The hyaluronan–protein complexes at low ionic strength: How the hyaluronidase activity is controlled by the bovine serum albumin

Hyaluronan (HA) hydrolysis catalysed by hyaluronidase (HAase) is strongly inhibited when performed at low HAase over HA concentration ratio and under low ionic strength conditions. The reason is the ability of long HA chains to form electrostatic and non-catalytic complexes with HAase. For a given HA concentration, low HAase concentrations lead to very low hydrolysis rates because all the HAase molecules are sequestered by HA, whilst high HAase concentrations lead to high hydrolysis rates because the excess of HAase molecules remains free and active. At pH 4, non-catalytic proteins like bovine serum albumin (BSA) are able to compete with HAase to form electrostatic complexes with HA, liberating HAase which recovers its catalytic activity. The general scheme for the BSA-dependency is thus characterised by four domains delimited by three noticeable points corresponding to constant BSA over HA concentration ratios. The existence of HA–protein complexes explains the atypical kinetic behaviour of the HA / HAase system. We also show that HAase recovers the Michaelis–Menten type behaviour when the HA molecule complexed with BSA in a constant complexion state, i.e. with the same BSA over HA ratio, is considered for substrate. When the ternary HA / HAase / BSA system is concerned, the stoichiometries of the HA–HAase and HA–BSA complexes are close to 10 protein molecules per HA molecule for a native HA of 1 MDa molar mass. Finally, we show that the behaviour of the system is similar at pH 5.25, although the efficiency of BSA is less.     

Electrostatic interactions between hyaluronan and proteins at pH 4: how do they modulate hyaluronidase activity

Hyaluronan (HA) hydrolysis catalyzed by hyaluronidase (HAase) is inhibited at low HAase over HA ratio and low ionic strength, because HA forms electrostatic complexes with HAase, which is unable to catalyze hydrolysis. Bovine serum albumin (BSA) was used as a model to study the HA-protein electrostatic complexes at pH 4. At low ionic strength, there is formation of (i) neutral insoluble complexes at the phase separation and (ii) small positively-charged or large negatively-charged soluble complexes whether BSA or HA is in excess. According to the ionic strength, different types of complex are formed. Assays for HA and BSA led to the determination of the stoichiometry of these complexes. HAase was also shown to form the various types of complex with HA at low ionic strength. Finally, we showed that at 0 and 150 mmol L(-1) NaCl, BSA competes with HAase in forming complexes with HA and thus induces HAase release resulting in a large increase in the hydrolysis rate. These results, in addition to data in the literature, show that HA-protein complexes, which can exist under numerous and varied conditions of pH, ionic strength and protein over HA ratio, might control the in vivo HAase activity.     

How hyaluronan–protein complexes modulate the hyaluronidase activity: The model

Hyaluronan (HA) is the substrate of hyaluronidase (HAase). In addition, HA is able to form electrostatic complexes with many proteins, including HAase. Experiments have shown the strong inhibition of the HA hydrolysis catalyzed by HAase when performed at low HAase over HA concentration ratio and under low ionic strength conditions. Non-catalytic P proteins are able to compete with HAase to form electrostatic complexes with HA and thus to modulate HAase activity. We have modeled the HA–HAase–P system by considering the competition between the two complex equilibria HA–P and HA–HAase, the Michaelis–Menten type behavior of HAase, and the non-activity of the electrostatically complexed HAase. Simulations performed by introducing experimental data produce a theoretical behavior similar to the experimental one, including all the atypical phenomena observed: substrate-dependence, enzyme-dependence and protein-dependence of HAase. This shows that our assumptions are sufficient to explain the behavior of the system and allow us to estimate unknown parameters and suggest new developments.     

Hyaluronidase activity is modulated by complexing with various polyelectrolytes including hyaluronan.

