Novel differential elimination method for determining kinetic coefficients under substrate self-inhibition

A differential elimination method (DEM) is developed to determine the kinetic coefficients for substrate self-inhibition. Finite differentiation of the equation eliminates either K_I or K_S, which enables the equation to be linearized so that [^(\textq)] {\hat{\text{q}}} , K_S, and K_I can be estimated without using nonlinear least square regression (NLSR). The DEM options that eliminate K_I or K_S computed the parameter values exactly when the data did not contain any errors. If one-point or random errors were not too large, both DEM options worked as well as NLSR when data were acquired with geometric intervals for substrate concentration. The DEM was more accurate for fitting the data for the smallest and largest values of S, but relatively weaker in estimating the observed maximum substrate utilization rate, q_max. The estimates for S_max, the concentration at which the maximum specific substrate utilization rate is observed, were relatively invariant among the methods, even when K_S and K_I differed. When the intervals were arithmetic (i.e., equal intervals of substrate concentration) and the data contained errors, the DEM and NLSR estimated the parameters poorly, indicating that collecting data with an arithmetic interval greatly increases the risk of poor parameter estimation. Parameter estimates by DEM fit very well experimental data from nitrification or photosynthesis, which were taken with geometric intervals of substrate concentration or light intensity, but fit poorly phenol-degradation data, which were obtained with arithmetic substrate intervals. Besides providing a reasonable substitute for NLSR, the DEM also can be used as a tool to diagnose the quality of experimental data by comparing its estimates between the DEM options, or, more rigorously, to those from NLSR.     

Accurate kinetic parameter estimation during progress curve analysis of systems with endogenous substrate production

We have identified an error in the published integral form of the modified Michaelis–Menten equation that accounts for endogenous substrate production. The correct solution is presented and the error in both the substrate concentration, S, and the kinetic parameters V_m, K_m, and R resulting from the incorrect solution was characterized. The incorrect integral form resulted in substrate concentration errors as high as 50% resulting in 7–50% error in kinetic parameter estimates. To better reflect experimental scenarios, noise containing substrate depletion data were analyzed by both the incorrect and correct integral equations. While both equations resulted in identical fits to substrate depletion data, the final estimates of V_m, K_m, and R were different and K_m and R estimates from the incorrect integral equation deviated substantially from the actual values. Another observation was that at R = 0, the incorrect integral equation reduced to the correct form of the Michaelis–Menten equation. We believe this combination of excellent fits to experimental data, albeit with incorrect kinetic parameter estimates, and the reduction to the Michaelis–Menten equation at R = 0 is primarily responsible for the incorrectness to go unnoticed. However, the resulting error in kinetic parameter estimates will lead to incorrect biological interpretation and we urge the use of the correct integral form presented in this study. Biotechnol. Bioeng. 2011;108: 2499–2503. © 2011 Wiley Periodicals, Inc.     

A steady state model of sequential irreversible enzyme reactions

Summary One of the questions which arises in the study of certain inborn errors of metabolism as well as in the field of enzyme kinetics is: what are the quantitative relationships between parameters of enzyme activity and substrate pool sizes in a metabolic pathway? A steady state model has been devised to answer this question for a homogeneous system of non-branched sequential irreversible enzyme reactions which follow Michaelis-Menten kinetics. The concentration of a substrate in such a pathway, [S_i], is a function of 5 variables: (a) the K_M of the enzyme which forms the substrate (K_{M (i–1)}), (b) the K_M of the enzyme which utilizes the substrate (K_{M i}), (c) the V_max of the enzyme which forms the substrate (V_{m (i–1)}), (d) the V_max of the enzyme which utilizes the substrate (V_{m i}) and (e) the immediate precursor concentration [S_(i–1)] where [S_i] = K_{M i} V_{m (i–1)} [S_(i–1)]/[S_(i–1)] (V_mi -V_{m (i–1)}) + K_{M (i–1)}) V_mi The model introduces and defines the concept of and conditions for amplification. An input in the form of a steady state concentration of precursor [S_(i–1)] may be amplified as an output in the form of an increased steady state concentration of product [S_i]. The model also defines the values of the above 5 parameters which do not allow attainment of a steady state for the type of pathway considered.From the Metabolism Branch, National Cancer Institute, National Institutes of Health, Bethesda, Maryland 20014.     

