”
• The phenomenon was first attributed to a ”vital force”
• Light particles (smoke, dust, small droplets of liquid) do similar
motion in air
György Vámosi
Diffusion
Brownian motion - explanation
• The particles of matter are in perpetual motion. Mean kinetic
energy of translating motion: E=3/2 kT
• Reminder: Maxwell’s velocity distribution in gases:
0.002
•Mean velocity:
v
3kT / m
f(v)~ n/n
T=20 C
2
T=500 C
0.001
0
0
N
500
1000 1500 2000
v (m/s)
Brownian motion is caused by the thermal motion and collisions
of the particles. Exchange of momentum and energy
Fluctuation of a macromolecule
(DNA)
Fick’s 1st law
Diffusion
•Adolf Fick’s experiment: the spontaneous dispersion of
Stationary diffusion (dc/dt = 0: concentration does not change with time)
dye molecules in water
0
dm
dt
c
c2
x
dc
water
c1
x1 x2
dx
dye
dc
dx
Diffusion is the net transport of particles due to the inhomogeneity
of concentration or other factors (temperature, solubility).
Diffusion is propelled by Brownian motion (thermal motion).
Fick’s 2nd law
•Non-stationary diffusion in 1D
c = c(x,t): the concentration changes both in time and in space
"c
"t
Temporal
rate
of
concentration change at a
fixed position x (partial
derivative with respect to t)
D
x
" 2c
"x 2 t
second partial derivative
of c with respect to x in a
given moment t
Partial differential equation, ”first order” with respect to time and ”second
order” in space.
The solution c(x,t) of the differential equation depends on the initial
conditions and the geometry. No general analytical solution exists.
c( x 2 ) ! c( x1 )
x 2 ! x1
dc
dx
-surface area
-diffusion coefficient
-diffusion current: amount of matter
crossing the surface per unit time
x
•liquids: mixing in several weeks
•gases: several seconds
•Diffusion: transport of particles
! DA
-concentration gradient
along the x axes
D: amount of substance crossing unit surface area in unit time at
unit concentration gradient
[D] = m2/s (cm2/s)
Solution of Fick’s 2nd law:
free diffusion in 1D
If at t=0 every molecule is in a narrow range of width l around
the origin, the spatial distribution of the molecules is a
(continuously widening) Gaussian:
c # x, t $
c0 l
4%Dt
e
!
c
c0
x2
4 Dt
2& 2
l
t1
Variance (mean squared
displacement):
t2
!2 = < x2> = 2Dt
0
x
Time dependence of diffusing
motion: logarithmic plot
Time dependence of diffusing
motion: linear plot
• Mean squared displacement, < x2> of the molecules at time “t”,
the Einstein – Smoluchowski equation:
< x2> = 2Dt
2
• Diffusion in 3D:
< r > = < x2> + < y2> + < z2> = 6Dt
'x
2Dt
'x
vt
time
•Diffusion: a “random walk”
time
- Fast for short distances
- Slow for longer distances
Diffusion in different phases.
What does D depend on?
•Gases: The mean kinetic energy for one translational degree of
freedom: ½ mvx2 = ½ kT
D ( vx ( T m
•Macromolecules, colloidal particles in liquids
Stokes-Einstein equation (for spherical molecules):
D
kT
f
kT
6%)r
• f: form factor or coefficient of
friction – for elongated molecules
larger than for spherical molecules
• r: hydrated molecular radius,
r ( 3 MW
• ": viscosity of the medium
temperature dependence of D:
-direct (kT)
-indirect through viscosity (T increases ! " decreases)
1 cm -2
log x (m)
• Displacement for
straight line motion
• Root mean square (rms) displacement
0
-4
1 "m -6
-8
1"s 1ms 1s
1 nm
-10
-10 -8 -6 -4 -2 0 2
log t (s)
1day 1 year
4
6
8
•Sufficiently fast for molecular/cellular events
•Too slow for macroscopic transport ! need for circulation
The diffusion coefficient of some
particles
Medium
D(m2/s)
H 2O
H 2O
2.26×10-9
2 cm
K+
H 2O
1.96×10-9
1.8 cm
H+
H 2O
9.3×10-9
4 cm
1.5 cm
Diffusing substance (MW)
x (per day)
ethanol
H 2O
1.24×10-9
glycine (75)
H 2O
1.05×10-9
1.3 cm
saccharose (342)
H 2O
5×10-10
0.9 cm
ribonuclease (13 000)
H 2O
1.1×10-10
0.4 cm
serum albumin (69 000)
H 2O
6.1×10-11
0.3 cm
H 2O
2.2×10-11
0.19 cm
tobacco mosaic virus (40
million)
H 2O
3×10-12
0.07 cm
H2
air
6.4×10-5
330 cm
Tropomyosin (93 000)
Why is there a net flow of molecules?
