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# Data-driven quantitative modeling of bacterial active nematics

Edited by Andrea J. Liu, University of Pennsylvania, Philadelphia, PA, and approved November 27, 2018 (received for review July 21, 2018)

## Significance

Active nematics are nonequilibrium fluids consisting of elongated units driven at the individual scale. They spontaneously exhibit complex spatiotemporal dynamics and have attracted the attention of scientists from many disciplines. Here, we introduce an experimental system (made of filamentous bacteria) and a type of microscopic model for active nematics. Simultaneous measurements of orientation and velocity fields yield comprehensive experimental data that can be used to identify optimal values for all important parameters in the model. At these optimal parameters, the model quantitatively reproduces all experimentally measured features. This, in turn, reveals key processes governing active nematics. Our versatile approach successfully combines quantitative experiments and data-driven modeling; it can be used to study other dense active systems.

## Abstract

Active matter comprises individual units that convert energy into mechanical motion. In many examples, such as bacterial systems and biofilament assays, constituent units are elongated and can give rise to local nematic orientational order. Such “active nematics” systems have attracted much attention from both theorists and experimentalists. However, despite intense research efforts, data-driven quantitative modeling has not been achieved, a situation mainly due to the lack of systematic experimental data and to the large number of parameters of current models. Here, we introduce an active nematics system made of swarming filamentous bacteria. We simultaneously measure orientation and velocity fields and show that the complex spatiotemporal dynamics of our system can be quantitatively reproduced by a type of microscopic model for active suspensions whose important parameters are all estimated from comprehensive experimental data. This provides unprecedented access to key effective parameters and mechanisms governing active nematics. Our approach is applicable to different types of dense suspensions and shows a path toward more quantitative active matter research.

Examples of active matter can be found at diverse length scales (1⇓⇓⇓⇓–6), from animal groups (7⇓⇓⇓–11) to cell colonies and tissues (12⇓⇓⇓⇓⇓–18) to in vitro cytoskeletal extracts (19⇓⇓⇓⇓⇓⇓–26) and manmade microscopic objects (27⇓⇓⇓⇓–32). Energy input at the level of the individual constituents drives active matter systems out of thermal equilibrium and leads to a wide range of collective phenomena, including flocking (7, 19, 28, 29, 33, 34), swarming (12, 13), clustering (14, 27, 30, 32), 2D long-range order (15, 35), giant number fluctuations (14, 15, 33, 35, 36), spontaneous flow (21, 24, 25, 37, 38), and synchronization (16).

Active matter systems consisting of elongated particles often lead to local nematic orientational order. This important active nematics class comprises experiments with vibrating granular rods (36), crawling cells (39⇓–41), swarming sperms (42), filamentous bacteria (15), and motor-driven microtubules (20, 21, 24), which, together with theoretical work, have shown that the interplay between orientational order, active stress, and particle and fluid flow leads to complex spatial–temporal dynamics and unusual fluctuations. The seminal work by Dogic and coworkers (21, 43) has been particularly influential. They experimentally observed spontaneous chaotic dynamics driven by topological defects, and their results triggered a large number of theoretical and modeling approaches. These are of two main types, particle-level “microscopic” models (43⇓–45) and continuous-level “hydrodynamic” descriptions (46⇓⇓⇓⇓⇓⇓–53), with the latter usually written phenomenologically or by complementing equilibrium liquid crystal theories with minimal active terms. These studies provided important insights into the multifaceted dynamics of active nematics, such as hydrodynamic instabilities, long-range correlations, anomalous fluctuations, defect dynamics, and spatial and temporal chaos. However, these models generally contain a large number of parameters. This has made comparisons between models and experiments semiquantitative at best.

