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INTRODUCTION

Analytical Ultracentrifugation (AUC) experiments give us a method for the direct measurement of basic hydrodynamic and thermodynamic properties of macromolecules in solution. Because sedimentation relies on the principal property of mass and the fundamental laws of hydrodynamics, it is a valuable technique for a wide variety of solution conditions. The information obtained does not depend upon assumed relationships to measurements of standard molecules. It is a biophysical method that does not depend on the interaction of the sample with a matrix or surface. The direct determination of boundary sedimentation or gradient formation can be further analyzed using curve fitting methods to rigorously determine sample purity, choose between binding models, detect and characterize conformational changes. Moreover, it is a rigorous method to measure equilibrium constants for self- and hetero-associating systems, and determine other thermodynamic parameters such as the second virial coefficient. The AUCRL


How to Do an AUC Experiment

1) What can be done with AUC to characterize a sample? 

 a)	Determine number of components and number of species; detection of impurities 
 b)	Molar mass of each species 
 c)	Distribution of s for complicated mixtures 
 d)	Kinds and stoichiometry of chemical reactions present in solution, including association with ligands, self-association. 
 e)	Equilibrium constants for rapid (reversible) chemical rxns, and kinetic constants for slow ones. 
 f)	Shape and charge of the molecules, as inferred from their sedimentation frictional behavior
 g)	Characterize thermodynamic nonideality

2) Experimental variables you will control, with the goal to optimize information content. 

  a)	Rotor
   (1) 4 hole
   (2) 8 Hole

  b)	Cells

  c)	Centerpieces
    1. Double sector
     a. standard DS
     b. synthetic boundary
     c. menicus matching
    2. High Speed Equilibrium
     a. 6 channel Beckman type (must be disassembled for blank run)
     b. 6 channel Yphantis external loading (remains assembled for blank run)
    3. Short Column Equilibrium

  d)	Optical systems
     1. absorbance
     2. intensity
     3. interference
     4. fluorescence

  e)	Windows
      a. quartz
      b. sapphire

  f)	Method

  g)	Sample concentration range
     1. absorbance  0.01 to 1.5 a.u. (12 mm cell)
     1a.  0.04 to 6.0 a.u. (3 mm cell)   
     2. interference  0.01 to >60 mg/mL

  h)	Temperature
     1. 0 - 40 degrees

  i)	Rotor speeds  (my machine doesn't calibrate at 1000 rpm ? I do it at 3K and then slow down to collect 1K data ?)  
     1. 4 hole rotor ->  1,000 to 60,000 RPM
     2. 8 hole rotor ->  1,000 to 50,000 RPM

  j)	Number of scans
     1. velocity method : 999 max
     2. Multi-speed velocity 9900 max (using eq. method)
     3. equilibrium method: 9900 max

  k)	Delay before scans  

  l)	Interval between scans
     1. velocity - as rapidly as possible; enter 0.
     2. equilibrium at least one every hour

3) Examples of molecules that can be analyzed: 

  a)	Proteins
  b)	Polysaccharides
  c)	Nucleic acids
  d)	Small molecules: drugs, ligands, gasses
  e)	Large aggregates: viruses, organelles

4) Kinds of buffers and additional solutes: 

  a)  	BME, DTT,
  b)   Triton X100. Tween-80
  b)	Nucleotides
  c)	Salts and neutral molecules that significantly affect density
  d)	PEG, glycerol affect viscosity and desnity strongly
  e)	6 M Gdn and 8 M Urea

5) Sample preparation 

  a)  	Gel filter sample prior to analysis, unless the question being addressed is “What’s in a solution”
  b)	Estimate concentration and volume
  c)	Bring sample to dialysis equilibrium with solvent (required for interference optics, not with others)

6) Methods 

  a)	Sedimentation velocity
  b)	Sedimentation equilibrium

7) Optical systems 

  a)	Absorbance and pseudo-absorbance (intensity)
  b)	Intensity
  c)	Interference
  d)	Fluorescence

Types of systems

1) Single Component, non-interacting (i.e. mono-disperse) 

     1. Sedimentation Coefficient
     2. Diffusion and Frictional Coefficient
     3. Molar Mass
          a. by sedimentation equilibrium
          b. by sedimentation velocity
     4. Molecular Shape Analysis
          a. Frictional coefficients and Stokes radius
          b. Perrin's Equations
       c. Bead Modeling
  5. Analysis of data
       a. SEDFIT
          1. C(S) ->  s and f/fo
          2. C(M)
          3. Lamm Equation fitting to c(r,t)
       b. SEDANAL
          1. Lamm Equation fitting to Δc(r,t)

2) Poly-disperse, non-interacting systems 

     1. Experimental setup
         .
     2. Analysis of data
        a. SEDFIT
           1. C(S) -> "weight average" \frac{f}{fo}
     b. DCDT
        1. \frac{\Delta c(r,t)}{\Delta t}

3) Rapidly Reversible, Interacting Systems 

     1. Experimental setup
        a. self-associating system  eg.   2A \rightleftharpoons A_2 
     b. hetero-associating system eg.   A + B \rightleftharpoons C 

  2. Analysis of data
     a. Determination of stoichiometries and equilibrium constants
     b. curve fitting of sedimentation velocity and equilibrium data to various reaction schemes.

