As early studies revealed that an unfolded protein can fold spontaneously to a three-dimensional structure under suitable environmental conditions 12, traditional approaches to understanding protein folding have focused on the prediction of the native structure. As more studies showed the existence of intermediates and interaction among residues during the protein folding process 34, there is substantial interest to understand the time order of events during the formation of the tertiary structure. From the free energy point of view, each conformation of a protein is associated with a free energy and the protein folds from the high-energy denatured conformation to its folded structure along a funnel-like energy landscape 56.

Although advances in experimental techniques allow the investigation of protein folding pathways at the microsecond timescale 78, experimental determination of protein folding pathways remains difficult. Most studies are only able to identify general characteristics of the folding pathway without much details and are limited to analyzing small proteins. Computational techniques are often used to simulate protein folding and the problem is transformed to energetic optimization problems, that is, computational search for global energy minimum over all possible conformations. The most accurate computational techniques utilize molecular dynamics to determine the order of events that lead to the tertiary structure through atomic-level simulations 9101112. Due to the extremely large conformation space, these approaches suffer from well-known problems accompanying high dimensionality, including computational expensiveness and ease of trapping in local minima, and are applicable only to small proteins in a short time course.

By omitting some details, proteins can be represented at the level of amino acids. Kolinski and Skolnick 13 performed Monte Carlo simulations of protein folding on a reduced lattice representation of the protein α-carbon backbone. Yue and Dill 14 limited the conformation space to a discrete subset of possibilities and used a branch-and-bound procedure to search for near-optimal conformations. Alm and Baker 15 and Muñoz and Eaton 16 further observed that the availability of the known native structure can dramatically reduce the search space. Alm and Baker 15 took into account only native interactions among residues and used a sequential binary collision model to predict protein folding mechanisms from the perspectives of free energy landscapes, while Muñoz and Eaton 16 used a slightly different approach of employing distinct free energy costs for different secondary structures. Amato and Song 17 represented a protein by the torsional angles of its residues and used the probabilistic roadmap technique with a biased sampling strategy around the native structure to predict folding pathways and secondary structure formation order. Liwo et al 18 and Kmiecik and Kolinski 1920 showed that the use of reduced models of proteins is highly successful in characterizing folding pathways for small proteins at the mesoscopic level. Although these techniques are able to predict folding pathways very accurately for proteins with up to about 100 residues, the majority of proteins in the Protein Data Bank (PDB) 21 are much larger (Figure 1).

The problem with representing a protein at the amino acid level is that even with the assumption that each residue has only two states (ordered or disordered), a protein with n residues still has 2ⁿ possible conformations 15. To overcome this problem, several recent approaches represent a protein at the level of secondary structure elements (SSEs), in which each element corresponds to one helix or one β-strand. By adopting the framework model in which secondary structures are thought to fold relatively independently of the tertiary structure 22, each SSE is treated as an indivisible unit that interacts with other SSEs as a whole. Since the number of SSEs in a protein is small (Figure 1), this model is much more tractable to simulate. Eyrich et al 23 assumed that the SSEs are fixed and used a branch-and-bound algorithm to search for near-optimal tertiary structures. Apaydin et al 24 assumed that each SSE of a protein is already in native conformation and moves as a unit, and used the probabilistic roadmap approach to predict folding pathways. Zaki et al 25 proposed an algorithm to predict unfolding pathways based on applying a minimum cut procedure to a weighted graph that represents a protein's contact map or interaction strength between SSEs. Although the underlying assumption that intermediate secondary structures are fully folded before the formation of tertiary structures is not satisfied for most proteins, these studies show that such a strategy is sufficient to study protein folding pathways at the mesoscopic level.

In this paper, our goal is to further reduce the conformation space without sacrificing prediction accuracy. This is achieved by assuming that SSEs that do not yet interact with each other are independent and can be treated separately. A conformation is represented by a collection of fully folded structures in which each of them contains a set of interacting SSEs. By using a steepest descent strategy, we show that it is possible to predict the most energetically favorable folding pathway of large proteins with hundreds of residues at the mesoscopic level and this model is detailed enough to distinguish between different folding pathways of structurally very similar proteins. In difference from the technique in 24, we do not consider the spatially moving process before the SSEs form native contacts, and thus we are able to achieve much better computational efficiency.

Assume that the native structure of a protein is known. The protein folding pathway prediction problem is to find an ordered sequence of intermediate conformations to fill the gap between the unfolded state and the native tertiary structure. At the secondary structure level, a protein can be viewed as an ordered sequence of secondary structure elements (SSEs) interspersed with irregular turns or loops, where each SSE is either a helix or a β-strand, and each β-sheet consists of a variable number of β-strands that are not necessarily consecutive on the primary sequence. We represent each protein by t₀s₁t₁⋯s_kt_k, where k is the number of SSEs, s_i denotes the ith SSE, t_j denotes the jth turn, and these elements are in the same order as they appear on the primary sequence. Given the three-dimensional structure of a protein, the assignment of SSEs can be obtained directly from the Protein Data Bank (PDB) 21 or using programs such as DSSP 26.

