Noise-guided design of synthetic protein waves in living cells

Abstract

Protein circuits organize cell biology, but synthetic dynamics are challenging to engineer due to stochastic genetic and biochemical variation. Genetically encoded oscillators (GEOs) built from bacterial MinDE-family ATPases and Activators generate synthetic protein waves that act as novel frequency-domain imaging barcodes in eukaryotic cells, providing a platform for understanding, engineering, and applying synthetic protein dynamics. Using budding yeast, we disentangle how expression levels and expression noise govern GEO waveform and encodability. While GEO amplitude is sensitive to extrinsic noise, GEO frequency is stably encoded by the Activator:ATPase ratio. By integrating GEO components into the yeast modular cloning toolkit, we developed different noise-guided expression strategies that act like filters on GEO waveform. We paired these filters with hundreds of biochemically distinct GEO variants to engineer clonal populations that oscillate at distinct frequencies and to design waveform libraries with customizable spectral features and tunable waveform variation. Our work establishes a robust platform for precision genetic encoding of synthetic GEO oscillations and highlights the utility of noise-guided strategies for dynamic protein circuit design.

Introduction.

Cells use networks of interacting genes and proteins to build molecular circuits that organize and regulate their dynamic behavior.^1-3 While genetic circuits can control expression over long timescales, protein circuits are essential to rapidly sense and manipulate biological activities with spatiotemporal precision.^4-6 Protein circuits implement many key dynamic cellular tasks like motility,^7,8 information processing,⁹ phagocytosis,¹⁰ polarized secretion,¹¹ and force generation.¹² As such, there is increasing interest in designing programmable protein circuits with fast predictable dynamics.^13-15

A protein circuit’s dynamic behavior is shaped at two levels. At one level, the activities of the protein components and their regulatory relationships establish a network structure that defines how their spatiotemporal dynamics unfold. For example, a coherent-feed forward network structure is well-suited to implement delays, while a mutual inhibition architecture facilitates polarization.^16,17 At a second level, protein circuits are affected by the concentrations of their components, which are controlled upstream by gene expression. Differences in expression levels can cause the same network architecture to produce different outcomes.¹⁸ As a result, engineering dynamic protein circuit behaviors can be challenging, as both the genetic and biochemical contributions to circuit performance must be carefully coordinated to realize a target behavior.

One source of expression variation in protein circuits arises from stochastic expression noise, which introduces uncertainty in the concentrations of circuit components of genetically identical cells.¹⁹ For an individual gene, its total expression noise can be partitioned into extrinsic and intrinsic components. Extrinsic noise arises from cell-to-cell variation in the concentrations of molecules required for gene expression, whereas intrinsic noise arises from the stochastic nature of the reactions needed for gene expression itself. From a circuit-design perspective, the intrinsic noise reflects the degree of non-correlation between equivalent copies of a gene within a single cell.¹⁹ These sources of noise create a distribution of component levels within the population, introducing cell-to-cell variation in circuit dynamics. Noise effects can be undesirable if a specific dynamic output is critical for function. Alternatively, structured variation can also be a feature that allows probabilistic behaviors to be encoded through a diversity of functional but quantitatively distinct outputs.²⁰ The effects and consequences of noise have been explored for many natural endogenous signaling networks.^21,22 Noise-guided strategies for synthetic protein circuit design could improve our ability to genetically encode specific dynamic outputs at a population level.

Recently, genetically encoded oscillators (GEOs) built from MinDE-family ATPase and Activator modules have been developed to generate fast orthogonal protein waves for spatiotemporal protein circuit design in eukaryotic cells.²³ These waves propagate through a reaction-diffusion mechanism: ATP-dependent assembly of the ATPase on membranes stimulates recruitment of its associated Activator, stimulating ATP hydrolysis and membrane release.^24-30 The resulting wave produces a robust oscillating signal in the cell whose frequency and amplitude can be quantified using Fast Fourier Transform (FFT).^23,31 GEO circuits have demonstrated a range of biotechnology applications including frequency-domain single-cell barcoding, AM/FM broadcasting of single-cell state dynamics, and sub-cellular patterning of biological activities.^23,32 The simplicity, orthogonality, and dynamic phenotypes of GEOs, combined with applications that rely on quantitative control of oscillations, makes them an ideal experimental testbed and model system for exploring genetic and biochemical coordination of synthetic protein dynamics.

Here, we implement GEOs in a budding yeast system to comprehensively characterize the genetic encodability of oscillations and engineer noise-guided circuit expression strategies that enable precise and tunable control over the resulting dynamics. We experimentally define and model phase portraits that reveal how the breadth of frequencies and amplitudes available to a GEO are genetically controlled by ATPase and Activator expression levels and noise arising from copy number, transcriptional, and translational variation. The GEO phase portrait reveals separable effects of extrinsic and intrinsic expression noise on GEO amplitude and frequency respectively. By integrating the GEO components into the yeast modular cloning kit³³, we could readily exploit these differences to engineer different GEO expression strategies that act as noise filters on the phase portrait, allowing us to construct clonal populations that robustly and stably oscillate at different frequencies or to build waveform libraries that independently tune frequency and amplitude spread within a single population. To expand GEO programmability, we characterized 169 biochemically distinct GEOs, providing a rich assortment of phase portraits spanning different frequency and amplitudes as starting points for waveform design and application. Our strategy of coordinating noise effects with circuit behavior may provide a more general strategy for improving precision control over protein dynamics in cells.

