computer modeling in molecular biology.pdf

6 Theory of TransDort in Zon Channels 157Tho trajectories are generated using the set of initial conditions taken from thebiased ensemble. The first one is propagated forwards in time with these initial conditions;the second one is propagated backwards in time, starting with the same initialconfiguration (this can be done by inverting the sign of all the initial velocitiesin the system at t = 0). The forwards and backwards trajectories are calculated from0 to +T and - r respectively, and joined together as a single activated trajectory.In this manner a trajectory starting at time -r ending at time +T and goingthrough the transition state at time t = 0 with a positive velocity is generated. Thesimulation time, r must be sufficiently long such that the barrier crossings eventsare completed [86]. In practice the dynamics at the barrier top relaxes rapidly andT is relatively short. The procedure is repeated many times to obtain a large numberof activated trajectories. In the present application 100 activated dynamics trajectorieswere generated from - 1.0 ps to + 1.0 ps. Typical examples of activated trajectoriesare shown in Figure 6-6; the fate of the trajectories is determined in 0.5 ps orless. From the position of the ion at time f T the activated trajectory can be assignedto one in four types: reactant to product, product to reactant, reactant to reactantand product to product. The transmission coefficient, K, is calculated as the netnumber of “reactive” trajectories over the total number of activated trajectories,To obtain the rate constant k, the transmission coefficient K is combined with kTsT~41.6.3.4.2 Analysis of the ResultsThe quantities relevant to the transition rate are summarized in Table 6-2. CombiningkTsT = 2.1 x lo9 s-l with K = 0.11, a transition rate of k = 2.3 X is obtained.The TST rate is thus reduced by one order of magnitude. From the calculatedtransition rate it is possible to obtain an estimate of the maximum conductance ofthe gramicidin channel. In the /3-helix there is one free energy barrier per (L, D) unit.This implies that the total number of barriers in the complete gramicidin channelis around 15. This number is confirmed by more recent calculations on the potentialof mean force of Na+ along the axis of the dimer channel [40]. From Eq. (6-9) themaximal conductance, Amax, is 7 pmho. Errors in the transition rate and in theestimated channel conductance are dominated by the activation energy calculatedfrom the potential of mean force technique [38]. For example, a plausible error onthe order of k,T = 2.49 kJ/mol in the activation energy of 18.9 kJ/mol leads to afactor of 3 in the estimated transport rate. Thus, it is less the absolute value than