Computational modeling problem Kinetic parameter

In biochemical engineering we are often faced with the problem of estimating average apparent growth or uptake/secretion rates. Such estimates are particularly useful when we compare the productivity of a culture under different operating conditions or modes of operation. Such computations are routinely done by analysts well before any attempt is made to estimate true kinetics parameters like those appearing in the Monod growth model for example. 

Once the kinetic parameters have been estimated, (3.60) becomes linear in the unknown parameters A Hr . Therefore, the errors between the total heat of reaction, computed via the detailed model, and the total heat, computed via each reduced model, can be minimized by resorting to the least squares solution of a linear regression problem, discussed in Sect. 3.4. The molar heats of reaction, included in the vector of parameters... 

In theory, by feeding the MWD and experimental rate data into a mathematical model containing a variety of polymerization mechanisms, it should be possible to find the mechanism which explains all the experimental phenomena and to evaluate any unknown rate constants. As pointed out by Zeman (58), as long as there are more independent experimental observations than rate parameters, the solution should, in principle, be unique. This approach involves critical problems in choice of experiments and in experimental as well as computational techniques. We are not aware of its having yet been successfully employed. The converse— namely, predicting MWD from different reactor types on the basis of mathematical models and kinetic data—has been successfully demonstrated, however, as discussed above. The recent series of interesting papers by Hamielec et al. is a case in point. 

Initially the model was compiled using both experimentally measured and theoretically calculated kinetic parameters. Then, the results of simulations were compared with the data of multiple experiments and sensitivity analysis was employed to select the parameters, which should be corrected for the better agreement between experimentally observed and simulated kinetic behavior. The computation routine can perform the modification of each kinetic parameter within the range of its initial uncertainty. Such an approach gives a serious cause for criticism, since the discrepancies with experimental data are eliminated (or minimized) by changing the values of multiple parameters. First, this makes all of them correlated. Next, an independent correction of just one parameter in the model, or just a slight modification of the micro-chemical scheme leads to the readjustment of the whole system of kinetic parameters. This is in a certain sense equal to the solution of the inverse kinetic task, which, as we mentioned above, is an ill-conditioned problem. 

Computer modeling of the kinetics of a reaction by solving rate equations is useful in the determination of mechanism and the estimation of rate parameters. Such analysis of kinetic data represents a higher-level approach to the problem of modeling than the molecular modeling discussed above. Here, we assume knowledge of the bulk properties of the system (the kinetic equations) and proceed to model the system comparing predictions to experimental measurements. 

Most problems associated with approximate kinetics are avoided when Michaelis Menten-type rate equations are utilized. Though this choice sacrifices the possibility of analytical treatment, reversible Michaelis Menten-type equations are straightforwardly consistent with fundamental thermodynamic constraints, have intuitively interpretable parameters, are computationally no more demanding than logarithmic functions, and are well known to give an excellent account of biochemical kinetics. Consequently, Michaelis Menten-type kinetics are an obvious choice to translate large-scale metabolic networks into (approximate) dynamic models. It should also be emphasized that simplified Michaelis Menten kinetics are common in biochemical practice almost all rate equations discussed in Section III.C are simplified instances of more complicated rate functions. 

A direct kinetic problem consists of calculating multi-component reaction mixture compositions and reaction rates on the basis of a given kinetic model (both steady-state and unsteady-state) with the known parameters. Reliable solution for the direct problem is completely dependent on whether these parameters, obtained either on theoretical grounds or from special experiments, have reliable values. Modern computers can solve high-dimensional problems. Both American and Soviet specialists have calculated kinetics for the mechanisms with more than a hundred steps (e.g. the reac-... 

Identification. In most cases, the mathematical models of interest in industry contain a few parameters whose values, essentially unknown a priori, must be computed on the basis of the available experimental data. In the case considered here, chemical kinetics is the main field in which this problem is of concern. Identification provides methods for obtaining the best estimates of those parameters and for choosing (i.e., identifying) the best mathematical model among different alternatives. 

Rawlings and co-workers proposed to carry out parameter estimation using Newton s method, where the gradient can be cast in terms of the sensitivity of the mean (Haseltine, 2005). Estimation of one parameter in kinetic, well-mixed models showed that convergence was attained within a few iterations. As expected, the parameter values fluctuate around some average values once convergence has been reached. Finally, since control problems can also be formulated as minimization of a cost function over a control horizon, it was also suggested to use Newton s method with relatively smooth sensitivities to accomplish this task. The proposed method results in short computational times, and if local optimization is desired, it could be very useful. 

---