Transport Processes

2005/4/24

Dept. Physics, Tunghai Univ.

Biophysics C. T. Shih‧

Movement Movement is one of the defining

characteristics of the biological world Movement of different scales:

Motion of the entire organisms Motion of the internal organs Motion of a cell Motion within a cell Molecular motions

Diffusion – Passive Transport

Motion of particles through space Mixing of particles amongst one another

– usually for biological systems Brownian motion:

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 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 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 For the random walk by m steps without directio

nal bias: Unit length of each step Mean distance <r>=0 Root mean square distance rrms=√m

For a stochastic Brownian motion,

D: diffusion coefficient, net flow of particles per unit time per unit area (perpendicular to the direction of flow)

Dtrrms

Diffusion Driven by Density Difference

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:

The particles move from the region of higher density toward the lower one

2/ ;/))()((2

1 2 fbDdxDdCbbxCxCfJ

J

Time-Dependent Diffusion

The net flux into the region between x and x+L is J(x+L)-J(x), where

Fick’s Second Law

constant is D if

/

/)/(//)()(~)(

/)(

2

2

x

CD

x

CD

xtC

dxdxDdCLddxDdCdxxLdJxJLxJ

dxDdCxJ

Solution of Fick’s 2nd Equation Separate the variable: C(x,t)=S(x)T(t) a

nd the equation becomes (let D=a2)

LFS: depends only on x; RHS: depends only on t

The only possibility is that both sides equal to a constant, say –a2k2

t

T

Tx

S

S

a

t

TS

x

STa

1

2

22

2

22

Solution of Fick’s 2nd Equation The general form of solution:

The coefficients A1k, A2k and A3k should be determined by the boundary conditions

)()()()(),(

)()exp()(

)exp()(

32

221

tTxdkStTxStxC

ikxAikxAxS

tkaAtT

kkk kk

kkk

kk

Example: Step Function

Dtx

dyyCtxC2/

0

20 ))exp(

21(

2

1),(

Poor Efficiency of Diffusion Typical values of D (in cm2s-1):

Solid: 10-9 Liquid: 10-5 Gas: 10-1

To reach the distance 10m: 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

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=E

What is the relation between the diffusion coefficient D and the mobility ?

Mobility and Diffusion Jelec: flux due to the external electric field = vdrift×C=EC

Choose a proper electric field such that Jelec+Jchem=0 → No particle flow

The electrical force on the cylinder is: Fe=EqCAdx

The mechanical force due to pressure difference is: Fc=A[(P+dP)-P]=AdP

ECdx

dCD

AP P+dP

x x+dx

E

Tk

qDDECTkdxTdCkEqC

TCkVTnkP

dxdPEqCAdPEqCdxFF

BBB

BB

ce

//

/

/

Einstein’s equation