Protein folded states are kinetic hubs

A Stochastic Landscape Approach for Protein Folding State Classification

Protein folding is a critical process that determines the functional state of proteins. Proper folding is essential for proteins to acquire their functional three-dimensional structures and execute their biological role, whereas misfolded proteins can lead to various diseases, including neurodegenerative disorders like Alzheimer's and Parkinson's. Therefore, a deeper understanding of protein folding is vital for understanding disease mechanisms and developing therapeutic strategies. This study introduces the Stochastic Landscape Classification (SLC), an innovative, automated, nonlearning algorithm that quantitatively analyzes protein folding dynamics. Focusing on collective variables (CVs) - low-dimensional representations of complex dynamical systems like molecular dynamics (MD) of macromolecules - the SLC approach segments the CVs into distinct macrostates, revealing the protein folding pathway explored by MD simulations. The segmentation is achieved by analyzing changes in CV trends and clustering these segments using a standard density-based spatial clustering of applications with noise (DBSCAN) scheme. Applied to the MD-based CV trajectories of Chignolin and Trp-Cage proteins, the SLC demonstrates apposite accuracy, validated by comparing standard classification metrics against ground-truth data. These metrics affirm the efficacy of the SLC in capturing intricate protein dynamics and offer a method to evaluate and select the most informative CVs. The practical application of this technique lies in its ability to provide a detailed, quantitative description of protein folding processes, with significant implications for understanding and manipulating protein behavior in industrial and pharmaceutical contexts.

Hierarchical consciousness: the Nested Observer Windows model

Foremost in our experience is the intuition that we possess a unified conscious experience. However, many observations run counter to this intuition: we experience paralyzing indecision when faced with two appealing behavioral choices, we simultaneously hold contradictory beliefs, and the content of our thought is often characterized by an internal debate. Here, we propose the Nested Observer Windows (NOW) Model, a framework for hierarchical consciousness wherein information processed across many spatiotemporal scales of the brain feeds into subjective experience. The model likens the mind to a hierarchy of nested mosaic tiles-where an image is composed of mosaic tiles, and each of these tiles is itself an image composed of mosaic tiles. Unitary consciousness exists at the apex of this nested hierarchy where perceptual constructs become fully integrated and complex behaviors are initiated via abstract commands. We define an observer window as a spatially and temporally constrained system within which information is integrated, e.g. in functional brain regions and neurons. Three principles from the signal analysis of electrical activity describe the nested hierarchy and generate testable predictions. First, nested observer windows disseminate information across spatiotemporal scales with cross-frequency coupling. Second, observer windows are characterized by a high degree of internal synchrony (with zero phase lag). Third, observer windows at the same spatiotemporal level share information with each other through coherence (with non-zero phase lag). The theoretical framework of the NOW Model accounts for a wide range of subjective experiences and a novel approach for integrating prominent theories of consciousness.

Revealing the conformational dynamics of UDP-GlcNAc recognition by O-GlcNAc transferase via Markov state model

The O-linked N-acetylglucosamine (O-GlcNAc) glycosylation is a critical post-translational modification and closely linked to various physiological and pathological conditions. The O-GlcNAc transferase (OGT) functions as the only glycosyltransferase of O-GlcNAc glycosylation by transferring GlcNAc from UDP-GlcNAc to serine or threonine residues on protein substrates. The interaction mode of UDP-GlcNAc against OGT has been preliminarily revealed by the crystal structures, yet an atomic-level comprehension for the conformational dynamics of the recognition process remains elusive. Here, we construct the Markov state model based on extensive all-atom molecular dynamics (MD) simulations with an aggregated simulation time of ∼9 μs, and reveal that the UDP-GlcNAc recognition process by OGT encompasses four key metastable states, occurring within an estimated timescale of ∼10 μs. During UDP-GlcNAc recognition process, we find the pyrophosphate moiety (P2O52-) initially anchors to the active pocket via salt bridge and hydrogen bonds, facilitating subsequent binding of the uridine and GlcNAc moieties. Furthermore, the functional roles of K842 involved in the salt bridge with P2O52- were evaluated through extra mutant MD simulations. Overall, our study provides valuable insights into the UDP-GlcNAc recognition mechanism by OGT, which could further aid in mechanistic studies of O-GlcNAc glycosylation and drug development targeting on OGT.

Folding@home: Achievements from over 20 years of citizen science herald the exascale era

Simulations of biomolecules have enormous potential to inform our understanding of biology but require extremely demanding calculations. For over 20 years, the Folding@home distributed computing project has pioneered a massively parallel approach to biomolecular simulation, harnessing the resources of citizen scientists across the globe. Here, we summarize the scientific and technical advances this perspective has enabled. As the project's name implies, the early years of Folding@home focused on driving advances in our understanding of protein folding by developing statistical methods for capturing long-timescale processes and facilitating insight into complex dynamical processes. Success laid a foundation for broadening the scope of Folding@home to address other functionally relevant conformational changes, such as receptor signaling, enzyme dynamics, and ligand binding. Continued algorithmic advances, hardware developments such as graphics processing unit (GPU)-based computing, and the growing scale of Folding@home have enabled the project to focus on new areas where massively parallel sampling can be impactful. While previous work sought to expand toward larger proteins with slower conformational changes, new work focuses on large-scale comparative studies of different protein sequences and chemical compounds to better understand biology and inform the development of small-molecule drugs. Progress on these fronts enabled the community to pivot quickly in response to the COVID-19 pandemic, expanding to become the world's first exascale computer and deploying this massive resource to provide insight into the inner workings of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus and aid the development of new antivirals. This success provides a glimpse of what is to come as exascale supercomputers come online and as Folding@home continues its work.

