The kinetics of nucleated polymerizations at high concentrations: amyloid fibril formation near and above the "supercritical concentration"

The formation of amyloid and other types of protein fibrils is thought to proceed by a nucleated polymerization mechanism. One of the most important features commonly associated with nucleated polymerizations is a strong dependence of the rate on the concentration. However, the dependence of fibril formation rates on concentration can weaken and nearly disappear as the concentration increases. Using numerical solutions to the rate equations for nucleated polymerization and analytical solutions to some limiting cases, we examine this phenomenon and show that it is caused by the concentration approaching and then exceeding the equilibrium constant for dissociation of monomers from species smaller than the nucleus, a quantity we have named the "supercritical concentration". When the concentration exceeds the supercritical concentration, the monomer, not the nucleus, is the highest-energy species on the fibril formation pathway, and the fibril formation reaction behaves initially like an irreversible polymerization. We also derive a relation that can be used in a straightforward method for determining the nucleus size and the supercritical concentration from experimental measurements of fibril formation rates.     

Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios

We extend the validity of the quasi-steady state assumption for a model double intermediate enzyme-substrate reaction to include the case where the ratio of initial enzyme to substrate concentration is not necessarily small. Simple analytical solutions are obtained when the reaction rates and the initial substrate concentration satisfy a certain condition. These analytical solutions compare favourably with numerical solutions of the full system of differential equations describing the reaction. Experimental methods are suggested which might permit the application of the quasi-steady state assumption to reactions where it may not have been obviously applicable before.     

Mechanistic limitations in the synthesis of polyesters by lipase-catalyzed ring-opening polymerization

Lipase-catalyzed polymerization of caprolactone (CL) in toluene with methoxy-poly(ethylene glycol) (MPEG) and water as initiators was characterized in detail for mechanistic insight. (1)H NMR analysis of polycaprolactone chains (PCL), dicaprolactone, degree of esterification of MPEG, and fractions of PCL chains initiated by MPEG and water were used to follow the reactions. The data were analyzed with the kinetic scheme involving formation of the acylenzyme and its consequent reaction with MPEG, water, or PCL to yield the MPEG- or water-initiated PCL chains, or increase in PCL length. A limit for MPEG initiator esterification in lipase-catalyzed CL polymerization was observed and was explained by preferential reaction of PCL propagation over MPEG esterification at long reaction times and low MPEG concentrations. Slower monomer conversion in concentrated monomer solutions was explained by decreased partitioning of PCL between the solvent and the enzyme. This effect resulted in inhibition of the lipase by the reaction product, PCL chains, and/or insufficient diffusion of monomer to the enzyme active site. High monomer/initiators ratio in these solutions did not yield longer polymer chains due to decreased monomer conversion and the corresponding decrease in product yields; lower yields were also observed for chain initiation by MPEG and water. A shift in the reaction rate-limiting step from formation of acylenzyme in dilute CL solutions to its deacylation in concentrated CL solutions yielded higher PCL polydispersity due to increased initiation by water. Enhanced intramolecular cyclization was also observed. Endgroup composition of PCL chains was influenced by the concentration of monomer, ratio of initiators (MPEG and water), and reaction time, yielding PCL chains initiated exclusively by MPEG at "infinite reaction times."     

Stereoselective polymerization of gamma-benzyl glutamate NCA

The polymerization of γ-benzyl glutamate NCAs has been investigated. The results indicate that a coil-to-helix transformation is not responsible for the kinetic data. The results were rationalized by assuming formation of polymer aggregates coupled with the stereoselective adsorption of monomer onto growing polymer prior to reaction. From kinetic data a ratio of equilibrium constants for adsorption was determined. Independent data confirmed the assumption that adsorption of monomer onto polymer occurred. Equilibrium constants were evaluated, and their ratio was found to be comparable with that obtained from the kinetic data.     

