Physics of biological membranes,

diffusion, osmosis

Dr. László Nagy

-Metabolic processes and transport processes.

- Macrotransport : transport of large amount of material

: through vessel systems

: in large distance

- Microtransport : small amount of material

: by diffusion

: in small distance

Name of the

transport process

What is

transported

Potential

Transport of

liquids and gases

macroscopic

material

pressure

difference(Dp)

Diffusion molecules concentration

difference(Dc)

Temperature heat temperature

difference (DT)

Electric current ions, electrons electric potential

(DU)

D

D

x

UJ

Current intensity:

- characteristic to the A surface

- m is: mass, volume, electric charge, energy, etc.

Unit can be: kg/s; m3/s; C/s (A); J/s…

dt

dmI

dm amount of material transported

in dt time across the A cross

section area

Current density:

dA

dIJ

dI is the transport intensity

perpendicular to dA surface

- vector quantity (direction = direction of the transport)

- it is defined at the point of the transport area

-differential quantity

Unit can be: kg/s/m2; m3/s/m2; C/s/m2 (A/m2); J/s/m2…

Specific conductivity:

U: - potential

: - negative gradient of U gives the driving force at any point of the transport

(potential energy, electric potential, temperature difference,

concentration gradient)

g: - generalized (specific) conductivity

Eg.: in the case of the diffusion the potential gradient is the gradient of the

concentration (dc/dx).

dx

dUgJ

potential gradient

specific conductivity

current density

Diffusion

c1 c2

c1> c2

a) Macroscopic approach:

dx

dcgJ

)tvA(cm Unit c and A:

,

dx

dcDJ

s

mD

2

b) Molecular approach:

Chemical potential of a single particle: m = m0 + kTlnc.

The gradient of the chemical potential: x

txc

c

Tk

x

txcTk

x

txf

),(),(ln),(m

dn moles are transported across the A surface in dt time: dn = cAū·dt.

dtx

txcADdtuAcdn

),(

x

txcD

dtA

dnJ

),(Fick’s I. law: The amount of transported material

across a unit area and unit time (the

rate of the diffusion) is proportional to

the gradient of the concentration. D is

the diffusion constant.

x

c

D

Dis constant at a given place and time.

Tkvm 2

3

2

1 2 m

Tk3v

2

1

2

1

mTD

D depends on:

1) temperature

4) geometry

r

TkD B

6

2) viscosity

3) mass

Einstein-Stokes

kB = Boltzmann constant

T = absolute temperature

= viscosity coefficient

r = radius of the particle

Fick’s IInd law

2

2 ),(

),(

),(

x

txcD

x

x

txc

Dt

txc

The change of the concentration in time at a given place is

proportional with the change of the concentration gradient with

the place at a given time.

Concentration gradient

x

txc

),(x

x x0

dc/dx

tD4

x2

etD2

M)t,x(c

tD2)t(x

In a special case:

e

MeMtxc

Dt

x 1

2

),(14

M is the amount of material at t=0

and x=0

D = diffusion coefficient

x = distance

t = time

The solution of Fick’s 2nd law and some consequences of it:

capillary capillary

tissues tissues

distance

Gas exchange across the alveocapillary membrane:

Doxigén 110-9 m2s-1; DCO2 610-9 m2s-1; x = 1 mm =110-6 m;

toxigén = 250 ms; tCO2 = 40 ms;

Pl.: D 10-9 m2s-1; x = 5 nm =510-9 m;

s

s

m

mt

9

18

29

229

104

1025

104

105

= 6.25 ns.

D 10-9 m2s-1; x = 50 mm =510-5 m; = 0.625 s.

D 10-9 m2s-1; x = 1 m = 7.9 év.

Diffusion is effective

at small distances!

