Kinetic bottlenecks

For our standard PES surface $\sigma$ is independent of l. However, a real PES is unlikely to show such uniformity. For example, it has been suggested that for some proteins the number of available pathways may decrease near to the native state leading to the observation of kinetically-determined intermediates[245].

In this section we investigate the effect of making $\sigma$ dependent on l. In particular, we reduce the number of connections between l=1 and l=2, and between l=2 and l=3. Such a PES is illustrated in Figure 5.8a; it has $\sigma_{12}=\sigma_{23}={1\over 3}$ and so only 1 in 3 of the minima in level 3 are connected to a minimum in level 2. These changes do not affect the thermodynamics in any way (including the Landau entropy) but instead produce a kinetic bottleneck. In Figure 5.8b, the dependence of the folding time for the modified surface upon $\sigma_{12}$ is shown. t_f increases as the number of connections is reduced, but at the most by only one to two orders of magnitude. The slowing down of the dynamics at the bottleneck leads first to local equilibrium above the bottleneck (Figure 5.8c) and to a build-up of probability in level 3 and to a lesser extent in levels 4 and 5 (Figure 5.8d).

These results show that kinetic bottlenecks may significantly affect the folding rate, but to a smaller extent than other factors, such as the slope of the funnel or the multiple funnels that we shall consider next. Kinetic bottlenecks may be important in rationalizing the detailed dynamic behaviour of a particular system, but are unlikely to represent the difference between, for example, folding and non-folding proteins.

  

Figure 5.9: (a) One-dimensional cross-section through a PES with two funnels where l_node=6 The values of l' for the minima in the secondary funnel are as labelled. (b) Dependence of t_f upon the energy and l_node. The values of l_node are as labelled, and the curve for our standard PES (dashed line) has been added for comparison.