Results.

Encodability of GEO protein oscillations is determined by phase portrait structure.

Genetically encoded oscillators (GEOs) derived from bacterial MinDE-family ATPase/activator pairs robustly generate synthetic orthogonal protein waves in vitro and in eukaryotes.^23,27,28,32 To systematically define the precision with which GEO oscillations can be genetically encoded and the impact of expression noise on oscillation behavior, we deployed GEOs in the budding yeast Saccharomyces cerevisiae (Fig. 1A). Budding yeast provide an ideal chassis for this inquiry, as a rich collection of expression vectors, promoter libraries, and single-copy integration strategies have been developed which allow for comprehensive manipulation of gene expression.³³

Figure 1: MinDE-family genetically encoded protein oscillations (GEOs) in budding yeast.

A. MinDE-family ATPase/activator pairs drive protein oscillations based on reaction-diffusion dynamics. MinD-family ATPases dimerize and binds the membrane in their ATP-bound state. MinE-family Activators catalyze the hydrolysis of ATP to drive components off the membrane. When the E. coli MinDE genes are expressed in yeast, circular waves form.

B. Each pixel of a cell has a temporal oscillation, so cells can be imaged at low magnification and high sample size. Cells are masked by fluorescence, and the time trace of each cell’s pixels is then processed with FFT to decode the underlying frequency. Applying this across all pixels in the image produces a multi-channel frequency-domain image in which individual cells can be identified based on their underlying GEO frequency. Peak height shows amplitude normalized to the cell’s mean fluorescence in signal channel. The two highlighted oscillating cells have peaks of 0.53 and 0.47. Non-oscillating cells have no discernable peak and are not visible on the frequency-colored image. Images shown represent 4.4% of total 10X image area. Scalebar = 20 microns.

We first expressed fluorescently tagged MinD ATPase and MinE Activator E. coli GEO components on separate Cen/ars plasmids. This expression strategy generates a wide range of ATPase and Activator expression levels that are uncoupled from one another on the population level, providing access to many different configurations of the GEO circuit. Imaging this population by timelapse fluorescence microscopy revealed a large proportion of oscillating cells, in which the GEO waves manifest as a “washing machine” phenomena circling the cell interior and developing bud (Video S1). Within this Cen/ars population, we observed considerable diversity in the waveforms across different expression levels: some cells oscillated slowly (7 mHz), some oscillated faster (13 mHz), and some cells did not oscillate at all (Fig. 1B). Thus, GEOs can realize a wide spectrum of waveforms when expressed in budding yeast.

To define the relationship between GEO expression levels and oscillation waveform, we took advantage of the fact that single cell oscillations could be robustly detected at the pixel level even at low (10x) magnification using Fast Fourier Transform (FFT).³¹ This allows the oscillation status (yes/no), waveform (frequency and amplitude) and associated ATPase and Activator expression levels (using fluorescence as a proxy) to be quantitatively analyzed for thousands of individual cells at once from a single field of view. To increase our coverage of the theoretically accessible expression space, we aggregated data from many fields of view for all pairwise combinations of high (2micron) and low (Cen/ars) copy expression plasmids, producing a comprehensive dataset consisting of more than 3500 cells (Fig. S3A).

From these single-cell data, we plotted phase portraits that map ATPase and Activator expression levels to the oscillation state (Fig. 2A), frequency (Fig. 2B) and amplitude (Fig. 2C). The structure of these portraits reveals the key programming logic controlling GEO waveform. Oscillations can only occur within a specific range of Activator:ATPase ratios. To quantify the boundaries between the functional and non-functional space, we developed and applied a machine-learning classifier based on two logistic regression models³⁴ to recover the minimum and maximum Activator:ATPase ratios that can support oscillations (Fig. S1). These parameters establish the GEO’s functional frequency bandwidth and the upper and lower limits of its dynamics (Fig. 2A). The model was highly accurate, precise, and sensitive in classifying cells as oscillating (accuracy: 0.938, precision: 0.944, sensitivity: 0.988; detailed in Fig. S1C).

Figure 2: Phase portrait analysis defines how ATPase and Activator expression controls GEO oscillation status, frequency, and amplitude.

Budding yeast single-cell GEO behavior, waveform metrics and component expression levels were aggregated from uncoupled plasmid expression data for visualization.

A. Oscillating cells fell between two Activator:ATPase ratios, which were classified with logistic regressions. The point color shows true status, while the background shading denotes ML-model predicted status. Blue= oscillating, red = non-oscillating.

