The kinetic pathway of folding of barnase

To search for folding intermediates, we have examined the folding and unfolding kinetics of wild-type barnase and four representative mutants under a wide range of conditions that span two-state and multi-state kinetics. The choice of mutants and conditions provided in-built controls for artifacts that might distort the interpretation of kinetics, such as the non-linearity of kinetic and equilibrium data with concentration of denaturant. We measured unfolding rate constants over a complete range of denaturant concentration by using by 1H/2H-exchange kinetics under conditions that favour folding, conventional stopped-flow methods at higher denaturant concentrations and continuous flow. Under conditions that favour multi-state kinetics, plots of the rate constants for unfolding against denaturant concentration fitted quantitatively to the equation for three-state kinetics, with a sigmoid component for a change of rate determining step, as did the refolding kinetics. The position of the transition state on the reaction pathway, as measured by solvent exposure (the Tanford beta value) also moved with denaturant concentration, fitting quantitatively to the same equations with a change of rate determining step. The sigmoid behaviour disappeared under conditions that favoured two-state kinetics. Those data combined with direct structural observations and simulation support a minimal reaction pathway for the folding of barnase that involves two detectable folding intermediates. The first intermediate, I(1), is the denatured state under physiological conditions, D(Phys), which has native-like topology, is lower in energy than the random-flight denatured state U and is suggested by molecular dynamics simulation of unfolding to be on-pathway. The second intermediate, I(2), is high energy, and is proven by the change in rate determining step in the unfolding kinetics to be on-pathway. The change in rate determining step in unfolding with structure or environment reflects the change in partitioning of this intermediate to products or starting materials.     

Is protein folding rate dependent on number of folding stages? Modeling of protein folding with ferredoxin-like fold

Statistical analysis of protein folding rates has been done for 84 proteins with available experimental data. A surprising result is that the proteins with multi-state kinetics from the size range of 50–100 amino acid residues (a.a.) fold as fast as proteins with two-state kinetics from the same size range. At the same time, the proteins with two-state kinetics from the size range 101–151 a.a. fold faster than those from the size range 50–100 a.a. Moreover, it turns out unexpectedly that usually in the group of structural homologs from the size range 50–100 a.a., proteins with multi-state kinetics fold faster than those with two-state kinetics. The protein folding for six proteins with a ferredoxin-like fold and with a similar size has been modeled using Monte Carlo simulations and dynamic programming. Good correlation between experimental folding rates, some structural parameters, and the number of Monte Carlo steps has been obtained. It is shown that a protein with multi-state kinetics actually folds three times faster than its structural homologs.     

Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics

The oxygen diffusion in a spherical cell is analyzed in the present work. An oxygen uptake kinetics of the Michaelis-Menten type is employed. The oxygen uptake kinetics predicts the oxygen uptake rates which agree fairly well with the observed data. It has been found that difference between the predicted steady state oxygen tension distribution using the previous simplified oxygen uptake kinetics and that using the present non-linear kinetics is very significant.     

Probing the folding and unfolding dynamics of secondary and tertiary structures in a three-helix bundle protein

Fast relaxation kinetics studies of the B-domain of staphylococcal protein A were performed to characterize the folding and unfolding of this small three-helix bundle protein. The relaxation kinetics were initiated using a laser-induced temperature jump and probed using time-resolved infrared spectroscopy. The kinetics monitored within the amide I' absorbance of the polypeptide backbone exhibit two distinct kinetics phases with nanosecond and microsecond relaxation times. The fast kinetics relaxation time is close to the diffusion limits placed on protein folding reactions. The fast kinetics phase is dominated by the relaxation of the solvated helix (nu = 1632 cm(-1)), which reports on the fast relaxation of the individual helices. The slow kinetics phase follows the cooperative relaxation of the native helical bundle core that is monitored by both solvated (nu = 1632 cm(-1)) and buried helical IR bands (nu = 1652 cm(-1)). The folding rates of the slow kinetics phase calculated over an extended temperature range indicate that the core formation of this protein follows a pathway that is energetically downhill. The unfolding rates are much more strongly temperature-dependent indicating an activated process with a large energy barrier. These results provide significant insight into the primary process of protein folding and suggest that fast formation of helices can drive the folding of helical proteins.     

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.     

