Relationship between the octanol-water partition coefficient of tertiary amines and their effect of ‘selective’ uncoupling of photophosphorylation

A series of tertiary amines was investigated for effects on the transmembrane proton potential difference ( H), on photophosphorylation and on electron-flux control related to the intrathylakoid proton potential ( H_I), using isolated chloroplasts ofSpinacia oleracea L. As indicated by 9-aminoacridine fluorescence and [14C]methylamine uptake, all amines studied inhibited a build-up of H and, in parallel, ATP synthesis. Even when H was low, strong H₁-dependent electron-flux control was observed under the influence of tertiary amines. The strength of flux control in the presence of low H and the effectiveness of inhibition of ATP synthesis linearly increased with the lipophilicity of the amines. The most effective of the amines tested caused 50% inhibition of ATP synthesis at a concentration of 6 M, which is about 1000-fold lower than the concentration required for inhibition by methylamine. The data presented indicate the existence of two proton domains in the thylakoid vesicles, one of them feeding the ATP-synthase, the other the sites of pH-dependent electron-flux control. It is concluded that tertiary amines develop their action in a lipophilic domain of the thylakoid membrane, in the vicinity of the ATP-synthase complex. A mechanism for selective uncoupling and for the maintenance of H_I-dependent electron flux control in the presence of low H is discussed.Abbreviations and symbols coefficient for pH-dependent electron flux control - 9-AA 9-aminoacridine - Chl chlorophyll - I₅₀ amine concentration producing 50% inhibition of ATP-synthesis - J_e flux of photosynthetic electron transport - k _H apparent rate constant for proton efflux - H₁ proton potential in the thylakoid lumen - H₁ transthylakoid proton potential difference - p partition coefficient - q _AA coefficient for 9-aminoacridine fluorescence quenching - PS photosystem - Q quantum flux of photosynthetically active light Dedicated to Professor Wilhelm Simonis, on the occasion of his 80th birthday     

The protonmotive force and phosphorylation potential developed by whole cells of the methylotrophic bacteriumMethylophilus methylotrophus

The and the Gp have been measured in whole cells ofMethylophilus methylotrophus during the oxidation of various respiratory chain substrates. The magnitude of the depended on the external pH and the composition of the assay medium, and varied from-109 to-165 mV. The relative contributions of the and the pH to the were found to vary with the external pH such that the internal pH remained constant; depending on the composition of the assay medium, this value was between 6.6 and 7.0. A Gp of approximately-46 kJ/mol was generated during the oxidation of methanol, and either the or pH alone was fully competent to drive ATP synthesis. Respiration and ATP synthesis were found to be poised far from equilibrium under the conditions of these experiments, and the value of the Gp was thus controlled kinetically. Comparison of the with the Gp yielded an H⁺/ATP quotient >2.6 g-ion H⁺/mol ATP.Abbreviations TMPD N,N,N,N-tetramethyl-p-phenylenediamine - FCCP carbonylcyanidep-trifluoromethoxyphenylhydrazone - DMO 5,5-dimethyloxazolidine-2,4-dione - TPMP⁺ triphenylmethylphosphonium (iodide salt); Tween 20, polyoxyethylenesorbitan monolaurate - TPP⁺ tetraphenylphosphonium (bromide salt) - bulk phase transmembrane electrochemical potential difference of protons ( ) - pH bulk phase transmembrane pH difference (pHin-pHout) - bulk phase transmembrane electrical potential difference (in-out) - p true protonmotive force (incorporating both bulk phase and localised protons; )     

Kinetic Modeling of Energy Metabolism and Superoxide Generation in Hepatocyte Mitochondria

