Treatment of Non-ideality in

coriandercultureMécanique

22 févr. 2014 (il y a 3 années et 4 mois)

52 vue(s)

Treatment of Non
-
ideality in
Sedimentation Velocity
Experiments


Walter Stafford,

Boston Biomedical Research Institute,


Watertown, MA 02472 U.S.A.

Hydrodynamic

vs

Thermodynamic

non
-
ideality


s

f
(
c
)
D

g
(
c
)

s

f
(
c
)
Generally empirical


backflow (A.J. Rowe …)


charge effects (Donnan equilibrium)

Except for large asymmetric proteins,

charge effects usually dominate backflow.


D

g
(
c
)
Both hydrodynamic and thermodynamic contributions

Except for large asymmetric proteins,

charge effects usually dominate excluded volume effects.


f
(
c
)

f
(
0
)
1

K
s
c


Frictional coefficient

HYDRODYNAMIC

Affects both s and D

Hydrodynamic c
-
dependence

= same for s and D


s

s
o
1

k
s
c


and
D

D
o
1

k
s
c


F
C
=gradient of centrifugal potential

F
D
=gradient of chemical potential

frictional resistance is the same for both

Thermodynamic

concentration dependence


1


ln(
y
)

ln(
c
)






Excluded volume and Donnan equilibrium

Colligative virial coefficients


ln(
y
)

Bc

Cc
2

Dc
3

...
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

k
s
c













ln(
y
i
)

B
i
c
i

C
i
c
i
2

D
i
c
i
3

...
c
i

ln(
y
i
)

c
i


ln(
y
i
)

ln(
c
i
)

2
Bc
i

3
Cc
i
2

4
Dc
i
3

...



D

D
o
1

2
BM
1
c


1

k
s
c






Expansion truncated after the 2nd virial term …

(Cf. Also Harding and Johnson, 1985;


and Alex Solovyova et al, 2001)




D

D
o
1

2
BM
1
c

k
s
c


Expansion truncated after the 2nd virial term …

And further approximation …

(both Harding and Johnson, 1985;


and Alex Solovyova et al, 2001)


1
1

ax

1

ax

...

for small
a


Multiple Species

Self
-
associating system


ln(
y
i
)

B
i
c
i

C
i
c
i
2

D
i
c
i
3

...
ln(
y
i
)

B
ii
c
i

B
ij
c
j
j

i
n


C
ii
c
i
2

C
ijk
c
j
c
k
j

i
n

k

i
n


D
ii
c
i
3

D
ijkl
c
j
c
k
c
l
l

i
n

k

i
n

j

i
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

and

frictional ratio

However …


D
i

D
i
o
1

2
BM
1
c
T


1

k
sj
c
j
j

1
n













g(s) 6

Non
-
ideal reversible monomer
-
dimer system

Protein
-
X

Lowest c

Intermediate c

Highest c

Sedanal

Global fitting with SEDANAL

l

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



Global fitting with SEDANAL

l

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

Future plans

Extend to higher concentrations

Higher order terms

Add higher order terms …



ln(
y
i
)

ln(
c
i
)

2
Bc
i

3
Cc
i
2

4
Dc
i
3

...
Donnan equilibrium

For uni
-
univalent electrolyte

e.g. NaCl


B

Z
2
4
M
2
2
m
3
C

0
D


Z
4
64
M
2
4
m
3
3
Roark and Yphantis 1971 Biochemistry


B

Z
2
4
M
2
2
m
3
D


B
2
4
m
3
"Significant" is >= 1% contribution

Excluded volume effects


BM
1

4
v
L
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 …


M
1

M
1
,
app
1


ln(
y
1
)

ln(
c
1
)








ln(
y
1
)

ln(
c
j
)
M
j
,
app
j

i
n

M
1


ln(
y
1
)

ln(
c
j
)
M
j
,
app
j

i
n


M
1
,
app
1


ln(
y
1
)

ln(
c
1
)






M
1
,
app

M
1
1


ln(
y
1
)

ln(
c
j
)
M
j
,
app
M
1
j

i
n









1


ln(
y
1
)

ln(
c
1
)






M
1
,
app

M
1
1


ln(
y
1
)

ln(
c
j
)
M
j
,
app
M
1
j

i
n









1


ln(
y
1
)

ln(
c
1
)







sRT



c






1


ln(
y
1
)

ln(
c
j
)
M
j
,
app
M
1
j

i
n









D
1


ln(
y
1
)

ln(
c
1
)






D
1
,
app

D
1
1


ln(
y
1
)

ln(
c
1
)






1


ln(
y
1
)

ln(
c
j
)
M
j
,
app
M
1
j

i
n









Poly
-
disperse system


D
1
,
app

D
1


ln(
y
1
)

ln(
c
1
)






1


ln(
y
1
)

ln(
c
j
)
M
j
,
app
M
1
j

i
n









Conclusions


Non
-
ideality in interacting systems


Simple linear approach


Expand to higher order terms of virial
expansion with cross
-
terms


Or use direct function

TMP


TURBO
-
MOLECULAR PUMP