A Simple Procedure for Constructing Experiment Designs with Incomplete Blocks of Sizes 2 and 3

In order to maximize control of heterogeneity within complete blocks, an experimenter could use incomplete blocks of size k = 2 or 3. In certain situations, incomplete blocks of this nature would eliminate the need for such spatial types of analyses as nearest neighbor. The intrablock efficiency factors for such designs are relatively low. However, with recovery of interblock information, FEDERER and SPEED (1987) have presented measures of design efficiency factors which demonstrate that efficiency factors approach unity for certain ratios of the intrablock and interblock variance components. Hence with recovery of interblock information, even incomplete block designs with k = 2 or 3 have relatively high efficiency factors. The reduction in the intrablock error variance over the complete block error variance in many situations will provide designs with high efficiency. A simple procedure for constructing incomplete blocks of sizes 2 and 3 is presented. It is shown how to obtain additional zero-one association confounding arrangements when v = 4 t, t an integer, and for v = pk, k ≤ p. It is indicated how to do the statistical analysis for these designs.     

The concentration dependence of sedimentation for polysaccharides

The relationship between the sedimentation coefficient s₀ and its concentration coefficient k_s obtained in experiments on velocity sedimentation for polysaccharides is discussed. The values of s₀, k_s and an independently determined molecular weight reported by different authors for different polysaccharides are considered. It was established that the scaling relation. k_s∼ s₀ ^v unambiguously relates to the scaling relation s₀∼ M^b. The values of the sedimentation parameter β _s introduced on the basis of Svedberg's equation for s₀ and on the basis of the expression k_s = B 〈h²〉^3/2M^–1 are discussed and the generalized Wales-van Holde-Rowe equation M_KS = (N_A/β _S)^3/2[s]^3/2 k_S ^1/2 is used for evaluation of the molecular weights of polysaccharides. The adequacy of this evaluation is illustrated by taking as an example the determination of the unit length weight of an extra-rigid polysaccharide chain and of the equilibrium rigidity of rigid-chain, semi-rigid-chain and flexible-chain polysaccharides. The pair of experimental values s₀ and k_S obtained in a single series of experiments give the same information as may be obtained from the other pairs of hydrodynamic values such as [η] and s₀ or [η] and D₀, where [η] is the intrinsic viscosity and D₀ is the translational diffusion coefficient. Accepted: 11 December 1996     

Construction and Analysis of an Augmented Lattice Square Design

Augmented designs are useful for screening experiments involving large numbers of new and untried treatments. Since resolvable row‐column designs are useful for controlling extraneous variation, it is desirable to use such designs for the check or standard treatments to construct augmented lattice square experiment designs. A simple procedure is described for constructing such designs using c = 2k and c = 3k check treatments and n = rk(k ‐— 2) and n = rk(k — 3) new treatments, respectively, r being the number of complete blocks. A trend analysis for these designs, which allows for solutions of fixed effects, is presented. The random effects case is also discussed. A SAS computer code and the output from this code illustrated with a small numerical example are available from the author.     

Designs Suited for Varietal Trials with Multiple Basals

DAS (1960) gave a method of construction of confounded balanced asymmetrical factorial designs of the type v × 2² by using BIB designs. In the present paper a method has been given for construction of balanced asymmetrical factorial designs of the type (v–t) × 2² by using truncated balanced incomplete block designs obtainable by omitting t treatments. Likewise, partially balanced asymmetrical factorial designs can also be obtained by omitting any particular treatment alongwith its first or second associate treatments from the v treatments of a PBIB design. We can get a large number of new designs not available in literature through this technique. These designs are well suited for varietal trials with multiple basals.     

Using a Normal Approximation to Test for the Binomial Distribution

The commonly used method to test for the binomial distribution is the x²-test. In this paper, we introduce an alternative method to test for the binomial distribution. Suppose N is the number of sample groups with n individuals each, x_ij is the jth sample in ith group, a Bernoulli variable with parameter and VVI=s²/[m(1 - m)/n]. Then it is well know that the asymptotic distribution of the statistic (N - 1) VVI is x²(N - 1) under the hypothesis p₁ = p₂ = … = p_N. Here we find that VVI has an asymptotic normal distribution N(1, 2(1 - 1/n)/(N - 1)). Unlike the x²-statistic, the variance of the normal test statistic is a function of n. This method is convenient in detecting spatial patterns and dispersion in the study of diseased organisms (e.g., plants) in field samples.     

