A Biophysical Limit for Quorum Sensing in Biofilms

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Ned and , b .. oiiefebc nA rdcinfo Isnig can sensing, AI from production AI itself, on autoinduction feedback where perspective, positive this conditions i.e., From other despite cost. production and that AI metabolic biofilms privilege the suggest must in cells results effective limiting, are pro- These be nutrients AI to QS. all if QS to employ for applies concentrations that limit limit AI systems biophysical biophysical of bacterial This novel (DR) nutrient-limited. a a is range to duction in dynamic result AI this the trace produce on over We which that advantage cells. no cells cells non-QS essentially QS QS have that manner While nutrient-dependent strategies, surprisingly, not. fixed found, all that do outcompete we could that strategies AI strategies produce matrix-production constitutively with QS comparing employ cells, forming nutrient by regu- limited in is advantage production an AI availability. provide if no still production matrix QS with lating whether production therefore ask were AI to We Notably, (20–23). inspired constitutive availability (21–23). nutrient assume on developed scenario dependence models further this advantage these and and further all replicated matrix reproduction, a been deactivate into yield has (20) to resources can al. redirecting QS biofilms et by employ mature Nadell in that work, production this strategies nonpro- upon that neighboring Building showed suffocating (19). by cells lineages ducing competitive cell strong a outward to provides advantage production descendants matrix O that push more found that a they to into found surface cells (19) a from allows Foster production and matrix Xavier effects, reaction-diffusion gene horizontal facilitating transfer. and predators, and chemicals against doi:10.1073/pnas.2022818118/-/DCSupplemental at online information supporting contains article This 1 ulse a 8 2021. 18, May Published BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This Submission.y Direct PNAS a is article This interest.y competing no declare tools.y authors computational A.V.N.The developed A.V.N. data; and analyzed paper; N.S.W. the and wrote A.V.N. 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A.V.N. research; research; designed N.S.W. and A.V.N. contributions: Author rvt drs,MpeodN,07040; NJ, Maplewood address, Private owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To insol eapie ucinta sntmetabolically not produc- is signal that limit, QS function this effective prized of most a on slaved. view be be limit In should to biophysical tion QS. that a conclude nutrient-limited discover we of to behave efficacy us among and the led competitions population abil- bacteria own of the biofilm their simulations limits assess which Numerical to QS, accordingly. bacteria to interior contribute of nutrient-deficient cannot ity biofilm the produc- a then signal of if nutrient-limited, However, is (QS). pro- tion sensing a in quorum signals called chemical cess via Within communicate community. bacteria bacterial of biofilms, form ubiquitous a are Biofilms Significance oti n,w iuae opttosaogbiofilm- among competitions simulated we end, this To realistic incorporate that biofilms of models simple In c,d,1 https://doi.org/10.1073/pnas.2022818118 . y 2 rc niomn.Consequently, environment. -rich raieCmosAttribution-NonCommercial- Commons Creative c eateto oeua Biology, Molecular of Department . y https://www.pnas.org/lookup/suppl/ | f6 of 1

