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# Stochastic Turing patterns in a synthetic bacterial population

Contributed by Nigel Goldenfeld, May 9, 2018 (sent for review November 29, 2017; reviewed by Raymond E. Goldstein, Eric Klavins, and Ehud Meron)

## Significance

In 1952, Alan Turing proposed that biological morphogenesis could arise from a dynamical process in reaction systems with a rapidly diffusing inhibitor and a slowly diffusing activator. Turing’s conditions are disappointingly hard to achieve in nature, but recent stochastic extension of the theory predicts pattern formation without such strong conditions. We have forward-engineered bacterial populations with signaling molecules that form a stochastic activator–inhibitor system that does not satisfy the classic Turing conditions but exhibits disordered patterns with a defined length scale and spatial correlations that agree quantitatively with stochastic Turing theory. Our results suggest that Turing-type mechanisms, driven by gene expression or other source of stochasticity, may underlie a much broader range of patterns in nature than currently thought.

## 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.

A central question in biological systems, particularly in developmental biology, is how patterns emerge from an initially homogeneous state (1). In his seminal 1952 paper “The chemical basis of morphogenesis,” Alan Turing showed, through linear stability analysis, that stationary, periodic patterns can emerge from an initially uniform state in reaction–diffusion systems where an inhibitor morphogen diffuses sufficiently faster than an activator morphogen (2). However, the requirements for realizing robust pattern formation according to Turing’s mechanism are prohibitively difficult to realize in nature. Although Turing patterns were observed in a chemical system in 1990 (3), the general role of Turing instabilities in biological pattern formation has been called into question, despite a few rare examples (ref. 4 and references therein).

Recently, Turing’s theory was extended to include intrinsic noise arising from activator and inhibitor birth and death processes (5⇓⇓–8). According to the resulting stochastic Turing theory, demographic noise can induce persistent spatial pattern formation over a wide range of parameters, in particular, removing the requirement for the ratio of inhibitor–activator diffusion coefficients to be large. Moreover, stochastic Turing theory shows that the extreme sensitivity of pattern-forming systems to intrinsic noise stems from a giant amplification resulting from the nonorthogonality of eigenvectors of the linear stability operator about the spatially uniform steady state (8). This amplification means that the magnitude of spatial patterns arising from intrinsic noise is not limited by the noise amplitude itself, as one might have thought naively. These developments imply that intrinsic noise can drive large-amplitude stochastic Turing patterns for a much wider range of parameters than the classical, deterministic Turing theory. In particular, it is often the case in nature that the activator and inhibitor molecules do not have widely differing diffusion coefficients; nevertheless, stochastic Turing theory predicts that, even in this case, pattern formation can occur at a characteristic wavelength that has the same functional dependence on parameters as in the deterministic theory.

To explore how global spatial patterns emerge from local interactions in isogenic cell populations, we present here a synthetic bacterial population with collective interactions that can be controlled and well-characterized (an introduction to this perspective is in ref. 9), where patterning is driven by activator–inhibitor diffusion across an initially homogeneous lawn of cells. Synthetic systems can be forward-engineered to include relatively simple circuits that are loosely coupled to the larger natural system into which they are embedded. This makes it easier to design and control the molecular underpinnings of the biological pattern phenomenon (10) and even front propagation phenomena (11). Previous pattern formation efforts in synthetic biology have focused on oscillations in time (12) or required either an initial template (13) or an expanding population of cells (14), neither of which show a Turing mechanism.

## Experimental Results

### Synthetic Biology of a Bacterial Community.

In our synthetic gene network design, which was guided by computational modeling (*SI Appendix*, section 1), we used two artificial diffusible morphogens: the small molecule *N*-(3-oxododecanoyl) homoserine lactone, denoted here as A*N*-butanoyl-l-homoserine lactone, denoted here as I*Pseudomonas aeruginosa las* and *rhl* quorum sensing pathways, respectively, in *P. aeruginosa* (15). A*A* and *B* and *SI Appendix*, section 1). A*rhl* and *las* hybrid promoters, respectively, to aid in experimental observation (*SI Appendix*, section 2).

