Diffusional effects on the heterogeneous kinetics of two-substrate enzymic reactions.

When a two-substrate reaction is catalyzed by a surface bound enzyme, the diffusion of both substrates can considerably modify the kinetic properties of the reaction. According to this theoretical analysis, limitations in substrate diffusion yield very different effects depending whether the two substrates have similar or different affinities for the enzyme. With two substrates of comparable affinities the diffusion of the two substrates can be limiting, and similar activity dependences on the two substrate concentrations are obtained. Under these conditions, diffusional limitations may only slightly influence half-maximal-activity substrate concentrations. With two substrates of widely different affinities, on the contrary, the rate of the enzymic reaction can only be limited by the diffusion of the high-affinity substrate, used at the lower concentration. Under these conditions, in the presence of diffusional limitations the activity dependence on the two substrate concentrations are highly different, and the half-maximal-activity concentration is increased and decreased for the high- and low-affinity substrates, respectively. The theoretical results are verified by experimental data previously obtained with collagen-bound aspartate aminotransferase and sorbitol dehydrogenase.     

Diffusion-controlled protein–DNA association: Influence of segemental diffusion of the DNA

The intrachain reaction theory of Wilemki and Fixman [(1974) J. Chem. Phys. 60 , 866–877] is used to assess the influence of internal DNA motions on various protein–DNA association schemes. It is found that, for large proteins, the diffusional association rate can be totally dominated by these motions rather than by the free-trans-lational diffusion rates. Also, the time required for the diffusion together of two DNA segments is estimated. This estimate can be used to provide an upper limit for the rate of intrachain cyclization, and also for the effective intrachain transfer rate of a protein bound to a DNA chain.     

Kinetic studies of alpha-galactosidase-containing mold pellets on PNPG hydrolysis.

Little is known about techniques for applying untreated microbial cells containing enzymes directly to industrial processes as a biocatalyst. The kinetic behavior of alpha-galactosidase-containing spherical pellets which are formed naturally under given conditions in a submerged culture of Mortierella vinacea was studied on the hydrolysis of PNPG (p-nitrophenyl-alpha-D-galactopyranoside). The effect on intraparticle diffusion on the overall reaction rate was assessed by the use of an effectiveness factor, which was calculated by the approximate solution to the equation derived from the mass balance within a pellet. The experimental effectiveness factors were found to be represented as a single function of the modified Thiele modulus, including such parameters as pellet size, enzyme concentration in the pellet, and substrate concentration. As the diffusional effect became more significant, the marked substrate inhibition as seen for a free enzyme disappeared gradually. The effect of product inhibition on the pellets was much weaker than that for a free enzyme at a given substrate concentration. In the region of diffusion controlled reaction, it was found that the rate is proportional to the square root of the enzyme concentration in the pellet. In addition, similarly to what was reported previously for a free enzyme,the reaction in a batch system was found to be approximately representable as simple first-order kinetics in which the rate constant was dependent on the initial substrate concentration.     

Brownian dynamics simulations of molecular recognition in an antibody-antigen system.

The crystal structure for an antibody-antigen system, that of the anti-hen egg lysozyme monoclonal antibody HyHEL-5 complexed to lysozyme, is used as the starting point for computer simulations of diffusional encounters between the two proteins. The investigation consists of two parts: first, the linearized Poisson-Boltzmann equation is solved to determine the long-range electrostatic forces between antibody and antigen, and then, the relative motion as influenced by these forces is modeled within Brownian motion theory. The effects of various point mutations on the calculated reaction rate are considered. It is found that charged residues close to the binding site exert the greatest influence in steering the proteins into a configuration favorable for their binding, while more distant mutations are qualitatively described by the Smoluchowski model for the mutual diffusion of two uniformly charged spheres. The antibody residues involved in forming salt links with the lysozyme, Glu-H35 and Glu-H50, appear to be particularly important in electrostatic steering, as neutralization of both of them yields reaction rates that are two to three orders of magnitude below those of wild-type rates. The relative rates obtained from the simulations can be tested through kinetic measurements on mutant protein complexes. Kinetically efficient partners can also be designed and constructed through directed mutagenesis.     

