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| '''Transition path sampling (TPS)''' is a [[Rare Event Sampling]] method used in [[computer simulation]]s of rare events: physical or chemical transitions of a system from one stable state to another that occur too rarely to be observed on a computer timescale. Examples include [[protein folding]], [[chemical reaction]]s and [[nucleation]]. Standard simulation tools such as [[molecular dynamics]] can generate the dynamical trajectories of all the atoms in the system. However, because of the gap in accessible time-scales between simulation and reality, even present supercomputers might require years of simulations to show an event that occurs once per microsecond without some kind of acceleration.
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| == Transition path ensemble ==
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| TPS focuses on the most interesting part of the simulation, ''the transition''. For example, an initially unfolded protein will vibrate for a long time in an open-string configuration before undergoing a transition and fold on itself. The aim of the method is to reproduce precisely those folding moments.
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| Consider in general a system with two stable states A and B. The system will spend a long time in those states and occasionally jump from one to the other. There are many ways in which the transition can take place. Once a probability is assigned to each of the many pathways, one can construct a [[Monte Carlo method|Monte Carlo]] random walk in the path space of the transition trajectories, and thus generate the ''ensemble'' of all transition paths. All the relevant information can then be extracted from the ensemble, such as the reaction mechanism, the transition states, and the [[rate constant]]s.
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| Given an initial path, TPS provides some algorithms to perturb that path and create a new one. As in all Monte Carlo walks, the new path will then be accepted or rejected in order to have the correct path probability. The procedure is iterated and the ensemble is gradually sampled.
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| A powerful and efficient algorithm is the so-called ''shooting move''.<ref>{{Cite journal|volume=108|pages=9236 |year=1998|doi=10.1063/1.476378|title=Efficient transition path sampling: Application to Lennard-Jones cluster rearrangements|issue=22|last1=Dellago|first1=Christoph|last2=Bolhuis|first2=Peter G.|last3=Chandler|first3=David|journal=The Journal of Chemical Physics}}</ref> Consider the case of a classical many-body system described by coordinates ''r'' and momenta ''p''. Molecular dynamics generates a path as a set of (''r''<sub>''t''</sub>, ''p''<sub>''t''</sub>) at discrete times ''t'' in [0,''T''] where ''T'' is the length of the path. For a transition from A to B, (''r''<sub>0</sub>, ''p''<sub>0</sub>) is in A, and (''r''<sub>''T''</sub>, ''p''<sub>''T''</sub>) is in ''B''. One of the path times is chosen at random, the momenta ''p'' are modified slightly into ''p'' + ''δp'', where ''δp'' is a random perturbation consistent with system constraints, e.g. conservation of energy and linear and angular momentum. A new trajectory is then simulated from this point, both backward and forward in time until one of the states is reached. Being in a transition region, this will not take long. If the new path still connects A to B it is accepted, otherwise it is rejected and the procedure starts again.
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| == Rate constant computation ==
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| In the Bennett–Chandler procedure <ref>{{Cite journal|volume=68|pages=2959 |year=1978|doi=10.1063/1.436049|title=Statistical mechanics of isomerization dynamics in liquids and the transition state approximation|issue=6|last1=Chandler|first1=David|journal=The Journal of Chemical Physics}}</ref><ref>{{Cite book|first=C. H.|last= Bennett|title=Algorithms for Chemical Computations, ACS Symposium Series No. 46|editor-first= R. |editor-last=Christofferson |publisher=American Chemical Society|location=Washington, D.C.|year= 1977|isbn=978-0-8412-0371-6}}</ref> the rate constant k<sub>AB</sub> for the transition from ''A'' to ''B'' is derived from the correlation function
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| : <math>
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| C(t) = \frac{\langle h_A(0) h_B(t) \rangle}{\langle h_A \rangle}
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| </math>
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| where ''h''<sub>''A''(''B'')</sub> is the characteristic function of state A(B), and ''h''<sub>''A''(''B'')</sub>(''t'') is either 1 if the system at time ''t'' is in state ''A''(''B'') or 0 if not. The time-derivative C'(''t'') starts at time 0 at the [[transition state theory]] (TST) value ''k''<sub>''AB''</sub><sup>''TST''</sup> and reaches a plateau ''k''<sub>''AB''</sub> ≤ ''k''<sub>''AB''</sub><sup>''TST''</sup> for times of the order of the transition time. Hence once the function is known up to these times, the rate constant is also available.
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| In the TPS framework ''C''(''t'') can be rewritten as an average in the path ensemble
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| : <math>
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| k_{AB}^{TPS}(t) = \frac{d}{dt}C(t) = \frac{\langle \dot{h_B(t)} \rangle_{AB}}{\langle h_B(t') \rangle_{AB}} C(t')
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| </math>
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| where the subscript AB denotes an average in the ensemble of paths that start in A and visit B at least once. Time ''t''' is an arbitrary time in the plateau region of ''C''(''t''). The factor ''C''(''t''<nowiki>'</nowiki>) at this specific time can be computed with a combination of path sampling and [[umbrella sampling]].