B. GEO frequency within the phase portrait. Frequency largely scales with Activator:ATPase ratio. Grey cells are non-oscillating.

C. GEO amplitude within the phase portrait. Amplitude largely scales with ATPase abundance.

D. Within vertical slices of the phase portrait, amplitude exhibits three distinct states: non-oscillating, stationary, and decay as expression approaches the upper boundary of oscillation. Frequency scales with Activator:ATPase ratio. N = 3643 cells.

Within the functional oscillation space, GEO frequency increases with increasing Activator:ATPase ratio and GEO amplitude scales with ATPase expression but drops off as the Activator levels approached the boundaries of the functional space (Fig. 2D). For a given slice of ATPase expression, increasing Activator toggles amplitude from background levels to an “on state”, which is maintained until a decay back to background at higher Activator levels.

Our phase portrait characterization agrees with previous observations of GEO behavior in mammalian cells, in vitro systems and mathematical models but provides new experimental clarity over the boundary and transition to non-oscillating states. Taken together, the GEO phase portrait provides a comprehensive description of oscillation behavior and its underlying expression constraints: where oscillations can arise, and how frequency and amplitude change as a function of ATPase and Activator concentration.

Independent effects of intrinsic and extrinsic expression noise on GEO frequency and amplitude.

The GEO phase portraits we constructed in Fig. 2 provide the ATPase and Activator expression levels required to realize a specific oscillation waveform. However, genetically identical cells will vary in component levels owing to expression noise, making it unclear how precisely a specific waveform can be genetically encoded in practice. This noise arises from two separable components: extrinsic noise, which arises from differences in the levels of factors needed for gene expression; and intrinsic noise, which arises from the stochastic nature of gene expression itself. Intuitively, extrinsic noise reflects variation between cells in a population, whereas intrinsic noise can be understood as variation between equivalent genes with identical promoters within the same cell.¹⁹

To experimentally determine the effects of intrinsic and extrinsic expression noise on GEO waveform, we generated a yeast strain harboring genomically integrated constructs that separately express each of the E. coli GEO ATPase and Activator components from identical copies of a strong Tef1 promoter. We imaged the dynamics of this genomically-locked strain (henceforth: int-GEO1) using timelapse fluorescence microscopy and applied our analysis pipeline to quantify the resulting expression levels and waveforms present in the population. Compared to Cen/ars, the observed GEO frequency within the int-GEO1 population was remarkably consistent (Fig. 3A, Video S2), with the inner 80% of the population (10^th-90^th percentiles) oscillating between 13 and 17 mHz, as compared to an inner 80% range of 8 to 20 mHz for the uncoupled Cen/ars population. Importantly, the stability of oscillations in the int-GEO1 strain persisted even as cells grew and expanded out over multiple generations (Fig. 3B).

Figure 3: Genomic integration reveals separable effects of intrinsic and extrinsic expression noise on GEO frequency and amplitude.

A. Integrating the GEO circuit reduced frequency bandwidth compared to the Cen/ars paradigm under the same promoters (N > 1200 for each population).

B. Integrated GEOs remain robust and predictable during culture expansion, while Cen/ars populations shuffle due to random plasmid copy number inheritance. Population spread is maintained as cells divide. Images represent 2.4% of total image area. N>1000 cells for each population at each timepoint. Scalebar = 20 microns.

C. Expression data from panel A plotted normalized by each population’s means along the given axes. Genomic integration has lower extrinsic and intrinsic noise compared to the Cen/ars population.

D. Simulated population structures with varying noise show that frequency noise is exclusively dependent on intrinsic noise, while amplitude noise is exclusively dependent on extrinsic noise. CV = Coefficient of Variation.

To clarify the origin of this robustness, we projected the genomically-locked single-cell int-GEO1 data onto the full GEO Cen/ars phase portrait, normalizing by separate population means along both axes (Fig. 3C). This revealed that expression levels of the ATPase and Activator were highly correlated, falling in a narrow band along a fixed Activator:ATPase ratio and consequently a specific GEO frequency. Thus, although extrinsic noise can introduce large variation in component expression levels, these differences are preserved across similar genes and promoters, resulting in low Activator:ATPase ratio variation.

To formalize our understanding of these experimental observations and gain intuition, we simulated waveforms for clonal populations facing different levels of intrinsic and extrinsic expression noise. Briefly, two random normal distributions were used to create expression profiles: the first for ATPase abundance, and the second for a set of “offsets” from the ATPase abundance to generate the Activator expression (Fig. S2A). The result was a set of expression profiles with varying intrinsic and extrinsic noise (Fig. S2B). Tuning the standard deviation of the ATPase distribution affected extrinsic noise, while the standard deviation of the offsets tuned intrinsic noise (Fig. S2C). Guided by the structure of the GEO phase portrait (Fig. 2), we modeled frequency as a linear scaling of Activator:ATPase ratio, and amplitude as a linear scaling of ATPase abundance. We then sampled ATPase and Activator expression levels from the simulated distributions and quantified the resulting frequency and amplitude variation within the resulting population. These simulations confirmed that a consequence of the GEO phase portrait is that frequency and amplitude variation are generally independent of one another, with frequency variation influenced by intrinsic noise and amplitude variation influenced by extrinsic noise.

