A two-state trajectory (also termed two-state time trajectory or a trajectory with two states) is a dynamical signal that fluctuates between two distinct values: ON and OFF, open and closed,

+/-

, etc. Mathematically, the signal

X(t)

has, for every

t,

either the value

X(t)=c_\mathrm

X(t)=c_\mathrm

. In most applications, the signal is

stochastic Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; i ...

; nevertheless, it can have

deterministic Determinism is the metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping mo ...

ON-OFF components. A completely deterministic two-state trajectory is a

square wave Square wave may refer to: *Square wave (waveform) A square wave is a non-sinusoidal waveform, non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same ...

. There are many ways one can create a two-state signal, e.g. flipping a coin repeatedly. A stochastic two-state trajectory is among the simplest stochastic processes. Extensions include: three-state trajectories, higher discrete state trajectories, and continuous trajectories in any dimension.

Two state trajectories in biophysics, and related fields

Two state trajectories are very common. Here, we focus on relevant trajectories in scientific experiments: these are seen in measurements in chemistry, physics, and the biophysics of individual molecules (e.g. measurements of protein dynamics and DNA and RNA dynamics, activity of

ion channel Ion channels are pore-forming membrane proteins that allow ions to pass through the channel pore. Their functions include establishing a resting membrane potential, shaping action potentials and other electrical signals by Gating (electrophysiol ...

enzyme activity Enzyme assays are laboratory methods for measuring enzyme, enzymatic activity. They are vital for the study of enzyme kinetics and enzyme inhibitor, enzyme inhibition. Enzyme units The quantity or concentration of an enzyme can be expressed in Mo ...

quantum dot Quantum dots (QDs) or semiconductor nanocrystals are semiconductor particles a few nanometres in size with optical and electronic properties that differ from those of larger particles via quantum mechanical effects. They are a central topic i ...

s). From these experiments, one aims at finding the correct model explaining the measured process. We explain about various relevant systems in what follows.

Ion channels

Since the

is either opened or closed, when recording the number of ions that go through the channel when time elapses, observed is a two-state trajectory of the current versus time.

Enzymes

Here, there are several possible experiments on the activity of individual

enzymes An enzyme () is a protein that acts as a biological catalyst by accelerating chemical reactions. The molecules upon which enzymes may act are called substrates, and the enzyme converts the substrates into different molecules known as pro ...

with a two-state signal. For example, one can create substrate that only upon the enzymatic activity shines light when activated (with a laser pulse). So, each time the enzyme acts, we see a burst of photons during the time period that the product molecule is in the laser area.

Dynamics of biological molecules

Structural changes of molecules are viewed in various experiments' type.

Förster resonance energy transfer Förster resonance energy transfer (FRET), fluorescence resonance energy transfer, resonance energy transfer (RET) or electronic energy transfer (EET) is a mechanism describing energy transfer between two light-sensitive molecules (chromophores). ...

is an example. In many cases one sees a time trajectory that fluctuates among several cleared defined states.

Quantum dots

Another system that fluctuates among an on state and an off state is a

. Here, the fluctuations are since the molecule is either in a state that emits photons or in a dark state that does not emit photons (the dynamics among the states are influenced also from its interactions with the surroundings).

References

{{reflist Statistical mechanics Stochastic processes