Microbial range expansions on liquid substrates
Severine Atis, Bryan T. Weinstein, Andrew W. Murray, David R. Nelson
MMicrobial range expansions on liquid substrates
Severine Atis, ∗ Bryan T. Weinstein, ∗ Andrew W. Murray,
3, 4 and David R. Nelson
1, 2, 3, 4 Department of Physics, Harvard University School of Engineering and Applied Sciences, Harvard University FAS Center for Systems Biology, Harvard University Department of Molecular and Cellular Biology, Harvard University (Dated: December 27, 2018)Despite the importance that fluid flow plays in transporting and organizing populations, fewlaboratory systems exist to systematically investigate the impact of advection on their spatial evo-lutionary dynamics. To address this problem, we study the morphology and genetic spatial structureof microbial colonies growing on the surface of a nutrient-laden fluid 10 to 10 times more viscousthan water in Petri dishes; the extreme but finite viscosity inhibits undesired thermal convectionand allows populations to effectively live at the air-liquid interface due to capillary forces. We dis-cover that S. cerevisiae (baker’s yeast) growing on a viscous liquid behave like “active matter”: theymetabolically generate fluid flows many times larger than their unperturbed colony expansion speed,and that flow, in return, can dramatically impact their colony morphology and spatial populationgenetics. We show that yeast cells generate fluid flows by consuming surrounding nutrients and de-creasing the local substrate density, leading to misaligned fluid pressure and density contours, whichultimately generates vorticity via a thresholdless baroclinic instability. Numerical simulations withexperimentally measured parameters demonstrate that an intense vortex ring is produced belowthe colony’s edge and quantitatively predict the observed flow. As the viscosity of the substrate islowered and the self-induced flow intensifies, we observe three distinct morphologies: at the highestviscosity, cell proliferation and movement produces compact circular colonies similar to those grownon hard agar plates except with a stretched regime of exponential expansion, intermediate viscositiesgive rise to compact colonies with “fingers” that are usually monoclonal and are ripped away tobreak into smaller cell clusters, and at the lowest viscosity, the expanding colony breaks up intomany genetically-diverse, mutually repelling, island-like fragments of yeast colonies that can colonizean entire 94 mm-diameter Petri dish within 36 hours. We propose a simple phenomenological modelin the spirit of the lubrication approximation that predicts the early colony dynamics. Our resultsprovide rich opportunities for future investigations and suggest that microbial range expansions onviscous fluids can provide a useful framework to examine the interplay between fluid flow and spatialpopulation genetics.
I. INTRODUCTION
The transport of living organisms by fluid flows playsan important part in the natural world. Hydrodynamictransport shapes and reorganizes populations across allscales [1], mixing populations to uniformity or leading tothe formation of spatial structures. For instance, turbu-lent mixing near the surface of oceans and lakes can clus-ter phytoplankton blooms into patchy, fractal-like spatialstructures [2, 3] that lead to ecological niches and geneticheterogeneity [4–6].Microbial populations expanding into unoccupied ter-ritory on agar plates, or range expansions, have been usedas a model system to investigate how population spatialstructure impacts evolution [7]. Range expansions de-velop spatial structure because a thin layer of cells at thepopulation front divide and generate genetically similardaughters who are not pushed very far away before theythemselves divide. As a result of this linear populationbottleneck at the frontier, the colony loses genetic diver-sity as the expansion progresses and quickly segregates ∗ Severine Atis and Bryan T. Weinstein contributed equally to thiswork: [email protected] into large monoclonal sectors that reveal the evolution-ary history of the colony in a process often referred to as“genetic demixing” [7]. Simplified stepping stone modelswith radial inflation have been used to describe the evo-lutionary dynamics of this process [8]. Microbial rangeexpansions revealed how various evolutionary forces, in-cluding selection [9–11], mutualism [12], competitive ex-clusion [13, 14], and irreversible mutation [15], impactthe dynamics of spatially structured populations.Microorganism growing on agar plates cannot be ad-vected as the underlying substrate is a solid, mimickingexpansions on land. Although investigated theoretically[16–19], few laboratory systems exist to systematicallystudy the interplay between the transport by fluid flowand spatial population dynamics. In this paper, we in-troduce a novel experimental system to grow microbialrange expansions on the surface of a nutrient-rich fluid10 to 10 times more viscous than water. The extremeviscosity of the liquid substrate enables capillary forces toconfine the cells over a macroscopic, quiescent air-liquidinterface, and typical settling velocities of isolated cellsthat leave the surface are less than a cell width per day.This unique system allows us to investigate microbialpopulation morphology and genetic segregation patternson liquid interfaces. a r X i v : . [ c ond - m a t . s o f t ] D ec To our surprise, even in the absence of externally im-posed flows [20], our experiments revealed that coloniesof the budding yeast,
Saccharomyces cerevisiae , inducedstrong outwards fluid flows in the surrounding substratemany times larger than the colony’s natural expansionvelocity. Remarkably, these flows arose from non-motile organisms, which do not possess, e. g. the flagellar-induced motilty of bacteria [21, 22]. In this paper, weshow how the induced fluid flow impacts colony morphol-ogy and genetic segregation patterns as the viscosity ofthe underlying substrate varies, and investigate the originof the induced flow.Section II summarizes our most important experimen-tal observations about the morphology and spatial popu-lation genetics of expanding yeast colonies on liquid sub-strates, and identifies three regimes: colonies behave ascompact circular colonies, circular colonies with fingers,or many solid-like repelling yeast fragments as the sub-strate viscosity is varied from high to low. In SectionIII, we describe our measurements of fluid flows gener-ated near the surface of growing colonies and identify twodistinct regimes. Experiments in Section IV argue thatthe fluid flow is not generated by surface tension gra-dients (Marangoni flows) but is instead generated whenyeast metabolism decreases the density of the surround-ing fluid, generating buoyant fluid flows via a baroclinicinstability due to the pressure and density contours cross-ing each other at an angle in the vicinity of the colony.Fluid-mechanics simulations calibrated to experiment inSection V provide further evidence that the buoyancy-driven baroclinic instability is the source of the fluid flow,as the simulations can quantitatively predict experimen-tal results. Finally, in Section VI, we present a simplephenomenological model in the spirit of the lubrication-approximation that combines colony growth, expansion,and thinning to predict the critical metabollically in-duced radial flow velocity at which colonies break apart.We compare predictions from the model to a phase dia-gram of yeast colony morphology over time as a functionof viscosity. The model displays a conventional Fisherpopulation wave in the absence of flow, but predicts ex-ponential growth of the colony radius in the presence ofa flow. When this radial flow is too strong, we find a“thinning catastrophe”, such that the colony thicknesstends to zero and breaks apart. Our work suggests manyinteresting avenues for future exploration, discussed inSection VII.