Molecular interpretation
c1, N1 , V1
c2, N2 , V2
Gibbs free energy
According to the 2nd law of thermodynamics the total entropy of a system +
its environment never decreases. How could we find out the direction of a
process based on a single parameter of the system?
' S s + ' S env , 0
' S env
Assume each molecule can move in any direction (±x, ±y, ±z)
with equal probability: p = 1/6
Number of molecules moving from volume 1 to 2 and vice versa:
N(1#2) = 1/6 N1 = 1/6 c1V
N(2#1) = 1/6 N2 = 1/6 c2V
Net molecule transport from 1 into 2:
Qenv / T
! Qs / T
!' H s / T
'S s ! 'H s / T , 0
'H s ! T 'S s * 0
Gs
H s ! TS s , 'Gs * 0
Gs is the Gibbs free energy of the system: it never increases
The decrease of G is equivalent to the increase of total entropy
(system+environment)
#N(1!2) - #N(2!1) = 1/6 (c1- c2) V > 0
Why is there a net flow?
Thermodynamic interpretation
At constant T and p the Gibbs free energy of the system decreases
in spontaneously occurring processes.
G(T,p,N) = E + pV - TS, G $ 0
The chemical potential (Gibbs free energy per one mole of
molecules) is:
" = "0(T,p) + RT ln c
["0(T,p): chem. pot. of 1M solution]
G of the system can decrease if the molecules from the place of
higher chemical potential (higher c or T) move to the place of lower
chemical potential.
This is a statistical driving force, the decrease of G is equivalent to
the increase of %S, the increase of disorder.
Diffusion has pivotal role in living
organisms
• At the end of the transport chain molecules (nutrients,
metabolites, O2, CO2) move from the capillaries to the cells and
back by diffusion.
• Transport of excreted, secreted substances, hormones, drugs to
target cells in the tissues
• Transmembrane diffusion through and
diffusion in the plane of cellular membranes
lateral/rotational
• Intracellular diffusion, molecular recognition processes
Passive diffusion through the
membrane
Mechanisms of transmembrane diffusion:
•Membranes
represent
barriers
of
differential
permeability for different molecules – transmembrane
diffusion is selective
•Passive diffusion:
concentration
”downhill”,
toward
the
lower
•The plasma membrane is permeable
for small uncharged molecules
Permeability for a molecule
• decreases with size
• decreases with polarity or charge
• increases with solubility in lipids
•Facilitated diffusion: ”downhill”, takes place with the
assistance of transporter molecules
•Active transport: ”uphill” transport against concentration
gradient - requires energy (ATP)
Passive diffuson across the
membrane
Lipid
H2 O
alveolus
c1,w
c1,l
>1 c
2,l
c2,w
&<1
x
!
cl
cw
Partition coefficient
(a measure of hydrophobicity:
solubility in olive oil vs. water)
O2 CO2
alveolar endothelium
interstitial space
capillary endothelium
blood
•Passive diffusion
•Diffusional contact with the
alveolar space: ~0.3 s
1 !m
concentration
H2 O
Gas exchange in erythrocytes in
the lung
$x2 2Dt % t $x2 / 2D
2
dm
dt
PA#c1, w ! c2, w $
~Fick I.
