Corynebacterium glutamicum - JUWEL - Forschungszentrum Jülich

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2.4. Fitting Models to Experimental Data In words: the possible error in the parameter estimates is caused by the possible error in the measured values and depends on how sensitive the predicted values are towards the parameter values. The covariance matrix of the parameters is a symmetric matrix with the variances of the parameters on the diagonal. When all measurements are independent, all other entries of the matrix are zeros. A very important indication of the accuracy of the parameter estimation is the comparison of the standard deviations of the parameter estimates with the estimated parameter values themselves. These standard deviations are calculated as the square root of the diagonal entries of the covariance matrix. From the covariance matrix of the parameter estimates, the correlation matrix of the parameter estimates Corθ can easily be calculated with element i, j of the correlation matrix defined as Corθ(i, j) = Covθ(i, j) � Covθ(i, i)Covθ(j, j) (2.33) The elements of this matrix, the correlation coefficients, are values between -1 and 1 where -1 indicates a complete negative correlation, 0 means no correlation was found and 1 indicates complete correlation. Correlation coefficients close to 1 or -1 also indicate that these parameters are not estimable uniquely. In some of these cases, it might be easily possible to simplify the model without much loss of accuracy of the model predictions, or an experiment could be planned aiming at estimating this pair of parameters. 2.4.3. Accuracy of the Fitted Model After the parameters of a model are estimated, the accuracy of the model fit to the available data needs to be evaluated. A couple of options for this evaluation are mentioned here. When the measured values have a normal distributed measurement error with zero mean and standard deviation σ, the difference between the predictions by an accurate unbiased model and the measured values is also expected to follow this distribution. The agreement between the distribution of the observed errors and the expected normal distribution can be tested by a chi squared test. A reduced chi squared, ˜χ 2 ,canbe defined as (Taylor, 1982) ˜χ 2 i = 1 d N� � �2 yi − ˆyi . (2.34) i=1 where yi indicates the measured value whereas ˆyi is the corresponding model prediction. The degrees of freedom d is used to scale the chi squared value. This degree of freedom can be calculated as the difference between the amount of estimated model parameters and the amount of measurements N. For good fitting models, ˜χ 2 is expected to be about 1. All models leading to lower values are accepted and models with much higher values do not fit properly. When values of slightly more than 1 are achieved, a more accurate σi 19
2. Theory evaluation can be made by comparing the achieved ˜χ 2 with values from standard chi squared tables. The strict use of this rule might be difficult for dynamic fermentation processes. Often, models are used with success although they probably do not meet the chi squared rule. As also discussed in paragraph 2.2 on page 9, a severe simplification of the complex system has to be made in order to get models which might be suitable for process optimization and control. Furthermore it is often difficult to estimate the real standard deviation of some of the measurements. The real measurement error of dilution steps and the measurement equipment can usually easily be determined. Possible errors by non-homogeneous samples, time delays in the off-line measurements et cetera, may be more difficult to estimate. These errors can cause higher standard deviations and a relatively high amount of outliers. Therefore, the choice of whether to reject a model is usually not done solely based on a little bit too high a ˜χ 2 . Still the reduced chi squared value forms a very valuable quantitative measure for the goodness of the fit and can also be used to compare the fit of different models. The weighting by the degree of freedom in the reduced form of the chi squared value provides some favouring of more simple models with less parameters. By using the reciprocal of ˜χ 2 , so that higher values represent better fits, and by subsequent scaling of the values for all m compared models to 1, a measure of the relative goodness of fit of different models can be calculated. So, the measure Gi for model i can be calculated as: Gi = 1 ˜χ 2 i m� i 1 ˜χ 2 i (2.35) Another method of comparing the fits of competing models is the Bayes approach of calculating relative model probabilities as mentioned for instance by Box and Hill (1967) or discussed by Stewart et al. (1996, 1998). Assuming a normal distribution of the measurement error, they formulated the probability density function pi of a model prediction ˆyi,n according to model i and measurement n as � � pi,n = 1 � 2π(σ 2 + σ 2 i,n ) exp − 1 2(σ2 + σ2 2 − ˆyi,n) i,n )(y (2.36) where σ2 is the variance of the measured value and σ2 i,n is the variance of the model prediction. This last variance is also called the model variance, which is the term that will be used further in the current work. This model variance is described in more detail in paragraph 2.5.1 on page 26. Then, the relative probability of model i out of m competing models after n measurements is calculated as Pi,n−1 · pi,n Pi,n = �m i=1 Pi,n−1 . (2.37) · pi,n 20