Dynamic conformational states of apo, ATP and cabozantinib bound TAM kinases to differentiate active-inactive kinetic models

The dynamically active and inactive conformations of kinases play a crucial role in the activation of intracellular downstream signaling pathways. The all-atom molecular dynamics (MD) simulations at microsecond (µs) timescale and longer provide robust insights into the structural details of conformational alterations in kinases that contribute to their cellular metabolic activities and signaling pathways. Tyro3, Axl and Mer (TAM) receptor tyrosine kinases (RTKs) are overexpressed in several types of human cancers. Cabozantinib, a small molecule inhibitor constrains the activity of TAM kinases at nanomolar concentrations. The apo, complexes of ATP (active state) and cabozantinib (active and inactive states) with TAM RTKs were studied by 1 µs MD simulations followed by trajectory analyses. The dynamic mechanistic pathways intrinsic to the kinase activity and protein conformational landscape in the cabozantinib bound TAM kinases are revealed due to the alterations in the P-loop, α-helix and activation loop that result in breaking the regulatory (R) and catalytic (C) spines, while the active states of ATP bound TAM kinases are retained. The co-existence of dynamical states when bound to cabozantinib was observed and the long-lived kinetic transition states of distinct active and inactive structural models were deciphered from MD simulation trajectories that have not been revealed so far.Communicated by Ramaswamy H. Sarma.

Molecular mechanisms of spontaneous curvature and softening in complex lipid bilayer mixtures

Membrane reshaping is an essential biological process. The chemical composition of lipid membranes determines their mechanical properties and thus the energetics of their shape. Hundreds of distinct lipid species make up native bilayers, and this diversity complicates efforts to uncover what compositional factors drive membrane stability in cells. Simplifying assumptions, therefore, are used to generate quantitative predictions of bilayer dynamics based on lipid composition. One assumption commonly used is that "per lipid" mechanical properties are both additive and constant-that they are an intrinsic property of lipids independent of the surrounding composition. Related to this is the assumption that lipid bulkiness, or "shape," determines its curvature preference, independently of context. In this study, all-atom molecular dynamics simulations on three separate multilipid systems were used to explicitly test these assumptions, applying methodology recently developed to isolate properties of single lipids or nanometer-scale patches of lipids. The curvature preference experienced by populations of lipid conformations were inferred from their redistribution on a dynamically fluctuating bilayer. Representative populations were extracted by both structural similarity and semi-automated hidden Markov model analysis. The curvature preferences of lipid dimers were then determined and compared with an additive model that combines the monomer curvature preference of both the individual lipids. In all three systems, we identified conformational subpopulations of lipid dimers that showed non-additive curvature preference, in each case mediated by a special chemical interaction (e.g., hydrogen bonding). Our study highlights the importance of specific chemical interactions between lipids in multicomponent bilayers and the impact of interactions on bilayer stiffness. We identify two mechanisms of bilayer softening: diffusional softening, driven by the dynamic coupling between lipid distributions and membrane undulations, and conformational softening, driven by the inter-conversion between distinct dimeric conformations.

Complex Inclusion Bodies and Defective Proteome Hubs in Neurodegenerative Disease: New Clues, New Challenges

A healthy physiological environment of cells represents the dynamic homeostasis of crowded molecules. A subset of cellular proteome forms protein quality control (PQC) machinery to maintain an uninterrupted synthesis of new polypeptides and targeted elimination of old or defective proteins. The process of PQC may get overwhelmed under specific genetic mutations, environmental stress conditions, and aging-associated perturbances. Many of these conditions may lead to the generation of various types of aberrant protein species that may or may not accumulate as large cellular inclusions. These proteinaceous formations, referred to as inclusion bodies (IBs), could be membrane-bound or membrane-less, cytoplasmic, or nuclear. Most importantly, they could either be toxic or protective. Under acute stress conditions, the formation of aggregates may cause proteostasis failure, leading to large-scale changes in the cellular proteome compositions. However, the large insoluble IBs may act as reservoirs for many soluble proteins with high aggregation propensities, which can overwhelm the cellular chaperoning capacity and protein degradation machinery. The kinetic equilibrium between folding and unfolding, misfolding, and refolding; aggregation and degradation is perturbed in one or many neurodegenerative disorders (NDDs) associated with dementia, cognitive impairments, movement, and behavioural losses. However, a detailed interplay of IBs into the manifestation of the NDDs is unknown, and a very primitive knowledge of structural compositions of amyloid inclusions is present. The present article presents a brief evolutionary background of IBs; their functional relevance for prokaryotes, plants, and animals; and associated involvement in neuronal proteostasis.

The importance of the compact disordered state in the fuzzy interactions between intrinsically disordered proteins

The intrinsically disordered C-terminal domain (CTD) of protein 4.1G is able to specifically bind a 26-residue intrinsically disordered region of NuMA, forming a dynamic fuzzy complex. As one of a few cases of extremely fuzzy interactions between two intrinsically disordered proteins/regions (IDPs/IDRs) without induced folding, the principle of the binding is unknown. Here, we combined experimental and computational methods to explore the detailed mechanism of the interaction between 4.1G-CTD and NuMA. MD simulations suggest that the kinetic hub states in the structure ensemble of 4.1G-CTD are favorable in the fuzzy complex. The feature of these hub states is that the binding 'hot spot' motifs βA and βB exhibit β strand propensities and are well packed to each other. The binding between 4.1G-CTD and NuMA is disrupted at low pH, which changes the intramolecular packing of 4.1G-CTD and weakens the packing between βA and βB motifs. Low pH conditions also lead to increased hydrodynamic radius and acceleration of backbone dynamics of 4.1G-CTD. All these results underscore the importance of tertiary structural arrangements and overall compactness of 4.1G-CTD in its binding to NuMA, i.e. the compact disordered state of 4.1G-CTD is crucial for binding. Different from the short linear motifs (SLiMs) that are often found to mediate IDP interactions, 4.1G-CTD functions as an intrinsically disordered domain (IDD), which is a functional and structural unit similar to conventional protein domains. This work sheds light on the molecular recognition mechanism of IDPs/IDRs and expands the conventional structure-function paradigm in protein biochemistry.

