Simulations of chaperone-assisted folding

We investigated a chaperone mechanism of protein folding using a 36-mer model on a cubic lattice. The mechanism simulates folding, which proceeds with repetitive cycles of binding, unfolding, and releasing of misfolded metastable states. We measured the yield enhancement due to this mechanism for sequences selected by evolutionary design and showed that the binding and releasing mechanism is efficient for the yield enhancement of folding for sequences that are poorly designed, i.e., where selection is not adequately strong. From this it follows that the chaperone mechanism can be considered as the evolutionary alternative to compensate for poor sequence design. On the other hand, random sequences show a decrease in yield and no effect on the total mean first passage time when the proposed chaperone mechanism is implemented, thus implying that sequence optimization is a necessary condition for the efficiency of the proposed mechanism. We qualitatively reproduced experimental results for folding in the presence of GroEL/GroES, fit our results with the aid of a double-exponential model of folding kinetics, and characterized the conditions under which this mechanism of chaperone action affects folding.

Modeling protein folding: the beauty and power of simplicity

It is argued that simplified models capture key features of protein stability and folding, whereas more detailed models may be more appropriate for protein structure prediction. A brief overview of experimental and theoretical results is presented that corroborates these points. I argue that statistical models capture the key principle of protein stability-cooperativity- and therefore provide a reasonable estimate of protein free energy whereas more detailed but less physically transparent calculations fail to do so. I also explain that the previously published claim that simple models give predictions that are inconsistent with experiments on polypeptide block-copolymers is based on incomplete analysis of such experiment.

Improved design of stable and fast-folding model proteins

BACKGROUND

A number of approaches to design stable and fast-folding sequences for model polypeptide chains have been based on the premise that optimization of the relative energy of the native conformation (or Z-score) is sufficient to yield stable and fast-folding sequences. Although this approach has been successful, for longer chains it often yielded sequences that failed to fold cooperatively, instead having multidomain folding behavior.

RESULTS

We show that one of the factors determining single-domain or multidomain folding behavior is the dispersion of energies of native contacts. So, we study folding of sequences optimized to have the same native conformation as a global energy minimum but having different dispersion of native contact energies. Our results suggest that under conditions at which native conformation is stable, the best-folding proteins are those that have smaller heterogeneity of native contact energies. For them, the folding transition is all-or-none. On the other hand, proteins with greater heterogeneity of native contact energies have more gradual multidomain folding transition and fold into stable native conformation much slower than those proteins with small dispersion of native contact energies.

Design of proteins with selected thermal properties

BACKGROUND

Methods of model protein design have until now been largely ad hoc, yielding sequences that are foldable only at some seemingly arbitrary simulation temperature. But real proteins exist and must fold within an imposed thermal environment. The need exists for a sequence design method based on statistical-mechanical first principles, thus containing a rigorous treatment of folding temperature.

RESULTS

In this work, we report a method of rational sequence design that takes a target structure and a desired optimal folding temperature TZ and generates a sequence that is predicted to be thermodynamically stable with respect to the target structure at a folding temperature TF approximately TZ. This 'cumulant design method' is based on a mean-field high temperature expansion of the molecular partition function. Folding simulations of the designed sequences confirm that sequences designed at TZ do indeed fold optimally when TF approximately TZ.

CONCLUSIONS

The cumulant method is highly successful in designing model proteins. It also provides some insight into the thermal properties of real proteins, illuminating the features that distinguish thermostable and psychotropic (cold-loving) sequences from their mesophilic counterparts.

Universality and diversity of the protein folding scenarios: a comprehensive analysis with the aid of a lattice model

BACKGROUND

The role of intermediates in protein folding has been a matter of great controversy. Although it was widely believed that intermediates play a key role in minimizing the search problem associated with the Levinthal paradox, experimental evidence has been accumulating that small proteins fold fast without any detectable intermediates.

RESULTS

We study the thermodynamics and kinetics of folding using a simple lattice model. Two folding sequences obtained by the design procedure exhibit different folding scenarios. The first sequence folds fast to the native state and does not exhibit any populated intermediates during folding. In contrast, the second sequence folds much slower, often being trapped in misfolded low-energy conformations. However, a small fraction of folding molecules for the second sequence fold on a fast track avoiding misfolded traps. In equilibrium at the same temperature the second sequence has a highly populated intermediate with structure similar to that of the kinetics intermediate.

CONCLUSIONS

Our analysis suggests that intermediates may often destabilize native conformations and derail the folding process leading it to traps. Less-optimized sequences fold via parallel pathways involving misfolded intermediates. A better designed sequence is more stable in the native state and folds fast without intermediates in a two-state process.

