To enable simultaneous, precise control and real-time visualization of Wnt signaling, we engineered a clonal HEK293T cell line with optogenetic control over Wnt pathway dynamics by fusing Cry2 (29) to the LRP6 co-receptor (30, 31) (from here on referred to as the opto-Wnt tool). To track pathway outputs at multiple levels, we used a CRISPR knock-in strategy to endogenously tag β-catenin (β-cat) with a custom tdmRuby2 fluorescent protein, enabling real-time observation of transcription factor accumulation and degradation. Additionally, to monitor Wnt target gene transcription, we integrated an 8X-TOPFlash-tdIRFP (32) reporter through lentiviral transduction. This engineered cell line, designated as the Wnt I/O (Input/Output), is the first clonal cell line to offer both temporal control and live visualization of Wnt signaling, providing a platform for studying dynamic decoding in this pathway (Fig. 1A). We confirmed that both reporters respond to 450nm illumination through imaging and FACS (Fig 1B, Supp Fig. 1A).

Next, given the observed timescales of Wnt-induced cell fate decisions, we performed a baseline ON-OFF experiment to understand both the dynamics and heterogeneity of the Wnt I/O line. We tracked single-cell β-catenin and TopFlash dynamics in over 300 single cells during 24 hrs of illumination to visualize pathway activation, followed by 8hrs of dark to visualize pathway de-activation (Fig. 1C, Supp. Mov 1). In comparison to the dark control, we noticed robust activation of the Wnt pathway on the population level, with variation in the activity patterns of individual cells. Notably, TopFlash reporter activity did not decay in the dark, while β-catenin showed relatively fast degradation in the dark. Wide variation in Wnt response among cell populations has been widely reported in both optogenetic and non-optogenetic cell lines (Supp Fig. 1B, C, 3, 26, 33). We attribute the observed cell population variation in both β-catenin and TopFlash to individual cells being in different phases of the cell cycle, which has been shown to influence Wnt signaling response (34, 35). To quantify our videos, individual cells were segmented and tracked between each frame of our videos using a custom image analysis pipeline that involved the CellPose-Trackmate (36) deep learning framework and focused on the nuclear fluorescence of β-catenin and TopFlash signals (Supp. Fig 1D, for details on tracking and segmentation see Methods). To reduce background noise in nuclear fluorescence signals, we first measured background fluorescence in the β-catenin and TopFlash channels for each frame. For each cell, raw nuclear β-catenin and TopFlash intensities were subtracted by the corresponding background value in that frame. The resulting traces were then normalized to the mean of all traces at t = 0. As a control, we also normalized the mean fluorescence of β-catenin and TopFlash in the light on condition to the light off condition (Supp Fig. 1E, F).

In addition to long duration Wnt signals (>24hrs), a wide range of shorter pulses of Wnt activity has been observed throughout mammalian development (21, 22). To understand the impact of these shorter Wnt pulses we performed a Wnt duration scan, varying the activation time between 6 and 21 hours, followed by a variable rest time for a total 26-hour experiment. We found that each cell population could be distinguished by either its maximum β-catenin concentration or the final level of TopFlash with statistical significance and that our choice of normalization had no impact on our results. (Fig. 1D, Supp. Table 1, Supp. Fig. 1G, H). We also observe that the TopFlash signal is delayed from β-catenin which responds earlier. Overall, these experiments demonstrate that our Wnt I/O line faithfully tracks and controls Wnt dynamics, allowing further interrogation of the signal processing capabilities.

Our quantifications reveal that β-catenin continues to accumulate in the presence of optogenetic stimulation, reaching a maximum at the end of optogenetic stimulation and decreases in the dark. Our statistical analysis shows a significant difference in the ‘light on’ maximum fluorescent intensity for β-catenin between most conditions with 18 and 21hrs showing no significance (Fig 1D, Supp. Fig. 1I). This observation is consistent with β-catenin saturating to a maximum level during a prolonged Wnt activation, as described by previous experiments and models of the Wnt pathways (37). Looking at the TopFlash response, we see that longer light durations led to higher TopFlash levels at the end of the experiment and that post-stimulus TopFlash levels increase monotonically with light duration. The results of our statistical analysis revealed that also all TopFlash conditions are statistically significant compared to one another (Fig 1D, Supp. Table 1, Supp. Fig. 1J). Overall, our data shows that our Wnt I/O cell line photo-switches on in light conditions and off in dark and can be used to quantify how temporal dynamics of Wnt signaling affect downstream gene expression.

