Markov State Research Articles

The αC-β4 loop controls the allosteric cooperativity between nucleotide and substrate in the catalytic subunit of protein kinase A

Allosteric cooperativity between ATP and substrates is a prominent characteristic of the cAMP-dependent catalytic subunit of protein kinase A (PKA-C). This long-range synergistic action is involved in substrate recognition and fidelity, and it may also regulate PKA’s association with regulatory subunits and other binding partners. To date, a complete understanding of this intramolecular mechanism is still lacking. Here, we integrated NMR(Nuclear Magnetic Resonance)-restrained molecular dynamics simulations and a Markov State Model to characterize the free energy landscape and conformational transitions of PKA-C. We found that the apoenzyme populates a broad free energy basin featuring a conformational ensemble of the active state of PKA-C (ground state) and other basins with lower populations (excited states). The first excited state corresponds to a previously characterized inactive state of PKA-C with the αC helix swinging outward. The second excited state displays a disrupted hydrophobic packing around the regulatory (R) spine, with a flipped configuration of the F100 and F102 residues at the αC-β4 loop. We validated the second excited state by analyzing the F100A mutant of PKA-C, assessing its structural response to ATP and substrate binding. While PKA-CF100A preserves its catalytic efficiency with Kemptide, this mutation rearranges the αC-β4 loop conformation, interrupting the coupling of the two lobes and abolishing the allosteric binding cooperativity. The highly conserved αC-β4 loop emerges as a pivotal element to control the synergistic binding of nucleotide and substrate, explaining how mutations or insertions near or within this motif affect the function and drug sensitivity in homologous kinases.

Pathways and mechanism of MRTX1133 binding to KRAS G12D elucidated by molecular dynamics simulations and Markov state models

KRAS G12D is the most common oncogenic mutation identified in several types of cancer. Therefore, design of inhibitors targeting KRAS G12D represents a promising strategy for anticancer therapy. MRTX1133 is a highly potent inhibitor (approximate experiment Kd ≈ 0.0002 nM) of KRAS G12D and is currently in Phase 1/2 study, however, pathways of the compound binding to KRAS G12D has remained unknown, and the mechanism underlying the complicated dynamic process are challenging to capture experimentally, which hinder the structure-based anti-cancer drug design. Here, using MRTX1133 as a probe, unbiased molecular dynamics (MD) was used to simulate the process of MRTX1133 spontaneously binding to KRAS G12D. In six of 42 independent MD simulation (a total of 99 μs), MRTX1133 was observed to successfully associate with KRAS G12D. The kinetically metastable states refer to the potential pathways of MRTX1133 binding to KRAS G12D were revealed by Markov state models (MSM) analysis. Additionally, 8 key residues that are essential for MRTX1133 recognition and tight binding at the preferred low energy states were identified by MM/GBSA analysis. In sum, this study provides a new perspective on understanding the pathways and mechanism of MRTX1133 binding to KRAS G12D.

The αC-β4 loop controls the allosteric cooperativity between nucleotide and substrate in the catalytic subunit of protein kinase A.

Allosteric cooperativity between ATP and substrates is a prominent characteristic of the cAMP-dependent catalytic subunit of protein kinase A (PKA-C). This long-range synergistic action is involved in substrate recognition and fidelity, and it may also regulate PKA's association with regulatory subunits and other binding partners. To date, a complete understanding of this intramolecular mechanism is still lacking. Here, we integrated NMR(Nuclear Magnetic Resonance)-restrained molecular dynamics simulations and a Markov State Model to characterize the free energy landscape and conformational transitions of PKA-C. We found that the apoenzyme populates a broad free energy basin featuring a conformational ensemble of the active state of PKA-C (ground state) and other basins with lower populations (excited states). The first excited state corresponds to a previously characterized inactive state of PKA-C with the αC helix swinging outward. The second excited state displays a disrupted hydrophobic packing around the regulatory (R) spine, with a flipped configuration of the F100 and F102 residues at the αC-β4 loop. We validated the second excited state by analyzing the F100A mutant of PKA-C, assessing its structural response to ATP and substrate binding. While PKA-CF100A preserves its catalytic efficiency with Kemptide, this mutation rearranges the αC-β4 loop conformation, interrupting the coupling of the two lobes and abolishing the allosteric binding cooperativity. The highly conserved αC-β4 loop emerges as a pivotal element to control the synergistic binding of nucleotide and substrate, explaining how mutations or insertions near or within this motif affect the function and drug sensitivity in homologous kinases.

