Markov State Research Articles

On the local limit theorems for linear sequences of lower psi-mixing Markov chains

In this paper we investigate the local limit theorem for partial sums of linear sequences of the form Xj=∑i∈Zaiξj−i. Here (ai)i∈Z is a sequence of constants satisfying ∑i∈Zai2<∞ and (ξi)i∈Z are functions of a stationary Markov chain with mean zero and finite second moment. The Markov chain is assumed to satisfy one-sided lower psi-mixing condition.

Unraveling motion in proteins by combining NMR relaxometry and molecular dynamics simulations: A case study on ubiquitin.

Nuclear magnetic resonance (NMR) relaxation experiments shine light onto the dynamics of molecular systems in the picosecond to millisecond timescales. As these methods cannot provide an atomically resolved view of the motion of atoms, functional groups, or domains giving rise to such signals, relaxation techniques have been combined with molecular dynamics (MD) simulations to obtain mechanistic descriptions and gain insights into the functional role of side chain or domain motion. In this work, we present a comparison of five computational methods that permit the joint analysis of MD simulations and NMR relaxation experiments. We discuss their relative strengths and areas of applicability and demonstrate how they may be utilized to interpret the dynamics in MD simulations with the small protein ubiquitin as a test system. We focus on the aliphatic side chains given the rigidity of the backbone of this protein. We find encouraging agreement between experiment, Markov state models built in the χ1/χ2 rotamer space of isoleucine residues, explicit rotamer jump models, and a decomposition of the motion using ROMANCE. These methods allow us to ascribe the dynamics to specific rotamer jumps. Simulations with eight different combinations of force field and water model highlight how the different metrics may be employed to pinpoint force field deficiencies. Furthermore, the presented comparison offers a perspective on the utility of NMR relaxation to serve as validation data for the prediction of kinetics by state-of-the-art biomolecular force fields.

Learning Markovian dynamics with spectral maps.

The long-time behavior of many complex molecular systems can often be described by Markovian dynamics in a slow subspace spanned by a few reaction coordinates referred to as collective variables (CVs). However, determining CVs poses a fundamental challenge in chemical physics. Depending on intuition or trial and error to construct CVs can lead to non-Markovian dynamics with long memory effects, hindering analysis. To address this problem, we continue to develop a recently introduced deep-learning technique called spectral map [J. Rydzewski, J. Phys. Chem. Lett. 14, 5216-5220 (2023)]. Spectral map learns slow CVs by maximizing a spectral gap of a Markov transition matrix describing anisotropic diffusion. Here, to represent heterogeneous and multiscale free-energy landscapes with spectral map, we implement an adaptive algorithm to estimate transition probabilities. Through a Markov state model analysis, we validate that spectral map learns slow CVs related to the dominant relaxation timescales and discerns between long-lived metastable states.

Forward starting options pricing under a regime-switching jump-diffusion model with Wishart stochastic volatility and stochastic interest rate

Vanilla options become effective immediately after they are entered, while some exotic options will only come to effective some time after they are bought or sold. Forward starting options are one kind of such exotic options started actually at some pre-specified future date. This paper presents an extension of regime-switching jump diffusion model, in which the parameters are driven by a continuous time and stationary Markov chain on a finite state space, by introducing the Wishart process into the instantaneous variance-covariance matrix of the risky asset price and stochastic interest rate. We derive the discounted conditional joint characteristic function and the forward characteristic function of the log-asset price and its the instantaneous variance-covariance process, and thereby the price of forward starting options are well evaluated by the probabilistic approach combined with the Fourier-cosine (COS) method. We also provide efficient Monte Carlo simulation of this proposed model, and simulated solutions to forward starting options pricing within a two-state regime switching framework. Numerical results show that the COS method is accurate and efficient for pricing forward starting options. Finally, we analyse impacts of some main parameters (especially, parameters in the Wishart stochastic volatility) in this proposed model on option prices and Δ values. Also, we consider the forward implied volatility. Furthermore, the forward starting options under the regime-switching jump diffusion model with Wishart stochastic volatility and stochastic interest rate which we derived are more generalized than those recently appeared in the derivatives pricing literature, and thus have wider application.

