Collective Space-Sensing Coordinates Pattern Scaling in Engineered Bacteria

Abstract

Scale invariance refers to the maintenance of a constant ratio of developing organ size to body size. Although common, its underlying mechanisms remain poorly understood. Here, we examined scaling in engineered Escherichia coli that can form self-organized core-ring patterns in colonies. We found that the ring width exhibits perfect scale invariance to the colony size. Our analysis revealed a collective space-sensing mechanism, which entails sequential actions of an integral feedback loop and an incoherent feedforward loop. The integral feedback is implemented by the accumulation of a diffusive chemical produced by a colony. This accumulation, combined with nutrient consumption, sets the timing for ring initiation. The incoherent feedforward is implemented by the opposing effects of the domain size on the rate and duration of ring maturation. This mechanism emphasizes a role of timing control in achieving robust pattern scaling and provides a new perspective in examining the phenomenon in natural systems.

Figure 1. Scale invariance in self-organized pattern formation in engineered bacteria

A. Circuit logic. The circuit consists of a T7 RNA polymerase that activates its own expression as well as the expression of LuxR and LuxI. Upon activation by T7RNAP (T7), LuxI mediates synthesis of AHL (A), which can diffuse across the cell membrane. When the global AHL concentration surpasses a threshold, intracellular AHL binds to LuxR to activate the synthesis of T7 lysozyme (L). Lysozyme then binds to the T7RNAP and forms a T7-lysozyme complex, therefore inhibiting the T7RNAP binding to the T7 promoter. This T7- lysozyme complex also inhibits T7RNAP transcription. In this process, the AHL concentration is affected by its initial concentration and the domain size. The expression rates of T7RNAP, lysozyme, and AHL are all controlled by the spatially dependent gene expression capacity. B. Self-organized pattern formation in engineered bacteria. Left: A composite fluorescent image. Right: mCherry image extracted by MATLAB code from left composite image. Images of a 1.2mm × 1.2mm field of colony at 2500 μm domain radius after 24 h of incubation. The experimental platform used here was described in Figure S1A. C. Scale invariance in self-organized pattern formation. The mCherry ring width (red circle) and the colony radius (green circle) are plotted as a function of the domain radius. Measurements were done in microcolonies after 32 h incubation. The error bars represent the standard error from ≥5 replicates for each domain radius. The solid lines represent the linear regression of the data points (green: colony radius; red: ring width) for intermediate domain radii (between 1500 μm and 7500 μm), where scale invariance emerges (the R-square value for ring width vs. domain radius linear regression is 0.9828; the R-square value for colony radius vs. domain radius linear regression is 0.9943). The purple block corresponds to domain radii<1500 μm; the yellow block corresponds to domain radii >7500 μm. The insets show mCherry images for domain radii of 1500 μm, 2500 μm, and 3750 μm, respectively. For all of the presented figures, if unnoted, filled circles represent the data where ring width vs. colony radius follows a linear regression with domain radius. D. Ratio of mCherry ring width to colony radius for different domain radii. The ratio was calculated from the data in Figure 1C. The dashed line shows the average ratio for data points within the range of domain radii where the scale invariance emerges. The standard deviation of the ratio values between 1500 and 7500 μm is 0.0448 (~10% of the total constant ratio).

