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II ‘ - I II' I i \K’ s K ‘ r" Illlllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 3 1293 10476 0461 ' LIBRARY Hichlgan State 1* University ' This is to certify that the dissertation entitled Kinetics of Hydrogen Consumption by Methanogenic Consortia and Cultures of Hydrogen—Consuming Anaerobes presented by Joseph Arlen Robinson has been accepted towards fulfillment of the requirements for PhoDo degreein MicrObiOlogy ! W 4444a,. Viajor professorO Date November 9, 1982 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 “If. MSU RETURNING MATERIALS: Place in book drop to remove this checkout from LIBRARIES . 51...... your record. FINES w1ll be charged if book is returned after the date stamped below. '2’?“ 3' \ ‘ {'4 : l 0133:}, SEP 2 719973 If” 7 157994 ~ I. KINETICS OF HYDROGEN CONSUMPTION BY METHANOGENIC CONSORTIA AND CULTURES OF HYDROGEN-CONSUMING ANAEROBES by Joseph Arlen Robinson A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Microbiology and Public Health 1982 ABSTRACT KINETICS OF HYDROGEN CONSUMPTION BY METHANOGENIC CONSORTIA AND CULTURES OF HYDROGEN-CONSUMING ANAEROBES by Joseph Arlen Robinson Hydrogen plays a central role in the breakdown of organic matter in anaerobic habitats, influencing the nature of the fermentation endproducts and possibly the rates of initial substrate degradation. Until recently, attention had focused primarily on the influence Hz-consumers exert on the types of endproducts generated from bacteria capable of reducing protons to H2 (facultative or obligate syntrophs). Currently, interest in H2 dynamics of anaerobic ecosystems has focused on (i) estimating in gi£u_rates of H2 consumption and turnover, and (ii) quantitating competition for H2 among Hz-consuming anaerobes, such as sulfate-reducing and methanogenic bacteria. I examined the kinetics of H2 consumption by samples from natural anaerobic habitats, pure cultures of Hz-consuming anaerobes, and co-cultures comprised of methanogenic and sulfate-reducing bacteria. In addition, I studied the turnover of ruminal H2 and the influence organic loading rates have on endogenous H2 and CH4 production in zigrg. These kinetic studies were performed using a gas-recirculation system that allowed precise measurements of gaseous phase H2 and CH4. Uptake and growth kinetic parameters were estimated for the natural samples and suspensions of H2~consumers by fitting H2 depletion (progress curve) Joseph Arlen Robinson data to integrated forms of Michaelis-Menten and Monod equations. H2 Km values for samples from eutrophic lake sediment, anaerobic digestor sludge and rumen fluid were similar, approximately 6 um. Maximum potential H2 consumption rates (Vmax estimates) suggested a ratio of activity for rumen fluid, sludge and sediment of about 100:10:1, presumably reflecting the relative densities of Hz-consuming bacteria-primarily methanogens-occurring in these anaerobic habitats. H2 Km estimates of the four methanogens studied ranged from 2 uM for Methanospirillum.PMl to 12 uM for Methanosarcina barkeri MS, while the other two methanogens-Methanospirillum hungatei JF-l and Methanobacterium.PM2-had mean half-saturation constants for H2 uptake of 5 uM. Average H2 Km estimates for the sulfate-reducers, Desulfovibrio strains 611 and PSI, were 1 and 0.7 uM, respectively. H2 Vmax values for the six H2 consumers assayed were not significantly different when normalized to total protein. The average half-saturation constant for growth (K3), yield _ coefficient (YHZ) and maximum specific growth rate (“max) of G11 were 3 uM, 0.8 g protein/mol H2 and 0.06/h, respectively. Initial studies with JF-l suggested that its K3 for growth On limiting Hz was higher than the ' mean value for Gll, falling in the range of 2-10 uM. There was an apparent correlation among uptake and growth half-saturation constants estimated for G11 and JP-l. This correlation was corroborated by the finding that the Km of Methanggpirillum.PM1 was lower than that of JF-l and that it grew significantly faster than JF-l. The above uptake and growth kinetic parameters predict that sulfate-reducers will outcompete methanogens in habitats where sulfate Joseph Arlen Robinson is not limiting.. The partitioning of H2, under resting conditions, was used to predict the fate of H2 when the densities of these competing bacterial groups are relatively fixed. On the other hand, Monod kinetic parameters must be used to predict the outcome of competition for H2 and the partitioning of this gaseous substrate when growth occurs. A consortium.of H2-consuming organisms having different affinities for H2 will exhibit an apparent Kn that depends on the fractions that the individual groups (e.g., sulfate-reducers, methanogens) comprise of the total Hz-consuming activity. This dependence was successfully predicted for various mixtures of Desulfovibrio G11 and Methanospirillum JF-l. Retention times for H2 turnover in giggg calculated from estimates of ruminal Michaelis-Henten parameters and CH4 production data were equivalent once the H2 pool attained steady-state. Retention times varied from 0.1-0.3 sec for strained rumen fluid and diluted whole contents. Retention times for ruminal H2 32:3322.were approximated to be 0.02 sec (mid-point of range) by correcting retention time estimates determined for diluted whole contents by the ratio of total particulates to strained rumen fluid used in these experiments. Endogenous H2 concentrations and CH4 production rates rapidly increased when strained rumen fluid samples were amended with finely ground hay. Additions of 5-10 g of milled hay resulted in increased H2 levels that peaked after several hours; 50 g or greater additions of ground hay typically resulted in extremely high H2 concentrations that never peaked. In the latter experiments, a decrease in pH accompanied the accumulation of high H2 concentrations. Further, calculated rates of H2 production were several times greater than Vmax estimates of H2 consumption obtained Joseph Arlen Robinson for strained rumen fluid. In all of the above kinetic studies, Hz was measured in the gas phase and liquid phase concentrations were calculated by multiplying the former by the appropriate Bunsen absorption coefficient. But this is only valid when the gaseous and aqueous phases are in equilibrium (i.e., when a phase-transfer limitation does not exist). In accord with this, I developed a mathematical model that predicted conditions leading to phase-transfer limited gaseous substrate consumption by resting and growing cells. For gaseous substrate (H2) consumption by resting bacteria the following criteria indicate that a phase-transfer limitation exists: (1) the apparent KIn shows a strong dependence on the initial substrate concentration and the magnitude of the sink (i.e., Vmax) in the aqueous phase; and (ii) the apparent Michaelis-Menten parameters exhibit dependence on stirring speed and the ratio of aqueous to gaseous phase volume. Similar principles apply to Michaelis-Menten gaseous product formation from either gaseous or nonegaseous substrates, and gaseous substrate consumption by growing cells. To Patricia-Marie and my loving parents, Russell and Nancy Robinson 11 In 1863 an unofficial visitor to the Whitehouse drew from Lincoln the remark that he had a great reverence for learning. ”This is not because I am not an educated man. I feel the need of reading. It is loss to a man not to have grown up among books”. The visitor replied with: ”Men of force can get along pretty well without books. They do their own thinking instead of adopting what other men think”. ”Yes”, said Lincoln, ”but books serve to show a man that those original thoughts of his aren't very new, after all”. ”Say not, 'I have found the truth', but rather, 'I have found a truth'”. Kahil Gibran iii ACKNOWLEDGEMENTS I am much indebted to Dr. James Tiedje. Not only did he provide a superb intellectual environment in which I could grow, but his continual support encouraged me to strive for scientific excellence. I am also indebted to the superb graduate students and post-docs that it was my pleasure to work with over the past five years. Any successes I have enjoyed thus far, I owe in large part to the high caliber of science maintained by Dr. Tiedje and the people who have worked for him in his laboratory. I thank the members of my guidance committee-Drs. Mike Klug, Clarence Suelter and C. A. Raddy. Special thanks go to Dr. King for supporting a summer's stay at the Kellogg Biological Station. I thank Walter J. Smolenski and Marilyn Boucher for their excellent technical assistance over the past three years. Their help in media preparation and routine culturing of the bacteria used in this study is much appreciated. I owe a great deal to my loving wife, Patricia-Marie. She has made many sacrifices since I became obsessed with microbial ecology, but her support never waned even through the long evenings I spent away from home. How can I ever say enough to thank my parents for their unrelenting support? They played a major role in making my years in graduate school the best ones of my life to date. The diversions Patti and I shared with them relieved many of the frustrations of being a graduate student. My only regret is I never learned how to make Dupont 3F; sorry Dad--I doubt if Jim knows either. iv TABLE OF CONTENTS LIST OF TABLES ............................................ LIST OF FIGURES ........................................... INTRODUCTION .............................................. LITERATURE CITED .................................. ARTICLE I (CHAPTER I). KINETICS OF HYDROGEN CONSUMPTION BY RUMEN FLUID, ANAEROBIC DIGESTOR SLUDGE AND SEDIMENT ABSTRACT 0.0.0.0000...00......OOOOOOOOOOOOOOOOOO... INTRODUCTION ooeooooooooooeeooeoooeeeoeooeoeoeooooo MATERIALS AND METHODS ooeooeoooeeoooeoooooeoooooooo PHASE-TRANSFER MODEL ooooseooeeoeoo0000000000000... RESULTS oeoeoooeoeooeoooeooooooooeooooooeoooooooeoo DISCUSSION oeooooeeooeoeeoooeoeoooooooeooeeoeeeoooo LITERATURE CITED ooooooooooeoo0.0000000000000000... ARTICLE II (CHAPTER II). TURNOVER OF HYDROGEN IN RUMEN FLUID 0.00.0.0...0.0.0.000...COCOOOOOOOOOOOCOCOOOOO ABSTRACT OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO INTRODUCTION COO...0.0.0.0....OOOOOOOIOOCOOOOOOOOOO MATERIALS AND METHODS .00...OCOCOOOOOOOOOOOOOCOOOOO RESULTS 0.0...OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO DISCUSSION O...0..OOOOOOOOOOOOOOOOOOOOOOO0.0.000... LITERATURE CITED O...O...0.0.0.0...OOOOOOOOOOOOOOOO ARTICLE III (CHAPTER III). RESOURCE COMPETITION AMONG SULFATE- REDUCERS AND METHANOGENS FOR HYDROGEN ............. ABSTRACT 00.0.0000...00.0.00...OOOOOOOOOOOOOOIOOOOO INTRODUCTION O0.0..0.0.0.0...OOOOOOOOOOOOOOOOOOOOOO MATERIALSANDETHODS O...0.00.00.00.00.00.0.0.0... RESULTS 0.0.0....0..00......OOOOOOOOOOOOOOOOOOOOOOO DISCUSSION OOOOOOOOOOOOOOOOOOOOCOCOOOOOOOOOOOOIOOOO LITERATURE CITED O...00......OOOOOOOOOOOCOOOOOIOOOO APPENDIX A. PHASE-TRANSFER KINETICS ...................... LITERATURE CITED .................................. APPENDIX B. PROGRESS CURVE ANALYSIS ...................... LITERATURE CITED ..............;................... APPENDIX C. COMPUTER PROGRAMS FOR DATA ANALYSIS .......... Page vii viii 11 17 21 42 49 52 53 54 55 58 71 76 78 79 80 82 89 112 122 126 153 154 195 197 LITERATIJRE CITED O...0.000000000000000000000000...O 225 vi Table LIST OF TABLES ARTICLE I Summary of Michaeliseuenten kinetic parameters for rumen fluid, Holt digestor sludge and Wintergreen sediment oooooooooooooooo00000000000000.0000.oooooo ARTICLE II Retention times (sec) of ruminal Hz calculated from endogenous Hz production versus Hz commtion rates 0.0.0.000...OOOOOOOOOOOOOOOOOOOOO ARTICLE III Summary of H2 kinetic parameters for methanogenic and sulfate-reducing bacteria ..................... Hz Km estimates for Desulfovibrio strains 611 and P81 at different initial H2 concentrations ...... Michaelis-Menten parameters for product appearance by methanogenic and sulfate~reducing bacteria ........ Monod growth kinetic parameters for Desulfovibrio Gll 0.0.0.0.0...0.0.0.0....OOOOOOOOOOOOOOOOOOOOOOOO APPENDIX A \ Comparison of nonlinear versus linear analysis of Michaelis-Menten progress curve data containing simple or relative errors ......................... vii Page 26 72 100 101’ 104 111 170 Figure 10 LIST OF FIGURES ARTICLE I Page Gas-recirculation system used for H2 consumption and CH4 pIOdUCtion “permeate 00.000000000000000 13 Theoretical curves dmr phase-transfer and non- tralsfer limited gaseous substrate consumption .... 20 Gaseous phase data for phase-transfer limited (first- order) H2 consumption by undiluted rumen fluid (URF) and Portland sludge (PS), and non-transfer limited (mixed order) Hz consumption by 20-fold diluted rumen f1u1d (DRE) cooooooosooosoooooooooooo 23 Calculated aqueous phase data for Michaelis-Menten Hz consumption by diluted rumen fluid (DEF), Holt sludge (HS) and Wintergreen sediment (WS) ......... 25 Dependence of apparent Hz Km on magnitude of the biological sink under phase-transfer and nonrtransfer limited conditions 0.0.0....OOOOOOOOOOOOOOOOOOOOOOO 29 Effect of an increase in endogenous substrate production (A) and an increase in Vmax_with time (B) on Michaelis-Menten progress curves .................. 31 Errors in apparent Vmax (A) and Km (B) resulting from concomitant endogenous H2 production during the course of progress curve experiments .............. 34 CH4 production data and calculated rates of Hz pr0ducr10n for undiluted mu flu1d o o o o o o o s s o o o o o 36 Relationship between endogenously produced H2 and H2 concentrations monitored during a progress curve using replicate suspensions of diluted rumen fluid (DRF) oosooooooooossoso...assess0.0000000000000000. 38 Relationship between endogenously produced Hz and H2 concentrations monitored during a progress curve using replicate samples of undiluted Holt 81ndge (HS) 000......OOOOOOOOOOOOOOOOOOOOOOOO0.0... 40 ARTICLE II Endogenous C34 production by two samples of strained rumen fluid (SRF) and diluted whole rumen contents (WC) oooooooooooooooosooosoo00000000000000 60 Endogenous dissolved H2 concentrations and viii calculated rates of HZ production for strained (squares) and diluted whole contents (circles) .... H2 retention thmes for strained rumen fluid and diluted whole contents estimated from total CH4 production rate data (Figure 2) ................... Dissolved H2 concentrations for strained rumen fluid amended with 50 (triangles), 10 (circles) or 1 (squares) g of finely milled hay .................. CH4 production by strained rumen fluid amended with 50 (triangles), 10 (circles) or 1 (squares) g of finely Milled my 0....O...0.00000000IOOOOOOOOOOOOOOOOOOOO ARTICLE III H2 progress curve data for Methanospirillum huggatei JF-l at two different initial H2 concentrations seesseooeeeooeeeooeooeoeeoeoeooeooeo H2 progress curves for Methanosarcina barkeri MS versus theoretical curves calculated from best- estimates of Michaelis-Menten parameters .......... Hz progress curve data for Methanospirillum.PM1 (squares) and Methanobacterium PMZ (triangles) versus theoretical curves calculated from best-estimates Of vm, “and so O...OOOOOOOOOOOOOOOIOOOOOO H2 progress curve data for Desulfovibrio 611 at two different initial concentrations of H2 plotted against theoretical curves ........................ H2 progress curve data for Desulfovibrio PSl plotted against theoretical Michaelis-Menten curves at two different initial H2 concentrations 0 o o e o e o o e o o s o CH4 and dissolved sulfide appearance data versus theoretical curves calculated using best-estimates of vmx 811de OOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOO Apparent H2 Km at different ratios of sulfate- reducing activity (organism l) to total Hz-consuming capaC1ty 0..OOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOO... Comparison of theoretical curve with measured H2 concentrations (squares) for growth of Desulfovibrio Gll growing on H2 as the sole electron donor .... Fifty-percent partitioning curves for mixtures of sulfate-reducing and methanogenic bacteria ........ ix 62 64 67 69 91 93 95 97 99 103 107 110 118 APPENDIX A Influence of Kla on gaseous substrate consumption not limited by phase-transfer OOOOOOOOOOOOOOOOOOOOOOOOO Influence of Kla on gaseous substrate consumption limited by mass-transport so...oeooooesooeeeooeeoee Gaseous substrate consumption by growing cells severly limited by mse‘trmport 00.0000.......OOOOOOOOOOO Gaseous substrate consumption by growing cells maderately limit“ by phase-transfer o o e o o e o e s o o e o o Gaseous substrate consumption by growing cells not limit“ by mas-transport 00.00....OOOOOOIOOOOOOOOO Influence of Kla on ratio of aqueous to gaseous phase concentrations of a gaseous product produced from a non-gaseous SUbStrate 00000000000000.0000...sooooes Gaseous product formation from a non-gaseous substrate for different values Of K18 .....OOOOOOOOOOOOOOOO Influence of gaseous substrate solubility on Km,app and Vmax,app estimated from aqueous phase data .. Influence of gaseous substrate solubility on Km ap and Vmax,a p estimated from for total mass (mol) 0? gaseous sugstrate being consumed with time ........ APPENDIX B Plots of Michaelis-Menten progress curve data transformed according to [3.3] (A), [3.4] (B) and [3.5] (C) with simple error bars of +/-0.01 units ....... Plots of Michaelis-Menten progress curve data transformed according to [3.3] (A), [3.4] (B) and [3.5] (C) with relative error bars of +/-0.005 units .... Sensitivity coefficients for Km and Vmax ...... Sensitivity coefficients for 80 derived from [3.1] versus [302] oeooeseeooeooeeooooo0.000000000000000. Sigmoidal substrate consumption (8) and biomass formation (X) for cells growing in batch .......... Linearized discretizEd “on“ data 0 e e s s e e o o o e o o o o o 0 Sensitivity coefficients for “max: K3 and Y ... 130 132 135 137 139 143 146 148 150 158 162 168 172 177 180 182 10 11 Residuals for H2 progress curve data fitted to [3.2] using nonlinear least-squares analysis ............ Simulated Honod progress curve data containing simple errors (standard deviation-0.01 units) fitted to the integrated form of the Michaelis-Menten equation [3.2] O.C.00......0....0.00.00.........OOOOIOOOOOO Residuals for Hz removal from an empty flask containing 82 due to sampling eeoeooooeoeesooooeo Residuals for 32 standard tank sampled with time xi 186 189 191 193 INTRODUCTION AND EXPERIMENTAL OBJECTIVES In the past 10-15 years, interest in H2 and the role it plays as an intermediate in the anaerobic dissimilation of organic matter has risen dramatically. This methanogenic precursor is believed to have a significant influence on pathways of anaerobic metabolism via interspecies 32 transfer (7,18,19). The earliest quantitative work on Hz kinetics and turnover in anaerobic habitats-specifically, the rumen-was published by Hungate and co-workers (5,6). Recently, several papers have appeared that document Hz's role and the kinetics of Hz consumption in other anaerobic ecosystems. The majority of these studies focused on 32 dynamics in lacustrine and marine sediments (1,2,4,11,13,14,16,17). Further, estimates of uptake constants for a methanogen and sulfate-reducer have appeared within the past several months (9) which agree with estimates obtained for methanogenic habitats. Between these two phases of activity, 32's importance in the anaerobic processing of organic matter was primarily assessed through studies on co-cultures of proton-reducing and Hz-consuming organisms (3,8,10,12,15,18). These studies dealt with products produced from organic matter dissimlation when Hz-consuming organisms (sulfate-reducers or methanogens) functioned in concert with a proton-reducing organism. Now there is renewed interest in estimating kinetic constants for Hz consumption in anaerobic ecosystems and additionally, in determining kinetic constants for pure cultures of methanogenic bacteria and other Hz-consuming anaerobes-specifically, sulfate-reducing bacteria. This interest stems from the need to estimate £3 situ rates of Hz consumption and the partitioning of Hz between competing populations of methanogenic and sulfate-reducing bacteria in habitats where sulfate is not limiting. The primary goals of my doctoral research was to estimate kinetic constants for Hz consumption by (1) samples of anaerobic ecosystems, (ii) pure cultures of methanogenic and sulfate-reducing bacteria in both resting and growing states, and (111) defined co-cultures of methanogens and sulfate-reducers. Knowledge of kinetic constants for Hz consumption by anaerobic habitats may be used and interpreted in several ways. Firstly, the affinities of various habitats--such as, eutrophic lake sediment, anaerobic digestor sludge and rumen fluid-for Hz can be compared using estimates of half-saturation constants (Km, K3). These half-saturation constants presumably reflect the overall affinity of the Hz-consuming organisms for Hz present in these habitats. Maximum potential rates of activity (Hz Vmax's) reflect the densities of methanogens (also sulfate reducers and Hz-consuming acetogens, if present) occurring in the same habitats and represent better estimates of the active Hz-consuming biomass than direct or indirect bacterial counts. Thirdly, estimates of Hz kinetic parameters, along with knowledge of the endogenous Hz concentration, can be used to predict in ‘gigg Hz retention times when this intermediate is at steady-state. In Chapter I, I present estimates of Michaelis-Menten kinetic parameters for Hz consumption by rumen fluid, anaerobic digestor sludge and eutrophic sediment. The kinetic constants for rumen fluid and data on endogenous C34 production are used in Chapter II to approximate endogenous Hz turnover and demonstrate the equivalence of Hz retention times calculated from Hz production and consumption data when the Hz pool has attained steady-state. Chapter II also contains data showing the influence of organic loading rates on endogenous ruminal Hz and CH4 production. In addition to presenting estimates of Hz uptake parameters for natural anaerobic ecosystems in Chapter I, I also describe criteria that may be used to assess whether gaseous substrate (e.g., Hz, 02) consumption is limited by mass-transport across an aqueous-gaseous phase interface and examine the influence endogenous Hz production has on estimates of Michaelis-Henten kinetic parameters. Mass-transport limitations must be circumvented before meaningful estimates of biological kinetic parameters can be obtained (see Appendix A). Chapter III contains estimates of Michaelis-Menten (uptake) and Monod (growth) kinetic parameters for pure cultures of methanogenic and sulfate-reducing bacteria. Estimates of Michaelis-Menten kinetic parameters can be used to predict the partitioning of limiting Hz between these two functional groups of bacteria and apparent uptake constants for mixtures of these microorganisms. Estimates of Monod growth parameters also predict the fate of an intial limiting amount of Hz when its consumed by a mixture of growing methanogens and sulfate-reducers. Further, Mbnod growth parameters, unlike uptake parameters, predict which Hz-consumer will eventually outgrow the other assuming Hz is the only limiting substrate. Appendices A through C contain information relevant to all three chapters. Appendix A provides more details than Chapter I on the influence of mass-transport on rates of gasesous substrate consumption by growing or resting bacterial cells. Further, Appendix A contains systems of differential equations that describe gaseous product formation from either gaseous or nongaseous substrates. In Appendix 3 are details on the nonlinear regression analyses applied to the progress curve (i.e., substrate depletion) data shown in Chapters I and III. Finally, Appendix C contains several computer programs written for analysis of the kinetic data presented in this dissertation. 1. 2. 3. 8. 9. 10. 11. 12. LITERATURE CITED Abram, J. W., and D. 3. Nedwell. 1978. Hydrogen as a substrate for methanogenesis and sulfate reduction in anaerobic saltmarsh sediment. Arch. Microbiol. 117:93-97. Abram, J. W., and D. B. Nedwell. 1978. Inhibition of methanogenesis by sulfate-reducing bacteria competing for transferred hydrogen. Arch. Microbiol. 117:89-92. Bryant, M. P., L. L. Campbell, C. A. Reddy, and M. R. Crabill. 1977. Growth of Desulfovibrio in lactate or ethanol media law in sulfate in association with Hz-utilizing methanogenic bacteria. Appl. Environ. Microbiol. 2251162-1169. Cappenberg, T. E. 1974. Interrelationships between sulfate-reducing and methane-producing bacteria in bottom deposits of a fresh-water lake. I. Field observations. Antonie van Leeuwenhoek J. Microbiol. Serol. 49;235-295. Hungate, R. E. 1967. Hydrogen as an intermediate in the rumen fermentation. Arch. Microbiol. 22; 158-164. Hungate, R. E., W. Smith, T. BauchOp, J. Yu, and J. C. Rabinawitz. 1970. Formate as an intermediate in the bovine rumen fermentation. J. Bacteriol. 102:389-397. Hungate, R. E. 1975. The rumen microbial ecosystem. Ann. Rev. ECOlo SYBte 2: 39‘66. Iannotti, E. L., D. Kafkewitz, M. J. Wolin, and M. P. Bryant. 1973. Glucose fermentation products of Ruminococcus albus grown in continuous culture with Vibrio succinogenes: Changes caused by interspecies transfer of— Hz. J. Bacteriol. 114: 1231-1240. Kristjansson, J. R., P. Schonheit, and R. K. Thauer. 1982. Different K, values for hydrogen of methanogenic bacteria and sulfate-reducing bacteria: An explanation for the apparent inhibition of methanogenesis by sulfate. Arch. Microbiol. _1§I:278-282. Latham, M. J., and M. J. Wolin. 1977. Fermentation of cellulose by Ruminococcus flavefaciens in the presence and absence of Methanobacterium ruminantium. Appl. Environ. Microbiol. 34: 297-301 a Lovely, D. R., D. Dwyer, and M. J. Klug. 1982. Kinetic analysis of competition between sulfate reducers and methanogens for hydrogen in sediments. Appl. Environ. Microbiol. 43: 1373-1379. McInerney, J. M., and M. P. Bryant. 1981. Anaerobic degradation of 13. 14. 15. 16. 17. 18. 19. lactate by synthrophic association of Methanosarcina barkeri and Desulfovibrio species and effect of Hz on acetate degradation. Appl. Environ. Microbiol. 413346-354. Mountfort, Do 00, Ra At “her, Ea La Maya, and Jo Ma Tiedjeo 1980. Carbon and electron flow in mud and sandflat intertidal sediments of Delaware inlet, Nelson, New Zealand. Appl. Environ. Microbial..32;686-694. Robinson, J. A., and J. M. Tiedje. Kinetics of hydrogen consumption by rumen fluid, anaerobic digestor sludge and sediment. Appl. Environ. Microbiol. in press (AEM no. 364). Scheifinger, C. C., B. Linehan, and M. J. Wolin. 1975. Hz production by Selenamonas ruminantium.in the presence and absence of methanogenic bacteria. Appl. Environ. Microbiol. 22: 480-483. Strayer, R. F., and J. M. Tiedje. 1978. Kinetic parameters of the conversion of methane precursors to methane in hypereutrophic lake sediment. Appl. Environ. Microbiol. 36:330-340. Winfrey, M. R., and J. G. Zeikus. 1977. Effect of sulfate on carbon and electron flow during microbial methanogenesis in freshwater sediments. Appl. Environ. Microbiol. 23:275-281. Wolin, M. J. 1974. Metabolic interactions among intestinal microorganisms. Amer. J. Clin. Nutr. 21: 1320-1328. Zehnder, A. J. B. 1978. Ecology of methane formation. In R. Mitchell (ed.), Water pollution microbiology. Vol. 2. John Wiley and Sons, Inc., New York. p. 349-376. ARTICLE I (CHAPTER I) KINETICS OF HYDROGEN CONSUMPTION BY RUMEN FLUID, ANAEROBIC DIGESTOR SLUDGE AND SEDIMENT by Joseph A. Robinson and James M. Tiedje ABSTRACT Michaelis-Menten kinetic parameters for H2 consumption by these three methanogenic habitats were determined from progress curve and initial velocity experiments. Additionally, the influences of mass-transfer resistance, endogenous H2 production and growth on apparent Michaelis-Menten kinetic parameters were investigated. H2 consumption by undiluted rumen fluid and sludge from a digestor with a typical retention time was limited by transfer of Hz across the gas-liquid interface; Hz consumption could not be saturated and apparent Km's were dependent on initial H2 concentrations and the magnitude of Vmax° Once rumen fluid was diluted 20-fold, H2 Km's became relatively constant and were independent of the initial Hz concentration and the magnitude of Vmax- Hz consumption by digestor sludge with a long retention time and hypereutrophic lake sediment was not phase-transfer limited, and exhibited Michaelis-Menten kinetics. Hz Kn's for these habitats were relatively constant with means of 5.8, 6.0 and 7.1 uM for rumen fluid, digestor sludge and sediment, respectively. Vméx estimates suggested a ratio of activity of approximately 100 (rumen fluid):10 (sludge):l (sediment); their ranges were rumen fluid (14-28 mM h‘l), Holt sludge (0.7-4.3 mM h‘l) and Wintergreen sediment (0.13-0.49 mM h‘l). Potential errors in the Michaelis-Menten kinetic parameters, resulting from endogenously produced Hz were less than 151 for rumen fluid and less than 101 for lake sediment or digestor sludge. Increases in Vmaxa during the course of progress curve experiments, were not great enough to produce systematic deviations from Michaelis-Menten kinetics. Conditions that create phase-transfer limitations for gaseous substrate consumption were elucidated using a mathematical model that combined mass-transport and Michaelis-Menten kinetics. 8 INTRODUCTION Hydrogen is a key intermediate in the degradation of organic matter, affecting both the rate of this process and the nature of the endproducts (4,12,34,35). Due to 82's central role in methanogenic habitats, the kinetics of H2 consumption has received considerable attention in the past few years, with most investigators concentrating on Hz kinetics in eutrophic lake (29,33) and marine (1,2,20) sediments, and anaerobic digestor sludge (15,26). Hz kinetics in the rumen has been studied to a lesser extent (7,10,13). Many investigators, particularly those studying Hz kinetics in digestor sludge (15,26), overlooked the influence mass-transfer resistance may have had on their experiments. That mass-transfer of gaseous substrates (e.g., 02) can limit microbial activity and growth has been understood for some time (3,22,23). But microbiologists investigating Hz kinetics in methanogenic habitats have often implicitly assumed their incubation systems circumvented potentially rate-limiting movement of Hz across the gas-liquid interface. Gross over-estimates of half-saturation constants (viz., Km, K3) result if data, obtained under mass-transfer (phase-transfer) limited conditions, are used to compute estimates of these parameters. Ngian, Lin and Martin (21) have demonstrated that mass-transfer resistance can significantly increase apparent transport Km's for organisms that form aggregates in biological reactors. They argue, as does Shieh (27), that mass-transfer resistances must be eliminated if biological parameters, independent of reaction vessel geometries, are to be estimated. Thus, we initially examined the influence inter—phase movement of Hz has on apparent kinetic parameters (viz., Km, Vmax) when consumption of Hz is monitored in the gaseous 9 10 phase, and determined the magnitude of errors associated with. Michaelis-Menten kinetic parameters calculated from data obtained under phase-transfer limited conditions. we developed a model (PHASIM) to evaluate effects of mass-transfer resistance on Michaelis-Menten kinetic parameters. The model is generally useful since it describes the influence mass-transfer resistance has on substrate consumption occurring in one phase of a multi-phase system, where the substrate partitions among the various phases (gas, liquid or solid). The other goals of the present study were to: (1) compare the kinetics of Hz consumption in rumen fluid, hypereutrophic lake sediment and anaerobic sludge; and (2) to assess the influence that endogenous H2 production and growth have on apparent Michaelis-Menten kinetic parameters for H2. MATERIALS AND METHODS Habitats sampled. Samples of rumen fluid were obtained from a fistulated cow that was fed daily 5.44 kg of hay ad libitum. Samples of rumen fluid were withdrawn from the cow according to a previously described procedure (23); a cheese-cloth screen reduced the proportion of particulate solids from that found in viva. Samples were usually obtained prior to feeding to minimize variability in endogenous H2 production rates. Rumen fluid was used within 30 min after it was collected. Anaerobic digestor sludge was obtained from municipal waste treatment plants in Holt and Portland, MI. Lake sediment samples were taken from the pelagic zone of hypereutrophic Wintergreen lake (Hickory Corners, MI) using an Eckman dredge. Mason jars were completely filled with the samples and sealed with metal canning lids. Samples were stored at 5-10°C for 1 week or less before being used. Experimental apparatus for measurement of H2 and CH4;_ All experiments were performed using the gas-recirculation system depicted in Figure 1, which is patterned after the one described by Kaspar and Tiedje (16). Five-hundred-milliter volumes of either undiluted rumen fluid, diluted rumen fluid, Wintergreen sediment, or digestor sludge were anaerobically transferred to a 2-1 flask, using a modification of the Hungate technique (11). The flask was stoppered, placed in a water bath (held at 39°C for rumen fluid, 30°C for sludge, or 9°C for sediment) and attached to the gas-recirculation system. Prior to the beginning of each experiment, the headspace of the flask was flushed out the vent with C02 11 12 Figure l. Gas-recirculation system used for H2 consumption and CH4 production experiments. The heavier line indicates the path of gas circulation between sample flask and H2 GC sampling loop. Components are detailed in Materials and Methods. O 3.3 no ...: 8 . L., as: H manure B oaaa 25% m =2. 2.2» 2.. 8i ‘_‘ '6‘ «8 a 14 (for rumen fluid), 30% C02/702 Ar (for sludge),or 52 C02/9SZ Ar (for sediment) for about 15-30 min. Trace levels of 02 were removed from the sparge gas by passing it through hot copper filings. After the flushing period, the gas-recirculation system was closed and the headspace of the flask was recirculated, via a bellows-type pump (Universal Electric Co., Owosso, MI), through a 3-ml gas sampling 100p in a Carle AGO-111 gas chromatograph (Carle Instruments Inc., Anaheim, CA). A magnetic stirrer set at near-maximal speed mixed the aqueous phase in the 2-1 flask. Argon, at a flow rate of 20 ml min'l, was used as the carrier gas for the Carle CC, and gas separation was made using two tandem columns held at 30°C; the first column was Porapak T (1.2-m length, 3.2-mm i.d.), while the second was Molecular Sieve 5A (2.7-m length, 3.2-mm i.d.). Injections were automatically performed with a timer (Carle Instruments Inc., Anaheim, CA) attached to the valve switching mechanism in the H2 60. The H2 retention time was ca. 3 min. After the H2 passed through the microthermistor detector, the gas flow was reversed by the timer and the rest of the gas sample (including the 084 in the original 3-ml injection) was backflushed to a Varian 600-D GC (Varian Instrument Group, Walnut Creek, CA) equipped with a flame ionization detector. The backflush stream of Ar, at a flow rate of 40 ml min’l, served as the carrier gas for the second (CHa-detecting) GC, and gas separation in this instrument was effected with a 1.5-m (3.2-mm i.d.) column of Porapak Q at room temperature. H2 and C34 peak areas were determined with integrators. The pressure transducer (Unimeasure, Grants Pass, OR) was used to check for leaks in the gas-recirculaton system. Bunsen coefficients, for the three temperatures of incubation, were calculated using an equation that describes the solubility of a gas as a 15 function of temperature (32). Initial velocity and progress curve experiments. For initial velocity experiments rumen fluid (diluted or undiluted) was incubated with five different initial H2 concentrations. Consumption of Hz was monitored at 10 min intervals for 2 h, and initial velocity estimates were obtained by evaluating the first-derivatives of the best-fit lines for H2 disappearance at t-O. Estimates of Ru and vmax for H2 consumption were determined from initial velocity-substrate concentration (v-s) pairs using (1) an unweighted Lineweaver-Burk analysis, (ii) the direct linear plot of Cornish-Bowden (9) and, (iii) a nonelinear regression analysis that involved fitting the v-s pairs directly to a rectangular hyperbola (6). For progress curve experiments, a saturating concentration of H2 [calculated from.IHOOOHD H021 Houms 15 as as Houms ' 10 S 1. k. 9w AU ?_ .1 «Id NI.UO>HOQOHO H021 Figure 4 26 Table 1. Summary of Michaelis-Menten kinetic parameters for rumen fluid, Holt digestor sludge and Wintergreen sedimenta Anaerobic habitat Vmax (mM h'l)b Km (uM)c Rumen fluid 28.3 +/- 0.37 9.08 +7“ 0.26 13.3 +/- 0.15 7.19 +/- 0.30 19.8 +/- 0.71 6.00 +/- 0.64 14.6 +/- 0.39 4.31 +/- 0.41 13.5 +/- 0.21 4.23 +/' 0.23 17.0 +/- 0.70 4.21 +/' 0.50 Holt digestor sludge 0.70 +/- 0.01 6.80 +/- 0.30 4.30 +/- 0.22 6.72 +/- 0.85 2.98 +/- 0.08 6.22 +/- 0.43 2.55 +/- 0.04 6.12 +/- 0.26 2.29 +/- 0.10 5.85 +/' 0.59 3.26 +/- 0.16 4.42 +/- 0.66 Wintergreen sediment 0.49 +/- 0.01 8.58 +/- 0.49 0.13 +/- 0.01 5.63 +/- 0.38 a Parameter estimate +/- SE*t-value (95% confidence interval); each pair of parameter estimates were determined from a minimum of 12 data pairs. Note SE's are approximate since the integrated form of the Michaelis- Menten model is nonlinear and were calculated assuming no correlation among the measurement errors. bValues are corrected for any dilutions required to overcome phase- transfer limitations and are for total consumption per unit volume of sample assayed. c Estimates are for H2 dissolved in the aqueous phase, calculated from gaseous phase data using Bunsen Absorption coefficients for pure water. 27 Wintergreen sediment were determined via nonlinear regression analysis. The goodness-of-fit of the non-transfer limited H2 depletion data to the integrated Michaelis-Menten equation is illustrated in Figure 4. Km estimates for the anaerobic habitats investigated ranged from 4-9 uM dissolved H2 (Table 1). Vmax's for H2 consumption by rumen fluid, eutrophic sediment and anaerobic digestor sludge varied from 0.1 to 28 mmol total H2 consumed 1‘1 h‘l, with rumen fluid having the highest potential Hz-consuming capacity and Wintergreen sediment the lowest (Table 1). Effect of phase-transfer limitation on HZ_§R:. The error associated with H2 Km estimates for a phase-transfer limited system was evaluated from initial velocity data obtained for undiluted and diluted rumen fluid. Each batch of rumen fluid was incubated with five different initial H2 concentrations. Substrate disappearance was monitored for 2 h, and initial velocity estimates were calculated from slopes of the best-fit curves at t-O. The apparent H2 Km estimates were markedly dependent on the rumen fluid concentration when H2 consumption was transfer limited (Figure 5). In addition, H2 Rm estimates were dependent on the initial H2 concentration; the higher the initial substrate concentration, the greater the apparent H2 Km. Once the phase-transfer limitation was overcome, H2 Ra's remained relatively constant with increasing dilution (Figure 5) and their values were independent of the initial H2 concentration as expected. Influence of endogenous H2 production °nlYmax and EM' The effect of endogenous H2 production on Vmax and Km estimates was investigated using the PHASIM model. Several simulations were run using the same Vmax and Km values, but with increasing linear rates of H2 production (Figure 28 Figure 5. Dependence of apparent H2 Km on magnitude of the biological sink under phase-transfer and non-transfer limited conditions. Points are Km estimates calculated from both linear and non-linear analyses of 5 initial velocity-H2 concentration data pairs. Along with apparent Km's determined from initial-velocity experiments are plotted Km estimates obtained from progress curves performed at dilutions of 20 or greater, to demonstrate equivalence of these two kinetic approaches once H2 consumption is not phase-transfer limited. (X) G Apparen+ H2 KM (pM) 4> a) 9 Q R) Q Q 29 Line represen+s avg. KM of 5.8 pM &————a en s 19 as so 40 so Dilu+ion Foc+or Figure 5 30 Figure 6. Effect of an increase in endogenous substrate production (A) and an increase in‘Vmax with time (B) on Michaelis-Menten progress curves. All theoretical curves were generated by PHASIM for gaseous substrate consumption under non-transfer limited conditions. The lowest curve in (A) is the case for no endogenous H2 production with the succeeding curves above this one representing linear endogenous H2 production rates equal to 5, 10, 15 and 20% of the Vmax for substrate consumption. The‘uppermost curve in (B) is identical to the lowest curve in (A) and represents the situation where me remains constant over the course of the progress curve. The lower succeeding curves depict the patterns of substrate consumption where Vmax doubles within 100, 75, 50 and 252 of the total time required for the control progress curve to be 99% complete. Concen+ho+ion Uni+s Concen+ha+ion Uni+s 31 PHASIM Model Do+o . 