I. Transport Processes

Study the movement of molecules in response to an applied
Force ==> F = m a (m = mass and a = acceleration)

A. Goals:

1. Gain information on size, shape and conformation of a macromolecule
2. Separate mixtures of macromolecules

B. Types of Forces

1. Concentration Gradient --> Diffusion
2. Gravitational --> Centrifugation
3. Electric Field --> Electrophoresis
4. Mobile vs. Stationary Phases --> Chromatography

First let's look at general properties of macromolecules

II. Volume and Hydration

A. Macromolecules (e.g. proteins, nucleic acids, etc.) interact with H₂O

Proteins, nucleic acids, polysaccharides all have polar functional groups capable of forming Hydrogen Bonds with H2O

1. can consider them to have some amount of bound water ==> H₂O molecules whose motion is restricted
2. most H₂O's aren't "stuck" to the macromolecule for a long time; they can trade places with bulk H₂O solvent
3. How does bound H₂O affect molecules during transport?

B. How much H₂O? Roughly, a monomolecular layer around a protein is strongly affected. H₂O's in the next layer out are affected somewhat.

• 0.3 gm H₂O/gm protein = d
• 0.6 gm H₂O/gm DNA

--> Nucleic acids have many more polar groups than proteins and a larger surface area ==> more bound water

C. Volume of a Hydrated Molecule -- must understand this to explain Transport Properties

1. here mass of solvent (m₁), temperature (T), and pressure (P) are held constant
   1. solute is component 2 of a 2 component system; component 1 is solvent
   2. units of V₂ are cm3/gm - this looks like the inverse of the units of density, but V₂ is not the inverse of density (even though this approximation works for some molecules) MgSO4 has a negative V₂ ==> solution volume decreases when MgSO₄ is dissolved in H2O
   3. Partial Molar Volume -- a similar property with units of cm³/mole
   4. V₂ depends upon: pH, concentrations of other solutes (salts etc.); won't worry about these effects in this course.
2. What is the volume of a hydrated molecule, V_h?

V_h = here M = molecular weight, No = Avogadro's number, d₁ is the hydration (grams of H₂O bound per gram of solute), and V₁ is the partial specific volume of H₂O (approx. 1)

1. Measuring V₂'s
   1. measure densitites of solutions with different solute concentrations ==> volume of solution with known gms of H2O and gms solute
      1. Pyncnometer -- vessel with very accurately known volume
      2. Density Gradient -- drop of solution will float at a point of equal density
      3. Microbalance -- weigh small volumes very accurately
   2. sum V's for each amino acid residue (or nucleotide in nucleic acids) ==> must know the amino acid composition. Assumes V's of amino acids don't change for different proteins. Actually works pretty well
   3. Sedimentation Equilibrium -- talk aobut this later
   4. Typical Values:

Proteins: 0.69 - 0.75 cm³/gm

Nucleic Acids: 0.54 cm³/gm with Na⁺ counterion

0.44 cm³/gm with Cs⁺ counterion (for CsCl density gradients)

III. Frictional Properties

A. What happens when a force, F, is applied to solute molecules?

1. F causes an acceleration, a: F = m a = m (dv/dt) (letters in bold are vectors)
2. Macromolecules moving through solution experience Frictional Drag --> a force proportional to v but in the opposite direction.
   Frictional Drag = f v (f is a constant of proportionality)
   

    

3. v increases rapidly until the two forces are equal in magnitude but opposite in direction
   ==> F - f v = 0 ==> v = F / f. This is like a skydivers "terminal velocity"
4. f is called the Frictional Coefficient and is determined by the size and shape of the macromolecule; these are things we might wish to know.
5. We will use this relationship, F = f v, repeatedly in studying different transport processes.

B. Relationship Between Friction and Size ==> use Dimensional Analysis to learn something about this relationship through common sense.

1. Friction must be propotional to:
   1. viscosity, h
   2. size, r = radius of hydrated sphere
   3. therefore -- f is propotional to h^x r^y (the question is what are x and y?)
2. f must have units of gm/s from F (gm-cm/s²) = f (gm/s) v (cm/s)
   1. h has units, gm/cm-s
   2. Therefore -- f is proportional to h¹ r¹ (gm/cm-s) cm = gm/s
3. For strong interactions between molecules and solvent:
   f_sphere = 6 p h r ==> Stokes Law and r = Stokes Radius
4. Often see Stokes Radius reported for molecular sizes measured by Gel Filtration Chromatography
5. In terms of volume:
   where V_h is the volume of the hydrated molecule
   ==> we need to know d₁ to calculate M
   this shows the relationship between f and M

C. Relationship Between Friction and Shape -- macromolecules aren't perfect spheres

1. Ellipsoids of Revolution -- solids produced by rotating ellipses around one of their two axes

1. Oblate Ellipsoid -- rotation around semi-minor axis, b ==> discus shaped. Vol = (4/3)pa²b
2. Prolate Ellipsoid -- rotation around semi-major axis, a ==> cigar-shaped. Vol = (4/3)pab²

Many macromolecules can often be approximated by an Ellipsoid of Revolution, either Prolate or Oblate.

