Local Energy Minima Research Articles

Myosin loop-4 is critical for optimal tropomyosin repositioning on actin during muscle activation and relaxation.

During force-generating steps of the muscle crossbridge cycle, the tip of the myosin motor, specifically loop-4, contacts the tropomyosin cable of actin filaments. In the current study, we determined the corresponding effect of myosin loop-4 on the regulatory positioning of tropomyosin on actin. To accomplish this, we compared high-resolution cryo-EM structures of myosin S1-decorated thin filaments containing either wild-type or a loop-4 mutant construct, where the seven-residue portion of myosin loop-4 that contacts tropomyosin was replaced by glycine residues, thus removing polar side chains from residues 366-372. Cryo-EM analysis of fully decorated actin-tropomyosin filaments with wild-type and mutant S1, yielded 3.4-3.6 Å resolution reconstructions, with even higher definition at the actin-myosin interface. Loop-4 densities both in wild-type and mutant S1 were clearly identified, and side chains were resolved in the wild-type structure. Aside from loop-4, actin and myosin structural domains were indistinguishable from each other when filaments were decorated with either mutant or wild-type S1. In marked contrast, the position of tropomyosin on actin in the two reconstructions differed by 3 to 4 Å. In maps of filaments containing the mutant, tropomyosin was located closer to the myosin-head and thus moved in the direction of the C-state conformation adopted by myosin-free thin filaments. Complementary interaction energy measurements showed that tropomyosin in the mutant thin filaments sits on actin in a local energy minimum, whereas tropomyosin is positioned by wild-type S1 in an energetically unfavorable location. We propose that the high potential energy associated with tropomyosin positioning in wild-type filaments favors an effective transition to B- and C-states following release of myosin from the thin filaments during relaxation.

Global and local energy minimizers for a nanowire growth model

We consider a model for vapor–liquid–solid growth of nanowires proposed in the physical literature. Liquid drops are described as local or global volume-constrained minimizers of the capillarity energy outside a semi-infinite convex obstacle modeling the nanowire. We first address the existence of global minimizers and then, in the case of rotationally symmetric nanowires, we investigate how the presence of a sharp edge affects the shape of local minimizers and the validity of Young’s law.

Inter-channel conversion between population-/coherence-channel dictates thermal transport in MAPbI3 crystals

Hybrid halide perovskites, with their favorable carrier recombination time and ultrashort phonon mean-free paths, are leading candidates for various energy conversion applications like photovoltaics and thermoelectrics. The origin of the ultralow thermal conductivity of the prototypical methylammonium lead triiodide (MAPbI3) is of intense research interest as it is critical for improving its energy conversion performance. So far, such an understanding remains elusive in the MAPbI3 above room temperatures (c-MAPbI3) despite numerous efforts due to its unstable phonon modes. Here, we report the discovery of several c-MAPbI3 local minimum energy structures that produce stable phonon dispersions to reveal an amorphous-like coherence-channel thermal transport mechanism in these crystals. Interestingly, an inter-channel conversion between the population- and coherence-channel of the phonons occurs when the c-MAPbI3 changes across these different structures at the same temperature. Such an effect is not yet observed in simple atomic crystals. Our work also shows that existing thermal transport intuitions based on the phonon gas model can be misleading in such hybrid crystals. Further, the dominance of the non-traditional coherence-channel of phonons will affect the interpretation of other phonon-mediated processes in MAPbI3 and other hybrid perovskites.

Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection–diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species.

Emergence of zero modes in disordered solids under periodic tiling.

In computational models of particle packings with periodic boundary conditions, it is assumed that the packing is attached to exact copies of itself in all possible directions. The periodicity of the boundary then requires that all of the particles' images move together. An infinitely repeated structure, on the other hand, does not necessarily have this constraint. As a consequence, a jammed packing (or a rigid elastic network) under periodic boundary conditions may have a corresponding infinitely repeated lattice representation that is not rigid or indeed may not even be at a local energy minimum. In this manuscript, we prove this claim and discuss ways in which periodic boundary conditions succeed in capturing the physics of repeated structures and where they fall short.

Energy-Shaping Controllers for Soft Robot Manipulators Through Port-Hamiltonian Cosserat Models

In this work, we discuss the application of energy-based controller design for under-actuated soft robot manipulators. The continuous dynamics of the soft robot are modeled through the differential geometry of Cosserat beams. Using a finite-dimensional truncation, the system can be written as a reduced port-Hamiltonian model that preserves the passivity condition. Then, a model-based controller is introduced that produces a local minimizer of closed-loop potential energy for the desired end-effector configuration. The stabilizing control utilizes an energy-based approach and exploits the passivity of the soft robotic system. The effectiveness of the energy-based controller is demonstrated through extensive simulations of various soft robotic systems that share a resemblance with biology. All software and numerical studies are provided in an open-access SOROTOKI toolkit written in Matlab.