The application of flow microcalorimetry to the study of enzyme kinetics

The use of flow microcalorimetry to measure velocities for enzyme-catalyzed reactions is described. Studies are presented involving the enzymes chymotrypsin, trypsin, and ribonuclease A. In determining the Michaelis-Menten parameters, V_m and K_m, for these systems complications arise due to substrate depletion and product inhibition. By employing an integrated Michaelis-Menten equation, V_m and K_m can be determined.     

Kinetic Behaviour of Free Lipase and Mica-Based Immobilized Lipase Catalyzing the Synthesis of Sugar Esters

The utilization of natural mica as a biocatalyst support in kinetic investigations is first described in this study. The formation of lactose caprate from lactose sugar and capric acid, using free lipase (free-CRL) and lipase immobilized on nanoporous mica (NER-CRL) as a biocatalyst, was evaluated through a kinetic study. The apparent kinetic parameters, K _m and V _max, were determined by means of the Michaelis-Menten kinetic model. The Ping-Pong Bi-Bi mechanism with single substrate inhibition was adopted as it best explains the experimental findings. The kinetic results show lower K _m values with NER-CRL than with free-CRL, indicating the higher affinity of NER-CRL towards both substrates at the maximum reaction velocity (V _max,app>V _max). The kinetic parameters deduced from this model were used to simulate reaction rate data which were in close agreement with the experimental values.     

To what extent is a constant volume design worse than a minimum volume design for a series of CSTR's?

The balance equations for substrate in a cascade of CSTR's undergoing an enzyme-catalyzed reaction following Michaelis-Menten kinetics are developed in dimensionless form. Analytical expressions relating the intermediate concentrations are independently obtained for the cases of minimum overall volume and constant volume. The fractional deviations between the overall volumes following these two design criteria are calculated and presented for several values of the relevant parameters. For situations of practical interest, the fractional deviation is below 10%. Increasing values of the Michaelis-Menten parameter, K _m(or decreasing values of the number of reactors in the cascade, N) lead to lower values of the maximum deviation; this maximum deviation is attained at lower conversions of substrate when K _mis increased or N decreased.List of Symbols C _{S, i}mol.m^–3 concentration of substrate at the outlet of the i-th reactor - C _* ^{S, i} normalized concentration of substrate at the outlet of the i-th reactor - C _* ^{S, i, eq} normalized concentration of substrate at the outlet of the i-th reactor using the design criterion of constant volume - C _* ^{S, i, opt} normalized concentration of substrate at the outlet of the i-th reactor using the design criterion of minimum overall volume - C _{S, 0} mol.m^–3 concentration of substrate at the inlet to the first reactor - Da _i Damköhler number for the i-th reactor - Da _eq constant Damköhler number for each reactor of the cascade - Da _{tot, eq} overall Damköhler number for the cascade assuming equal-sized reactors - Da _{tot, min} minimum overall Damköhler number for the cascade - Er fractional deviation between the overall volumes using the two different design criteria - K _mmol. m^–3 Michaelis-Menten constant - K _* ^M dimensionless Michaelis-Menten constant - N number of reactors of the cascade - Q m³. s^–1 volumetric flow rate - V _im³ volume of the i-th reactor - v _max mol. m^–3. s^–1 reaction rate under saturation conditions of the enzyme with substrate - V _{tot, opt} m³ minimum overall volume of the cascade - V _{tot, eq} m³ overall volume of the cascade assuming equal-sized reactors     

Kinetic study of the cellobiase activity of Trichoderma reesei cellulase complex at high substrate concentrations

At high cellobiose concentrations, the cellobiase activity of a Trichoderma reesei cellulase preparation does not follow Michaelis–Menten kinetics and shows substrate inhibition. Several rate equations were fitted to the initial rate-cellobiose concentration data. The best fit is obtained for a rate equation corresponding to partial substrate inhibition of cellobiase. In this case, the K_m, V_max and K_I values obtained are 1.1 mM, 16 IU ml^–1 and 26 mM, respectively.     

