Frequency dependent growth of bacteria in living materials

The fusion of living bacteria and man-made materials represents a new frontier in medical and biosynthetic technology. However, the principles of bacterial signal processing inside synthetic materials with three-dimensional and fluctuating environments remain elusive. Here, we study bacterial growth in a three-dimensional hydrogel. We find that bacteria expressing an antibiotic resistance module can take advantage of ambient kinetic disturbances to improve growth while encapsulated. We show that these changes in bacterial growth are specific to disturbance frequency and hydrogel density. This remarkable specificity demonstrates that periodic disturbance frequency is a new input that engineers may leverage to control bacterial growth in synthetic materials. This research provides a systematic framework for understanding and controlling bacterial information processing in three-dimensional living materials.

Where µ represents the growth rate (dimensionless), t represents time (dimensionless), , , and are dimensionless parameters that control the bistable state of the system, is a dimensionless parameter that controls the amplitude of a weak oscillatory signal, Ω is the period of a weak oscillatory signal (dimensionless), and controls the noise magnitude (dimensionless).
The first two terms on the right-hand side of Eq. S1 represent bistable growth caused by the feedback between importation, degradation, and translation-inhibition activity of antibiotics as described by Deris et. al. 2 . To verify the bistability of the equation, we plotted the output of the positive and negative function terms to identify the stable and unstable fixed points (Supp. Fig. 11A). We also implemented a simpler equation (Eq. S2) to represent non-bistable (i.e. monostable) growth of bacteria in the absence of tetracycline. Where * and * are dimensionless parameters. A parameter set for Eq. S2 was empirically derived to quantitatively match the integral of the growth rates from Eq. S1. The stability diagram of Eq. S2 was plotted under the chosen parameters, demonstrating that it is not bistable (Supp. Fig. 11B).
Next, Eq. S1 and S2 were used to simulate bacterial growth with weak periodic perturbation of growth rates. Hydrogel encapsulation was modeled as increasing the noise in bacterial growth by causing noisy TetX expression 3 . Simulations showed that at a narrow range of amplitude and frequency, the relative bacterial growth rate over a range of simulated gel densities was enhanced (Supp. Fig. 11C). Simulated bacterial growth without bistability did not experience frequency and noise specific enhancements (Supp. Fig. 11D).
The simulated growth of bacteria experiencing frequency-dependent enhancement was plotted against the range of simulated gel densities (Supp. Fig. 11E black line). Under conditions where relative growth enhancement was reported, simulated growth was found to be biphasic as simulated hydrogel densities increased (Supp. Fig. 11E black line). Removing the weak periodic force (Supp. Fig. 11E dark grey), changing the underlying bistability (Supp. Fig. 11E mid grey), or altering the periodic forcing frequency (Supp. Fig. 11E light grey) eliminated the biphasic growth with respect to simulated hydrogel density. These simulations demonstrate that stochastic resonance may explain the biphasic, frequency-dependent bacterial growth curves that we experimentally observed in tetracycline-resistant bacteria.

Supplementary Figure Legends
Supplementary Figure 1. Frequency spectra of cells for A490/A405 ratio Graphs representing Fourier transformations of the oscillating ratio of absorbance at 490nm and absorbance at 405nm. The ratio of A490/A405 has been shown to represent the ratio of ATP to ADP in cells expressing Perceval. These results show that the frequency spectra of cells with Perceval and cells without Perceval can be distinguished at 0.0% and 0.25% gel, but not 0.5% or 0.75% gel. Furthermore, the frequency spectra of cells expressing Perceval and shaken at 10s/min can be distinguished from cells expressing Perceval and shaken at 0s/min. All graphs show 3 biological replicates, except for DH5 cells in 0.0% gel with 10s/min shaking, which shows 2 biological replicates. Metabolic activity of cells in media with .5% low-melting-temperature agarose as measured by cells expressing luciferase, which uses an ATP-and oxygen-dependent reaction to catalyze the production of light. Periodic shaking induces an approximately two-fold periodic change in luminescence (Methods-section D). Along with fluctuations in Perceval A490/A405 levels, these results suggest that periodic shaking of cells imparts periodic perturbation of metabolism in those cells. Points represent mean values, and error bars represent standard error of the mean. Mean and SEM are calculated using six biological replicates. (A)  unstable.
(A-E) Cell grown in LB with 0, 0.25, 0.5, or 0.75% gel along with 0 µg/mL, 0.3125 µg/mL, or 0.625 µg/mL tetracycline. Points represent mean values, and error bars represent standard error of the mean. Mean and SEM are calculated using six biological replicates except for (C), which is calculated using four biological replicates. (D) Heat plot showing the simulated growth of bacteria that cannot achieve bistable growth at different frequencies and amplitudes of periodic perturbation. Color indicates the maximum growth of bacteria over a range of simulated hydrogel densities normalized by the growth of bacteria at the lowest gel density. The results indicate that stochastic resonance does not occur under these conditions. Surface plot values represent mean values. The mean is calculated using two hundred simulation replicates. (E) Simulated bacterial growth in different hydrogel densities. Black line represents simulation with the amplitude, frequency, and bistable growth required for stochastic resonance. Black line shows growth under stochastic resonance conditions causes a biphasic relationship between the total growth in the system and the simulated gel density, similar to the 10 s/min shaking conditions in Fig. 3. Dark grey represents simulation with a periodic amplitude of zero, similar to the 0 s/min shaking conditions. Intermediate grey represents a simulation with a monostable system. Light grey represents a simulation with a reduction in the frequency of periodic perturbation. Results demonstrate that changing periodic forcing amplitude, frequency, or the underlying bistability of growth eliminates biphasic growth with respect to simulated gel density. Points represent mean values, and error bars represent standard error of the mean. Mean and SEM are calculated using two hundred simulation replicates. The change in basal growth rate associated with inducible TetX production is sufficient to shift biphasic relative growth curve (A) OD600 values of constitutive and inducible TetX producing cells normalized by initial OD600. Comparison was performed at 0.625 µg/mL tetracycline, 0% gel, and 0 s/min shaking to characterize the difference in basal growth rate between the two lines under tetracycline treatment. A single-tailed t-test shows that the constitutive TetX producers have a significantly higher growth rate than the inducible TetX producers (p-value: 3x10 -5 ). Points represent mean values, and error bars represent standard error of the mean. Mean and SEM are calculated using six biological replicates.

Supplementary
(B) Simulation of stochastic resonance governing bacterial growth with different basal growth rates. Basal growth rate is controlled by changing the value of in Eq. S1. Lines are labeled with the value of used in the simulation. Results show that decreasing the basal growth rate of a bacterial strain affected by stochastic resonance is sufficient to increase the optimal gel density for bacterial growth. These results suggest that the change in basal growth rate shown in Supp. Fig. 12A is responsible for the shifted biphasic relative growth curve seen in Fig. 3G (red lines). AU = arbitrary units.
Points represent mean values, and error bars represent standard error of the mean. Mean and SEM are calculated using two hundred simulation replicates.