Hyaluronidase (HAase) plays an important role in the control of the size and concentration of hyaluronan (HA) chains, whose biological properties strongly depend on their length. Our previous studies of HA hydrolysis catalyzed by testicular HAase demonstrated that, whilst the substrate-dependence curve has a Michaelis-Menten shape with a 0.15 mol L(-1) ionic strength, at low ionic strength (5 mmol L(-1)), a strong decrease in the initial hydrolysis rate is observed at high substrate concentrations; the HA concentration for which the initial rate is maximum increases when the HAase concentration is increased. After examination of various hypotheses, we suggested that this could be explained by the ability of HA to form non-specific complexes with HAase, which thus becomes unable to catalyze HA hydrolysis. In order to verify this hypothesis, we first showed from turbidimetric measurements that HAase, like albumin, is able to form electrostatic complexes with HA. Albumin then was used as a non-catalytic protein able to compete with HAase for the formation of non-specific complexes with HA, allowing HAase to be free and catalytically active. The kinetic results showed that the HA-HAase non-specific complex inhibits HAase catalytic activity towards HA. Depending on the albumin concentration with respect to the HAase and HA concentrations, albumin can either remove this inhibition or induce another type of inhibition. Finally, the extent of such non-specific interactions between polyelectrolytes and proteins in HAase inhibition or activation, in particular under in vivo conditions, is discussed.     

Temporal and spatial analysis of hyaluronidase activity during development of the embryonic chick limb bud

The glycosaminoglycan hyaluronate (HA) appears to play an important role in limb cartilage differentiation. The large amount of extracellular HA accumulated by prechondrogenic mesenchymal cells may prevent the cell-cell and/or cell-matrix interactions necessary to trigger chondrogenesis, and the removal of extracellular HA may be essential to initiate the crucial cellular condensation process that triggers cartilage differentiation. It has generally been assumed that HA turnover during chondrogenesis is controlled by the activity of the enzyme hyaluronidase (HAase). In the present study we have performed a temporal and spatial analysis of HAase activity during the progression of limb development and cartilage differentiation in vivo. We have separated embryonic chick wing buds at several stages of development into well-defined regions along the proximodistal axis in which cells are in different phases of differentiation, and we have examined HAase activity in each region. We have found that HAase activity is clearly detectable in undifferentiated wing buds at stage 18/19, which is shortly following the formation of a morphologically distinct limb bud rudiment, and remains relatively constant throughout subsequent stages of development through stage 27/28, at which time well-differentiated cartilage rudiments are present. Moreover, HAase activity in the prechondrogenic distal subridge regions of the limb at stages 22/23 and 25 is just as high as, or even slightly higher than, it is in proximal central core regions where condensation and cartilage differentiation are progressing. We have also found that limb bud HAase is active between pH 2.2 and 4.5 and is inactive above pH 5.0. This suggests that limb HAase is a lysosomal enzyme and that extracellular HA would have to be internalized to be degraded. These results indicate that the onset of chondrogenesis is not associated with the appearance or increase in activity of HAase. We suggest that possibility that HA turnover may be regulated by the binding and endocytosis of extracellular HA in preparation for its intracellular degradation by lysosomal HAase. Finally, we have found that the apical ectodermal ridge (AER)-containing distal limb bud ectoderm possesses a relatively high HAase activity. We suggest the possibility that a high HAase activity in the AER may ensure a rapid turnover and remodeling of the disorganized HA-rich basal lamina of the AER that might be essential for limb outgrowth.     

A dual‐readout nanosensor based on biomass‐based C‐dots and chitosan@AuNPs with hyaluronic acid for determination of hyaluronidase

A dual‐signal strategy is proposed based on fluorescent biomass‐based carbon dots (BC‐dots) and chitosan stabilized AuNPs (CS@AuNPs) to determine hyaluronidase (HAase). BC‐dots can induce aggregation of CS@AuNPs nanoparticles with a colour change from red to blue. Positively charged CS@AuNPs interacted with the negatively charged hyaluronic acid (HA) through electrostatic adsorption, and CS@AuNPs maintained stability due to the semirigid coil conformation of HA. However, in the presence of HAase, due to enzymatic hydrolysis of HA by HAase, the CS@AuNPs agglomerated. Based on the change of fluorescence and colour, quantitative analysis of HAase was achieved. Linear ranges for the fluorometric and colorimetric determinations were 2.0–70 U mL^?1 and 8–60 U mL^?1, respectively, with a detection limit of 0.27 U mL^?1. This dual‐signal sensing system possesses high potential for determination of HAase in biological matrices.     

Further studies on the contribution of electrostatic and hydrophobic interactions to protein adsorption on dye-ligand adsorbents.