Dissecting the molecular mechanism of ion-solute cotransport: Substrate specificity mutations in theputP gene affect the kinetics of proline transport

Summary Rare mutations that alter the substrate specificity of proline permease cluster in discrete regions of theputP gene, suggesting that they may replace amino acids at the active site of the enzyme. IfputP substrate specificity mutations directly alter the active site of proline permease, the mutants should show specific defects in the kinetics of proline transport. In order to test this prediction, we examined the kinetics of threeputP substrate specificity mutants. One class of mutation increases theK _m over 120-fold but only decreases theV _max fourfold. SuchK _m mutants may be specifically defective in substrate recognition, thus identifying an amino acid critical for substrate binding. Another class of mutation decreases theV _max 80-fold without changing theK _m .V _max mutants appear to alter the rate of substrate translocation without affecting the substrate binding site. The last class of mutation alters both theK _m andV _max of proline transport. These results indicate that substrate specificity mutations alter amino acids critical for Na⁺/proline symport.     

NMR for direct determination of Km and Vmax of enzyme reactions based on the Lambert W function-analysis of progress curves

¹H NMR spectroscopy was used to follow the cleavage of sucrose by invertase. The parameters of the enzyme's kinetics, K_m and V_max, were directly determined from progress curves at only one concentration of the substrate. For comparison with the classical Michaelis-Menten analysis, the reaction progress was also monitored at various initial concentrations of 3.5 to 41.8 mM. Using the Lambert W function the parameters K_m and V_max were fitted to obtain the experimental progress curve and resulted in K_m = 28 mM and V_max = 13 μM/s. The result is almost identical to an initial rate analysis that, however, costs much more time and experimental effort. The effect of product inhibition was also investigated. Furthermore, we analyzed a much more complex reaction, the conversion of farnesyl diphosphate into (+)-germacrene D by the enzyme germacrene D synthase, yielding K_m = 379 μM and k_cat = 0.04 s^− 1. The reaction involves an amphiphilic substrate forming micelles and a water insoluble product; using proper controls, the conversion can well be analyzed by the progress curve approach using the Lambert W function.     

Temperature sensitivity of soil enzyme kinetics under N‐fertilization in two temperate forests

Soil microbes produce extracellular enzymes that degrade carbon (C)‐containing polymers in soil organic matter. Because extracellular enzyme activities may be sensitive to both increased nitrogen (N) and temperature change, we measured the effect of long‐term N addition and short‐term temperature variation on enzyme kinetics in soils from hardwood forests at Bear Brook, Maine, and Fernow Forest, West Virginia. We determined the V_max and K_m parameters for five hydrolytic enzymes: α‐glucosidase, β‐glucosidase, β‐xylosidase, cellobiohydrolase, and N‐acetyl‐glucosaminidase. Temperature sensitivities of V_max and K_m were assessed within soil samples subjected to a range of temperatures. We hypothesized that (1) N additions would cause microbial C limitation, leading to higher enzyme V_max values and lower K_m values; and (2) both V_max and K_m would increase at higher temperatures. Finally, we tested whether or not temperature sensitivity of enzyme kinetics is mediated by N addition. Nitrogen addition significantly or marginally significantly increased V_max values for all enzymes, particularly at Fernow. Nitrogen fertilization led to significantly lower K_m values for all enzymes at Bear Brook, but variable K_m responses at Fernow Forest. Both V_max and K_m were temperature sensitive, with Q₁₀ values ranging from 1.64–2.27 for enzyme V_max and 1.04–1.93 for enzyme K_m. No enzyme showed a significant interaction between N and temperature sensitivity for V_max, and only β‐xylosidase showed a significant interaction between N and temperature sensitivity for K_m. Our study is the first to experimentally demonstrate a positive relationship between K_m and temperature for soil enzymes. Higher temperature sensitivities for V_max relative to K_m imply that substrate degradation will increase with temperature. In addition, the V_max and K_m responses to N indicate greater substrate degradation under N addition. Our results suggest that increasing temperatures and N availability in forests of the northeastern US will lead to increased hydrolytic enzyme activity, despite the positive temperature sensitivity of K_m.     

