Transcript Transport Processes - Tunghai University

Transport Processes
2005/4/24
Dept. Physics, Tunghai Univ.
Biophysics‧C. T. Shih
Movement
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Movement is one of the defining
characteristics of the biological world
Movement of different scales:
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Motion of the entire organisms
Motion of the internal organs
Motion of a cell
Motion within a cell
Molecular motions
Diffusion – Passive Transport
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Motion of particles through space
Mixing of particles amongst one
another – usually for biological systems
Brownian motion:
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Discovered by botanist Robert Brown in
1828 – pollen particles in water
Explained by Albert Einstein in 1905
Brown, R. "A Brief Account of
Microscopical Observations
Made in the Months on June,
July, and August, 1827, on the
Particles Contained in the
Pollen of Plants; and on the
General Existence of Active
Molecules in Organic and
Inorganic Bodies." Phil. Mag. 4,
161-173, 1828.
Einstein, A. "Über die von
der molekularkinetischen
Theorie der Wärme
geforderte Bewegung von in
ruhenden Flüssigkeiten
suspendierten Teilchen."
Ann. Phys. 17, 549, 1905.
Brownian Motion
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Einstein showed that Brownian motion is caused
by the impacts on the pollen particles of water
molecules
The process can be characterized by a series of
random walk
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The trajectory of a particle can be divided into small
straight paths
The particle jumps from point to point
Each jump is independent of the history of
movement – the process is stochastic (Markov chain)
Random Walk
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For the random walk by m steps without
directional bias:
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Unit length of each step
Mean distance <r>=0
Root mean square distance rrms=√m
For a stochastic Brownian motion,
rrms  Dt
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D: diffusion coefficient, net flow of particles per
unit time per unit area (perpendicular to the
direction of flow)
Diffusion Driven by
Density Difference
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J
One dimension diffusion, at a certain time
b: the length of a single diffusive jump
C(x): number of particles per unit length
f: frequency of jumps
J: flux of particles (per unit area per unit time)
Fick’s First Law:
1
J  f (C ( x)  C ( x  b)) b   DdC / dx; D  b 2 f / 2
2
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The particles move from the region of higher density
toward the lower one
Time-Dependent Diffusion
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The net flux into the region between x and
x+L is J(x+L)-J(x), where
J ( x)   DdC / dx
J ( x  L) ~ J ( x)  LdJ ( x) / dx   DdC / dx  Ld ( DdC / dx) / dx
  C 
C / t   D

x  x 
 2C
 D 2 if D is constant
x
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Fick’s Second Law
Solution of Fick’s 2nd Equation
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Separate the variable: C(x,t)=S(x)T(t)
and the equation becomes (let D=a2)
 S
T
a  S 1 T
a T 2 S


2
x
t
S x
T t
2
2
2
2
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LFS: depends only on x; RHS: depends
only on t
The only possibility is that both sides
equal to a constant, say –a2k2
Solution of Fick’s 2nd Equation
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The general form of solution:
Tk (t )  A1k exp(a k t )
2
2
S k ( x)  A2 k exp(ikx)  A3k (ikx)
C ( x, t )  k S k ( x)Tk (t )   dkSk ( x)Tk (t )
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The coefficients A1k, A2k and A3k should
be determined by the boundary
conditions
Example: Step Function
1
2
C ( x, t )  C0 (1 
2
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
x / 2 Dt
0
exp( y 2 )dy)
Poor Efficiency of Diffusion
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Typical values of D (in cm2s-1):
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To reach the distance 10m:
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Solid: 10-9
Liquid: 10-5
Gas: 10-1
It takes 4 months for gas
For liquid, reaches 10 cm only
For solid, 0.1 cm
Some other methods necessary for transport in
organisms
For charged particles, the transport can be
accelerated by an external electric field
Passive and Active Transport
Electric Field Driven Transport
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If the molecules are charged, they can
be drifted by an external electric field
Let the diffusion flux contributed by the
density gradient denoted by Jchem
The drift velocity of the charged particle
is vdrift=mE
What is the relation between the
diffusion coefficient D and the mobility
m?
Mobility and Diffusion
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Jelec: flux due to the external electric field = vdrift×C=mEC
Choose a proper electric field such that Jelec+Jchem=0 →
dC
No particle flow
D
  mEC
dx
The electrical force on the cylinder is: Fe=EqCAdx
 The mechanical force due to pressure difference is:
Fc=A[(P+dP)-P]=AdP
E
Fe  Fc  EqCdx  AdP  EqC  dP / dx
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P  nkBT / V  Ck BT 
Einstein’s equation
qD
EqC  k BTdC / dx  k BTmEC / D  m 
k BT
P
A
x
P+dP
x+dx