Following 24 and 25, we consider each SSE as an indivisible unit that folds independently of the others according to the contacts present in the native structure. This is based on the framework model that assumes that extensive intermediate secondary structures exist before they are assembled into the tertiary structure 22, and our goal is to predict the interaction order of SSEs during folding. Based on the observation in 15 and 16 that a model using only native interactions can explain most experimental results, we assume that the interactions between SSEs or turns are the same as the ones present in the native structure. Although these assumptions are often not satisfied as there are many proteins in which there are no clear secondary structures before the formation of tertiary structures or there are no clear preservations of secondary structures throughout folding, such a strategy is sufficient for studying folding pathways at the mesoscopic level and is often the only feasible choice for large proteins.

We represent a conformation of a protein on the folding pathway by C = {S₁, ..., S_k}, where each S_i represents a structure consisting of a set of fully folded SSEs and there are no interactions between two different sets S_j and S_j' (see Figure 2 for an illustration). Since our focus is on the SSEs, turns are not included in the conformation but will be utilized when computing energies (see below). The protein folding problem is transformed to identifying a sequence of conformational changes that start from an initial state with fully folded SSEs but no interactions between SSEs through some intermediate conformations and ending in the native structure (Figure 2). Each conformational change corresponds to finding a new pair of interactions that merges two smaller structures of SSEs into a bigger one. Figure 2 illustrates the folding pathway prediction on the B1 domain of the streptococcal protein G (GB1). In the prediction, β₃ and β₄ interact first, then α₁ is added, followed by β₁ and β₂.

Folding pathway predictions are obtained through the computation of free energies of intermediate conformations. For an intermediate conformation C = {S₁, ..., S_k}, the free energy E(C) of C is defined as:

E ( C ) = ∑ i = 1 k E ( S i ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyrauKaeiikaGIaem4qamKaeiykaKIaeyypa0ZaaabCaeaacqWGfbqrcqGGOaakcqWGtbWudaWgaaWcbaGaemyAaKgabeaakiabcMcaPaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaem4AaSganiabggHiLdGccqGGSaalaaa@3E5E@

where each S_i is viewed as an isolated entity and each E(S_i) is obtained separately by extracting the three-dimensional coordinates of its residues from the Protein Data Bank (PDB) 21 and using the Rosetta software 27 to compute its free energy. The original Rosetta energy function is used, which is obtained by representing each side chain by a centroid that is located at the center of mass, and computing a weighted sum of the binned probability descriptions of multiple effects, including the solvation and electrostatic effects based on observed distributions in known protein structures, the secondary structure packing effects that include strand pairing, strand arrangement into sheets and helix-strand packing, and the effects of steric repulsion and Van der Waals interactions (more details are available in 28 and in Table I of 27). To take the backbone into consideration, a turn t_j is included in the computation of E(S_i) if both of its adjacent SSEs s_j (if it exists) and s_j+1 (if it exists) are included in S_i.

Since the interactions that favor folding usually decrease the free energy while the interactions that destabilize the native structure increase the free energy, our goal is to find the most energetically favorable folding pathway by identifying the conformational change that decreases the free energy the most in each step so that the protein can get to lower energy states as quickly as possible. Figure 3 illustrates our SSFold algorithm that uses a steepest descent strategy to choose a new pair of interactions that leads to a conformation with the lowest free energy in each iteration. This procedure is very efficient since only k - 1 iterations are needed. Within each iteration, O(k²) comparisons are needed to find the best pair of interactions that results in the lowest free energy. This leads to an overall time complexity of O(k³t), where k is the number of SSEs in a protein and t is the time to compute the free energy of a potentially partial protein that contains only some of the SSEs and turns.

We test our strategy on proteins from the Protein Data Bank (PDB) 21 that have known intermediate folding states from experimental data. We illustrate that our model is detailed enough to distinguish between subtle differences in the folding pathways of the streptococcal protein G, the peptostreptococcal protein L, and variants NuG1 and NuG2 of protein G, which are all structurally very similar proteins. We demonstrate that our approach is applicable to large proteins with hundreds of residues by testing it on the 416 residue pig muscle phosphoglycerate kinase (PGK). We further test it on proteins studied in 29 and 25 to validate that our model has very good accuracy.

While our strategy corresponds most closely to the diffusion-collision model that allows folding to proceed independently in different parts of a protein 54, it is possible to use a modified strategy for other models. For example, to simulate the nucleation-propagation model 55 or the nucleation-condensation model 56, in which the existence of a nucleus facilitates further folding, one can iteratively add a SSE that results in the lowest free energy to the nucleus. Since energy computations can still be slow and can take hours, which account for significant amount of computation time in our algorithm, it is also possible to use lower resolution methods to compute energy.

While our strategy finds the most energetically favorable protein folding pathway, there are evidences that multiple folding pathways exist 557. The ability to analyze multiple folding pathways will also allow the study of protein misfolding 58. Our approach can be generalized to study the entire free energy landscape 5 as follows: construct a graph in which each vertex represents a biologically plausible conformation and each edge represents a feasible conformation change, which is similar to the roadmap graph in 24 and 17 and the protein folding network in 59 except that we consider each SSE as an indivisible unit. Various graph-theoretic algorithms can then be used to generate predictions of alternative folding pathways.

We have shown that our procedure has sufficient accuracy to distinguish between subtle differences and our strategy can be applied to large proteins due to its speed. An important future direction is to consider cooperative folding of secondary structures without too much sacrifice in speed, that is, when folding in one secondary structure affects folding in others.

QY performed the research and implemented the algorithm. S–HS supervised the research. All authors read and approved the final manuscript.