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.     

The complex kinetics of protein folding in wide temperature ranges

The complex protein folding kinetics in wide temperature ranges is studied through diffusive dynamics on the underlying energy landscape. The well-known kinetic chevron rollover behavior is recovered from the mean first passage time, with the U-shape dependence on temperature. The fastest folding temperature T0 is found to be smaller than the folding transition temperature Tf. We found that the fluctuations of the kinetics through the distribution of first passage time show rather universal behavior, from high-temperature exponential Poissonian kinetics to the relatively low-temperature highly non-exponential kinetics. The transition temperature is at Tk and T0 < Tk < Tf. In certain low-temperature regimes, a power law behavior at long time emerges. At very low temperatures (lower than trapping transition temperature T < T0/(4 approximately 6)), the kinetics is an exponential Poissonian process again.     

Theory of transport noise in membrane channels with open-closed kinetics

A theoretical approach to transport noise in kinetic systems, which has recently been developed, is applied to electric fluctuations around steady-states in membrane channels with different conductance states. The channel kinetics may be simple two state (open-closed) kinetics or more complicated. The membrane channel is considered as a sequence of binding sites separated by energy barriers over which the ions have to jump according to the usual single-file diffusion model. For simplicity the channels are assumed to act independently. In the special case of ionic movement fast compared with the channel open-closed kinetics the results agree with those derived from the usual Master equation approach to electric fluctuations in nerve membrane channels.For the simple model of channels with one binding site and two energy barries the coupling between the fluctuations coming from the open-closed kinetics and from the jump diffusion is investigated.     

Cooperativity and the origins of rapid, single-exponential kinetics in protein folding

The folding of naturally occurring, single-domain proteins is usually well described as a simple, single-exponential process lacking significant trapped states. Here we further explore the hypothesis that the smooth energy landscape this implies, and the rapid kinetics it engenders, arises due to the extraordinary thermodynamic cooperativity of protein folding. Studying Miyazawa-Jernigan lattice polymers, we find that, even under conditions where the folding energy landscape is relatively optimized (designed sequences folding at their temperature of maximum folding rate), the folding of protein-like heteropolymers is accelerated when their thermodynamic cooperativity is enhanced by enhancing the nonadditivity of their energy potentials. At lower temperatures, where kinetic traps presumably play a more significant role in defining folding rates, we observe still greater cooperativity-induced acceleration. Consistent with these observations, we find that the folding kinetics of our computational models more closely approximates single-exponential behavior as their cooperativity approaches optimal levels. These observations suggest that the rapid folding of naturally occurring proteins is, in part, a consequence of their remarkably cooperative folding.     

Structural relaxation and nonexponential kinetics of CO-binding to horse myoglobin. Multiple flash photolysis experiments.

The geminate recombination kinetics of CO-myoglobin strongly deviates from single exponential behavior in contrast to what is expected for unimolecular reactions (1). At low temperatures, this result was attributed to slowly exchanging conformational states which differ substantially in barrier height for ligand binding. Above 160 K the kinetics apparently slow down with temperature increase. Agmon and Hopfield (2) explain this result in terms of structural relaxation perpendicular to the reaction coordinate, which enhances the activation energy. In their model, structural relaxation homogenizes the kinetic response. Recently, Steinbach et al. (3) proposed a relaxation model which conserves the kinetic inhomogeneity. Below we test these conjectures by single and multiple excitation experiments. This method allows for discrimination between parallel (inhomogeneous) and sequential (homogeneous) kinetic schemes. The kinetic anomaly above 160 K is shown to result from a homogeneous, structurally relaxed intermediate. However a second anomaly is found above 210 K concerning the inhomogeneous phase which may indicate either a shift in activation energy or entropy.     

Quantifying the kinetic paths of flexible biomolecular recognition

Biomolecular recognition often involves large conformational changes, sometimes even local unfolding. The identification of kinetic pathways has become a central issue in understanding the nature of binding. A new approach is proposed here to study the dynamics of this binding-folding process through the establishment of a path-integral framework on the underlying energy landscape. The dominant kinetic paths of binding and folding can be determined and quantified. The significant coupling between the binding and folding of biomolecules often exists in many important cellular processes. In this case, the corresponding kinetic paths of binding are shown to be intimately correlated with those of folding and the dynamics becomes quite cooperative. This implies that binding and folding happen concurrently. When the coupling between binding and folding is weak (strong), the kinetic process usually starts with significant folding (binding) first, with the binding (folding) later proceeding to the end. The kinetic rate can be obtained through the contributions from the dominant paths. The rate is shown to have a bell-shaped dependence on temperature in the concentration-saturated regime consistent with experiment. The changes of the kinetics that occur upon changing the parameters of the underlying binding-folding energy landscape are studied.     