Oxygen transfer to slurries treated in a rotating drum operated at atmospheric pressure

The objective of this work was to determine (1) the effect of rotational speed (N) and lifters on the oxygen transfer coefficient (k _L) of a mineral solution and (2) the effect of solids concentration of a slurry soil-mineral solution on k _L, at a fixed value N (0.25 s⁻¹); in both cases the treatment was carried out in an aerated rotating drum reactor (RDR) operated at atmospheric pressure. First, the k _L for the mineral solution was in the range 6.38 × 10⁻⁴–7.69 × 10⁻⁴ m s⁻¹, which was of the same order of magnitude as those calculated for closed rotating drums supplied with air flow. In general, k _L of RDR implemented with lifters was superior or equal to that of RDR without lifters. For RDR implemented with lifters, k _L increased with N in the range 6.65 × 10⁻⁴–10.51 × 10⁻⁴ m s⁻¹, whereas k _L of RDR without lifters first increased with N up to N = 0.102 s⁻¹, and decreased beyond this point. Second, regarding soil slurry experiments, an abrupt fall of k _L (ca. 50%) at low values of the solid concentration (C _v) and an asymptotic pattern at high C _v were observed at N = 0.25 s⁻¹. These results suggest that mass transfer phenomena were commanded by the slurry properties and a semi-empirical equation of the form Sh = f(Re, Sc) seems to corroborate this finding.     

Minimum sample sizes for identifying chromosomal fragile sites from individuals: Monte Carlo estimation

A Monte Carlo simulation procedure was used to estimate the exact level of the standardized X ² test statistic (X _s ²) for randomness in the FSM methodology for the identification of fragile sites from chromosomal breakage data for single individuals. A random-number generator was used to simulate 10 000 chromosomal breakage data sets, each corresponding to the null hypothesis of no fragile sites for numbers of chromosomal breaks (n) from 1 to 2000 and at three levels of chromosomal band resolution (k). The reliability of the test was assessed by comparisons of the empirical and nominal α levels for each of the corresponding values of n and k. These analyses indicate that the sparse and discrete nature of chromosomal breakage data results in large and unpredictable discrepancies between the empirical and nominal α levels when fragile site identifications are based on small numbers of breaks (n < 0.5 k). With n≥ 0.5 k, the distribution of X _s ² appears to be stable and non-significant differences in the empirical and nominal α levels are generally obtained. These results are inherent to the nature of the data and are, therefore, relevant to any statistical model for the identification of fragile sites from chromosomal breakage data. For FSM identification of fragile sites at α = 0.05, we suggest that n≥ 0.5 k is the minimum reliable number of mapped chromosomal breaks per individual. Received: 28 April 1997 / Accepted: 1 July 1997     