Markov State Models Reveal How Folding Kinetics Influence Absorption Spectra of Foldamers.

Self-assembly of platinum(II) complex foldamers is an essential approach to fabricate advanced luminescent materials. However, a comprehensive understanding of folding kinetics and their absorption spectra remains elusive. By constructing Markov state models (MSMs) from large-scale molecular dynamics simulations, we reveal that two largely similar dinuclear alknylplatinum(II) terpyridine foldamers, Pt-PEG and Pt-PE with slightly different bridges, exhibit distinctive folding kinetics. Particularly, Pt-PEG bears bridge-dominant, plane-dominant, and cooperative pathways, while Pt-PE only prefers the plane-dominant pathway. Such preference originates from their difference in intrabridge electrostatic interactions, leading to contrastive distributions of metastable states. We also found that the bridge-dominant pathway for Pt-PEG becomes more favorable when lowering the temperature. Interestingly, based on the comprehensive conformation ensembles from our MSMs, we reveal the conformation-dependent absorption spectra of Pt-PEG and Pt-PE. Our theoretical spectra not only align with experimental results but also reveal the contributions of diverse conformations to the overall absorption bands explicitly, facilitating the rational design of stimuli-responsive smart luminescent molecules.

Identification of a Cryptic Pocket in Methionine Aminopeptidase-II Using Adaptive Bandit Molecular Dynamics Simulations and Markov State Models.

Methionine aminopeptidase-II (MetAP-II) is a metalloprotease, primarily responsible for the cotranslational removal of the N-terminal initiator methionine from the nascent polypeptide chain during protein synthesis. MetAP-II has been implicated in angiogenesis and endothelial cell proliferation and is therefore considered a validated target for cancer therapeutics. However, there is no effective drug available against MetAP-II. In this study, we employ Adaptive Bandit molecular dynamics simulations to investigate the structural dynamics of the apo and ligand-bound MetAP-II. Our results focus on the dynamic behavior of the disordered loop that is not resolved in most of the crystal structures. Further analysis of the conformational flexibility of the disordered loop reveals a hidden cryptic pocket that is predicted to be potentially druggable. The network analysis indicates that the disordered loop region has a direct signaling route to the active site. These findings highlight a new way to target MetAP-II by designing inhibitors for the allosteric site within this disordered loop region.

Markov decision process framework of optimal energy dispatch in a smart data center with uninterruptible power supplies

This paper examines the use of an Uninterruptible Power Supply (UPS) to enhance the operational efficiency of data centers. It focuses on developing an optimal energy scheduling strategy for a data center equipped with UPS, using the Markov decision process (MDP) framework. The MDP framework simulates the decision-making process involved in minimizing energy costs. Each unit’s available power output in the data center is treated as a Markov state, taking into account the uncertainty associated with renewable distributed generation. This uncertainty drives the system to transition to other Markov states in subsequent decision times. A recursive optimization model is established for each Markov state at each decision time to guide state-based operations, which includes determining the unit output while considering both current and future costs. The challenge of dealing with high dimensionality, arising from a substantial number of states and actions in the model, is effectively addressed by adopting an approximate dynamic programming (ADP) method. This approach incorporates decision-state and forwards dynamic algorithms to tackle the complexity of the MDP-based model. By employing ADP, the computational burden is reduced, enabling efficient and practical solutions to be obtained.

