Microbial competition in porous environments can select against rapid biofilm growth
- aDepartment of Zoology, University of Oxford, Oxford OX1 3PS, United Kingdom;
- bWolfson Centre for Mathematical Biology, University of Oxford, Oxford OX2 6GG, United Kingdom;
- cOxford Centre for Integrative Systems Biology, University of Oxford, Oxford OX1 3QU, United Kingdom;
- dComputational Biology, Memorial Sloan Kettering Cancer Center, New York, NY 10065;
- eInstitut de Physique de Rennes, Université de Rennes 1, 35042 Rennes, France;
- fDepartment of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom
See allHide authors and affiliations
Edited by Howard A. Stone, Princeton University, Princeton, NJ, and approved November 1, 2016 (received for review December 21, 2015)

Significance
The overwhelming majority of bacteria live in porous environments, like soil, aquifers, and sediments, where they facilitate many important processes. Despite their importance, we understand little about how these complex environments shape the composition of the microbial communities that live within them. Here, we combine two diverse bodies of theory—fluid dynamics and game theory—to shed light on how bacteria evolve in these habitats. We show that bacteria in porous environments face a fundamental dilemma: they rely on flow for nutrients and dispersal; however, as cells grow, they tend to reduce their access to flow. A fast growing strain can, therefore, choke off its own nutrient supply, diverting it instead to competitors. In contrast with classical theory, our results suggest that cells within a biofilm can obtain a competitive advantage by growing more slowly.
Abstract
Microbes often live in dense communities called biofilms, where competition between strains and species is fundamental to both evolution and community function. Although biofilms are commonly found in soil-like porous environments, the study of microbial interactions has largely focused on biofilms growing on flat, planar surfaces. Here, we use microfluidic experiments, mechanistic models, and game theory to study how porous media hydrodynamics can mediate competition between bacterial genotypes. Our experiments reveal a fundamental challenge faced by microbial strains that live in porous environments: cells that rapidly form biofilms tend to block their access to fluid flow and redirect resources to competitors. To understand how these dynamics influence the evolution of bacterial growth rates, we couple a model of flow–biofilm interaction with a game theory analysis. This investigation revealed that hydrodynamic interactions between competing genotypes give rise to an evolutionarily stable growth rate that stands in stark contrast with that observed in typical laboratory experiments: cells within a biofilm can outcompete other genotypes by growing more slowly. Our work reveals that hydrodynamics can profoundly affect how bacteria compete and evolve in porous environments, the habitat where most bacteria live.
Modern microbiology relies on growing cells in liquid cultures and agar plates. Although these conditions offer high throughput and repeatability, they lack the complex physical and chemical landscapes that microbes experience in their natural environments. This environmental heterogeneity is increasingly recognized to exert a powerful influence on microbial ecology across a wide diversity of habitats, ranging from the ocean to the human gut (1⇓⇓–4). Although advances in sequencing technology now allow us to resolve how the genetic composition of microbial communities changes in response to environmental conditions (5, 6), we often lack a mechanistic understanding of the underlying processes. Novel empirical approaches, which simulate the conditions found in realistic microbial habitats, are needed to understand the strategies that cells use to gain an advantage over their competitors (7).
The overwhelming majority of bacteria live in porous environments between the particles that compose soil, aquifers, and sediments, and cumulatively comprise roughly half of the carbon within living organisms globally (8). Cells in porous environments typically reside in surface attached structures known as biofilms (9), in which diverse bacterial genotypes live under intense competition for limited resources (10, 11). Recent efforts have identified specialized mechanisms that cells use to gain advantage over competing genotypes in biofilms, ranging from the secretion of toxins to polymer production and metabolic regulation (12⇓⇓⇓⇓⇓–18). Whereas genotypic competition is most frequently studied in biofilms growing on simple flat surfaces (19⇓–21), biofilms growing in the interstitial spaces within porous structures face additional constraints. In porous environments, space is much more limited, and biofilm growth tends to attenuate the fluid flow that supplies cells with nutrients and facilitates dispersal.
