Entrainment of a Bacterial Synthetic Gene Oscillator through

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Entrainment of a Bacterial Synthetic Gene Oscillator through Proteolytic Queueing Nicholas C. Butzin, Philip Hochendoner, Curtis T. Ogle, and William H. Mather ACS Synth. Biol., Just Accepted Manuscript • DOI: 10.1021/acssynbio.6b00157 • Publication Date (Web): 09 Dec 2016 Downloaded from http://pubs.acs.org on December 11, 2016

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ACS Synthetic Biology

Entrainment of a Bacterial Synthetic Gene Oscillator through Proteolytic Queueing Nicholas C. Butzin,1, ∗ Philip Hochendoner,1, ∗ Curtis T. Ogle,1 and William H. Mather1, 2, † 1

2

Department of Physics, Virginia Tech, 50 West Campus Dr., Blacksburg, VA 24061-0435 Department of Biological Sciences, Virginia Tech, 1405 Perry St., Blacksburg, VA 24061-0406 (Dated: October 28, 2016)

Internal chemical oscillators (chemical clocks) direct the behavior of numerous biological systems, and maintenance of a given period and phase between many such oscillators may be important for their proper function. However, both environmental variability and fundamental molecular noise can cause biochemical oscillators to lose coherence. One solution to maintaining coherence is entrainment, where an external signal provides a cue that resets the phase of oscillators. In this work, we study the entrainment of gene networks by a queueing interaction established by competition between proteins for a common proteolytic pathway. Principles of queueing entrainment are investigated for an established synthetic oscillator in E. coli. We first explore this theoretically using a standard chemical reaction network model and a map-based model, which both suggest that queueing entrainment can be achieved through pulsatile production of an additional protein competing for a common degradation pathway with the oscillator proteins. We then use a combination of microfluidics and fluorescence microscopy to verify that pulse trains modulating the production rate of a fluorescent protein targeted to the same protease (ClpXP) as the synthetic oscillator can entrain the oscillator. Keywords: entrainment, queueing, synthetic biology, microfluidics, stochastic modeling, gene networks

I.

INTRODUCTION

Biological rhythms are ubiquitous in nature and are ultimately rooted in biochemical oscillators (clocks). These oscillators can function through a variety of different molecular mechanisms [1], but all biochemical oscillators are imperfect in the sense that they are fundamentally noisy due to the probabilistic nature of chemical reactions in small volumes [2]. This noise and any environmental variability will inevitably lead to decoherence among a population of oscillators, which could be disastrous for biological function. Evolutionary pressure has correspondingly modified native biological networks to contain mechanisms that prevent asynchronous behavior. One major mechanism is spontaneous synchronization due to cell-cell communication. For example, in the context of early development, synchronous division of cells is maintained by active chemical signaling that influences the cell cycle oscillator [3]. Entrainment provides an additional major route to coherence, where synchronous behavior can be enforced by an external reference signal. A number of natural systems depend on entrainment, including the locking of circadian rhythms to the light-dark cycle set by the Sun [4]. The general principles behind both spontaneous synchronization and entrainment have long been studied [5–9], and significant research in particular has been dedicated to understanding synchronization and entrainment in biological networks. For example, entrainment has been explored for circadian rhythms [4, 10], glycolytic oscillations [11], cell cycle oscillations [12], and even tidal rhythms [13]. Unfortunately, the complexity of natural oscillators often acts as a barrier to understanding their underlying mechanism. Synthetic biology provides an alternative approach to this problem, where custom synthetic gene circuits can be designed to test whether a given molecular scheme is sufficient to generate a given behavior [14]. Recently, synthetic gene oscillators have provided insight into the inner workings of biological rhythms [15–17]. These synthetic oscillators have been used as experimental models for synchronization via cell-cell communication and inter-circuit coupling [18–20] and for entrainment by both periodic and aperiodic chemical signals [21, 22], the last of which is of interest due to the fluctuating nature of many environmental signals. One major feature found in virtually all synthetic oscillators to date is the use of active degradation of proteins (proteolysis) to ensure strong oscillations [17]. More specifically, most existing synthetic oscillators are in E. coli

