Renewal Theory and its Ramifications

Renewal-reward processes are used to provide a framework for the mathematical description of single-molecule bead-motor assays for processive motor proteins. The formulation provides a more powerful, general approach to the fluctuation analysis of bead-motor assays begun by Svoboda et al. (Proc. Natl. Acad. Sci. USA 91(25):11782, 1994). Fluctuation analysis allows one to gain insight into the mechanochemical cycle of motor proteins purely by measuring the statistics of the displacement of the cargo (e.g., bead) the protein transports. The statistical parameters of interest are shown to be the steady-state slopes (in time) of the cumulants of the bead (the cumulant rates). The first two cumulant rates are the steady-state velocity and slope of the variance. The cumulant rates are shown to be insensitive to experimental disturbances such as the initial state of the enzyme and from the viewpoint of modeling, unaffected by substeps. Two existing models--Elston (J. Math. Biol. 41(3):189-206, 2000) and Peskin and Oster (Biophys. J. 68(4):202S-211S, 1995)--are formulated as renewal-reward processes to demonstrate the insight that the formulation affords. A key contribution of the approach is the possibility of accounting for wasted hydrolyses and backward steps in the fluctuation analysis. For example, the randomness parameter defined in the first fluctuation analysis of optical trap based bead-motor assays (Svoboda et al. in Proc. Natl. Acad. Sci. USA 91(25):11782, 1994), loses its original purpose of estimating the number of rate-determining steps in the chemical cycle when backward steps and wasted hydrolyses are present. As a simple application of our formulation, we extend the randomness parameter's scope by showing how it can be used to infer the presence of wasted hydrolyses and backward steps with certainty. A more powerful fluctuation analysis using higher cumulant rate measurements is proposed: the method allows one to estimates the number of intermediate reactions, the average chemical rate, and the probability of stepping backward or forward. The stability of the method in the presence of measurement errors is demonstrated numerically to encourage its use in experiments.

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“…Let ρ represent their correlation coefficient. Then the longtime variance of R(t) takes the form (Smith 1958) …”

Section: Cumulants Of the Bead-motor Systemmentioning

confidence: 99%

Renewal-Reward Process Formulation of Motor Protein Dynamics

Krishnan

Epureanu

2011

Bull Math Biol

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“…We suppose that the process is regenerative, where, following Smith [8], we define a regenerative process as follows. Let T be a positive random variable defined on some probability space (Q, .91, P) and X a random function defined on the same space with index set [0, T).…”

Section: The Supremum In a Class Of Stochastic Processesmentioning

confidence: 99%

Extreme value theory for a class of discrete distributions with applications to some stochastic processes

Anderson

1970

J. Appl. Probab.

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Let ξn be the maximum of a set of n independent random variables with common distribution function F whose support consists of all sufficiently large positive integers. Some of the classical asymptotic results of extreme value theory fail to apply to ξn for such F and this paper attempts to find weaker ones which give some description of the behaviour of ξn as n → ∞. These are then applied to the extreme value theory of certain regenerative stochastic processes.

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“…In 1958 an elegant summation of the known mathematical results in renewal theory was presented by Smith [68]. As has been mentioned, this theory has many applications to replacement problems.…”

Section: The Formative Yearsmentioning

confidence: 97%