Biofilms typically reduce the flow through porous environments by orders of magnitude at the Darcy scale (22), a macroscopic scale that measures the flow averaged over many pore spaces. Harnessing this effect, biofilms can be used to limit the transport of pollutants that have leaked into groundwater aquifers and to facilitate the extraction of petroleum from recalcitrant regions of reservoirs (23, 24). However, biofilm-induced clogging also generates unwanted effects: for example, it severely limits the efficiency of porous filtration systems (25) and curtails the rate at which water infiltrates into aquifers (26), exacerbating droughts. Due to its importance, the attenuation of flow by biofilms has long been studied at the Darcy scale (27, 28), and more recent works have sought to resolve how this process, in turn, is mediated by biofilm–hydrodynamic interactions at the microscopic pore scale (29⇓–31). However, it is largely unknown how these interactions influence the ecology and evolution of the bacteria themselves. Here, we combine experiments and models to show that porous media hydrodynamics can dramatically affect the principles of bacterial competition and evolution.
Results
A Conceptual Model to Study Hydrodynamic Interactions Between Competing Biofilms.
Bacteria within biofilms tend to form patches of genetically identical cells, even when the cells from which they are founded are initially mixed. This genotypic patchiness occurs because in situ growth, combined with the low mobility of cells within biofilms, means that clone mates tend to remain in close proximity to one another (32, 33). Moreover, genotypic patchiness in biofilms is enhanced by population bottlenecks, which occur more frequently in nutrient-limited conditions, and when biofilms are initiated from a sparse distribution of attached cells (34⇓–36). Based on these observations, we focus here on the competition between localized biofilm “patches” that each comprise a single genotype and assume that competing patches occupy different pore spaces.
To investigate how biofilm growth influences the flow through a porous environment, we calculated the Stokes flow through a representative network of pore spaces that is driven by a difference in pressure at the boundaries (Fig. 1A and Materials and Methods). The addition of a small impermeable biofilm patch sharply reduces the flow through the pore in which the biofilm resides, while concurrently increasing the flow through neighboring pores (Fig. 1 B and C). Although the magnitude of this flow diversion depends on the specific geometry of the pore space, this simulation shows that as a biofilm patch proliferates, it tends to decrease its access to flow, while increasing the flow to patches of biofilm that reside along other flow paths. This diversion of flow introduces a way in which biofilms can interact: genotypes inhabiting a porous environment can affect one another via modulating their respective access to flow. This “hydrodynamic interaction” differs from interactions observed in classical biofilm assays, where different genotypes growing together on flat surfaces typically have to be in close proximity to interact, for example, through capturing one another’s nutrients or via cell secretions. Rather, here we see that in porous environments, biofilms can influence one another over much larger distances, by either curtailing or increasing one another’s ability to capture flow.
A growing biofilm tends to decrease its access to flow while increasing the flow to its competitors. (A) Viscosity dominates inertia in most porous environments, owing to the relatively small pore spaces (10
Although porous substrates harbor many biofilm patches that can simultaneously perturb one another’s flow environment, we idealize this network of interactions as a collection of its constituent pairwise interactions. We then resolve the dynamics of competition between a single pair of biofilm patches, each of which is composed of a different genotype. In this pairwise approximation, the proportion of the total volumetric flow rate,
Microfluidic Experiments Show That Rapidly Expanding Biofilms Tend to Divert Flow to Biofilms That Increase in Thickness More Slowly.
We next developed a microfluidic version of our pairwise flow model to experimentally test how pore-scale hydrodynamics affects the competition between genotypes that form biofilms at different rates (Fig. 2 and Fig. S1). Our experiments used a well-studied Escherichia coli experimental system. Specifically, we competed wild-type E. coli cells with
Microfluidic competition experiments show biofilms that rapidly increase in thickness tend to divert flow to biofilms that expand more slowly. (A and B) The left pore of each device was seeded with wild-type cells (green), whereas the right pore was inoculated with
Schematic of microfluidic device that simulates the hydrodynamic interactions between patches of biofilm within a porous environment. (A) A syringe pump was used to pull fluid through the bottom outlet of the device at a constant volumetric flow rate,
Wild-type biofilms form at a faster rate than RpoS-null biofilms. We inoculated relatively wide straight microfluidic channels (
In porous environments biofilm growth is opposed by flow-induced detachment, which reduces the thickness of biofilms by shearing away cells from its surface (40⇓–42). To simulate different ambient flow conditions in our experiment, and thus the relative amount of detachment, we applied a total flow rate of either
Three independent repeats of our competition experiment yielded the same result at steady state. In the low-flow treatment (
A Mathematical Model of Flow–Biofilm Interaction Reveals a Diversity of Competitive Regimes and Enables Prediction of How Cell Dispersal Varies in Experiments.