Kinetics of L-valine uptake in tobacco leaf discs. Comparison of wild-type,the digenic mutant Valr-2, and its monogenic derivatives

Uptake rates of L-valine in epidermis-free leaf discs of tobacco (Nicotiana tabacum L. cv. Xanthi) were measured over the concentration range 0.1 M to 50 mM. Wild-type tobacco was compared with the digenic mutant Val^r-2 (genotype vr2/vr2; vr3/vr3), and with the monogenic mutant strains h9 and h10 (genotype +/+; vr3/vr3) and h17 and h23 (genotype vr2/vr2; +/+). Rate equations consisting of one to three Michaelis-Menten terms, possibly in combination with a linear term were fitted to the kinetic data. These rate equations are equivalent to rational polynomials which may be regarded as the general type of mathematical function describing the kinetics of enzymes and carriers. Kinetic data of the four genotypes conformed to the sum of three Michaelis-Menten terms. Accordingly, three kinetic components could be distinguished. In the wild-type the approximate K_ms were 40 M, 1mM, and 40 mM, respectively. In Val^r-2 a component with a very low K_m (about 4 M) was found which may represent either the modified low-K_m component of the wild-type or a fourth component which is undetectable in the wild-type by kinetic analysis. The V_max of the low-K_m component in Val^r-2 was at least a 100-fold lower than in the wild-type. In the presence of one of the mutant genes the calculated V_max of the low-K_m component was 48% (strains h9 and h10) or 40% (strains h17 and h23) of the corresponding V_max in the wild-type. It is reasoned that the mutations have no effect on the activity of the other two kinetic components, though the evidence for this is circumstantial. Autoradiographs of leaf discs showed that in Val^r-2 the uptake of ¹⁴C-labelled valine in both mesophyll and minor veins was strongly reduced as compared with the wild-type.Abbreviations CCCP carbonylcyanide m-chlorophenylhydrazone - DW dry weight - TPP⁺ tetraphenylphosphonium ion A preliminary account of part of this work has been presented (Borstlap 1986)     

Kinetic changes in surface phosphatase activity of Synedra acus (Bacillariophyceae) in relation to pH variation

• 1 The kinetics of surface phosphatase activity (PA) in the diatom Synedra acus were studied at low substrate concentrations that are frequently encountered in freshwaters and pH variations typical of hardwaters, ranging from pH 7 to pH 9. Higher pH resulted in higher values for the half saturation constant (K_m) and the maximal velocity (V_max).
• 2 The pH optimum of PA was shown to be a linear function of the logarithm of substrate concentration.
• 3 Similar slopes of Michaelis-Menten curves at different pHs in the range of low substrate concentrations indicate that the species is well adapted to hardwaters. The rate of release of phosphate from enzymaticaliy hydrolysable phosphorus was calculated from Michaelis-Menten kinetics for lake-water samples dominated by S. acus. Algal surface phosphatases were responsible for the hydrolysis of 1.54–1.64 nm min^?1 although incubation was performed at a lake temperature of only 7.3°C.

