Folding Transition Temperature Research Articles

Roles of native topology and chain-length scaling in protein folding: A simulation study with a Gō-like model

We perform folding simulations on 18 small proteins with using a simple Gō-like protein model and analyze the folding rate constants, characteristics of the transition state ensemble, and those of the denatured states in terms of native topology and chain length. Near the folding transition temperature, the folding rate kF scales as kF ∼ exp(−c RCO N0.6) where RCO and N are the relative contact order and number of residues, respectively. Here the topology RCO dependence of the rates is close to that found experimentally (kF ∼ exp(−c RCO)), while the chain length N dependence is in harmony with the predicted scaling property (kF ∼ exp(−c N2/3)). Thus, this may provides a unified scaling law in folding rates at the transition temperature, kF ∼ exp(−c RCO N2/3). The degree of residual structure in the denatured state is highly correlated with RCO, namely, proteins with smaller RCO tend to have more ordered structure in the denatured state. This is consistent with the observation that many helical proteins such as myoglobin and protein A, have partial helices, in the denatured states. The characteristics of the transition state ensemble calculated by the current model, which uses native topology but not sequence specific information, are consistent with experimental φ-value data for about half of proteins.

Characterization of the folding kinetics of a three-helix bundle protein via a minimalist Langevin model

We use a simple off-lattice Langevin model of protein folding to characterize the folding and unfolding of a fast-folding, 46 residue three-helix bundle. Under conditions at which the C-terminal helix is 30 % stable, we observe a clear three-state folding mechanism. In the on-pathway intermediate state, the middle and C-terminal helices are folded and in contact with each other, while the N-terminal region remains disordered. Nevertheless, under these conditions this intermediate is thermodynamically unstable relative to its unfolded state. The first and highest folding barrier corresponds to the organization of the hinge between the middle and C-terminal helices. A subsequent major barrier corresponds to the organization of the hinge between the middle and N-terminal helices. Hyperstabilizing the hinge regions leads to twice the folding rate that is obtained from hyperstabilizing the helices, even though much fewer contacts are involved in hinge hyperstabilization than in helix hyperstabilization. Unfolding follows single-exponential kinetics, even at temperatures only slightly above the folding transition temperature.

Extraction of interaction potentials between amino acids from native protein structures

We discuss methods for the determination of the effective pairwise interactions between amino acids in globular proteins in order to be able to easily recognize the native state conformation of any protein sequence among a set of decoy structures. The first method entails the application of a numerical strategy to a training set of proteins that maximizes the native fold stability with respect to alternative structures. The extracted parameters are shown to be very reliable for identifying the native states of proteins (unrelated to those in the training set) among thousands of conformations. Folding transition temperatures are estimated for a few proteins for which reliable alternative structures have recently been generated. The only poor performers are proteins with stabilizing heme groups whose complexity cannot be captured by standard pairwise energy functionals. The key ingredient of this technique is the knowledge of viable decoys for each protein sequence in the training set. We then present a second strategy which circumvents this difficulty. This method relies on the fact that protein sequences are special compared to random heteropolymers and are characterized by high thermodynamic stability in their native conformations. We validate the technique on a lattice model of proteins, we apply it to real proteins and carry out tests of the quality of the extracted interaction parameters. We find that this novel technique leads to good results that are comparable to those obtained with the first method.

Linking rates of folding in lattice models of proteins with underlying thermodynamic characteristics

We investigate the sequence-dependent properties of proteins that determine the dual requirements of stability of the native state and its kinetic accessibility using simple cubic lattice models. Three interaction schemes are used to describe the potentials between nearest neighbor nonbonded beads. We show that, under the simulation conditions when the native basin of attraction (NBA) is the most stable, there is an excellent correlation between folding times τF and the dimensionless parameter σT=(Tθ−TF)/Tθ, where Tθ is the collapse temperature and TF is the folding transition temperature. There is also a significant correlation between τF and another dimensionless quantity Z=(EN−Ems)/δ, where EN is the energy of the native state, Ems is the average energy of the ensemble of misfolded structures, and δ is the dispersion in the contact energies. In contrast, there is no significant correlation between τF and the Z-score gap ΔZ=EN−Ems. An approximate relationship between σT and the Z-score is derived, which explains the superior correlation seen between τF and σT. For two state folders τF is linked to the free energy difference (not simply energy gap, however it is defined) between the unfolded states and the NBA.

