Simple Two-state Kinetics Research Articles

Direct Observation of Insulin Association Dynamics with Time-Resolved X-ray Scattering.

Biological functions frequently require protein-protein interactions that involve secondary and tertiary structural perturbation. Here we study protein-protein dissociation and reassociation dynamics in insulin, a model system for protein oligomerization. Insulin dimer dissociation into monomers was induced by a nanosecond temperature-jump (T-jump) of ∼8 °C in aqueous solution, and the resulting protein and solvent dynamics were tracked by time-resolved X-ray solution scattering (TRXSS) on time scales of 10 ns to 100 ms. The protein scattering signals revealed the formation of five distinguishable transient species during the association process that deviate from simple two-state kinetics. Our results show that the combination of T-jump pump coupled to TRXSS probe allows for direct tracking of structural dynamics in nonphotoactive proteins.

Markov State Models of Millisecond Folder ACBP Reveals New Views of the Folding Reaction

A combination of large-scale distributed computing efforts and Markov State Model (MSM) approaches have recently been used to simulate the folding of proteins on the millisecond timescale. These models predict new views of the folding reaction--a complex set of metastable intermediates, multiple pathways, a hub-like network structure, and compact unfolded states with residual structuring--yet, experimental observables often report simple two-state kinetics. To study the events that precede folding, we used MSM approaches to model the folding reaction of ACBP (acyl-CoA binding protein), an 86-residue helix-bundle protein that folds on the ∼10 ms timescale, with an ∼80 µs collapse phase that can be probed both by simulation and experiment. Our combined results suggest that the fast kinetic phase is not due to barrier-limited formation of a well-defined intermediate, but rather the surprisingly slow acquisition of unfolded-state structure.

Energy Changes at the Acetylcholine Receptor Gate Region

Abstract The neuromuscular acetylcholine receptor (AChR) alternatively switches between an inactive (closed) and active (open) conformation. during the gating isomerization (R↔R∗) process. The ‘gate’ region is formed by the equatorial residues (9’-16’). We and others have speculated that channel gating involves a wetting/de-wetting of this region of the pore, based on measurements of gating equilibrium changes caused by mutations, phi value analysis and molecular dynamics simulations. We are studying the coupling energies between 9’-13’ residues in different subunits, by estimating the effect of mutations, separately vs. in combination, on the unliganded gating equilibrium constant E0. The end states and conformational pathway for AChR gating is fundamentally the same with and without agonists, and E0 can be estimated without obfuscation by channel block. We characterized two background mutants (αA96N/W149S and αA96H/Y96H) that greatly increase spontaneous gating and exhibit simple two-state kinetics (E0 = 0.076 and 0.031). These E0 values predict an E0wt value that is similiar to previous estimates (∼6.5x10−7). With these spontaneously-opening backgrounds, the individual free energy changes for βV13A and δV13'L (−4.1 kcal/mol and −1.9 kcal/mol) are similar to previous estimates obtained by using diliganded AChRs. Initial results suggest that these 13’ mutants are only weakly coupled (∼-1.2 kcal/mol). We expect these studies will help unravel the structural changes in protein and water that occur at the AChR gate region.

Rapid Collapse into a Molten Globule Is Followed by Simple Two-State Kinetics in the Folding of Lysozyme from Bacteriophage λ

Stopped-flow fluorescence and circular dichroism spectroscopy have been used in combination with quenched-flow hydrogen exchange labeling, monitored by two-dimensional NMR and electrospray ionization mass spectrometry, to investigate the folding kinetics of lysozyme from bacteriophage λ (λ lysozyme) at pH 5.6, 20 °C. The first step in the folding of λ lysozyme occurs very rapidly (τ < 1 ms) after refolding is initiated and involves both hydrophobic collapse and formation of a high content of secondary structure but only weak protection from (1)H/(2)H exchange and no fixed tertiary structure organization. This early folding step is reflected in the dead-time events observed in the far-UV CD and ANS fluorescence experiments. Following accumulation of this kinetic molten globule species, the secondary structural elements are stabilized and the majority (ca. 88%) of refolding molecules acquire native-like properties in a highly cooperative two-state process, with τ = 0.15 ± 0.03 s. This is accompanied by the acquisition of substantial native-like protection from hydrogen exchange. A double-mixing experiment and the absence of a denaturant effect reveal that slow (τ = 5 ± 1 s) folding of the remaining (ca. 12%) molecules is rate limited by the cis/trans isomerization of prolines that are trans in the folded enzyme. In addition, native state hydrogen exchange and classical denaturant unfolding experiments have been used to characterize the thermodynamic properties of the enzyme. In good agreement with previous crystallographic evidence, our results show that λ lysozyme is a highly dynamic protein, with relatively low conformational stability (ΔG°(N-U) = 25 ± 2 kJ·mol(-1)).

