Togashi and Mikhailov (PNAS, 2007) showed that elastic-network models of the motor proteins F1-ATPase and myosin relax from large arbitrary deformations onto low-dimensional attractive manifolds, that random networks lack this property, and that networks with a single dominant soft mode can be designed by evolutionary spectral-gap optimization - including a 64-node ligand-powered machine performing robust cyclic hinge motions under additive noise. The paper left the physical completion of this concept open along five directions. The thermal-noise and hydrodynamics direction was later substantially addressed (mesoscale-solvent simulations in 2008; stochastic designed-motor operation under load in 2016), while four components remain open: the nonequilibrium thermodynamics - efficiency, energy transduction per ligand-binding cycle, and ligand-link parameter dependence - of designed machines; design algorithms that prescribe a given number and spatial pattern of soft collective modes with controlled spectral gaps; single-molecule experimental validation of attractive relaxation manifolds in real proteins; and physical realization of designed architectures in fabricable molecular systems.

Background

Molecular machines such as the motor proteins myosin and F1_1-ATPase convert chemical energy, released in discrete portions whenever a fuel ligand binds or detaches, into ordered cyclic conformational motions. A standard coarse-grained description of such proteins is the elastic network model (ENM): each amino-acid residue is a node placed at its position in the experimentally determined native structure, and every pair of nodes closer than a cutoff distance is connected by a harmonic spring. Linearizing the dynamics about the native structure yields normal modes - collective vibration and relaxation patterns - whose slowest, smallest-eigenvalue members are called soft modes; in machine proteins these are typically separated from the rest of the spectrum by a spectral gap, so slow relaxation is dominated by a few collective coordinates.

Togashi and Mikhailov (2007) went beyond this linear picture and simulated the fully nonlinear relaxation of the ENMs of the F1_1-ATPase beta-subunit and of myosin from large, essentially arbitrary initial deformations. They found that trajectories quickly converge to a low-dimensional attractive manifold - an attracting set in conformation space, spanned by the soft modes, that the relaxation flow approaches from a broad basin of initial states - so that each protein responds to very different perturbations with the same well-defined internal mechanical motion, robustly and unlike randomly generated networks. They then showed, by evolutionary optimization (a Metropolis-type loop of random structural mutations, accepted with a probability favoring an increase of the spectral gap g=log⁡10(λ2/λ1)g=\log_{10}(\lambda_2/\lambda_1) between the two slowest modes), that artificial 64-node networks with one dominant soft mode and a large gap can be designed; one such network, two dense clusters joined by a hinge, performs regular cyclic hinge motions driven by repeated stochastic binding and detachment of a model ligand, only weakly perturbed by additive random forces standing in for thermal noise. A footnote of the paper notes that this algorithm constructs only single-soft-mode networks, although it 'can be, however, easily modified' to design networks with two or more soft modes separated by a gap from the rest of the spectrum.

The paper's Discussion points forward: using such designed networks, 'fundamentals of molecular machine operation, including energetic aspects, the role of thermal noise and hydrodynamic interactions, can be discussed'; the predictions 'can be verified' by special single-molecule experiments monitoring conformational relaxation after arbitrary initial deformations; and the approach provides 'a systematic approach for the design of proteins with prescribed (programmed) robust conformational motions' and, prospectively, non-protein machine-like nanodevices. No thermodynamic efficiency, energy budget per cycle, or ligand-link parameter dependence was computed there.

