Determination the rate constants of some biexponential reactions

Reactions that are described by biexponential functions are typical for many biological processes. The kinetics of these reactions is described by transcendental irrational equations interconnecting the reagent concentrations, time and rate constants. Meantime, their graphical representation in the semi-logarithmic coordinates can be decomposed into two straight lines that intercept at some angle. New simple methods for asymptotic numerical solution of the equations describing these reactions are suggested. These methods permit determining the rate constants using the kinetic data of initial substance concentration, which transform into final product according to a two-component model, a sequential model or a competitive model.     

A mathematical model for radiation-induced myelopoiesis

A model for damage, repair, killing, and repopulation of myelopoietic marrow is presented. Evaluation produces time and dose-rate profiles during and following any complex irradiation. Equations model variable dose rates, multiple exposures, different sources, and arbitrary intervals between treatments. If factors which dominate the control of biological processes can be demonstrated, an option is to set biological rate constants to experimentally determined values. Previously, knowledge did not permit identification of dominating biological processes and their temporal rates. But a unique feature of this study is that unspecified lesions for killing and injury of cells are evaluated from mortality data on the animal species of choice. "Unspecified" is used to indicate a condition of assumption-free modeling of molecular processes, whereby rate constants for cellular effects are simply computed directly from animal mortality data. Coefficients (estimated by maximum-likelihood methods for nonspecific processes) are compared with experimental values for specific processes. The model has many uses, including modeling of the myelopoietic potential as a function of time. Another option is to calculate the whole-body survival curve for cells that control myelopoiesis as a result of the treatment schedule. Also through simple extensions of the model, an extremely complex protocol can be identified with an equivalent prompt dose value--even for partial-body, fractionated exposures.     

A curvilinear approach to the kinetic analysis of linoleate peroxidation in aqueous liposomes by 2,2'azobis(2-amidoinopropane) dihydrochloride

Lipid peroxidation is a common feature of many chemical and biological processes, and is governed by a complex kinetic scheme. A fundamental stage in kinetic investigations of lipid peroxidation is the accurate determination of the rate of peroxidation, which in many instances is heavily reliant on the method of finite differences. Such numerical approximations of the first derivative are commonly employed in commercially available software, despite suffering from considerable inaccuracy due to rounding and truncation errors. As a simple solution to this, we applied three empirical sigmoid functions (viz. the Prout-Tompkins, Richards & Gompertz functions) to data obtained from the AAPH-mediated peroxidation of aqueous linoleate liposomes in the presence of increasing concentrations of Trolox, evaluating the curve fitting parameters using the widely available Microsoft Excel Solver add-in. We have demonstrated that the five-parameter Richards' function provides an excellent model for this peroxidation, and when applied to the determination of fundamental rate constants, produces results in keeping with those available in the literature. Overall, we present a series of equations, derived from the Richards' function, which enables direct evaluation of the kinetic measures of peroxidation. This procedure has applicability not only to investigations of lipid peroxidation, but to any system exhibiting sigmoid kinetics.     

Kinetic mechanism of the fastest motor protein, Chara myosin

Chara corallina class XI myosin is by far the fastest molecular motor. To investigate the molecular mechanism of this fast movement, we performed a kinetic analysis of a recombinant motor domain of Chara myosin. We estimated the time spent in the strongly bound state with actin by measuring rate constants of ADP dissociation from actin.motor domain complex and ATP-induced dissociation of the motor domain from actin. The rate constant of ADP dissociation from acto-motor domain was >2800 s(-1), and the rate constant of ATP-induced dissociation of the motor domain from actin at physiological ATP concentration was 2200 s(-1). From these data, the time spent in the strongly bound state with actin was estimated to be <0.82 ms. This value is the shortest among known values for various myosins and yields the duty ratio of <0.3 with a V(max) value of the actin-activated ATPase activity of 390 s(-1). The addition of the long neck domain of myosin Va to the Chara motor domain largely increased the velocity of the motility without increasing the ATP hydrolysis cycle rate, consistent with the swinging lever model. In addition, this study reveals some striking kinetic features of Chara myosin that are suited for the fast movement: a dramatic acceleration of ADP release by actin (1000-fold) and extremely fast ATP binding rate.     