Protein Folding Dynamics as Diffusion on a Free Energy Surface: Rate Equation Terms, Transition Paths, and Analysis of Single-Molecule Photon Trajectories

The rates of protein (un)folding are often described as diffusion on the projection of a hyperdimensional energy landscape onto a few (ideally one) order parameters. Testing such an approximation by experiment requires resolving the reactive transition paths of individual molecules, which is now becoming feasible with advanced single-molecule spectroscopic techniques. This has also sparked the interest of theorists in better understanding reactive transition paths. Here we focus on these issues aiming to establish (i) practical guidelines for the mechanistic interpretation of transition path times (TPT) and (ii) methods to extract the free energy surface and protein dynamics from the maximum likelihood analysis of photon trajectories (MLA-PT). We represent the (un)folding rates as diffusion on a 1D free energy surface with the FRET efficiency as a reaction coordinate proxy. We then perform diffusive kinetic simulations on surfaces with two minima and a barrier, but with different shapes (curvatures, barrier height, and symmetry), coupled to stochastic simulations of photon emissions that reproduce current SM-FRET experiments. From the analysis of transition paths, we find that the TPT is inversely proportional to the barrier height (difference in free energy between minimum and barrier top) for any given surface shape, and that dividing the TPT into climb and descent segments provides key information about the barrier's symmetry. We also find that the original MLA-PT procedure used to determine the TPT from experiments underestimates its value, particularly for the cases with smaller barriers (e.g., fast folders), and we suggest a simple strategy to correct for this bias. Importantly, we also demonstrate that photon trajectories contain enough information to extract the 1D free energy surface's shape and dynamics (if TPT is >4-5-fold longer than the interphoton time) using the MLA-PT directly implemented with a diffusive free energy surface model. When dealing with real (unknown) experimental data, the comparison between the likelihoods of the free energy surface and discrete kinetic three-state models can be used to evaluate the statistical significance of the estimated free energy surface.

Multiscale Approach for Computing Gated Ligand Binding from Molecular Dynamics and Brownian Dynamics Simulations

We develop an approach to characterize the effects of gating by a multiconformation protein consisting of macrostate conformations that are either accessible or inaccessible to ligand binding. We first construct a Markov state model of the apo-protein from atomistic molecular dynamics simulations from which we identify macrostates and their conformations, compute their relative macrostate populations and interchange kinetics, and structurally characterize them in terms of ligand accessibility. We insert the calculated first-order rate constants for conformational transitions into a multistate gating theory from which we derive a gating factor γ that quantifies the degree of conformational gating. Applied to HIV-1 protease, our approach yields a kinetic network of three accessible (semi-open, open, and wide-open) and two inaccessible (closed and a newly identified, "parted") macrostate conformations. The parted conformation sterically partitions the active site, suggesting a possible role in product release. We find that the binding kinetics of drugs and drug-like inhibitors to HIV-1 protease falls in the slow gating regime. However, because γ = 0.75, conformational gating only modestly slows ligand binding. Brownian dynamics simulations of the diffusional association of eight inhibitors to the protease─having a wide range of experimental association constants (∼10⁴-10¹⁰ M^-1 s^-1)─yields gated rate constants in the range of ∼0.5-5.7 × 10⁸ M^-1 s^-1. This indicates that, whereas the association rate of some inhibitors could be described by the model, for many inhibitors either subsequent conformational transitions or alternate binding mechanisms may be rate-limiting. For systems known to be modulated by conformational gating, the approach could be scaled computationally efficiently to screen association kinetics for a large number of ligands.

Molecular dynamics study of conformation transition from helix to sheet of Aβ42 peptide

Aβ42 peptides can form helix and sheet structure under different conditions. The conformational conversion is closely associated with Aβ peptides aggregation and their neurotoxicity. But the transition from helix to sheet is not be clearly understood. In this study we performed microsecond timescale MD simulations of Aβ42 peptide to investigate the conformation transition from α-helix to β-sheet. Markov state model (MSM) was built to facilitate identification of crucial intermediate states and possible transition pathway. Based on the analysis, we found that the region Y10-A21 in the middle of Aβ42 peptide plays an initial role in this transition. MSM model revealed that the collapse of helical structure in this region might trigger the formation of sheet structure. Moreover, we further simulated the aggregation of Aβ42 peptides with different conformations. We found that the Aβ42 peptides forming sheet structure have higher aggregation potential compared with peptides with helix structure. These results demonstrate that we can prevent the aggregation of Aβ42 peptides by stabilizing the helix structure in the region of Y10-A21. In addition, this study provides new insight into better understanding the conformational transition and aggregation of Aβ42 peptides.

Unsupervised Learning Methods for Molecular Simulation Data

Unsupervised learning is becoming an essential tool to analyze the increasingly large amounts of data produced by atomistic and molecular simulations, in material science, solid state physics, biophysics, and biochemistry. In this Review, we provide a comprehensive overview of the methods of unsupervised learning that have been most commonly used to investigate simulation data and indicate likely directions for further developments in the field. In particular, we discuss feature representation of molecular systems and present state-of-the-art algorithms of dimensionality reduction, density estimation, and clustering, and kinetic models. We divide our discussion into self-contained sections, each discussing a specific method. In each section, we briefly touch upon the mathematical and algorithmic foundations of the method, highlight its strengths and limitations, and describe the specific ways in which it has been used-or can be used-to analyze molecular simulation data.

Thermodynamic stability versus kinetic accessibility: Pareto fronts for programmable self-assembly

A challenge in designing self-assembling building blocks is to ensure the target state is both thermodynamically stable and kinetically accessible. These two objectives are known to be typically in competition, but it is not known how to simultaneously optimize them. We consider this problem through the lens of multi-objective optimization theory: we develop a genetic algorithm to compute the Pareto fronts characterizing the tradeoff between equilibrium probability and folding rate, for a model system of small polymers of colloids with tunable short-ranged interaction energies. We use a coarse-grained model for the particles' dynamics that allows us to efficiently search over parameters, for systems small enough to be enumerated. For most target states there is a tradeoff when the number of types of particles is small, with medium-weak bonds favouring fast folding, and strong bonds favouring high equilibrium probability. The tradeoff disappears when the number of particle types reaches a value m*, that is usually much less than the total number of particles. This general approach of computing Pareto fronts allows one to identify the minimum number of design parameters to avoid a thermodynamic-kinetic tradeoff. However, we argue, by contrasting our coarse-grained model's predictions with those of Brownian dynamics simulations, that particles with short-ranged isotropic interactions should generically have a tradeoff, and avoiding it in larger systems will require orientation-dependent interactions.