Estimation of polymerization efficiency in the formation of polyacrylamide gel,using continuous optical scanning during polymerization

The Ferguson plot and ‘quantitative’ gel electrophoresis (based on the Ferguson plot) depend on a knowledge of accurate gel concentrations. The easiest way to estimate accuracy of gel concentrations, in terms of the degree of completion of the polymerization reaction which gives rise to a gel, is by spectrophotometry. Making use of the apparatus for continuous optical scanning of polyacrylamide gels, the extent and rate of polymerization of cross-linked polyacrylamide were estimated by measuring the absorbance at 275 mm of the reaction mixture subsequent to free radical initiation of polymerization. Under appropriate conditions of monomer concentration, initiator levels and temperature, absorbance decreased monotonically after alag period of 10 min, and after 20–30 min of reaction the absorbance reached a plateau value which provided a measure of polymerization efficiency. Application of a standard curve of absorbance vs. monomer concentration allowed one to quantitate concentrations of residual monomer throughout the course of polymerization. Under a set of arbitrary polymerization conditions (e.g. 6–20% total gel concentration), the reaction went to 63–96% completion. The rate of polymerization was approximately proportional to the square of the monomer concentration (2nd-order reaction kinetics). Absorbance decrease subsequent to the initiation of the polymerization reaction appeared suitable as a measure of efficiency of polymerization since: (1) absorbance spectra of monomers at 0.5%T and residual monomers in a 10%T gel, at a time when polymerization seemed terminated, coincided; (b) values of residual monomer obtained were reasonable (10–30%); (c) bimolecular reaction kinetics were found, in agreement with expectation; and (c) absorbance of incomplete polymerization mixtures, deficient in either initiators or monomers, was constant with time.     

Systems approach to the study of drug transport across membranes using suspension cultures of mammalian cells V. Simultaneous passive transport and biosynthesis

A physical model is described for the simultaneous enzymatic bioconversion of a nonelectrolyte solute and the passive transport of both the solute and product of the enzymatic reaction out of cells in culture suspension. The plasma membrane is assumed to be the rate-determining transport barrier. This model provides the basis for the experimental design and analysis of the Michaelis-Menten kinetic parameters of simple enzymatic reactions in situ, the phenomenological transport parameters and other factors. The primary set of differential equations describing the quasisteady state rate of change in the concentration of the solute and product within the cell due to enzyme reaction and transport are given. These are nonlinear and must be solved by numerical methods. However, analytical mathematical expressions have been derived for various cases in the limit when the rate of enzymatic reaction is first or zero order.     

Kinetic analysis of enzyme systems with suicide substrate in the presence of a reversible competitive inhibitor, tested by simulated progress curves

The use of suicide substrates remains a very important and useful method in enzymology for studying enzyme mechanisms and designing potential drugs. Suicide substrates act as modified substrates for the target enzymes and bind to the active site. Therefore the presence of a competitive reversible inhibitor decreases the rate of substrate-induced inactivation and protects the enzyme from this inactivation. This lowering on the inactivation rate has evident physiological advantages, since it allows the easy acquisition of experimental data and facilitates kinetic data analysis by providing another variable (inhibitor concentration). However despite the importance of the simultaneous action of a suicide substrate and a competitive reversible inhibition, to date no corresponding kinetic analysis has been carried out. Therefore we present a general kinetic analysis of a Michaelis-Menten reaction mechanism with double inhibition caused by both, a suicide substrate and a competitive reversible inhibitor. We assume rapid equilibrium of the reversible reaction steps involved, while the time course equations for the reaction product have been derived with the assumption of a limiting enzyme. The goodness of the analytical solutions has been tested by comparison with the simulated curves obtained by numerical integration. A kinetic data analysis to determine the corresponding kinetic parameters from the time progress curve of the product is suggested. In conclusion, we present a complete kinetic analysis of an enzyme reaction mechanism as described above in an attempt to fill a gap in the theoretical treatment of this type of system.     