The potential public health impact of the respiratory syncytial virus prefusion F protein vaccine in people aged ≥60 years in Japan: results of a Markov model analysis

ABSTRACT Background Respiratory syncytial virus (RSV), a common respiratory pathogen, can lead to severe symptoms, especially in older adults (OA). A recently developed RSV prefusion F protein (RSVPreF3 OA) vaccine confers high protection against RSV lower respiratory tract disease (LRTD) over two full RSV seasons. The aim of this study was to assess the potential public health impact of RSVPreF3 OA vaccination in the Japanese OA population. Research design and methods A static Markov model was used to estimate the number of symptomatic RSV cases, hospitalizations and deaths in the Japanese population aged ≥ 60 years over a 3-year time horizon. Japan-specific RSV epidemiology and healthcare resource use parameters were used; vaccine efficacy was derived from a phase 3 randomized study (AReSVi-006, NCT04886596). Vaccination coverage was set to 50%. Results Without vaccination, >5 million RSV acute respiratory illness (ARI) would occur (2.5 million LRTD and 2.8 million upper respiratory tract infections) leading to ~ 3.5 million outpatient visits, >534,000 hospitalizations and ~ 25,500 RSV-related deaths over 3 years. Vaccination could prevent > 1 million RSV-ARI cases, 728,000 outpatient visits, 143,000 hospitalizations and 6,840 RSV-related deaths. Conclusions RSVPreF3 OA vaccination is projected to have a substantial public health impact by reducing RSV-related morbidity and mortality in the OA population.

On Inactivation of the Coronavirus Main Protease.

A deeper understanding of the inactive conformations of the coronavirus main protease (MPro) could inform the design of allosteric drugs. Based on extensive molecular dynamics simulations, we built a Markov State Model to investigate structural changes that can inactivate the SARS-CoV-2 MPro. In a subset of structures, one subunit of the homodimer assumes an inactive conformation that resembles an inactive crystal structure. However, contradicting the widely held half-of-sites activity hypothesis, the most populated enzyme structures have two active subunits. We then used transition path theory (TPT) and the Jensen-Shannon Divergence (JSD) to pinpoint residues involved in the inactivation process. A π stack between Phe140 and His163 is a key feature that can distinguish active and inactive conformations of MPro. Each subunit has unique inactive conformations stabilized by π stacking interactions involving residues Phe140, Tyr118, His163, and His172, a hydrogen bonding network centered around His163 and His172, and a modified network of interactions in the dimer interface. The importance of these residues in maintaining an active structure explains the sensitivity of enzymatic activity to site-directed mutagenesis.

Cost-Effectiveness of Late Endovascular Thrombectomy vs. Best Medical Management in a Clinical Trial Setting and Real-World Setting.

To assess cost-effectiveness of late time-window endovascular treatment (EVT) in a clinical trial setting and a "real-world" setting. Data are from the randomized ESCAPE trial and a prospective cohort study (ESCAPE-LATE). Anterior circulation large vessel occlusion patients presenting > 6 hours from last-known-well were included, whereby collateral status was an inclusion criterion for ESCAPE but not ESCAPE-LATE. A Markov state transition model was built to estimate lifetime costs and quality-adjusted life-years (QALYs) for EVT in addition to best medical care vs. best medical care only in a clinical trial setting (comparing ESCAPE-EVT to ESCAPE control arm patients) and a "real-world" setting (comparing ESCAPE-LATE to ESCAPE control arm patients). We performed an unadjusted analysis, using 90-day modified Rankin Scale(mRS) scores as model input and analysis adjusted for baseline factors. Acceptability of EVT was calculated using upper/lower willingness-to-pay thresholds of 100,000 USD/50,000 USD/QALY. Two-hundred and forty-nine patients were included (ESCAPE-LATE:n = 200, ESCAPE EVT-arm:n = 29, ESCAPE control-arm:n = 20). Late EVT in addition to best medical care was cost effective in the unadjusted analysis both in the clinical trial and real-world setting, with acceptability 96.6%-99.0%. After adjusting for differences in baseline variables between the groups, late EVT was marginally cost effective in the clinical trial setting (acceptability:49.9%-61.6%), but not the "real-world" setting (acceptability:32.9%-42.6%). EVT for LVO-patients presenting beyond 6 hours was cost effective in the clinical trial setting and "real-world" setting, although this was largely related to baseline patient differences favoring the "real-world" EVT group. After adjusting for these, EVT benefit was reduced in the trial setting, and absent in the real-world setting.