The adsorption equilibria of bovine serum albumin (BSA), gamma-globulin, and lysozyme to three kinds of Cibacron blue 3GA (CB)-modified agarose gels, 6% agarose gel-coated steel heads (6AS), Sepharose CL-6B, and a home-made 4% agarose gel (4AB), were studied. We show that ionic strength has irregular effects on BSA adsorption to the CB-modified affinity gels by affecting the interactions between the negatively charged protein and CB as well as CB and the support matrix. At low salt concentrations, the increase in ionic strength decreases the electrostatic repulsion between negatively charged BSA and the negatively charged gel surfaces, thus resulting in the increase of BSA adsorption. This tendency depends on the pore size of the solid matrix, CB coupling density, and the net negative charges of proteins (or aqueous - phase pH value). Sepharose gel has larger average pore size, so the electrostatic repulsion-effected protein exclusion from the small gel pores is observed only for the affinity adsorbent with high CB coupling density (15.4 micromol/mL) at very low ionic strength (NaCl concentration below 0.05 M in 10 mM Tris-HCl buffer, pH 7.5). However, because CB-6AS and CB-4AB have a smaller pore size, the electrostatic exclusion effect can be found at NaCl concentrations of up to 0.2 M. The electrostatic exclusion effect is even found for CB-6AS with a CB density as low as 2.38 micromol/mL. Moreover, the electrostatic exclusion effect decreases with decreasing aqueous-phase pH due to the decrease of the net negative charges of the protein. For gamma-globulin and lysozyme with higher isoelectric points than BSA, the electrostatic exclusion effect is not observed. At higher ionic strength, protein adsorption to the CB-modified adsorbents decreases with increasing ionic strength. It is concluded that the hydrophobic interaction between CB molecules and the support matrix increases with increasing ionic strength, leading to the decrease of ligand density accessible to proteins, and then the decrease of protein adsorption. Thus, due to the hybrid effect of electrostatic and hydrophobic interactions, in most cases studied there exists a salt concentration to maximize BSA adsorption.     

Adsorption of BSA on strongly basic chitosan: Equilibria

Equilibrium isotherms for adsorption of bovine serum albumin (BSA) on a new adsorbent, a strongly basic crosslinked chitosan (Chitopearl 2503), which is hard and is not compressed by pressure in a column, have been presented and compared with diethylaminoethyl (DEAE) Sepharose Fast Flow (hard gel). In Chitopearl 2503, when only buffer existed in the BSA solution, the isotherm was not affected by the initial concentration of BSA but it was affected by pH considerably. The isotherm was favorable when pH >/= pl ( congruent with 4.8). When NaCl existed in the BSA solution, the amount of BSA absorbed on the resin decreased with increasing concentration of NaCl. When the concentration of NaCl was 200 mol/m(3), the resin did not adsorb BSA at all. The equilibrium data were correlated by the Langmuir equation reasonably well. The BSA may be adsorbed mainly by electrostatic attraction between negatively charged BSA and positively charged quanternary ammonium groups at pH > pl and by protonation reaction of the primary ammonium groups by weak acid groups of BSA at pH = pl. These are confirmed by measuring the amount of inorganic ion exchanged for BSA. In DEAE Sepharose Fast Flow, the isotherm was favorable when pH > pl but unfavorable ar pH = pl. The saturation capacity of BSA on Chitopearl 2503 is about 1.3 to 2.2 times larger than that on DEAE Sepharose Fast Flow. (c) 1994 John Wiley & Sons, Inc.     

Protein partitioning in aqueous two-phase systems composed of a pH-responsive copolymer and poly(ethylene glycol)

The effect of pH and salt concentration on the partitioning behavior of bovine serum albumin (BSA) and cytochrome c in an aqueous two-phase polymer system containing a novel pH-responsive copolymer that mimics the structure of proteins and poly(ethylene glycol) (PEG) was investigated. The two-phase system has low viscosity. Depending on pH and salt concentration, the cytochrome c was found to preferentially partition into the pH-responsive copolymer-rich (bottom) phase under all conditions of pH and salt concentrations considered in the study. This was caused by the attraction between the positively charged protein and negatively charged copolymer. BSA partitioning showed a more complex behavior and partitioned either to the PEG phase or copolymer phase depending on the pH and ionic strength. Extremely high partitioning levels (partition coefficient of 0.004) and very high separation ratios of the two proteins (up to 48) were recorded in the new systems. This was attributed to strong electrostatic interactions between the proteins and the charged copolymer.     

