Design of a minimal, allosteric, and ATPase-like machine using mechanical linkages

Reviewer #1 (Public review):

Summary:

This paper presents a creative physics-based model of an ATPase-like molecular machine using mechanically coupled linkages to mimic allosteric cycles. The authors construct the complete reaction network by enumerating all mechanically allowed binding configurations and transitions between them. The resulting system contains hundreds of microstates connected through node-level binding, dissociation, intramolecular rearrangements, cleavage, and ligation reactions. Stochastic simulations are then used to study how the machine cycles between ligand-bound and substrate-bound states. Overall, the manuscript presents an interesting and creative mechanochemical framework for modeling ATPase-like allosteric cycles integrating multivalent binding, geometric exclusion through rigidity arising from binding of substrates and ligands, with stochastic simulations.

Strengths:

(1) The manuscript presents a creative mechanochemical framework that combines multivalent binding, geometric exclusion, rigidity-based coupling, stochastic kinetics, and catalysis within a unified model.

(2) The use of geometric exclusion and rigidity to generate negative allosteric coupling is elegant and provides an intuitive physical mechanism for coordinated molecular behavior.

(3) The interpretation of catalysis as a transient release of mechanical constraints is conceptually interesting and offers a novel perspective on how energy-consuming reactions can regulate state transitions.

(4) The distinction between productive and futile cycles is insightful and provides a useful framework for understanding pathway selection in molecular machines.

(5) The explicit construction of a stochastic state network allows the authors to connect microscopic binding events with emergent cyclic behavior.

(6) The work provides a conceptual platform for exploring how simple mechanical principles may give rise to allosteric regulation and mechanochemical transduction in synthetic molecular systems.

Weaknesses:

(1) The manuscript is very dense and difficult to follow. The notation and microstate labels (e.g., {S/L}:{10,5}) obscure the central ideas, and the stochastic model is not explained clearly enough. The authors should provide a simpler schematic, a mapping of state labels, and a step-by-step example of a productive cycle. The supplementary videos would also benefit from additional explanation.

(2) The framework appears most applicable to mechanically gated motor proteins and may not generalize to allosteric enzymes that operate through conformational ensembles, dynamic coupling, or entropy-driven regulation. The scope of the model should be discussed more carefully.

(3) The reported behavior appears highly dependent on specific parameter choices and rate hierarchies. A broader sensitivity analysis is needed to demonstrate robustness.

(4) The binary treatment of states as either rigid or flexible oversimplifies the continuous energy landscapes and fluctuations observed in real biomolecules. The limitations of this approximation should be discussed.

(5) The role of nonequilibrium thermodynamics is underdeveloped. The relationship between the model, ATP chemical potential, free-energy dissipation, entropy production, and the energetic cost of futile cycles should be discussed more explicitly.

(6) Although a large number of microstates are enumerated, it remains unclear which states and pathways dominate the dynamics. A more coarse-grained analysis highlighting the key states and transitions would improve interpretability and facilitate comparison with experimental systems.

Reviewer #2 (Public review):

This is an interesting study aiming to capture the fundamental principles of ATPase-like machines with an elementary model of rigid bars. While molecular motors have been the subject of many studies in statistical physics, taking very simplified approaches, these past studies generally abstract away from geometrical constraints and do not account for allosteric mechanisms. In turn, several simple physical models of allostery are now available, but most consider only long-range effects without reference to reactivity or the conversion of chemical to mechanical energy. The essential ingredient of the present model is a form of negative allostery that stems from geometrical constraints. These constraints impose the presence of hidden microstates and the connections between states that form a reaction network. Nontrivial tradeoffs can then be derived, e.g., on the catalytic rate.

One limitation of the approach is that energetic constraints, which are equally important, are not themselves derived from physical principles. This includes the different kinetic rates, the binding constants, and the mechanisms of reactivity, even though, in principle, they should follow from basic interaction energies in the context of thermal fluctuations. This is a legitimate choice of modeling level, handled notably by imposing a hierarchy between rates. It would be appreciated, however, if the overall logic of the derivation were clarified, presenting more clearly from the beginning what the fundamental hypotheses of the model are, which aspects are derived from these hypotheses, and which require additional assumptions. It seems indeed that the model involves both fundamental physical assumptions that are used to derive some emerging kinetic features (e.g., geometry imposes futile cycles) and global kinetic assumptions that are used to derive some microscopic features (e.g., a productive cycle imposes the relative values of the kinetic rates).

