Pith. sign in

REVIEW 3 major objections 3 minor 27 references

Funnel-like protein energy landscapes emerge from functional evolution under thermal fluctuations

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Selecting only for a local active-site motif generates funnel-like folding landscapes in a lattice protein model, but only at intermediate environmental temperatures.

desk verdict Direct landscape construction supports the claim that functional selection alone can build a funnel, but the low-temperature leg is tested only up to f<0.9, and the authors explicitly defer the f→1 tail. read the letter →

arxiv 2608.01143 v1 pith:BU56VHR5 submitted 2026-08-02 physics.bio-ph cond-mat.stat-mechq-bio.BM

Norifumi Maruyama , Macoto Kikuchi This is my paper · ORCID
classification physics.bio-phcond-mat.stat-mechq-bio.BM
keywords proteinfoldingenergylandscapefunctionalselectionlatticemodelmulticanonicalMonteCarlothermalfluctuationstwo-statenativecontacts
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether foldability is a direct evolutionary target or a by-product of selection for function. In a 20-residue two-dimensional lattice protein model with four amino-acid types, the authors define fitness as the equilibrium probability that a prescribed local active-site motif is populated at the environmental temperature $T$, and they sample sequences across the full fitness range with a sequence-space multicanonical Monte Carlo method. They find that high-fitness sequences evolved at an intermediate temperature ($T = 1.0$) spontaneously acquire funnel-like energy landscapes and cooperative two-state free-energy landscapes, although foldability never enters the fitness definition. At low temperature ($T = 0.1$), similarly fit sequences instead show rugged, glass-like landscapes with nearly one native conformation per sequence, and at high temperature ($T = 1.4$) no highly functional sequences exist. The contribution is a concrete, structural demonstration that local functional constraints, when maintained under thermal fluctuations, can organize the entire protein energy landscape.

What carries the argument

The load-bearing object is the fitness function $$f(T) = \frac{\sum_C \varphi(C) $e^{{-E(C)/T}}$}{\sum_C $e^{{-E(C)/T}}$},$$ the equilibrium probability that a conformation $C$ contains the prescribed active-site motif, defined as two exposed polar residues backed by a hydrophobic pair with neighboring lattice sites empty; $\varphi(C)$ is 1 for functional conformations and 0 otherwise. This fitness contains no term for the native structure, thermodynamic stability, or foldability. The method pairs this fitness with sequence-space multicanonical Monte Carlo using entropic sampling, which performs a random walk over fitness bins and yields representative sequences across the entire fitness range, and then reconstructs energy and free-energy landscapes from the equilibrium conformational ensemble, with the number of native contacts serving as the structural order parameter.

What would settle it

Repeat the same sequence-space sampling in a three-dimensional or off-lattice model with the same fitness definition and check whether intermediate-temperature high-fitness sequences still display funnel-like energy spectra and cooperative two-state free-energy profiles; if they do not, the proposed thermodynamic mechanism fails to generalize. Alternatively, run explicit folding kinetics on the $T = 1.0$ high-fitness sequences: if they relax slowly and non-cooperatively despite funnel-shaped equilibrium landscapes, the paper's identification of funnel landscapes with foldability would be contradicted.

Watch

Extended reading notes

Core claim

The central discovery is that selection acting only on the equilibrium probability of a single prescribed local active-site motif is sufficient to generate funnel-like protein energy landscapes and cooperative two-state free-energy landscapes, but only in an intermediate range of environmental temperatures. At $T = 1.0$, the lowest-energy spectra of high-fitness sequences decrease systematically with the number of native contacts, and the free-energy landscape shows two minima separated by a barrier, the signature of two-state folding; the 100 highest-fitness sequences share the same overall organization, and 500 independent sequence patterns collapse onto only four native folds. At $T = 0.1$, high-fitness sequences achieve comparable fitness through glass-like, rugged energy landscapes, and essentially every sequence adopts its own native conformation. The paper concludes that foldability can emerge as a thermodynamic consequence of maintaining function under thermal fluctuations: the requirement that many thermally populated states preserve the active-site geometry converts a local functional constraint into a global constraint on the energy landscape, providing a common origin for the consistency principle and minimal frustration.

Load-bearing premise

The argument would collapse if real protein function cannot be represented by the equilibrium probability of a single local active-site motif in a 20-residue two-dimensional lattice model with four amino-acid types and the hand-set contact energies of Table I; under that representation, the claim that foldability emerges purely from functional selection may not generalize to real proteins.

