@MAKE(Report) @DEVICE(lpt) @STYLE(PAPERWIDTH=33CM,LINEWIDTH=31CM,PAPERLENGTH=36CM,INDENTATION=10) @BEGIN(TITLEPAGE) Advanced Technology Research Center | Silab Division | Samani International Enterprises FERMENTATION GROWTH STAGE DETERMINATION: Fourier Series Fit via Marquardt-Levenberg Algorithm VASOS-PETER JOHN PANAGIOTOPOULOS II [President, Samani International Enterprises] @VALUE(DATE) @begin(researchcredit) @center(@b(ABSTRACT)) Bacteria are extensively utilised in some newer chemical processes. Often, one would wish to approximate the stage of their growth, especially so that one can monitor and control by computer. A numerical algorithm is proposed for such determination. @end(researchcredit) @END(TITLEPAGE) @PAGEHEADING( left "Fermentation Growth Stage Determination", CENTER "@value(page)", RIGHT "Vasos-Peter John Panagiotopoulos II") @PAGEFOOTING( left "@value(sectiontitle)",CENTER "@value(date)", RIGHT "Samani Silab ATRC" ) @Chap(Determination of the Characteristic Curve ) Fitting the expected growth curve @foot{T.J.Parry & R.K.Fausey, @i,p.19, UK:Hutchison Educational,1973} to an even Fourier cosine series yields the following coeffiecients: @equation(a@-<0>=9.1325;a@-<2>=-.1756;a@-<4>=-.1496) @foot[Hewlett-Packard HP41C @i, Program Library ROM, Fourier Series Program,Corvallis,OR:HP,1979.] These are taken as initial guesses to fit the arbitrarily scaled data and to determine the arbitrary period, L. This is performed with Marquardt-Levenberg optimisation methods applied to curvefittng [coefficients are the variables over which the optimisation of error form data occurs] @foot{ cf: "Iterative Solution of Nonlinear Boundary Value Problems in Ordinary Differential Equations", Kenneth J. Anselmo and Jordan L. Spence, Columbia University Department of Chemical Engineering and Applied Chemistry. Received from Prof. Spencer in March, 1982.} on a modeling package. @Foot[Knott, G. D., & Reece, D. K., MLAB: A Civilized Curve-Fitting System,@i,@b<1>,p.497-526,Brunel,UK(1972); Knott, G.D.,& Reece, D. K.,@i, interactive program in SAIL, for DEC PDP-10 systems, Laboratory of Statistical and Mathematical Methodology, DCRT, NIH, Bethesda, Md.(1980). The curve fitting performed later in this document were obtained via MLAB's Marquardt-Levenberg error optimisation capabilities in which: (1) The standard errors are standard deviations when the model is linear in its coefficients; (2) Dependency values positively measure intercoefficional dependency and uniqueness of fit, i.e., the sharpness of the stationary point; (3) Sum of squares are a chi-squared statistic with degrees of freedom being the number of observations less the number of coefficients; (4) RMS error is a dimensional goodness of fit measure equal to the square root of the chi-square/degrees-of-freedom; (5) t-statistic may be calculated by dividing the coefficient by the standard error; (6) r-squared may be proxied via 1-RMS-error/Best-data-value. ] @BEGIN(FIGURE) @BEGIN(VERBATIM) {EDITED FOR BREVITY} FUNCTION FOUR(T) = A0+A2*COS((2*PI*T)/L)+A4*COS((4*PI*T)/L)+A6*COS((6*PI*T)/L) *A0=9.1325;A4=-.1496;A2=-.1756;A6=A4/10;L=25 *FIT(A0,A2,A4,A6,L)FOUR TO M BEGIN ITERATION 1.1, SUM OF SQUARES= 19393.1 BEGIN ITERATION 2.1, SUM OF SQUARES= 1738.55 