Folding Speed Limit Research Articles

Robust membrane protein tweezers reveal the folding speed limit of helical membrane proteins.

Single-molecule tweezers, such as magnetic tweezers, are powerful tools for probing nm-scale structural changes in single membrane proteins under force. However, the weak molecular tethers used for the membrane protein studies have limited the observation of long-time, repetitive molecular transitions due to force-induced bond breakage. The prolonged observation of numerous transitions is critical in reliable characterizations of structural states, kinetics, and energy barrier properties. Here, we present a robust single-molecule tweezer method that uses dibenzocyclooctyne cycloaddition and traptavidin binding, enabling the estimation of the folding 'speed limit' of helical membrane proteins. This method is >100 times more stable than a conventional linkage system regarding the lifetime, allowing for the survival for ~12 hr at 50 pN and ~1000 pulling cycle experiments. By using this method, we were able to observe numerous structural transitions of a designer single-chained transmembrane homodimer for 9 hr at 12 pN and reveal its folding pathway including the hidden dynamics of helix-coil transitions. We characterized the energy barrier heights and folding times for the transitions using a model-independent deconvolution method and the hidden Markov modeling analysis, respectively. The Kramers rate framework yields a considerably low-speed limit of 21 ms for a helical hairpin formation in lipid bilayers, compared to μs scale for soluble protein folding. This large discrepancy is likely due to the highly viscous nature of lipid membranes, retarding the helix-helix interactions. Our results offer a more valid guideline for relating the kinetics and free energies of membrane protein folding.

Threading proteins at the speed limit of folding

Mutations in proteins can speed-up folding/unfolding (1,2), by optimizing the population of intermediate states (3), by stabilizing transition states (1) or by tuning the ruggedness of the energy landscape (4). For fast folders, even the transition path time between states can be resolved (5). However, this process has yet to be fully resolved in single-molecule experiments (6). One way to quantify the ruggedness of a protein's energy landscape is to count the total number of conformational states of mutants (5).

Loop formation and translational diffusion of intrinsically disordered proteins.

The conformational flexibility and dynamics of unfolded peptide chains is of major interest in the context of protein folding and protein functioning. The rate with which amino acids at different positions along the peptide chain meet sets an upper speed limit for protein folding. By using single-molecule photo-induced energy transfer spectroscopy, we have systematically measured end-to-end and end-to-internal site contact formation rates for several intrinsically disordered protein fragments (IDPs) (11 to 41 amino acids) and have also determined their hydrodynamic radius using dual-focus fluorescence correlation spectroscopy. For interpreting the measured values, we have developed a Brownian dynamics model (based on bead-rod chain dynamics in a thermal bath including hydrodynamic interactions) which quantitatively reproduces all measured data surprisingly well while requiring only two fit parameters. The model provides a complete picture of the peptides' dynamics and allows us to translate the experimental rates and radii into molecular properties of the peptides: We find a persistence length of l_{P}=5.2±1.9Å, a hydrodynamic radius of a=3.5±0.7Å per amino acid, and that excluded volume effects play an important role in the dynamics of IDPs.

Ultrafast folding kinetics of WW domains reveal how the amino acid sequence determines the speed limit to protein folding

Protein (un)folding rates depend on the free-energy barrier separating the native and unfolded states and a prefactor term, which sets the timescale for crossing such barrier or folding speed limit. Because extricating these two factors is usually unfeasible, it has been common to assume a constant prefactor and assign all rate variability to the barrier. However, theory and simulations postulate a protein-specific prefactor that contains key mechanistic information. Here, we exploit the special properties of fast-folding proteins to experimentally resolve the folding rate prefactor and investigate how much it varies among structural homologs. We measure the ultrafast (un)folding kinetics of five natural WW domains using nanosecond laser-induced temperature jumps. All five WW domains fold in microseconds, but with a 10-fold difference between fastest and slowest. Interestingly, they all produce biphasic kinetics in which the slower phase corresponds to reequilibration over the small barrier (<3 RT) and the faster phase to the downhill relaxation of the minor population residing at the barrier top [transition state ensemble (TSE)]. The fast rate recapitulates the 10-fold range, demonstrating that the folding speed limit of even the simplest all-β fold strongly depends on the amino acid sequence. Given this fold's simplicity, the most plausible source for such prefactor differences is the presence of nonnative interactions that stabilize the TSE but need to break up before folding resumes. Our results confirm long-standing theoretical predictions and bring into focus the rate prefactor as an essential element for understanding the mechanisms of folding.

When fast is better: protein folding fundamentals and mechanisms from ultrafast approaches.