∗ co-lead † corresponding

author

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and target one or more proteins to the native ClpXP protease by a genetically encoded SsrA degradation tag on the C-terminus of the protein [23]. A major though largely unanticipated consequence of this design is queueing coupling, where proteins compete for proteolytic resources and thus modulate each other’s degradation rate [19, 24– 28]. The label “queueing” is apt, since this competition between molecular species mimics classical queueing systems, where one or more types of jobs compete for the processing by one or more types of servers. This indirect queueing coupling is not expected to be unique to E. coli or even to bacteria, but it is rooted in the finite bandwidth of proteolytic resources, which applies just as well to the proteasome in mammalian cells. Recent studies of queueingbased synchronization between two oscillators suggest that queueing synchronization is quite robust [19, 28], i.e. that synchronization can occur for a wide range of system parameters and environmental conditions. This prompted us to consider whether queueing would be similarly effective in the context of entrainment. If this is indeed the case, our results would establish a deeper understanding of a potentially common mode of entrainment in natural biological oscillators. The field of synthetic gene oscillators would also benefit, since most if not all modern synthetic gene oscillators are predicted to be strongly influenced by queueing effects [17, 26]. In the present work, we explore the principles of queueing entrainment, which may be applicable to a variety of organisms spanning the tree of life. Queueing entrainment is here defined to be when an externally-controlled protein can entrain an oscillator by competing for the same degradation resources as the oscillator. We use a well-studied synthetic gene oscillator in E. coli to illustrate queueing entrainment [29] (see Fig. 1). More precisely, we investigated a variant of this oscillator that was recently used in a study of aperiodic entrainment [22]. In this case, queueing entrainment is achieved by periodically expressing an additional synthetic protein, CFP-LAA, which competes for degradation with the oscillator proteins, LacI-LAA, AraC-LAA, and GFP-LAA. A theoretical study of this system using both conventional chemical reaction network models and map-based models suggests that entrainment can be achieved for pulsatile driving of CFP-LAA production with a period that is approximately the natural (mean) period of the synthetic oscillator. Our theoretical results then motivate an experimental study of queueing entrainment using a combination of microfluidics and microscopy, where we apply a time-dependent concentration of the quorum sensing chemical AHL to control CFP-LAA production. We find experimentally that queueing coupling can produce a coherent response across a population of oscillators, and we provide an early investigation that shows the frequency response of our system is consistent with entrainment. Our work can be distinguished from other studies on how queueing affects oscillators. In an early work [25], it was noticed that an oscillator co-expressed with CFP-LAA would propagate an oscillatory signal to CFP-LAA. This result established that an oscillator could be coupled to and drive an otherwise simple system (a fluorescent protein) via queueing, but this result did not address how CFP-LAA could reciprocally propagate signal and thus entrain the oscillator. In a more recent work [19], it was seen that two oscillators could robustly couple via queueing and thus form a synchronized hybrid oscillator, despite the characteristic periods of the two oscillators being quite different. This study of synchronization does not preclude a study of entrainment, since synchronization and entrainment are distinct concepts. Indeed, our theoretical results and experimental data suggest that queueing entrainment is not guaranteed to be strong for all choices of drive signal, e.g. with respect to drive period, which contrasts with the robustness of queueing synchronization. Thus, a proper study of queueing entrainment is required if we are to understand how it might be leveraged in natural and synthetic systems. To this end, we have introduced two related but distinct modeling analyses that reveal key theoretical principles for queueing entrainment, and we have provided experimental data that demonstrates queueing entrainment in living cells.

II. A.

RESULTS AND DISCUSSION Choice of a Pulsatile Drive Signal

We first committed ourselves to a theoretical study of queueing entrainment. We considered models for systems similar to that in Fig. 1, where externally-controlled chemical signal AHL controls expression of a protein CFP-LAA, the latter of which can compete with the oscillator (including proteins LacI-LAA, AraC-LAA, and GFP-LAA) for the ClpXP protease. In the absence of CFP-LAA protein, the oscillator is free running and obeys degrade-and-fire (DF) dynamics [29, 30], where pulses of protein production (firing phase) are followed by subsequent degradation of protein (degradation phase). The mechanism of oscillation hinges on rapid proteolytic degradation and delayed negative feedback, and the loss of either of these can lead to a loss of oscillations. In particular, loss of rapid enzymatic