BIOPHYSICS AND COMPUTATIONAL BIOLOGY be viewed as one way that cells decouple AI production from Biomass production in biofilms requires nutrients. For exam- metabolism. ple, aerobic biofilms depend on oxygen (O2), which usually diffuses in from a source located far away (15, 19, 25, 26). In Results our simulations, we consider a single limiting nutrient, taken Agent-Based Model. For simplicity and ease of visualization, we to be O2, which diffuses from the top boundary of the simula- performed simulations with agent-based models (ABMs) on tion domain at a constant flux, mimicking a distant source (SI a two-dimensional square lattice (Fig. 1A). ABMs represent Appendix). We assume strong O2 uptake by bacterial cells to a system as an ensemble of autonomous agents which inter- allow for a well-defined surface-growth layer within our small act with one another according to predefined behaviors (24). simulation domain. Since the timescale for the O2 concentration In our simulations, a square can be occupied by a cell or an to come to a quasi-steady state (∼20 s for our simulation domain) equivalent volume of matrix or be unoccupied. Cells start at is much shorter than the timescale of biomass production (∼1 h), the bottom of the simulation domain, which is taken as the we assume a separation of the two timescales. substrate to which the biofilm adheres. Cells may reproduce Biomass production in the simulated biofilm is limited by O2 and form identical copies of themselves or produce matrix uptake, which we assume to be proportional to local O2 con- (see Fig. 1 B and C). Matrix itself performs no actions but centration. Thus, if the uptake of O2 is rapid, only cells in fills space. the upper layers of the biofilm have access to O2 and produce Both reproduction and matrix production may require shoving biomass. We define the fraction of O2 uptake used for matrix to make an adjacent site available. Shoving is performed by first production to be the matrix bias. For the same amount of O2 choosing a nearest vacant site and a shortest path to the chosen taken up, a bacterium can produce a much greater volume of vacant site (both of which may not be unique); then, all occupants matrix than of new cells [we take the cost of matrix produc- of the squares in the path are displaced along the path toward the tion to be 1/14 of the cost of reproduction on a per-volume vacant site. In our simulations, cells are assumed to be immotile basis (27)]. and thus only move when shoved. Thus, the biofilm, composed of cells and the matrix they produce, increases in biomass and Bacterial Competitions. To estimate the optimal matrix bias for grows upward. Each simulation ends when 50% of the lattice bacteria in our model, we performed pairwise competitions sites become occupied or a cell reaches the top of the simulation between different matrix-bias strategies (Fig. 1 D–F). Starting domain. with two strategies at a 1:1 ratio, we compared the cell counts of

Fig. 1. Simulated competitions of matrix-producing biofilms that grow on a submerged surface toward an O2 source. For details see Materials and Methods and SI Appendix.(A) All simulations are performed on a two-dimensional square lattice of width 128 sites and height 256 sites. Red squares are bacterial cells, yellow squares are extracellular matrix, and cyan squares are unoccupied. White arrows indicate diffusion of O2 from above. (B and C) Schematic of cell division and matrix production, shown for a blue cell surrounded by red cells. Cell division results in an identical cell’s being placed in an adjacent site. If no adjacent site is available, cells are shoved out of the way to make room for the new cell. Similarly, matrix production results in filling of an adjacent site with matrix. (D) Snapshot of a pairwise competition after 500 simulation timesteps. Red cells have matrix bias of 0.6 while blue cells have matrix bias of 0.5. Shade of cyan squares indicates normalized O2 concentration (normalized by the highest O2 concentration recorded for the entire simulation). (E) Snapshot of the same competition in C after 1,500 timesteps. (F) Mean of the natural logarithm of the final ratio of number of cells with matrix bias A to number of cells with matrix bias B. Between 75 and 350 simulations were performed for each competition.

