Unconditional Confidence Interval for the Difference between Two Proportions

In applied statistics it is customary to have to obtain a one‐ or two‐tail confidence interval for the difference d = p₂ – p₁ between two independent binomial proportions. Traditionally, one is looking for a classic and non‐symmetric interval (with respect to zero) of the type d ∈ [δ_L;δ_U], d ≤ δ₀ or d ≥ δ₀. However, in clinical trials, equivalence studies, vaccination efficacy studies, etc., and even after performing the classic homogeneity test, intervals of the type |d| ≤ Δ₀ or |d| ≥ Δ₀, where Δ₀ > 0, may be necessary. In all these cases it is advisable to obtain the interval by inverting the appropriate test. The advantage of this procedure is that the conclusions obtained using the test are compatible with those obtained using the interval. The article shows how this is done using the new exact and asymptotic unconditional tests published. The programs for performing these tests may be obtained at URL http://www.ugr.es/~bioest/software.htm.     

Unconditional Non-Asymptotic One-Sided Tests for Independent Binomial Proportions When the Interest Lies in Showing Non-Inferiority and/or Superiority

For two independent binomial proportions Barnard (1947) has introduced a method to construct a non-asymptotic unconditional test by maximisation of the probabilities over the ‘classical’ null hypothesis H₀= {(θ₁, θ₂) ∈ [0, 1]²: θ₁ = θ₂}. It is shown that this method is also useful when studying test problems for different null hypotheses such as, for example, shifted null hypotheses of the form H₀ = {(θ₁, θ₂) ∈ [0, 1]²: θ₂ ≤ θ₁ ± Δ } for non-inferiority and 1-sided superiority problems (including the classical null hypothesis with a 1-sided alternative hypothesis). We will derive some results for the more general ‘shifted’ null hypotheses of the form H₀ = {(θ₁, θ₂) ∈ [0, 1]²: θ₂ ≤ g(θ₁ )} where g is a non decreasing curvilinear function of θ₁. Two examples for such null hypotheses in the regulatory setting are given. It is shown that the usual asymptotic approximations by the normal distribution may be quite unreliable. Non-asymptotic unconditional tests (and the corresponding p-values) may, therefore, be an alternative, particularly because the effort to compute non-asymptotic unconditional p-values for such more complex situations does not increase as compared to the classical situation. For ‘classical’ null hypotheses it is known that the number of possible p-values derived by the unconditional method is very large, albeit finite, and the same is true for the null hypotheses studied in this paper. In most of the situations investigated it becomes obvious that Barnard's CSM test (1947) when adapted to the respective null space is again a very powerful test. A theorem is provided which in addition to allowing fast algorithms to compute unconditional non-asymptotical p-values fills a methodological gap in the calculation of exact unconditional p-values as it is implemented, for example, in Stat Xact 3 for Windows (1995).     

Binding of the Bovine and Porcine Pancreatic Secretory Trypsin Inhibitor (Kazal) To Human Leukocyte Elastase: A Thermodynamic Study

Abstract

The effect of pH and temperature on the apparent association equilibrium constant (K_a) for the binding of the bovine and porcine pancreatic secretory trypsin inhibitor (Kazal-type inhibitor, PSTI) to human leukocyte elastase has been investigated. At pH8.0, values of the apparent thermodynamic parameters for human leukocyte elastase: Kazal-type inhibitor complex formation are: bovine PSTT – K_a = 6.3 × 10⁴M^?1, δ5G° = -26.9kJ/mol, δH° = +11.7kJ/mol, and δS° = +1.3 × 10² entropy units; porcine PSTI –K_a = 7.0 × 10³M^?1,δG° = -21.5kJ/mol, δH° = +13.0kJ/mol, and δS° = +1.2 × 10² entropy units (values of K_a δG° and δS° were obtained at 21.0°C; values of δH° were temperature independent over the range (between 5.0°C and 45.0°C) explored). On increasing the pH from 4.5 to 9.5, values of K_a for bovine and porcine PSTI binding to human leukocyte elastase increase thus reflecting the acidic pK-shift of the His57 catalytic residue from ?7.0, in the free enzyme, to ?5.1, in the serine proteinase: inhibitor complexes. Thermodynamics of bovine and porcine PSTI binding to human leukocyte elastase has been analyzed in parallel with that of related serine (pro)enzyme/Kazal-type inhibitor systems. Considering the known molecular models, the observed binding behaviour of bovine and porcine PSTI to human leukocyte elastase was related to the inferred stereochemistry of the serine proteinase/inhibitor contact region(s).     

