Energy landscape

Energy landscape
A hypothetical energy landscape, the x,y-plane serves to locate the various conformational states of the folding protein, with the Gibbs free energy shown on the vertical z-axis, a decrease in free energy indicative of increased stability, the native protein being the normal state or most stable configuration. [4]
In thermodynamics, energy landscape, as they are referred to in protein thermodynamics, or
"free energy map", as they are called in drug-receptor thermodynamics, is a computer-generated map that shows the three-dimensional regions or landscapes of local free energy minimums in systems of particles (atoms, ions, molecules, chemical species, proteins, human molecules, etc.). [1] Free energy maps often show that the most populated cluster of entities often occur around regions of the lowest free energy local minimums. [2]

An energy landscape, in simple terms, is a mapping of all possible conformations of a molecular entity (or the spatial positions of interacting molecules in a system) and their corresponding energy levels, typically Gibbs free energy, on a two- or three-dimensional Cartesian system.

It seems likely that the free energy map (landscape) concept arose following the 1931 development of the "potential energy surface", developed during the work of hydrogen atom - hydrogen atom interactions, by Mexican-born American theoretical chemist Henry Eyring and Hungarian-born English physical chemist Michael Polanyi.

In this sense, the energy landscape conception was originally proposed to study potential energy in physical systems and had its first rigorous development in the context of Hamiltonian systems (such as possibly summarized in the work of Vladimir Arnold). [9]

In the mid 1980s, energy landscape models were being employed in the study of protein folding. The theory of RNA free energy landscapes was being utilized as early as 1993.

Potential energy surface vs free energy surface
The first unified textbook on the subject of energy landscapes was the 2003 Energy Landscapes: Applications to Clusters, Biomolecules, and Glasses by David Wales. In it Wales explains the difference between a potential energy surface and a free energy surface: [13]

Potential energy surfaceFree energy surface
The potential energy surface, where V, the potential energy, is an explicit function of just two internal coordinates for the linear hydrogen atom exchange reaction, shown above, namely the internuclear distances RAB and RBC, in Angstroms. A trajectory that runs close to the valley bottom is marked in red. The free energy surface schematic for lysozyme, shown above, is constructed to match experimental observations by averaging over all the solvent and protein coordinates except for Q and P. G is the Gibbs free energy, Q is the fraction of native contacts present, and P represents some measure of compactness. The three folding pathways are superimposed on the surface: the yellow trajectory represents the fast folding, the green represents a slower path involving a higher free energy barrier, and for the red pathway the system first explorers a partly folded state before escaping to the route for fast folding.

Protein folding
In protein thermodynamics, the free energy landscape model was employed beginning in the late 1980s to understand protein folding through the work of Ken Dill (1985), followed by Byrngelson and Wolynes (1987), among others. [5]

To construct an energy landscape protein conformations are sampled by iteratively perturbing existing conformations, starting typically with a confirmation of the known native state or “target conformation” (below left), and then sampling neighboring conformations (middle two), of which resultantly there tends to be a denser coverage near the target. These samples yield a three dimensional free energy graph of the different geometries of the molecular structure of the protein (below right): [7]

energy landscapes (sampling)

Landscape types
The following are various landscapes made by Danny Heap of the University of California, at San Francisco, for the 1998 article “Protein Folding in the Landscape Perspective” by Hue Chan and Ken Dill: [5]

energy landscape figure 5
energy landscape figure 12
General shapes of landscapes: (a) The HP landscape is shown pictorially as having a kinetic trap, A is a throughway folding trajectory whereas path B passes through the trap; (b) the HP+ landscape is smooth, unfolding paths are simply the reverses of the folding paths shown; (c) shows a landscape on which all folding molecules must pass through an obligatory folding intermediate, represented by the "moat" in the figure.
Different folding scenarios: The vertical axis is internal free energy. Each conformation is represented as a point on the landscape. The two horizontal axes represent the many chain degrees of freedom. (a) shows a rugged landscape with hills and traps, folding kinetics is likely multi-exponential. (b) shows a landscape in which folding is faster than unfolding. A is a throughway folding path, whereas unfolding chains (path B) must surmount a barrier in order to reach the most stable denatured conformations. (c) shows a landscape in which folding is slower than unfolding. Most folding paths (path A) pass through a kinetic trap, whereas some low-lying denatured conformations are readily accessible from the native state during unfolding (path B).