Purification of the basic protein avidin using Gradiflow technology

The Gradiflow, a preparative electrophoresis instrument, which separates proteins on the basis of charge or size, was used to purify the basic protein avidin, pI 10, from chicken egg white. Using a charge based separation at pH 9.0, the high pI of avidin and lysozyme (pI 10.7) allows them to be easily separated from remaining egg white proteins, as these are the only positively charged proteins. In a second step at pH 10.2, the negatively charged avidin is separated from the positively charged lysozyme. This sequential two-step protocol was complete within 4.5h. Enzyme immunoassay of avidin fractions obtained indicated recoveries of 60-65% from one egg white with minimal lysozyme activity detected.     

Electrotransfer of basic proteins from nondenaturing polyacrylamide acid gels to nitrocellulose: detection of enzymatic and inhibitory activities and retention of protein antigenicity.

We have developed a method for electrotransfer of strongly basic proteins (lysozyme, pI 11; mucus proteinase inhibitor, pI greater than 10; bovine pancreas trypsin inhibitor; pI 10.5; human leukocyte elastase, pI greater than 9) from nondenaturing acid gels (pH 4.5) to nitrocellulose sheets. Buffers were those used in a discontinuous system for transfer from sodium dodecyl sulfate (SDS)-containing polyacrylamide gels with one modification in the cathode buffer which contained 0.1% SDS. This method was compared to electrotransfer performed in 0.7% acetic acid. The basic proteins studied, which were positively charged in the gel, formed with SDS negative complexes which migrated toward the anode and were efficiently transferred to the nitrocellulose. Moreover, their biological properties were preserved: inhibitory activity, enzyme activity, and antigenicity. This method is advantageous because it is simple, is sensitive, and can be applied to various biological fluids to detect inhibitors, enzymes, and other proteins which have a basic character, after electrophoretic separation under their native forms.     

Interaction between acidic polysaccharides and proteins

There was an ionic interaction between acidic polysaccharides (APS) and proteins at the pH range in which APS were negatively charged and proteins were positively charged, and in enzymes the interaction was detected as a change in the enzyme activity. At pH 4.7, acid phosphatase (pI, 5.4), alpha-glucosidase (pI, 5.7), and beta-glucosidase (pI, 7.3) were inhibited by APS to various extents. On the other hand, alpha-glucosidase and alkaline phosphatase (pI, 4.5) were not inhibited by APS at pH 6.8 and 9.8, respectively, most of these two enzymes being negatively charged at the respective pHs. Sulfated polysaccharides combined with hemoglobin (pI, 6.8 to approximately 7.0) by an ionic bond at pH 2 to make hemoglobin unsusceptible to proteolysis by pepsin, but polyuronides which were not charged at this pH did not affect hydrolysis of hemoglobin.     

Formation of Multilayers at the Interface of Oil-in-Water Emulsion via Interactions between Lactoferrin and β-Lactoglobulin

The interactions between negatively charged β-lactoglobulin and the positively charged lactoferrin at the droplet surface to form a multi-protein surface layer were examined. Addition of lactoferrin to the aqueous phase of emulsions formed with β-lactoglobulin at pH 7.0 caused an increase in the ζ-potential of emulsion droplets, and the ζ-potential became positive as the concentration of added lactoferrin was higher than 1% in the system. It is found that lactoferrin binds to adsorbed β-lactoglobulin at droplet surface probably via electrostatic interactions. The amount of lactoferrin at interface increased with increasing the concentration of added lactoferrin, but it decreased with a decrease in the pH. No lactoferrin was observed at interface at pH 3 and 4. By contrast, when β-lactoglobulin was added in the emulsions formed with lactoferrin at pH 7.0, the ζ-potential of emulsions changed from positive to negative as the concentration of added β-lactoglobulin increased. The amount of β-lactoglobulin at surface increased correspondingly with increasing the concentration of added β-lactoglobulin. However, in this case, β-lactoglobulin remained bound at interface even at pH 3 and 4 where both lactoferrin and β-lactoglobulin are positively charged. The association of lactoferrin or β-lactoglobulin with the surface proteins that have oppositely charge is probably mainly through electrostatic interactions between the two proteins. It appears that alternative layers of these proteins could be created at the droplet surface.     

