A novel topology for representing protein folds

Abbreviations:

RCO, relative contact order; ECO, effective contact order; 3D, three dimensional; DT, Delaunay tessellation; Kτ-NN, Kendall's tau—nearest neighbor.

Introduction

The protein folding problem, or problems,1 despite considerable recent headway, remains one of the greatest challenges facing computational biology. The interrelated folding problems can be described as2: (a) the folding code: given a protein's amino acid sequence, how does the thermodynamic interplay of interatomic forces determine the protein's structure? (b) structure prediction: how can a protein's (native, 3D) structure be computationally predicted from its amino acid sequence? and (c) the folding process: given the vast number of conformational possibilities embodied in a protein's amino acid sequence, how does it fold so quickly to its native state? This last problem, often referred to as the Levinthal3 paradox, has been addressed using a spectrum of theoretical and experimental approaches. Some remarkable4 and striking5 findings to emerge in recent years pertain to the fact that folding rates of two-state proteins (those folding without observable intermediates), which can vary over more than eight orders of magnitude, from microseconds6 to hours,7 and include a wide range of folds and functions, are largely determined by the topology of the native structure, with relative insensitivity to features such as the details of inter-atomic interactions and protein length.8, 9

Making such inferences—prediction of folding rates based on protein topology—requires a quantification of topology and a number of derived summaries have been advanced for this purpose. Generally, these summaries are employed as “bulk” properties—aggregated over the protein structure—so as to relate to (overall) folding rate. However, more locally defined topological summaries may prove informative with respect to local attributes such as folding initiation and propagation sites.10

It has been noted that (tertiary) native structures ought reflect their folding path histories, at least for some folding mechanisms.11 This motivates our framing of a novel topological characterization of a folded protein. It is based on the permutation representation of nearest neighbors, with subsequent use of Kendall's tau distance metric to capture perturbation from the unfolded polypeptide chain. We contrast the performance of leading topologies in predicting two-state protein folding and unfolding rates, demonstrating significant prediction gains for our new measure. This performance is all the more notable since it has achieved without implicit or explicit optimization, the new topology being devoid of tuning parameters. Some preliminary exploration of local properties is also proffered.

The importance of topology in terms of prediction of folding rates was first established for relative contact order (RCO).8 Let a_i designate the i^th residue in a protein primary sequence of length n. A (nonlocal) contact between two residues a_i, a_j, separated by at least l_cut residues along the sequence, is defined as occurring if there are two heavy (nonhydrogen) atoms, one from each residue, within a cutoff distance of R_cut. Standard values for l_cut and R_cut are 2 (sequence positions) and 6 Å, respectively. Assume there are n_c contacting residue pairs. Then RCO is defined as

where the sum is over all contacting pairs (a_i, a_j).

Several variants of RCO have been proposed, with emphasis on sensitivity to the cutoff parameters and the scope (short-, mid-, or long-range) of contacts, as discussed later. An alternate formulation, termed effective contact order (ECO),11, 12 operationalizes contacts and scope in terms of shortest path lengths between residues that can be achieved in the presence of existing (covalent or topological) links. This formulation attempts to capture effective loop size, and hence the size of the conformational search space necessary to form a conditional (on preexisting links) contact and, as such, is postulated to relate to search (folding) speed.2 Our new topology captures such constructs but in a distinct framework.

What constitutes folding from a topological (rather than mechanistic) perspective? Clearly, any definition must be dependent on the underlying polypeptide chain, since purely 3D coordinate based definitions would give rise to a multitude of (irrelevant) “folds”. One widely used primitive is based on backbone dihedral angles as depicted in Ramachandran plots.13 These plots have been used for crystallographic quality control purposes to detect angular outliers since many angle combinations do not occur due to steric hindrance. Further, by modeling sequential angular dependencies along the chain using dynamic Bayesian networks, successful generative models of local protein structure have been devised.14, 15 However, unlike RCO above, topological summaries derived from angular representations have not been employed in relation to predicting attributes such as folding rates.

As an alternative to backbone angles and contact orders, we can conceptualize folding as resulting in some residues being brought closer together relative to their positions in a denatured random coil. Although related to the underpinnings of contact order, it is by operationalizing this notion, without invoking contact distances that we arrive at our new topology.

Let u_i = (x_i, y_i, z_i) denote the 3D coordinates of the Cα atom of residue a_i. We compute the n × n matrix of Euclidean distances between all Cα pairs:

Then, for each residue (i.e., for each row of D) we map its Euclidean distance to a “nearest neighbor” ranking:

()

This enables use of cycle structure to capture topology with respect to an underlying chain, which is not available using Euclidean distances directly. A useful byproduct of such rank based approaches is their relative insensitivity to noise, a known concern with regard experimental (X-ray crystallography or NMR) determination of atomic coordinates.16

As a first step we treat r₁. as a permutation of {1, 2, …, n} and, as such, an element of the symmetric group S_n. This provides access to a wealth of techniques and theory, some of which is germane to folding. Every permutation can be written as a product of disjoint cycles. For example, corresponding to the permutation that takes the red (sequence) ordering to the blue (nearest neighbor) ordering in Figure 1(A) we have

where (b₁b₂…b_k) means b₁ → b₂, b₂ → b₃ … b_k → b₁. Intuitively, we expect fold topology to relate to cycle structure. If we (simplistically) regard a highly denatured (unfolded) protein as an unstructured molecule, and focus on the N-terminal residue a₁, the above process gives the identity permutation composed of n 1-cycles. When folded, as depicted in the two dimensional cartoon in Figure 1(A), we obtain a cycle structure that captures the loops.