Interplay between phosphorylation and oligomerization tunes the conformational ensemble of SWEET transporters.

SWEET sugar transporters are desirable biotechnological targets for improving plant growth. One engineering strategy includes modulating how SWEET transporters are regulated. Phosphorylation and oligomerization have been shown to positively regulate SWEET function, leading to increased sugar transport activity. However, constitutive phosphorylation may not be beneficial to plant health under basal conditions. Structural and mechanistic understanding of the interplay between phosphorylation and oligomerization in functional regulation of SWEETs remains limited. Using extensive molecular dynamics simulations coupled with Markov state models, we demonstrate the thermodynamic and kinetic effects of SWEET phosphorylation and oligomerization using OsSWEET2b as a model. We report that the beneficial effects of these SWEET regulatory mechanisms bias outward-facing states and improved extracellular gating, which complement published experimental findings. Our results offer molecular insights to SWEET regulation and may guide engineering strategies throughout the SWEET transport family.

Implementation of Girsanov Reweighting in OpenMM and Deeptime.

Classical molecular dynamics (MD) simulations provide invaluable insights into complex molecular systems but face limitations in capturing phenomena occurring on time scales beyond their reach. To bridge this gap, various enhanced sampling techniques have been developed, which are complemented by reweighting techniques to recover the unbiased dynamics. Girsanov reweighting is a reweighting technique that reweights simulation paths, generated by a stochastic MD integrator, without evoking an effective model of the dynamics. Instead, it calculates the relative path probability density at the time resolution of the MD integrator. Efficient implementation of Girsanov reweighting requires that the reweighting factors are calculated on-the-fly during the simulations and thus needs to be implemented within the MD integrator. Here, we present a comprehensive guide for implementing Girsanov reweighting into MD simulations. We demonstrate the implementation in the MD simulation package OpenMM by extending the library openmmtools. Additionally, we implemented a reweighted Markov state model estimator within the time series analysis package Deeptime.

Information Bottleneck Approach for Markov Model Construction.

Markov state models (MSMs) have proven valuable in studying the dynamics of protein conformational changes via statistical analysis of molecular dynamics simulations. In MSMs, the complex configuration space is coarse-grained into conformational states, with dynamics modeled by a series of Markovian transitions among these states at discrete lag times. Constructing the Markovian model at a specific lag time necessitates defining states that circumvent significant internal energy barriers, enabling internal dynamics relaxation within the lag time. This process effectively coarse-grains time and space, integrating out rapid motions within metastable states. Thus, MSMs possess a multiresolution nature, where the granularity of states can be adjusted according to the time-resolution, offering flexibility in capturing system dynamics. This work introduces a continuous embedding approach for molecular conformations using the state predictive information bottleneck (SPIB), a framework that unifies dimensionality reduction and state space partitioning via a continuous, machine learned basis set. Without explicit optimization of the VAMP-based scores, SPIB demonstrates state-of-the-art performance in identifying slow dynamical processes and constructing predictive multiresolution Markovian models. Through applications to well-validated mini-proteins, SPIB showcases unique advantages compared to competing methods. It autonomously and self-consistently adjusts the number of metastable states based on a specified minimal time resolution, eliminating the need for manual tuning. While maintaining efficacy in dynamical properties, SPIB excels in accurately distinguishing metastable states and capturing numerous well-populated macrostates. This contrasts with existing VAMP-based methods, which often emphasize slow dynamics at the expense of incorporating numerous sparsely populated states. Furthermore, SPIB's ability to learn a low-dimensional continuous embedding of the underlying MSMs enhances the interpretation of dynamic pathways. With these benefits, we propose SPIB as an easy-to-implement methodology for end-to-end MSM construction.

Computational Study on the Wealth Conformations of Hepatitis C Virus Envelope Glycoprotein 2 and Role of Its Glycan Coat.