4) Kinetically limited interacting systems 

     1. Experimental setup
     2. Analysis of data
          a. Determination of stoichiometries and accounting for kinetics of slow reactions

          A  +  B  \rightleftharpoons  C  

     for which    K_{eq} = \frac{k_f}{k_r}

5) Ligand induced and Ligand Mediated interactions 

     1. Experimental setup
     2. Analysis of data
          a. Determination of stoichiometries, equilibrium and ligand binding constants
  
           A + L \rightleftharpoons AL 
        2AL  \rightleftharpoons (AL)_{2}

6) Non-ideal 

      1. McMillan & Mayer Theory
      2. l'n(y) = Bc + C'c<b>2 + D'''c3 + ...</span> this equation is incorrect - you fix it.

.

Analysis Methods

SEDANAL

Introduction to SEDANAL

SEDANAL is a program for the analysis of data from the Beckman XL Analytical Ultracentrifuge.

SEDANAL was originally written as "ABCD_Fitter" to analyze heterologous interacting systems of the type

A + B \rightleftharpoons C       Ka = [C] / [A][B]

and

A + B \rightleftharpoons C       K1 = [C] / [A][B]
C + B \rightleftharpoons D       K2 = [D] / [C][B]


SEDANAL has evolved to be able to handle any arbitrary reaction scheme with up to 28 components and/or 28 species related by up to 27 chemical reactions.


Both isodesmic and isoenthalpic indefinite self-associations are also included.

A Model Editor program is used to maintain a small database of models that are used by the main SEDANAL fitting program.

SEDANAL can process both sedimentation velocity and sedimentation equilibrium data.

SEDANAL can be down loaded from the RASMB at Boston Biomedical Research institute. BBRI

SEDVIEW

Real time g(s) for sedimentation velocity experiments

\left( \frac{\partial c}{\partial s^*} \right) _t = \left[ \left( \frac{\partial c}{\partial t} \right) _r - \left( \frac{\partial c}{\partial t} \right)_{s^*} \right] \left( \frac{\partial t}{\partial s^*} \right)

dln(s^*) = \frac{ds^*}{s^*} => ds^*=s^*(dln(s^*))

\left(\frac{\partial c}{\partial ln(s*)}\right)_t = \left[\left(\frac{\partial c}{\partial t}\right)_r - \left(\frac{\partial c}{\partial t}\right)_{s^*}\right]\left(\frac{\partial t}{\partial ln(s*)}\right)


\left(\frac{\partial c}{\partial ln(s^*)}\right)_t = \left(\frac{\partial c}{\partial ln(s^*)}\right)_r - \left(\frac{\partial c}{\partial t}\right)_{s^*} \left(\frac{\partial t}{\partial ln(s^*)}\right)_r


\left(\frac{\partial c}{\partial ln(s^*)}\right)_t = \left(\frac{\partial c}{\partial ln(s^*)}\right)_r - \left[2{\omega}^2t \left(\int^{s=s^*}_{s=0}{s\frac{\partial c}{\partial s}ds}\right)\right]\left[\left(\frac{\partial t}{\partial ln(s^*)}\right)_r\right]


\left(\frac{\partial c}{\partial ln(s^*)}\right)_t = \left(\frac{\partial c}{\partial ln(s^*)}\right)_r - 2\omega^2t \int^{ln(s^*)=+\infty}_{ln(s^*)=-\infty}s^*{\left(\frac{\partial c}{\partial ln(s^*)}\right)_t dln(s^*)}

... in progress. MORE TO COME. .

SEDFIT 

Link to SEDFIT  http://www.analyticalultracentrifugation.com/
  Limitations http://www.analyticalultracentrifugation.com/systematic_noise_analysis.htm#Limitation

UltraScan 

Link to UltraScan http://www.ultrascan.uthscsa.edu/


Lamm

Joachim Behlke- fitting to various analytical approximations to the Lamm equation

Svedberg Equation 

 { \frac{s}{D} } = \frac{M(1-v\rho)}{RT}

Lamm Equation: sedimentation velocity 

 { \left(\frac{\partial c}{\partial t}\right)_r = \frac{\partial}{r\partial r} \left\{rD\left(\frac{\partial c}{\partial r}\right)_t - r^{2}\omega^2sc\right\}_t }

Sums of exponentials: sedimentation equilibrium 

 j(\xi) = j_o(\xi) + \sum c_{i,ref} \exp\left(\sigma \xi -2B\sigma(j-j_o)\right) 

where j is signal and jo is the offset (zero level for that signal).

ci is the concentration of species i at a reference radius, B is the colligative second virial coefficient; 

where \sigma \equiv \frac {\omega^{2}s}{D}= \frac{M(1-v\rho)\omega^2}{RT} and \xi = \left\{\frac{r^2}{2} - \frac {r^{2}_{ref}}{2}\right\}

Time derivative Equation 

  g(s^*)=\left(\frac{\partial c}{\partial s^*}\right)_t  =\left[ \left(\frac{\partial c}{\partial t}\right)_r - \left(\frac{\partial c}{\partial t}\right)_{s^*}\right]\left[\left(\frac{\partial t}{\partial s^*}\right)_r\right]   
(exact: but does not result in baseline elimination)


  g(s^*)=\left(\frac{\partial c}{\partial s*}\right)_t  =\left[ \left(\frac{\partial c}{\partial t}\right)_r +2\omega ^2 \int^{s*}_0 s^*\left(\frac{\partial c}{\partial s*}\right)_t ds^*\right]\left[\left(\frac{\partial t}{\partial s*}\right)_r\right]  
(approximate: but gives complete time independent baseline elimination)
(becomes more exact in the limit of large σ )


testing

\left ( 1 - \bar v \rho_o \right )


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