Nonlinear Regression Analysis of the time Course of Ligand Binding Experiments

Abstract

The binding and release of hormones and growth factors are often relatively slow processes under biological conditions. Consequently, a knowledge of the underlying rate constants may be of greater physiological relevance than the equilibrium constant. Here we show how, by following a single time course of binding, the rate constants for both binding and release can be determined. The ratio of these rate constants allows the binding constant to be calculated. A nonlinear regression computer program is described which facilitates these calculations and which provides estimates and standard errors of the constants determined. The method is illustrated by the binding of human growth hormone to the human growth hormone binding protein, and the binding of ovine prolactin to the rabbit prolactin receptor.     

Analyzing complicated protein folding kinetics rapidly by analytical Laplace inversion using a Tikhonov regularization variant

Kinetic experiments provide much information about protein folding mechanisms. Time-resolved signals are often best described by expressions with many exponential terms, but this hinders the extraction of rate constants by nonlinear least squares (NLS) fitting. Numerical inverse Laplace transformation, which converts a time-resolved dataset into a spectrum of amplitudes as a function of rate constant, allows easy estimation of the rate constants, amplitudes, and number of processes underlying the data. Here, we present a Tikhonov regularization-based method that converts a dataset into a rate spectrum, subject to regularization constraints, without requiring an iterative search of parameter space. This allows more rapid generation of rate spectra as well as analysis of datasets too noisy to process by existing iterative search algorithms. This method's simplicity also permits highly objective, largely automatic analysis with minimal human guidance. We show that this regularization method reproduces results previously obtained by NLS fitting and that it is effective for analyzing datasets too complex for traditional fitting methods. This method's reliability and speed, as well as its potential for objective, model-free analysis, make it extremely useful as a first step in analysis of complicated noisy datasets and an excellent guide for subsequent NLS analysis.     

In vivo significance of kinetic constants of protein proteinase inhibitors

We describe the in vivo significance of the kinetic parameters which characterize the interaction between proteinases and protein proteinase inhibitors. Knowledge of the second-order association rate constant kass and in vivo inhibitor concentration allows the calculation of the delay time of inhibition, i.e., the time required for complete inhibition of a proteinase in vivo. The influence of biological substrates on the delay time is also analyzed. The extent of substrate breakdown during the delay time of inhibition may be computed from the various constants describing the proteinase/substrate/inhibitor interactions and the biological concentrations of proteinase and inhibitor. The in vivo partition of a proteinase between two inhibitors may be calculated if the kinetic parameters are known. We define a stability time for enzyme-inhibitor complexes as a minimal time during which the complexes may be considered as stable. This time is related to kdiss the dissociation rate constant of the reversible enzyme-inhibitor complex or to k, the breakdown rate constant of the complex formed with temporary inhibitors. The overall stability of the complex depends upon the ratio between the inhibitor concentration and Ki, the equilibrium dissociation constant of the complex. If this ratio is higher than 1000, a reversible inhibitor behaves like an irreversible one in vivo whatever the enzyme concentration.     

Monte Carlo simulation of diffusion and reaction in two-dimensional cell structures.

Diffusion and reaction processes control the dynamics of many different biological systems. For example, tissue respiration can be limited by the delivery of oxygen to the cells and to the mitochondria. In this case, oxygen is small and travels quickly compared with the mitochondria, which can be considered as immobile reactive traps in the cell cytoplasm. A Monte Carlo theoretical investigation quantifying the interplay of diffusion, reaction, and structure on the reaction rate constant is reported here for diffusible particles in two-dimensional, reactive traps. The placement of traps in overlapping, nonoverlapping, and clustered spatial arrangements can have a large effect on the rate constant when the process is diffusion limited. However, under reaction-limited conditions the structure has little effect on the rate constant. For the same trap fractions and reactivities, nonoverlapping traps have the highest rate constants, overlapping traps yield intermediate rate constants, and clustered traps have the lowest rate constants. An increase in the particle diffusivity in the traps can increase the rate constant by reducing the time required by the particles to reach reactive sites. Various diffusive, reactive, and structural conditions are evaluated here, exemplifying the versatility of the Monte Carlo technique.     

A simple function for maternal-age-specific rates of Down syndrome in the 20-to-49-year age range and its biological implications.

