Diffusion
Part 2
Diffusion part 1 recap
Recap
The diffusion equation in 1D is
∂2c
∂c
=D 2
∂t
∂x
This can be derived from either
kinetic gas theory, by thinking about flux and conservation of mass (continuity):
Jx = −D
dc
dx
∂Jx
∂c
=−
∂t
∂x
random walk theory, by considering N left- or rightward steps, taken with probability k, of size Δx)
t = 2N k
D ≡ k∆x2
This equation has free-space solutions that are spreading Gaussians (normal distributions):
c(x, t) = !
N
2πσ 2 (t)
2
e−x
/2σ 2 (t)
σ 2 (t) ≡ σ02 + 2Dt
Recap
These free-space solutions look like
c
1.0
0.8
0.6
0.4
0.2
!10
!5
Snapshots in time
5
10
x
Animation
Recap
Advection-diffusion
If there is drift in addition to diffusion, the equation becomes
∂c
∂2c
∂(vc)
=D 2 −
∂t
∂x
∂x
That drift may be caused by a driving force (for instance, a potential energy). This gives the
Smoluchowski equation:
!
∂p
∂ p
1 ∂
=D
+
∂t
∂x2
kB T ∂x
2
"
∂G
p
∂x
#$
Time scales in diffusion
Time scales in diffusion
Diffusion is an efficient mode of transport only over small distance scales
rms displacement !m"
1.
long neuron
ll
ma
typical neuron
ion
s
ein
0.001
t
pro
capillary
bacterium
1. ! 10"6
synaptic cleft
1.
1000.
1. ! 106
time !s"
r
s
ur
m
a
ye
0.001
y
da
1. ! 10"6
ho
1. ! 10"9
1. ! 10"9
Time scales in diffusion
Yet another derivation of law of diffusion
The Langevin equation
ma = −γv + f (t)
Newton’s 2nd law with a drag force and a random (thermal) force:
This has a solution with mean-square displacement
x2
!
"
#
x (t) = 2D (t − t0 ) + t0 e
2
kB T
D=
γ
−t/t0
m
t0 ≡
γ
$
Dt0
5
!
" D
x2 = t2
t0
4
3
2
!
"
x2 = 2D(t − t0 )
1
0
1
2
3
4
For t << t0, this gives ballistic motion: 〈x2〉 ∝ t2
For t >> t0, this gives regular diffusion solution 〈x2〉 ∝ t1
For a typical protein in water at room temperature, t0 ~ 1 ps, so the ballistic regime is totally negligible.
5
t!t0
Time scales in diffusion
Rotational Brownian motion
There are random (thermal) torques on a molecule as well as random (thermal) forces
Iα = −γR ω + τ (t)
By analogy to the Langevin equation, this causes rotational diffusion with a diffusion coefficient DR.
Ignoring t0, translation and rotational diffusion go like
! 2"
x = 2DT t
! 2"
θ = 2DR t
Consider a sphere of radius a
The translation and rotational drag coefficients γT and γR can be calculated
Fdrag = γT v = 6πηav
τdrag = γR ω = 8πηa3 ω
This gives expressions the translational and rotational diffusion coefficients
kB T
kB T
DT =
=
γT
6πηa
DR =
kB T
kB T
=
γR
8πηa3
We will use rotational diffusion to understand polarization anisotropy techniques
Time scales in diffusion
Membrane diffusion
An object diffusing in a membrane does regular 2D diffusion:
! 2"
ρ = 4DM t
Since it has contact with both the membrane and the bulk fluid, its diffusion coefficient
depends on both
!
"
ηM h
kB T
ln
− 0.5722
DM =
4πηM h
ηwater a
membrane thickness = object height
object radius
This assumes the object spans the membrane.
DM depends primarily on the membrane viscosity and the membrane thickness.
