Effect of mixing on reaction-diffusion kinetics for protein hydrogel-based microchips

Protein hydrogel-based microchips are being developed for high-throughput evaluation of the concentrations and activities of various proteins. To shorten the time of analysis, the reaction-diffusion kinetics on gel microchips should be accelerated. Here we present the results of the experimental and theoretical analysis of the reaction-diffusion kinetics enforced by mixing with peristaltic pump. The experiments were carried out on gel-based protein microchips with immobilized antibodies under the conditions utilized for on-chip immunoassay. The dependence of fluorescence signals at saturation and corresponding saturation times on the concentrations of immobilized antibodies and antigen in solution proved to be in good agreement with theoretical predictions. It is shown that the enhancement of transport with peristaltic pump results in more than five-fold acceleration of binding kinetics. Our results suggest useful criteria for the optimal conditions for assays on gel microchips to balance high sensitivity and rapid fluorescence saturation kinetics.     

A kinetic study of analyte-receptor binding and dissociation, and dissociation alone, for biosensor applications: a fractal analysis

A fractal analysis is presented for (a) analyte-receptor binding and dissociation kinetics and (b) dissociation kinetics alone for biosensor applications. Emphasis is placed on dissociation kinetics. Data taken from the literature may be modeled, in the case of binding, using a single-fractal analysis or a dual-fractal analysis. The dual-fractal analysis represents a change in the binding mechanism as the reaction progresses on the surface. A single-fractal analysis is adequate to model the dissociation kinetics in the examples presented. Predictive relationships developed for the dissociation rate coefficient(s) as a function of the analyte concentration are of particular value since they provide a means by which the dissociation rate coefficients may be manipulated. Relationships are also presented for the binding and dissociation rate coefficients as a function of their corresponding fractal dimension, D(f), or the degree of heterogeneity that exists on the surface. When analyte-receptor binding or dissociation is involved, an increase in the heterogeneity on the surface (increase in D(f)) leads to an increase in the binding and in the dissociation rate coefficient.     

Limit behaviour of spatially growing cell cultures

Growing cell cultures often exhibit branch patterns. The cultures are usually modelled as free boundary problems. Here, we review some of these models and investigate the appearance of branch patterns using methods from asymptotic analysis. Two extreme cases--large kinetics and small kinetics--are considered. For large kinetics the models reduce to the well-studied Stefan problem, which is known to exhibit a diffusive-instability and hence shows branch patterns. Considering small kinetics, small perturbations are smoothed out resulting in regular shapes of the cultures. The branching phenomenon in general can be understood as somewhere between these two extremes. Not relying on special assumptions, the presented analysis is very general.     

Kinetics analysis methods for approximate folding landscapes

MOTIVATION: Protein motions play an essential role in many biochemical processes. Lab studies often quantify these motions in terms of their kinetics such as the speed at which a protein folds or the population of certain interesting states like the native state. Kinetic metrics give quantifiable measurements of the folding process that can be compared across a group of proteins such as a wild-type protein and its mutants. RESULTS: We present two new techniques, map-based master equation solution and map-based Monte Carlo simulation, to study protein kinetics through folding rates and population kinetics from approximate folding landscapes, models called maps. From these two new techniques, interesting metrics that describe the folding process, such as reaction coordinates, can also be studied. In this article we focus on two metrics, formation of helices and structure formation around tryptophan residues. These two metrics are often studied in the lab through circular dichroism (CD) spectra analysis and tryptophan fluorescence experiments, respectively. The approximated landscape models we use here are the maps of protein conformations and their associated transitions that we have presented and validated previously. In contrast to other methods such as the traditional master equation and Monte Carlo simulation, our techniques are both fast and can easily be computed for full-length detailed protein models. We validate our map-based kinetics techniques by comparing folding rates to known experimental results. We also look in depth at the population kinetics, helix formation and structure near tryptophan residues for a variety of proteins. AVAILABILITY: We invite the community to help us enrich our publicly available database of motions and kinetics analysis by submitting to our server: http://parasol.tamu.edu/foldingserver/.     

Schematic methods for probabilistic enzyme kinetics.

The theory of steady-state enzyme processes which avoids using the mass action law of chemical kinetics and consistently describes catalytic mechanisms by probabilistic concepts has recently been proposed (Mazur, 1991, J. theor. Biol. 148, 229-242). To facilitate the analysis of complex reaction graphs by this theory the possibility of constructing schematic rules similar to those used in classical kinetics is studied. It is found that due to the similarity of algebraic procedures the popular method of King & Altman can be applied in probabilistic kinetics in addition to the earlier proposed rule based on enumeration of cycles of the reaction graph. This similarity also allows one to adapt many other shortcut methods of classical kinetics for probabilistic reaction graphs. The paper considers separately the possibility of transforming reaction mechanisms so that the initial graph is replaced by a simpler but equivalent one. It is shown that there are few cases when a group of states can be replaced by one united state, with earlier known rules such as the rule of Cha for equilibrium stages being particular cases of a more general procedure. In addition a novel method is proposed which performs step-by-step reduction of any reaction graph. All the new methods can be adapted for traditional kinetics as well. The results obtained demonstrate that many schematic rules of classical kinetics are of probabilistic origin.     