Kinetic sedimentation of Rhizobium-aggregates produced by leguminous lectins

Lectins from Canavalia brasiliensis (CnBr), Cratylia floribunda (CFL), Vatairea macrocarpa (VML) and Phaseolus vulgaris (PHA) aggregate Rhizobiumbacteria. The relationship between specific sedimentation rate, (based on bacterial dry biomass) of bacterial aggregates and lectin concentrations was hyperbolic and showed bacterial surface affinity by lectins. R. tropici (Rt), R. leguminosarum bv. phaseoli (Rlp) and R. etli (Re) surfaces showed predominantly receptors of galactosidic nature. The Rt surfaces showed very high affinities (k _s = ±8.6 × 10^–8 ag lectin protein ml^–1) by Gal-specific lectins (PHA and VML), and very low affinities (k_s=± 4.9 × 10^–6) by Glc-specific lectins (CnBr and CFL). The Rlp surface had intermediate affinities by lectins. The Re surface showed high affinities by PHA (k_s= ±1.26 × 10^–8) and intermediate affinities by VML, CnBr and CFL. The relationship between sedimentation specific (based on lectin weight) and bacterial density was a sigmoid and showed lectin affinity by Rt surfaces. The bacterial sedimentation showed positive cooperative binding of lectins. The V_max induced by Glc-specific lectins was ±20 of that produced by Gal-specific lectins. The PHA affinity (k_s= 1.19 mg dry biomass ml^–1) was larger than VML (k_s = 1.23). The Glc-specific lectin affinities were smaller than those of Gal-specific. The apparent binding site number of lectins (n_app) was: 2.7-PHA; 2.2-VML; 3.2-CFL and 3.2-CnBr. The dissociation constant, k_s, of lectin-binding kinetics decreased with sugar-hapten treatment (10 M). The n_app decreased in PHA and CFL, increasing in VML + sugar-hapten treatment. This study showed that there is a difference in Rhizobium surfaces for lectin binding.     

Fractional Plans to Estimate Main Effects and Two Factor Interactions Inclusive of a Specific Factor

Balanced incomplete block designs are used to construct non‐geometric 2ⁿ fractional factorial plans to estimate all n main effects and n – 1, 2 factor interactions with a specific factor included in each interaction. When the balanced incomplete block design is of Family (A), the resulting fractional factorial plan has the same number of runs as a fold‐over Hadamard matrix giving same variances for the estimates; however, some new designs are shown to be non‐isomorphic to the fold‐over Hadamard matrix plans.     

Immobilization of lactase for the continuous hydrolysis of whey permaete

The production of lactose-based sweeteners is considered very promising. Fungal lactase has been immobilized on crosslinked chitin to develop a process for the continuous hydrolysis of demineralized whey permaete. The optimization of lactase immobilization on chitin and chitosan was performed, activities of 4 · 10⁵ and 2.2 · 10⁵ u/kg at yields of 33 and 23% were obtained for both supports, respectively. The chitin based catalyst was selected for further studies and a procedure was developed for in-situ enzyme immobilization. The kinetic behaviour of the catalyst was determined to propose a kinetic model for the initial rate of lactose hydrolysis. Pseudo steady-state and long term operation of packed bed reactors with chitin-immobilized lactase ranging from small laboratory to pre-pilot unit was carried out. The results are discussed and compared with commercial immobilized lactases. Preliminary economic evaluation for the production of ultrafiltered whey protein and hydrolyzed lactose syrup, within a dairy industry in Chile, was satisfactory in terms of profitability, both for the chitin immobilized lactase developed and for a commercial immobilized lactase.List of Symbols a moles/m³ glucose concentration in Eq. (1) - C _i US$ total annual cost (without considering plant depreciation) - D US$ annual depreciation - F m³/h flowrate - h m³/h volumetric mass transfer coefficient - i moles/m³ galactose concentration in Eqs. (1) and (2) - K _A moles/m³ dissociation constant for glucose in Eq. (1) - K _A moles/m³ dissociation constant for glucose in Eq. (1) - K _I moles/m³ inhibition constant for galactose in Eqs. (1) and (2) - K _m moles/m³ Michaelis constant for substrate in Eqs. (1) and (2) - k _D h^–1 first-order thermal deactivation constant - P kg dry weight of catalyst - PV US$ net present value - R % discounted cash-flow rate of return - s moles/m³ substrate concentration - s₀ moles/m³ feed substrate concentration - S _n US$ annual sales income - TC US$ total capital income - t _1/2 h catalyst half-life - v moles/h · kg initial rate of reaction - V _MAX moles/h · kg maximum reaction rate in Eqs. (1) and (2) - V _MAX moles/h · kg maximum reaction rate in Eq. (1) - ¯V _max moles/h initial rate of reaction - V _R m³ reaction volume free of catalyst particles - X substrate degree of conversion = s₀–s/s₀ - Damkoehler number = ¯V _MAX /h k _m - moles/(m³ · h) reactor productivity in Eq. (3)     