BEGIN ITERATION 5.1, SUM OF SQUARES= 26.3939 FINAL NORMAL ERROR DEPENDENCY PARAMETER VALUES: STANDARD ERRORS: VALUES: 4.28745 .110259@@-1 .126308 A0 -5.05075 .143704@@-1 .257494@@-1 A2 .464362@@-1 .153142@@-1 .127523 A4 .832312 .146220@@-1 .223575@@-1 A6 23.4808 .198168@@-1 .195328 L RMS ERROR= .230681 FINAL SUM OF SQUARES= 26.3939 {TSTAT=389,-351,3,57,1185 (ALL SIGNIFICANT), CHISQ=.21} @caption(Fitting of Fourier series to characteristic curve) @END(VERBATIM) @END(FIGURE) @begin(figure) @blankspace(73 mm) @caption(Fourier-Characteristic Graph) @end(figure) We can therefore conclude that the characteristic function can be approximated as @equation{Fourier(t)=.126+.0256cos(32.17t) +.128cos(64.33t)+.0224cos(96.5t)} @chap(Stage Determination of Expiremental Data) Thereafter,the experimental data is fitted to the expected data for the determination of scale factors. This shall permit determination of the growth stage involved. @foot[@i] The data is fitted to the equation @equation{f(t')=c*Fourier(a*t'+b) + [d=0],} for the determination of the scaling factors, a and c, and the time shifting, b. It is assumed that both data have a common ordinate origin @foot[d=0, elsewise fit d as well]. One should plot the Fourier function and choose what times delimits the stage change, t@-, t@-, and t@-, hereafter indicated t@-=a*t'+b. And the control and analysis program can use them as set points, @equation{ t'=(t@--b)/a}. @begin(figure) @begin(verbatim) {EDITED FOR BREVITY} *FCT FOUR(T)=A0+A2*COS(2*PI*T/L)+A4*COS(4*PI*T/L)+A6*COS(6*PI*T/L) *L=23.4808;A0=4.28745;A2=-5.05075;A4=.0464362;A6=.832312 *A=1;B=1;C=1;FCT F(T)=C*FOUR(A*T+B);PI=ATAN(1)*4 *M=LIST(12.5,13.5,14,14.5,15,15.5,16) *"ABOVE ARE TIME,DY OF EXPT-BELOW,ABSORBANCE" *M COL 2=LIST(.05,.055,.055,.055,.06,.1,.12) *FIT(A,B,C)F TO M BEGIN ITERATION 1.1, SUM OF SQUARES= 391.777 BEGIN ITERATION 25.6, SUM OF SQUARES= .123202@@-2 FINAL NORMAL ERROR DEPENDENCY PARAMETER VALUES: STANDARD ERRORS: VALUES: -.233554 3.40511 .999998 A 23.0022 27.3330 .999993 B .819191@@-1 2.25473 .999990 C FINAL SUM OF SQUARES= .123149@@-2 BEGIN ITERATION 1.1, SUM OF SQUARES= .123149@@-2 BEGIN ITERATION 1.4, SUM OF SQUARES= .123149@@-2 FINAL NORMAL ERROR DEPENDENCY PARAMETER VALUES: STANDARD ERRORS: VALUES: -.233554 3.34405 .999998 A 23.0022 26.7837 .999993 B .819191@@-1 2.32294 .999990 C CONVERGED RMS ERROR= .175463@@-1 FINAL SUM OF SQUARES= .123149@@-2 *DRAW POINTS(F,-10:10:.5);DRAW M,LINETYPE 0,POINTTYPE 4;DISPLAY=T4010;DRAW POINTS(F,0:100:.3) *DRAW 0:100&'0;DRAW 0&'0:1;DELETE AXESBOX @end (verbatim) @caption(Fitting of Expirement to Determine Stage) @end(figure) @begin(figure) @blankspace(141 mm) @caption(Fitted Data) @end(figure) @appendix(Synopsis of BioChemical Engineering) Yeast cells thrive best at temperatures between thirty and fourty centigrade degrees. Water must be about seventeen twentieths pure to avoid osmotic distortions such as crenation and bursting. While the yeast are saccharolytic and thus saccharophagic, they need various nutrients. They are composed of proteins, derived from amino acids. Thus potassium, phosphorus, sulphur, magnesium @i(et seq.) must be present in the food supply. Growth begins similarly to the start of an exponential curve, after which it becomes logarithmic. The former is classified as the lag phase during which adjustment to the new environment occurs. The latter is the logarithmic phase, in which rapid division occurs. Then the stationary phase begins as the population remains numerically constant and the living cells obtain nutrients from the dead cells. This sorry state of affairs leads them to the aptly named death or decline phase, which is approximable via a mirrored logarithmic function.