Chapter 6 Relationship between the protein dynamics and perme-
6.2 Issues in the calculations of the free-energies
6.2.2 Analysis of the issues
A generalisation of the issues encountered in the investigation of permeation helps to determine the appropriate strategy to approach the task. Biological systems are governed by complex free-energy landscapes with barriers of varying sizes, from the very low to the practically insuperable within the sampling. FES are computed to address specific processes that involve transitions which are generally small with
B A
D C
Non-conductive conformation
b) Evolution on the orthogonal degrees of freedom a) Projections of the FES
Non-conductive conformation Starting point B A D C B A D C B A D C R64-D80 interaction
Figure 6.3: Multi-dimensional well-tempered metaD calculation performed over
the CVs zK12,zK3andxyK3. No additional restrains applied.
respect to the complete dynamics of the system. Even relatively complex processes usually require the sampling of specific subspaces of the phase space. Moreover, a complete sampling would result in undesired FESs.
An example is suggested in Fig. 6.4, which considers the simple case of an umbrella sampling applied to explore a two-dimensional FES where only one degree of freedom is biased. Umbrella sampling is used because it allows a simple view but, as said, similar problems must be considered common to most of the methods used to compute free energies. As described in Chap.3, umbrella sampling and the other biased methods involve a dimensional reduction which theoretically requires a com- plete ergodicity and mixing of the degrees of freedom orthogonal to the biased CVs. Even when exists in theory, ergodicity is not true in practice for biological systems, due the limited length of the sampling imposed by the computational resources. This lack of ergodicity does not necessary result in a loss of accuracy in the event that it does not produce non-stationary processes. Therefore it is often exploited in free- energy calculations, for instance when one among the many activities of a protein is investigated without sampling denaturation or unfolding. In the case of permeation in KcsA, an extended sampling of the protein phase space, even without includ-
ing the extreme cases such as unfolding, would result in an overall non-conductive FES, given the low probability of the conductive state. It means that this lack of ergodicity is usually exploited in free-energy calculations to confine the sampling to within an arbitrary region of interest which is eventually able to deliver as many features of the investigated process as possible. This implicit approach does not al- ways work, especially when the barriers involved in the process are comparable with undesired events, such as in the case of permeation and inactivation in KcsA. When this happens important issues can easily arise which, in turn, can lead to undesired results. Three common cases are described in the Fig. 6.5, again referring to the simple example of a one-dimensional US over a two-dimensional system:
A: Issues in the initialisation of the subsequent umbrella sampling simulations; B: Risk of being trapped outside the region of interest during the sampling; C: Incoherent FES coming from improper sampling.
Case ``A'' is delivered by the diabatic displacements induced by external potentials associated with the different methods. In the example, the potential which was used to move the system between two subsequent windows. This step requires particu- lar care for a good application of a biased method on biological systems because a displacement along the direction of the CVs is usually accompanied by unknown displacements along the remaining degrees of freedom, due to the high correlations which characterise these systems. When a previous extensive knowledge of the FES is missing, a common way of initialising new windows in US is to apply an external potential strong enough to guarantee the system will move the free-energy along an upwards gradient in the direction of the CV, and then relax. Very likely, this is the method that has been applied in most of the investigations on permeation and se- lectivity performed on ion channels.20,36,78,121,141 However, this additional potential
can perturb the system and push it towards unknown positions of the phase space when the barriers along the CVs and the orthogonal degrees of freedom have compa- rable heights. The displacements described can have dramatic effects, even if these are very small in the phase space due to the huge dimensionality of the investigated systems and the high correlations. Another common procedure used in the investi- gation of ion channels39,54is to perform the displacements by manual modification of
the coordinates of the elements of the system. While, depending on the experience of the researcher, it can helps to avoid huge inconsistencies, it does not guarantee that the starting points of subsequent windows will belong to consistent pathways. This means that it is practically impossible to completely prevent the occurrence of issues similar to case ``A''. Thus, a similar approach does not have a general application.
Cases ``B'' and ``C'' derive from the natural complexity of the underlying FES, and their identification can be very difficult. The FESs appear consistent and the calculations are very likely reproducible (especially for case ``C'').
Several approaches can be applied to avoid the described issues. The sim- plest way to address issues similar to case ``A'' is to initialise the free-energy cal- culations according to extensive unbiased simulations. However, risks of experienc- ing issues similar to case ``C'' remain unchanged. Moreover, typically an extensive preliminary knowledge of the process and, especially, of the pathways which con- nect the different states is absent.86,149 This is usually the case for research on ion
channels, in which the recognition of detailed pathways is one of the ultimate ob- jectives.18,36,45,54,69,78,92,131,141 An alternative approach is the identification of the
adiabatic pathways, followed by the reconstruction of the FES.45,86,137,143,153 Well-
temperad metaD (sec.2.2.3) permit the reconstruction of quasi-adiabatic pathways and a straightforward derivation of the FES, which is theoretically true, without additional steps.13 In metaD the construction of the biased potential is not truly
adiabiatic, since the deposition of the Gaussians is not continuous and small pertur- bations occur when a new Gaussian is added. Preliminary tests demonstrated that the initial height of the Gaussiansw0 is the most important parameter for avoiding
the occurrence of similar events, because higher w0 caused problems similar to case
``A''. For this reason very small w0 (0.005 kcal/mol) was selected. In any event, as
previously shown, a straightforward application of metaD also failed because the method is very sensitive to being trapped in deep minima located outside the region of interest, similar to case ``B''.
In general, issues similar to cases ``B'' and ``C'' are observed in FES calculations aiming to complex systems by using any biased method and they related to the fact that the investigations often require to be focused on a small subspace of the whole phase space. The confinement represents a layer on the top of the free-energy calculations and it is strictly system-dependent.
In ion channels, similar procedures have been widely applied to study per- meation and selectivity mechanisms by means of restraints on the dynamics of the ions4,5,69,92,109,131, and sometimes to specifically investigate the behaviours of partic-
ular states of the proteins.18,20 However similar transparent approaches have rarely
been applied over the conductive states to improve the coherence of the samplings which, in consequence, were very often kept short. It was mainly due to a lack of understanding of the channel behaviour and the inactivation mechanisms. An at- tempt to confine the sampling to a conductive state was made in recent works via restraints on the V76 dynamics, which have been suggested being related to inacti-
Starting point of the window Displacements to initialise the following window
Legend
Sampled region
FES of the system
Biased CV
Orthogonal degree of freedom
Subspace of interest
Biased CV
Ener
gy
Expected FES
Ideal Umbrella Sampling
US windows
Biased CV
Orthogonal degree of freedom
Subspace of interest
A
Biased CV Ener gy Obtained FES US windows Biased CVOrthogonal degree of freedom
Subspace of interest
B
Biased CV Ener gy Obtained FES US windows Biased CVOrthogonal degree of freedom
Subspace of interest
C
Biased CV Ener gy Obtained FES US windows Biased CVOrthogonal degree of freedom
Subspace of interest
vation.77,78,141 In any event, these local restraints do not address the fundamental
problem of conductive/non-conductive equilibrium in the SF, with the consequence that the related complex dynamics of the pore are mostly maintained. Moreover, they directly limit the flexibility of the SF in such a way that important properties of the SF might be hidden during the sampling.
In the present work, the analysis on permeation and selectivity was preceded by an extensive investigation of the molecular determinants of inactivation. The confinement of the sampling to a conductive conformation of the channel was trans- parently implemented by restraining the dynamics of the network behind the SF. Since the restrained residues do not belong to the SF, most of the pore flexibility and dynamics could be preserved, including the variability of V76 residues and their flipping.