18 A 12‘ 8‘ 4. O l - V 0 59 100 150 Time Uni+s 1 6 PHAS I M Model Data 18 a so 109 150 Time Uni+s Figure 6 32 6A). After the simulated data were generated they were analyzed, using the same non-linear regression program employed in analyzing experimental data, to obtain estimates of the kinetic parameters. Rates of endogenous production that were a greater percentage of the maximum rate of substrate consumption produced less error in estimates of Vmax than in Km (Figure 7). The apparent Km's increased with increasing rates of endogenous substrate production, whereas apparent Vmax's decreased with increasing rates of substrate production. In the absence of endogenous substrate production, the kinetic parameters calculated from simulated data equalled the parameter values plugged into the PHASIM model for that particular simulation. In order to assess the error in calculated Vmax and Km estimates for filtered rumen fluid, arising from endogenously produced H2, rates of H2 production were determined for this methanogenic habitat. These rates were subsequently used to calculate potential errors in the kinetic parameters using Figure 7. undiluted rumen fluid (500 ml) was incubated in the gas-recirculation system, and H2 and CH4 concentrations were monitored (Figure 8).Best-fit curves were calculated for the CH4 data and the first-derivatives of these curves were evaluated at times when endogenous H2 concentrations were measured. Endogenous rates of ruminal H2 production were calculated by multiplying calculated CH4 production rates by 4, and then assuming that 1002 of total CH4 production derives from chemolithotrophic methanogenesis (10). Rates of H2 production for undiluted rumen fluid, over the period of time the H2 pool size was relatively constant (Figure 9), ranged from 0.7-0.9 uM h"1 (Figure 8). These rates equal 41 and 52, respectively, of the average ruminal Vmax for H2 consumption. Thus, potential errors 33 Figure 7. Errors in apparent Vmax (A) and Km (B) resulting from concomitant endogenous H2 production during the course of progress curve experiments. Each theoretical curve was generated from 60 separate simulations, all of which were run under non-transfer limited conditions. 34 XONA +ue4oddv UT 40443 x n unawwm XOZ> m0 N 00 0+Om COH+ODDOL¢ m OdocomOUCu 0+ mm mm mm ow m4 9 m o o a) (O .—u V' N N 0') 8 ¢ XOZ> CH LOLLU EM CH LOLLM O+OQ H0302 EHm " I. C. 0 :II 1’ 1: ® 0 II [> 1: ‘ LO 0 I.II 1’ 0 (Y) I. C, o I: [> 4) .) I [> 1: Q 0 I; [> 3 (ad 0 g [:> I: 0) 0 ll. E> :: O ‘19 o ,'.' 1’ 1: a E I. 9’ 0 ll 1.> 0 "'1 0 II > 1: 2 0 I'll: " 40 $ , . 0 u I j: .—4 \‘ " ' 0’ u 0 .‘a‘ " 3 o 1"“ P f,’ 1 l‘ u v. e 0 n ” 5i \ I’ g ’l p, A :I ‘2 ‘4’ (SD 6) ® ‘9 0 Lo 0 m H H I-T 2H P9AT°SSTP T°“U Figure 2 63 Figure 3. H2 retention times for strained rumen fluid and diluted whale contents estimated from total CH4 production rate data (Figure 2). Straight lines parallel to x-axis encompass range of retention times calculated using previously published H2 kinetic parameters. 64 owe m muomwm . 0¢+DCH2 8m 8m 8m on; E. xoz>xz¥ 23 a xoz>xz¥ Ems: .m; (S) 9NT+ 49A044n+ 8H 65 consumption rate. Retention times for endogenously produced Hz were calculated by dividing measured H2 concentrations by estimated rates of H2 production, caculated from the CH4 data, or by H2 consumption calculated using previously determined Vmax and Km estimates (13). [Once the H2 pool approximately attained steady-state, H2 retention times calculated in the above two ways agreed reasonably well for strained rumen fluid (Figure 3). 0n the other hand, H2 retention times for diluted whale rumen contents were slightly shorter than those predicted from estimates of H2 kinetic parameters for strained rumen fluid. Influence of organic loadingon H2 production. The response of endogenous H2 production to organic loading was examined by incubating filtered rumen fluid with 1, 10 or 50 g of ground hay. The endogenous H2 level in the flask containing 10 g of hay rose approximately S-fold above endogenous H2 concentrations measured for unamended rumen fluid (both for diluted whale contents and strained rumen fluid) (Figure 4), and the rate of CH4 production was correspondingly higher (Figure 5). The endogenous H2 cancentraton rose to a maximum (0.57 uM dissolved) and subsequently declined. In contrast, when 50 g of hay was incubated with strained rumen fluid the endogenous H2 concentration increased dramatically and did not peak during the course of the experiment (Figure 4). This particular experiment was prematurely halted because of excess pressure due to autgassing of C02 from the liquid phase (the pH had dropped to 5.6). CH4 production stopped abruptly in the flask containing 50 g of hay (Figure 5). The steady-state H2 concentration in rumen fluid amended with with 1 g of milled hay was comparable (0.12-0.13 uM dissolved) with endogenous H2 concentrations for unamended strained rumen fluid and whole rumen contents (Figure 3). Additional 66 Figure 4. Dissolved H2 concentrations for strained rumen fluid amended with 50 (triangles), 10 (circles) or 1 (squares) g of finely milled hay. 67 “+50 9 P [.0 C0 I-T 2H pGATOSSTp Towd Minu+es Figure 4 68 Figure 5. CH4 production by strained rumen fluid amended with 50 (triangles), 10 (circles) or 1 (squares) g of finely milled hay. 69 n munmwm m0+3CH2 ®m¢ ®mm ®mm QHN 9; on unnuuhuuuWJ-“ni.“nun-Fnunou-uuuuu".J.,Q..\ H N m om m on t Q <— (I) (I.I .—1 70 experiments performed using various organic loading rates showed similar trends; additions of 50 or more g of milled hay always resulted in extremely high H2 concentrations that failed to return to steady-state concentrations. DISCUSSION Under steady-state conditions, the rates of H2 consumption and production must be equal if the H2 pool does not vary with time (i.e., de/dt - 0). We observed that the H2 concentration in unamended strained rumen fluid and diluted whole rumen contents reached approximate steady-state conditions (range: 0.1-0.2 uM) in a relatively short period of time (viz., 2 h). The steady-state was approximate because the H2 level declined slightly with time. This presumably reflects the gradual depletion of organic substrates in the samples assayed. Qualitatively similar patterns have been observed for endogenous H2 concentrations measured in fistulated cows maintained solely on a roughage diet; the dissolved H2 concentration was approximately constant (range: 0.4-2 uM), declining slowly with time after feeding (unpublished results). The retention times for H2 we measured ignzitgg ranged from 0.2-1 sec for strained rumen fluid, and agree well with those calculated from the CH4 production and dissolved H2 data of Hungate (6) (Table 1). In a later publication, Czerkawski et. al. studied the kinetics of H2 consumption and CH4 production in strained rumen fluid obtained from fistulated sheep. We calculate from their CH4 production rate and dissolved H2 data, H2 retention times ranging from 9-28.4 sec (4). These values are rather high given the low solubility of this gaseous substrate and the characteristically high rates of methane production for undiluted rumen fluid. The H2 retention times for diluted whole rumen contents were shorter than those for strained rumen fluid presumably reflecting the higher levels of Hz-consuming biomass present 71 72 .Ammv 4 edema ow nwsau soapy voououuo you muouoaouoo ouuoaax «m Baum voumauummo .Nm maven new: nouonsoon you on: panda mossy amass muooawuooxo you on N Canoe ow mowuuoo soon vouoaaoamoe .vonvo uom onus mououunnsn moon? muaoawuooxo you Amv _ manna mu mowuumo Baum vouoanoamo noawu uo>oousao .m.xoa> a: mo nonmawuoo he noowumuumooooo N: maooowovoo mafia m.aM um vaumswuoo mouva>wv an vouoasoaoon .Ac a mouse nowuoavouo amonuoav mouse mouuoovoun am he cowumuumoocoo N: no>Hommqv mm4n4>wn he nouoanoaouo ma~4v afieoae was caucuses ~ua.o meson summons 41~.o u: u: shoe .H« .u. «emasxuuuo nuns Ase shaman: mo.o cine assess: ~m.o mmmmmmmmm. nooquoaamooo Nu oooouomom moowuosvoua N: .mouou cowuoasmooo a: msmuo> soauosvouo mm moosowovso aoum vouoasaaoo N: Hacaasu mo Aoomv moawu soauaouom .4 manna 73 in the former. The above retention time estimates are for H2 production, calculated from CH4 production data assuming that 1002 of the ruminal CH4 derives from chemolithrophic methanogenesis. H2 retention times may be alternately estimated by calculating endogenous rates of H2 consumption and dividing endogenous H2 concentrations by these calculated rates. Using the derivative form of the Michaelis-Menten equation, estimates of Vmax and Km for ruminal H2 consumption previously obtained (13), and H2 concentrations measured in ziggg, we calculated H2 retention times similar to those estimated from our iguziggg_CH4 production data (Table 1). In a review on the rumen microbial ecosystem (7), Hungate calculated a turnover rate costant for H2 of 710 min'l, which is equivalent to a retention time of 0.08 sec. This value is lO-fold less than the average retention time calculated from Hungate's CH4 production and H2 concentration data (6), and about 2-fold below our lowest in zitgg estimates. But his retention time estimate is comparable in magnitude to retention times (about 0.3 sec) for diluted whole contents once the latter values have been corrected for dilution. Several investigators (9,10,16) have shown that ruminal fermentative bacteria are nearly evenly distributed between the particulate and fluid phases of whole rumen contents. Assuming this is true for Hz-consumers as well, then our estimates of total H2 production rates and calculated retention times for strained rumen fluid were underestimated and overestimated, respectively, by about 502. Taking this into account, the retention time of H2 in whole rumen contents should be 0.2-0.3 sec. Its interesting to note that these values are similar to H2 retention times estimated for diluted whole contents, but 74 are almost 10-fold greater than retention times (0.03 sec) for diluted whole contents corrected for a dilution factor of four. Taking the above into account, The retention time of ruminal H2 in 2252.13 likely between 0.03 and 0.3 sec. The endogenous H2 pool responds rapidly to increases in rates of H2 production. We found in zigrg that finely ground hay can increase the rate of H2 production to such an extent that methanogenic activity is saturated and the H2 pool size increases. The rate of H2 production for strained rumen fluid amended with 50 g milled hay (65 mM h'l) was several fold higher than H2 Vmax's (mean - 18 mM hfl) estimated for rumen fluid obtained from grain-fed cows before feeding (13). Hz-consuming activity may become transiently saturated in ruminants fed once daily rations relatively high in carbohydrates; the H2 pool size increases within several hours past-feeding, but declines to pre-feeding steady-state levels within a few hours, presumably after readily degradable materials have been consumed (12). Czerkawski and Breckenridge found that the dissolved H2 concentration rapidly rose and then peaked in sheep fed a concentrate diet (5). We believe the above observed spikes in the H2 concentrations result from increased rates of H2 production that temporarily saturate Hz-consuming activity. Fifty g of milled hay added to 500 m1 of rumen fluid (0.1 Kg 1-1) is equivalent to a loading rate 1/4 of that the fistulated cows received at feeding (0.4 Kg hay 1’1), assuming a total rumen fluid volume of 50 1. As stated above, the endogenous H2 pool measured igugitg for fistulated cows fed a roughage ration did not spike after feeding, remaining approximately constant during the day. In contrast, the endogenous H2 concentration dramatically increased in strained rumen 75 fluid amended with 10 g or more of milled hay. (The colonizable surface area of the hay used in our loading experiments was much greater than that for unmilled hay, and this greater surface area would be expected to significantly increase the availability of readily degradable materials resulting in greater endogenous H2 production rates. We have found endogenous CH4 production rates and steady-state H2 concentrations in strained rumen fluid amended with unmilled bay to not be significantly different from values for strained rumen fluid alone (data not shown). If the level of organic loading is high enough (50 g of finely ground hay in this case), then endogenous H2 levels may exceed by several hundred-fold the normal endogenous H2 concentrations. In such a case, H2 is primarily derived from reactions that are exergonic at Hz concentrations much greater than those required for interspecies H2 transfer (15). Although we did not measure them directly, rates of acid production were probably high since the pH typically dropped to values of 5.6-6 when 50 g of hay or more was incubated with strained rumen fluid. During the onset of lactic acid acidosis, a high production rate of lactic acid results in a lowering of the ruminal pH to 5.5, which in turn halts methanogensis (2). ACKNOWLEDGEMENTS The authors thank R. E. Hungate and R. B. Hespell for discussions on ruminal H2 kinetics and ruminant metabolism. This work was supported by National Science Foundation grants DEB 78-05321 and DEB 81-09994. 1. 4. 5. 9. 10. 11. 12. LITERATURE CITED 0 Cappenberg, T. E., and R. A. Prins. 1974. Interrelationships between sulfate-reducing and methane-producing bacteria in bottom deposits of a freashwater lake. III. Experiments with 14C labeled substrates. Antonie van Leewenhoek J. Microbiol. Serol . 42: 457-469 . Counotte, G. H. M., and R. A Prins. 1978/1979. Regulation of rumen lactate metabolism and the role of lactic acid in nutritional disorders of ruminants. Vet. Sci. Commun. 2: 277-303. Czerkawski, J. W., and G. Breckenridge. 1971. Determination of concentration of hydrogen and some other gases dissolved,in bio- logical flaidso Lab. PraCte 32: [603-405, 4130 Czerkawski, J. W., C. G. Harfoot, and G. Breckenridge. 1972. The relationship between methane production and concentrations of hydrogen in the aqueous and gaseous phases during rumen fermen- tation in_vitro. J. Appl. Bacterial. 25: 537-551. Flett, R. J., R. D. Hamilton, and N. E. R. Campbell. 1976. Aquatic acetylene-reduction techniques: solutions to several problems. Can. J. Microbiol. 32; 43-51. Hungate, R. E. 1967. Hydrogen as an intermediate in the rumen fermentation. Arch. Mikrabiol. 22: 158-164. Hungate, R. E. 1975. The rumen microbial ecosystem. Ann. Rev. Ecol. syste 2': 39-650 Hungate, R. E., W. Smith, T. Bauchop, I. Yu, and J. C. Rabinowitz. 1970. Formate as an intermediate in the rumen fermentation. J. Bacteriol. 102: 384-397. Leedle, J. A. 2., and R. B. Hespell. 1980. Differential carbohydrate media and anaerobic replica plating techniques in delineating carbohydrate-utilizing subgroups in rumen bacterial populations. Appl. Environ. Microbiol. 32: 709-719. Leedle, J. A. 2., M. P. Bryant, and R. B. Hespell. 1982. Diurnal variations in bacterial numbers and fluid parameters in ruminal contents of animals fed low- or high-forage diets. Appl. Environ. Microbiol. 44: 402-412. Lovely, D. R., and M. J. Klug. 1982. Intermediary metabolism in a entraphic lake. Appl. Environ. Microbial. 43: 552-560. Robinson, J. A., R. F. Strayer, and J. M. Tiedje. 1981. Method for measuring dissolved hydrogen in anaerobic ecosystems: Application to the rumen. Appl. Environ. Microbiol. 41: 545-548. 76 13. 14. 15. 16. 17. 18. 77 Robinson, J. A., and J. M. Tiedje. Kinetics of hydrogen consumption by rumen fluid, anaerobic digestor sludge and sediment. Appl. Environ. Microbial. in press. Smith, P. H., and R. A. Mah. 1966. Kinetics of acetate metabolism during sludge digestion. Appl. Microbial. 14: 368-371. Thauer, R. R., K. Jungermann, and K. Decker. 1977. Energy conservation in chemotrOphic anaerobic bacteria. Bact. Revs. 41: 100-180. Warner, A. C. I. 1962. Some influencing the rumen microbial population. J. Gen. Microbiol. 28: 129-146. Wilhelm, E., R. Battino, and R. J. Wilcack. 1977. Low-pressure solubility of gases in liquid water. Chem. Rev. 11: 219- 262. Wolin, M. J. 1979. The rumen fermentation: A.model for microbial interactions in anaerobic ecosystems. Adv. Microb. Ecol. 3: 49-77. ARTICLE III (CHAPTER III) RESOURCE COMPETITION AMONG SULFATE-REDUCERS AND METHANOGENS FOR HYDROGEN by Joseph A. Robinson and James M. Tiedje 78 ABSTRACT Michaelis-Menten (uptake) parameters (Vmax and Km) were estimated for various genera of methanogens and strains of sulfate-reducing bacteria from substrate depletion and product appearance data. In addition, Monod growth kinetic parameters (“max: K3 and YHZ) were determined for Desulfovibrio G11 and approximate values obtained for Methanospirillum hungatei JF-l. Km estimates for the methanogenic bacteria ranged from a low of 2 uM (Methanospirillum PMl) to 12 uM for Methanosarcina barkeri MS; Methanospirillum hungatei JF-l and Methanobacterium.PMl had intermediate H2 Km values of about 5 uM. Estimates of Km for H2 of Desulfovibrio P31 and G11 were lower than values for the methanogens, with means of 0.7 and 1.0 uM, respectively. Significant differences were not observed among the H2 Vmax estimates of methanogenic and sulfate-reducing bacteria when these were normalized to total protein. A two-term Michaelis-Menten equation demonstrated that apparent uptake affinities by resting mixtures of sulfate-reducers plus methanogens depended on the ratio of sulfate-reducing activity to total Hz-consuming capacity. Half-saturation constants for growth-KB-of 011 (2-4 uM) and JF-l (2-10 uM) were comparable in magnitude to their Km values for H2 uptake. Maximum specific growth rates and YHZ values appeared to be greater for 011 than for JF-l. These kinetic investigations suggest that sulfate-reducers outcompete methanogens for H2 whether in a resting or growing state when sulfate is not limiting, and suggest that a correlation exists between uptake and growth kinetic parameters for the bacteria studied. 