We show how to localize and quantify the functional evolutionary constraints on natural proteins. Protein folding has been one of the strongest constraints in sequence evolution. The method we propose compares the perturbations caused by local sequence variants to the energetics of the protein folding process and to the corresponding change to the apparent selection landscape of sequences over the evolutionary time scale. The difference between the physical folding free energies and the evolutionary free energies can be called a "dark energy." We analyze various protein sets and thereby show that dark energy is largely localized at functional sites, which are also often energetically frustrated from the point of view of folding. Overall, we find that about 25% of the positions of the folded globular proteins display some significant dark energy. When a function relies on a free energy that can be thermodynamically quantified, such as the binding energy to a partner, the relationship of this physical free energy with dark energy can be used to define a functional selection temperature, just as there is a selection temperature for folding. We show that selection for folding and binding functions bear similar weights in specific protein-protein interactions.

Within the framework of the random first-order transition theory of glasses, we discuss the statistics of thermal avalanches, the large scale rearrangements in driven amorphous systems near their instability. Stringy excitations yield nonPoisson waiting time statistics. Embedding these statistics in a generalized Master equation captures the nonMarkovian, aging dynamics of avalanche clusters. We apply this framework to analyze nonequilibrium signatures of thermal avalanches, auto correlation functions and effective temperatures, under both quasi static shear and stochastic shaking protocols. We use full counting statistics to derive the complete distribution of both the avalanche magnitudes and avalanche counts, uncovering the intermediate time behavior.

The processivity of Structural Maintenance of Chromosome complexes defines the characteristic run length and lifetime of loop extrusion events, which set up the large-scale architecture of chromosomes. We introduce an active, non-Markovian mechanistic model that explicitly incorporates motor processivity to provide a statistical mechanical treatment that identifies the nontrivial effects induced by the processive character of such active motors. At low activity, in interphase, processive loop extrusion generates effective cooperative multibody interactions, which lead to the so-called "chromatin jets." Upon increasing activity, symmetry breaking occurs, as seen in the characteristic, cylindrically anisotropic mitotic chromosome organization. The strength of the motor processivity determines whether the symmetry breaking transition leads to crystalline ordering or to liquid-crystalline architectures for the mitotic chromosome.

Fluorescent protein fusions with environmentally sensitive fluorophores have been widely used to investigate changes in the protein microenvironment. Unfortunately, these techniques often rely on bulky fluorescent proteins or tags to the N terminus or C terminus of the target protein, which can disrupt the behavior of the target protein and may limit their ability to investigate microenvironment changes with high spatial resolution. Here we develop a strategy to visualize microenvironment changes of protein substructures in real time by genetically incorporating environment-sensitive noncanonical amino acids (ncAAs) containing rotor-based fluorophores at specific positions of the target protein. Through computational redesign of aminoacyl-tRNA synthetase, we successfully incorporated these rotor-based ncAAs into several proteins in mammalian cells. Precise placement of these ncAAs at specific sites of proteins enables the detection of microenvironmental changes around individual residues during events such as aggregation, clustering, cluster dissociation and others.

Gene regulation often entails a cooperative dynamic interplay among several protein molecules and several distinct DNA segments. Intersegment transfer of the Fis protein stimulates DNA inversion during DNA recombination. Individual DNA segments have been found to facilitate the dissociation of Fis proteins already bound to DNA and also allow for the transfer of the Fis between segments. Here, we use the hybrid coarse-grained AWSEM/3SPN.2C model to simulate the Fis protein intersegment transfer and explore its mechanism. We show that entropic effects within the Fis protein-DNA complex dictate the transfer pathway through specific structural configurations, involving specific grooves and consequent orientation constraints on the DNA segments. Multiple copies of the Fis protein facilitate intersegment transfer, explaining how changes in protein-DNA stoichiometry and concentration influence how the Fis-DNA complex architecture is established. This orientational dependence indicates that the assembly of the Fis-DNA complex mimics the interlocking of screws, functioning as a molecular machine that may couple to DNA supercoiling and torsional stress in the DNA generated by motor proteins, thus offering a potential regulatory mechanism for chromosomal organization and gene expression.

During mitosis, near-spherical chromosomes reconfigure into rod-like structures to ensure their accurate segregation to daughter cells. We explore here, the interplay between the nonequilibrium activity of molecular motors in determining the chromosomal organization in mitosis and its characteristic symmetry-breaking events. We present a hybrid motorized chromosome model that highlights the distinct roles of condensin I and II in shaping mitotic chromosomes. Guided by experimental observations, the simulations suggest that condensin II facilitates large-scale scaffold formation, while condensin I is paramount in local helical loop arrangement. Together, these two distinct grappling motors establish the hierarchical helical structure characteristic of mitotic chromosomes, which exhibit striking local and, sometimes global, chirality and contribute to the robust mechanical properties of mitotic chromosomes. Accompanying the emergence of rigidity, the model provides mechanisms of forming defects, including perversions and entanglements, and shows how these may be partially resolved through condensin activity and topoisomerase action. This framework bridges coarse-grained energy landscape models of chromosome dynamics and non-equilibrium molecular dynamics, advancing the understanding of chromosome organization during cell division and beyond.

