Diffusion and Transport 13. Passive and Active Transport Passive transport refers to diffusion for which thermal energy is the only source of motion.
x = 0  r 2 1/2  6Dt
rrms  t For spherical diffuser
D  k BT / 6R

 r 2 1/2  R 1/2 In biological systems, diffusive transport may work
on a short scale, but for long ranges, we need
directed transport
x
Directed
Diffusive
r21/2 for small protein moving in water
~10 nm →10-7 s
t
~10 μm → 10-1 s
Active transport refers to directed motion:
r  vt
r t
This requires an input of energy into the system, however, we must still deal with random
thermal fluctuations.
How do you speed up transport? 


Reduce dimensionality: Facilitated diffusion
Free energy (chemical potential) gradient: Diffusion with drift
Directional: Requires input of energy, which drives the switching between two
conformational states of the moving particle tied to translation.
Dimensionality Reduction One approach that does not require energy input works by
recognizing that displacement is faster in systems with reduced
dimensionality.
Example 2b
L
How fast do molecules find a small target with radius b centered in
a spherical volume with radius L, where L ≫ b. If the molecules are
initially uniformly distributed, the mean time to reach the target is1
1.
O. G. Berg and P. H. von Hippel, Diffusion-controlled macromolecular interactions, Annu. Rev. Biophys.
Biophys. Chem. 14, 131-158 (1985); H. C. Berg and E. M. Purcell, Physics of chemoreception, Biophys. J. 20,
193-219 (1977).
Andrei Tokmakoff 8/9/2016 3 D 
L2  L 
 
3D3  b 
2 D 
L2
L
ln  
2 D2  b 
1D 
L2
3D1
Here Dn are the diffusion constants in n dimensions (cm2/sec), and are not very different in
magnitude. Note the big difference in the last factor L/b ≫ ln(L/b) > 1. Based on the volume that
needs searching, there can be a tremendous advantage to lowering the dimensionality.
Facilitated Diffusion2 Example: E. coli Lac repressor Experiments by Riggs et al. showed that E. coli Lac repressor finds its binding site about
one hundred times faster than expected by 3D diffusion.3 They measured ka=7×109 M-1 s1
, and calculated the diffusion limited association rate from the Smoluchowski equation
as ka≈108 M-1 s-1 using estimated values of D≈5×10-7 cm2 s-1 and R≈5×10-8 cm. Berg and
von Hippel theoretically described the possible ways in which non-specific binding to
DNA enabled more efficient one-dimensional motion coupled to three dimensional
transport.4
Many possibilities for locating targets diffusively: Coupled 1D + 3D Diffusion 1)
2)
3)
4)
2.
3.
4.
Sliding (1D diffusion along chain as a result of nonspecific interaction)
Microhop (local translocation with free diffusion)
Macrohop (...to distal segment via free diffusion)
Intersegmental transfer at crossing—varies with DNA dynamics
P. H. von Hippel and O. G. Berg, Facilitated target location in biological systems, J. Biol. Chem. 264 (2), 675678 (1989).
A. D. Riggs, S. Bourgeois and M. Cohn, The lac represser-operator interaction, J. Mol. Biol. 53 (3), 401-417
(1970); Y. M. Wang, R. H. Austin and E. C. Cox, Single molecule measurements of repressor protein 1D
diffusion on DNA, Phys. Rev. Lett. 97 (4), 048302 (2006).
O. G. Berg, R. B. Winter and P. H. Von Hippel, Diffusion-driven mechanisms of protein translocation on
nucleic acids. 1. Models and theory, Biochemistry 20 (24), 6929-6948 (1981).
2
Consider Coupled Sliding and Diffusion The transcription factor diffuses in 1D (or 2D) along DNA with the objective of locating a
specific binding site. The association of the protein and DNA at all points is governed by a
nonspecific interaction. Sliding requires a balance of non-specific attractive forces that are not
too strong (or the protein won’t move) or too weak (or it won’t stay bound). The nonspecific
interaction is governed by an equilibrium constant and exchange rates between the bound and
free forms:
ka