An evolutionary approach to enzyme kinetics: Optimization of ordered mechanisms

A theoretical investigation is presented which allows the calculation of rate constants and phenomenological parameters in states of maximal reaction rates for unbranched enzymic reactions. The analysis is based on the assumption that an increase in reaction rates was an important characteristic of the evolution of the kinetic properties of enzymes. The corresponding nonlinear optimization problem is solved taking into account the constraint that the rate constants of the elementary processes do not exceed certain upper limits. One-substrate-one-product reactions with two, three and four steps are treated in detail. Generalizations concern ordered uni-uni-reactions involving an arbitrary number of elementary steps. It could be shown that depending on the substrate and product concentrations different types of solutions can be found which are classified according to the number of rate constants assuming in the optimal state submaximal values. A general rule is derived concerning the number of possible solutions of the given optimization problem. For high values of the equilibrium constant one solution always applies to a very large range of the concentrations of the reactants. This solution is characterized by maximal values of the rate constants of all forward reactions and by non-maximal values of the rate constants of all backward reactions. Optimal kinetic parameters of ordered enzymic mechanisms with two substrates and one product (bi-uni-mechanisms) are calculated for the first time. Depending on the substrate and product concentrations a complete set of solutions is found. In all cases studied the model predicts a matching of the concentrations of the reactants and the corresponding Michaelis constants, which is in good accordance with the experimental data. It is discussed how the model can be applied to the calculation of the optimal kinetic design of real enzymes.     

Enzyme kinetics at high enzyme concentration

We re-visit previous analyses of the classical Michaelis-Menten substrate-enzyme reaction and, with the aid of the reverse quasi-steady-state assumption, we challenge the approximation d[C]/dt ≈ 0 for the basic enzyme reaction at high enzyme concentration. For the first time, an approximate solution for the concentrations of the reactants uniformly valid in time is reported. Numerical simulations are presented to verify this solution. We show that an analytical approximation can be found for the reactants for each initial condition using the appropriate quasi-steady-state assumption. An advantage of the present formalism is that it provides a new procedure for fitting experimental data to determine reaction constants. Finally, a new necessary criterion is found that ensures the validity of the reverse quasi-steady-state assumption. This is verified numerically.     

Kinetic modeling and fitting software for interconnected reaction schemes: VisKin

Reaction kinetics for complex, highly interconnected kinetic schemes are modeled using analytical solutions to a system of ordinary differential equations. The algorithm employs standard linear algebra methods that are implemented using MatLab functions in a Visual Basic interface. A graphical user interface for simple entry of reaction schemes facilitates comparison of a variety of reaction schemes. To ensure microscopic balance, graph theory algorithms are used to determine violations of thermodynamic cycle constraints. Analytical solutions based on linear differential equations result in fast comparisons of first order kinetic rates and amplitudes as a function of changing ligand concentrations. For analysis of higher order kinetics, we also implemented a solution using numerical integration. To determine rate constants from experimental data, fitting algorithms that adjust rate constants to fit the model to imported data were implemented using the Levenberg-Marquardt algorithm or using Broyden-Fletcher-Goldfarb-Shanno methods. We have included the ability to carry out global fitting of data sets obtained at varying ligand concentrations. These tools are combined in a single package, which we have dubbed VisKin, to guide and analyze kinetic experiments. The software is available online for use on PCs.     

Determining rate constants for irreversible polymerization where the initial step and propagation steps have different rate constants: Consideration of polyadenylate polymerase

A new relationship is derived between the amount of monomer incorporated and the amount of initiated primer in an irreversible polymerization where the first step, initiation, has a rate constant differing from the elongation rate constants. It is valid for template directed and template independent polymerization. This relationship can be used in kinetic simulation. It suggests a simpler curve fitting technique to attain rate constants from a relatively small data set. Our analysis reveals some limitations of the model of irreversible polymerization; these limitations have not been obvious previously. For example, the initiation rate constant is not attainable from simple monomer incorporation data. Reliable rate constants can be obtained with minimal time course studies.     