Hepatitis C virus (HCV) is a major cause of chronic liver disease and hepatocellular carcinoma. Antibody development efforts mainly revolve around HCV envelope glycoprotein 2 (E2), which mediates host cell entry by interacting with several cell surface receptors, including CD81. We still have limited knowledge about the structural ensembles and the dynamic behavior of both the CD81 binding sites and the glycans on E2. Here, multiple microsecond-long, all-atom molecular dynamics (MD) simulations, as well as a Markov state model (MSM), were performed to provide an atomistic perspective on the dynamic nature of E2 and its glycans. End-to-end accessibility analyses outline a complete overview of the vulnerabilities of the glycan shield of E2, which may be exploited in therapeutic efforts. Additionally, the Markov state model built from the simulation maps four metastable states for AS412 and three metastable states for the front layer in CD81 binding sites, while binding with HEPC3 would induce a conformation selection for both of them. Overall, this work presents hitherto unseen functional and structural insights into E2 and its glycan coat, providing a new theoretical foundation to control the conformational plasticity of E2 that could be harnessed for vaccine development.

L1-gain control for 2D delayed positive continuous Markov jumping systems

This study is focused on analyzing the L1-stochastic internal stability and the design of an L1-gain controller for two-dimensional (2D) continuous positive Markov jumping systems (PMJSs) with constraints on the inputs and states. An algorithm is developed to explicitly determine the state feedback control law with the optional L1-gain performance. First, by constructing a co-positive stochastic Lyapunov function for the positive system and establishing the equation for the Markov states' mathematical expectation, several sufficient and necessary conditions for L1-stochastic internal stability are derived. This analysis indicates that 2D PMJSs with directional delays are influenced by sizes of the system matrices and magnitude of delays, and transition matrix. Second, a method for calculating the L1-gain is formulated utilizing linear programming (LP), thus, an L1-gain controller is designed to guarantee that the closed-loop system is positive and L1-stochastically internally stable, while achieving optimal L1-gain performance. Two numerical and practical examples are considered, and the influence of delays and transition probability matrices on the L1-gain is specified to validate the preceding theoretical findings.

Accounting for fast vs slow exchange in single molecule FRET experiments reveals hidden conformational states.

Proteins are dynamic systems whose structural preferences determine their function. Unfortunately, building atomically detailed models of protein structural ensembles remains challenging, limiting our understanding of the relationships between sequence, structure, and function. Combining single molecule Förster resonance energy transfer (smFRET) experiments with molecular dynamics simulations could provide experimentally grounded, all-atom models of a protein's structural ensemble. However, agreement between the two techniques is often insufficient to achieve this goal. Here, we explore whether accounting for important experimental details like averaging across structures sampled during a given smFRET measurement is responsible for this apparent discrepancy. We present an approach to account for this time-averaging by leveraging the kinetic information available from Markov state models of a protein's dynamics. This allows us to accurately assess which timescales are averaged during an experiment. We find this approach significantly improves agreement between simulations and experiments in proteins with varying degrees of dynamics, including the well-ordered protein T4 lysozyme, the partially disordered protein apolipoprotein E (ApoE), and a disordered amyloid protein (Aβ40). We find evidence for hidden states that are not apparent in smFRET experiments because of time averaging with other structures, akin to states in fast exchange in NMR, and evaluate different force fields. Finally, we show how remaining discrepancies between computations and experiments can be used to guide additional simulations and build structural models for states that were previously unaccounted for. We expect our approach will enable combining simulations and experiments to understand the link between sequence, structure, and function in many settings.

