Statistical methods For Fermentation Optimization Edwin O. Geiger 1.0 INTRODUCTION A common problem for a biochemical engineer is to be handed a microorganism and be told he has six months to design a plant to produce th new fermentation product. Although this seems to be a formidable task, with the proper approach this task can be reduced to a manageable level. There are many ways to approach the problem of optimization and design of a fermentation process. One could determine the nutritional requirements of the organism and design a medium based upon the optimum combination of each nutrient, i.e., glucose, amino acids, vitamins, minerals, etc. This approach has two drawbacks. First, it is very time-consuming to study each nutrient and determine its optimum level, let alone its interaction with other nutrients. Secondly, although knowledge of the optimal nutritional require- ments is useful in designing a media, this knowledge is difficult to apply when economics dictate the use of commercial substrates such as corn steep liquor, soy bean meal, etc, which are complex mixtures of many nutrie 2.0 TRADITIONAL ONE- VARIABLE-AT-A-TIME METHOD The traditional approach to the optimization problem is the one- variable-at-a-time method In this process, all variables but one are held constant and the optimum level for this variable is determined. Using this
Statistical Methods For Fermentation Optimization Edwin 0. Geiger 1.0 INTRODUCTION A common problem for a biochemical engineer is to be handed a microorganism and be told he has six months to design a plant to produce the new fermentation product. Although this seems to be a formidable task, with the proper approach this task can be reduced to a manageable level. There are many ways to approach the problem of optimization and design of a fermentation process, One could determine the nutritional requirements of the organism and design a medium based upon the optimum combination of each nutrient, i.e., glucose, amino acids, vitamins, minerals, etc. This approach has two drawbacks. First, it is very time-consuming to study each nutrient and determine its optimum level, let alone its interaction with other nutrients. Secondly, although knowledge of the optimal nutritional requirements is useful in designing amedia, this knowledge is difficult to apply when economics dictate the use of commercial substrates such as corn steep liquor, soy bean meal, etc., which are complex mixtures of many nutrients. 2.0 TRADITIONAL ONE-VARIABLE-AT-A-TIME METHOD The traditional approach to the optimization problem is the onevariable-at-a-time method. In this process, all variables but one are held constant and the optimum level for this variable is determined. Using this 161
162 Fermentation and Biochemical Engineering Handbook optimum, the second variable's optimum is found, etc. This process works if, and only if, there is no interaction between variables. In the case shown in Fig. l, the optimum found using the one-variable-at-a-time approach was 85%, far from the real optimum of 90%. Because of the interaction between the two nutrients, the one-variable-at-a-time approach failed to find the true optimum. In order to find the optimum conditions, it would have been necessary to repeat the one-variable-at-a-time process at each step to verify that the true optimum was reached. This requires numerous sequential experimental runs, a time-consuming and ineffective strategy, especially when many variables need to be optimized. Because of the complexity of microbial metabolism, interaction between the variables is inevitable, espe- cially when using commercial substrates which are a complex mixture of many nutrients. Therefore, since it is both time-consuming and inefficient, the one-variable-at-a-time approach is not satisfactory for fermentation development. Fortunately, there are a number of statistical methods which will find the optimum quickly and efficiently. 3.0 EVOLUTIONARY OPTIMIZATION An alternative to the one-variable-at-a-time approach is the technique of evolutionary optimization. Evolutionary optimization(EVOP), also known as method of steepest ascent, is based upon the techniques developed by Spindle, et al. Q] The method is an iterative process in which a simplex figure is generated by running one more experiment than the number of variables to be optimized. It gets its name from the fact that the process slowly evolves toward the optimum. A simplex process is designed to find the slope, i.e., path with greatest increase in yield The procedure starts by the generation of a simplexfigure. The simplex figure is triangle whentwo variables are optimized, a tetrahedron when three ariables are optimized, increasing to an n+l polyhedron, where n is the number of variables to be optimized. The experimental point with the poorest response is eliminated and a new point generated by reflection of the eliminated point through the centroid of the simplex figure. This process is continued until an optimum is reached. In Fig. 2, experimental points 1, 2 and 3 form the vertices of the original simplex figure. Point I was found have the poorest yield, and therefore was eliminated from the simplex figure and a new point(B )generated. Point 3 was then eliminated and the new point ( C)generated. The process was continued until the optimum was reached The EVOP process is a systematic method of adjusting the variables until an optimum is reached
162 Fermentation and Biochemical Engineering Handbook optimum, the second variable's optimum is found, etc. This process works if, and only if, there is no interaction between variables. In the case shown in Fig. 1, the optimum found using the one-variable-at-a-time approach was 85%, far from the real optimum of 90%. Because of the interaction between the two nutrients, the one-variable-at-a-time approach failed to find the true optimum. In order to find the optimum conditions, it would have been necessary to repeat the one-variable-at-a-time process at each step to verify that the true optimum was reached. This requires numerous sequential experimental runs, a time-consuming and ineffective strategy, especially when many variables need to be optimized. Because of the complexity of microbial metabolism, interaction between the variables is inevitable, especially when using commercial substrates which are a complex mixture of many nutrients. Therefore, since it is both time-consuming and inefficient, the one-variable-at-a-time approach is not satisfactory for fermentation development. Fortunately, there are a number of statistical methods which will find the optimum quickly and efficiently. 3.0 EVOLUTIONARY OPTIMIZATION An alternative to the one-variable-at-a-time approach is the technique of evolutionary optimization. Evolutionary optimization (EVOP), also known as method of steepest ascent, is based upon the techniques developed by Spindley, et al.