Our microfluidic competition experiments suggest that hydrodynamic interactions between biofilms can profoundly affect genotypic competition. To understand this process better, we next developed a model that couple two competing biofilms with a model of flow, enabling us to explore a much wider range of competitive scenarios. Whereas the two pores in our experiment are strongly coupled, such that flow diverted from one pore is fully absorbed by the other pore, in a network of pores, the strength of the hydrodynamic coupling between two competing biofilms will vary depending on the geometry of the pore space and their relative proximity to one another (Fig. 1 A–C). To account for this variability, we consider two identical fluid pathways of width
A wide range of physical and biological processes can affect biofilm development (43); however, the thickness of biofilms in flowing environments is chiefly governed by the balance between cell division and flow induced detachment (44, 45). Cell division in biofilms is often confined to a layer at the exterior of the biofilm, where substrates are exposed to nutrients from the flow (46, 47). The characteristic thickness,
We used our model to simulate the development of two biofilms, which grow at rates
Our model predicted a diversity of different ecological outcomes (Fig. 3 and Fig. S4). When flow was relatively weak (
Diverse ecological regimes emerge from a model of biofilm competition where two strains are coupled by flow. (A) The phase space formed by
The impact of flow on biofilm competition. As a biofilm grows, it increases the hydrodynamic resistance of its pore space, which tends to decrease both its access to flow (Fig. 1) and flow-induced detachment. These dynamics create a positive-feedback loop because decreased detachment further increases its hydrodynamic resistance. When flow is weak (A), this process can lead to the faster-growing genotype (green;
The results from the model are in broad agreement with the two distinct flow regimes observed in our microfluidic experiments, which show that the wild-type biofilm growth tends to reduce its access to flow at smaller flow rates (equivalent to smaller
Combining experimental measurements with a mechanistic model allows us to infer how the normalized biofilm dispersal rate,
The Impact of Flow on the Evolution of Bacterial Growth Rate.
Our model shows that a biofilm’s fate depends not only on its growth rate but also on the behavior of other biofilms elsewhere within the porous network. However, how do hydrodynamic interactions between genotypes impact bacterial evolution? Over evolutionary timescales, it is expected that biofilm patches will repeatedly form and dissipate as a result of both natural processes and human intervention [e.g., predation (56, 57), enzymatic decay (58), and the periodic flushing of a porous filtration systems (59)]. This continual turnover of biofilm patches means that if new genotypes are introduced into a network of pore spaces—whether through in situ mutation or immigration—then they will be able to form new patches and potentially compete with the resident genotype over many iterated rounds of competition. Our model can then be used as a tool to measure the competitive ability of a newly introduced genotype, allowing us to infer how its frequency will change in the population over time.
To resolve how the bacterial growth rate would evolve over many successive rounds of competition, we embedded our mechanistic model of flow–biofilm interaction within a game theoretical framework known as adaptive dynamics (60). Specifically, this invasion analysis tests whether a novel genotype that grows at rate
A game-theoretical analysis of the coupled biofilm model predicts an evolutionary stable growth rate. (A) We used adaptive dynamics to construct a pairwise invasion plot, which maps the region of parameter space where a mutant that grows at rate
We find that mutants can invade only when their growth rate is slightly larger than the resident population (Fig. 4A). However, over time, successive invasions (Fig. 4A, arrows) of new genotypes are predicted to systematically increase the growth rate of the resident population until it reaches a evolutionary stable value,
Accounting for Potential Covariance Between Rates of Bacterial Growth and Flow-Induced Detachment Does Not Qualitatively Affect Our Predictions.