Abnormal behaviour of proline in the isothiocyanate degradation.

It has been observed that proline residues often initiate overlaps during sequenator analysis. The cause has been shown to be an abnormally slow cleavage reaction. The kinetics of the cleavage reaction has been studied and found to obey pseudo-first-order kinetics. There are considerable differences in reaction rates depending on the position of proline in the sequence, as demonstrated for the four prolines in the N-terminal section of the H2B histone from chicken.     

Collection and analysis of kinetic data from a stopped-flow spectrophotometer using a microcomputer

A method of interfacing an inexpensive microcomputer to a stopped-flow kinetics spectrophotometer is described. It allows software-selectable sampling frequencies between 0.1 ms and 8 s and large numbers of data points to be collected. Machine language routines to use the interface are described and these allow the sampling frequency to be altered during data collection to ensure adequate numbers of points in critical regions of the kinetic profile. BASIC programs for collection and analysis of multicomponent kinetic data using this system are also described. Due to the large number of data points that can be collected and the ability to selectively sample transmittance values in regions where the signal is rapidly changing with time, relatively unsophisticated methods of data analysis can be used. These methods are suitable for use by microcomputers and mean that data analysis and acquisition can be performed on the same microcomputer in real time. To illustrate this, multicomponent analysis of kinetic transients is performed on simulated data and on the dissociation kinetics of the ethidium-DNA complex.     

Product Inhibition in Native-State Proteolysis

The proteolysis kinetics of intact proteins by nonspecific proteases provides valuable information on transient partial unfolding of proteins under native conditions. Native-state proteolysis is an approach to utilize the proteolysis kinetics to assess the energetics of partial unfolding in a quantitative manner. In native-state proteolysis, folded proteins are incubated with nonspecific proteases, and the rate of proteolysis is determined from the disappearance of the intact protein. We report here that proteolysis of intact proteins by nonspecific proteases, thermolysin and subtilisin deviates from first-order kinetics. First-order kinetics has been assumed for the analysis of native-state proteolysis. By analyzing the kinetics of proteolysis with varying concentrations of substrate proteins and also with cleavage products, we found that the deviation from first-order kinetics results from product inhibition. A kinetic model including competitive product inhibition agrees well with the proteolysis time course and allows us to determine the uninhibited rate constant for proteolysis as well as the apparent inhibition constant. Our finding suggests that the likelihood of product inhibition must be considered for quantitative assessment of proteolysis kinetics.     

A new approach in the kinetics of biological transport the potential of reversible inhibition studies

Kinetic equations are derived for reversible inhibition of both active and facilitated transport systems for seven common experimental arrangements. It is shown that the unique features of transport kinetics may be exploited to give new kinds of information. It is also shown that the familiar rules of enzyme kinetics, though often applied to transport, can be seriously misleading. The analysis leads to the following general conclusions: (1) A competitive mechanism frequently gives rise to non-competitive kinetics, depending on the experimental design, but a non-competitive mechanism never produces competitive kinetics. (2) Inhibition studies on exchange diffusion at equilibrium in non-active systems or in the final steady state in active systems are the only unambiguous kinetic tests to distinguish competitive from non-competitive mechanisms. (3) Substrate analogs that are bound to the carrier and transported are readily distinguished by inhibition kinetics from those not transported, even though both may rapidly enter the cell by another route. (4) Even in non-active systems competitive inhibitors commonly have far different affinities for the substrate sites on the two membranes faces: where sufficient non-polarity allows their penetration into the cell, inhibition kinetics readily establish such sidedness in their action. (5) Inhibition kinetics of the mixed competitive and non-competitive type result from moderately asymmetrical binding of inhibitor at the substrate site. (6) Asymmetry is a necessary feature of active transport; hence studies of inhibition kinetics should provide important insights into its mechanism.     