Theoretical analysis of a translocation-like model with saturable kinetics.

A theoretical analysis of the initial rates of product appearance in both compartments of a specifically designed diffusion cell separated by an asymmetrical enzyme membrane is presented. Variable substrate concentrations and different substrate diffusional limitations were considered. Our analysis shows that, under specific conditions, not only a product accumulation occurs in the compartment opposite to that in which the reaction takes place, but that substrate saturable kinetics can be obtained. These product translocation-like kinetics appear similar to those observed with translocation processes reported for biological situations. For such phenomena, a key role of the diffusion layer surrounding a bioactive surface is proposed.     

Immobilized luciferase of Luciola mingrelica fireflies. The kinetic properties and thermostability of luciferase immobilized on cellulose films

Luciferase of fireflies Luciola mingrelica was immobilized on cellulose films activated by cyanuric chloride or sodium periodate. Kinetic properties and the contribution of diffusional obstacles to the kinetics of the immobilized enzyme were examined. External and internal diffusion were found to influence the kinetic parameters. The stability of the enzyme was investigated at 25 degrees C and pH 7.8. Thermoactivation of the immobilized enzyme was shown to proceed in two stages: fast and slow. Dithiotreitol and cystein stabilized the enzyme at the fast stage while salt supplements at both stages. The fast thermoinactivation stage was apparently associated with the oxidation of luciferase SH-groups. It is demonstrated that the immobilized enzyme of Luciola mingrelica can be employed to measure ATP traces with the detection limit 0.1 mM. The enzyme immobilized on cellulose films can be used repeatedly.     

Mass transfer studies on immobilized alpha-chymotrypsin biocatalysts prepared by deposition for use in organic medium

Mass transfer limitations were studied in enzyme preparations of alpha-chymotrypsin made by deposition on different porous support materials such as controlled pore glasses, Celite, and polyamides of different particle sizes. It is the onset of mass transfer limitations that determines the position of the activity optimum with respect to enzyme loading on each support. The evidence of various experiments indicates that internal diffusional limitations are the important mechanism for the observed mass transfer limitations. External diffusion was not found to play an important role under the conditions used, and it was also found that when immobilizing multilayers of enzyme the buried enzyme molecules are active to a large extent. An extreme situation is observed on Celite at very high loadings. Under these conditions, this support is expected to have its pores completely filled with packed enzyme molecules, and then it is the diffusion within the enzyme layer that determines the observed rate. As the enzyme loading increases, the area of contact between the deposited enzyme layers and the liquid solution inside the pores diminishes, causing a decrease on the observed rate of an intrinsically fast reaction which apparently is incongruous with the presence of more enzyme in the system. This work shows that mass transfer limitations can be an important factor when working with immobilized enzymes in organic media, and its study should be carried out in order to avoid undesired reduced enzyme activities and specificities.     

Diffusional influences on the parametric sensitivity of immobilized enzyme catalysts

Immobilization of an enzyme within an insoluble material permeable to substrate can change the apparent behavior of the enzyme. In particular, external mass transfer and intraparticle diffusion effects can significantly influence the dependence of observed reaction rate on operating parameters such as temperature and pH. This analysis shows that, under very general conditions, the influence of diffusion alone is to diminish the sensitivity of the observed rate to any parameter that is uniform throughout the catalyst particle. However, the optimal value of the parameter is not changed because of intrapellet diffusional effects.     

Effect of internal diffusion in heterogeneous enzyme systems: Evaluation of true kinetic parameters and substrate diffusivity

The effect of internal diffusion on the overall reaction rate in spherical particles and membranes containing immobilized enzymes has been investigated theoretically. Since they represent open systems, the MichaelisMenten kinetics is obeyed in the absence of diffusional effects at steady state even at high enzyme concentrations. When internal diffusion perturbs the reaction, the system can not be described any more by K_M and V_max? alone, but is conveniently characterized by the modulus. Assuming that only internal diffusion interferes with the enzyme reaction, the effect of the modulus on the overall rate of reaction is illustrated by the results of computer calculations. Plots of the overall reaction rate against the substrate concentration are hyperbolas at various moduli for both membranes and spherical particles and no sigmoidal curves are obtained with immobilized enzyme systems. Since the conventional plots of enzyme kinetics do not yield straight lines under such conditions, a graphical method is proposed to determine K_M and V_max? as well as the substrate diffusivity in the enzymic medium.     

