Implication of correlations among some common stability statistics — a Monte Carlo simulations

Stability analysis of multilocation trials is often based on a mixed two-way model. Two stability measures in frequent use are the environmental variance (S _i ² )and the ecovalence (W _i). Under the two-way model the rank orders of the expected values of these two statistics are identical for a given set of genotypes. By contrast, empirical rank correlations among these measures are consistently low. This suggests that the two-way mixed model may not be appropriate for describing real data. To check this hypothesis, a Monte Carlo simulation was conducted. It revealed that the low empirical rank correlation amongS _i ² and W _i is most likely due to sampling errors. It is concluded that the observed low rank correlation does not invalidate the two-way model. The paper also discusses tests for homogeneity of S _i ² as well as implications of the two-way model for the classification of stability statistics.     

An Unbalanced Nested Model with Random Effects Having Unequal Variances

Consider the model Y_ijk=μ + a_i + b_ij + e_ijk (i=1, 2,…, t; j=1,2,…, B_i; k=1,2…,n_ij), where μ is a constant and a¹,b_ij and e_ijk are distributed independently and normally with zero means and variances σ²_ad_ij and σ², respectively, where it is assumed that the d_i's and d_ij's are known. In this paper procedures for estimating the variance components (σ², σ²_a and σ²_b) and for testing the hypothesis σ²_b = 0 and σ²_a = 0 are presented. In the last section the mixed model y_ijk, where x_ijkkm are known constants and β_m's are unknown fixed effects (m = 1, 2,…,p), is transformed to a fixed effect model with equal variances so that least squares theory can be used to draw inferences about the β_m's.     

Determination of the optimum operating time for batch isothermal performance of enzyme-catalyzed multisubstrate reactions

This communication consists of a mathematical analysis encompassing the maximization of the average rate of monomer production in a batch reactor performing an enzymatic reaction in a system consisting of a multiplicity of polymeric substrates which compete with one another for the active site of a soluble enzyme, under the assumption that the form of the rate expression is consistent with the Michaelis-Menten mechanism. The general form for the functional dependence of the various substrate concentrations on time is obtained in dimensionless form using matrix terminology; the optimum batch time is found for a simpler situation and the effect of various process and system variables thereon is discussed. The reasoning developed here emphasizes, in a quantitative fashion, the fact that the commonly used lumped substrate approaches lead to nonconservative decisions in industrial practice, and hence should be avoided when searching for trustworthy estimates of optimum operation.List of Symbols O 1/s row vector of zeros - a 1/s row vector of rate constants k _i(i = 2,...,N) - A 1/s matrix of rate constants k _i and k_–i (i=2,...,N) - b 1/s row vector of rate constant k ₂ and zeros - C mol/m³ molar concentration of S - C mol/m³ vector of molar concentrations of C _i (i=0, 1, 2, ..., N) - C ₀ mol/m³ column vector of initial molar concentrations of C _i(i=0, 1, 2,.., N) - C _–01 mol/m³ column vector of initial molar concentrations of C _i(i=2,..., N) - C _{E, tot} mol/m³ total molar concentration of enzyme molecules - C _i mol/m³ molar concentration of S _i (i=0,1,2,...,N) - C _{i, o} mol/m³ initial molar concentration of S _i(i=0, 1, 2, ..., N) - E enzyme molecule - I identity matrix - K 1/s matrix of lumped rate constants - k _i 1/s pseudo-first order lumped rate constant associated with the formation of S _i -1 (i=1, 2, ...,N) - k _{cat, i} 1/s first order rate constant associated with the formation of S _i-1 (i=1, 2, ..., N) - K _m mol/m³ Michaelis-Menten constant - L number of distinct eigenvalues - M _i multiplicity of the i-th eigenvalue - N maximum number of monomer residues in a single polymeric molecule - r ₁ mol/m³ s rate of formation of S ₀ - r _i mol/m³ s rate of release of S _i -1 - r _opt maximum average dimensionless rate of production of monomer S₀ - S lumped, pseudo substrate - S₁ inert moiety - S _i substrate containing i monomer residues, each labile to detachment as - S₀ by enzymatic action (i=1,2,...,N) - t s time elapsed since startup of batch reaction - t _lag s time interval required for cleaning, loading, and unloading the batch reactor - t _opt s time interval leading to the maximum average rate of monomer production - v _ij s^1-j eigenvectors associated with eigenvalue imi (i=1, 2, ..., L; j =1, 2, ..., M_i) Greek Symbols _ij mol/m³ arbitrary constant associated with eigenvalue _i (i=1, 2, ..., L; j=1, 2, ..., M _i ) - 1/s generic eigenvalue - _i 1/s i-th eigenvalue     