Direct nonenzymatic oxidation of semiquinone by oxygen is one of the main sources of superoxide radicals in mitochondria. Using all the known data on hepatocyte mitochondria, we have revealed the correlation between the rate of superoxide generation by the bc ₁complex and the transmembrane potential (). Assuming that the main electrogenic stage of the Qcycle is the electron transfer between the cytochrome bhemes, then the rate of superoxide generation sharply increases when grows from 150 to 180 mV. However, this interrelation is ambiguous. Indeed, the increase of the generation rate with the growth of the potential can occur faster when succinate dehydrogenase is inhibited by malonate than when external ADP is exhausted. When the potential is changed by adding phosphate or potassium (K⁺), the rate of production remains constant, although the comparison of the rates at the same reveals the effect of phosphate or potassium. It turned out that the rate of generation is a function of rather than any of its components. Phosphate and K⁺have practically no influence on , since the change in is compensated by pH. The rate of superoxide generation by the bc ₁complex is a multiple function of the electron-transfer activity of enzymes, the processes determining the membrane potential (e.g., loading), and the oxygen concentration. The kinetic model proposed in this work may serve to understand how the superoxide production is regulated.     

Electrochemical potential of protons in vesicles reconstituted from purified,proton-translocating adenosine triphosphatase

Summary Measurements were made of the difference in the electrochemical potential of protons ( ) across the membrane of vesicles reconstituted from the ATPase complex (TF ₀ ·F ₁) purified from a thermophilic bacterium and P-lipids. Two fluorescent dyes, anilinonaphthalene sulfonate (ANS) and 9-aminoacridine (9AA) were used as probes for measuring the membrane potential () and pH difference across the membrane ( pH), respectively.In the presence of Tris buffer the maximal and no pH were produced, while in the presence of the permeant anion NO ₃ ^– the maximal pH and a low were produced by the addition of ATP. When the ATP concentration was 0.24mm, the was 140–150 mV (positive inside) in Tris buffer, and the pH was 2.9–3.5 units (acidic inside) in the presence of NO ₃ ^– . Addition of a saturating amount of ATP produced somewhat larger and pH values, and the attained was about 310 mV.By trapping pH indicators in the vesicles during their reconstitution it was found that the pH inside the vesicles was pH 4–5 during ATP hydrolysis.The effects of energy transfer inhibitors, uncouplers, ionophores, and permeant anions on these vesicles were studied.     

Ventilatory responses to exercise and carbon dioxide in elderly and younger humans

The present investigation examined the relationship between CO₂ sensitivity [at rest (S _R) and during exercise (S _E)] and the ventilatory response to exercise in ten elderly (61–79 years) and ten younger (17–26 years) subjects. The gradient of the relationship between minute ventilation and CO₂ production ( _E/ CO₂) of the elderly subjects was greater than that of the younger subjects [mean (SEM); 32.8 (1.6) vs 27.3 (0.4); P<0.01]. At rest, S _R was lower for the elderly than for the younger group [10.77 (1.72) vs 16.95 (2.13) 1 · min^–1 · kPa^–1; 1.44 (0.23) vs 2.26 (0.28) 1 · min^–1 · mmHg^–1; P<0.05], but S _E was not significantly different between the two groups [17.85 (2.49) vs 19.17 (1.62) l · min^–1 · kPa^–1; 2.38 (0.33) vs 2.56 (0.21) 1 · min^–1 · mmHg^–1]. There were significant correlations between both S _R and S _E, and _E/ CO₂ (P<0.05; P<0.001) for the younger group, bot none for the elderly. The absence of a correlation for the elderly supports the suggestion that _E/ CO₂ is not an appropriate index of the ventilatory response to exercise for elderly humans.     

Thermodynamics of Mn2+-binding to goat α-lactalbumin

By means of reaction calorimetry we measured the apparent enthalpy change, H^app, of the binding of Mn²⁺-ions to goat -lactalbumin as a function of temperature. The observed H^app can be written as the sum of contributions resulting from a conformational and a binding process. In combination with the thermal unfolding curve of goat -lactalbumin, we succeeded in separating the complete set of thermodynamic parameters (H, G, S, C_p) into the binding and conformational contributions. By circular dichroism we showed that NH ₄ ⁺ -ions, upon binding to bovine a-lactalbumin, induce the same conformational change as do Na⁺ and K⁺: the binding constant equals 98 ± 9 M^–1.Abbreviations BLA bovine -lactalbumin - GLA goat -lactalbumin - HLA human -lactalbumin - CD circular dichroism Offprint requests to: H. Van DaelDeceased     