Bacteria are widely used as model systems to study active matter (12⇓⇓⇓–16). A recent study showed that elongated *E. coli* cells strongly confined between two glass plates can display the long-range nematic order and anomalous fluctuations typical of dry, dilute active nematics systems (15). However, so far, almost no bacterial system has been reported to exhibit the phenomenology of dense, wet active nematics, as reported first by Dogic and coworkers (21, 43). One exception is a study of motile bacteria dispersed in a nontoxic lyotropic nematic liquid crystal (54, 55). When bacteria concentration is high enough, active stress destabilizes the ordered nematic state of this biosynthetic system, leading to a state where topological defects in the liquid crystal evolve chaotically in a manner closely resembling that of the Dogic system. Here, we show that the typical phenomenology of wet, dense, active nematics can be experimentally realized in colonies of filamentous bacteria and show how to build a data-driven quantitatively faithful theoretical description of it. To this aim, we introduce a type of microscopic model for active suspensions, and we use simultaneous experimental measurements of both orientation and velocity fields to estimate all its parameters.

## Experimental Results

Our experiments are carried out with *Serratia marcescens* bacteria. At the edge of growing colonies, two to three layers of cells actively swim by rotating flagella in a micrometer-thick, millimeters-wide film of liquid on the agar surface (Fig. 1*A*). Apart from a narrow (*SI Appendix*, Fig. S1 *A*). Bacteria are labeled with a green fluorescent protein, which allows to record their motion under the microscope. In the dense, thin layer of interest, cells are almost always in close contact and nearly cover the whole surface. Our elongated cells are also frequently nematically aligned, as testified by the presence of *B*). (Standard cells cultivated without antibiotic drug do not give rise to any significant local order.) Our images do not allow to distinguish the current polarity of each cell, i.e., in which direction it is currently swimming with respect to the fluid. In fact, the swimming of most bacteria is strongly hampered at such high density. Nevertheless, our cells move collectively, mainly advected by the fluid they have set in motion, in a spatiotemporally chaotic manner strongly reminiscent of other active nematics systems (21, 54) (Movies S1 and S2). From each image, we extract a nematic orientation field *C* and *D* and *SI Appendix*, Fig. S2). Movie S1 shows the typical evolution of the obtained coarse-grained orientation and velocity fields. This dynamics is fast. Typical correlation times are of the order of seconds (see below). In each experiment, we record images for 30 s, which is significantly shorter than the cell division time (20 min). Therefore, contributions of cell growth to active stress are negligible in our work (56, 57).

### Global Measurements.

We first measure global statistical properties of our velocity and orientation fields. The average cell speed *SI Appendix*, Fig. S1 *B and C*).

Next, we compute spatial and temporal two-point correlation functions, which are defined and shown in Fig. 2 *A*–*D*. The spatial/temporal separations corresponding to a correlation value of *A* and *B*). Such a systematic variation is only observed for correlation times *C* and *D*). Correlation functions from various experiments with different drug concentrations collapse onto each other when space and time are rescaled by correlation lengths and times (insets in Fig. 2 *A*–*D*). Moreover, all of these quantities are linearly related to each other. Strikingly, transforming correlation times into correlation lengths using the mean speed v, we find that *E*). This indicates that our experiments are characterized by a single lengthscale and the mean flow speed (58, 59). Because our bacteria are too closely packed to measure their length, we use *SI Appendix*, Fig. S1).

### Defect Properties.

To go beyond the reduction of the complex spatiotemporal dynamics of our bacterial system to just a lengthscale and the mean speed, we now focus on the

We identify the location of *C* and Movie S1 for typical results). From the trajectories of defect cores, we measure *SI Appendix*, Fig. S2 *D*). We finally determine the intrinsic orientation *SI Appendix*, *SI Text* and Fig. S2 for details).

As in other active nematics systems (21, 36, 41), defects are created in ± pairs via the bending of ordered regions (Movie S1). Upon generation, *A*–*C*). For defects of opposite sign,

Restricting our analysis to “isolated” defects from now on, i.e., whose distance from nearest neighbors is larger than nematic correlation length *E* and *G*). We also find that the defect orientation *H*–*K*). Note that a small but finite velocity in the fluid frame *Discussion*.

