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to SST. Furthermore, using the simulations SST/C and

SST/D we will study the effects of the chosen exchange

scheme (SEO vs DEO) and of the (overall) average exchange

probability P

) 〈P

〉 on the round trip rates τ

-1

and the

sampling speeds.

Sampling Speed. The main objective of tempering

methods is to enhance the sampling speed of the simulation.

A corresponding measure for the sampling speed is given

by an algorithm recently suggested by Lyman and Zucker-

man,

which we will denote as LZA. LZA integrates the

“volume” in conﬁgurational space sampled by a trajectory.

The average volume sampled during a given simulation time

span provides a measure for the sampling speed.

For 8ALA we deﬁne the conformational space by the eight

dihedral angles ψ

spanned by the backbone uni ts N

- C

- N

. From a trajectory of the eight-dimensional

tuples (ψ

,...,ψ

), LZA randomly chooses one tuple and

removes it from the trajectory together with all other tuples

lying within the sphere of predeﬁned radius r around the

chosen tuple. This procedure is repeated until all tuples of

the initial trajectory have been removed. The number of steps

required is a dimensionless measure for the conﬁgurational

volume V

sampled by the trajectory. Because this algorithm

is nondeterministic, it is repeated m times, and the corre-

sponding average number n

lza

of required steps is calculated.

For our analyses we choose r ) 25°8

and m ) 50. For a

fair comparison between ST and RE, we compute the

sampling speed per replica, i.e. one for ST and N for RE.

Results and Discussion

At the start of an ST simulation , initial estimates for the

weight parameters w

are needed. We determined these

estimates from preparatory simulations using both the

approximate trapezoid rule eq 12 and the asymptotically

unbiased WHAM formula eq 14. Now, a ﬁrst issue is the

reliability of the trapezoid rule, which we check using the

100 ps CST simulation.

Reliability of the Trapezoid Rule for CST. Table 2

compares the initial CST weights w

determined by the

trapezoid rule eq 12 from the preparatory CRE simulation

with the asymptotically unbiased values calculated by the

WHAM formula eq 14. For all T

the w

obtained by the two

formulas agree quite well. The errors ∆w

of eq 12 never

exceed 12% of k

, and, correspondingly, the uniformity

measures χˆ

are all close to 1.0. Hence, one expects a nearly

uniform sampling even if the weights are determined by eq

12. Thus, adaptation schemes, which are based on the

trapezoid rule and on the WHAM formula, respectively,

should be nearly equivalent for the given system. In the CST

simulation we, therefore, applied the trapezoid rule for the

periodical recomputation of the w

Representative for the eighteen weights, Figure 1(a) shows

the deviation of the weights w

and w

from their initial

values as a function of the simulation time. The exceptional

ﬁrst update w

new

) w

old

+ ln(t

), which can be seen as a

special case of the update scheme proposed by Zhang and

Ma,

sizably reduces both weights and reﬂects the nonuni-

form sampling within the preceding ﬁrst nanosecond of the

CST simulation. Here, the temperatures T

) 378 K and T

) 500 K apparently have been visited more frequently than

) 300 K. The following updates, which rely on eq 12,

lead to considerable changes of the weights, which, however,

become smaller toward the end of th e simulation. After

27 ns the weights seem to be converged within roughly (

0.1. This is approximately the same magnitude of error as

the one introduced by the trapezoid rule.

Figure 1(b) shows the measured (circles) and expecte d

(squares) uniformity measures χ

and χˆ

extracted from the

last 25 ns of the CST simulation as functions of the

temperature. In contrast, the uniformity data shown in Table

Table 2. Weights Determined from the CRE Simulation

i T

(trapezoid) w

(WHAM) ∆w

χˆ

0 300 K 0.0 0.0 -0.00 1.06

1 308 K 482.58 482.59 -0.01 1.05

2 317 K 991.97 991.99 -0.02 1.04

3 326 K 1468.66 1468.69 -0.03 1.03

4 336 K 1963.47 1963.49 -0.02 1.04

5 346 K 2425.14 2425.17 -0.03 1.03

6 356 K 2856.65 2856.68 -0.03 1.03

7 367 K 3299.53 3299.56 -0.03 1.03

8 378 K 3712.36 3712.41 -0.05 1.00

9 390 K 4131.64 4131.73 -0.09 0.97

10 402 K 4521.49 4521.56 -0.07 0.99

11 415 K 4913.99 4914.07 -0.08 0.98

12 428 K 5278.33 5278.44 -0.11 0.95

13 442 K 5642.42 5642.50 -0.08 0.98

14 456 K 5980.14 5980.21 -0.07 0.99

15 470 K 6294.02 6294.09 -0.07 0.99

16 485 K 6606.46 6606.55 -0.09 0.97

17 500 K 6896.68 6896.80 -0.12 0.94

The weights w

determined by the trapezoid rule eq 12 and by

the WHAM formula eq 14 together with the deviations ∆

and the

correspondingly predicted (cf. eq 16) uniformity measures χˆ

. The

WHAM weights were employed as starting values for the CST

simulation.

Figure 1. Uniformity of the temperature sampling in the CST

simulation. (a) Time evolution of the weights

and

with

respect to their initial values. (b) Uniformity measures ob-

served (χ

, eq 15, circles) and predicted (χˆ

, eq 16, squares)

after 27 ns at the temperatures

. The standard deviations

were estimated from MC trial simulations (see text for further

details). The dotted lines serve as a guide for the eye.

2852 J. Chem. Theory Comput., Vol. 5, No. 10, 2009 Denschlag et al.

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