Figure 2. Pattern formation dynamics in engineered bacteria

A. Parameter search. Each spike represents the range of a parameter, ranging from 0 to its maximum value (the intersect with the outer circle) (Von Dassow et al., 2000). G₄ was explored between 0 and 10; other parameters’ search ranges are listed in Table S1. Each polygon represents a parameter set. We started with 18,231 parameter sets (light blue). Dark blue lines indicate 409 parameter sets that generated core-ring patterns for varying domain radii (from 1 to 3). A local search on each of these 409 sets led to an optimal parameter set that generated scale invariance (red curves). α and β are the inhibition factors of T7RNAP and T7 lysozyme on cell growth, respectively. n is the Hill coefficient for distance-dependent gene expression capacity. K_φ is the half activation distance for gene expression capacity. α_T and α_L are synthesis rates of T7RNAP and T7 lysozyme, respectively. K_T and K_P are half-inhibition concentration of T7RNAP and T7-lyszome complex, respectively. G₄ is the synthesis rate of AHL, after non-dimensionalized. B. Conditions underlying scale invariance. Five parameters shifted systematically during the optimization step: α and β characterize circuit mediated metabolic burden; n and K_φ determine the shape of gene expression capacity; and α_T characterize the strength of T7RNAP feedback. In contrast, the four other parameters, G₄, α_L, K_T and K_P did not change significantly during optimization. The blue histograms represent values from the 409 pattern-forming sets in Figure 2A before optimization. The red histograms represent values of the 409 sets resulting from the optimization. The light red is the overlap between blue and red histogram. C. Simulated spatio-temporal profiles of CFP and mCherry intensity. Left: Average CFP intensity over colony cross-section from 1 to 150 in arbitrary unit. Right: Average mCherry intensity over colony cross-section from 1 to 150 in arbitrary unit. D. Experimentally measured spatio-temporal profiles of CFP and mCherry. Left: average CFP intensity of bacterial colony at different radii from 8 h to 43 h. Right: Average mCherry intensity of bacterial colony at different radii from 8 h to 43 h. The experiments were measured at domain radius of 2500 μm.

Figure 3. A collective space sensing mechanism underlies scale invariance

A. A collective space sensing mechanism. The domain size controls the pace and extent of colony expansion by determining nutrient availability. The domain size also controls the timing of ring initiation by determining the accumulation rate of AHL. The interplay of the gene expression capacity (magenta curve), T7RNAP profile (blue curve), and AHL concentration (orange curve) leads to the scale invariance of ring width to colony size. The gene expression capacity and T7RNAP profiles are drawn as functions of the distance to the colony center at the final time point. The red arrow indicates the time window for ring maturation. B. Simulated emergence of a core-ring pattern. Top: AHL (orange curve) and nutrient (black curve) concentrations over time. Bottom: Simulated mCherry distributions at different time points. The time points were labeled on x-axis. The mCherry ring initiates when AHL reaches its maximum concentration (t₁), which coincides with a pause of colony expansion (t₂). The ring matures during the time window, ΔT, between ring initiation and system stabilization (red arrow). C. Emergence of scale invariance during core formation. When the colony is at fast expanding phase, the pattering process is mainly governed by an integral feedback topology which is similar to the expansion-repression model (Ben-Zvi and Barkai, 2010). In our model, the T7RNAP can be considered the expander that drives morphogen (AHL) synthesis, whereas the T7 lysozyme serves as the repressor. At time point t₁, the mCherry profile scales with the colony radius. D. Emergence of scale invariance during ring maturation. All the x-axes are normalized to the domain radius. The units of y-axes are all on a per cell basis. Left: At different domain radii, the mCherry profile (pink) at the ring initiation time (t₁) approximately scales with domain size. R_C is the colony radius, and it is a function of domain radius. Middle: mCherry accumulation during maturation time (ΔT) is mainly determined by the gene expression capacity. The x-axis represents the distance from the colony edge. D₁ and D₂ are two different domain radii, D₁ < D₂. R_{C_D1} and R_{C_D2} are the colony radius for domain radii D₁ and D₁, respectively. The y-axis is ΔT, the maturation time window. ΔT_D1 and ΔT_D2 are the maturation time windows when the domain radii are D₁ (light magenta) and D₂ (dark magenta). The intersect of x- and y- axes represent that the mCherry accumulation rate is 0 at the colony edge at ring initiation time, t₁. The z-axis indicates the mCherry accumulation during the given time window. Right: mCherry at t₂ is a combination of that at t₁ (pink) and that accumulated during ΔT (color code is the same with that in middle panel). Its minimum is at the same relative location on the normalized axis (red pointer), indicating proportionality between the inner edge position and the domain radius. The outer edge of the ring pattern is roughly the colony edge, which is also proportional to the domain radius. Combining these two aspects leads to a ring width that scales with the colony size.

