Transient response of encapsulated enzymes in hollow-fiber reactor

A mathematical model for the transient response of encapsulated enzymes is developed showing the effects of the outer boundary layer, the encapsulating membrane, the partition coefficient, and diffusion with reaction within the encapsulating medium. The model incorporates both first-order kinetics and Michaelis-Menten kinetics for the reaction rate. Using typical hollow-fiber or microcapsule parameters, the model shows that (a) the partition coefficient affects the overall rate only when the rate-limiting step is diffusion through the membrane, (b) the transient overall effectiveness factor rises sharply with time and approaches an asymptotic value for most situations, and (c) the first-order approximation to Michaelis-Menten kinetics is not valid when the initial outside bulk concentration is higher than the Michaelis constant and the overall rate is reaction limited. The model is compared with experimental data using uricase in a hollow-fiber enzyme reactor configuration. Batch assay and CSTUER (continuous-stirred ultrafiltration enzyme reactor) studies were conducted on the free enzyme to provide some of the parameters used in the model. The CSTUER data fit the case of substrate inhibition kinetics with the apparent Michaelis constant approaching zero. The hollow-fiber reactor was conducted with uricase dissolved in both a buffer solution and a concentrated hemoglobin solution. Diffusivities of the solute were measured in both solutions as was the osmotic pressure of the hemoglobin solution. While experimental data for uricase in buffer solution could easily be matched by the model, that in the concentrated hemoglobin solution could not.     

Saturation of the internal sodium site of the sodium pump can distort estimates of potassium affinity.

The Na+/K+ exchange pump in cardiac Purkinje strands has been well studied with the voltage clamp and Na+-selective microelectrodes. Models describing the observed results suggest that the pump rate can be considered proportional to [Na+]i over the range examined and depends on external [K+] in accordance with Michaelis-Menten kinetics. Estimates of the external [K+] that achieves a half-maximal pump rate (Km) range from 0.9 to 6.3 mM depending on the preparation and method of estimation. Here we show that much of the variability in the estimates of the Km can be eliminated when saturation of the internal Na+ pump site is taken into account. If the half-activation concentration for saturation of this Na+ site is sufficiently high (greater than 20 mM), removal of intracellular Na+ in response to a Na+ load will approximate first-order kinetics. Under these conditions however, Na+ saturation will nevertheless cause large systematic errors in estimates of the K+ dependence of pump activity.     

The uptake and metabolism of urea by Chara australis: IV. Symport with sodium—A slip model for the high and low affinity systems

We have previously investigated the electrogenic influx of urea in Chara, and the urea- and sodium-dependent membrane current. We have shown that there is a sodium-stimulated component of urea influx and a urea-stimulated component of sodium influx, and that these are of the same size. We conclude that the electrogenic inward transport of urea, and of its analogues acetamide and acrylamide, is by sodium symport, with a stoichiometric ratio of 1∶1. The kinetics of the fluxes and currents show two different K _M values for sodium in different cells and two different kinds of kinetics for the effect of urea on membrane current, one of which fits the Michaelis-Menten equation, while the other shows a maximum and fits the difference of two Michaelis-Menten terms, suggesting a phenomenon like cis-inhibition. Similarities in kinetic characteristics between the inhibitory site and the electrically silent uptake site (System II) lead us to suggest that the same protein may be responsible for both the low-K _M, electrogenic influx of urea (System I) and the high-K _M, electrically silent influx by System II. We suggest a “slip” model for urea uptake in Chara.     

A Model for Single-Substrate Trimolecular Enzymatic Kinetics

We developed a kinetic model for a single-substrate trimolecular enzymatic system, where a receptor binds and stretches a substrate to expose its cleavage site, allowing an enzyme to bind and cleave it into product. We demonstrated that the general kinetics of the trimolecular enzymatic system is more complex than the Michaelis-Menten kinetics. Under a limiting condition when the enzyme-substrate binding is in fast equilibrium, the enzymatic kinetics of the trimolecular system reduces to the Michaelis-Menten kinetics. In another limiting case when the receptor dissociates negligibly slowly from the substrate, the trimolecular system is simplified to a bimolecular system, which follows the Michaelis-Menten equation if and only if there is no enzyme-substrate complex initially. We applied this model to a particular trimolecular system important to hemostasis and thrombosis, consisting of von Willebrand factor (substrate), platelet glycoprotein Ibα (receptor), and ADAMTS13 (enzyme). Using parameters from independent experiments, our model successfully predicted published data from two single-molecule experiments and fitted/predicted published data from an ensemble experiment.     