79 INTRODUCTION The ability of sulfate-reducers to outcompete methanogens in both natural habitats and defined anaerobic consortia has been suggested in a number of previous investigations (1,2,6,7,16,17,18,19,32). Greater affinities for H2 or acetate (1,2,16,18,31), the effect of H28 on methanogenic bacteria (7), and bioenergetic reasons (17,34) have all been invoked to explain the apparent competitive advantage sulfate-reducers have over methanogens. Until recently, quantitative data supporting any of these hypotheses were lacking. In the past year, Kristjansson et. al. (15) and Robinson and Tiedje (J. A. Robinson and J. M. Tiedje, Annu. Meet. Am. Soc. Microbiol., 1982, I95, p. 110) reported on H2 Km's for resting cultures of sulfate-reducing and methanogenic bacteria. Additionally, Lovely et. al. (16) determined H2 kinetic parameters for lake sediments containing sulfate-reducing and methanogenic activites. All of these studies suggest that sulfate-reducers outcompete methanogens for H2 under resting conditions, given similar levels of biomass, because of their lower KIn for Hz. This manuscript extends the above findings by including more varied groups of Hz-consuming anaerobic bacteria. Our results substantiate the claim that sulfate-reducers generally will outcompete methanogens for H2 when sulfate is not limiting. Using a two-term Michaelis-Menten model we were able to predict apparent uptake affinities for H2 by several mixtures of sulfate-reducing and methanogenic bacteria. Further, this model predicts that differences in H2 Km's between these two functional groups of bacteria significantly affect which group processes the 80 81 greater proportion of a given quantity of H2 only in the mixed- and first-order regions. Michaelis-Menten paramters may only be used to evaluate competition between sulfate-reducers and methanogens for H2 when these bacteria are not growing. In order to evaluate competition among growing bacteria, Monod-type kinetic parameters must be estimated. Therefore, we also determined Monod growth kinetic parameters of sulfate-reducers and methanogens growing in batch on limiting H2. Robinson and Tiedje (submitted, AEM no. 723) recently demonstrated the feasibility and advantages of using the integrated Monod equation for estimating growth kinetic parameters, particularly for gaseous substrate consumption. We fitted H2 depletion data to the integrated Monod equation and the estimated parameters predict that sulfate-reducers can outgrow and hence outcompete methanogens for growth-limiting H2. These results correlate with conclusions drawn from estimates of uptake parameters determined for these two groups of bacteria. MATERIALS AND METHODS Organisms. Methanospirillum hungatei JF-l, Methanosarcina barkeriMS and Desulfovibrio G11 were obtained from the culture collection of Marvin P. Bryant. Methanospirillum PMl, Methanobacterium PM2 and Desulfovibrio P82 were gifts from Dan R. Shelton, who isolated these organisms from an anaerobic aromatic-degrading consortium. Media andfigrowth conditions. The percentage composition of the growth medium was: KQHP04, 0.15; KH2P04, 0.3; (N34)2804, 0.3; NaCl, 0.6; MgSO4 x 7H20, 0.12; CaClz x 2H20, 0.08; trypticase (BBL), 0.2; resazurin, 0.0001; vitamins, 0.01; trace metals, 0.01; cysteine-Nazs, 0.038; and Na2003, 0.2. The vitamin solution contained in mg/l: biotin, 2; folic acid, 2; pyridoxal-HCl, 10; riboflavin, 5; thiamin, 5; nicotinic acid, 5; pantothenic acid, 5; cyanocobalamin, 5; p-aminobenzoic acid, 5; and lipoic acid, 6. The trace metal solution contained the following compounds in g/l: ZnSO4 x 7H20, 0.01; MnClz x 4H20, 0.003; H3303, 0.03; CaClz x 6H20, 0.02; CuClz x 2H20, 0.001; NiClz x 6H20, 0.002; NazMo04 x 2H20, 0.003; FeClz x 4H20, 0.15; and Na23e03, 0.01. For growth of the two sulfate-reducers (viz., G11 and P31) the above medium was supplemented with 3.4 g of Na2804/l. On occasion, autoclaved clarified rumen fluid (52) and yeast extract (Difco; final concentration, 0.2 2) were added to the medium to increase cell yields. The growth medium was prepared under Ar according to the Hungate anaerobic technique (12) and bacteria were routinely cultured in 160-ml serum vials containing 50 ml of medium. Serum vials were sealed with butyl rubber stoppers held in place with aluminum crimp seals. Before 82 83 serum bottles were inoculated the headspaces were evacuated and refilled to 2.5 atm several times with an 802 H2-202 C02 gas mixture. The gas mixture was passed over hot copper filings to remove trace amounts of 02. Serum bottles were incubated on their sides on a rotary shaker at 37°C, and the headspaces of the bottles were repressurized approximately every other day. Methanogenic bacteria were transferred weekly, whereas sulfate-reducers were transferred every 2 or 3 days. Anaerobic diluent. The anaerobic diluent had the following composition in g/l: anpoa, 0.9; N301, 0.9; CaClz x 2H20, 0.027; MgClz x 6H20, 0.02; MnClz x 4H20, 0.01; CoClz x 6H20, 0.001; resazurin, 0.001; Na2C03, 4; cysteine-H01, 0.56; and Na2804, 3.4. It was prepared anaerobically under 002 and was pipetted into both 160-ml serum vials (100 ml) and 2-1 flasks (300 or 500 ml). In some experiments, the Na2804 was omitted. Resting cell suspensions. Cells were harvested in mid- to late-exponential growth (25-50 ml) using an Amicon ultrafiltration cell (Amicon Corp., Lexington, MA). Millipore filters having a 47-mn diameter (type HA; pore size - 0.45 um) were cut to fit the sterile filtration cell (dia. - 45 mm), and the entire assembly flushed with C02 for ca. 30 min prior to the addition of cells. Cultures were anaerobically transferred to the filtration cell using pre-flushed plastic syringes. After the 25-50 ml of culture medium was filtered, the cells trapped on the filter were resuspended in an equal volume of the diluent and filtered again. This was repeated once more and the cells were finally resuspended in 10-25 ml of the anaerobic diluent. This cell suspension was then anaerobically transferred to a 2-1 flask containing more anaerobic diluent, typically 300 or 500 ml. 84 Progress curve experiments. Progress curve experiments were initiated by first attaching the 2-1 flask to the gas-recirculation used in previous studies (25). At time zero, ultra-high purity H2 (Matheson, Joliet, IL) was injected into the headspace of the flask. Its concentration, and the concentration of CH4 (when measured), were monitored at 20, 30 or 60 min intervals. Each progress curve was terminated when the H2 concentration in the recirculating gas-stream had reached 1-52 of the H2 concentration at the first time point . For sulfide appearance experiments, a 2-1 flask with a stopper pierced by an ll-gauge maleable connector to which was attached a one-way stainless steel stopcock (Popper and Sons, New Hyde Park, NY) was used. The distal end of the maleable connector was immersed several mm below the aqueous-gaseous phase interface in the flask, and samples were withdrawn from the aqueous phase at 60 min intervals using a sterile 3-ml glass syringe. To avoid absorption of H28 by the rubber stapper, thin layers of teflon and then Saran were wrapped over that part of the stapper in contact with the headspace of the flask. All progress curves were run at 37°C. Hz, CH4 and H28 measurements. H2 and CH4 were measured in the recirculating gas-stream.using gas chromatographs equipped with microthermistor and flame-ionization detectors, respectively. The columns used in these GC's and the chromatographic conditions employed have been previously outlined (25). The liquid phase concentrations and total amounts of both H2 and CH4 were calculated using equations that have appeared elsewhere (10). Sulfide dissolved in the aqueous phase was measured using a modification of the Pachmayr method (22). One-ml samples withdrawn from 85 the 2-1 flask were injected into 25-ml serum vials each containing 20 ml of a 22 Zn acetate solution. Each vial was sealed with a teflon-backed silicon septum, which was in turn held in place with an aluminum crimp seal. Ten ml of the resulting ZnS suspension was transferred to an 18.5-ml screw cap tube, to which was added 1 ml of acidic 0.22 phenylenediamine reagent and 0.1 ml of an acidic 102 solution of FeNH4(SO4)2 x 12H20. The screw cap tube was then sealed with a teflon-lined cap and the absorbence of the solution read after 30 min. Sulfide concentrations were determined by comparing absorbsnce readings with those for a standard curve prepared using a ZnS solution. The amount of H28 in the gaseous phase was estimated by first calculating the fraction of total sulfide in the aqueous phase present as H28 (aq) (the pH of the anaerobic diluent was 6.7) and then dividing this by the Bunsen absorption coefficient for H28 at 37°C (31). Michaelis-Menten progress curve analysis. H2 depletion curves were fitted to the integrated form of the Michaelis-Menten equation (8) using nonlinear regression analysis. In addition to estimating Km and Vmax in this way, we also treated the initial H2 concentration as another parameter (initial state) to be estimated. Details on fitting substrate depletion data to the integrated form of the Michaelis-Menten equation when the initial substrate concentration is unknown are provided in J. A. Robinson's Ph. D. thesis, Michigan State University, East Lansing, 1982, D.A. no. 00-00000. CH4 and sulfide appearance data were also fitted to the integrated form of the Michaelis-Menten equation that describes product appearance in the presence of background levels of the product (i.e., progress curves of unknown origin). Equations required for this analysis have 86 appeared elsewhere (9,20). Provisional estimates of KIn and Vmax for CH4 and sulfide were calculated from least-squares analysis using a linearized version of the integrated Michaelis-Menten equation for progress curves of unknown origin (20). An initial estimate of the displacement term P0 (background concentration of product at t - 0) was determined by fitting a straight line to the first 3-6 product concentration-t data pairs and extrapolating back to the product concentration axis. The final concentration of sulfide or CH4 was taken as an estimate of So; So was not updated. All Km estimates are for the concentrations of H2, CH4 or sulfide in the aqueous phase that half-saturate activity, whereas Vmax estimates are calculated for the total amount of the substrate or product present in the 2-1 flask, normalized to the volume of sample assayed. Co-culture kinetic experiments. Substrate consumption (H2) by two competing organisms having different affinities and maximum potential a°91V1t33 (Vmax's) cannot be described by a one-term Michaelis-Menten model, but may be predicted using a two-term Michaelis-Menten equation. The latter expression only describes substrate consumption when the total number of catalytic units (i.e., cells) is fixed. This expression has the farm -dS/dt - V13/(K1‘+ S) + st/(KZ‘+ 3) where, V1 and V2 are the maximum rates of substrate consumption by organisms one (Desulfovibrio G11) and two (Methalospirillum JF-l), respectively. The constants K1 and K2 correspond to the half-saturation constants for the two organisms. For the co-culture kinetic experiments, G11 and JF-1 were grown in the above described medium and filtered separately. Washed cell 87 suspensions were either (1) mixed in various ratios and added to 2-f1asks containing anaerobic diluent or (ii) one organism was added in increasing increments to a flask already containing the other bacterium. For each mixture of G11 plus JF-l, H2 and CH4 progress curve data were obtained and analyzed using the same nonlinear analyses applied to progress curve data obtained using G11 or JF-l alone. Theoretical curves describing total H2 consumed and CH4 plus sulfide produced for resting suspensions of G11 plus JF-l were calculated by integrating the above equation along with two one-term Michaelis-Menten equations for formation of the products using numerical integration. Parameter values were those estimated from H2 depletion data for these two Hz-consuming anaerobes. Growth kinetic experiments. Estimates of YHZ, “max and K3 for H2 consumption by Desulfovibrio G11 and Methanospirillum JF-l growing in batch with limiting Hz were estimated by fitting sigmoidal H2 depletion data to the integrated Monod equation. Details on fitting substrate depletion data to the integrated Monod equation using nonlinear least-squares analysis has been previously described (Robinson and Tiedje, AEM submitted, ms no. 723). Estimates of Ks and Y32’ analogous with Km and Vmax: were calculated from aqueous H2 concentration data and the total amount of H2 consumed with time, respeCtively. The parameter “max may be estimated from aqueous or gaseous data or the sum of these two, since its value does not depend on the solubility of the gaseous substrate (J. A. Robinson, Ph.D. thesis, Michigan State University, East Lansing, 1982). For growth kinetic studies, 10-25 ml of a mid-exponential culture of either G11 or JF-l were filtered as above and added to 300 ml of the 88 growth medium, in which rumen fluid and trypticase were replaced with 0.22 Na acetate. Monod progress curves were initiated and terminated in the same fashion as those progress curves designed to estimate Michaelis-Menten parameters for H2. Biomass formation was followed at hourly intervals by measuring protein content of the growth medium, determined using the Lawry assay with BSA as the standard. Cell suspensions were hydrolyzed in dilute alkali (33) before assays were performed. RESULTS H9 Michaelis-Mentengparameters for methanogenic and sulfate-reducingfpacteria. Goodness-af-fit of the H2 progress curve data, obtained under resting conditions, to the integrated Michaelis-Menten equation is illustrated in Figures 1-5. H2 Km estimates for the sulfate-reducing bacteria were consistently lower (0.7-1 uM) than those for the methanogens which ranged from 12 uM (Methanosarcina barkeri MS) to approximately 2 uM.(Methanospirillum PMl) (Table 1). H2 Vmax values for the organisms assayed, when normalized to total protein content per flask, exhibited considerable overlap. For all organisms except Methanosarcina MS, Hz-consuming activity (Vmax) was stable for three days or greater; MS activity gradually declined over this same period of time. Additionally, substitution of cysteine-HCl with cysteine-Nazs as reductant or elimination of NazSO4 from the anaerobic diluent did not affect apparent Km estimates for H2 consumption by the methanogenic bacteria. Influence of initial H2 concentration an apparent Km- Under phase-transfer limited conditions, the apparent Km for H2 will depend on the initial substrate concentration (80) (25), which is inconsistent with the Michaelis-Menten model. To be certain the H2 KIn estimates in Table 1 were not obtained under mass-transport limited conditions and to check for product inhibition of substrate consumption we performed H2 progress curves at different initial H2 concentrations. No apparent dependence of half-saturation constants for H2 uptake on the initial H2 concentration was found for Desulfovibrio strains G11 and P81 (Table 2); 89 90 Figure 1. H2 progress curve data for Methanospirillum hungatei JF-l at two different initial H2 concentrations. Continuous curves are theoretical progress curves calculated from best-estimates of Van“, Q and SO obtained via nonlinear regression analysis. 91 368 480 600 Minu+es 248 120 I-I 3H pGAIOSSTP Towd Figure 1 92 Figure 2. H2 progress curves for Methanosarcina barkeri MS versus theoretical curves calculated from best-estimates of Michaelis-Menten parameters. l 93 1800 Q G) (D G) ‘ G <- V- (.0 (1) Q (U .—a I-T 8H peATossTp Towd Minu+es Figure 2 94 Figure 3. H2 progress curve data for Methanospirillum PMl (squares) and Methanobacterium PM2 (triangles) versus theoretical curves calculated from best-estimates of me, Km and So. 95 g 889 A. i Minu+es Figure 3 400 _ (0 ® peATOSSTP TONH 96 Figure 4. H2 progress curve data for Desulfovibrio G11 at two different initial concentrations of H2 plotted against theoretical curves. 97 611 4 1‘ \r‘w u! 540 Minu+es 360 100 o to a}; I-TBH pGATOSSTp TowH Figure 4 98 Figure 5. H2 progress curve data for Desulfovibrio PSl plotted against theoretical Michaelis-Menten curves at two different initial H2 concentrations. 99 540 720 900 Minu+es 360 I-T 3H peATOSSTP Figure 5 100 Table 1. Summary of H2 kinetic parameters for methanogenic and sulfate-reducing bacteria.a Organism Vmax (uM min'l)b KIn (uM)c Methanospirillum JF-l 9.05 5.00 hungatei 12.5 4.94 13.2 4.36 16.3 5.51 Methanospirillum PMl 19.6 3.24 16.5 1.75 Methanobacterium PM2 13.3 4.1 Methanosarcina MS 21.6 12.8 barkeri 22.6 12.5 Desulfovibrio 011 6.89 1.19 6.29 1.01 6.54 1.06 6.75 1.03 Desulfovibrio PSl 4.53 0.69 1.82 0.70 3.46 0.65 aMichaelis-Menten parameters were estimated by fitting H2 disappearance data directly to the integrated Michaelis-Menten equation, given initial Km and Vmax estimates obtained via linear analysis of transformed data. bvmax values are for total amount of H2 consumed per volume of aqueous phase (300 or 500 ml) in which the bacteria were suspended. Protein content was 0.04-0.09 g/flask, except for MS progress curves where total g protein was about twice this value. cKmestimates are for H2 dissolved in the aqueous phase. 101 Table 2. H2 Km estimates for Desulfovibrio strains 011 and P81 at different initial H2 concentrations.a Organism So (uM)b Km (uM)c PSI 10.6 0.70 9.69 1.50 8.42 0.69 2.92 1.18 G11 20.6 0.74 16.1 1.31 6.89 1.01 8So and Km estimates were obtained by fitting the H2 progress curve data to the intergated Michaelis-Menten equation using nonlinear least-squares analysis. Initial estimates of Km and $0 for nonlinear analysis were calculated using a linearized version of the integrated Michaelis-Menten equation. Total protein content per flask was 0.08 g. Initial H2 concentrations are mol H2 dissolved in the aqueous phase att-OO CK“ estimates are for H2 dissolved in the aqueous phase. 102 Figure 6. CH4 and dissolved sulfide appearance .data versus theoretical curves calculated using best-estimates of Vmax and Km The initial substrate concentration was not updated and taken as being equal to the final concentration of product, either sulfide or CH4. 103 I-T 2_3 peATossTg Toww [D [D .—u P; Q o e a Q Q l n r 7 ® u (I) II II II II '1: O :I ‘ ® '1' (o I 1'. V “I I‘ q- " ‘5 I ® 3' m KL.) ‘ Q N v “ fl‘ l “a (D v a“ a v- L >1 <9 fix 1 Q \s v (U ‘4: \s N! 4...‘ :1 . L Q <0 x- (u <9 I-T 4H3 PGATOSSTG TowH Minu+es Figure 6 104 Table 3. Michaelis-Menten parameters for product appearance by methanogenic and sulfate-reducing bacteria.a Organism - vm (uM min’1)b Km (uM)° Methanosarcina MS 5.52 4.37 barkeri Methanospirillum PMl 4.41 1.57 Methanobacterium.PM2 2.97 1.15 Desulfovibrio 011d 1 3 1 8 1.2 3 aProduct (sulfide and CH4) appearance data were fitted to the integrated Michaelis-Menten equation for progress curves of unknown origin (8,19). bTotal mol H2 consumed within each flask, normalized to the volume of anaerobic diluent in which the bacteria were suspended. Protein content per flask ranged from 0.04-0.09 g. CK“ estimates are mol H2 1'1 dissolved in the aqueous phase. dParameter estimates were calculated from fits of transformed data to a linearized version of the integrated Michaelis-Menten equation describing progress curves of unknown origin (19). 105 similar results were observed for Methanospirillum JF-l. Michaelis-Menten parameters for product appearance by methanogens and sulfate-reducers. The fit of CH4 and sulfide appearance data to the integrated Michaelis-Menten equation for product formation is illustrated in Figure 6. CH4 progress curve data generally fit the integrated Michaelis-Menten equation better than did sulfide appearance data. The Km values for CH4 were about equal to or slightly less than those calculated from H2 progress curve data (Table 3). Sulfide Km estimates for Desulfovibrio G11 were generally higher than those determined from H2 depletion data (Table 3). H2 Vmax estimates determined from product appearance were approximately 1/4 of the H2 Vmax values for the same suspension of cells. Apparent H2 Kmvalues for mixtures of G11 and JF-l. To examine the influence different ratios of Desulfovibrio G11 to total Hz-consuming activity have on the apparent Km for H2 consumption, we performed H2 progress curves on bacterial suspensions containing various mixtures of Gll plus JF-l. In the absence of the other organism, the Km for H2 uptake agreed with estimates for G11 and JF-l in Table 1. In contrast, the apparent Km was shifted in a nonlinear way from those values for the pure cultures when one organism (G11 versus JF-l) dominated total H2-consuming activity (Figure 7). Qualitatively similar results were obtained with mixtures of G11 plus Methanosarcina MS. The theoretical curve in Figure 7 was constructed by numerical integration of the two-term Michaelis-Menten equation shown above. All simulations were run using an initial substrate (H2) concentration of 4 x the Km for the methanogenic population (viz., 5 uM). The resulting simulated data were fitted to the integrated Michaelis-Menten equation 106 Figure 7. Apparent H2 Km at different ratios of sulfate-reducing activity (organism 1) to total Hz-consuming capacity. See text for experimental details and how theoretical curve was constructed. 107 m ouzmfim xm>+4>cxe> E. ... a. er er D D D 0>L30 H00H+0L00£ (U 0 V‘ (W0) “9 +Ueu°ddv 108 using nonlinear regression analysis to obtain apparent H2 Km values at different ratios of sulfate-reducing to total Hz-consuming activity. Estimates of Monod_growth kineticgparameters. Estimates of “max: K3 and Y32 for H2 consumption by Desulfovibrio G11 were obtained by fitting sigmoidal substrate depletion data to the integrated Monod relation, assuming constant yield. The fitting of sigmoidal H2 progress curve data obtained for this bacterium has been illustrated in another publication (Robinson and Tiedje, AEM submitted, ms no. 723). Estimates of Monod growth kinetic parameters for G11 are shown in Table 4. The half-saturation constant (K8) for Hz-limited growth of G11 was 2-4 times its half-saturation constant for uptake (Table 1). Preliminary experiments with Methanospirillum.JF-l indicate that its K3 for H2 is in the range 2-10 uM. 109 Figure 8 . Comparison of theoretical curve with measured H2 concentrations (squares) for growth of Desulfovibrio Gll growing on H2 as the sole electron donor. The initial substrate concentration (SO) was estimated by extrapolating back to t-O on the H2 concentration axis. 110 m ouswwh OLDOI 9.. mm ...m S m o 56 :1 mums rxmo.