Protein folding research stalled for decades because conventional experiments indicated that proteins fold slowly and in single strokes, whereas theory predicted a complex interplay between dynamics and energetics resulting in myriad microscopic pathways. Ultrafast kinetic methods turned the field upside down by providing the means to probe fundamental aspects of folding, test theoretical predictions and benchmark simulations. Accordingly, experimentalists could measure the timescales for all relevant folding motions, determine the folding speed limit and confirm that folding barriers are entropic bottlenecks. Moreover, a catalogue of proteins that fold extremely fast (microseconds) could be identified. Such fast-folding proteins cross shallow free energy barriers or fold downhill, and thus unfold with minimal co-operativity (gradually). A new generation of thermodynamic methods has exploited this property to map folding landscapes, interaction networks and mechanisms at nearly atomic resolution. In parallel, modern molecular dynamics simulations have finally reached the timescales required to watch fast-folding proteins fold and unfold in silico. All of these findings have buttressed the fundamentals of protein folding predicted by theory, and are now offering the first glimpses at the underlying mechanisms. Fast folding appears to also have functional implications as recent results connect downhill folding with intrinsically disordered proteins, their complex binding modes and ability to moonlight. These connections suggest that the coupling between downhill (un)folding and binding enables such protein domains to operate analogically as conformational rheostats.

Proteome folding kinetics is limited by protein halflife.

How heterogeneous are proteome folding timescales and what physical principles, if any, dictate its limits? We answer this by predicting copy number weighted folding speed distribution – using the native topology – for E.coli and Yeast proteome. E.coli and Yeast proteomes yield very similar distributions with average folding times of 100 milliseconds and 170 milliseconds, respectively. The topology-based folding time distribution is well described by a diffusion-drift mutation model on a flat-fitness landscape in free energy barrier between two boundaries: i) the lowest barrier height determined by the upper limit of folding speed and ii) the highest barrier height governed by the lower speed limit of folding. While the fastest time scale of the distribution is near the experimentally measured speed limit of 1 microsecond (typical of barrier-less folders), we find the slowest folding time to be around seconds (8 seconds for Yeast distribution), approximately an order of magnitude less than the fastest halflife (approximately 2 minutes) in the Yeast proteome. This separation of timescale implies even the fastest degrading protein will have moderately high (96%) probability of folding before degradation. The overall agreement with the flat-fitness landscape model further hints that proteome folding times did not undergo additional major selection pressures – to make proteins fold faster – other than the primary requirement to “sufficiently beat the clock” against its lifetime. Direct comparison between the predicted folding time and experimentally measured halflife further shows 99% of the proteome have a folding time less than their corresponding lifetime. These two findings together suggest that proteome folding kinetics may be bounded by protein halflife.

Why and how does native topology dictate the folding speed of a protein?

Since the pioneering work of Plaxco, Simons, and Baker, it is now well known that the rates of protein folding strongly correlate with the average sequence separation (absolute contact order (ACO)) of native contacts. In spite of multitude of papers, our understanding to the basis of the relation between folding speed and ACO is still lacking. We model the transition state as a gaussian polymer chain decorated with weak springs between native contacts while the unfolded state is modeled as a gaussian chain only. Using these hamiltonians, our perturbative calculation explicitly shows folding speed and ACO are linearly related when only the first order term in the series is considered. However, to the second order, we notice the existence of two new topological metrics, termed COC(1) and COC(2) (COC stands for contact order correction). These additional correction terms are needed to properly account for the entropy loss due to overlapping (nested or linked) loops that are not well described by simple addition of entropies in ACO. COC(1) and COC(2) are related to fluctuations and correlations among different sequence separations. The new metric combining ACO, COC(1), and COC(2) improves folding speed dependence on native topology when applied to three different databases: (i) two-state proteins with only α∕β and β proteins, (ii) two-state proteins (α∕β, β and purely helical proteins all combined), and (iii) master set (multi-state and two-state) folding proteins. Furthermore, the first principle calculation provides us direct physical insights to the meaning of the fit parameters. The coefficient of ACO, for example, is related to the average strength of the contacts, while the constant term is related to the protein folding speed limit. With the new scaling law, our estimate of the folding speed limit is in close agreement with the widely accepted value of 1 μs observed in proteins and RNA. Analyzing an exhaustive set (7367) of monomeric proteins from protein data bank, we find our new topology based metric (combining ACO, COC(1), and COC(2)) scales as N(0.54), N being the number of amino acids in a protein. This is in remarkable agreement with a previous argument based on random systems that predict protein folding speed depends on exp (-N(0.5)). The first principle calculation presented here provides deeper insights to the role of topology in protein folding and unifies many parallel arguments, seemingly disconnected, demonstrating the existence of universal mechanism in protein folding kinetics that can be understood from simple polymer physics based principles.