Editorial extensions

If this is right

  • Foldability can be viewed as an emergent thermodynamic property of maintaining a local functional motif, not as an independent target of selection.
  • There is a nontrivial temperature window for the emergence of foldability: low thermal noise removes the evolutionary pressure to organize the whole spectrum, while high thermal noise prevents high fitness from being realized at all.
  • Functional selection can strongly restrict the accessible fold space, since 500 independently sampled sequence patterns collapse onto only four native conformations at $T = 1.0$.
  • The consistency principle and minimal frustration can be derived from the same mechanism: competing low-energy conformations are selectively disfavored because they reduce the equilibrium probability of maintaining the active site.
  • The maximum-target-probability design criterion becomes a special case of this framework, suggesting a complementary function-oriented design strategy in which only the functional motif is specified.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension would select on two or more functional motifs simultaneously and test whether funnel organization strengthens, weakens, or breaks; the paper only treats one motif.
  • If the mechanism scales to larger chains, one would expect a quantitative link between the environmental temperature during selection and the folding temperature of the resulting proteins; that link is a prediction, not a result of the paper.
  • The collapse of many sequences onto a few folds hints at a thermodynamic contribution to the limited fold repertoire of natural proteins, but the 20-residue lattice is too small to establish that claim for real proteins.
  • Replacing equilibrium fitness with a kinetic fitness, such as the probability of reaching the active site within a fixed time, could narrow or shift the intermediate-temperature window; this is an untested extension of the paper's logic.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. This manuscript asks whether funnel-like protein energy landscapes can arise solely from selection for a local functional motif, without any explicit selection for foldability. Using a 20-residue two-dimensional lattice protein with four residue types, the authors define fitness at environmental temperature T as the equilibrium probability that a prescribed four-residue active-site motif is realized (Eq. 3). They sample the sequence space with a sequence-space multicanonical Monte Carlo method and construct energy and free-energy landscapes of high-fitness sequences. The main finding is that at an intermediate temperature (T = 1.0), high-fitness sequences show funnel-like energy landscapes and two-state free-energy landscapes, whereas at a low temperature (T = 0.1), sequences with fitness near 0.9 show rugged, glass-like landscapes. At T = 1.4 no highly functional sequences exist. The authors conclude that functional selection alone can generate foldability as a thermodynamic consequence of maintaining function under thermal fluctuations, and that environmental temperature is a key determinant. The manuscript includes consistency checks: four independent multicanonical runs, a complete-ensemble check for representative sequences, explicit evolutionary simulations at T = 1.0, and an analysis of native-structure degeneracy.

Significance. If the central claim holds, the paper is significant: it offers a concrete statistical-mechanical mechanism by which local functional constraints can organize a global energy landscape, connecting functional selection to the funnel picture of protein folding. The study goes beyond earlier spin-model work by constructing genuine structural order parameters and landscape profiles for a chain molecule. The authors are explicit that they are not claiming direct selection for foldability, and they identify the low-temperature extreme-fitness regime as an open question in the Supplemental Information. The computational approach is well suited to the question, and the inclusion of explicit evolutionary simulations and exhaustive enumeration checks strengthens the evidence. However, the main temperature-dependent comparison is not made at matched fitness values, and the funnel/glass distinction is assessed visually rather than quantitatively; these issues bear directly on the paper's headline claim.

