Discrete-state stochastic modeling

Lente proposed a discrete-state stochastic modeling approach in which chiral amplification could be described by a quadratic autocatalytic model without considering cross-inhibition [67,68]. However, the discrepancy between the usually employed deterministic kinetic approach, which reinforces the need for cross-inhibition, and the discrete-state stochastic approach is only apparent. The discrete approach considers the repetitive reproduction of single molecules which, in the case of a chiral system, obviously are individually all enantiomerically pure. Hence, basically no amplification of the ee occurs at all during the discrete scenario. It has been indicated that deter-... 

Continuous time discrete state stochastic models... 

Why are the arguments for the usual continuous time discrete state stochastic model so good for reaction kinetics here ... 

To define the problem of nonlinear filtering (Ristic, Arulampalam, Gordon, 2004), let us consider the state vector ii , where n is the dimension ofthe state vectorand k e N. Here the index k is attributed to a continuous-time instant tf.. The state vector evolves according to the following discrete-time stochastic model ... 

Discrete-state stochastic Markov jump model from MD trajectories... 

If we consider the example described at the beginning of this chapter, the element of study in stochastic modelling is the particle which moves in a trajectory where the local state of displacement is randomly chosen. The description for this discrete displacement and its associated general model, takes into consideration the... 

When a stochastic model is described by a continuous polystochastic process, the numerical transposition can be derived by the classical procedure that change the derivates to their discrete numerical expressions related with a space discretisation of the variables. An indirect method can be used with the recursion equations, which give the links between the elementary states of the process. 

One possibility to simplify a continuous stochastic system is the reduction to a description in terms of a few discrete states. The system s behavior is then specified by the transition times between these discrete states. For example when investigating a neurons behavior, the important aspect are often only the times when a spike is emitted and not the complex evolution of the membrane potential [16]. In a doublewell potential system, depending on the questions asked, it may be sufficient to know in which of the two wells the system is located, neglecting the fluctuations in the wells as well as the actual dynamics when crossing from one well to the other. In these cases a reduction to a discrete description can be considered as an appropriate simplification. We first review the two state description of bistable systems [10] and then introduce a phenomenological discrete model for excitable dynamics. 

The simulation of lignin liquefaction combined a stochastic interpretation of depolymerization kinetics with models for catalyst deactivation and polymer diffusion. The stochastic model was based on discrete mathematics, which allowed the transformations of a system between its discrete states to be chronicled by comparing random numbers to transition probabilities. The transition probability was dependent on both the time interval of reaction and a global reaction rate constant. McDermott s ( analysis of the random reaction trajectory of the linear polymer shown in Figure 6 permits illustration. 

The stator of rotary Fi motor is composed of six proteins. Three of them catalyze the hydrolysis of ATP, which drives the rotation of a shaft. The shaft of this Fi complex is glued to a proton turbine called Fq, which is located in the internal membrane of mitochondria. The whole FoFi-ATPase synthesizes ATP using the proton flow across the inner membrane. The Fi protein complex can function in reverse and serve as a motor performing mechanical work. These motors are modeled as stochastic systems with random jumps between the chemical states. If the rotation follows discrete steps and substeps, then the shaft has motions between well-defined orientations corresponding to the chemical states of the motor leading to a stochastic system based on discrete states. The result still will be the transition rates of the random jumps between the discrete states. These transition rates depend on the mass action law of chemical kinetics. 

Discrete state space stochastic models of chemical reactions can be identified with the Markovian jump process. In this case the temporal evolution can be described by the master equation ... 

Equation (5.27) is an ordinary difierential equation in function space. Giving an appropriate interpretation of the state-space it can describe spatiotemporal phenomena . Thinking of chemical applications we might set the state-space as the unification of the real three-dimensional space and the w-dimensional component space. However, the formulation of stochastic models of chemical reactions accompanied by difiusion is not easy, and practically all of the applications treat both the component space and the real space by discrete methods. Although it is quite natural to apply the notion of discrete state-space for chemical reactions, at least from the mesoscopic point of view, it might be better if diflTusion were described in terms of continuous models. In Chapter 6 we return to this question. 