Maximal oxygen uptake,sweating and tolerance to exercise in the heat

The purpose of this experiment was to determine if tolerance to exercise in the heat is related to maximal oxygen uptake (max ₀₂) and sweating. Seven men with max ₀₂ between 42 and 66 ml/(min·kg) underwent one 2-hr exposure at 24°C T_q while working on a bicycle ergometer at rel ₀₂ of 28% ( ₀₂ = 1.23 1/min). In the hot exposures the high capacity subjects had maximal sweat rates of 800 to 1,000 g/(hr·m²) while the lower capacity men sweated 300 to 400 g/(hr·m²). These differences in sweating were not related to neuromuscular stimuli, ₀₂ (metabolic rate), T_re, T_re, _s, _s or tolerance time. Tolerance to exercise in the heat was not related to maximal ₀₂ capacity when the subjects worked at the same relative load in spite of large differences in sweating. These results question the importance of the rate of sweating for predicting work performance in hot environments.

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Zusammenfassung Das Ziel dieser Untersuchung war, zu prüfen, ob die Toleranz bei Arbeit in der Hitze in einer Beziehung steht zur maximalen O₂-Aufnahme und Schwitzen. Sieben Männer mit V₀₂ zwischen 42 – 66 ml/(min·kg) wurden belastet während 2 Stunden bei T_a 24°C und 3 × 2 Stunden bei 47°C mit Arbeit auf dem Fahrrad-Ergometer bei im Mittel von 28% V₀₂ = 1.23 1/min. Während der Hitzebelastung zeigten die leistungsfähigen Personen Schweissekretionsraten von 800 – 1000 g/(hr·m²) und die wenig leistungsfähigen 300 – 400 g/(hr·m²). Diese Unterschiede waren ohne Beziehung zu neuromuskulären Stimuli, Stoffwechselrate, T_re, T_re, _s, _s oder der Toleranzzeit. Ausdauer bei Arbeit in der Hitze war ohne Beziehung zur maximalen V₀₂-Kapazität, wenn die Personen bei der gleichen relativen Belastung arbeiteten tro grosser Unterschiede im Schwitzen. Die Ergebnisse stellen den Wert der Schweissekretionsrate zur Voraussage der Arbeitsleistung in der Hitze in Frage.

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Resume Dans cette étude, on a cherché à voir si la tolérance au travail sous contrainte de chaleur était en relation avec la possibilité maximum d'absorption de O₂ ( ₀₂) d'une part, de transpirer d'autre part. 7 hommes présentant des ₀₂ compris entre 42 et 66 ml/(min · kg) ont pédalé sur un ergomètre pendant 2 heures par une T_a de 24°C et 3 × 2 heures par 47°C et cela par une ₀₂ relative de 28% ( ₀₂ = 1,25 1/min). Durant l'effort sous contrainte de chaleur, les plus actifs ont eu des sécrétions de sueur de 800 à 1.000 g h^–1 m^–2 et les moins actifs de 300 à 400 g/h · m². Ces différences étaient sans rapport avec les stimulus neuro-musculaires, le taux de métabolisme, T_re, T_re, T_s et T_s ou la durée de tolérance. L'endurance au travail sous contrainte de chaleur n'a pas été fonction de la capacité maximum de ₀₂, lorsque les personnes travaillaient dans des conditions analogues, même si l'on a noté de grandes différences dans la transpiration. Ces résultats mettent en doute la représentativité du taux de sécrétion de sueur comme indicatif des possibilités de travailler en atmosphère chaude.