A comparison of five distance‐based methods for spatial pattern analysis

Abstract. The behaviour of five statistics (extensions of Pielou's, Clark and Evansapos;, Pollard's, Johnson & Zimmer's, and Eberhardt's statistics, which are denoted as P_i, C_e, P_o, J_z and E_b respectively) that involve the distance from a random point to its jth nearest neighbour were examined against several alternative patterns (lattice‐based regular, inhomogeneous random, and Poisson cluster patterns) through Monte Carlo simulation to test their powers to detect patterns. The powers of all the five statistics increase as distance order j increases against inhomogeneous random pattern. They decrease for P_i and C_e and increase for P_o, J_z, and E_b against regular and Poisson cluster patterns. P_o, J_z, and E_b can reach high powers with the third or higher order distances in most cases. However, P_o is recommended because no extra information is needed, it can reach a high power with the second or third distance even though the sample size is not large in most cases, and the test can be performed with an approximate χ² distribution associated with it. When a regular pattern is expected, J_z is recommended because it is more sensitive to lattice‐based regular pattern than P_o and E_b, especially for the first distance. However, simulation tests should be used because the speed of convergence of J_z to normal distribution is very slow.     

Integration of continuous production and recovery of solvents from whey permeate: use of immobilized cells of Clostridium acetobutylicum in a flutilized bed reactor coupled with gas stripping

An investigation was performed into the operation of an integrated system for continuous production and product recovery of solvents (acetone-butanol-ethanol) from the ABE fermentation process. Cells of Clostridium acetobutylicum were immobilized by adsorption onto bonechar, and used in a fluidized bed reactor for continuous solvent production from whey permeate. The reactor effluent was stripped of the solvents using nitrogen gas, and was recycled to the reactor. This relieved product inhibition and allowed further sugar utilization. At a dilution rate of 1.37 h^–1 a reactor productivity of 5.1 kg/(m³ · h) was achieved. The solvents in the stripping gas were condensed to give a solution of 53.7 kg/m³. This system has the advantages of relieving product inhibition, and providing a more concentrated solution for recovery by distillation. Residual sugar and non-volatile reaction intermediates are not removed by gas stripping and this contributes to high solvent yields.List of Symbols C kg/m³ Lactose concentration in reactor effluent - C _b kg/m³ Lactose concentration in bleed stream - C _c kg/m³ Lactose concentration in whey permeate feed - C _i kg/m³ Lactose concentration at reactor inlet - C _p kg/m³ Lactose concentration in condensed solvent stream (=0) - C _r kg/m³ Lactose concentration in recycle line (C _b=C _r) - C kg/h Amount of lactose utilized during certain time period - D h¹ Dilution rate of reactor, F _i/D=F/D - F dm³/h, m3/h F _i = Rate of feed flow to the reactor - F _b dm ³/h, m³/h Rate of bleed - F _c dm³/h, m³/h Rate of feed of whey permeate solution - F _p dm³/h, m³/h Rate of concentrated product removal - F _r dm³/h, m³/h Rate of recycle of stripped effluent to the reactor - P _l % Percent lactose utilization - R _l kg/(m³ · h) Overall lactose utilization rate - R _p kg/(m³ · h) Overall reactor (solvent) productivity - R _sl kg/h Rate of solvent loss - S kg/m³ Solvent concentration in reactor effluent - S _b kg/m³ Solvent concentration in bleed - S _c kg/m³ 0; Solvent concentration in concentrated whey permeate solution - S _i kg/m³ Solvent concentration at inlet of reactor - S _p kg/m³ Solvent concentration in concentrated product stream - S _r kg/m³ Solvent concentration in stripped effluent, S _r=S_b - S kg/h Amount of solvent produced from C amount of lactose in a particular time - ds/dt kg/(m³ · h) Rate of accumulation of solvents in the stripper - t h Time - V dm³, m³ Total reactor volume - V ¹ dm³, m³ Liquid volume in stripper - Y _P/S Solvent yield     

Variable responses of mesophyll conductance to substomatal carbon dioxide concentration in common bean and soybean