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degradation is closely associated with the elimination of strong oscillations [30]. A consequence then is that queueing competition can destroy oscillations, since if highly expressed CFP-LAA fully occupies the protease ClpXP, this is theoretically equivalent to a loss of enzymatic degradation. Thus, the mean level of expression for CFP-LAA should be kept sufficiently low to not destroy oscillations. Refer to the Supplemental Information for numerical simulations demonstrating this effect. We tested whether conventional sinusoidal variation of AHL would theoretically be a good candidate for queueing entrainment of the synthetic oscillator. A sinusoidal signal to the synthetic oscillator was successfully used previously, where entrainment was achieved via modulation of arabinose, which modulated transcriptional activity of all proteins [21]. In the present case, a sinusoidal drive would imply that AHL would be given a fixed mean value and varied about this mean value using the shape of a sine function, whose amplitude about the mean defines the amplitude of drive. We found sinusoidal variation to be suboptimal for queueing entrainment, primarily because zero amplitude drive would correspond to a nonzero constant AHL concentration, which could strongly disrupt oscillations due to effectively slowing proteolytic degradation for the oscillator due to background production of CFP-LAA (see the previous paragraph for the importance of fast degradation). Thus, the oscillatory quality of free running oscillators could be quite poor, or worse, the gene network may cease to oscillate at zero drive amplitude. A further reason not to use sinusoidal drive is that if the transcriptional response to AHL concentration is cooperative, as expected, then the underlying gene network would sense a highly distorted version of the sinusoidal signal. In fact, the transmitted signal would appear pulsatile in nature. Instead of a sinusoidal AHL signal, we only explored square pulses that jump between a low value (exactly zero theoretically, approximately zero experimentally) to a higher value of AHL concentration. These signals are defined by their amplitude (upper value of AHL concentration), period (time to repeat), and duty cycle (fraction of the time the signal is at the high value). The response of a network to square pulses is much less sensitive to cooperative gene activity and other molecular details, which allows pulses to reliably propagate information [31]. For example, if the amplitude of drive varies between inducing (saturating) and non-inducing (effectively zero) values for AHL concentration, then the square pulse is theoretically propagated as a square pulse of transcriptional activity that is relatively insensitive on the drive amplitude. Thus, the protein produced during a pulse should be simply proportional to the duty cycle and relatively insensitive to amplitude.

B.

Theoretical Investigation

We hypothesized based on our investigation of a suitable drive signal that pulsatile production of CFP-LAA could entrain the oscillator. The basic elements of entrainment in general are positive and negative control of an oscillator’s period and phase, e.g. as can be investigated with a phase response curve [32]. Simply stated, entrainment arises when an oscillator that slightly leads the drive has its period shortened to compensate, while an oscillator that slightly lags the drive has its period lengthened. We find that queueing coupling of the oscillator to CFP-LAA can both shorten and lengthen the period of an oscillator, making queueing a candidate for entrainment. Shortening of the period can be achieved by an effectively slower degradation velocity of repressor by ClpXP due to CFP-LAA competition, which is expected lower the amplitude of the next oscillator firing event [30], thus decreasing the time to the subsequent firing event according to the principles of DF dynamics. Intuitively, delay in the negative feedback coupled to rapid degradation leads to “overcorrection” by the oscillator [17], thus producing a pulse of protein shortly after repressor reaches low concentrations, but a low degradation rate of oscillator proteins due to queueing competition with CFPLAA allows the delayed feedback to better sense protein concentrations and avoid overcorrection, thus lowering firing amplitude. Lengthening of the period is also a possibility, since adding extra CFP-LAA to the system lengthens the “queue” of proteins waiting for degradation until the next oscillator firing event. The timing of the CFP-LAA pulse determines whether shortening or lengthening of the period dominates, and entrainment can follow when oscillators are stably locked by this period control. These qualitative arguments are supported using two similar but distinct quantitative models (see Supplementary Information for details). The first model of interest is a chemical reaction network model that includes reactions for gene activity, proteolytic degradation, and time-dependent production of CFP-LAA according to a pulsatile function of time. Our model is similar to other models for this oscillator [21, 30], though AraC-LAA (the positive feedback transcription factor) was not modeled explicitly. The parameters of the model were tuned to have a free running period close to 30 min for the purposes of this theoretical investigation. While this period is close to that of the