Haematococcus pluvialis cultivation in split-cylinder internal-loop airlift photobioreactor under aeration conditions avoiding cell damage

The effects of superficial gas velocity in the riser (U_Gr) and gas entrance velocity (v) on the growth of Haematococcus pluvialis cultivated in a split-cylinder internal-loop airlift photobioreactor were investigated. Cell growth decreased when U_Gr and v were increased above 12 mm s^–1 and 22.8 m s^–1, respectively. The maximum cell density of H. pluvialis was 110×10⁴ vegetative cells ml^–1 and the chlorophyll-a titer was 7 mg l^–1. The cell damage in the photobioreactor was greater when v was increased by an increase in U_Gr rather than by a decrease in sparger internal diameter. The overall volumetric mass transfer coefficient (k_La) of the photobioreactor was measured at the same U_Gr (6–24 mm s^–1) and v (12–80 m s^–1). The k_La values reached in the airlift photobioreactor were between 10 h^–1 and 32 h^–1.     

Liquid-phase mass transfer in fixed-bed of polyurethane foam matrices containing immobilized anaerobic sludge

This paper reports on the results of liquid-phase mass transfer experiments carried-out in a bed of immobilized anaerobic sludge. The liquid superficial velocity (v_s) was found to deeply affect the liquid-phase volumetric mass transfer coefficient (k_sa) in the v_s range of 0.007 to 0.075 cm.s^–1. Moreover, k_sa increased exponentially with v_s in that range, due to the decrease of the external mass transfer resistance.     

Purification and Characterization of β-N-Acetylhexosaminidase from the Liver Prawn,Penaeus japonicus

β-N-Acetvlhexosaminidase (EC 3.2.1.52) was purified from the liver of a prawn, Penaeus japonicus, by ammonium sulfate fractionation and chromatography with Sephadex G-100, hydroxylapatite, DEAE-Cellulofine, and Cellulofine GCL-2000-m. The purified enzyme showed a single band keeping the potential activity on both native PAGE and SDS–PAGE. The apparent molecular weight was 64,000 and 110,000 by SDS–PAGE and gel filtration, respectively. The pI was less than 3.2 by chromatofocusing. The aminoterminal amino acid sequence was NH₂-Thr-Leu-Pro-Pro-Pro-Trp-Gly-Trp-Ala-?-Asp-Gln-Gly-VaI-?-Val-Lys-Gly-Glu-Pro-. The optimum pH and temperature were 5.0 to 5.5 and 50°C, respectively. The enzyme was stable from pH 4 to 11, and below 55°C. It was 39% inhibited by 10mM HgCl₂.

Steady-state kinetic analysis was done with the purified enzyme using N-acetylchitooligosaccharides (GlcNAc_n, n = 2 to 6) and p-nitrophenyl N-acetylchitooligosaccharides (pNp-β-GlcNAc_n, n= 1 to 3) as the substrates. The enzyme hydrolyzed all of these substrates to release monomeric GlcNAc from the non-reducing end of the substrate. The parameters of K_m and k_cat at 25°C and pH 5.5 were 0.137 mM and 598s^–1 for pNp-β-GlcNAc, 0.117 mM and 298s^–1 for GlcNAc₂, 0.055 mM and 96.4s^–1 for GlcNAc₃, 0.044 mM and 30.1 s^–1 for GlcNAc_4, 0.045 mM and 14.7 s^–1 for GlcNAc₅, and 0.047 mM and 8.3 s^–1 for GlcNAc₆, respectively. These results suggest that this β-N-acetylhexosaminidase is an exo-type hydrolytic enzyme involved in chitin degradation, and prefers the shorter substrates.     