A tree search algorithm towards solving Ising formulated combinatorial optimization problems

Simulated annealing (SA) attracts more attention among classical heuristic algorithms because many combinatorial optimization problems can be easily recast as the ground state search problem of the Ising Hamiltonian. However, for practical implementation, the annealing process cannot be arbitrarily slow and hence, it may deviate from the expected stationary Boltzmann distribution and become trapped in a local energy minimum. To overcome this problem, this paper proposes a heuristic search algorithm by expanding search space from a Markov chain to a recursive depth limited tree based on SA, where the parent and child nodes represent the current and future spin states. At each iteration, the algorithm selects the best near-optimal solution within the feasible search space by exploring along the tree in the sense of “look ahead”. Furthermore, motivated by the coherent Ising machine (CIM), the discrete representation of spin states is relaxed to a continuous representation with a regularization term, which enables the use of the reduced dynamics of the oscillators to explore the surrounding neighborhood of the selected tree nodes. We tested our algorithm on a representative NP-hard problem (MAX-CUT) to illustrate the effectiveness of the proposed algorithm compared to semi-definite programming (SDP), SA, and simulated CIM. Our results show that with the primal heuristics SA and CIM, our high-level tree search strategy is able to provide solutions within fewer epochs for Ising formulated combinatorial optimization problems.

Energetic stability and spatial inhomogeneity in the local electronic structure of relaxed twisted trilayer graphene

We study the energetic stability and the local electronic structure of the general twisted trilayer graphene (TTG) with the top and bottom layers rotated with respect to the middle layer respectively by $\theta$ and $\theta'$. Approximate supercells of the moir\'{e}-of-moir\'{e} superlattices with $\theta$ and $\theta^{\prime}$ within $1^{\circ}\sim 2^{\circ}$ are established to describe the structural and electronic properties of relaxed TTG with the periodic boundary condition. Full relaxation demonstrates that the commensurate TTG with $\theta=\theta^{\prime}$ has the local minimum total energy ($E_{tol}$) at a fixed $\theta$, while $E_{tol}$ first reaches a local maximum and begins to drop with decreasing $\theta^{\prime}$ for $\theta^{\prime} < \theta$. Some regions exhibit enhanced in-plane relaxation in the top and bottom layers but suppressed relaxation in the middle layer and form a hexagonal network with the moir\'{e}-of-moir\'{e} length scale. The stacking configurations with the atoms in the three layers vertically aligned at the origin of the relaxed TTG supercells at $\theta$ around $1.6^{\circ}$ and $\theta^{\prime}$ around $1.4^{\circ}$ have a high density of states (DOS) near the Fermi level ($E_F$), which can reach that of the mirror symmetric TTG with equal twist angles of about $1.7^{\circ}$. In contrast, some other stackings can have rather low DOS around $E_F$. The significant stacking dependence of DOS for some TTG supercells demonstrates that the local electronic structure of TTG can exhibit strong spatial inhomogeneity when the twist angles are slightly away from those of the small supercells with large variations of DOS among different stackings. Moreover, the structural relaxation of TTG plays a crucial role in the high DOS and its strong stacking dependence.

A Smörgåsbord of Skyrmions

We study static solutions of the standard Skyrme model with a pion mass. Using approximately 105 pseudo-random initial configurations made of single Skyrmions in the non-symmetrized product Ansatz and an automatic detection of repeated solutions, we find 409 local energy minimizers (Skyrmions) of the model with baryon numbers 1 through 16, of which 383 are new. In particular, we find new solutions for baryon numbers 5, 8, 9, 10, 11, 12, 13, 14, 15, and 16. Our results for the number of solutions per baryon number suggest that this number could grow either polynomially or exponentially. We identify new families of solutions: sheets of Skyrmions in synchronized and antisynchronized hexagonal layers (which we call graphene); chains of 2- and 3-tori; chain-like solutions containing a hinge and many clustered Skyrmions. Contrary to common lore, only the B = 12 global energy minimizer is made of alpha particles or some chunk of a cubic crystal, whereas the B = 9, 11, 14, 15 minimizers contain the B = 7 icosahedrally symmetric Skyrmion as a component. The B = 10, 13, 16 are symmetric graphene-like solutions. We find B = 5 and B = 8 minimizers with numerically indistinguishable energies. The B = 8 candidates are the chain of two cubes, which is a chunk of the cubic Skyrme crystal and the fullerene-type ball found originally by the rational map approximation. The B = 5 global minimizer is either the well-known D2d symmetric fullerene or a new C2v symmetric solution. Finally, our findings show a large number of solutions have no discrete symmetries or just one symmetry, contrary to the common lore that Skyrmions are highly symmetric configurations.

