ch 24 pages 627-632 lecture 7 – diffusion and molecular shape and size

Ch 24pages 627-632

Lecture 7 – Diffusion and Molecular Shape and Size

Diffusion was introduced to describe the net transport of material in the direction opposite a concentration gradient

Fick’s First Law of diffusion relates the flux to the concentration gradients

Summary of lecture 6

D is a phenomenological property called the diffusion coefficient

22 ( )

dCJ x D

dx

The diffusion coefficient is related to the temperature and to the frictional coefficient f. This quantity depends on the solvent property (viscosity) and on the molecular property of the solute (size, shape and hydration). From Stokes equation:


This last equation is Einstein’s PhD thesis

f R 6

fTkD b /

The diffusion equation (Fick’s second law) relates changes in concentration over time to the second derivative of the concentration:


The diffusion equation has the following general solution:

22

2

222

22

x

CD

x

C

f

Tk

t

C b

tDxetD

CtxC 2

2 4/

2

02

4),(

Although the average displacement is 0, the mean squared displacement and root mean square displacement are not:


The rms displacement is related to the width of the gaussian at half height:

tDxx rms 22 2

2ln4 Dt

Friction expresses the counterforce acting on a particle moving in a viscous medium

F=fu


The frictional force F increases with speed, so that the speed of the particle will only increase up to a point, until it reaches a steady state value. The viscous drag defines a certain velocity as the steady state speed at which the particles move under the influence of an acting external force and of the viscosity of the medium

The radius of a spherical protein can be calculated from the diffusion coefficient (i.e. the Stokes radius)

However, biological molecules are always hydrated and solvation effectively increases the hydrodynamic volume of a molecule and therefore its frictional coefficient

Let us introduce the concept of partial specific volume, i.e. the volume change in the solution when w2 grams of solute are added

(it expresses essentially the volume of solution occupied per gram of un-hydrated solute, e.g. protein).

Hydration and Size of Biological Molecules

1,,22

wPTw

VV

The partial specific volume can be measured as the definition implies, by measuring a change in volume when a certain weight of solute is added

For an impenetrable object (e.g. glass bead), the partial specific volume is simply the specific volume (volume per gram) of the bead

For biological molecule, it is found experimentally to depend on conditions such as T, P, but also concentration of solute, pH etc


The hydrodynamic volume of an un-hydrated molecule is simply the partial specific volume multiplied by the weight per molecule

For an hydrated protein or DNA molecule, we have to include hydration as well and account for all water molecules bound to the solute, which effectively increase the size of the molecule as it diffuses

The hydrodynamic volume can then be calculated by adding the partial specific volume to the volume of bound waters per gram of macromolecule:


is the solvent density and 1 the number of grams of water

bound per gram of solute, typically 0.2-0.6 for proteins and 0.5-0.7 for DNA or RNA)

/1

The degree of hydration of the protein (i.e. grams of hydrated waters per gram of protein) may be determined by comparison of the Stokes radius with the radius calculated assuming an un-hydrated protein

If we define f0 as the frictional coefficient expected for an un-

hydrated molecule and f the frictional coefficient for a fully, hydrated system, then:


The denominator is the volume of an un-hydrated molecule, and the term above is the total hydrodynamic volume, including hydration

2

112

3

0 V

VV

f

f

To a good approximation, the partial specific volume of water is simply the inverse of the density of water, 1 cm3/g. If the molecule can be assumed to be approximately spherical (as is the case for many globular proteins), then:


3

0

3

0

r

r

f

f

Example: The diffusion coefficient and specific volume of bovine pancreas ribonuclease (an enzyme that digests RNA) have been measured in dilute buffer at T=293K and are found to be:

D=13.1x10-11 m2/s


The molecular weight of the protein is 13,690 g/mole

We can calculate the frictional coefficient f from the diffusion coefficient D and the temperature T

V cm g230 707 . / .

16 2 2 18

7 2 1

1.38 10 2933.08 10 /

13.1 10bk T x gcm s K K

f x g sD x cm s

If we assume that the protein is an un-hydrated sphere, we can calculate its frictional coefficient. Its radius is related to the partial specific volume by the following equation:


Where we are simply expressing the volume of a sphere from its radius (remember, the hydrodynamic volume of an un-hydrated molecule is simply the specific volume times the weight per molecule)

AN

VMWxr 23

03

4

33 2

0 23

3 3 13,690 / 0.707 /

4 4 6.02 10 /A

MWxV g molex cm gr

N x molecules mole

70 1.57 10 15.7r x cm Angstroms

We can now use Stokes’ equation to calculate the frictional coefficient:


From which it follows that:

Therefore:

sgxxsgcmrf cm /1095.21057.101.066 87110

2

112

3

0

3

0

05.1V

VV

r

r

f

f

3 3 38

80 0

3.09 101.05

2.95 10

f r x

f r x

From which we can calculate:


(grams water per gram/protein)

From this we can estimate how many water molecules hydrate each protein molecule:

moles H O g g mole moles210105 18 0 0058 . / .

moles protein g g mole moles 1 13 690 0 0000731/ , .

H O protein2 0 0058 0 000073 79 5 80/ . / . .

105.0/707.0/707.005.1/0.1

1 33

31 gcmgcmxgcm

The frictional coefficient of non-spherical molecules will in general be large than that of a spherical molecule of the same volume, simply because there will be a larger surface in contact with solvent and this will of course increase the hydrodynamic drag. It is possible to calculate frictional coefficient for any shape by computational methods, and for certain simple shapes, the calculation can be executed analytically

The Shape of non Spherical Biological Molecules

In many cases, the shapes of rigid, non-spherical molecules can be modeled in terms of some idealized shape. For example, virus particles can be modeled as rods or ellipsoids of rotation.


We can use some simple consideration to calculate the frictional coefficient for rod-like particles

Suppose a rod-like particle has a length 2a and radius b. Its volume is given by the formula:


The axial ratio P is defined as:

For a rod-like particle, the frictional coefficient can be found to be:

22 abVrod

baP /

30.0)2ln(

3/2 3/23/1

0

P

Pff

The term f0 in the equation above is defined as


where R0 is defined as the radius of a sphere that has a volume

equal to the volume of the rod with axial ratio P=a/b, while the remainder of the expression accounts for a ‘shape’ correction, i.e. represents the effect of the particular rod-like shape on diffusion

00 6 Rf


R0 is determined as follows:

2

3 230

abR

rodsphere VV

230 2

3

4abR


Another good example is the frictional coefficient for a prolate ellipsoid (cigar-shaped particle):

The volume of a prolate ellipsoid is:

where the lengths a and b are called the major and minor semi-axis lengths

2

3

4abV pro


For a prolate ellipsoid:

Where again:

R0 is again the radius of a sphere which has a volume equal to the

volume of a prolate ellipsoid for which P=a/b

In this case:

00 6 Rf

The frictional coefficient ratio f/f0 for a

prolate ellipsoid does not vary markedly with aspect ration P. For example, for P ranging from 1 to 20, f/f0 only varies from 1 to about 2

1ln

12

23/1

0

PP

PPff

3/120 abR

ch 24 pages 627-632 lecture 7 – diffusion and molecular shape and size

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