Time-resolved fluorescence spectroscopy of spinach chloroplast

Picosecond fluorescent kinetics and time-resolved spectra of spinach chloroplast were measured at room temperature and low temperatures. The measurement is conducted with 530 nm excitation at an average intensity of 2 · 10¹⁴ photons/cm², pulse and at a pulse separation of 6 ns for the 100 pulses used. The 685 nm fluorescent kinetics was found to decay with two components, a fast component with a 56 ps lifetime, and a slow component with a 220 ps lifetime. The 730 nm fluorescent kinetics at room temperature is a single exponential decay with a 100 ps lifetime. The 730 nm fluorescence lifetime was found to increase by a factor of 6 when the temperature was lowered from room temperature to 90 K, while the 685 and 695 nm fluorescent kinetics were unchanged. The time-resolved spectra data obtained within 10 ps after excitation is consistent with the kinetic data reported here. A two-level fluorescence scheme is proposed to explain the kinetics. The effect of excitation with high light intensity and multiple pulses is discussed.     

Probing the kinetics of single molecule protein folding

We propose an approach to integrate the theory, simulations, and experiments in protein-folding kinetics. This is realized by measuring the mean and high-order moments of the first-passage time and its associated distribution. The full kinetics is revealed in the current theoretical framework through these measurements. In the experiments, information about the statistical properties of first-passage times can be obtained from the kinetic folding trajectories of single molecule experiments (for example, fluorescence). Theoretical/simulation and experimental approaches can be directly related. We study in particular the temperature-varying kinetics to probe the underlying structure of the folding energy landscape. At high temperatures, exponential kinetics is observed; there are multiple parallel kinetic paths leading to the native state. At intermediate temperatures, nonexponential kinetics appears, revealing the nature of the distribution of local traps on the landscape and, as a result, discrete kinetic paths emerge. At very low temperatures, exponential kinetics is again observed; the dynamics on the underlying landscape is dominated by a single barrier. The ratio between first-passage-time moments is proposed to be a good variable to quantitatively probe these kinetic changes. The temperature-dependent kinetics is consistent with the strange kinetics found in folding dynamics experiments. The potential applications of the current results to single-molecule protein folding are discussed.     

A general model of reaction kinetics in biological systems

Dynamic mathematical models in biotechnology require, besides the information about the stoichiometry of the biological reaction system, knowledge about the reaction kinetics. Modulation phenomena like limitation, inhibition and activation occur in different forms of competition with the key enzymes responsible for the respective metabolic reaction steps. The identification of a priori unknown reaction kinetics is often a critical task due to the non-linearity and (over-) parameterization of the model equations introduced to account for all the possible modulation phenomena. The contribution of this paper is to propose a general formulation of reaction kinetics, as an extension of the Michaelis-Menten kinetics, which allows limitation/activation and inhibition effects to be described with a reduced number of parameters. The versatility of the new model structure is demonstrated with application examples.     

Non-photochemical quenching kinetics during the dark to light transition in relation to the formation of antheraxanthin and zeaxanthin

Nonlinear regression analysis (NLR) is applied to quantify the dynamic response of non-photochemical fluorescence quenching (NPQ) of Trifolium repens cv. Regal upon dark to light transition. Commonly, only steady-state levels of NPQ are evaluated, ignoring transient kinetics. Experimental NPQ kinetics are fitted best with a sum of two functions: a sigmoidal Hill function plus a transient logarithmic normal function. It is shown that not only steady-state level of NPQ, but also the speed at which steady state is reached, increased with light intensity. The question is raised which biological processes cause the induction of the components of NPQ kinetics. The NPQ kinetics are found to resemble the kinetics of antheraxanthin and zeaxanthin formation during a dark to light transition. Furthermore, both molecules are known to induce NPQ. The hypothesis is put forward that a transient phase of NPQ (0-2 min after transition) is dependent upon concentrations of antheraxanthin, whereas the saturating phase corresponds with the production of zeaxanthin. A mathematical model, based on the presented hypothesis, predicts the effect of increasing light intensity on concentrations of antheraxanthin and zeaxanthin which correspond with experimental results. Implications of the hypothesis are discussed as well as the role of NLR in evaluating chlorophyll a fluorescence kinetics.     