Effect of A New Ionic Pair on the Unfolding Activation Barrier of &amp;#945;- Glucosidase B

Thermal unfolding kinetics of beta-glucosidase B from Paenibacillus polymyxa and its thermoresistant mutant H62R were determined from far-UV circular dichroism (CD) measurements at different temperatures. The unfolding of both enzymes followed simple two-state kinetics. The new ionic pair formed between Arg62 and Glu429 in the H62R variant did not change substantially the enzyme structure as judged by far-UV CD and fluorescence spectra, but produced an increase in the unfolding activation barrier of 0.95 +/- 0.10 kcal mol(-1), in good agreement with the energetic contribution reported for surface salt bridges in proteins. Eyring's analysis of the unfolding kinetic constants showed that the activation enthalpies for thermal denaturation of both enzymes were essentially the same. Thus, the greater kinetic stability rendered by the salt bridge seems to be due to a reduction in the activation entropy.

Protein Folding Rates Estimated from Contact Predictions

Folding rates of small single-domain proteins that fold through simple two-state kinetics can be estimated from details of the three-dimensional protein structure. Previously, predictions of secondary structure had been exploited to predict folding rates from sequence. Here, we estimate two-state folding rates from predictions of internal residue-residue contacts in proteins of unknown structure. Our estimate is based on the correlation between the folding rate and the number of predicted long-range contacts normalized by the square of the protein length. It is well known that long-range order derived from known structures correlates with folding rates. The surprise was that estimates based on very noisy contact predictions were almost as accurate as the estimates based on known contacts. On average, our estimates were similar to those previously published from secondary structure predictions. The combination of these methods that exploit different sources of information improved performance. It appeared that the combined method reliably distinguished fast from slow two-state folders.

Class-specific correlations between protein folding rate, structure-derived, and sequence-derived descriptors.

Small single-domain proteins that fold by simple two-state kinetics have been shown to exhibit a wide variation in their folding rates. It has been proposed that folding mechanisms in these proteins are largely determined by the native-state topology, and a significant correlation between folding rate and measures of the average topological complexity, such as relative contact order (RCO), has been reported. We perform a statistical analysis of folding rate and RCO in all three major structural classes (alpha, beta, and alpha/beta) of small two-state proteins and of RCO in groups of analogous and homologous small single-domain proteins with the same topology. We also study correlation between folding rate and the average physicochemical properties of amino acid sequences in two-state proteins. Our results indicate that 1) helical proteins have statistically distinguishable, class-specific folding rates; 2) RCO accounts for essentially all the variation of folding rate in helical proteins, but for only a part of the variation in beta-sheet-containing proteins; and 3) only a small fraction of the protein topologies studied show a topology-specific RCO. We also report a highly significant correlation between the folding rate and average intrinsic structural propensities of protein sequences. These results suggest that intrinsic structural propensities may be an important determinant of the rate of folding in small two-state proteins.

Simple two-state protein folding kinetics requires near-levinthal thermodynamic cooperativity.

Simple two-state folding kinetics of many small single-domain proteins are characterized by chevron plots with linear folding and unfolding arms consistent with an apparent two-state description of equilibrium thermodynamics. This phenomenon is hereby recognized as a nontrivial heteropolymer property capable of providing fundamental insight into protein energetics. Many current protein chain models, including common lattice and continuum Gō models with explicit native biases, fail to reproduce this generic protein property. Here we show that simple two-state kinetics is obtainable from models with a cooperative interplay between core burial and local conformational propensities or an extra strongly favorable energy for the native structure. These predictions suggest that intramolecular recognition in real two-state proteins is more specific than that envisioned by common Gō-like constructs with pairwise additive energies. The many-body interactions in the present kinetically two-state models lead to high thermodynamic cooperativity as measured by their van't Hoff to calorimetric enthalpy ratios, implying that the native and denatured conformational populations are well separated in enthalpy by a high free-energy barrier. It has been observed experimentally that deviations from Arrhenius behavior are often more severe for folding than for unfolding. This asymmetry may be rationalized by one of the present modeling scenarios if the effective many-body cooperative interactions stabilizing the native structure against unfolding is less dependent on temperature than the interactions that drive the folding kinetics.