     

Mitochondrial substrate specificity in beetle flight muscle: assessing respiratory oxygen flux in small samples from Dermestes maculatus and Tenebrio molitor

In the present study, the permeabilized fibre approach is adapted to investigate substrate utilization patterns in the flight muscle mitochondria of Dermestes maculatus (Coleoptera: Dermestidae; a carrion scavenger beetle) and Tenebrio molitor (Coleoptera: Tenebrionidae; a phytophagous scavenger beetle). Respiration in saponin‐permeabilized fibres is measured during titration of palmitoyl‐carnitine (Palm‐C), pyruvate (Pyr) or l ‐glycerol 3‐phosphate (G3‐P). Michaelis–Menten‐type enzyme kinetics for oxygen consumption are observed as a function of substrate concentration in Pyr and G3‐P, from which substrate‐specific apparent K_m (sensitivity) and V_max (capacity) are determined. Compared with D. maculatus, the apparent K_m in T. molitor is lower (P < 0.001) for Pyr, and V_max is greater for G3‐P (P < 0.001). In D. maculatus, the apparent K_m for G3‐P is greater (P < 0.001), and respiratory V_max is lower (P < 0.001), than kinetics for Pyr. Robust respiration with l ‐proline (Pro) is also observed in both beetle species tested; however, it is over 2.5‐fold greater in D. maculatus than T. molitor (P < 0.05). These results demonstrate that respiration in beetle flight muscle mitochondria can be assessed in small samples (i.e. at the individual beetle level) using the approach adapted for the present study. The results of the present study also highlight the substrate oxidative capacity patterns in both D. maculatus and T. molitor, which rank Palm‐C < G3‐P < Pyr < Pro.     

The influence of salinity on the kinetics of NH inf4 sup+ uptake in Spartina alterniflora

Summary The effects of short- and long-term exposure to a range in concentration of sea salts on the kinetics of NH _inf4 ^sup+ uptake by Spartina alterniflora were examined in a laboratory culture experiment. Long-term exposure to increasing salinity up to 50 g/L resulted in a progressive increase in the apparent K_m but did not significantly affect V_max (mean V_max=4.23±1.97 mole·g^–1·h^–1). The apparent K_m increased in a nonlinear fashion from a mean of 2.66±1.10 mole/L at a salinity of 5 g/L to a mean of 17.56±4.10 mole/L at a salinity of 50 g/L. These results suggest that the long-term effect of exposure to total salt concentrations within the range 5–50 g/L was a competitive inhibition of NH _inf4 ^sup+ uptake in S. alterniflora. No significant NH _inf4 ^sup+ uptake was observed in S. alterniflora exposed to 65 g/L sea salts. Short-term exposure to rapid changes in salinity significantly affected both V_max and K_m. Reduction of solution salinity from 35 to 5 g/L did not change V_max but reduced K_m by 71%. However, exposing plants grown at 5 g/L salinity to 35 resulted in an decrease in V_max of approximately 50%. Exposure of plants grown at 35 g/L to a total sea salt concentration of 50 g/L for 48h completely inhibited uptake of NH _inf4 ^sup+ . For both experiments, increasing salinity led to an increase in the apparent K_m similar to that found in response to long-term exposure. Our data are consistent with a conceptual model of changes in the productivity of S. alterniflora in the salt marsh as a function of environmental modification of NH _inf4 ^sup+ uptake kinetics.     

An algebraic model to determine substrate kinetic parameters by global nonlinear fit of progress curves

We propose that the time course of an enzyme reaction following the Michaelis-Menten reaction mechanism can be conveniently described by a newly derived algebraic equation, which includes the Lambert Omega function. Following Northrop's ideas [Anal. Biochem.321, 457–461, 1983], the integrated rate equation contains the Michaelis constant (K_M) and the specificity number (k_S≡k_cat/K_M$k_{\text{S}}\equiv k_{\text{cat}}/K_{\text{M}}$) as adjustable parameters, but not the turnover number k_cat. A modification of the usual global-fit approach involves a combinatorial treatment of nominal substrate concentrations being treated as fixed or alternately optimized model parameters. The newly proposed method is compared with the standard approach based on the “initial linear region” of the reaction progress curves, followed by nonlinear fit of initial rates to the hyperbolic Michaelis-Menten equation. A representative set of three chelation-enhanced fluorescence EGFR kinase substrates is used for experimental illustration. In one case, both data analysis methods (linear and nonlinear) produced identical results. However, in another test case, the standard method incorrectly reported a finite (50–70 μM) K_M value, whereas the more rigorous global nonlinear fit shows that the K_M is immeasurably high.     

