In biophysics

Biophysics

Biophysics is an interdisciplinary science that uses the methods of physical science to study biological systems. Studies included under the branches of biophysics span all levels of biological organization, from the molecular scale to whole organisms and ecosystems...

 and related fields, reduced dimension forms (RDFs) are unique on-off mechanisms for random walk

Random walk

A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...

s that generate two-state trajectories

Two-state trajectory

A two-state trajectory is a dynamical signal that fluctuates among two distinct values . In most applications, the signal is stochastic; nevertheless, it can have deterministic ON-OFF components...

 (see Fig. 1 for an example of a RDF and Fig. 2 for an example of a two-state trajectory). It has been shown that RDFs solve two-state trajectories, since only one RDF can be constructed from the data, where this property does not hold for on-off kinetic schemes, where many kinetic schemes can be constructed from a particular two-state trajectory (even from an ideal on-off trajectory). Two-state time trajectories are very common in measurements in chemistry, physics, and the biophysics of individual molecules  (e.g. measurements of protein dynamics and DNA and RNA dynamics

Force spectroscopy

Force spectroscopy is a dynamic analytical technique that allows the study of the mechanical properties of single polymer molecules or proteins, or individual chemical bonds. It is performed by pulling on the system under scrutiny with controlled forces...

, activity of ion channel

Ion channel

Ion channels are pore-forming proteins that help establish and control the small voltage gradient across the plasma membrane of cells by allowing the flow of ions down their electrochemical gradient. They are present in the membranes that surround all biological cells...

s, enzyme activity

Enzyme

Enzymes are proteins that catalyze chemical reactions. In enzymatic reactions, the molecules at the beginning of the process, called substrates, are converted into different molecules, called products. Almost all chemical reactions in a biological cell need enzymes in order to occur at rates...

, quantum dot

Quantum dot

A quantum dot is a portion of matter whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei...

s ) thus making RDFs an important tool in the analysis of data in these fields.

Since RDFs are uniquely obtained from the data, they have many advantages over other mathematical and statistical methods that were developed for solving two-state trajectories.

Desciption of RDF

A RDF is a lattice of substates, each substate represents either the on state or the off state, and has a particular number (see Figure 1). The connections are only among substates of different states.
A simulation of an on-off trajectory from a RDF is made with a generalized Gillespie algorithm

Gillespie algorithm

In probability theory, the Gillespie algorithm generates a statistically correct trajectory of a stochastic equation. It was created by Joseph L...

, where here a random jumping time is first taken from density functions that are (usually) not exponential using the rejection method

Rejection sampling

In mathematics, rejection sampling is a basic pseudo-random number sampling technique used to generate observations from a distribution. It is also commonly called the acceptance-rejection method or "accept-reject algorithm"....

, and then the specific next substate is chosen according to the jumping probabilities that are determined from the jumping time probability density functions.
A RDF can have irreversible connections, yet, it generates an on-off trajectory that has the property of microscopic reversibility

Microscopic reversibility

The principle of Microscopic reversibility in physics and chemistry is twofold:* First, it states that the microscopic detailed dynamics of particles and fields is time-reversible because the microscopic equations of motion are symmetric with respect to inversion in time ;* Second, it relates to...

, meaning that the physical system fluctuates around equilibrium.

Two-state trajectories

A two-state trajectory is a fluctuating signal made of on periods and off periods; an on period, and then an off period, and so on (see, Fig. 2). In most cases where this signal appears in applications in science, the trajectory is random; that is, the length of the on and off periods changes, and is a random quantity. There may be correlations in the trajectory; e.g., when we see a short off period and the next on period is relatively long (that is, long with a large probability), we say that there are off-on correlations. In principle, there are 4 independent types of correlations in two-state trajectories: on-on, on-off, off-on, and on-on. Two-state trajectories can be obtained from on-off kinetic scheme

Kinetic scheme

In physics and chemistry and related fields, a kinetic scheme is a network of states and connections among the states representing the scheme of a dynamical process. Usually, a kinetic scheme represents a Markovian process, where when the process is not Markovian, the scheme is a generalized...

s, RDFs, or any other stochastic equation of motion (with a clear on-off definition). In experiments from individual molecules, two-state trajectories are common, where from the trajectory we aim at finding the right model of the process.

Properties of RDFs in solving two-state trajectories

It was shown in Ref. 1 that RDFs are unique is the sense that a particular RDF generates a particular time trajectory (in a statistical sense), and a time trajectory is associated with only one RDF. This property does not hold for on-off kinetic schemes, where from a trajectory several kinetic schemes can be constructed ; see for example,. RDFs are also constructed more reliably from the data than kinetic schemes. Figure 3 illustrates RDFs, kinetic schemes and two-state trajectories, and the relations among these. Given a two-state trajectory (generated from any mechanism), it is safer to go from the data and construct a RDF, rather than trying to construct the kinetic scheme from the data directly. With the constructed RDF, one can find several possible kinetic schemes very accurately (usually, one eventually tries constructing a kinetic scheme from the data), where these kinetic schemes are all equivalent (with regard to the data).

The software RDF

• Based on RDFs, software for deducing the correct mechanisms from real data (e.g. two-state trajectories) is designed. See Figure 4 for an illustration of the aims of the software. The software is named RDF.