To further quantify the structure of defects, we average, over time and many defects, the orientation and velocity fields around their core, sitting in their intrinsic reference frame. The familiar mushroom-shape and threefold symmetry of, respectively, the *A* and *B*). The flow field around the *C* and *D*), in agreement with previous work (53, 60).

Because of the chaotic collective dynamics, the magnitude of these averaged fields decays away from the defect core. We define defect core sizes *E* and *F*, we plot profiles of the angle of the nematic director calculated at three different radii around the defect cores. These profiles show clear systematic deviations from the linear variation predicted in one-constant equilibrium liquid crystals theory (61). The velocity orientation profiles, as well as the profiles of the magnitude of orientation and velocity fields, show also systematic variations reflecting the fine structure of defects (Fig. 4 *I*–*L*).

We have performed the above analysis of the dynamics and fine structure of defects on a large set of experiments. We now describe how the main defect properties vary with our two effective control parameters, the correlation length *A*). In the steady state, the density of defects is statistically constant. From this steady density, one can extract an interdefect lengthscale *B*). We also find that the speed of defects relative to the local flow speed at their core decreases with *C* and *D*; see a discussion of this below). Remarkably, the detailed spatial structure of defects does not vary significantly between experiments with different characteristic lengths: after rescaling spatial coordinates by defect core size, or, equivalently, correlation length, averaged director and velocity fields from different datasets overlap nicely. We further confirm this by comparing defect angular profiles at 0.6*E*–*L*).

## Quantitative Modeling.

A microscopically faithful model of our dense, thin bacterial system where cells and their many flagella are in constant contact with each other and with the gel substrate is a formidable task well beyond current numerical power. Besides, this would require the knowledge of many specific details that are unknown. Here, we adopt a radically different approach: we treat the collisions and local interactions between cell bodies and their flagella at some effective level, where, we assume, they amount to a combination of steric repulsion and alignment. In addition, the far-field interactions and other effects due to the incompressible fluid surrounding bacteria are taken into account by solving the Stokes equation for the fluid flow. All of this also allows us to build an efficient, streamlined, but comprehensive model in two space dimensions.

### Description of the Numerical Model.

Recall that most cells in our dense system are not able to swim freely, simply because nearby cells prevent them from doing so (see Movie S2). These crowded cells mostly exert force dipoles on the fluid, which is then set in motion by their collective action. Cells, in turn, are advected and rotated by the fluid. Our model thus consists of nonswimming force dipoles immersed in an incompressible fluid film and differs significantly from the common choice of using a dynamic equation for a director field (64⇓⇓⇓⇓–69). As shown by a schematic diagram in *SI Appendix*, Fig. S3, each dipole represents the local cell body orientation and active forcing.

The fluid flow

Our dipoles are point particles with position **3**, the first term on the right-hand side, with strength **2**, the right-hand side term **2** because our system is crowded.

The force field F in Eq. **1** is assumed to be dominated by the gradient of the active stress tensor field. (A small, residual contribution from the short-range repulsion force between neighboring dipoles exists but can usually be neglected; see *SI Appendix*, Eq. **S2** for details about this point.) The active stress tensor is itself assumed, as usual in wet active nematics studies (50, 51), to be proportional to the gradient of the orientation field:

The full system constituted by Eqs. **1**–**4** can be seen as a minimal Vicsek-style model (71, 72) incorporating the main mechanisms at play in our bacterial active nematics. One thus expects a basic interplay between alignment and noise: if the alignment strength

It is relatively easy to find parameter values such that the dynamics of our model closely resembles the experimental observations. As a matter of fact, the region of parameter space where spatiotemporally chaotic active nematics behavior occurs is rather large. To go beyond such qualitative agreement, we have systematically investigated the effects of parameters. We now show that for each experimental dataset, there exists a unique set of parameter values at which the model optimally matches the experiment, in the sense that all quantities studied in the previous section are in quantitative agreement.

