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.

--> 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, Vh?


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 hx ry (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 > fsphere
    1. Perrin Shape Factor: F = f / fsphere
    2. can calculate F for either oblate or prolate ellipsoids of known axial ration, a/b.
    3. Stokes Law becomes: f = 6 p h rsphere 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 Ni = number of particles/cm3 and if dt = 1 s; then J = Ni 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 = (Ni / f)F = L F and ==> L = Ni / 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)


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 Go + 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)
    1. Do = (R T) / (Nof) = (k T) / f; Boltzman's Constant, k = R/No ==> R for a single molecule

  1. Do is the diffusion coefficient when the concentration is very small because
    g --> 1 and (d ln g / dc)-->0; Do 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 rsphere
  1. What does Do tell us? ==> how fast molecules move.

Do 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:

(dc / dt)r = D (d2c / dr2)

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

co = 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).