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.     

Robustness of Sequential Tests for Means

Three sequential tests of the null hypothesis, μ = μ₀, versus an alternative of the form (μ ? μ₀)²= =σ²d² are compared for different forms of violation of the underlying normal assumption. 10000 samples were simulated for d=0.6; 1.0 and 1.6 (σ² = 1), four (α, β)-combinations and seven alternative distributions. The results show that for small d-values one test is robust for α and another for β. For large d all tests can be used.     

Inhibition of α-, β-, γ-, and δ-carbonic anhydrases from bacteria and diatoms with N′-aryl-N-hydroxy-ureas

The inhibition of α-, β-, γ-, and δ-class carbonic anhydrases (CAs, EC 4.2.1.1) from bacteria (Vibrio cholerae and Porphyromonas gingivalis) and diatoms (Thalassiosira weissflogii) with a panel of N’-aryl-N-hydroxy-ureas is reported. The α-/β-CAs from V. cholerae (VchCAα and VchCAβ) were effectively inhibited by some of these derivatives, with K_Is in the range of 97.5?nM – 7.26?µM and 52.5?nM – 1.81?µM, respectively, whereas the γ-class enzyme VchCAγ was less sensitive to inhibition (K_Is of 4.75 – 8.87?µM). The β-CA from the pathogenic bacterium Porphyromonas gingivalis (PgiCAβ) was not inhibited by these compounds (K_Is?>?10?µM) whereas the corresponding γ-class enzyme (PgiCAγ) was effectively inhibited (K_Is of 59.8?nM – 6.42?µM). The δ-CA from the diatom Thalassiosira weissflogii (TweCAδ) showed effective inhibition with these derivatives (K_Is of 33.3?nM – 8.74?µM). As most of these N-hydroxyureas are also ineffective as inhibitors of the human (h) widespread isoforms hCA I and II (K_Is?>?10?µM), this class of derivatives may lead to the development of CA inhibitors selective for bacterial/diatom enzymes over their human counterparts and thus to anti-infectives or agents with environmental applications.     

Hydrogen isotope fractionation during H2/CO2 acetogenesis: hydrogen utilization efficiency and the origin of lipid-bound hydrogen

Hydrogen metabolism was studied in the anaerobic bacterium, Sporomusa sp. strain DMG 58, by measuring natural abundance levels of deuterium in H₂, H₂O, and individual fatty acids during acetogenic growth on H₂/CO₂. Four cultures were grown, each in medium with a distinct hydrogen‐isotopic composition (δD‐H₂O). The δD value of H₂ was quantified in the residual gas exiting the growth chambers and found to decrease concurrently with net H₂ consumption, indicating rapid isotope exchange between H₂ and H₂O. An isotopic mass balance was used to constrain the efficiency with which H₂ was activated by the cell and the reducing equivalents catabolized, which we term the H₂ utilization efficiency. Results indicate that H₂ utilization efficiency in these cultures is less than 20% during the growth phase, and less than 2% after the growth phase. The gross rate of cellular H₂ activation was similar in the growth phase and afterward. Biomass harvested at the end of each experiment was used to analyse the D/H of individual membrane lipids. Values of δD were highly correlated between lipids and water (δD‐lipids = 0.59 × δD‐water – 381‰; R² = 0.995), indicating the source of lipid hydrogen is in isotopic equilibrium with water. Results are consistent with two possibilities: (i) water is the sole source of hydrogen to lipids, and the fractionation during biosynthesis is significantly larger than previously observed (α = 0.59), or (ii) hydrogen from H₂ is incorporated into lipids, but only after reaching isotopic equilibrium with H₂O. Fatty acids were strongly depleted in deuterium relative to all other organisms studied thus far, and such large depletions may prove useful as biomarkers for studying H₂ cycling in anoxic environments as well as in the geological record.     