Predictions from an Ising-like Statistical Mechanical Model on the Dynamic and Thermodynamic Effects of Protein Surface Electrostatics.

Charged residues on the surface of a protein are known hot-spots for post-translational modification, protein/ligand-binding, and tuning conformational stabilities. Recent experimental evidence points to the fact that surface electrostatics can also modulate thermodynamic barriers and hence folding mechanisms. To probe for this behavior across different proteins, we develop a novel version of the Wako-Saitô-Muñoz-Eaton (WSME) model in which we include an electrostatic potential term in the energy function while simplifying the treatment of solvation free energy. Both of the energy terms are obtained by quantitatively fitting the model to differential scanning calorimetry (DSC) experiments that carry critical information on the protein partition function. We characterize four sets of structural/functional homologues (HEWL/BLA, CspB, engrailed, α-spectrin) either by fitting the experimental data of a single domain in the homologous set and predicting the conformational behavior of the rest with the same set of parameters or by performing semiblind predictions. The model with the added electrostatic term is able to successfully reproduce the order of thermodynamic stabilities and relaxation rates of most of the homologues. In parallel, we predict diverse conformational features including a wide range of thermodynamic barriers (∼9-40 kJ/mol), broad native ensembles in helical proteins, structured unfolded states and intermediates, rugged folding landscapes, and further provide an independent protein-specific estimate of the folding speed limit at 298 K (1/(7-300 μs)). Our results are evidence that protein surface electrostatics can be tailored to not only engineer stabilities but also folding mechanisms and the ruggedness of the underlying landscape.

Chain Length Determines the Folding Rates of RNA

We show that the folding rates (kFs) of RNA are determined by N, the number of nucleotides. By assuming that the distribution of free-energy barriers separating the folded and the unfolded states is Gaussian, which follows from central limit theorem arguments and polymer physics concepts, we show that kF≈k0exp(−αN0.5). Remarkably, the theory fits experimental rates spanning over 7 orders of magnitude with k0∼1.0(μs)−1. Our finding suggests that the speed limit of RNA folding is ∼1 ms, just as it is in the folding of globular proteins.

Speed limit of protein folding evidenced in secondary structure dynamics

As the simplest and most prevalent motif of protein folding, α-helix initiation is the starting point of macromolecular complexity. In this work, helix initiation was directly measured via ultrafast temperature-jump spectroscopy on the smallest possible helix nucleus for which only the first turn is formed. The rate's dependence on sequence, length, and temperature reveals the fastest possible events in protein folding dynamics, and it was possible to separate the rate-limiting torsional (conformational) diffusion from the fast annealing of the helix. An analytic coarse-grained model for this process, which predicts the initiation rate as a function of temperature, confirms this picture. Moreover, the stipulations of the model were verified by ensemble-converging all-atom molecular dynamics simulations, which reproduced both the picosecond annealing and the nanosecond diffusion processes observed experimentally.

The dynamical contact order: Protein folding rate parameters based on quantum conformational transitions

Protein folding is regarded as a quantum transition between the torsion states of a polypeptide chain. According to the quantum theory of conformational dynamics, we propose the dynamical contact order (DCO) defined as a characteristic of the contact described by the moment of inertia and the torsion potential energy of the polypeptide chain between contact residues. Consequently, the protein folding rate can be quantitatively studied from the point of view of dynamics. By comparing theoretical calculations and experimental data on the folding rate of 80 proteins, we successfully validate the view that protein folding is a quantum conformational transition. We conclude that (i) a correlation between the protein folding rate and the contact inertial moment exists; (ii) multi-state protein folding can be regarded as a quantum conformational transition similar to that of two-state proteins but with an intermediate delay. We have estimated the order of magnitude of the time delay; (iii) folding can be classified into two types, exergonic and endergonic. Most of the two-state proteins with higher folding rate are exergonic and most of the multi-state proteins with low folding rate are endergonic. The folding speed limit is determined by exergonic folding.