Improved neuronal models for studying neural networks

Previous neuronal models used for the study of neural networks are considered. Equations are developed for a model which includes: 1) a normalized range of firing rates with decreased sensitivity at large excitatory or large inhibitory input levels, 2) a single rate constant for the increase in firing rate following step changes in the input, 3) one or more rate constants, as required to fit experimental data for the adaptation of firing rates to maintained inputs. Computed responses compare well with the types of neuronal responses observed experimentally. Depending on the parameters, overdamped increases and decreases, damped oscillatory or maintained oscillatory changes in firing rate are observed to step changes in the input. The integrodifferential equations describing the neuronal models can be represented by a set of first-order differential equations. Steady-state solutions for these equations can be obtained for constant inputs, as well as the stability of the solutions to small perturbations. The linear frequency response function is derived for sufficiently small time-varying inputs. The linear responses are also compared with the computed solutions for larger non-linear responses.     

Influence of transition rates and scan rate on kinetic simulations of differential scanning calorimetry profiles of reversible and irreversible protein denaturation.

The thermodynamic parameters characterizing protein folding can be obtained directly using differential scanning calorimetry (DSC). They are meaningful only for reversible unfolding at equilibrium, which holds for small globular proteins; however, the unfolding or denaturation of most large, multidomain or multisubunit proteins is either partially or totally irreversible. The simplest kinetic model describing partially irreversible denaturation requires three states: Formula [see text] We obtain numerical solutions for N, U, and D as a function of temperature for this model and derive profiles of excess specific heat (Cp) in terms of the reduced variables v/ki and k1/k3, where v is the scan rate. The three-state model reduces to the two-state reversible or irreversible models for very large or very small values of k1/k3, respectively. The apparent transition temperature (Tapp) is always reduced by the irreversible step (U-->D). For all values of k3, Tapp is independent of v/k1 at sufficiently slow scan rates, even when denaturation is highly irreversible, but increases identically for all models at fast scan rates in which case the excess specific heat profile is determined by the rate of unfolding. Accurate values of delta H and delta S can be obtained for the reversible step only when k1 is more than 2000-50,000 times greater than k3. In principle, approximate values for the ratio k1/k3 can be obtained from plots of fraction unfolded vs fraction irreversibly denatured as a function of temperature; however, the fraction irreversibly denatured is difficult to measure accurately by DSC alone.(ABSTRACT TRUNCATED AT 250 WORDS)     

On the analysis of oxygen diffusion and reaction in biological systems

An analysis of oxygen diffusion and reaction in multiregion biological systems is presented. This analysis considers a time-dependent flux boundary condition and oxygen consumption governed by Michaelis-Menten kinetics. The mathematical problem is developed in a uniform fashion, so as to include both the single cell and anisotropic systems with distinct regions which are characteristic of either a multicell spheroid or a tumor mass. Both transient and steady-state solutions are obtained, based on orthogonal collocation. Literature results on single-cell analysis are corroborated, and detailed transient solutions are presented for the oxygenation of a multicell spheroid, and for systemic oxygenation of both small and large tumors.     

An Optimization Algorithm for a Distributed-Loop Model of an Avian Urine Concentrating Mechanism

To better understand how the avian kidney’s morphological and transepithelial transport properties affect the urine concentrating mechanism (UCM), an inverse problem was solved for a mathematical model of the quail UCM. In this model, a continuous, monotonically decreasing population distribution of tubes, as a function of medullary length, was used to represent the loops of Henle, which reach to varying levels along the avian medullary cones. A measure of concentrating mechanism efficiency – the ratio of the free-water absorption rate (FWA) to the total NaCl active transport rate (TAT) – was optimized by varying a set of parameters within bounds suggested by physiological experiments. Those parameters include transepithelial transport properties of renal tubules, length of the prebend enlargement of the descending limb (DL), DL and collecting duct (CD) inflows, plasma Na⁺ concentration, length of the cortical thick ascending limbs, central core solute diffusivity, and population distribution of loops of Henle and of CDs along the medullary cone. By selecting parameter values that increase urine flow rate (while maintaining a sufficiently high urine-to-plasma osmolality ratio (U/P)) and that reduce TAT, the optimization algorithm identified a set of parameter values that increased efficiency by ∼60% above base-case efficiency. Thus, higher efficiency can be achieved by increasing urine flow rather than increasing U/P. The algorithm also identified a set of parameters that reduced efficiency by ∼70% via the production of a urine having near-plasma osmolality at near-base-case TAT.In separate studies, maximum efficiency was evaluated as selected parameters were varied over large ranges. Shorter cones were found to be more efficient than longer ones, and an optimal loop of Henle distribution was found that is consistent with experimental findings.     