We then continued to analyze the ability of the Wnt signaling pathway to process dynamical inputs by developing a quantitative model. Although the Wnt pathway has been previously modeled, we sought to build a model of the Wnt pathway that captures our experimental observations with fewer parameters. We reduced the total number of differential equations in a previously established model of Wnt signaling (27, 38, 39) by describing only interactions between the DC and β-catenin (represented as state variables c(t) and b(t), respectively) (Fig. 2A). The complex formed when β-catenin binds to the DC is represented by state variable c_b(t). We represent optogenetic Wnt activation (denoted as l(t)) as directly increasing disassociation of the DC, omitting receptor-level dynamics of the pathway and instead modeling the optogenetic response through the activation and deactivation of Dvl (30) (represented as the state variable d_a(t) when active and d_i(t) when inactive). In total, we have nine {k₁₋₇, d₀, c₀} parameters governing DC and β-catenin dynamics, with k₁₋₇ denoting rates and d₀and c₀denoting the conserved concentrations of Dvl and DC respectively (Methods and Supp. Text). Since our experiments do not track all state variables directly, we use literature values (28, 40–42) to eliminate five parameters, and constrain the remaining four parameters of the model using the experimental data (Methods and Supp. Text). We model TopFlash transcription (represented as the state variable g(t)) as a sigmoidal function of β-catenin accumulation. We also use experimental data to constrain the four free parameters {r_max, τ, n, K} governing TopFlash expression, whose expression activates non-linearly, with a Hill function with parameters n and K, based on β-catenin levels at time t − τ. We arrived at the following differential equations to describe Wnt signalling dynamics: Figure 2.

(A) Schematic of ordinary differential equation (ODE) model of Wnt signaling. Dotted lines represent light dependent parameters. For information on model variables and parameters, refer to Methods, ODE Model for Wnt Signaling. (B) ODE model predictions (solid) of β-cat mean fluorescent intensity (MFI) for 6, 15 and 24hrs compared against our unsmoothed experimental results (light) from Figure 1D. Post-26 hours, experimental data was corrected for over confluency effects in both β-cat and TopFlash. (C) ODE model predictions (solid) of TopFlash MFI compared against our unsmoothed experimental results (light) from Figure 1D. (D) Visualization of duty cycle and frequency. Left: Constant frequency with varying duty cycle. Right: Constant duty cycle with varying frequency. (E) ODE model generated heatmap of endpoint TopFlash MFI for various combinations of duty cycle and frequency conditions. F) Line graph of 45-75% duty cycles vs frequency with 1/24, 1/3 and 4 cycles/hr labeled as A, B and C.

Next, we tested if the anti-resonant frequencies predicted by the model occur in the Wnt I/O cell-line. To simultaneously scan through a large range (=96) of unique duty cycle and frequency combinations, we utilized a high-throughput light stimulation device, the LITOS plate (44), that can deliver unique light patterns to all individual wells of a 96-well plate (Fig. 3A). Using this device, we performed the first ever optogenetic screen of Wnt pathway dynamics over a 48-hour period.

We find that β-catenin is more uniformly expressed across frequencies than our model predicted, with some increase of β-catenin fluorescence for high frequency inputs (Supp. Fig. 2A, 3A, B). Based on our model, the β-catenin heatmap can be understood by the fast degradation times of β-cat (∼10 minutes) relative to the timepoint of the last pulse of this experiment, meaning that β-catenin fluorescence after 48h depends also on the pulse arrangement in a particular frequency/duty cycle combination (Fig. 3B, C, Supp. Fig. 3A, B). In addition to quantifying fluorescence, we confirmed that there were no statistical differences due to cell number or total integrated light exposure (Supp. Fig. 3C, D). Next, we look at TopFlash expression to verify the presence of the anti-resonance.

TopFlash displayed a clear anti-resonant effect with minimum activity between 1/3 and 1 cycles/hr, demonstrating that cells with the same duty cycle but different frequencies have varying pathway outputs (Fig. 3D, Supp. Fig. 3E). For example, the 55% duty cycle stimulation resulted in a clear decrease in TopFlash at the ∼1/2 cycles/hr frequency (Fig. 3E). We further verified robustness of our results by performing a replicate experiment where we rearranged the order of the duty cycle and frequency conditions, demonstrating that the anti-resonance is independent of the experimental arrangement (Supp. Fig. 3F). We quantified the TopFlash mean fluorescent intensity (MFI) of our replicate experiment and averaged them with our original experiment’s TopFlash MFI. Plotting the average of our two experiments as a heatmap, we observe a strong agreement between the results of our two experiments (Fig. 3F, Supp. Fig 3G, F). These experimental results confirm anti-resonance in Wnt signaling and further confirm the predictions by our computational model of the Wnt pathway.