Cooperativity in protein folding: from lattice models with sidechains to real proteins

Over the past few years novel folding mechanisms of globular proteins have been proposed using minimal lattice and off-lattice models. The factors determining the cooperativity of folding in these models and especially their explicit relation to experiments have not been fully established, however. We consider equilibrium folding transitions in lattice models with and without sidechains. A dimensionless measure, omega c, is introduced to quantitatively assess the degree of cooperativity in lattice models and in real proteins. We show that larger values of omega c resembling the values seen in proteins are obtained in lattice models with sidechains. The enhanced cooperativity of such models results from possible denser packing of sidechains in the interior of the model polypeptide chain. We also establish that omega c correlates extremely well with sigma T = (T o - T f) /T o, where T o and T f are collapse and folding transition temperatures, respectively. These theoretical ideas are used to analyze folding transitions in two-state folders (RNase A, chymotrypsin inhibitor 2, fibronectin type III modules and tendamistat) and three-state folders (apomyoglobin and lysozyme). The values of omega c extracted from experiments show a correlation with sigma T (suitably generalized when folding is induced by denaturants or acid). A quantitative description of the cooperative transition of real proteins can be made by lattice models with sidechains. The degree of cooperativity in minimal models and real proteins can be expressed in terms of the single parameter sigma, which can be estimated from experimental data.

A free energy analysis by unfolding applied to 125-mers on a cubic lattice

A common approach to the protein folding problem involves computer simulation of folding using lattice models of amino acid sequences. Key factors for good performance in such models are the correct choice of the temperature and the average interaction energy between residues. In order to push the lattice approach to its limit it is important to have a method to adjust these parameters for optimal folding that is not limited by our ability to successfully simulate folding in a reasonable time. In this study, we adopt a simple cubic-lattice model and present a method for calculating the free energy of a chain as a function of the number of native contacts. This does not require that we are able to fold the sequence by simulation and it provides a method of estimating the folding transition temperature. For a given set of parameters, the free energy analysis also allows an estimate of foldability. By applying the method to sequences with 27 and 125 residues, we show that optimal folding occurs near the folding transition temperature and at either zero or small negative average interaction energy. We find ourselves able to fold only 125-mers that have significant short-range native contacts. A free energy analysis during unfolding is a useful tool for the study of foldability and should be applicable to a variety of folding models. In this way we are able to fold some 125-mer designed sequences and our results confirm the finding that short-range contacts contribute to foldability.

Kinetic partitioning mechanism as a unifying theme in the folding of biomolecules

We present a unified framework for folding kinetics of proteins and RNA. The basis for this framework relies on the notion of topological frustration, which gives rise to several competing basins of attraction (CBA) in addition to the native basin of attraction (NBA) on the free energy surface. A rough free energy surface results in direct and indirect pathways to the NBA, i.e., a kinetic partitioning mechanism (KPM). The unified framework for folding kinetics allows us to propose a foldability principle, according to which fast folding sequences are characterized by the folding transition temperature $T_{F}$ being close to the collapse transition temperature $T_{\theta }$. Biomolecules, for which foldability principle is satisfied, such as small proteins and tRNAs, are expected to fold rapidly with two-state kinetics. Estimates for the multiple time scales in KPM are also given.

Dynamics of Random Hydrophobic-Hydrophilic Copolymers with Implications for Protein Folding.

We study the dynamics of a single amphiphilic random copolymer chain. The dynamical treatment shows that in addition to the collapse temperature, ${T}_{\ensuremath{\Theta}}$, there exists a scale dependent glass transition temperature ${T}_{G}(l)$. Distinct scenarios for the glassy behavior emerges depending on the relative values of ${T}_{\ensuremath{\Theta}}$ and ${T}_{G}(l)$. The possible implications of our results for protein folding, including an estimate of the dependence of the ratio of the folding transition temperature to ${T}_{G}(l)$ on the length of the chain, are sketched.

Kinetics and Thermodynamics of Folding of a de NovoDesigned Four-helix Bundle Protein

A simple continuum model of a de novodesigned model of a four-helix bundle is presented. The thermodynamics and kinetics of the model are studied using Langevin simulations. We use a three-letter minimal off- lattice representation of a de novodesigned four-helix bundle protein. The native state of the model, which can be thought of as an alpha-carbon representation of the peptide chain, is a caricature of the sequence designed by Ho and Degrado and shows several characteristics found in the naturally occurring four-helix bundles. These include the structural aspects and the relative stability of the native conformation. The model four-helix bundle shows two characteristic temperatures T θand T f. The former is the temperature above which the structure resembles that of the random coil. Below the first-order folding transition temperature T fthe chain adopts the native conformation corresponding to the four-helix bundle. It is shown that in order to obtain a unique native structure a proper free energy balance between secondary and tertiary interactions is needed. The thermal denaturation starting from the unique native conformation indicates that at least a three-state analysis is required. The intermediates in the equilibrium thermal denaturation are all found to be native-like. The kinetics of refolding starting from an ensemble of denatured states shows that the acquisition of the native conformation takes place viaa kinetic partitioning mechanism. A fraction of molecules, Φ, reaches the native state by a topology inducing nucleation collapse mechanism, while the remainder (1 −Φ) follows a complex three-stage multipathway process. We suggest, in accord with our earlier studies, that Φ is essentially determined by the intrinsic temperature scales T θand T f. Our studies indicate that better design of proteins can be achieved by making T θas close to T fas possible. Experimental implications for de novodesign of proteins are briefly discussed.

