The slope of the funnel

In the energy landscape model of protein folding, the slope of the funnel (potential energy gradient) is a key parameter[246,250]. Here we analyse the effect of the slope by varying $\Delta E$. To keep E_f constant we must also vary E₂. For the case when $\Delta E$ is zero, i.e. a flat surface, E_l=10 for $l\ge 2$.It can be seen from Figure 5.7a that the slope has a dramatic effect on the folding time, and that a sufficiently steep slope is necessary for reasonably rapid relaxation to the global minimum. This behaviour can be explained from the relative rates of uphill and downhill transitions in our model. The barrier for going downhill is b, and for going uphill $\Delta E+b$.Therefore, the downhill transition rate is not significantly affected by the slope, but the uphill rate decreases rapidly as the slope is increased. For the flat PES taking a step to higher l is, ignoring the vibrational contributions to the rate, g times more likely than to lower l, causing the probability distribution to remain concentrated at large values of l. As a slope is introduced into the PES the uphill rate decreases relative to the downhill rate and the folding time begins to decrease rapidly. However, once the PES is sufficiently steep to make the uphill rate small compared to the downhill rate, further increasing the slope only has a limited effect. At large $\Delta E$ the folding time even begins to increase again because the level l=2 begins to have a low enough energy for it to have a significant equilibrium population.

  

Figure 5.7: (a) Variation of t_f with the slope of the PES. (b) Variation of the magnitude of the Landau entropy bottlenecks (solid lines) with $\Delta E$.The dashed line gives the magnitude of the Landau entropy difference between the two stable states. (c) Equilibration tree and (d) time evolution of the probabilities, P_l, for the flat PES ($\Delta E$=0) from an initial distribution P₁₀=1. All results are for E=26.41.

An equivalent explanation for the behaviour can be provided in terms of the Landau entropy. In order to keep E_f constant as the slope is decreased the energy of the intermediate levels must be increased. This has the effect of reducing the equilibrium probability for occupation of these levels. Therefore, as $\Delta E$ is decreased a Landau entropy bottleneck develops and then increases in depth (Figure 5.7b). The folding time increases rapidly as the bottleneck deepens, but has only a weak dependence on $\Delta E$ when there is a single Landau entropy maximum.

The flat PES would correspond to the Levinthal limit where the states are searched randomly, but for the variation of the vibrational frequency with l, which makes the minima in higher levels slightly more probable. For this PES, equilibration occurs initially at large values of l and proceeds away from the Landau entropy maximum. There is a large separation (of the order of 10¹³) between the time scales for equilibration within the large l entropy maximum and between the entropy maxima (Figure 5.7c), because of the deep Landau entropy bottleneck. This is evidenced by the probability flows across the PES (Figure 5.7d); P₁₀ and P₉ rapidly reach local equilibrium values, which then decay away as the ground state is slowly populated.

These results help to explain why potassium chloride clusters are able to relax to the rock-salt structures so rapidly[91]. Reaction pathways leading down to the rock-salt structures generally exhibit[252] $\Delta E\gg b$.

As $\Delta E$ is decreased E₂, which is simply the energy gap between the global minimum and the next lowest energy state, has to be increased to maintain E_f. Therefore, the efficiency with which the global minimum is found is reduced as E₂ is increased. This result does not necessarily contradict the view of Sali , however, since we can decouple factors in our model in a way that may not be possible for real PES's. However, it does show that even if the energy gap view were correct, it would require further explanation; it is not self-evident why an energy gap should guarantee kinetic foldability.

  

Figure 5.8: (a) Schematic depiction of a PES with a kinetic bottleneck. g=3, and $\sigma=1$except for $\sigma_{12}=\sigma_{23}={1\over 3}$.(b) Dependence of t_f on $\sigma_{12}$.(c) Equilibration tree and (d) time evolution of the probabilities, P_l, from an initial distribution P₁₀=1 for a PES with $\sigma_{12}=0.1$. All results are for E=26.41.