Kinetics of proton diffusion in the regimes of fast and slow exchange between the membrane surface and the bulk solution

The phenomenological model developed in our recent publications [9,10] is used to investigate the kinetics of proton diffusion from a source to a detector on the membrane surface. In most cases the observed kinetics shows a single diffusional maximum with the exponential ascending front and the power-law descending tail. The kinetics depends on the distance between the source and the detector. If the detector is located inside the proton collecting antenna, the kinetics corresponds to the surface diffusion at the times near the maximum and shortly thereafter, and it turns into the bulk diffusion kinetics at longer times, after the equilibrium is established between the membrane surface and the bulk solution. If the detector is located outside the antenna, the kinetics corresponds to the bulk diffusion at all times where the signal is nonvanishing. What is seen at locations near the antenna radius depends on the exchange regime. In the regime of fast exchange between the surface and the bulk as compared to the bulk diffusion, the kinetics shows a single peak whose location is intermediate between the peaks for the surface and bulk diffusion. In the regime of slow exchange there are two maxima corresponding to the surface and bulk diffusion. In buffered solutions the antenna radius decreases with increasing buffer concentration, which changes the kinetics from the surface to bulk diffusion. The theory is applied to interprete recent experiments on a phospholipid membrane [25]. It is found that (i) the fast exchange regime is operating since only a single maximum is observed; (ii) the shift of the maximum toward longer times with increasing buffer concentration is a manifestation of the transition from the surface to bulk diffusion kinetics. The authors are grateful to Yu. N. Antonenko and A. I. Kotelnikov for helpful discussions. This work was supported by the Russian Foundation for Basic Research (05-03-32104), U.S. Civilian Research and Development Foundation (RUC2-2658-MO-05), U. S. National Science Foundation, and U. S. National Institutes of Health.     

Modeling of biofilm reactors with consecutive reactions

Modeling of biofilm reactors has been carried out by several authors. Most of the models use first-order or zero-order kinetics, because of the simplicity of the solution of the mathematical problem. However, the reaction kinetics for many practical situations is a non-linear Monod kinetics, which requires numerical solutions. This paper deals with the modeling of biofilm reactors and effectiveness factor calculations for a biofilm particle with Monod kinetics and two consecutive reactions. The model is applied to biological denitrification in a fluidized bed bioreactor, in which the liquid phase is assumed to be in plug flow. Effectiveness factors of biofilm are numerically calculated by solving the system of ODES by orthogonal collocation. Axial concentration profiles of nitrate and nitrite species are calculated and compared with experimental results.     

Growth and cometabolic reduction kinetics of a uranium- and sulfate-reducing Desulfovibrio/Clostridia mixed culture: Temperature effects

Bioremediation of contaminated soils and aquifers is subject to spatial and temporal temperature changes that can alter the kinetics of key microbial processes. This study quantifies temperature effects on the kinetics of an ethanol-fed sulfate-reducing mixed culture derived from a uranium-contaminated aquifer subject to seasonal temperature fluctuations. The mixed culture contains Desulfovibrio sp. and a Clostridia-like organism. Rates of growth, ethanol utilization, decay, and uranium reduction decreased with decreasing temperature. No significant uranium reduction was observed at 10 degrees C. While both Monod saturation kinetics and pseudo second-order kinetics adequately described the rates of growth and utilization of electron donor (ethanol), model parameters for the pseudo second-order expression had smaller uncertainties. Uranium reduction kinetics were best described by pseudo second-order kinetics modified to include a term for inactivation/death of cells.     

Evidence for complexity at the nanometer scale of radiation-induced DNA DSBs as a determinant of rejoining kinetics

The rejoining kinetics of double-stranded DNA fragments, along with measurements of residual damage after postirradiation incubation, are often used as indicators of the biological relevance of the damage induced by ionizing radiation of different qualities. Although it is widely accepted that high-LET radiation-induced double-strand breaks (DSBs) tend to rejoin with kinetics slower than low-LET radiation-induced DSBs, possibly due to the complexity of the DSB itself, the nature of a slowly rejoining DSB-containing DNA lesion remains unknown. Using an approach that combines pulsed-field gel electrophoresis (PFGE) of fragmented DNA from human skin fibroblasts and a recently developed Monte Carlo simulation of radiation-induced DNA breakage and rejoining kinetics, we have tested the role of DSB-containing DNA lesions in the 8-kbp-5.7-Mbp fragment size range in determining the DSB rejoining kinetics. It is found that with low-LET X rays or high-LET alpha particles, DSB rejoining kinetics data obtained with PFGE can be computer-simulated assuming that DSB rejoining kinetics does not depend on spacing of breaks along the chromosomes. After analysis of DNA fragmentation profiles, the rejoining kinetics of X-ray-induced DSBs could be fitted by two components: a fast component with a half-life of 0.9+/-0.5 h and a slow component with a half-life of 16+/-9 h. For alpha particles, a fast component with a half-life of 0.7+/-0.4 h and a slow component with a half-life of 12+/-5 h along with a residual fraction of unrepaired breaks accounting for 8% of the initial damage were observed. In summary, it is shown that genomic proximity of breaks along a chromosome does not determine the rejoining kinetics, so the slowly rejoining breaks induced with higher frequencies after exposure to high-LET radiation (0.37+/-0.12) relative to low-LET radiation (0.22+/-0.07) can be explained on the basis of lesion complexity at the nanometer scale, known as locally multiply damaged sites.     

