Sampling of Minima and Saddlepoints
Many problems in computational chemistry require a concise description of the large-scale geometry and topology of a high-dimensional potential surface. Usually, such a compact description will be statistical, and many questions arise as to the appropri ate ways of characterizing such a surface. Often such concise descriptions are not what is sought; rather, one seeks a way of fairly sampling the surface and uncovering a few representative examples of situations on the surface that are relevant to the a ppropriate chemistry. Some specific examples include finding snapshots of crucial or typical configurations or movies of kinetic pathways. This allows what one might call an artistic description of the chemical situation. Such a description is often lo oked down upon by quantitative scientists as being "anecdotal," but it is important not to cut ourselves off from any route to understanding. To make this point one might compare the kinds of understanding of ancient cultures that are obtained from the n umerous scholarly statistical studies of bookkeeping accounts and what we learn from the great paintings of the same periods, which give us different perspectives on social life. The main danger of such artistic representations is that one must have some guarantee that they do not simply represent a kind of beautiful propaganda for an incorrect qualitative viewpoint. Clearly, statistics must be used to validate such individual samples of a system's behavior.
| Statistical mechanics is the study of the collective behavior of large numbers of interacting particles. Properties of interest include those describing time-dependent, irreversible process. The basic principles of this discipline were laid d own in the nineteenth century by Ludwig Boltzmann, James Clerk Maxwell, and Josiah Willard Gibbs. |
Several chemical problems truly demand the solution of these mathematical problems connected with the geometry of the potential surface. Such a global understanding is needed to be able to picture long time scale complex events in chemical systems. On e area in which this is clearly essential is the understanding of conformational transitions of biological molecules. The regulation of biological molecules is quite precise and relies on sometimes rather complicated motions of a biological molecule. Th e most well studied of these is the so-called allosteric transition in hemoglobin, but indeed, the regulation of most genes also relies on these phenomena. These regulation events involve rather long time scales from the molecular viewpoint. Their under standing requires navigating through the complete configuration space. Another such long time scale process that involves complex organization in the configuration space is biomolecular folding itself. By what process is the structure of a biological mol ecule determined? In order to function, enzymes require a fairly precise three-dimensional positioning of different chemical groups in the protein molecule. To achie ve this precise positioning of only a few groups, the collective interactions of the rest of the molecule must conspire to form such a fairly rigid construction. Although the three-dimensional structures of protein molecules exhibit some symmetries, they are exquisitely complex, and in addition, the architectures of folded protein are formed from molecules that have no simple pattern in their one-dimensional sequence.
| Ergodicity is the capacity of a dynamical system spontaneously to sample all of its phase space. |
The very deepest minima of systems can be characterized by using techniques superficially similar to those of thermodynamics and equilibrium statistical mechanics. Generalization of mean field theory for random Hamiltonians is used. The low-lying states of heteropolymeric biological macromolecules have been studied in this way. There is a very clear analogy to the phenomenon of broken ergodicity studied in spin glasses by the quasi-equilibrium statistical mechanical methods. The problem of broken ergodicity is one that is central to understanding the global topology of poten tial energy surfaces for such "random" systems. This problem plays a role both in the issues discussed here of biological macromolecules and amorphous materials, and in other optimization problems as well. There are deep connections with the theory of N P-completeness, a fundamental question in theoretical computer science. The formal questions of broken ergodicity in spin glasses (i.e., the topology of low-energy states) have not been answered entirely unambiguously by experiment, and the question of t he nature of the low-lying states is one that is still hotly debated. An important route to understanding this sort of broken ergodicity has been by the methods of rigorous statistical mechanics pioneered by mathematicians. It has been shown rigorousl y in some higher-dimensional problems that the broken ergodicity imagined in simple phenomenological theories of protein folding can, in fact, occur. It is still an open question, however, how ergodicity is broken for three-dimensional systems, spin gla ss systems, or for the random heteropolymers themselves.
One of the most interesting results of the theory of broken ergodicity based on quasi-equilibrium statistical mechanics is that the low-energy states of a typical Hamiltonian are related to each other in a fashion that is characterized by an ultrame tric distance. This ultrametricity concept arose earlier in the study in pure mathematics. The ultrametric organization may well play a role in the dynamics on such surfaces, and ultrametric hopping models have been widely discussed.
While the use of statistical energy surface topography is now coming to be accepted in the context of biomolecules, there is a still deeper mathematical question in its application to glasses. This question is, How does a Hamiltonian that is perfectly regular, having no explicit randomness, possess solutions that appear to be totally irregular and aperiodic? A long-standing issue for the purist has been whether even hard spheres have, as their most dense state, the simple regular packing characterist ic of face-centered cubic (FCC) crystals. Recently a proof of this was announced, but it has apparently been retracted. In fact, for the three-dimensional situation there is little doubt from the experimental input that the dense state is in fact period ic. The question of the closest packings in high-dimensional systems has many contacts with group theory and the theory of optimal coding.
The existence of quasicrystals has made the problem an even richer one since even the existence of such quasi-periodic structures was ruled out by "folk theorems" of physicists. It is likely that truly aperiodic crystals can, in principle, exist in three dimensions. An important argument for this is based on tiling theory. It has been shown that certain tiling problems are NP-complete. This implies that it is cer tainly very difficult to figure out whether a periodic packing of such a tiling is possible. Thus, it seems that the problem of totally aperiodic crystal phases for regular structures may be itself tied to the NP-completeness question.
The problem of transitions between minima on such a high-dimensional surface is in a still more primitive state than the characterization of minima. The search for minima is itself a relatively stable computational problem. The search for saddlepoints that connect individual minima is computationally much more difficult. This is certainly a consequence of the unstable mode at such saddlepoints. Despite numerous efforts, there are no entirely reliable methods for carrying out such a search. For many simple problems, finding a reasonably good transition state is possible, but these techniques become still more complicated and less reliable as system size increases. On the purely theoretical side, very simple models that relate the heights of barrier s to the statistics of minima have been developed, but almost no truly rigorous work has been done. Simply characterizing the minima and the saddlepoints connecting a few of them does not give an entire description of significant processes on a complex energy landscape. It is clear that one must understand something more about the basin of attraction of any given minimum. If the nearby minima are not entirely uncorrelated, this basin of attraction will depend on their structure as well. A characteri zation of the size of such funnels in biomolecular problems is essential to understanding protein folding.
Similarly, in many such complex problems it has been imagined that specific kinetic pathways are important. Again some work has already been done on the question of how specific pathways can emerge on a statistical energy landscape. These ideas are, ho wever, based on the quasi-equilibrium statistical mechanics of such systems, and there are many questions about the rigor of this approach. Similarly, a good deal of work has been carried out to characterize computationally pathways on complicated realistic potential energy surfaces.
Techniques based on path integrals have been used to good effect by Elber in studying the recombination of ligands in biomolecules and in the folding events involved in the formation of a small helix from a coiled polypeptide. These techniques tend to focus on individual optimal pathways, but it is also clear that sets of pathways are very important in such problems. How these pathways are related to each other and how to discover them and count them is still an open computational challenge.
NAS Home Page | NAP Home Page | Reading Room | Report Home Page