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Section 5: Free Energy Landscapes

A certified unsolved problem in physics is why the fundamental physical constants have the values they do. One of the more radical ideas that has been put forward is that there is no deeper meaning. The numbers are what they are because an unaccountable number of alternate universes are forming a landscape of physical constants. We just happen to be in a particular universe where the physical constants have values conducive to form life and eventually evolve organisms who ask such a question.

This idea of a landscape of different universes actually came from biology, evolution theory, in fact, and was first applied to physics by Lee Smolin. Biology inherently deals with landscapes because the biological entities, whether they are molecules, cells, organisms, or ecologies, are inherently heterogeneous and complex. Trying to organize this complexity in a systematic way is beyond challenging. As you saw in our earlier discussion of the protein folding problem, it is easiest to view the folding process as movement on a free energy surface, a landscape of conformations.

Glasses, spin glasses, landscapes

There is a physical system in condensed matter physics that might provide a simpler example of the kind of landscape complexity that is characteristic of biological systems. Glasses are surprisingly interesting physical systems that do not go directly to the lowest free energy state as they cool. Instead, they remain frozen in a very high entropy state. For a physical glass like the windows of your house, the hand-waving explanation for this refusal to crystallize is that the viscosity becomes so large as the system cools, that there is not enough time in the history of the universe to reach the true ground state.

As a glass cools, the viscosity increases so rapidly that the atoms get frozen in a disordered state.

Figure 18: As a glass cools, the viscosity increases so rapidly that the atoms get frozen in a disordered state.

Source: © OHM Equipment, LLC. More info

A more interesting glass, and one more directly connected to biology, is the spin glass. It actually has no single ground state, which may be true for many proteins as well. The study of spin glasses in condensed matter physics naturally brings in the concepts of rough energy landscapes, similar to those we discussed in the previous section. The energy landscape of a spin glass is modified by interactions within the material. These interactions can be both random and frustrated, an important concept that we will introduce shortly. By drawing an analogy between spin glasses and biological systems, we can establish some overriding principles to help us understand these complex biological structures.

A spin glass is nothing more than a set of spins that interact with each other in a certain way. At the simplest level, a given spin can be pointing either up or down, as we saw in Unit 6; the interaction between two spins depends on their relative orientation. The interaction term Jij specifies how spin i interacts with spin j. Magically, it is possible to arrange the interaction terms between the spins so that the system has a large set of almost equal energy levels, rather than one unique ground state. This phenomenon is called "frustration."

This simple system of three spins is frustrated, and has no clear ground state.

Figure 19: This simple system of three spins is frustrated, and has no clear ground state.

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For a model spin glass, the rule that leads to frustration is very simple. We simply set the interaction term to be +1 if the two spins point in the same direction, and -1 if they point in different directions. If you go around a closed path in a given arrangement of spins and multiply all the interaction terms together, you will find that if the number is +1, the spins have a unique ground state; and if it is -1, they do not. Figure 19 shows an example of a simple three-spin system that is frustrated. The third spin has contradictory commands to point up and point down. What to do? Note that this kind of a glass is different from the glass in your windows, which would find the true ground state if it just had the time. The spin glass has no ground state, and this is an emergent property.

Frustration arises when there are competing interactions of opposite signs at a site, and implies that there is no global ground energy state but rather a large number of states with nearly the same energy separated by large energy barriers. As an aside, we should note that this is not the first time we've encountered a system with no unique ground state. In Unit 2, systems with spontaneously broken symmetry also had many possible ground states. The difference here is that the ground states of the system with broken symmetry were all connected in field space—on the energy landscape, they are all in the same valley—whereas the nearly equal energy levels in a frustrated system are all isolated in separate valleys with big mountains in between them. The central concept of frustration is extremely important in understanding why a spin glass forms a disordered state at low temperatures, and must play a crucial role in the protein problem as well.

Hierarchical states

A Rubik's Cube is a familiar example of a hierarchical distribution of states.

Figure 20: A Rubik's Cube is a familiar example of a hierarchical distribution of states.

Source: © Wikimedia Commons, GNU Free Documentation License 1.2. Author: Lars Karlsson (Keqs), 5 January 2007. More info

Take a look at a Rubik's cube. Suppose you have some random color distribution, and you'd like to go back to the ordered color state. If you could arbitrarily turn any of the colored squares, going back to the desired state would be trivial and exponentially quick. However, the construction of the cube creates large energy barriers between states that are not "close" to the one you are in; you must pass through many of the allowed states in some very slow process in order to arrive where you want to be. This distribution of allowed states that are close in "distance" and forbidden states separated by a large distance is called a hierarchical distribution of states. In biology, this distance can mean many things: how close two configurations of a protein are to each other, or in evolution how far two species are apart on the evolutionary tree. It is a powerful idea, and it came from physics.

To learn anything useful about a hierarchy, you must have some quantitative way to characterize the difference between states in the hierarchy. In a spin glass, we can do this by calculating the overlap between two states, counting up the number of spins that are pointing the same way, and dividing by the total number of spins. States that are similar to one another will have an overlap close to one, while those that are very different will have a value near zero. We can then define the "distance" between two states as one divided by the overlap; so states that are identical are separated by one unit of distance, and states that are completely different are infinitely far apart.

Here, we see two possible paths across an energy landscape strewn with local minima.

Figure 21: Here, we see two possible paths across an energy landscape strewn with local minima.

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Knowing that the states of a spin glass form a hierarchy, we can ask what mathematical and biological consequences this hierarchy has. Suppose we ask how to pass from one spin state to another. Since the spins interact with one another, with attendant frustration "clashes" occurring between certain configurations, the process of randomly flipping the spins hoping to blunder into the desired final state is likely to be stymied by the high-energy barriers between some of the possible intermediate states. A consistent and logical approach would be to work through the hierarchical tree of states from one state to another. In this way, one always goes through states that are closely related to one another and hence presumably travels over minimum energy routes. This travel over the space is movement over a landscape. In Figure 21, we show a simulated landscape, two different ways that system might pick its way down the landscape, and the local traps which can serve as metastable sticking points.

In some respects, this landscape picture of system dynamics is more descriptive than useful to the central problems in biological physics that we are discussing in this course. For example, in the protein section, we showed the staggering complexity of the multiple-component molecular machines that facilitate the chemical reactions taking place within our bodies, keeping us alive. The landscape movement we have described so far is driven by pre-existing gradients in free energy, not the time-dependent movement of large components. We believe that what we observe there is the result of billions of years of evolution and the output of complex biological networks.


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