Hypothesis formulation and testing in aquatic bioassays: a deterministic model approach

The paper examines the significance of toxicant kinetics information obtained from aquatic toxicity bioassays and bioconcentration tests. The data, bioconcentration kinetics and acute mortality versus exposure-duration information for juvenile American flagfish (Jordanella foridae) exposed to 1,4-dichlorobenzene, are interpreted in terms of a one-compartment, first-order kinetics model. The output of the model is used to formulate a testable hypothesis regarding the comparison of toxicant kinetics derived from both bioconcentration test exposures and toxicity bioassays. The model's estimates of the toxicant body burden attained at mortality are compared with theoretical and observed body burdens from literature sources. The use of a simple, deterministic residue-based, one-compartment, first-order kinetics model to evaluate existing data, as well as to formulate hypotheses to direct experimental designs, is examined.     

Steady state kinetics of an enzyme reaction with one substrate and one modifier

This study examines the steady state kinetics of a reaction involving an enzyme, a substrate and a modifier when the reaction follows Michaelis-Menten kinetics. Conditions for Michaelis-Menten kinetics are deduced, and it is shown that an analogue of detailed balance determines the complexity of the rate equations in these cases. A scheme to distinguish many cases of Michaelis-Menten kinetics is presented. It is shown that steady state kinetics are, in general, insufficient to specify the mechanism of a reaction, since different effects of a modifier can give identical steady state kinetic data.     

Gating of IsK expressed in Xenopus oocytes depends on the amount of mRNA injected

IsK is a K+ channel of the delayed rectifier type widely distributed throughout both excitable and nonexcitable cells. Its structure is different from other cloned K+ channels and molecular details of its gating remain obscure. Here we show that the activation kinetics of IsK expressed in Xenopus oocytes depend upon the amount of its mRNA injected, with larger amounts resulting in slower activation kinetics with a longer initial delay during activation. Similar changes in activation kinetics occur with time after a single injection of IsK mRNA. We present two kinetic schemes which illustrate how our experimental results could arise. Both imply an interaction among individual channel proteins during IsK activation. The dependence of channel gating on mRNA concentration provides a novel mechanism for long term regulation of ion current kinetics.     

Kinetics of the immobilization of trypsin on activated silichrome

Trypsin is studied for kinetics of its immobilization on the surface of a porous spheric inorganic carrier with the grafted aldehyde groups in the surface layer. This process is found to be controlled by the enzyme diffusion. It is shown possible to use a body of mathematics known for kinetics of physical adsorption on the porous adsorbents to describe kinetics of protein chemosorption on the analogous carriers. A simple method is suggested for plotting a kinetic curve of the enzyme immobilization on the matrix with any sizes of particles from the experimentally obtained kinetic curve of its binding on the carrier with a definite diameter of particles.     

Differential antigen presentation kinetics of CD8+ T-cell epitopes derived from the same viral protein

The kinetics of peptide presentation by major histocompatibility complex class I (MHC-I) molecules may contribute to the efficacy of CD8+ T cells. Whether all CD8+ T-cell epitopes from a protein are presented by the same MHC-I molecule with similar kinetics is unknown. Here we show that CD8+ T-cell epitopes derived from SIVmac239 Gag are presented with markedly different kinetics. We demonstrate that this discrepancy in presentation is not related to immunodominance but instead is due to differential requirements for epitope generation. These results illustrate that significant differences in presentation kinetics can exist among CD8+ T-cell epitopes derived from the same viral protein.     

