Biophysics I. 2015/2016
Lecture 15.
Related literature:
Related multimedia material:
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Topics of the lecture:
1.Definition of diffusion
2. Biological importance of diffusion
3. Molecular motion
3.1. Brownian motion
4. What does the „strength” of diffusion depend on?
4.1. Quantifying diffusion in space
4.1.1. Describing the diffusion according to the temporal distribution of the amount
of
substance
4.1.2. Diffusion described by the spatial distribution of concentration
5. Fick’s 1st law
6. Fick’s 2nd law
7. Diffusion through the cell membrane
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Diffusion
(1) Definition of diffusion
Certain substances sooner or later disperse in the available
fluid of gas-space (e.g. sugar disperse in coffe after dissolving,
the scent of flowers can fill the whole room). This spreading
process is called diffusion.
(2) Biological importance of diffusion
•
microscopic matter transport processes
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•
•
•
•
•
•
transport through the cell membrane
metabolism
 gas exchange between blood and the lungs
stimuli
absorption of medicines
chemical reactions
(inter- and intracellular molecular movements)
(3) Molecular motion
Most of the particles of biological systems are in constant motion
→mainly in fluid phase – aqueous medium (55 – 60 % of the human body consists of water)
→ occasionally in lipid phase – cell membrane (exhibit higher order organisation)
3.1. Brownian motion
Robert Brown (scottish botanist, 1827):
experiment: microscopic investigation of pollen in water
observation: random, zig-zag motion of pollen particles, similar to gas particles
The thermal motion of particles forms the basis of diffusion (= Brownian motion – the basis of
diffusion)
(The description of molecular motion is much more complicated in fluid phase than in gases, because
interactions between molecules are more substantial and complicated than fluids. In the following
we will present the basic diffusion laws for gases, but the results are applicable under certain
conditions to fluid too.)
Model of the perfect gas: (equipartition law, see Thermodynamics).
1
3
𝜀̅kin= 2 𝑚𝑣̅ 2 = 2 𝑘𝑇~𝑇
Due to the non-uniform (inhomogeneous) distribution of particles  net transport (Brownian
motion) of the particles occurs  from regions of higher concentration to regions of lower
concentration. Which continues until the distribution of the particles is uniform (homogeneous).
(4) What does the „strength” of diffusion depend on?
4.1. Quantifying diffusion in space
Fick (German physiologist): examined the distribution of dye molecules in a container:
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Two particle fluxes occur: upwards (INup) and downwards (INdown) In the beginning of the process:
INup>INdown,  INup– INdown net particle flux proceeds upward.
Analyse diffusion in a simplified model in 1D along the axis X (analogy of Fick’s experiment):
change of particle density (number of particles in a unit volume) along axis X as a function of time
4.1.1. Describing the diffusion according to the temporal distribution of the amount of substance
The rate of diffusion can be described by the particle flux (IN): during Δt time, ΔN pieces of particles
(= number of particles) travels through a surface of A (perpendicular to the X axis /direction of the
flow/)
𝐼𝑁 =
Δ𝑁
,
Δ𝑡
unit:
1
.
𝑠𝑒𝑐
It is more convenient to use the matter flow rate (IV) (particle flux divided by the Avogadro’s number
/NA/):
𝐼𝑉 =
Δ𝑛
Δ𝑡
, unit:
𝑚𝑜𝑙
𝑠𝑒𝑐
,and ∆𝑛 =
Δ𝑁
.
𝑁𝐴
Due to diffusion during Δt time Δn amount of particles (mole fraction) travels through a surface of A.
Particle flux and matter flow rate are depend on the surface (A) it is practical to introduce a
quantity which is independent form the surface (A): matter flow density (Jv):
𝐽𝑉 =
Δ𝐼𝑣
Δ𝐴
=
Δ𝑛
,
Δ𝐴∆𝑡
unit:
𝑚𝑎𝑡𝑡𝑒𝑟 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒
𝑠𝑢𝑟𝑓𝑎𝑐𝑒
𝑚𝑜𝑙
.
𝑚2 𝑠𝑒𝑐
Matter flow density: number of moles of substance travelling through a unit surface during a time
interval of unity. This is proportional with the strength of diffusion.
4.1.2. Diffusion described by the spatial distribution of concentration
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A surface of A perpendicular to the axis x at x0, and a Δt
time-interval short enough to ensure that n(x) /=c(x)/
that n(x) can be assumed constant.
In a very short Δt time concentration n(x) is constant in
the region of x0: the slope of the line is
Δ𝑛
,
Δ𝑥
Δ𝑐
i.e. Δ𝑥 =
concentration gradient (the concentration drop over a
unit-length; spatial variation of the concentration along
the x axis is decreasing.
(5) Fick’s 1st law
𝚫𝒄
𝑱 = −𝑫 𝚫𝒙.
Matter flow density (J) is linearly proportional to the drop in concentration (−
Δ𝑐
)
Δ𝑥
Negative sign: particles diffuse from the high concentration regions to the low concentration regions.
(See General description of transport processes: Onsager’s linear equation. Thermodynamics)
D: diffusion coefficient
𝑚2