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IAppendix SI .(C ). nphto h aecmeiinin competition same the of Snapshot ) 2 efrsbet- performs ) ifssfo bv si i.1 u h oo hd o niae oa Icnetain(in concentration AI local indicates now shade color the but 1, Fig. in as above from diffuses 2 n could and b max = tzr Idw to down AI zero at 0.9 nphto ariecmeiinatr100smlto ietp.Green timesteps. simulation 1,000 after competition pairwise a of Snapshot ) where uretlmtdQ i o rvd usata competitive substantial a provide not did QS that now found nutrient-limited cells, we Strikingly, QS production. AI and nutrient-dependent with cells fixed-matrix-bias between local on competitions linearly depend production AI O let investi- further we To question, environment? merits this nutrient-limited gate a that in beneficial cells still QS QS for pres- their effect the the on Allee to suggests capitalize weak bias This investigation. to a nutrient. matrix them of diffusive high allow ence the initial would over that their these monopoly bias from Without AI sufficient matrix switch fractions. produce low QS to to cells initial fail mag- QS low cells its of found very advantage, number we minimal competitive at While the a reduced ). S1 enjoy was always Fig. cells nitude to Appendix, QS cells of (SI QS ratio cells the seeding strategy initial fixed the to varying by frequency-dependent for effects con- tested assuming a selection also and employing We (19) by production. Foster AI (20) and stitutive al. Xavier simi- et to Nadell A similar by framework strategies. reached fixed-matrix-bias was conclusion all lar than better performed varying by chose bias, matrix function, the modeled We 2 1 (b 2 oee,wa fA rdcini uretdpnet .. is i.e., nutrient-dependent, is production AI if what However, min ocnrto.A hw nFg ,w efre pairwise performed we 3, Fig. in shown As concentration. h B + K 10 = eno h aua oaih fthe of logarithm natural the of Mean (D) timesteps. 000 3, after b max steA ocnrto twhich at concentration AI the is b b min )= ([AI]) afa ewe t iiu n aiu.We maximum. and minimum its between halfway ), oyedana wthlk epnet I Indeed, AI. to response switch-like near a yield to , b b max min and , = b min thg I(e Eq. (see AI high at 0.1 K (b + efudmlil Ssrtge that strategies QS multiple found we max b https://doi.org/10.1073/pnas.2022818118 fteQ atraa Hill a as bacteria QS the of , − b min ) K h .NnQ el rd do (red) cells Non-QS 1). [AI] [AI] + b tan h value the attains h h PNAS , | f6 of 3 [1]

BIOPHYSICS AND COMPUTATIONAL BIOLOGY Fig. 3. Simulated biofilm competitions between nutrient-limited QS cells that produce AI proportional to local O2 concentration and non-QS cells. (A) QS cells shown in green produce AI at a rate proportional to local O2 concentration (arrows highlight AI diffusion). Bacterial cells shown in red do not produce any AI. (B) Snapshot of a pairwise competition after 200 simulation timesteps. Green QS cells adjust their matrix bias based on local AI concentration as in Fig. 2. Red cells do not produce AI and have a fixed matrix bias of 0.7. Shade of squares indicates local AI concentration (in arbitrary units as described in SI Appendix), and O2 diffuses from above. (C) Snapshot of the same competition in B after 1,000 timesteps. (D) Mean of the natural logarithm of the final ratio of number of O2-dependent QS cells to number of fixed-strategy cells. The error bars indicate standard deviations of log ratios. Over 60 simulations were performed for each competition.

benefit. Specifically, nutrient-limited QS strategies had to be bacteria whose AI production is limited by uptake of a diffusible highly fine-tuned to ever perform better overall than fixed strate- nutrient (derivation in SI Appendix). Specifically, there is an gies, and at best they did not perform nearly as well as QS upper limit on the DR of possible AI concentrations experienced strategies with constitutive AI production. Notably, although by cells for a given source of diffusible nutrient. For a diffusible, the nutrient-limited QS strategies initially switched from high nondecaying AI, the minimum AI concentration, [AI]min, is that matrix bias to low matrix bias the cells later switched back to experienced by a single isolated cell, which senses only its own a high matrix bias and thus failed to capitalize on the lack of AI production. We prove that no matter how cells are arranged competition. in three dimensions the maximum AI concentration that any cell What is it that prevents nutrient-limited QS bacteria that have can experience has an upper bound specified by achieved dominance from switching to a low matrix bias? We observed that only the cells at the edge of the biofilm produce [AI]max 4πDO r0 DR ≡ = 2 + 1, [2] substantial amounts of AI (as O2 penetration into the biofilm [AI]min γ was designed to be low) and so the total AI production remains