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experimental oscillator, we do not claim that the model’s mean period is the same as the mean experimental period. An ensemble of oscillators defined by our model exhibited coherent oscillations in response to pulsatile driving of CFP-LAA (see Fig. 2). Importantly, individual oscillators mostly retained their overall amplitude and time profile compared to that of free running oscillators, but the phase of oscillators drifted to a common collective phase in a manner characteristic of entrainment. We then densely explored a range of parameter space to characterize queueing entrainment. A particular metric was chosen for entrainment response: the standard deviation (over time) of the oscillator’s ensemble mean GFP-LAA, i.e. how widely the mean amplitude of oscillators varies as a function of time [22]. While other perhaps more rigorous measurements exist [21], our approach is applicable to the many experiments where only batch measurements are available, and thus our approach should be relevant to a number of existing experimental techniques that do not depend on single cell trajectories. A summary of our results appears in Fig. 3. We find classical Arnold tongues emanating at primary resonances (when the drive period is some rational multiple of the natural period of the oscillator). Furthermore, for stronger drive amplitude, these tongues bend to higher periods, which is likely due to the additional CFP-LAA dilating the period of the oscillator for reasons already discussed. One weakness of the chemical reaction network model, like many such complex models, is that little can be said concerning the origin of entrainment. To this end, we also explored a map-based model for the oscillator that allows normally emergent properties to be defined explicitly as parameters, e.g. the mean amplitude and amplitude variability of the free running oscillator can be tuned independently. This treatment allows for a more intuitive explanation of how queueing entrainment can come about (see the Supplemental Information), and similar simplifications with similar motives appear in other fields, such as the integrate-and-fire model in neuroscience [33]. Importantly, we assume a simple functional form for the coupling between CFP-LAA and the oscillator that matches our intuitive discussion of coupling earlier in this section. This map-based model simplifies oscillations to idealized DF oscillations [22, 34], where each oscillator trajectory appears approximately sawtooth due to impulse-like firing events connected by zeroth order degradation (linear decay as a function of time). In the absence of CFP-LAA, the oscillator runs freely due to a core negative feedback component (LacI-LAA) that degrades deterministically to zero concentration, at which point LacI-LAA instantaneously increases according to a random variable with a given distribution, and the process repeats. CFP-LAA appears in the system due to periodic bursts of CFP-LAA production, and the produced CFP-LAA affects the DF oscillator in two well-defined ways. CFP-LAA influences oscillator firing amplitude via a decreasing function of the CFP-LAA concentration just before firing events (technically, this is a function of the ratio of CFP-LAA to LacI-LAA just prior to firing), such that the presence of CFP-LAA in the system depresses oscillation firing amplitude. A pulse of CFP-LAA can also dilate the time to the next firing event due to extra protein that must be degraded. These transparent coupling mechanisms between CFP-LAA and the oscillator are apparently sufficient to qualitatively reproduce many of the features in the more complicated model (see Fig. 4). Additional comments on the map-based model, including a more analytic approach to entrainment, appear in the Supplemental Information.

C.

Experimental Entrainment of the Oscillator in a Microfluidic Device

Our theoretical investigations suggested that pulsatile driving of CFP-LAA near the natural period of the experimental synthetic oscillator should lead to a strong oscillatory response, i.e. lead to oscillator coherence. We tested this by loading the synthetic oscillator into a microfluidic device, perturbing the oscillator with time-dependent levels of AHL, and observing the response using fluorescence microscopy. The oscillator itself is tuned by two primary chemical species, arabinose (ARA) and isopropyl β-D-1-thiogalactopyranoside (IPTG) [29]. In all experiments, these were set to background levels of 1.0% ARA and 2.0 mM IPTG, which we observed to provide the oscillators with a characteristic oscillator period in the range of 40 to 50 min, with the period being variable from cell to cell and across time. Several different microfluidic traps containing a monolayer of cells were perturbed with precisely-defined AHL chemical signals (see Supplementary Table 1, and Supplementary Video 1). We calculated the time-dependent response (the filtered mean fluorescence, similar as in Ref. [22], see Methods) for measured GFP-LAA and CFP-LAA fluorescence across each microfluidic trap, and we found that while single cells readily oscillated, coherent collective oscillations only emerged in response to time-dependent AHL drive signals (see Fig. 5A). Two signals in this experiment (labeled 1 and 2) were pulse trains with a period close to that of the oscillator, and in both cases a strong collective oscillator response was observed, even though the amplitude of individual oscillators did not change appreciably. Evidence of