Boundary conditions
Boundary conditions (1D)
Non-free boundary conditions lead to new solutions
Absorbing wall on left. Reflecting wall on right.
c
c
1.0
1.0
0.8
0.6
0.8
flux in = flux out
0.6
∂c
=0
∂x
0.4
0.4
flux out = 0
0.2
c=0
!4
!2
2
4
0.2
x
!4
!2
Since there is an absorbing wall (at x=-5), the concentration will eventually vanish: c=0
2
4
x
Sources and sinks
Sources and sinks
Everything we’ve seen so far was headed to c=const (and actually, to c=0) eventually.
If you have an infinitely large reservoir however, diffusion won’t smear everything to zero.
If the concentration is pegged at zero, it’s a sink.
If the concentration is pegged at a positive value, it’s a source
Steady-state solution is
c=1
c
∂c
= 0 → J" = const → c ∼ kx
∂t
1.0
0.5
c will be linear in space but not constant.
c=0
!4
!2
2
!0.5
!1.0
In a strange geometry (like in a funnel) J=const but c ≠ kx
4
x
Diffusion to capture
The gambler’s ruin or Martingale
Diffusion to capture
Random walk with boundary conditions
Imagine an unbiased random walk with boundary conditions on the LH and RH sides:
The effect of these boundary conditions can be captured by setting c=0 on both boundaries: both act like perfectly
absorbing walls.
The boundaries are not equivalent to the drunk (one is death and one is a good night’s sleep), but encountering
either one will remove the walker permanently
The walker is removed the first time he encounters a boundary: this is called a first passage process.
Diffusion to capture
As before, the solution can be derived using a random walk model or by solving the
diffusion PDE.
I use the PDE solver in MATLAB to generate a movie of the probability distribution (or concentration) evolving as a
function of time between absorbing boundary conditions at x=0 and x=1.
Unfortunately, this doesn’t tell you whether your walker is absorbed to the left (death) or to the right (bed).
You can take your solution and integrate up the flux D ∂c/∂x at each boundary to find the total number absorbed.
Diffusion to capture
Instead of an absorbing wall, we can put a very deep well and then run the Smoluchowski
equation:
This effectively traps anything that hits the “wall” at x=0 or x=1 in the corresponding well.
It’s useful to see the total trapped population: here, 10:1 death:escape because the walk started at x=0.1
This doesn’t seem particularly convenient mathematically, but we will return to this idea when we talk about
molecular motors.
Diffusion to capture
Image solution
You can also derive the solution by using an “image
charge” type of procedure
c!x,0"
1.0
Start with free-space diffusion
To make a c=0 boundary, place an image diffusion profile
outside the range.
0.5
If you only have one boundary, you’re finished.
If you have two boundaries, the image of one boundary will perturb
the solution at the other boundary.
!4
!2
2
Place an image of the image to cancel this perturbation ...
x
4
!0.5
!1.0
Sum the contributions of each of these spreading normal
distributions to give the solution:
1.0
0.5
1.0
0.0
0.05
x
0.5
0.10
0.15
0.0
0.20
t
c!x,t"
Diffusion to capture
Gambler’s ruin
The odds of winning decline in direct proportion to the house limit
The expected take is constant (no net gain or loss) for a 50:50 game.
In particular, if you start with $1 to the house’s $N, the odds of going bust asymptote to (N-1)/N and the odds of winning
asymptote to 1/N.
If the house is infinitely rich, you cannot win (though if you did win, you’d become infinitely rich).
expected take
probability
1000:1 house:gambler
1.0
100:1 house:gambler
10:1 house:gambler
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
time
probability of winning/losing for an unbiased game
If the game is biased, as most games are, you have an overall drift towards going bust in addition to this.
Diffusion to capture
Roulette again
Roulette with only red/black (or similar) bets:
Biased random walk with probabilities pleft (losing) and pright (winning) for each round of betting.
pleft=20/38 and pright=18/38
After t rounds of betting, there is a probability p(n,t) that you have $n.