Our analyses above assume that a biofilm’s growth rate can vary independently from its other phenotypic characteristics. However, previous experiments have shown that faster-growing biofilms are more susceptible to flow-induced detachment (62⇓⇓–65). This dependency may occur because fast-growing genotypes invest less in secretions of exopolymeric substances that glue cells together (63) or because rapidly growing genotypes form biofilms with more fragile morphologies, rendering them more susceptible to detachment (55, 66). To model how covariance between growth and detachment influences bacterial competition, we extended our model using the parameterization of Speitel and DiGiano (62), who empirically quantified this coupling in porous environments using radiolabeled carbon sources. This parameterization measures the strength of the coupling between growth and detachment with the nondimensional parameter
A model in which a biofilm’s rate of detachment is coupled to its rate of growth generates the same qualitative result as a model that omits this dependency. Whereas
SI Text
Control Experiments.
We performed separate experiments to ascertain if the dye used in the competition experiments or the different fluorescent markers had an effect on biofilm development. We injected cells into straight microfluidic channels (1-mm width; 75-
Microscopy and Image Processing.
We imaged microfluidic experiments using a Zeiss Axio Observer inverted microscope with an AxioCam MRm camera and a Definite Focus system. A Zeiss Plan Apochromat 20× objective was used for competition experiments (Fig. 2), whereas a Zeiss EC Plan Neofluar 10× objective was used for control experiments (Fig. S2). We used the software package Zen Blue (Zeiss) to automatically record bright-field, GFP, and RFP images at each time point. Because our devices were larger than a single field of view, we recorded multiple adjacent images and stitched them together in post processing. To clarify the presentation of Fig. 2A and B, we plotted each strain’s fluorescence only in the arm of the device in which it was localized, so that the dye interface was more visible downstream.
To quantify the proportion of flow moving through each arm of the competition device over time, we manually tracked the location of the dye interface using the image analysis software Fiji (97) and applied a moving-average filter to reduce sampling noise (Fig. 2 A–C). This analysis neglects the thickness of the biofilm growing in the downstream channel of the device, which acts to further reduce the thickness of the dye stream,
Using a Hydrodynamic Model to Resolve How Flow Distributes in Competition Experiments.
Here, we develop a physical model to convert measurements of the position of the dye interface,
In the absence of molecular diffusion of dye, the interface would remain sharp and precisely track the streamline that forms the boundary between two fluid streams coming from either inlet. In this limit, we could measure the flow past the wild-type biofilm, even as it becomes arbitrarily small. In reality, however, molecular diffusion of the dye causes a gradient of dye to form in the direction traverse to flow, causing the dye interface to become more diffuse as one moves further downstream from where the two streams meet one another. The width of the dye gradient is quantified by the diffusive length scale,
Molecular diffusion of the dye thus places a key limitation on the smallest
Assuming pressure driven Poiseuille flow through a channel of square cross-section, the volumetric flow rate can be obtained by integrating the velocity profile
The mean fluid velocity along the dye interface can be calculated as
Here, we consider our slow-flow-rate experiments where
The dye interface will not be altered significantly (i.e., to leading order of magnitude) by the influence of diffusion provided
The fraction of flow through the wild-type biofilm’s pore space
Inferring the Rates of Biofilm Dispersal in Competition Experiments Using Empirical Measurements.
To estimate how the sloughing rate,
In this analysis, we cannot directly compare the dispersal rates between the genotypes because a biofilm’s intrinsic propensity to detach,
To determine the dispersal per unit length along the channel occupied by the wild-type biofilm, we add the dispersal per unit area along each face, weighted by the length of the face:
The normalized dispersal rates are plotted in Fig. S5 for both flow treatments. This analysis indicates that the wild-type biofilm sharply reduces its dispersal rate in the low-flow-rate treatment experiment, whereas the RpoS mutant sharply increases its dispersal rate. In contrast, in the high-flow-rate experiment, both genotypes initially increase their dispersal rate, but then begin to plateau. These results resemble regimes
Estimation of the Shear Stresses in Rectangular Channels.
The mean shear stresses within our microfluidic devices can be derived from the equations for Poiseuille flow through a rectangular channel (92). For a channel of width
For the competition channels (Fig. 2), this expression yields a mean shear stress
Extensions to Incorporate Dependencies Between Growth Rate and Dispersal.