A multienzyme bioluminescent time-resolved pyrophosphate assay

We have developed a high-sensitivity assay for measurement of inorganic pyrophosphate (PPi) in adenosine 5'-triphosphate (ATP)-contaminated samples. The assay is based on time-resolved measurements of the luminescence kinetics and implements multiple enzymes to convert PPi to ATP that is, in turn, utilized to produce light and to hydrolyze PPi for measurement of the steady state background luminescence. A theoretical model for describing luminescence kinetics and optimizing composition of the assay detection mixture is presented. We found that the model is in excellent agreement with the experimental results. We have developed and evaluated two algorithms for PPi measurement from luminescence kinetics acquired from ATP-contaminated samples. The first algorithm is considered to be the method of choice for analysis of long, i.e., 3-5 min, kinetics. The activity of enzymes is controlled during the experiment; the sensitivity of PPi detection is about 7 pg/ml or 15 pM of PPi in ATP-contaminated samples. The second algorithm is designed for analysis of short, i.e., less than 1-min, luminescence kinetics. It has about 20 pM PPi detection sensitivity and may be the better choice for assays in microplate format, where a short measurement time is required. The PPi assay is primarily developed for RNA expression analysis, but it also can be used in various applications that require high-sensitivity PPi detection in ATP-contaminated samples.     

Ion Uptake in Higher Plants and KCl Stimulation of Plasmalemma Adenosine Triphosphatase: Comparison of Models

Criteria for the acceptance or rejection of the dual, the cooperative, and the multiphasic model of ion uptake are given and are used to evaluate the models on the basis of previously published analyses. In addition, mathematical representations of the models are fitted to concentration-dependence data for various ions and plant tissues by a general curve fitting program. The calculated parameters are evaluated for biological relevance, and the fit is compared by statistical analysis. The kinetics of ion uptake in higher plants are found to be consistent with the concept of multiphasic uptake mechanisms, while the dual and the cooperative model must be rejected. KCl stimulation of plasmalemma-bound ATPases is also shown to obey multiphasic kinetics, thus strengthening the correlation between ion uptake and membrane-bound ATPases.     

Aggregation kinetics of extended porphyrin and cyanine dye assemblies

The kinetics of J-aggregate formation has been studied for two chromophores, tetrakis-4-sulfonatophenylporphine in an acid medium and pseudoisocyanine on a polyvinylsulfonate template. The assembly processes differ both in their sensitivity to initiation protocols and in the reaction profiles they produce. The porphyrin's assembly kinetics, for example, displays an induction period unlike that of the cyanine dye. Two kinetic models are presented. For the porphyrin, an autocatalytic pathway in which the formation of an aggregation nucleus is rate-determining appears to be applicable; for the pseudoisocyanine dye, an equation derived for diffusion-limited aggregation of a fractal object satisfactorily fits the data. These models are shown to be useful for the analysis of kinetic data obtained for several biologically important aggregation processes.     

Effects of mutation, truncation, and temperature on the folding kinetics of a WW domain

The purpose of this work is to show how mutation, truncation, and change of temperature can influence the folding kinetics of a protein. This is accomplished by principal component analysis of molecular-dynamics-generated folding trajectories of the triple β-strand WW domain from formin binding protein 28 (FBP28) (Protein Data Bank ID: 1E0L) and its full-size, and singly- and doubly-truncated mutants at temperatures below and very close to the melting point. The reasons for biphasic folding kinetics [i.e., coexistence of slow (three-state) and fast (two-state) phases], including the involvement of a solvent-exposed hydrophobic cluster and another delocalized hydrophobic core in the folding kinetics, are discussed. New folding pathways are identified in free-energy landscapes determined in terms of principal components for full-size mutants. Three-state folding is found to be a main mechanism for folding the FBP28 WW domain and most of the full-size and truncated mutants. The results from the theoretical analysis are compared to those from experiment. Agreements and discrepancies between the theoretical and experimental results are discussed. Because of its importance in understanding protein kinetics and function, the diffusive mechanism by which the FBP28 WW domain and its full-size and truncated mutants explore their conformational space is examined in terms of the mean-square displacement and principal component analysis eigenvalue spectrum analyses. Subdiffusive behavior is observed for all studied systems.     

An Electrostatic Model for DNA Surface Hybridization

DNA hybridization at surfaces is a crucial process for biomolecular detection, genotyping, and gene expression analysis. However, hybridization density and kinetics can be strongly inhibited by electric fields from the negatively charged DNA as the reaction proceeds. Here, we develop an electrostatic model to optimize hybridization density and kinetics as a function of DNA surface density, salt concentrations, and applied voltages. The electrostatic repulsion from a DNA surface layer is calculated numerically and incorporated into a modified Langmuir scheme, allowing kinetic suppression of hybridization. At the low DNA probe densities typically used in assays (<10¹³/cm²), electrostatics effects are largely screened and hybridization is completed with fast kinetics. However, higher hybridization densities can be achieved at intermediate DNA surface densities, albeit with slower kinetics. The application of positive voltages circumvents issues resulting from the very high DNA probe density, allowing highly enhanced hybridization densities and accelerated kinetics, and validating recent experimental measurements.     

