Correctly validating results from single molecule data: the case of stretched exponential decay in the catalytic activity of single lipase B molecules

The question of how to validate and interpret correctly the waiting time probability density functions (WT-PDFs) from single molecule data is addressed. It is shown by simulation that when a stretched exponential WT-PDF, with a stretched exponent alfa and a time scale parameter tau, generates the off periods of a two-state trajectory, a reliable recovery of the input WT-PDF from the trajectory is obtained even when the bin size used to define the trajectory, dt, is much larger than the parameter tau. This holds true as long as the first moment of the WT-PDF is much larger than dt. Our results validate the results in an earlier study of the activity of single Lipase B molecules and disprove recent related critique

The rule for a subdiffusive particle in an extremely diverse environment

The dynamics of a subdiffusive continuous time random walker in an inhomogeneous environment is analyzed. In each microscopic jump, a random time is drawn from a waiting time probability density function (WT-PDF) that decays as a power law: phi(t;k)~k/(1+kt)^(1+beta), 0<beta<1. The parameter k, which is the diffusion coefficient for the jump, is a random quantity also; in each jump, it is drawn from a PDF, p(k)~1/k^gamma (0<gamma<1). We show that this system exhibits a transition in the scaling law of its effective WT-PDF, psi(t), which is obtained when averaging phi(t;k) with p(k). psi(t) decays as a power law, psi(t)~1/t^(1+mu), and mu is given by two different formula. When 1-gamma> beta;, mu=beta, but when 1-gamma<beta, mu=1-gamma. The transition in the scaling of psi(t) reflects the competition between two different mechanisms for subdiffusion: subdiffusion due to the heavily tailed phi(t;k) for microscopic jumps, and subdiffusion due to the collective effect of an environment made of many slow local regions. These two different mechanisms for subdiffusion are not additive, and compete each other. The reported transition is dimension independent, and disappears when the power beta is also distributed, in the range, 0<bate<1. Simulations exemplified the transition, and implications are discussed

Kinetic Path Summation, Multi--Sheeted Extension of Master Equation, and Evaluation of Ergodicity Coefficient

We study the Master equation with time--dependent coefficients, a linear kinetic equation for the Markov chains or for the monomolecular chemical kinetics. For the solution of this equation a path summation formula is proved. This formula represents the solution as a sum of solutions for simple kinetic schemes (kinetic paths), which are available in explicit analytical form. The relaxation rate is studied and a family of estimates for the relaxation time and the ergodicity coefficient is developed. To calculate the estimates we introduce the multi--sheeted extensions of the initial kinetics. This approach allows us to exploit the internal ("micro")structure of the extended kinetics without perturbation of the base kinetics.Comment: The final journal versio

Renewal-anomalous-heterogeneous files

Renewal-anomalous-heterogeneous files are solved. A simple file is made of Brownian hard spheres that diffuse stochastically in an effective 1D channel. Generally, Brownian files are heterogeneous: the spheres' diffusion coefficients are distributed and the initial spheres' density is non-uniform. In renewal-anomalous files, the distribution of waiting times for individual jumps is exponential as in Brownian files, yet obeys: {\psi}_{\alpha} (t)~t^(-1-{\alpha}), 0<{\alpha}<1. The file is renewal as all the particles attempt to jump at the same time. It is shown that the mean square displacement (MSD) in a renewal-anomalous-heterogeneous file, , obeys, ~[_{nrml}]^{\alpha}, where _{nrml} is the MSD in the corresponding Brownian file. This scaling is an outcome of an exact relation (derived here) connecting probability density functions of Brownian files and renewal-anomalous files. It is also shown that non-renewal-anomalous files are slower than the corresponding renewal ones.Comment: Accepted for publication (August, 2010

Translocation of a Single Stranded DNA Through a Conformationally Changing Nanopore

We investigate the translocation of a single stranded DNA through a pore which fluctuates between two conformations, using coupled master equations. The probability density function of the first passage times (FPT) of the translocation process is calculated, displaying a triple, double or mono peaked behavior, depending on the interconversion rates between the conformations, the applied electric field, and the initial conditions. The cumulative probability function of the FPT, in a field-free environment, is shown to have two regimes, characterized by fast and slow timescales. An analytical expression for the mean first passage time of the translocation process is derived, and provides, in addition to the interconversion rates, an extensive characterization of the translocation process. Relationships to experimental observations are discussed.Comment: 8 pages, 5 figures, Biophys. J., in pres

Toolbox for analyzing finite two-state trajectories

In many experiments, the aim is to deduce an underlying multi-substate on-off kinetic scheme (KS) from the statistical properties of a two-state trajectory. However, the mapping of a KS into a two-state trajectory leads to the loss of information about the KS, and so, in many cases, more than one KS can be associated with the data. We recently showed that the optimal way to solve this problem is to use canonical forms of reduced dimensions (RD). RD forms are on-off networks with connections only between substates of different states, where the connections can have non-exponential waiting time probability density functions (WT-PDFs). In theory, only a single RD form can be associated with the data. To utilize RD forms in the analysis of the data, a RD form should be associated with the data. Here, we give a toolbox for building a RD form from a finite two-state trajectory. The methods in the toolbox are based on known statistical methods in data analysis, combined with statistical methods and numerical algorithms designed specifically for the current problem. Our toolbox is self-contained - it builds a mechanism based only on the information it extracts from the data, and its implementation on the data is fast (analyzing a 10^6 cycle trajectory from a thirty-parameter mechanism takes a couple of hours on a PC with a 2.66 GHz processor). The toolbox is automated and is freely available for academic research upon electronic request

Insight into Resonant Activation in Discrete Systems

The resonant activation phenomenon (RAP) in a discrete system is studied using the master equation formalism. We show that the RAP corresponds to a non-monotonic behavior of the frequency dependent first passage time probability density function (pdf). An analytical expression for the resonant frequency is introduced, which, together with numerical results, helps understand the RAP behavior in the space spanned by the transition rates for the case of reflecting and absorbing boundary conditions. The limited range of system parameters for which the RAP occurs is discussed. We show that a minimum and a maximum in the mean first passage time (MFPT) can be obtained when both boundaries are absorbing. Relationships to some biological systems are suggested.Comment: 5 pages, 5 figures, Phys. Rev. E., in pres