Switches, Oscillators, Yeast Cells, Magnets, Earthquakes, Cancer Cells, Brains….

I don’t often write about my own research, partly because it’s either a little too arcane to make an easy post, or because I’m still working on it and not ready to talk about it. However, I think my latest paper, written in collaboration with my friend Elana Fertig at Johns Hopkins, may be interesting to a broader audience, with or without the math. The paper itself is obviously aimed at a technical audience, so I’m not going to just repeat what we wrote, but try to put it into context and explain why I think it’s potentially exciting to others besides us. If you are interested in learning more, our paper is published in PLoS ONE, an open-access journal from the Public Library of Science (PLoS), so anyone can download and read it without paying a huge access fee.

In my earlier post Sync or Swim, I described how networks of oscillators can spontaneously synchronize, adjusting their rate of oscillation to match each other. An “oscillator” in this context can mean a wide variety of things: runners on a track, quantum-mechanical systems, fireflies flashing, rhythms in cells. The main thing is that when left alone, oscillators repeat themselves in a predictable way, cycling through the same motions or energy flows or whatever. When they are allowed to interact (runners adjusting their pace to match others, for example), they can fall into sync with each other – assuming the interaction is strong enough.

To quote from Sync or Swim:

In fact, a vast number of systems in physics and biology can be treated as interacting oscillators, with various types of behaviors arising based on the strength of coupling (or revulsion). If cell growth and death cycles get out of whack, cancer can be the result; the ordinary interaction is non-functional. Many other systems in the body have their own natural frequencies, and must synchronize to work properly for good health.

Static friction can make things act somewhat like a switch: you have to build up enough force to get a table moving across a carpet. Below a threshold force, the table won't move, but above that threshold it can. The unmoving table is the switch in its "off" state, and the moving table is "on".

Oscillators are a very important type of system in physics and biology, but they aren’t the only type. Other things flip rapidly: electron spins inside materials can suddenly align under certain conditions, magnetizing the system. Another type of system acts like a switch, which we can understand with an analogy: imaging pushing a heavy table across carpeting. If you push gently, the table refuses to move because of static friction; it might take a heavy push just to get it started moving, and that push might make it give way suddenly. We’ll think about your effort as a potential that builds up, and the motion of the table as a switch that “turns on” only above a certain threshold. This kind of thing also happens inside nerve cells, where a build-up of chemical signals is needed before the cell triggers and passes a message to its neighboring cell. A boundary where two tectonic plates meet is similar to our original table example: pressure builds up until the plates slip suddenly, then stick again, which we experience as a type of earthquake.

As with oscillators, we can have networks of switches, with some kind of flow of energy (or analogous thing) flipping the switches on or off, depending on how they interact. In the tectonic example, if the pressure isn’t sufficient to budge the plate, then it will tend to dissipate, so we’ll think of the “natural” state of a switch to be “off”: nothing happening. Left alone, even an “on” switch will eventually turn off; we’ll refer to this as a Glass model for the scientist who first described it. However, if the coupling between switches is strong, they can feed energy into each other, making the entire network turn on and stay on. (I suppose one could also develop a model where switches like to force each other to turn off, but I don’t know of an example in science where that kind of thing happens.)

Many things in nature aren’t strictly oscillators or switches, but combine features of both. Cell growth and interaction has features that are both oscillatory and switch-like, in particular, and that’s where my research with Elana comes in. We developed a model combining the Kuramoto model of oscillators (which I wrote about in Sync or Swim) with a Glass network of switches, so that switches and oscillators interact with each other in a particular way. Oscillators can feed energy into switches during a certain part of their cycle, possibly kicking them into their “on” state if the interaction is strong enough; switches can kill oscillations by throttling the flow of energy into the oscillators.

As you can probably guess, this is a very complex set of possibilities, so we didn’t try to examine all of them. Our first simplification was to connect every switch and oscillator to every other switch and oscillator, which isn’t particularly realistic, but frees us from worrying about how the network is “wired”. We made some other simplifying assumptions that I won’t catalog here, but which would need to be looked at if we want to examine many specific applications. We were mostly interested in studying the general features of our theory, which should occur even when our assumptions are relaxed.

Plots for the four possible behaviors of our model, taken from the published paper. Each pair of plots (a, b, c, and d) goes with the corresponding item in the list from the text. The lines in the top plot are correlations: a value of 1 means the switches are all "on" and the oscillators are acting together, while 0 means "off" switches and oscillators not in concert. The bottom plot is the phase of the oscillators: if it's a constant value, the oscillators have frozen out, but if it changes in a regular way, they are synchronized.

Our model was programmed into a computer, and run multiple times to make sure everything is repeatable. (Depending on how many network components we included, this process could take several days, even on a fairly powerful computer!) The switches and oscillators were started in a random configuration, so that a particular switch might be “on” or “off” and the phase of an oscillator could be in any part of the cycle. Over time, the interactions produced four basic types of behavior, depending on the strength of the various coupling parameters:

  1. all switches turn on and stay on, while oscillators synchronize;
  2. switches turn off and stay off, while oscillators freeze out: they stop cycling and stay at some phase forever;
  3. switches flip on and off periodically in synchrony with the oscillators; or
  4. things begin to synchronize as in #3, but after some time the oscillators freeze out and switches turn off.

(Links above go to AVI animations for each case.) We didn’t allow the system’s parameters to change during the simulation, so if the state of the system settled into behavior #1, it wouldn’t transition to #2.

Probably the most interesting result was #3, where the switches acted like oscillators and the whole network acted in synchrony. This never happens when switches are alone in the system! Even this behavior can be fragile, since if the oscillators’ natural frequency is too high, the switches don’t have time to react to the change and die off, like #2. In addition, if the natural frequency is too low, the switches will tend to turn on and stay on, as in #1. Hopefully you can see how this might apply to (for example) biological systems: if the metabolic parameters are at their proper values, cycles of production of proteins for cell division happen as they should, and everything is synchronized. If things are out of place, however, runaway cell division could happen (cancer) or at the other extreme, cell death could result. In computers, oscillators and switches control communication between the various parts of the system, and if all is working, everything is efficient.

I’m not claiming our theory as it stands is a perfect model for cancer or any other specific system, because it’s not. However, we’ve shown in principle that a network composed of two very different types of components can give rise to behavior not present in systems that have only switch-like or oscillator-like components. Also, Elana’s group performed some simple experiments on yeast, looking at how the cells divided under certain conditions, with and without external signals, and found that our model gives the same sorts of behaviors–without anything that looks precisely like a switch or oscillator, or even a “network” in the electrical sense.

The power of a theory often lies in how well it can cope with diverse phenomena in nature. You can easily come up with an idea that works for one particular example, but has no explanatory power or use outside that example. A hallmark of pseudoscience is the appeal to special cases: models that work only in limited applications, “just-so” stories, and so forth. If Elana’s and my model is to be truly useful, it must continue to be extended and tested against real-world data. We hope this taste has given you an idea of why we think it’s exciting and worth looking at in more detail.

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