Page 1 and 2: Lebenswissenschaften Life Sciences
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Page 6: ’Therefore, mathematical modeling
Page 9 and 10: Contents 3.4. Application: L-Valine
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Page 13 and 14: 1. Introduction need for improved m
Page 15 and 16: 2. Theory In a batch process, all s
Page 17 and 18: 2. Theory 1997; Goncalves et al., 2
Page 19 and 20: 2. Theory For batch fermentations,
Page 21 and 22: 2. Theory with Yps being the yield
Page 23 and 24: 2. Theory Inhibition Some catalytic
Page 25: 2. Theory The method of least squar
Page 29 and 30: 2. Theory We define the measured or
Page 31 and 32: 2. Theory build in the model. An ob
Page 33 and 34: 2. Theory nations, the researcher i
Page 35 and 36: 2. Theory W = J · Covθ · J T (2.
Page 37 and 38: 2. Theory arg max ξ � m� m�
Page 39 and 40: 2. Theory with all measured values
Page 41 and 42: 2. Theory which can be estimated us
Page 43 and 44: 2. Theory One option to account for
Page 45 and 46: 2. Theory θ 2 $θ 2 0 area ≅ det
Page 47 and 48: 2. Theory designed using a paramete
Page 49 and 50: 2. Theory proteome14 (Hermann et al
Page 51 and 52: 2. Theory 44
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Page 55 and 56: 3. Dynamic Modeling Framework 3.3.
Page 57 and 58: 3. Dynamic Modeling Framework 3.3.3
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Page 63 and 64: 3. Dynamic Modeling Framework For d
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4. Simulative Comparison of Model D
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5. L-Valine Production Process Deve
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A. Nomenclature Table A.1.: Symbols
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Table A.1.: Symbols and Abbreviatio
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Table A.1.: Symbols and Abbreviatio
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B. Ratio between OD600 and Dry Weig
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C. The Influence of the Oxygen Supp
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The measurements of amino acids and
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kiv [mM] lac [mM] 15 10 5 0 0 10 20
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OTR, CTR 0.06 0.04 0.02 0 0 10 20 3
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DO properly at a low value in dynam
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D. Dilution for Measurement of Amin
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E. Models used in the First Model D
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Model 1.6 Model 1.6 is similar to m
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Table E.1: Model Parameters of the
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Table E.2: Model Parameters of the
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161
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Model 2.2 Model 2.2 is similar to m
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Table F.1.: Model Parameters of the
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Table F.2.: Estimated Lag-Times of
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Table F.3: Model Parameters of the
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Table F.4.: Estimated Lag-Times of
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G. Alternative Off-line Optimized P
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G.2. Overall Yield of Product on Su
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Summary Fast development of product
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Zusammenfassung Es ist wichtig, die
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Acknowledgements This thesis result
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Bibliography K. Akashi, H. Shibai,
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Bibliography A. Bodalo-Santoyo, J.L
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Bibliography A.J. Cooper, J.Z. Gino
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Bibliography A.G. Fredrickson, R.D.
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Bibliography W.G. Hunter and A.M. R
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Bibliography A.C. McHardy, A. Tauch
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Bibliography E. Radmacher, A. Vaits
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Bibliography K.J. Versyck, J.E. Cla
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Corynebacterium glutamicum - JUWEL - Forschungszentrum Jülich

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