major comments (3)
  1. [Results and Supplemental Information] The central temperature comparison is made between unmatched fitness regimes. At T = 1.0 the analysis uses the highest-fitness sequence and the 100 highest-fitness sequences, whereas at T = 0.1 the text and Fig. 5 analyze the sequence with the highest fitness among those with f < 0.9, and the 100 highest among those with f < 0.9. The Supplemental Information explicitly states that 'Whether the energy landscapes in this regime nevertheless acquire partial funnel-like character remains to be examined.' Because the abstract and Discussion conclude that funnel-like landscapes emerge only within an appropriate temperature range, the untested f → 1 tail at T = 0.1 is load-bearing: if maximal-fitness low-temperature sequences turn out to be funnel-like, the claim must be weakened to a statement about typical, rather than maximal, high-fitness sequences. Please analyze sequences at the true maximum fitness (or at least at matched fitness values) at T = 0.1 and report the corresponding landscapes.
  2. [Results, Figs. 4 and 5] The classification of energy landscapes as 'funnel-like' versus 'glass-like' is made by visual inspection of energy spectra and free-energy profiles, with no quantitative order parameter. The central claim is precisely that funnel-like organization emerges at intermediate temperature and not at low temperature, so a quantitative measure is needed: for example, the correlation between energy and the number of native contacts, a roughness or frustration statistic, or a Z-score against randomized sequences. Such a metric would also make the deferred question of 'partial funnel-like character' in the Supplemental Information well posed. Please compute a quantitative funnel measure for all analyzed sequences at both temperatures.
  3. [Results and Discussion] The claim that funnel-like landscapes emerge 'only within an appropriate range of environmental temperatures' is supported by only three temperature values (T = 0.1, 1.0, 1.4), with exactly one temperature in the purported intermediate regime. While the three points bracket a nonmonotonic behavior, the phrase 'range' implies a more systematic dependence than the data establish. A scan over additional intermediate temperatures, or otherwise a clear statement that only the three values were examined, would make the temperature-dependence claim commensurate with the evidence.
minor comments (3)
  1. [Title/header] The full-text header contains an apparent line-break typo, 'functio nal evolution', which should be corrected.
  2. [Methods, Eq. (3) and Table I] The sentence 'The environmental temperature is the only parameter of the present model' is imprecise: the contact-energy matrix in Table I and the restriction to conformations with 9–12 contacts are also model parameters and approximations. Please rephrase to clarify that T is the only evolutionary-selection parameter varied in the study.
  3. [Results, Fig. 4 and Fig. 5] The term 'native conformation' is used for low-temperature sequences whose ground states are described as one of many arbitrary members of a degenerate set; please state explicitly how the native conformation was chosen for the T = 0.1 cases, since the low-temperature sequences are said to be individually diverse.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: funnel-like landscapes are computed outputs, not imposed inputs.

full rationale

Fitness is defined by Eq. (3) as the Boltzmann-weighted probability of realizing a prescribed local active-site motif; this is an input assumption, not the target claim. The targets—funnel-like energy landscapes and two-state free-energy profiles—are computed from Eq. (1) contact energies and the equilibrium conformational ensemble, with no term in the fitness or sampling that rewards native contacts, thermodynamic stability, or foldability. The result that high-fitness T=1.0 sequences show a systematic decrease of energy with native contacts (Fig. 4) is therefore a nontrivial computational finding rather than a consequence of the fitness definition. The temperature dependence is partly anticipated by the form of Eq. (3): at low T fitness is dominated by ground-state occupancy, while at intermediate T many thermally populated conformations must carry the motif; but whether this forces global energetic organization is not guaranteed, and the low-temperature contrast is an empirical result. The authors' own multicanonical sequence-space method (refs 15-18) is cited only as a sampling tool; no load-bearing claim depends on an unverified theorem from those papers. The clearest caveat is the supplement's admission that at T=0.1 the very-high-fitness tail (f approaching 1) was not examined: 'Whether the energy landscapes in this regime nevertheless acquire partial funnel-like character remains to be examined, and a quantitative characterization of funnel-like organization as a function of fitness and temperature will be reported elsewhere.' This is a matched-fitness comparison limitation, not a circular reduction: the reported landscapes at f<0.9 are still computed rather than fitted, and the omitted tail does not make the presented result definitionally equivalent to the fitness input. Thus no circular step is identifiable, and the derivation is self-contained with respect to the stated model assumptions.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. It uses a hand-set contact energy matrix and a prescribed active-site motif, which are model inputs rather than new claims about nature.

free parameters (2)
  • Contact energy matrix entries (Table I) = See Table I (e.g., H1-H1: -3.5, H2-H2: -2.3, P2-P2: 0.0)
    Chosen by hand with reference to the Miyazawa-Jernigan matrix; not fitted to produce funnels, but they set the energetic ordering on which all results depend.
  • Fitness cutoff at T=0.1 = f < 0.9
    Selected to represent typical high-fitness sequences because maximal-fitness sequences are degenerate at low T; affects the comparison but is disclosed in the text.
assumptions (5)
  • domain assumption The 20-residue, four-letter 2D lattice model is a sufficient representation of protein folding physics.
    Model used throughout; based on earlier studies showing five types are sufficient, with four chosen for computational efficiency.
  • domain assumption Biological fitness is the equilibrium probability of a prescribed local active-site motif at temperature T (Eq. 3).
    This defines the fitness landscape; no selection on foldability or native structure is included.
  • domain assumption Restricting the fitness sum to conformations with 9-12 contacts does not change conclusions.
    Stated in Model and Methods; verified for representative sequences in Supplemental Fig. S1, but not exhaustively.
  • domain assumption The peak of the specific heat identifies the folding temperature.
    Used operationally to define Tf because conformational entropy is not a continuous function of energy.
  • standard math Multicanonical sequence-space sampling converges and accurately represents the sequence distribution.
    Established method (Lee 1993, Wang-Landau 2001) assumed to be correctly applied; four independent runs were checked for consistency.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Funnel-like protein energy landscapes emerge from functional evolution under thermal fluctuations." pith.science (2026). https://pith.science/paper/BU56VHR5