A rugged energy landscape mechanism for trapping of transmembrane receptors during endocytosis

Efficient clathrin-mediated endocytosis of transmembrane receptors requires that clathrin-coated pits retain the receptors long enough to allow vesicle formation and internalization. In many cases, however, the receptors can exhibit mean lifetimes in coated pits much shorter than the lifetime of the pit at the plasma membrane. A rugged energy landscape for binding, which produces a broad distribution of residence times, ensures a significant probability of times much greater than the mean and would allow efficient trapping of proteins in these cases. We used fluorescence correlation spectroscopy and total internal reflection microscopy to measure the kinetics of movement of a C5a receptor-yellow fluorescent protein fusion in living cells. These experiments demonstrate that clusters of trapped receptors exhibit fluctuations in fluorescence intensity that vary in time scale over 2 orders of magnitude. Most of the variation in intensity is likely due to the motion of the receptors in the plane of the plasma membrane, although it is not possible to rule out a small contribution from motion orthogonal to the plane of the membrane. The broad time scale distribution of the intensity fluctuations is consistent with a rugged energy landscape mechanism for trapping of the receptors. This mechanism, which allows efficient trapping to coexist with rapid exchange, may also be relevant to other biological processes involving binding in heterogeneous chemical environments.     

Direct measurement of protein energy landscape roughness

The energy landscape of proteins is thought to have an intricate, corrugated structure. Such roughness should have important consequences on the folding and binding kinetics of proteins, as well as on their equilibrium fluctuations. So far, no direct measurement of protein energy landscape roughness has been made. Here, we combined a recent theory with single-molecule dynamic force spectroscopy experiments to extract the overall energy scale of roughness epsilon for a complex consisting of the small GTPase Ran and the nuclear transport receptor importin-beta. The results gave epsilon > 5k(B)T, indicating a bumpy energy surface, which is consistent with the ability of importin-beta to accommodate multiple conformations and to interact with different, structurally distinct ligands.     

Conformational dynamics of partially denatured myoglobin studied by time-resolved electrospray mass spectrometry with online hydrogen-deuterium exchange

This study demonstrates the use of electrospray mass spectrometry in conjunction with rapid online mixing ("time-resolved" ESI-MS) for monitoring protein conformational dynamics under equilibrium conditions. The hydrogen/deuterium exchange (HDX) kinetics of mildly denatured myoglobin (Mb) at pD 9.3, in the presence of 27% acetonitrile, were studied with millisecond time resolution. Analytical ultracentrifugation indicates that the average protein compactness under these solvent conditions is similar to that of native holomyoglobin (hMb). The mass spectrum shows protein ions in a wide array of charge and heme binding states, indicating the presence of multiple coexisting conformations. The experimental approach used allows the HDX kinetics of all of these species to be monitored separately. A combination of EX1 and EX2 behavior was observed for hMb ions in charge states 7+ to 9+, which predominantly represent nativelike hMb in solution. The EX1 kinetics are biphasic, indicating the presence of two protein populations that undergo conformational opening events with different rate constants. The EX2 kinetics observed for nativelike hMb are biphasic as well. All other charge and heme binding states represent non-native protein conformations that are involved in rapid interconversion processes, thus leading to monoexponential EX2 kinetics with a common rate constant. Burst phase labeling for these non-native proteins occurs at 125 sites. In contrast, the nativelike protein conformation shows burst phase labeling only for 88 sites. A kinetic model is developed which is based on the assumption of three distinct (un)folding units in Mb. The model implies that the free energy landscape of the protein exhibits a major barrier. The crossing of this barrier is most likely associated with slow, cooperative opening/closing events of the heme binding pocket. Rapid conformational fluctuations on either side of the barrier give rise to the observed EX2 kinetics. Simulated HDX kinetics based on this model are in excellent agreement with the experimental data.     

Energetics of folding subtilisin BPN'.