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| == Transition interface sampling ==
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| The TPS rate constant calculation can be improved in a variation of the method called Transition interface sampling (TIS).<ref>{{Cite journal |volume=118|pages= 7762 |year=2003|doi=10.1063/1.1562614|title=A novel path sampling method for the calculation of rate constants|issue=17 |last1=Van Erp |first1=Titus S. |last2=Moroni |first2=Daniele |last3=Bolhuis |first3=Peter G. |journal=The Journal of Chemical Physics}}</ref> In this method the transition region is divided in subregions using interfaces. The first interface defines state A and the last state B. The interfaces are not physical interfaces but hypersurfaces in the [[phase space]].
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| The rate constant can be viewed as a flux through these interfaces. The rate k<sub>AB</sub> is the flux of trajectories starting before the first interface and going through the last interface. Being a rare event, the flux is very small and practically impossible to compute with a direct simulation. However, using the other interfaces between the states, one can rewrite the flux in terms of transition probabilities between interfaces
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| <math>
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| k_{AB} = \Phi_{1,0} \prod_{i=1}^{n-1} P_A (i+1|i)
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| </math>
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| where ''P''<sub>''A''</sub>(''i'' + 1|''i'') is the probability for trajectories, coming from state ''A'' and crossing interface i, to reach interface ''i'' + 1. Here interface 0 defines state ''A'' and interface n defines state B. The factor Φ<sub>1,0</sub> is the flux through the interface closest to ''A''. By making this interface close enough, the quantity can be computed with a standard simulation, as the crossing event through this interface is not a rare event any more.
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| Remarkably, in the formula above there is no Markov assumption of independent transition probabilities. The quantities ''P''<sub>''A''</sub>(''i'' + 1|i) carry a subscript ''A'' to indicate that the probabilities are all dependent on the history of the path, all the way from when it left ''A''. These probabilities can be computed with a path sampling simulation using the TPS shooting move. A path crossing interface i is perturbed and a new path is ''shot''. If the path still starts from A and crosses interface ''i'', is accepted. The probability ''P''<sub>''A''</sub>(''i'' + 1|''i'') follows from the ratio of the number of paths that reach interface ''i'' + 1 to the total number of paths in the ensemble.
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| Theoretical considerations show that TIS computations are at least twice as fast as TPS, and computer experiments have shown that the TIS rate constant can converge up to 10 times faster. A reason for this is due to TIS using paths of adjustable length and on average shorter than TPS. Also, TPS relies on the correlation function ''C''(''t''), computed by summation of positive and negative terms due to recrossings. TIS instead computes the rate as an effective positive flux, the quantity ''k''<sub>''AB''</sub> is directly computed as an average of only positive terms contributing to the interface transition probabilities.
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| == Time Dependent Processes ==
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| TPS/TIS as normally implemented can be acceptable for [[non-equilibrium]] calculations provided that the interfacial fluxes are time-independent ([[stationary process|stationary]]). To treat non-stationary systems in which there is time dependence in the dynamics, due either to variation of an external parameter or to evolution of the system itself, then other [[Rare Event Sampling|rare event]] methods may be needed, such as [[Stochastic process rare event sampling|Stochastic Process Rare Event Sampling]].<ref>{{Cite journal|volume=133|pages= 244101 |year=2010|doi=10.1063/1.3525099|pmid=21197970|title=Sampling rare events in nonequilibrium and nonstationary systems|issue=24|last1=Berryman|first1=Joshua T.|last2=Schilling|first2=Tanja|journal=The Journal of Chemical Physics}}</ref>
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| == Cited references ==
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| {{reflist}}
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| == More references ==
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| For a review of TPS
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| * {{Cite journal|volume=123|issue=1|pages=1–84 |year=2002|doi=10.1002/0471231509.ch1|title=Advances in Chemical Physics|isbn=0-471-21453-1|last1= Dellago|first1= Christoph|last2= Bolhuis|first2= Peter G.|last3= Geissler|first3= Phillip L.|chapter=Transition Path Sampling}}
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| * {{Cite journal|journal=Annual Review of Physical Chemistry|volume=53|pages=291–318|year=2002|doi=10.1146/annurev.physchem.53.082301.113146|title=TRANSITION PATH SAMPLING: Throwing Ropes Over Rough Mountain Passes, in the Dark|pmid=11972010|last1=Bolhuis|first1=Peter G.|last2=Chandler|first2=David|last3=Dellago|first3=Christoph|last4=Geissler|first4=Phillip L.}}
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| For a review of TIS
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| * {{cite thesis|first=D.|last= Moroni|title=Efficient sampling of rare event pathways: from simple models to nucleation|degree= Ph.D. |publisher= Universiteit van Amsterdam |year=2005|url=http://dare.uva.nl/record/149505 |chapter=DARE}}
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| == External links ==
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| * [http://www.berrymanscience.com/downloads.html C++ source code of an S-PRES wrapper program], with optional parallelism using [[OpenMP]].
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| {{DEFAULTSORT:Transition Path Sampling}}
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| [[Category:Computational chemistry]]
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| [[Category:Monte Carlo methods]]
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| [[Category:Molecular dynamics]]
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| [[Category:Theoretical chemistry]]
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Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. Some of the Wordpress development services offered by us are:. It sorts the results of a search according to category, tags and comments. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. Akismet is really a sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites.
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