Markovian state models

Singhal, N. Snow, C. D. Pande, V. S., Using path sampling to build better Markovian state models predicting the folding rate and mechanism of a tryptophan zipper beta hairpin, J. Chem. Phys. Jul 2004,121, 415—425. 

Markovian State Models and Stochastic Road Maps... 

The PPTIS method coarse-grains the transition pathways to a one-dimensional Markovian state model . In principle, there is no need to limit oneself to only one dimension. It is possible to coarse-grain the folding pathways using a Markovian state network model, also called stochastic road map [91-93] or... 

Fig. 8 Time-evolving XT state populations obtained from quantum dynamical (MCTDH) calculations for the 2-state model of Sec. 5.1, for different levels of the HEP hierarchy as compared with the full-dimensional (24-mode) result. Panel (a) shows the H approximation (3 modes) as compared with the II1 1 1 approximation (6 modes) and the II1 2 1 approximation (6 modes). Panel (b) presents a comparison with the approximation including a Markovian closure as described...

Markovian queueing models, 2158-2159 M/G/1 queue, 2159-2160 notation for, 2157-2158 steady state, rate of convergence to, 2162-2163... 

The configurational structure (stereoregularity) of 1-butene and of the higher polyolefins up to 1-nonene has been studied by NMR spectroscopy in solution [38, 39], interpreted with the aid of chemical shift calculations, consideration of the y effect and of the rotational isomeric state model of Flory. The evaluation of the results favors the bicatalytic sites model of polymerization [40] over simple Markovian statistics. In contrast to polypropylene, side-chain conformation also has to be considered. Comparison with alkane model compounds indicates that in meso-units of poly-1-butene, trans conformation of backbone is less favored than in isotactic polypropylene because of contiguous ethyl group interactions. Introduction of racemic units in both... 

There is a fundamental relationship between d-dimensional PCA and d + 1)-dimensional Ising spin models. The simplest way to make the connection is to think of the successive temporal layers of the PCA as successive hyper-planes of the next higher-dimensional spatial lattice. Because the PCA rules (at least the set of PCA rules that we will be dealing with) are (1) Markovian (i.e. the probability of a state at time t + T depends only on a set of states at time t, and (2) local, one can always define a Hamiltonian on the higher-dimensioned spatial lattice such that the thermodynamic weight of a configuration 5j,( is equal to the probability of a corresponding space-time history Si t). ... 

Under current treatment of statistical method a set of the states of the Markovian stochastic process describing the ensemble of macromolecules with labeled units can be not only discrete but also continuous. So, for instance, when the description of the products of living anionic copolymerization is performed within the framework of a terminal model the role of the label characterizing the state of a monomeric unit is played by the moment when this unit forms in the course of a macroradical growth [25]. 

This is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

In this Section we introduce a stochastic alternative model for surface reactions. As an application we will focus on the formation of NH3 which is described below, equations (9.1.72) to (9.1.76). It is expected that these stochastic systems are well-suited for the description via master equations using the Markovian behaviour of the systems under study. In such a representation an infinite set of master equations for the distribution functions describing the state of the surface and of pairs of surface sites (and so on) arises. As it was told earlier, this set cannot be solved analytically and must be truncated at a certain level. The resulting equations can be solved exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. This procedure is well-suited for the description of surface reaction systems which includes such elementary steps as adsorption, diffusion, reaction and desorption.The numerical part needs only a very small amount of computer time compared to MC or CA simulations. 

First of all, we define the transition rates for our stochastic model using an ansatz of Kawasaki [39, 40]. In the following we use the abbreviation X for an initial state (07 for mono- and oion for bimolecular steps), Y for a final state (ct[ for mono- and a[a n for bimolecular steps) and Z for the states of the neighbourhood ( cr f 1 for mono- and a -1 a -1 for bimolecular steps). If we study the system in which the neighbourhood is fixed we observe a relaxation process in a very small area. We introduce the normalized probability W(X) and the corresponding rates 8.(X —tY Z). For this (reversible) process we write down the following Markovian master equation... 

Computer simulations (see Section III) reveal, however, that in the liquid state the variable velocity is neither Markovian nor Gaussian, thereby making it necessary to discard modeling approaches that are linear in nature. 

These results leave several basic questions open How to derive a non-Markovian master equation (ME) for arbitrary time-dependent driving and modulation of a thermally relaxing two-level system Would the two-level system (TLS) model hold at all for modulation rates, that are comparable to the TLS transition frequency u)a (between its states e) and g)) which may invalidate the standard rotating-wave approximation (RWA), [to hen-Tannoudji 1992] Would temperature effects, which are known to incur upward g) —> e) transitions, [Lifshitz 1980], further complicate the dynamics and perhaps hinder the suppression of decay How to control decay in an efficient, optimal fashion We address these questions by outlining the derivation of a ME of a TLS that is coupled to an arbitrary bath and is driven by an arbitrary time-dependent field. 

A series of probable transitions between states can be described with the Markov chain. A Markovian stochastic process is memoryless, and this is illustrated subsequently. We generate a sequence of random variables, (yo, yi, yi, ), so that each time t > 0, the next state yt+i would be sampled from a distribution P(y,+ily,), which would depend only on the current state of the chain, y,. Thus, given y, the next state y,+i would not depend additionally on the history of the chain (yo, yi, yi,---, y i). The name Markov chain is used to describe this sequence, and the transition kernel of the chain is i (.l.) does not depend on t if we assume that the chain is time homogeneous. A detail description of the Markov model is provided in Chapter 26. 

A series of probable transitions between states can be described with Markov modeling. The natural course of a disease, for example, can be viewed for an individual subject as a sequence of certain states of health (12). A Markovian stochastic process is memoryless. To predict what the future state will be, knowledge of the current state is sufficient and is independent of where the process has been in the past. This is termed the strong Markov property (13). 

Markov models are used to describe disease as a series of probable transitions between health states. The methodology has considerable appeal for use in phar-macometrics since it offers a method to evaluate patient compliance with prescribed medication regimen, multiple health states simultaneously, and transitions between different sleep stages. An overview of the Markov model is provided together with the Markovian assumption. The most commonly used form of the Markov model, the discrete-time Markov model, is described as well as its application in the mixed effects modeling setting. The chapter concludes with a discussion of a hybrid Markov mixed effects and proportional odds model used to characterize an adverse effect that lends itself to this combination modeling approach. 

Here, we used the shorthand notation qt to denote the value of q at time t as q t) in an attempt to simplify the subsequent expressions. Here, t is used to denote the traditional use of the time instant in the Markovian modeling literature. The notation P x y) indicates the conditional probability of observing x, premised on the presence of y. The change of the states is captured by transition probabilities, ajj, given by... 

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