Some reports indicate that mesophyll conductance (g _m) to carbon dioxide varies greatly with the substomatal carbon dioxide concentration (C _i) during the measurement, while other reports indicate little or no change in g _m with C _i. I used the oxygen sensitivity of photosynthesis to determine the response of g _m to C _i over the range of about 100 to 300 μmol mol⁻¹ C _i at constant temperature in common bean (Phaseolus vulgaris) and soybean (Glycine max) grown over a range of temperatures and photosynthetic photon flux densities (PPFD). In soybean grown and measured at high PPFD there was only a slight, approximately 15% decrease in g _m with C _i over the range of 100 to 300 μmol mol⁻¹. With lower PPFD during the measurement of g _m, and especially with low PPFD during plant growth, there was a larger decrease in g _m with C _i in soybean. In common bean, the same range in C _i resulted in about a 60% decrease in g _m for plants grown and measured at high PPFD, with an even larger decrease for plants at low growth or measurement PPFD. Growth temperatures of 20 to 30°C had little influence on the response of g _m to C _i or its absolute value in either species. It is concluded that these two species differed substantially in the sensitivity of g _m to C _i, and that PPFD but not temperature during leaf development strongly affected the response of g _m to C _i.     

Photosynthesis of Ectocarpus siliculosus in red light and after pulses of blue light at high pH – evidence for bicarbonate uptake

Pulses of blue light cause stimulation of red light saturated photosynthesis in Ectocarpus siliculosus, because blue light activates the operation of a pathway for inorganic carbon (C_i) acquisition by inducing the mobilization of CO₂ from an intermediate metabolite. In the absence of exogenous C_i, photosynthetic rates roughly equal those of CO₂ release by respiration. In seawater of pH 9·5 (2·3 mol m^–3 total C_i, but concentrations of free CO₂ below 0·2 mmol m^–3), photosynthesis was clearly above these rates, although they were only ≈ 30% of those in normal seawater (≈ pH 8). The degree and the time course of the stimulations of photosynthesis by pulses of blue light were unaltered at high pH. Essentially the same characteristics were found after buffering or in the presence of acetazolamide, an inhibitor of extracellular carbonic anhydrase activity. Therefore, it is concluded that Ectocarpus is able to directly take up HCO₃^– in addition to CO₂ (uptake of CO₃^2– cannot be excluded). The dependence of photosynthesis on C_i at pH 9·5 was biphasic, with C_i below 0·2 mol m^–3 having no effect at all. In C_i-free seawater, the shapes of the stimulations after blue light pulses differed for pH 6, pH 8 and pH 9·5. At low pH, only the fast peak (maximum ≈ 5 min after blue light) was detected, whereas at high pH mainly the slow peak (maximum ≈ 20 min after blue light) was observed. At the intermediate pH 8, both peaks were present. As inhibition of total carbonic anhydrase by ethoxyzolamide brought out the fast peak of the stimulations at pH 9·5 it is concluded that the fast component was due to a transient disequilibrium of an intracellular pool of C_i which, after blue light, was fed by CO₂ released from the postulated storage intermediate.     

The CO2-concentrating mechanism is absent in the green alga Coccomyxa: a comparative study of photosynthetic CO2 and light responses of Coccomyxa, Chlamydomonas reinhardtii and barley protoplasts

Photosynthesis was characterized for the unicellular green alga Coccomyxa sp., grown at low inorganic carbon (C_i) concentrations, and compared with Chlamydomonas reinhardtii, which had been grown so that the CO₂ concentrating mechanism (CCM) was expressed, and with protoplasts isolated from the C₃ plant barley (Hordeum vulgare). Chlamydomonas had a significantly higher C_i-use efficiency of photosynthesis, with an initial slope of the C_i-response curve of 0.7 mol(gChl)⁻¹ h⁻¹ mmol C_im⁻³)⁻¹, as compared to 0.3 and 0.23 mol(gChl)⁻¹ h⁻¹ (mmol C_im⁻³)⁻¹ for Coccomyxa and barley, respectively. The affinity for C_i was also higher in Chlamydomonas, as the half maximum rate of photosynthesis [K_0.5 (C_i)] was reached at 0.18 mol m⁻³, as compared to 0.30 and 0.45 mol m⁻³ for Coccomyxa and barley, respectively. Ethoxyzolamide (EZ), an inhibitor of the enzyme carbonic anhydrase (CA) and the CCM, caused a 17-fold decrease in the initial slope of the photosynthetic Cj-response curve in Chlamydomonas, but only a 1.5- to two-fold decrease in Coccomyxa and barley. The photosynthetic light-response curve showed further similarities between barley and Coccomyxa. The rate of bending of the curve, described by the convexity parameter, was 0.99 (sharp bending) and 0.81–0.83 (gradual bending) for cells grown under low and high light, respectively. In contrast, the maximum convexity of Chlamydomonas was 0.85. The intrinsically lower convexity of Chlamydomonas is suggested to result from the diversion of electron transport from carbon fixation to the CCM. Taken together, these results suggest that Coccomyxa does not possess a CCM and due to this apparent lack of a CCM, we propose that Coccomyxa is a better cell model system for studying C₃ plant photosynthesis than many algae currently used.     