Kinetics of Hydrogen Consumption by Rumen Fluid, Anaerobic Digestor Sludge, and Sediment

Michaelis-Menten kinetic parameters for H₂ consumption by three methanogenic habitats were determined from progress curve and initial velocity experiments. The influences of mass transfer resistance, endogenous H₂ production, and growth on apparent parameter estimates were also investigated. Kinetic parameters could not be determined for undiluted rumen fluid and some digestor sludge from gas-phase measurements of H₂, since mass transfer of H₂ across the gas-liquid interface was rate limiting. However, accurate values were obtained once the samples were diluted. H₂ consumption by digestor sludge with a long retention time and by hypereutrophic lake sediment was not phase transfer limited. The K_m values for H₂ uptake by these habitats were similar, with means of 5.8, 6.0, and 7.1 μM for rumen fluid, digestor sludge, and sediment, respectively. V_max estimates suggested a ratio of activity of approximately 100 (rumen fluid):10 (sludge):1 (sediment); their ranges were as follows: rumen fluid, 14 to 28 mM h⁻¹; Holt sludge, 0.7 to 4.3 mM h⁻¹; and Wintergreen sediment, 0.13 to 0.49 mM h⁻¹. The principles of phase transfer limitation, studied here for H₂, are the same for all gaseous substrates and products. The limitations and errors associated with gas phase determination of kinetic parameters were evaluated with a mathematical model that combined mass transport and Michaelis-Menten kinetics. Three criteria are described which can be used to evaluate the possibility that a phase transfer limitation exists. If it does not exist, (i) substrate consumption curves are Michaelis-Menten and not first order, (ii) the K_m is independent of initial substrate concentration, and (iii) the K_m is independent of biomass (V_max) and remains constant with dilution of sample. Errors in the Michaelis-Menten kinetic parameters are caused by endogenously produced H₂, but they were <15% for rumen fluid and 10% for lake sediment and digestor sludge. Increases in V_max during the course of progress curve experiments were not great enough to produce systematic deviations from Michaelis-Menten kinetics.     

Mathematical modelling of dynamics and control in metabolic networks: VI. Dynamic bifurcations in single biochemical control loops

A dynamic stability analysis of an extended form of the Goodwin equations is presented. The Goodwin equations are extended to include Michaelis-Menten kinetics for the removal of the end-product. Inclusion of saturation kinetic behavior substantially increases the likelihood of dynamic instability in this model control loop. Oscillations are found for reaction chains of low order, as low as second order, and low degrees of co-operativity, as low as v = 2, simultaneously, thus indicating that dynamic instability in this system exists for physiologically realistic parameter values. The branches of bifurcated solutions are computed numerically and unstable Hopf bifurcations are found. Further, solution branches from stable Hopf bifurcation points are found to "fold back", i.e. have periodic limit points, producing situations where multiple stable limit cycles exist.     

The effect of solution non-ideality on membrane transport in three-dimensional models of the renal concentrating mechanism

Previous models of the renal concentrating mechanism employ ideal approximations of solution thermodynamics for membrane transport calculation. In three-dimensional models of the renal medulla, predicted urine concentrations reach levels where there idealized approximations begin to break down. In this paper we derive equations that govern membrane transport for non-dilute solutions and use these equations in a three-dimensional model of the concentrating mechanism. New numerical methods were employed that are more stable than those employed previously. Compared to ideal solution models, the urea non-ideality tends to increase predicted osmolarities, whereas NaCl non-ideality decreases predictions.     

The influx of isoleucine into the cerebral hemispheres and cerebellum: carrier-mediated transport and diffusion

The influx of L-isoleucine from the blood into the cerebral hemispheres, and into the cerebellum, of the living rat was studied. When various raised levels of L-isoleucine were maintained in the circulation, influx did not increase in proportion to the raised blood levels, which showed that a carrier-mediated transport was being saturated. It was found that the values for influx could not be explained precisely by Michaelis-Menten kinetics. The divergence of the observed values for influx, from those which were obtained by the application of Michaelis-Menten kinetics, could be explained if part of the influx was due to a small, non-saturable component. This minor component of influx may be due to diffusion or to a second carrier-mediated transport system that is not readily saturated.     