A great amount of stochastic physics investigates the approximation of jump processes by diffusion processes, i.e. of the master equation by a Fokker-Planck equation, since the latter is easier to solve. The rationale behind this procedure is the fact that the usual deterministic (CCD) and stochastic (CDS) models differ from each other in two aspects. The CDS model offers a stochastic description with a discrete state space. In most applications, where the number of particles is large and may approach Avogadro s number, the discreteness should be of minor importance. Since the CCD model adopts a continuous state-space, it is quite natural to adopt CCS model as an approximation for fluctuations. 

Other kinds of Fokker-Planck equations can be also derived. The continuous state-space stochastic model of a chemical reaction, which considers the reaction as a diffusion process , neglects the essential discreteness of the mesoscopic events. However, some shortcomings of (5.65) have been eliminated by using a direct Fokker-Planck equation obtained by means of nonlinear transport theory (Grabert et al., 1983). 

Relations between the usual deterministic and stochastic model have been studied since the start of the subject. Early investigators gauged the quality of a stochastic model by the proximity of its behaviour to that of the corresponding deterministic one. If one considers that the CDS model takes into consideration the discrete character of the state-space and it does not neglect fluctuations then the appropriate question nowadays seems to be in what sense and to what extent can the deterministic model be considered a good approximation of the stochastic one ... 

The proposed procedure is shown in Figure 1 and consists of five basic steps. First a standard ODE model is derived from first engineering principles and the constitutive equations containing unknown functional relations are identified. The ODE model is then translated into a stochastic state space model consisting of a set of SDE s describing the dynamics of the system in continuous time and a set of discrete time measurement equations signifying how the available experimental data was obtained. 

In the second step of the procedure the ODE model is translated into a stochastic state space model with r as an additional state variable. This is straightforward, as it can simply be done by replacing the ODE s with SDE s and adding a set of discrete time measurement equations, which yields a model of the following type ... 

The degradation mechanisms affecting industrial systems, structures and components (SSCs) can be modeled as discrete-state transport processes (e.g., Baraldi et al., 2013 a and b Moghaddas Zuo, 2011). The stochastic transitions between the states obey known models (e.g., WeibuU distribution, exponential distribution, etc.), whose parameters (e.g., the scale and shape factors for the WeibuU distribution, the failure rate for the exponential distribution, etc.) need to be estimated. Degradation models are used in support to maintenance decision making, for example to optimize the inspection period in a preventive maintenance approach or to estimate the remaining useful life of a SSC in a predictive maintenance approach (Zio Compare, 2013). 

Any of the modeling/simulation approach can be used in the reliability analysis when the stochastic nature of the system components is represented and when the target system state is identified and assessed. Vulnerability analysis and system planning rely on the characterization of system functionality hybridization of CN theory, SD, DS, IIM and ABM approaches is suitable in this step. Resilience analysis and optimization of the operations require modeling time-dynamics, transitions between discrete states, adaptation and recovery in the systems hybridization of CN theory, IIM and ABM approaches is suitable in this step. 

Nassar et al. [10] employed a stochastic approach, namely a Markov process with transient and absorbing states, to model in a unified fashion both complex linear first-order chemical reactions, involving molecules of multiple types, and mixing, accompanied by flow in an nonsteady- or steady-state continuous-flow reactor. Chou et al. [11] extended this system with nonlinear chemical reactions by means of Markov chains. An assumption is made that transitiions occur instantaneously at each instant of the discretized time. 

It is very important to remark that an AVT provides information only on the vibration responses of a structure excited by unmeasured inputs. Consequently, it is impossible to distinguish the input term /k from the noise terms iPk and Vk in Eq. 17. This results in the following discrete-time stochastic state-space model ... 

Models can be characterized in many ways, in what might be called dimensions. Some dimensions are a matter of degree. These include ranges such as simple to complex, phenomenological to mechanistic, descriptive to predictive, and quantitative to qualitative. Other dimension types are discrete and either/or steady-state or dynamic, deterministic or stochastic. Using these descriptive dimensions facilitates understanding the differences between models and their fitness for specific uses. 

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]. 

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 ... 

Figure 11.3 State transition diagram for stochastic discrete model for the single unimolecular reaction of Equation (11.5). It is assumed that there is a total of N molecules in the system k is the number of molecules in state A. The intrinsic rate constants are Ai>2.

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