Two-dimensional Energy Landscape Research Articles

CLEAPA: a framework for exploring the conformational landscape of cryo-EM using energy-aware pathfinding algorithm.

Cryo-electron microscopy (cryo-EM) is a powerful technique for studying macromolecules and holds the potential for identifying kinetically preferred transition sequences between conformational states. Typically, these sequences are explored within two-dimensional energy landscapes. However, due to the complexity of biomolecules, representing conformational changes in two dimensions can be challenging. Recent advancements in reconstruction models have successfully extracted structural heterogeneity from cryo-EM images using higher-dimension latent space. Nonetheless, creating high-dimensional conformational landscapes in the latent space and then searching for preferred paths continues to be a formidable task. This study introduces an innovative framework for identifying preferred trajectories within high-dimensional conformational landscapes. Our method encompasses the search for the minimum energy path in the graph, where edge weights are determined based on the energy estimation at each node using local density. The effectiveness of this approach is demonstrated by identifying accurate transition states in both synthetic and real-world datasets featuring continuous conformational changes. The CLEAPA package is available at https://github.com/tengyulin/energy_aware_pathfinding/.

Diffusion in a two-dimensional energy landscape in the presence of dynamical correlations and validity of random walk model.

Diffusion in a multidimensional energy surface with minima and barriers is a problem of importance in statistical mechanics and it also has wide applications, such as protein folding. To understand it in such a system, we carry out theory and simulations of a tagged particle moving on a two-dimensional periodic potential energy surface, both in the presence and absence of noise. Langevin dynamics simulations at multiple temperatures are carried out to obtain the diffusion coefficient of a solute particle. Friction is varied from zero to large values. Diffusive motion emerges in the limit of a long time, even in the absence of noise. Noise destroys the correlations and increases diffusion at small friction. Diffusion thus exhibits a nonmonotonic friction dependence at the intermediate value of the damping, ultimately converging to our theoretically predicted value. The latter is obtained using the well-established relationship between diffusion and random walk. An excellent agreement is obtained between theory and simulations in the high-friction limit but not so in the intermediate regime. We explain the deviation in the low- to intermediate-friction regime using the modified random walk theory. The rate of escape from one cell to another is obtained from the multidimensional rate theory of Langer. We find that enhanced dimensionality plays an important role. To quantify the effects of noise on the potential-imposed coherence on the trajectories, we calculate the Lyapunov exponent. At small friction values, the Lyapunov exponent mimics the friction dependence of the rate.

A fully discrete Peierls model for the screw dislocation in Ta

ABSTRACT The discrete Peierls model is generalised to be able to study the nonplanar dislocations as well as the planar dislocations. In the discrete model, the six-fold screw dislocation that is widely studied in numerical simulation and analytical theory splits into two different cores: the six-fold B core and the six-fold O core. The six-fold B core has the lowest energy and is the most stable while the six-fold O core has about 38 meV more energy in a period length. Because local dislocation hopping can only occur along the folds and hopping to sites out of the folds is not allowed, the dislocation movement is realised by transforming itself into a planar one in the model. The Peierls stress associated with this transition tunnel is , which is much higher than that obtained in simulations. Limitations of the model can be removed by using the two-dimensional energy landscape that is a natural consequence of the concept of the nonplanar dislocation. In the leading terms approximation of the two dimensional energy landscape, the non-Schmid behaviour of the screw dislocation in Ta is quantitatively predicted based on the model.

Amyloid assembly is dominated by misregistered kinetic traps on an unbiased energy landscape