In our experimental setup, the A*SI Appendix*, section 3). The estimated diffusion coefficient for A*C*). Within these red domains, both A*SI Appendix*, section 4 and Fig. S8), GFP is attenuated (18). The faster diffusion rate of I

### Experimental Patterns and Controls.

To study pattern-forming behavior, engineered cells are first cultured in liquid media, and then, they are concentrated and plated on a petri dish to form an initially homogeneous “lawn” of cells (*Materials and Methods*). After plating, the petri dish is incubated for 24 h at 30 °C, and microscope fluorescence images are captured as needed. Before the self-activation of the A*A*). Time series microscopy reveals that patterns begin to emerge after approximately 16 h (*SI Appendix*, Fig. S12).

In control experiments, we show that our patterns are not simply a result of the outward growth of clusters of differentially colored cells (Fig. 2 *B* and *C* and *SI Appendix*, section 5). Also, by performing an experiment with cells that harbor independent bistable green/red toggle switches, we test whether observable patterns would emerge if individual cells autonomously made cell fate decisions at some point after plating (*SI Appendix*, section 5). The fluorescence fields after 24 h of incubation at 30 °C in both control experiments are uniform, showing no emergence of patterns.

Next, we examine how changes in the strengths of localized interactions lead to different global outcomes in our pattern-forming gene circuit. In our system, IPTG can be used to modulate the inhibitory efficiency of I*A* and *C*). In addition to offering further support that our gene circuit gives rise to emergent patterns, these results show how pattern formation characteristics can potentially be tuned to fit future application needs.

## Theoretical Results

Having established that our system forms emergent patterns, we proceeded to study the mechanisms driving these patterns. We formulated deterministic and stochastic models and analyzed our data to assess agreement with the theory of stochastic Turing patterns.

### Deterministic Model.

We first developed a detailed deterministic reaction–diffusion model (*SI Appendix*, section 7). The model explicitly describes chemical reactions for the LasI and RhlI synthases, regulatory protein CI, and synthesis and diffusion of the morphogens A

We model the hybrid promoters using the following Hill functions:*SI Appendix*, Table S5, and definitions of the rate constants are in *SI Appendix*, Tables S6 and S7. Hill functions used in this model have a shared form of

We initially ran simulations of this model using a high diffusion rate ratio (*D*), suggesting that the underlying dynamics of our system are Turing-like, with the potential for Turing instabilities. Deterministic simulations of IPTG modulations also correlate well with the trends of the experimental results (Fig. 3*E* and *SI Appendix*, section 6).

While the overall behavior of our system is reminiscent of classical Turing patterns (19), there are key differences. In particular, when we ran simulations at the measured diffusion rate ratio of *F*). For some two-node implementations of Turing systems, this rate would be sufficient for pattern formation (20). In addition, certain networks with more nodes can allow small or even equal morphogen diffusion rate ratios to generate Turing instabilities (21). However, a practical biological implementation imposes certain dynamics, such as delays associated with protein production, that can strongly impact pattern formation (22, 23). Indeed, our deterministic modeling results suggest that the ratio of diffusion constants for the activator and inhibitor in our system is either barely within the range required for a Turing instability or even outside the range, depending on the precise medium in which signal diffusion is measured (*SI Appendix*, section 7). In addition, whereas in the deterministic simulation, spots are identical and evenly distributed, those in the experimental systems vary in size, shape, fluorescence intensity, and the intervals between them.

### Stochastic Model.

The deterministic modeling results indicate that our system may be beyond the regime where classical Turing patterns are formed but still within the regime where stochastic Turing patterns occur (5⇓⇓–8). Indeed, gene expression in microbes is inherently noisy due to the small volume of cells and the fact that many reactants are present in low numbers, suggesting that stochastic Turing patterns could be present in our system (24).