Risk quantification as an epidemiological analysis strategy: Analysis and application to bud rot in oil palm

AbstractBud rot (BR), caused by Phytophthora palmivora, limits growth and development of oil palms in the American continent, particularly in Colombia. Due to the absence of systematic epidemiological analyses and determination of risk factors associated with BR, this study aimed to employ deterministic mathematical models and frequentist and Bayesian statistical methods to quantify the genetic response and edaphoclimatic variables as risk indicators of BR. From 2011 to 2014, the severity of BR in oil palm crops was recorded monthly in two locations: Tumaco (hot spot) and Villavicencio (cold spot), determining the edaphoclimatic variables at each site. Using the area under the disease progress stairs curve, temporal models were applied to determine the rate of disease development (R0) to quantify risk at locality and genotype levels. Subsequently, the observed intensity and severity levels of BR were adjusted to models such as survival curves, Cox proportional hazard risk and transition probabilities, or Markov states, with the aim of quantifying and characterizing risk factors associated with genotype and edaphoclimatic variables at each location. The risk in Tumaco was three times higher than in Villavicencio, and the R0 values were differential by genotype, with Tenera more susceptible than Hybrids. Moreover, the risk of BR increased when periods of 2–3 months occurred with successive instances of precipitation and relative humidity greater than 150 mm/month and 90%, respectively, and when manganese and zinc levels were below optimal. This approach allowed us to characterize epidemiological factors that cause plant diseases, allowing quantification of the risk of BR.

Constrained Markov Decision Processes with Non-constant Discount Factor

This paper studies constrained Markov decision processes under the total expected discounted cost optimality criterion, with a state-action dependent discount factor that may take any value between zero and one. Both the state and the action space are assumed to be Borel spaces. By using the linear programming approach, consisting in stating the control problem as a linear problem on a set of occupation measures, we show the existence of an optimal stationary Markov policy. Our results are based on the study of both weak-strong topologies in the space of occupation measures and Young measures in the space of Markov policies.

CoVAMPnet: Comparative Markov State Analysis for Studying Effects of Drug Candidates on Disordered Biomolecules

CoVAMPnet: Comparative Markov State Analysis for Studying Effects of Drug Candidates on Disordered Biomolecules

Secondary Structure Stabilization of Macrocyclic Antimicrobial Peptides via Cross-Link Swapping.

Lactam cross-links have been employed to stabilize the helical secondary structure and enhance the activity and physiological stability of antimicrobial peptides; however, stabilization of β-sheets via lactamization has not been observed. In the present study, lactams between the side chains of C- and N-terminal residues have been used to stabilize the β-sheet conformation in a short ten-residue analogue of chicken angiogenin-4. Designed using a combination of molecular dynamics simulations and Markov state models, the lactam cross-linked peptides are shown to adopt stabilized β-sheet conformations consistent with simulated structures. Replacement of the peptide side-chain Cys-Cys disulfide by a lactam cross-link enhanced the broad-spectrum antibacterial activity compared to the parent peptide and exhibited greater propensity to induce proinflammatory activity in macrophages. The combination of molecular simulations and conformational and biological analyses of the synthetic peptides provides a useful paradigm for the rational design of therapeutically active peptides with constrained β-sheet structures.

Unsupervised detection of large-scale weather patterns in the northern hemisphere via Markov State Modelling: from blockings to teleconnections

Detecting recurrent weather patterns and understanding the transitions between such regimes are key to advancing our knowledge of the low-frequency variability of the atmosphere and have important implications in terms of weather and climate-related risks. We adopt an analysis pipeline inspired by Markov State Modelling and detect in an unsupervised manner the dominant winter mid-latitude Northern Hemisphere weather patterns in the Atlantic and Pacific sectors. The daily 500 hPa geopotential height fields are first classified in about 200 microstates. The weather dynamics are then represented on the basis of these microstates and the slowest decaying modes are identified from the spectral properties of the transition probability matrix. These modes are defined on the basis of the nonlinear dynamical processes of the system and not as tentative metastable states, as often done in Markov state analysis. When focusing on a shifting longitudinal window of 60∘, we find that the longitude-dependent estimate of the longest relaxation time is smaller where stronger baroclinic activity is found. In the Atlantic and Pacific sectors slow relaxation processes are mainly related to transitions between blocked regimes and zonal flow. We also find strong evidence of a dynamical regime associated with the simultaneous Atlantic-Pacific blocking. When the analysis is performed on a broader geographical region of the Atlantic sector, we discover that the slowest relaxation modes of the system are associated with transitions between dynamical regimes that resemble teleconnection patterns like the North Atlantic Oscillation and weather regimes like the Scandinavian and Greenland blocking, yet have a much stronger dynamical foundation than classical methods based e.g. on EOF analysis. Our method clarifies that, as a result of the lack of a time-scale separation in the atmospheric variability of the mid-latitudes, there is no clear-cut way to represent the atmospheric dynamics in terms of few, well-defined modes of variability. The approach proposed here can be seamlessly applied across different regions of the globe for detecting regional modes of variability, and has a great potential for intercomparing climate models and for assessing the impact of climate change on the low-frequency variability of the atmosphere.