### Data-Driven Parameter Optimization.

We proceed in two steps. First, simultaneous measurements of velocity and orientation fields allow us to pinpoint the parameters in Eq. **1** without resorting to the “microscopic” part of the model, i.e., Eqs. **2** and **3**.

Dividing both sides of Eq. **1** by α, we are left with two independent parameters, **1**. We then compare *A*, where *B* and *C* and Movie S3).

After fluid parameters are fixed, we proceed to the second step and match the full model with experiments. Eqs. **2** and **3** contain six parameters. We first evaluate their influence by varying them individually around a reference point (see *SI Appendix*, *SI Text* and Fig. S4 for details). We find that angular noise level

We performed a systematic scan of this restricted parameter space, running the model for many sets of parameter values, and extracting from each of these runs the quantities of interest, i.e., those measured also in the experiment. To quantify the match between model and experiment, we found that using three independent quality functions is sufficient. Here, we use *D*. Perfect matching (*D*. For the particular experiment considered, we find

Finally, we performed two “consistency checks.” We verified that choosing a different value of *SI Appendix*, Fig. S6). In short, *SI Appendix*, Fig. S7). This confirms that our procedure, for a given experiment, yields a unique set of model parameters at which model dynamics optimally matches spatiotemporal data.

### Variation of Model Parameters.

We have successfully applied our matching procedure to a large set of experiments with drug concentration above 15 *SI Appendix*, Table S1). This provides us with a wealth of information about our experimental system.

We first discuss the effect of the mean speed v at fixed correlation length. Choosing a subset of experiments yielding approximately the same correlation length, we observe that v almost exclusively influences *A*). The other parameters remain constant with the exception of the interaction range *B*–*E*). The clear linear growth of

The variation of optimal model parameters with correlation length, at fixed mean speed v, is presented in Fig. 7 *F*–*J*. From the extracted “fluid” parameters, we can construct two length scales that are proportional to **1** with the friction term *F*. We can also balance friction with the viscous force *G*. (We show that the two scalings above are verified for all our data points in *SI Appendix*, Fig. S8.) These findings provide a physical understanding of the factors contributing to the correlation length and, in particular, of how it is connected to the fluid effective parameters. The vorticity coupling parameter *H*). This is in agreement with Jeffery’s theory, which shows that *E*–*L*) and that *SI Appendix*, Fig. S5). On the other hand, the strain coupling parameter *I*), at odds with Jeffery’s results, which show that longer objects have higher *J*), which shows that the correlation length increases linearly with the area where nematic alignment takes place, i.e., the number of aligning neighbors, in our model.

## Discussion

To summarize, we presented a systematic study of collective motion and defect properties in a dense, wet active nematic system composed of filamentous bacteria and introduced a minimal microscopic model to account for our experiments. We have shown that using both orientation and velocity measurements enables to determine a unique, optimal set of parameter values at which our Vicsek-style model for active suspensions accounts quantitatively for many, if not all, quantities that one can extract from experimental data. Because the collective dynamics of our bacterial active nematics is always chaotic, we have used topological defects to estimate these optimal parameter values. As a matter of fact, it is sufficient to use a small subset of the various quantities we measured to determine all optimal parameter values, after which the remaining subset is “automatically” matched too. The existence of a unique optimum at which matching is nearly perfect constitutes, in retrospect, evidence of the quality of our model.

Thanks to quantitative match at a remarkable level of detail, the interplay between experiments and model provides a deeper understanding of our system. This is, in particular, the case for the dynamics and structure of topological defects. Fig. 5*C* demonstrates that *C*), which shows two vortices above and below the strong jet advecting the defects. The shape of defects is essentially governed by the vorticity coupling constant *C* and *H*) but is shown by simulations of our quantitatively faithful model (*SI Appendix*, Fig. S5). Thus, the vortices can destabilize orientational order ahead of the core, causing the defect to move faster than the background flow. The accelerating effect weakens when particles are less sensitive to flow vorticity or when nematic interaction becomes stronger, as shown in simulation (*SI Appendix*, Fig. S4 *C*) and experiments (Fig. 5 *C* and *D*).