Simulation of glucose isomerase reactor: optimum operating temperature

A design equation for immobilized glucose isomerase (IGI) packed bed reactor is developed assuming enzyme deactivation and substrate protection. The developed equation is used to simulate the performance of the reactor at various temperatures (50–80 °C). Enzyme deactivation is significant at high temperature. Substrate protection showed to have significant effect in reducing enzyme deactivation and increasing the enzyme half-life. Factors affecting the optimum operating temperature are discussed. The optimum operating temperature is greatly influenced by the operating period and to a lesser extent with both initial glucose concentration and glucose conversion.Two modes of reactor operation are tested i.e., constant feed flow rate and constant conversion. Reactor operating at constant conversion is more productive than reactor operating at constant flow rate if the working temperature is higher than the optimum temperature. Although at lower temperatures than the optimum, the two modes of operation give the same result.List of Symbols a residual enzyme activity - E [mg/l] concentration of active enzyme - E _a [kJ/mole] activation energy - E ₀ [mg/l] initial concentration of active enzyme - k [Specific] kinetic parameter - k _d [h^–1] first order thermal deactivation rate constant - k _e equilibrium constant - k _m [mole/l] apparent Michaelis constant - k _p [mole/l] Michaelis constant for product - k _s [mole/l] Michaelis constant for substrate - k ₀ [Specific] pre-exponential factor - Q [1/h] volumetric flow rate - ¯Q [1/h] average volumetric flow rate - R [kJ/mol·k] ideal gas constant - s [mole/l] apparent substrate concentration - s [mole/l] substrate concentration - s _e [mole/l] substrate concentration at equilibrium - s ₀ [mole/l] substrate concentration at reactor inlet - p [mole/l] product concentration - p _e [mole/l] product concentration at equilibrium - P _r [mole fructose/l·h] reactor productivity - T [k] temperature - t [h] time - t _p [h] operating time - V [l] reactor volume - v [mole/l·h] reaction rate - v [mole/l] reaction rate under enzyme deactivation and substrate protection - v _m [mole/l·h] maximum apparent reaction rate - v _p [mole/l·h] maximum reaction rate for product - v _s [mole/l·h] maximum reaction rate for substrate - x substrate fractional conversion - x _e substrate fractional conversion at equilibrium Greek Symbols effectiveness factor - mean effectiveness factor - substrate protection factor - [h] residence time - [h] average residence time - ₀ [h] initial residence time     

Assessment of photosynthetic performance in the two life history stages of Alaria crassifolia (Laminariales,Phaeophyceae)

Understanding of the physiological responses of kelp to environmental parameters is crucial, especially in the context of environmental change that may have contributed to the decline of kelp forests all over the world. The current study presents the photosynthetic characteristics of the macroscopic sporophyte and microscopic gametophyte stages of the brown alga Alaria crassifolia from Hokkaido, Japan, as determined by examining their photosynthetic responses over a range of temperature and irradiance using dissolved oxygen and chlorophyll fluorescence measurements. Net photosynthetic rates of the sporophyte were consistently higher than those of gametophyte across temperature gradients and irradiance levels. Photosynthesis–irradiance curves at 8°C, 16°C, and 20°C revealed similar initial slopes (α = 0.4–0.9) on the two life history stages, but higher compensation (E _c = 4–7 μmol photons m^?2 s^?1) and saturation irradiances (E _k = 53–103 μmol photons m^?2 s^?1) for the sporophyte than for the gametophyte (E _c = 0–7 μmol photons m^?2 s^?1; E _k = 7–10 μmol photons m^?2 s^?1). Both stages exhibited chronic photoinhibition, as shown by the failure of recovery in their maximum quantum yields (F _v/F _m) following high irradiance stress, with greater possibility of photodamage at low temperature. Gametophytes were less sensitive to low temperatures than sporophytes, given their relatively stable F _v/F _m response. Nevertheless, temperature optima for photosynthesis of both stages coincide with each other at 20–23°C, which correspond to the growth and maturation periods of A. crassifolia in Japan. This species is also likely to suffer from thermal inhibition as both GP rates and F _v/F _m decreased above 24°C.     

