In numerous systems in biophysics and related fields, scientists measure (with very smart methods) individual molecules (e.g. biopolymers (proteins, DNA, RNA, etc), nano - crystals, ion channels), aiming at finding a model from the data. But the nois
e is not solved accurately in not so few cases and this may lead to misleading models. Here, we solve the noise. We consider two state photon trajectories from any on off kinetic scheme (KS): the process emitting photons with a rate {gamma}on when it is in the on state, and emitting with a rate {gamma}off when it is in the off state. We develop a filter that removes the noise resulting in clean data also in cases where binning fails. The filter is a numerical algorithm with various new statistical treatments. It is based on a new general likelihood function developed here, with observable dependent form. The filter can solve the noise, in the detectable region of the rate space: that is, we also find a region where the data is too noisy. Consistency tests will find the regions type from the data. If the data is ruled too noisy, binning obviously fails, and one should apply simpler methods on the raw data and realizing that the extracted information is partial. We show that not applying the filter while cleaning results in erroneous rates. This filter (with minor adjustments) can solve the noise in any discrete state trajectories, yet extensions are needed in tackling the noise from other data, e.g. continuous data and FRET data. The filter developed here is complementary with our previous projects in this field, where we have solved clean two state data with the development of reduced dimensions forms (RDFs): only the combined procedures enabling building the most accurate model from noisy trajectories from single molecules
Models that explain the economical and political realities of nowadays societies should help all the worlds citizens. Yet, the last four years showed that the current models are missing. Here we develop a dynamical society-deciders model showing that
the long lasting economical stress can be solved when increasing fairness in nations. fairness is computed for each nation using indicators from economy and politics. Rather than austerity versus spending, the dynamical model suggests that solving crises in western societies is possible with regulations that reduce the stability of the deciders, while shifting wealth in the direction of the people. This shall increase the dynamics among socio-economic classes, further increasing fairness.
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.
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 a
nd 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, <r^2>, obeys, <r^2>~[<r^2>_{nrml}]^{alpha}, where <r^2 >_{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.
Normal dynamics in a quasi-one-dimensional channel of length L (toinfty) of N hard spheres are analyzed. The spheres are heterogeneous: each has a diffusion coefficient D that is drawn from a probability density function (PDF), W D^(-{gamma}), for sm
all D, where 0leq{gamma}<1. The initial spheres density {rho} is non-uniform and scales with the distance (from the origin) l as, {rho} l^(-a), 0leqaleq1. An approximation for the N-particle PDF for this problem is derived. From this solution, scaling law analysis and numerical simulations, we show here that the mean square displacement for a particle in such a system obeys, <r^2>~t^(1-{gamma})/(2c-{gamma}), where c=1/(1+a). The PDF of the tagged particle is Gaussian in position. Generalizations of these results are considered.
Deducing an underlying multi-substate on-off kinetic scheme (KS) from the statistical properties of a two-state trajectory is the aim from many experiments in biophysics and chemistry, such as, ion channel recordings, enzymatic activity and structura
l dynamics of bio-molecules. Doing so is almost always impossible, as the mapping of a KS into a two-state trajectory leads to the loss of information about the KS (almost always). Here, we present the optimal way to solve this problem. It is based on unique forms of reduced dimensions (RD). RD forms are on-off networks with connections only between substates of different states, where the connections can have multi-exponential waiting time probability density functions (WT-PDFs). A RD form has the simplest toplogy that can reproduce a given data. In theory, only a single RD form can be constructed from the full data (hence its uniqueness), still this task is not easy when dealing with finite data. For doing so, a toolbox made of known statistical methods in data analysis and new statistical methods and numerical algorithms develped for this problem is presented. Our toolbox is self-contained: it builds a mechanism based only on the information it extracts from the data. The implementation of the toolbox on the data is fast. The toolbox is automated and is available for academic research upon electronic request.
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 abo
ut 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.
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.
We report controlled local manipulation of single vortices by low temperature magnetic force microscope (MFM) in a thin film of superconducting Nb. We are able to position the vortices in arbitrary configurations and to measure the distribution of lo
cal depinning forces. This technique opens up new possibilities for the characterization and use of vortices in superconductors.
The diffusion process of N hard rods in a 1D interval of length L (--> inf) is studied using scaling arguments and an asymptotic analysis of the exact N-particle probability density function (PDF). In the class of such systems, the universal scaling
law of the tagged particles mean absolute displacement reads, <|r|>~ <|r|>_{free}/n^mu, where <|r|>_{free} is the result for a free particle in the studied system and n is the number of particles in the covered length. The exponent mu is given by, mu=1/(1+a), where a is associated with the particles density law of the system, rho~rho_0*L^(-a), 0<= a <=1. The scaling law for <|r|> leads to, <|r|>~rho_0^((a-1)/2) (<|r| >_{free})^((1+a)/2), an equation that predicts a smooth interpolation between single file diffusion and free particle diffusion depending on the particles density law, and holds for any underlying dynamics. In particular, <|r|>~t^((1+a)/2) for normal diffusion, with a Gaussian PDF in space for any value of a (deduced by a complementary analysis), and, <|r|>~t^((beta(1+a))/2), for anomalous diffusion in which the systems particles all have the same power-law waiting time PDF for individual events, psi~t^(-1-beta), 0<beta<1. Our analysis shows that the scaling <|r|>~t^(1/2) in a standard single file is a direct result of the fixed particles density condition imposed on the system, a=0.