Atomic resolution of short-range sliding dynamics of thymine DNA glycosylase along DNA minor-groove for lesion recognition

Thymine DNA glycosylase (TDG), as a repair enzyme, plays essential roles in maintaining the genome integrity by correcting several mismatched/damaged nucleobases. TDG acquires an efficient strategy to search for the lesions among a vast number of cognate base pairs. Currently, atomic-level details of how TDG translocates along DNA as it approaches the lesion site and the molecular mechanisms of the interplay between TDG and DNA are still elusive. Here, by constructing the Markov state model based on hundreds of molecular dynamics simulations with an integrated simulation time of ∼25 μs, we reveal the rotation-coupled sliding dynamics of TDG along a 9 bp DNA segment containing one G·T mispair. We find that TDG translocates along DNA at a relatively faster rate when distant from the lesion site, but slows down as it approaches the target, accompanied by deeply penetrating into the minor-groove, opening up the mismatched base pair and significantly sculpturing the DNA shape. Moreover, the electrostatic interactions between TDG and DNA are found to be critical for mediating the TDG translocation. Notably, several uncharacterized TDG residues are identified to take part in regulating the conformational switches of TDG occurred in the site-transfer process, which warrants further experimental validations.

A kinetic ensemble of the Alzheimer's Aβ peptide

The conformational and thermodynamic properties of disordered proteins are commonly described in terms of structural ensembles and free energy landscapes. To provide information on the transition rates between the different states populated by these proteins, it would be desirable to generalize this description to kinetic ensembles. Approaches based on the theory of stochastic processes can be particularly suitable for this purpose. Here, we develop a Markov state model and apply it to determine a kinetic ensemble of Aβ42, a disordered peptide associated with Alzheimer's disease. Through the Google Compute Engine, we generated 315-µs all-atom molecular dynamics trajectories. Using a probabilistic-based definition of conformational states in a neural network approach, we found that Aβ42 is characterized by inter-state transitions on the microsecond timescale, exhibiting only fully unfolded or short-lived, partially folded states. Our results illustrate how kinetic ensembles provide effective information about the structure, thermodynamics and kinetics of disordered proteins.

Optimal dimensionality reduction of Markov chains using graph transformation

Markov chains can accurately model the state-to-state dynamics of a wide range of complex systems, but the underlying transition matrix is ill-conditioned when the dynamics feature a separation of timescales. Graph transformation (GT) provides a numerically stable method to compute exact mean first passage times (MFPTs) between states, which are the usual dynamical observables in continuous-time Markov chains (CTMCs). Here, we generalize the GT algorithm to discrete-time Markov chains (DTMCs), which are commonly estimated from simulation data, for example, in the Markov state model approach. We then consider the dimensionality reduction of CTMCs and DTMCs, which aids model interpretation and facilitates more expensive computations, including sampling of pathways. We perform a detailed numerical analysis of existing methods to compute the optimal reduced CTMC, given a partitioning of the network into metastable communities (macrostates) of nodes (microstates). We show that approaches based on linear algebra encounter numerical problems that arise from the requisite metastability. We propose an alternative approach using GT to compute the matrix of intermicrostate MFPTs in the original Markov chain, from which a matrix of weighted intermacrostate MFPTs can be obtained. We also propose an approximation to the weighted-MFPT matrix in the strongly metastable limit. Inversion of the weighted-MFPT matrix, which is better conditioned than the matrices that must be inverted in alternative dimensionality reduction schemes, then yields the optimal reduced Markov chain. The superior numerical stability of the GT approach therefore enables us to realize optimal Markovian coarse-graining of systems with rare event dynamics.

EspcTM: Kinetic Transition Network Based on Trajectory Mapping in Effective Energy Rescaling Space

The transition network provides a key to reveal the thermodynamic and kinetic properties of biomolecular systems. In this paper, we introduce a new method, named effective energy rescaling space trajectory mapping (EspcTM), to detect metastable states and construct transition networks based on the simulation trajectories of the complex biomolecular system. It mapped simulation trajectories into an orthogonal function space, whose bases were rescaled by effective energy, and clustered the interrelation between these trajectories to locate metastable states. By using the EspcTM method, we identified the metastable states and elucidated interstate transition kinetics of a Brownian particle and a dodecapeptide. It was found that the scaling parameters of effective energy also provided a clue to the dominating factors in dynamics. We believe that the EspcTM method is a useful tool for the studies of dynamics of the complex system and may provide new insight into the understanding of thermodynamics and kinetics of biomolecular systems.

The Objectivity of Organizational Functions

We critique the organizational account of biological functions by showing how its basis in the closure of constraints fails to be objective. While the account treats constraints as objective features of physical systems, the number and relationship of potential constraints are subject to potentially arbitrary redescription by investigators. For example, we show that self-maintaining systems such as candle flames can realize closure on a more thorough analysis of the case, contradicting the claim that these "simple" systems lack functional organization. This also raises problems for Moreno and Mossio's associated theory of biological autonomy, which asserts that living beings are distinguished by their possession of a closed system of constraints that channel and regulate their metabolic processes.

Investigation of noncovalent interactions between peptides with potential intrinsic sequence patterns by mass spectrometry

RATIONALE

The conformation of a protein largely depends on the interactions between peptides. Specific and intrinsic sequence peptide patterns, such as DNA double helix backbones, may be present in proteins. A computational statistical deep learning method has supported this assumption, but it has not been experimentally proven. Mass spectrometry, as a fast and accurate experimental method, could be used to evaluate the interaction of biomolecules. The results would be of great value for further study of the mechanism of protein folding.

METHODS

Several potential intrinsic peptides were chosen by the deep learning method, including seven groups of pentapeptides and five groups of nonapeptides. The noncovalent interactions between mixed polypeptides were investigated by electrospray ionization mass spectrometry (ESI-MS) in full-scan and collision-induced dissociation (CID) modes. Molecular dynamics and molecular mechanics Poisson-Boltzmann surface area (MD-MM/PBSA) analyses were also performed to support the results.