Impact of local and non-local interactions on thermodynamics and kinetics of protein folding

To address the question of how the geometry of a protein's native conformation affects its folding and stability, we studied three model 36-mers on a cubic lattice. The native structure of one of these model 36-mers consisted mostly of local contacts, while that of a second consisted mostly of non-local contacts. The third native structure had a typical compact native conformation, and served as our reference. For each protein, the amino acid sequence was designed to have a pronounced energy minimum at its native conformation. We observed dramatic differences in folding, dependent on the presence or absence of non-local contacts. For the proteins with a typical large number of non-local contacts, the folding transition was all-or-none, whereas for the one with mostly local contacts, it was not. Although the maximum rate of folding was similar for all three proteins, we found that under conditions at which each native conformation was stable, the structure with mostly non-local contacts folded two orders of magnitude faster than the one with mostly local contacts. The statistical analysis of protein structure agrees fully with the implications of the theory. We discuss the importance of cooperativity in protein folding for its stability.

Exhaustive enumeration of protein conformations using experimental restraints

We present an efficient new algorithm that enumerates all possible conformations of a protein that satisfy a given set of distance restraints. Rapid growth of all possible self-avoiding conformations on the diamond lattice provides construction of alpha-carbon representations of a protein fold. We investigated the dependence of the number of conformations on pairwise distance restraints for the proteins crambin, pancreatic trypsin inhibitor, and ubiquitin. Knowledge of between one and two contacts per monomer is shown to be sufficient to restrict the number of candidate structures to approximately 1,000 conformations. Pairwise RMS deviations of atomic position comparisons between pairs of these 1,000 structures revealed that these conformations can be grouped into about 25 families of structures. These results suggest a new approach to assessing alternative protein folds given a very limited number of distance restraints. Such restraints are available from several experimental techniques such as NMR, NOESY, energy transfer fluorescence spectroscopy, and crosslinking experiments. This work focuses on exhaustive enumeration of protein structures with emphasis on the possible use of NOESY-determined distance restraints.

Domains in folding of model proteins

By means of Monte Carlo simulation, we investigated the equilibrium between folded and unfolded states of lattice model proteins. The amino acid sequences were designed to have pronounced energy minimum target conformations of different length and shape. For short fully compact (36-mer) proteins, the all-or-none transition from the unfolded state to the native state was observed. This was not always the case for longer proteins. Among 12 designed sequences with the native structure of a fully compact 48-mer, a simple all-or-none transition was observed in only three cases. For the other nine sequences, three states of behavior-the native, denatured, and intermediate states-were found. The contiguous part of the native structure (domain) was conserved in the intermediate state, whereas the remaining part was completely unfolded and structureless. These parts melted separately from each other.

Is burst hydrophobic collapse necessary for protein folding?

Folding of the lattice model of proteins is studied using Monte Carlo simulation. The amino acid sequence is designed to have a pronounced energy minimum for a given target (native) conformation. Our simulations reveal two possible scenarios. When the overall attraction between residues dominates, we find that folding to the native conformation is preceded by a rapid collapse into a burst intermediate which is a compact but structureless globule. Then, after a much longer time, an all-or-none transition from the globule to the native conformation occurs. In contrast, when the overall attraction is not strong, we do not observe a burst collapse stage. Instead, we find an all-or-none transition directly from the coil to the native conformation. Both scenarios yield comparable rates of folding. On the basis of these findings we discuss the role of intermediates in thermodynamics and kinetics of protein folding.

Evolution-like selection of fast-folding model proteins

We propose an algorithm providing sequences of model proteins with rapid folding into a given target (native) conformation. This algorithm is applied to a chain of 27 residues on a cubic lattice. It generates sequences with folding 2 orders of magnitude faster than that of the practically random starting sequence. Thermodynamic analysis shows that the increase in speed is matched by an increase in stability: the evolved sequences are much more stable in their native conformation than the initial random sequence. The unfolding temperature for evolved sequences is slightly higher than the simulation temperature, bearing direct correspondence to the relatively low stability of real proteins.

A test of lattice protein folding algorithms

We report a blind test of lattice-model-based search strategies for finding global minima of model protein chains. One of us (E.I.S.) selected 10 compact conformations of 48-mer chains on the three-dimensional cubic lattice and used their inverse folding algorithm to design HP (H, hydrophobic; P, polar) sequences that should fold to those "target" structures. The sequences, but not the structures, were sent to the UCSF group (K.Y., K.M.F., P.D.T., H.S.C., and K.A.D.), who used two methods to attempt to find the globally optimal conformations: "hydrophobic zippers" and a constraint-based hydrophobic core construction (CHCC) method. The CHCC method found global minima in all cases, and the hydrophobic zippers method found global minima in some cases, in minutes to hours on workstations. In 9 out of 10 sequences, the CHCC method found lower energy conformations than the 48-mers were designed to fold to. Thus the search strategies succeed for the HP model but the design strategy does not. For every sequence the global energy minimum was found to have multiple degeneracy with 10(3) to 10(6) conformations. We discuss the implications of these results for (i) searching conformational spaces of simple models of proteins and (ii) how these simple models relate to proteins.