The second step is to move beyond the extreme N-terminal residue and to define permutations for each residue from the perspective of its position in the chain. Thus, instead of regarding r_i. as a permutation of {1, …, n} it is treated as a permutation of

()

where the top row in (2), termed the referent N-terminal re-ordering and designated ρ_i, represents nearest neighbor ordering for residue a_i in the unfolded state. This schema is illustrated in Figure 1(B). Note that r_ii = 1 by definition (each Cα is closest to itself), so that every cycle representation will contain the 1-cycle (1). Of course, we could equally utilize a C-terminal based re-ordering, as discussed later.

In this manner, we obtain a permutation and its attendant cycle representation for each residue. We can then entertain characterizing a folded protein structure using properties or summaries of this collection. For example, we could summarize each residue by maximal cycle length, and then further summarize a structure by the maximum (over all residues) of these maxima. However, this summary proves to be not very useful. At the residue level, maximal cycle length is strongly dependent on whether a referent N-terminal or C-terminal reordering is employed, an arbitrariness to avoid. And, on the protein level, we obtain maximal cycle lengths of ≈ (n - 2) across a wide range of structures. This is consistent with modal cycle length under random permutation.

So, we take a more direct approach to capturing the difference between 3D nearest neighbor and sequence orderings. The referent N-terminal reordering is obviously a permutation of {1, …, n}, and so an element of S_n. Now, a variety of metrics have been defined on S_n.17 Here, as recommended,17 we focus on Kendall's tau (Kτ) which, for π, σ permutations in S_n, is defined as

the inverses being used to make the metric right invariant. Then, as our third and final step to operationalizing a nearest neighbor: sequence position-based topology, we define our Kendall's τ-nearest neighbor (Kτ-NN) summary for residue i, designated Γ_i, as where division by standardizes such that Γ_i ∈ [0,1]. We do not employ optimization in arriving at a bulk summary, but simply employ the average over all residues:

Importantly, Γ_i is insensitive as to whether a N-terminal or C-terminal referent reordering is employed. This is a simple consequence of the triangle inequality: let η_i represent the referent C-terminal reordering as given by the top row of (3)

()

Then by the triangle inequality . This agreement is exemplified in Figure 2 which showcases near perfect agreement for N- and C-terminal reorderings, and illustrates the dependence on sequence length, the structures possessing 65 and 294 Cα's, respectively.

Some distinctions between Kτ-NN and RCO and related topologies are worth highlighting. It is important to recognize the complete absence of tuning (parameters) in computing Γ. Although refinements involving incorporation of tuning possibilities are discussed subsequently, all results presented herein use the (untuned) formulation described above. In contrast, use of RCO requires specification as to what physical (Euclidean) distance constitutes a contact, and what minimal sequence separation should be imposed. For the former a value of 6 Å is commonly used, this being the original specification,8 but other choices (e.g., 8 Å,18, 19) have been advocated. Although some studies indicate that results are insensitive to this specification,8, 20 others suggest that the choice has a strong influence.21 Additionally, there are disparate ways of operationalizing contact order scope: local-, mid-, and long-range contacts being distinguished.18, 22 Similarly, for example, the geometric distance based on Delaunay tessellation (DT)23 requires specification of sequence and spatial separation parameters.

Kτ-NN topology attempts to capture chain deformation/structural information between the referent and contacting residues, whereas this is ignored in computing RCO and DT. Conversely, Kτ-NN also incorporates such information beyond contacting residues. It could be argued that inclusion of such remote (from the referent residue) nearest neighbor rankings is at best irrelevant, and at worst distorting, for a topological summary. The following considerations are relevant: (i) applications of topologies are typically to bulk protein attributes, as opposed to residue specific. Summarization over the entire chain serves to downweight these distant contributions; (ii) a constraint on the range of sequence positions prior to ranking (1) is a putative tuning parameter; and (iii) transforming Γ_i to corresponding proximities via prox(Γ_i) = 1 − exp(−Γ_i) can be used to achieve such downweighting24 without specifying tuning parameters. Notably, despite these concerns, the performance of Γ_i-based summaries in predicting two-state folding rates exceeds that of alternates as described next.