CYANOBACTERIAL ACCLIMATION TO RAPIDLY FLUCTUATING LIGHT IS CONSTRAINED BY INORGANIC CARBON STATUS1

Acclimation to rapidly fluctuating light, simulating shallow aquatic habitats, is altered depending on inorganic carbon (C_i) availability. Under steady light of 50 μmol photons·m^?2·s^?1, the growth rate of Synechococcus elongatus PCC7942 was similar in cells grown in high C_i (4 mM) and low C_i (0.02 mM), with induced carbon concentrating mechanisms compensating for low C_i. Growth under fluctuating light of a 1‐s period averaging 50 μmol photons·m^?2·s^?1 caused a drop in growth rate of 28%±6% in high C_i cells and 38%±8% in low C_i cells. In high C_i cells under fluctuating light, the PSI/PSII ratio increased, the PSII absorption cross‐section decreased, and the PSII turnover rate increased in a pattern similar to high‐light acclimation. In low C_i cells under fluctuating light, the PSI/PSII ratio decreased, the PSII absorption cross‐section decreased, and the PSII turnover remained slow. Electron transport rate was similar in high and low C_i cells but in both was lower under fluctuating than under steady light. After acclimation to a 1‐s period fluctuating light, electron transport rate decreased under steady or long‐period fluctuating light. We hypothesize that high C_i cells acclimated to exploit the bright phases of the fluctuating light, whereas low C_i cells enlarged their PSII pool to integrate the fluctuating light and dampen the variation of the electron flux into a rate‐restricted C_i pool. Light response curves measured under steady light, widely used to predict photosynthetic rates, do not properly predict photosynthetic rates achieved under fluctuating light, and exploitation of fluctuating light is altered by C_i status.     

Artefactual responses of mesophyll conductance to CO2 and irradiance estimated with the variable J and online isotope discrimination methods

Studies with the variable J method have reported that mesophyll conductance (g_m) rapidly decreases with increasing intercellular CO₂ partial pressures (C_i) or decreasing irradiance. Similar responses have been suggested with the online isotope discrimination method, although with less consistency. Here we show that even when the true g_m is constant, the variable J method can produce an artefactual dependence of g_m on C_i or irradiance similar to those reported in previous studies for any of the following factors: day respiration and chloroplastic CO₂ photocompensation point are estimated with Laisk method; C_i or electron transport rate is positively biased; net photosynthetic rate is negatively biased; insufficient NADPH is assumed while insufficient ATP limits RuBP regeneration. The isotopic method produces similar artefacts if fractionation of carboxylation or C_i is positively biased or Δ¹³ negatively biased. A non‐zero chloroplastic resistance to CO₂ movement results in a qualitatively different dependence of g_m on C_i or irradiance and this dependence is only sensitive at low C_i. We thus cannot rule out the possibility that previously reported dependence of g_m on C_i or irradiance is a methodological artefact. Recommendations are made to take advantage of sensitivities of the variable J and isotopic methods for estimating g_m.     

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     

An Unbalanced Two-way Model With Random Effects Having Unequal Variances

Consider the model y_ijk=u ± a_i ± b_i ± c_ij ± e_ijk i=1, 2,…, t; j=1, 2,…b; k=1, 2,…,n_ij where μ is a constant and a_i, b_i, c_ij are distributed independently and normally with zero means and variances Δ₂ Δ2/bd_ij and δ₂ respectively. It is assumed that d_i's, and d_ij's are known (positive) constants (for all i and j). In this paper procedures for estimating the variance components (Δ₂, Δ²_b and Δ²_a) and for testing the hypothesis H_oc:Δ²_c/Δ² = y₃ and H_oa:Δ²_b/Δ² = y₄ (where y₂, y₃, and y₄, are specified constants) are presented. A generalization for the mixed model case is discussed in the last section.     