Emulsion formed in bovine serum album/anionic surfactant/H2O system under acidic condition

At neutral pH, bovine serum album (BSA) conformation is "heart-shaped", and with decreasing pH, BSA may adopt the fast "F" form below a pH of about 4, and expanded "E" form at pH lower than about 3. However, as far as we know, the researches on the interaction between protein and surfactant are all carried out at pH higher than its isoelectric point (pI, which is 4.9), which means only the information about how the "heart-shaped" BSA interacts with surfactant is understood so far. In this paper, we studied the interaction between BSA and anionic surfactant at pH lower than its pI and hope to help understand the role of protein conformation in its interaction with surfactant. We found that BSA and anionic surfactant could form emulsion only when pH value was lower than or about 3, suggesting that the "E" formed BSA was an important criterion for emulsion formation. Moreover, the emulsion formation was companied with increased α-helix content and decreased β-sheet content for BSA. In addition, the emulsion formation was closely dependent on the anionic surfactant content; only a moderate anionic surfactant could make emulsion formed.     

Electrostatic selectivity in protein-nanoparticle interactions

The binding of bovine serum albumin (BSA) and β-lactoglobulin (BLG) to TTMA (a cationic gold nanoparticle coupled to 3,6,9,12-tetraoxatricosan-1-aminium, 23-mercapto-N,N,N-trimethyl) was studied by high-resolution turbidimetry (to observe a critical pH for binding), dynamic light scattering (to monitor particle growth), and isothermal titration calorimetry (to measure binding energetics), all as a function of pH and ionic strength. Distinctively higher affinities observed for BLG versus BSA, despite the lower pI of the latter, were explained in terms of their different charge anisotropies, namely, the negative charge patch of BLG. To confirm this effect, we studied two isoforms of BLG that differ in only two amino acids. Significantly stronger binding to BLGA could be attributed to the presence of the additional aspartates in the negative charge domain for the BLG dimer, best portrayed in DelPhi. This selectivity decreases at low ionic strength, at which both isoforms bind well below pI. Selectivity increases with ionic strength for BLG versus BSA, which binds above pI. This result points to the diminished role of long-range repulsions for binding above pI. Dynamic light scattering reveals a tendency for higher-order aggregation for TTMA-BSA at pH above the pI of BSA, due to its ability to bridge nanoparticles. In contrast, soluble BLG-TTMA complexes were stable over a range of pH because the charge anisotropy of this protein at makes it unable to bridge nanoparticles. Finally, isothermal titration calorimetry shows endoenthalpic binding for all proteins: the higher affinity of TTMA for BLGA versus BLGB comes from a difference in the dominant entropy term.     

Unusual secondary specificity of prolyl oligopeptidase and the different reactivities of its two forms toward charged substrates.

Prolyl oligopeptidase belongs to a new family of serine proteases which contains both exo- and endopeptidases, and this suggests that the enzyme binds its substrate in a special manner. Its secondary specificity, i.e., its interaction with the other residues linked to the proline that accounts for the primary specificity, has been investigated by using peptide substrates of various length and charge. Elongation of the classic dipeptide substrate Z-Gly-Pro-2-naphthylamide with 1-3 residues (Gln, Ala-Gln, Ala-Ala-Gln, and Ala-Lys-Gln) resulted in decreased specificity rate constants. This indicated a limited binding site for prolyl oligopeptidase, a major difference from the finding with other serine endopeptidases. Insertion of charged residues into the substrates, such as lysine or aspartic acid, considerably affected the rates and the pH-rate profiles. The rate constants were higher with the positively charged peptides and lower with the substrates bearing a negative charge. These electrostatic effects were reduced at high ionic strength. The results can be interpreted in terms of a negatively charged active site, which exists at high pH and exerts electrostatic attraction or repulsion toward charged substrates. The pH dependencies of the rate constants with neutral substrates exhibited roughly bell-shaped curves, whereas with charged substrates the existence of two active enzyme forms was clearly demonstrated. The physiologically competent high pH form preferred positively charged substrates (Z-Lys-Pro-2-(4-methoxy)naphthylamide, Z-Ala-Lys-Gln-Gly-Pro-2-naphthylamide), whereas the low pH form reacted faster with the negatively charged substrate (Z-Asp-Gly-Pro-2-naphthylamide).(ABSTRACT TRUNCATED AT 250 WORDS)     