Figure 4. Modulation of scaling property by environmental factors

A. Simulated scale invariance (base case). Left: Illustration of T7RNAP and gene expression capacity shape in the scale invariance range. Middle: Dependence of the ring width (red circles) and the colony radius (green circles) on the domain radius. The solid lines represent the linear regression of the colony radius and the ring width with respect to the domain radius in the white region. Solid circles represent the linear range of the dependence of the ring width and colony radius on the domain radius. Right: The ratio of mCherry ring width to the colony radius for different domain radii. The dashed line shows the average ratio for the values in the white region. Top left: Illustration of gene expression capacity profile when the domain is too small (purple shaded regions in Top and Middle panels). Top right: Illustration of the AHL profile over time when the domain is too large (yellow shaded regions in Top and Middle panels). The thinner gray curve corresponds to the AHL profile in the left panel; the thicker orange curve corresponds to a larger domain. These results correspond to time point t₂. B. Modulating the scaling property by adding exogenous AHL. Left: T7RNAP and gene expression capacity profiles were the same with those in base case. Adding exogenous AHL changes the temporal cue. The thinner gray line indicates the AHL concentration over time in the base case; the thicker orange line indicates the AHL concentration over time in the presence of initial exogenous AHL. The ring maturation window is longer (red arrow) compared with the base case. Right top: Simulated scaling property when the parameter of initial exogenous AHL (ahl₀) were set to 10 and 25 (from left to right). The two figures show relationships of the colony radius and of the ring width to the domain radius with different initial AHL concentrations. The insets represent the ratio of the ring width to the colony radius. The x-axes for the insets are on the same scale as the corresponding figure panels; the y-axes range from 0 to 1. Right bottom: Experimentally measured scaling property in the presence of 20 nM AHL and 50 nM AHL (from left to right). The two figures are plotted as in Figures 1C, D. The error bars represent the standard error or range of 2–5 replicates. Each data point was obtained at 32h after start of experiment. C. Modulating the scaling property by having a higher metabolic burden. Cell growth is slowed down at a high metabolic burden, so AHL reaches its peak value later than in the base case. At that time, T7RNAP distribution over space is not flat yet; instead, it has a higher distribution near the center of the colony, which will increase the mCherry core accumulation during the ring maturation stage. Because the maturation time is shorter, the ring cannot catch up with the core formation and the eventual ring width is smaller. The figure symbols are the same as those in Figure 4B. Simulated scaling properties with γ, the metabolic burden from an effector gene, at a value of 5 (γ = 0 for the base case). D. Modulating the scaling property by using weak T7RNAP positive feedback. Due to the weak feedback, the T7RNAP distribution over space is no longer flat but instead reflects that of the gene expression profile. The weak feedback also slows down accumulation of AHL, thus delaying the ring initiation in comparison with the base case. Therefore, the ring maturation time (red arrow) is shorter than that in the base case. The figure symbols are the same as those in Figure 4B. The scaling property was simulated with varying positive-feedback strength (the T7 promoter rate is 0.25 or 0.1 fold of promoter rate when circuit is fully induced).

Figure 5. Sequential actions of integral feedback and incoherent feedforward underlie the scale invariance

The overall system input is the domain size; the output is the mCherry pattern in a colony scaled with respect to the domain size. The integral feedback underlies a scale-invariant mCherry distribution at ring initiation. In our system, the activation module represents the strong T7RNAP positive feedback; morphogen represents AHL; the repression module represents T7 lysozyme. The incoherent feedforward controls the mCherry increment during maturation time window of ring formation. ΔT is a proportional function of domain size; rate represents mCherry accumulation rate, which is dominated by gene expression capacity during ring maturation. At the same relative location, a larger domain results in smaller gene expression capacity and thus a reduced accumulation rate in mCherry. This reduction is compensated by the increase in ΔT, leading to the same increment in mCherry at the same relative position for different domain sizes. The sum of the mCherry distribution at ring initiation and the increment during maturation leads to the final mCherry ring (red) that scales with the domain size. The color code is the same as in previous figures.