Noise in stochastic Turing patterns expands the range of parameters in which patterns form, in contrast to the usual expectation that noise serves as a destabilizing agent. The patterns observed in stochastic Turing systems correspond to the slowest decaying mode of the fluctuations. Similar noise stabilization phenomena can be observed in other systems that are out of equilibrium. For example, in predator–prey systems, fluctuations can drive temporal oscillations of populations (25, 26). Noise-driven stabilization has also been recently discovered in the clustering of molecules on biological membranes (27, 28) and in models that exhibit Turing-like pattern formation (7). In particular, whereas spatial symmetry breaking and pattern formation via the original Turing design require two morphogens with diffusion rates that differ by a large factor on the order of 10 or 100 (1), the requirements to form stochastic Turing patterns are less stringent. For example, in a pattern-forming plankton–herbivore ecosystem, the noise associated with discrete random birth and death processes reduces the required ratio of diffusion constants for pattern formation from a threshold of 27.8 for normal Turing patterns to a threshold of 2.48 for stochastic Turing patterns (5⇓⇓–8).

To determine whether noise in the chemical reactions underlying gene expression and morphogen diffusion in our system can cause the emergence of patterns over a wider range of parameters than a deterministic model, we constructed a stochastic spatiotemporal model using the same biochemical reactions, diffusion, and rate constants used in our deterministic model (*SI Appendix*, section 8). This model captures stochastic effects in the production and degradation of the proteins and morphogens in our system but approximates diffusion as deterministic. Simulations of the stochastic model generically produce patterns with large variability in spot size, shape, intensity, and intervals, which are similar to the patterns observed in our experiments and different from those predicted for the deterministic model (Fig. 3*F*). We have compared the experimental patterns with stochastic simulations in both real space and in 2D Fourier transform (2DFT) space (*SI Appendix*, section 8). Neither the experimental 2DFT nor the simulated 2DFT contain pronounced peaks that would be present in a deterministic honeycomb Turing pattern. Moreover, as the IPTG concentration is increased, both experimental and simulated patterns become more regular (Fig. 3*G*).

To further test the hypothesis that we are observing stochastic Turing patterns, we measured the power spectrum for both of our fluorescent reporters. Theory predicts that the power spectrum will have a power law tail as a function of wavenumber, k, for large wavenumbers, with an exponent characteristic of the noise source (7, 25). The exponent values are −2 and −4 for stochastic Turing patterns and deterministic Turing patterns with additive noise, respectively, and can be interpreted simply as follows. The −2 arises, because at small frequency or wavenumber, the random variable (i.e., concentration) is simply diffusing and therefore, follows the behavior of a random walk, which has a power spectrum that exhibits a −2 power law. The −4 arises, because for a system that is executing deterministic damped periodic motion but driven by additive white noise, the response of the random variable is a Lorentzian, with an asymptotic behavior for the power spectrum that exhibits a −4 exponent.

For the GFP channel, we observe a power law tail with an exponent of *B* and *SI Appendix*, section 9A). For the RFP channel, we also observe a power law tail with an exponent of *SI Appendix*, section 9A). To better understand the implications of these tails, we examined our detailed stochastic model of the genetic systems and also developed a reduced stochastic model that explicitly includes only the morphogens (*SI Appendix*, section 9B). Both models predict that our experimental parameters will produce a stochastic pattern with a power law tail of *C* and *SI Appendix*, section 9C). However, in the range of parameters likely to correspond to the experiments (Fig. 4*A* and *SI Appendix*, section 9C), the detailed stochastic model predicts that the exponent of the power law tail for the activator will be *SI Appendix*, Fig. S24). This behavior once again agrees with our experimental data and supports our identification of stochastic Turing patterns. In summary, spectral analysis of the patterns of activator and inhibitor is consistent with a model in which fluctuations in the amount of signaling morphogens drive stochastic Turing patterns.