Atomistic description of protein fibril formation has been elusive due to the complexity and long time scales of the conformational search. Here, we develop a multiscale approach combining numerous atomistic simulations in explicit solvent to construct Markov State Models (MSMs) of fibril growth. The search for the in-register fully bound fibril state is modeled as a random walk on a rugged two-dimensional energy landscape defined by β-sheet alignment and hydrogen-bonding states, whereas transitions involving states without hydrogen bonds are derived from kinetic clustering. The reversible association/dissociation of an incoming peptide and overall growth kinetics are then computed from MSM simulations. This approach is applied to derive a parameter-free, comprehensive description of fibril elongation of Aβ16-22 and how it is modulated by phenylalanine-to-cyclohexylalanine (CHA) mutations. The trajectories show an aggregation mechanism in which the peptide spends most of its time trapped in misregistered β-sheet states connected by weakly bound states twith short lifetimes. Our results recapitulate the experimental observation that mutants CHA19 and CHA1920 accelerate fibril elongation but have a relatively minor effect on the critical concentration for fibril growth. Importantly, the kinetic consequences of mutations arise from cumulative effects of perturbing the network of productive and nonproductive pathways of fibril growth. This is consistent with the expectation that nonfunctional states will not have evolved efficient folding pathways and, therefore, will require a random search of configuration space. This study highlights the importance of describing the complete energy landscape when studying the elongation mechanism and kinetics of protein fibrils.

Engulfment of ellipsoidal nanoparticles by membranes: full description of orientational changes

We study the engulfment of ellipsoidal nanoparticles by membranes. It has been previously predicted that wrapping by the membrane can induce reorientation of the particle, however, previous studies only considered the wrapping process constrained to either side-oriented or tip-oriented particles. In contrast, we consider here the full two-dimensional energy landscape for engulfment, where the two degrees of freedom represent (i) the amount of wrapping and (ii) the particle orientation. In this way, we obtain access to the stability limits of the differently-oriented states, as well as to the energy barriers between them. We find that prolate and oblate particles undergo qualitatively different engulfment transitions, and show that the initial orientation of the particle at first contact with the membrane influences its fate.

VR-SCOSMO: A smooth conductor-like screening model with charge-dependent radii for modeling chemical reactions.

To better represent the solvation effects observed along reaction pathways, and of ionic species in general, a charge-dependent variable-radii smooth conductor-like screening model (VR-SCOSMO) is developed. This model is implemented and parameterized with a third order density-functional tight binding quantum model, DFTB3/3OB-OPhyd, a quantum method which was developed for organic and biological compounds, utilizing a specific parameterization for phosphate hydrolysis reactions. Unlike most other applications with the DFTB3/3OB model, an auxiliary set of atomic multipoles is constructed from the underlying DFTB3 density matrix which is used to interact the solute with the solvent response surface. The resulting method is variational, produces smooth energies, and has analytic gradients. As a baseline, a conventional SCOSMO model with fixed radii is also parameterized. The SCOSMO and VR-SCOSMO models shown have comparable accuracy in reproducing neutral-molecule absolute solvation free energies; however, the VR-SCOSMO model is shown to reduce the mean unsigned errors (MUEs) of ionic compounds by half (about 2-3 kcal/mol). The VR-SCOSMO model presents similar accuracy as a charge-dependent Poisson-Boltzmann model introduced by Hou et al. [J. Chem. Theory Comput. 6, 2303 (2010)]. VR-SCOSMO is then used to examine the hydrolysis of trimethylphosphate and seven other phosphoryl transesterification reactions with different leaving groups. Two-dimensional energy landscapes are constructed for these reactions and calculated barriers are compared to those obtained from ab initio polarizable continuum calculations and experiment. Results of the VR-SCOSMO model are in good agreement in both cases, capturing the rate-limiting reaction barrier and the nature of the transition state.

Simulations of thermodynamics and kinetics on rough energy landscapes with milestoning.

We investigated by computational means the kinetics and stationary behavior of stochastic dynamics on an ensemble of rough two-dimensional energy landscapes. There are no obvious separations of temporal scales in these systems, which constitute a simple model for the behavior of glasses and some biomaterials. Even though there are significant computational challenges present in these systems due to the large number of metastable states, the Milestoning method is able to compute their kinetic and thermodynamic properties exactly. We observe two clearly distinguished regimes in the overall kinetics: one in which diffusive behavior dominates and another that follows an Arrhenius law (despite the absence of a dominant barrier). We compare our results with those obtained with an exactly-solvable one-dimensional model, and with the results from the rough one-dimensional energy model introduced by Zwanzig. © 2015 Wiley Periodicals, Inc.

