The Energy Landscape

The next step in the prediction of metastable compounds consists in the exploration of this energy landscape. For T = 0 K, it is very clear what constitutes a (meta)stable structure It is the one, and only one, configuration that is associated Avith a local minimum of the energy hypersurface. Thus, for T = 0 K our task is to determine all local minima of the energy landscape. In contrast, for T 0 K, equili- 

Z(R) is the partition function for the region R of phase space, and 

If Xesc(R) Xeq(R), it would be possible to equilibrate the system within R before such a transition takes place. Thus one could replace the time-average of any measured observable along some trajectory within R by an ensemble average within R, and we define the system to be locally ergodic 5 in R on the time scale 

Xeq(R). If now the time scale on which we perform our measurements (e.g. a powder diffraction measurement), tobs is smaller that Xesc(R)  

In some cases, a shortcut is possible, if the region R is separated by energy barriers of height Eb (measured with respect to the minimum of R) from the rest of the energy landscape. Then, one often finds an Arrhenius law for the escape time 

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Wolynes P G 1996 Symmetry and the energy landscape of biomolecules Proc. Natl Acad. Sci. (USA) 93 14 249-55... 

Bryngelson J D, Onuchic J N, Socci N D and Wolynes P G 1995 Funnels, pathways, and the energy landscape of protein folding a synthesis Profe/ns 21 167-95... 

Hans Prauenfelder, Sthephen G. Sligar, and Peter G. Wolynes. The energy landscape and motions of proteins. Science., 254 1598-1603, 1991. 

Fig. 10.27 Schematic representation of the energy landscape for protein folding. (Figure adapted from Onuchic ] N, Z Luthcy-Schulten and P Wolynes 1997. Theory of Protein Folding The Energy Landscape Perspective. Annual Reviews in Physical Chemistry 48 545-600.)...

Bryngelson J D, J N Onuchic, N D Socci and P G Wolynes 1995. Funnels, Pathways, and the Energy Landscape of Protein Folding A Synthesis. Proteins Structure, Function and Genetics 21 167-195. 

Figure 1 A schematic view of (a) a low temperature simulation that is confined by high energy baiTiers to a small region of the energy landscape and (b) a high temperature simulation that can overcome those barriers and sample a larger portion of conformational space.

Solving the master equation for the minimally frustrated random energy model showed that the kinetics depend on the connectivity [23]. Eor the globally connected model it was found that the resulting kinetics vary as a function of the energy gap between the folded and unfolded states and the roughness of the energy landscape. The model... 

Figure 4 The energy landscape of the pnon protein (Pi P) (residues 124-226) in vacuum, obtained by principal coordinate analysis followed by the minimal energy envelope procedure. Two large basins are seen. One basin is associated with the native Pi P conformation the other is associated with partially unfolded conformations.

JN Onuchic, Z Luthey-Schulten, PG Wolynes. Theory of protein folding The energy landscape perspective. Annu Rev Phys Chem 48 545-600, 1997. 

In the previous section we discussed how a Hopfield net can sometimes converge to a local minimum that docs not correspond to any of the desired stored patterns. The problem is that while the dynamics embodied by equation 10.7 steadily decreases the net s energy (equation 10.9), because of the general bumpiness of the energy landscape (see figure 10.5), whether or not such a steady decrease eventually lands the system at one of the desired minima depends entirely on where the system begins its descent, or on its initial state. There is certainly no general assurance that the system will evolve towards the desired minimum. 

Levy, Y, Jortner, J., and Becker, O. M. (2001). Solvent effects on the energy landscapes and folding kinetics of polyalanine. Proc. Natl. Acad. Sci. USA 98, 2188-2193. 

Fig. 6. Schematic energy landscape for protein folding (folding funnel). The approximate regions of the energy landscape that correspond to the various partly folded states of apoMb are indicated on the right.

Straub, J.E. Thirumalai, D., Exploring the energy landscape in proteins, Proc. Natl Acad. Sci. USA 1993, 90, 809-813... 

Garcia, A. E. Sanbonmatsu, K. Y., Exploring the energy landscape of a beta hairpin in explicit solvent., Proteins 2001, 42, 345-354. 

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