Bio-inspired programmable multi-stable origami

This Letter presents a programmable path to multi-stability of the Kresling origami by introducing bio-inspired nonlinear creases. The origami mathematical model is proposed for the bio-inspired Kresling with the validations by uniaxial compression experiment and finite element analysis. Mono-stability, bi-stability, and tri-stability are demonstrated in one Kresling origami cell. Local energy minimum of multi-stability is found to arise from asymmetric energy barriers. Stable state bifurcation can be tunable under different nonlinearity of the creases and free-stress dihedral angles. Position of stable equilibria can be programmable by varying free-stress dihedral angle of the crease. This work provides a strategy to design programmable multi-stable origami structures.

Dynamic and static stability of a drop attached to an inhomogeneous plane wall

This article concerns the stability of a drop on a wall for which the contact angle, \(\theta _\mathrm{{{w}}}\), varies from place to place. Such a wall may allow unstable equilibria of the drop, i.e. ones for which small perturbations to equilibrium grow, making the equilibrium unrealisable in practice. This will be referred to as dynamic instability and is one of the two versions of instability considered. The other arises from consideration of potential energy, which is the sum of surface (liquid/gas, liquid/solid and solid/gas) components and the gravitational potential energy. Equilibria are extrema of the potential energy with respect to variations of drop geometry which preserve its volume. An equilibrium is said to be statically stable if it is a local minimum of the potential energy for volume-preserving perturbations of the drop. The relationship between static and dynamic stability is the main subject of this paper. The liquid flow is governed by the incompressible Navier–Stokes equations. To allow for the moving contact line, a Navier slip condition with slip length \(\lambda \) is used at the wall, as is a prescribed contact angle, \(\theta _\mathrm{{{w}}} =\theta _\mathrm{{{w}}} ({x,y})\), at the contact line, where x, y are Cartesian coordinates on the wall. The perturbation is assumed small, allowing linearisation of the governing equations and, in the usual manner of stability analysis, complex modes having the time dependency \(e^\mathrm{{{st}}}\) are introduced. This leads to an eigenvalue problem with eigenvalue s, the sign of whose real part determines dynamic stability/instability. A quite different eigenvalue problem, which describes static stability/instability is also derived. It is shown that, despite this difference, the conditions for dynamic and static instability are in fact the same. This conclusion is far from evident a priori but should be good news for interested numerical analysts because determination of static stability is much less numerically costly than a dynamic stability study, whereas it is the latter which gives a true determination of stability.

Machine learning the metastable phase diagram of covalently bonded carbon

Conventional phase diagram generation involves experimentation to provide an initial estimate of the set of thermodynamically accessible phases and their boundaries, followed by use of phenomenological models to interpolate between the available experimental data points and extrapolate to experimentally inaccessible regions. Such an approach, combined with high throughput first-principles calculations and data-mining techniques, has led to exhaustive thermodynamic databases (e.g. compatible with the CALPHAD method), albeit focused on the reduced set of phases observed at distinct thermodynamic equilibria. In contrast, materials during their synthesis, operation, or processing, may not reach their thermodynamic equilibrium state but, instead, remain trapped in a local (metastable) free energy minimum, which may exhibit desirable properties. Here, we introduce an automated workflow that integrates first-principles physics and atomistic simulations with machine learning (ML), and high-performance computing to allow rapid exploration of the metastable phases to construct “metastable” phase diagrams for materials far-from-equilibrium. Using carbon as a prototypical system, we demonstrate automated metastable phase diagram construction to map hundreds of metastable states ranging from near equilibrium to far-from-equilibrium (400 meV/atom). We incorporate the free energy calculations into a neural-network-based learning of the equations of state that allows for efficient construction of metastable phase diagrams. We use the metastable phase diagram and identify domains of relative stability and synthesizability of metastable materials. High temperature high pressure experiments using a diamond anvil cell on graphite sample coupled with high-resolution transmission electron microscopy (HRTEM) confirm our metastable phase predictions. In particular, we identify the previously ambiguous structure of n-diamond as a cubic-analog of diaphite-like lonsdaelite phase.

Mapping Magnetic Signals of Individual Magnetite Grains to Their Internal Magnetic Configurations Using Micromagnetic Models.