Accurate estimates of dynamical statistics using memory.

Many chemical reactions and molecular processes occur on time scales that are significantly longer than those accessible by direct simulations. One successful approach to estimating dynamical statistics for such processes is to use many short time series of observations of the system to construct a Markov state model, which approximates the dynamics of the system as memoryless transitions between a set of discrete states. The dynamical Galerkin approximation (DGA) is a closely related framework for estimating dynamical statistics, such as committors and mean first passage times, by approximating solutions to their equations with a projection onto a basis. Because the projected dynamics are generally not memoryless, the Markov approximation can result in significant systematic errors. Inspired by quasi-Markov state models, which employ the generalized master equation to encode memory resulting from the projection, we reformulate DGA to account for memory and analyze its performance on two systems: a two-dimensional triple well and the AIB9 peptide. We demonstrate that our method is robust to the choice of basis and can decrease the time series length required to obtain accurate kinetics by an order of magnitude.

Machine Learning Deciphered Molecular Mechanistics with Accurate Kinetic and Thermodynamic Prediction.

Time-lagged independent component analysis (tICA) and the Markov state model (MSM) have been extensively employed for extracting conformational dynamics and kinetic community networks from unbiased trajectory ensembles. However, these techniques may not be the optimal choice for elucidating transition mechanisms within low-dimensional representations, especially for intricate biosystems. Unraveling the association mechanism in such complex systems always necessitates permutations of several essential independent components or collective variables, a process that is inherently obscure and may require empirical knowledge for selection. To address these challenges, we have implemented an integrated unsupervised dimension reduction model: uniform manifold approximation and projection (UMAP) with hierarchy density-based spatial clustering of applications with noise (HDBSCAN). This approach effectively generates low-dimensional configurational embeddings. The hierarchical application of this architecture, in conjunction with MSM, reveals global kinetic connectivity while identifying local conformational states. Consequently, our methodology establishes a multiscale mechanistic elucidation framework. Leveraging the benefits of the uniform sample distribution and a denoising approach, our model demonstrates robustness in preserving global and local data structures compared to traditional dimension reduction methods in the field of MD analysis area. The interpretability of hyperparameter selection and compatibility with downstream tasks are cross-validated across various simulation data sets, utilizing both computational evaluation metrics and experimental kinetic observables. Furthermore, the predicted Mcl1-BH3 association kinetics (0.76 s-1) is in close agreement with surface plasmon resonance experiments (0.12 s-1), affirming the plausibility of the identified pathway composed of representative conformations. We anticipate that the devised workflow will serve as a foundational framework for studying recognition patterns in complex biological systems. Its contributions extend to the exploration of protein functional dynamics and rational drug design, offering a potent avenue for advancing research in these domains.

Diffusive dynamics of a model protein chain in solution.

A Markov state model is a powerful tool that can be used to track the evolution of populations of configurations in an atomistic representation of a protein. For a coarse-grained linear chain model with discontinuous interactions, the transition rates among states that appear in the Markov model when the monomer dynamics is diffusive can be determined by computing the relative entropy of states and their mean first passage times, quantities that are unchanged by the specification of the energies of the relevant states. In this paper, we verify the folding dynamics described by a diffusive linear chain model of the crambin protein in three distinct solvent systems, each differing in complexity: a hard-sphere solvent, a solvent undergoing multi-particle collision dynamics, and an implicit solvent model. The predicted transition rates among configurations agree quantitatively with those observed in explicit molecular dynamics simulations for all three solvent models. These results suggest that the local monomer-monomer interactions provide sufficient friction for the monomer dynamics to be diffusive on timescales relevant to changes in conformation. Factors such as structural ordering and dynamic hydrodynamic effects appear to have minimal influence on transition rates within the studied solvent densities.