®uxozl HOZ\.+OLQ m an» aim oaroa>ooanmooddmwmmw. ® .—I Q (U Q ('0 ® I-T 2H peATossTp Towd G) V" 111 Table 4. Monod growth kinetic parameters for Desulfovibrio G11.a 10"2 x “max (h’l)b . K8 (uM)c Ynz (g prot. mol’l)d 6.54 4.17 0.72 4.87 2.42 0.99 8Data were fitted to the integrated Monod equation according to procedures described by Robinson and Tiedje (AEM submitted, ms no. 723). bCalculated from aqueous phase H2 data. cHalf-saturation coefficients for growth are for mol H2 1'1 dissolved in the aqueous phase. ineld coefficients are expressed as g protein produced per total H2 consumed in the flask. DISCUSSION Our H2 Km estimates for sulfate-reducing bacteria agree with a value recently reported for Desulfovibrio vulgaris Marburg determined by Kristjansson et. al. (15). They estimated an H2 Km of 1.3 uM from disappearance of H2 in the aqueous phase using an Hz-electrode. The agreement between this value and our H2 Km estimates for Desulfovibrio G11 and P81 is noteworthy since ours were determined by following non-transfer limited H2 consumption in the gas phase, while theirs was calculated from direct measurements of aqueous phase H2 concentrations. This lends additional support to the validity of determining half-saturation coefficients by monitoring depletion of the substrate from the gas phase (25). Its interesting to note the close agreement between Kristjansson et. a1.'s H2 KIn estimate and our values given they estimated this parameter by calculating slopes at different points along a progress curve and using these as initial velocity estimates, a statistically questionable procedure (8). The H2 Km estimates we obtained for the methanogens, with the exception of Methanosarcina barkeri MS are higher on average (mean - 5 uM) than previously determined estimates. Hungate et. al. (13) and Ferry et. al. (27) reported H2 Km values for Methanobrevibacter ruminantium.and Methanobacterium formicicum of 1 and 2 uM, respectively. Kristjansson et. al. (15) reported an H2 Km for Methanobrevibacter arboriphilus AZ of 6.6 uM, similar to values we found for Methanospirillum hungatei JF-l and Methanobacterium PM2. The two strains of Methanospirillum.we studied-namely, JF-l and G11-possessed different half-saturation constants for H2 uptake; PMl's H2 Km was about 112 113 2 uM and similar to KIn estimates for the sulfate-reducers, whereas JF-l had an average H2 KIn of approximately 5 uM. The uptake parameter estimates for PMl correlated with the observation that it grew more rapidly than did JF-l; in fact, its growth rate was on a par with that of Desulfovibrio Gll. The KIn for H2 consumption by Methanosarcina MS was at least 2-fold greater thanKm estimates for the other methanogens. Two factors may potentially account for MS's apparently lower affinity for H2: (1) a decrease in activity gradually over the course of the progress curve or (ii) mass-transport limitations between the aqueous phase and the surfaces of MS aggregates. A slight decline in Hz-consuming activity with time is the more likely of these two, since the initial H2 concentration and differences in stirring rates did not influence the apparent Km, factors which are expected to alter apparent KIn values for gaseous substrate consumption if a mass-transport limitation exists (25). Our H2 Km estimates for sulfate-reducers and methanogens are similar to values found by other workers for methanogenic habitats and sediments in which sulfate-reducers are the dominant H2 consumers. Robinson and Tiedje (25) determined H2 KIn values for rumen fluid, anaerobic digestor sludge and eutrophic sediment ranging from 4-9 uM, overlapping the range we found for methanogenic bacteria. Additionally, Lovely et. al. (16) observed a shift in the apparent H2 Km when FeSO4-amended sediments were treated with CHCl3 to inhibit methanogenesis comparable in magnitude to the difference between average K, values for the methanogens and sulfate reducers studied by us. Estimates for rumen fluid of 1 uM and eutrophic sediments of 2 uM 114 reported by Hungate et. a1. (13) and Strayer and Tiedje (29), respectively, are low but comparable in magnitude to the H2 Km estimates in Table 1. In addition to determining Km values for H2 consumption, we also estimated Michaelis-Menten parameters for CH4 and sulfide appearance. This was done to (i) check if Vmax estimates for product appearance were 1/4 of values for H2 consumption (the expected result) and (ii) determine whether uptake parameters for mixtures of bacteria with different substrate affinities can be estimated by following the individual products produced. Although the estimated uptake half-saturation constants for product appearance were similar to Km values obtained for H2 depletion, they must be cautiously interpreted since the final concentration of either sulfide or CH4 was taken as equalling the initial substrate concentration. Its not possible to use the initial H2 concentration to estimate Michaelis-Menten parameters for sulfide or CH4 appearance because a one-to-one stoichiometry between H2 consumption and production of either product does not exist. An important result of the product appearance experiments is that Vmax estimates for CH4 and sulfide production were about 1/4 of the H2 Vmax values obtained for a given flask experiment, in accordance with the stoichiometries for chemolithotrophic methanogenesis and sulfate-reduction. The more significant result, however, is the finding that uptake constants for mixtures of bacteria having different substrate affinities-such as sulfate-reducers and methanogens-cannot be estimated by following appearance of the individual products (CH4 and sulfide); significant underestimates of the half-saturation constant will be obtained depending on the fraction the organism producing the 115 product (CH4, for example) comprises of total substrate-consuming activity. ' A one-term Michaelis-Menten model cannot be used to aCcurately predict rates of substrate depletion by bacterial consortia in which individual papulations possess different uptake parameters. One manifestation of the inadequacy of a one-term model for such cases is the dependence of the apparent H2 Km on the relative amounts of Desulfovibrio G11 and Methanospirillum hungatei JF-l for mixtures of these organisms. On the other hand, a two-term Michaelis-Menten equation does predict accurately total H2 consumption by two resting bacterial populations and further, predicts (i) apparent H2 affinities for particular combinations of sulfate-reducing plus methanogenic activities and (ii) the partitioning of a given initial H2 concentration between these two anaerobic HZ-consumers. Lovely et. al. (16) observed that a two-term Michaelis-Menten model accurately predicted total H2 consumption by sulfate-reducers and methanogens in eutrophic sediments when Michaelis-Menten parameters independently determined for these two populations were plugged into a two-term Michaelis-Menten expression. We found apparent H2 Km values for several ratios of sulfate-reducing to total Hz-consuming activity agreed with values predicted from solutions of a two-term model, illustrating that apparent uptake constants for substrate consumption (or product appearance) depend on the respective densities of the organisms catalyzing this process when they possess dissimilar substrate affinities. Michaelis-Menten parameters for the organisms we studied (assuming that no significant difference really exists among Vmax estimates normalized to protein) predict that sulfate-reducers will outcompete 116 methanogens (with the possible exception of Methanospirillum PM1) for H2 if sulfate is not limiting. Our results are corroborated by the finding that sulfate-reduction dominates total Hz-consuming activity in habitats where sulfate concentrations are high (14,21). 0n the other hand, methanogens may effectively compete for H2 under sulfate-limited conditions, a situation that presumably occurs in many eutrOphic lake sediments. Solutions to the two-term Michaelis-Menten equation predict that methanogenic bacteria may process more of an initially saturating H2 concentration even when sulfate-reducers are saturated for sulfate (Figure 9). Further, when the initial H2 concentration is in the saturating region (10 x Km) with respect to methanogenic activity (also saturating for sulfate-reduction) Km differences of 10-fold between competing bacterial populations influence little the partitioning of H2. 0n the other hand, organisms like JF-l must comprise a significantly greater fraction of total Hz-consuming activity when the initial substrate concentration is in the mixed- to first-order region in order for them to process an equal share of the H2 (Figure 9), in analogy with conclusions reached by Healey regarding competition among growing bacterial populations (11). Experiments performed with mixtures of JF-l and Gll at different initial H2 concentrations corroborate predictions made by Figure 9; the difference in the half-saturation constant between sulfate-reducers and methanogens becomes improtant when the initial H2 concentration (or steady-state H2 pool in a habitat) does not saturate Hz-consuming activity. Thus, methanogens could compete well with sulfate-reducers in sufate-rich natural habitats if their densities are relatively high and the steady-state H2 concentration is saturating. 117 Figure 9 . Fifty-percent partitioning curves for mixtures of sulfate-reducing and methanogenic bacteria. Each curve predicts what proportion of the total H2-consuming activity an organism (2) with a lower uptake affinity for H2 (or any substrate) must comprise in order for it to process an equal share (502) of a given initial substrate concentration. Curves were constructed by numerically integrating a two-term Michaelis-Menten expression at a given ratio of Km values (Km 2/Km,1) for various initial substrate concentrations and ratios of methanogenic to total Hz-consuming activity. Initial substrate concentrations were 100 (diamonds), 10 (circles), 1 (triangles) and 0.1 (squares). 118 .—. I A ,5) 4.4 II «I o l‘ “ 0 4 m u 4 o a D . u 1 . <> *CO H 54 IN «3 (9 <<>| \\‘ (\J u x‘ o < .¢¥ E \‘ o 0 E <\o 0‘ (U 0') (I) F-(D-U) (IA+3A)/3A Figure 9 119 But in most anaerobic ecosystems $2 EEEE,HZ concentrations are in the first- to mixed-order regions (26,29) and thus, sulfate-reducers would be expected to account for an appreciable prOportion of total Hz-consuming activity due to their lower Km for H2, given comparable levels of methanogen and sulfate-reducer biomass. When growth (or an increase in activity) occurs during a progress curve the resulting substrate depletion curve is sigmoidal (23,25), an observation inconsistent with Michaelis-Menten kinetics. Thus Michaelis-Menten kinetic parameters cannot be used to assess competition for H2 between sulfate-reducers and methanogens when growth occurs. In a natural habitat, Michaelis-Menten kinetics will predict the partitioning of a given substrate among competing bacterial populations only if the biomass levels are in steady-state or approximately so. If biomass levels of the competing populations increase (252 or greater) then Monod kinetics predicts the outcome of substrate partitioning. We estimated half-saturation constants (K3) for batch growth of Desulfovibrio G11 on Hz were 2-4 uM, similar in magnitude to this bacterium's H2 Km. Our YHZ Estimates (avg. - 0.86 g protein mol"l H2), when converted into the equivalent of g cells produced per mol sulfate reduced, are similar in magnitude to those reported for Desulfovibrio Marburg and Madison strains reported by Badziong et. al. (3). Results of preliminary growth studies with Methanospirillum JF-l suggest its K8 for H2 is 2-10 uM [> 10-fold below a K3 value reported by Schonheit et. al. (28) for Methanobacterium thermoautotrophicum probably obtained under phase-transfer limited conditions], and that its “max and Y32 are lower than those for G11. Differences in‘YH2 for these two bacteria are expected since sulfate-reduction yields more energy than does 120 chemolithotrphic methanogenesis per mol of H2 consumed (30). In summary, Monod kinetic parameters predict, in analogy with uptake H2 parameters, that sulfate-reducers will outcompete methanogens H2 and in so doing, process a greater fraction of the H2 available to these two bacterial groups. One final item must be considered in evaluating the competition for H2 among sulfate-reducers and methanogens. The latter group of Hz-consumers are metabolic specialists; that is, the number of different electron donors they may metabolize appears to be quite limited (4). In contrast, sulfate-reducers are generalists; they are capable of anaerobically respiring a diversity of electron donors including H2 and additionally, can ferment numerous organic compounds (24). Thus, sulfate-reducers not limited to growth on H2 would be expected to grow to significant cell densities in habitats where sulfate is limiting and then if sulfate is suddenly added (either naturally or artificially) they can effectively compete with chemautotrophic methanogens for limiting H2. Lovely et. al. (16) observed that sulfate-reducers comprised 502 of the overall Vmax for H2 consumption in eutrophic sediments amended with 20 mM sulfate. Typically these sediments are sulfate-depleted and the large population of sulfate-reducers capable of oxidizing H2 presumably developed as a result of their fermentative growth on the cadre of organic substrates occurring in eutraphic sediments. ACKNOWLEDGMENTS We thank Dr. Marvin P. Bryant and Dan R. Shelton for supplying cultures used in this study. We also thank Walter Smolenski and Marilyn 121 Boucher for expert technical assistance. This research was supported by National Science Foundation grants DEB 78-05321 and DEB 81-09994. 1. 2. 3. LITERATURE CITED Abram, J. W., and D. B. Nedwell..1978. Hydrogen as a substrate for methanogenesis and sulfate reduction in anaerobic saltmarsh sediment. Arch. Microbiol. 117:93-97. Abram, J. W., and D. B. Nedwell. 1978. Inhibition of methanogenesis by sulfate-reducing bacteria competing for transferred hydrogen. Arch. Microbiol. 117:89-92. Badziong, W., R. K. Thauer, and J. G. Zeikus. 1978. Isolation and characterization of Desulfovibrio growing on hydrogen plus sulfate as the sole energy source. Arch Microbiol. 116:41-49. 40 men, We Ea, Ge Ea Fox, Le Jo mam, Ce Re Woese, 311d Re So 5. 6. 8. 9. 10. 11. 12. 13. Wolfe. 1979. Methanogens: Reevaluation of a unique biological group. Microbiologic. Revs. 43: 260-296. Beck, J. V., and K. J. 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Different K8 values for hydrogen of methanogenic bacteria and sulfate-reducing bacteria: An explanation for the apparent inhibition of methanogenesis by sulfate. Arch. Microbiol. .l31:278-282. Lovely, D. R., D. Dwyer, and M. J. Klug. 1982. Kinetic analysis of competition between sulfate reducers and methanogens for hydrogen in sediments. Appl. Environ. Microbiol. 43; 1373-1379. McInerney, J. M., and M. P. Bryant. 1981. Anaerobic degradation of lactate by synthrophic association of Methanosarcina barkeri and Desulfovibrio species and effect of H2 on acetate degradation. Appl. Environ. Microbial. 41:346-354. McInerney, M. J., M. P. Bryant, and N. Pfennig. 1979. Anaerobic bacterium that degrades fatty acids in syntrophic association with methanogens. Arch. Microbiol. 122: 129-135. muntfort, Do Do, Ro Ao Asher, Eo Lo Mays, and Jo Mo Tiedjeo 1980o Carbon and electron flow in mud and sandflat intertidal sediments of Delaware inlet, Nelson, New Zealand. Appl. Environ. Microbiol.‘39:686-694. 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Gerhardt (ed.), Manual of methods for general bacteriology. American Society for Microbiology, Washington, D.C. Zehnder, A. J. B. 1978. Ecology of methane formation. 22 R. Mitchell (ed.), Water pollution microbiology. Vol. 2. John Wiley and Sons, Inc., New York. p. 349-376. APPENDICES 125 APPENDIX A PHASE-TRANSFER KINETICS Phase-boundaries commonly occur in the natural habitats of microoganisms. They are regions where dissimilar phases (e.g., gas, solid, liquid) come together and examples can be found in terrestrial as well as aquatic ecosystems. Any boundary between phases potentially limits microbial activity when substrates required for metabolism must pass from one phase (gas or solid) into the phase where the microorganisms reside (liquid). Mass-transport across these interfaces dominates the overall kinetic pattern if the rate of phase-transfer cannot supply the biological demand (2). But if phase-transfer exceeds the rate of microbial substrate consumption, then mathematical relations defining the nature of this consumption can be discerned, and the parameters in these equations estimated (3,7). Thus, any kinetic investigation of microbial activity in situations where phase-boundaries occur must first assess whether mass-transport is limiting apparent rates of microbial activity or not. The majority of the work presented in this thesis involved estimating kinetic parmaters for H2 consumption by suspensions of bacteria and samples obtained from methanogenic habitats. I wanted to estimate H2 kinetic parameters by monitoring substrate depletion in the gaseous phase, since this is generally easier than following substrate consumption in the aqueous phase. But concentrations of a gaseous substrate in the gas phase may be multiplied by a partition coefficient to obtain aqueous phase concentrations only when a mass-transport 126 127 limitation does not exist (5). Thus, it was essential to understand the influence transport of H2 from the gaseous into the aqueous phase exerts an the kinetic pattern of H2 consumption. In accord with this, I constructed a model (PHASIM) to predict conditions leading to mass-transport limitations and more importantly, how these limitations influence apparent H2 kinetic paramters. In this appendix I not only include the PHASIM.model found in chapter I, but also additional systems of differential equations that describe (1) gaseous substrate consumption by growing cells and (ii) gaseous product formation from gaseous and nongaseous substrates. This appendix concludes with recommendations on how mass-transport limitations may be experimentally detected and the consequences for kinetic investigations if they are ignored. The systems of equations that appear below are nonlinear and lack analytical solutions. Thus, solution curves (viz., substrate or product concentration versus time) for these equations must be approximated using numerical integration. There are many techniques available to solve these types of initial-value problems, but I chose the Runge-Kutta procedure since its self-starting and provides good approximations to even 'stiff' nonlinear equations (1). Gaseous substrate consumption by resting cells. Gaseous substrate consumption by cells in the aqueous phase of a two-component system, comprised of gaseous and aqueous phases, may be described by the two equations shown below. These expressions are identical to those in the PHASIM model (chapter I) and are presented here again for the sake of completeness. 128 ng/dt--K1a(BSg-Sa) [A.l ] dsa/dt.Kla(BSg-Sa)-VmaxSa/(Kni'sa) [AoZ] The variables 83 and 33 equal the gaseous and aqueous phase concentrations of the gaseous substrate, respectively. The parameters Kla, B, Vmax and Km are the volumetric transfer coefficient, the Bunsen absorption coefficient, the maximum rate of gaseous substrate consumption, and the Sn at which the rate of consumption is half-maximal. Solutions to the above system of equations predict that gaseous substrate consumption follows Michaelis-Menten kinetics when the Kla is greater than Vmax and that further increases in the Kla do not alter substrate depletion (Figure 1). In contrast, the mass-transport term in [A.2] controls substrate consumption when Vmax is greater than the K1a. In this case, the kinetic pattern is approximately first-order and increases in the K1a increase rates of substrate depletion (Figure 2). In addition to these, solutions to [A.l] and [A.2] predict that apparent Km's for gaseous substrate consumption are markedly dependent on Vmax and the inital substrate concentration under mass-transport limited conditions. Thus, only when a phase-transfer limitation does not exist do apparent saturation kinetic parameters have biological meaning, and this holds whether gaseous substrate depletion is monitored in the gaseous or aqueous phase. Gaseous substrate consumption by growigg cells. When gaseous substrate consumption results in microbial growth, then the Michaelis-Menten model no longer applies. This is particularly evident when the initial substrate concentration is greater than the 129 Figure 1. Influence of Kla on gaseous substrate consumption not limited by phase-transfer. Equations [A.1] and [A.2] were numerically integrated for the following parameter values and initial conditions: B-0.02, Vmax‘O-l: Km-S, 83,0-20, and Sg,0-0. Kla equalled 10 (squares), 5 (circles) and 2 (triangles). (8) 130 150 200 250 Minu+es 100 so 20 9 8 7 uoT+04+ueouoo 9+04+sqn3 Figure l 131 Figure 2. Influence of Kla on gaseous substrate consumption limited by mass-transport. All parameter values and initial conditions are identical with those used for Figure 1, except Vmax was increased to 10. 