Enhancement of product selectivity via enzyme immobilization in sequential degradation reactions of polymeric substrates

The balance equations pertaining to the modelling of a slap-shaped bead containing immobilized enzyme uniformly distributed which catalyzes the sequential reactions of degradation of a polymeric substrate were written and analytically solved in dimensionless form. The effect of the Thiele modulus on the selectivity of consumption of each multimeric product was studied for a simple case. Whereas plain diffusional regime leads to lower selectivities than plain kinetic regime, improvements in selectivity of species A _i relative to species A_i+1 may be obtained at the expense of higher Thiele moduli within a limited range when the diffusivity of A _i is larger than that of A _i +1, or when the pseudo first order kinetic constant describing the rate of consumption of A _i is lower than that of A_i+1.List of Symbols A _i polymeric substrate containing i monomeric subunits - C _i mol·m^–3 normalized counterpart of C _i - C _i mol·m^–3 concentration of substrate A _i - C _i,0 mol·m^–3 initial concentration of substrate A _i - C _i,0 normalized counterpart of C _i,0 - D _ap,i m²·s^–1 apparent diffusivity of substrate A _i - k _i s^–1 pseudo-first order rate constant - K _m,i mol·m^–3 Michaelis-Menten constant associated with substrate A _i - L m half-thickness of the catalyst slab - N number of monomeric subunits of the largest substrate molecule - Th Thiele modulus - V _i mol·m^–3·s^–1 rate of rection of substrate A _i - V_max,i mol·m^–3·s^–1 maximum rate of reaction under saturating conditions of substrate A _i - x m longitudinal coordinate - S _i,i+1 selectivity of enzyme with respect to substrates with consecutive numbers of monomeric subunits Greek Symbols _i ratio of maximum rates of reaction - _i ratio of apparent diffusivities     

Random median sampling to enhance the precision of population estimates

We describe the characteristics of a sampling procedure called random median sampling that was proposed to enhance the precision of population estimates. In performing random median sampling, we first select a sampling item at random from the sampling area. We roughly compare the abundance of individuals in the selected item with that of the adjacent two items in order to identify the item that has median abundance, i.e., the item that has the second largest abundance among the three items. We count the number of individuals of the item having the median abundance. This procedure is repeated n times in the sampling area (i = 1, 2, ..., n). Let m _i be the ith median abundance. The estimates of the mean abundance per sampling item and the variance of estimates are given by Σm _i /n and Σ(m _i –Σm _i /n)²/n(n – 1), respectively. This method is a local application of the median ranked set sampling that was proposed by Muttlak (J Appl Stat Sci 6:245–255, 1997). Random median sampling is effective when the correlation coefficient between adjacent items is small. If the correlation coefficient is close to zero, random median sampling reduces the variance of estimates to 45 or 32% of that in simple random sampling when the distribution follows a normal distribution or a Laplace distribution, respectively. The sample size required to achieve a given precision of estimate decreases accordingly. The effectiveness of random median sampling, however, is small if the correlation coefficient is large. The condition in which random median sampling is superior to simple random sampling is also discussed.     

Lactose-H(OH) transport system of Escherichia coli Multistate gated pore model based on half-sites stoichiometry for high-affinity substrate binding in a symmetrical dimer

A model is proposed for the d-galactoside-H⁺(⁻OH) transporter of Escherichia coli that accounts for essentially all the experimental observations established for this system to date. In this model, the functional unit is postulated to be a dimer (consisting of two copies of lacY-specified polypeptide) which spans the membrane with a 2-fold symmetry axis in the membrane plane (Lancaster, J.R. (1978) J. Theor. Biol. 75, 35–50). The functional dimer is assumed to possess a single pore flanked by an inner gate (g_i) and an outer gate (g_o) and encompassing two oppositely oriented galactoside binding sites, designated m and μ. When g_o is open and g_i is closed under non-energized conditions, binding site m adopts a configuration defined as State A (i.e., m_o^A) exhibiting high affinity toward Class G^a galactosides (thiodigalactoside, melibiose, α-p-nitrophenylgalactoside) but low affinity for Class G^b galactosides (lactose, β-o-nitrophenylgalactoside, β-isopropylthiogalactoside), whereas binding site μ adopts State B (i.e., μ_o^B) displaying relatively high affinity toward Class G^b galactosides but comparatively low affinity for Class G^a galactosides; further, each m_o^A : μ_o^B dimer contains one thiol group whose reaction with N-ethylmaleimide inactivates the transporter unless blocked by galactoside binding at site m_o^A, while the second homologous thiol of the dimer is unreactive toward thiol reagents. Translocation of the m_o^A : μ_o^B dimer involves closing of g_o followed by opening of g_i, and causes the two thiols (as well as sites m and μ) to interchange roles in a symmetrical fashion: m_o^A : μ_o^B ↔ m_i^B : μ_i^A. In the presence of a substantial (negative) transmembrane Δμ~_H⁺, the m : μ dimer is postulated to undergo an electrogenic protein conformational change to a second form, ^*(m : μ), in which both sites m and μ possess low affinity toward internal Class G^b substrates; galactoside transport in both m : μ and ^*(m : μ) is assumed to be coupled to H⁺-symport (⁻OH-antiport) with a stoichiometry of approximately 1 : 1. Finally, five characteristic predictions of the half-sites model are outlined for further tests of its validity.     

