where
= partial specific volume,| Chem 432 |
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Fall 2002 |
| Lecture Notes:: February 21 |
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Size and Shape: A variety of hydynamic properties are widely used to to determine size and shape. We will look at the most common and important. (We will not discuss viscosity or osmometry - you should be familiar with basic osmometry from General Chemistry and Introductory Biology.)
Major Techniques:
Sedimentation methods are the standards to which others are traditionally compared. Generally give the best estimate of MW aside from calculations based on amino acid analysis for proteins. Based on density differences between biopolymers and water - proteins, nucleic acids and carbohydrates are denser and should thus settle out (lipids are less dense and will float).
Complication: the energy of gravity on earth, Egrav = 1.7 x 10-23 j/cm for a 105 dalton protein, while the energy due to temperature, ET = kT = 4 x 10-21 j (about 200 x's Egrav). Thus use a high speed centrifuge to give a high gravitational force.
Two basic types:
Using both methods together provides a definite MW and mixed infromation about shape and the extent of hydration.
The mathematics of sedimentation is difficult except under two conditions: a) zero net force, and b) zero net flow.
In this method we look at zero net force. First, the force due to centrifugation is:
where
= partial specific volume,
For a sedimenting particle this force will be opposed by friction:
where f
= the frictional coefficient (varies with shape)For sedimentation velocity the particle will accelerate until Fcent = Ffric or Fcent - Ffric = 0.
= svedberg
= sedimentation rate/unit centrifugal force (units = 10-13sec.Note that s µ Mf for similar molecules.
Finally know that f= kT/D where D is the diffusion constant, and R = kN = the Gas Constant in energy units (8.31 Jmol-1K-1). Substituing we get the Svedberg Equation:
Can find s values by rearranging s equation: ,
to
If we now plot ln r (or log r) vs. t the slope = w2s. Note that values for s are always given for standard conditions (s20,w) = 20 °C in pure water. Data gathered under other conditions must be extrapolated or corrected to give values for standard conditions.
Values of s can be used to give the shape, size and extent
of hydration when additional information such as D is known from
other experiments.
Sedimentation equilibrium takes advantage of the conditions no net flow where the math is again relatively easy. Consider a solution of protein (or particles) in a gravitational field, as in a centrifuge cell. If the protein is denser than the solvent there will be a net flow toward the bottom of the cell driven by the force due to acceleration:
As a gradient builds up, and if the gravitational field is not excessive, there will be a counter-flow due to diffusion from the concentrated solution near the cell bottom towards the top. This flow will be driven by the force of diffusion:
For no net flow: flowa + flowd= 0, or flowa = flowd. Substituting:
Substituting kT/D for f then R for Nk, canceling terms (A and D), and integrating from r1 -> r2 ([P]1 -> [P2]) then gives:
Experimentally just measure the concentration of protein at various values of r and plot ln [P] vs r. The slope is then proportional to the molecular weight of the protein, M! Notice that unlike with sedimentation velocity we do not need to know D.
Sedimentation equilibrium is very accurate, but takes much time to attain equilibrium (1-3 days at 6000-7000 rpm - can't spin too fast or the protein precipitates!). A variety of special techniques have been developed (see any biophysics text) to get around this limitation and get information faster (Boundary methods), but we won't discuss them.
Density Gradient centrifugation has also been used in the past for crude MW determinations. In these methods the unknown particle is run with standards (usually in a sucrose gradient) and fractionated to obtain an approximate MW by comparison.
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Last modified 21 February 2002