Recombination of S-peptide with S-protein during folding of ribonuclease S. I. Folding pathways of the slow-folding and fast-folding classes of unfolded S-protein.

The refolding kinetics of ribonuclease S have been measured by tyrosine absorbance, by tyrosine fluorescence emission, and by rapid binding of the specific inhibitor 2′CMP ² to folded RNAase S. The S-protein is first unfolded at pH 1.7 and then either mixed with S-peptide as refolding is initiated by a stopped-flow pH jump to pH 6.8, or the same results are obtained if S-protein and S-peptide are present together before refolding is initiated. The refolding kinetics of RNAase S have been measured as a function of temperature (10 to 40 °C) and of protein concentration (10 to 120 μm). The results are compared to the folding kinetics of S-protein alone and to earlier studies of RNAase A. A thermal folding transition of S-protein has been found below 30 °C at pH 1.7; its effects on the refolding kinetics are described in the following paper (Labhardt &; Baldwin, 1979).In this paper we characterize the refolding kinetics of unfolded S-protein, as it is found above 30 °C at pH 1.7, together with the kinetics of combination between S-peptide and S-protein during folding at pH 6.8. Two classes of unfolded S-protein molecules are found, fast-folding and slow-folding molecules, in a 20: 80 ratio. This is the same result as that found earlier for RNAase A; it is expected if the slow-folding molecules are produced by the slow cis-trans isomerization of proline residues after unfolding, since S-protein contains all four proline residues of RNAase A.The refolding kinetics of the fast-folding molecules show clearly that combination between S-peptide and S-protein occurs before folding of S-protein is complete. If combination occurred only after complete folding, then the kinetics of formation of RNAase S should be rather slow (5 s and 100 s at 30 °C) and nearly independent of protein concentration, as shown by separate measurements of the folding kinetics of S-protein, and of the combination between S-peptide and folded S-protein. The observed folding kinetics are faster than predicted by this model and also the folding rate increases strongly with protein concentration (apparent 1.6 order kinetics). The fact that RNAase S is formed more rapidly than S-protein alone is sufficient by itself to show that combination with S-peptide precedes complete folding of S-protein. Computer simulation of a simple, parallel-pathway scheme is able to reproduce the folding kinetics of the fast-folding molecules. All three probes give the same folding kinetics.These results exclude the model for protein folding in which the rate-limiting step is an initial diffusion of the polypeptide chain into a restricted range of three-dimensional configurations (“nueleation”) followed by rapid folding (“propagation”). If this model were valid, one would expect comparable rates of folding for RNAase A and for S-protein and one would also expect to find no populated folding intermediates, so that combination between S-peptide and S-protein should occur after folding is complete. Instead, RNAase A folds 60 times more rapidly than S-protein and also combination with S-peptide occurs before folding of S-protein is complete. The results demonstrate that the folding rate of S-protein increases after the formation, or stabilization, of an intermediate which results from combination with S-peptide. They support a sequential model for protein folding in which the rates of successive steps in folding depend on the stabilities of preceding intermediates.The refolding kinetics of the slow-folding molecules are complex. Two results demonstrate the presence of folding intermediates: (1) the three probes show different kinetic progress curves, and (2) the folding kinetics are concentration-dependent, in contrast to the results expected if complete folding of S-protein precedes combination with S-peptide. A faster phase of the slow-refolding reaction is detected both by tyrosine absorbance and fluorescence emission but not by 2′CMP binding, indicating that native RNAase S is not formed in this phase. Comparison of the kinetic progress curves measured by different probes is made with the use of the kinetic ratio test, which is defined here.     

A steady-state random-order mechanism for the oxidative deamination of norvaline by glutamate dehydrogenase.

The kinetic mechanism of glutamate dehydrogenase with the monocarboxylic substrate norvaline was examined by using initial-rate steady-state kinetics and inhibition kinetics. To a first approximation the reaction mechanism can be described as a rapid-equilibrium random-order one. Binding synergism between the monocarboxylic substrate and coenzyme is not observed. Dissociation constants for NAD+ and 2-oxoglutarate calculated from the kinetic data assuming a rapid-equilibrium random-order model are in good agreement with independently obtained estimates. Lineweaver-Burk plots with varied norvaline concentration are not strictly linear, and it is concluded that a steady-state random-order model more accurately reflects the observed kinetics with norvaline as substrate.