One very interesting area of application for ideas and techniques from nonlinear dynamics is the study of biological cycles. The circadian rhythm, the internal 24 hour clock that a very large number of organisms have, has been a particular object of study over the years. Naturwissenschaften‘s list of 100 most cited papers includes a classic paper by Aschoff and Pohl on phase relations between a circadian rhythm and a zeitgeber (a stimulus that entrains the clock). The most prominent zeitgeber is of course the day-night cycle, but other stimuli can reset your circadian clock, including meal times and social interaction. In this particular paper, the authors examine the relationship between the circadian phase (e.g. the time of maximum observed activity relative to start of day) and the day length. Studies like these often use ideas from nonlinear dynamics on the entrainment of oscillators to derive insights into the workings of the clock based on how the phase changes as the difference between the natural frequency of the clock and the entraining (zeitgeber) frequency increases. In this case however, the authors focused on quantitative differences between the phase responses of different groups of organisms. We now know that there are several, evolutionarily distinct circadian oscillators operating in different groups of organisms to which the results of Aschoff and Pohl could likely be correlated.

Coupled oscillators are a recurring theme in nonlinear dynamics, and the Naturwissenschaften list also includes a paper by Hübler and Lüscher on the possibility of controlling one oscillator using a driving signal derived from a second oscillator. Although this is very much a fundamental study, this kind of work has found a number of applications over the years. I have already mentioned the use of such studies to understand biological oscillators. Coupled oscillators show up all over the place, both in natural and in engineered systems. One example that has been the focus of a lot of research is the use of coupled oscillators in secure communications. The problem here is that you want an authorized receiver to get your message, but you don’t want anyone else to be able to eavesdrop. I’m not sure what the current status of this research is, but there have been a number of proposals over the years to use coupled chaotic oscillators for this purpose. The original idea was (relatively) simple: If two chaotic oscillators have the same parameters, they can be made to synchronize by introducing a driving signal that increases with the difference between the transmitted signal and the computed signal at the receiver. Even small differences in parameters are enough to ruin the synchronization because of the sensitive dependence property of chaotic systems. If you add a low-amplitude signal to the transmitted signal, the receiver will still synchronize to the transmitted chaotic “carrier”. The computed chaotic trajectory at the receiver can be subtracted from the incoming signal, the difference then being the superimposed message. The key to make this work is to share a set of parameters for the chaotic system via a private channel. Easy in principle, but there are lots of technical conditions that have to be met to make this work, and lots of variations to be explored to find the most secure means of encoding the message within the transmitted signal.

The control of a process by an oscillator sometimes also has spatial manifestations. An example of this is provided in the paper on slime-mold aggregation by Gerisch in the Naturwissenschaften top-100 list. When they are well fed, slime molds live as single cells. Starve them, and they aggregate, form a fruiting body, and disperse spores. How do they know where to go during the aggregation process? The answer turns out to involve periodic cyclic AMP (cAMP) signaling. Starving cells put out periodic pulses of cAMP. The cells don’t all signal at the same rate, and they tend to synchronize to and move toward the fastest signaler. Note again the importance of coupled oscillators: This works in part because the cells “listen” to each other’s cAMP signals and adjust the frequency of their own oscillator to match the fastest frequency they “hear”.

Well, those are the things that caught my attention in the Naturwissenschaften list. There is lots of other wonderful stuff in there. I would love to hear what caught everyone else’s fancy.

The journal Naturwissenschaften turns 100 this year. Naturwissenschaften translates as “The Science of Nature”. It’s a journal that publishes papers in all areas of the biological sciences, broadly conceived. As many other journals have done, Naturwissenschaften is celebrating its 100th anniversary by posting a list of its 100 most cited papers. As with all such lists, especially with generalist journals like this one, what you find interesting may be different from what I find interesting, so it’s worth taking a look at the list yourself. However, if you’re reading this blog, perhaps we share some interests.

The first thing I noticed was that the list contained several of Manfred Eigen’s papers on biological self-organization, including his classic papers on the hypercycle. These papers were intended to address the problem of how biological organisms may have gotten started. The emphasis of this work tended to be on self-replicating molecular systems, such as the hypercycle, which is a family of models consisting of networks of autocatalytic units coupled in a loop. I’m not sure how large a contribution these papers made to the problem of the origin of life, but they sure caught people’s imaginations when they were written, and they led to interesting questions about the dynamics of systems with loops which are still being actively studied. If you have never read anything about hypercycles and have an interest either in theories on the origin of life or in nonlinear dynamics, you should track down these papers and read them. They will likely seem a little dated—they were written in the 1970s—but I think they’re still interesting.

Also near the top of the list, we see a paper by Karplus and Schulz on the “Prediction of chain flexibility in proteins”. Protein dynamics is all the rage these days. Everybody wants to think about how their favorite protein moves. This wasn’t always so. In the 1980s when this paper was published, we were starting to see a steady flow of high-quality x-ray protein structures. People were making very good use of these structures to understand protein function, and of course that is still the case. However, there was a tendency for biochemists back then to think of protein structure as an essentially static thing. This tendency was so pronounced that I remember attending a seminar in the mid-1990s at which the speaker made a point of talking about how cool it was that part of his enzyme could be shown to have a large-scale motion as part of its working cycle! The Karplus and Schulz paper therefore has to be understood in this context. At the time it was written, it wasn’t so easy to recognize flexible parts of proteins, and there was a lot of skepticism that flexibility was important to protein function. Needless to say, things have changed a lot.

The Naturwissenschaften list also includes a paper by Bada and Schroeder on the slow racemization of amino acids and its use for dating fossils. Living organisms mostly use the L isomers of the amino acids. Over time though, amino acids tend to racemize to a mixture of the L and D forms. While an organism is alive, this process is, in most tissues, completely insignificant since proteins are turned over relatively rapidly. After an organism dies, turnover starts, and we can use the D to L ratio to date fossil materials. There are other interesting applications of this technique, including its use to determine the ages of recently deceased organisms using the eye lens nucleus, a structure formed in utero. I wrote about this dating technique in my book.

I’ll come back to Naturwissenschaften‘s list in a few days. There are a number of other papers in there that I think are interesting.