Our analysis of the stochastic Turing model predicts that stochastic patterns form over a wide range of parameters (*SI Appendix*, section 8). Indeed, our stochastic model predicts that stochastic Turing patterns are possible at the measured ratio of diffusion rates for A*F* and 4*A*). In addition, to determine the sensitivity of the stochastic model to the parameters chosen, we individually varied parameters from 0.5× their nominal value to 1.5× their nominal value while keeping all other parameters fixed at their best estimated value. For each set of parameters, we calculate the analytical power spectrum and the eigenvalues of the Jacobian (linear stability matrix) of the stochastic model evaluated at a fixed point found numerically. Based on this analysis (*SI Appendix*, section 8), we classify each set of parameters as producing an unstable homogeneous state at wavenumber *SI Appendix*, Fig. S22 and illustrate the significant ranges for each parameter that can lead to stochastic Turing patterns. Indeed, the estimated parameter values yield stochastic Turing patterns and variation of

To quantify the way in which stochasticity enlarges the pattern-forming regime of parameter space, we simultaneously varied all model parameters and performed the classification used above. Specifically, we used Latin hypercube sampling to randomly generate 500 parameter sets, where all of the parameters were allowed to vary between 0.5× and 1.5× their nominal value. For this analysis, we found that

## Discussion

### Alternative Hypotheses.

Now, we consider alternative hypotheses to our claim that the theory of stochastic Turing patterns explains our experimental observations. We consider the duration and dynamics of our pattern formation experiments. One may expect to observe early events in Turing pattern formation, such as splitting of clusters or increases in intercluster distances. These processes may be, in fact, be taking place but may be difficult to observe due to weak reporter expression in the earlier stages. In addition, we must consider the limited duration of our experiments and the possibility that, theoretically, longer observations may result in different patterns if nonlinear processes eventually began to dominate dynamics. Indeed, we do not feed fresh nutrients to sustain the system for extremely long durations. However, as confirmed by analysis of the dynamics in *SI Appendix*, Fig. S12, cluster size growth and spacing between clusters appear to be stabilizing toward the end of the experiment. In addition, domains are neither created nor destroyed in the later time periods. Essentially, it appears that the patterns are close to stabilizing within the 32-h observation period.

Another alternative hypothesis is that cell growth dynamics primarily drive the observed pattern formation. Our control experiments with mixtures of red and green cell populations (*SI Appendix*, Fig. S9) along with our bistable switch control (*SI Appendix*, Fig. S10) suggest that cell growth does not explain our patterns. Moreover, our ability to tune pattern characteristics offers support for the fact that our patterns are not a simple consequence of natural biofilm growth morphologies but rather, are driven by our genetic circuit. However, growth may indeed impact regularity and may likely explain the fact that our experimental patterns are less regular than those observed in our stochastic models (Fig. 3*F*). Indeed, future experiments to show different classes of patterns (e.g., labyrinth patterns) would offer further support, but collectively, our experiments strongly support our hypothesis that a Turing mechanism driven by our genetic circuit explains our observed patterns.

### Summary of Evidence for Stochastic Turing Patterns.

We summarize our evidence for showing stochastic Turing patterns and not showing amplification of random noise as follows. Our control experiments with mixtures of red and green cells (Fig. 2*B* and *SI Appendix*, Fig. S9) along with a bistable switch (*SI Appendix*, Fig. S10) did not produce the patterns that we observe with our genetic circuit (Fig. 2*A*). Our ability to tune pattern characteristics offers further support that pattern formation is driven by our genetic circuit. In addition to our experimental controls, we identify patterns in the stochastic model but not in the deterministic model of our system for the experimentally observed ratio of diffusion rates (Fig. 3*F*). These model patterns resemble the experimentally observed patterns in real space, exhibit no peaks in the 2DFT (*SI Appendix*, Figs. S13 and S20), and recapitulate the observed trend with IPTG variation. Analyses of our experimental data are also in accord with the theory of stochastic Turing patterns. The exponents in the tails of the experimental radial power spectra agree with theoretical predictions (Fig. 4*B* and *SI Appendix*, Fig. S13). In addition, although spatial regularity is weak, we observe a radial spectral peak for our experimental patterns (Fig. 4*B* and *SI Appendix*, Fig. S13), indicating a characteristic length scale. Furthermore, exploration of the large parameter space of the stochastic model indicates that the experimental parameters are most likely to be in the regime where only stochastic patterns can form (*SI Appendix*, section 8). Collectively, this body of evidence suggests that our experiments indeed exhibit stochastic Turing pattern formation.