132 30 40 50 Minu+es 20 16 I . A Q 0 LO ® 10 Q (U H .—I (6) uoT+04+ueouoo 9+04+sqns Figure 2 133 half-saturation constant (K3) for growth, since this results in sigmoidal substrate depletion (4,5). This type of kinetic pattern never occurs if Michaelis-Menten kinetics controls the rate of gaseous substrate consumption. To account for growth [A.2] can be changed to dSa/dt"Kla(Bsg‘sa)'1“maxSa/(Ks+5a)1X/Y [A031 where “max: KB and Y equal the maximum growth rate, the $3 at which the growth rate is half-maximal and the yield coefficient, respectively. In order to simultaneously solve [A.3] and [A.1] using numerical integration a third equation for updating the biomass concentration (X) is needed. This equation has the form dx/dt'lllmaxSa/(Kg+33)lx [A‘4] The gaseous substrate depletion curve for growing cells is approximately first-order when mass-transport is severly limiting, even when So is greater than Rs (Figure 3). This is reminicent of gaseous substrate consumption by resting cells (5) and substantial overestimates of K8 would probably result if apparent growth rates were used together with [A.4] under steady-state conditions to estimate this parameter. This probably accounts for the unrealsitically high H2 Ks's for methanogenic bacteria grown in continuous culture appearing in the literature (6). When gaseous substrate depletion by growing cells is moderately mass-transport limited, it becomes difficult to qualitatively discern this situation from gaseous substrate consumptiont affected by phase-transfer (Figure 4 versus Figure 5). But if Sa/Sg ratiosare calculated and plotted versus time, the difference between these two simulatons becomes obvious. When gaseous substrate consumption is not phase-transfer limited, Sa/Sg ratios equal the partition coefficient and do not change during substrate depletion (Figure 5). In contrast, Sa/Sg 134 Figure 3. Gaseous substrate consumption by growing cells severly limited by mass-transport. Equations [A.1], [A.3] and [A.4] were numerically integrated for the following parameter values and initial conditions: Kla-O.1, “max’O-Z: Ks-l, Y-0.05, B-0.02, Sg,o-200, Sa’o-O and Xo-l. Note the ratio of Sa/Sg should equal B if phase-transfer is not rate-limiting. 135 (8) 3/-K1a' (Pa-mg) [A-8] and dPg/dt-Kla' (Pa-B' P8) [A-9] In [A.7], S equals the concentration of the non-gaseous substrate. Experimental detection ofgphase-transfer limitations. A mass-transport limitation influences apparent saturation kinetic parmaters in a number of predictable ways. How this limitation influences the Km for gaseous substrate consumption has been extensively discussed elsewhere (3,5,7). In summary, apparent Km's for gaseous substrate consumption depend on the (1) initial substrate concentration, (11) magnitude of the biological sink in the aqueous phase, and (iii) K1a under phase-transfer limited conditions. Most significantly, the biological activity cannot be saturated if phase-transfer rates are less than the microbial demand. Similarly, the production of a gaseous product reflects the kinetics of gaseous substrate consumption and hence, kinetic parameters for product appearance will be erroneous if the mass-transport rate is less than Vmax' Experiments should be designed to check for these behaviors before biological significance can 142 Figure 6. Influence of K1a on ratio of aqueous to. gaseous phase concentrations of a gaseous product produced from a non-gaseous substrate. Parameter values and initial conditions were me-l, Km-S, B-0.02, So-ZO, Pa 0-0 and Pg’o-O. For curves A, B and C the K1a respectively was 100, 0.5 and 0.1. Note the aqueous phase becomes supersaturated with the gaseous product when the Kla is less than Vmax° 143 (8) uoT+O4+ueouoo lonp04d O 1.0 Q t - , (S) ‘¢ f 32 24 16 Time G <9 ' to <9 00 <9 N ‘_" s-c uoT+04+ueouoo 9+04+sqns Figure 6 144 be ascribed to the estimated Km's. Ascertaining whether or not mass-transport limits gaseous product appearance in the gas phase,when this product is derived from a non-gaseous substrate, is complicated because sigmoidal product appearance may result from (i) substrate depletion under phase-transfer limited conditions and (ii) growth-linked product formation. As for the cases mentioned above, kinetic parameters for product formation will depend on So and Vmax under mass-transport limitations. But the only way to check for the obfuscating influence of growth is by following substrate depletion. If substrate consumption is non-sigmoidal and product appearance is sigmoidal, then chances are a mass-transport limitation exists (Figure 7). This assumes that no lag occurs before Michaelis-Menten substrate depletion commences. In summary, both the nan-gaseous substrate and gaseous product should be monitored in order to verify that gaseous product formation is not limited by its movement from the aqueous into the gaseous phase. Only then may estimates of microbial kinetic parmaters be obtained that are independent of physical paramters that affect the Kla, such as stirring speed and inter-facial area. Should only aqueous phase data be used to calculate saturation kinetic parameters fo£_gaseous substrate consumption? After extensively simulating gaseous substrate consumption using the PHASIM model, I found apparent Vmax'3 for substrate depletion depend on the partition coefficient (B) when aqueous phase data are used to estimate this kinetic parameter. In contrast, the estimated Km equals the Km used for a given simulation when this parameter is calculated from aqueous phase substrate concentrations (Figure 8). PHASIM also predicts that Vmax 145 Figure 7. Gaseous product formation from a non-gaseous substrate for different values of Kla. Same parameter values and initial conditions used for Figure 6. 146 n ounwam OZHH m®.®nL3U re. ‘8' o-d a <8) d/ (be) _ d 147 Figure 8. Influence of gaseous substrate solubility on Km,app and Vmax,app estimated from aqueous phase data. Vmax and Km values were each 5, and the Kla was 50 for all simulations. Note Km’app (aq) equals Km and Vmax,app (aq) does not equal Vmax- 148 (b0) XONA +ueqoddv LO .23 .2 .‘1 (o <}- a (50) “>1 lueqoddv .4 Bunsen coeFFicien+ Figure 8 149 Figure 9. Influence of gaseous substrate solubility on Km ap 9 ‘Vmax,app estimated from for total mass (mol) of gaseous substrate Be and ing consumed with time. Same parameter values used for Figure 8. Note that Vmax,app (tot) equals Vmax: whereas Km,app (tot) does not equal Km. 120 150 <+°+) “N +Uedoddv ® ® ® 0') (0 (Y) Q .1 é- ch <9 <+°+> XONA +Uedoddv .3 .2 Bunsen coeFFicien+ Figure 9 151 should be calculated from the sum, 88 plus 88, since only when this is done do apparent Vmax's equal the Vmax'3 used to simulate the data (Figure 9). These predictions are supported by the observation that theoretical Sg curves approximate actual Hz progress curves only when Km's calculated from aqueous phase data and Vmax'B estimated from total quantities of H2 consumed are plugged into PHASIM. Using the Vmax estimated from aqueous Hz consumption data yields theoretical progress curves far too long compared to Hz depletion curves presented in chapters I and III. The above findings may be generalized as follows: half-saturation kinetic parameters should be estimated from substrate concentrations occurring in the phase containing the microorganisms, while vmax estimates should be calculated from the total amount of the partitioning substrate consumed in the closed system. Of course, this assumes that a mass-transport limitation is absent. A preliminary analysis of gaseous substrate consumption for growing cells suggest that (1) like Km, K3 should be estimated from aqueous phase data, (ii) in analogy with Vmax: Y should be calculated from the total quantity of partitioning substrate being consumed, and (iii) “max may be estimated from gaseous or aqueous phases substrate concentrations, or the sum of these two. These statements need some explanation. The K3 is computed using only aqueous phase data because its the liquid phase concentration of the growth-limiting substrate that determines to what extent the bacteria are saturated. Since biomass-C (or electrons) may be derived from the gaseous substrate present in either of the 2 phases, then Y should be estimated using data for the total amount of the substrate present in the closed system. Finally, the cells do not partition and estimation of their growth rate requires 152 no knowledge of the partitioning substrates' fate therefore, “max will be identical no matter which phase (or both) substrate depletion is monitored in. 1. 2. 3. 5. 6. LITERATURE CITED Burden, R, L., J. D. Faires, and A. C. Reynolds. 1978. Numerical analysis. Prindle, Weber and Schmidt. Boston, Massachusetts. p. 239-245. Charles, M. 1980. Technical aspects of the rheological properties of microbial cultures. Advances in biochemical engineering. Q; 1-62. N81“, R. Fe, 80 Ho Lin, and W. R. Bo Martin. 19770 EffeCt 0f mass transfer resistance on the Lineweaver-Burk plots for floculating microorganisms. Biotech. Bioeng. 12: 1773-1784. Pirt, S. J. 1975. Principles of cell and microbe cultivation. John Wiley and Sons, Inc., New York, New York. p. 22-28. Robinson, J. A., and J. M. Tiedje. Kinetics of hydrogen consumption by rumen fluid, anaerobic digestor sludge and sediment. Appl. Environ. Microbiol. in press. Schonheit, P. J., J. Moll, and R. K. Thauer. 1980. Growth parameters (K3, “max and Y3) of Methanobacterium thermoautotrophicum. Arch. Microbiol. 127: 59-65. Shieh, W. K. 1979. Theoretical analysis of the effect of mass- transfer resistances on the Lineweaver-Burk plot. Biotech. Bioeng. .El‘ 503-504. 153 154 APPENDIX B PROGRESS CURVE ANALYSIS Michaelis-Menten kinetics. Either the derivative or intergated forms of the Michaelis-Menten equation can be used to estimate Km and Vmax° Both of these forms are examples of nonlinear models that may be algebraically transformed into linear models. Linearized versions of the derivative form of the MichaeliséMenten equation, most typically the Lineweaver-Burk equation, have received widespread use for estimation of the above two kinetic parameters from initial velocity data (4). To a lesser extent, product appearance data obtained from progress curve experiments have been fitted to a linearized version of the integrated Michaelis-Menten equation to estimateKm and Vmax (11). Use of the integrated form has the advantage that estimates of Km and Vmax can be obtained from a single experiment in which substrate depletion or product formation is monitored, assuming that product inhibition is absent. In contrast, multiple experiments are required if the derivative form of the equation is employed and thus, the integrated Michaelis-Menten equation is preferable when routine estimates of these parameters are needed. thwithstanding use of linearized versions of Michaelis-Menten equations, its always preferable to directly fit data to nonlinear models since transforming data for fitting to linearized versions of these models concomitantly transforms the measurement errors (4). The inappropriateness of using ordinary least-squares analysis to fit data to linearized forms of Michaelis-Menten expressions has been previously pointed out (4). This practice should only be used to obtain 155 provisional estimates of the MichaeliseMenten parameters for subsequent improvement via nonlinear regression analysis. But a disadvantage to using nonlinear extremization techniques is these are iterative procedures (2), which can require appreciable computing power. The dramatic increase in availability of inexpensive microcomputers now makes it possible to take full advantage of these parameter estimation techniques when fitting data to integrated and derivative forms of equations describing enzyme-catalyzed reactions. Thus, linearized versions of these nonlinear models may be relegated to providing initial estimates of parameters and for illustrative purposes only. After integration, the Michaelis-Menten expression for product appearance is vmxc-me1n[ so/ (So-P) ] [B - 1 1 where, Vmax-the maximum rate of product appearance (or substrate consumption), t-time, P-product concentration at time t, Knesubstrate concentration at which rate of product appearance is half-maximal, and So-the initial substrate concentration. Since the rate of product formation equals the rate of substrate consumption, [8.1] can be rewritten to give met-So-SfiqnlMSo/S) ”‘21 in which s-the substrate concentration at time t. The integrated Michaelis-Menten equation cannot be explicitly solved for either P or S, but solutions to it can be approximated using either (i) Newton's method for finding roots to implicit functions or by (ii) numerically integrating the derivative form of this expression (3). Linear analysis. As alluded to previously, the integrated Michaelis-Menten equation can be linearized. Three different 156 linearizations, taken from a monograph by CornishPBOwden (4), are shown below. t=/1m<$o/S>'-(l/Vmax)[(so-S)/1n/+ 161 Figure 2. Plots of Michaelis-Menten progress curve data transformed according to [3.3] (A), [3.4] (3) and [3.5] (C) with relative error bars of +/-0.005 units. Same Km, Vmax and 30 used for Figure l. 162 m N ouswwm Amxomvca\amloms mma .H m. a . mm®®.®t mm®®w®+ fi rut A1 ;. <3/°$>UI/+ (Y) 163 vmamwusoo N ouzwwm +\mm\omcma m. m .... nm®®.®| w a mm®®.®+ .mm. .mm. +/cs-°s> 164 woaefiuaoo N muswfim amnomcxamxomcca .... .3 nm®®.®+ r a mm®®.®a (s-°S)/+ 165 NOnlinear analysis. The use of iterative techniques for fitting data to nonlinear equations requires that the sensitivity of the dependent variable to changes in each of the parameters be calculable. The first derivatives of P or S with respect to KIn and Vmax satisfy this requirement and these expressions may be derived from [3.1] or [3.2] using implicit differentiation (14). The sensitivity equations for Km and Vmax are only different in sign when derived from [3.1] versus [3.2]. The sensitivity of S to changes in Vmax is described by dS/deaX-t/(1+Km/S) [3.7] and the sensitivity of S to Km is given by dS/defln(80/S)/(1+Km/S) [3.8] Note that [3.7] and [3.8] are both functions of Km. By definition then, the integrated Michaelis-Menten equation is a nonlinear model since its sensitivity equations are not independent of the parameters. Linear models have sensitivity equations that are functions of only the independent variable or equal unity (2). In addition to their being required for nonlinear regression, the sensitivity equations predict (1) whether the parameters may be uniquely estimated or not, (ii) the relative precision of the estimated parameters, and (iii) the range of the independent variable over which the model is most sensitive to changes in the parameters. The last item is useful in designing optimal experiments for parameter estimation (2). If the sensitivity equations are proportional, then its impossible to obtain unique estimates of the paramters from the data using least-squares analysis (2). Unique parameter estimates may be obtained when the sensitivity equations are very similar, but they are highly correlated. This situation is undesirable since it implies that several 3": . \,l 166 combinations of parameter estimates may describe the same data set and is somewhat true for the integrated Michaelis-Menten equation, depending on So. The sensitivity equations for Km [3.8] and Vmax [3.7] yield similar curves with the former being numerically dominated by the latter (Figure 3). These facts explain the strong correlation among errors in these 2 parameters and the tendency of standard error estimates for Vmax to be less than those for Km (4,13). Lower standard errors for Vmax versus Km run contrary to intuition since the optimal initial substrate concentration for progress curve experiments is in the mixed-order region (2-4 x Km). But the explanation lies in the behavior of the integrated Michaelis-Menten equation itself and the sensitivity equations derived from it. There exist a number of techniques that may be used to fit S-t data pairs to [3.2] in the least-squares sense. I used the Gaussian method (2) to fit the fig consumption data presented in chapters I and III to this nonlinear model. This technique uses the equation S-So-Avmde/dvm+AK,ndS/dlsw .Ao xqeuuaaouoam mesa: _~.m_ on some soauoamov ouuuumnsm osu newsman hauouuuv he wouumuno one: moumauumo assume ..m.m_ ou unavuooou eoEHOMonuuu pump «0 mauhassm soamuouwou nausea scum woumasoflmo one: «masseuse HmuuaoHo .ouoanaoo Nam Hana: can one: use condom must assume Adamsvo cm vengeance o>u=u moouwoun scum .mouuuwanou o>uso muoumoue o~ no names one mosau> youosuuumn .Amv souoonu eased Huuusoo may scum eo>uuov soauusvo so mean: vuuuuosow one: muouuo Asouuuauu> mo usouoamuooo assumsoov o>«uuaou use Aeouuoa>ov vuuvsuun assumeoov nausea .N no om so use a mo Noa> was 6M o you _N.m. wsq>~oa an vouuuosow mums mo>uso umoumoumu mo.~ mh.N m~.~ “5.x am.~ am.~ mummm\«mme ma.o ~o.~ oc.~ no.~ ~o.~ _o.~ vu.xaa> os.o ma.o es.o sa.o ms.o so.o 03.xsa> ma.o oo.a oo.~ ma.~ no.n no._ um.au es.o sm.o no.0 mm.o ~m.o oa.o oH.ax No.0 I >0 ~0.0 I >0 no.0 I >0 «0.0 I am No.0 I am ~0.o I am nAom I av duo: ] muouuo o>wuuaom muouuo oamaqm «.muouuo o>wuudou no pauses wsucumusoo some u>uso amoumoua cousozImaHousoez mo ufimhausm unused msmuo> uuoaaasos mo eomuuueaoo .~ «Home 171 Figure 4. Sensitivity coefficients for SO derived from [3.1] versus [3.2]. Same Km, Vmax and So used for Figure 1. 172 m .0 q madman @EHH mnm mum on m. a 33%.. 1 [q fi omnxmn LO (U LO LO 1‘ Osp/sp do 0era/ap- 173 experiments, So was estimated by fitting a straight line to the first 3-6 data points and extrapolating back to the S axis. Its possible to treat an inital state, such as So, as another 'parameter' to be estimated using nonlinear regression analysis (2). But a sensitivity equation for So is required before this can be done. The sensitivity of S to changes in 30, derived from [3.2], is given by dS/dSo-(1+Il/Y [3.141 which may be integrated to give C11u[[Y(So-S)+Xol/Xo}-Czln(S/So)-umxt [3.151 where, cl-(x,r+sor+xo)/(Yso+xo) and Cz-KsY/(YSQ+X0). Equation [3.15] gives the familiar S-shaped curve for substrate depletion during batch growth. A relation similar to [3.15] describing the increase of biomass during batch growth has been derived by integrating the expression obtained after eliminating S from [3.12] using the mass-balance relation [3.13] (10.12). 175 Equation [3.15] like [3.2] is implicit in S and its solution must be numerically approximated. Though Newton's method works for approximating solutions to [3.2], I have been unable to solve [3.15] using this technique. But solution curves may be estimated by solving [3.12] simultaneously with dx/dt-[umxs/(K,+s)lx Ill-161 using numerical integration. An example of a solution is depicted in Figure 5, which was solved for the following initial conditions and parameter values: “max9001: Ks-S, Y-0.2, Xo-l and 80-20. Linear analysis. Its not possible to linearize the integrated Monod equation for the purpose of obtaining provisional K3, “max and Y estimates from transformed S-t data pairs. But initial estimates of these paramters may be obtained using -AS/At-[umxS/ (K3+s) ]X/Y [B-17] and AX/At-[umaxS/(Ks+8) ]x [3.181 Equations [3.17] and [3.18] are derived by replacing infinitesimal time (dt) with finite time (At) and are approximately correct if At is relatively small (14). These finite-difference equations are nonlinear, but may be linearized by taking the reciprocals of both sides giving -At/AS'(K5Y/um)X/S+YX/umx [3-19] and At/AX-(KS/Umax)X/S+X/um [3020] If both sides of [3.19] and [3.20] are multiplied by x then these equations become 176 Figure 5. Sigmoidal substrate consumption (S) and biomass formation (X) for cells growing in batch. Equation [3.14] was numerically integrated for the following parameter values and initial conditions: “max'oel: Ks-S, Y-0.2, 30-20 and Xo-l. 177 UOT+OJ+UOOUOO SSONOT g (o <- . 