Treatment of Non-ideality in Sedimentation Velocity

Experiments

Walter Stafford,Boston Biomedical Research Institute,

Watertown, MA 02472 U.S.A.

Hydrodynamicvs

Thermodynamicnon-ideality

s f (c)Dg(c)

s f (c)Generally empirical

backflow (A.J. Rowe …)charge effects (Donnan equilibrium)

Except for large asymmetric proteins,charge effects usually dominate backflow.

Dg(c)Both hydrodynamic and thermodynamic contributions

Except for large asymmetric proteins,charge effects usually dominate excluded volume effects.

f (c) f (0) 1K sc

Frictional coefficient

HYDRODYNAMIC

Affects both s and D

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

Thermodynamicconcentration dependence

1 ln(y)

ln(c)

Excluded volume and Donnan equilibrium

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)

Combined hydrodynamic and thermodynamic concentration

dependence of D

D D o

1 ln(y)

ln(c)

1 ksc

ln(yi ) Bici Cici2 Dic i

3 ...

ci ln(yi )

ci

ln(yi )

ln(ci )2Bci 3Cci

2 4Dci3 ...

DDo12BM1c

1 ksc

Expansion truncated after the 2nd virial term …

(Cf. Also Harding and Johnson, 1985; and Alex Solovyova et al, 2001)

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

Multiple Species

Self-associating system

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)

Self-associating system all species must have the same charge to mass ratio

andfrictional ratio

However …

Di Dio 12BM1cT

1 ksjc jj1

n

g(s) 6

Non-ideal reversible monomer-dimer system

Protein-X

Lowest cIntermediate c

Highest c

Sedanal

Global fitting with SEDANAL

= 280 nm and 220 nm; path length = 12mm and 3 mm

Global fitting with SEDANAL

= 280 nm; path length = 12mm and 3 mm

Future plans

Extend to higher concentrations

Higher order terms

Add higher order terms …

ln(yi )

ln(ci )2Bci 3Cci

2 4Dci3 ...

Donnan equilibriumFor uni-univalent electrolyte

e.g. NaCl

BZ 2

4M22m3

C 0

D Z 4

64M24m3

3

Roark and Yphantis 1971 Biochemistry

BZ 2

4M22m3

DB2

4m3

"Significant" is >= 1% contribution

Excluded volume effects

BM1 4vL

d

Hard sphere approx.or some other model

ln(i) f (c,{a})

Hard Sphere models

Minton 2007 J. Pharm Sci.

Minton 2007, Biophys J.

Scattering …

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

D1,app D 1 ln(y1)

ln(c1)

1 ln(y1) ln(c j )

M j ,app

M1ji

n

Conclusions

Non-ideality in interacting systemsSimple linear approachExpand to higher order terms of virial

expansion with cross-termsOr use direct function

TMP

TURBO-MOLECULAR PUMP