nearly constant. Thus, despite the increasing total population of where DO2 is the diffusion constant for O2 (which we take to QS bacteria, the AI concentration at the growing front of the be the limiting nutrient), r0 is the cell radius, and γ is the rate of biofilm does not increase over time. (Note that we assume a intake of O2 per cell per concentration of O2. Intuitively, the bio- slow decay of AI, yielding a decay length of ∼100 µm, to avoid physical limit expressed by Eq. 2 comes from recognizing that in artifacts associated with the finite simulation domain size.) As and around a biofilm the O2 concentration and AI concentration a result, the nutrient-limited QS bacteria are not able to distin- are effectively mirror images (Fig. 4). This follows because O2 is guish between being at the edge of a large “successful” biofilm linearly converted to AI, so local O2 consumption translates to and being part of the initial seeding density of bacteria, still in local AI production, and both O2 and AI satisfy corresponding competition with other species. This contrasts with the case of diffusion equations. This means that the local AI concentration constitutive AI production where the total AI production and can never be higher than a limit set by the minimum local O2 concentration both increase with the total population of QS concentration, which is zero. Since a single isolated cell already bacteria. experiences a finite AI concentration due to its own AI production, this upper limit on AI concentration implies an absolute A Biophysical Limit. In our simple two-dimensional simulations upper bound on the DR. we found that nutrient-limited QS strategies provided little or Under what conditions can DR be large? Intuitively, large DR no benefit to cells competing for a diffusible resource. Does this requires a small [AI]min, so a cell on its own must be a rela- conclusion apply in more realistic settings? Perhaps surprisingly, tively weak producer of AI, i.e., it must be a weak consumer

we found that the answer is yes: There exists a corresponding of O2. Indeed, the combination of parameters DO2 r0/γ in Eq. biophysical limit for the efﬁcacy of QS in three dimensions for 2 is large if a single cell only weakly perturbs the local O2

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AI of DR limited the illustrating Schematic 3 r [O 0 3 2 sarsl,Eq. result, a As ρ. 2 ] ydfuin oee,teecniin r not are conditions these However, diffusion. by 0 atra ooy osmn O consuming colony, bacterial A (B) . 2–4. 2 λ ocnrto ed oteuprlmto h Ro AI of DR the on limit upper the to leads concentration = DR n p = [O DR D (48π = λρ 2 O ] ∞ 2 ∝ 4 = γ 1/3 where , /γρ n h O the and , hc mle atreplenishment fast implies which , [ O 3 πρλ n h ims ouefraction volume biomass the and , 2 2 ] a lob rte as written be also can ∞ φ) hsrlto ewe h Icon- AI the between relation This . 2 1/3 r 0 n 1 + 2 2 ρ ocnrto ntevicinity the in concentration 1 + stelclcl density. cell local the is . . ρ 2 −1/3 n ertn I If AI. secreting and h number the , D O 2 sin- A A) and 4 [4] [3] r 2 is 0 2 , l eldniy Icnetainwudhtismxmma given as maximum its Eq. thresh- hit by would a concentration Above AI nutrient-limited: density, cell were old be production not AI would feed- if presumption positive in this correct for However, result pre- production. need necessarily AI the be on would obviating back could cells concentration, is It of AI (2, se density. higher density a local systems per higher their a autoinduction QS sense that why sumed to many understood cells fully of for not sensing, desirable feature is AI well-established from It production 30–33). a AI is not of is feedback which production positive i.e., AI tion, if greater much advan- be the to Nevertheless, seem nutrient-limited. cells. vicinity QS of the of in number cells tages total growing the the information of than of number relevant the rather direction the be that might Alternatively, the bacteria signal nutrient. gradient, for chemotactic the negative consolidated of its single source via a indicate, as would act nutrient-limited densities, nutri- could cell large lower AI a at imply even would Further, concentra- source. concentration ent AI AI depleted high local locally a a the with source. that nutrient so AI, its concentration, nondecaying nutrient at the to a mirrors nutrient QS tion for diffusible employ because, could the is bacteria of This biofilm high, concentration a is the in cells example, infer of For density ways. the several where in bacteria by exploited in QS involving scenarios is all limit to the environments. applies limit, nutrient-limited and biophysical mecha- general this concrete of more a effects much the provides of case example illustrative nistic our While source. usata oto rdcino oeAs(6 7,clsare production cells AI 37), regulate (36, and processes AIs two some are the decouple of the QS to production and able 35), and of links 34, 17, strong cost production (16, the substantial production despite AI AI and that metabolism that and between metabol- functions propose entirely bacterial be We cell privileged not local should slaved. infer production reliably ically AI to envi- QS, that their via is about density information conclusion of main types our some ronment, infer to bacteria by other infor- of of presence pieces the vital and availability two cells. nutrient integrates local cell its the mation: autoinduction how and reveal limitation nutrient physiological would of effect the under- joint Further, the reveal conditions. standing growth would different in QS limitation of importance nutrient growth function. under with prized scales a rate production is AI QS how that of suggest investigation would Generally, it nutrient- rate, proportion- growth and strict to a ality in nutrient-limited by higher predicted privileged, is than in production conditions be AI nutrient-limited If production must compared. be AI can production conditions replete AI nutrient-limited of that rates in prediction the our cell test the to by prioritized is metabolic conditions. prized a that is QS that pro- function suggest AI would of nutrients dependence on nonlinear duction a that or area any autoinduction In an either study. is case, experimental this including and investigation two, dependence. further the linear merits between strictly fall and might an systems nonlinearity) be Real a would such (which of production AI example two the constitutive contrast of to of cases chosen importance extreme have we the QS, highlight at in high to dependence is order metabolic consumed In nutrient availability. of nutrient unit low per availability, be production nutrient AI would on that dependence nonlinearly such depend this to breaking production of AI for way order (Eqs. Another in limit S1A). availability biophysical tion nutrient the on evade production to AI the breaking of of way dependence one represent simply may autoinduction tive, u eut ugs eaoi nepeaino autoinduc- of interpretation metabolic a suggest results Our be still could production AI nutrient-limited principle, In hl togyntin-iie Ipouto ol eused be could production AI nutrient-limited strongly While example, For experimentally. tested be can predictions Our 2 n rvd ofrhrifrain rmti perspec- this From information. further no provide and https://doi.org/10.1073/pnas.2022818118 2–4 and IAppendix, SI PNAS | f6 of 5 sec-