5..“— 5’9 1m JG xi '9 .6) <- o .82 00°F] l.— .6) U)! N .6) v-I G: U) ... .. :9. m o uoTioqiueouoo eloq+sqns Figure 5 178 -AtX/AS-(KsY/umax)l/S+Y/umax [3.21] and AtX/Ax-(Ks/umaxM/S-I-l/umax [Ii-22] which yield straight lines when AtX/AS and Atx/AX are plotted against l/S (Figure 6). Thus, [3.21] and [3.22] may be used to estimate K8, “max and Y given 8- and Kat data pairs. Nonlinear analysis. As was the case for [3.2], S-t data pairs may be fitted to [3.15] using the Gaussian technique. The required sensitivity equations for the parameters of this nonlinear model are dS/sz-(Y/Cs)[ln(X/Xo)-ln(S/So)]/06 [3.24] dS/dY-{C3(So-S)/X+ln(X/Xo)/C5[Kg+(1-C3)So] -ln(S/So)/05(Kg-C480)}/C6 lB-ZSI In the above three equations the terms C3, C4, C5 and C5 equal (xsr+rso+xo)/(rso+xo), KEY/(YSO+X0), YSo+xo and CgY/X+04/S, respectively. Although nonlinear analysis is possible for [3.15] and superior to analysis of linearized data, the graphical similarity among [3.23], [3.24] and [3.25] (Figure 7) predicts that estimates of “max: K3 and Y will be highly correlated. . The integrated form of the Monod relation {[3.15]} is superior to the derivative forms {[3.12] and {3.161} for estimating K3, “max and Y from gaseous substrate consumption data. A sensitivity analysis of derivative forms of Monod equations (like those used for continuous culture) predict that the best information about “max is achieved at the highest rates of substrate consumption. But mass-transport can easily limit gaseous substrate consumption by cells reproducing at high growth 179 Figure 6. Linearized discretized Monod data. Simulated data plotted in Figure 5 were linearized according to [3.21] and [3.22]. - S/ t and X/ t values were calculated by fitting cubic splines to the S- and X-t data pairs and then evaluating the first derivatives of the cubic polynomials. 208 SP/+PX- 180 ® (9 H xi (1)1 580 XP/+PX G) [O m Figure 6 181 Figure 7. Sensitivity coefficients for “max: K3 and Y. Same parameter values and initial conditions used for Figure 5. Jami? 182 XONdP/SP— (S) Q (S) Q LO ® Q (U H s—a LO Q a Q or) 0) L0 ‘ .. M‘ >- (U .® (0 m? X, 4H.F+ O [— Z 1 .® H -m . i . <9 ® G) G) G) ® CU 0) (O (‘0 H SNP/SP X 3-01 40 AP/SP X 8'8 Figure 7 183 rates, thus preventing the estimation of biological kinetic parameters (Appendix A). This holds for cells growing in continuous culture or batch. In contrast, good information about “max is obtainable using [3.15] at substrate concentrations in the mixed-order region, where mass-transport influences estimates of this parameter to a lesser extent. Indeed the sensitivity equations for [3.15] predict that (i) the optimal substrate concentration for estimation of K8, “max and Y using [3.15] is no greater than four times K3 and (ii) the precision of the estimated growth parameters is greatest for “max and least for Rs (Figure 7). These predictions intuitively seem incorrect, but a similar situation holds for substrate consumption controlled by Michaelis-Menten kinetics. Although one expects a Michaelis-Menten progress curve to yield better estimates of Km than Vmax: since optimal initial substrate concentrations lie in the mixed-order region, the opposite is true in practice (compare the estimated standard errors of Km with those of Vmax in chapters I and III). In Appendix C is listed MONODCRV which is a program that uses S-So-AKgdS/dKS-i-Aumxd8/dumx-I-AYdS/dY 03-26) to fit substrate disappearance data to [3.15]. It requires inital estimates of K8, “max and Y plus initial substrate (So) and biomass (X0) concentrations. MDNODCRV does not calculate provisional values for these parameters and must be supplied with independently determined estimates. In order to obtain these for data shown in chapter III, I fitted cubic splines (3) to the S- and Xet data pairs using a computer program (SPLINE) written by David D. Myrold which calculates the spline interpolants plus their respective first-derivatives. This information can be used in conjunction with [3.22] and [3.23] to estimate Ks, “max 184 and Y. MONODCRY does not update 80 nor X0. So could not be directly measured because of the previously stated reasoned and had to be estimated by extrapolating back to the S axis. Its possible to derive a sensitivity equation for updating So, but this was not done because I wanted to restrict the number of estimated parameters 3. As with PROGCRVI and PROGCRVZ, MONODCRV estimates the standard errors of the parameters assuming no correlation among measurement errors. Correlated measurement errors. Underestimates of standard errors for parameters are obtained if least-squares analysis is applied to data that lack statistical independence (i.e., possess correlated measurement errors). This situation worsens as the number of data points increases and has been shown to be true for linear (2,7) as well as nonlinear models (2). The easiest means of checking for lack or presence of correlated measurement errors is by examining the residuals (2,5,7). These equal the observed values of the dependent variable minus values predicted using the fitted equation and contain all of the information not explained after fitting data to a given model {e.g., [3.2], [3.15]}. In Figure 8 are shown the residuals for Hz consumption by Desulfovibrio Gll. Ideally these should be random and give the impression of a horizontal band. But its obvious they do not and in fact all the residuals calculated after fitting [3.2] to the Michaelis-Menten progress curves shown in chapters I and III exhibit essentially the same pattern. Thus, the measurement errors are correlated and the standard errors estimated for Km and Vmax by PROGCRV1 are too low. Notwithstanding the presence of correlated measurement errors, I was able to reduce their influence on standard error estimates for Km and 185 Figure 8. Residuals for Hz progress curve data fitted to [3.2] using nonlinear least-squares analysis. Residuals were calculated by subtracting predicted HZ concentrations (calculated from the best parameter estimates) from the observed Hz concentrations. Residuals are all expressed as percentages of the predicted Hz values. 186 w ouowem uoTioTAeg % 187 Vmax by increasing the sampling interval from 10 to 30 min. This reduced the degree to which the standard errors for Km and Vmax were underestimated since the magnitude of this error is proportional to the number of measurements (7). The correlated mesurement errors observed for Hz consumption by resting cells could have resulted from (1) fitting data to the wrong model for biological Hz consumption or (ii) the influence of physical factors on the circulation of gas through the sampling loop of the 32 GC. The first cause is serious since it questions the significance of the Km estimates presented in chapters I and III. If So is less (e.g., 2- to 5-fold) than K8, then [3.15] yields substrate depletion data very similar to what is expected for [3.2], regardless of the values of “max and Y (Figure 9). When the simulated data in Fig. 9 are fitted to [3.2] using PROGCRV1, the residuals exhibit a pattern similar to that seen in Figure 8. But the apparent Km bears no relation to the K3 and depends upon So. The Km's for 32 consumption by rumen fluid, anaerobic sludge, eutrophic lake sediment, Desulfovibrio strains and the methanogenic bacteria were independent of So and thus, the correlated measurement errors are probably not a result of incorrectly fitting the fig consumption data to [3.2] instead of an integrated Monod-type equation. This is supported by the pattern of residuals obtained after fitting a straight line to Hz disappearance data that resulted from sampling an empty flask containing H2 (Figure 10). These systematic residuals were not a result of adsorption occurring within the sampling loop, as has been previously observed for volatile organics (9), since repeated injections of Hz standards exhibited no statistical dependence (Figure 11), but likely resulted from incomplete mixing of Hz in the 188 Figure 9. Simulated Monod progress curve data containing simple errors (standard deviation-0.01 units) fitted to the integrated form of the Michaelis-Menten equation [3.2] . Parameter values and initial conditions were “max'0°1: Ra's, Y=0.2, 80-4 and Xo-l. Inset shows residuals calculated for data fitted to [3.2]. Note that although theoretical curve passes smoothly through all the data points, the residuals exhibit systematicity indicating the data have actually been fitted to the wrong model. 189 LO (U P .O (U .1!) '_q + .8 + q—l & m 0 Z M . . . . ® [.0 ‘1' (‘0 (U H 8 uoT+oq+ueouoo e+oq+sqn3 Time Figure 9 190 Figure 10. Residuals for Hz removal from an empty flask containing 32 due to sampling. The raw data were fitted to a linear model and this used to estimate predicted values which were in turn subtracted from the observed Hz concentrations. The percentage deviation of each residual is plotted versus injection number. Note the systematicity exhibited by’ the residuals reminicent of that seen in Figure 8. 191 5'0 UOT+OTA90 % Figure 10 192 Figure 11. Residuals for Hz standard tank sampled with time. A 5.98 x lO‘Satm (6.05 Pa) was sampled with time and residuals calculated by subtracting the mean Hz concentration from all measured values. Residuals are plotted versus injection number. Note the lack of a systematic trend among the residuals indicating statistical independence of the H2 measurements. 193 ®> an gusset UOT+OTA90 % 194 recirculating gas stream. In summary, the correlated measurement errors observed for the Hz progress curves were caused by slight heterogeneities in the recirculating 32 gas and not by inappropriate use of the integrated Michaelis-Menten equation. 1. 2. 3. 4. 5. 6. 10. 11. 12. 13. LITERATURE CITED Atkins, G. L., and I. A. Nimmo. 1973. The reliability of Michaelis constants and maximum velocities estimated by using the integrated Michaelis-Menten equation. Biochem. J. 135: 779-784. Beck, J. V., and K. J. Arnold. 1976. Parameter estimation in engineering and science. John Wiley and Sons, Inc., New York, New York. p. 334-3500 Burden, R. L., J. D. Faires, and A. C. Reynolds. 1978. Numerical analysis Prindle, Weber and Schmidt. Boston, Massachusetts. p. 116-128 and 239-245. Cornish-Bowden, A. 1979. Fundamentals of enzyme kinetics. Butterworth, Inc., Boston, Massachusetts. p. 200-210. Draper, N. R., and H. Smith. 1981. Applied regression analysis. John Wiley and Sons, Inc., New York, New York. p. 459. Duggleby, R. G., and J. F. Morrison. 1977. The analysis of progress curves for enzyme-catalyzed reactions by nonlinear regression. Biochem. BiOphys. Acta. 481: 297-312. Esener, A. A., I. A. Roels, and N. F. W. Kossen. 1981. On the statistical analysis of batch data. Biotech. Bioeng. 23:2391- 2396. Harbaugh, J., and G. Bonham-Carter. 1970. Computer simulation in geology. John Wiley and Sons, Inc., New York, New York. p. 61-97. Jonsson, J. A., J. Vejrosta, and J. Novak. 1982. Systematic errors occurring with the use of gas-sampling loop injectors in gas chromatography. J. Chromatogr. 236: 307-312. Knowles, G., A. L. Downing, and M. J. Barrett. 1965. Determination of kinetic constants for nitrifying bacteria in mixed culture, with the aid of an electronic computer. J. Gen. Microbiol. 28: 263-278. Nimmo, I. A., and G. L. Atkins. 1974. A comparison of two methods for fitting the integrated Michaelis-Menten equation. Biochem. J. 141: 913-914. Pirt, S. J. 1975. Principles of microbe and cell cultivation. John Wiley and Sons, Inc., New York, New York. p. 22-28. Robinson, J. A., and J. M. Tiedje. Kinetics of hydrogen consumption by rumen fluid, anaerobic digestor sludge and sediment. Appl. 195 196 Environ. Microbiol. in press. 14. Thomas, G. 3., Jr. 1972. Calculus and analytical geometry. Addison- Wesley, Inc., Reading, Massachusetts. p. 76-81. APPENDIX C COMPUTER PROGRAMS FOR DATA ANALYSIS Five computer programs (FILEMAN, PROGCRVI, PROGCRVZ, MONODCRV and PHASIM) are contained within this appendix and were written during the course of my graduate studies. All 5 are in North Star BASIC (North Star Computers Inc., CA) and were written for an IMSAI 8080 microcomputer, linked to 2 minifloppy diskette drives (North Star disk operating system), a black-and-white monitor (Sanyo Electric Inc., CA), Decwriter II (Digital Equipment Corp., MA) and digital plotter (Houston Instrument, TX). FILEMAN and the progress curve analysis programs (PROGCRVI, PROGCRVZ and MONODCRV) were written to facilitate data reduction and analysis, whereas PHASIM was written to simulate the kinetic behavior of substrate consumption by cells in a liquid phase. Data filing;system. FILEMAN serves as the main link in a chain of programs that include PROGCRVI, PROGCRVZ and MONODCRV, as well as other programs not included in this appendix (e.g., REGRESSl (regression routine for fitting data to one-term linear, exponential, logarithmic or power models), AUTOPLOT (routine for plotting data using digital plotter)]. I wrote all of FILEMAN except for the VIDPLOT subroutine (lines 3070-3910) which was written by Russell M. Edwards and Timothy B. Perkin. FILEMAN is used to create data arrays that are stored on minifloppy diskettes, using the Nerth Star random access filing system, and can be retrieved later using FILEMAN or read into other programs (e.g., PROGCRVI, REGRESSI) for analysis. All the programs listed in this appendix were written so that they may read arrays created by 197 198 FILEMAN (PROGCRVl, PROGCRVZ and~MONODCRV) or create simulated data arrays (PHASIM) that may be scanned and transformed using FILEMAN. There are 2 levels of control in FILEMAN, with the following commands available to the user at the highest level: (1) GEN, (2) SCAN, (3) PROOF, (4) MODIFY, (5) FILE, (6) DESTROY, (7) CHAIN, (8) MERGE, (9) PLOT and (10) QUIT. GEN and SCAN are the only 2 commands encountered when FILEMAN is first run (lines 30-70). Thus, the user may initially create a new data array (lines 150-330) or scan an already existing one (lines 1070-1220 and 340-540). After this, the user has access to these 2 commands plus the additional 8 listed above (lines 80-140). The maximum size an array may be is 250 rows by 7 columns. But this may be increased or decreased, depending on the amount of computer memory available, by changing the dimensions of the doubly subscripted variable X(i,j) in line 20. The PROOF command allows the data file to be viewed on either the monitor or Decwriter (lines 340-540) and if the user is satisfied with its contents, the array can be stored on minifloppy diskettes using the FILE command (lines 720-1060). Before a data array can be stored, a filename and identification string having no more than 8 and 80 alphanumeric characters, respectively, must be specified. The identification string is stored along with the elements of the data array under the specified file name on the dikette. Erasure of unwanted data files on diskettes is accomplished via the DESTROY command (lines 1230-1290). The CHAIN command allows the user to load and execute other programs for analysis of arrays created and stored using FILEMAN (lines 1300-1410). Concatenation of data arrays having the same number of columns can be done using the MERGE command (lines 2280-2500). Again, the array generated using this command must have no more than 250 rows, 199 otherwise a 'dimension error' will result. In addition to outputting the array to either the monitor or Decwriter, it may be graphically displayed on these devices via the PLOT command (lines 3070—3910). The PLOT subroutine uses low-resolution graphics and scales the x-y axes to the largest elements in the two columns plotted. Only data in the first quadrant of the Cartesian co-ordinate system (i.e., positve x and y values) may be plotted. Finally, the QUIT command terminates execution of FILEMAN. The second level of control in FILEMAN is linked to the first through the MODIFY and RETURN commands. The former provides the user with the following options: (1) CORRECT, (2) ADD, (3) DELETE, (4) TRANSFORM and (5) RETURN. The CORRECT command allows elements of the data array, including the identification string, to be corrected (lines 580-710). Secondly, data arrays may have elements added (lines 2510-2700) or deleted (lines 2710-3060) using the ADD and DELETE commands, respectively. They also may be mathematically transformed in various ways (delineated below) using the TRANSFORM command (lines 1420-2270). Lastly, the RETURN command permits the user to return to the first or highest command level. When the TRANSFORM command is executed the user is first faced with 2 Options, either a column within the data array may be mathematically transformed (lines 1760-2270) or 1 column of numbers can be used to transform a second (lines 1480-1750). If the first option is chosen the user may add, subtract, multiply or divide the specified column of numbers by a constant. Additionally, the natural logarithm of the numbers may be taken or they may be raised to a particular power. Lastly, the user may operate on the specified column of numbers with a 200 user-defined function entered at line 3940 before program execution. If the user chooses Option 2 instead then 1 column may be added to or subtracted from, or multiplied or divided by, a second column of numbers. After each of the above transformations, control passes back to the highest level and combinations of the above commands may once more be executed. The above commands encountered by the user when FILEMAN is executed are generally self-explanatory and the entire data analysis package, with FILEMAN as its core, is designed to be 'user-friendly'. FILEMAN LISTING 10 ICER$(26) 20 DIM X(250,7),I$(80) 30 INPUT ”GEN(1),SCAN(2)? ",c 40 IF c<1 OR c>2 THEN 30 50 INPUT "OUTPUT DEVICE (TYPE '0' FOR CRT OR '1' FOR DECWRITER)? ”,D 60 IF 0<>0 AND 0<>1 THEN 50 70 ON c GOTO 150.1070 80 PRINT ”GEN(1),SCAN(2),PROOF(3),MODIFY(4),FILE(5),DESTROY(6),CHAIN(7)" 90 INPUT "MEROE(8),PL0T(9),QUIT(10) ",c 100 IF c<1 OR C>10 THEN 80 110 IF c>2 THEN 140 120 INPUT ”OUTPUT DEVICE (TYPE '0" FOR CRT OR '1' FOR DECWRITER)? ",D 130 IF 0<>0 AND 0<>1 THEN 120 140 ON 0 GOTO 150,1070,340,550,720,1230,1300,2280,3070,3920 150 INPUT "ENTER THE NUMBER OF ROWS OF DATA YOU WISH To STORE ",N 160 IF N<-190 THEN 190 170 PRINT#O ”MAXIMUM NO. OF ROWS ALLOWED IS 250; SEE LINE 10" 180 GOTO 3770 190 INPUT ”ENTER THE NUMBER OF COLUMNS OF DATA YOU WISH TO STORE ”,M 200 IF M<-7 THEN 230 210 PRINT#O ”MAXIMUM No. OF COLUMNS ALLOWED IS 7; SEE LINE 10" 220 GOTO 3770 230 PRINT#O CHR$(26)240 FOR I-1 TO N 250 PRINT ”ROW #",23I,I, 260 FOR J-l To M 270 PRINT#O TAD(J*10). 280 INPUTl x(I,J) 290 IF J<>M THEN 310 300 310 320 330 335 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760 770 780 790 800 810 201 PRINT#O NEXT J NEXT I GOTO 80 REM ':' - North Star BASIC's 'backslash' PRINT#O:PRINT#O:PRINT#O IS," Ni ”,Z3I,N," M! ",Z3I,M:PRINT#O Il-l FOR KPI TO M PRINT#O,TAB(10*K-2),ZZI,K, NEXT K:PRINT#O FOR I'l TO N PRINT#0 231, I, FOR J-l TO M IF X(I,J)"99 THEN 430 ELSE 450 PRINT#O ” ", GOTO 460 PRINTFO XIOEZ,X(I,J), IF J<>M THEN 480 PRINT#O NEXT J IF 0'1 THEN 530 IF I<>Il*15 THEN 530 INPUT "PRESS 'RETURN' TO CONTINUE ",Z7$ Il-Il+1 NEXT I GOTO 80 INPUT "CORRECT(1),ADD(2),DELETE(3),TRANSFORM(4),RETURN(5)? ",Cl IF Cl<1 OR Cl>5 THEN 550 ON Cl GOTO 580,2510,2710,1420,80 INPUT ”ENTER THE NO. OF ENTRYS TO BE CORRECTED ",N3 FOR L-l T0 N3 PRINT "ENTER MATRIX NOTATION OF ENTRY NO. ”,L INPUT I,J IF I>N OR J)M THEN 580 IF I<>0 and J<>0 THEN 680 PRINT "INCORRECT IDENTIFICATION STRING IS ",IS PRINT "CORRECT IDENTIFICATION STRING?" INPUT I$ GOTO 700 PRINT "INCORRECT VALUE IS ",212E4,X(I,J) INPUT "CORRECT VALUE? ",X(I,J) NEXT L GOTO 80 INPUT "DO YOU TO OVERHRITE AN EXISTING DATAFILE(1-YES;O-NO)? ",A2 IF A2-0 THEN 820 IF A2<>l THEN 720 INPUT "ENTER THE NAME OF THE DATAFILE TO BE OVERWRITTEN ",N$ Q8-EILE(N$) ‘ IF 08-3 THEN 800 PRINT "THIS FILE DOES NOT EXIST ON THE DISK" GOTO 750 DESTROY N$ GOTO 870 820 830 840 850 860 870 880 890 900 910 920 930 940 950 960 970 980 990 1000 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 INPUT ”ENTER THE NAME OF THE DATAFILE YOU WISH TO CREATE L8-LEN(N$) IF L8<-8 THEN 870 PRINT ”FILENAMES MAY CONSIST OF 8 ALPHANUMERIC CHARCATERS OR LESS" GOTO 820 ' Ql-FILE(N$) IF 01-1 THEN 910 PRINT ”THIS FILE ALREADY EXISTS ON THE DISK” GOTO 820 IF A2-1 THEN 940 PRINT ”ENTER IDENTIFICATION STRING FOR YOUR DATAFILES(80 CHRS. OR LESS)” INPUT IS B-INT((7*M*N+100)/256)+1 CREATE N$,B,3 0PEN#0,N$ WRITE#Ozo,I$,N,M,N0ENDMARX z-O FOR 1-1 TO N 202 ".N$ FOR J-l TO M WRITE#oz(z*7+95),X(I,J),N0ENDMARR z-z+1 NEXT J NEXT I CLOSE#0 GOTO 80 INPUT "ENTER THE FILENAME YOU WISH TO SCAN 04-FILE(N$) IF 04-3 THEN 1120 PRINT "THIS FILE DOES NOT EXIST ON THE DISK" GOTO 1070 0PEN#O,N$ READ#ozo,I$,N,M z-O FOR I-1 To N FOR J-1 TO M READ#OZ(z*7+95).X(I,J) z-z+1 NEXT J NEXT I CLOSE#0 GOTO 340 » INPUT ”ENTER THE NAME OF THE FILE YOU WISH TO DESTROY ",N$ 06-FILE(N$) IF 06-3 THEN 1280 PRINT ”THIS FILE DOES NOT EXIST ON THE DISK" GOTO 1230 DESTROY N$ GOTO 80 PRINT "YOU MAY CHAIN THE FOLLOWING ".N$ PROCRAMS:RECRESS(1),PROCCURv(2),“ 1310 1320 PRINT "VIDPLOT(3),AUTOPLOT(4),REGANOV(5),NONLIN(6),DTLNPT(7)” INPUT "ENTER THE NUMBER OF THE PROGRAM YOU WISH TO CHAIN TO ”,Hl 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 COL-" 1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 1540 COLUMN(1'YES;O-NO)? 