Later work developed parts of this program. Cressman, Togashi, Mikhailov and Kapral (2008) coupled the elastic-network machine to a particle-based mesoscale solvent, thereby including thermal solvent fluctuations and hydrodynamic coupling consistently, and found the cycle timescale strongly affected. Sarkar, Flechsig and Mikhailov (2016) attached a filament to a designed machine and demonstrated, in extensive stochastic simulations, operation as a deterministic ratchet (power stroke: mechanical work produced by a conformational relaxation strongly correlated with ligand chemistry) or as a Brownian ratchet (transport rectified from thermal fluctuations by biased binding), with a continuous transition between regimes, operation under load approaching stall, and cycle statistics over thousands of periods. For real motors, Flechsig and Mikhailov (2010) simulated entire chemomechanical cycles of hepatitis C virus helicase in the ENM approximation, and Togashi, Yanagida and Mikhailov (2010) showed that thermal fluctuations already violate linear normal-mode descriptions of functional motions in myosin V and KIF1A. Hydrodynamics of machine models was analyzed by Sakaue, Kapral and Mikhailov (2010) and Huang, Kapral, Mikhailov and Chen (2013). Flechsig (2017) later evolved networks with optimized long-range allosteric coupling (communication between distant sites) in which one or a few soft modes emerge, though neither their number nor their spectral separations are prescribed. Adjacent to the direct line, Rocks et al. (2017) and Yan, Ravasio, Brito and Wyart (2018) designed two-dimensional disordered spring networks with prescribed strain responses, and Vossel, de Groot and Godec (2025) identified a mode-coupling ('allosteric lever') principle for such responses. Albaugh and Gingrich (2022) simulated a chemically fueled catenane motor by nonequilibrium molecular dynamics with chemostats (reservoir schemes holding chemical potentials fixed in a nonequilibrium steady state). Ito, Sato and Toyabe (2024) designed a rotary motor fabricable by DNA origami (self-assembled nanostructures built from designed DNA strands), validated by simulation with realistic parameters. What no later work supplies is a quantitative thermodynamic characterization of the designed elastic-network machines, an algorithm prescribing several soft modes with controlled gaps, the proposed single-molecule test, or a physical realization of a designed architecture; the 2018 and 2019 reviews by Flechsig, Togashi and Mikhailov explicitly frame designed networks as model systems whose realization 'is not yet' addressable and whose nonlinear dynamics needs 'further analytical and numerical investigations'.

Problem Statement

Advance the physical completion of the designed elastic-network machine concept along the directions left open by Togashi and Mikhailov (2007). The common objective uniting these directions is to turn the 2007 proof of concept - a ligand-powered elastic network that performs robust, repeatable cyclic machine-like motion - into a quantitative, experimentally grounded framework for the design principles of molecular machines. Each direction is stated for the designed elastic-network machine class (the cyclic hinge machine of Togashi and Mikhailov and motors derived from it) and stands as a separate open goal.

(1) Energetics. Determine the nonequilibrium thermodynamics of machine operation: the energy supplied per ligand-binding event, the elastic energy stored and dissipated per cycle, the mechanical work performed when the machine drives a load, the resulting thermodynamic efficiency, and how these quantities depend on the ligand-link parameters (stiffness, natural length, binding geometry), on noise intensity, and on operating frequency. Hydrodynamic and fluctuation effects on cycle dynamics are known to be strong (mesoscale-solvent simulations in Cressman et al. 2008; stochastic designed-motor operation under load in Sarkar et al. 2016), which is relevant background for any such computation.

(2) Prescribed multi-mode design. Construct and demonstrate a design algorithm that yields elastic networks with a prescribed number and spatial pattern of soft collective modes, at controlled spectral separations from one another and from the rest of the spectrum.

(3) Experimental validation. The source suggests that the attractive-manifold prediction 'can be verified' by special single-molecule experiments monitoring conformational relaxation after arbitrary initial deformations. The open deliverable is to specify and analyze such a single-molecule protocol for a named real machine protein - the observable to be measured, how arbitrary initial deformations are prepared, the convergence statistic, and the protein selected - and to state which experimental outcome would confirm and which would refute the manifold prediction in that real protein.

(4) Physical realization. The source Discussion looks forward to 'a systematic approach for the design of proteins with prescribed (programmed) robust conformational motions' and, prospectively, to the engineering of non-protein machine-like nanodevices. The open direction is whether a designed node-and-link architecture can be mapped to a fabricable molecular system - a protein sequence, DNA-origami construct, or other nanoscale structure - whose mechanics and cyclic dynamics preserve the designed behavior.

Each direction is separately meaningful; together they form the framework envisaged in the source's Discussion: the energetics quantifies what the designs achieve, the multi-mode algorithm what they can be programmed to do, the experiments whether real proteins obey the same principle, and the realization whether the designs are physically buildable. The directions are separable open goals sharing the common objective stated above.

References

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