Single-stranded DNA-binding proteins (SSBs) protect transiently exposed ssDNA, yet how DNA polymerase (DNAp) displaces them during replication remains unclear. Using single-molecule force spectroscopy, dual-color imaging, and molecular dynamics simulations on bacteriophage T7 DNAp and SSB, we investigated molecular mechanisms underlying SSB displacement. T7 SSB modulates replication in a force-dependent manner: enhancing it at low tension by preventing secondary structures while impeding it at high tension. Dual-color imaging shows SSBs remain stationary as DNAp advances, supporting a sequential displacement model. Molecular dynamics suggests that DNAp actively lowers the SSB dissociation energy barrier through interactions mediated by the SSB C-terminal tail. FRET confirms close protein proximity during encounters. Optimal replication requires SSB saturation of ssDNA, establishing a delicate balance between protection and efficiency. This spatiotemporal coordination between DNAp and SSB is critical for resolving molecular collisions and may represent a general mechanism for resolving molecular collisions, ensuring both processivity and genomic integrity.

The aggregation pathways of Aβ42 peptides are complex and can lead to both amyloids and nonamyloid aggregates. We use in situ atomic force microscopy imaging to monitor the assembly of aggregate structures and their dynamics. Two aggregation pathways emerge, one leading to amyloid fibrils and a second one that includes the formation of oligomers and apparently amorphous aggregates, which we identify as nonamyloid. Whereas the fibrils seem to require elevated peptide concentration to nucleate and grow, oligomers and amorphous aggregates form at near-physiological peptide concentrations. On the time scales of the experiments, the two aggregation pathways do not cross: the oligomers and aggregates do not participate in the fibrillization pathway and, analogously, secondary nucleation assisted by mature fibrils does not produce misfolded aggregates. We show that distinct Aβ42 fibril polymorphs form and coexist under identical conditions. Mature fibrils serve as substrates for secondary nucleation that leads to forked, branched, and thicker fibrils and, importantly, produces new fibril fragments. Aβ42 fibrils accumulate structural defects, with more defects generated at higher peptide concentrations. The defects lead to substantial variations of growth rate both over time and between different fibrils. The average growth rates of Aβ42 fibrils are about 50-fold faster than those of Aβ40 fibrils. Our findings are consistent with the basic premise of the polymorph selection hypothesis, according to which the late onset of Alzheimer's disease, its high clinical variability, and the presence of amyloid plaques in healthy individuals have their origins in differing toxicities and aggregation kinetics of distinct Aβ structural polymorphs.

The concept that proteins are selected to fold into a well-defined native state has been effectively addressed within the framework of energy landscapes, underpinning the recent successes of structure prediction tools like AlphaFold. The amyloid fold, however, does not represent a unique minimum for a given single sequence. While the cross-β hydrogen-bonding pattern is common to all amyloids, other aspects of amyloid fiber structures are sensitive not only to the sequence of the aggregating peptides but also to the experimental conditions. This polymorphic nature of amyloid structures challenges structure predictions. In this paper, we use AI to explore the landscape of possible amyloid protofilament structures composed of a single stack of peptides aligned in a parallel, in-register manner. This perspective enables a practical method for predicting protofilament structures of arbitrary sequences: RibbonFold. RibbonFold is adapted from AlphaFold2, incorporating parallel in-register constraints within AlphaFold2's template module, along with an appropriate polymorphism loss function to address the structural diversity of folds. RibbonFold outperforms AlphaFold2/3 on independent test sets, achieving a mean TM-score of 0.5. RibbonFold proves well-suited to study the polymorphic landscapes of widely studied sequences with documented polymorphisms. The resulting landscapes capture these observed polymorphisms effectively. We show that while well-known amyloid-forming sequences exhibit a limited number of plausible polymorphs on their "solubility" landscape, randomly shuffled sequences with the same composition appear to be negatively selected in terms of their relative solubility. RibbonFold is a valuable framework for structurally characterizing amyloid polymorphism landscapes.