Kinetics of Hydrogen Consumption by Rumen Fluid, Anaerobic Digestor Sludge, and Sediment

Michaelis-Menten kinetic parameters for H₂ consumption by three methanogenic habitats were determined from progress curve and initial velocity experiments. The influences of mass transfer resistance, endogenous H₂ production, and growth on apparent parameter estimates were also investigated. Kinetic parameters could not be determined for undiluted rumen fluid and some digestor sludge from gas-phase measurements of H₂, since mass transfer of H₂ across the gas-liquid interface was rate limiting. However, accurate values were obtained once the samples were diluted. H₂ consumption by digestor sludge with a long retention time and by hypereutrophic lake sediment was not phase transfer limited. The K_m values for H₂ uptake by these habitats were similar, with means of 5.8, 6.0, and 7.1 μM for rumen fluid, digestor sludge, and sediment, respectively. V_max estimates suggested a ratio of activity of approximately 100 (rumen fluid):10 (sludge):1 (sediment); their ranges were as follows: rumen fluid, 14 to 28 mM h⁻¹; Holt sludge, 0.7 to 4.3 mM h⁻¹; and Wintergreen sediment, 0.13 to 0.49 mM h⁻¹. The principles of phase transfer limitation, studied here for H₂, are the same for all gaseous substrates and products. The limitations and errors associated with gas phase determination of kinetic parameters were evaluated with a mathematical model that combined mass transport and Michaelis-Menten kinetics. Three criteria are described which can be used to evaluate the possibility that a phase transfer limitation exists. If it does not exist, (i) substrate consumption curves are Michaelis-Menten and not first order, (ii) the K_m is independent of initial substrate concentration, and (iii) the K_m is independent of biomass (V_max) and remains constant with dilution of sample. Errors in the Michaelis-Menten kinetic parameters are caused by endogenously produced H₂, but they were <15% for rumen fluid and 10% for lake sediment and digestor sludge. Increases in V_max during the course of progress curve experiments were not great enough to produce systematic deviations from Michaelis-Menten kinetics.     

Phenol degradation by a psychrotrophic strain of Pseudomonas putida

Summary Cell growth and phenol degradation kinetics were studied at 10°C for a psychrotrophic bacterium, Pseudomonas putida Q5. The batch studies were conducted for initial phenol concentrations, S_o, ranging from 14 to 1000 mg/1. The experimental data for 14<=S_o<=200 mg/1 were fitted by non-linear regression to the integrated Haldane substrate inhibition growth rate model. The values of the kinetic parameters were found to be: _m=0.119 h^–1, K _S=5.27 mg/1 and K _I=377 mg/1. The yield factor of dry biomass from substrate consumed was Y=0.55. Compared to mesophilic pseudomonads previously studied, the psychrotrophic strain grows on and degrades phenol at rates that are ca. 65–80% lower. However, use of the psychrotrophic microorganism may still be economically advantageous for waste-water treatment processes installed in cold climatic regions, and in cases where influent waste-water temperatures exhibit seasonal variation in the range 10–30°C.Nomenclature K _S saturation constant (mg/l) - K _I substrate inhibition constant (mg/l) - specific growth rate (h^–1) - _m maximum specific growth rate without substrate inhibition (h^–1) - _max maximum achievable specific growth rate with substrate inhibition (h^–1) - S substrate (phenol) concentration (mg/l) - S_o initial substrate concentration (mg/l) - S_max substrate concentration corresponding to _max (mg/l) - t time (h) - X cell concentration, dry basis (mg DW/l) - X_f final cell concentration, dry basis (mg DW/l) - X_o initial cell concentration, dry basis (mg DW/l) - Y yield factor (mg DW cell produced/mg substrate consumed)     