RESULTS

The ESI-MS spectra showed that 11 of the 12 groups of mixed polypeptides formed binary and ternary complexes with relatively high stability. The binding between nonapeptide groups was stronger than that between pentapeptide groups according to the relative intensity. The binding energies calculated by the MM/PBSA binding energy tool also provided strong evidence for the combination of the complexes. Electrostatic interactions, hydrophobic interactions, and van der Waals forces were thought to stabilize the complexes according to the binding models.

CONCLUSIONS

The results implied the formation of stable complexes between polypeptides and identified their noncovalent interactions, proving that specific sequences and combinations with relatively strong binding ability exist in potential intrinsic sequences of peptides in protein structures.

Global and Kinetic Profiles of Substrate Diffusion in Candida antarctica Lipase B: Molecular Dynamics with the Markov-State Model

Profiling substrate diffusion pathways with kinetic information, which accounts for the dynamic nature of enzyme-substrate interaction, can enable molecular reengineering of enzymes and process optimization of enzymatic catalysis. Candida antarctica lipase B (CALB) is extensively used for producing various chemicals because of its rich catalytic mechanisms, broad substrate spectrum, thermal stability, and tolerance to organic solvents. In this study, an all-atom molecular dynamics (MD) combined with Markov-state models (MSMs) implemented in pyEMMA was proposed to simulate diffusion pathways of 4-nitrophenyl ester (4NPE), a commonly used substrate, from the surface into the active site of CALB. Six important metastable conformations of CALB were identified in the diffusion process, including a closed state. An induced-fit mechanism incorporating multiple pathways with molecular information was proposed, which might find unprecedented applications for the rational design of lipase for green catalysis.

Are RNA networks scale-free?

A network is scale-free if its connectivity density function is proportional to a power-law distribution. It has been suggested that scale-free networks may provide an explanation for the robustness observed in certain physical and biological phenomena, since the presence of a few highly connected hub nodes and a large number of small-degree nodes may provide alternate paths between any two nodes on average—such robustness has been suggested in studies of metabolic networks, gene interaction networks and protein folding. A theoretical justification for why many networks appear to be scale-free has been provided by Barabási and Albert, who argue that expanding networks, in which new nodes are preferentially attached to highly connected nodes, tend to be scale-free. In this paper, we provide the first efficient algorithm to compute the connectivity density function for the ensemble of all homopolymer secondary structures of a user-specified length—a highly nontrivial result, since the exponential size of such networks precludes their enumeration. Since existent power-law fitting software, such as powerlaw, cannot be used to determine a power-law fit for our exponentially large RNA connectivity data, we also implement efficient code to compute the maximum likelihood estimate for the power-law scaling factor and associated Kolmogorov–Smirnov p value. Hypothesis tests strongly indicate that homopolymer RNA secondary structure networks are not scale-free; moreover, this appears to be the case for real (non-homopolymer) RNA networks.

Explicit Characterization of the Free-Energy Landscape of a Protein in the Space of All Its Cα Carbons

By using an approach that allows computing the free energy in high-dimensional spaces together with a clustering technique capable of identifying kinetic attractors stabilized by conformational disorder, we analyze a molecular dynamics trajectory of the Villin headpiece from Lindorff-Larsen, K.; et al. How fast-folding proteins fold. Science 2011, 334, 517-520. We compute its free-energy landscape in the space of all its C_α carbons. This landscape has the shape of a 12-dimensional funnel with the free energy decreasing monotonically as a function of the native contacts. There are no significant folding barriers. The funnel can be partitioned in five regions, three mainly folded and two unfolded, which behave as Markov states. The slowest relaxation time among these states corresponds to the folding transition. The second slowest time is only twice smaller and corresponds to a transition within the unfolded state. This indicates that the unfolded part of the funnel has a nontrivial shape, which induces a sizable kinetic barrier between disordered states.

Mean first passage times reconstruct the slowest relaxations in potential energy landscapes of nanoclusters

Relaxation modes are the collective modes in which all probability deviations from equilibrium states decay with the same relaxation rates. In contrast, a first passage time is the required time for arriving for the first time from one state to another. In this paper, we discuss how and why the slowest relaxation rates of relaxation modes are reconstructed from the first passage times. As an illustrative model, we use a continuous-time Markov state model of vacancy diffusion in KCl nanoclusters. Using this model, we reveal that all characteristics of the relaxations in KCl nanoclusters come from the fact that they are hybrids of two kinetically different regions of the fast surface and slow bulk diffusions. The origin of the different diffusivities turns out to come from the heterogeneity of the activation energies on the potential energy landscapes. We also develop a stationary population method to compute the mean first passage times as mean times required for pair annihilations of particle-hole pairs, which enables us to obtain the symmetric results of relaxation rates under the exchange of the sinks and the sources. With this symmetric method, we finally show why the slowest relaxation times can be reconstructed from the mean first passage times.

The dynamic conformational landscape of the protein methyltransferase SETD8

Elucidating the conformational heterogeneity of proteins is essential for understanding protein function and developing exogenous ligands. With the rapid development of experimental and computational methods, it is of great interest to integrate these approaches to illuminate the conformational landscapes of target proteins. SETD8 is a protein lysine methyltransferase (PKMT), which functions in vivo via the methylation of histone and nonhistone targets. Utilizing covalent inhibitors and depleting native ligands to trap hidden conformational states, we obtained diverse X-ray structures of SETD8. These structures were used to seed distributed atomistic molecular dynamics simulations that generated a total of six milliseconds of trajectory data. Markov state models, built via an automated machine learning approach and corroborated experimentally, reveal how slow conformational motions and conformational states are relevant to catalysis. These findings provide molecular insight on enzymatic catalysis and allosteric mechanisms of a PKMT via its detailed conformational landscape.

Our cells contain thousands of proteins that perform many different tasks. Such tasks often involve significant changes in the shape of a protein that allow it to interact with other proteins or ligands. Understanding these shape changes can be an essential step for predicting and manipulating how proteins work or designing new drugs. Some changes in protein shape happen quickly, whereas others take longer. Existing experimental approaches generally only capture some, but not all, of the different shapes an individual protein adopts.

A family of proteins known as protein lysine methyltransferases (PKMTs) help to regulate the activities of other proteins by adding small tags called methyl groups to specific positions on their target proteins. PKMTs play important roles in many life processes including in activating genes, maintaining stem cells and controlling how organs develop.