Optimal investor life cycle decisions with time-inconsistent preferences

We examine the lifetime decisions of an investor with time-inconsistent preferences, obtaining optimal time-consistent strategies for investment, consumption, and life insurance premiums explicitly under a stationary Markov perfect equilibrium framework. We find that the investor increases consumption to seek immediate gratification, and simultaneously increases life insurance purchasing to fulfill a legacy need. However, at a later stage in the life cycle, the investor confronts slower wealth accumulation and reduces consumption and life insurance purchasing accordingly. We also investigate the effects of naivete, degree of time inconsistency, risk preferences, legacy weight, insurance cost, death duty, and labor income on the investor's decision making. Our study reveals the balance of consumption, bequest, and terminal wealth motives in life cycle decisions featuring microfounded bequest utility and stochastic labor income.

The Role of Force Fields and Water Models in Protein Folding and Unfolding Dynamics.

Protein folding is a fascinating, not fully understood phenomenon in biology. Molecular dynamics (MD) simulations are an invaluable tool to study conformational changes in atomistic detail, including folding and unfolding processes of proteins. However, the accuracy of the conformational ensembles derived from MD simulations inevitably relies on the quality of the underlying force field in combination with the respective water model. Here, we investigate protein folding, unfolding, and misfolding of fast-folding proteins by examining different force fields with their recommended water models, i.e., ff14SB with the TIP3P model and ff19SB with the OPC model. To this end, we generated long conventional MD simulations highlighting the perks and pitfalls of these setups. Using Markov state models, we defined kinetically independent conformational substates and emphasized their distinct characteristics, as well as their corresponding state probabilities. Surprisingly, we found substantial differences in thermodynamics and kinetics of protein folding, depending on the combination of the protein force field and water model, originating primarily from the different water models. These results emphasize the importance of carefully choosing the force field and the respective water model as they determine the accuracy of the observed dynamics of folding events. Thus, the findings support the hypothesis that the water model is at least equally important as the force field and hence needs to be considered in future studies investigating protein dynamics and folding in all areas of biophysics.

Log-periodic oscillations as real-time signatures of hierarchical dynamics in proteins.

The time-dependent relaxation of a dynamical system may exhibit a power-law behavior that is superimposed by log-periodic oscillations. D. Sornette [Phys. Rep. 297, 239 (1998)] showed that this behavior can be explained by a discrete scale invariance of the system, which is associated with discrete and equidistant timescales on a logarithmic scale. Examples include such diverse fields as financial crashes, random diffusion, and quantum topological materials. Recent time-resolved experiments and molecular dynamics simulations suggest that discrete scale invariance may also apply to hierarchical dynamics in proteins, where several fast local conformational changes are a prerequisite for a slow global transition to occur. Employing entropy-based timescale analysis and Markov state modeling to a simple one-dimensional hierarchical model and biomolecular simulation data, it is found that hierarchical systems quite generally give rise to logarithmically spaced discrete timescales. By introducing a one-dimensional reaction coordinate that collectively accounts for the hierarchically coupled degrees of freedom, the free energy landscape exhibits a characteristic staircase shape with two metastable end states, which causes the log-periodic time evolution of the system. The period of the log-oscillations reflects the effective roughness of the energy landscape and can, in simple cases, be interpreted in terms of the barriers of the staircase landscape.

Toward physics-based precision medicine: Exploiting protein dynamics to design new therapeutics and interpret variants.