## Materials and Methods

### Strains and Conditions.

Our patterning system was constructed using two plasmids that correspond to the upper and lower portions of the circuit diagram in Fig. 1*B*: pFNK512 and pFNK806 in Fig. 2*A* and pFNK512 and pFNK804lacOlacI in Fig. 3. The two-color bistable toggle switch plasmid pTOG-1 was constructed from plasmid pIKE-107. All plasmids were constructed using standard cloning and DNA recombination techniques. Plasmid construction details are described here and in *SI Appendix*, section 2. *Escherichia coli* strain MG1655 was used for all experiments.

### Code Availability.

Custom code used in this manuscript is currently available at https://www.dropbox.com/sh/di3hbaaubx5qd0q/AADpSOMfJtm_F_lEDRFoch0sa?dl=0.

### Experimental Procedure.

Cells harboring appropriate plasmids were initially grown in LB liquid media with corresponding antibiotics at 30 °C until OD at 600 nm was reached

### Data Analysis.

Fluorescence density plots, power spectrum of green fluorescence, averaged green fluorescence, total area of red spots, collectivity metric, and Moran’s I were all computed by analyzing the experimental time-lapse microscopy data with custom Matlab software.

### Mathematical Modeling.

The patterning system was simulated by numerically integrating differential equations using in house-developed C software. We also developed stochastic spatiotemporal models using a hybrid stochastic simulation algorithm (30). *SI Appendix* has details about the models and the simulation environments.

### Note Added in Proof.

After the completion and acceptance of this work, an independent observation of stochastic Turing patterns in the cyanobacteria colonies of *Anabaena* sp. was reported (31).

## Acknowledgments

We thank D. Volfson and T. Danino for help in mathematical modeling and E. Andriananto for help in plasmid construction. We also thank R. Mehreja and J. Ku for help in creation of the RhlR mutant. We thank members of the laboratory of R.W., especially N. Davidsohn, J. Sun, S. Gupta, M. Chen, C. Grecu, L. Wroblewska, and Y. Li, for discussions. This work was supported by the NIH and National Science Foundation Grants CCF-1521925 and CNS-1446474. D.K. acknowledges support from the Independent Research and Development Program of the Johns Hopkins University Applied Physics Laboratory. K.M.M. acknowledges partial support from the Center for the Physics of Living Cells through National Science Foundation Physics Frontiers Center Program PHY 1430124.

## Footnotes

↵

^{1}D.K., K.M.M., and T.L. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: nigel{at}illinois.edu or rweiss{at}mit.edu.

Author contributions: D.K., K.M.M., T.L., N.G., and R.W. designed research; D.K., K.M.M., T.L., N.G., and R.W. performed research; D.K., T.L., and R.W. contributed new reagents/analytic tools; D.K., K.M.M., T.L., N.A.D., N.G., and R.W. analyzed data; D.K., K.M.M., T.L., N.A.D., N.G., and R.W. wrote the paper.

Reviewers: R.E.G., University of Cambridge; E.K., University of Washington; and E.M., Ben-Gurion University, Blaustein Institutes for Desert Research.

The authors declare no conflict of interest.

Data deposition: The DNA sequences of plasmids reported in this paper have been deposited in the GenBank database (accession nos. MH300673–MH300677).

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1720770115/-/DCSupplemental.

Published under the PNAS license.

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