It is important for cells to properly control the activity of PKMTs because too much, or too little, activity can promote cancers and neurological diseases. For example, genetic mutations that increase the levels of a PKMT known as SETD8 appear to promote the progression of some breast cancers and childhood leukemia. There is a pressing need to develop new drugs that can inhibit SETD8 and other PKMTs in human patients. However, these efforts are hindered by the lack of understanding of exactly how the shape of PKMT proteins change as they operate in cells.

Chen, Wiewiora et al. used a technique called X-ray crystallography to generate structural models of the human SETD8 protein in the presence or absence of native or foreign ligands. These models were used to develop computer simulations of how the shape of SETD8 changes as it operates. Further computational analysis and laboratory experiments revealed how slow changes in the shape of SETD8 contribute to the ability of the protein to attach methyl groups to other proteins.

This work is a significant stepping-stone to developing a complete model of how the SETD8 protein works, as well as understanding how genetic mutations may affect the protein’s role in the body. The next step is to refine the model by integrating data from other approaches including biophysical models and mathematical calculations of the energy associated with the shape changes, with a long-term goal to better understand and then manipulate the function of SETD8.

Commonly used FRET fluorophores promote collapse of an otherwise disordered protein

The dimensions that unfolded proteins, including intrinsically disordered proteins (IDPs), adopt in the absence of denaturant remain controversial. We developed an analysis procedure for small-angle X-ray scattering (SAXS) profiles and used it to demonstrate that even relatively hydrophobic IDPs remain nearly as expanded in water as they are in high denaturant concentrations. In contrast, as demonstrated here, most fluorescence resonance energy transfer (FRET) measurements have indicated that relatively hydrophobic IDPs contract significantly in the absence of denaturant. We use two independent approaches to further explore this controversy. First, using SAXS we show that fluorophores employed in FRET can contribute to the observed discrepancy. Specifically, we find that addition of Alexa-488 to a normally expanded IDP causes contraction by an additional 15%, a value in reasonable accord with the contraction reported in FRET-based studies. Second, using our simulations and analysis procedure to accurately extract both the radius of gyration (R_g) and end-to-end distance (R_ee) from SAXS profiles, we tested the recent suggestion that FRET and SAXS results can be reconciled if the R_g and R_ee are "uncoupled" (i.e., no longer simply proportional), in contrast to the case for random walk homopolymers. We find, however, that even for unfolded proteins, these two measures of unfolded state dimensions remain proportional. Together, these results suggest that improved analysis procedures and a correction for significant, fluorophore-driven interactions are sufficient to reconcile prior SAXS and FRET studies, thus providing a unified picture of the nature of unfolded polypeptide chains in the absence of denaturant.

Enspara: Modeling molecular ensembles with scalable data structures and parallel computing

Markov state models (MSMs) are quantitative models of protein dynamics that are useful for uncovering the structural fluctuations that proteins undergo, as well as the mechanisms of these conformational changes. Given the enormity of conformational space, there has been ongoing interest in identifying a small number of states that capture the essential features of a protein. Generally, this is achieved by making assumptions about the properties of relevant features-for example, that the most important features are those that change slowly. An alternative strategy is to keep as many degrees of freedom as possible and subsequently learn from the model which of the features are most important. In these larger models, however, traditional approaches quickly become computationally intractable. In this paper, we present enspara, a library for working with MSMs that provides several novel algorithms and specialized data structures that dramatically improve the scalability of traditional MSM methods. This includes ragged arrays for minimizing memory requirements, message passing interface-parallelized implementations of compute-intensive operations, and a flexible framework for model construction and analysis.

Choice of Adaptive Sampling Strategy Impacts State Discovery, Transition Probabilities, and the Apparent Mechanism of Conformational Changes

Interest in atomically detailed simulations has grown significantly with recent advances in computational hardware and Markov state modeling (MSM) methods, yet outstanding questions remain that hinder their widespread adoption. Namely, how do alternative sampling strategies explore conformational space and how might this influence predictions generated from the data? Here, we seek to answer these questions for four commonly used sampling methods: (1) a single long simulation, (2) many short simulations run in parallel, (3) adaptive sampling, and (4) our recently developed goal-oriented sampling algorithm, FAST. We first develop a theoretical framework for analytically calculating the probability of discovering select states on simple landscapes, where we uncover the drastic effects of varying the number and length of simulations. We then use kinetic Monte Carlo simulations on a variety of physically inspired landscapes to characterize the probability of discovering particular states and transition pathways for each of the four methods. Consistently, we find that FAST simulations discover each target state with the highest probability, while traversing realistic pathways. Furthermore, we uncover the potential pathology that short parallel simulations sometimes predict an incorrect transition pathway by crossing large energy barriers that long simulations would typically circumnavigate. We refer to this pathology as "pathway tunneling". To protect against this phenomenon when using adaptive-sampling and FAST simulations, we introduce the FAST-string method. This method enhances sampling along the highest-flux transition paths to refine an MSMs transition probabilities and discriminate between competing pathways. Additionally, we compare the performance of a variety of MSM estimators in describing accurate thermodynamics and kinetics. For adaptive sampling, we recommend simply normalizing the transition counts out of each state after adding small pseudocounts to avoid creating sources or sinks. Lastly, we evaluate whether our insights from simple landscapes hold for all-atom molecular dynamics simulations of the folding of the λ-repressor protein. Remarkably, we find that FAST-contacts predicts the same folding pathway as a set of long simulations but with orders of magnitude less simulation time.

Small-World Networks and RNA Secondary Structures

Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \cal S}_n}$$ \end{document} denote the network of all RNA secondary structures of length n, in which undirected edges exist between structures s, t such that t is obtained from s by the addition, removal, or shift of a single base pair. Using context-free grammars, generating functions, and complex analysis, we show that the asymptotic average degree is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$O ( n )$$ \end{document} , and that the asymptotic clustering coefficient is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$O ( 1 / n )$$ \end{document} , from which it follows that the family \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \cal S}_n}$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$n = 1 , 2 , 3 , \ldots$$ \end{document} of secondary structure networks is not small world.