@misc{pith2026260801143,
  author       = {Pith},
  title        = {Pith review of: Funnel-like protein energy landscapes emerge from functional evolution under thermal fluctuations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BU56VHR5}},
  note         = {Machine review of arXiv:2608.01143}
}
abstract

Proteins perform biological functions by folding into specific native structures that are stabilized by funnel-like energy landscapes shaped through evolution. We investigate whether such foldable proteins can emerge solely from selection for function, without any direct selection for foldability. Using a two-dimensional lattice protein model with four amino-acid types, we define protein function as the equilibrium probability that a prescribed local structure forming an active site is realized at environmental temperature $T$, and use this probability as the evolutionary fitness. Amino-acid sequences spanning the entire fitness range are sampled by multicanonical Monte Carlo method and their energy and free-energy landscapes are constructed from the equilibrium conformational ensemble. We find that only within an appropriate range of environmental temperatures do high-fitness sequences spontaneously acquire funnel-like energy landscapes, despite foldability never being included in the fitness definition. High-fitness sequences are furthermore supported by only a limited number of native conformations. At low temperatures, in contrast, high-fitness sequences exhibit rugged, glass-like energy landscapes and much greater structural diversity. These results suggest that foldable proteins need not be direct targets of evolution, but can emerge as a thermodynamic consequence of maintaining function under thermal fluctuations, whereby local functional constraints organize the global fold and give rise to funnel-like energy landscapes.

Figures

Figures reproduced from arXiv: 2608.01143 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic representations of protein energy and fre [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic illustration of the two-dimensional latt [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Genotypic entropy as a function of fitness for envi [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Energy and free-energy landscapes of high-fitness se [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Energy and free-energy landscapes of high-fitness se [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    C. J. Epstein, R. F. Goldberger, and C. B. Anfinsen, The genetic control of tertiary pro- tein structure: Studies with model systems, Cold Spring Harb Symp Quant Biol 28, 439 (1963)

    work page 1963

  2. [2]

    C. B. Anfinsen, Principles that govern the folding of pro- tein chains, Science 181, 223 (1973)

    work page 1973

  3. [3]

    Since low-energy confor- mations are expected to possess many contacts, this ap- proximation primarily excludes high-energy states

    are restricted to conformations with 9–12 noncovalent contacts. Since low-energy confor- mations are expected to possess many contacts, this ap- proximation primarily excludes high-energy states. For representative sequences, calculations using the complete conformational ensemble yielded energy and free-energy landscapes that are nearly indistinguishable...

    work page

  4. [4]

    G¯ o, Theoretical studies of protein folding, Ann Rev Biophys Bioeng 12, 183 (1983)

    N. G¯ o, Theoretical studies of protein folding, Ann Rev Biophys Bioeng 12, 183 (1983)

    work page 1983

  5. [5]

    J. D. Bryngelson and P. G. Wolynes, Spin glasses and the statistical mechanics of protein folding., P Natl Acad Sci USA 84, 7524 (1987)

    work page 1987

  6. [6]

    J. D. Bryngelson, J. N. Onuchic, N. D. Socci, and P. G. Wolynes, Funnels, pathways, and the energy landscape of protein folding: A synthesis, Proteins 21, 167 (1995)

    work page 1995

  7. [7]

    Saito, M

    S. Saito, M. Sasai, and T. Yomo, Evolution of the folding ability of proteins through functional selection, P Natl Acad Sci USA 94, 11324 (1997)

    work page 1997

  8. [8]

    T. Yomo, S. Saito, and M. Sasai, Gradual development of protein-like global structures through functional sele c- tion, Nat Struct Biol 6, 743 (1999)

    work page 1999

Show all 27 references
  1. [9]