Analysis of the ionization constants and heats of ionization of reduced and oxidized horse heart cytochrome c

Simultaneous curve fitting for the ionization parameters of oxidized and reduced horse heart cytochrome c in 0.15M KCl and 20°C yields values for the ionization constants (as pK′) and the heats of ionization (ΔH_i) which can reconstruct either the potentiometric or thermal titration curves. Reduced cytochrome c requires 8 sets of groups, whereas oxidized cytochrome c requires 10 sets of groups. The additional groups in the oxidized preparation appear to involve the ferriheme (pK′, 9.25; ΔH_i, 13.7 kcal/mol) and a tyrosine (pK′ ? 10.24) that is not present in the reduced form. The potentiometric and thermal difference curves (reduced – oxidized) involve the appearance of 17 kcal/mol centered at pH 9.7 and 5.8 kcal/mol centered at pH 4.9. The carboxyl groups in both species appear to be normal for the hydrogen-bonded form. Only one histidine has normal ionization properties (pK′, 6.7; ΔH_i, 7.5 kcal/mol), as do 17 of the lysine residues (pK′, 10.8; ΔH_i, 11.5 kcal/mol).     

Thermodynamic Parameters of Ligand-Receptor Interactions: Computation and Error Margins

Abstract

A semiempirical relationship describing the temperature function of ligand-receptor dissociation constants (K_d), derived from heat capacities of the system in equilibrium, is suggested for computation of the standard enthalpy (δH°) and standard entropy (δS°) changes in equilibrium. The use of the linear expression (called Gibbs-van't Hoff equation) may lead to inaccurate results when heat capacity C_p displays a considerable temperature dependence. The accuracy of K_d, δH° and δS° has been studied by simulation experiments. In the case of K_d, deviations of computed from “true” values are determined by both the accuracy of experimental data used for its estimation, and by the shape of the binding isotherm (for instance, by Hill coefficients or by the presence of low affinity sites). As a rule, if errors of bound ligand measurements are greater than 20 per cent, K_d estimates ought to be considered as less reliable. However, computations of δH° and δS° that use such K_d values, are more correct, probably due to an error compensation. The suggested nonlinear temperature function of K_d enables an estimate of the heat capacity of the system and its temperature dependence.     

Tables for Testing the Equality of Two Proportions when Prior Information on their Common Value May be Available

In testing the equality of two proportions, one may define a rejection point T? such that if the test statistic T (in this paper, the ordinary Pearsonian chi-squared) exceeds T? then the hypothesis may safely be rejected, whatever the common value p of the two proportions may be; and similarly define an acceptance point A? such that if T ≦ T? then one may safely accept. These points may be refined if prior information is available, for example that p must lie in the central interval (c, 1 – c) or one of the extremal intervals (v, w) and (1 – w, 1 – v). Smallsample tables are provided both for the unrestricted case and for situations where one has such prior information.     

On the Analysis and Synthesis of a Four-Field-Table

The real four-field-table measure k is calculable as follows: K₁ for V(a, b, c, d) with k₁ = (arc K₁)/100, K₁₁ for V(a, b, 0, 0) with k₁₁ = (arc K₁₁)/100, K₁₂ for V(c, d, 0, 0) with k₁₂ = (arc K₁₂)/100 and K₁₂₁ for V(0, 0, c, d) with k₁₂₁ = – k₁₂. The equation k₁ = k₁₁ – k₁₂ holds good. It is possible to calculate the probability of error of a four-field-table with small frequencies, indirectly from component values. Two examples are given.     

Asparagine and glutamine differ in their propensities to form specific side chain-backbone hydrogen bonded motifs in proteins

Short range side chain‐backbone hydrogen bonded motifs involving Asn and Gln residues have been identified from a data set of 1370 protein crystal structures (resolution ≤ 1.5 Å). Hydrogen bonds involving residues i ? 5 to i + 5 have been considered. Out of 12,901 Asn residues, 3403 residues (26.4%) participate in such interactions, while out of 10,934 Gln residues, 1780 Gln residues (16.3%) are involved in these motifs. Hydrogen bonded ring sizes (C_n, where n is the number of atoms involved), directionality and internal torsion angles are used to classify motifs. The occurrence of the various motifs in the contexts of protein structure is illustrated. Distinct differences are established between the nature of motifs formed by Asn and Gln residues. For Asn, the most highly populated motifs are the C₁₀ (CO^δ_i …NH_{i + 2}), C₁₃ (CO^δ_i …NH_{i + 3}) and C₁₇ (N^δH_i …CO_{i ? 4}) structures. In contrast, Gln predominantly forms C₁₆ (CO^ε_i …NH_{i ? 3}), C₁₂ (N^εH_i …CO_{i ? 2}), C₁₅ (N^εH_i …CO_{i ? 3}) and C₁₈ (N^εH_i …CO_{i ? 4}) motifs, with only the C₁₈motif being analogous to the Asn C₁₇structure. Specific conformational types are established for the Asn containing motifs, which mimic backbone β‐turns and α‐turns. Histidine residues are shown to serve as a mimic for Asn residues in side chain‐backbone hydrogen bonded ring motifs. Illustrative examples from protein structures are considered. Proteins 2012; © 2011 Wiley Periodicals, Inc.     