Membrane transporters are essential for cell homeostasis. Among them P-type ATPases, which couple ATP hydrolysis to solute transport, play a key role. Despite extensive research, the fine mechanisms by which this coupling occurs are not completely understood. In this work, we analyzed the effect of substrate, temperature, and urea on the steady state ATPase activity, tryptophan fluorescence and far-UV ellipticity of the catalytic domain of a thermophilic Cu(I) transport ATPase. Through local frustration analysis in combination with AlphaFold2 prediction, we identified a novel conformation, enabling molecular dynamics simulations of an open/close transition. Our results revealed a 'cracking' mechanism as a key step in the catalytic cycle. Furthermore, we developed a model that fully describes all the experimental observations. These findings reinforce the idea that local unfolding is involved in enzyme catalysis and suggest a key role in the regulation of enzyme activity. ### Competing Interest Statement The authors have declared no competing interest.

During mitosis, there are significant structural changes in chromosomes. We used a maximum entropy approach to invert experimental Hi-C data to generate effective energy landscapes for chromosomal structures at different stages during the cell cycle. Modeled mitotic structures show a hierarchical organization of helices of helices. High-periodicity loops span hundreds of kilobases or less, while the other low-periodicity ones are larger in genomic separation, spanning several megabases. The structural ensembles reveal a progressive decrease in compartmentalization from interphase to mitosis, accompanied by the appearance of a second diagonal in prometaphase, indicating an organized array of loops. While there is a local tendency to form chiral helices, overall, no preferential left-handed or right-handed chirality appears to develop on the time scale of the cell cycle. Chromatin thus appears to be a liquid crystal containing numerous defects that anneal rather slowly.

Molecules provide the ultimate language in terms of which physiology and pathology must be understood. Myriads of proteins participate in elaborate networks of interactions and perform chemical activities coordinating the life of cells. To perform these often amazing tasks, proteins must move and we must think of them as dynamic ensembles of three dimensional structures formed first by folding the polypeptide chains so as to minimize the conflicts between the interactions of their constituent amino acids. It is apparent however that, even when completely folded, not all conflicting interactions have been resolved so the structure remains "locally frustrated". Over the last decades it has become clearer that this local frustration is not just a random accident but plays an essential part of the inner workings of protein molecules. We will review here the physical origins of the frustration concept and review evidence that local frustration is important for protein physiology, protein-protein recognition, catalysis and allostery. Also, we highlight examples showing how alterations in the local frustration patterns can be linked to distinct pathologies. Finally we explore the extensions of the impact of frustration in higher order levels of organization of systems including gene regulatory networks and the neural networks of the brain.

Complex gene regulatory networks often display emergent simple behavior. Sometimes this simplicity can be traced to a nearly equivalent energy landscape, but not always. Here, we show how a topological theory for stochastic and biochemical networks can predict phase transitions between dynamical regimes, where the simplest landscape paradigm would fail. We demonstrate the utility of this topological approach for a simple gene network, revealing a new oscillatory regime in addition to previously recognized multimodal stationary phases. We show how local winding numbers predict the steady-state locations in the single-mode and bimodal phases, and a flux analysis predicts the respective strengths of the steady-state peaks.

Quantum simulation enables studies of open-system dynamics in non-perturbative regimes by programming electronic, vibrational, and environmental interactions on comparable energy scales. Trapped ions offer this capability, combining spins, phonons, and tunable dissipation on one platform. We demonstrate an open-system quantum simulation of charge and exciton transfer in a multi-mode linear vibronic coupling model. Using tailored spin-phonon interactions with reservoir engineering, we emulate a system with two dissipative vibrational modes coupled to donor and acceptor sites and track its non-equilibrium dynamics. We continuously tune the system from the charge transfer regime to the vibrationally assisted exciton transfer regime and find that degenerate modes enhance transfer rates at large energy gaps, while non-degenerate modes activate pathways that reduce the energy-gap dependence. Thus, the presence of one additional vibration introduces interfering pathways and reshapes non-perturbative excitation transfer. Our results establish a scalable, hardware-efficient route to simulate vibronic processes with engineered environments.

Single-stranded DNA-binding proteins (SSBs) play a crucial role in stabilizing and protecting transiently exposed single-stranded DNA (ssDNA), yet the mechanisms governing their displacement by DNA polymerase (DNAp) during replication remain largely unexplored. Using bacteriophage T7 DNAp and its SSB, T7 gp2.5, we investigated the molecular mechanisms and visualized the dynamic process underlying SSB displacement. Our single-molecule force spectroscopy demonstrates that T7 SSB modulates DNA replication in an ssDNA conformation-dependent manner, regulated by tension applied to the DNA template. By integrating dual-color single-molecule imaging, we observe that T7 SSB remains stationary as DNAp approaches, indicating that SSB molecules are sequentially displaced rather than pushed forwards. Molecular dynamics (MD) simulations revealed reduced energy barriers for SSB dissociation in the presence of DNAp. This finding, combined with the detected FRET signals when DNAp approaches an SSB-bound ssDNA region and observations of faster replication rates compared to relative slow intrinsic SSB dissociation, collectively support an active displacement mechanism. Using both ensemble and single-molecule analyses, we demonstrated that SSB saturation of ssDNA is critical for optimal replication efficiency, with each SSB molecule contributing positively to the process. Taken together, the uncovered spatial-temporal coordination between SSB and DNAp is necessary for resolving molecular collisions during DNA replication, and may represent a universal strategy employed by other DNA translocating motors to ensure genomic integrity.