Transition-state structure for a conformation change of ribonuclease

The rate constants k₁₂ⁿ for isomerization of the E₁H isomer (pK_a 8 in H₂O) of ribonuclease-A to the E₂H isomer (pK_a = 6.1 in H₂O), determined from proton-uptake measurements by the temperature-jump technique, in mixtures of protium and deuterium oxides (atom fraction of deuterium n), are described by the equation k₁₂ⁿ = (733 ± 16)(1 − n + [0.46 ± 0.04]n)(1 − n + 0.69n)²sec⁻¹ at 25°C. On the basis of the absolute magnitude of the rate constant, the magnitude of the solvent isotope effect and the proton inventory, it appears that the rate-determining step is proton transfer to a water molecule from the imidazolium form of a histidine residue, with a product-like activated complex resembling a hydronium ion. The subsequent motion of the protein structure to generate the new isomer (conformation change) must then occur in a time approaching a vibrational period. Alternative but less likely mechanisms include rate-limiting protein reorganization concerted with proton transfer to water, rate-limiting diffusion of hydronium ion away from the enzyme, or “solvation catalysis” of protein reorganization.     

Characteristic parameters and polydispersity elimination factors for hydrodynamic quantities of semirigid polymers, especially of DNA

Relations are given allowing the calculation of intrinsic viscosity, diffusion constant, and related molecular weights of monodisperse subfractions of polydisperse samples from the measured averages of the hydrodynamic quantities and the individual (reduced) sedimentation constant distribution. The peculiarities resulting from the special behavior of semirigid polymers like DNA are treated. Expressions are derived which describe the dependence of the exponents a_s(a_ηs = a_η/a_s), and a_η of the relations s^o = k_sM^as [η] = k_ηs(s^o)^aηs, and [η] = k_ηM^aη on s^o and [η], respectively, and which are useful for practical and theoretical discussions of the wormlike chain. Furthermore, a generalized exponent rule for a_η and a_s has been suggested, considering the dependence of the Mandelkern-Flory-Scheraga parameter β(?Φ^1/3P^?1) on molecular weight. The results of this paper are applied extensively in the following paper to molecular weight and hydrodynamic properties of homogeneous DNA.     

Energetics of swimming at maximal speeds in humans

The energy cost per unit of distance (C _s, kilojoules per metre) of the front-crawl, back, breast and butterfly strokes was assessed in 20 elite swimmers. At sub-maximal speeds (v), C _s was measured dividing steady-state oxygen consumption (V˙O₂) by the speed (v, metres per second). At supra-maximal v, C _s was calculated by dividing the total metabolic energy (E, kilojoules) spent in covering 45.7, 91.4 and 182.9 m by the distance. E was obtained as: E = E _an+V˙O_2max t _p−V˙O_2max(1−e^{−( t p/)}), where E _an was the amount of energy (kilojoules) derived from anaerobic sources, V˙O_2max litres per second was the maximal oxygen uptake, α (=20.9 kJ · l O₂ ⁻¹) was the energy equivalent of O₂, τ (24 s) was the time constant assumed for the attainment of V˙O_2max at muscle level at the onset of exercise, and t _p (seconds) was the performance time. The lactic acid component was assumed to increase exponentially with t _p to an asymptotic value of 0.418 kJ · kg⁻¹ of body mass for t _p ≥ 120 s. The lactic acid component of E _an was obtained from the net increase of lactate concentration after exercise (Δ[La]_b) assuming that, when Δ[La]_b = 1 mmol · l⁻¹ the net amount of metabolic energy released by lactate formation was 0.069 kJ · kg⁻¹. Over the entire range of v, front crawl was the least costly stroke. For example at 1 m · s⁻¹, C _s amounted, on average, to 0.70, 0.84, 0.82 and 0.124 kJ · m⁻¹ in front crawl, backstroke, butterfly and breaststroke, respectively; at 1.5 m · s⁻¹, C _s was 1.23, 1.47, 1.55 and 1.87 kJ · m⁻¹ in the four strokes, respectively. The C _s was a continuous function of the speed in all of the four strokes. It increased exponentially in crawl and backstroke, whereas in butterfly C _s attained a minimum at the two lowest v to increase exponentially at higher v. The C _s in breaststroke was a linear function of the v, probably because of the considerable amount of energy spent in this stroke for accelerating the body during the pushing phase so as to compensate for the loss of v occurring in the non-propulsive phase. Accepted: 14 April 1998     