Mesophyll CO2 conductance and leakiness are not responsive to short‐ and long‐term soil water limitations in the C4 plant Sorghum bicolor

Breeding economically important C₄ crops for enhanced whole‐plant water‐use efficiency (WUE_plant) is needed for sustainable agriculture. WUE_plant is a complex trait and an efficient phenotyping method that reports on components of WUE_plant, such as intrinsic water‐use efficiency (WUE_i, the rate of leaf CO₂ assimilation relative to water loss via stomatal conductance), is needed. In C₄ plants, theoretical models suggest that leaf carbon isotope composition (δ¹³C), when the efficiency of the CO₂‐concentrating mechanism (leakiness, ?) remains constant, can be used to screen for WUE_i. The limited information about how ? responds to water limitations confines the application of δ¹³C for WUE_i screening of C₄ crops. The current research aimed to test the response of ? to short‐ or long‐term moderate water limitations, and the relationship of δ¹³C with WUE_i and WUE_plant, by addressing potential mesophyll CO₂ conductance (g_m) and biochemical limitations in the C₄ plant Sorghum bicolor. We demonstrate that g_m and ? are not responsive to short‐ or long‐term water limitations. Additionally, δ¹³C was not correlated with gas‐exchange estimates of WUE_i under short‐ and long‐term water limitations, but showed a significant negative relationship with WUE_plant. The observed association between the δ¹³C and WUE_plant suggests an intrinsic link of δ¹³C with WUE_i in this C₄ plant, and can potentially be used as a screening tool for WUE_plant in sorghum.     

Synthesis and activity of di- or trisubstituted N-(phenoxyalkyl)- or N-{2-[2-(phenoxy)ethoxy]ethyl}piperazine derivatives on the central nervous system

Aim of the study was evaluation of anxiolytic, antidepressant, anticonvulsant and analgesic activity in a series of a consistent group of compounds. A series of eleven new N-(phenoxyalkyl)- or N-{2-[2-(phenoxy)ethoxy]ethyl}piperazine derivatives has been obtained. Their affinity towards 5-HT_1A, 5-HT_2A, 5-HT₆, 5-HT₇, D₂ and α₁ receptors has been assessed, and then functional assays were performed. The compounds were evaluated in mice, i.p. for their antidepressant-like (forced swim test), locomotor, anxiolytic-like (four-plate test) activities as well as – at higher doses – for anticonvulsant potential (MES) and neurotoxicity (rotarod). Two compounds (3, 6) were also evaluated for their analgesic activity in neuropathic pain models (streptozocin test, oxaliplatin test) and they were found active against allodynia in diabetic neuropathic pain at 30?mg/kg. Among the compounds, anxiolytic-like, anticonvulsant or analgesic activity was observed but antidepressant-like activity was not. One of the two most interesting compounds is 1-{2-[2-(2,4,6-trimethylphenoxy)ethoxy]ethyl}-4-(2-methoxyphenyl)piperazine dihydrochloride (9), exhibiting anxiolytic and anticonvulsant activity in mice, i.p. 30 min after administration (at 2.5?mg/kg and ED₅₀?=?26.33?mg/kg, respectively), which can be justified by the receptor profile: 5-HT_1A K_i?=?5?nM (antagonist), 5-HT₇ K_i?=?70?nM, α₁ K_i?=?15?nM, D₂ K_i?=?189?nM (antagonist). Another interesting compound is 1-[3-(2,4,6-trimethylphenoxy)propyl]-4-(4-methoxyphenyl)piperazine dihydrochloride (3), exhibiting anxiolytic, anticonvulsant and antiallodynic activity in mice, i.p., 30?min after administration (at 10?mg/kg, ED₅₀?=?23.50?mg/kg, at 30?mg/kg, respectively), which can be related with 5-HT_1A weak antagonism (K_i?=?146?nM), or other possible mechanism of action, not evaluated within presented study. Additionally, for the most active compound in the four-plate test (7), molecular modeling was performed (docking to receptors 5-HT_1A, 5-HT_2A, 5-HT₇, D₂ and α_1A).     