     

A simple method to quantitatively measure polypeptide JHNHα coupling constants from TOCSY or NOESY spectra

A simple linear relationship between the J coupling constant and the linewidth (_1/2) of in-phase NMR peaks has been identified. This relationship permits the rapid and accurate determination of polypeptide J coupling constants from a simple inspection of amide cross peaks in homonuclear ¹H TOCSY or ¹H NOESY spectra. By using the appropriate set of processing parameters we show that J = 0.5(_1/2) – MW/5000 + 1.8 for TOCSY spectra and J = 0.6(_1/2) – MW/5000 – 0.9 for NOESY spectra, where _1/2 is the half-height linewidth in Hz and MW is the molecular weight of the protein in Da. The simplicity of this relationship, combined with the ease with which _1/2 measurements can be made, means that J coupling constants can now be rapidly determined (up to 100 measurements in less than 30 min) without the need for any complex curve-fitting algorithms. Tests on 11 different polypeptides involving more than 650 separate J measurements have shown that this method yields coupling constants with an rmsd error (relative to X-ray data) of less than 0.9 Hz. Furthermore, the correlation coefficient between the predicted NMR coupling constants and those derived from high-resolution X-ray crystal structures is typically better than 0.89. These simple linear relationships have been found to be valid for peptides as small as 1 kDa to proteins as large as 20 kDa. Despite the method's simplicity, these results are comparable to the accuracy and precision of the best techniques published to date.     

Foam behavior of biological media

Summary The influence of the alcohol concentration on the foaminess, , of-BSA-solutions is considered. This effect is calculated by means of the function (C_BSA . f), where f=1 for pure protein solutions and f>1 for alcohol solutions. f is calculated by f = 2^TT_eff. Here, where T_T is the turbidity temperature change due to solvent structure effects and T_D, the temperature correction due to alcohol-protein interaction. The constants necessary to calculate T_T and T_D are tabulated. The agreement between the calculated and measured foaminess , as a function of the n-propanol concentration is satisfactory and for methanol or ethanol excellent.     

The generation of the proton electrochemical potential and its role in energy transduction

Summary The evidence that all energy transducing membranes can generate a proton electrochemical potential difference, _H, across the membrane and that this potential can be used to transfer energy among energy transducing units and to generate ATP, has increased the interest for the view that _H plays an obligatory role in energy transduction and ATP synthesis. In the present article we shall concentrate on two experimental questions related with the generation and role of _H: (a) the charge/site ratio; (b) the relation between the proton electrochemical potential on one side and the cation electrochemical potential, the phosphate potential and the redox potential on the other. We shall then discuss the view that energy transduction corresponds to a molecular energy machine rather than to a fuel cell.     

A study on Na+-coupled oxidative phosphorylation: ATP formation supported by artificially imposed ΔpNa and ΔpK inVibrio alginolyticus cells

Addition of Na⁺ to the K⁺-loadedVibrio alginolyticus cells, creating a 250-fold Na⁺ gradient, is shown to induce a transient increase in the intracellular ATP concentration, which is abolished by the Na⁺/H⁺ antiporter, monensin. The pNa-supported ATP synthesis requires an additional driving force supplied by endogenous respiration or, alternatively, by a K⁺ gradient (high [K⁺] inside). In the former case, ATP formation is resistant to the protonophorous uncoupler. Dicyclohexylcarbodiimide and diethylstilbestrol, but not vanadate, completely inhibit Na⁺ pulse-induced ATP formation. The data agree with the assumption that Na⁺-ATP-synthase is involved in oxidative phosphorylation inV. alginolyticus. Interrelation of H⁺ and Na⁺ cycles in bacteria is discussed.Abbreviations and electrochemical gradients of H⁺ and Na⁺, respectively - transmembrane electric potential difference - pH, pNa, and pK concentration gradients of H⁺, Na⁺, and K⁺, respectively - CCCP carbonyl cyanidem-chlorophenylhydrazone - DCCD N,N-dicyclohexylcarbodiimide - DES diesthylstilbestrol - HQNO 2-heptyl-4-hydroxyquinolineN-oxide - Tricine N[2-hydroxy-1,1-bis(hydroxymethyl)ethyl]glycine     

Relationships among violaxanthin deepoxidation,thylakoid membrane conformation,and nonphotochemical chlorophyll fluorescence quenching in leaves of cotton (Gossypium hirsutum L.)