The averaged orientation and velocity fields around *B* and *D*) show an approximate threefold rotational symmetry, which is consistent with the conventional, equilibrium picture: this symmetry implies that active and elastic stresses are balanced around the defect core and that *E*, *I*, *G*, and *K* and 5*D*). Moreover, instantaneous fields around *SI Appendix*, Fig. S9). Such deviations break stress balance around the core and give *SI Appendix*, Figs. S9 and S10) indeed show that the degree of deviation from threefold symmetry correlates with this velocity.

Our work also explains the multiple effects of cell length (under the influence of cephalexin). Cell length directly, and not surprisingly, governs all length scales in our system and does so nearly identically (Figs. 2*E*, 5 *A* and *B*, and 7 *F*–*J*). More surprising is the observation that the relative speed of defects decreases with cell length (Fig. 5 *C* and *D*) and that the strain coupling constant *I*).

These findings are just a subset of all those illustrating how, thanks to the quantitative modeling, one cannot only determine key effective parameters (such as the strength of flagella or the effective viscosity of our suspension) but also “read” important physical mechanisms from observing how model parameters change in experiments or are changed in simulations.

Our data-driven quantitative matching was made possible thanks to the relative simplicity of our Vicsek-style model: even though it deals with wet active suspensions, it possesses a relatively small number of parameters and is numerically efficient. Treating near-field interactions only effectively, it is also versatile, and we believe the same approach can be applied to other active suspensions and extended to include other effects, such as external field and polar order.

The simplicity of our model should also allow for derivation of continuous, hydrodynamic equations. Works on hydrodynamic theories of wet active nematics abound, but they typically lack a direct connection to microscopic mechanisms. Thus, deriving a faithful hydrodynamic theory from our quantitatively valid model is a very promising step. That would, in particular, allow to estimate how far our active nematics deviates from elastic theory predictions, something hinted by the structure of defects (Fig. 4 *E* and *F*).

## Materials and Methods

### Bacteria Strain and Colony Growth.

We use wild-type *S. marcescens* strain American Type Culture Collection 274 labeled with green fluorescent protein p15A-eGFP. Bacteria colonies are grown on a soft (

### Imaging Procedure.

After a growth time of 8–9 h, collective motion is observed for as long as 2 h near the expanding edge of a colony, in an active region about *S. marcescens* colonies quickly change from monolayer to three-layer within

## Acknowledgments

H.P.Z. thanks Julia Yeomans for useful discussions at the initial stage of this work. X.-q.S., H.C., and H.P.Z. acknowledge financial support from National Natural Science Foundation of China Grants 11422427 and 11774222 (to H.P.Z.), 11635002 (to X.-q.S. and H.C.), and 11474210 and 11674236 (to X.-q.S.). H.P.Z. thanks the Program for Professor of Special Appointment at Shanghai Institutions of Higher Learning (Grant GZ2016004). H.C. thanks the French Agence Nationale de la Recherche project “Baccterns.”

## Footnotes

↵

^{1}H.L. and X.-q.S. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: hugues.chate{at}cea.fr or hepeng_zhang{at}sjtu.edu.cn.

Author contributions: H.L., X.-q.S., H.C., and H.P.Z. designed research; H.L., X.-q.S., M.H., X.C., M.X., C.L., H.C., and H.P.Z. performed research; H.L., X.-q.S., H.C., and H.P.Z. analyzed data; and H.L., X.-q.S., H.C., and H.P.Z. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1812570116/-/DCSupplemental.

Published under the PNAS license.

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