@foot{B.Faber,@I,pp.20-30,unpublished, Columbia University, Fall,1979,C1501.} More advanced life forms use oxygen in their metabolic reactions. In a muscle, sugar is converted to carbon dioxide and water, via oxygen, or aerobically. Energy is derived from this process. Electron transfer is via the interconversion of adosine tri- and di- phosphate. The latter half of this is named the citric acid cycle. Hexacarbonous glucose is coverted to pyruvic acid and follows to citrate, alpha-ketoglutarate, succinate, fumarate, malate, oxaloacetate, acetyl, and back to citrate. Most life forms follow this example.@foot{J.D.Watson,@i,3rd.ed., pp.66-81,Reading,Massachusetts:W.A.Benjamin,1976.} In the absence of oxygen, however, metabolism may proceed anaerobically. Glucose is coverted to pyruvic acid. Some bacteria, and muscle cells, metabolise this to tricarbonous lactic acid, which results in muscular fatigue. Yeasts, alternatively, proceed via acetaldehyde to dicarbonous ethyl alcohol. This metabolism is not as complete, yet provides energy. @foot{J.J.W.Baker & G.E.Allen, @i,3rd.ed., pp.209-222,Reading, Massachusetts:Addison-Wesley,1977} In engineering work, the Monod Equation may be applied as a primary approximation. It is of a form similar to the famed Michaelis-Menton Equation, as well as to the Langmuir adsorption isotherm. It states that the growth rate, @equation [@ovp[)]\=@ovp[)]\@-(max)c@-(i)/(K@-+c@-)|@ovp[)]\@-(max)=2@ovp[)]\|K@-(i)] with individual nutrient concentration, @equation[c@-(i)=e@+{-@ovp<)>\t}] @foot[J.E.Bailey & D.F.Ollis,@i, pp.346,NY:McGrawHill,1977;J.Monod,The Growth of Bacterial Cultures, @i:371(1949);Dr. Monod received the 1965 Nobel Prize for this and other work on enzymatic adaptation, which he began during WW2. @i O' Sullivan, D., G., Quantitative Potentialities in Enzyme Cytochemistry. Modified Michaelis-Menten Rate Law Applicable when a Substrate Diffuses Slowly into an Enzyme Site,@i,@b<2>, pp.119,123,124(1962) ] Prof. Levenspiel restates this as @equation{r@-=kC@-C@-/(C@-+C@-),} where c indicates cells and A, substrate. C@- is, as above, the Monod constant, or the substrate concentration at the semi-maximal rate. Due to inhibitory product,R, he proposes the more generalised form, @equation{k:=k@-=k(1-C@-/C@-@+<*>)@+.} The asterisk is indicative of the state at which reproduction is fully inhibited. @foot[O.Levenspiel, The Monod Equation: A Revisit and a Generalization to Product Inhibition Situations, @i,pp.1671-1687(1980)] For a CFSTR, @equation{t@-=(C@-+C@-)/C@-=([C/A](C@-+C@-)-C@-)/([C/A]C@--C@-) |C@-=0 & kt@- @b(>)1.} This is the equation that was developed simultaneously by Monod @foot[Monod, Jacques, @i,@b<79>,390(1950).] and by Novick and Szilard @foot[Novick, A., & Szilard, L., @i,@b<36>,708(1950).]