A New Approach to Equivalence Assessment in Standard Comparative Bioavailability Trials by Means of the Mann-Whitney Statistic

By a suitable transformation of the pairs of observations obtained in the successive periods of the trial, bioequivalence assessment in a standard comparative bioavailability study reduces to testing for equivalence of two continuous distributions from which unrelated samples are available. Let the two distribution functions be given by F(x) = P[X ≤ x], G(y) = P[Y ≤ y] with (X, Y) denoting an independent pair of real-valued random variables. An intuitively appealing way of putting the notion of equivalence of F and G into nonparametric terms can be based on the distance of the functional P[X > Y] from the value it takes if F and G coincide. This leads to the problem of testing the null hypothesis H_o P[X > Y] ≤ 1/2 - ε₁ or P[X > Y] ≥ 1/2 + ε₂ versus H₁ : 1/2 ? ε₁ < P[X > Y] < 1/2 + ∈₂, with sufficiently small ε₁, ε₂ ∈ (0, 1/2). The testing procedure we derive for (₀, H₁) and propose to term Mann-Whitney test for equivalence, consists of carrying out in terms of the U-statistics estimator of P[X > Y] the uniformly most powerful level a test for an interval hypothesis about the mean of a Gaussian distribution with fixed variance. The test is shown to be asymptotically distribution-free with respect to the significance level. In addition, results of an extensive simulation study are presented which suggest that the new test controls the level even with sample sizes as small as 10. For normally distributed data, the loss in power as against the optimal parametric procedure is found to be almost as small as in comparisons between the Mann-Whitney and the t-statistic in the conventional one or two-sided setting, provided the power of the parametric test does not fall short of 80%.     

Calcium elevation-dependent and attenuated resting calcium-dependent abscisic acid induction of stomatal closure and abscisic acid-induced enhancement of calcium sensitivities of S-type anion and inward-rectifying K+ channels in Arabidopsis guard cells

Stomatal closure in response to abscisic acid depends on mechanisms that are mediated by intracellular [Ca²⁺] ([Ca²⁺]_i), and also on mechanisms that are independent of [Ca²⁺]_i in guard cells. In this study, we addressed three important questions with respect to these two predicted pathways in Arabidopsis thaliana. (i) How large is the relative abscisic acid (ABA)‐induced stomatal closure response in the [Ca²⁺]_i‐elevation‐independent pathway? (ii) How do ABA‐insensitive mutants affect the [Ca²⁺]_i‐elevation‐independent pathway? (iii) Does ABA enhance (prime) the Ca²⁺ sensitivity of anion and inward‐rectifying K⁺ channel regulation? We monitored stomatal responses to ABA while experimentally inhibiting [Ca²⁺]_i elevations and clamping [Ca²⁺]_i to resting levels. The absence of [Ca²⁺]_i elevations was confirmed by ratiometric [Ca²⁺]_i imaging experiments. ABA‐induced stomatal closure in the absence of [Ca²⁺]_i elevations above the physiological resting [Ca²⁺]_i showed only approximately 30% of the normal stomatal closure response, and was greatly slowed compared to the response in the presence of [Ca²⁺]_i elevations. The ABA‐insensitive mutants ost1‐2, abi2‐1 and gca2 showed partial stomatal closure responses that correlate with [Ca²⁺]_i‐dependent ABA signaling. Interestingly, patch‐clamp experiments showed that exposure of guard cells to ABA greatly enhances the ability of cytosolic Ca²⁺ to activate S‐type anion channels and down‐regulate inward‐rectifying K⁺ channels, providing strong evidence for a Ca²⁺ sensitivity priming hypothesis. The present study demonstrates and quantifies an attenuated and slowed ABA response when [Ca²⁺]_i elevations are directly inhibited in guard cells. A minimal model is discussed, in which ABA enhances (primes) the [Ca²⁺]_i sensitivity of stomatal closure mechanisms.     

The role of 5-arylalkylamino- and 5-piperazino- moieties on the 7-aminopyrazolo[4,3-d]pyrimidine core in affecting adenosine A1 and A2A receptor affinity and selectivity profiles

New 7-amino-2-phenylpyrazolo[4,3-d]pyrimidine derivatives, substituted at the 5-position with aryl(alkyl)amino- and 4-substituted-piperazin-1-yl- moieties, were synthesized with the aim of targeting human (h) adenosine A₁ and/or A_2A receptor subtypes. On the whole, the novel derivatives 1–24 shared scarce or no affinities for the off-target hA_2B and hA₃ ARs. The 5-(4-hydroxyphenethylamino)- derivative 12 showed both good affinity (K_i =?150?nM) and the best selectivity for the hA_2A AR while the 5-benzylamino-substituted 5 displayed the best combined hA_2A (K_i =?123?nM) and A₁ AR affinity (K_i =?25?nM). The 5-phenethylamino moiety (compound 6) achieved nanomolar affinity (K_i =?11?nM) and good selectivity for the hA₁ AR. The 5-(N⁴-substituted-piperazin-1-yl) derivatives 15–24 bind the hA₁ AR subtype with affinities falling in the high nanomolar range. A structure-based molecular modeling study was conducted to rationalize the experimental binding data from a molecular point of view using both molecular docking studies and Interaction Energy Fingerprints (IEFs) analysis.     

