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[[File:Two state trajectory.jpg|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 process|stochastic]]; nevertheless, it can have [[Deterministic system|deterministic]] ON-OFF components. A completely deterministic two-state trajectory is a [[Square wave|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 [[Physics:Single-molecule experiment|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. E. |journal=Science |volume=283 |issue=5408 |pages=1670–6 |pmid=10073924 |last2=Orrit |first2=M|bibcode = 1999Sci...283.1670M }}</ref><ref>{{cite journal |doi=10.1126/science.283.5408.1676 |title=Fluorescence Spectroscopy of Single Biomolecules |year=1999 |last1=Weiss |first1=Shimon |journal=Science |volume=283 |issue=5408 |pages=1676–83 |pmid=10073925|bibcode = 1999Sci...283.1676W }}</ref> (e.g. measurements of protein dynamics and [[Physics:Force spectroscopy|DNA and RNA dynamics]],<ref>{{cite journal |doi=10.1038/nature01060 |title=Probing the free-energy surface for protein folding with single-molecule fluorescence spectroscopy |year=2002 |last1=Schuler |first1=Benjamin |last2=Lipman |first2=Everett A. |last3=Eaton |first3=William A. |journal=Nature |volume=419 |issue=6908 |pages=743–7 |pmid=12384704|bibcode = 2002Natur.419..743S |s2cid=1356830 |url=https://zenodo.org/record/1233253 }}</ref><ref>{{cite journal |doi=10.1126/science.1086911 |title=Protein Conformational Dynamics Probed by Single-Molecule Electron Transfer |year=2003 |last1=Yang |first1=Haw |journal=Science |volume=302 |issue=5643 |pages=262–6 |pmid=14551431 |last2=Luo |first2=Guobin |last3=Karnchanaphanurach |first3=Pallop |last4=Louie |first4=Tai-Man |last5=Rech |first5=Ivan |last6=Cova |first6=Sergio |last7=Xun |first7=Luying |last8=Xie |first8=X. 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 |pages=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 [[Biology:Ion channel|ion channel]]s,<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 |first4=David W. |last4=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> [[Biology:Enzyme|enzyme activity]],<ref>{{cite journal |doi=10.1126/science.282.5395.1877 |title=Single-Molecule Enzymatic Dynamics |year=1998 |last1=Lu |first1=H. P. |journal=Science |volume=282 |issue=5395 |pages=1877–82 |pmid=9836635 |last2=Xun |first2=L |last3=Xie |first3=XS|bibcode=1998Sci...282.1877P }}</ref><ref>{{cite journal |doi=10.1016/S0301-0104(99)00098-1 |title=The fluctuating enzyme: A single molecule approach |year=1999 |last1=Edman |first1=Lars |last2=Földes-Papp |first2=Zeno |last3=Wennmalm |first3=Stefan |last4=Rigler |first4=Rudolf |journal=Chemical Physics |volume=247 |issue=1 |pages=11–22 |bibcode=1999CP....247...11E}}</ref><ref>{{cite journal |doi=10.1002/anie.200460625 |title=Single-Enzyme Kinetics of CALB-Catalyzed Hydrolysis |year=2005 |last1=Velonia |first1=Kelly |last2=Flomenbom |first2=Ophir |last3=Loos |first3=Davey |last4=Masuo |first4=Sadahiro |last5=Cotlet |first5=Mircea |last6=Engelborghs |first6=Yves |last7=Hofkens |first7=Johan |last8=Rowan |first8=Alan E. |last9=Klafter |first9=Joseph |last10=Nolte |first10=Roeland J. M. |last11=De Schryver |first11=Frans C. |journal=Angewandte Chemie International Edition |volume=44 |issue=4 |pages=560–4 |pmid=15619259|display-authors=8 }}</ref><ref>{{cite journal |doi=10.1073/pnas.0409039102 |title=Stretched exponential decay and correlations in the catalytic activity of fluctuating single lipase molecules |year=2005 |last1=Flomenbom |first1=O. |journal=Proceedings of the National Academy of Sciences |volume=102 |issue=7 |pages=2368–72 |pmid=15695587 |last2=Velonia |first2=K |last3=Loos |first3=D |last4=Masuo |first4=S |last5=Cotlet |first5=M |last6=Engelborghs |first6=Y |last7=Hofkens |first7=J |last8=Rowan |first8=AE |last9=Nolte |first9=RJ |last10=Van Der Auweraer |first10=M. |last11=De Schryver |first11=F. C. |last12=Klafter |first12=J. |pmc=548972|bibcode = 2005PNAS..102.2368F |display-authors=8 |doi-access=free }}</ref><ref>{{cite journal |doi=10.1038/nchembio759 |title=Ever-fluctuating single enzyme molecules: Michaelis-Menten equation revisited |year=2005 |last1=English |first1=Brian P |last2=Min |first2=Wei |last3=Van Oijen |first3=Antoine M |last4=Lee |first4=Kang Taek |last5=Luo |first5=Guobin |last6=Sun |first6=Hongye |last7=Cherayil |first7=Binny J |last8=Kou |first8=S C |last9=Xie |first9=X Sunney |journal=Nature Chemical Biology |volume=2 |issue=2 |pages=87–94 |pmid=16415859 |s2cid=2201882 }}</ref> [[Physics:Quantum dot|quantum dot]]s<ref>{{cite journal |doi=10.1126/science.7973650 |title=Probing individual molecules with confocal fluorescence microscopy |year=1994 |last1=Nie |first1=S |last2=Chiu |first2=D. |last3=Zare |first3=R. |journal=Science |volume=266 |issue=5187 |pages=1018–21 |pmid=7973650|bibcode = 1994Sci...266.1018N }}</ref><ref>{{cite journal |doi=10.1063/1.3582336 |title=Anomalous diffusion of oligomerized transmembrane proteins |year=2011 |last1=Schmidt |first1=Ulrich |last2=Weiss |first2=Matthias |journal=The Journal of Chemical Physics |volume=134 |issue=16 |pages=165101 |pmid=21528980|bibcode = 2011JChPh.134p5101S }}</ref><ref>{{cite journal |doi=10.1103/PhysRevLett.93.260601 |title=Recurrence and Photon Statistics in Fluorescence Fluctuation Spectroscopy |year=2004 |last1=Zumofen |first1=Gert |last2=Hohlbein |first2=Johannes |last3=Hübner |first3=Christian |journal=Physical Review Letters |volume=93 |issue=26 |bibcode=2004PhRvL..93z0601Z |pmid=15697961 |pages=260601}}</ref><ref>{{cite journal |doi=10.1073/pnas.0509976103 |title=Suppressing Brownian motion of individual biomolecules in solution |year=2006 |last1=Cohen |first1=Adam E. |journal=Proceedings of the National Academy of Sciences |volume=103 |issue=12 |pages=4362–5 |bibcode=2006PNAS..103.4362C |jstor=30048946 |pmid=16537418 |last2=Moerner |first2=WE |pmc=1450176|doi-access=free }}</ref><ref>{{cite journal |bibcode=1997Natur.388..355D |doi=10.1038/41048 |year=1997 |last1=Moerner |first1=W. E. |last2=Dickson |first2=Robert M. |last3=Cubitt |first3=Andrew B. |last4=Tsien |first4=Roger Y. |journal=Nature |volume=388 |issue=6640 |pages=355–8 |pmid=9237752 |title=On/off blinking and switching behaviour of single molecules of green fluorescent protein|s2cid=4313830 |doi-access=free }}</ref><ref>{{cite journal |doi=10.1103/PhysRevB.70.165304 |title=Relationship between single quantum-dot intermittency and fluorescence intensity decays from collections of dots |year=2004 |last1=Chung |first1=Inhee |last2=Bawendi |first2=Moungi |journal=Physical Review B |volume=70 |issue=16 |pages=165304 |bibcode=2004PhRvB..70p5304C}}</ref>). From these experiments, one aims at finding the correct model explaining the measured process.