Our initial model assumes that both competing genotypes can equally resist detachment, however, previous studies have shown the rate of detachment can depend on both the flow environment and the biofilm’s growth rate (62, 63). To capture this additional dependency, we extended our model to take into account growth rate-dependent detachment, using an empirically derived formulation first proposed by Speitel and DiGiano (62), who examined biofilms growing under different nutrient conditions. This dependency adds an additional term to our model where detachment is proportional to the biofilm growth rate. The new model reads
Here, as in the work of Speitel and DiGiano (62),
More generally, one can understand the relationship between the original model and its extension via the continuous bijection for the dimensional model
Discussion
Biofilms growing in porous environments facilitate a wide range of important processes in the natural environment and industry (8, 24⇓–26, 67⇓⇓⇓–71). Our proof-of-principle experiments, mathematical modeling, and game-theoretical analyses show that the feedback between biofilm proliferation and porous media hydrodynamics can dramatically affect how different genotypes compete. We find that relatively strong and weak flow conditions favor fast- and slow-growing biofilms, respectively, whereas intermediate flow rates allow biofilms with different growth rates to maintain access to flow (Fig. 3).
In industrial settings, these principles could be exploited to engineer microbial systems to favor a bacterial species with a particular growth rate or keep multiple species with different growth rates active over longer time scales. For example, in porous wastewater reactors, relatively fast-growing species of bacteria convert ammonia to nitrite, but it is desirable to inhibit often slower-growing species that further oxidize these products into nitrate, a potent environmental contaminant (72). Our work predicts then that using a larger flow rate may be a way to favor the former species of bacteria over the latter. In contrast, the remediation of mercury-contaminated wastewater in porous reactors can be enhanced by maintaining multiple species of bacteria that grow at different rates (73). Moreover, our findings suggest that inoculating porous substrates with a community of cells from the effluent of a porous system would favor the growth of biofilms that do not block their pore space, whereas inoculating cells from communities that have evolved in homogeneous laboratory conditions would promote blocking. Such information has implications for the design of effective water treatment systems, where blocking reduces efficiency, or in the design of biobarriers to stifle the movement of groundwater contaminants, where blocking is the main objective.
Our results may also shed light on how cells compete in natural environments. We expect that temporal fluctuations in flow and heterogeneity in pore size will promote diversity. This variablity is expected to be common due to episodic patterns of rainfall and geological processes that mix different particle sizes (74). However, some groundwater aquifers and packed-bed bioreactors have nearly constant rates of flow and a more uniform distribution of pore spaces, which may promote competitive exclusion. In systems where blocking does occur, natural selection may favor cells periodically detaching en masse to regain access to flow. Broadly consistent with this hypothesis, increased detachment has been observed empirically in response to nutrient deprivation and quorum sensing (75, 76).
Bacteria are the subject of intense empirical and theoretical study. However, the vast majority of work on bacteria focuses on their behavior in liquid cultures or in simple biofilm assays. Here, we have combined diverse bodies of theory, including fluid dynamics and game theory, to understand how bacteria compete and evolve within the complex porous environments where most bacteria live. Our assumptions greatly simplify the complexity of these systems, so there is considerable potential for extensions to our work. Many microbial traits can influence biofilm formation, including the strength of initial cell adhesion, which may itself be a function of the hydrodynamic or nutrient conditions (21, 77), production of extracellular polysaccharides (78), streamer formation (30, 79), quorum sensing (80), motility (81), and cell metabolism (82, 83). Further work will be needed to resolve how the wide diversity of microbial traits impact the processes described here.
Future efforts will also be required to resolve how the specific structure of the pore space and the distribution of different genotypes within them affect microbial competition. While our work predicts that bacteria can benefit from making less biofilm when growing in a clonal patch, this may change in mixed-genotype biofilms, where the priority may shift to locally outgrowing competitors (84). It is interesting, then, that bacteria are known to respond to competing strains by increasing their investment into biofilm (85).