The familial increase in the rate of Down syndrome with maternal age can be represented by a simple equation, consisting of the sum of a constant term plus an exponential term that is a first-order function of masternal age: y = a + exp (b + cx), where y is the rate in live births, x is maternal age, and a, b, and c are constants. Unlikely analyses in which two separate equations were derived from different segments of the 20 to 49 maternal age range, this single, simple equation can be applied to the entire range. An unlike previous complex equations that were derived by regression analysis for the entire age range, the component terms can be readily understood as contributions by different etiologic categories. This model fits the data recently available by 1-year intervals about as well as the approach that used separate equations, but it has fewer parameters and requires no ad hoc division of the age range. However, it does not postulate a sharp transition in biological processes around maternal age 30, but, rather, a process continuously accumulating at a constant exponential rate (analogous to that produced by an infectious mechanism), superimposed upon a constant background rate.     

Determination of rate distributions from kinetic experiments.

Rate processes in proteins are often not adequately described by simple exponential kinetics. Instead of modeling the kinetics in the time domain, it can be advantageous to perform a numerical inversion leading to a rate distribution function f(lambda). The features observed in f(lambda) (number, positions, and shapes of peaks) can then be interpreted. We discuss different numerical techniques for obtaining rate distribution functions, with special emphasis on the maximum entropy method. Examples are given for the application of these techniques to flash photolysis data of heme proteins.     

Binding, internalization, and intracellular processing of protein ligands. Derivation of rate constants by computer modeling

Information about ligand binding, dissociation, internalization, and intracellular processing and about receptor turnover, processing, and insertion into the membrane is contained in the time-dependent changes in concentrations of membrane-associated and internalized ligand. Single experiments similar in design to those typically performed for Scatchard analyses of binding data conducted at physiological temperature and in the absence of inhibitors of ligand-receptor complex internalization and degradation can provide kinetic data sufficient to permit derivation of all the respective rate constants by numerical methods. We developed an analytical solution of the kinetic model which assumes that all of these processes follow first order kinetics. The model represents interactions of surface receptors (R)s, the surface ligand-receptor complex (LR)s and internalized receptor-ligand complex (LR)I: d[R]S/dt = Vr - kt[R]S - ka[L] [R]S + kd [LR]S; d[LR]S/dt = ka[L] [R]S - kd[LR]S - ke[LR]S; d[LR]I/dt = ke[LR]S - kh[LR]I; Vr is the constant rate of insertion of receptors into the membrane, kt is the internalization rate constant for free receptors, ka and kd are association and dissociation rate constants for ligand-surface receptor interaction, ke is the internalization rate constant for ligand-receptor complexes, and kh is the intracellular ligand decomposition rate constant. The interaction of radioiodinated human recombinant interferon-alpha 2a with the human alveolar lung carcinoma cell line, A549, was adequately accounted for by the model. The rate constants, numerically derived from time-dependent concentrations of surface-bound and internalized ligand of other systems taken from the literature, were in agreement with values of these rate constants individually measured by steady-state experiments. In cases where the fate of internalized radioactivity was more complex than assumed by the model, the parameters ka, kt, (kd + ke) and Vr could be derived from the time dependence of [LR]S.     

A Theoretical and Experimental Study of Competition Between Solution and Surface Receptors for Ligand in a Biacore Flow Cell

Rate constants that characterize the kinetics of binding and dissociation between biomolecules carry fundamental information about the biological processes these molecules are involved in. An instrument that is widely used to determine these rate constants is the Biacore. In a Biacore experiment, one of the reactants, which we will call the receptor, is immobilized on a sensor chip. During the binding phase of the experiment the other reactant flows past the chip. After binding, buffer alone is introduced into the flow cell and dissociation is monitored. Often surface-based binding assays are influenced by the transport of the reactant in solution, complicating the determination of the chemical rate constants from the observed binding kinetics. We propose a new way to determine the dissociation rate constant by adding soluble receptor during dissociation. The method is tested first on simulated data and then on Biacore experiments where the lac repressor protein binds and dissociates from a stretch of double stranded DNA containing the lac repressor binding site. With this method we find a dissociation rate constant k_d=0.075 ± 0.005s^-1, a value that is faster than previously obtained from Biacore experiments. In developing our method to analyze these experiments we obtain an expression for the transport limited rate constant for a Biacore experiment when soluble receptor is present during dissociation.     

