# Spectral Rate Theory for Two-State Kinetics.

@article{Prinz2014SpectralRT, title={Spectral Rate Theory for Two-State Kinetics.}, author={Jan-Hendrik Prinz and John D. Chodera and Frank No{\'e}}, journal={Physical review. X}, year={2014}, volume={4 1} }

Classical rate theories often fail in cases where the observable(s) or order parameter(s) used is a poor reaction coordinate or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate…

## Figures from this paper

## 21 Citations

### Projected metastable Markov processes and their estimation with observable operator models.

- PhysicsThe Journal of chemical physics
- 2015

It is shown that the observed dynamics of a PMM can be exactly described by an observable operator model (OOM) and derive aPMM estimator based on the OOM learning.

### Identification of slow molecular order parameters for Markov model construction.

- Computer ScienceThe Journal of chemical physics
- 2013

The variational principle of conformation dynamics is used to derive an optimal way of identifying the "slow subspace" of a large set of prior order parameters - either generic internal coordinates or a user-defined set of parameters.

### Kinetic distance and kinetic maps from molecular dynamics simulation.

- BiologyJournal of chemical theory and computation
- 2015

A kinetic distance metric for irreducible Markov processes that quantifies how slowly molecular conformations interconvert is defined and the total kinetic variance (TKV) is an excellent indicator of model quality and can be used to rank different input feature sets.

### Investigating Molecular Kinetics by Variationally Optimized Diffusion Maps.

- Computer ScienceJournal of chemical theory and computation
- 2015

This work has applied the Diffusion Map approach with the recently proposed Variational Approach for Conformation Dynamics to ultra-long MD simulations of the Fip35 WW domain and found that the first DCs are indeed a good approximation to the true reaction coordinates of the system, but they could be further improved using the VAC.

### Hierarchical Time-Lagged Independent Component Analysis: Computing Slow Modes and Reaction Coordinates for Large Molecular Systems.

- PhysicsJournal of chemical theory and computation
- 2016

A hierarchical TICA (hTICA) method is derived that approximates the full TICA solution by a hierarchical, divide-and-conquer calculation and is used to identify previously unknown relaxation processes in the bovine pancreatic trypsin inhibitor.

### Galerkin approximation of dynamical quantities using trajectory data.

- Computer ScienceThe Journal of chemical physics
- 2019

This work presents a general framework for calculating dynamical quantities by approximating boundary value problems using dynamical operators with a Galerkin expansion, and shows that delay embedding can reduce the information lost when projecting the system's dynamics for model construction.

### A Bayesian Nonparametric Model for Spectral Estimation of Metastable Systems

- Computer ScienceUAI
- 2014

This paper analyzes the difficulties of spectral estimation for infinite HMMs, and presents a new nonparametric model called stick-breaking half-weighted model (SB-HWM) that defines a sparse prior of eigenvalues and can be applied to Bayesian inference of dominant eigenpairs of metastable systems in a non Parametric manner.

### Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules.

- PhysicsThe Journal of chemical physics
- 2013

It is shown how various molecular observables of interest that are often computed from MSMs can be computed from HMMs/PMMs, and a practically feasible approximation via Hidden Markov Models (HMMs) is derived.

### Dynamic neutron scattering from conformational dynamics. I. Theory and Markov models.

- PhysicsThe Journal of chemical physics
- 2013

It is shown how Markov models can be used to connect structural changes observed in molecular dynamics simulation directly to the relaxation processes probed by scattering experiments, using a conformational dynamics theory of dynamical neutron and X-ray scattering.

### Addressing the Embeddability Problem in Transition Rate Estimation

- Computer SciencebioRxiv
- 2019

Six different methods to overcome the embeddability problem of MSM are described and a one-dimensional toy model is used to show the workings of these methods and discuss the robustness of each method in terms of its dependence in lag time and trajectory length.

## References

SHOWING 1-10 OF 67 REFERENCES

### A robust approach to estimating rates from time-correlation functions

- Chemistry
- 2011

A modified version of this theory which does not require numerical derivatives, allowing rate constants to be robustly estimated from the time-correlation function directly directly is presented.

### Markov models of molecular kinetics: generation and validation.

- PhysicsThe Journal of chemical physics
- 2011

An upper bound for the approximation error made by modeling molecular dynamics with a Markov chain is described and it is shown that this error can be made arbitrarily small with surprisingly little effort.

### Bayesian hidden Markov model analysis of single-molecule force spectroscopy: Characterizing kinetics under measurement uncertainty

- Physics
- 2011

The utility of a Bayesian extension that allows the experimental uncertainties to be directly quantified and build in detailed balance to reduce uncertainty through physical constraints is illustrated incharacterizing the three-state kinetic behavior of an RNA hairpin in a stationary optical trap.

### Trajectory analysis of a kinetic theory for isomerization dynamics in condensed phases

- Physics
- 1979

We use an exact microscopic formalism to study the implications of a stochastic model of isomerization dynamics in liquids. In the model, a reaction coordinate moves in a multistable potential and is…

### Statistical mechanics of isomerization dynamics in liquids and the transition state approximation

- Chemistry
- 1978

In this article, time correlation function methods are used to discuss classical isomerization reactions of small nonrigid molecules in liquid solvents. Molecular expressions are derived for a…

### A Variational Approach to Modeling Slow Processes in Stochastic Dynamical Systems

- MathematicsMultiscale Model. Simul.
- 2013

A variational principle is derived that is based on the maximization of a Rayleigh coefficient that forms a basis for the development of adaptive and efficient computational algorithms for simulating and analyzing metastable Markov processes while avoiding the sampling problem.

### On the Approximation Quality of Markov State Models

- MathematicsMultiscale Model. Simul.
- 2010

A sharp error bound for the difference in propagation of probability densities between the MSM and the original process on long timescales is provided for a rather general class of Markov processes ranging from diffusions in energy landscapes to Markov jump processes on large discrete spaces.

### Using the histogram test to quantify reaction coordinate error.

- Computer ScienceThe Journal of chemical physics
- 2006

The mean and variance of the intrinsic committor distribution is derived in terms of the mean and volatility of the histogram of Committor estimates, enabling the first quantitative calculations of reaction coordinate error for complex systems.

### Rate theories for biologists

- BiologyQuarterly Reviews of Biophysics
- 2010

An attempt is made to clear up a number of misconceptions in the literature regarding popular rate theories, including the appearance of Planck's constant in the transition-state theory and the Smoluchowski result as an upper limit for protein–protein and protein–DNA association rate constants.

### Analysis of single-molecule FRET trajectories using hidden Markov modeling.

- PhysicsBiophysical journal
- 2006

An analysis scheme is developed that casts single-molecule time-binned FRET trajectories as hidden Markov processes, allowing one to determine, based on probability alone, the most likely FRET-value distributions of states and their interconversion rates while simultaneously determining the mostlikely time sequence of underlying states for each trajectory.