In sum, our approaches indicate that porous habitats, and the flows within them, can have a profound impact on bacterial evolution. Although rapid division gives a microbe an evolutionary advantage in typical laboratory environments, our results suggest that this paradigm does not extend to many bacterial habitats.
Materials and Methods
Modeling Stokes Flow Through a Representative Network of Pore Spaces.
The geometry of the pore space (Fig. 1 A–C) was obtained using Particle Flow Code in Two Dimensions (Itasca), which models the mechanical processes that form many porous substrates. The particle locations were then imported into COMSOL Multiphysics to model incompressible Stokes flow within the pore spaces between the particles using the finite element method. Zero-flux, no-slip boundary conditions were used at the left and right boundaries of the computational domain as well as on the surfaces of all of the particles. At the top and bottom boundaries of the computational domain, the pressure was fixed at two different values, such that the resulting pressure gradient was responsible for driving flow. The Stokes equations were solved with and without the presence of a biofilm patch in one of the pore spaces. Results were then exported into Matlab 2015a (MathWorks) for further analysis and plotting.
Bacterial Strains and Culturing.
Our experiments used E. coli strain K12-W3110 and a mutant with a rpoS819 allele insertion (86, 87). Each strain was labeled with either green fluorescent protein (GFP) or red fluorescent protein (RFP). Cell cultures were grown overnight in tryptone broth (1× TB) (10 g of Bacto Tryptone per 1 L of water) at 37°C, diluted to an optical density of 0.1 (at 600 nm), and then grown for a further hour at 37°C so that cells were in exponential phase when they were first introduced into the microfluidic devices.
Competition Experiments.
Microfluidic device masters were fabricated from SU-8 on silicon wafers using standard soft lithography techniques (88) and were cast with polydimethylsiloxane (PDMS) (Sylgard 184; Dow Corning). The depth of these devices was
For the high-flow-rate treatments (
After the microfluidic device and tubing were primed with 1× TB to remove air from the system, cells were introduced into the device by pulling cultures of the wild-type and
After the initial inoculation phase, we applied a flow rate of
Mechanistic Model of Biofilm Competition.
Our mathematical model of biofilm competition simulates the two processes that are predicted to dominate biofilm development in flowing environments: bacterial growth and the flow induced detachment (45, 52). The differential equations that govern the thicknesses of the two biofilms
All three flow paths experience the same difference in pressure,
Our equation for hydrodynamic resistance assumes pressure-driven, planar flow between two parallel plates separated by a distance
Combining Eqs. 1, 4, and 5 yields the coupled differential equations that govern
Game Theoretical Analysis.
To examine how hydrodynamic interactions impact biofilm evolution, we embedded the mechanistic model presented above within an adaptive dynamics framework (60, 93). This analysis examines whether a new genotype is able to invade and displace a porous environment already colonized by “resident” genotype. Specifically, adaptive dynamics assumes that the rate at which novel genotypes are introduced into a group of interacting pore spaces—either through in situ mutation or immigration—is small compared with the rate at which a new genotype can displace the resident population (94). This assumption means that we can consider the pairwise interaction of a novel genotype that grows at
We assumed that the fitness of a genotype is directly proportional to its rate of cell dispersal at steady state, which is equal to the rate of biofilm growth at steady state (Eq. 1). In our model, the fitness of a biofilm,
Acknowledgments
We thank Juan Keymer for bacterial strains. K.Z.C. was funded by the Engineering and Physical Sciences Research Council, K.R.F. was funded by European Research Council Grant 242670, and W.M.D. was funded by Human Frontier Science Program Fellowship LT001181/2011L.
Footnotes
- ↵1To whom correspondence may be addressed. Email: kevin.foster{at}zoo.ox.ac.uk or w.m.durham{at}sheffield.ac.uk.
Author contributions: K.Z.C., H.T., and W.M.D. designed and performed the microfluidic experiments; K.Z.C., E.A.G., K.R.F., and W.M.D. developed the mathematical models; and K.Z.C., E.A.G., K.R.F., and W.M.D. 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.1525228113/-/DCSupplemental.
References
- ↵
- ↵
- ↵
- ↵.
- Horner-Devine MC,
- Carney KM,
- Bohannan BJM
- ↵
- ↵
- ↵
- ↵.