Origins of chevron rollovers in non-two-state protein folding kinetics.

Chevron rollovers of some proteins imply that their logarithmic folding rates are nonlinear in native stability. This is predicted by lattice and continuum Gō models to arise from diminished accessibilities of the ground state from transiently populated compact conformations under strongly native conditions. Despite these models' native-centric interactions, the slowdown is due partly to kinetic trapping caused by some of the folding intermediates' non-native topologies. Notably, simple two-state folding kinetics of small single-domain proteins are not reproduced by common Gō-like schemes.

Solvation Effects and Driving Forces for Protein Thermodynamic and Kinetic Cooperativity: How Adequate is Native-centric Topological Modeling?

What energetic and solvation effects underlie the remarkable two-state thermodynamics and folding/unfolding kinetics of small single-domain proteins? To address this question, we investigate the folding and unfolding of a hierarchy of continuum Langevin dynamics models of chymotrypsin inhibitor 2. We find that residue-based additive Gō-like contact energies, although native-centric, are by themselves insufficient for protein-like calorimetric two-state cooperativity. Further native biases by local conformational preferences are necessary for protein-like thermodynamics. Kinetically, however, even models with both contact and local native-centric energies do not produce simple two-state chevron plots. Thus a model protein's thermodynamic cooperativity is not sufficient for simple two-state kinetics. The models tested appear to have increasing internal friction with increasing native stability, leading to chevron rollovers that typify kinetics that are commonly referred to as non-two-state. The free energy profiles of these models are found to be sensitive to the choice of native contacts and the presumed spatial ranges of the contact interactions. Motivated by explicit-water considerations, we explore recent treatments of solvent granularity that incorporate desolvation free energy barriers into effective implicit-solvent intraprotein interactions. This additional feature reduces both folding and unfolding rates vis-à-vis that of the corresponding models without desolvation barriers, but the kinetics remain non-two-state. Taken together, our observations suggest that interaction mechanisms more intricate than simple Gō-like constructs and pairwise additive solvation-like contributions are needed to rationalize some of the most basic generic protein properties. Therefore, as experimental constraints on protein chain models, requiring a consistent account of protein-like thermodynamic and kinetic cooperativity can be more stringent and productive for some applications than simply requiring a model heteropolymer to fold to a target structure.

Pattern formation of ion channels with state-dependent charges and diffusion constants in fluid membranes.

A model of mobile, charged ion channels in a fluid membrane is studied. The channels may switch between an open and a closed state according to a simple two-state kinetics with constant rates. The effective electrophoretic charge and the diffusion constant of the channels may be different in the closed and in the open state. The system is modeled by densities of channel species, obeying simple equations of electrodiffusion. The lateral transmembrane voltage profile is determined from a cable-type equation. Bifurcations from the homogeneous, stationary state appear as hard-mode, soft-mode, or hard-mode oscillatory transitions within physiologically reasonable ranges of model parameters. We study the dynamics beyond linear stability analysis and derive nonlinear evolution equations near the transitions to stationary patterns.

Concepts and misconcepts in the analysis of simple kinetics of protein folding.

Unusually simple two-state kinetics characterizes the folding of a number of small proteins possessing a variety secondary structures. This limits dramatically the number of experimentally resolvable parameters that may characterize this process and also suggests the possibility to describe it based on simple theories borrowed from the field of ordinary chemical reactions. An attempt is made to critically evaluate the basic concepts, which are in the background of this approach. We demonstrate their limitations, which may cast doubt on the interpretation of experimental data. It is shown also that, in contrast to provisions of transition state theory, the simple kinetics of protein folding does not correlate with folded state stability or with the size of the folding unit. Moreover, the folding kinetics exhibits anomalous dependence on temperature and pressure and surprisingly strong dependence on solvent viscosity. The possible role in folding of fluctuations, relaxations and gradient dynamics is discussed. Being overlooked or underestimated, these mechanisms may determine the rate and specificity of the process.

Conversion of two-state to multi-state folding kinetics on fusion of two protein foldons

Chymotrypsin inhibitor 2 (CI2) is the archetypal single-foldon protein that folds in simple two-state kinetics without the accumulation of a folding intermediate. To model the effects of fusion of single foldons to give a multi-foldon protein, we engineered a “double-CI2“ protein, in which another CI2 polypeptide was inserted into the loop region of the parent CI2. CD and HSQC spectra demonstrated that while the double-CI2 protein adopted two kinds of native conformations, CI2-like structure was almost preserved in both the domains of double-CI2. In the folding kinetic studies, double-CI2 exhibited a remarkable rollover of the observed folding rates at low denaturant concentrations, indicating that double-CI2 accumulated a kinetic folding intermediate. The different folding mechanisms between WT-CI2 and double-CI2 support the present view that protein size or number of domains is an important determinant for formation of folding intermediates.