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collective oscillator synchrony was observed in revivals of collective oscillator response after the last pulse of AHL in a pulse train (see Fig. 5B,C). This is the “echo” due to oscillators continuing to keep a common oscillatory phase for some time beyond the entraining signal’s influence. These observations for short pulse trains were consistent with additional experiments, the result of which are relegated to the Supplementary Information. The final signal (labeled 3) in the experiment presented in Fig. 5A was a chirp signal, which had an oscillatory period that changes linearly in time from 5 min. to 80 min. (see Methods). This signal was designed to try and probe the Arnold tongue structure observed in our theoretical investigations, in lieu of a more direct measurement of the Arnold tongue using a wide array of separate experiments. We find that oscillators begin to exhibit a coherent response near an instantaneous drive period close to 40 min., which is consistent with what would be seen for entrainment. Response appears to diminish for characteristic periods approaching 80 min., though this effect is less pronounced. Our results suggest that a chirp signal may be an effective means to quickly scan entrainment behavior in future experiments, though a full investigation is pending. In all cases, the signal was well above background, i.e. the response in the absence of time-dependent drive was clearly lower in magnitude than the response with drive. High signal to noise is linked to the high cell count for our measurements: ∼ 1000 cells are in each microfluidic trap, and we report the statistical median of each trap’s response across 6 different traps. Our study of collective mean fluorescence dynamics in Fig. 5 was complemented with corresponding single cell tracking analysis (see Fig. 6). Single cell results allow us to better determine whether our observations were more consistent with entrainment as compared to alternative explanations for oscillator synchrony. For example, a collective oscillator response in mean GFP fluorescence could arise by strongly modulating oscillator amplitude. We find that oscillators continue to oscillate with roughly the same amplitude throughout the whole experiment, and we find that the phase of oscillators can align during CFP-LAA pulse trains. Our observations are consistent with the viewpoint that oscillators are being entrained by our external signal. Finally, we performed a control experiment to determine whether entrainment could be reproduced by modulating AHL for a control construct that expresses untagged CFP instead of CFP-LAA but is otherwise the same as the construct in Fig. 5. The pronounced entrainment response witnessed for the CFP-LAA construct was not observed for the untagged CFP construct. This result supports that entrainment arises due to queueing competition. See the Supplementary Information for further details.

III.

CONCLUSION

We have shown through simulation of our models and microfluidics-microscopy experiments that a synthetic gene oscillator can be strongly entrained through competition for the cell’s degradation machinery. This alternative approach to entrainment does not require external control of the production rate of proteins within the network, as is often assumed in circadian rhythm systems [35–37], nor does queueing entrainment directly modulate the degradation rate of a substrate via chemical modification, e.g. as is the case for active degradation of cyclins during the eukaryotic cell cycle [38, 39]. Instead, queueing entrainment allows all substrates targeted to a single protease (or more generally, to a single enzyme) to modulate each other’s concentration by the action of an enzymatic bottleneck. We explored this using simple yet informative measures of oscillator coherence, which allows our results to be used by those studying batch cultures. Also, we provided early investigations into an alternative method to measure Arnold tongues using carefully structured chirp signals, which provides an efficient probe for oscillator entrainment response. The results of our work were made possible by a synthetic biology approach [14]. Leveraging gene networks with known molecular parts and interactions greatly simplifies the analysis and interpretation of these networks. Native networks, in contrast, have a host of unknown parts and interactions that obfuscate the origin of network dynamics. In the present case, we were able to construct a system where two relatively small networks couple through mutual competition for a protease, and we were able to demonstrate that this queueing interaction could relay an entraining signal. It is unclear how easily this conclusion could have been reached in native networks. We also believe our work will contribute back to synthetic biology as a field, since queueing entrainment should be able to be implemented for essentially all existing synthetic gene oscillators in bacteria, owing to the dependence of these oscillators on rapid protein degradation by ClpXP [17], as in our current work. Our work suggests that oscillator entrainment and synchronization may readily arise in natural systems. Periodic modulation of a bottleneck (queue) for degradation could entrain and thus synchronize numerous chemical oscillators



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(or damped oscillators) that share the same proteolytic pathway. In many mammalian oscillators, protein turnover is integral to maintaining oscillations. For example, two circadian regulatory proteins PER1 and PER2 go through a daily cycle of production and degradation to maintain circadian rhythms [40]. Circadian rhythms are entrainable by a number of stimuli, including light, but our results suggest that an oscillatory network that itself is not directly entrainable by these signals could be indirectly regulated if it shares a proteolytic pathway with the circadian network. This could be of particular importance for understanding oscillators associated with human diseases. For example, the tumor suppressor p53 and Mdm2 oscillate following DNA damage, and this oscillation is dependent on protein degradation [41]. We anticipate that future work will reveal surprising synchrony between these and other natural networks, and that the source of synchrony will linked to queueing synchronization and entrainment.