P = D / x permeability coefficient
What does P depend on?
• (hydrophobic/hydrophilic character)
•D (molecular size and shape)
• x (thickness of the membrane)
DO2 10#9 m
erythrocyte
DCO2
s
2
6 &10#9 m
s
t O2
500"s
t CO2
80"s
Facilitated diffusion
Facilitated diffusion
Selective transport of specific molecules/ions with low lipid solubility
by transporter molecules (proteins called permeases, and ion
channels).
binding
Characteristics
•Faster than passive diffusion
•Selective
•Saturable
•Inhibitable
•No external energy is required, transport occurs from the high to the low
concentration
translocation
vmax
facilitated
½ vmax
passive
KM
v
Examples:
•D-glucose transporter in the erythrocyte membrane
•water transporter in kidney and bladder cells
•ion channels, though they are not saturable
vmax c
KM ' c
KM: Michaelis-constant
(conc. at which v = ½ vmax)
c
Measurement of lateral diffusion I.
Diffusion in the plane of the cell
membrane
•FRAP: (Fluorescence Recovery after Photobleaching)
•proteins are tagged with fluorescent antibodies, or fluorescent lipid analogues are
introduced into the membrane by incubation
•laser beam illuminates small area of the membrane and excites dye molecules; the
fluoresence intensity is measured by a CCD camera/PMT
•1000-fold laser intensity partially bleaches the fluorescence
•The time course of the recovery of fluorescence due to the diffusion of molecules
with intact dye labels into the observed area is recorded
•time constant: # ~1/D
•R: recovery fraction
(mobile fraction)
Dynamics of the cell membrane
•Membrane lipids and proteins perform
lateral (and rotational) diffusion
•viscosity: "membrane >> "water (200-1000×)
•Lipid domains of gel and liquid phase, of
different lipid and protein composition
•Membrane proteins can be associated to
one another, the extracellular matrix, the
cytoskeleton, or be sterically hindered in
motion by the underlying meshwork of
the
cytoskeleton:
the
“membrane
skeleton” – diffusion can be obstructed
•Proteins: D ~ 10-9-10-13 cm2/s
•Lipids: D ~ 10-8 cm2/s
Experimental techniques
measure lateral diffusion:
to
•Fluorescence
Recovery
After
Photobleaching (FRAP)
•Single particle tracking, single dye
tracking (SPT, SDT)
•Fluorescence correlation spectroscopy
(FCS)
Fluorescence intensity
v transport rate
dissociation
Fi
F(
Fi -F0
R
F( # F0
Fi # F0
F0
#
time
Measurement of lateral diffusion in
the cell membrane II.
SPT: (Single Particle Tracking)
•The proteins of interest are labeled with “Membrane
fluorescent antibodies/beads or 10-40 nm skeleton”
colloidal gold beads
•The trajectory of the particle is recorded by
CCD camera
•Membrane domains can be mapped
•Different types of diffusion can be
distinguished:
•free diffusion, <$r2> = 4Dt
•directed diffusion, <$r> ~ vt, <$r2>~v2t2
•hindered diffusion: <$r2> has a negative
deviation from linear time dependence
Single particle tracking 2.
Gold bead
$r 2 ~ v2t2
$r 2
Receptor
directed diffusion
free diffusion
$r 2 < 4Dt,
if t is large
“hindered diffusion”
1!m
$t (s )
See movie on the hindered diffusion of transferrin receptor (© Prof. Akihiro Kusumi, Nagoya University, Japan)
Axonal transport: directional motion
Axonal transport II.
Transport of vesicles in
the giant axon of squid
Fast: vesicles, 250 mm/day
Interm.: mitochondria, 50 mm/day
Slow: polymerized cytoskeletal
proteins, 0,2-1 mm/day
$r 2 = 4Dt