Comparison of two experimental methods for the determination of Michaelis-Menten kinetics of an immobilized enzyme

For the application of immobilized enzymes, the influence of immobilization on the activity of the enzyme should be Known. This influence can be obtained by determining the intrinsic kinetic parameters of the immobilized enzyme, and by comparing them with the kinetic parameters of the suspended enzyme. This article deals with the determination of the intrinsic kinetic parameters of an agarose-gel bead immobilized oxygen-consuming enzyme: L-lactate 2-monooxygenase. The reaction rate of the enzyme can be described by Michaelis-Menten kinetics. Batch conversion experiments using a biological oxygen monitor, as well as steady-state profile measurements within the biocatalyst particles using an oxygen microsensor, were performed. Two different mathematical methods were used for the batch conversion experiments, both assuming a pseudosteady-state situation with respect to the shape of the profile inside the bead. One of the methods used an approximate relation for the effectiveness factor for Michaelis-Menten kinetics which interpolates between the analytical solutions for zero- and first-order kinetics. The other mathematical method was based on a numerical solution and combined a mass balance over the reactor with a mass balance over the bead. The main difference in the application of the two methods is the computer calculation time; the completely numerical calculation procedure was about 20 times slower than the other calculation procedure.The intrinsic kinetic parameters resulting from both experimental methods were compared to check the reliability of the methods. There was no significant difference in the intrinsic kinetic parameters obtained from the two experimental methods. By comparison of the kinetic parameters for the suspended enzyme with the intrinsic kinetic parameters for the immobilized enzyme, it appeared that immobilization caused a decrease in the value of V(m) by a factor of 2, but there was no significant difference in the values obtained for K(m).     

A region-based model framework for the rat urine concentrating mechanism

The highly structured organization of tubules and blood vessels in the outer medulla of the mammalian kidney is believed to result in preferential interactions among tubules and vessels; such interactions may promote solute cycling and enhance urine concentrating capability. In this study, we formulate a new model framework for the urine concentrating mechanism in the outer medulla of the rat kidney. The model simulates preferential interactions among tubules and vessels by representing two concentric regions and by specifying the fractions of tubules and vessels assigned to each of the regions. The model equations are based on standard expressions for transmural transport and on solute and water conservation. Model equations, which are derived in dynamic form, are solved to obtain steady-state solutions by means of a stable and efficient numerical method, based on the semi-Lagrangian semi-implicit method and on Newton’s method. In this application, the computational cost scales as O(N ²), where N is the number of spatial subintervals along the medulla. We present representative solutions and show that the method generates approximations that are second-order accurate in space and that exhibit mass conservation.     

Utilization and transport of glucose in Olea Europaea cell suspensions

Cell suspensions of Olea europaea var. Galega Vulgar grown in batch culture with 0.5% (w/v) glucose were able to transport D-[(14)C]glucose according to Michaelis-Menten kinetics associated with a first-order kinetics. The monosaccharide carrier exhibited high affinity (K(m) approximately 50 micro M) and was able to transport D-glucose, D-fructose, D-galactose, D-xylose, 2-deoxy-D-glucose and 3-O-methyl-D-glucose, but not D-arabinose, D-mannitol or L-glucose. D-[(14)C]glucose uptake was associated with proton uptake, which also followed Michaelis-Menten kinetics. The transport of 3-O-methyl-D-glucose was accumulative (40-fold, at pH 5.0) and the protonophore carbonyl cyanide m-chlorophenylhydrazone strongly inhibited sugar accumulation. The results were consistent with the involvement of a monosaccharide: proton symporter with a stoichiometry of 1 : 1. When cells were grown with 3% (w/v) glucose, the uptake of D-[(14)C]glucose followed first-order kinetics and monosaccharide:proton symporter activity was not detected. The value obtained for the permeability coefficient of hexoses in O. europaea cells supported the hypothesis that the first-order kinetics observed in 0.5% and 3% sugar-grown cells was produced exclusively by passive diffusion of the sugar. The results indicate that in O. europaea cells sugar levels have a regulatory effect on sugar transport, because the activity for monosaccharide transport was repressed by high sugar concentrations.     