The kinetics and temperature dependencies of development and relaxation of light-induced absorbance changes caused by deepoxidation of violaxanthin to antheraxanthin and zeaxanthin (Z; peak at 506 nm) and by light scattering (S; peak around 540 nm) as well as of nonphotochemical quenching of chlorophyll fluorescence (NPQ) were followed in cotton leaves. Measurements were made in the absence and the presence of dithiothreitol (DTT), an inhibitor of violaxanthin deepoxidase. The amount of NPQ was calculated from the Stern-Volmer equation. A procedure was developed to correct gross AS (S_g) for absorbance changes around 540 nm that are due to a spectral overlap with Z. This protocol isolated the component which is caused by light-scattering changes alone (S_n). In control leaves, the kinetics and temperature dependence of the initial rate of rise in S_n that takes place upon illumination, closely matched that of Z. Application of DTT to leaves, containing little zeaxanthin or antheraxanthin, strongly inhibited both S_n and NPQ, but DTT had no inhibitory effect in leaves in which these xanthophylls had already been preformed, showing that the effect of DTT on A_n and NPQ results solely from the inhibition of violaxanthin deepoxidation. The rates and maximum extents of S_n and NPQ therefore depend on the amount of zeaxanthin (and/or antheraxanthin) present in the leaf. In contrast to the situation during induction, relaxation of Z upon darkening was much slower than the relaxation of S_n and NPQ. The relaxation of S_n and NPQ showed quantitatively similar kinetics and temperature dependencies (Q₁₀=2.4). These results are consistent with the following hypotheses: The increase in lumen-proton concentration resulting from thylakoid membrane energization causes deepoxidation of violaxanthin to antheraxanthin and zeaxanthin. The presence of these xanthophylls is not sufficient to cause S_n or NPQ but, together with an increased lumen-proton concentration, these xanthophylls cause a conformational change, reflected by S_n. The conformational change facilititates nonradiative energy dissipation, thereby causing NPQ. Membrane energization is prerequisite to conformational changes in the thylakoid membrane and resultant nonradiative energy dissipation but the capacity for such changes in intact leaves is quite limited unless zeaxanthin (and/or antheraxanthin) is present in the membrane. The sustained S_n and NPQ levels that remain after darkening may be attributable to a sustained high lumen-proton concentration.Abbreviations A antheraxanthin - DTT dithiothreitol - F, F_m chlorophyll fluorescence yield at actual, full closure of the PSII centers - NPQ nonphotochemical chlorophyll fluorescence quenching - PFD photon flux density - PSII photosystem II - V violaxanthin - Z zeaxanthin - S_n, Z spectral absorbance change caused by light-scattering, violaxanthin deepoxidation We thank Connie Shih for skillful assistance in growing the plants, and for conducting HPLC analyses. A Carnegie Institution Fellowship and a Feodor-Lynen-Fellowship by the Alexander von Humboldt-Foundation to W. B. is gratefully acknowledged. This work was supported in part by Grant No. 89-37-280-4902 of the Competitive Grants Program of the U.S. Department of Agriculture to O.B. This is C. I. W. — D. P. B. Publication No. 1094.     

Metabolic cost of sodium transport in toad urinary bladder

Summary The metabolic cost of active sodium transport was determined in toad bladder at different gradients of transepithelial potential, , by continuous and simultaneous measurements of CO₂ production and of transepithelial electric current. Amiloride was used to block active sodium transport in order to assess the nontransport-linked, basal, production of CO₂ and the passive permeability of the tissue. From these determinations active sodium transport,J _Na, and suprabasal CO₂ production, , were calculated. Since large transients inJ _Na and frequently accompanied any abrupt change in , steady state conditions were carefully defined.Some 20 to 40 min were required after a change in before steady state of transport activity and of CO₂ production were achieved. The metabolic cost of sodium transport proved to be the same whether the bladder expended energy moving sodium against a transepithelial electrical potential grandient of +50 mV or whether sodium was being pulled through the active transport pathway by an electrical gradient of –50 mV. In both cases the value of the ratio averaged some 20 sodium ions transported per molecule of CO₂ produced.When the Na pump was blocked by 10^–2 m ouabain, the perturbations of the transepithelial electrical potential did not elicit changes ofJ _Na nor, consequently, of .The independence of the ratio from over the range ±50 mV indicates a high degree of coupling between active sodium transport and metabolism.     

