Rare events 333distribution by performing a short run with a thermostat coupling, even if such a
relaxation process has been shown to be formally unnecessary (Jarzynski, 2004).
8.5 The problem of rare events
In Section 8.3, we alluded to the rare-event problem associated withlarge barriers
separating important minima on a potential energy surface. Such energy surfaces are
known asrough energy landscapesand characterize, for example, proteins, glasses,
and polymers. As we noted in Section 8.3, when high barriers separate the minima,
the probability that a fluctuation capable of driving the system fromone minimum to
another over such a barrier will occur becomes exponentially small with the ratio of
the barrier height tokT. Consequently, such an event is described as arare event(see
Fig. 7.5). In the remainder of this chapter, we will discuss this problem at length to-
gether with methods for enhancing conformational sampling on rough potential energy
surfaces.
In order to illustrate the concept of “roughness,” consider the alanine dipeptide,
shown in Fig. 3.8. This small dipeptide can exist in a number of stable conformations,
which can be characterized by two backbone dihedral anglesφandψknown as the
Ramachandranangles (see Fig. 3.8). Fig. 8.5 shows the two-dimensional free energy
surface in these two angles obtained by “integrating out” the remaining degrees of
freedom (we will discuss methods for generating such surfaces later in this chapter).
The figure shows that there are four pronounced minima on the surface for the par-
ticular model employed, indicating four stable conformations with different relative
free energies. These relative free energies can be used to rank the minima in terms
of their thermodynamic importance. The full free energy surfacealso contains the lo-
cations and heights of barriers, from which information about transition states and
rates of conformational changes. If we now consider that the free energy surface of just
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A(y,f) [kcal/mol]y fA(y,f) [kcal/mol]Fig. 8.5Free energy surface in the Ramachandran angles for an alanine dipeptide in solution.