EPSRC Portfolio Partnership in Complex Fluids and Complex Flows

Use Of Protein Structure Data For The Prediction Of Ultrafiltration Separation Processes

Complete Amino Acid Sequence for BSA

Knowing the amino acid sequence of the protein enables the calculation of the total molecular weight and the specific volume. From these parameters it is possible to calculate the hydrodynamic size of the protein.

For BSA the following quantities are obtained:

Dry hard sphere radius 2.69 nm

Hydrodynamic radius 3.20 nm

The latter allows for a modified Gouy-Chapman model of the protein/solution interface with the Outer Helmholtz Plane at the distance of closest approach of hydrated sodium ions.

Amino Acid

Number of Amino Acids in

Molecule

Mass of Amino Acid

(Daltons)

Total Weight

(Daltons)

Weight % Specific Volume of Amino Acid

(Weight % x Sp. Vol) /

100

Ala (A) 46 71.09 3270.14 4.93 0.74 0.0365Arg (R) 23 156.19 3592.37 5.42 0.7 0.0379Asn (N) 13 114.11 1483.43 2.24 0.59 0.0132Asp (D) 41 115.09 4718.69 7.12 0.6 0.0427Cys (C) 35 103.15 3610.25 5.45 0.61 0.0332Gln (Q) 20 128.14 2562.8 3.87 0.67 0.0259Glu (E) 59 129.12 7618.08 11.49 0.66 0.0759Gly (G) 16 57.05 912.8 1.38 0.64 0.0088His (H) 17 137.14 2331.38 3.52 0.67 0.0236Ile (I) 14 113.16 1584.24 2.39 0.9 0.0215Leu (L) 61 113.16 6902.76 10.41 0.9 0.0937Lys (K) 59 128.17 7562.03 11.41 0.82 0.0935Met (M) 4 131.19 524.76 0.79 0.75 0.0059Phe (F) 27 147.18 3973.86 5.99 0.77 0.0462Pro (P) 28 97.12 2719.36 4.1 0.76 0.0312Ser (S) 28 87.08 2438.24 3.68 0.63 0.0232Thr (T) 34 101.11 3437.74 5.19 0.7 0.0363Trp (W) 2 186.21 372.42 0.56 0.74 0.0042Tyr (Y) 19 163.18 3100.42 4.68 0.71 0.0332Val (V) 36 99.14 3569.04 5.38 0.86 0.0463Total 582 - 66 284.81 100 - 0.7329

Amino acid sequence from Brown and Shockley (1982).

Theory – BSA Surface Charge

To describe the specific binding of chloride ions to a BSA molecule, a modification of an expression developed by Scatchard et al. (1950) has been used:

Z = 440 Cl exp -ze

kT

1 + 44 Cl exp -zekT

+ 33 Cl exp -ze

kT

1 + 1.1 Cl exp -zekT

Cl

- o

- o

- o

- o-

The total charge due to the amino acids on the surface of the BSA molecule may be calculated using the acid base equilibria data in the table above,

-+AB Z - Z = Z

The charge distribution is described by the non-linear PBE in spherical co-ordinates,

From the solution of the PBE, the charge density can be determined, 

 

The surface potential of the particle, may also then be determined from,

 

 

From electroneutrality,

kTezsinh ze2n

drd

r2

drd

ro

o

2

2

drord dr

d

dro

oo

d) (a ad

2T

do a 4eZ -

pH

4 5 6 7 8 9 10 11

Zeta

Pot

entia

l /(m

V)

-60

-50

-40

-30

-20

-10

0

10

Measured ValuesPredicted Values

-log [Cl-]

0.5 1.0 1.5 2.0 2.5 3.0

Zeta

Pot

entia

l /(m

V)

-60

-50

-40

-30

-20

-10

Measured ValuesPredicted Values

Dependence on pH at an ionic strength of 0.03M NaCl.

Dependence on ionic strength of NaCl at pH 8.0.

Theory – Osmotic Pressure1) Electrostatic interactions

2) London-van der Waals interactions

  The attractive force, may then simply be calculated from:

3) Entropic interactions

1 -

kTze

coshkTnS 31 = (D) f o

2

2

2

2

2

2H

A 2a + D4a - 1ln +

2a + D2a +

4aD + D2a

6A- = (D)V

dDdV

-= F AATT

32

65432

3ENT

1.04305 - 3 + 3 - 1X - 0.5 - - 0.67825 - + + 1

a43 =

kTP 3.9Y))- exp(Y(7.9 6.028 X

- 62 Y

where f (D) is the configurational electrostatic force

FATT is the dispersion force

PENT is the entropic pressure

Ah is the effective area per particle

(as suggested by Evans and Napper (1978))

This is defined by:

 

 Thus the overall interparticle interactions may be represented by the osmotic pressure.