Erratum:Ab initiomodeling of the two-dimensional energy landscape of screw dislocations in bcc transition metals [Phys. Rev. B89, 024104 (2014)

Erratum:<i>Ab initio</i>modeling of the two-dimensional energy landscape of screw dislocations in bcc transition metals [Phys. Rev. B<b>89</b>, 024104 (2014)

Ab initiomodeling of the two-dimensional energy landscape of screw dislocations in bcc transition metals

A density functional theory (DFT) study of the 1/2$\ensuremath{\langle}111\ensuremath{\rangle}$ screw dislocation was performed in the following body-centered cubic transition metals: V, Nb, Ta, Cr, Mo, W, and Fe. The energies of the easy, hard, and split core configurations, as well as the pathways between them, were investigated and used to generate the two-dimensional (2D) Peierls potential, i.e. the energy landscape seen by the dislocation as a function of its position in the (111) plane. In all investigated elements, the nondegenerate easy core is the minimum energy configuration, while the split core configuration, centered in the immediate vicinity of a $\ensuremath{\langle}111\ensuremath{\rangle}$ atomic column, has a high energy near or above that of the hard core. This unexpected result yields 2D Peierls potentials very different from the usually assumed landscapes. The 2D Peierls potential in Fe differs from the other transition metals, with a monkey saddle instead of a local maximum located at the hard core. An estimation of the Peierls stress from the shape of the Peierls barrier is presented in all investigated metals. A strong group dependence of the core energy is also evidenced, related to the position of the Fermi level with respect to the minimum of the pseudogap of the electronic density of states.

Particle dynamics in two-dimensional random-energy landscapes: Experiments and simulations

The dynamics of individual colloidal particles in random potential energy landscapes was investigated experimentally and by Monte Carlo simulations. The value of the potential at each point in the two-dimensional energy landscape follows a Gaussian distribution. The width of the distribution, and hence the degree of roughness of the energy landscape, was varied and its effect on the particle dynamics studied. This situation represents an example of Brownian dynamics in the presence of disorder. In the experiments, the energy landscapes were generated optically using a holographic setup with a spatial light modulator, and the particle trajectories were followed by video microscopy. The dynamics is characterized using, e.g., the time-dependent diffusion coefficient, the mean squared displacement, the van Hove function, and the non-Gaussian parameter. In both experiments and simulations the dynamics is initially diffusive, showing an extended subdiffusive regime at intermediate times before diffusive motion is recovered at very long times. The dependence of the long-time diffusion coefficient on the width of the Gaussian distribution agrees with theoretical predictions. Compared to the dynamics in a one-dimensional potential energy landscape, the localization at intermediate times is weaker and the diffusive regime at long times reached earlier, which is due to the possibility to avoid local maxima in two-dimensional energy landscapes.