Reactor design for the enzymatic isomerization of glucose to fructose

A comprehensive methodology is presented for the design of reactors using immobilized enzymes as catalysts. The design is based on material balances and rate equations for enzyme action and decay and considers the effect of mass transfer limitations on the expression of enzyme activity. The enzymatic isomerization of glucose into fructose with a commercial immobilized glucose isomerase was selected as a case study. Results obtained are consistent with data obtained from existing high-fructose syrup plants. The methodology may be extended to other cases, provided sound expressions for enzyme action and decay are available and a simple flow pattern within the reactor might be assumed.List of Symbols C kat/kg specific activity of the catalyst - D m²/s substrate diffusivity within the catalyst particle - Dr m reactor diameter - d d operating time of each reactor - E kat initial enzyme activity - E _i kat initial enzyme activity in each reactor - F m³/s process flowrate - F _i m³/s reactor feed flowrate at a given time - F ₀ m³/s initial feed flowrate to each reactor - H number of enzyme half-lives used in the reactors - K mole/m³ equilibrium constant - K _S mole/m³ Michaelis constant for substrate - K _P mole/m³ Michaelis constant for product - K _m mole/m³ apparent Michaelis constant f(K, K _s, K_p, s₀) - k mole/s · kat reaction rate constant - k _d d^–1 first-order thermal inactivation rate constant - L m reactor height - L _r m height of catalyst bed - N _R number of reactors - P _i kg catalyst weight in each reactor - p mole/m³ product concentration - R m particle radius - R _P ratio of minimum to maximum process flowrate - r m distance to the center of the spherical particle - s mole/m³ substrate concentration - s _0i mole/m³ substrate concentration at reactor inlet - s ₀ mole/m³ bulk substrate concentration - s mole/m³ apparent substrate concentration - T K temperature - t d time - t _i d operating time for reactor i - t _s d time elapsed between two successive charges of each reactor - V m³ reactor volumen - V _m mole/m³ s maximum apparent reaction rate - V _p mole/m³ s maximum reaction rate for product - V _R m³ actual volume of catalyst bed - V _r m³ calculated volume of catalyst bed - V _S mol/m³ s maximum reaction rate for substrate - v mol/m³ s initial reaction rate - v _i m/s linear velocity - v _m mol/m³ s apparent initial reaction rate f(K_m, s,V_m) - X substrate conversion - X _eq substrate conversion at equilibrium - =s/K dimensionless substrate concentration - ₀=s₀/K bulk dimensionless substrate concentration - _eq=s_eq/K dimensionless substrate concentration at equilibrium - local effectiveness factor - mean integrated effectiveness factor - Thiéle modulus - =r/R dimensionless radius - _s kg/m³ hydrated support density - substrate protection factor - s residence time     

A rapid assay for the binding of actinomycin to DNA.

The theory of countercurrent distribution (CCD) was reviewed and extended. The separation function for the fundamental distribution of CCD was presented in the form n = t²(αk₁+β)²/αk₁(β?1)² where n is the number of transfers, t the abscissa of the standard normal distribution, α = v_m/v₈ the phase ratio, β = k₁/k₂≥ 1 the separation factor, and k₁ the partition coefficient of the more radidly moving component; n was found to be minimal on the condition αk₁ = β. The separation function for the single withdrawal of CCD was obtained in the form N = u + 1 = t²{(αk₁ + 1)^1/2 + [β(αk₁ + β)]^1/2}²/(β ? 1)²+ 1, where N is the number of partition units. From this equation it appears that N is minimal when αk₁ = 0. Compared with the former separation functions presented in the literature, these separation functions have the advantage of giving directly the relationships among the phase ratio, the absolute partition coefficient, the separation factor, the resolution degree, and the number of transfers or partition units required. In addition, the dependencies of the elution volumes and the widths of the elution curves on α, β, and the partition coefficients were considered mathematically by means of differential calculus. The elution volumes were found to have minima at certain αk₁ values. The standard deviations, on the contrary, did not have minima in respect to αk₁. The theory presented can be used for selecting proper operating conditions while separating chemical compounds.