Bubble column bioreactors

Summary The photographic and electrical conductivity methods to measure the structure of two phase flow, especially bubble size, bubble frequency, local gas hold-up and, for the latter, the bubble velocity are described.Symbols specific interfacial area - a gas/liquid interfacial area - B constant in Eq. (4) - d diameter of the bubbles - d mean diameter of the bubbles - d_S Sauter diameter - E_G relative gas hold up - I current - k_L mass transfer coefficient across the gas/liquid interface - k_L local k_L - LT^–1 - LT^–1 - 1 longitudinal distance between the start and stop sensors - 1_B pierced length of the bubble - t time - t₁ length of the square-wave signal at the start sensor - t₂ length of the square-wave signal at the stop sensor - t₁₂ time delay between start and stop signals - V volume of the bubbling layer - V_L volume of the bubble free layer - V_B bubble volume - v_B bubble velocity     

Clinical characteristics of 16 cystic fibrosis patients with the missense mutation R334W,a pancreatic insufficiency mutation with variable age of onset and interfamilial clinical differences

We present the genotype/phenotype correlation analysis for 16 cystic fibrosis (CF) patients who carry mutation R334W. Current age and age of diagnosis was significantly higher in the R334W/any-mutation group (P < 0.05 and P < 0.01), compared with the 508/508 group. A slightly, but not significantly, worse lung function was found in the R334W/any-mutation group, when compared with the 508/508 patients. The proportion of patients with lung colonization with bacterial pathogens was slightly, but not significantly, higher in the R334W/any-mutation group (71.4%), compared with the 508/508 or R334W/508 groups (55.5%). None of the R334W patients had meconium ileus but 60% were pancreatic insufficient (PI), a significantly lower proportion (P 0.001) than 508/508 patients. Two R334W/N1303K compound heterozygous sisters were PI but discrepant for lung function. Two groups of three sibs with genotype R334W/508 showed interfamilial discordant clinical data for lung and pancreatic function. The data provided here for mutation R334W demonstrate that this mutation is responsible for a less severe form of CF than 508. Interfamilial differences for PI and lung function suggest that other factors, viz. genetic, environmental and medical, contribute to the wide spectrum of clinical differences observed in CF patients with the same CF transmembrane conductance regulator genotypes.     

The electrochemical H+ gradient in the yeastRhodotorula glutinis

The electrochemical gradient of protons, , was estimated in the obligatory aerobic yeastRhodotorula glutinis in the pH₀ range from 3 to 8.5. The membrane potential, , was measured by steady-state distribution of the hydrophobic ions, tetraphenylphosphonium (TPP⁺) for negative above pH₀ 4.5, and thiocyanate (SCN^–) for positive below pH₀ 4.5. The chemical gradient of H⁺ was determined by measuring the chemical shift of intracellular P_i by³¹P-NMR at given pH₀ values. The values of pH_i increased almost linearly from 7.3 at pH₀ 3 to 7.8 at pH₀ 8.5. In the physiological pH₀ range from 3.5 to 6, was fairly constant at values between 17–18 KJ mol^–1, gradually decreasing at pH₀ above 6. In deenergized cells, the intracellular pH_i decreased to values as low as 6, regardless of whether the cell suspension was buffered at pH₀ 4.5 or 7.5. There was no membrane potential detectable in deenergized cells.     