The goal of precision medicine is to utilize our knowledge of the molecular causes of disease to better diagnose and treat patients. However, there is a substantial mismatch between the small number of food and drug administration (FDA)-approved drugs and annotated coding variants compared to the needs of precision medicine. This review introduces the concept of physics-based precision medicine, a scalable framework that promises to improve our understanding of sequence-function relationships and accelerate drug discovery. We show that accounting for the ensemble of structures a protein adopts in solution with computer simulations overcomes many of the limitations imposed by assuming a single protein structure. We highlight studies of protein dynamics and recent methods for the analysis of structural ensembles. These studies demonstrate that differences in conformational distributions predict functional differences within protein families and between variants. Thanks to new computational tools that are providing unprecedented access to protein structural ensembles, this insight may enable accurate predictions of variant pathogenicity for entire libraries of variants. We further show that explicitly accounting for protein ensembles, with methods like alchemical free energy calculations or docking to Markov state models, can uncover novel lead compounds. To conclude, we demonstrate that cryptic pockets, or cavities absent in experimental structures, provide an avenue to target proteins that are currently considered undruggable. Taken together, our review provides a roadmap for the field of protein science to accelerate precision medicine.

Stable laws for random dynamical systems

Abstract In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $\nu $ . We consider a class of non-square-integrable observables $\phi $ , mostly of form $\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$ , where $x_0$ is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index $\alpha \in (0,2)$ . The two types of maps we concatenate are a class of piecewise $C^2$ expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and $\alpha $ , we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by $\nu $ . This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.

Global oil price and stock markets in oil exporting and European countries: Evidence during the Covid-19 and the Russia-Ukraine war

The relationship between oil price movements and stock markets during the COVID-19 pandemic and the geopolitical crisis like the ongoing Russian-Ukraine war is yet unexplored extensively. This study therefore examines the return-correlation effects of oil prices on stock markets and their spillover effects in oil-exporting and European countries using daily closing data. After estimating the GARCH process, we employ the static and dynamic Markov Switching model that allow the relationship between oil price and stock market to switch between two regimes coined the COVID-19 and the Russia-Ukraine war periods. The static model shows stock price returns to respond positively and significantly to oil price returns in Italy, Germany and the US during the Covid-19 period while the response is significantly positive only for US in the Russia-Ukraine war period. As regards the volatility spillover, significant spillover is found from stock to oil market for Nigeria, vice versa for Saudi Arabia and bi-directional volatility spillover found for the US, Italy and Germany during the COVID-19 period. The policy implication is that Nigeria and Saudi Arabia should prioritize financial policy and energy policy respectively while US, Italy and Germany should adopt policy coordination to stabilize oil-stock market volatility during low oil price period like the COVID-19 period.

Unraveling the Interplay of Extracellular Domain Conformational Changes and Parathyroid Hormone Type 1 Receptor Activation in Class B1 G Protein-Coupled Receptors: Integrating Enhanced Sampling Molecular Dynamics Simulations and Markov State Models.

Parathyroid hormone (PTH) type 1 receptor (PTH1R), as a typical class B1 G protein-coupled receptor (GPCR), is responsible for regulating bone turnover and maintaining calcium homeostasis, and its dysregulation has been implicated in the development of several diseases. The extracellular domain (ECD) of PTH1R is crucial for the recognition and binding of ligands, and the receptor may exhibit an autoinhibited state with the closure of the ECD in the absence of ligands. However, the correlation between ECD conformations and PTH1R activation remains unclear. Thus, this study combines enhanced sampling molecular dynamics (MD) simulations and Markov state models (MSMs) to reveal the possible relevance between the ECD conformations and the activation of PTH1R. First, 22 intermediate structures are generated from the autoinhibited state to the active state and conducted for 10 independent 200 ns simulations each. Then, the MSM is constructed based on the cumulative 44 μs simulations with six identified microstates. Finally, the potential interplay between ECD conformational changes and PTH1R activation as well as cryptic allosteric pockets in the intermediate states during receptor activation is revealed. Overall, our findings reveal that the activation of PTH1R has a specific correlation with ECD conformational changes and provide essential insights for GPCR biology and developing novel allosteric modulators targeting cryptic sites.

Free energy profile of the substrate-induced occlusion of the human serotonin transporter.