Determination of ultrafast protein folding rates from loop formation dynamics

Quenching of the triplet state of tryptophan by contact with cysteine can be used to measure the kinetics of loop formation in unfolded proteins. Here we show that cysteine quenching dynamics also provide a novel method for measuring folding rates when the exchange between folded and unfolded states is faster than the unquenched triplet lifetime (approximately 100 micros). We use this technique to investigate folding/unfolding kinetics of the 35 residue headpiece subdomain of the protein villin, which contains a single tryptophan residue and was engineered to contain a cysteine residue at the N terminus. At intermediate concentrations of denaturant the time-course of the triplet decay consists of two relaxations, the rates and amplitudes of which reveal the fast kinetics for folding and unfolding of this protein. The folding rates extracted using a simple kinetic model are close to those reported previously from laser-induced temperature-jump experiments that employ the change in tryptophan fluorescence as a probe. However, the results differ significantly from those reported from dynamic NMR line shape analysis on a variant with methionine at the N terminus, an issue that remains to be resolved. The analysis of the triplet quenching kinetics also shows that the quenching rates in the unfolded state increase with decreasing denaturant concentration, indicating a compaction of the unfolded protein.     

Applications of graph theory to enzyme kinetics and protein folding kinetics. Steady and non-steady-state systems

Graphic methods have proved to be very useful in enzyme kinetics, as reflected in both raising the efficiency of performing calculations and aiding in the analysis of catalytic mechanisms. The kinetic relations among protein folding states are very similar to those between enzyme-catalyzed species. Therefore, it should be equally useful to provide a visually intuitive relation between kinetic calculations and folding mechanisms for protein folding kinetics, as manifested by the graphic rules in enzyme kinetics. It can actually be anticipated that, due to increasing interest in protein folding, the graphic method will become an important tool in folding kinetics as well. Based on the recent progress made in graphic methods of enzyme kinetics, in this review four graphic rules are summarized, which can be used to deal with protein folding systems as well as enzyme-catalyzed systems. Rules 1-3 are established for deriving the kinetic equations for steady-state processes and Rule 4 for those in the case of non-steady-state processes. In comparison with conventional graphic methods, which can only be applied to a steady-state system, the current rules have the following advantages: (1) Complicated and tedious calculations can be greatly simplified. (2) A lot of wasted labor can be turned away. (3) Final results can be double-checked by a formula provided in each of the graphic rules. (4) Transient kinetic systems can also be treated. The mathematical proof of Rules 1-4 is given in appendices A-D, respectively.     

Determination of Michaelis parameters from differentials of progress curves.

First differentials of progress curves are easily obtainable in many enzyme assay systems. Such curves may be more readily applicable to kinetic analysis than are the usual progress curves. The theory for this approach is developed, and simple graphical procedures for the determination of Michaelis parameters are indicated. By using an electronic differentiator device the application of the method is demonstrated on the kinetics of three different serine proteinases with various synthetic substrates. Whenever the steady-state concentration of an intermediate of the reaction is proportional to the rate, the transition of this intermediate in substrate-depletion experiments may be analysed in similar terms. This is demonstrated with cytochrome c oxidase kinetics. A number of other possible applications are discussed.     

Relationship between extreme pathways and structurally minimal pathways

The determination of reaction pathways is one of the most important functions that should be performed in exploring the kinetics of catalyzed chemical reactions or biochemical reactions, the latter being generally catalyzed by enzymes. It is proven that the terms, “type-I extreme pathway” and “structurally minimal pathway”, both introduced to characterize the kinetics of a catalyzed reaction are equivalent. These two terms are based on two distinct methodologies, one mainly rooted in convex analysis and the other in graph theory. The equivalence promises further even more effective methods for reaction-pathway identification by synergistic integration of existing ones.     

Mathematical models for hollow-fiber enzyme reactors

Two analytically solved mathematical models are presented for a reactor ystem employing immobilized whole cells as a biocatalyst. The whole cells are entrapped or pumped through the shell side of the dialyzer reactor unit. The reactant mixture is circulated through the cialyzer tube side. Nutrient diffuses across the hollow fiber membrane from the tube side to the shell side, where it reacts to form product, which then back diffuses into the reactant mixture stream. The use of a high recirculation ratio of nutrient through the dialyzer tubes to nutrient feed rate to the entire system, allows the system to be modeled as a continuous-flow stirred-tank reactor. The first analysis details the development of an effectiveness-factor correlation for first- and zero-order kinetics. The second analysis presents the solution to an unsteady-state-system mass balance with Michaelis–menten kinetics.