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

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.List of Symbols 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 k₂ and zeros - C mol/m³ molar concentration of S - C mol/m³ vector of molar concentrations of C_i (i=0, 1, 2, ..., N) - C₀ 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 - r₁ mol/m³ s rate of formation of S₀ - 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 s^1-j eigenvectors associated with eigenvalue imi (i=1, 2, ..., L; j =1, 2, ..., M_i)Greek Symbols _ij mol/m³ arbitrary constant associated with eigenvalue _i (i=1, 2, ..., L; j=1, 2, ..., M_i) - 1/s generic eigenvalue - _i 1/s i-th eigenvalue