Biologically Inspired Surface Physics: The HP Protein Model - Nanophenomena at Surfaces - page 174

,we
obtained the radius of gyration R g [ 49 ] and the Jaccard index q [ 50 ], here mea-
suring the similarity between any conformation and the ground states of an HP
sequence, i.e.,

By means of multicanonical sampling [ 48 ] given our estimates for g ( E )

max

E g = E min

C s , g
C s , g +

q = (7.1)

C s + C g

C s , g denotes the number of common (native) H-H contacts between a conforma-
tion s and the ground state g, and C s , C g are the numbers of H-H contacts found
only in s and g, respectively (the maximum stems from the degeneracy of ground
states). Figures 7.2 and 7.3 also show the temperature averages

for
sequences 2D100b and 3D103 and illustrates the complementary information of
these two quantities. While

R g and q

identifies
the folding transition to the native state and thus may serve as a suitable structural
order parameter for these kinds of systems. In case of sequence 3D103, the ground
state ( E

R g indicates the coil-to-globule collapse, q

=−

58) was excluded from the sampling (due to the difficulty of finding
this state) which results in only rather small Jaccard indices for T 0. This shows

that there are still large structural differences between conformations with E

→

=− 57

and the ground state with E 58.

Table 7.2 compares results obtained using various methods, and, if available,
the density of states g

=−

for common benchmark HP sequences. We also include
results from methods which were focused on the low-temperature range only, i.e.,
FRESS [ 34 ] and the variants of PERM (pruned-enriched Rosenbluth method) [ 30 ,
31 ] and do not provide the entire density of states. Except for the longest sequence
(3D136), we could confirm all minimum energy states found previously.

( E )

Table 7.2 Energy minima ( E min ) found by several methods for benchmark HP sequences in 2D
and 3D. The first column names the sequence (dimension and length); see Table 7.1 . Dashes mean
no data available. For abbreviations, see text
Seq. FRESS a PERM a

WLS EES MCCG MSOE

2D50 − 21 − 21 - - − 21 -

2D60 − 36 − 36 - - − 36 − 36

2D64 − 42 − 42 - − 42 − 42 − 42

2D85 − 53 − 53 - - − 53 − 53

2D100a − 48 − 48 - − 47 − 48 − 48

50 b

2D100b − 50 − 49 - − − 50 − 50

3D42 − 34 - − 34 − 34 - -

3D48 − 34 - − 34 - − 34 -

3D67 − 56 - - - − 56 − 56

3D88 − 72 - - - − 72 − 69

3D103 − 58 - − 56 - − 57 − 55

3D124 − 75 - - - − 75 − 71

3D136 − 82 - - - − 83 − 80

a Ground state search only
b g

( E ) not attained