Micromagnetic tomography (MMT) is a technique that combines X‐ray micro computed tomography and scanning magnetometry data to obtain information about the magnetic potential of individual grains embedded in a sample. Recovering magnetic signals of individual grains in natural and synthetic samples provides a new pathway to study the remanent magnetization that carries information about the ancient geomagnetic field and is the basis of all paleomagnetic studies. MMT infers the magnetic potential of individual grains by numerical inversion of surface magnetic measurements using spherical harmonic expansions. The magnetic potential of individual particles in principle is uniquely determined by MMT, not only by the dipole approximation, but also more complex, higher order, multipole moments. Here, we show that such complex magnetic information together with both particle shape and mineral properties severely constrains the internal magnetization structure of an individual grain. To this end, we apply a three dimensional micromagnetic model to predict the multipole signal from magnetization states of different local energy minima. We show that for certain grains it is even possible to uniquely infer the magnetic configuration from the inverted magnetic multipole moments. This result is crucial to discriminate single‐domain particles from grains in more complex configurations such as multi‐domain or vortex states. As a consequence, our investigation proves that by MMT it is feasible to select statistical ensembles of magnetic grains based on their magnetization states, which opens new possibilities to identify and characterize stable paleomagnetic recorders in natural samples.

Adsorption Free Energy of Cellulose Nanocrystal on Water-Oil Interface.

To investigate the amphiphilicity of cellulose, a series of molecular dynamics simulations were performed with a cellulose nanocrystal and a water–octane interfacial system. Assuming that the axis of cellulose is parallel to the water–octane interface, the freedoms of motion of the nanocrystal were restricted to two, the distance from the interface and the orientation around the axis. The mean force and the mean torque on the nanocrystal were evaluated with sufficiently long simulation at each crystal configuration, and their numerical integration gave a smooth free energy surface as the potential of mean force. The cellulose sample used here was found to be much more hydrophilic than oleophilic with the free energy difference kcal/mol. Three adsorption states with local minimum of adsorption free energy are distinguished in the free energy surface—the direct contact type which is similar to previously reported one, the hydrophilic-surface/water/octane type where a thin water layer is sandwiched between the surface and the octane phase, and the oleophilic/water/octane type where a thin water layer also exists. Water molecules in these water layers contribute to stabilize the adsorption states by taking a special orientational order and slow self-diffusion.

Skyrmion pinning by disk-shaped defects

High precision skyrmion pinning by an intentionally created magnetic structure is important in skyrmion manipulation. Here, we consider skyrmion pinning by various types of disks. Among other findings, we clarify that, in terms of pinning position and pinning potential landscape, one needs to distinguish thin-wall skyrmions from thick-wall skyrmions because of fundamental differences. A skyrmion wall prefers areas with a weaker exchange stiffness, a larger DMI constant, or a smaller magnetic anisotropy while a skyrmion core experiences only the magnetic anisotropy, not exchange stiffness and DMI. Depending on disk type, skyrmion type, and the relative size of the disk (to skyrmion), a skyrmion can be pinned at the symmetric (center) or asymmetric (off-center) point of a disk. In case a skyrmion is pinned by a local energy minimum, thermal agitation can depin the skyrmion, and the pinning lifetime follows an Arrhenius law. Interestingly, when the disk size is comparable to the skyrmion size, the skyrmion deforms greatly such that the skyrmion will shrink or expand to fill the whole disk. These findings should be important for skyrmion manipulations.

Programming Soft Shape-Morphing Systems by Harnessing Strain Mismatch and Snap-Through Bistability: A Review.

Multi-modal and controllable shape-morphing constitutes the cornerstone of the functionalization of soft actuators/robots. Involving heterogeneity through material layout is a widely used strategy to generate internal mismatches in active morphing structures. Once triggered by external stimuli, the entire structure undergoes cooperative deformation by minimizing the potential energy. However, the intrinsic limitation of soft materials emerges when it comes to applications such as soft actuators or load-bearing structures that require fast response and large output force. Many researchers have explored the use of the structural principle of snap-through bistability as the morphing mechanisms. Bistable or multi-stable mechanical systems possess more than one local energy minimum and are capable of resting in any of these equilibrium states without external forces. The snap-through motion could overcome energy barriers to switch among these stable or metastable states with dramatically distinct geometries. Attributed to the energy storage and release mechanism, such snap-through transition is quite highly efficient, accompanied by fast response speed, large displacement magnitude, high manipulation strength, and moderate driving force. For example, the shape-morphing timescale of conventional hydrogel systems is usually tens of minutes, while the activation time of hydrogel actuators using the elastic snapping instability strategy can be reduced to below 1 s. By rationally embedding stimuli-responsive inclusions to offer the required trigger energy, various controllable snap-through actuations could be achieved. This review summarizes the current shape-morphing programming strategies based on mismatch strain induced by material heterogeneity, with emphasis on how to leverage snap-through bistability to broaden the applications of the shape-morphing structures in soft robotics and mechanical metamaterials.