unit: 𝑠𝑒𝑐,

characterizes the mobility of a diffusing particle - tells us how ‘fast’ a given substance diffuses

gives the amount of substance that diffuses through a surface unit during a time unit if the
concentration drop was unity
depends on both the diffusing particle and the medium in which the particle diffuses
can be calculated according to the Einstein-Stokes equation:
for shperical particles (r: radius) in a viscous medium (η) at T temperature:


𝐷=
𝑘𝑇
,
6𝜋η𝑟
𝐽𝑜𝑢𝑙𝑒

k: Boltzmann constant, k=1.38 x 10-23𝐾𝑒𝑙𝑣𝑖𝑛,


T: temperature, the higher the temperature, the stronger the thermal motion
η: viscosity of the medium, diffusion is faster in low viscosity media than in high
viscosity media; gases>liquids
r: radius of globular particle

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(6) Fick’s 2nd law
Fick’s 1st law: quantitated diffusion
considering the spatial variations in the
concentration (c(x), did not consider
that the concentration changes with
time.  Fick 2nd law: spatial and
temporal description of diffusion (c(x,t).
Graphical illustration of the concentration (c(x)) at time point t1 and t2 (t1<t2) according to the simplified Fick experiment.
Δ𝑐
( Δ𝑡 = 𝐷
∆(
Δc
)
Δx
Δ𝑥
)
The diffusion time (t) is proportional to the square of the diffusion distance (R) Diffusion is
relatively fast (< seconds) over a short distance (100 μm) and exceptionally slow (> days) over a long
distance (1 cm).
𝑡~
𝑅2
2𝐷
Problem (1)
Gas exchange between the blood and the lungs
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(7) Diffusion through the cell membrane
1. PASSIVE DIFFUSION
passive transport, without mediator





direction of transport: ELECTRO-CHEMICAL POTENTIAL
GRADIENT
o chemical potential gradient (concentration)
o electric potential gradient (charge)
rate of diffusion: Fick’s laws
mediator: no
energetic requirement: no
examples:
o hydrophobic molecules: O2, N2
o small polar molecules: CO2, water, alcohol, urea, glycerol
2. FACILITATED DIFFUSION
a.) Passive transport, with mediator: ION-CHANNEL




direction of transport: chemical or electro-chemical potential gradient
rate of diffusion: faster than that expected from Fick’s laws
mediator: ION-CHANNEL PROTEIN
o transmembrane proteins (porous structure)
o closed / open state: no transport / transport
o regulation of the closed/open state:
 mechanically-gated (mechanical tension)
 voltage-gated (potential difference between the sides of the membrane, see:
Action potential)
 ligand-gated (ligand-binding)
o selectivity: size & charge of the ions
energetic requirement: no
b.) Passive transport, with mediator: CARRIER PROTEINS
 direction of transport: chemical or electro-chemical potential gradient
 rate of diffusion: faster than that expected from Fick’s laws
 mediator: CARRIER PROTEIN (transporter)
o specifically binds the ions or molecules and promotes their transport
(conformational change)
 energetic requirement: no
3. ACTIVE TRANSPORT
Passive transport, with mediator: CARRIER PROTEINS
 direction of transport: AGAINST the chemical or electro-chemical potential gradient!
ENERGY IS REQUIRED
 mediator: CARRIER PROTEIN
o uniporter (one molecule-one direction)
o symporter/antiporter (one molecule-same/opposite direction)
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

energetic requirement: yes
o ATPase transporter (ATP hydrolysis)
o photo transporter (light energy)
o coupled transporter (energy from another transport)
example: Na+-K+ pump (see Action potential)