Universal folding pathways of polyhedron nets

Low-dimensional objects such as molecular strands, ladders, and sheets have intrinsic features that affect their propensity to fold into 3D objects. Understanding this relationship remains a challenge for de novo design of functional structures. Using molecular dynamics simulations, we investigate the refolding of the 24 possible 2D unfoldings ("nets") of the three simplest Platonic shapes and demonstrate that attributes of a net's topology-net compactness and leaves on the cutting graph-correlate with thermodynamic folding propensity. To explain these correlations we exhaustively enumerate the pathways followed by nets during folding and identify a crossover temperature [Formula: see text] below which nets fold via nonnative contacts (bonds must break before the net can fold completely) and above which nets fold via native contacts (newly formed bonds are also present in the folded structure). Folding above [Formula: see text] shows a universal balance between reduction of entropy via the elimination of internal degrees of freedom when bonds are formed and gain in potential energy via local, cooperative edge binding. Exploiting this universality, we devised a numerical method to efficiently compute all high-temperature folding pathways for any net, allowing us to predict, among the combined 86,760 nets for the remaining Platonic solids, those with highest folding propensity. Our results provide a general heuristic for the design of 2D objects to stochastically fold into target 3D geometries and suggest a mechanism by which geometry and folding propensity are related above [Formula: see text], where native bonds dominate folding.

A Probabilistic Framework for Constructing Temporal Relations in Replica Exchange Molecular Trajectories

Knowledge of the structure and dynamics of biomolecules is essential for elucidating the underlying mechanisms of biological processes. Given the stochastic nature of many biological processes, like protein unfolding, it is almost impossible that two independent simulations will generate the exact same sequence of events, which makes direct analysis of simulations difficult. Statistical models like Markov chains, transition networks, etc. help in shedding some light on the mechanistic nature of such processes by predicting long-time dynamics of these systems from short simulations. However, such methods fall short in analyzing trajectories with partial or no temporal information, for example, replica exchange molecular dynamics or Monte Carlo simulations. In this work, we propose a probabilistic algorithm, borrowing concepts from graph theory and machine learning, to extract reactive pathways from molecular trajectories in the absence of temporal data. A suitable vector representation was chosen to represent each frame in the macromolecular trajectory (as a series of interaction and conformational energies), and dimensionality reduction was performed using principal component analysis (PCA). The trajectory was then clustered using a density-based clustering algorithm, where each cluster represents a metastable state on the potential energy surface (PES) of the biomolecule under study. A graph was created with these clusters as nodes with the edges learned using an iterative expectation maximization algorithm. The most reactive path is conceived as the widest path along this graph. We have tested our method on RNA hairpin unfolding trajectory in aqueous urea solution. Our method makes the understanding of the mechanism of unfolding in the RNA hairpin molecule more tractable. As this method does not rely on temporal data, it can be used to analyze trajectories from Monte Carlo sampling techniques and replica exchange molecular dynamics (REMD).

Slowest kinetic modes revealed by metabasin renormalization

Understanding the slowest relaxations of complex systems, such as relaxation of glass-forming materials, diffusion in nanoclusters, and folding of biomolecules, is important for physics, chemistry, and biology. For a kinetic system, the relaxation modes are determined by diagonalizing its transition rate matrix. However, for realistic systems of interest, numerical diagonalization, as well as extracting physical understanding from the diagonalization results, is difficult due to the high dimensionality. Here, we develop an alternative and generally applicable method of extracting the long-time scale relaxation dynamics by combining the metabasin analysis of Okushima et al. [Phys. Rev. E 80, 036112 (2009)PLEEE81539-375510.1103/PhysRevE.80.036112] and a Jacobi method. We test the method on an illustrative model of a four-funnel model, for which we obtain a renormalized kinematic equation of much lower dimension sufficient for determining slow relaxation modes precisely. The method is successfully applied to the vacancy transport problem in ionic nanoparticles [Niiyama et al., Chem. Phys. Lett. 654, 52 (2016)CHPLBC0009-261410.1016/j.cplett.2016.04.088], allowing a clear physical interpretation that the final relaxation consists of two successive, characteristic processes.

Computational Exploration of Conformational Transitions in Protein Drug Targets

Protein drug targets vary from highly structured to completely disordered; either way dynamics governs function. Hence, understanding the dynamical aspects of how protein targets function can enable improved interventions with drug molecules. Computational approaches offer highly detailed structural models of protein dynamics which are becoming more predictive as model quality and sampling power improve. However, the most advanced and popular models still have errors owing to imperfect parameter sets and often cannot access longer timescales of many crucial biological processes. Experimental approaches offer more certainty but can struggle to detect and measure lightly populated conformations of target proteins and subtle allostery. An emerging solution is to integrate available experimental data into advanced molecular simulations. In the future, molecular simulation in combination with experimental data may be able to offer detailed models of important drug targets such that improved functional mechanisms or selectivity can be accessed.

Phosphatidylserine-Induced Conformational Modulation of Immune Cell Exhaustion-Associated Receptor TIM3

In the face of chronic cancers and protracted viral infections, human immune cells are known to adopt an exhausted state in which their effector functions are lost. In recent years, a number of inhibitory receptors have been connected to the immune cell exhaustion phenotype; furthermore, ligands capable of activating these receptors have been discovered. The molecular mechanisms by which these ligands affect the exhausted states of immune cells, however, are largely unknown. Here, we present the results of molecular dynamics simulations of one potential exhaustion-associated system: the complex of human inhibitory receptor TIM3 (hTIM3) and its ligand phosphatidylserine (PSF). We find that PSF fundamentally alters the electrostatic environment within hTIM3's Ca²⁺ binding site, facilitating the formation of a salt bridge and freeing a tyrosine-containing strand. This liberated tyrosine then collapses into a nearby hydrophobic pocket, anchoring a modified conformational ensemble typified by a β-strand rearrangement. The "electrostatic switching/hydrophobic anchoring" mechanism of conformational modulation reported here suggests a new type of process by which TIM3 activation might be achieved. This work also highlights strategies by which PSF-mediated conformational change could be controlled, either through administration of small molecules, execution of mutations, or modification of receptor phosphorylation states.