Optimal temperature and concentration profiles in a cascade of CSTR's performing Michaelis-Menten reactions with first order enzyme deactivation

A necessary condition is found for the intermediate temperatures and substrate concentrations in a series of CSTR's performing an enzyme-catalyzed reaction which leads to the minimum overall volume of the cascade for given initial and final temperatures and substrate concentrations. The reaction is assumed to occur in a single phase under steady state conditions. The common case of Michaelis-Menten kinetics coupled with first order deactivation of the enzyme is considered. This analysis shows that intermediate stream temperatures play as important a role as intermediate substrate concentrations when optimizing in the presence of nonisothermal conditions. The general procedure is applied to a practical example involving a series of two reactors with reasonable values for the relevant five operating parameters. These parameters are defined as dimensionless ratios involving activation energies (or enthalpy changes of reaction), preexponential factors, and initial temperature and substrate concentration. For negligible rate of deactivation, the qptimality condition corresponds to having the ratio of any two consecutive concentrations as a single-parameter increasing function of the previous ratio of consecutive concentrations.List of Symbols C _E,0 mol.m^–3 Initial concentration of active enzyme - C _E,i mol.m^–3 Concentration of active enzyme at the outlet of the i-th reactor - C _S,0 mol.m^–3 Initial concentration of substrate - C _S,i mol.m^–3 Concentration of substrate at the outlet of the i-th reactor - Da _i Damköhler number associated with the i-th reactor ((V _i.k_v,0.C_E,0)/(Q.C_S,0)) - Da _min Minimum value of the overall Damköhler number - Da _tot Overall Damköhler number - E _d J.mol^–1 Activation energy of the step of deactivation of the enzyme - E _m J.mol^–1 Standard enthalpy change of the step of binding of substrate to the enzyme - E _v J.mol^–1 Activation energy of the step of enzymatic transformation of substrate - i Integer variable - j Dummy integer variable - k Dummy integer variable - k _d,i s^–1 Kinetic constant associated with the deactivation of enzyme in the i-th reactor (k _d,o·exp{–E _d/(R.T _i}) - k _d,0 s^–1 Preexponential factor of the kinetic constant associated with the deactivation of the enzyme - K _m,i mol.m^–3 Equilibrium constant associated with the binding of substrate to the enzyme in the i-th reactor, (k _m,o·exp{–E _m}(R.T _i}) - K _m,0 mol.m^–3 Preexponential factor of the Michaelis-Menten constant associated with the binding of substrate to the enzyme - k _v,i s^–1 Kinetic constant associated with the transformation of the substrate by the enzyme in the i-th reactor (k _v,o·exp{–E _v/(R.T _i})) - k _v,0 s^–1 Preexponential factor of the kinetic constant associated with the transformation of the substrate by the enzyme - N Number of reactors in the series - Q m³.s^–1 Volumetric flow rate of reacting liquid through the reactor network - R J.K^–1.mol^–1 Ideal gas constant - T _i K Absolute temperature at the outlet of the i-th reactor - T ₀ K Initial absolute temperature - V _i m³ Volume of the i-th reactor - v _max mol.m^–3.s^–1 Maximum rate of reaction under saturation conditions of substrate - x _i Normalized concentration of substrate (C_S,i/C_{S, 0}) - x _i,opt Optimum value of the normalized concentration of substrate - y _i Dimensionless temperature (exp{–T ₀/T _i}) - y _i,opt Optimum value of the dimensionless temperature Greek Symbols Dimensionless preexponential factor associated with the Michaelis-Menten constant (K _m,0/C_s,0) - Dimensionless activation energy of the step of enzymatic transformation of substrate (E _v/R.T₀)) - Dimensionless standard enthalpy change of the step of binding of substrate to the enzyme (E _m/(R.T₀)) - Dimensionless activation energy of the step of deactivation of the enzyme (E _d/(R.T₀)) - Dimensionless deactivation preexponential factor ((k _d,0.C_S,0)/(k_v,0.C_E,0)     