Criterion that determines the foldability of proteins.

We point out that the correlation between folding times and $\sigma = (T_{\theta } - T_{f})/T_{\theta }$ in protein-like heteropolymer models where $T_{\theta }$ and $T_{f}$ are the collapse and folding transition temperatures was already established in 1993 before the other presumed equivalent criterion (folding times correlating with $T_{f}$ alone) was suggested. We argue that the folding times for these models show no useful correlation with the energy gap even if restricted to the ensemble of compact structures as suggested by Karplus and Shakhnovich (cond-mat/9606037).

From Minimal Models to Real Proteins: Time Scales for Protein Folding Kinetics

The multipathway mechanism discovered using minimal protein models in conjunction with scaling arguments are used to obtain time scales for the various processes in the folding of real proteins. We consider pathways involving low energy native-like structures as well as direct pathways that proceed via a nucleation mechanism. The average activation barrier separating the low energy structures and the native state is predicted to scale as √N where N is the number of aminoacids in the proteins. In addition estimates of folding times for direct pathways in which collapse and folding are (almost) synchronous are given. It is argued folding sequences whose folding transition temperature is very close to the collapse transition temperature are likely to reach the native conformation rapidly.

Simple model of protein folding kinetics.

A simple model of the kinetics of protein folding is presented. The reaction coordinate is the "correctness" of a configuration compared with the native state. The model has a gap in the energy spectrum, a large configurational entropy, a free energy barrier between folded and partially folded states, and a good thermodynamic folding transition. Folding kinetics is described by a master equation. The folding time is estimated by means of a local thermodynamic equilibrium assumption and then is calculated both numerically and analytically by solving the master equation. The folding time has a maximum near the folding transition temperature and can have a minimum at a lower temperature.

Folding kinetics of proteins: A model study

We study the kinetics of folding of a heteropolymer containing a specific sequence of hydrophobic, hydrophilic, and neutral residues. The heteropolymer, representing the alpha carbon sequence of a model protein, folds into a β-barrel structure below a characteristic folding transition temperature. Several measures are introduced to probe the dynamics of approach to a folded state with particular emphasis on the kinetics of formation of nonbonded contacts starting from a high temperature random configuration. The measures of compactness are used to argue that even below the folding temperature, nonbonded contacts are formed at a very slow rate. This result is further corroborated by analyzing the dynamics of relaxation of the nonbonded interactions which are responsible for the formation of folded structures at low temperatures. We also show that below the folding transition temperature and for short times, the heteropolymer undergoes large structural fluctuations. Examination of the time dependence of the radius of gyration Rg shows that the heteropolymer is trapped in one minimum (characterized by a roughly constant Rg ) for arbitrary times followed by sudden excursions to neighboring minima. In this regime, multistate kinetics is appropriate. However, at longer times when the heteropolymer becomes trapped in one of the minima corresponding to the folded conformation, the folding process appears to be an all or none process. The implication of our results for the kinetics of in vitro folding of proteins are discussed.

The nature of folded states of globular proteins.

We suggest, using dynamical simulations of a simple heteropolymer modelling the alpha-carbon sequence in a protein, that generically the folded states of globular proteins correspond to statistically well-defined metastable states. This hypothesis, called the metastability hypothesis, states that there are several free energy minima separated by barriers of various heights such that the folded conformations of a polypeptide chain in each of the minima have similar structural characteristics but have different energies from one another. The calculated structural characteristics, such as bond angle and dihedral angle distribution functions, are assumed to arise from only those configurations belonging to a given minimum. The validity of this hypothesis is illustrated by simulations of a continuum model of a heteropolymer whose low temperature state is a well-defined beta-barrel structure. The simulations were done using a molecular dynamics algorithm (referred to as the "noisy" molecular dynamics method) containing both friction and noise terms. It is shown that for this model there are several distinct metastable minima in which the structural features are similar. Several new methods of analyzing fluctuations in structures belonging to two distinct minima are introduced. The most notable one is a dynamic measure of compactness that can in principle provide the time required for maximal compactness to be achieved. The analysis shows that for a given metastable state in which the protein has a well-defined folded structure the transition to a state of higher compactness occurs very slowly, lending credence to the notion that the system encounters a late barrier in the process of folding to the most compact structure. The examination of the fluctuations in the structures near the unfolding----folding transition temperature indicates that the transition state for the unfolding to folding process occurs closer to the folded state.