Graph-based machine learning beyond stable materials and relaxed crystal structures

There has been a recent surge of interest in using machine learning to approximate density functional theory in materials science. However, many of the most performant models are evaluated on large databases of computed properties of, primarily, materials with precise atomic coordinates available, and which have been experimentally synthesized, i.e., which are thermodynamically stable or metastable. These aspects provide challenges when applying such models on theoretical candidate materials, for example for materials discovery, where the coordinates are not known. To extend the scope of this methodology, we investigate the performance of the crystal graph convolutional neural network on a data set of theoretical structures in three related ternary phase diagrams (Ti,Zr,Hf)-Zn-N, which thus include many highly unstable structures. We then investigate the impact on the performance of using atomic positions that are only partially relaxed into local energy minima. We also explore options for improving the performance in these scenarios by transfer learning, either from models trained on a large database of mostly stable systems, or a different but related phase diagram. Models pretrained on stable materials do not significantly improve performance, but models trained on similar data transfer very well. We demonstrate how our findings can be utilized to generate phase diagrams with a major reduction in computational effort.

Stable anisotropic capillary hypersurfaces in a wedge

&lt;abstract&gt;&lt;p&gt;We study a variational problem for hypersurfaces in a wedge in the Euclidean space. Our wedge is bounded by a finitely many hyperplanes passing a common point. The total energy of each hypersurface is the sum of its anisotropic surface energy and the wetting energy of the planar domain bounded by the boundary of the considered hypersurface. An anisotropic surface energy is a generalization of the surface area which was introduced to model the surface tension of a small crystal. We show an existence and uniqueness result of local minimizers of the total energy among hypersurfaces enclosing the same volume. Our result is new even when the special case where the surface energy is the surface area.&lt;/p&gt;&lt;/abstract&gt;

1H NMR spectra, structure, and conformational exchange of S-n-alkyl-tetrahydrothiophenium cations of some ionic liquids

S-n-alkyl-tetrahydrothiophenium cations [R-THT]+ in some iodide or tetraphenylborate ion-pairs [R-THT][X] (R = methyl to n-decyl, X−= I−, [BPh4]−, [NTf2]−) are characterized in the solid state by puckered THT rings in twist forms. In contrast, solution 1H NMR spectra of compounds [R-THT][X] indicate a rapid conformational exchange leading to dynamic [ABCD]2 spin systems due to eight protons of the THT ring. These strongly coupled spectra obtained at high magnetic fields were analyzed by iterative methods and simulated successfully. In addition, molecular modeling and gauge-including atomic orbitals (GIAOs) based calculations of proton NMR chemical shifts for two such ion-pair compounds in solution were performed at the density functional theory (DFT) level . As a result, conformational equilibrium between two different pairs of isoenergetic twist forms of the THT ring in [Me-THT][I] (global and local energy minima) was found to exist. These theoretical results are discussed in view of experimental 1H NMR spectral data.

Microenvironment-mediated cancer dormancy: Insights from metastability theory

Dormancy is an evolutionarily conserved protective mechanism widely observed in nature. A pathological example is found during cancer metastasis, where cancer cells disseminate from the primary tumor, home to secondary organs, and enter a growth-arrested state, which could last for decades. Recent studies have pointed toward the microenvironment being heavily involved in inducing, preserving, or ceasing this dormant state, with a strong focus on identifying specific molecular mechanisms and signaling pathways. Increasing evidence now suggests the existence of an interplay between intracellular as well as extracellular biochemical and mechanical cues in guiding such processes. Despite the inherent complexities associated with dormancy, proliferation, and growth of cancer cells and tumor tissues, viewing these phenomena from a physical perspective allows for a more global description, independent from many details of the systems. Building on the analogies between tissues and fluids and thermodynamic phase separation concepts, we classify a number of proposed mechanisms in terms of a thermodynamic metastability of the tumor with respect to growth. This can be governed by interaction with the microenvironment in the form of adherence (wetting) to a substrate or by mechanical confinement of the surrounding extracellular matrix. By drawing parallels with clinical and experimental data, we advance the notion that the local energy minima, or metastable states, emerging in the tissue droplet growth kinetics can be associated with a dormant state. Despite its simplicity, the provided framework captures several aspects associated with cancer dormancy and tumor growth.