[Paper Review] Protein structure prediction software generate two different sets of conformations. Or the study of unfolded self-avoiding walks

Jacques M. Bahi, Christophe Guyeux+2 more|arXiv (Cornell University)|Jun 6, 2013

Protein Structure and Dynamics3 citations

TL;DR

This paper investigates the limitations of protein structure prediction (PSP) software that fold straight-line conformations via pivot moves, revealing that not all self-avoiding walks (SAWs) can be reached—specifically, some 'unfoldable' SAWs exist. The authors define and analyze subsets of SAWs, prove the existence of infinitely many unfoldable walks, and show that folded SAWs remain exponentially numerous, highlighting critical implications for PSP algorithm design and conformational sampling accuracy.

ABSTRACT

Self-avoiding walks (SAW) are the source of very difficult problems in probabilities and enumerative combinatorics. They are also of great interest as they are, for instance, the basis of protein structure prediction in bioinformatics. Authors of this article have previously shown that, depending on the prediction algorithm, the sets of obtained conformations differ: all the self-avoiding walks can be reached using stretching-based algorithms whereas only the folded SAWs can be attained with methods that iteratively fold the straight line. A first study of (un)folded self-avoiding walks is presented in this article. The contribution is majorly a survey of what is currently known about these sets. In particular we provide clear definitions of various subsets of self-avoiding walks related to pivot moves (folded or unfoldable SAWs, etc.) and the first results we have obtained, theoretically or computationally, on these sets. A list of open questions is provided too, and the consequences on the protein structure prediction problem is finally investigated.

Motivation & Objective

• To investigate why certain protein conformations cannot be generated by pivot-based folding algorithms despite being valid self-avoiding walks.
• To define and analyze new subsets of self-avoiding walks, including folded, unfoldable, and k-foldable walks, under pivot move dynamics.
• To clarify the mathematical and computational implications of these subsets for protein structure prediction (PSP) software.
• To highlight the risk of chaotic dynamics in folding algorithms and the consequences of model choice on prediction accuracy.

Proposed method

• The authors define four subsets of SAWs: folded SAWs (reachable via pivot moves from the straight line), unfoldable SAWs (not reachable by any sequence of ±90° pivot moves), SAWs foldable at least once, and SAWs foldable k times for k > 1.
• They use theoretical analysis and computational enumeration to bound the cardinality of folded SAWs and verify equivalence with full SAW sets up to n ≤ 14.
• The study leverages known results from statistical mechanics and combinatorics, particularly Madras and Sokal’s ergodicity theorem for pivot algorithms.
• A counterexample walk of 223 steps was analyzed to demonstrate the existence of non-ergodic SAWs under standard pivot moves.
• The authors compare two PSP software paradigms: (1) sequential construction of SAWs and (2) iterative folding of the straight line, showing they generate different conformational sets.
• They explore the biological plausibility of folding during synthesis and question whether a third dynamic model—continuous folding while extending—might yield a distinct subset of SAWs.

Experimental results

Research questions

• RQ1 Can all self-avoiding walks be generated by iterating pivot moves on the straight-line conformation, or are there inherent topological obstructions?
• RQ2 What is the cardinality and structural nature of the set of unfoldable self-avoiding walks, and do they form infinite families?
• RQ3 How do the sets of conformations generated by different PSP algorithms (construction vs. folding) compare, and what are the implications for conformational sampling?
• RQ4 Is the complexity of the protein folding problem reduced when restricted to folded SAWs, or does it remain NP-hard?
• RQ5 Can a dynamic model that folds continuously during chain extension produce a different set of conformations than the standard folding or construction methods?

Key findings

• There exists an infinite number of self-avoiding walks that cannot be reached by any sequence of ±90° pivot moves, proving the existence of 'unfoldable' SAWs.
• The number of folded SAWs (reachable via pivot moves) remains exponential in the number of steps, despite being a strict subset of all SAWs.
• A shorter example of an unfoldable SAW was found, reducing the previous known minimal length from 223 to 107 steps.
• The set of folded SAWs is equal to the full set of SAWs for n ≤ 14, as verified computationally, indicating that non-ergodicity emerges only at larger n.
• The cardinality of folded SAWs is bounded, suggesting a non-trivial subset of SAWs is accessible under pivot dynamics.
• The study reveals that PSP software based on folding the straight line may miss biologically plausible conformations, raising concerns about model accuracy and chaotic sensitivity in prediction algorithms.