Exploiting noise insights to engineer GEO waveform structure within a population.

Our analysis has so far revealed that 1) GEOs can generate a diversity of waveforms across different expression levels programmed by an underlying phase portrait; and 2) variation in GEO frequency and amplitude are independently controlled by intrinsic and extrinsic expression noise respectively. From an engineering point of view, this suggested that GEO circuit expression strategies that introduce or reduce expression variation along the extrinsic-like or intrinsic-like axes of the population could be used to tune or filter GEO waveform. We hypothesized that these noise-guided strategies would enable construction of different clonal strains that robustly oscillate at unique frequencies, or waveform libraries that independently vary frequency or amplitude for barcoding or dynamic perturbation applications.

To develop different noise-guided GEO circuit expression strategies, we domesticated the E. coli GEO ATPase and Activator genes as parts for use in the MoClo modular golden gate cloning system.³³ This allowed GEO components to be easily connected to different promoters to build diverse transcriptional units (Fig. 4A). These transcriptional units can be further combined in different ways to introduce copy-number variation or drive coupled or uncoupled component expression within a population, thus tuning the effective intrinsic or extrinsic expression noise.

Figure 4: Noise-guided expression strategies for controlling GEO waveform.

A. GEO ATPase and activator components were domesticated for MoClo toolkit logic, allowing for multipart assembly of transcriptional units (TU) and different co-expression strategies. Scatterplot shows example population distributions for specified expression strategies.

B. Genomic integration results in frequency and amplitude tunable populations. The means and standard deviations are shown. For specific promoter combinations disambiguation, see Fig. S3B.

C. Expressing GEO activator and ATPase on one plasmid results in populations with low frequency noise, but high amplitude variation. For specific promoter combinations disambiguation, see Fig. S3C.

D. A mixed expression strategy with integrated ATPase and plasmid Activator results in frequency-varied, amplitude-locked populations. For specific promoter combinations disambiguation, see Fig. S3D.

E. Population-level noise depends on expression methods, with distinct clusters forming along the intrinsic and extrinsic axes depending on the expression strategy. Expression noise translates directly to GEO oscillation noise in amplitude and frequency. CV = Coefficient of Variation. Here, the terms “extrinsic” and “intrinsic” refer to their phenomenological decomposition along the indicated axes.

Because the int-GEO1 strain produced a tight, near uniform frequency at the population level, we reasoned that genomically integrated populations with different Activator:ATPase ratios would access different frequencies. To this end, we generated ATPase and Activator transcriptional units driven by a range of different promoter strengths and used single-copy integration to deliver these into the same two genomic locations for comparison (Fig.4B, Fig. S3B). In this context, different promoter combinations reliably established different Activator:ATPase ratios within the population, resulting in specific frequencies locked with population standard deviations of ~2 mHz (Fig. 4B). Amplitude variation was also low, due to the lack of DNA-level variation in ATPase gene copy number.

Seeing that different GEO frequencies can effectively be locked by different genomically integrated promoter combinations, we next sought to create population structures that maintained a consistent frequency but introduced large variation in wave amplitude. To do this, we expressed both the ATPase and Activator transcriptional units with specific promoter combinations on single Cen/ars plasmids. Like their genomically integrated counterparts, these single-plasmid expression strategies showed tight Activator:ATPase ratios (low intrinsic-like noise) within the population and thus a narrow band of GEO frequencies (Fig. 4C). However, because the copy number of these plasmids varied among individuals within the clonal population, there was extensive variation in total expression and amplitude due to high extrinsic-like noise (Fig. 4C, Fig. S3C). Thus, coupled expression strategies – whether realized through variable-copy plasmids or genomic integration – serve as effective frequency-locking strategies by maintaining consistent expression ratios even in the presence of extrinsic noise. Because of this immunity to extrinsic noise, as a signal, the frequency varies less on a population level than the fluorescence of the ATPase channel, regardless of expression strategy (Fig. S3F).

Finally, we aimed to create population structures that introduced variation in GEO frequency while maintaining a consistent GEO amplitude. We accomplished this using a mixed strategy design in which ATPase expression was locked by genomic integration, but Activator was expressed from a Cen/ars plasmid. This expression approach aimed to increase population variation in the Activator:ATPase ratio, scrambling GEO frequency, while maintaining a relatively fixed ATPase level, locking in GEO amplitude. Using this strategy with different promoter combinations and number of ATPase genome integrations, we could effectively establish different populations with similar, broad GEO frequency bandwidths but whose oscillations were centered about different amplitudes with low variation (Fig. 4D, Fig. S3D).