Selective adsorption of endotoxin inside a polycationic network of flat-sheet microfiltration membranes

For the removal of remaining amounts of endotoxin, sorbents with high selectivity for endotoxin are required. Typically, particulate sorbents with positively charged ligands, such as histidine, polymyxin B poly-L-lysine and poly(ethyleneimine) (PEI), display moderate to high removal efficiencies in an environment of low ionic strength. It was found that polycationic ligands are most suitable to meet an endotoxin concentration which is below the threshold level required for parenteralia. Furthermore, protein recoveries close to 100% are obtained if the decontamination is performed at a pH close to the pI of acidic proteins. The high selectivity is probably caused by complexation of the polycationic ligand with the polyanionic endotoxin, leading to interactions with KD < 10(-9) M using PEI and assuming M(r) = 10 kDa for monomeric endotoxin; with BSA the same ligand reveals only KD = 4 x 10(-6) M. Using polymer-coated microfiltration membranes, immobilization of positively charged ligands leads to membrane adsorbers which are generally superior to chromatographic adsorbers and allow faster processing. Since immobilization takes place at polymer chains, low-molecular-weight ligands mainly add positive charges to the hydrophilic polymer. Consequently, membrane adsorbers with low-molecular-weight ligands, even DEAE, demonstrate similar selectivity to PEI or poly-L-lysine.     

Electrostatic effect upon association of reduced nicotinamide adenine dinucleotide and equine liver alcohol dehydrogenase

The rate of association of equine liver alcohol dehydrogenase and its coenzymes exhibits a large pH dependence with slower rates at basic pH and an observed kinetic pKa value of approximately 9-9.5. This pH dependence has been explained by invoking local active site electrostatic effects which result in repulsion of the negatively charged coenzyme and the ionized hydroxyl anion form of the zinc-bound water molecule. We have examined a simpler hypothesis, namely, that the pH dependence results from the electrostatic interaction of the coenzyme and the enzyme which changes from an attractive interaction of the negatively charged coenzyme and the positively charged enzyme to a repulsive interaction between the two negatively charged species at the isoelectric point for the enzyme (pH 8.7). We have tested this proposal by examining the ionic strength dependence of the association rate constant at various pH values. These data have been interpreted by using the Wherland-Gray equation, which we have shown can be applied to the kinetics of enzyme-coenzyme association. Our results indicate that the shielding of the buffer electrolyte changes from a negative to a positive value as the charge on the protein changes at the isoelectric point. This result is exactly that which is predicted for electrostatic effects that depend on the charge of the protein molecule and is not consistent with predictions based upon the local active site effects. At low ionic strength values of 10 mM or less, approximately 75% of the observed pH dependence results from the enzyme electrostatic effects; the remaining pH dependence may result from active site effects.(ABSTRACT TRUNCATED AT 250 WORDS)     

Chain-length dependence of the kinetics of the hyaluronan hydrolysis catalyzed by bovine testicular hyaluronidase

Hyaluronan (HA) has various biological functions that are strongly dependent on its chain length. In some cases, as in inflammation and angiogenesis, long and short chain-size HA effects are antagonistic. HA hydrolysis catalyzed by hyaluronidase (HAase) is believed to be involved in the control of the balance between longer and shorter HA chains. Our studies of native HA hydrolysis catalyzed by bovine testicular HAase have suggested that the kinetic parameters depend on the chain size. We thus used HA fragments with a molar mass ranging from 8x10(2) g mol(-1) to 2.5x10(5) g mol(-1) and native HA to study the influence of the chain length of HA on the kinetics of its HAase-catalyzed hydrolysis. The initial hydrolysis rate strongly varied with HA chain length. According to the Km and Vm/Km values, the ability of HA chains to form an efficient enzyme-substrate complex is maximum for HA molar masses ranging from 3x10(3) to 2x10(4) g mol(-1). Shorter HA chains seem to be too short to form a stable complex and longer HA chains encounter difficulties in forming a complex, probably because of steric hindrance. The hydrolysis Vm values strongly suggest that as the chain length decreases the HAase increasingly catalyses transglycosylation rather than hydrolysis. Finally, two HA chain populations, corresponding to HA chain molar masses lower and higher than approximately 2x10(4) g mol(-1), are identified and related to the bi-exponential character of the model we have previously proposed to fit the experimental points of the kinetic curves.