We have shown that a biochemical model based on established models of the canonical Wnt pathway correctly predicts the anti-resonance observed in optogenetic Wnt activation. Next, we abstract our model further to establish a minimal model that explains this non-monotonic behavior. Rather than explicitly modelling the interactions of proteins upstream of β-catenin, we abstract the time-dependent response of the Wnt pathway into a single a “hidden variable” a(t) (Fig. 4A). This variable is activated optogenetically until saturation at a rate k_on, and deactivated in absence of light at a rate k_off. We use equivalent β-catenin dynamics to the ones earlier, with a degradation and a synthesis term, consistent with Goentoro et al (38). The degradation of β-catenin dynamics is inhibited by the hidden variable, such that the β-catenin accumulates during optogenetic stimulation as before. We describe TopFlash transcription using the same Hill-type activation as the biochemical model in Fig. 2 and note that the coefficient of the Hill-function has no significant effect on the presence of the anti-resonance.

If the dynamics of the hidden variable occur at shorter timescales than the dynamics of β-catenin, we can reproduce the dynamics of β-catenin and TopFlash from Fig. 2 as well as the anti-resonance (see Supplemental Text). Our more abstract model allows us to see that the anti-resonance arises because of the interplay of the timescales involved. In the low frequency regime, the timescale of a(t) becomes irrelevant, and only the slower β-catenin dynamics are important. When the frequency is increased, the pulse duration is no longer long enough for β-catenin to saturate to a steady-state value. Especially when coupled with non-linear Hill-type activation, this leads to less TopFlash being produced. As the frequency increases further towards the high-frequency regime, the dynamics of a(t) become important. In this regime, we obtain increasing TopFlash expression with frequency as long as k_off < (1 + k₂)k_on is satisfied. We confirm this bound analytically in Sec. 1.3 of the Supplemental Text. In Fig. 4B and D, we quantify the concavity of the anti-resonance and the corresponding anti-resonant frequency, respectively. As k_onincreases with k_off fixed, we observe that the minimum in TopFlash expression becomes sharper, while the location of the minimum shifts to the left. For five different points on the (k_on, k_off)-plane, we plot the final TopFlash level as a function of the frequency of the light cycles (Fig. 4C). We observe in Fig. 4B and 4C that the anti-resonant frequency indeed only appears once we enter the region k_off < (1 + k₂)k_on.

The explanation of anti-resonant behavior through two timescales in the hidden component(s) of the pathway suggest that the anti-resonance is a generic feature of the canonical Wnt pathway: it arises from differences in the timescale of activation and deactivation of the signaling cascade upstream of β-catenin. Mechanistically, the anti-resonant behavior is possible when the timescale of activation of the hidden variable is faster than its deactivation. In our full biochemical model, this can correspond, for example, to the phosphorylation timescale of Dvl protein at the receptor, which is thought to be fast for HEK293T cells (45).

Our minimal model suggests that the anti-resonant suppression of regulatory responses at pulses of intermediate frequencies does not require fine-tuning to specific rates, but it does require these rates to satisfy a loose bound. Therefore, the generality of the model suggests that this behavior may also occur in other cells.

To investigate whether Wnt pathway anti-resonance influences stem cell fate decisions in the human embryo, we explored its role in the differentiation of H9 human embryonic stem cells (hESCs). Indeed, Wnt signaling pulses, oscillations, and waves have all been observed in organoid models of human development (e.g. gastruloids, somatoids, embryoid bodies) (46–48). Accordingly, we chose to address two questions: 1) Does anti-resonance affect the Wnt signals of human embryonic stem cells and, if so, (2) does it have a significant impact on developmental cell fate decisions?

We began by constructing a clonal H9 human embryonic stem cell (hESC) line containing our optoWnt tool and CRISPR-tagged tdmRuby β-catenin (Fig 5A). First, we CRISPR-tagged the N-terminus of β-catenin in H9 hESCs with tdmRuby. We next used the piggyBac transposon system to introduce our opto-Wnt tool into these H9 cells which were then selected for integrands and generated a clonal line (see Methods for details of cell line construction). Finally, we verified the opto-response by illuminating Wnt I/O H9s for 24 hours using 405nm light and staining for Brachyury (BRA), a mesodermal cell fate marker (49). Optogenetic stimulation of our Wnt I/O H9 cell line resulted in efficient mesodermal differentiation (Fig 5B). From these observations, we conclude that our Wnt I/O H9 line responds to optogenetic activation and that this activation can control the cell fate decision of mesoderm differentiation.

