nanoHUB-U Biological Engineering: Cellular Despgn Principles/Biological Oscillations I
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[Slide 1 L4.1] Welcome back. I'm Professor Rickus and this lecture will introduce biological oscillations. 

[Slide 2] We'll look at some examples of biological oscillators. Introduce limit cycles, and look at the four basic design principles of biochemical oscillators in living cells, including negative feedback, sufficient delay on that feedback, sufficient non-linearity in the system and appropriately balanced rate parameters. 

[Slide 3] So if we think of all the possible responses as time goes to infinity of an unperturbed dynamic system composed of a set of variables. In this case, we'll name X1, X2 through to Xn. And these X values could be for example are values or levels of protein or gene expression. And if we consider the dynamic system that defines that, a set of ordinary differential equations, such that the rates of changes of these variabls or gene products, are all dependent on the state at that time of those product levels. And we think of all the possible responses. What could happen to our X values or our g product values? Well it could blow up to infinity. It could settle to a steady state value, that could be a zero or non zero constant steady state. But our system could also oscillate and have stable oscillations that go onto infinity unless perturbed. Our system could also take on chaotic behavior or a periodic behavior that never repeats. But we'll save that one for a different day. And today we're going focus on oscillations and particularly stable oscillations inside cells. 

[Slide 4] So there are quite a few oscillations or rhythms and stable rhythms that occur in nature. And they occur at a very wide range of time scales. From ecological and population cycles that can have a frequency time scale on the order of years. We have circadian rhythms in us. These are rhythms that have a period of about a day. There also could be circadian rhythms down in single cells and in gene expression. And we'll look more closely at some models of that in the next lecture. Biochemicals inside cells, for example metabolites, oscillate within glycolysis and other systems with time scales can be on the order of minutes. You have cardiac rhythms with rhythms in the order of seconds. Neuronal oscillations are off much faster than that millisecond to second time scales. Hormonal oscillations from minutes to hours to the day. And you can also have a range of interestingly of synchronized oscillations via communication in animals as well as cells. Bacteria can synchronize within a population. Fireflies can synchronized their flashing. You have synchronized oscillations in your neurons in your brain even. 

[Slide 5] So, some of the first biological oscillations that were discovered were in the brain and were very periodic in nature. So here's a trace of the first EEG published in the US in 1935 that observed these so-called alpha waves. And so, here's this oscillation that you see that's being recorded in the brain through scalp type electrodes that measure this EEG. In certain regions and conditions, you can observe these nice oscillations in the EEG. They're defined by their frequency and there's a range of frequencies. And that are observed in the brain that range from very ultrafast from 200 to 600 hertz down to much slower rhythms, including these alpha waves that you, we see here. 

[Slide 6] So oscillations maybe also be more complex. Many biological oscillations may have bursting behavior. For example, this is common again in neuronal systems and excitable cells. So here we see some recordings from some three different neurons that are responsible for the rhythmic chewing in a crab. And the crab is a really interesting model system here as well because the temperature independence of these oscillations can be observed. Which is an interesting feature that comes up, we'll talk more about later, in a lot of our biological oscillations. So here what you're seeing are the membrane potential recordings of these three neurons that are in this small neuronal circuit or network that they're looking at.   Again, in this chewing crab. We also have down here from a human patient. This is an EEG from a patient with epilepsy. Now the top trace shows a recording between seizures and the bottom trace shows recordings during seizure. And again, this sort of bursting in different complex rhythm patterns can be often seen and are important in characterizing different seizure states in patients with epilepsy. 

[Slide 7] So, looking at the phase plane now. So, if we were to look at a dynamic system that consists of two differential equations. So we've got two variables or state variables here, X and Y, which could be two different gene products. And the rate of change of those depends on each other's level at that instant in time. And if we look at the trajectory. So if we were to look at these system were to oscillate. And if both x and y were oscillating in time. So this is now plotting x and or y versus time. And we're to look in the phase plane and we plot x versus y. An oscillation now appears as a closed trajectory in that phase plane. 

[Slide 8] And this gives us a chance to introduce the concept of a limit cycle. So a limit cycle is an isolated closed trajectory in the phase plane. And these really only occur in non-linear systems. And so, because this is an isolated closed trajectory, we see that neighboring trajectories, if this limit cycle is stable, will be attracted to the limit cycle. In other words, any, if your system starts outside the limit cycle, it will spiral in to your oscillating state. If it's within that limit cycle, it will spiral out until it reaches that stable oscillation. For an unstable limit cycle, neighboring trajectories are repelled away from the limits of whether inside or outside that limit cycle. So this unstable, then if you're inside the limits cycle and started in that state in the phase plane, then you would circle into an approach steady state value in there. You can also have half limit cycles where trajectories on one side are attracted whereas trajectories on the other side are repelled. And this concept of a limit cycle is important. 

