Geometry and Topology of Folding Landscapes - Biological Networks - page 106

energy whose basins of attraction are separated from the rest of the conguration
space by energy barriers suciently high to determine a separation of time scales
between the exploration of the set itself and the average time necessary to leave it. In
the limit of extremely low temperature each minimum of the potential corresponds
to a single metastable state. The rate of the transition between two potential energy
minima is dictated by the energy and the shape of the highest energy point on the
minimal energy path connecting them, which, for C 2 potentials, can be shown to
be a saddle of the rst order of the potential. The rate of the passage from the
i-th to the j-th minimum can be approximated, for realistic friction coecients
and suciently low temperatures, by the Langer estimate [15] which corrects the
standard Arrhenius term by an entropic factor depending on the curvature of the
potential both in the saddle and the starting minimum:

Q

N 0

k=1 ! (k)

i;j = ! ki;j V s i;j V i

k B T

i

exp (6.1)

Q

N 0 1

k=1 ! (k) ?i;j

where the ! (k)

i 's are the N 0 = dN3 non zero eigenfrequencies of the minimum
i (d is the spatial dimension), ! (k)

? 's are the N 0 1 non zero frequencies of the
saddle lying on the border between the basins of attraction of i and j, while ! k
is associated with the only expanding direction. Finally, is the dissipation rate,
while the exponential factor depends on the height of the energy barrier, V s i;j V i ,
normalized to the reduced temperature k B T, k B being the Boltzmann constant.
The analysis of saddles and their crossing rates becomes then essential to the study
of protein folding, because they summarize the only relevant dynamic contribution
as far as large conformational changes are concerned. In [16] it has been shown that
distant minima are connected with saddles characterized by higher jumping rates
in the case of a fast-folding hetero-polymer. This observation nicely ts with the
scenario of the diusion on the energy landscape of a fast-folder being helped by
the existence of fast connections between distant congurations.
Given this scenario it is then natural to study the folding dynamics of a protein
as a diusion process on a connectivity graph, that is a graph whose N min nodes
represent the basins of attraction of the minima of the energy landscape of the
protein and whose N sad edges represent connections between such basins, i.e., rst
order saddles. For a realistic description connections must be weighted according
to their dynamical relevance, which is well represented by the corresponding rate
of barrier crossing. In such a framework the entire system can be summarized by
a non{symmetric N min N min connectivity matrix whose element i;j equals
the jumping rate from the the i-th to the j-th minimum, if the two are directly
connected, and is 0 otherwise.
According to this formulation the probability that the protein resides in the
basin of attraction of the minimum i obeys the following master equation:

N min

X

N min

X

P i =

P j j;i P i i;j (6.2)

j=1 j=1