Effect of pH on the production of lipolyzed butter oil by a calf pregastric esterase immobilized in a hollow-fiber reactor: I. uniresponse kinetics

A calf pregastric esterase immobilized in a hollow-fiber reactor was employed to hydrolyze milkfat, thereby producing a lipolyzed butteroil. The reaction kinetics can be modeled by a two-parameter model of the general Michaelis-Menten form based on a ping-pong bi-bi mechanism; the rate of enzyme deactivation can be modeled as a first-order reaction. The initial concentration of accessible glyceride bonds, [G](O), was estimated by complete saponification of the substrate butteroil as 2400 mM. An extra sum of squares test indicated that not only the parameters of the kinetic generalized Michaelis-Menten model, but also the deactivation-rate constant varied significantly with pH. The optimum pH, for lypolysis is near 6.0 at a temperature of 40 degrees C because at this pH the rate of deactivation of the esterase is minimized.     

An efficient numerical method for distributed-loop models of the urine concentrating mechanism

In this study we describe an efficient numerical method, based on the semi-Lagrangian (SL) semi-implicit (SI) method and Newton's method, for obtaining steady-state (SS) solutions of equations arising in distributed-loop models of the urine concentrating mechanism. Dynamic formulations of these models contain large systems of coupled hyperbolic partial differential equations (PDEs). The SL method advances the solutions of these PDEs in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The SI approach controls stiffness arising from transtubular transport terms by averaging these terms in time along flow trajectories. An approximate SS solution of a dynamic formulation obtained via the SLSI method can be used as an initial guess for a Newton-type solver, which rapidly converges to a highly accurate numerical approximation to the solution of the ordinary differential equations that arise in the corresponding SS model formulation. In general, it is difficult to specify a priori for a Newton-type solver an initial guess that falls within the radius of convergence; however, the initial guess generated by solving the dynamic formulation via the SLSI method can be made sufficiently close to the SS solution to avoid numerical instability. The combination of the SLSI method and the Newton-type solver generates stable and accurate solutions with substantially reduced computation times, when compared to previously applied dynamic methods.     

Kinetic Explanation for Accumulation of Nitrite, Nitric Oxide, and Nitrous Oxide During Bacterial Denitrification

The kinetics of denitrification and the causes of nitrite and nitrous oxide accumulation were examined in resting cell suspensions of three denitrifiers. An Alcaligenes species and a Pseudomonas fluorescens isolate characteristically accumulated nitrite when reducing nitrate; a Flavobacterium isolate did not. Nitrate did not inhibit nitrite reduction in cultures grown with tungstate to prevent formation of an active nitrate reductase; rather, accumulation of nitrite seemed to depend on the relative rates of nitrate and nitrite reduction. Each isolate rapidly reduced nitrous oxide even when nitrate or nitrite had been included in the incubation mixture. Nitrate also did not inhibit nitrous oxide reduction in Alcaligenes odorans, an organism incapable of nitrate reduction. Thus, added nitrate or nitrite does not always cause nitrous oxide accumulation, as has often been reported for denitrifying soils. All strains produced small amounts of nitric oxide during denitrification in a pattern suggesting that nitric oxide was also under kinetic control similar to that of nitrite and nitrous oxide. Apparent K_m values for nitrate and nitrite reduction were 15 μM or less for each isolate. The K_m value for nitrous oxide reduction by Flavobacterium sp. was 0.5 μM. Numerical solutions to a mathematical model of denitrification based on Michaelis-Menten kinetics showed that differences in reduction rates of the nitrogenous compounds were sufficient to account for the observed patterns of nitrite, nitric oxide, and nitrous oxide accumulation. Addition of oxygen inhibited gas production from ¹³NO₃⁻ by Alcaligenes sp. and P. fluorescens, but it did not reduce gas production by Flavobacterium sp. However, all three isolates produced higher ratios of nitrous oxide to dinitrogen as the oxygen tension increased. Inclusion of oxygen in the model as a nonspecific inhibitor of each step in denitrification resulted in decreased gas production but increased ratios of nitrous oxide to dinitrogen, as observed experimentally. The simplicity of this kinetic model of denitrification and its ability to unify disparate observations should make the model a useful guide in research on the physiology of denitrifier response to environmental effectors.     

Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, is presented. Korzuhin's Theorem. which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem ‘for beginners’ is also given.