An exact procedure for the analysis of heterogeneous kinetics in polyclonal antibody populations

Antibody populations with heterogeneous binding properties exhibit complex first-order dissociation kinetics. An analytical method has been developed to determine the average dissociation rate constant and the heterogeneity index of a specific antibody population. This procedure was based on Laplace transformation of the gamma distribution function, which yielded an exact, macroscopic rate law for the entire antibody population. Linearization of the macroscopic rate law is achieved by plotting data points versus their numerical derivatives using log-log axes. Linear regression of such plots yields the average dissociation rate constant from the Y-intercept, and heterogeneity index from the slope. This analytic method is transparent to the antibody system and kinetic assay employed, requiring only a programmable calculator to perform the necessary calculations. The usefulness of this analytic method was demonstrated by the evaluation of dissociation kinetics in murine monoclonal and rabbit polyclonal anti-fluorescyl-IgG antibody populations.     

Oligomerization-dependent changes in the thermodynamic properties of the TPR-MET receptor tyrosine kinase

Although oligomerization of receptor tyrosine kinases (RTKs) is necessary for receptor activation and signaling, a quantitative understanding of how oligomerization mediates these critical processes does not exist. We present a comparative thermodynamic analysis of functionally active dimeric and functionally inactive monomeric soluble analogues of the c-MET RTK, which clearly reveal that oligomerization regulates the binding affinity and binding kinetics of the kinase toward ATP and tyrosine-containing peptide substrates. Thermodynamic binding data for oligomeric c-MET were obtained from the dimeric TPR-MET oncoprotein, a functionally active fusion derivative of the c-MET RTK. This naturally occurring oncoprotein contains the cytoplasmic domain of c-MET fused to a coiled coil dimerization domain from the nuclear pore complex. Comparative data were obtained from a soluble monomeric kinase compromising the c-MET cytoplasmic domain (cytoMET). Significantly, under equilibrium binding conditions, the oligomeric phosphorylated kinase showed a significantly lower dissociation constant (K(d,dimer) = 11 microM) for a tyrosine-containing peptide derived from the C-terminal tail of the c-MET RTK when compared to the phosphorylated monomeric kinase cytoMET (K(d,monomer) = 140 microM). Surprisingly, equilibrium dissociation constants measured for the kinase and ATP were independent of the oligomerization state of the kinase (approximately 10 microM). Stopped-flow analysis of peptide substrate binding showed that the association rate constants (k(2)) differed 2-fold and dissociation rate constants (k(-2)) differed 10-fold when phosphorylated TPR-MET was compared to phosphorylated cytoMET. ATP binding abrogated the differences in k(2) rates observed between the two oligomeric states of the c-MET cytoplasmic domain. These results clearly imply that oligomerization induces important thermodynamic and conformational changes in the substrate binding regions of the c-MET protein and provide quantitative mechanistic insights into the necessary role of oligomerization in RTK activation.     

Complex formation between the copper protein, azurin and the cytochrome c peroxidase of Pseudomonas aeruginosa.

Reduced azurin reacts with the resting, oxidized cytochrome c peroxidase of Pseudomonas aeruginosa to yield time courses observed at 420 nm, which consist of the sum of two exponential processes. Each process exhibits a hyperbolic dependence of the observed rate constant on the reduced azurin concentration. The fraction of the total optical density change which each process contributes is found to be dependent on the reduced azurin concentration. This pattern of reactivity is maintained at pH values between 5.5 and 8.0. The data has been analyzed in terms of a complex formation between the two proteins followed by an intramolecular electron exchange reaction. This analysis yields values for the binding constants at each pH value. The intramolecular exchange reaction is independent of pH, whilst the pH dependence of the binding reaction suggests the involvement of a histidine residue in this process.     

Model calculations on the kinetics of oligonucleotide double helix coil transitions. Evidence for a fast chain sliding reaction

