Transition state ensembles

Select a random point (here isobutcine molecule) of a transition path. For this point [x, Ukinl. we generate ui random momenta p from a Maxwell-Boltzmcinn distribution. These momenta are rescaled in such a way that the toted initial kinetic energy of the system (Ukin) is a constant. The value of ni should be large enough to have some reasonable initial statistics on the probability of reaching regions A and B. 

For each ni momenta, the equations of motion are integrated. If the system reaches A or B, or when the time of the integration exceeds t, the integration is stopped. 

From the ui integrations, we obtain estimates for pa and pb, which are respectively pa and Pb  

---

Figure 5 The transition state ensemble is the set of trajectories that are crossing the transition state from reactant to product at equilibrium (shown as black dots). There are four types of trajectories, shown top to bottom in the diagram. (1) Starting as reactant, the trajectory crosses and recrosses the transition state and is deactivated as reactant. It does not add to the reactive flux. (2) Starting as reactant, the trajectory is deactivated as product. It adds +1 to the reactive flux. (3) Starting as product, the trajectory crosses and recrosses the transition state and is deactivated as product. Such a trajectory must be subtracted from the ensemble, so it counts —1 to the reactive flux. (4) Starting as product, the trajectory is deactivated as reactant. It does not contribute to the reactive flux.

Fig. 3.5 Wrapping patterns for chain conformations occurring during the burst phase (cf. Fig. 3.4) of protein-G variant generated at 6.4 x 10 4 s upperpanel) and 6.5 x 10 4 s lowerpanel). These conformations commit the chain to fold are part of the transition state ensemble, and do not contain exclusively native interactions, as it becomes apparent in the upper panel. Reprinted from [35], with permission from Elsevier...

The path ensemble, as created by the transition path sampling methodology, is a statistically representative collection of trajectories leading from a reactant region to a product region. Further analysis of this ensemble of pathways is necessary to obtain rate constants, reaction mechanisms, reaction coordinates, transition state structures etc. In this section we will describe how to analyze the path ensemble by determining transition state ensembles, and how to test proposed reaction coordinates using committor distributions. 

Fig. 9. All configurations on transition pathways that have a committor of ps = 0.5 are members of the transition state ensemble. Each transition path contributes at least one configuration but possibly more to the transition state ensemble. The dashed line is the separatrix (or isocommittor 1/2 surface)...

Here, P pb) is the probability (density) for finding the committor ps in the ensemble g = g. If this distribution is peaked around pb = 0.5, the constraint ensemble g = g is located on the separatrix and coincides with the transition state ensemble. In this case, g is a good reaction coordinate, at least in the neighborhood of the separatrix. This is illustrated in Fig. 11a. Other possible scenarios for the underlying free energy landscape result in different committor distributions, and are also illustrated in Fig. 11. The committor distribution can thus be used to estimate how far a postulated reaction coordinate is removed from the correct reaction coordinate. An application of this methodology can be found in [33], where the reaction coordinate of the crystallization of a Lennard-Jones fluid has been resolved by analysis of committor distributions. 

Many Folding Pathways the Search for the Transition State Ensemble... 

As mentioned in the introduction, there is not a single folding pathway. The flexible protein might choose from many different pathways via many different low populated intermediates or transition states, all leading eventually to the native state. Hence, there is not only one transition state, but instead a whole ensemble of states the transition state ensemble (TSE). To explore the... 

In stage 2, one calculates a transmission coefficient for the stage-1 rate constant. This transmission coefficient is calculated by an ensemble average over CVT//rOMT calculations for various reaction paths (labeled a = 1,2,. ..,A ) that pass through samples from stage-1 transition state ensemble. In particular, the final ensemble-averaged... 

Clement C, Nymeyer H, Onuchic JN. 2000. Topological and energetic factors What determines the structural details of the transition state ensemble and en-route intermediates for protein folding An investigation for small globular proteins. Journal of Molecular Biology 298 937. 

For each intergronp transition state ensemble in turn, merge the two directly connected groups together if their forward and backward free energy barriers both lie below a fixed threshold value,... 

Calculate the free energy of each changed group, and of the changed intergroup transition state ensembles. 

If the latter is the case, the time sUce is accepted as a member of the transition state ensemble. Otherwise it is rejected. 

Advanced Topics The Transition State Ensemble for Folding 201... 

ADVANCED TOPICS THE TRANSITION STATE ENSEMBLE FOR FOLDING... 

In this section on advanced folding topics, we provide a broad overview of computational approaches used to determine the transition state ensemble (TSE) for folding. Comprehensive reviews are available elsewhere on closely related topics, such as experimental characterization of the transition states and free energy landscape approaches to the transition states. 

---