We can directly compare these different GEO circuit expression strategies in the context of noise by decomposing their expression variation into its extrinsic-like and intrinsic-like components. We emphasize that this noise decomposition only reflects the phenomenological contributions of extrinsic and intrinsic expression variation, which are nonetheless quite useful for design. Using this approach, we found genomically integrated GEOs had the lowest intrinsic and extrinsic noise, while expressing components from separate plasmids produced the highest intrinsic and extrinsic noise (Fig. 4E). As intended, our plasmid-coupled expression designs had low intrinsic-like noise but high extrinsic-like noise; and our mixed expression strategies had low extrinsic-like noise but high intrinsic-like noise. Critically, these differences in how expression noise was partitioned directly corresponded with the degree of amplitude or frequency variation observed within a population (Fig. 4E). Taken together, we were able to exploit noise effects to design different GEO expression strategies that establish a spectrum of waveforms with independent tunability over frequency and amplitude variation on the population level.

Evolutionarily diverse ATPases and Activators expand noise-guided design of GEO oscillations and frequency-domain imaging barcodes.

The noise-guided waveform encoding strategies we demonstrated for the E. coli GEO system in budding yeast operate at the level of gene expression, acting like filters on the GEO phase portrait that isolate specific subsets of expression space. As a result, the waveforms accessible and programmable by these approaches are still constrained by the biochemistry of the GEO ATPase and Activator components that underlie that phase portrait. An expanded set of alternative GEO ATPase/Activator pairs would provide additional phase portraits as starting points for application of our GEO expression strategies in waveform design.

To this end, we isolated and cloned 13 evolutionarily diverse MinDE-family ATPase and Activator modules³² into separate Cen/ars plasmids and used a yeast mating strategy to generate all 169 pairwise GEO combinations (Fig. 5A). These constructs expressed MinD ATPase variants N-terminally tagged with mCherry, and MinE Activator variants C-terminally tagged with EGFP. We imaged each system and defined its associated phase portrait using our analysis pipeline. Of the 169 combinations, 66 had >5% oscillating cells, which we defined as the threshold for considering a GEO as functional. From this library we identified new and useful oscillation properties for many of these GEO pairs, where identical expression levels program different frequencies (Fig. 5B, Video S3). To compactly visualize the diversity of GEO behaviors available, we plotted the mean frequency and amplitude for all functional members (Fig. 5C). This allows ready identification of GEO systems that oscillate faster (P. damselae MinDE, 44 mHz), or show significantly greater amplitude (P. poae MinD, Dickeya dadantii MinE, 47 AU) than the original E. coli GEO system (mean frequency 12 mHz; mean amplitude 18 AU). Hybrid pairings can also be functional, with the Activator module from fast species boosting oscillations of slower ATPase modules (E. coli MinD, P. damselae MinE, mean frequency 30 mHz).

Figure 5: Evolutionary diversity of GEO components uncovers new phase portraits for noise-guided waveform design and application.

A. 13 MinD-family ATPases and MinE-family Activators from across bacteria were cloned for uncoupled plasmid expression and tested using a mating cross to generate 169 GEO combinations.

B. Different GEOs had vastly different phase portraits, with the same expression levels resulting in markedly different frequencies and amplitudes. Gray cells are non-oscillating.

C. GEO phase portraits had diverse mean frequencies and amplitudes, with the P. damselae native pair having the highest mean frequency, and the P. poae MinD + D. dadantii MinE having the highest mean amplitude.

D. Radial “speedometer” plot of modeled fit parameters, visualizing the range of Activator:ATPase ratios that support oscillation colored by frequency for all well-fit GEO library members side-by-side. The white line denotes the existence of a minimum Activator:ATPase ratio. * = E. coli GEO; ** = P. damselae GEO.

E. Genomic integration of the fast P. damselae MinDE proteins creates fast and tunable clonal populations. The same promoter combinations for E. coli GEO are shown in grey (see figure 4B). Error bars denote standard deviation.

F. Iso-frequency populations enable single-fluorophore barcoding of clonal populations. These cells exist in similar expression regimes, but distinct frequencies, permitting multiplexed experiments on pooled populations barcoded by frequency. Here, four populations are decoded with frequency filters at 10, 20, and 35 mHz. Histograms represent distribution of fluorescence and frequency in entire field of view. Scalebar = 20 microns.

To further analyze the phase portrait structure of each GEO library member, we used our ML-classifier to define the upper limit on Activator:ATPase ratio for each GEO’s functional oscillation space. These models returned good classifications (see supplement) and upper boundary slopes ranging from 1.25 to 10.67 depending on GEO. We then modeled how frequency is programmed within the functional space by fitting frequency to a linear regression of the Activator:ATPase ratio (Fig. S4A). The metrics for all functional and well-fit GEO pairs can be visualized side-by-side using a radial “speedometer plot”, which shows the range of ratios that support oscillation for each GEO pair along the radial axis color-coded by the frequency they generate at that ratio (Fig. 5D). Here, the same Activator: ATPase fluorescence ratio resulted in markedly different results depending on the GEO components, both in oscillation state and frequency. This toolbox of phase portraits suggests new waveform programming strategies that take advantage of differences in the binary state of oscillation and frequency bandwidth of different GEO systems. Some GEOs have a wide operable expression range, offering many subsets of expression for frequency locking. Others, like the P. damselae ATPase + P. sp. Mhis Activator, scale frequency rapidly (Fig. S4A) but have a low upper bound, resulting in a sensitive but narrow operating range.