Subtilisin is an unusual example of a monomeric protein with a substantial kinetic barrier to folding and unfolding. Here we document for the first time the in vitro folding of the mature form of subtilisin. Subtilisin was modified by site-directed mutagenesis to be proteolytically inactive, allowing the impediments to folding to be systematically examined. First, the thermodynamics and kinetics of calcium binding to the high-affinity calcium A-site have been measured by microcalorimetry and fluorescence spectroscopy. Binding is an enthalpically driven process with an association constant (Ka) equal to 7 x 10(6) M-1. Furthermore, the kinetic barrier to calcium removal from the A-site (23 kcal/mol) is substantially larger than the standard free energy of binding (9.3 kcal/mol). The kinetics of calcium dissociation from subtilisin (e.g., in excess EDTA) are accordingly very slow (t1/2 = 1.3 h at 25 degrees C). Second, to measure the kinetics of subtilisin folding independent of calcium binding, the high-affinity calcium binding site was deleted from the protein. At low ionic strength (I = 0.01) refolding of this mutant requires several days. The folding rate is accelerated almost 100-fold by a 10-fold increase in ionic strength, indicating that part of the free energy of activation may be electrostatic. At relatively high ionic strength (I = 0.5) refolding of the mutant subtilisin is complete in less than 1 h at 25 degrees C. We suggest that part of the electrostatic contribution to the activation free energy for folding subtilisin is related to the highly charged region of the protein comprising the weak ion binding site (site B).(ABSTRACT TRUNCATED AT 250 WORDS)     

Equilibrium and kinetic analysis of nucleotide binding to the DEAD-box RNA helicase DbpA

The Escherichia coli DEAD-box protein A (DbpA) is an RNA helicase that utilizes the energy from ATP binding and hydrolysis to facilitate structural rearrangements of rRNA. We have used the fluorescent nucleotide analogues, mantADP and mantATP, to measure the equilibrium binding affinity and kinetic mechanism of nucleotide binding to DbpA in the absence of RNA. Binding generates an enhancement in mant-nucleotide fluorescence and a corresponding reduction in intrinsic DbpA fluorescence, consistent with fluorescence resonance energy transfer (FRET) from DbpA tryptophan(s) to bound nucleotides. Fluorescent modification does not significantly interfere with the affinities and kinetics of nucleotide binding. Different energy transfer efficiencies between DbpA-mantATP and DbpA-mantADP complexes suggest that DbpA adopts nucleotide-dependent conformations. ADP binds (K(d) approximately 50 microM at 22 degrees C) 4-7 times more tightly than ATP (K(d) approximately 400 microM at 22 degrees C). Both nucleotides bind with relatively temperature-independent association rate constants (approximately 1-3 microM(-1) s(-1)) that are much lower than predicted for a diffusion-limited reaction. Differences in the binding affinities are dictated primarily by the dissociation rate constants. ADP binding occurs with a positive change in the heat capacity, presumably reflecting a nucleotide-induced conformational rearrangement of DbpA. At low temperatures (<22 degrees C), the binding free energies are dominated by favorable enthalpic and unfavorable entropic contributions. At physiological temperatures (>22 degrees C), ADP binding occurs with positive entropy changes. We favor a mechanism in which ADP binding increases the conformational flexibility and dynamics of DbpA.     

Epigenetic landscape of interacting cells: A model simulation for developmental process

We propose a physical model for developmental process at cellular level to discuss the mechanism of epigenetic landscape. In our simplified model, a minimal model, the network of the interaction among cells generates the landscape epigenetically and the differentiation in developmental process is understood as a self-organization. The effect of the regulation by gene expression which is a key ingredient in development is renormalized into the interaction and the environment. At earlier stage of the development the energy landscape of the model is rugged with small amplitude. The state of cells in such a landscape is susceptible to fluctuations and not uniquely determined. These cells are regarded as stem cells. At later stage of the development the landscape has a funnel-like structure corresponding to the canalization in differentiation. The rewinding or stability of the differentiation is also demonstrated by substituting test cells into the time sequence of the model development.     