The dataset used to assess performance of the competing topologies was obtained from a recent compilation.25 Critically, this article was the first to derive and assemble folding (k_f), and unfolding (k_u), rate constants obtained under standard experimental conditions, necessary for meaningful comparisons. The data provides rate constants for 30 proteins, 27 of which have PDB identifiers. RCO for these 27 structures was obtained using the Baker lab Perl script http://depts.washington.edu/bakerpg/contact_order/, as well as an online calculator http://www.copredictor.ca/. Calculation of Γ_i and DT made recourse to custom R26 code.

Associations between the respective topologies and log folding rates are presented in Figure 3. Note that the (absolute) correlations attained using are substantially and significantly greater than those achieved by RCO for both folding and unfolding rates. Delaunay tessellations have been used in several contexts to capture protein structural attributes, so it is natural to relate correspondingly defined topologies to folding rates. Doing so for both two- and multistate proteins23 yielded impressive results, with DT (Nα in Ref. 23) outperforming RCO in both settings. However, DT is strongly correlated with chain length. For multi-state proteins length is known to be a significant determinant of k_f.27 But, for two-state proteins, results generally show no association between folding rates and length.4, 8 Indeed, for the present set of two-state proteins, with folding rates determined under standardized conditions, neither length nor DT are well-correlated with k_f: absolute correlations being 0.19 and 0.41 respectively, again significantly less than for .

So far, we have utilized the new topology only in terms of a bulk property: its average, , over a given protein. Examination of numerous traces of individual (residue level) Γ_i versus sequence position reveals notable fine structure and variation. To illustrate, we showcase behavior for the two proteins from25 with extreme k_f values: PBD IDs 1APS and 1LMB. Figure 4 contrasts profiles of three topological summaries over sequence position for the two proteins. The superposed smooths (1APS in red, 1LMB in green) were obtained using lowess.26 The measures are, respectively, Kτ-NN (Γ_i), residue level relative contact order (RCO_i), and average area buried under folding (AABUF,10, 28), a refinement of hydrophobicity incorporating residue size. For both Γ_i and RCO_i, we see clear differences in terms of overall level between the two proteins, indicative of their bulk (mean) summaries ability to predict folding rate. No such separation is evidenced for AABUF which, indeed, is not associated with folding rates, at least for the proteins considered here (not shown). Further, AABUF does not behave smoothly [Fig. 4(C)] with respect to sequence position. This is in contrast to RCO_i [Fig. 4(B)] and, to a much greater extent, Γ_i [Fig. 4(A)], which exhibits well-defined local minima and maxima.

The existence of these well-defined Γ_i extremes begs the question as to whether they relate to attributes of the folding process or properties of the 3D protein structure. Unfortunately, there is limited data, at the residue level, for making such assessments. Features of interest, but for which insufficient data is available, include nucleation as measured by a residue's Φ value,21, 29 which provides a measure of the extent to which the residue participates in native-like interactions during the rate limiting folding step, and folding initiation sites. We speculate that residues with small Φ values (not part of folding nuclei) will have small Γ_i values, the logic being that these small values of Γ_i correspond to residues that are relatively “unperturbed” by the folding process. This pertains for Villin 14T30 for which we observe [Fig. 5(A)] a significant correlation of 0.66 between Γ_i and Φ values after trimming four negative and near zero instances for which Φ is less than its standard error.31 The Γ_i sequence position profile [Fig. 5(B)] reveals considerable fine structure. A ribbon diagram of Villin 14T, colored according to Γ_i value [Fig. 5(C)], with select extreme Φ and Γ_i residues (3, 7, 43, 84: see 5A) highlighted, showcases the positive association. Similar to the toy example (see Fig. 1), we generally observe lower Γ_i values within helices and strands, and higher values at loop inflections, consistent with zipping and assembly mechanisms.2 However, it is important to recognize the highly presumptive nature of these putative associations, in some part attributable to the considerable uncertainties in, and sparsity of, measured Φ values.

Now, focusing exclusively on the protein backbone and disregarding side chains, the folding process that transforms a highly denatured random coil, even containing residual sequence-local structure,32 can be coarsely viewed as a mapping that is a contraction: there is a real q, 0 ≤ q < 1 such that d(Tu_i, Tu_j) ≤ q · d(u_i, u_j) for all residues i, j where d(·,·) is Euclidean distance, and u_i gives the coordinates of the i^th Cα atom. Then, from the Banach fixed point theorem,33 we have that T has a unique fixed point u* such that Tu* = u*. Now, consequences of this result are moot since T is unknown, as are atomic coordinates in the unfolded state. However, by mapping from to S_n and invoking the Kτ-NN topology, we can identify the residue closest to the fixed point as i*: Γ_i* = 0. So, applying this speculation to 1LMB [green trace, Fig. 4(A)] we predict that residue 38 (Γ minima) and neighbors (in view of smoothness of Γ) are (relatively) fixed and, accordingly, are removed from nucleation or initiation sites, as could be determined via Φ analysis. Additional future possibilities include incorporation of such predictions into structure prediction algorithms,19 developing improved characterizations, refinements, and applications of the newly proposed topology, and exploring whether residues predicted as less critical for structural/folding roles are less well-conserved in related sequences.