First principles prediction of protein folding rates

Experimental studies have demonstrated that many small, single-domain proteins fold via simple two-state kinetics. We present a first principles approach for predicting these experimentally determined folding rates. Our approach is based on a nucleation-condensation folding mechanism, where the rate-limiting step is a random, diffusive search for the native tertiary topology. To estimate the rates of folding for various proteins via this mechanism, we first determine the probability of randomly sampling a conformation with the native fold topology. Next, we convert these probabilities into folding rates by estimating the rate that a protein samples different topologies during diffusive folding. This topology-sampling rate is calculated using the Einstein diffusion equation in conjunction with an experimentally determined intra-protein diffusion constant. We have applied our prediction method to the 21 topologically distinct small proteins for which two-state rate data is available. For the 18 beta-sheet and mixed alpha-beta native proteins, we predict folding rates within an average factor of 4, even though the experimental rates vary by a factor of ∼4×104. Interestingly, the experimental folding rates for the three four-helix bundle proteins are significantly underestimated by this approach, suggesting that proteins with significant helical content may fold by a faster, alternative mechanism. This method can be applied to any protein for which the structure is known and hence can be used to predict the folding rates of many proteins prior to experiment.

Equilibria and kinetics of folding of gelsolin domain 2 and mutants involved in familial amyloidosis-Finnish type.

Mutations D187N and D187Y in domain 2 of the actin-regulating protein gelsolin cause familial amyloidosis-Finnish type (FAF). We have constructed and expressed a recombinant version of gelsolin domain 2 that is sufficiently stable for kinetic and equilibrium measurements. The wild-type domain and the two amyloidogenic mutants fold via simple two-state kinetics without the accumulation of an intermediate. Unfolding kinetics exhibits significant curvature with increasing urea concentration, indicating that the transition state for unfolding becomes more native-like under increasingly denaturing conditions in accordance with the Hammond postulate. Mutations D187N and D187Y destabilize gelsolin domain 2 by 1.22 and 2.16 kcal. mol(-1) (1 kcal = 4.18 kJ) respectively. The mutations do not prevent disulfide bond formation despite their direct contiguity with a cysteine residue involved in disulfide linkage. The destabilization conferred on gelsolin domain 2 by the FAF mutations is sufficient to predict that an appreciable fraction is unfolded and, therefore, extremely susceptible to proteolysis at body temperature.

Folding Pathway of FKBP12 and Characterisation of the Transition State

The folding pathway of human FKBP12, a 12 kDa FK506-binding protein (immunophilin), has been characterised. Unfolding and refolding rate constants have been determined over a wide range of denaturant concentrations and data are shown to fit to a two-state model of folding in which only the denatured and native states are significantly populated, even in the absence of denaturant. This simple model for folding, in which no intermediate states are significantly populated, is further supported from stopped-flow circular dichroism experiments in which no fast “burst” phases are observed. FKBP12, with 107 residues, is the largest protein to date which folds with simple two-state kinetics in water ( k F=4 s −1 at 25 °C). The topological crossing of two loops in FKBP12, a structural element suggested to cause kinetic traps during folding, seems to have little effect on the folding pathway. The transition state for folding has been characterised by a series of experiments on wild-type FKBP12. Information on the thermodynamic nature of, the solvent accessibility of, and secondary structure in, the transition state was obtained from experiments measuring the unfolding and refolding rate constants as a function of temperature, denaturant concentration and trifluoroethanol concentration. In addition, unfolding and refolding studies in the presence of ligand provided information on the structure of the ligand-binding pocket in the transition state. The data suggest a compact transition state relative to the unfolded state with some 70 % of the surface area buried. The ligand-binding site, which is formed mainly by two loops, is largely unstructured in the transition state. The trifluoroethanol experiments suggest that the α-helix may be formed in the transition state. These results are compared with results from protein engineering studies and molecular dynamics simulations (see the accompanying paper).

How do small single-domain proteins fold?

Many small, monomeric proteins fold with simple two-state kinetics and show wide variation in folding rates, from microseconds to seconds. Thus, stable intermediates are not a prerequisite for the fast, efficient folding of proteins and may in fact be kinetic traps and slow the folding process. Using recent studies, can we begin to search for trends which may lead to a better understanding of the protein folding process?