The N-Terminal Domain of Ribosomal Protein L9 Folds via a Diffuse and Delocalized Transition State

The N-terminal domain of L9 (NTL9) is a 56-residue mixed α-β protein that lacks disulfides, does not bind cofactors, and folds reversibly. NTL9 has been widely used as a model system for experimental and computational studies of protein folding and for investigations of the unfolded state. The role of side-chain interactions in the folding of NTL9 is probed by mutational analysis. ϕ-values, which represent the ratio of the change in the log of the folding rate upon mutation to the change in the log of the equilibrium constant for folding, are reported for 25 point mutations and 15 double mutants. All ϕ-values are small, with an average over all sites probed of only 0.19 and a largest value of 0.4. The effect of modulating unfolded-state interactions is studied by measuring ϕ-values in second- site mutants and under solvent conditions that perturb unfolded-state energetics in a defined way. Neither of these alterations significantly affects the distribution of ϕ-values. The results, combined with those of earlier studies that probe the role of hydrogen-bond formation in folding and the burial of surface area, reveal that the transition state for folding contains extensive backbone structure and buries a significant fraction of hydrophobic surface area, but lacks well developed side-chain-side-chain interactions. The folding transition state for NTL9 does not contain a specific "nucleus" consisting of a few key residues; rather, it involves extensive backbone hydrogen bonding and partially formed structure delocalized over almost the entire domain. The potential generality of these observations is discussed.

RNA folding kinetics using Monte Carlo and Gillespie algorithms

RNA secondary structure folding kinetics is known to be important for the biological function of certain processes, such as the hok/sok system in E. coli. Although linear algebra provides an exact computational solution of secondary structure folding kinetics with respect to the Turner energy model for tiny ([Formula: see text]20 nt) RNA sequences, the folding kinetics for larger sequences can only be approximated by binning structures into macrostates in a coarse-grained model, or by repeatedly simulating secondary structure folding with either the Monte Carlo algorithm or the Gillespie algorithm. Here we investigate the relation between the Monte Carlo algorithm and the Gillespie algorithm. We prove that asymptotically, the expected time for a K-step trajectory of the Monte Carlo algorithm is equal to [Formula: see text] times that of the Gillespie algorithm, where [Formula: see text] denotes the Boltzmann expected network degree. If the network is regular (i.e. every node has the same degree), then the mean first passage time (MFPT) computed by the Monte Carlo algorithm is equal to MFPT computed by the Gillespie algorithm multiplied by [Formula: see text]; however, this is not true for non-regular networks. In particular, RNA secondary structure folding kinetics, as computed by the Monte Carlo algorithm, is not equal to the folding kinetics, as computed by the Gillespie algorithm, although the mean first passage times are roughly correlated. Simulation software for RNA secondary structure folding according to the Monte Carlo and Gillespie algorithms is publicly available, as is our software to compute the expected degree of the network of secondary structures of a given RNA sequence-see http://bioinformatics.bc.edu/clote/RNAexpNumNbors .

Exploring the free energy gain of phase separation via Markov state modeling

The gain of free energy upon unmixing is determined via application of Markov state modeling (MSM), using an Ising model with a fixed number of up- and down-spins. MSM yields reasonable estimates of the free energies. However, a closer look reveals significant differences that point to residual non-Markovian effects. These non-Markovian effects are rather unexpected since the typical criteria to study the quality of Markovianity indicate complete Markovian behavior. We identify the sparse connectivity between different Markov states as a likely reason for the observed bias. By studying a simple five state model system, we can analytically elucidate different sources of the bias and thus explain the different deviations that were observed for the Ising model. Based on this insight, we can modify the determination of the count matrix in the MSM approach. In this way, the estimation of the free energy is significantly improved.

Imaging Metastable States and Transitions in Proteins by Trajectory Map

It has been a long-standing and intriguing issue to develop robust methods to identify metastable states and interstate transitions from simulations or experimental data to understand the functional conformational changes of proteins. It is usually hard to define the complicated boundaries of the states in the conformational space using most of the existing methods, and they often lead to parameter-sensitive results. Here, we present a new approach, visualized Trajectory Map (vTM), to identify the metastable states and the rare interstate transitions, by considering both the conformational similarity and the temporal successiveness of conformations. The vTM is able to give a nonambiguous description of slow dynamics. The case study of a β-hairpin peptide shows that the vTM can reveal the states and transitions from all-atom MD trajectory data even when a single observable (i.e, one-dimensional reaction coordinate) is used. We also use the vTM to refine the folding/unfolding mechanism of HP35 in explicit water by analyzing a 125 μs all-atom MD trajectory and obtain folding/unfolding rates of about 1/μs, which are in good agreement with the experimental values.

Properties of kinetic transition networks for atomic clusters and glassy solids

Small-world and scale-free properties are analysed for kinetic transition networks of clusters and glassy systems.

Knotting and unknotting of a protein in single molecule experiments

Spontaneous folding of a polypeptide chain into a knotted structure remains one of the most puzzling and fascinating features of protein folding. The folding of knotted proteins is on the timescale of minutes and thus hard to reproduce with atomistic simulations that have been able to reproduce features of ultrafast folding in great detail. Furthermore, it is generally not possible to control the topology of the unfolded state. Single-molecule force spectroscopy is an ideal tool for overcoming this problem: by variation of pulling directions, we controlled the knotting topology of the unfolded state of the 52-knotted protein ubiquitin C-terminal hydrolase isoenzyme L1 (UCH-L1) and have therefore been able to quantify the influence of knotting on its folding rate. Here, we provide direct evidence that a threading event associated with formation of either a 31 or 52 knot, or a step closely associated with it, significantly slows down the folding of UCH-L1. The results of the optical tweezers experiments highlight the complex nature of the folding pathway, many additional intermediate structures being detected that cannot be resolved by intrinsic fluorescence. Mechanical stretching of knotted proteins is also of importance for understanding the possible implications of knots in proteins for cellular degradation. Compared with a simple 31 knot, we measure a significantly larger size for the 52 knot in the unfolded state that can be further tightened with higher forces. Our results highlight the potential difficulties in degrading a 52 knot compared with a 31 knot.