Ch 24 pages 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

 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: Summary of lecture 6 This last equation is Einstein’s PhD thesis

 The diffusion equation (Fick’s second law) relates changes in concentration over time to the second derivative of the concentration: Summary of lecture 6 The diffusion equation has the following general solution:

Although the average displacement is 0, the mean squared displacement and root mean square displacement are not: Summary of lecture 6 The rms displacement is related to the width of the gaussian at half height:

Friction expresses the counterforce acting on a particle moving in a viscous medium F=fu Summary of lecture 6 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 w 2 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

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 Hydration and Size of Biological Molecules

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: Hydration and Size of Biological Molecules  is the solvent density and  1 the number of grams of water bound per gram of solute, typically for proteins and for DNA or RNA)

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 f 0 as the frictional coefficient expected for an un- hydrated molecule and f the frictional coefficient for a fully, hydrated system, then: Hydration and Size of Biological Molecules The denominator is the volume of an un-hydrated molecule, and the term above is the total hydrodynamic volume, including hydration

To a good approximation, the partial specific volume of water is simply the inverse of the density of water, 1 cm 3 /g. If the molecule can be assumed to be approximately spherical (as is the case for many globular proteins), then: Hydration and Size of Biological Molecules

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=13.1x m 2 /s Hydration and Size of Biological Molecules 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

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: Hydration and Size of Biological Molecules 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)

We can now use Stokes’ equation to calculate the frictional coefficient: Hydration and Size of Biological Molecules From which it follows that: Therefore:

From which we can calculate: Hydration and Size of Biological Molecules (grams water per gram/protein) From this we can estimate how many water molecules hydrate each protein molecule:

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. The Shape of non Spherical Biological Molecules 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 Shape of non Spherical Biological Molecules The axial ratio P is defined as: For a rod-like particle, the frictional coefficient can be found to be:

The term f 0 in the equation above is defined as The Shape of non Spherical Biological Molecules where R 0 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

The Shape of non Spherical Biological Molecules R 0 is determined as follows:

The Shape of non Spherical Biological Molecules 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

The Shape of non Spherical Biological Molecules For a prolate ellipsoid: Where again: R 0 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: The frictional coefficient ratio f/f 0 for a prolate ellipsoid does not vary markedly with aspect ration P. For example, for P ranging from 1 to 20, f/f 0 only varies from 1 to about 2