Diffusion and Macromolecular Movement

Diffusion and Macromolecular
Movement
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Diffusion: in dead and living
Role of diffusion
Microscopic and continuum descriptions
FRAP
Receptors and signalling: physical limits
Ion Channel Opening
•  Flow of ions into cell
•  Distribution of ions within
cell
Diffusion times
L2
t≈
D
Diffusion vs. Directed Motion
Protein in water D=100 µm2/s
Scales of E. coli Swimming
Diffusion Times
D=100
µm2/s
Passive vs. Active Transport
Kinesin transport rates
~ 1 µm/s
Single myosin speeds
~ 10 nm/s
FRAP
FRAP of KDEL-GFP
White & Stelzer (1999)
FLIP of KDEL-GFP
White & Stelzer (1999)
FCS Concept
Macroscopic Diffusion:
Concentration Field
ΔV small
volume
c(r,t)
Conc. Profile
Flux
Deriving Fick’s Law
Fick’s Law
First Equation
1D (along x) concentration profile
∂c
j = −D
∂x
j = current density per unit time
D = diffusion coefficient €
Fick’s Law
Second Equation
2
∂C
∂C
=D 2
∂t
∂x
€
Fick’s Law
In 3D
J = −D∇C
€
€
∂C
2
= D∇ C
∂t
Micro-Trajectories in Simple
Diffusion
a = Lattice spacing
Particle Displacement
Mean
Total steps N=t/Δt
Displacement Δxi , i=1,2,…N
Total displacement Δxtot=Δx1+Δx2+…ΔxN
<Δx>=a*kΔt+(-a)*kΔt+(0)*(1-2kΔt)=0
Variance (avg. sq. displacement)
<Δx2>=a2*kΔt+(-a)2*kΔt+(0)2*(1-2kΔt) = 2(a2k)Δt
Total displacement for N=t/Δt,
<Δx2tot> = 2(a2k)t
D=a2k
Probability density p(x,t) can be calculated
P(x,t+Δt)=(1-2kΔt)*p(x,t) + Stay put
kΔt*p(x-a,t) + Jump right
Jump left
kΔt*p(x+a,t)+
Markov Process
Can be used to arrive at Ficks Equations
Solutions of Fick’s Law
•  Carslaw and Jaeger (1959)
•  Crank (1975)
•  Jost (1960)
Pipette Experiment
Zigmond et al. (2001) Curr. Prot. Cell. Biol.
Point Source
Spike of concentration at origin at time t=0
C(r,t) =
N
−r 2 4 Dt
e
4 πDt
Gaussian distribution
Eg. Micropipette experiment
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Concentration Profile
FRAP of GFP in Bacterium
Mullineaux et al. (2006) First Postbleach image
t = 4s
Difference = IPostbleach-IPrebleach
Time Evolution of FRAP
a=L/2, setting different t values Solution to FRAP
FRAP curve
Normalized by
2c 0 a(1− a L)
Nf in the
bleached area
in the long
time limit
Predictions
Recovery is fastest when 2a = L/2
Recovery curves are identical for bleached
regions of fractional size a/L and (1-a/L)