Δx=$1 and Δt=1 round
The mapping to PDE coefficients is
D
=
v
=
1 $2
pleft + pright ∆x2
=
2
∆t
2 round
2
2
$
∆x
(pright − pleft )
=−
∆t
38 round
Assume $1 bets, a starting stake of $S, and that the player will beat the house (or retire) at $H.
1 ∂2p
2 ∂p
∂p
=
−
∂t
2 ∂n2
38 ∂n
To compare different scenarios, use a normalized n’ = n/H, so that 0 < n’ < 1, with bust and break on the left and
right and a normalized t’= t/H2. Then the PDE is
1 ∂2p
2H ∂p
∂p
=
−
∂t!
2 ∂n!2
38 ∂n!
with initial conditions n’=S/H.
Diffusion to capture
Solutions look like:
distributions
break/bust for even odds
0.8
0.7
0.7
0.6
0.6
0.2
0.3
0.4
0.5
n'=$/H
0.6
0.7
0.8
0.9
1
fraction
probability
probability distribution
0.1
0.5
18:20 odds
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.5
1
rounds / H2
1.5
1:1 odds
999:1001 odds
0.5
0.4
0
bust
break
still playing
0.9
0.8
0
0
1
bust
break
still playing
0.9
Probability distributions after (1, 500, 1000, 1500, 2000) rounds of betting
$10 initial stake trying to make $100
gambler's ruin for $100 stake and $1000 cashout with 0, 2/38 and 0.1% bias
gambler's ruin for a 50:50 game with $400, $100 and $10 stakes and $1000 cashout
1
even odds
roulette odds
break/bust for biased games
0
0
0.1
0.2
0.3
0.4
0.5
0.6
2
rounds / H
0.7
0.8
0.9
1
Diffusion to capture
Time to capture
The unbiased gambler’s ruin is the same as diffusion to capture
Capture probability → 100% for a 1D semi-infinite interval.
The majority of captures occur pretty quickly, but t → ∞ for the last fraction.
Searching in N dimensions
1D
Searcher always finds point target
2D search
Searcher always find point target
3D search
Finite chance that searcher will never find point target, even given infinite time.
Proteins searching for a specific site on DNA (e.g. restriction enzymes, RNA polymerase) break a 3D search into
two lower-D searches
First bind nonspecifically to DNA (approximately a 2D search)
Then slide along the DNA (1D search) to find the site.
This is ~100x faster than 3D search for typical cell conditions
Since 1D search becomes less efficient in time (it keeps revisiting the same locations), in practice proteins fall off, diffuse to new part of
DNA, and rebind.
This search/release strategy is particularly efficient if the DNA is randomly coiled, because sections of DNA that are distant in 1D might
be physically close in 3D.
Diffusion-limited uptake
model cell
Figure 13.21: Cartoon representing the chemoreception process. The
is peppered with receptors. For the purposes of simple analytical c
we idealize a bacterium as a sphere with a uniform density of recept
Diffusion-limited uptake
Diffusion can impose fundamental limits on microscopic processes.
Diffusive uptake by spherical “bacterium”:
Model the cell as a perfectly absorbing sphere of radius a, consuming nutrient from a very large bath.
Overall picture is very insensitive to exact shape of bacterium.
A “very large” bath has c → c0 as r → ∞.
The diffusion equation in 3D is
!
"
∂c
" · −D∇c
"
= −∇
∂t
Steady-state solutions must have constant flux through any spherical surface:
"
!
2 ∂c
= F0
F = −D 4πr
∂r
c(r)
c0
which has the solution
c(r, t) = c0
a"
1−
r
!
F = −4πDac0
c(a)
r
cell
signaling
molecule
receptor
Figure 13.22: Concentration profile in the neighborhood of a sph
The concentration profile c(r) is spherically symmetric and charac
concentration of ligands as a function of distance from the cell surfa
Diffusion-limited uptake
This is a fundamental limit.
No surface chemistry / transport process / metabolic process can increase it.
Swimming can ...