SIZE-DEPENDENT PHOSPHORUS UPTAKE KINETICS AND CELL QUOTA IN PHYTOPLANKTON1

The allometric equation, y = aX^b, described the interspecific variation of phosphate uptake kinetics and cell quota with phytoplankton cell size and showed that smaller cells are superior in uptake rate to large. Species-specific measurements, made by track autoradiography in phosphorus deficient cultures of communities from a phosphorus-limited lake, revealed that eight different species did not differ significantly in the Michaelis-Menten half-saturation constant, K_m. However, both saturated uptake rates (V_max) and the initial slope of the uptake curve (V_max:K_m) decreased per unit biomass with increasing cell size. Biomass-specific cell phosphorus quotas also decreased with increasing cell volume, but less rapidly than did V_max or V_max: K_m. Comparable data from the literature showed that marine species were superior in phosphorus uptake to freshwater species of similar size, but allometric variation of kinetics appeared to exist within both groups. Together with a variable internal stores model of phosphorus-limited growth, the allometric relationships of uptake kinetics and quotas predicted competition to favor smaller cells, with a differential in growth rate diminishing as competitive intensity increased.     

“Exchange diffusion”: Rate equations for the influx of α-aminoisobutyric acid into mouse cerebrum slices containing this amino acid

Rate equations for the gross influx of -aminoisobutyric acid (AIB) into mouse cerebrum slices containing AIB have a first-order term for unsaturable concentrative influx, identical to the corresponding term for unloaded slices, and a modified Michaelis-Menten term,V_max/(1+K _t /S), for saturable concentrative influx. [V_max v _L (1+K _t /S), wherev _L =saturable component of influx,S=AIB concentration in medium, andK _t =Michaelis constant for unloaded slices.] Below a tissue AIB (T) of 19 µmol/g final wet weight,V_max increases linearly followingV_max=V ₁+m ₁ T; above that value,V _max is virtually constant. The transition is sharp. This equation is consistent with a carrier model for active transport. At the transition, intracellular AIB is about 1 molecule for every 70 amino acid residues of tissue protein, vastly more than could be accommodated by AIB-binding sites in cell membranes. The transition may come from a slow process that does not fill all sites when the tissue AIB is below the transition concentration, or from an AIB-induced phase transition in the membrane.Nomenclature AIB -aminoisobutyric acid - A radioactivity of reference; unspecified amino acid - C counts in tissue sample; carrier for transport - C _i carrier in form that reacts with intracellular substrate - C _o carrier in form that reacts with extracellular substrate - C _R counts in reference - CS complex of substrate with carrier - (CS) _i complex of substrate with carrier in formC _i - (CS) _o complex of substrate with carrier in formC _o - G counts per gram of tissue - HEPES N-2-Hydroxyethylpiperazine-N-2-ethanesulfonic acid - k _u rate constant for first-order unsaturable uptake - K,K ,K ,K ,K _d adjustable parameters in Eqs. (9)–(13) for v, analogous to the Michaelis constant - K _d dissociation constant - K _t Michaelis constant for saturable uptake - K _t Michaelis constant for gross saturable uptake by tissue containing substrate - m ₁,m ₂ slope in Eq. (5) or (6) expressing dependence ofV_max onT orT _i ^w in Region 1 or 2 - M binding site for amino acid A - n number of data points - P number of parameters to be determined; parameter in Stein's (1981) equation, Eq. (17) in this paper - P ₁,P ₂,P ₁₂ property of tissue with unoccupied binding sites, property of tissue with occupied binding sites, property of tissue with both unoccupied and occupied binding sites, respectively - Q parameter in Stein's (1981) equation, Eq. (17) in this paper - r Pearson's correlation coefficient - Relative error RE =100{[(observed quantity – calculated quantity)/calculated quantity]²/(n –P)}^1/2 - S concentration of substrate in medium; transport substrate - S _i intracellular transport substrate - S _int AIB in medium corresponding to intracellular AIB at intersection - S _o extracellular transport substrate - T observed concentration of substrate in tissue including substrate in extracellular space and adherent fluids - T _i intracellular concentration of substrate - T _int tissue AIB corresponding to intracellular AIB at intersection - T _i ^w ,T _i ^/30 intracellular concentration of substrate withw% (30%) extracellular and adherent fluids - U observed uptake of labeled substrate by incubated tissue including substrate in extracellular and adherent fluids - U _R observed uptake of labeled substrate referred to concentration of substrate in medium - U _max adjustable parameter in Eqs. (9)–(15) for v, analogous to the Michaelis-Menten maximum rate,V _max - v influx of substrate - v _L gross influx of substrate into tissue containing substrate - v _L contribution of saturable component to gross influx into tissue containing substrate - v incremental influx, that is, gross influx into tissue that contains substrate minus influx under the same conditions into tissue that does not contain substrate - V ₁,V ₂ intercept in Eq. (5) or (6) expressing dependence ofV_max onT orT _i ^w in Region 1 or 2, respectively - V _max maximum rate in Michaelis-Menten equation - V_max apparent maximum rate defined byV_maxv_max(1+K _t /S) - V_{max 1},V_{max 2} apparent maximum rate in Region 1 or 2, respectively - V_int apparent maximum rate at intersection defining boundary between Regions 1 and 2 - w weight of incubated tissue - W _d dry weight of tissue expressed as fraction by weight - W _e extracellular and surface space of incubated tissue expressed as percent by weight - , , adjustable parameters in modified expressions for gross unsaturable influx into tissue containing substrate - , , , exponents ofS orT in Eqs. (9)–(13) for v - parameter in Stein's (1981) equation, Eq. (17), corresponding more or less tom ₁ For my wife, Lynn.     