1550 1560 1570 1580 1590 1600 1610 1620 1630 1640 1650 1660 1670 1680 1690 1700 1710 1720 1730 1740 1750 1760 1770 203 Hl-l Hl-Z H1-3 H1-4 H1-5 H1-6 THEN HS'"REGRESSI" THEN Hs-"PROGCURV" THEN HS'"VIDPLOT” THEN Hs-"AUTOPLOT" THEN HS-"REGANOV" THEN HS-"NONLIN" H1-7 THEN Hs-"DTLNPT" H1<1 OR H1>7 THEN 1300 CHAIN H$+”,2" PRINT "YOU MAY TRANSFORM A SINGLE COLUMN OF NUMBERS(OPTION 1) OR" PRINT "YOU MAY TRANSFORM ONE COLUMN OF NUMBERS USING A SECOND QQQGGEGQ PRINT ”UMN OF NUMBERS(OPTION 2)" INPUT ”OPTION? ”,P IF P<>1 AND P<>2 THEN 1420 IF P-1 THEN 1760 PRINT "THE FOLLOWING TRANSFORMATIONS MAY BE PERFORMED: ADD(l),” PRINT "SUBTRACT(2),MULTIPLY(3),DIVIDE(4)” INPUT ”WHICH TRANSFORMATION DO YOU WISH TO USE? IF T1<1 OR T1>4 THEN 1480 INPUT "ENTER THE FIRST AND SECOND COLUMNS IF C1)M 0R CZ>M THEN 1520 . INPUT "SHOULD THE TRANSFORMED NUMBERS FORM A NEW ” A3 3 IF A3<>1 AND A3<>0 THEN 1540 IF A3-1 THEN M1-M+1 ELSE M1-03 FOR I-1 TO N IF X(I,C3)-99 0R X(I,C4)--99 THEN 1590 ELSE 1610 GOTO 1720 IF T1<>1 THEN 1640 X(I.M1)-X(I,C3)+X(I,C4) GOTO 1720 IF T1<>2 THEN 1670 GOTO 1720 IF T1<>3 THEN 1710 X(I.M1)-X(I.ca)*x(1.c4) GOTO 1720 IF T1<>4 THEN 1720 X(I,M1)-X(I,C3)/X(I,C4) NEXT I IF A3-0 THEN 1750 M?M1 GOTO 80 PRINT ”THE FOLLOWING TRANSFORMATIONS MAY BE PERFORMED: ADD(1),” PRINT ”,Tl ”,C1,C2 "SUBTRACT(2),MULTIPLY(3),DIVIDE(4),POWER(5),LN(6),OPERATE(7)" 1780 1790 1800 1810 1820 INPUT "WHICH TRANSFORMATION DO YOU WISH TO USE? IF T2<1 OR T2>7 THEN 1760 INPUT "WHICH COLUMN OF NUMBERS DO YOU WISH TO TRANSFORM? IF C>M THEN 1800 INPUT "SHOULD THE TRANSFORMED NUMBERS FORM A NEW ",T2 OO’C 204 COLUMN(l-YES;O-NO)? ”,A3 1830 1840 1850 1860 ”,A4 1870 1880 1890 IF A3<>1 AND A3<>0 THEN 1820 IF A3-1 THEN M1-M+1 ELSE Ml-C APO:S.0:L'1:D‘1311'1:N1'N INPUT ”DO YOU WISH TO TRANSFORM THE ENTIRE COLUMN?(1-YES;0-NO)? IF A4-1 THEN 1930 IF A4<>O THEN 1860 PRINT "ENTER THE FIRST AND LAST ELEMENTS OF THE COULMN YOU WISH TO TRANSFORM” 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2110 2120 2130 2140 2150 2160 2170 2180 2190 2200 2210 2220 2230 2240 2250 2260 2270 2280 2290 2300 2310 2320 INPUT Il,N1 IF Il>N1 THEN 1890 IF I1>N OR N1>N THEN 1890 IF T2<>1 THEN 1950 INPUT "HOW MUCH DO YOU WISH TO ADD TO EACH ENTRY? ",A IF T2<>2 THEN 1970 INPUT "HOW MUCH DO YOU WISH TO SUBTRACT FROM EACH ENTRY? ",8 IF T2<>3 THEN 1990 INPUT ”BY WHAT DO YOU WISH TO MULTIPLY EACH ENTRY? ",L IF T2<>4 THEN 2010 INPUT ”BY WHAT DO YOU WISH TO DIVIDE EACH ENTRY? ”,D IF T2<>5 THEN 2030 INPUT ”BY WHAT POWER DO YOU WISH TO RAISE EACH ENTRY T0? ”,P9 FOR I-Il T0 N1 IF X(I,C)--99 THEN 2050 ELSE 2070 GOTO 2170 IF T2<>5 THEN 2100 X(I,M1)-X(I,C)°P9 GOTO 2170 IF T2<>6 THEN 2130‘ X(I,M1)-LOG(X(I,C)) GOTO 2170 IF T2<>7 THEN 2160 X(I,M1)-FNA(X(I,C)) GOTO 2170 X(I,M1)-L*X(I,c)/D+A-S NEXT I IF A3-O OR A4-1 THEN 2550 FOR I-1 TO I1-1 X(I,Mu)-X(I,C) NEXT I FOR I-N1+l TO N X(I,M1 )-X(IDC) NEXT I IF A3-o THEN 2270 MhMl GOTO 80 INPUT "HOW MANY FILES DO YOU WISH To MERGE? ”,F z9-0 FOR I-1 T0 F PRINT ”ENTER THE NAME OF DATAFILE NO. ",1 INPUT N$ 2330 2340 2350 2360 2370 2380 2390 2400 2410 2420 2430 2440 2450 2460 2470 2480 2490 2500 2510 2515 2520 2530 2540 2550 2560 2570 2580 2590 2600 2610 2620 2630 2640 2650 2660 2670 2680 2690 2700 2710 2720 2730 2740 2750 2760 2770 2780 2790 2800 2810 2820 2830 2840 205 07-FILE(N$) IF 07-3 THEN 2370 PRINT ”THIS FILE DOES NOT EXIST ON THE DISK" GOTO 2310 OPEN#0,NS READ#Ozo,I$,N,M z-O FOR J-1+Z9 To N+79 FOR K91 To M READ #oz(z*7+95),X(J,R) z-z+1 NEXT K NEXT J CLOSE#0 z9-N+z9 NEXT I N-z9 GOTO 80 - INPUT ”DO YOU WISH TO ADD ROW(S)(1) 0R COLUMN(S)(2)? IF A5<1 OR A5>2 THEN 2510 IF A5-1 THEN 2530 ELSE 2620 INPUT “ENTER NO. OF ROW(S) TO BE ADDED FOR I-N+1 T0 N+R FOR J-1 TO M PRINT ”DATUM(”,I,”,",J,")?“ INPUT X(I,J) NEXT J NEXT I NhN+R GOTO 80 INPUT ”ENTER NO. OF COLUMN(S) TO BE ADDED? FOR I-1 TO N FOR J-M+1 T0 M+Cl PRINT ”DATUM(",I,“,”,J,")?" INPUT X(I,J) NEXT J NEXT I M9M+C1 GOTO 80 INPUT ”DELETE ROW(S)(1),COLUMN(S)(2) OR ENTRYS(3)? IF C2<1 OR cz>3 THEN 2710 ON 02 GOTO 2740,2860,2980 INPUT "ENTER FIRST AND LAST ROW(S) TO BE DELETED N1-N-(R2-R1+1) IF N1>O THEN 2790 PRINT ”ARE YOU SURE YOU WANT TO DELETE ALL OF THE ROWS?" GOTO 2740 FOR I-R1 T0 N1 FOR J-1 TO M X(I,J)-X(I+N-N1,J) NEXT J NEXT I OI’R ",Cl n’cz ",R1,R2 ,N-Nl 2850 2860 2870 2880 2890 2900 2910 2920 2930 2940 2950 2960 2970 2980 2990 3000 3010 3020 3030 3040 3050 3060 3070 3080 3090 3100 3110 3120 3130 3140 3150 3160 3170 3180 3190 3200 3210 3220 3230 3240 3250 3260 3270 3280 3290 3300 3310 3320 3330 3340 3350 3360 3370 206 GOTO 80 INPUT ”ENTER FIRST AND LAST COLUMN(S) TO BE DELETED ",C1,C2 Ml-M-(CZ-Cl+1) IF M1>0 THEN 2910 PRINT ”ARE YOU SURE YOU WANT TO DELETE ALL OF THE COULMNS?" GOTO 2860 FOR I-1 TO N FOR J-C1 TO M1 X(I,J)-X(I,J+M-M1) NEXT J NEXT I Mer GOTO 80 INPUT ”ENTER NO. OF ENTRY(S) TO BE DELETED ",N2 IF N2>N*M THEN 2980 FOR K91 T0 N2 PRINT ”MATRIX NOTATION 0F ENTRY N0. ",K INPUT I,J IF I>N OR J>M THEN 3010 NEXT K GOTO 80 FILL 63487,8:PRINT CHR$(26) PRINT CHR$(16):PRINT CHR$(30) FILL 63487,12 z-1:P-1.74532E-2 FOR I9-61608 TO 63178 STEP 80:FILL 19,197 NEXT 19 FILL 63208.195 FOR I9-63209 TO 63276:FILL I9,202 NEXT 19 IF U3-1 THEN 3170 GOSUB 3320 GOSUB 3230 IF T>60 THEN 3230:IF E>145 THEN 3230 NEXT E:STOP IF E<1o 0R E>145 THEN 329O:IF T<4 0R T>70 THEN 3300 IF z0 THEN z-1 L1-(23-INT(T/3))*80+61284+INT(E/2+320 M1-2°((1-E+INT(E/2)*2)*3+T-INT(T/3)*3) v1-128 vs-INT(v1/M1/2)*2*M1+v1—INT(V1*2/M1)*M1/2+M1*z FILL L1,v5 RETURN. PRINT "X OUT OF RANGE: ”,X:RETURN PRINT "Y OUT OF RANGE: ”,Y:RETURN FILL 93487,8:STOP INPUT ”COLUMN I PLOTTED 0N X-AXIS,Y-AXIS? ”,v2,v3 IF v2>N THEN 3320 ELSE 3340 IF v3>M THEN 3320 PRINT CHR$(30) PRINT CHR$(26),CHR$(11), PRINT "FILE ”,NS,” ","COLUMN ”,vz," vs. COLUMN ",v3 3380 3390 3400 3410 3420 3430 3440 3450 3460 3470 3480 3490 3500 3510 3520 .3530 3540 3550 3560 3570 3580 3590 3600 3610 3620 3630 3640 3650 3660 3670 3680 3690 3700 3710 3720 3730 3740 3750 3760 3770 3780 3790 3800 207 PRINT CHR$(10). F1-X(1,v2) F3-X(1,v2) FOR A-z To N F2-X(A,v2) IF F2>F1 THEN F1-F2 IF F2Gl THEN 01-02 IF G2<><><><><><><><><><><><><><><><><><><><><><><><><><><><> <><><><><><><>" 30 PRINT#O TAB(3),"<>NOTE_THIS VERSION OF PROGCURV IS FOR ANALYSIS OF SUBSTRATE DATA ONLY<>" 4O PRINTFO TAB(3),”<><><><><><><><><><><><><><><><><><><><><><><><><><><><><> <><><><><><><>” 50 6O 7O 8O 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 DIM.8(160),T(160),SO(160),I$(80),X(3,3),C(3,3),B(3),D(3) DIM R(3,3) INPUT ”OUTPUT T0 CRT(0) OR DECWRITER(1) ",0 INPUT ”WHAT IS THE NAME OF THE DATAFILE? ",NS 0PEN#0,N$ READ#ozo,I$,N,M V8-7*M INPUT "WHICH COLUMN IS THE INDEPENDENT VARIABLE? ",V2 INPUT "WHICH COLUMN IS THE DEPENDENT VARIABLE? ”,v3 V0-V2*7+88 V1-v3f7+88 z9-0 FOR I-1 TO N READ#OZ(z9*V8+V0),T(I) READ#OZ(Z9*V8+V1).S(I) z9-z9+1 NEXT I CLOSE #0 Y1-0:X1-0:Y2-0:X2-0:X9-O FOR I-1 TO 6 Y1-Y1+S(I) X1-X1+T(I) Y2-Y2+S(I)*S(I) X2-X2+T(I)*T(I) X9-X9+S(I)*T(I) NEXT I B9-(X9-X1*Y1/6)/(x2-X1*X1/6) $0-Y1/6-39*X1/6 Y1-0:X1-0:Y2-0:X2-O:X9-0 FOR I-1 TO N SO(I)-S(I) Y-T(I)/LOG(S0/S(I)) x-(SO-S(I))/LOG(SOIS(I)) Y1-Y1+Y X1-X1+X Y2-Y2+Y*Y X2-X2+X*X X9-X9+Y*X NEXT I B9-(x9-X1*Y1/N)/(X2-X1*X1/N) 211 450 A9-Y1/N-B9*X1/N:X-A9/B9:V-1/B9 460 R-B9*(X9-X1*Y1/N)/(Yz-Y1*Y1/N) 470 PRINT#O:PRINT#O:PRINT#O 480 PRINT#O ”**************************************************************** *****” 490 PRINTIO "PROGCURV ANALYSIS Y-COLUMN NO. ”,V3,” x-COLUMN NO. ”,V2 .500 PRINT#0 "FILENAME-",N$:PRINT#O "ID STRING- ",Is 510 PRINT#0 "FIRST ESTIMATE FOR SO IS",212E4,SO 520 PRINT#O "FIRST ESTIMATE FOR K IS",212E4,X," FOR V IS",z12E4,v 530 PRINT#0 ”COEFFICIENT OF DETERMINATION IS ",112E6,R 540 E0-1E-8 550 E1-1E-4 560 GOSUB 1090 570 GOSUB 1180 580 IF ABS(V9)-E1 THEN 560 610 PRINT#O " +F+F+H+hH+l-H-H—H+H+H+H—H—H++H+H+H—H—i—H+H-” 620 PRINT#O " + KM ESTIMATE IS”,ZlZE4,K,” +/-”,212E4,SQRT(C(1,1)*32)," +" 630 PRINT#0 ” +TVMAX ESTIMATE IS",z12E4,V," +/-".212E4,SQRT(C(2,2)*32),' +” 640 PRINT#0 " + so ESTIMATE IS",ZlZE4,SO," +/-”.11284,SQRT(C(3,3)*32),” +" 650 PRINT#0 " +H+H-H++-H+PH++-H-H++H-H—H+H—++hH—H+H—H+H4+" 660 PRINT#0 670 PRINT#O 680 PRINT#O "APPROXIAMTE PARAMETER COVARIANCE MATRIX" 690 PRINT#O 700 FOR I-l TO 3 710 FOR J-l TO 3 715 C(I,J)-SZ*C(I,J) 720 PRINT#O 112E4,C(I,J),:IF J-3 THEN PRINT#O 730 NEXT J 740 NEXT I 750 C9-C(1,1)*C(2,2)*C(3,3)+2*C(1,2)*C(1,3)*C(2,3) 760 C9-C9-(C(1,1)*C(2,3)*C(2,3)+C(2,2)*C(1,3)*C(1,3)+ C(3.3)*C(1.2)*C(l.2)) 770 PRINTFO "DETERMINANT 0F APPROXIMATE COVARIANCE MATRIX",212E4,C9 780 PRINTFO 790 PRINT#O "APPROXIMATE PARAMETER CORRELATION MATRIX" 800 PRINTfO 810 FOR I-1 TO 3 820 830 840 850 860 870 212 FOR J-1 T0 3 R(I.J)-C(I.J)/SQRT(C(I.I)*c(J.J)) PRINT#0 z1zE4,R(I,J),:IF J-3 THEN PRINT#0 NEXT J NEXT I PRINT#0 "**************************************************************** *****” ' 880 890 900 910 920 930 940 950 960 970 980 990 1000 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270‘ 1280 1290 1300 1310 1320 INPUT ”DO YOU WANT TO GENERATE A THEORETICAL CURVE? IF 01-0 THEN 1050:IF 01<>0 AND Ql<>1 THEN 88o INPUT ”ENTER NAME OF DATAFILE TO BE CREATED? PRINT#0 "ENTER AN ID STRING FOR YOUR FILE” INPUT I$ B-INT((49*N+144)/256)+1 CREATE N$,B,3 OPEN#0,N$ WRITE#Ozo,I$,N+1,7,N0ENDMARX WRITE#oz95,0,So,o,-99,o,0,SO,N0ENDMARX:z-O FOR I-l To N D-1+R/S(I):x1-XELOG(s0/S(I))/D:X2-V*-T(I)/D:X3-SO*(1+X/SO)/D WRITE#Oz(z*49+144),T(I),S(I),So-S(I),s0(I)-S(I),X1,X2,X3,NOENDMARX z-z+1 PRINT#0 210E2,T(I),21032,S(I),210E2,SO-S(I),210E2,SO(I)-S(I) NEXT I CLOSE#0 INPUT "DO YOU WISH TO CHAIN BACK TO FILEMAN? IF 03-0 THEN 1080:IF 03<>0 AND 03<>1 THEN 1050 CHAIN "FILEMAN,2" END Sl-SO:P1-0 FOR I-1 TO N P2-P1-(P1+RELOG(Sol(so-P1))-V*T(I))/(1+K/(So-P1)) IF ABS(P2-P1)<-E0 THEN 1140 P1-P2:GOT0 1110 Pl-P2:Sl-SO-P1:S(I)-Sl NEXT I RETURN END DATA 0,0,0,0,0,0,0,o,0,0,0,o,0,0 READ A1,A2,A3,Bl,32,33,C1,C2,C3,Dl,DZ,D3,D4,D5 RESTORE FOR I-1 TO N D-1+K/S(I) Y-SO(I)-S(I) X1-LOG(80/S(I))/D X2--T(I)/D X3-(1+X/SO)/D A1-A1+X1 A2-A2+X2 A3-A3+x3 81-81+X1*X1 Bz-Bz+X2*X2 33-B3+x3*x3 ".Ql ”’Ns '0’Q3 213 1330 01-C1+X1*X2 1340 02-02+X2*X3 1350 C3-C3+X1*X3 1360 D1-D1+Y 1370 DE-D2+Y*Y 1380 D3-D3+X1*Y 1390 04-D4+X2*Y 1400 05-D5+X3*Y 1410 NEXT I 1420 X(1,1)-81:X(1,2)-01:X(1,3)-c3 1430 x(2,1)-X(1,2):X(2,2)-B2:X(2,3)-C2 1440 X(3,1)-X(1,3):X(3,2)-X(2,3):X(3,3)-B3 1450 D(1)-D3:D(2)-D4:D(3)-05 1460 D9-X(1,1)*X(2,2)*X(3,3)+2*X(1,2)*X(1,3)*X(2,3) 1470 D9-D9-(X(1,1)*X(2,3)*X(2,3)+X(2,2)*X(1,3)*X(1,3)+ X(3,3)*X(1,2)*X(1,2) 1480 C(1,1)-X(2,2)*X(3,3)-X(2,3)*X(2,3) 1490 C(l,2)-X(1,3)*X(2,3)-X(1,2)*X(3,3):C(2,1)-C(1,2) 1500 C(1,3)-X(l,2)*X(2,3)-X(1,3)*X(2,2):C(3,1)-C(1,3) 1510 C(2,2)-X(1,1)*X(3,3)-X(1,3)*X(1,3) 1520 C(2,3)-X(1,2)*x(1,3)-X(1,1)*X(2,3):C(3,2)-C(2,3) 1530 C(3,3)-X(1,1)*X(2,2)-X(1,2)*X(1,2) 1540 C(1,1)-C(1,1)/D9:C(1,2)-C(1,2)/D9:C(1,3)-C(1,3)/D9 1550 C(2,1)-C(1,2):c(2,2)-C(2,2)/D9:C(2,3)-C(2,3)/D9 1560 C(3,1)-C(1,3):c(3,2)-C(2,3):C(3,3)-C(3,3)/D9 1570 FOR I-1 TO 3 ' 1580 B(I)-o 1590 NEXT I 1600 FOR I-1 T0 3 1610 FOR J-1 T0 3 1620 B(I)-C(I,J)*D(J)+B(I) 1630 NEXT J 1640 NEXT I 1650 R9-B(1):v9-B(2):s9-B(3) 1660 KPK+K9:V-V+V9:SO-SO+S9 1670 Sz-(Dz-D1*01/N-R9*(03-A1*D1/N)-V9*(D4-A2*D1/N)- S9*(D5-A3*D1/N))/(N-3) 1680 PRINT#0 ”INTERMEDIATE VALUES, X",z12E4,R," V",z12E4,V,“ SO”,z12E4,SO 1690 RETURN 1700 END PROGCRVZ fits product concentration-time data pairs to the integrated form of the Michaelis-Menten equation when the origin of the progress curve is unknown. It estimates Km and Vmax plus Po, which is defined as a positive displacement above the origin on the product concentration axis. A progress curve of unknown origin arises when 214 product is present at the start of a progress curve in which product appearance is monitored. The concentration of the product present as background must be lower (e.g., 5-fold) than that derivable from 80 if accurate estimates of Km and Vmax are to be obtained. Like PROGCRVI, PROGCRVZ calculates initial estimates of the above parameters which are then improved via nonlinear regression. Initial estimates of Km and Vmax are obtained from linear regression analysis of the P-t data pairs transformed according to equation [3.6] (lines 350-490). An inital estimate of P0 is calculated by fitting the first 3 data pairs to a straight line and extrapolating back to the product concentration axis (lines 250-340). Unlike PROGCRVl, 30 is not updated and assumed to be known. Analagous to PROGCRVl, PROGCRVZ approximates the standard errors of the estimated parameters along with the covariance and parameter correlation matrices. The sensitivity equations (lines 1240 and 1260-1280) are not included in Appendix 3 since they have appeared elsewhere (4,5). Many of the lines in PROGCRVZ are identical to those occurring in PROGCRVl. In the listing below, a line number followed by an "-" indicates that this line of code is identical in both programs. When a line number occurs after an "." this line of code in PROGCRVZ is identical with that line number of PROGCRVI following the ”-". LISTING 0F PROGCRVZ 10 - 20 - 30 PRINT#0 TAB(3),"><>NOTE THIS VERSION OF PROCCURV IS FOR ANALYSIS OF PRODUCT DATA 0NLY<><" "' 40 - 50 6O 7O 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 500 510 520 530 540 550 560 570 580 215 DIM P(160),T(160),PO(160),I$(80),X(3,3),C(3,3),B(3),D(3) READ#OZ(Z9*V8+V1),P(I) PO(I)-P(I) -220 INPUT ”ENTER INITIAL SUBSTRATE CONCENTRATION ”,80 -230 FOR I-l TO 3 Y1-Y1+P(I) X1-X1+T(I) Y2-Y2+P(I)*P(I) X2-X2+T(I)*T(I) X9-X9+P(I)*T(I) NEXT I 39-(X9-X1*Y1/3)/(X2-X1*X1/3) P0-Y1/3-39*X1/3 Y1-0:X1-0:Y2-0:X2-0:X9-0:N9-N -340 IF SO>(P(I)+PO) THEN 390 N9-I-1:EXIT 470 Y-T(I)/LOG(SO/(SO-P(I)-P0)) x-(P(I)-P0)/LOG(SO-P(I)_P0)) -380 -390 -400 -410 -420 -430 B9-(X9-X1*Y1/N9)/(X2-X1*X1/N9) A9-Y1/N9-B9*X1/N9:K-A9/B9:V-l/B9 -470 -480 -490 -500 PRINT#O "INITIAL SUBSTRATE CONC. IS",112E4,SO -520 -530 -540 -550 590 600 610 620 630 640 650 660 670 216 GOSUB 1110 GOSUB 1200 IF ABS(V9)0 AND Ql<>1 THEN 630 -900 I910 I920 BIINT((28*N+123)/256)+1 I930 -940 WRITE#OZO,I$,N+1,4,NOENDMARK WRITE#OZ95,0,S0,0,-99 ,NOENmARX:z-0 I980 WRITE#OZ(Z*28+123),T(I),S(I),SO-S(I),SO(I)-S(I),NOENDMARK I1010 -1020 -1030 -1040 I1050 I1060 I1070 I1080 X1IX0:CZI2:C6I6:D0-.01 FOR IIl T0 T(N) FOR JIl TO l/DO K1I(V*(X0-X1+Y*SO)*X1)/(Y*K+XO-X1+Y*SO)*DO K2I(V*(X0-(K1/C2+X1)+Y*SO)*(K1/C2+Xl))/Y*K+X0-(K1/CZ+X1)+Y*SO)*DO K3I(V*(X0-(K2/C2+X1)+Y*SO)*(K2/C2+X1))/Y*K+X0-(K2/CZ+X1)+Y*SO)*D0 890 900 910 920 930 940 950 960 970 980 990 1000 1010 1020 1030 1040 1050 1060 221 K4-(V*(X0-(K3+X1)+Y*SO)*(K3+X1))/Y*K+X0-(K3+X1)+Y*SO)*D0 X1IX1+(K1+C2*K2+C2*K3+K4)/C6 NEXT J X1(I)-X1:S(I)-(X0-x1)/Y+SO !#o 26I,T(I),212E4,S(I).212E4,SO(I)-S(I),212E4,X1(I) -1150 -1160 -1170 -1180 -1190 -1200 P9-(K*Y+YSO+X0)/(Y*SO+X0):Q9IK*Y/(Y*SO+X0):R9IY*SO+X0 -1210 D9-P9*Y/X1(I)+Q9/S(I) EISO(I)-S(I) X1IY/R9*(LOG(X1(I)/X0)-LOG(S(I)/SO))/D9 XZI-T(I)/D9 X3I(P9*(SO-S(I))/X1(I)+LOG(X1(l)/X0)/R9*(K+(1-P9)*SO) -LOG(S(I)/SO)/R9*(K-Q9*SO))/D9 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 I1270 I128O I129O I13OO I1310 I1320 I133O I1340 I1350 D1IDl+E D2ID2+E*E D3ID3+X1*E D4ID4+X2*E D5ID5+X3*E I1410 I1420 I1430 I1440 I1450 I1460 I1470 I1480 I1490 I1500 I1510 I1520 I1530 I1540 I1550 I1560 I157O I1580 I1590 I1600 222 1410 -1610 1420 I1620 1430 I1630 1440 -1640 1450 K9I3(1):V9IB(2):Y9IB(3) 1460 K-K+K9:VIV+V9:YIY+Y9 1470 SZ-(DZ-Dl*Dl/N-K9*(D3-A1*D1/N)-V9*(D4-A2*D1/N)-Y9*(D5-A3*D1/N))/ 09-3) 1480 PRINT#O ”INTERMEDIATE VALUES, K”,ZlZE4,K," V”,ZIZE4,V,” Y”, ZlZE4,Y 1490 RETURN 1500 END Numerical integration of PHASIM model equations. PHASIM simultaneously numerically integrates the equations [A.1], [A.2] for gaseous substrate consumption by resting cells in a liquid phase described in Appendix A. A third equation for first-order increase in Vmax with time is numerically integrated concomitant with the above 2. The user has the Option of storing the gaseous and aqueous phase concentrations of the gaseous substrate on a minifloppy diskette, in a data file that may be read by FILEMAN (lines 10-40, 370-390 and 420-430). The initial gaseous and aqueous phase concentrations of the gaseous substrate along with the Kla, Vmax: Km, K (first-order coefficient for increase in Vmax with time) and R (linear rate of endogenous substrate production) are entered at lines 50-80. The user then enters the Bunsen absorption coefficient plus the time-step for integration (delta t) and length of the simulation (Tmax) at lines 90-100. The time-step for integration must be a number whose reciprocal is an integer. A fourth-order Runge-Kutta procedure (2) is used to estimate the solution curves (i.e., C vs. t and A vs. t) for the PHASIM system of ordinary differential equations (lines 170-350, 400 and 480-520). The time and concentrations of the substrate in the gaseous and aqueous phases is then outputted to the Decwriter (line 360). 223 Lastly, the user may decrease the delta t (lines 440-450) and re-run the simulation or a new set of parameter values may be entered (lines 460-470) and the solution curves estimated once more. PHASIM LISTING 10 DIM I$(80) 20 INPUT ”IF YOU WISH TO SAVE DATA ON A FILE TYPE 'SAVE' ",88 30 INPUT "ENTER THE NAME OF THE DATAFILE ”,NS 40 INPUT ”ENTER AN IDENTIFICATION STRING FOR YOUR FILE "IS 50 PRINT "ENTER INITIAL CONCENTRATIONS OF THE GAS IN THE GASEOUS AND AQUEOUS PHASES” 60 INPUT G0,AO 70 PRINT ”WHAT ARE THE VALUES 0F XLA,VMAX,XM,X, AND R?" 80 INPUT K1,V,K3,K2,R 90 INPUT ”ENTER THE BUNSEN ABSORPTION COEFFICIENT ”,B9 100 INPUT ”ENTER VALUES FOR 'IMAX AND DELTAT ”,T9,D 110 Dl-l/D:G1-GO:V1IV:A1-A0:Z-0 120 IF S$<>"SAVE" THEN 170 130 BIINT(7*T9*3+100)/256)+1 140 CREATE N$,3,3 150 OPEN,N$ 160 WRITE #020,I$,T9,3,NOENDMARX 170 FOR I-1 T0 T9 180 FOR J-1 TO D1 190 G2--K1*(B9*G1-A1)*D 200 V2IK2*V1*D 210 A2I(K1*(B9*G1-A1)+R-V1*A1/(K3+A1))*D 220 GSI(-K1*(B9*(G2/2+Gl)-(A2/2+A1)))*D 230 V3IK2*(V2/2+V1)*D 240 A3I(K1*(B9*(G2/2+G1)-(A2/2+A1))+R-(V2/2+V1)*(A2/2+A1)/(K3+ (A2/2+A1)))*D 250 G4-(-K1*(B9*(G3/2+Gl)-(A3/2+A1)))*D 260 V4-X2*(v3/2+V1)*D 270 A4I(K1*(B9*(G3/2+Gl)-(A3/2+Al))+R-(V3/2+V1)*(A3/2+A1)/(K3+ (A3/2+A1)))*D 280 G5I(-K1*(B9*(G4+G1)-(A4+A1)))*D 29o V5IK2*(V4+V1)*D 300 A5I(K1*(B9*(G4+Gl)-(A4+A1))+R-(V4+V1)*(A4+A1)/(R3+(A4+A1)))*D 310 G6IFNA(G1,G2,G3,G4,G5) 320 V6IFNA(V1,V2,V3,V4,V5) 330 A6IFNA(A1,A2,A3,A4,A5) 340 GlIG6:V1-V6:A1IA6 350 NEXT J 360 PRINT#0 241,1,212E4,G1,212E4,A1 370 IF ss<>"SAVE” THEN 400 380 WRITE#oz(z*21+95),I,01,A1,N0ENDMARX 390 400 410 420 430 440 450 460 470 480 490 500 510 224 z-z+1 NEXT I PRINT#0:PRINT#0:PRINT#0 IF S$<>”SAVE” THEN 440 CLOSE #0 INPUT ”HOW ABOUT ANOTHER DELTAT? (YES OR NO) ",AS IF A$-”YES" THEN 130 INPUT ”HOW ABOUT ANOTHER SET OF PARAMTERS? (YES OR NO) IF F$I”YES” THEN 10 ELSE END DEF FNA(QI.Q2.Q3.Q4.QS) VIQl+(02+2*Q3+2*Q4+05)/6 RETURN V FNEND ",FS 1. 2. 3. 6. LITERATURE CITED Beck, J. V., and K. J. Arnold. 1977. Parameter estimation in engineering and science. John Wiley and Sons, New York. Burden, R. L., J. D. Faires, and A. C. Reynolds. 1978. Numerical analysis. Prindle, Schmidt, and Weber, Boston. Cornish-Bowden, A. 1976. Principles of enzyme kinetics. Butterworths, Boston. Duggleby, R. G., and J. F. Morrison. 1977. The analysis of progress curve data for enzyme-catalyzed reactions by non-linear analysis. Biochem. Biophys. Acta. 481: 297-312. Nimmo, I. A., and G. L. Atkins. 1974. A comparison of two methods for fitting the integrated Michaelis-Menten equation. Biochem. J. 141: 913-914. - Pirt, S. J. 1975. Principles of microbe and cell cultivation. John Wiley and Sons, New York. 225 "‘TIFFAJLITITRT‘TTWES