To determine whether anti-resonance affects mesoderm differentiation, we conducted a light stimulation screen targeting various duty cycles and frequencies, as described earlier (Fig. 3). We focused on a 24-hour differentiation window (50) as it is the literature reported timescale for Wnt driven mesoderm differentiation, and we found that using a 48-hour window caused all conditions to fully differentiate into mesodermal fates (Supp. Fig. 4A, B). We applied the same stimulation parameters previously tested in the HEK293T line and following the stimulation period, cells were fixed and stained (51) for BRA. We then fluorescently imaged our cells for both β-catenin and BRA (Fig 5C, Supp 4C, D, E). Quantification of β-catenin agrees well with our model and HEK293T results (Supp. Fig. 4F). Remarkably, quantification of BRA demonstrated a striking anti-resonant effect (Fig 5D). Namely, for cells with the same duty cycle (e.g. 55%), we observe that high and low frequencies induce total differentiation, while frequencies around 1/3 cycles per hour show markedly reduced BRA (Fig 5E). Together with our computational findings, these results demonstrate that anti-resonance is conserved across cell lines and that dynamic activation of the Wnt pathways influences stem cell differentiation in anti-resonant ways.

Our work establishes an anti-resonant suppression of gene regulatory outputs for oscillatory signals at a frequency range. We connect this suppression to a hidden layer in the signaling network, which introduces an additional timescale to the problem. This consideration of different timescales is abundant in signaling cascades and the door to applying our approach to other signaling cascades with well-known dynamical responses, such as Erk (55–57), NF-kB (58), BMP (59), and Notch (66), which also exhibit oscillatory behavior and waves which can determine differentiation. Future work will explore whether anti-resonance is a universal feature of signaling networks and leverage this knowledge to uncover new regulatory mechanisms that may only become apparent in the context of highly dynamic signaling inputs.

Human 293T cells were cultured at 37°C and 5% CO2 in Dulbecco’s Modified Eagle Medium, high glucose GlutaMAX (Thermo Fisher Scientific, 10566016) medium supplemented with 10% fetal bovine serum (Atlas Biologicals, F-0500-D) and 1% penicillin-streptomycin. Experiments in human Embryonic Stem Cell (hESC) lines were performed using the H9 hESC cell line purchased from the William K. Bowes Center for Stem Cell Biology and Engineering at UCSB. Cells were grown in mTeSR™ Plus medium (Stem Cell Technologies) on Matrigel® (Corning) coated tissue culture dishes and tested for mycoplasma in 2-month intervals.

Clonal 293Ts containing CRISPR tdmRuby3-β-cat and oLRP6_Puro were obtained from Dr. Ryan Lach (26). Cells were co-transfected with pPig_8X-TOPFlash-tdIRFP_Puro obtained from Dr. Ryan Lach and Super PiggyBac Transposase (System Biosciences cat#: PB210PA-1) using manufacturers recommendations and standard PEI-based transfection procedures. Cells were incubated for 24hrs before replacing with fresh media. Cells were then subject to 12hrs of continuous 405nm light activation on the LITOS and single-cell FACS sorted for tdmRuby3+, tdIRFP+ into a 96-well plate. Cells were monitored for growth over 14 days, only wells containing single colonies (arising from attachment of a single clone) were kept for subsequent processing. Prospective clonal populations were then imaged pre- and post 12hr light activation to screen for low baseline expression of β-catenin and TOPFlash and medium-high expression post activation.

Clonal β-catenin reporter lines were generated through CRISPR/Cas9-mediated homology directed repair using analogous methods to HEK293T counterparts. Accutase digested single hESCs were seeded onto Matrigel coated 12 well plates and transfected with Lipofectamine™ Stem Transfection Reagent (Invitrogen, STEM00015) according to manufacturer recommendations. Once the cells grew to confluency, they were selected with 2µg/mL puromycin in mTeSR plus. Clonal populations were isolated through single cell sorting with the SH800 Sony Cell Sorter and expanded in mTeSR™ Plus medium. The resulting population was screened and periodically treated with puromycin to ensure they mimicked canonical Wnt signaling upon optogenetic stimulation.