Energy landscape (model)energy landscape figure 14
energy landscape figure 15
Funnelscape for a fast folding protein: Folding is limited by the rate of meandering downhill.
Champagne glass landscape: Illustrates how conformational entropy can cause "free energy barriers" to folding. The "bottleneck" or rate limit to folding is the aimless wandering on the flat plateau as the chain tries to find its way downhill (b) Serpin scenario shows a landscape with a deep kinetic trap on the left (A), which is easily accessible from the open conformations. Chain trapped in this deep local minima anneal to the global minimum (B, in the middle) only very slowly. This corresponds to the folding of some serpins such as PAI-1.

Drug-receptor encounter complexs
In drug-receptor thermodynamics, free energy maps are employed to map the free energy changes or values involved in an encounter complex between to molecules, such as a drug and a receptor. The relative orientation of two molecules in an encounter complex can be specified, it is said, by five Euler angles, whereby the free energy map or landscape is a continuous function in space of these fiver variables, obtained by smoothing the discrete free energy values on a grid of on the surface of the receptor. [1]
fitness landscape
Two-dimensional fitness landscape, where X1 and X2 are coordinates in a space; the global peak is the highest fitness in the space; a local peak is a point of all whose neighbors, defined by some neighbor relationship, are of lower fitness. [11]

Fitness landscapes
Some, such as Hungarian biochemist Peter Csermely (and possibly Stuart Kauffman), have argued that the concept of an ‘energy landscape’ was introduced, underlying, in the 1932 work of American geneticist Sewell Wright, who viewed evolution as an optimization process on a "fitness landscape". [3] This, however, seems to be a retrospect appropriation. There doesn’t seem to be an actual formal proof or article equating free energy with fitness, but by 1997 statements can be found, in discussions of Wright’s fitness landscape models, to the effect that: [11]

“In physics, fitness is analogous to free energy, the minimum of which determines the configuration of a system.”

Wright introduced the metaphorical concept of adaptive landscapes, visualizing evolution as a hill-climbing (local optimization) process. With the development of formal thermodynamic methods, in the years to follow, such as finding the minimum free energy in spin glasses, some began to develop evolutionary dynamic models using a mix of both theories (adaptive landscapes + energy landscapes = fitness landscapes), to the logic that an analogy can be employed whereby just as a physical system minimizes its free energy as it climbs down a an energy surface, so to does an evolving population maximize its fitness as it climbs a “landscape” to an adaptive peak. In the latter model, a fitness landscape or fitness function is defined by assigning a reproduction rate to every point in genotype space. [10]

Free energy landscape (2000)
South African physical chemist Adriaan De Lange's 2001 free energy landscape of evolution, employing a mixture of chaos theory, Prigoginean bifurcation theory, order-disorder logic, time (past vs future), free energy barrier, path functions, and discussions of high and low values of entropy change.

Human chemistry | thermodynamics
See main: Human chemistry
In 2001, South African physical chemist Adriaan de Lange developed a Gibbs free energy theory of human evolution (with, to note, underlying spirituality implications) and drew out various free energy landscape diagrams (possibly culling logic from Stuart Kauffman, whom de Lange often cites), such as the one pictured adjacent, which he says is "a simplified version of the image in my mind", wherein the vertical axis represents free energy, the 'Urphaenomen' or prototype of all functions having limits. In explaining his plot, de Lange states:

“All fitness functions, how imaginative we may create them, depend on free energy as the mother of them all. No change is possible without free energy changing somewhere in the universe, whether in the system SY or in the surroundings SU. The free energy F is not merely a theoretical concept of the imagination. It is a quantity based on innumerous measurements and calculations in the realm of physical chemistry. It is a quantity of bewildering consequences, the nemesis of many a student in physical chemistry.”

In describing his graph further he explains:

“Please notice the two shaded regions, designated past and future digestions. (Forget for a moment the thickest lines called A, B and AB as well as the strange barrier in the unshaded region where the two lines join.) The free energy in both shaded regions increases (the landscape bulges upwards to a summit). The difference is that the "hill of the past" is often lower than the "hill of the future". (I have actually drawn the future hill much higher than the past hill so that you can easily observe it.) Looking towards all the free energy hills of the future, there is a gradual elevation along the future hills. It is as if the system is gradually climbing the rugged landscape called free energy F from sea level towards a high mountain "Everest" beyond the horizon. "Steigerung" (staggering) is necessary to do so. Specialization, on the other hand, will cause the system to stay meandering within in a "patch" (region) containing only some hills this side of the horizon.”
Moral landscape (thermodynamics)
American neuroscience philosopher Sam Harris's 2010 visual conception of "moral landscapes" posited to be explainable by science, shown with the standard criterion for constitutes "natural" and "unnatural" for earth-bound reactions and processes. [15]