[Slide 9] Because many of the biological examples that we're looking at are stable oscillations. So, in the phase plane, a stable limit cycle represents a self-sustained stable oscillation, and there's many important biological examples of this. And the consequence of this phase plane of trajectories, spiraling in or starting in the middle and spiraling out toward this limit cycle is that any small perturbations in the system decay. Decay back to the stable oscillation and the oscillations persist. You can imagine this is very important in biological, if that oscillation is very important. Biological and living organisms are under constant inputs and changing environments all the time. So, to have self-sustaining oscillations, any small perturbations away from that state need to decay in order to preserve that oscillation. So, the stable limit cycle in the phase plain is often something we're looking for when we're looking to model or looking for the source of oscillations in a biological system. 

[Slide 10] So, let's look at some different biochemical oscillators. So, I said in a cellular level. Right? So, we said that gene systems, proteins, and metabolites are their examples . All of which that oscillate in time. I mentioned metabolism before, including components that include glucose and ATP. There are many signalling oscillation systems such as a very important one in cancer, the P53, MDM2 system. As well as others involved in egg cycles, and circadian rhythms. This genetic oscillator, the period timeless and other protein genes such as clock and cycle. These are ones that we'll look at in our next lecture and go and dive into a little bit more deeply on. 

[Slide 11] So, there are some basic design requirements for the underlying system to create biochemical oscillation. So, what are the design requirements for a biochemical system to operate? Well, within that system, there must be some negative feedback. There also must be some sufficient time delay in that feedback. Two other important characteristics of biochemical oscillators, the kinetic rate laws in that system must be sufficiently non-linear and the rates of events. So, you can imagine rates of events in our biochemical system. This could be gene expression versus protein degradation, the expression of one a gene versus another, but these relative rates of events must be on appropriate time scales relative to each other. From a mathematical sense, this is another way of saying. Saying that there are important parameter sets that can generate oscillations while other parameters might not. So those parameter values must be tuned in order to generate and have persisting oscillations. 

[Slide 12] So let's look a little bit more detail from a biochemical perspective. So what is some ways that negative feedback could be exists in our biochemical system. Well one way, in a genetic network, that we're looking at includes negative auto regulation.  Right? So this is one way that we looked at previously. Where a gene product inhibits its own synthesis, either directly or indirectly. So you have some expression of a protein And that protein then comes back and negatively affects its own expression. All right? And so looking at having a sufficient time delay in that negative feedback, one way that this could happen would be, for example, time for phosphorylation or transport into the nucleus in the case of eukaryotic cells. For that negative effector to have an effect on its own expression. So that would be a way to insert some delay in that negative feedback. So, imagine our same system here. If that protein If you're looking in the nucleus, that mRNA has to get out into the cytosol in order to be generated as protein. Now there could be some activation steps, perhaps phosphorylation or some other modification that's required to activate it. That would introduce some time and time delay. And that protein needs to come back into the nucleus in order to affect its own transcription at the DNA level. So any of these steps, if sufficiently long, could insert the type of delay that's needed to generate the oscillation. 

[Slide 13] So some other basic requirements. We talked about the kinetic rate laws being sufficiently non-linear. One example of this is from our familiar Hill function, the Hill coefficient on our Hill function is one way to create non-linearity such as in our gene activation or gene repression systems We also said that the rates of events must be on appropriate times scales relative to each other. An example of this is a balancing of the perimeters of the synthesis rate relative to decay coming from degradation or dilution of our protein product as we looked at previously. 

[Slide 14] So within negative feedback there are some other common motifs that are found in biochemical oscillators.  Besides just simple negative auto-regulation. There are delayed negative feedback loops. OK, so we have this feedback loop where another gene is inserted inside this process creating a delay between the effect of our orange, say X protein In gene product on our Z which can come back and inhibit. So we've added some additional steps, via additional, if this were a genetic network transcription events of multiple genes. Other common motifs include incoherent amplified negative feedback loops. And amplified negative feedback loops. 