Spatial patterning of light during timelapse fluorescent imaging sessions was accomplished via purpose-built microscope-mounted LED-coupled digital micromirror devices (DMDs) triggered via Nikon NIS Elements software. Stimulation parameters (brightness levels, duration, pulse frequency) were optimized to minimize phototoxicity while maintaining continuous activation of Cry-2. For DMD-based stimulation on the microscope, the final settings for ‘Light ON’ were 25% LED power (λ = 455nm), 2s duration pulses every 30s. For experiments that did not require frequent confocal imaging, cells were stimulated via a benchtop LED array purpose-built for light delivery to cells in standard tissue culture plates (‘LITOS’) (44). The same light delivery parameters were used for LITOS-based stimulation as for microscope mounted DMDs. Light was patterned to cover the entire surface of intended wells of plates used, rather than a single microscope imaging field.

CHIR and Wnt3a Stimulation. HEK293T cells were treated using 10µM of CHIR99201 (Stem Cell Technologies, 72052) or 2.5nM of Wnt3a ligand (R&D Systems, 5036-WN) added into Dulbecco’s Modified Eagle Medium, high glucose GlutaMAX (Thermo Fisher Scientific, 10566016) medium supplemented with 10% fetal bovine serum (Atlas Biologicals, F-0500-D) and 1% penicillin-streptomycin. Cells were then imaged with DAPI in the same media 16hrs post addition of CHIR99201 and Wnt3a ligand.

Primary antibodies used to stain for Brachyury and Sox2 in H9s were α-Sox2 (Cell Signaling 3579, 1:500 dil.) and α-Brachury (RnD AF2085, 1:500). Secondary antibodies used were α-Rbt-Alexa-488 (Thermofisher A21206, 1:1000) and α-Gt-Alexa-647 (Thermofisher A21447, 1:1000). Tissue fixation and staining was carried out using standard protocols using cold methanol (51). Immunofluorescent samples were imaged using confocal microscopy (see below). Nuclear stains were carried out using NucBlue Live ReadyProbes (Hoescht 33342, R37605) according to manufacturer’s instructions.

All live and fixed cell imaging experiments were carried out using a Nikon W2 SoRa spinning-disk confocal microscope equipped with incubation chamber maintaining cells at 37°C and 5% CO2. Glass-bottom culture plates (Cellvis # P96-1.5H-N) were pre-treated with bovine fibronectin (Sigma #F1141) in the case of 293Ts or Matrigel in the case of H9s, and cells were allowed to adhere to the plate before subsequent treatment or imaging. HEK293T cells were imaged in Dulbecco’s Modified Eagle Medium, high glucose GlutaMAX (Thermo Fisher Scientific, 10566016) medium supplemented with 10% fetal bovine serum (Atlas Biologicals, F-0500-D) and 1% penicillin-streptomycin. H9s were imaged in mTeSR™ Plus medium.

All quantification of raw microscopy images was carried out using the same general workflow: background subtraction > classification > measurement > normalization > statistical comparison. When possible, subcellular segmentation of nuclear fluorescence was performed via context-trained deep learning-based Cellpose 2.0 algorithm derived from the ‘nuclei’ or ‘cyto2’ pretrained models pre-packed with the current Cellpose software distribution available here: (1) (36). Single-cell tracking and raw measurements were performed with the ‘LAP Tracker’ function in the TrackMate plugin for ImageJ available here: https://imagej.net/plugins/trackmate/ (36, 67). Tracks containing fewer than 50 contiguous frames (spurious or exited camera field of view) were omitted from subsequent analysis. Mean fluorescent intensity of regions of interest were measured and subsequently processed. Raw measurements were compiled, processed, and plotted via custom Matlab (ver. R2024b) and Python (ver. 3.9.13) scripts, available upon request.

We begin by using the segmented and tracked cells produced by our Image Analysis method above. Each segmented and tracked cell contains information of its nuclear β-catenin and TopFlash at each timepoint during our timelapse experiment. In order to reduce background noise in nuclear fluorescence signals, we first measured background fluorescence in the β-catenin and TopFlash channels for each frame. For each cell, raw nuclear β-catenin and TopFlash intensities were subtracted by the corresponding background value in that frame. The resulting traces were then normalized to their values at t = 0 such that all traces being at a normalized fluorescent intensity of 1.