IV. A.

METHODS

Model Details and Simulation

Two similar but qualitatively distinct models where used to investigate queueing entrainment. The first can be described by a set of coupled chemical reactions simulated using the Gillespie algorithm [42]. These equations describe the evolution in time of the intracellular count a number of components, including transcriptional repressor LacI-LAA (R), GFP-LAA (Y ), and CFP-LAA (C). AraC-LAA is not modeled explicitly. All reaction velocities include the standard mass action terms, as in Ref. [42]. The map-based degrade-and-fire model was used to attempt to isolate certain details related to queueing entrainment [30]. The map-based model is similar to the model used in Refs. [22, 34]. Further details for these theoretical models appear in the Supplementary Information.

B.

Reagents

All reagents were reagent grade and purchased from Sigma-Aldrich Co., Fisher Scientific, Inc., or Thomas Scientific unless otherwise stated. The chemical form of AHL used was 3-oxohexanoyl-homoserine lactone (3OC6HSL from Sigma-Aldrich Co.).

C.

Strains and Plasmids

E. coli NB002 and NB010 cells were used for all in vivo experiments. All plasmid DNA sequences can be found in a single Zip file in GenBank format (Supplemental Data 1). NB002 was constructed from NB001 [22] by introducing plasmids pTDCL7 and p15aCm01. NB010 was constructed from NB001 by introducing plasmids pTDCL7 and p31CmNB96. The plasmid p15aCm01 was constructed by cloning a region (contains PluxI cfp-LAA and PluxR luxR) from pBlunt plux::cfp into p31Cm XhoI-BamHI restriction sites. The plasmid p31Cm was constructed by PCR amplifying and cloning a T1 terminator into KpnI-ClaI restriction sites of pZA31MCS (purchased from Dr. Rolf Lutz) [43]. The pTDCL7 and pBlunt plux::cfp plasmids were gifts from Dr. Jeff Hasty from University of California, San Diego. The p31CmNB96 plasmid was constructed by cloning a region (containing PluxR luxR and PluxI cfp) from pMK-RQ SDC39 into p31Cm AatII-PmeI restriction sites. The pMK-RQ SDC39 plasmid was purchased from ThermoFisher. The plasmids pTDCL7, pBlunt plux::cfp, and pMK-RQ SDC39 were maintained using kanamycin (Km, 25 µg/ml), while p31Cm, p15aCm01, pZA31MCS, and p31CmNB96 were maintained using chloramphenicol (Cm, 10 µg/ml). The cultures were grown in Lysogeny Broth (LB).

D.

Microfluidics and Microscopy

The images were taken using a Nikon Ti microscope with CFP, YFP, and mCherry fluorescence cubes at 1000x magnification (a 100x objective coupled to additional 10x magnification). A YFP optics cube was used to image GFP-LAA within the cells, to avoid the overlap in excitation and emission frequencies between GFP and CFP cubes.


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The inducer concentration was monitored using 5 µg/mL of sulforhodamine dye, a red fluorescent dye. Phase contrast images were taken every 30 seconds while fluorescence images were taken every 3 minutes. The cells were grown in LB overnight and used as inculum (1:100 dilution) for a culture grown for 4 h before loading cells into the microfluidic device. Microfluidic experiments used LB with 10 µg/mL chloramphenicol, 25 ug/mL kanamycin, 1% arabinose, 2 mM IPTG, and 0.2% Tween-80 surfactant (to prevent cell adhesion), and timevarying levels of AHL.

E.

Microfluidics Devices

The microfluidic device used for the experiment in this main text is labeled Dial-a-Wave (DAW) 5.0. The traps have a dimension of 50 µm x 71 µm x 1 µm. Our control experiment that tested the influence of untagged CFP production on the oscillator used a similar microfluidic design (DAW 8.0). The design of DAE 5.0 is included in the Supplementary Information.

F.

Chirp Signal Generation

The chirp signal used in Fig. 5 was generated such that the instantaneous period varies linearly in time. This was achieved by defining the phase variable φ(t), which controls the phase of a simple square wave signal S, where S = 1 for 0 ≤ φ < π, S = 0 for π ≤ φ < 2π, and so on with period 2π. An instantaneous period T (t) is defined to vary linearly in time (from 5 min. to 80 min. in our experiment), and this affects φ through the relationship dφ/dt = 2π/T (t), which is integrated using a simple first order method in Excel.