Pseudodihedrals: simplified protein backbone representation with knowledge-based energy

Pairwise contact energies do not explicitly take protein secondary structure into account, and so provide an incomplete description of conformational energy. In order to construct a Hamiltonian that specifically relates to protein backbone conformations, a simplified backbone angle is used. The pseudodihedral angle (the torsion angle between planes defined by 4 consecutive alpha-carbon atoms) provides a simplified backbone representation and continues to manifest information about secondary-structure elements: the pseudo-Ramachandran plot contains helical and sheetlike regions. The distribution of pseudodihedral angles is highly sensitive to the identity of the central pair of amino acids. Therefore, a sequence-dependent, knowledge-based potential energy was found according to a quasichemical approximation. These functions form complementary additions to the contact potentials currently in use. This pseudodihedral potential greatly enhances the ability to design sequences that are specific to a given conformation and also improves the ability to discriminate a native conformation from many other conformations.

Specific nucleus as the transition state for protein folding: evidence from the lattice model

We have studied the folding mechanism of lattice model 36-mer proteins. Using a simulated annealing procedure in sequence space, we have designed sequences to have sufficiently low energy in a given target conformation, which plays the role of the native structure in our study. The sequence design algorithm generated sequences for which the native structures is a pronounced global energy minimum. Then, designed sequences were subjected to lattice Monte Carlo simulations of folding. In each run, starting from a random coil conformation, the chain reached its native structure, which is indicative that the model proteins solve the Levinthal paradox. The folding mechanism involved nucleation growth. Formation of a specific nucleus, which is a particular pattern of contacts, is shown to be a necessary and sufficient condition for subsequent rapid folding to the native state. The nucleus represents a transition state of folding to the molten globule conformation. The search for the nucleus is a rate-limiting step of folding and corresponds to overcoming the major free energy barrier. We also observed a folding pathway that is the approach to the native state after nucleus formation; this stage takes about 1% of the simulation time. The nucleus is a spatially localized substructure of the native state having 8 out of 40 native contacts. However, monomers belonging to the nucleus are scattered along the sequence, so that several nucleus contacts are long-range while other are short-range. A folding nucleus was also found in a longer chain 80-mer, where it also constituted 20% of the native structure. The possible mechanism of folding of designed proteins, as well as the experimental implications of this study is discussed.

A new approach to the design of stable proteins

We propose a simple algorithm to design a sequence which fits a given protein structure with a given energy. The algorithm is a modification of the Metropolis Monte Carlo scheme in sequence space with an evolutionary temperature which sets the energy scale. There is a one to one correspondence between this optimization scheme and the Ising model of ferromagnetism. This analogy implies that the design algorithm does not encounter multiple-minima problems and is very fast. The algorithm is tested by 'predicting' the primary structures of four proteins. In each case the calculated primary structures had statistically significant homology with the natural structures.

Engineering of stable and fast-folding sequences of model proteins

The statistical mechanics of protein folding implies that the best-folding proteins are those that have the native conformation as a pronounced energy minimum. We show that this can be obtained by proper selection of protein sequences and suggest a simple practical way to find these sequences. The statistical mechanics of these proteins with optimized native structure is discussed. These concepts are tested with a simple lattice model of a protein with full enumeration of compact conformations. Selected sequences are shown to have a native state that is very stable and kinetically accessible.

Influence of point mutations on protein structure: probability of a neutral mutation

We investigate the probability X that a random mutation (i.e. the substitution in a random site of one amino acid residue by randomly chosen residue) will be a neutral one, i.e. it will not lead to a change in structure. Using a random energy model for the description of protein energy "levels" we show that this probability depends only on stiffness of a chain which is characterized by the number of conformations per peptide bond gamma. The result is X approximately gamma -8. The application of this result for protein engineering experiments and for possible scenario of neutral evolution is discussed.

Implications of thermodynamics of protein folding for evolution of primary sequences

Natural proteins exhibit essentially two-state thermodynamics, with one stable fold that dominates thermodynamically over a vast number of possible folds, a number that increases exponentially with the size of the protein. Here we address the question of whether this feature of proteins is a rare property selected by evolution or whether it is in fact true of a significant proportion of all possible protein sequences. Using statistical procedures developed to study spin glasses, we show that, given certain assumptions, the probability that a randomly synthesized protein chain will have a dominant fold (which is the global minimum of free energy) is a function of temperature, and that below a critical temperature the probability rapidly increases as the temperature decreases. Our results suggest that a significant proportion of all possible protein sequences could have a thermodynamically dominant fold.