Direct Observation of Villin Folding in a Microfluidic Mixer

Villin headpiece subdomain (HP35) is a 35 residues polypeptide that folds on the microsecond timescale. Therefore, it has been an attractive object of protein folding for both computational and experimental studies, with some disagreement between the two methods. In this work, we observed folding of this protein after dilution of high denaturant in an ultrarapid microfluidic mixer far below the thermal midpoint. Tryptophan fluorescence changes were observed at various times beyond mixing from 8 ∼1500 μs at temperatures between 10 ∼ 40 °C. It is similar with T-jump that fluorescence relaxes can be fit with two exponential decays. Most of the amplitude decays within the mixing time, in agreement with the T-jump observation. But a small population decays on the tens of microsecond timescale. This slow phase is in rough agreement with MD simulations by Ensign et al. (1) and suggests a simple two-state model is not adequate to describe the folding pathway.1. Ensign DL, PM Kasson, VS Pande. 2007. Heterogeneity even at the speed limit of folding: Large-scale molecular dynamics study of a fast-folding variant of the villin headpiece. Journal of Molecular Biology 374(3):806-816.

Coordinate-dependent diffusion in protein folding

Diffusion on a low-dimensional free-energy surface is a remarkably successful model for the folding dynamics of small single-domain proteins. Complicating the interpretation of both simulations and experiments is the expectation that the effective diffusion coefficient D will in general depend on the position along the folding coordinate, and this dependence may vary for different coordinates. Here we explore the position dependence of D, its connection to protein internal friction, and the consequences for the interpretation of single-molecule experiments. We find a large decrease in D from unfolded to folded, for reaction coordinates that directly measure fluctuations in Cartesian configuration space, including those probed in single-molecule experiments. In contrast, D is almost independent of Q, the fraction of native amino acid contacts: Near the folded state, small fluctuations in position cause large fluctuations in Q, and vice versa for the unfolded state. In general, position-dependent free energies and diffusion coefficients for any two good reaction coordinates that separate reactant, product, and transition states, are related by a simple transformation, as we demonstrate. With this transformation, we obtain reaction coordinates with position-invariant D. The corresponding free-energy surfaces allow us to justify the assumptions used in estimating the speed limit for protein folding from experimental measurements of the reconfiguration time in the unfolded state, and also reveal intermediates hidden in the original free-energy projection. Lastly, we comment on the design of future single-molecule experiments that probe the position dependence of D directly.

Folding kinetics of two-state proteins based on the model of general random walk in native contact number space

We investigated the folding kinetics of a series of two-state proteins by using the model of general random walk in native contact number space, and derive the observed linear relationship between the logarithms of the folding rate constants and the numbers of native contacts from “kinetic viewpoint”. The protein folding speed limit and stability in this model are consistent with experimental observations.

Slowing Down Downhill Folding: A Three-Probe Study

The mutant Tyr22Trp/Glu33Tyr/Gly46Ala/Gly48Ala of λ repressor fragment λ6−85 was previously assigned as an incipient downhill folder. We slow down its folding in a cryogenic water-ethylene-glycol solvent (−18 to −28°C). The refolding kinetics are probed by small-angle x-ray scattering, circular dichroism, and fluorescence to measure the radius of gyration, the average secondary structure content, and the native packing around the single tryptophan residue. The main resolved kinetic phase of the mutant is probe independent and faster than the main phase observed for the pseudo-wild-type. Excess helical structure formed early on by the mutant may reduce the formation of turns and prevent the formation of compact misfolded states, speeding up the overall folding process. Extrapolation of our main cryogenic folding phase and previous T-jump measurements to 37°C yields nearly the same refolding rate as extrapolated by Oas and co-workers from NMR line-shape data. Taken together, all the data consistently indicate a folding speed limit of ∼4.5 μs for this fast folder.

Reaching the protein folding speed limit with large, sub-microsecond pressure jumps

Biomolecules are highly pressure-sensitive, but their dynamics upon return to ambient pressure are often too fast to observe with existing approaches. We describe a sample-efficient method capable of large and very fast pressure drops (<1 nanomole, >2,500 atmospheres and <0.7 microseconds). We validated the method by fluorescence-detected refolding of a genetically engineered lambda repressor mutant from its pressure-denatured state. We resolved barrierless structure formation upon return to ambient pressure; we observed a 2.1 +/- 0.7 microsecond refolding time, which is very close to the 'speed limit' for proteins and much faster than the corresponding temperature-jump refolding of the same protein. The ability to experimentally perform a large and very fast pressure drop opens up a new region of the biomolecular energy landscape for atomic-level simulation.

Dynamics of one-state downhill protein folding.