Validity of quasi-steady-state and transfer-function representations for input-output relation in a Michaelis-Menten reaction

In relation to the input-output characteristics of enzymatic reactions in the cellular metabolism and biochemical reactors, the validity of the quasi-steady-state and transfer-function representations of reaction velocity has been examined for a basic Michaelis-Menten reaction employing computer simulation, that is, numerical integration of the rate equation. The well-known S-v relationship (relationship between substrate concentration and reaction velocity)derived on the quasi-steady-state assumption is found to be in general a good approximation to the actual velocity throughout the temporal progress of the reaction. The validity of the approximation depends on a ratio of the Michaelis constant to the total enzyme concentration in the reaction system rather than on the individual rate constants. A transfer-function representation is derived on assuming an exponential change in the reaction velocity for the indicial response to the substrate influx rate. The representation has a wider valid region with a decrease in influx rate than with an increase in the influx rate. The validity is most dependent on a ratio of total enzyme concentration to the steady-state concentration of the substrate. The analysis of the linear sensitivity of the reaction velocity to rate constants reveals that the characteristics of these valid representations in systems analysis change according to the phase of the reaction.     

Competitive and uncompetitive inhibitors simultaneously acting on an autocatalytic zymogen activation reaction

The time course of the residual enzyme activity of a general model consisting of an autocatalytic zymogen activation process inhibited by an irreversible competitive inhibitor and an irreversible uncompetitive inhibitor has been studied. Approached analytical expressions which furnish the time course of the residual enzyme activity from the onset of the reaction depending on the rate constants and initial concentration have been obtained. The goodness and limitations of the analytical equations were checked by comparing with the results obtained from the numerical integration, i.e. with the simulated progress curves. A dimensionless parameter giving the relative contributions of both the activation and the inhibitions routes is suggested, so that the value of this parameter determines whether the activation or the inhibitions routes prevail or if both processes are balanced during the time for which the analytical expressions are valid. The effects of the initial zymogen, free enzyme and inhibitors concentrations are analysed. Finally an experimental design and kinetic data analysis is proposed to evaluate simultaneously the kinetic parameters involved and to discriminate between different zymogen activation processes which can be considered particular cases of the general model.     

Competitive and uncompetitive inhibitors simultaneously acting on an autocatalytic zymogen activation reaction

The time course of the residual enzyme activity of a general model consisting of an autocatalytic zymogen activation process inhibited by an irreversible competitive inhibitor and an irreversible uncompetitive inhibitor has been studied. Approached analytical expressions which furnish the time course of the residual enzyme activity from the onset of the reaction depending on the rate constants and initial concentration have been obtained. The goodness and limitations of the analytical equations were checked by comparing with the results obtained from the numerical integration, i.e. with the simulated progress curves. A dimensionless parameter giving the relative contributions of both the activation and the inhibitions routes is suggested, so that the value of this parameter determines whether the activation or the inhibitions routes prevail or if both processes are balanced during the time for which the analytical expressions are valid. The effects of the initial zymogen, free enzyme and inhibitors concentrations are analysed. Finally an experimental design and kinetic data analysis is proposed to evaluate simultaneously the kinetic parameters involved and to discriminate between different zymogen activation processes which can be considered particular cases of the general model.     

Kinetic analysis of enzyme systems with suicide substrate in the presence of a reversible, uncompetitive inhibitor.

We present a general kinetic analysis of enzyme catalyzed reactions evolving according to a Michaelis-Menten mechanism, in which an uncompetitive, reversible inhibitor acts. Simultaneously, enzyme inactivation is induced by an unstable suicide substrate, i.e. it is a Michaelis-Menten mechanism with double inhibition: one originating from the substrate and another originating from the reversible inhibitor. Rapid equilibrium of the reversible reaction steps involved is assumed and the time course equations for the reaction product have been derived under the assumption of limiting enzyme. The goodness of the analytical solutions has been tested by comparison with simulated curves obtained by numerical integration. A kinetic data analysis to determine the corresponding kinetic parameters from the time progress curve of the product is suggested.     

Parameter estimation for ligand binding systems kinetics applied to 1,25-dihydroxycholecalciferol

A mathematical analysis of the kinetics of the hormone-receptor interaction was applied to the 1,25-dihydroxycholecalciferol-intestinal receptor system. The exact analytical solution and the numerical integration of the kinetic equation were installed in a Statistical Analysis System (SAS) computer program to estimate the rate constants of the reaction. Estimates of the parameters obtained by these two methods are similar, demonstrating that the numerical integration can be combined with the nonlinear regression procedure for least-squares parameter fitting using a simple SAS program. This enables estimation of kinetics rate constants when the kinetic equation cannot be solved analytically. The ratio of the rate constants (ka/kd) found by the nonlinear procedure is close to the independently determined equilibrium (Scatchard) constant in the nonlinear analysis.     