Batch reactor performance for the enzymatic synthesis of cephalexin: influence of catalyst enzyme loading and particle size

A mathematical model is presented for the kinetically controlled synthesis of cephalexin that describes the heterogeneous reaction-diffusion process involved in a batch reactor with glyoxyl-agarose immobilized penicillin acylase. The model is based on equations considering reaction and diffusion components. Reaction kinetics was considered according to the mechanism proposed by Schro?n, while diffusion of the reacting species was described according to Fick's law. Intrinsic kinetic and diffusion parameters were experimentally determined in independent experiments. It was found that from the four kinetic constants, the one corresponding to the acyl-enzyme complex hydrolysis step had the greatest value, as previously reported by other authors. The effective diffusion coefficients of all substances were about 5×10(-10)m(2)/s, being 10% lower than free diffusion coefficients and therefore agreed with the highly porous structure of glyoxyl-agarose particles. Simulations made from the reaction-diffusion model equations were used to evaluate and analyze the impact of internal diffusional restrictions in function of catalyst enzyme loading and particle size. Increasing internal diffusional restrictions decreases the Cex synthesis/hydrolysis ratio, the conversion yield and the specific productivity. A nonlinear relationship between catalyst enzyme loading and specific productivity of Cex was obtained with the implication that an increase in catalyst enzyme loading will not increase the volumetric productivity by the same magnitude as it occurs with the free enzyme. Optimization of catalyst and reactor design should be done considering catalyst enzyme loading and particle size as the most important variables. The approach presented can be extended to other processes catalyzed by immobilized enzymes.     

An annular bound-enzyme reactor

A column reactor with an annular cross section was formed by rolling up DEAE cellulose paper and a screening spacer. Glucoamylase was attached by ion adsorption. For the spacer used, pressure drop was very low, suggesting that this form may be useful with feed streams that are not completely particle-free. Tests of this reactor at the high substrate concentrations characteristic of commercial reactors showed very little diffusional resistance, exhibiting zero-order behavior over most of the concentration range. At low concentrations, the reactor had an apparent “half-order” behavior caused by diffusional limitation in the paper. In this range, flow rate influenced the reaction rate, showing that mass transfer in the main stream also is a contributing factor in this range. Because of the high concentrations and the low Michaelis constant (0.0011 M) the reactor does not show first-order behavior, even at very high conversions. The design of a plant-scale reactor was formulated from these data. The increase in the quantity of enzyme necessary to compensate for the effects of diffusion was only a few percent. Two reactors were formed with sheets nonporous to the enzyme, binding the enzyme with cyanogen bromide after forming the reactor. The amount of enzyme bound was about one monolayer, and there appeared to be no diffusional limitations, even at low substrate concentrations.     

Engineering aspects of immobilized lipases on esterification: A special emphasis of crowding,confinement and diffusion effects

Cross‐linked enzyme crystal (CLEC) and sol‐gel entrapped pseudomonas sp. lipase were investigated for the esterification of lauric acid with ethanol by considering the effects of reaction conditions on reaction rate. The activation energy for the reaction was estimated to be 1097.58 J/mol and 181.75 J/mol for sol‐gel and CLEC entrapped lipase respectively. CLEC lipase exhibited a marginal internal diffusion effect on reaction rate over sol‐gel lipases and found to be interesting. The overall reaction mechanism was found to conform to the Ping Pong Bi Bi mechanism. The higher efficiency of sol‐gel lipases over CLEC lipases in esterification reaction is mainly due to the combined effects of crowding, confinement and diffusional limitations.     

