Oscillatory behaviors, especially in regards to the field of biology or biochemistry, can be are very useful in many regards. For example, oscillations can be used to transfer information quickly through the use of waves. Traveling waves of charge that originate at one’s brain and travel to a muscle, for instance, control the movement of that muscle. Here we see another useful property of waves: that the movement of energy can be faster than the average movement of a particle within the solution in which the wave propagates. If muscles were not controlled by oscillatory mechanisms in the nerves, then the relay of information between one’s brain and muscles would either be much slower (as the body would have to transport chemicals from the brain to the muscle every time a muscle movement was needed) or there would have to be in place a very fast chemical network that could carry said chemicals from the brain to the muscle in a matter of milliseconds. This is one of many oscillatory behaviors that is useful in biology. A circadian rhythm, glycolysis, AMP production, and the horseradish peroxidase reaction are other examples of biological oscillations.
Novák et al states that biological oscillators have to have four basic requirements in order to function properly: “negative feedback, time delay, sufficient ‘nonlinearity’ of the reaction kinetics and proper balancing of the timescales of opposing chemical reactions.” [1, 2] Nonlinearity of the system refers to how the characteristic equation for the system should be structured, specifically, differential equation used to describe the reaction mechanism ought to follow the general form . If the equation did not possess sufficient ‘nonlinearity’ (ie u(t) oscillates much closer to equilibrium than x(t)) then the system would lose amplitude over time, and essentially die down. With sufficient nonlinearity (ie u(t)’s amplitude is sufficiently large compared to x(t), perhaps even larger) the oscillation will be continued. One way to achieve nonlinearity in an oscillation is to entertain the oscillation. Entertainment is defined as the systematic process of synchronization of the biological network. Oscillators often respond to external stimuli at some point in their cycle, and if these stimuli are strong compared to the strength of the oscillation, then the oscillation can be driven by the stimulus.
The other three requirements for a biological oscillator are a lot less “abstract” in nature. Negative feedback means that at some point in a biological system that oscillates, there needs to be some sort of mechanism that brings the system back to its original state. Time delay oscillators are a class of oscillators in which there is a biological equivalent of an inverting amplifier that has some sort of time delay mechanism coupled along with it. “Balancing of time scales” simply means that the frequency of oscillations should remain stable after a certain period of time.
Michael B. Elowitz and Stanislas Leibler were the first to design a synthetic network that produces oscillatory behaviors, termed “the Repressilator.” Their paper on the study, “A synthetic oscillatory network of transcriptional regulators,” was published in Nature Magazine in 2000. This paper was groundbreaking to the field of synthetic biology as it was one of the first synthetic networks designed to execute a particular function. As shown in figure 1, the Repressilator has three the repressor genes (tetR-lite, lac-lite, and λ cl-lite) coded for on a plasmid. The system oscillates between three states. In one state, the tetR gene binds to the tet01 binding site, inhibiting the production of λ cl-lite. This state therefor immediately produces the second state of the system, for without the presence of λ cl-lite, lacl-lite can be produced as the λ promoter site is left uninhibited. Lacl then represses the transcription of tetR by binding to the PL lac01 promoter site. This then leads to the third state of the system, for since tetR is not coded for, the PLtet01 promoter site is left uninhibited, and λ ct-lite can be coded for and repress the production of lacl-lite. This brings the system back to the first state described. The period of oscillation in the Repressilator was far greater than the time between cell divisions. This, however, did not hinder the ability of the oscillator to function properly, as the cell clones essentially picked up where the parent cell left off and continued the oscillation. While this study was significant, in that it was among the first synthetically devised biological system that performed a predicted function, the oscillation itself was shown to be unstable and produced a lot of noise. .
Dual Feedback Oscillator Circuit
Before discussing the study, it is important to define a few key terms. A positive feedback loop is one in which the output of the system further induces the system to perform its function (for instance, if a few sheep in a heard start running, they will cause other sheep to start running, which will cause more sheep to start running, and so on). A negative feedback loop is the opposite, in which the output of the system inhibits the system from repeating the function.
A study published in Nature by Sticker et al demonstrated a more robust oscillator that is both tunable and has fast oscillatory periods (around 13 minutes in length). It achieves this robustness by implementing both a positive feedback loop and a negative feedback loop. The intricacy of the design of the network lies in the implementation of a hybrid promoter site that is used to promote gene expression of all three genes used in the oscillator.
The araC gene produces Arabinose, which promotes the transcription of araC, yemGFP and lacI. Note that the fact that Arabinose is used to promote its own synthesis is a positive feedback loop. LacI then transcribes IPTG, which represses the expression of all three genes (including itself). Note that IPTG represses LacI transcription, which is an example of a negative feedback loop. SsrA degradation tags were added to each gene to decrease the lifetime of proteins.
Oscillatory periods could be controlled by temperature, as at lower temperatures the system produced higher oscillatory periods, while at higher temperatures the system produced lower oscillatory periods. Further control over the period of oscillation could be done by increasing the presence of Arabinose or IPTG. 
Basics behind mathematical models for oscillations
(Image on left: http://hyperphysics.phy-astr.gsu.edu/hbase/images/oscda13.gif) The right image is text I wrote (the equations were not transferring well into the wiki page). The image on the left shows what a damped, over damped, and critically damped system look like. Note that the oscillations depicted in the graph are not driven and thus are non-linear.
This image shows what a driven oscillation (an oscillation with nonlinearity) generally looks like.
One can work out the components of an oscillator influence the frequency of that oscillation, the damping factor, and so on. How fast the DNA is transcribed then translated into a protein, and how fast that protein can get and bind to its binding site or do its function will obviously influence how the frequency of oscillation of either of the two studies mentioned above. If all processes governing the movement of all parts in a biological oscillator are fast, then the angular frequency coefficient would be high, while if said mechanisms are slow then the angular frequency coefficient would be low. The damping factor (mu) could come into effect if there is a lack of amino acids available (if there is not a constant supply). Deviations from ideal oscillatory behavior in the Repressilator model are due to inhibiting proteins remaining bound to the GFP plasmid.
Creating stable biological oscillators has advanced the field of synthetic biology, and further efforts to increase complexity and robustness of oscillators will enable the creation of biological circuits that can execute complex functions. Biochemical oscillations are steady in nature, and thus ought to remain steady when synthetically derived.
iGEMThe 2012 Fudan Lux team created a biological oscillator that was able to synchronize oscillations between multiple cells by the use of light. This is useful, as it allowed for cell to cell communication and synchronization which is present in the behavior of a lot of organisms. 
The 2013 KU team modeled an oscillating system in which all bacterium in the culture experience synchronized oscillations. Using the concept quorum sensing mechanisms, they were able to create a model for a system in which all the cells in a population produce beta-farnesenesynthase in periodic intervals, through the use of extracellular biochemicals. 
- ák2008 Novak, B. "Design principles of biochemical oscillators." Nat Rev Mol Cell Biol. (2008)
Requirements for biological systems to oscillate.
- Elowitz, M. "A synthetic oscillatory network of transcriptional regulators." Nature 403, 335-338 (2000)
- Sticker, J. "A fast, robust and tunable synthetic gene oscillator." Nature 456, 516-519 (2008)
Dual Feedback Oscillator
Fudan 2012 iGEM project
The KU synthetic oscillator design