Kinetic Parameters of the Conversion of Methane Precursors to Methane in a Hypereutrophic Lake Sediment

The kinetic parameters K_m, V_max, T_t (turnover time), and v (natural velocity) were determined for H₂ and acetate conversion to methane by Wintergreen Lake sediment, using short-term (a few hours) methods and incubation temperatures of 10 to 14°C. Estimates of the Michaelis-Menten constant, K_m, for both the consumption of hydrogen and the conversion of hydrogen to methane by sediment microflora averaged about 0.024 μmol g⁻¹ of dry sediment. The maximal velocity, V_max, averaged 4.8 μmol of H₂ g⁻¹ h⁻¹ for hydrogen consumption and 0.64 μmol of CH₄ g⁻¹ h⁻¹ for the conversion of hydrogen to methane during the winter. Estimated natural rates of hydrogen consumption and hydrogen conversion to methane could be calculated from the Michaelis-Menten equation and estimates of K_m, V_max, and the in situ dissolved-hydrogen concentration. These results indicate that methane may not be the only fate of hydrogen in the sediment. Among several potential hydrogen donors tested, only formate stimulated the rate of sediment methanogenesis. Formate conversion to methane was so rapid that an accurate estimate of kinetic parameters was not possible. Kinetic experiments using [2-¹⁴C]acetate and sediments collected in the summer indicated that acetate was being converted to methane at or near the maximal rate. A minimum natural rate of acetate conversion to methane was estimated to be about 110 nmol of CH₄ g⁻¹ h⁻¹, which was 66% of the V_max (163 nmol of CH₄ g⁻¹ h⁻¹). A 15-min preincubation of sediment with 5.0 × 10⁻³ atm of hydrogen had a pronounced effect on the kinetic parameters for the conversion of acetate to methane. The acetate pool size, expressed as the term K_m + S_n (S_n is in situ substrate concentration), decreased by 37% and T_t decreased by 43%. The V_max remained relatively constant. A preincubation with hydrogen also caused a 37% decrease in the amount of labeled carbon dioxide produced from the metabolism of [U-¹⁴C]valine by sediment heterotrophs.