Chromatin is partially structured through the effects of biological motors. “Swimming motors” such as RNA polymerases and chromatin remodelers are thought to act differentially on the active parts of the genome and the stored inactive part. By systematically expanding the many-body master equation for chromosomes driven by swimming motors, we show that this nonuniform aspect of motorization leads to heterogeneously folded conformations, thereby contributing to chromosome compartmentalization.

Traditional methods, such as the use of fluorescent protein fusions and environment-sensitive fluorophores, have limitations when studying protein microenvironment changes at the finest spatial resolution. These techniques often rely on bulky proteins or tags restricted to the N- or C-terminus, which can disrupt the natural behavior of the target protein and dramatically limit the ability of their method to investigate noninvasively microenvironment effects. To overcome these challenges, we have developed an innovative strategy to visualize microenvironment changes of protein substructures in real-time by genetically incorporating environment-sensitive noncanonical amino acids (ncAAs) containing rotor-based fluorophores (RBFs) at specific positions within a protein of interest. Through computational redesign of aminoacyl-tRNA synthetase, we successfully incorporated these rotor-based ncAAs into various proteins in mammalian cells. By site-specifically placing these ncAAs in distinct regions of proteins, we detected microenvironmental changes of several different protein domains during events such as aggregation, clustering, aggregation disassembly, and cluster dissociation.

Analyses of ancient DNA typically involve sequencing the surviving short oligonucleotides and aligning to genome assemblies from related, modern species. Here, we report that skin from a female woolly mammoth (†Mammuthus primigenius) that died 52,000 years ago retained its ancient genome architecture. We use PaleoHi-C to map chromatin contacts and assemble its genome, yielding 28 chromosome-length scaffolds. Chromosome territories, compartments, loops, Barr bodies, and inactive X chromosome (Xi) superdomains persist. The active and inactive genome compartments in mammoth skin more closely resemble Asian elephant skin than other elephant tissues. Our analyses uncover new biology. Differences in compartmentalization reveal genes whose transcription was potentially altered in mammoths vs. elephants. Mammoth Xi has a tetradic architecture, not bipartite like human and mouse. We hypothesize that, shortly after this mammoth’s death, the sample spontaneously freeze-dried in the Siberian cold, leading to a glass transition that preserved subfossils of ancient chromosomes at nanometer scale.

While significant advances have been made in predicting static protein structures, the inherent dynamics of proteins, modulated by ligands, are crucial for understanding protein function and facilitating drug discovery. Traditional docking methods, frequently used in studying protein-ligand interactions, typically treat proteins as rigid. While molecular dynamics simulations can propose appropriate protein conformations, they’re computationally demanding due to rare transitions between biologically relevant equilibrium states. In this study, we present DynamicBind, a deep learning method that employs equivariant geometric diffusion networks to construct a smooth energy landscape, promoting efficient transitions between different equilibrium states. DynamicBind accurately recovers ligand-specific conformations from unbound protein structures without the need for holo-structures or extensive sampling. Remarkably, it demonstrates state-of-the-art performance in docking and virtual screening benchmarks. Our experiments reveal that DynamicBind can accommodate a wide range of large protein conformational changes and identify cryptic pockets in unseen protein targets. As a result, DynamicBind shows potential in accelerating the development of small molecules for previously undruggable targets and expanding the horizons of computational drug discovery.

ABSTRACT According to the Principle of Minimal Frustration, folded proteins can have only a minimal number of strong energetic conflicts in their native states. However, not all interactions are energetically optimized for folding but some remain in energetic conflict, i.e. they are highly frustrated. This remaining local energetic frustration has been shown to be statistically correlated with distinct functional aspects such as protein-protein interaction sites, allosterism and catalysis. Fuelled by the recent breakthroughs in efficient protein structure prediction that have made available good quality models for most proteins, we have developed a strategy to calculate local energetic frustration within large protein families and quantify its conservation over evolutionary time. Based on this evolutionary information we can identify how stability and functional constraints have appeared at the common ancestor of the family and have been maintained over the course of evolution. Here, we present FrustraEvo, a web server tool to calculate and quantify the conservation of local energetic frustration in protein families. The webserver is freely available at URL: https://frustraevo.qb.fcen.uba.ar