Titration Properties and Thermodynamics of the Transition State for Folding: Comparison of Two-state and Multi-state Folding Pathways

CI2 folds and unfolds as a single cooperative unit by simple two-state kinetics, which enables the properties of the transition state to be measured from both the forward and backward rate constants. We have examined how the free energy of the transition state for the folding of chymotrypsin inhibitor 2 (CI2) changes with pH and temperature. In addition to the standard thermodynamic quantities, we have measured the overall acid-titration properties of the transition state and its heat capacity relative to both the denatured and native states. We were able to determine the latter by a method analogous to a well-established procedure for measuring the change in heat capacity for equilibrium unfolding: the enthalpy of activation of unfolding at different values of acid pH were plotted against the average temperature of each determination. Our results show that the transition state of CI2 has lost most of the electrostatic and van der Waals' interactions that are found in the native state, but it remains compact and this prevents water molecules from entering some parts of the hydrophobic core. The properties of the transition state of CI2 are then compared with the major folding transition state of the larger protein barnase, which folds by a multi-state mechanism, with the accumulation of a partly structured intermediate (D physor I). CI2 folds from a largely unstructured denatured state under physiological conditions viaa transition state which is compact but relatively uniformly unstructured, with tertiary and secondary structure being formed in parallel. We term this an expanded pathway. Conversely, barnase folds from a largely structured denatured state in which elements of structure are well formed through a transition state that has islands of folded elements of structure. We term this a compact pathway. These two pathways may correspond to the two extreme ends of a continuous spectrum of protein folding mechanisms. Although the properties of the two transition states are very different, the activation barrier for folding (D phys→‡ ) is very similar for both proteins.

The Structure of the Transition State for Folding of Chymotrypsin Inhibitor 2 Analysed by Protein Engineering Methods: Evidence for a Nucleation-condensation Mechanism for Protein Folding

The 64-residue protein chymotrypsin inhibitor 2 (CI2) is a single module of structure. It folds and unfolds as a single co-operative unit by simple two-state kinetics viaa single rate determining transition state. This transition state has been characterized at the level of individual residues by analysis of the rates and equilibria of folding of some 100 mutants strategically distributed at 45 sites throughout the protein. Only one residue, a helical residue (Ala16) buried in the hydrophobic core, has its full native interaction energy in the transition state. The only region of structure which is well developed in the transition state is the α-helix (residues 12 to 24). But, the interactions within it are weakened, especially at the C-terminal region. The rest of the protein has varying degrees of weakly formed structure. Thus, secondary and tertiary interactions appear to form concurrently. These data, reinforced by studies on the structures of peptide fragments, fit a "nucleation-condensation" model in which the overall structure condenses around an element of structure, the nucleus, that itself consolidates during the condensation. The high energy transition state is composed of the whole of the molecule making a variety of weak interactions, the nucleus being those residues that make the strongest interactions. The nucleus here is part of the α-helix and some distant residues in the sequence with which it makes contacts. The remainder of the protein has to be sufficiently ordered that it provides the necessary interactions to stabilize the nucleus. The nucleus is only weakly formed in the denatured state but develops in the transition state. The onrush of stability as the nucleus consolidates its local and long range interactions is so rapid that it is not yet fully formed in the transition state. The formation of the nucleus is thus coupled with the condensation. These results are con- sistent with a recent simulation of the folding of a computer model protein on a lattice which is found to proceed by a nucleation-growth mechanism. We suggest that the mechanism of folding of CI2 may be a common theme in protein folding whereby fundamental folding units of larger proteins, which are modelled by the folding of CI2, form by nucleation-condensation events and coalesce, perhaps in a hierarchical manner.

Calcium-Permeable Channel in Sarcoplasmic Reticulum of Rabbit Skeletal Muscle

Membrane vesicles from sarcoplasmic reticulum of rabbit skeletal muscle were incorporated into a bilayer lipid membrane. With this system, single current fluctuation was observed in the presence of 50 mM Ba-gluconate. This channel activity was observed only in vesicles from terminal cisternae. The single channel conductance was 14.1 pS, and the channel state was almost wholly open. The open-close transition of the channel obeyed simple two-state kinetics and was voltage-independent. The ionic selectivity was also studied, and the channel showed no selectivity among Ba, Ca, Mn, and Mg. On the other hand, it was less permeable to Cs than to Ba. Based on these results, the relation of the Ca channel to excitation-contraction coupling is discussed.