Sodium ions are necessary for growth and energy transduction in the marine cyanobacterium Oscillatoria brevis

The maximal growth rate of the marine cyanobacterium Oscillatoria brevis was reached at 200–400 mM NaCl and pH 9.0–9.6. NaCl was found (i) to stimulate the rate of the light-supported generation across the cytoplasmic membrane of the cells and (ii) to decrease the sensitivity of level and motility of the O. brevis trichomes to protonophorous uncouplers. The Na⁺/H⁺ antiporter, monensin, increased both and the uncoupler sensitivity of the cells. The data obtained agree with the assumption that O. brevis possesses a primary Na⁺ pump in its cytoplasmic membrane.Abbreviations ATP adenosine-5-triphosphate - TTFB tetrachlortrifluoromethylimidazol - CCCP carbonyl cyanide m-chlorophenylhydrazone - Na⁺ transmembrane electrochemical potential differences of Na⁺ - transmembrane electric potential difference - pNa transmembrane pNa difference     

On the appropriateness of use of a continuous formulation for the modelling of discrete multireactant systems following Michaëlis–Menten kinetics

The possibility of solving the mass balances to a multiplicity of substrates within a CSTR in the presence of a chemical reaction following Michaelis-Menten kinetics using the assumption that the discrete distribution of said substrates is well approximated by an equivalent continuous distribution on the molecular weight is explored. The applicability of such reasoning is tested with a convenient numerical example. In addition to providing the limiting behavior of the discrete formulation as the number of homologous substrates increases, the continuous formulation yields in general simpler functional forms for the final distribution of substrates than the discrete counterpart due to the recursive nature of the solution in the latter case.List of Symbols C{N. M} mol/m³ concentration of substrate containing N monomer residues each with molecular weight M - {N, M} normalized value of C{N. M} - C {M} mol/m³ da concentration of substrate of molecular weight M - _in normalized value of C {M} at the i-th iteration of a finite difference method - {M} normalized value of C {M} - C ₀{N.M} mol/m³ inlet concentration of substrate containing N monomer residues each with molecular weight M - {N ·M} normalized value of C₀{N. M} - _{0 i} normalized value of C ₀ {M} at the i-th iteration of a finite difference method - C ₀ {M} mol/m³ da initial concentration of substrate of molecular weight M - C _tot mol/m³ (constant) overall concentration of substrates (discrete model) - C _tot mol/m³ (constant) overall concentration of substrates (continuous model) - D deviation of the continuous approach relative to the discrete approach - i dummy integer variable - I arbitrary integration constant - j dummy integer variable - k dummy integer variable - K _m mol/m³ Michaëlis-Menten constant for the substrates - l dummy integer variable - M da molecular weight of substrate - M normalized value of M - M da maximum molecular weight of a reacting substrate - N number of monomer residues of a reacting substrate - N maximum number of monomer residues of a reacting substrate - N total number of increments for the finite difference method - Q m³/s volumetric flow rate of liquid through the reactor - S inert product molecule - S _i substrate containing i monomer residues - V m³ volume of the reactor - v _max mol/m³ s reaction rate under saturating conditions of the enzyme active site with substrate - v _max{N. M} mol/m³ s reaction rate under saturating conditions of the enzyme active site with substrate containing N monomer residues with molecular weight M - _max{N · M} dimensionless value of v_max{N. M} (discrete model) - _max{M} dimensionless value of v _max {M} (continuous model) - mol/m³ s molecular weight-averaged value of v_max (discrete model) - mol.da/m³s molecular weight-averaged value of v_max (continuous model) - v _max {M} mol.da/m³s reaction rate under saturating conditions of the enzyme active site with substrate with molecular weight M - _max {M} dimensionless value of v_max{M} - _{max, (i)} dimensionless value of v_max{M} at the i-th iteration of a finite difference method - v _max mol/m³ s reference constant value of v _max Greek Symbols dimensionless operating parameter (discrete distribution) - dimensionless operating parameter (continuous distribution) - M da (average) molecular weight of a monomeric subunit - M selected increment for the finite difference method - auxiliary corrective factor (discrete model)