To demonstrate how this expanded collection of GEO phase portraits can be combined with the precision genetic control strategies we developed earlier, we analyzed the oscillation behavior of the fast and robust P. damselae GEO (mean 44 mHz) using the same promoter combinations we previously analyzed for the E. coli GEO. Since we domesticated these parts in our toolkit, we returned to the original FP designation of mNeongreen-MinD and MinE-mRuby2. While the expression levels of the GEO components are similar between matched promoter combinations (Fig. S3B,E), the different GEOs give rise to clonal populations that oscillate at unique, well-resolved frequencies (Fig. 5E). We used E. coli and P. damselae int-GEOs to spectrally barcode four separate clonal populations and deconvolute clone identity in mixtures based solely on single-cell frequency in a single fluorescent channel (Fig. 5F).

The diverse portfolio of GEO systems we characterized provide a rich assortment of phase portraits for noise-informed waveform programming. By coordinating differences in component biochemistry with the different GEO circuit expression strategies we developed earlier, we expand the space of oscillations that can be precisely encoded within a population.

Discussion.

In this work, we demonstrate noise-guided control strategies for genetically encodable oscillators (GEOs) and frequency-domain imaging barcodes that leverage evolutionary diversity and different expression methods. Key engineering insights came from the construction and quantitative analysis of GEO phase portraits, which revealed the spectrum of waveforms accessible to a given GEO circuit and how ATPase and Activator expression levels, noise, and biochemical features control oscillation status and waveform properties. These phase portraits allowed us to model and experimentally define how different sources of expression noise independently control the variation in a GEO’s frequency and amplitude. Using this knowledge, we developed a suite of GEO expression strategies that act like subtractive filters on the broad “white noise”-like distribution of waveforms within a GEO’s phase portrait.

GEO frequency is ratiometrically encoded in the relative amounts of ATPase and Activator, making it naturally buffered against extrinsic noise. Target waveforms could thus be reliably encoded using single copy integration circuits that combine different promoter strengths with GEO components from different species. Clonal populations could be barcoded by frequency within a single fluorescent channel, allowing multiplexed phenotypic analysis or interrogation of the spatial structure of populations of mixed-cell types or lineages. A dedicated biochemical carrier signal can thus be hard-wired in a cell’s genome for AM or FM encoding of dynamic single cell data using more elaborate circuit designs.²³ Not only does frequency provide an added dimension of data encoding, but the time-resolved unit (mHz) is concrete and transferable compared to the arbitrary units (AU) of fluorescence alone. Constructs that encode specific GEO waveforms could potentially be pooled and DNA-barcoded for use in sequencing-based readouts that relate specific dynamics to phenotypic outcomes.

Alternatively, we also found we could harness the separable effects of intrinsic and extrinsic noise on GEO waveform variability. Inherently shuffled amplitude libraries can be generated within specific frequencies (using coupled, single plasmid expression), or diverging frequency libraries can be standardized within specific amplitude ranges (using mixed, uncoupled expression). These expression paradigms could potentially be combined with additional strategies that operate downstream of gene expression to further buffer against noise, such as feedback regulation of transcription,³⁵ protein release from condensates,^36,37 or cell cycle synchronization.³⁸ Conversely, a genome-integrated GEO could potentially provide a dynamic reporter of expression noise, using the frequency and amplitude variation as a single-fluorophore population reporter of noise.

By using different methods of expressing GEO components, waveforms can be generated and applied to a diverse range of experimental needs, not only in budding yeast but also other eukaryotic systems. While not all of the strategies that we implemented in yeast are directly transferable to other systems, similar functional outcomes can likely be achieved by implementing the same overall logic but adhering to organism-specific constraints. For example, because coupled GEO component expression tightens frequency variation, similar outcomes could likely be achieved in mammalian cells using strategies like P2A/T2A, IRES elements, or tandem transcriptional units expressed from a single-construct^39,40 When considering how to transfer yeast results to other systems, it is also important to note that the specific mapping between expression and waveform we define here is based on fluorescence intensity, which depends both on the choice of fluorophore and the instrumentation at play. Going forward, more quantitative mappings between systems would benefit from direct measurement of GEO component concentrations within the cell, using techniques like FLIM or FPCountR.^41,42 Nonetheless, our work showcases the utility of noise-guided approaches to circuit design and the yeast chassis provide a powerful engineering and screening platform for designing new cellular dynamics.

