## 5 Results  
As discussed above, the general method of arriving at the values of \(K_{C}\) and \(\tau_{I}\) is the open-loop protocol of [PERSON] and [PERSON] [15]. From our earlier experience [6], however, we had already reasonable values. Accordingly, the final values of \(K_{C}\) and \(\tau_{I}\) and the weight factor \(w_{\rm a}\) of Eq. (4) were selected by runs using combinations of \(K_{C}=1\) and \(K_{C}=10\), \(\tau_{I}=2000\) and \(\tau_{I}=20,000\), \(w_{\rm a}=0.99\), \(w_{\rm a}=0.999\), and \(w_{\rm a}=0.9999\) and monitoring the fluctuation of the acceptance rates and of the tuned stepsizes. In
Legend: (a) \(N_{\rm MC}\) is the number of Monte Carlo steps. One million (1M) steps correspond to approximately 1000 move attempts of each torsion angle. (b) \(K_{C}\) and \(\tau_{I}\) refer to the parameters in the PID equations (Eq. (1-3)). (c) \(w_{\rm a}\) is the acceptance rate weight factor in Eq. (4). (d) The subscript \(a\) refers to acceptance rates in %. (e) The subscript \(\Delta\) refers to torsion stepsize in degrees: the change in a torsion angle is in the range [\(-\Delta/2\),\(\Delta/2\)]. (f) The standard deviations SD are calculated separately for each torsion angle sampled, from the start of each run. (g) The superscripts L and T indicate averages over torsion with local (concerted) moves and simple torsions, respectively. (h) The SD's are averaged over the respective torsion angle set and over the three separate runs (\(\langle\,\rangle\))  
\begin{table}
\begin{tabular}{|c c c c c c c c|} \hline \(N_{\rm MC}\) & \(K_{C}\) & \(\tau_{I}\) & \(w_{\rm a}\) & \(\langle\langle\) SD\({}_{\rm a}^{\rm T}\rangle\rangle\) & \(\langle\langle\) SD\({}_{\Delta}^{\rm T}\rangle\rangle\) & \(\langle\langle\) SD\({}_{\rm a}^{\rm L}\rangle\rangle\) & \(\langle\langle\) SD\({}_{\Delta}^{\rm L}\rangle\rangle\) \\ \hline  
engineering control scenarios, PIC performance is typically robust over a wide range of operating conditions [14]. By analogy, the parameters determined here are expected to be effective in a variety of simulation systems without modification. Three runs were performed with each parameter combination whose initial conformations were extracted from preliminary runs, at least \(5*10^{8}\) MC steps apart. The fluctuations were averaged separately for torsions that were sampled with local moves and those sampled as simple torsions. Table 1 shows the parameter combinations used, and the calculated standard deviations, of the acceptance rates and stepsizes after 25M (1M=\(10^{6}\)) and 50M MC steps. Standard deviations were calculated from 5M long block averages. Based on the calculated fluctuations we selected \(K_{C}=10\), \(\tau_{I}=2000\) for the controller parameters. As for the weight to be used for the calculation of the 'current' acceptance rate, \(w_{\rm a}=0.999\) appeared to work best for simple torsions and \(w_{\rm a}=0.9999\) for the local torsional moves. The evolution of the acceptance rates and step sizes in one of the three runs during tuning is given in Tables 2 and 3, resp., for the \(\phi\), \(\psi\) and \(\chi_{1}\) angles of the two middle residues (LYS\({}_{3}\) and Ac-LYS\({}_{4}\)). The backbone torsion angles (\(\phi\) and \(\psi\)) were sampled with local moves while the side chain torsion angles (\(\chi_{1}\)) were sampled as simple torsion. Note that the convergence characteristics of the side chains are expected to be similar to those of the terminal residue(s). The results show that the targeted acceptance rate (30%) is reached reasonably quickly and stays steady. torsions, adequately tuned stepsizes were obtained in the 25M long run, using \(w_{\Delta}=0.999\). However, the backbone torsions require both longer tuning runs and longer stretch to average. Using Eq. (6), at least 50M long tuning run is necessary with \(w_{\Delta}=0.99999\). Similarly, using simple averaging, averaging over the shorter length used (10M) produced by far the largest acceptance rate spread (after the 50M tuning runs).