Stochastic Turing patterns in a synthetic bacterial population

Abstract

The origin of biological morphology and form is one of the deepest problems in science, underlying our understanding of development and the functioning of living systems. In 1952, Alan Turing showed that chemical morphogenesis could arise from a linear instability of a spatially uniform state, giving rise to periodic pattern formation in reaction-diffusion systems but only those with a rapidly diffusing inhibitor and a slowly diffusing activator. These conditions are disappointingly hard to achieve in nature, and the role of Turing instabilities in biological pattern formation has been called into question. Recently, the theory was extended to include noisy activator-inhibitor birth and death processes. Surprisingly, this stochastic Turing theory predicts the existence of patterns over a wide range of parameters, in particular with no severe requirement on the ratio of activator-inhibitor diffusion coefficients. To explore whether this mechanism is viable in practice, we have genetically engineered a synthetic bacterial population in which the signaling molecules form a stochastic activator-inhibitor system. The synthetic pattern-forming gene circuit destabilizes an initially homogenous lawn of genetically engineered bacteria, producing disordered patterns with tunable features on a spatial scale much larger than that of a single cell. Spatial correlations of the experimental patterns agree quantitatively with the signature predicted by theory. These results show that Turing-type pattern-forming mechanisms, if driven by stochasticity, can potentially underlie a broad range of biological patterns. These findings provide the groundwork for a unified picture of biological morphogenesis, arising from a combination of stochastic gene expression and dynamical instabilities.

Keywords: Turing patterns; biofilm; signaling molecules; stochastic gene expression; synthetic biology.

Fig. 1.

Design of a synthetic multicellular system for emergent pattern formation. (A) Abstractly, the system consists of two signaling species A$3\text{OC}12\text{HSL}$ and I$\text{C}4\text{HSL}$. A$3\text{OC}12\text{HSL}$ is an activator catalyzing synthesis of both species, while I$\text{C}4\text{HSL}$ is an inhibitor repressing their synthesis, with additional repression by A$3\text{OC}12\text{HSL}$ via competitive binding. (B) Genetic circuit implementation. Promoter regions are indicated by white boxes, while protein coding sequences are indicated by colored boxes. IPTG is an external inducer modulating system dynamics. (C, Top) Illustration of signaling species concentrations in 1D space. The dashed orange and blue lines correspond to A$3\text{OC}12\text{HSL}$ and I$\text{C}4\text{HSL}$, respectively. (C, Middle) Spatial profiles of reporter proteins. RFP expression (red line) correlates with A$3\text{OC}12\text{HSL}$ concentrations, while GFP expression (green line) roughly mirrors RFP expression. (C, Bottom) A vertical slice of cell lawn. Cells express fluorescence proteins according to the profiles above and produce a global multicellular pattern.

Fig. 2.

Experimental observations of emergent pattern formation measured by green fluorescence intensity (GFI) and red fluorescence intensity (RFI). (A) Representative microscope images (based on six technical replicates) of a typical field of view showing a fluorescent pattern formed by an initially homogeneous isogenic lawn of cells harboring the Turing circuit with no IPTG. Spots and voids appear in the red fluorescence (RF) and green fluorescence (GF) channels, respectively. (Scale bar: 100 μm.) (B) Microscope images of cell lawns with constitutive expression of fluorescent proteins. (Left) Cells expressing RFP, (Center) cells expressing GFP, and (Right) mixed population of red and green cells. (C) Fluorescence density plots computed from the images above (from left to right: red, green, red/green, and Turing). Color intensity is in log scale [arbitrary units (a.u.)].

Fig. 3.

Mathematical modeling and correlation between pattern modulation experiments and simulations. (A) Experimental results for IPTG modulation of pattern formation with microscopy images corresponding to specific IPTG concentrations in B. The same display mappings were used for all images in A. (B) Collectivity, metric parameter $\mathrm{\Theta }$ is influenced by IPTG modulation. (C) Pattern statistics over IPTG modulation for experimental results. (D) Pattern obtained from simulating a deterministic reaction–diffusion model with $D_v/D_u=100$. (E) Pattern statistics over IPTG modulation for deterministic modeling. (F) Patterns obtained from simulating our deterministic model (Upper) and stochastic spatiotemporal model (Lower) at the measured diffusion ratio of $D_v/D_u=21.6$. (G) Pattern statistics over IPTG modulation for stochastic modeling. a.u., arbitrary unit.

Fig. 4.

Spectral analysis and parameter analysis. (A) Pattern-forming regimes in parameter space and estimated parameters for our system. Parameters above the green surface of neutral stochastic stability can form stochastic patterns, and parameters above the blue surface of deterministic neutral stability can form deterministic Turing patterns. The ratio of the diffusion coefficients $\nu$/$\mu$, the ratio of degradation rate to production rate $d/p$, and the ratio of production rates are estimated for our system by the yellow ellipsoid. The parameters for our system are mostly in the regime where stochastic patterns form and outside the region where deterministic Turing patterns form. Example stochastic simulations are shown for parameters drawn from a deterministic parameter region with $D_{\nu }/D_{\mu }=100$ (Upper Right) and a stochastic region with $D_{\nu }/D_{\mu }=21.6$ (Lower Right). (B) Radial power spectrum of green fluorescence and best fit power law tail with an exponent of $-2.3\pm 0.2$. (C) Radial power spectrum for eight trials of our stochastic simulation, their mean, and the best fit power law tail.