Regulation of the mechanism for HCO 3 − use by the inorganic carbon level in Porphyra leucosticta Thur. in Le Jolis (Rhodophyta)

The capacity for HCO₃ ⁻ use by Porphyra leucosticta Thur. in Le Jolis grown at different concentrations of inorganic carbon (C_i) was investigated. The use of HCO₃ ⁻ at alkaline pH by P. leucosticta was␣demonstrated by comparing the O₂ evolution rates measured with the O₂ evolution rates theoretically supported by the CO₂ spontaneously formed from HCO₃ ⁻ . Both external and internal carbonic anhydrase (CA; EC 4.2.1.1) were implied in HCO₃ ⁻ use during photosynthesis because O₂ evolution rates and the increasing pH during photosynthesis were inhibited in the presence of azetazolamide and ethoxyzolamide (inhibitors for external and total CA respectively). Both external and internal CA were regulated by the C_i level at which the algae were grown. A high C_i level produced a reduction in total CA activity and a low C_i level produced an increase in total CA activity. In contrast, external CA was increased at low C_i although it was not affected at high C_i . Parallel to the reduction in total CA activity at high C_i is a reduction in the affinity for C_i, as estimated from photosynthesis versus C_i curves, was found. However, there was no evident relationship between external CA activity and the capacity for HCO₃ ⁻ use because the presence of external CA became redundant when P. leucosticta was cultivated at high C_i. Our results suggest that the system for HCO₃ ⁻ use in P. leucosticta is composed of different elements that can be activated or inactivated separately. Two complementary hypotheses are postulated: (i) internal CA is an absolute requirement for a functioning C_i-accumulation mechanism; (ii) there is a CO₂ transporter that works in association with external CA. Received: 20 April 1996 / Accepted: 5 August 1996     

The effect of light quality on the induction of efficient photosynthesis under low CO2 conditions in Chlamydomonas reinhardtii and Chlorella pyrenoidosa

The effect of blue and red light on the adaptation to low CO₂ conditions was studied in high-CO₂ grown cultures of Chlorella Pyrenoidosa (82T) and Chlamydomonas reinhardtii(137⁺) by measuring O₂ exchange under various inorganic carbon (Cⁱ) concentrations. At equal photosynthetic photon flux density (PPFD), blue light was more favourable for adaptation in both species, compared to red light. The difference in photosynthetic oxygen evolution between cells adapted to low C_iunder blue and red light was more pronounced when oxygen evolution was measured under low C_i compared to high C_i conditions. The effect of light quality on adaptation remained for several hours. The different effects caused by blue and red light was observed in C. pyrenoidosa over a wide range of PPFD with increasing differences at increasing PPFD. The maximal difference was obtained at a PPFD above 1 500 μmol m^?2s^?1. We found no difference in the extracellular carbonic anhydrase activity between blue- and red light adapted cells. The light quality effect recorded under C_i-limiting conditions in C. reinhardtii cells adapted to air, was only 37% less when instead of pure blue light red light containing 12.5% of blue light (similar PPFD as blue light) was used during adaptation to low carbon. This indicates that in addition to affecting photosynthesis, blue light affected a sensory system involved in algal adaptation to low C_i conditions. Since the affinity for C_i of C. Pyrenoidosa and C. reinhardtii cells adapted to air under blue light was higher than that of cells adapted under red light, we suggest that induction of some component(s) of the C_i accumulating mechanism is regulated by the light quality.     

Modelling of enzyme hydrolysis of maltose in a single pellet of immobilized biocatalyst

