{{Distinguish|Two-state solution}} thumb|right|'''Figure 1''': Two-state trajectories
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, <math>+/-</math>, etc. Mathematically, the signal <math>X(t)</math> has, for every <math>t,</math> either the value <math>X(t)=c_\mathrm{off}</math> or <math>X(t)=c_\mathrm{on}</math>.
In most applications, the signal is stochastic; nevertheless, it can have deterministic ON-OFF components. A completely deterministic two-state trajectory is a square wave. 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.<ref>{{cite book | author= Erhan Cinlar | title=Introduction to Stochastic Processes | publisher=Prentice Hall Inc, New Jersey |year=1975 |isbn=978-0-486-49797-6}}</ref>
== 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<ref>{{cite journal |doi=10.1126/science.283.5408.1670 |title=Illuminating Single Molecules in Condensed Matter |year=1999 |last1=Moerner |first1=W. 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Sunney|bibcode = 2003Sci...302..262Y |s2cid=18706150 }}</ref><ref>{{cite journal |doi=10.1103/PhysRevLett.94.198302 |title=Observation of a Power-Law Memory Kernel for Fluctuations within a Single Protein Molecule |year=2005 |last1=Min |first1=Wei |last2=Luo |first2=Guobin |last3=Cherayil |first3=Binny J. |last4=Kou |first4=S. C. |last5=Xie |first5=X. Sunney |journal=Physical Review Letters |volume=94 |issue=19 |pmid=16090221 |article-number=198302 |bibcode=2005PhRvL..94s8302M}}</ref><ref>{{cite journal |doi=10.1073/pnas.2628068100 |title=Watching proteins fold one molecule at a time |year=2003 |last1=Rhoades |first1=Elizabeth |journal=Proceedings of the National Academy of Sciences |volume=100 |issue=6 |pages=3197–202 |pmid=12612345 |last2=Gussakovsky |first2=Eugene |last3=Haran |first3=Gilad |pmc=152269 |bibcode=2003PNAS..100.3197R |jstor=3139336|doi-access=free }}</ref><ref>{{cite journal |doi=10.1126/science.1069013 |title=Correlating Structural Dynamics and Function in Single Ribozyme Molecules |year=2002 |last1=Zhuang |first1=X. |journal=Science |volume=296 |issue=5572 |pages=1473–6 |pmid=12029135 |last2=Kim |first2=H |last3=Pereira |first3=MJ |last4=Babcock |first4=HP |last5=Walter |first5=NG |last6=Chu |first6=S|bibcode = 2002Sci...296.1473Z |s2cid=9459136 }}</ref> activity of ion channels,<ref>{{cite journal |doi=10.1038/260799a0 |title=Single-channel currents recorded from membrane of denervated frog muscle fibres |year=1976 |last1=Neher |first1=Erwin |last2=Sakmann |first2=Bert |journal=Nature |volume=260 |issue=5554 |pages=799–802 |pmid=1083489|bibcode = 1976Natur.260..799N |s2cid=4204985 }}</ref><ref>{{cite journal |first1=John J. |last1=Kasianowicz |first2=Eric |last2=Brandin |first3=Daniel |last3=Branton |author3-link=Daniel Branton |first4=David W. |last4=Deamer |author4-link=David Deamer |pmid=8943010 |pmc=19421 |jstor=40976 |bibcode=1996PNAS...9313770K |doi=10.1073/pnas.93.24.13770 |title=Characterization of individual polynucleotide molecules using a membrane channel |year=1996 |journal=Proceedings of the National Academy of Sciences |volume=93 |issue=24 |pages=13770–3|doi-access=free }}</ref> enzyme activity,<ref>{{cite journal |doi=10.1126/science.282.5395.1877 |title=Single-Molecule Enzymatic Dynamics |year=1998 |last1=Lu |first1=H. 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J. |journal=Physical Review E |volume=78 |issue=6 |bibcode=2008PhRvE..78f6105F |pmid=19256903 |article-number=066105|arxiv = 0802.1520 |s2cid=16196911 }}</ref><ref>{{cite book |doi=10.1002/9781118131374.ch13 |chapter=Making it Possible: Constructing a Reliable Mechanism from a Finite Trajectory |chapter-url=https://books.google.com/books?id=vRCrDHyyNUgC&pg=PT372 |title=Single-Molecule Biophysics: Experiment and Theory, Volume 146 |series=Advances in Chemical Physics |year=2011 |last1=Flomenbom |first1=Ophir |volume=146 |isbn=978-1-118-13137-4 |pages=367–93 |arxiv=0912.3952 |s2cid=15743989 |editor1-first=Tamiki |editor1-last=Komatsuzaki |editor2-first=Masaru |editor2-last=Kawakami |editor3-first=Satoshi |editor3-last=Takahashi |editor4-first=Haw |editor4-last=Yang |editor5-first=Robert J. |editor5-last=Silbey}}</ref> We explain about various relevant systems in what follows.
===Ion channels=== Since the ion channel 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 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 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 quantum dot. 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).
==See also== * Single-molecule experiment * Reduced dimensions form * Kinetic scheme * Master equation * Wave
==References== {{reflist}}
Category:Statistical mechanics Category:Stochastic processes