The conclusion ends with a proposal to generalize to more elaborate models. I was wondering, however, if the opposite would not be desirable. The current model is already quite involved (as the 7 pages of SI listing transitions between states testify). Wouldn't it be possible to obtain some of the main results, e.g., the trade-offs defining an optimal cleavage rate, from an even simpler model?

Reviewer #3 (Public review):

Summary:

In "Design of a minimal, allosteric, and ATPase-like machine using mechanical linkage," Omabegho introduces a simplified model of an allosterically-regulated molecular machine. The model machine takes inspiration from simple ATPase motors, specifically myosin or dynein, and is meant to capture the interactions between enzyme, ligand, and substrate that empower these enzymatic biomolecular machines. The model system attempts to specifically model the inhibitory allosteric regulation relevant to machine operation, following work on mechanical features of allostery in protein-analogue elastic network models. After the introduction of the model machine, the manuscript discusses the primary cycle by which the substrate serves to displace and then cyclically replace the ligand (roughly corresponding to a "recovery" and "power" stroke, respectively, in their biomolecular machine inspiration), as well as cataloguing other futile cycles or individual chemical pathways the machine may traverse. After an exhaustive enumeration of all possible states and transitions of the model machine, the actual mechano-chemical system is mapped to an abstracted stochastic chemical reaction network, which is finally studied numerically in more detail. We believe this model system is novel, interesting, and, importantly, interpretable and could prove to be useful in future modeling and rational design of simple bio-mimetic nanoscopic systems.

Strengths:

(1) Omabegho introduces a novel chemo-mechanical model system that captures the core behavior of a simple ATPase molecular machine. The model is relatively simple in construction, but, as the manuscript demonstrates, it can display very rich behavior depending on various competing chemical timescales and mechanisms. These features suggest that this system could, in fact, serve as a useful starting model if adopted by the wider community.

(2) A key feature to highlight is the mechanistic interpretability of the model. By cutting to the seemingly core functional details of an ATPase-like machine, Omebegho is able to algorithmically produce an exhaustive listing of all allowed chemical states and all possible transitions between them, enabling the study of all possible mechanistic pathways and the relative frequency of each observed path. Further, Omebegho produced very clear visualizations of these states, transitions, pathways, and cycles that again facilitate the interpretability and utility of the model. This interpretability is key to our suggestion that this model could be more widely adopted as a clear ATPase analogue model for study by the broader biophysical community.

Weaknesses:

There are some issues to note.

(1) First, although the model is inspired by ATPases, the cycle in question is not a model even for the significantly abstracted form presented in Supplemental Section 1.1, as noted in the manuscript. The allosteric regulation utilized in the model does not mandate that the ligand (a stand-in for actin, for example) displaces both products (stand-ins for ADP and P). Rather, one product (P) dissociates spontaneously, and the other, larger one (ADP) is allosterically displaced. We recognize mandating this small further detail would presumably complexify the model (i.e., it may require an additional tile in the enzyme construction), but it would also lead to a more accurate, while still simple and interpretable, picture of the biological molecular machines in question.

(2) Much as we recognize and applaud the model's structure as simple and interpretable, when trying to actually study the chemical dynamics and observed dynamical behaviors, individual chemical rates of various binding and unbinding processes appear to be quite finely tuned. There is some attempt at studying the behavior of this model in various parameter tuning regimes, but the large number of model parameters makes a truly complete numerical study prohibitive. Given this difficulty, it would be instructive to know if a comparison could be made to the actual motivating biological systems and any observed chemical rates in the experiment. Further, given the desired cyclic behavior, it would be quite interesting to see to what degree this behavior is robust to various parameter choices. Again, preliminary work was done in the manuscript by biasing rates or numerical siloing experiments, but a far more exhaustive study is certainly worthwhile.

(3) Perhaps our biggest critique of the manuscript is that, although it incorporates both mechanical and chemical aspects into the model system construction, all mechanical aspects of the model simply function to limit allowed state transitions. From our understanding, all mechanical aspects of the model are, in fact, abstracted away during any simulations. This modeling choice, of course, retains some mechanical inspiration while making the resulting system more tractable, but it is ultimately not an actual mechanical model. ATPases, especially motors such as myosin and dynein, are inherently mechanical, their fundamental features being the conversion of chemical fuel into mechanical motion. A fuller treatment, perhaps in future work, should include these physical degrees of freedom in simulation, thus truly tracking the interplay between physical enzyme mechanics and chemistry. The absence of actual mechanics as yet, beyond a strict allosteric restriction, is an inherent limitation of the model.