The reversible hydrolysis of maltose to glucose by immobilized glucoamylase entrapped in spherical solid particles is studied theoretically. For this purpose a known kinetic model taking into account these reversible reactions and the competitive synthesis of iso-maltose was adopted. The mass transfer limitations in the bulk liquid and in the pores of the particles containing the enzyme are considered, using Fick's law. On the basis of mathematical modelling the optimum conditions for biocatalyst performance are established. An appropriate combination of particle size and initial substrate concentration may lead to reduction of undesirable mass transfer resistance and therefore product inhibition and to an improved selectivity of the biocatalyst with respect of glucose formation.List of Symbols C _i kmoles/m³ current concentration ofi-th component along the radius - C _oi kmoles/m³ bulk concentration ofi-th component - C _i ^* kmoles/m³ concentrations ofi-th component on the pellet surface - D _si ,D _i m²/s internal and molecular diffusion coefficient ofi-th component - W _M kmoles/m³·s reaction rate of maltose hydrolysis - W _IM kmoles/m³·s reaction rate of iso-maltose formation - W _G kmoles/m³·s reaction rate of glucose production - R ₀ m pellet radius - r m current radius of the pellet - t s time coordinate - r ₀ ratio of the time step to the square of the radial coordinate - Re Reynolds number =w·R/v - Sc Schmidt number =v/D - Bi Biot number = R/D - A _j ,B, C _j coefficients in the system of linear equations, Eq. (8) - X _i dimensionless degree of transformation - NR number of independent reactions - N number of division sections of the pellet radius - G kmoles/m³ concentration of glucose - M kmoles/m³ concentration of maltose - IM kmoles/m³ concentration of isomaltose - K _m kmoles/m³ Michaelis constant - V _max kmoles/m³·s maximum reaction rate in Eq. (6) - K _i kmoles/m³ inhibition constant - K _1eq ,K _2eq equilibrium constants in Eq. (6) - , h steps along the time and radial coordinate in the pellet - m/s mass transfer coefficient - dimensionless radius of the pellet - computation accuracy Indices i number of reaction component - j index along the radius of the pellet - k index along the time coordinate This work was accomplished with thanks to the financial support of the Bulgarian National Fund for Scientific Investigations —Grant No. MU-1-BE/93.     

Optimum temperature policy for batch reactors performing biochemical reactions in the presence of enzyme deactivation

A necessary condition is found for the optimum temperature policy which leads to the minimum reaction time for a given final conversion of substrate in a well stirred, enzymatic batch reactor performing an enzyme-catalyzed reaction following Michaelis-Menten kinetics in the presence of first order enzyme decay. The reasoning, which is based on Euler's classical approach to variational calculus, is relevant for the predesign steps because it indicates in a simple fashion which temperature program should be followed in order to obtain the maximum advantage of existing enzyme using the type of reactor usually elected by technologists in the fine biochemistry field. In order to highlight the relevance and applicability of the work reported here, the case of optimality under isothermal operating conditions is considered and a practical example is worked out.List of Symbols C _E mol.m^–3 concentration of active enzyme - C _E ^* dimensionless counterpart of C_E - C _E,0 mol.m^–3 initial concentration of active enzyme - C _E,b mol.m^–3 final concentration of active enzyme - C _E,opt ^* optimal dimensionless counterpart of C_E - C _smol.m^–3 concentration of substrate - C _S ^Emphasis>/* dimensionless counterpart of C_S - C _S,0mol.m^–3 initial concentration of substrate - C _S,bmol.m^–3 final concentration of substrate - E enzyme in active form - E ₃ ^* dimensionless counterpart of E_a,3 - E _a,1J.mol^–1 activation energy associated with k₁ - E _a,3J.mol^–1 activation energy associated with k₃ - E _d enzyme in deactivated form - ES enzyme/substrate complex - k ₁ s^–1 kinetic constant associated with the enzyme-catalyzed transformation of substrate - k _1,0 s^–1 preexponential factor associated with k₁ - k ₂ mol^–1.m³s^–1 kinetic constant associated with the binding of substrate to the enzyme - k _–2 s^–1 kinetic constant associated with the dissociation of the enzyme/substrate complex - K _2,0 mol.m^–3 constant value of K₂ - K _2,0 ^* dimensionless counterpart of K_2,0 - k ₃ s^–1 kinetic constant associated with the deactivation of enzyme - k _3,0 s^–1 preexponential factor associated with k₃ - k _3,0 ^* dimensionless counterpart of k_3,0 - P product - R J.K^–1.mol^–1 ideal gas constant - S substrate - t s time since start-up of reaction - T K absolute temperature - T ^* dimensionless absolute temperature - T _i,opt ^* optimal dimensionless isothermal temperature of operation - T _opt ^* optimal dimensionless temperature of operation - t _b s time of a batch - t _b ^* dimensionless counterpart of t_b - t _b,min ^* minimum value of the dimensionless counterpart of t_b Greek Symbols dimensionless counterpart of C_E,0 - dimensionless counterpart of C_E,b - dummmy variable of integration - dummy variable of integration - auxiliary dimensionless variable - ^* dimensionless variation of k₁ with temperature - _i ^* dimensionless value of k₁ under isothermal conditions - _opt ^* optimal dimensionless variation of k₁ with temperature