These engineering strategies emulate how natural systems exploit noise as a tunable parameter – quenching it when precision is needed or exploiting it to introduce population-level variation during stress or division of labor.^43-45 While our study focused on oscillatory dynamics, we anticipate that noise-guided circuit design principles will be valuable for understanding and engineering dynamic protein circuits more generally. For instance, synthetic networks of interacting kinase/phosphatases,⁴⁶ GEF/GAPs,^47,48 receptors/ligands,^15,49,50 or protease/substrates^14,51,52 have recently been established as tools for dynamic protein circuit design. Because expression level effects and component ratios will impact the responsiveness and timescales of these circuits, aligning gene expression and noise effects with dynamic outputs would improve robustness and performance. Our work highlights the value of considering the impact and effects of noise in protein circuit dynamics, not as something detrimental to their performance but as a powerful tool for distributing different dynamics within a population.

Methods

Gene synthesis and cloning.

The modular yeast toolkit³³ was obtained from the McClean lab at UW-Madison. Further part plasmids were generated using the golden gate procedures described by Lee et. al. (2015) and NEB. The ymNeongreen fluorescent marker was cloned from addgene #125704 from Botman et. al (2019)⁵³ using a PCR product with forward primer 5’ -GCATCGTCTCATCGGTCTCATATGGTATCTAAGGGTGAAGAA -3’ and reverse primer 5’ – ATGCCGTCTCAGGTCTCAAGAACCCTTGTACAATTCGTCCATACCC – 3’ which introduces a point silent mutation at the N terminus to allow compatibility with the Moclo workflow. All remaining MinD, MinE proteins were cloned or ordered from Twist or IDT following Moclo syntax, with flexible linkers at the N or C termini, respectively. MinD genes were cloned as parts 3b and MinE as parts 3a. 3b MinD parts contained a stop codon to keep the C-terminus membrane-targeting sequence unobstructed by a GS linker. DNA was prepared from KCM transformation into DH5alpha or XL10-Gold cells following overnight growth and miniprep with Qiagen kits.

Yeast culturing, transformation, and imaging methods.

S. cerevisiae strains BY4741 (MATa his3Δ1 leu2Δ0 met15Δ0 ura3Δ0) and BY4742 (MATɑ his3Δ1 leu2Δ0 lys2Δ0 ura3Δ0) were obtained from the Craig and Weeks lab at UW-Madison. Budding yeast were transformed with plasmids and integration vectors following a standard LiOAc + ssDNA method.⁵⁴ YPD was used for preparing cells for transformation, which contains 1% (w/v) Bacto Yeast Extract, 2% (w/v) Bacto Peptone, 2% (w/v) Dextrose. Transformation products were plated on auxotrophic selection agar plates containing 0.67% (w/v) Yeast Nitrogen Base Without Amino Acids (Sigma #Y0626), 2% (w/v) Bacto Agar, 2% (w/v) Dextrose, and 0.2% appropriate amino acid supplement (with specific auxotrophic selection). The individual amino acids concentrations are defined in Yeast Synthetic Drop-out Media Supplements (Sigma-Aldrich, Saint Louis, Missouri).

Yeast GEO variant screen.

We constructed Cen/ars plasmids expressing MinD and MinE variants from the mammalian vectors in Rajasekaran et al. (2025)³². Briefly, the mCherry-MinD and MinE-EGFP genes were isolated with restriction digestion (BamHI and NotI), then ligation cloned into custom Cen/ars vectors with the pTef1 promoter and Adh1 terminator flanked by BsaI cut sites compatible with the overhangs of the genes. The MinD plasmids had the leucine selection marker, and the MinE plasmids had the histidine selection marker. We transformed the MinD and MinE plasmids into S. cerevisiae strains BY4741 (MATa his3Δ1 leu2Δ0 met15Δ0 ura3Δ0) and BY4742 (MATɑ his3Δ1 leu2Δ0 lys2Δ0 ura3Δ0)⁵⁵ respectively using the usual LiOAc/ssDNA transformation protocol⁵⁴. Single colonies were streaked in stripes on the appropriate selection media plates, then a mating cross was performed by replica-plating these stripes first onto YPD, growing them overnight, then onto double-selection plates (-his -leu) to select for diploid cells carrying both plasmids.

Imaging preparation.

Low fluorescence media (LFM) was made consisting of 0.17% Yeast Nitrogen Base without Ammonium Sulfate, Folic Acid, or Riboflavin (#MP114030512, Thermo Fisher Scientific, Waltham, Massachusetts), 2% Dextrose, 0.5% Ammonium Sulfate, 0.2% appropriate amino acid supplement (with specific auxotrophic selection). The individual amino acids concentrations are defined in Yeast Synthetic Drop-out Media Supplements (Sigma-Aldrich, Saint Louis, Missouri).