Just moving around won’t help much. If you’re living in a medium with uniform concentration of nutrient, then swimming with a speed
v can boost your total intake (flux) to
Acrivos and Taylor 1962:
1
1
Sh = 1 + Pe + Pe2 ln Pe
2
2
Sh is the “Sherwood number”:
Sh ≡
F
4πDac0
Pe is the “Peclet number”:
Pe ≡
va
D
For a typical small cell, Pe ~ 0.01, so the movement-enhanced nutrient flux is only 0.5% higher than the “free” flux from simply sitting
and waiting for food to diffuse to you. This doesn’t seem worth it.
... but you need to actively seek out regions of higher c0 if you want a better life.
This is called chemotaxis.
Alternatively, you can position yourself next to a fresh, high-c stream of nutrients (like blood)
Diffusion-limited uptake
Actually, a cell can absorb close to Flimit = 4πDac0 using only a fraction of its surface area.
Physics says that
A small patch of radius s absorbs a flux F1 = 4Dsc0.
You’d think that N such patches would absorb a flux N F1, but nearby patches interfere with each other, so
1
FN
! "
=
π
Flimit
1 + N as
These patches occupy a fractional area
N ! s "2
AN
=
Atotal
4 a
Since FN ∝ N s–1 but AN ∝ N s–2, many small patches will give close to Flimit absorption but occupy only a small
fraction of the total surface area.
The cell can
use the rest of the surface for other types of receptors
make do with imperfectly absorbing receptors at very little cost in terms of uptake rate.
This argument applies adsorption (pores, transport channels, receptors) and emission (vesicle release):
Any compact object can achieve close to maximum diffusive exchange by using many small sites covering only a fraction of its
total surface area.
Diffusion-limited uptake
With reasonable numbers,
s=10 Å and a=0.5 µm (typical pore and small bacterium)
F!N!Flimit
1.0
0.8
0.6
0.4
0.2
1
10
100
1000
104
105
1000 pores = 0.1% area
N
Diffusion-limited uptake
Why is partial coverage so efficient?
A typical random walk explores local space thoroughly before moving off, so
a small target works almost as well as a large target.
to avoid competition, it’s better to have several small targets “far” (in diffusive terms) apart.
This is the same reason why it’s better to explore randomly for a while and then move on a large distance before
searching again.
Chemotaxis
Chemotaxis
Peritrichously flagellated bacteria (such as Escherichia coli and Salmonella) typically have:
cell body: 1 x 4 µm
6-10 flagella: 20 nm x 10 µm
individual reversible rotary motors
Salmonella typhimurium
Electron micrograph from
Ingraham/Low/Magasanik/Schaechter/Umbarger
Escherichia coli and Salmonella Typhimurium
Escherichia coli
roughly constant swimming speed
Chemotaxis
Cells actively control their bias while performing a biased random walk
early experiments were done on populations (like real diffusion)
now usually done with individuals (like random walk)
Bacterial behavior
Berg and Brown, Nature, (1972)
Chemotaxis
4
5
3
6
2
1
7
CCW
Old run
CW
Tumble
CCW
New run
8
Chemotaxis
Distribution of run and
tumble intervals:
ADAPTATION IN BACTERIAL CHEMOTAXIS
VOL. 154, 1983
Run and tumble intervals are
exponentially distributed.
]
60ao B
300 A
l
VI
>
200-
Attractant lengthens the average run.
cw
ccw
400-
kA
K
x
-
Since v~constant, step size is
exponentially distributed.
315
E 100
z
200-
-bS
K
O
I
2
3
.