Competitive inhibition by substrates of the esterification reaction between l-phenylalanine and d-glucose catalysed by the lipases of Rhizomucor miehei and Candida rugosa

A kinetic study on esterification between d-glucose and l-phenylalanine catalysed by lipases from Rhizomucor miehei (RML) and Candida rugosa (CRL) in organic media investigated in detail showed that both the lipases followed a Ping-Pong Bi-Bi mechanism with two distinct types of competitive inhibitions. Graphical double reciprocal plots and computer simulation studies showed that competitive double substrate inhibition took place at higher concentrations leading to dead-end inhibition in the case of RML and in the case of CRL, inhibition only by d-glucose at higher concentrations leading to dead-end lipase–d-glucose complexes. An attempt to obtain the best fit of these kinetic models through curve-fitting yielded in good approximation, the apparent values of important kinetic parameters, RML: k _cat = 2.24 ± 0.23 mM h⁻¹ (mg protein)⁻¹, K _{m l-phenylalanine} = 95.6 ± 9.7 mM, K _{m d-glucose} = 80.0 ± 8.5 mM, K _{i l-phenylalanine} = 90.0 ± 9.2 mM, K _{i d-glucose} = 13.6 ± 1.42 mM; CRL: k _cat = 0.51 ± 0.06 mM h⁻¹ (mg protein)⁻¹, K _{m l-phenylalanine} = 10.0 ± 0.98 mM, K _{m d-glucose} = 6.0 ± 0.64 mM, K _{i d-glucose} = 8.5 ± 0.81 mM.     

Binding of the Recombinant Proteinase Inhibitor Eglin c,of the Soybean Bowman-Birk Proteinase Inhibitor and of its Chymotrypsin and Trypsin Inhibiting Fragments to Leu-Proteinase,the Leucine Specific Serine Proteinase from Spinach (Spinacia oleracea L.) Leaves: Thermodynamic Study

Abstract

The effect of pH and temperature on the apparent association equilibrium constant (K_a) for the binding of the recombinant proteinase inhibitor eglin c (eglin c), of the soybean Bowman-Birk proteinase inhibitor (BBI) and of its chymotrypsin and trypsin inhibiting fragments (F-C and F-T, respetively) to Leuproteinase, the leucine specific serine proteinase from spinach (Spinacia oleracea L.) leaves, has been investigated. On lowering the pH from 9.5 to 4.5, values of K_a (at 21°C) for complex formation decrease thus reflecting the acidic pK-shift of the hystidyl catalytic residue from ～6.9, in the free Leu-proteinase, to ～5.1, in the enzyme: inhibitor adducts. At pH 8.0, values of the apparent thermodynamic parameters for the proteinase:inhibitor complex formation are: Leu-proteinase:eglin c - K_a = 2.2 × 10¹¹ M^-1, δG°= - 64kJ/mol, δH° = + 5.9kJ/mol, and δS° = + 240J/molK; Leu-proteinase:BBI - K_a = 3.2 × 10¹⁰ M^-1, δG° = - 59kJ/mol, δH°= + 8.8kJ/mol, and δS° = + 230J/molK; and Leu-proteinase:F-C - K_a = 1.1 × 10⁶ M^-1, δG°= - 34kJ/mol, δH° = + 18J/mol, and δS° = + 180J/molK (values of K_a, δG° and δS° were obtained at 21.0°C; values of δH° were temperature-independent over the range explored, i.e. between 10.0°C and 40.0°C). F-T does not inhibit Leu-proteinase up to an inhibitor concentration of 1.0 × 10^-3 M, suggesting that the upper limit of K_a is 1 × 10² M^-1. Considering the known molecular models, the observed binding behaviour of eglin c, BBI, F-C and F-T to Leu-proteinase has been related to the inferred stereochemistry of the enzyme/inhibitor contact region