Hierarchic finite level energy landscape model: to describe the refolding kinetics of phosphoglycerate kinase

One of the most intriguing predictions of energy landscape models is the existence of non-exponential protein folding kinetics caused by hierarchical structures in the landscapes. Here we provide the strongest evidence so far of such hierarchy and determine the time constants and weights of the kinetic components of the suggested hierarchic energy landscape. To our knowledge, the idea of hierarchical folding energy barriers has never been tested over such a broad timescale. Refolding of yeast phosphoglycerate kinase was initiated from the guanidine-unfolded state by stopped-flow or manual mixing and monitored by tryptophan fluorescence from 1 ms to 15 min. The strategy to build a model that describes folding of yeast phosphoglycerate kinase was to start from the simplest paradigm and modify it stepwise to the necessary minimal extent after repeated comparisons with the experiments. We made no a priori assumptions about the folding landscape. The result was a hierarchic finite level landscape model that quantitatively describes the refolding of yeast phosphoglycerate kinase from 1 ms to 15 min. The early steps of the folding process happen in the upper region of the landscape, where the surface has a hierarchic structure. This leads to stretched kinetics in the early phase of the folding. The lower region of the energy landscape is dominated by a trap that reflects the accumulation of molten globule intermediate state. From this intermediate, the protein can reach the global energy minimum corresponding to the native state through a cross-barrier folding step.     

Kinetics of a finite one-dimensional spin system as a model for protein folding

Simple spin models are used to analyze the kinetic nature of lowest energy state formation of the spin systems as models of protein folding kinetics. The models employed in the present paper are based on the spin systems as models of biopolymers previously proposed by the author for the analysis of the equilibrium nature of transitions [T. Kikuchi, Biophys. Chem. 65 (1997) 109]. In particular, the effect of frustrations on the kinetics is investigated with the Monte Carlo simulations in this study. The results show that the kinetics of the present systems are characterized by the ratio of foldables (pathways on the energy landscape that to lead to the lowest energy state) and the temperature dependence of the mean first passage time of foldables. We also discuss the free energy profile of the present models and the relation of the present results to the kinetics of actual proteins.     

Removal of kinetic traps and enhanced protein folding by strategic substitution of amino acids in a model alpha-helical hairpin peptide

The presence of non-native kinetic traps in the free energy landscape of a protein may significantly lengthen the overall folding time so that the folding process becomes unreliable. We use a computational model alpha-helical hairpin peptide to calculate structural free energy landscapes and relate them to the kinetics of folding. We show how protein engineering through strategic changes in only a few amino acid residues along the primary sequence can greatly increase the speed and reliability of the folding process, as seen experimentally. These strategic substitutions also prevent the formation of long-lived misfolded configurations that can cause unwanted aggregations of peptides. These results support arguments that removal of kinetic traps, obligatory or nonobligatory, is crucial for fast folding.     

The Off-rate of Monomers Dissociating from Amyloid-β Protofibrils

The interconversion of monomers, oligomers, and amyloid fibrils of the amyloid-β peptide (Aβ) has been implicated in the pathogenesis of Alzheimer disease. The determination of the kinetics of the individual association and dissociation reactions is hampered by the fact that forward and reverse reactions to/from different aggregation states occur simultaneously. Here, we report the kinetics of dissociation of Aβ monomers from protofibrils, prefibrillar high molecular weight oligomers previously shown to possess pronounced neurotoxicity. An engineered binding protein sequestering specifically monomeric Aβ was employed to follow protofibril dissociation by tryptophan fluorescence, precluding confounding effects of reverse or competing reactions. Aβ protofibril dissociation into monomers follows exponential decay kinetics with a time constant of ∼2 h at 25 °C and an activation energy of 80 kJ/mol, values typical for high affinity biomolecular interactions. This study demonstrates the high kinetic stability of Aβ protofibrils toward dissociation into monomers and supports the delineation of the Aβ folding and assembly energy landscape.     

Gating kinetics of Ca2+-activated K+ channels from rat muscle incorporated into planar lipid bilayers. Evidence for two voltage- dependent Ca2+ binding reactions

The gating kinetics of a Ca2+-activated K+ channel from adult rat muscle plasma membrane are studied in artificial planar bilayers. Analysis of single-channel fluctuations distinguishes two Ca2+- and voltage-dependent processes: (a) short-lived channel closure (less than 1 ms) events appearing in a bursting pattern; (b) opening and closing events ranging from one to several hundred milliseconds in duration. The latter process is studied independently of the first and is denoted as the primary gating mode. At constant voltage, the mean open time of the primary gating mode is a linear function of the [Ca2+], whereas the mean closed time is a linear function of the reciprocal [Ca2+]. In the limits of zero and infinite [Ca2+], the mean open and the mean closed times are, respectively, independent of voltage. These results are predicted by a kinetic scheme consisting of the following reaction steps: (a) binding of Ca2+ to a closed state; (b) channel opening; (c) binding of a second Ca2+ ion. In this scheme, the two Ca2+ binding reactions are voltage dependent, whereas the open-closed transition is voltage independent. The kinetic constant derived for this scheme gives an accurate theoretical fit to the observed equilibrium open-state probability. The results provide evidence for a novel regulatory mechanism for the activity of an ion channel: modulation by voltage of the binding of an agonist molecule, in this case, Ca2+ ion.     