4
5
6
Time(sec)
78
96 o
2
3
4
5
6
Time (sec)
7
8
9
0
FIG. 2. CW and CCW interval distributions of adapted cells. Histograms for each record were scaled,
combined, and fit by an exponential, as described in the text. (A) CW interval distribution computed from the
5,237 events longer than 0.4 s in 108 records on 24 cells; 4 events are off the scale. Range of adjusted means for
each record, 0.14 to 4.4 s; global adjusted mean, 1.06 s; decay time for the exponential fit, 1.33 s; reduced x2'
1.06; 51 degrees of freedom; P value, 36%. (B) CCW interval distribution computed from the 7,255 events longer
than 0.4 s in 108 records on 24 cells; 13 events are off the scale. Range of adjusted means for each record, 0.47 to
VOL.s; 154,
5.2
global1983
adjusted mean, 1.20 s; decay time forADAPTATION
the exponential IN
1.22 s; reducedCHEMOTAXIS
fit, BACTERIAL
of
x2, 1.05; 54 degrees 317
Berg 1983
freedom; P value, 37%.
Baseline
cw
8
cw
spline-fit smoothing routine (24, 25), which smooths tion,
a type of behavior specified by the Weber6~0
the fit curve in a-4 least-squares sense and generates Fechner law (11). We tested exponential ramps
coefficients that define the derivative. For the deriva- of the form exp(at), with ramp rates, a, of either
4
tive to be well -8behaved, the smoothing must span
and exponentiated sine waves of the form
sign,
several adjacent Erotation intervals. Thus, although the
.0I
exp[sin(wt)].
These stimuli generate changes in
z an estimate of the rotational bias at
derivative provides
2
time t, its value depends on the behavior of the cell at P that are linear and sinusoidal, respectively.
Behavior at fixed concentration. The cells were
adjacent times. As a result, abrupt changes in bias are
allowed
to2 adapt
rounded off.
3 4 to
0
5 C0ow
6 7 or
8 Chigh
9 10 for at least 5
2 3
The reversal rate was computed Time
from(sec)
the density of min before dataTimve
were
(sec) taken. They were monidata points, as described by Block et al. (8). The tored for 3 to 5 min before and immediately after
FIG. 5. CW
and CCW
interval
distributions ofmodcells during an exponential ramp up. Histograms for 13 records
Montecarlo
simulation
of the
response-regulator
eachcombined,
ramp. The
interval
distributions
for data
cells
exposed
to ramps
of rate 0.013
as described
in
s-1 were scaled,
and fit
by exponentials,
elfrom
wassix
done
by
for
a counter
arranging
(representing
obtained
before
the
ramps
are
shown
in
Fig. for
2.
the
text.
(A)
CW
interval
distribution
computed
from
534
events
than
s.
of
means
longer
0.4
Range
adjusted
the amount of regulator, X) to be incremented by one These distributions were accurately fit by single
each record,process
0.30 toand
1.5decremented
s; global adjusted
mean,with
0.69 s; decay time for the exponential fit, 0.78 s, reduced x2',
exponential
another,
exponentials.
0.913; 17 degrees
interval
of freedom;
P value,by56%.
than
(B)generafromdistributions
795 events longer
CCW interval
at Clow
distribution The
computed
probabilities
obtained
from
random-number
0.4 s;The
32 events
are kept
off thetrack
scale.ofRange
adjustedformeanswere
for each
record, 0.75 to 12.0
indistinguishable
froms; global
those adjusted
at Chigh mean,
(data
tion.
program
those of
intervals
2.38 s;thedecay
timewas
for above
the exponential
1.99 s;value
reducednot
1.18; 19 degrees
of freedom;
27%.adapt.
value,
(Not
x2, shown),
that the Pcells
indicating
fully
which
counter
or below afit,
critical
shown) CW and
CCW and
interval
distributions
for the same The
each ramp.
CW globalperfused
cells before
adjusted mean
cells were
continuously
with
to XCnt)
(corresponding
the correspondcompiled
computed
740 events longer than 0.04 s in 13 records,buffer
meanenergy
from [1])
948
0.76 s; containing
CCW global lactate
adjusted (an
computed
ing
intervalfrom
histograms.
source
events longer than 0.4 s in 13 records, 0.89 s. Exponential
to these data.
were not made (required
andfits L-methionine
for adaptation in
S!
