The Protein Folding Problem: The Role of Theory

Abstract

The protein folding problem was first articulated as question of how order arose from disorder in proteins: How did the various native structures of proteins arise from interatomic driving forces encoded within their amino acid sequences, and how did they fold so fast? These matters have now been largely resolved by theory and statistical mechanics combined with experiments. There are general principles. Chain randomness is overcome by solvation-based codes. And in the needle-in-a-haystack metaphor, native states are found efficiently because protein haystacks (conformational ensembles) are funnel-shaped. Order-disorder theory has now grown to encompass a large swath of protein physical science across biology.

Keywords: coarse-grained modeling; disordered proteins; protein aggregation; protein folding theory; statistical mechanics.

Fig. 1.

The first view of a protein structure. Myoglobin at 6 Å resolution, published in 1958 by Kendrew and coworkers. Its basic chain trace (white) did not have the simple symmetries or regularities previously seen in crystals of smaller molecules. [With permission from the Medical Research Council Laboratory of Molecular Biology].

Fig. 2.

Unfolded-state radii depend on chain length, charge and hydrophobicity. (A) Unfolded proteins in concentrated denaturant are highly expanded like polymers in Flory good solvents (R_g ~ N^v, where N is number of monomers and v ≈ 0.6). (B,C) In low denaturant, the compactnesses of unfolded proteins depend on hydrophobicity and charge. Data from Refs. ,

Fig. 3.

The helix-coil transition (Zimm and Bragg, 1959). The model was among the first explanations of disorder-order in helices, which are a key component of folded proteins. It’s not a full explanation for general folding, however, because β-sheets also fold cooperatively, and because these helix forces are weak (here, 1500-mer helices are required to show the same level of folding cooperativity as typical 100-mer proteins).

Fig. 4.

Experimental validation of the Thirumalai glassy folding kinetics model. Protein folding rates as a function of size. The folding rates were computed as the inverse of folding times from the data in Ref. confirming the theoretically derived relation log k_f ~ N^1/2 over 10 orders of magnitude. The black line is a linear fit to the data.

Fig. 5.

The Foldon Funnel Model describes the thermodynamics and kinetics of folding. (A) The energy landscape of folding is volcano-shaped for a 4-helix-bundle. As the protein progresses along the folding pathway it accumulates local secondary structures that are not stable by themselves (uphill landscape). Tertiary folds result from the cooperative associations of secondary structural elements that ultimately confer the protein its stable native fold at the last step (downhill landscape). (B) The Foldon Funnel Model describes state populations versus time. A single helix is meta-stable long enough to form helix dimers, then trimers, and so on. The black line shows the native state rising late and becoming fully populated with time as dimers and trimers progress on the folding landscape. Reproduced from Ref.

Fig. 6.

IDP conformations depend on sequence patterns, as explained by models and simulations. (A) Two sequence patterns: blocky (left) and strictly alternating (right). SCD measures the blockiness of anionic and cationic residues, while SHD measures the blockiness of amino acids of differing hydropathy values. (B,C) Analytical theory of SCD predicts IDP conformations (x) (where $x=\langle R_e^2\rangle /\langle R_{e,\text{randomcoil}}^2\rangle$) for (B) wild type and (C) its two phosphorylated variants. Theory (lines) agrees well with all-atom simulations (points) (Ref. 159). (D,E) Modeling predicts the effects of charge and hydropathy patterning on the scaling exponent and R_g from experiments^,,,,,, (Ref. 165).

Fig. 7.

Antibody liquid phase equilibria computed from Wertheim liquid theory. (A) Isolated beads first stick together as 7-bead Y-shaped (covalent) molecules. These can then come together into (noncovalent) assemblies in solution. (B) Phase diagram of temperature T versus antibody concentration from experiments (points) and theory (line). Adapted from Ref.

Fig. 8.

Kinetic theory for amyloid fibril formation. (A) The number of hydrogen bonds made at the templating fibril is a quantity that undergoes a random walk in the range [0,N] where N = 6 is number of monomers in the incoming peptide. The time for a chain to fully attach (or release) from the fibril is computed from the rates of forming (or breaking) all hydrogen bonds in this simplified scheme. The growth rate for insulin fibrils computed by this theory qualitatively matches that measured by experiments as a function of (B) peptide concentration, (C) temperature and (D) denaturant concentration (Ref. 228).

Fig. 9.

The dominant mechanism of forming amyloid oligomers. (A) Free monomers attach to an existing fibril surface and rearrange into new oligomers with rate k₂. This is known as secondary nucleation. (B) By solving the population balance equations under different scenarios, the likely mechanism dominating fibril nucleation, the secondary pathway shown here, can be inferred by fitting to experimental data at different monomer concentrations (colored lines). The kinetic data were obtained from Ref. .

Fig. 10.

A single-chain collapse property predicts a multi-chain phase equilibrium property of IDPs. (A) Collapse transition of an IDP (B) The critical temperature and Θ temperature of an IDP were determined in coarse-grained simulations, and show good agreement. (C) The agreement is consistent over a wide range of IDPs. Reproduced from Ref. (D) Information from single chain simulations can be used to predict coexistence concentrations (+ symbols) and show agreement with concentrations obtained with coarse-grained phase-separation simulations (o symbols). Temperatures are in reduced units. Reproduced from Ref. .

Fig. 11.

Viral capsids result from simple geometry rules. (Left) Different capsid sizes come from different geometric triangulations T, between the centers of polyhedra on a hexagonal lattice. (Middle) Stable capsids result by placing pentamers on the 12 icosahedral vertices and hexamers in between. (Right) The individual pentameric and hexameric protein complexes of the capsid are formed from the individual proteins shown as dot bundles at the center of each pentamer/hexamer. Reproduced from Ref.