67 research outputs found
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
On the relationships between kinetic schemes and two-state single molecule trajectories
Trajectories of a signal that fluctuates between two states which originate
from single molecule activities have become ubiquitous. Common examples are
trajectories of ionic flux through individual membrane-channels, and of photon
counts collected from diffusion, activity, and conformational changes of
biopolymers. By analyzing the trajectory, one wishes to deduce the underlying
mechanism, which is usually described by a multi-substate kinetic scheme. In
previous works, we divided kinetic schemes that generate two-state trajectories
into two types: reducible schemes and irreducible schemes. We showed that all
the information in trajectories generated from reducible schemes is contained
in the waiting time probability density functions (PDFs) of the two states. It
follows that reducible schemes with the same waiting time PDFs are not
distinguishable. In this work, we further characterize the topologies of
kinetic schemes, now of irreducible schemes, and further study two-state
trajectories from the two types of scheme. We suggest various methods for
extracting information about the underlying kinetic scheme from the trajectory
(e. g., calculate the binned successive waiting times PDF and analyze the
ordered waiting times trajectory), and point out the advantages and
disadvantages of each. We show that the binned successive waiting times PDF is
not only more robust than other functions when analyzing finite trajectories,
but contains, in most cases, more information about the underlying kinetic
scheme than other functions in the limit of infinitely long trajectories. For
some cases however, analyzing the ordered waiting times trajectory may supply
unique information about the underlying kinetic scheme
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
Clustering in anomalous files of independent particles
The dynamics of classical hard particles in a quasi one-dimensional channel
were studied since the 1960s and used for explaining processes in chemistry,
physics and biology and in applications. Here we show that in a previously
un-described file made of anomalous, independent, particles (with jumping times
taken from, {\psi}_{\alpha} (t) t^(-1-{\alpha}), 0<{\alpha}<1), particles form
clusters. At steady state, the percentage of particles in clusters is about,
\surd(1-{\alpha}^3), only for anomalous {\alpha}, characterizing the
criticality of a phase transition. The asymptotic mean square displacement per
particle in the file is about, log^2(t). We show numerically that this exciting
phenomenon of a phase transition is very stable, and relate it with the
mysterious phenomenon of rafts in biological membranes, and with regulation of
biological channels.Comment: main text (13 pages, 4 figures), plus supplementary material (15
pages, 6 figures
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
Comment on 'Path Summation Formulation of the Master Equation'
Comment on 'Path Summation Formulation of the Master Equation
Path probability density functions for semi-Markovian random walks
In random walks, the path representation of the Green's function is an
infinite sum over the length of path probability density functions (PDFs). Here
we derive and solve, in Laplace space, the recursion relation for the n order
path PDF for any arbitrarily inhomogeneous semi-Markovian random walk in a
one-dimensional (1D) chain of L states. The recursion relation relates the n
order path PDF to L/2 (round towards zero for an odd L) shorter path PDFs, and
has n independent coefficients that obey a universal formula. The z transform
of the recursion relation straightforwardly gives the generating function for
path PDFs, from which we obtain the Green's function of the random walk, and
derive an explicit expression for any path PDF of the random walk. These
expressions give the most detailed description of arbitrarily inhomogeneous
semi-Markovian random walks in 1D
Single Stranded DNA Translocation Through A Nanopore: A Master Equation Approach
We study voltage driven translocation of a single stranded (ss) DNA through a
membrane channel. Our model, based on a master equation (ME) approach,
investigates the probability density function (pdf) of the translocation times,
and shows that it can be either double or mono-peaked, depending on the system
parameters. We show that the most probable translocation time is proportional
to the polymer length, and inversely proportional to the first or second power
of the voltage, depending on the initial conditions. The model recovers
experimental observations on hetro-polymers when using their properties inside
the pore, such as stiffness and polymer-pore interaction.Comment: 7 pages submitted to PR
Semi-Markov Graph Dynamics
In this paper, we outline a model of graph (or network) dynamics based on two
ingredients. The first ingredient is a Markov chain on the space of possible
graphs. The second ingredient is a semi-Markov counting process of renewal
type. The model consists in subordinating the Markov chain to the semi-Markov
counting process. In simple words, this means that the chain transitions occur
at random time instants called epochs. The model is quite rich and its possible
connections with algebraic geometry are briefly discussed. Moreover, for the
sake of simplicity, we focus on the space of undirected graphs with a fixed
number of nodes. However, in an example, we present an interbank market model
where it is meaningful to use directed graphs or even weighted graphs.Comment: 25 pages, 4 figures, submitted to PLoS-ON
Utilizing the information content in two-state trajectories
The signal from many single molecule experiments monitoring molecular
processes, such as enzyme turnover via fluorescence and opening and closing of
ion channel via the flux of ions, consists of a time series of stochastic on
and off (or open and closed) periods, termed a two-state trajectory. This
signal reflects the dynamics in the underlying multi-substate on-off kinetic
scheme (KS) of the process. The determination of the underlying KS is difficult
and sometimes even impossible due to the loss of information in the mapping of
the mutli dimensional KS onto two dimensions. Here we introduce a new procedure
that efficiently and optimally relates the signal to all equivalent underlying
KSs. This procedure partitions the space of KSs into canonical (unique) forms
that can handle any KS, and obtains the topology and other details of the
canonical form from the data without the need for fitting. Also established are
relationships between the data and the topology of the canonical form to the
on-off connectivity of a KS. The suggested canonical forms constitute a
powerful tool in discriminating between KSs. Based on our approach, the upper
bound on the information content in two state trajectories is determined.Comment: The file contains: main text (+4 figures), supporting information (+9
figures), poster (1 page
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