With attractant
RESULTS
-j-ofree-swimIt is1.0known from earlier work with
A
cells that do not synthesize it [27]). They were
in theoxygenated
mean CCWbut
interval
was of
greater
the
well
deprived
otherthan
amino
Downloaded from jb.asm.org at Harvard Libraries on June
Downloaded
18, 2007from jb
A
Chemotaxis
v2 τ
The effective diffusion coefficient would be D =
3
if the trajectory were equivalent to a random walk.
2800
TURNER ET AL.
J. BACTERIOL.
θ
However, the tumble doesn’t
fully randomize direction:
compliance might make the curly filament bend (literally) to
the will of the other filaments, and thus prevent it from interfering with the forward motion of the cell body that occurs
after the cell has altered course, but before the bundle has
consolidated.
We have not studied S. enterica serovar Typhimurium as
extensively as E. coli, but tumbles appear to involve the same
transformations in either organism. In particular, we do see
normal-to-semicoiled transformations in Salmonella. These
were not observed in dark field in the initial work of Macnab
and Ornston (26) or in later studies in which video recordings
were made with a silicon-intensified-target camera (17). We
2
suspect that this discrepancy is due to the fact that the normalto-semicoiled transformation shortens the filament (compare
fields 2 and 10 in Fig. 6) so that the semicoiled form is hidden
by light scattered by the cell body.
S. enterica serovar Typhimurium does appear to have more
flagella than E. coli. Iino (14) shows distributions for cells of
this species with a mean of between 6 and 7; for cells of E. coli
labeled with Alexa Fluor 532, we found a mean of 3.4. If the
trend evident in Fig. 12D holds for Salmonella, then one might
expect to see fewer tumbles that involve relatively small numbers of filaments. This might have contributed to the impression that tumbles involve all of the filaments in a bundle.
Other vistas. We have not looked at the behavior of cells
with reduced numbers of filaments, with mutant flagellar filaments, or with normal filaments in highly viscous media. For
example, more might be learned about correlations between
changes in the direction of the cell body and filament polymorphic form if there were fewer filaments to complicate the issue.
Nor have we looked beyond E. coli, S. enterica serovar Typhi2 strain. Since labeling with
murium, or a motile Streptococcus
the Alexa Fluor succinimidyl esters rendered the latter cells
immotile, presumably because the gram-positive cell wall is
permeable to these reagents, the use of this technique in such
species might be limited to determinations of flagellar number
and morphology. But even this is useful. The labeling technique is so simple, and the images are so vivid, even when seen
with an ordinary fluorescence microscope, that the world of the
flagellum is now more accessible.
so D =
FIG. 13. Polar plots of the change in direction from run to run as a function
of fraction of filaments out of the bundle (A) and fraction of filaments remaining
in the bundle (B) (both plotted radially). Data are for cells with one to six
filaments.
v τ
3(1 − "cos θ#)
With v ~ 30 um/s, τ ~ 1 sec and <cosθ> ~ 0.33, D ~ 400 µm /s
68 ! 36° rather than 90 ! 39°, with the distribution for sudden
changes in direction peaking even more sharply at 62 ! 26°).
Later, it was recognized that runs occur when flagella spin
CCW, and tumbles occur when they spin CW (19). However,
the latter experiments were done with tethered cells, where
one looks at only one flagellar motor at a time. Following the
seminal work of Macnab and Ornston (26) described in the
introduction, it was commonly thought that all of the motors
Chemotaxis
Active control over diffusion coefficient allows cells to swim up a
gradient
Drift velocity is only 1/10 swimming velocity, so there is only a small excess of cells
swimming in “right” direction compared to “wrong” directions.
Chemotaxis is a statistical phenomenon: not deterministic
Rotational Brownian motion predicts decorrelation of direction over ~3 s, so it’s impossible
for a bacterium to swim in a straight line for more than a few seconds anyway
Many other bacteria do run-tumble chemotaxis, though the details may differ.
To be added
To be added
Diffusion-limited aggregation
collapse / rescue of actin filaments (PBoC)