Single colonies of yeast were inoculated in 1mL appropriate auxotrophic LFM in Nunc^™ 96-Well Polypropylene DeepWell^™ plates (Thermofisher #95040462) and grown overnight in a shaking incubator (180RPM) at 30 °C. The following morning OD600 values were measured, and cells were diluted to 0.01 OD, then allowed to grow for 4-6 hours. A 96-well glass plate was prepared by pipetting 100 μL ConA (2mg/mL) into each well and was incubated for 1 minute, then removed. Yeast cultures were transferred to 1.5 mL Eppendorf tubes and vortexed vigorously. 200 μL of yeast culture was then added to the well, then removed after 10-15 minutes of settling. 100 μL of fresh LFM media was carefully added on top of the cells.

Cells were imaged with a Tokai stage top incubator. All temperatures were set to 30 °C except for the top heater, which was set to 37 °C to prevent condensation.

Fluorescence microscopy.

Cells were imaged using on a Nikon Ti2-Eclipse. Most data were imaged using the Photometrics 95B 25mm camera at 10X magnification with the Optivar. Data in figure 3B and 5F were imaged using the Hamamatsu Orca-fire camera at 10X magnification without the Optivar. Time-series were collected in the green and red channels for 10 minutes with 3 second intervals (2 second interval in the MinDE variants experiments (figure 5)).

Data analysis and modeling

Image processing.

Timelapse fluorescence images were processed with custom-written Python scripts. For cell segmentation, the red and green channels were time-averaged and blurred with a gaussian blur. An adaptive threshold was applied to identify fluorescent pixels and create a mask (openCV package)⁵⁶. Cells were identified in the mask using scipy ndimage functions⁵⁷. Fluorescence, size, and location data were extracted using numpy⁵⁸, and frequency information was extracted as previously described⁵⁹. Cells were filtered from background abberations using size filters. Cell oscillation state was determined using filters on within-cell pixel-level frequency variance and corrected amplitude (amplitude / mean MinD fluorescence). Frequency colored images were made using the real component of the FFT image stack using the temporal color code function in Fiji⁶⁰.

Logistic regression models.

Analysis was performed using custom-written Python scripts. A median ratio of MinE to MinD was calculated per hybrid pair using green and red fluorescence within the oscillating class. Cells were labeled as above or below the median and as oscillating or not. The data was then split and stratified by these two classes into a training set, reserving one third of the data for testing. Two logistic regression models were then instantiated using the clogistic Python library.⁶¹ One logistic regression model was trained on cells above the median ED ratio to categorize them as oscillating or non-oscillating. The second logistic regression model was then trained on this filtered set in the same manner. Binary predictions from the two models were combined by elementwise multiplication. These predictions were evaluated on the hold-out test set. The decision boundaries produced by each model were also plotted together.

GEO variant library functional boundary models.

Models were trained as in the previous case, with the following alterations. Oscillating cells that fell below the median MinE:MinD ratio were excluded from analysis, as there were insufficient non-oscillating data for classifying the bottom boundary. A single model was trained for each oscillating MinDE pair, detecting the upper functional MinDE ratio. With this strategy, we were able to create a well-fitting model for each of the 66 oscillating hybrid pairs.

GEO variant frequency linear regression.

For each frequency bin within a variant, the mean Activator:ATPase ratio was calculated. A 0-intercept linear regression (sklearn.linear_model.LinearRegression) was fit to these mean values with the Activator:ATPase ratio on the x-axis and Frequency in mHz on the y axis (Fig. S4). Bins with fewer than 10 cells were removed from analysis. The slope of these lines was used for generating Fig 5D, in addition to the upper boundary slope for each GEO variant determined by the logistic regressions described above.

Noise analysis.

Prior to calculating noise, expression was normalized to a standard range by dividing cell expression by the population mean for that sample along the two axes separately. Populations with different expression paradigms were analyzed to determine the extrinsic and intrinsic noise, using the formulas from Elowitz et al¹⁹. Only the oscillating cells were considered for direct comparison to GEO signal noise. The noise within the frequency and amplitude parameters were calculated by dividing population standard deviation by the population mean.

Noise simulation.

A standard normal distribution was used to create populations with defined means and standard deviation in one dimension. Another standard normal dimension was used to create a set of offsets of the first. Noise was calculated over a range of standard deviations in the simulated MinD and simulated MinE distributions, as well as a predicted frequency and amplitude noise parameter. (Fig. S2) These results were then binned by intrinsic and extrinsic noise, and the means per bin of the frequency and amplitude noise were plotted (figure 3D).

Supplementary Material

• ML workflow and performance evaluation; workflow for generating distributions with different expression noise parameters; raw data showing population structures of different expression strategies; quantitative description of GEO variant phase portraits.(Supplementary materials PDF)

• Materials used in this study. (Table S1)

• Yeast strains generated in this study. (Table S2)

• GEO species MinDE sequences and abbreviation disambiguation. (Table S3)

• Expression methods summary statistics. (Table S4)

• GEO variant summary statistics and fit parameters (Table S5)

• Representative Cen/ars E. coli GEO behavior (Video S1)

• Representative genomically integrated E. coli GEO behavior (Video S2)

• Side-by-side comparison of different GEO variants (Video S3)