Relaxation data obtained previously for the double helix coil transition of oligoriboadenylates and oligoribouridylates are compared to the results of numerical calculations according to various models. In these models the helix coil transition is described by individual rate constants for the first steps of helix formation, whereas the rate constants of the following steps of helix chain growth are assumed to be uniform. The existence of various helix intermediates containing the same number of base pairs is accounted for by statistical factors. First a quasistationary treatment of a zipper model is used for an analysis of the influence of various model parameters. Then relaxation spectra are calculated including helix coil intermediates explicitly without any assumption of quasistationarity. The relaxation spectrum calculated for any chain length N comprises N—1 fast processes with time constants in the range of 0.1 to 0.5 μs and one slow process with a time constant τ depending upon the nucleotide concentration (τ is usually in the ms time range). The fast processes are associated mainly with the unzippering at helix ends and are usually characterized by relatively small amplitudes, whereas the slow process represents the overall helix coil transition usually characterized by a very large amplitude.Consideration of staggered helix series (where the different helix scries are coupled to each other by the single stranded state) leads to a spectrum of slow relaxation processes with one separate relaxation process for each helix series. It is shown that this “non-sliding” staggering zipper model is not consistent with the experimental results. The measured relaxation curves can be represented by single exponentials for nucleotide chain lengths 8 to 11 (within experimental accuracy). This is also true for conditions where several, clearly separated time constants should be expected according to the theoretical model. The experimental data suggest the existence of a direct coupling between different series of staggered helices by a chain sliding mechanism with a time constant < 1ms. Chain sliding may be explained by diffusion of helix defects along the double helix such as diffusion of small loops. A simple model calculation for the diffusion of a bulge loop assuming quasistationarity suggests a sliding time constant around 100 μs for a helix comprising 10 base pairs.Finally some thermodynamic and kinetic parameters are evaluated according to the “sliding” staggering zipper model: The negative activation enthalpy observed for helix recombination can he described using a series of nucleation parameters indicating reduced stability constants for the first three base pairs. Nucleation may usually be achieved with the formation of the third or fourth base pair depending upon the magnitude of the chain growth parameter. The rate constant of helix chain growth is around 10⁶ s^?1 at 0.05 M [Na⁺] and increases to about 4 × 10⁶ s^?1 at 0.17 M [Na⁺].     

A soluble form of the avian hepatitis B virus receptor. Biochemical characterization and functional analysis of the receptor ligand complex

Avian hepatitis B virus infection is initiated by the specific interaction of the extracellular preS part of the large viral envelope protein with carboxypeptidase D (gp180), the primary cellular receptor. To functionally and biochemically characterize this interaction, we purified a soluble form of duck carboxypeptidase D from a baculovirus expression system, confirmed its receptor function, and investigated the contribution of different preS sequence elements to receptor binding by surface plasmon resonance analysis. We found that preS binds duck carboxypeptidase D with a 1:1 stoichiometry, thereby inducing conformational changes but not oligomerization. The association constant of the complex was determined to be 2.2 x 10(7) M-1 at 37 degreesC, pH 7.4, with an association rate of 4.0 x 10(4) M-1 s-1 and a dissociation rate of 1.9 x 10(-3) s-1, substantiating high affinity interaction of avihepadnaviruses with their receptor carboxypeptidase D. The separately expressed receptor-binding domain, comprising about 50% of preS as defined by mutational analysis, exhibits similar constants. The domain consists of an essential element, probably responsible for the initial receptor contact and a part that contributes to complex stabilization in a conformation sensitive manner. Together with previous results from cell biological studies these data provide new insights into the initial step of hepadnaviral infection.     

A method for determining a rate constant for biexponential processes using data on dynamics of accumulating the end product of a reaction

A new method the rate constant determination of some biexponential processes is suggested. The method is based on the asymptotic solution of transcendental irrational equations, described such processes. This method can be used when kinetics of the final product of reaction is known. The values of the rate constants obtained by suggested method are precise, if the data used for calculation are also precise. In the other case methods of mathematical statistics should be used for evaluation of the slopes of kinetics curves in the semi-logarithmic coordinates.     

Single turnover kinetics of the reaction between oxycytochrome P-450cam and reduced putidaredoxin

A study of the single turnover kinetics of the reaction between oxycytochrome P-450cam and reduced putidaredoxin was performed using the inhibitor metyrapone to trap the cytochrome immediately after release of the product, 5-exo-hydroxycamphor. EPR determinations of the concentrations of reduced putidaredoxin and ferric metyrapone-bound cytochrome at the same time points showed that there is no time lag between the oxidation of reduced putidaredoxin and the appearance of metyrapone-bound cytochrome. This implies that the rate constant for electron transfer is smaller than the rate constant for the later processes involved in product formation and release, lumped into a single step. Taking this restriction into account and doing computer simulation of absorbance versus time curves, previously obtained at various putidaredoxin concentrations using stopped-flow spectrophotometry, allowed bounds to be determined for rate constants of the processes within the reaction. At 4 degrees C in buffer at pH 7.4 with 0.50 M KCl, the rate constant for the bimolecular association of the two enzymes is between 3 and 20/microM.s; the rate constant for dissociation is between 12 and 600/s; the rate constant for electron transfer is between 60 and 100/s; and the rate constant for the later processes is at least 200/s.