Calibration of mechanistic kinetic models describing microorganism growth and secondary metabolite production on solid substrates is a difficult problem owing to model complexity and the sheer number of parameters needing to be estimated. We show how advanced non-linear programming techniques can be applied to achieve fast and accurate calibration of a complex kinetic model describing the growth of Gibberella fujikuroi and production of gibberellic acid on an inert solid support in glass columns. Experimental culture data was obtained under several temperature and water activity conditions. The model’s differential equations were discretized using efficient mathematical methods while a special purpose optimization code (IPOPT) was applied to solve the resulting large-scale non-linear program. Convergence proved much faster, a better fit of certain variables was achieved over previous approaches and statistical analysis showed most parameter estimates to be accurate.

Solid substrate fermentation (SSF) is best defined as the cultivation of microorganisms on solid substrates devoid of or deficient in free water (Pandey, 2003). SSF has several advantages (Holker and Lenz, 2005) over the more conventional submerged fermentation. Many promising lab-scale SSF processes are reported periodically; unfortunately they seldom enter commercial production due to the magnitude of the technical difficulties in operating and optimizing large scale SSF bioreactors. Since modern process control and optimization engineering techniques are model-based, mathematical modelling should significantly improve the likelihood of getting a laboratory-scale SSF process into commercial production. Nevertheless, a number of factors make modelling SSF processes particularly trying: the absence of reliable on-line measurements of crucial cultivation variables and inherent system complexity. In addition, dynamic models are highly complex involving numerous parameters that have to be estimated from extensive, good quality experimental data. Acquiring such data is costly and time consuming, yet even when available, establishing accurate parameter estimates from the data is far from trivial (Gelmi et. al. 2002).

Kinetic model

The model describes the temporal evolution of concentrations of total and active fungi, urea, assimilable nitrogen, starch and gibberellins, as well as carbon dioxide production and oxygen consumption rates. It was calibrated under four culture conditions: Temperature = 25ºC, Water Activity = 0.992; Temperature = 25ºC, Water Activity = 0.999; Temperature = 31ºC, Water Activity = 0.985; and Temperature = 31ºC, Water Activity = 0.992.

Parameter estimation

The simultaneous (SimSO) approach was adopted to solve the parameter estimation problem. Here, the set of differential equations were discretized using a fast and efficient mathematical method (orthogonal collocation on finite elements) and model parameters were estimated by weighted least squares. This resulted in a large set of non-linear algebraic equations, which were solved with IPOPT (Biegler et al. 2002), a special purpose non-linear programming package especially designed to deal with very large problems.

In a comparison of our calibration results with the fit Gelmi et al (2002) obtained, we were specifically interested in fit quality, performance of the optimization procedure and the value and accuracy of the estimated parameters. Statistical analysis showed that most of parameter estimates with the SimSO method were accurate. Most parameter estimates obtained using the two methods differed significantly. However, our fit quality was better for crucial variables, such as biomass, gibberellins and starch, as illustrated in Figure 1.

Comparing cultures at 25ºC and 31ºC, keeping water activity levels constant, we found that twice as much biomass was produced at 25ºC, while at 31ºC slightly more gibberellins were produced. Also, at 25ºC carbon dioxide production and oxygen consumption were a bit higher. Comparing cultivations at different water activities (a_w = 0.992 and a_w = 0.999) holding the temperature at 25ºC, we found that around 30% more biomass and twice as much GA₃ were produced for a_w = 0.992.

The key contribution of this study is the significant reduction in convergence time achieved using SimSO. The approach simplified the parameter estimation problem enormously. Typical calibration using the standard approach - described in Gelmi et al. (2002) - took over a week to complete. The standard procedure entailed many runs of the optimization program, each taking several hours to converge and required substantial use of heuristics. In this regard, the method described here is a valuable tool that should contribute appreciably to the development and testing of complex SSF models.

The authors thank Alex Crawford for his assistance in improving the style of the text.

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