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6.2. The Connectivity Graph of the Energy Landscape
We can think of the energy landscape as the collection of the basins of attraction
of several local minima of the potential. A basin of attraction of a local minimum
is the collection of points in phase space, whose overdamped dynamics converges to
that minimum. Accordingly, any trajectory in phase space evolves by going through
contiguous basins of attractions of dierent local minima. In this sense we can think
of the dynamics as a sequence of transitions between dierent minima separated by
energy barriers corresponding to saddles. This essentially amounts to approximate
the dynamics as a sequence of thermally activated transitions between dierent
local minima. We expect that such an approximation is a suitable one when the
temperature of the system (expressed in units of the Boltzmann constant) is small
enough to be comparable with the typical height of the saddles separating nearby
minima. Relying upon these considerations, we can assume that for suciently
(but no too) small temperatures the molecular dynamics can be eectively replaced
by a stochastic dynamics dened onto a connected graph. This approach has been
put forward in [6], where the graph has been dened such as its nodes are the
local minima and their local connectivity is determined by the existence of a saddle
separating them from other local minima. The transition rate between connected
minima can be determined by purely geometric features of the energy landscape
as a suitable generalization of the Arrhenius law to a high{dimensional space, i.e.,
Langer's formula, see Eq. (6.1).
Reconstructing the energy landscape of a protein model amounts then to rst
identify all the local minima of the potential energy. Since the number stationary
points of the potential energy typically grows exponentially with the number of
degrees of freedom, such a task is practically unfeasible for accurate all-atom po-
tential energies, but a reasonable sampling of the minima may become accessible
for minimalistic potentials. Minimalistic models are those where the polymer is
described at a coarse-grained level, as a chain of N beads where N is the number of
aminoacids; no explicit water molecules are considered and the solvent is taken into
account only by means of eective interactions among the monomers. Minimalistic
models can be relatively simple, yet in some cases yield very accurate results which
compare well with experiments [7, 8]. The local properties of the energy landscape
of minimalistic models have been recently studied (see e.g. Refs. [9{13]) and very in-
teresting clues about the structure of the folding funnel and the dierences between
protein-like heteropolymers and other polymers have been found: in particular, it
has been shown that a funnel-like structure is present also in homopolymers, but
what makes a big dierence is that in protein-like systems jumps between minima
corresponding to distant congurations are much more favoured dynamically [14].
As stated above, at suciently low temperatures, the dynamics of many systems
characterized by a rough energy landscapes can be summarized as a quick random
wandering inside metastable states intertwined by thermally activated jumps to a
new state. Typically a metastable state consists in a set of minima of the potential