The serotonin transporter (SERT) is a member of the Solute Carrier 6 (SLC6) family and is responsible for maintaining the appropriate level of serotonin in the brain. Dysfunction of SERT has been linked to several neuropsychiatric disorders, including depression, anxiety and obsessive-compulsive disorder. Therefore, an in-depth understanding of the mechanism on an atomistic level, coupled with a quantification of transporter dynamics and the associated free energies is required. Here, we constructed Markov state models (MSMs) from extensive unbiased molecular dynamics simulations to quantify the free energy profile of serotonin (5HT) triggered SERT occlusion and explored the driving forces of the mechanism of occlusion. Our results reveal that SERT occludes via multiple intermediate conformations and show that the motion of occlusion is energetically downhill for the 5HT-bound transporter. Force distribution analyses show that the interactions of 5HT with the bundle domain are crucial. During occlusion, attractive forces steadily increase and pull on the bundle domain, which leads to SERT occlusion. Some interactions become repulsive upon full occlusion, suggesting that SERT creates pressure on 5HT to promote its movement towards the cytosol.

Identifying the generator matrix of a stationary Markov chain using partially observable data.

Given that most states in real-world systems are inaccessible, it is critical to study the inverse problem of an irreversibly stationary Markov chain regarding how a generator matrix can be identified using minimal observations. The hitting-time distribution of an irreversibly stationary Markov chain is first generalized from a reversible case. The hitting-time distribution is then decoded via the taboo rate, and the results show remarkably that under mild conditions, the generator matrix of a reversible Markov chain or a specific case of irreversibly stationary ones can be identified by utilizing observations from all leaves and two adjacent states in each cycle. Several algorithms are proposed for calculating the generator matrix accurately, and numerical examples are presented to confirm their validity and efficiency. An application to neurophysiology is provided to demonstrate the applicability of such statistics to real-world data. This means that partially observable data can be used to identify the generator matrix of a stationary Markov chain.

Cost-Effectiveness of Diffusion-Weighted Magnetic Resonance Imaging Versus Second-Look Surgery in Treating Cholesteatoma: A Modeling Study.

To evaluate whether canal wall-up (CWU) tympanomastoidectomy with diffusion-weighted magnetic resonance imaging (DW-MRI) is a cost-effective method of treating cholesteatoma compared with CWU with second-look surgery. Cost-effectiveness analysis was conducted using a Markov state transition model. The simulation model adhered to the Panel Recommendations on Cost-Effectiveness in Health and Medicine established by the US Public Health Service. One-way and Monte Carlo probability sensitivity analyses were conducted for validation. Tympanomastoidectomy with DW-MRI versus tympanomastoidectomy with second-look surgery. Effectiveness and health utility were measured using quality-adjusted life years (QALYs). Costs were derived from Medicare reimbursement using the perspective of the payer. Probabilities for outcomes and complications were taken from existing literature. Cost-effectiveness was assessed using the incremental cost-effectiveness ratio. With base case analysis, the total cost was $15,069 when treated with CWU and second-look surgery versus $13,126 when treated with CWU and DW-MRI. The second-look treatment pathway yielded 17.05 QALYs, whereas the DW-MRI pathway yielded 16.91 QALYs in terms of health benefit accrued across the lifetime of the patient. The cost-effectiveness incremental cost-effectiveness ratio was $21,800/QALY. Using the conventional $50,000 willingness-to-pay threshold, second-look surgery was the more cost-effective approach 63.7% of the time by simulation. Both treatment pathways were found to be cost-effective, with second-look surgery incrementally cost-effective 63.7% of the time. Assumptions were validated by one-way and Monte Carlo probability sensitivity analysis. There is ample variation in treatment pathways regarding usage of DW-MRI and second-look surgery for cholesteatoma. To evaluate the cost-effectiveness of DW-MRI and second-look surgery approaches, accounting for health-related quality-of-life outcomes and costs for the duration of the patient lifetimes. To inform the discussion on the treatment for cholesteatoma given emergent noninvasive technologies.Level of Evidence: Level III.Indicate IRB or IACUC: Exempt.