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Concatenative nonmonotonicity and optimal links in HP protein folding models

Bjørn Kjos-Hanssen

TL;DR

This work shows that in the $HP$ protein folding framework, the optimal fold energy is nonmonotone under concatenation across several standard lattices, contradicting the idea that adding sequence is always beneficial. It uses $Z$-optimality and isoperimetric arguments to construct explicit counterexamples in 2D and 3D rectangular, triangular, and hexagonal lattices, and demonstrates analogous nonmonotonicity via a Berger-Leighton–style construction. Beyond nonmonotonicity, the paper establishes that closed-chain folds can realize nontrivial knots and links as uniquely optimal folds for certain inputs, including a trefoil knot example, and demonstrates, under the levels-of-hydrophobicity model, that certain links are indispensable for optimality. Together, these results reveal rich topological structure in HP-folding models and connect computational protein folding questions to knot/link theory, with implications for how concatenation can nontrivially affect fold quality and for topological constraints in lattice-based protein models.

Abstract

The hydrophobic-polar (HP) model represents proteins as binary strings embedded in lattices, with fold quality measured by an energy score. We prove that the optimal fold energy is not monotonic under concatenation for several standard lattices, including the 2D and 3D rectangular, hexagonal, and triangular lattices. In other words, concatenating two polymers can produce a fold with strictly worse optimal energy than one of the polymers alone. For closed chains, we show that under the levels-of-hydrophobicity model of Agarwala et al. (1997), proper links can arise as uniquely optimal folds, revealing an unexpected connection between HP models and knot/link theory.

Concatenative nonmonotonicity and optimal links in HP protein folding models

TL;DR

This work shows that in the protein folding framework, the optimal fold energy is nonmonotone under concatenation across several standard lattices, contradicting the idea that adding sequence is always beneficial. It uses -optimality and isoperimetric arguments to construct explicit counterexamples in 2D and 3D rectangular, triangular, and hexagonal lattices, and demonstrates analogous nonmonotonicity via a Berger-Leighton–style construction. Beyond nonmonotonicity, the paper establishes that closed-chain folds can realize nontrivial knots and links as uniquely optimal folds for certain inputs, including a trefoil knot example, and demonstrates, under the levels-of-hydrophobicity model, that certain links are indispensable for optimality. Together, these results reveal rich topological structure in HP-folding models and connect computational protein folding questions to knot/link theory, with implications for how concatenation can nontrivially affect fold quality and for topological constraints in lattice-based protein models.

Abstract

The hydrophobic-polar (HP) model represents proteins as binary strings embedded in lattices, with fold quality measured by an energy score. We prove that the optimal fold energy is not monotonic under concatenation for several standard lattices, including the 2D and 3D rectangular, hexagonal, and triangular lattices. In other words, concatenating two polymers can produce a fold with strictly worse optimal energy than one of the polymers alone. For closed chains, we show that under the levels-of-hydrophobicity model of Agarwala et al. (1997), proper links can arise as uniquely optimal folds, revealing an unexpected connection between HP models and knot/link theory.

Paper Structure

This paper contains 12 sections, 23 theorems, 26 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Automatic complexity.
  3. Overview of the paper.
  4. Nonmonotonicity
  5. 2D rectangular lattice
  6. 3D rectangular lattice
  7. Berger-Leighton inspired method
  8. Knots and links
  9. An optimal link.
  10. An optimal trefoil knot.
  11. The indispensability of links
  12. Topological conclusions

Key Result

Theorem 3

For all $w$, $J_{\mathrm{rect}}(1w1)\le Z(w)$, where $Z(w)$ is the number of zeros in $w$.

Figures (7)

  • Figure 1: Illustration of the Induced Edge Problem solution for the triangular lattice.
  • Figure 2: Folds that demonstrate nonmonotonicity in the triangular model.
  • Figure 3: The fold $x_5$ in the triangular lattice of a word $w_5$ having no occurrence of $0^{11}$.
  • Figure 4: An optimal configuration (b) in the 2D rectangular lattice that is not a ball (a) in the graph metric. The edges shown are the edges in the symmetric difference between the two graphs.
  • Figure 5: Folding to obtain nonmonotonicity in the 2D rectangular model.
  • ...and 2 more figures

Theorems & Definitions (56)

  • Definition 1
  • Conjecture 2: Stecher's cul-de-sac conjecture
  • Theorem 3: Agarwala et al. ABDDFHMS97
  • proof
  • Theorem 4: AICHHOLZER2003139
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • proof
  • ...and 46 more