The small helical protein BBL has been shown to fold and unfold in the absence of a free energy barrier according to a battery of quantitative criteria in equilibrium experiments, including probe-dependent equilibrium unfolding, complex coupling between denaturing agents, characteristic DSC thermogram, gradual melting of secondary structure, and heterogeneous atom-by-atom unfolding behaviors spanning the entire unfolding process. Here, we present the results of nanosecond T-jump experiments probing backbone structure by IR and end-to-end distance by FRET. The folding dynamics observed with these two probes are both exponential with common relaxation times but have large differences in amplitude following their probe-dependent equilibrium unfolding. The quantitative analysis of amplitude and relaxation time data for both probes shows that BBL folding dynamics are fully consistent with the one-state folding scenario and incompatible with alternative models involving one or several barrier crossing events. At 333 K, the relaxation time for BBL is 1.3 micros, in agreement with previous folding speed limit estimates. However, late folding events at room temperature are an order of magnitude slower (20 micros), indicating a relatively rough underlying energy landscape. Our results in BBL expose the dynamic features of one-state folding and chart the intrinsic time-scales for conformational motions along the folding process. Interestingly, the simple self-averaging folding dynamics of BBL are the exact dynamic properties required in molecular rheostats, thus supporting a biological role for one-state folding.

Addendum to “Heterogeneity Even at the Speed Limit of Folding: Large-Scale Molecular Dynamics Study of a Fast-Folding Variant of the Villin Headpiece” [J. Mol. Biol. 374 (2007) 806–816

Addendum to “Heterogeneity Even at the Speed Limit of Folding: Large-Scale Molecular Dynamics Study of a Fast-Folding Variant of the Villin Headpiece” [J. Mol. Biol. 374 (2007) 806–816

Folding Kinetics of a Naturally Occurring Helical Peptide: Implication of the Folding Speed Limit of Helical Proteins

The folding mechanism and dynamics of a helical protein may strongly depend on how quickly its constituent alpha-helices can fold independently. Thus, our understanding of the protein folding problem may be greatly enhanced by a systematic survey of the folding rates of individual alpha-helical segments derived from their parent proteins. As a first step, we have studied the relaxation kinetics of the central helix (L9:41-74) of the ribosomal protein L9 from the bacterium Bacillus stearothermophilus , in response to a temperature-jump ( T-jump) using infrared spectroscopy. L9:41-74 has been shown to exhibit unusually high helicity in aqueous solution due to a series of side chain-side chain interactions, most of which are electrostatic in nature, while still remaining monomeric over a wide concentration range. Thus, this peptide represents an excellent model system not only for examining how the folding rate of naturally occurring helices differs from that of the widely studied alanine-based peptides, but also for estimating the folding speed limit of (small) helical proteins. Our results show that the T-jump induced relaxation rate of L9:41-74 is significantly slower than that of alanine-based peptides. For example, at 11 degrees C its relaxation time constant is about 2 micros, roughly seven times slower than that of SPE(5), an alanine-rich peptide of similar chain length. In addition, our results show that the folding rate of a truncated version of L9:41-74 is even slower. Taken together, these results suggest that individual alpha-helical segments in proteins may fold on a time scale that is significantly slower than the folding time of alanine-based peptides. Furthermore, we argue that the relaxation rate of L9:41-74 measured between 8 and 45 degrees C provides a realistic estimate of the ultimate folding rate of (small) helical proteins over this temperature range.

Understanding the mechanism of β‐sheet folding from a chemical and biological perspective

Perturbing the structure of the Pin1 WW domain, a 34-residue protein comprised of three beta-strands and two intervening loops has provided significant insight into the structural and energetic basis of beta-sheet folding. We will review our current perspective on how structure acquisition is influenced by the sequence, which determines local conformational propensities and mediates the hydrophobic effect, hydrogen bonding, and analogous intramolecular interactions. We have utilized both traditional site-directed mutagenesis and backbone mutagenesis approaches to alter the primary structure of this beta-sheet protein. Traditional site-directed mutagenesis experiments are excellent for altering side-chain structure, whereas amide-to-ester backbone mutagenesis experiments modify backbone-backbone hydrogen bonding capacity. The transition state structure associated with the folding of the Pin1 WW domain features a partially H-bonded, near-native reverse turn secondary structure in loop 1 that has little influence on thermodynamic stability. The thermodynamic stability of the Pin1 WW domain is largely determined by the formation of a small hydrophobic core and by the formation of desolvated backbone-backbone H-bonds enveloped by this hydrophobic core. Loop 1 engineering to the consensus five-residue beta-bulge-turn found in most WW domains or a four-residue beta-turn found in most beta-hairpins accelerates folding substantially relative to the six-residue turn found in the wild type Pin1 WW domain. Furthermore, the more efficient five- and four-residue reverse turns now contribute to the stability of the three-stranded beta-sheet. These insights have allowed the design of Pin1 WW domains that fold at rates that approach the theoretical speed limit of folding.