Transient interactions between biomolecules

The ultimate goal of cellular biology and biochemistry is to penetrate into and comprehend the complexity of the molecular interaction networks existing in the living cell so as to provide a molecular rationale for understanding the details of cell (mis)functioning and diseases. Indeed, specific protein–protein and protein–nucleic acid interactions underlie virtually every process in the cell, regulating cell metabolism and determining cell life and death. Such interactions between biomolecules occur on a wide range of timescales. Stable complexes, with lifetime ranging from minutes to days, involve high-affinity and high-specificity binding. Amongst many others, examples are irreversible enzyme inhibition by inhibitors and assembly of proteins supporting the cell ultrastructure. Weak complexes, with lifetime in the second–microsecond range, are formed when fast turnover or reversible multicomponent assembly is required. The topic of this Special Issue of European Biophysics Journal, entitled ‘‘Transient Interactions between Biomolecules’’, is focussed on short-lived complexes, in which a fine balance between specificity of binding and high turnover rate is sought. Actually, most of these complexes are characterized by equilibrium dissociation constants within the lM or even mM range. An important question arises regarding how such weak interactions guarantee specificity between partners high enough to be relevant from the biological point of view. Different systems may rely on different solutions. In some cases, multiple interactions may be involved, while in others a kinetic trap is possible, as in the recognition between the ribosome and transfer RNA (tRNA). Whatever the solution is, the transient interaction has to reach a compromise between specificity and fast dissociation. Intriguingly, contrary to what one might think, these molecular recognition mechanisms are not uncommon and play a key role in many biological processes, namely the electron transfer chains in respiration and photosynthesis and the cell signalling cascades involving kinases and phosphatases. Our better molecular understanding of the different cellular processes has led to the growing realization that transient and reversible interactions are necessary for a very broad range of processes, such as assembling post-transcriptional regulatory complexes and regulating messenger RNA (mRNA) metabolism from splicing to nuclear export. Communication between biomolecules through specific binding interfaces is essential to transfer information in an efficient way. However, such interfaces represent a small fraction of the total surface area of the interacting partners, and so their encounter and further proper interaction become a non-trivial task. To reduce the conformational search space on the path to the specific complex, it has recently proposed that complex formation, especially that involving transient interactions, proceeds via a two-step mechanism. The first step yields the so-called encounter complex, which is the result of the weak association between partners via diffusion-controlled intermolecular collisions. During the short-lived encounter phase, the two partners remain largely solvated and adopt an ensemble of different orientations located in multiple local free energy minima of a two-dimensional energy landscape on the surface. Electrostatics dominates this state. The subsequent Special Issue: Transient interactions in biology.

RNA polymerase pushing

Molecular motors can exhibit Brownian ratchet or power stroke mechanisms. These mechanistic categories are related to transition state position: An early transition state suggests that chemical energy is stored and then released during the step (stroke) while a late transition state suggests that the release of chemical energy rectifies thermally activated motion that has already occurred (ratchet). Cellular RNA polymerases are thought to be ratchets that can push each other forward to reduce pausing during elongation. Here, by constructing a two-dimensional energy landscape from the individual landscapes of active and backtracked enzymes, we identify a new pushing mechanism which is the result of a saddle trajectory that arises in the two-dimensional energy landscape of interacting enzymes. We show that this mechanism is more effective with an early transition state suggesting that interacting RNAPs might translocate via a power stroke.

A Thermo-Magneto-Mechanical Free Energy Model for NiMnGa Single Crystals

AbstractThe paper extends the authors' recent model for one-dimensional rate-dependent magneto-mechanical behavior of NiMnGa single crystals to account for temperature-dependent effects including austenite/martensite and ferro-/paramagnetic phase transitions. The magneto-mechanical model is based on the Helmholtz free energy landscape constructed for a meso-scale lattice element with strain and magnetization as order parameters. This two-dimensional energy landscape includes three paraboloidal wells representing the two easy-axis and one hard-axis martensite variants relevant for the structurally one-dimensional case. Phase transformations resulting from applied stresses and magnetic fields follow from a system of evolution laws based on the Gibbs free energy equations and the theory of thermally activated processes, which in the low-thermal-activation limit appropriately reproduce the athermal transformation behavior observed in these materials. The phase fractions subsequently determine the macroscopic strain and magnetization of a sample of NiMnGa by means of a standard averaging procedure. To account for the first-order phase transitions to austenite, additional temperature-dependent wells representing the stable states of austenitic NiMnGa are introduced into the Helmholtz energy landscape. The transition from ferromagnetic to paramagnetic states is modeled as a second order transformation based on the gradual degeneration of the ferromagnetic wells with increasing temperature.

Stochastic resonance in the perceptual interpretation of ambiguous figures: A neural network model

We describe the results of computer simulations of the dynamical behavior of an autoassociative network with a two-dimensional energy landscape. Such a network can model some aspects of the phenomenon of perceptual bistability in the presence of ambiguous figures. The network can be operated at either zero or nonzero temperatures which represent an internal system noise. Our results show that, under the influence of a weak periodic external signal, the network exhibits a maximum in the signal-to-noise ratio at an optimum noise level: the characteristic signature of stochastic resonance.