BIOPHYSICS AND COMPUTATIONAL BIOLOGY largely independently of cell metabolism. More broadly, we hope the cell that produced it. After all matrix production and reproduction has that the results of this study will spur theoretical and experimen- taken place, the reaction-diffusion equations are again solved for the next tal interest in the roles of metabolic dependency of QS signal timestep. This alternating procedure is repeated until the simulation halts production. at a prespeciﬁed halt condition. For additional details see SI Appendix.

Materials and Methods Data Availability. There are no data underlying this work. All simulations were performed via agent-based modeling using Nanoverse ACKNOWLEDGMENTS. We thank Bonnie Bassler, Farzan Beroz, Daniel (24). At each timestep, the reaction-diffusion equations for the O2 and AI Greenidge, Matthew Jemielieta, Ameya Mashruwala, Yigal Meir, and Jing concentrations speciﬁed by their production, consumption, and decay (if Yan for stimulating discussions and input. A.V.N. acknowledges support any) are solved to obtain their steady-state concentrations. This steady-state from the Princeton University Physics Department, Ofﬁce of the Dean of Research at Princeton University, University of California San Diego (UCSD) concentration determines the matrix production strategy and the probabil- Physics Department, and the qBio program at UCSD. This research was sup- ities in each timestep of matrix production and/or reproduction. If a cell ported, in part, by NIH Grant R01-GM082938 (to N.S.W.). This work was produces matrix and/or reproduces in a timestep, then the positions of supported in part by the NSF, through the Center for the Physics of Biologi- some surrounding matrix and bacterial cells are “shoved” as necessary to cal Function (PHY-1734030). This work was done on land considered part of allow the newly produced matrix/cell to occupy a lattice site adjacent to the ancient homelands of the Lenni-Lenape and the Kumeyaay peoples.

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