F 
B
kd
The protein stays bound for a period of time dictated by the dissociation rate kd. It can then
diffuse in 3D until reaching a contact with DNA again, at a point which may be short range in
distance but widely separated in sequence.
1/2
 4D 
R*   1 
 kd 
3
The target for the transcription factor search can be much larger that the physical size of the
operator. Since the 1D sliding is the efficient route to finding the binding site, the target size is
effectively covered by the mean 1D diffusion length of the protein, that is, the average distance
over which the protein will diffuse in 1D before it dissociates. Since one can express the average
time that a protein remains bound as kd1 , the target will have length along the DNA contour of
1/2
 4D 
R*   1 
 kd 
If the DNA is treated as an infinitely long cylinder with radius b, and the protein is considered to
a uniform probability of non-specifically associating along any part of the DNA. Then one can
solve for the steady-state solution for diffusion equation, assuming a completely absorbing
target. The rate constant for specific binding to the operator has been determined as
kbind
 D K   D K  
 4D3b  1 ln  1  
 D3b  D3b  
1/2
where K  is the equilibrium constant for nonspecific binding per unit surface area of the cylinder.
The association rate will be given by the product of kbind and the concentration of protein.
Diffusion with Drift and Biased Random Walk As we saw in our discussion of diffusion with drift, if there is a force acting on the system, it is
possible to move quasi-deterministically. That is, diffusive motion is superimposed on an overall
drift velocity for the concentration gradient.
C  x, t  
2
1
e( xvt ) 4 Dt 4Dt
The peak of the distribution moves as x =vt and the variance in
the width grows as x2 1/2 = 2Dt.
The diffusion with drift equation emerges from a biased
random walk. Extending earlier model—we have a walker on a 1D lattice which can step left or
right by an amount distance Δx for every time interval Δt. However, there is unequal probability
(P±) of walking right (+) or left (–) during Δt. The change in position for a given time interval is
x  t  t   x  t   x P
 x  t   x t k 
(0)
Here we have defined k± as the rate constant for stepping: P± = k±Δt with k  k– . Since
P  P  1
k  k – 
1
t
(0)
4
How does the average position evolve? Performing an ensemble average over eq. (0)
x  t   t   x (t )  ( k   k  )  t  x
(0)
 x (t )  v  t
where the drift velocity is given by difference in hopping rates
v   k  k  x Eq. (0) says that the mean of the distribution has an equation of motion that looks like traditional
linear motion: x(t) = x0 + vt.
What about the variance?
2
x 2  t  t   x 2 (t )  2x t k x(t )   k  k  x 2 t 2
 x 2 (t )  2vt x(t )   k  k  x 2 t 2
2
(0)
Comparing eq. (0) with x2 1/2 = 2Dt, remembering from our earlier random walk result that 2D
= ∆x2/∆t, and using (k+ + k–)∆t = 1 leads to the conclusion that the diffusion coefficient is
D
1
 k  k  x 2
2
What leads to biased diffusion? Gradient in free energy in direction x is a force: f  G / x (or more accurately
f  i / x where μi is the chemical potential for species i. Let’s look at the energy
landscape for the biased random jump model
Let’s approximate:
G = –kBT ln P(x,t)
f 
G
P( x  x)
 kBT ln
 kBT ln K step
x
P( x)
Kstep 
k P
 k– P–
5
A free energy gradient cannot exist in an equilibrium system. It is best to consider this a steady
state solution (dP(x)/dt = 0). Active Transport Active transport requires the input of energy to get directed motion. Many proteins act as
molecular motors/engines using an energy source to move themselves or cargo in space. These
take energy in chemical bonds, charge transfer or concentration gradients. Even with this input of
energy, fluctuations and Brownian motion remain very important.
Chemical Energy Sources Chemical energy 

High energy bonds
Redox cofactors
ATP  ADP  Pi
NADH  NAD  2 H   2e
FADH 2  FAD  2 H   2e
Chemical potential gradients 