['] The method is an iterative process in which a simplex $figure is generated by running one more experiment than the number of variables to be optimized. It gets its name from the fact that the process slowly evolves toward the optimum. A simplex process is designed to find the optimum by ascending the reaction surface along the lines of the steepest slope, Le., path with greatest increase in yield. The procedure starts by the generation of a simplex figure. The simplex figure is atriangle when two variables are optimized, a tetrahedron when three variables are optimized, increasing to an n+l polyhedron, where n is the number ofvariables to be optimized. The experimental point with the poorest response is eliminated and a new point generated by reflection of the eliminated point through the centroid of the simplex figure. This process is continued until an optimum is reached. In Fig. 2, experimental points 1 , 2, and 3 form the vertices of the original simplex figure. Point 1 was found to have the poorest yield, and therefore was eliminated from the simplex figure and a new point (B) generated. Point 3 was then eliminated and the new point (C) generated. The process was continued until the optimum was reached. The EVOP process is a systematic method of adjusting the variables until an optimum is reached
5,89 2 3.75 2,5 Apparent optimum 9 1,25 1,25 2.59 3,75 5即 NUTRIENT 1 Figure 1. Example of one-variable-at-a-time approach. Contour plot of yield
Statistical Methods for Fermentation Optimization I63 i 0 (u a c( 0 c)
5,80 4 3,8 98 7 22,9 1,8 1,9 2,8 4,8 5, NUTRIENT 1 Figure 2. Example of evolutionary optimization contour plot of yield
164 Fermentation and Biochemical Engineering Handbook
Statistical Methods for Fermentation Optimization 165 Numerous modifications have been made to the original simplex method. One of the more important modifications was made by Nelder and Mead( 2 who modified the method to allow expansions in directions which are favorable and contractions in directions which are unfavorable. This modification increased the rate at which the optimum is found important modifications were made by Brissey 3] who describes a high algorithm, and Keefer 4 who describes a high speed algorithm and methods dealing with bounds on the independent variables Additional modifications were reported by Nelson, 51 Bruley, [61 Deming, 9]and Ryan I8 For reviews on the simplex methods see papers by Deming et al.[9]-[ll] evoP does have its limitations. first because of its iterative nature it is a slow process which can require many steps. Secondly, it provides only limited information about the effects of the variables. Upon completion of the EvoP process only a limited region of the reaction surface will have been explored and therefore, minimal information will be available about the effects of the variables and their interactions. This information is necessary to determine the ranges within which the variables must be controlled to insure optimal operation. Further, EVOP approaches the nearest optimum It is unknown whether this optimum is a local optimum or the optimum for the entire process Despite the limitations, EVoP is an extremely useful optimization technique. EVOP is robust, can handle many variables at the same time, and will always lead to an optimum. Also, because of its iterative nature, little needs to be known about the system before beginning the process. Most important, however, is the fact that it can be useful in plant optimization where the cost of running experiments using conditions that result in low ields or able product cannot be tolerated. In theory, the proce improves at each step of the optimization scheme, making it ideal for a production situation, For application of EvoP to plant scale operations, see Refs.12-14. The main difficulty with using EVOP in a plant environment is performing the initial experimental runs. Plant managers are reluctant to run at less than optimal conditions. Attempts to use process data as the initial experiments in the simplex is, in general, not successful because of confound- ng. Confounding occurs because critical variables are closely cor and therefore, the error in measuring the conditions and results tend to be greater than theeffect of the variables. Because of this, operating data usually gives a false perspective as to which variables are important and the changes to be made for the next step
Statistical Methods for Fermentation Optimization I65 Numerous modifications have been made to the original simplex method. One of the more important modifications was made by Nelder and Mead[?] who modifiedthe method to allow expansions in directions which are favorable and contractions in directions which are unfavorable. This modification increased the rate at which the optimum is found. Other important modifications were made by Bris~ey[~I who describes a high speed algorithm, and KeeferL4I who describes a high speed algorithm and methods dealing with bounds on the independent variables. Bruley,I6I Deming,ig] and Ryan.[8] For reviews on the simplex methods see papers by Deming et al.[9]-[11] EVOP does have its limitations. First, because of its iterative nature, it is a slow process which can require many steps. Secondly, it provides only limited information about the effects ofthe variables. Upon completion ofthe EVOP process only a limited region of the reaction surface will have been explored and therefore, minimal information will be available about the effects of the variables and their interactions. This information is necessary to determine the ranges within which the variables must be controlled to insure optimal operation. Further, EVOP approaches the nearest optimum. It is unknown whether this optimum is a local optimum or the optimum for the entire process Despite the limitations, EVOP is an extremely usefbl optimization technique. EVOP is robust, can handle many variables at the same time, and will always lead to an optimum. Also, because of its iterative nature, little needs to be known about the system before beginning the process. Most important, however, is the fact that it can be useful in plant optimization where the cost of running experiments using conditions that result in low yields or unusable product cannot be tolerated. In theory, the process improves at each step of the optimization scheme, making it ideal for a production situation. For application of EVOP to plant scale operations, see Refs. 12-14. The main difficulty with using EVOP in a plant environment is performing the initial experimental runs. Plant managers are reluctant to run at less than optimal conditions. Attempts to use process data as the initial experiments in the simplex is, in general, not successful because of confounding. Confounding occurs because critical variables are closely controlled, and therefore, the error in measuring the conditions and results tend to be greater than the effect ofthe variables. Because ofthis, operating data usually gives a false perspective as to which variables are important and the changes to be made for the next step. Additional modifications were reported by