Diffusion in the cell
Single particle (random walk)
Microscopic view
Macroscopic view
Measuring diffusion
Diffusion occurs via Brownian
motion (passive)
Ex.: D = 7 μm2/s for GFP in E. coli
!
t ~ L2/D
time to cross cell (1 μm): ~0.14 s (very
slow!)
PBoC 13.1.2
Measuring diffusion
1D: <r2> = 2Dt
FRAP experiment
2D: <r2> = 4Dt
FRAP experiment
3D: <r2> = 6Dt
PBoC 13.1.2
Fick’s First Law
∂c
j = −D
∂x
dc/dx < 0 ➔ j > 0
!
particles flow from high to low
concentration!
PBoC 13.2
Diffusion Equation
∂c
∂Nbox
=
∆x∆y∆z
∂t
∂t
∂j
∂c
=−
∂t
∂x
1D diffusion equation (Fick’s
second law)
conservation
of mass
∂2c
∂c
=D 2
∂t
∂x
PBoC 13.2
Stochastic approach: Brownian motion
slow, deterministic forces
Langevin Equation
(1908)
rapid, stochastic forces
mẍ = −γ ẋ + σξ(t)
⟨ξ(t)⟩ = 0
Gaussian “white”
⟨ξ(t1 )ξ(t0 )⟩ = δ(t1 − t0 ) noise
Going from stochastic to deterministic
mẍ = −γ ẋ + σξ(t)
|γ ẋ| >> |mẍ| → γ ẋ = σξ(t)
(Limit of strong friction)
Every stochastic process has
2
2
∂
σ
a corresponding Fokker∂t p(x, t|x0 , t0 ) = 2 2 p(x, t|x0 , t0 )
Planck equation on its
2γ ∂x
probability
conditional probability (Markov process)
initial condition: p(x, t → t0 |x0 , t0 ) = δ(x − x0 )
boundary condition: p(x → ∞, t|x0 , t0 ) = 0
Solution:
!
1
(x−x0 )2
− 4D(t−t )
0
p(x, t|x0 , t0 ) = !
e
4πD(t − t0 )
(Green’s function)
σ2
(D = 2 )
2γ
Diffusion Equation
N
−x2 /4Dt
c(x, t) = √
e
4πDt
PBoC 13.2.2
Diffusion Equation
Fluorescence recovery after photobleaching (FRAP)
PBoC 13.2.3
Diffusion in the presence of a force
PBoC 13.2.5
dc
J2 = −D
dx
F
J1 = vc = c
γ
F
dc
+ c
J = −D
dx
γ
F
dc
= c
D
dx
γ
steady state, J(x) = 0 (no flux)
−(U (x)−U (0))/γD
c(x) = c(0)e
Einstein
D = kT /γ relation
Fluctuation-dissipation theorem
Langevin equation
γ ẋ = F (x) + σξ(t)
∂ F (x)
∂2
)p(x, t|x0 , t0 )
∂t p(x, t|x0 , t0 ) = (D 2 −
∂x
∂x γ
corresponding Fokker-Planck equation
(Smoluchowski Equation)
D = kT /γ Einstein relation (previous slide)
σ2
D= 2
2γ
Therefore:
2
= 2kT
from Langevin equation
Dissipative force (friction) derives from
random fluctuations!
Ex: membrane-protein insertion
Model growth of nascent protein as a freely-jointed
chain
r
2
2
= N r L ; N r = t/τ
Range of TM helix in membrane due to ribosome
tether grows with √t as a function of synthesis rate,
subject to potential:
αr 2
3kT τ
U ( r, t ) = U 1 ( r ) + U 2 ( r, t ) = U 1 ( r ) +
;α =
t
2 L2
U1(r) comes from thermodynamics
U2(r,t) comes from ribosome tether
diffusion “simulations”: ∂ t p(r,t) = ∇·De−βU(r,t)∇eβU(r,t)p(r,t)
irreversible
“COMMITMENT”
Novel diffusion-elongation model of
thermodynamically AND kinetically driven
membrane insertion
J. Gumbart et al. (2013) JACS. 135:2291-2297.
Ex: membrane-protein insertion
force?
Recent cryo-EM based structure
shows part of the nascent protein
unfolded between ribosome and
channel, just as predicted
Structure of the SecY channel during initiation
of protein translocation. E Park, JF Ménétret, JC
Gumbart, SJ Ludtke, W Li, A Whynot, TA Rapoport
and CW Akey. Nature, 506:102-106, 2014.
force?
(Perfectly) Absorbing sphere
dn
Derive
= 4πDac0 assuming
dt
PBoC 13.3.1
c(a) = 0 (perfect absorber)
∂c
c(∞) = c0
=0
∂t
(Finitely) absorbing sphere
# of receptors
absorption rate
dn
= M kon c(a)
dt
Finite rate of absorption
c0
c(a) =
1 + M kon /4πDa
kon → 0 ?
kon → ∞ ?
M kon c0
dn
=
≤ 4πDac0
dt
1 + M kon /4πDa
rate limited
PBoC 13.3.1
Optimal number of receptors
Assume 90% of maximum (diffusionlimited) is sufficient for the cell
dn
= 0.9(4πDac0 )
dt
M kon c0
dn
= 0.9(4πDac0 ) =
dt
1 + M kon /4πDa
0.1M kon = 0.9(4πDa)
If each receptor is 10 nm2 and the
cell surface is 1200 μm2...
4πDa
5
M =9
≈ 9 × 10 receptors
kon
only 0.0075 of the surface needs
to be covered! (more receptors
doesn’t help)
PBoC 13.3.1
Real chemoreceptors
cryo tomogram of
receptor array
MDFF fit of atomic
structure to averaged map
Briegel, Ames, Gumbart et al. The mobility of two kinase domains in the Escherichia coli
chemoreceptor array varies with signalling state (2013) Mol Microbio. 89:831-841.