ࡱ> :<9a Sbjbj *`hGbhGbf-jjjjj~~~8dt~o(|(v'x'x'x'x'x'x'$*, 'j'jj)( jjv' v' %&@j&b'?(0o(&<-x-0&-j&x ''ro(-B 0:  NCMH PhD Tutorial 4: Sedimentation Equilibrium Fundamental equation of sedimentation equilibrium: From thermodynamic considerations or by equating sedimentation and diffusive forces, for an ideal single solute at sedimentation equilibrium: {1/c(r)}.dc(r)/dr = w2rM(1- EMBED Equation.2 r)/{RT} (1) ( EMBED Equation.2 = partial specific volume of the solute, ( = solvent density, ( = angular velocity, T = thermodynamic temperature, R = gas constant) Integrated form (between selected radial positions r1 and r2) c(r2) = c(r1)exp[w2M(1- EMBED Equation.2 r)(r12-r22)/2RT] (2) For systems other than ideal single-solute, either we can represent the data by specific model equations or we can more generally replace M by Mw,app, where Mw,app is the apparent weight average molecular weight of the material between positions r1 and r2 (we use the term "apparent" because of non-ideality)  To describe the apparent weight average over all the material (from meniscus, r=a and cell base, r=b): c(b) = c(a)exp[w2Mw,app(1- EMBED Equation.2 r)(b2-a2)/2RT] (3) For ideal monodisperse systems the Mw,app s in eq(2) and eq(3) will be the same, otherwise only eq (3) represents the entire distribution of macromolecular solute. The programmes IDEAL1, NONLIN etc are based around selected distributions (between r1 and r2) in the ultracentrifuge cell. MSTAR refers to the whole distribution. Eq. 3 can be manipulated in a number of ways 1. Logarithmic form over the whole distribution Mw,app = ln[c(b)/c(a)] . {2RT/[w2(1- EMBED Equation.2 r)(b2-a2)]} (4) 2. Integral form over the whole distribution (co = initial loading concentration) Mw,app = {[c(b) - c(a)]/co}. {2RT/[w2(1- EMBED Equation.2 r)(b2-a2)]} (5) n.b. at low loading concentration (<0.5 mg/ml for proteins) Mw,app ~ Mw. Since data near the meniscus or base are not measurable, some form of extrapolation of the c(r) data to c(a) and c(b), or ln[c(r)] to ln[c(a)] and ln[c(b)] is necessary. (i) For homogenous/ monodisperse systems determination of c(b) is not too difficult. For heterogeneous sytems (polydisperse, self-associating), extrapolation can be difficult through strong curvature of c(b) or ln[c(b)] (ii) Extraction of c(a) for uv absorption records is not difficult, since curvature is not strong near the meniscus. NB for interference optics, the optical record is NOT c(r) vs r but [c(r) - c(a)] vs r. The M* function The M* method was developed to address (i). From manipulations of equation (1), the function M*(r), at a radial position r is defined by  EMBED Equation.2  (6) where k =  EMBED Equation.2 , The M*(r) function has several interesting properties. The most important: M*(r) extrapolated to the cell base = Mw,app (over the whole distribution). {the analogy is with the computer "Pacman" game, with the M*(r) gradually homing in on the true Mw,app as eventually the whole distribution of solute in the ultracentrifuge cell has been considered by the time r=b has been reached}. The extrapolation of M*(r) to find Mw,app is less steep than the c(r) or ln[c(r)] extrapolations to the cell base and has been tested by extensive simulation Two component mixtureVery polydisperseSelf-association (isodesmic, no error)Self-association (isodesmic, random error)Single solute, random errorln[c(r)]659000473000245000215000499000M*(r)664000487000277000263000500000True value667000487000302000302000500000 Please see the SEDFIT-MSTAR manual for full operating instructions References: Creeth JM & Harding SE (1982)  HYPERLINK "http://www.nottingham.ac.uk/ncmh/documents/papers/Paper12.PDF" Some observations on a new type of point average molecular weightJ. Biochem. Biophys. Methods.7,25-34 Schuck, P., Gillis, R.B., Besong, D., Almutairi, F.M., Adams, G.G., Rowe, A.J. & Harding, S.E.,2014. HYPERLINK "http://pubs.rsc.org/en/content/articlepdf/2013/an/c3an01507f?page=search" SEDFIT-MSTAR: Molecular weight and molecular weight distribution analysis of polymers by sedimentation equilibrium in the ultracentrifuge.Analyst.139,79-92 Questions Give the relation between concentration c (mg/ml) (i) and absorbance A and (ii) and interference fringe displacement J Give the relation between the relative radial displacement function x(r) and r, a and b. 3. 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The buffer had a density ro of 1.0021 g/ml and the partial specific volume  EMBED Equation.2  was 0.739 ml/g (calculated from the amino acid sequence). The operating speed was 20410 rev/min and temperature was 20.0oC. R=8.31434 x 107 c.g.s. units. The molecular weight from protein sequencing was 24123 Da. What can you conclude about this particular protein? Is this surprising? 4. For ribonuclease M = 13683 Da,  EMBED Equation.2  = 0.694 ml/g In a sedimentation equilibrium experiment with a solution of ribonuclease containing 0.780 mg/ml in a very dilute buffer at 25.00oC and a rotor speed of 18,600 rpm, calculate the equilibrium concentrations at the cell bottom (r = b) and meniscus (r=a), if b and a = 7.013 and 6.750 cm respectively [Hint: Calculate C(b) - C(a) and C(b)/C(a); combine]. What is the exact concentration half-way down the solution column? Ribonuclease is not self-associating within this concentration range. 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