1. For either, their surface area is greater than for a sphere of the same volume
   ==> f_ellipsoid > f_sphere
   1. Perrin Shape Factor: F = f / f_sphere
   2. can calculate F for either oblate or prolate ellipsoids of known axial ration, a/b.
   3. Stokes Law becomes: f = 6 p h r_sphere F

Prolate ellipsoids always have larger F's than oblate ellipsoids of the same volume because they have larger surface areas

IV. Macromolecular Diffusion

A. Mass Transort as a Flux of Particles (2 components)

1. General Equation -- will be used in discussion of other Transport Processes: J = L F
   [J = flux or no. particles passing through area A per second; L = Conductivity; F = force causing the flow of molecules]
   

   This equation applies to all sorts of movements; for example

   Ohms's Law: i = V / R Here i = current = flux of electrons;
   R = resistance ==> 1/R = conductivity; V = voltage, the applied force

2. What types of forces cause Mass Transport?
   1. A system which is not at equilibrium will move toward equilibrium.
   2. A system which is not at equilibrium has a gradient of potential energy: dU/dr
   3. Net movement of particles is down the potential energy gradient (like a ball rolls down a hill) to lower U

Thus: F = - (dU / dr) and J = -L (dU / dr)

B. What is L? Does it tell us something about the macromolecule?

1. During a length of time, dt, how many particles will pass through the area, A?

1. Let's say that during dt, particles travel on average, a distance dr.
2. Then, the number of particles which pass through area, A, will be the number in the volume V = A dr

1. Let's work with this equation
   1. V = A dr = A (dr/dt) dt = A v dt = (1 cm²) v dt ==> cm³
   2. If N_i = number of particles/cm³ and if dt = 1 s; then J = N_i v = L F
      Here we have a general expression relating Flux to particle density (concentration)
   3. Recall that: F = f v ==> v = F / f then substituting into 2.b gives:
      J = (N_i / f)F = L F and ==> L = N_i / f
   4. Let's change units to moles of particles. Divide twice by Avogadro's number; first time to change concentration to moles and second time to convert frictional coefficient, f, to mole units:
      
      (This eq.can be used for different transport processes arising from different F's)

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C. What is F for diffusion?

1. The driving force in diffusion is a concentration gradient, (dc/dr) which is described thermodynamically as a Gradient of Chemical Potential, µ.
2. The chemical potential of a solute is defined as the change in free energy, G, of the system caused by a small change in the amount of that solute: µ₂ = (dG / dn₂) with everything else held constant
3. µ₂ is, therefore, a function of concentration
   µ = µo + R T ln (g c): c is the concentration of solute; g is the activity coefficient which corrects for the non-ideal behavior of solutes in solution, the fact that molecules of solute interact with one another
   Recall: DG = D G_o + R T ln [products] / [reactants] This is just like the equation above for chemical potential summed for all species in a chemical reaction.
   
4. We want to know: F = - (dU / dr) = - (dU / dc) (dc / dr) <-- applying the chain rule
   or, when U is chemical potential: (dµ / dr) = - (dµ / dc) (dc / dr)
   
5. 
   1. D_o = (R T) / (N_of) = (k T) / f; Boltzman's Constant, k = R/No ==> R for a single molecule

1. D_o is the diffusion coefficient when the concentration is very small because
   g --> 1 and (d ln g / dc)-->0; D_o is what we want to know ==> measure D and different concentrations and extrapolate to c = 0
2. from Do we can calculate f = and f gives us information about size and shape
   ==> f_sphere = 6 p h r_sphere

1. What does D_o tell us? ==> how fast molecules move.

D_o has units cm²/s ==> during time t s, a molecule will move (diffuse) on average cm

Typical Values:

Glycine	75 gm/mole	D20,w = 93.3 x 10-7 cm2/s
Hemoglobin	68,000 gm/mole	D20,w = 6.9 x 10-7 cm2/s
Tomato Bushy Stunt Virus	10,700,000 gm/mole	D20,w = 1.15 x 10-7 cm2/s

D. How does one actually measure D?

1. most obvious way is to layer solvent on top of a macromolecular solution and watch diffusion occur

How do we interpret this? Fick's Second Law of Diffusion:

a differential equation which can be solved to give: c(r,t) = c_o (4 p D t)^1/2 exp (r²/4Dt)

c_o = initial concentration. Thus, one can measure dc/dr for a particular time, t, and determine D from this equation. The experiment is more difficult than it looks because:

1. It takes days for appreciable diffusion to occur. and
2. The system is very sensitive to changes in temperature and to vibration which causes mixing

1. Quasi-Elastic Light-Scattering or Dynamical Light Scattering: Measure wavelengths of scattered laser light

1. The incident light is very monochromatic (a characteristic of laser light)
2. The scattered light has a broader distribution of wavelengths (is less monochromatic) owing to the Doppler Effect. Light scattered from moving particles has its wavelength shifted with respect to a stationary detector; photons scattered from particles moving away from the detector have a lower frequency (longer wavelength) than the incident photon and photons scattered from particles moving toward the detector have a higher frequency (shorter wavelength).