We begin by using the segmented cells produced by our Image Analysis method. We do not calculate tracks here since duty cycle and frequency screen experiments only include end point image so there is no temporal data. Each segmented and tracked cell contains information of its nuclear β-catenin and TopFlash. We measured background fluorescence in the β-catenin and TopFlash channels for each condition in the duty cycle and frequency experiment. Each segmented cell was background-subtracted. We then calculated the mean β-catenin and TopFlash fluorescence for each condition. Finally, we normalized each condition to the maximum mean β-catenin or TopFlash fluorescence, making the range of normalized fluorescent intensities between 0-1. For Figure 3F, background subtraction was performed on both the original and replicate experiments. Subsequently, the TopFlash fluorescence for each well was scaled by a factor of log(N), where N represents the cell count in that well. After applying this scaling, we calculated the mean between the original and replicate datasets to generate a combined dataset. Finally, the combined dataset was normalized to its maximum value before being visualized as a heatmap.

A simple model describing β-catenin accumulation in response to Wnt activation using a single differential equation, derived by Goentoro et al. (38), did not capture the results here. Adapting a more detailed model, for example Lee et al. (27) involving over 20 parameters. Thus, after parameter considerations detailed in the supplemental text, we arrived at the following differential equations describing β-catenin and TopFlash dynamics. Variables and parameters are defined in the main text and in Tables 1.1 and 1.2 below. The dynamics of β-catenin for a given Wnt input is governed by: Table 1.1. Table 1.2.

In addition, we have the following conserved quantities:

We model the dynamics of TopFlash using a simple Hill-type activation function: where we have added a time delay τ to represent the time delay between β-cat accumulation and TopFlash transcription.

We coarse-grain the model from Fig 2, described by Eqs. 1-3, to identify the core features of the model that give rise to the anti-resonance. The details of the signaling cascade upstream of β-catenin are coarse-grained into a single hidden variable a(t). We denote the presence of the stimulus at time t with l(t) ∈ {0,1}, with l(t) = 1 during “light on” and l(t) = 0 during “light off”. The variable a(t) is activated at rate k_on when l(t) = 1, and deactivated at rate k_off when l(t) = 0, i.e.: Note that we have normalized a(t) such that the steady state when l(t) = 1 is a(t) = 1. Next, we couple the hidden variable to the β-catenin dynamics b(t). We take it to be of the form derived by Goentero et al. (38): where b(t) has been rescaled so that the steady state when l(t) = 0 is b(t) = 1. Finally, we describe the TopFlash expression g(t) via the Hill-type activation function in Eq. 3 from the previous section (ODE model for Wnt Signaling).

The dynamical model described by Eqs. 3, 4, and 5 are summarized by the schematic in Fig. 4 in the main text. We confirm that this abstract model also fits the single-pulse data and refer to the Supplemental Text for more details. We find that parameters k_b and k_a dictate the coarse-grained β-catenin dynamics in Eq. 5, while the parameters k_on and k_offgovern the existence and shape of the anti-resonance. In the numerical simulations we observe that the anti-resonance occurs in a parameter regime k_off < (1 + k_a)k_on. We derive this bound in the Supplemental Text and sketch out the procedure here.

A periodic light input, like in our experiments, is determined by the duty cycle δ and frequency f, with period T = 1/f. For low frequencies T∼ 1/k_b. we can ignore the comparatively fast changes in a(t) and replace it in Eq. 5 with l(t): In this regime, the mean β-catenin level during the experiment strictly decreases as a function of frequency (see Supplemental Text). Since TopFlash expression approximately tracks mean β-catenin levels, it also decreases with frequency in this regime. We find that the exact functional form of TopFlash expression (Eq. 3), and the Hill parameter, influences the slope.

Next, we consider the high-frequency limit where T ≪ 1/k_b. In this regime, the duration of each light pulse is short compared to the timescale in which significant changes occur in β-catenin levels. Hence, we can solve for small oscillations of beta-catenin Δb(t) around a constant value b̃: For a TopFlash reduction at intermediate frequencies, TopFlash should increase as the frequency increases towards the high-frequency limit. As the TopFlash activation function in Eq. 3 is strictly increasing in b(t), and Δb(t) is small, it is sufficient to show that b̃ increases as a function of frequency.

To first order in Δb(t), the β-catenin dynamics in Eq. 5 can be written as: which we can solve for Δb(t). Then, by imposing periodic boundary conditions, we can solve for the constant b̃ (see Supplemental Text for details). For b̃ to increase as the frequency increases towards the high-frequency limit, we require that: Substituting the above limits (derived in the Supplemental Text) yields the following condition: in agreement with results from the numerical simulations. All modeling code is available at is available at https://github.com/olivierwitteveen/wnt_antiresonance_model (68).