D

xttDtx

42)(

2

In the air: Doxigén 210-5 m2s-1;

DCO2 1.610-5 m2s-1

Swimming of E. coli:

rvdt

dvm 6

dtm

rv

v

dv 6

t

evtv

0)(

s7102

0

0)()( vtdtvd

d=4∙10-10 cm = 0.04Å

diameter of H-atom!

tD2)t(x

nmmssmtx 28108,2102102)( 87129

Distance travelled by the oxygen:

Fs=6πμrv F=m∙a=m(dv/dt)

velocity friction

The membrane equilibrium of electrically neutral

particles

osmosis

Definition of osmosis

phydr = rgh = RTc= posm V=1/c (dilution)

pV = RT

van’t Hoff’s law: posm = RTc

Sugar in water.

Distilled water.

For diluted solutions.

Semipermeable

membrane.

pVxTR ln0mm

The chemical potential of water:

In equilibrium:

21 mm

The difference in two pressure values:

2

112 ln

x

x

V

TRpppozm

For non diluted solutions:

2

2

watersugar

sugarosm

ccRTp

v

RTcpthancWhen osm

water

sugar v

2

vwater: molar

volume of water

Osmotic pressure is additive:

1 osmolal is the osmotic pressure of 1 molal material.

0.1 molal NaCl → 0.2 osmolal

0.1 molal CaCl2 → 0.3 osmolal

Estimation of the magnitude:

RT20oC 2.44 MPa M-1

if c = 0.3 molal than

posm = 2.44 MPaM-1 0.3 M-1 0.73 MPa (7.3 bar)

in see water: 2.6 MPa 260 m high water column!

in tyres: 0.2 MPa

iosm cRTp

solventg

solutemoleMolality

1000

kgmoleunitionconcentratRault :;

How to measure it?

1. By using the van’t Hoff’s law: Pfeffer osmometer

2. By using the Rault’s law

increase in boiling point: mT

M

GT DD

' G’: g solute in 1000g

solvent

M: molar mass

DTm: molal boiling point increase

phydr = rgh = RTc= posm

depends on the solvent

Isoosmotic ↔ Isotonic

???

Calculated ↔ Measured

Rejection coefficient: 0 < s < 1

s = 1: solute is completely excluded

osmolality = tonicity

s = 0: solute is not excluded

osmolality ≠ tonicity (e.g. in biological

membranes rather tonicity)

Importance of osmosis in biology

- „Epsom salt” („bath salt”, MgSO4)

- iso-, hyper-, hypotonic solutions

hemolysis

physiological saline:0.9 m/m%

(~300 mOsm) NaCl

(sea water: 3.5%)

- dialysis,

haemodialysis,

peritoneal

dialysis

-reverse

osmosis

-Starling effect: the equilibrium between the blood

plasma and the intersticium

unbalance → oedema

35 Hgmm

(4,67 kPa)

Plasm hydrostatic

pressure

2 Hgmm

(0,27 kPa)

Interstitium

hydrostatic pressure

0 Hgmm Interstitium colloid

osmotic pressure

25 Hgmm

(3,33 kPa)

8 Hgmm

(1,07 kPa) Resulting pressure

15 Hgmm

(2,00 kPa)

Plasm hydrostatic

pressure

Interstitium

hydrostatic

pressure

1 Hgmm

(0,27 kPa)

3 Hgmm

(0,27 kPa)

Interstitium colloid

osmotic pressure

8 Hgmm

(1,07 kPa) Resulting pressure

25 Hgmm

(3,33 kPa) Colloid osmotic pressure Arterial end

Vein end

Transport across

biological membranes

Comparison Passive

diffusion

Facilitated

diffusion

Active transport

Mediator Membrane lipids Ionofores, proteins

(permeases)

Membrane proteins

Direction of the

flux

In the concentration

gradient.

In the concentration

gradient.

Against the

concentration gradient

as well.

Connection to cell

energy supply

None None, or indirect Direct connection

Specificity None Large Large

Saturation None Possible Possible

Specific inhibition None Possible Possible

Reversibility Reversible Reversible Irreversible

Fick’s laws are

valid

Yes No, Michaelis-

Menten kinetics

No, Michaelis-Menten

kinetics

Transzported

materials

Lipid soluble, small

molecular mass

neutral molecules.

Ions, polar

compound

Large variety of

compounds, ions,

proteins, etc.

Comparison of transport processes across the mambrane

Good luck for your studies!