ENTATTh

P + F (D) fA

6 =

2h D/2) (a 32 A

Comparison of the osmotic pressure predictions with the experimental data

of Vilker et al. (1981) for BSA at pH 5.4 and 7.4 in 0.15M NaCl.

It is assumed that the electrostatically stabilised dispersion exists in a structurally regular packing form of minimum energy, hexagonal close packing. Using this assumption, the osmotic pressure at any point in the dispersion may be calculated via the summation of the colloidal interactions occurring.

Theory – Gradient Diffusion Coefficient

1) S(), the thermodynamic coefficient:This may be calculated from the osmotic pressure which in turn depends on the particle-particle interactions.

a6kT = D0

D( ) = D K( )S( )

= D K( ) ( ) 4 a3kT0 0

3

3kT

a4 )(

)S(

1 3

2) K(), the hydrodynamic interaction coefficient:

Disordered system equation:

Ordered system equation:

- 1 )K(

2a eff dr kT)]exp(-V(r)/ - [1 2a d

dr (r)F - V(r) TOT

r

(r)F f(r) (r)F ATTTOT

3

effo/d d

2a 0.55

Comparison of theoretical predictions with experimental results for BSA at

pH 6.5 and various ionic strengths.

Theory - Viscosity

where is the relative viscosity is the viscosity of the solution is the viscosity of the solvent is the intrinsic viscosity (= 2.5 for spheres)

 

)O( ][ 1 o

sr

3

effmax d

2a 0.7404

and

Perturbation Theory Method:

From the interaction energy it is possible to calculate an effective hard sphere diameter for charged particles,

2 0.3079 - 0.4842 0.6233 - 1.2989 0.0602 - 1.7876 - 1 )K( 3

53

43

23

1

rso][

Comparison of the viscosity predictions with the experimental data of Kozinski and Lightfoot (1972) for BSA at pH 6.7 in a 0.1M and 0.5M

buffer.

 Krieger-Dougherty Equationmax ]-[

maxr - 1

where max is the maximum packing value

Einstein

For a face centred cubic array of hard spheres the maximum packing fraction is 0.7404. Therefore the index the Krieger-Dougherty equation is 1.85 for hard spheres.

Affect of Colloidal Interactions on ViscosityThe new maximum packing fraction may be determined from:  

The intrinsic viscosity, [], is also affected by the colloidal interactions so this must be taken into account when using the Krieger-Dougherty equation.

Dilute Limit:The Stokes-Einstein equation:

Concentrated suspension:The generalised Stokes-Einstein equation:

K() is the hydrodynamic interaction coefficientS() is the thermodynamic coefficient

Knowing o/d, may be evaluated by equating the disordered and ordered equations.

The new order/disorder transition may now be determined:

Where

Theory – Frontal Filtration

Starting from a Darcy type expression and taking interparticle interactions into account gives rise to three equations that describe the time course of filtration:

HAPPAob

mm

2

K / - 1 ) - (1

1 - C1RAV

dtdV

dtdV

dtd

dVdt

= z zA

R + p = (t)pm

mo32

where V is the total volume filtered, t is the time, Cb is the particle bulk concentration, Rm is the membrane resistance, Am is the membrane area, p3 is the pressure on the reverse side of the membrane, p2(t) is the pressure at the interface between the membrane and filter cake, is the local voidage of the cake, KHAPP

is the local value of the Kozeny coefficient, oA accounts for the electroviscous effects.

Comparison of predictions with experiment for various conditions of ionic strength and pH.

At membrane wall, y = 2h

y)(x,u

yx

membrane

layerionconcentratdeveloping

(x)vw

suspensionbulk

wallsolid

0 yv x

u

yu

y xp1- y

u v xuu

oc c 0 v

2

2

o hh) -(y - 1 1.5u u

0 u 0 v 0 yc

0 u w

ww

Dcv

yc

ycDy y

c v xcu

msw r

- p v

Continuity equation:

Momentum equation:

Mass balance:

At inlet, x = 0:

At non-porous wall, y = 0:

This boundary condition represents no slip condition as well as the solute transport at the non-porous wall.

Theory – Cross Flow filtration

Permeate flux versus applied pressure for a 10g l-1 solution of BSA at pH 10.0 in 0.03M NaCl.

We are concerned with the development of predictive methods that require no adjustable parameters – that is ab initio methods that move from physics to process description. We have several projects developing methods for microfiltration, ultrafiltration and nanofiltration.

For ultrafiltration, the osmotic pressure, gradient diffusion coefficient and solution viscosity are key parameters required for these predictions.

The pertinent question to ask is: how can these three key parameters be theoretically predicted from knowledge of physical property information available in the biochemical literature?

We will use these predictions to model frontal (dead-end) and cross flow ultrafiltration. In all cases the theoretical predictions obtained are compared to experimental data.

Introduction

PRIFYSGOL CYMRU ABERTAWE

UNIVERSITY OF WALES SWANSEA