<ref>{{cite journal |bibcode=1987BpJ....52..961B |doi=10.1016/S0006-3495(87)83289-7 |title=Theory of the kinetic analysis of patch-clamp data |year=1987 |last1=Bauer |first1=R.J. |last2=Bowman |first2=B.F. |last3=Kenyon |first3=J.L. |journal=Biophysical Journal |volume=52 |issue=6 |pages=961–78 |pmid=2447973 |pmc=1330095}}</ref><ref>{{cite journal |doi=10.1098/rspb.1989.0024 |title=Equivalence of Aggregated Markov Models of Ion-Channel Gating |year=1989 |last1=Kienker |first1=P. |journal=Proceedings of the Royal Society B: Biological Sciences |volume=236 |issue=1284 |pages=269–309 |bibcode=1989RSPSB.236..269K |jstor=2410562 |pmid=2471201|s2cid=29761646 }}</ref><ref>{{cite journal |first1=Donald R. |last1=Fredkin |first2=John A. |last2=Rice |year=1986 |title=On Aggregated Markov Processes |journal=Journal of Applied Probability |volume=23 |issue=1 |pages=208–14 |jstor=3214130 |doi=10.2307/3214130|s2cid=123503233 }}</ref><ref>{{cite journal |doi=10.1098/rstb.1982.0156 |title=On the Stochastic Properties of Bursts of Single Ion Channel Openings and of Clusters of Bursts |year=1982 |last1=Colquhoun |first1=D. |last2=Hawkes |first2=A. 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J. |journal=Proceedings of the National Academy of Sciences |volume=102 |issue=18 |pages=6326–31 |bibcode=2005PNAS..102.6326B |jstor=3375322 |pmid=15843461 |last2=Yang |first2=J |last3=Pearson |first3=JE |pmc=1088360|doi-access=free }}</ref><ref name=first>{{cite journal |doi=10.1073/pnas.0604546103 |title=Utilizing the information content in two-state trajectories |year=2006 |last1=Flomenbom |first1=O. |journal=Proceedings of the National Academy of Sciences |volume=103 |issue=29 |pages=10907–10 |jstor=30049381 |bibcode=2006PNAS..10310907F |pmid=16832051 |last2=Silbey |first2=RJ |pmc=1544147|arxiv = q-bio/0703013 |doi-access=free }}</ref><ref>{{cite journal |doi=10.1529/biophysj.104.055905 |title=What Can One Learn from Two-State Single-Molecule Trajectories? |year=2005 |last1=Flomenbom |first1=Ophir |last2=Klafter |first2=Joseph |last3=Szabo |first3=Attila |journal=Biophysical Journal |volume=88 |issue=6 |pages=3780–3 |pmid=15764653 |pmc=1305612|arxiv = q-bio/0502006 |bibcode = 2005BpJ....88.3780F }}</ref><ref>{{cite journal |doi=10.1103/PhysRevE.78.066105 |title=Toolbox for analyzing finite two-state trajectories |year=2008 |last1=Flomenbom |first1=O. |last2=Silbey |first2=R. J. |journal=Physical Review E |volume=78 |issue=6 |bibcode=2008PhRvE..78f6105F |pmid=19256903 |pages=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 |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 [[Biology:Ion channel|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 [[Biology:Enzymes|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. [[Biology:Förster resonance energy transfer|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 [[Physics:Quantum dot|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==
* [[Physics:Single-molecule experiment|Single-molecule experiment]]
* [[Physics:Reduced dimensions form|Reduced dimensions form]]
* [[Physics:Kinetic scheme|Kinetic scheme]]
* [[Physics:Master equation|Master equation]]
* [[Wave]]

==References==
{{reflist}}

[[Category:Statistical mechanics]]
[[Category:Stochastic processes]]

{{Sourceattribution|Two-state trajectory}}