Influence of substrate and product diffusion on the heterogeneous kinetics of enzymic reversible reactions

When a reversible reaction is catalyzed by a surface bound enzyme, the diffusion of both substrate and product can considerably modify the kinetic properties of the reaction. According to this theoretical analysis, the enzyme activity is decreased due to the presence of substrate and product concentration gradients in the enzyme microenvironment, and the relative kinetic importance of the two diffusion steps mainly depend on the value of a dimensionless criterion inversely proportional to the equilibrium constant. Moreover diffusional effects increase with increasing bound enzyme activity, but decrease with increasing substrate and product concentration. Analytical expressions are established for the limiting values of substrate and product concentrations in the enzyme microenvironment, as well as for the increase in half-maximal-activity substrate concentration in the presence of substrate and product diffusional limitations.     

Brownian dynamics study of the influences of electrostatic interaction and diffusion on protein-protein association kinetics.

A unified model is presented for protein-protein association processes that are under the influences of electrostatic interaction and diffusion (e.g., protein oligomerization, enzyme catalysis, electron and energy transfer). The proteins are modeled as spheres that bear point charges and undergo translational and rotational Brownian motion. Before association can occur the two spheres have to be aligned properly to form a reaction complex via diffusion. The reaction complex can either go on to form the product or it can dissociate into the separate reactants through diffusion. The electrostatic interaction, like diffusion, influences every step except the one that brings the reaction complex into the product. The interaction potential is obtained by extending the Kirkwood-Tanford protein model (Tanford, C., and J. G. Kirkwood. 1957. J. Am. Chem. Soc. 79:5333-5339) to two charge-embedded spheres and solving the consequent equations under a particular basis set. The time-dependent association rate coefficient is then obtained through Brownian dynamics simulations according an algorithm developed earlier (Zhou, H.-X. 1990. J. Phys. Chem. 94:8794-8800). This method is applied to a model system of the cytochrome c and cytochrome c peroxidase association process and the results confirm the experimental dependence of the association rate constant on the solution ionic strength. An important conclusion drawn from this study is that when the product is formed by very specific alignment of the reactants, as is often the case, the effect of the interaction potential is simply to scale the association rate constant by a Boltzmann factor. This explains why mutations in the interface of the reaction complex have strong influences on the association rate constant whereas those away from the interface have minimal effects. It comes about because the former mutations change the interaction potential of the reaction complex significantly and the latter ones do not.     

Pilot plant production of glucose with glucoamylase immobilized to porous silica.

Glucoamylase was immobilized to porous silica and its kinetics and stability were observed with acid- and alpha-amylase-hydrolyzed dextrin as feed. The enzyme was found to be extremely stable in both laboratory and pilot plant operations. When the feed had been previously only lightly hydrolyzed, pore diffusion limitation caused appreciable decreases in glucose production rate. The severity of starch hydrolysis to dextrin markedly affected ultimate glucose yields. The diffusional gradients present in the carrier pores caused the immobilized enzyme to yield lower glucose concentrations than the free enzyme at similar feed conditions.     

Quantifying Protein Diffusion and Capture on Filaments

The functional relevance of regulating proteins is often limited to specific binding sites such as the ends of microtubules or actin-filaments. A localization of proteins on these functional sites is of great importance. We present a quantitative theory for a diffusion and capture process, where proteins diffuse on a filament and stop diffusing when reaching the filament’s end. It is found that end-association after one-dimensional diffusion is the main source for tip-localization of such proteins. As a consequence, diffusion and capture is highly efficient in enhancing the reaction velocity of enzymatic reactions, where proteins and filament ends are to each other as enzyme and substrate. We show that the reaction velocity can effectively be described within a Michaelis-Menten framework. Together, one-dimensional diffusion and capture beats the (three-dimensional) Smoluchowski diffusion limit for the rate of protein association to filament ends.     