Population analyses of kinetic partitioning in protein folding

A simple lattice model of protein folding is studied in order to analyze the kinetic partitioning phenomena in the energy landscape perspective. By restricting the area of conformational space, it becomes possible to follow many Monte Carlo trajectories until they reach equilibrium. Alteration of population of trajectories is monitored and the relations between the energy landscape and kinetics are examined. Kinetic partitioning phenomena are categorized into different types in terms of characteristic time constants and partitioning ratio. In a specific partitioning process, refolding proceeds along the parallel pathways; the time constants have a temperature dependence similar to that observed in hen lysozyme. High-energy conformations are classified into groups according to the probability that the trajectories starting from those conformations will reach each energy valley. The partitioning ratio is determined by the way in which the conformational space is organized into these groups.     

Kinetics of gonadotropin binding by receptors of the rat testis. Analysis by a nonlinear curve-fitting method.

The kinetics of the reaction between human chorionic gonadotropin (hCG) and specific gonadotropin receptors in the rat testis were determined at 24 and 37 degrees, over a wide range of hormone concentrations. Hormone concentrations were corrected for the binding activity of the (-125I)hCG tracer preparations. Analysis of the experimental data was performed with an interactive nonlinear curve fitting program, based upon the second-order chemical kinetic differential equation. The mean values for the association rate constant (k1) were 4.7 x 10-7 M-1 min-1 at 24 degrees, and 11.0 x 10-7 M-1 min-1 at 37 degrees. At both temperatures, the values of kl were independent of hormone concentration. Initial dissociation rates were consistent with first order kinetics, with dissociation rate constant (k2) of 1.7 x 10 minus -3 and 4.6 x 10 minus -3 min minus -1 at 24 and 37 degrees, respectively. When studied over longer periods at 24 degrees, the dissociation process appeared to be multiexponential. The kinetics of degradation of (-125I)hCG and receptors were determined at both temperatures, and a mathematical model was developed by modification of the second-order chemical kinetic differential equation to take these factors into account. The application of such a model to hCG kinetic binding data demonstrated that reactant degradation had little significant effect on the derivation of the association rate constant (k1), but caused significant overestimation of the dissociation rate constant (k2) values derived from association experiments. The model was also applied by computer simulation to a theoretical analysis of the effects of degradation of free hormone and receptor sites upon kinetic and steadystate binding data. By this method, the initial velocities of hormone binding were shown to be less affected by degradation than the steady-state levels of hormone-receptor complex. Also, reactant degradation in simulated steady-state experiments caused an underestimate of the apparent equilibrium association constant, but had relatively less effect on the determination of binding site concentration.     

Stochastic roadmap simulation: an efficient representation and algorithm for analyzing molecular motion.

Classic molecular motion simulation techniques, such as Monte Carlo (MC) simulation, generate motion pathways one at a time and spend most of their time in the local minima of the energy landscape defined over a molecular conformation space. Their high computational cost prevents them from being used to compute ensemble properties (properties requiring the analysis of many pathways). This paper introduces stochastic roadmap simulation (SRS) as a new computational approach for exploring the kinetics of molecular motion by simultaneously examining multiple pathways. These pathways are compactly encoded in a graph, which is constructed by sampling a molecular conformation space at random. This computation, which does not trace any particular pathway explicitly, circumvents the local-minima problem. Each edge in the graph represents a potential transition of the molecule and is associated with a probability indicating the likelihood of this transition. By viewing the graph as a Markov chain, ensemble properties can be efficiently computed over the entire molecular energy landscape. Furthermore, SRS converges to the same distribution as MC simulation. SRS is applied to two biological problems: computing the probability of folding, an important order parameter that measures the "kinetic distance" of a protein's conformation from its native state; and estimating the expected time to escape from a ligand-protein binding site. Comparison with MC simulations on protein folding shows that SRS produces arguably more accurate results, while reducing computation time by several orders of magnitude. Computational studies on ligand-protein binding also demonstrate SRS as a promising approach to study ligand-protein interactions.