Energy landscapes - Past, present, and future

What is it about?

Energy landscapes and their closely related cousins, the cost function and fitness landscapes, represent a supremely valuable tool to analyze highly complex systems in a multitude of fields that range from the natural sciences (physics, chemistry, biology) and mathematics over engineering, medicine and economics, to the humanities. This perspective paper presents a broad overview of the history of the field and outlines many directions of active research in a multitude of fields where energy landscape investigations are going to be of relevance. For those not familiar with energy landscapes, perhaps the most important aspect to note is that all these complex systems have in common that they exhibit many stable states in which we can observe them. Such a stable state corresponds to a whole region on the energy landscape. These regions are often analogous to important features of the physical landscape we live in on earth, such as a valley or a mountain - this is the reason why one commonly uses the term "energy landscape" instead of just the mathematical notion of "energy function". Each of these stable states exhibits different properties, and one of the major tasks of scientists - and everyone in their everyday lives ! - is to observe these properties, classify them, find the state with the "best" properties and figure out a way to reach this state (and stay there for as long as possible, perhaps). Some examples of such complex systems are: a) A molecule, which can exhibit different shapes. Think of a protein, which can fold and unfold as part of its performing its functions inside the human body, or (hopefully not) misfold - the latter case possibly leading to health issues. Clearly, understanding how this process happens, and how we can ensure that no misfolding occurs is of great importance both on a basic research and on an application level. In particular, each individual arrangement of the atoms of the protein can be characterized by an important number, its energy. A stable atom arrangement of the protein molecule corresponds to the atom arrangement for which the energy is a local minimum. More generally, a stable state corresponds to a whole set of closely related atom arrangements - imagine that the atoms are slightly wobbling around the minimum energy arrangement (this might correspond to a folded or a misfolded state) or drift in more languid larger motions inside a larger region of the landscape (corresponding to the unfolded state). b) While a single molecule is built up of a small number of atoms, we usually deal with large macroscopic numbers of atoms that appear to us as rigid unchangeable entities, such as a rock or a diamond. A typical rock tends to be a baked-together assembly of tiny crystals, once we take a closer look, while the diamond we see in the jewelry store is usually just one such crystal. But even though each of these crystals seems to have an unchangeable periodic structure corresponding to a stable state of the system of atoms, the atoms involved can actually arrange themselves in a multitude of such structures, each of which corresponds to a different stable state of the system. A classical example are actually the carbon atoms that form the diamond structure: if we arrange the atoms in the layered graphite structure (familiar from old-style pencils) instead, this stable state has actually a lower energy than the state corresponding to the seemingly immutable diamond structure ! One should note that if we change the environment that the carbon atoms experience, then the relevant energy can change: for example, if we apply very high pressures, then the relevant "energy" (now actually corresponding to the so-called enthalpy) of the diamond atom arrangement is lower than the one of the graphite arrangement, and the graphite transforms into diamonds ! c) The modeling of evolution involves another class of energy landscapes, the so-called fitness landscapes. Here, each species or individual of a species is assigned a fitness - usually corresponding to the likelihood that this individual or species will be able to procreate or continue to exist in competition with other individuals or related species, respectively, competing for the same niche in the living world. Trying to maximize one's fitness by evolving useful traits that fit the world one lives in or the species exists in, corresponds to finding stable states of traits. One should note that external changes of the environment change the importance of certain traits for the survival/procreation ability of individuals and species - a cute example is the change in the color of certain moths and butterflies in response to the darkening of walls due to air pollution during the industrial revolution (wing color got darker to allow for better camouflage), and the return to brighter wing colors once the air and the walls were being cleaned up. d) The classic example of the use of a cost function is the attempt to find the optimal set-up of a business: given a certain amount of money to invest, and a certain spectrum of possibilities to invest the money in, what are stable optimal solutions, and which one of these investment strategies will yield the most profit or involves the lowest cost for a given amount of revenue flow. While one often only looks for the strategy with the greatest profit at the end of the year and assumes that the economy one operates in is constant, one might want to consider, whether other strategies are more robust, especially against changes in the business environment, and which thus would yield profits also years down the road.

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Photo by Dmitry Ant on Unsplash

Photo by Dmitry Ant on Unsplash

Why is it important?

As noted above, this perspective paper presents a broad overview of the history of the field and outlines many directions of active research in a multitude of fields where energy landscape investigations are going to be of relevance. These include, but are not restricted to, the development of new tailored materials, understanding of neural networks employed in artificial intelligence research and applications, optimizing the use of scarce resources in business and society, understanding the response of nature to catastrophic events, and, last but not least, finding ways to adapt to changing environments - both natural and societal ones - with minimal discomfort for people and optimal employment of finite resources while respecting basic human rights in the process. Being aware of the long history of the field and its extremely broad applicability, together with the many techniques developed over the years to study such energy, fitness and cost function landscapes, allows us to gain many new insights that are relevant to our own fields of research and also to the way we look at the world around us, and, furthermore, this awareness provides us with new tools and lines of approach to deal with challenges we encounter in the world outside of science.

Perspectives