Hydrodynamic vs Thermodynamic non-ideality
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Transcript of Hydrodynamic vs Thermodynamic non-ideality
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Treatment of Non-ideality in Sedimentation Velocity
Experiments
Walter Stafford,Boston Biomedical Research Institute,
Watertown, MA 02472 U.S.A.
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Hydrodynamicvs
Thermodynamicnon-ideality
s f (c)Dg(c)
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s f (c)Generally empirical
backflow (A.J. Rowe …)charge effects (Donnan equilibrium)
Except for large asymmetric proteins,charge effects usually dominate backflow.
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Dg(c)Both hydrodynamic and thermodynamic contributions
Except for large asymmetric proteins,charge effects usually dominate excluded volume effects.
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f (c) f (0) 1K sc
Frictional coefficient
HYDRODYNAMIC
Affects both s and D
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Hydrodynamic c-dependence= same for s and D
sso
1 ksc and D
Do1 ksc
FC=gradient of centrifugal potential FD=gradient of chemical potential
frictional resistance is the same for both
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Thermodynamicconcentration dependence
1 ln(y)
ln(c)
Excluded volume and Donnan equilibrium
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Colligative virial coefficients
ln(y) BcCc2 Dc3 ...
B, C and D are the 2nd, 3rd and fourth virial colligative coefficients, respectively.(McMillan and Mayer, 1945, statistical thermodynamics)
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Combined hydrodynamic and thermodynamic concentration
dependence of D
D D o
1 ln(y)
ln(c)
1 ksc
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ln(yi ) Bici Cici2 Dic i
3 ...
ci ln(yi )
ci
ln(yi )
ln(ci )2Bci 3Cci
2 4Dci3 ...
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DDo12BM1c
1 ksc
Expansion truncated after the 2nd virial term …
(Cf. Also Harding and Johnson, 1985; and Alex Solovyova et al, 2001)
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DDo 12BM1c ksc
Expansion truncated after the 2nd virial term …And further approximation …
(both Harding and Johnson, 1985; and Alex Solovyova et al, 2001)
1
1ax1 ax ... for small a
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Multiple Species
Self-associating system
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ln(yi ) Bici Cici2 Dic i
3 ...
ln(yi ) Biici Bijc jji
n
Ciici2 Cijkc jck
ji
n
ki
n
Diic i3 Dijklc jckcl
li
n
ki
n
ji
n
...
Cross terms …
Single species (no cross terms)
Multiple species (cross terms)
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Self-associating system all species must have the same charge to mass ratio
andfrictional ratio
However …
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Di Dio 12BM1cT
1 ksjc jj1
n
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g(s) 6
Non-ideal reversible monomer-dimer system
Protein-X
Lowest cIntermediate c
Highest c
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Sedanal
Global fitting with SEDANAL
= 280 nm and 220 nm; path length = 12mm and 3 mm
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Global fitting with SEDANAL
= 280 nm; path length = 12mm and 3 mm
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Future plans
Extend to higher concentrations
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Higher order terms
Add higher order terms …
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ln(yi )
ln(ci )2Bci 3Cci
2 4Dci3 ...
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Donnan equilibriumFor uni-univalent electrolyte
e.g. NaCl
BZ 2
4M22m3
C 0
D Z 4
64M24m3
3
Roark and Yphantis 1971 Biochemistry
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BZ 2
4M22m3
DB2
4m3
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"Significant" is >= 1% contribution
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Excluded volume effects
BM1 4vL
d
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Hard sphere approx.or some other model
ln(i) f (c,{a})
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Hard Sphere models
Minton 2007 J. Pharm Sci.
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Minton 2007, Biophys J.
Scattering …
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M1 M1,app 1 ln(y1)
ln(c1)
ln(y1)
ln(c j )M j ,app
ji
n
M1 ln(y1)
ln(c j )M j ,app
ji
n
M1,app 1 ln(y1)
ln(c1)
M1,app
M1 1 ln(y1) ln(c j )
M j ,app
M1ji
n
1 ln(y1) ln(c1)
M1,app
M1 1 ln(y1) ln(c j )
M j ,app
M1ji
n
1 ln(y1) ln(c1)
sRTc
1 ln(y1) ln(c j )
M j ,app
M1ji
n
D 1 ln(y1) ln(c1)
D1,app D1 1 ln(y1)
ln(c1)
1 ln(y1) ln(c j )
M j ,app
M1ji
n
Poly-disperse system
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D1,app D 1 ln(y1)
ln(c1)
1 ln(y1) ln(c j )
M j ,app
M1ji
n
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Conclusions
Non-ideality in interacting systemsSimple linear approachExpand to higher order terms of virial
expansion with cross-termsOr use direct function
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TMP
TURBO-MOLECULAR PUMP