G.

Analysis of Mean Fluorescence

In Videos 1-2, and in Fig. 5, we define fluorescence “response” (for GFP-LAA and CFP-LAA proteins) in the following way. Define filtered mean fluorescence as mean fluorescence (across the region of the image containing cells) with its 30.0 min time-symmetric mean subtracted. Fluorescence response is defined as filtered mean fluorescence divided by its standard deviation (taken across the time of the experiment). Thus, fluorescence response has a standard deviation (taken across all time) of 1.0. This approach conveniently normalizes response. The response used in Fig. 5 is in fact the median fluorescence response across 6 microfluidic traps.

H.

Image Analysis

The bulk of image analysis was performed using custom scripts leveraging the SciPy and OpenCV packages for Python 2.7, similarly as in Ref. [22]. We have also developed a custom pipeline for single cell segmentation and tracking using machine learning techniques. Segmentation was done in three steps. (1) Phase contrast images were automatically classified into cell and non-cell pixels by a classifier trained using the Trainable Weka Segmentation tool in Fiji [44], thus generating a mask that can be used for cell identification. Cells ideally would be identified by connected regions of like-colored pixels. The classifier was a FastRandomForest classifier with 200 trees, a maximum depth of 11, and 2 randomly selected features. The classifier was applied automatically and in parallel using Fiji scripting techniques. (2) Since the resulting cell mask did not always perfectly identify single cells, e.g. having two cells being identified as a single cell due to contact with each other in the mask, we trained another classifier to automatically correct small defects in the mask. This new classifier was used to generate new masks from the first set of masks. (3) Custom code leveraging the Python scipy library rapidly identified cells using these final masks. Simultaneously, the mean fluorescence was measured over corresponding pixels in fluorescence images, which allowed us to determine mean cell fluorescence. Tracking then followed by comparing cells in adjacent (in time) mask images. With mild constraints to detect error, the cells in adjacent frames with maximum overlap of their identified pixels were identified as either being the same cell at two different times or as being mother-daughter pairs due to cell division. More details concerning our segmentation and tracking algorithm will appear in a future publication.



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V.

SUPPORTING INFORMATION

The Supporting Information is available free of charge on the ACS Publications website. Supporting Information. Supplementary details and discussion, videos of key microfluidic experiments, DNA sequences of synthetic constructs. Queueing Entrainment SI.pdf. Additional details, discussion, and results, including details of the mathematical models. Supplementary Video 1.qt. Video of the microfluidic experiment in Fig. 5. Video of the microfluidic experiment in Fig. S5 in the document QueueSupplementary Video 2.qt. ing Entrainment SI.pdf. Plasmid Files Q-entrain.zip. DNA sequences for the constructs used.


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2500

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(B)

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drive period = 31.25 min

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(D) drive period = 31.25 min

1400 1200 1000 800 600 400 200 0 0 50 100 150 200 time (min)

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Y C R

50 100 150 200 time (min)

FIG. 2: Theoretical results for the chemical reaction network model of queueing entrainment. Shown are results for GFP-LAA (Y), CFP-LAA (C), and LacI-LAA (R). The quantitative model was driven by a pulse train with a duty cycle of 25% and with an amplitude of 50 min−1 . A single realization of the oscillator was driven with either (A) a very short period (0.25 min) or (B) a period (31.25 min) near the natural period of the oscillator (approximately 30 min.). In each case, the qualitative shape of the oscillator’s trajectory is similar, suggesting that the oscillator is not simply being driven by the CFP-LAA signal. However, for ensembles of size 10,000, the ensemble mean of each component exhibits substantially different behavior for a period of (C) 0.25 min vs. (D) 31.25 min. The shorter period leads to virtually no collective variation in the ensemble mean for any component, while the longer resonant period leads to significant oscillation in the ensemble mean. Additional parameters are relegated to the Supplementary Information.


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Oscillator

IPTG

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GFP

ACS Synthetic Biology Oscillator Response

Queueing Feedback

LacI ClpXP

ARA

AHL Signal


CFP Response

Φ

Queueing Feedback

Entrainment of a Bacterial Synthetic Gene Oscillator through

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