    Sasaki and M

    T. Sasaki and M. Sasai, Correlation between the confor- mation space and the sequence space of peptide chain, J Biol Phys 28, 483 (2002)

    2002

  2. [10]

    Nagao, T

    C. Nagao, T. P. Terada, T. Yomo, and M. Sasai, Correlation between evolutionary structural development and protein folding, P Natl Acad Sci USA 102, 18950 (2005)

    2005

  3. [11]

    Sakata, K

    A. Sakata, K. Hukushima, and K. Kaneko, Funnel landscape and mutational robustness as a result of evolution under thermal noise, Phys. Rev. Lett. 102, 148101 (2009)

    2009

  4. [12]

    P. G. Wolynes, As simple as can be?, Nat Struct Biol 4, 871 (1997)

    1997

  5. [13]

    D. S. Riddle, J. V. Santiago, S. T. Bray-Hall, N. Doshi, V. P. Grantcharova, Q. Yi, and D. Baker, Functional rapidly folding proteins from simplified amino acid se- quences, Nat Struct Biol 4, 805 (1997)

    1997

  6. [14]

    Wang and W

    J. Wang and W. Wang, A computational ap- proach to simplifying the protein folding alphabet, Nat Struct Biol 6, 1033 (1999)

    1999

  7. [15]

    Miyazawa and R

    S. Miyazawa and R. L. Jernigan, Self-consistent esti- mation of inter-residue protein contact energies based on an equilibrium mixture approximation of residues, Proteins 34, 49 (1999)

    1999

  8. [16]

    Saito and M

    N. Saito and M. Kikuchi, Robustness leads close to the edge of chaos in coupled map networks: toward the understanding of biological networks, New J Phys 15, 053037 (2013)

    2013

  9. [17]

    Nagata and M

    S. Nagata and M. Kikuchi, Emergence of cooperative bistability and robustness of gene regulatory networks, PLoS Comput Biol 16, e1007969 (2020)

    2020

  10. [18]

    Kaneko and M

    T. Kaneko and M. Kikuchi, Evolution enhances muta- tional robustness and suppresses the emergence of a new phenotype: A new computational approach for studying evolution, PLoS Comput Biol 18, e1009796 (2022)

    2022

  11. [19]

    Kikuchi, Phenotype selection due to mutational ro- 8 bustness, PLoS ONE 19, e0311058 (2024)

    M. Kikuchi, Phenotype selection due to mutational ro- 8 bustness, PLoS ONE 19, e0311058 (2024)

    2024

  12. [20]

    B. A. Berg and T. Neuhaus, Multicanonical ensemble: A new approach to simulate first-order phase transitions, Phys. Rev. Lett. 68, 9 (1992)

    1992

  13. [21]

    B. A. Berg and T. Celik, New approach to spin-glass simulations, Phys. Rev. Lett. 69, 2292 (1992)

    1992

  14. [22]

    Lee, New Monte Carlo algorithm: Entropic sampling, Phys

    J. Lee, New Monte Carlo algorithm: Entropic sampling, Phys. Rev. Lett. 71, 211 (1993)

    1993

  15. [23]

    Wang and D

    F. Wang and D. P. Landau, Efficient, multiple-range ran- dom walk algorithm to calculate the density of states, Phys. Rev. Lett. 86, 2050 (2001)

    2001

  16. [24]

    Wang and D

    F. Wang and D. P. Landau, Determining the den- sity of states for classical statistical models: A ran- dom walk algorithm to produce a flat histogram, Phys. Rev. E 64, 056101 (2001)

    2001

  17. [25]

    J. M. Deutsch and T. Kurosky, New algorithm for protein design, Phys. Rev. Lett. 76, 323 (1996)

    1996

  18. [26]

    Y. Iba, K. Tokita, and M. Kikuchi, Design equa- tion: A novel approach to heteropolymer design, J Phys Soc Jpn 67, 3985 (1998)

    1998

  19. [27]

    Takahashi, G

    T. Takahashi, G. Chikenji, and K. Tokita, Lat- tice protein design using Bayesian learning, Phys. Rev. E 104, 014404 (2021) . 9 Supplemental Information: Funnel-like Protein Energy Landscapes Emerge from Functio nal Evolution under Thermal Fluctuations Norifumi Maruyama ( 丸山恭史...

    2021

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.