Thermodynamics and kinetics of actin filament nucleation.

We have performed computer simulations and free energy calculations to determine the thermodynamics and kinetics of actin nucleation and thus identify a probable nucleation pathway and critical nucleus size. The binding free energies of structures along the nucleation pathway are found through a combination of electrostatic calculations and estimates of the entropic and surface area contributions. The association kinetics for the formation of each structure are determined through a series of Brownian dynamics simulations. The combination of the binding free energies and the association rate constants determines the dissociation rate constants, allowing for a complete characterization of the nucleation and polymerization kinetics. The results indicate that the trimer is the size of the critical nucleus, and the rate constants produce polymerization plots that agree very well with experimental results over a range of actin monomer concentrations.     

Oxygen and carbon monoxide kinetics of Glycera dibranchiata monomeric hemoglobin.

Oxygen and carbon monoxide kinetics of Glycera dibranchiata monomeric hemoglobin have been studied using laser photolysis, air flash, and stopped flow techniques. The reactions of this hemoglobin with both ligands were found to be more rapid than the corresponding reactions involving myoglobin and were also biphasic in nature, the rate constants being approximately an order of magnitude different for the fast and slow phases in each case. No pH or hemoglobin concentration dependence of the pseudo-first order rate constants was apparent between pH 6 and 9 and in the concentration range of 1.25 to 40 muM heme. Both fast and slow pseudo-first order oxygen combination rate constants varied linearly with oxygen concentration between 16 and 1300 muM. A first order slow relaxation was also noted which was linearly dependent on heme concentration and inversely dependent on oxygen concentration. This reaction has been shown to be due to a replacement of oxygen by carbon monoxide. The presence of this reaction is a result of the high affinity of Glycera monomer for carbon monoxide as shown by the partition coefficient Mr = approximately 20,000 ana an equilibrium dissociation constant of the order L = 1.1 X 10(-9) M.     

Seeded isothermal batch crystallization of lysozyme

The kinetics of lysozyme crystallization under seeded isothermal batch conditions was followed by measurement of the decline in solution concentration versus time. Kinetics were measured for five different values of the seed crystal mass. The data were analyzed using a recently proposed mathematical model. For each seed mass, the model fit the kinetic data well. Growth rate constants determined using the model were approximately constant over a sixfold increase in the seed crystal mass, and fell well within the range of values reported in the literature, but obtained using entirely different experimental techniques. These results confirmed the utility of the proposed model. The proposed model can be used to analyze crystallization kinetics using absorbance measurements only, without the need to characterize the crystal size, thus avoiding the need for expensive laser light scattering and digital microscopy instrumentation. Thus, the model offers a low-cost straightforward method to analyze and simulate the effects of changes in operating parameters such as the seed crystal mass, solution volume, initial protein concentration, pH, temperature, salt concentration, and time.     

A strategy to determine operating parameters in tissue engineering hollow fiber bioreactors

The development of tissue engineering hollow fiber bioreactors (HFB) requires the optimal design of the geometry and operation parameters of the system. This article provides a strategy for specifying operating conditions for the system based on mathematical models of oxygen delivery to the cell population. Analytical and numerical solutions of these models are developed based on Michaelis–Menten kinetics. Depending on the minimum oxygen concentration required to culture a functional cell population, together with the oxygen uptake kinetics, the strategy dictates the model needed to describe mass transport so that the operating conditions can be defined. If c_min ≫ K_m we capture oxygen uptake using zero‐order kinetics and proceed analytically. This enables operating equations to be developed that allow the user to choose the medium flow rate, lumen length, and ECS depth to provide a prescribed value of c_min. When , we use numerical techniques to solve full Michaelis–Menten kinetics and present operating data for the bioreactor. The strategy presented utilizes both analytical and numerical approaches and can be applied to any cell type with known oxygen transport properties and uptake kinetics. Biotechnol. Bioeng. 2011; 108:1450–1461. © 2011 Wiley Periodicals, Inc.