Effect of ethylmaleimide on the transport of Ca+ and K+ ions across mitochondrial membranes

As already reported, it has been found that the gradient of protons, set up across the inner membrane during the Ca2+ uptake by rat liver mitochondria, can be completely reversed by the addition of NEM. Identical results have been obtained by following the energy dependent K+ uptake. In these last conditions, the rate of H+ efflux supported by succinate oxidation is greatly enhanced only when NEM is added after rotenone. It is proposed that the increased rate other than to the inhibition of Pi uptake, as suggested by Reynafarje and Lehninger, could also be ascribed to a further decrease in the energetic level of the membrane as well as to an increased rate of succinate-Pi exchange diffusion reaction induced by NEM. A possible direct effect of NEM on succinate oxidation has been also considered to account for the inhibition observed when it is added before rotenone.     

Sequence specificity of exonuclease III from E. coli.

The influence of the nucleotide sequence on the digestion of deoxyribonuclease from E. coli, has been investigated. It was found that the rate at which mononucleotides are released varies in a sequence dependent fashion. C-residues are cleaved off rapidly and G-residues slowly while A and T are released at an intermediate rate. Quantitative analyses of digestion experiments with synthetic DNA fragments made it possible to determine rate constants for the cleavage of several dinucleotide bonds by exonuclease III. These values were found to differ by up to a factor of 3. Summation of the differences can lead to appreciable variation in the overall rate of digestion of a DNA strand. The nucleotide specificity of exonuclease III leads to a transient appearance of a series of discrete DNA fragments intermediate in digestion and a stable set of fragments in limit digests, i.e. at the point when all DNA has become single-stranded. This property of exonuclease III needs to be taken into account for the application of the enzyme in the analysis of nucleoprotein complexes.     

Theoretical study of mechano-chemical couplings in a compartmental enzyme system. I. Analytical treatment

This paper deals with theoretical aspects of the volume changes of a system in which diffusion, convection and reaction processes are coupled. This study involves a material able to swell in the presence of a chemical effector produced by an enzyme reaction. Three limiting factors of volume change rate were considered: fluid flow, diffusion or reaction limitations. Dimensionless diffusion-reaction and diffusion-convection parameters were introduced to allow quantitative predictions in limit cases. The steady states appear to be independent of convection processes; however, the transient states depend on diffusion, convection and reaction processes.     

The kinetics of hemolytic plaque formation. II. Inhibition of plaques by hapten.

We present a mathematical theory of hapten inhibition of hemolytic plaque formation. The treatment is based upon the mathematical model for plaque growth presented by DeLisi &; Bell (1974). The lymphocyte under consideration is embedded in an infinite three-dimensional medium, and is secreting antibodies isotropically at a constant rate. As the antibodies diffuse from the source they can bind reversibly to hapten, and in the most general case reversibly to red blood cell (RBC) epitope. The model leads to a non-linear diffusion equation coupled to a set of first order differential equations. The system must, in general, be solved numerically. However, in many cases of experimental interest simplifications arise which permit closed form solutions to be obtained. In this paper we have developed solutions for three special cases.In the first example antibodies can bind only univalently to RBCs, as would be expected if the epitopes are sparsely distributed. In this case reaction between antibody site and RBC epitope is rapid ( ⪆ 1 sec) and reversible and local equilibrium is assumed. This leads to a “pure” diffusion equation in the free antibody concentration, but with a reduced diffusion coefficient.In another example univalent attachment of an antibody site to a RBC epitope is followed by a rapid irreversible intramolecular reaction. This might be expected for example if the epitope density is large. An exact solution to the resulting diffusion equation was also found in this case. In order to assess an intermediate situation, we also solved the equations for a model in which intramolecular reaction is slow and irreversible.The theory predicts that the type of information one can obtain from inhibition experiments depends critically upon the preparation of the RBC. If the cell is sufficiently haptenated so that rapid irreversible multivalent attachment is favorable, a differential plot of the inhibition curve will reflect the affinity distribution of antibody sites for free hapten. If only univalent attachment with RBCs is possible, so that antibody sites bind to RBC hapten in the same way they bind to free hapten, then a differential plot of the inhibition curve will reflect the secretion rate distribution.