- Whitman WB,
- Coleman DC,
- Wiebe WJ
- ↵
- ↵.
- Torsvik V,
- Øvreås L,
- Thingstad TF
- ↵.
- Rothman DH,
- Forney DC
- ↵
- ↵.
- Xavier JB,
- Foster KR
- ↵
- ↵.
- Rao D,
- Webb JS,
- Kjelleberg S
- ↵
- ↵
- ↵.
- Kim W,
- Racimo F,
- Schluter J,
- Levy SB,
- Foster KR
- ↵.
- Stoodley P, et al.
- ↵
- ↵.
- Zhang W,
- Sileika T,
- Packman AI
- ↵
- ↵.
- Cunningham AB,
- Sharp RR,
- Hiebert R,
- James G
- ↵.
- Raiders RA,
- Knapp RM,
- McInerney MJ
- ↵
- ↵
- ↵.
- Cunningham AB,
- Characklis W,
- Abedeen F,
- Crawford D
- ↵.
- Shafahi M,
- Vafai K
- ↵.
- Graf von der Schulenburg DA,
- Pintelon TRR,
- Picioreanu C,
- Van Loosdrecht MCM,
- Johns ML
- ↵.
- Drescher K,
- Shen Y,
- Bassler BL,
- Stone HA
- ↵.
- Dupin HJ,
- Kitanidis PK,
- McCarty PL
- ↵
- ↵
- ↵.
- Mitri S,
- Foster KR
- ↵.
- Hallatschek O,
- Hersen P,
- Ramanathan S,
- Nelson DR
- ↵
- ↵
- ↵.
- Adams JL,
- McLean RJ
- ↵
- ↵.
- Trulear MG,
- Characklis WG
- ↵
- ↵
- ↵
- ↵.
- van Loosdrecht MCM, et al.
- ↵
- ↵
- ↵.
- Werner E, et al.
- ↵.
- Pirt S
- ↵.
- Horn H,
- Lackner S
- ↵
- ↵.
- Abbas F
- ↵
- ↵
- ↵.
- Brovelli A,
- Malaguerra F,
- Barry DA
- ↵.
- Ebigbo A,
- Helmig R,
- Cunningham AB,
- Class H,
- Gerlach R
- ↵
- ↵
- ↵.
- Allison DG,
- Ruiz B,
- SanJose C,
- Jaspe A,
- Gilbert P
- ↵
- ↵.
- Brännström Å,
- Johansson J,
- von Festenberg N
- ↵
- ↵.
- Speitel GE,
- DiGiano FA
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Leite JAC,
- Fernandes BS,
- Pozzi E,
- Barboza M,
- Zaiat M
- ↵.
- Michalakos GD,
- Nieva JM,
- Vayenas DV,
- Lyberatos G
- ↵
- ↵.
- Von Canstein H,
- Kelly S,
- Li Y,
- Wagner-Döbler I
- ↵.
- Rehfeldt KR,
- Boggs JM,
- Gelhar LW
- ↵
- ↵.
- Sauer K, et al.
- ↵
- ↵.
- Danese PN,
- Pratt LA,
- Kolter R
- ↵
- ↵
- ↵
- ↵.
- Bjergbæk LA,
- Haagensen JAJ,
- Reisner A,
- Molin S,
- Roslev P
- ↵
- ↵.
- Nadell CD,
- Drescher K,
- Foster KR
- ↵.
- Oliveira NM, et al.
- ↵.
- Keymer JE,
- Galajda P,
- Lambert G,
- Liao D,
- Austin RH
- ↵.
- Lambert G,
- Liao D,
- Vyawahare S,
- Austin RH
- ↵
- ↵
- ↵.
- Moller S, et al.
- ↵
- ↵.
- Stone HA
- ↵.
- Rudnicki R
- Diekmann ODO
- ↵.
- Nowak MA,
- Sigmund K
- ↵
- ↵
- ↵
- ↵
Citation Manager Formats
This article has a Letter. Please see:
- Relationship between Research Article and Letter - March 24, 2017
See related content:
- Useful models are simple and extendable- Mar 24, 2017