Membrane potential
Proton gradients
Motor Proteins Create directed motion by coupling energy use to conformational change.
Motor Classes 1) Translational
 Cytoskeletal motors that step along filaments (actin, microtubules)
 Helicase translation along DNA
2) Rotary
 ATP synthase
 Flagellar motors
3) Polymerization
 Cell motility
4) Translocation
 DNA packaging in viral capsids
 Transport of polypeptides across membranes
6
Translational Motors Processivity 

Some motors stay on fixed track for numerous cycles
Others bind/unbind often—mixing stepping and diffusion
Cytoskeletal motors 



Used to move vesicles and displace one filament relative to another
Move along filaments—tracks have polarity (±)
Steps of fixed size
Use free energy of ATP hydrolysis
o ~50 kJ/mol
o ~20 kT
o ~80 pN/nm
Classes 


Dynein
Kinesin
Myosin
moves on
Microtubules (+ → ‒)
Microtubules (mostly ‒ → +)
Actin Brownian Ratchet5 A useful example for understanding motor proteins stepping from one position to another. There
are many examples of Brownian ratchets coming from all families of motor proteins.
Directed motion requires switching between at least two states that are coupled to the
motion, and for which the exchange is driven by input energy. Switching between states results
in biased diffusion. Some people consider this cycle as very deterministic, whereas others as
quite random and noisy. More and more evidence that noise/Brownian motion is being exploited
to an advantage.
We’ll consider an example relevant to the ATP fueled stepping of cytoskeletal motors.
The motor cycles between two states: (1) a bound state, for which the protein must bind ATP in
order to associate to a particular site on the filament, and (2) a free state resulting from ATP
hydrolysis, for which the protein freely diffuses along the filament. The bound state is described
by a periodic, spatially asymmetric energy profile UB(x), for which the protein localizes to a
particular energy minimum along the filament. Although the energy varies periodically in space
along the direction of transport, the shape of this periodic curve are determined by the binding of
ATP to the motor. In the free state, there are no barriers to motion and the protein diffuses freely.
When the free protein binds another ATP, it returns to UB(x) and drops to the nearest energy
minimum.
5.
K. Dill and S. Bromberg, Molecular Driving Forces: Statistical Thermodynamics in Biology, Chemistry,
Physics, and Nanoscience. (Taylor & Francis Group, New York, 2010); R. Phillips, J. Kondev, J. Theriot and
H. Garcia, Physical Biology of the Cell, 2nd ed. (Taylor & Francis Group, New York, 2012).
7
Let’s calculate the velocity of the Brownian ratchet. The important parameters for our model are:
 The distance between binding sites is Δx
 The position of the forward barrier relative to the binding site is xf. A barrier for
backward diffusion is at –xr, so that
x f  xr  x
The asymmetry of UB is described by
  ( x f – xr ) / x

The average time that a ratchet stays free or bound are F and B . Therefore, the average
time per bind/release cycle is
t  F  B

The diffusion length is determined from the time that the protein is free
 0  4 D F
Although the free state results in most of the forward motion, we don’t want the protein
to stay free too long. If  0  x , the influence of the asymmetry in UB is greatly
2
reduced. We would like F  x / D .
We can now calculate the probability of diffusing forward over the barrier during the free
interval:
P 

1
4D F


xf
e x
2
 20
 xf 
1
erfc  
2
 0 
And, similarly, the probability for diffusing backward over the barrier at x‒xr is
8
x 
1
P–  erfc  r 
2
 0 
Now we can determine the average velocity of the protein by calculating the average
displacement in a given time step. The average displacement is the difference in probability for
taking a forward versus a reverse step, times the step size. This displacement occurs during the
time interval Δt. Therefore
v

Px
t
 P  P  x
 B  F 
We can also come up with a kinetic scheme: Bn:
Fn :
Bound at site n with ATP
Free at site n following hydrolysis
k
Hy
Bn 
Fn  ADP  P
k
diff
Fn 
n 1 Fn 1  n Fn  n 1 Fn 1
kbind
Fn  ATP 
 Bn

So how does the ATP hydrolysis influence the free energy gradient? Here free energy gradient is
GHyd .
.
x

k  A e
 Gbarrier Ghydrolysis
k  A e
 Gbarrier  kT

kT
v   k   k   x
9