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Determination of the optimum operating time for batch isothermal performance of enzyme-catalyzed multisubstrate reactions

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Abstract

This communication consists of a mathematical analysis encompassing the maximization of the average rate of monomer production in a batch reactor performing an enzymatic reaction in a system consisting of a multiplicity of polymeric substrates which compete with one another for the active site of a soluble enzyme, under the assumption that the form of the rate expression is consistent with the Michaelis-Menten mechanism. The general form for the functional dependence of the various substrate concentrations on time is obtained in dimensionless form using matrix terminology; the optimum batch time is found for a simpler situation and the effect of various process and system variables thereon is discussed. The reasoning developed here emphasizes, in a quantitative fashion, the fact that the commonly used lumped substrate approaches lead to nonconservative decisions in industrial practice, and hence should be avoided when searching for trustworthy estimates of optimum operation.

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Abbreviations

O 1/s:

row vector of zeros

a 1/s:

row vector of rate constants k i(i = 2,...,N)

A 1/s:

matrix of rate constants k i and k−i (i=2,...,N)

b 1/s:

row vector of rate constant k 2 and zeros

C mol/m3 :

molar concentration of S

C mol/m3 :

vector of molar concentrations of C i (i=0, 1, 2, ..., N)

C 0 mol/m3 column vector of initial molar:

concentrations of C i(i=0, 1, 2,.., N)

C −01 mol/m3 :

column vector of initial molar concentrations of C i(i=2,..., N)

C E, tot mol/m3 :

total molar concentration of enzyme molecules

C i mol/m3 :

molar concentration of S i (i=0,1,2,...,N)

C i, o mol/m3 :

initial molar concentration of S i(i=0, 1, 2, ..., N)

E:

enzyme molecule

I :

identity matrix

K 1/s:

matrix of lumped rate constants

k i 1/s:

pseudo-first order lumped rate constant associated with the formation of S i -1 (i=1, 2, ...,N)

k cat, i 1/s:

first order rate constant associated with the formation of S i-1 (i=1, 2, ..., N)

K m mol/m3 :

Michaelis-Menten constant

L :

number of distinct eigenvalues

M i :

multiplicity of the i-th eigenvalue

N :

maximum number of monomer residues in a single polymeric molecule

r 1 mol/m3 s:

rate of formation of S 0

r i mol/m3 s:

rate of release of S i -1

r ⋆ opt :

maximum average dimensionless rate of production of monomer S0

S:

lumped, pseudo substrate

S1 :

inert moiety

S i :

substrate containing i monomer residues, each labile to detachment as

S0 :

by enzymatic action (i=1,2,...,N)

t s:

time elapsed since startup of batch reaction

t lag s:

time interval required for cleaning, loading, and unloading the batch reactor

t opt s:

time interval leading to the maximum average rate of monomer production

v ij s1-j :

eigenvectors associated with eigenvalue λimi (i=1, 2, ..., L; j =1, 2, ..., Mi)

αij mol/m3 :

arbitrary constant associated with eigenvalue λ i (i=1, 2, ..., L; j=1, 2, ..., M i )

λ 1/s:

generic eigenvalue

λ i 1/s:

i-th eigenvalue

References

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Authors and Affiliations

  1. Escola Superior de Biotecnologia, Universidade Católica Portuguesa, Rua Dr. Antonio Bernardino de Almeida, 4200, Porto, Portugal

    F. X. Malcata

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  1. F. X. Malcata
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Malcata, F.X. Determination of the optimum operating time for batch isothermal performance of enzyme-catalyzed multisubstrate reactions. Bioprocess Engineering 10, 173–178 (1994). https://doi.org/10.1007/BF00387526

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  • DOI: https://doi.org/10.1007/BF00387526

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