Using Gaussian accelerated molecular dynamics combined with Markov state models to explore the mechanism of action of new oral inhibitors on Complex I

In this study, our focus was on investigating H-1,2,3-triazole derivative HP661 as a novel and highly efficient oral OXPHOS inhibitor, with its molecular-level inhibitory mechanism not yet fully understood. We selected the ND1, NDUFS2, and NDUFS7 subunits of Mitochondrial Complex I as the receptor proteins and established three systems for comparative analysis: protein-IACS-010759, protein-lead compound 10, and protein-HP661. Through extensive analysis involving 500 ns Gaussian molecular dynamics simulations, we gained insights into these systems. Additionally, we constructed a Markov State Models to examine changes in secondary structures during the motion processes.The research findings suggest that the inhibitor HP661 enhances the extensibility and hydrophilicity of the receptor protein. Furthermore, HP661 induces the unwinding of the α-helical structure in the region of residues 726–730. Notably, key roles were identified for Met37, Phe53, and Pro212 in the binding of various inhibitors. In conclusion, we delved into the potential molecular mechanisms of triazole derivative HP661 in inhibiting Complex I. These research outcomes provide crucial information for a deeper understanding of the mechanisms underlying OXPHOS inhibition, offering valuable theoretical support for drug development and disease treatment design.

MGF-based SNC for stationary independent Markovian processes with localized application of martingales

MGF-based SNC for stationary independent Markovian processes with localized application of martingales

Conformational Dynamics in Corynebacterium glutamicum Diaminopimelate Epimerase: Insights from Ligand Parameterization, Atomistic Simulation, and Markov State Modeling.

The microbial enzyme diaminopimelate epimerase (DapF), a vital enzyme in the lysine biosynthetic pathway, catalyzes the conversion of L, L-diaminopimelate (L, L-DAP) to D, L-diaminopimelate (D, L-DAP) using a catalytic cysteine dyad with one cysteine in thiol state and another in thiolate. Under oxidizing conditions, the catalytic cysteines of apo DapF form a disulfide bond that alters the structure and function of DapF. Given its potential as a target for antimicrobial resistance treatments, understanding DapF's functional dynamics is imperative. In the present work, we employ microsecond-scale all-atom molecular dynamics simulations of product-bound DapF and apo-DapF under oxidized and reduced conditions. We employ a polarized charge model for the ligand and the active site residues, which was necessary to preserve the electrostatic environment in the active site and retain the ligand in the active site. The product-bound DapF and apo-DapF in oxidized and reduced conditions exhibit a closed, semi-open, and open conformation, respectively, as identified using the internal coordinates of the dimeric enzyme and the principal component analysis. The conformational switch is guided by the dynamic catalytic (DC) loop, loop II, and loop III movements in the active site. The time scale of the close-to-open conformational transition is estimated to be 0.8 μs through Markov state modeling (MSM) and transition path theory (TPT). The present study explains the role of various active site residues and loops in ligand binding and protein dynamics in the DapF enzyme under different redox conditions. Such information will be helpful in future inhibitor design studies targeting the DapF enzyme.