In 2007, American chemical engineer Libb Thims argued that the logic of free energy maps can be employed in the study of encounter complexes between humans, such as in the male-female reaction. It can be shown, for instance, that the map of the estate that the Captain makes in Goethe's celebrated novella Elective Affinities, is in fact a prototype free energy map or energy landscape. [8]

Further reading
● Yoo, Soohaeng. (2004). Energy Landscapes: Application to Silicon Nanoclusters and Protein Stabilities. University of Nebraska.
● Janke, Wolfhard. (2008). Rugged Free Energy Landscapes: Common Computational Approaches in Spin Glasses, Structural Glasses, and Biological Macromolecules. Springer.

1. (a) Raffa, Robert B. (2001). Drug-Receptor Thermodynamics - Introduction and Applications (ch. 27: Thermodynamic Maps of Receptor-Ligand Pairs Reveal How Some Proteins Bond, pgs. 581-92, subsection: “Free Energy Maps”, pgs. 584-85). New York: John Wiley & Sons.
(b) Camacho, C.J. Weng Z., Vajda S., et al. (1999). “Free Energy Landscapes of Encounter Complexes in Protein-Protein Association.” Biophys. J. 76: 1168-78.
2. Floudas, Christodoulos A. and Paradalos, Panos M. (2002). Optimization in Computational Chemistry and Molecular Biology, (pg. 252). Springer.
3. (a) Wright, Sewell. (1932). “The Role of Mutation, Inbreeding, Crossbreeding, and Selection in Evolution”, in: Proc. Sixth International Congress on Genetics, pgs. 355-66.
(b) Kauffman, Stuart. (1989). “Adaption of Rugged Fitness Landscapes”, in: Lectures on the Sciences of Complexity, ed. Daniel L. Stein, Addison-Wesly, 1: 527-618.
4. Csermely, Peter. (2006). Weak Links: the Universal Key to the Stability of Networks and Complex Systems (5.2: energy landscapes, pgs. 121-24). Springer. 2009, 2nd ed.
5. Dill, Ken A. (1985). “Theory for the Folding and Stability of Globular Proteins”, Biochemistry, 24: 1501-09.
6. Chan, Hue S. and Dill, Ken A. (1998). “Protein Folding in the Landscape Perspective: Chevron Plots and Non-Arrrhenious Kinetics” (abstract), Proteins: Structure, Function, and Genetics, 30(1): 2-33.
7. Protein folding FAQ – Department of Computer Science and Engineering, Texas A&M University.
8. Thims, Libb. (2007). Human Chemistry (Volume One) (key term: “free energy map”, pgs. 143,-44, 153, 252). (preview), (Google books). Morrisville, NC: LuLu.
9. See: Vladimir Arnold, Mathematical Methods of Classical Mechanics (Springer, 1978); Ralph Abraham and Christopher Shaw, Dynamics: the Geometry of Behavior (Aerial Press, 1983); Gregoroire Nicolis and Ilya Prigogine, Exploring Complexity: an Introduction (Freeman, 1989).
10. Crutchfield, James P. and Schuster, Peter. (2003). Evolutionary Dynamics (6.1: Landscapes, pgs. xxiv-vi). Oxford University Press.
11. Moos, W.H., Moss, Walker H., Pavia, Michael R., Kay, B.K., and Ellington, A.D. (1997). Annual Reports in Combinatorial Chemistry and Molecular Diversity (fitness landscapes, pgs. 126-). Springer.
12. De Lange, A.M. (2001). “Fitness Landscape and other Landscapes” (threads: LO27222), 09/03/01 –
13. Wales, David. (2003). Energy Landscapes: Applications to Clusters, Biomolecules, and Glasses. Cambridge University Press.
14. (a) Harris, Sam. (2010). The Moral Landscape (pg. moral landscape definition, pg. 7). Free Press.
(b) Moral landscape (video image) -
(c) Guggenheim, Eduard, A. (1933). Modern Thermodynamics by the Methods of Willard Gibbs (pgs. 5, 17). London: Methuen & Co.

Further reading
● Dill, Ken A. and Chan, Hue S. (1997). “The New View of Folding: from Levinthal to Pathwas to Funnels” (abstract), Nature Structural Biology, 4(1).

External links
Energy landscapes – Wikipedia.

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