[Slide 15] So, looking at our sufficient source of nonlinearity, we've already mentioned one way of getting a nonlinearity, and that's. Through for example are Hill coefficient on our Hill function, and if you recall in our Hill function one way to change that in is through the stoichiometry of binding of our activator or repressor at our DNA. So the stoichiometry are a little more binding, in other words, requiring more than one of our activator or repressor to bind in this case, to our gene site, creates the sigmoidal response as we saw in our Hill function. That was one way there are other ways as well. Cooperativity and allostery in enzyme reactions can also. Again, so we've got some stoichiometry going on and some cooperativity. You can put, again, this curvature in our transfer function, essentially. Multiple phosphorylation sites. We will see this in our next lecture, when we look at the period timeless model. And how multiple phosphorylation sites play a role in this case, in generating non linearity as well as some delay. You could also have stoichiometric, other forms of stoichiometric inhibition where the concentration of our free inhibitor here is sigmoidal with some S signal value. 

[Slide 16] So the important point here is just to note that there's multiple ways at the biochemical level, the biochemistry, that can generate nonlinearity in the mathematics when we generate mathematical representations of what's happening mechanistically at the biochemistry level. So in this dependence on parameters. And here we're going to go back to our limit cycle, to sort of think about this a little bit more. So we looked at some motif structures that are important underlying our oscillations such as negative feedback, and different types of negative feedback. So the motif structure itself may be necessary, but not sufficient. The parameters within that, so the structure will affect the form and structure of our equations, but the parameters within those equations must also be tuned. Not all parameter sets will necessarily give oscillations. And previously we talked about bifurcations. In an isolating system, there is something known as the supercritical Hopf bifurcation which happens quite often in biological systems that oscillate. And so, what happens here is that under one set of parameters, our system may have a stable, fixed point, or a constant or steady state value of variable levels. And so and in a certain parameter set, some of those trajectories may start to spiral in. Which gives us some hint that an oscillation or a stable oscillation might be possible. And if we tune those parameters, here we're just representing by the canonical example that's often given for the supercritical Hopf bifurcation. Where this mu represents our parameter value, in our r and theta state space that we're looking at in our phase plane here. And for this set of mu in our canonical example of mu less than zero, in this case, we get this stable fix points where nearby trajectories spiral in. But if we tune that mu value now, and in this case our critical value and our canonical example is zero. When mu gets bigger than 0, that fixed point loses its stability and a stable limit cycle emerges. And so now, this fixed point in here is unstable, and trajectories near it spiral away, now instead of in and towards this stable limit cycle. And trajectories outside the stable limit cycle spiral in towards it. So what this would look like if we were to look at r theta versus time. Right? For mu less than that critical value, in this case, zero, we would see, basically, damped oscillations. Right? These trajectories that are spiraling in, towards a steady state, stable steady state value. Whereas stable oscillations, if we were to start at the same point now that we've tuned mu greater than that critical value, in this case, zero. And we start inside somewhere in our phase plain near this unstable state. We would see oscillations that grow until they settle into stable oscillations that would continue indefinitely unless the system were further perturbed. So the existence of a supercritical Hopf bifurcation is often something that we're looking at in the structure of mathematically of our system. And then what we wanna do is find those critical parameter values which on one side of which we get stable oscillations, on the other side of which we don't. And so this is where the tuning of parameter values, often in our biochemical examples are rate parameters. Where the tuning of those rate parameters in some cases can result in oscillations and in others not. So the structure isn't enough. The parameters, and often rate parameters must also be tuned. 

[Slide 17] So for further reading on these topics I've listed a couple of really great books. So, understanding more about biochemical oscillators, I really recommend this book by Goldbeter who's really been a pioneer in this field. And for more into some of the nonlinear dynamics side, and the mathematical side of the oscillations. I highly recommend Strogatz's book, Nonlinear Dynamics and Chaos. As well as, looking at the design principles of biochemical oscillators. Again, another pioneer in this work by Tyson, has a nice nature review on this topic as well. 

[Slide 18] So, coming up. We're going to move into circadian rhythms, a particular type of biological oscillator that I talked about today. And we're going to dive a little more deeply into a really nice example of this. And that's the period timeless model that happens at the genetic level. This is a genetic circadian oscillator that was first identified and characterized in drosophila. But that has homologues in us and affects our wake-night cycle, sleep-wake cycle, with the day and night. So it's something that you have very close personal relationship, whether you know it or not. But it's also a really great example from a mathematical and biochemistry perspective because it exemplifies quite nicely the four design principals of biochemical oscillators that we just discussed. So I hope to see you next time.