Tuning a synthetic in vitro oscillator using control-theoretic toolsChristopher Sturk, Elisa Franco and Richard M. MurrayAbstract— This paper demonstrates the effectiveness of sim-ple control-theoretic tools in generating simulation-guided ex-periments on a synthetic in vitro oscillator. A theoreticalanalysis of the behavior of such system is motivated by highcost, time consuming experiments, together with the excessivenumber of tuning parameters. A simplified model of thesynthetic oscillator is chosen to capture only its essentialfeatures. The model is analyzed using the small gain theoremand the theory of describing functions. Such analysis revealswhat are the parameters that primarily determine when thesystem can admit stable oscillations. Experimental verificationof the theoretical and numerical findings is carried out andconfirms the predicted results regarding the role of productionand degradation rates.I. INTRODUCTIONSynthetic biology has two main objectives: engineeringnew biological systems out of characterized parts, and, bysystematic modification of existing systems, improving ourunderstanding of design principles. This interdisciplinaryfield attracts scientists from the areas of biology, mathemat-ics, physics and engineering: a strong theoretical analysisof experimental data increases our ability to interpret andpredict the behavior of engineered biological devices.When operating in an in vitro environment with a limitednumber of biological parts, scientists have the opportunityto program and deeply investigate the molecular interactionsthat produce overall designed behaviors. This is one of themost important features of the field of DNA nanotechnol-ogy [12], and is a key to the development of molecularcomputation [11]. But even in a controlled environment,there are cases in which the behavior of the system underobservation needs a thorough theoretical and experimentalanalysis to be correctly interpreted.A recently proposed synthetic in vitro oscillator [5], com-posed only of nucleic acids and two enzyme species, presentsseveral challenges regarding its dynamics and tuning. Inparticular, detailed modeling of the underlying chemicalreaction network offers a poor qualitative understanding of itsbehavior: the complex model is in turn of little help in aidingexperiments to modulate the frequency/amplitude character-istics of the circuit. Additionally, the cost and duration of theexperiments make it well worth looking for a better modelingresource that could qualitatively predict the features of theoscillator.Christopher Sturk is with the Department of Automatic Control, LTH,Lund University, Sweden. [email protected] Franco and Richard M. Murray are with the Division of Engineeringand Applied Sciences, California Institute of Technology, Pasadena, CA91125. [email protected], [email protected] authors would like to thank Erik Winfree for helpful discussions andadvise regarding the design and synthesis of in vitro genetic circuits.In this paper we consider a simplified model of thementioned synthetic oscillator [7] and study its characteris-tics using classical control-theoretic tools rather than solelysimulations of its differential equations. In particular, weused the small gain theorem for monotone systems [2], [1] toderive parametric conditions for which the system can admitan oscillatory regime. Such results are numerically refinedusing the method of the describing functions, which is anappropriate tool for systems presenting static nonlinearitiesand delays.Under certain simplifying assumptions, we found thatRNA production and degradation, together with the Hillfunctions thresholds, confine the region of the parameterspace where oscillations are achievable. We focused on therole of production and degradation, mapping it qualitativelyto the amounts of enzymes used in the experiments. Byvarying the production/degradation enzyme ratio and totalenzyme volume, we collected data that confirm the main fea-tures of the model predictions. The theoretical and numericalanalysis were therefore useful in guiding the experimentalchoices and allowed us to obtain a tuning methodology.The paper is organized as follows. The synthetic oscillatorwe consider is described in Section II, where we outlineits biological features and introduce the chosen simplifiedmathematical model. In Section III we report the small-gain theorem analysis and the numerical results based onthe describing function method. Finally, experimental resultsare reported in Section IV.II. A SYNTHETIC in vitro OSCILLATORThe in vitro genetic circuits considered in this paperconsist only of nucleic acids (DNA and RNA) and twoenzyme species, RNA polymerase (RNAP) and RNase H.RNAP binds to DNA double-stranded promoter regions andtranscribes the downstream sequence into RNA. The RNaseH instead hydrolyses and degrades RNA in RNA-DNAduplexes, releasing the DNA strand.The genes are synthetic and can be designed using ex-isting software packages [8], [9]. Each gene state can beswitched on and off by designing its promoter with a nickon the coding template (the strand complement-transcribedby RNAP). Exploiting the mechanism of toehold-mediatedbranch migration [13], the promoter can be covered oruncovered, effectively reducing the rate of transcription byRNAP. For example: a synthetic gene is in an off state whenthe promoter is incomplete, i.e. partly single stranded. Apromoter-complementary activating DNA strand in solutionwill switch the gene on. This activator is designed to presentan exposed overhang, i.e. is longer than the promoter bindingSubmitted, 2010 Conference on Decision and Controlhttp://www.cds.caltech.edu/~murray/papers/sfm10-cdc.htmlFig. 1: Scheme representing the main chemical reactionsoccurring in the considered biochemical oscillator. This isa two-node oscillator: the synthetic genes T12and T12respectively repress or activate one another. Single strandednucleic acids dA1, dA2and dI1are DNA species whoseconcentration is controlled; rA1and rI2are RNA species.Arrows on the nucleic acid strands indicate the 50to 30orientation. The molecular domains are assigned a color toemphasize their different roles. Areas that are complementaryhave the same color. In orange and red we indicate the activa-tion/inhibition domains, while the branch migration initiationtoeholds are colored in blue, cyan and light green. Gray anddark gray indicate promoter and transcribed regions.region. An inhibiting strand fully