# Multistage affinity cross-flow filtration: mathematical modelling and analysis

## Abstract

A multistage affinity cross-flow filtration (mACFF) process for protein purification is proposed. The process is mathematically modelled taking into account a case of rapid equilibrium binding of a target protein to its macroligand. The process performance, i.e., dimensionless breakthrough volume (Q + b )and recovery yield (REC) to obtain a desired purity is analysed by computer simulations. The results indicate that Q + b increases with the increase of stage number (n) due to the increase of affinity binding efficiency. In addition, REC also increases with the increase of n, especially for lower affinity systems, even though the feed loading is the same as the corresponding breakthrough volume that increases with n. Thus both feed loading and recovery yield can be enhanced by raising the stage number. Incompletely permeable membranes reject the target and contaminant proteins. So they delay the appearance of the breakthrough point and compromise the contaminant washing efficiency. Hence although Q + b increases with the increase of membrane rejection coefficient (R), REC decreases when the feed loading equals that of Q + b . However, when the feed loading is kept unchanged and equals Q + b at R=0, REC does not decrease, but slightly increases with the increase of R. This result indicates that incompletely permeable membranes may also be employed for the mACFF process. In general, the model gives a predictive evaluation of the mACFF process successfully.

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## Abbreviations

C mol/l:

contaminant concentration

F l/min:

flow rate

f :

macroligand volume fraction

j :

index for stage number

K d mol/l:

dissociation constant

n :

number of stages

P %:

purity

Q 1:

feeding or washing volume

q mol/l:

binding density

q m mol/l:

binding capacity

R :

membrane rejection coefficient

REC %:

recovery yield

T mol/l:

target protein concentration

t min:

time

V 1:

sum of contactor volumes

V j 1:

volume of the jth stage contactor

b :

breakthrough

C :

contaminant proteins

f :

feeding step

in :

feeding broth into the 1st stage

j :

the jth contactor

o :

out of contactor (filtrate)

T :

target protein

w :

washing step

+:

dimensionless quantity

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## Author information

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Correspondence to Y. Sun.

## Additional information

This work was supported by the National Natural Science Foundation of China.

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### Cite this article

Dong, X.Y., Li, L.Y. & Sun, Y. Multistage affinity cross-flow filtration: mathematical modelling and analysis. Bioprocess Engineering 15, 99–104 (1996). https://doi.org/10.1007/BF00372984

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### Keywords

• Waste Water
• Affinity Binding
• Permeable Membrane
• Process Performance
• Protein Purification