II. RANGE EXPANSIONS ON LIQUIDSUBSTRATES
To ensure a macroscopic quiescent liquid surface, weperformed experiments with fluids 10 − times moreviscous than water. The viscosity of the fluid is con-trolled by adding 2-hydroxyethyl cellulose, a long chainpolymer, to YPD (yeast extract, peptone, dextrose (glu-cose)) microbial growth medium; see Appendix A for additional experimental details. Characteristic polymerconcentrations used in our experiments and correspond-ing substrate viscosities are given in Table I. Althoughthe fluid has shear-thinning properties for shear rates˙ γ (cid:38) − s − [23], as discussed in Appendix B, theflow typical shear rates were of the order of ˙ γ = u/H ∼ − − − s − , where u is the characteristic surfaceflow velocity and H is the substrate fluid height, suchthat non-Newtonian effects were negligible in our experi-ment. In contrast with plates filled with hard agar, whichform a gel substrate with a shear modulus, cellulose poly-mers do not form a three-dimensional mesh, allowing thegrowth medium to flow.Polymer % (w/v) η (Pa · s)2.0 54 ± ± ± ± ± ± TABLE I. Newtonian approximation to the liquid substrate’sviscosity at a shear rate of ˙ γ ∼ − s − (Appendix B)at various concentrations 24 hours after mixing it with 2-hydroxyethyl cellulose. Our large substrate viscosity prevented thermal gradi-ents in the environment from driving undesired convec-tion under our laboratory conditions; no substrate fluidmotion was observed due to stray thermal gradients inthe absence of a colony growing on the surface. After de-position on the substrate, droplets containing yeast cellsspread unformely, allowing a dilute concentration of cellsto be held at the air-liquid interface by capillary forces.The cells rapidly aggregate due to attractive forces: cap-illary forces at the interface [24] for large distances, andVan der Waals forces between the cells for short dis-tances, in a process resembling spinodal decompositionor nucleation and growth [25]; see supplementary FigureA.1 and supplemental movie 0. Capillary forces werelarge enough to keep the cells on the surface of the fluiddespite their slightly higher density than the media, al-lowing the colony to grow at the air-liquid interface overthe typical several days time scale of our experiments.The large substrate viscosity also leads to extremely slowsedimentation velocities of any small clumps of cells thatbreak through the surface.We followed the segregation of two
S. cerevisiae strains, genetically identical except for constitutively ex-pressing different fluorescent proteins. The experimentswere initiated by depositing cells in a 2 µ L droplet ofsaturated overnight culture at the center of a 94 mmdiameter circular Petri dish filled with 40 mL of our vis-cous medium. The resulting colony expansion was thenmonitored over 5 days with a stereoscope (Appendix A).Shortly after cell growth and division begin, the microor-ganisms exhibit dramatically different growth dynamics
FIG. 1. Selected yeast colony morphologies on a) a hard agar plate after 72h of growth, and on the surface of the viscoussubstrate with decreasing viscosities: b) for η = 600 ±
90 Pa · s after 72h of growth, c) η = 450 ±
70 Pa · s after 84h of growth,d) η = 300 ±
45 Pa · s after 36h of growth, and e) magnification of a single representative finger from regime c). Qualitativelysimilar morphologies were observed in the range of viscositites indicated in b)-d). The figure shows merged brightfield andfluorescent images. White: transmitted brightfield, red: YFP strains, and cyan: mCherry strains. The scale bars in a) and b)correspond to 5 mm, the scale bars in c) and d) to 10 mm, and to 2 mm in e). relative to the well-studied hard agar plates, and exhibita rich variety of morphologies depending on the mediaviscosity. We systematically varied the polymer concen-tration in the medium, allowing us to investigate the mi-crobial population behavior over a range of dynamic vis-cosities η from 54 ± · s to 600 ±
90 Pa · s (correspondingto 2% to 3% w/v polymer, Table I). Figure 1 shows exam-ples of yeast colonies after 72 hours of growth on a hardagar gel plate, compared to growth on liquid substratesfor three different viscosities.At the highest viscosity studied, η = 600 ±
90 Pa · s,the yeast cells formed a single, compact, circular colonywhich expands radially over time (see supplementalmovie 1). However, unlike colonies on solid media wheregenetic drift dominates very close to the original frontierof the inoculation [7], colonies on the substrate had astretched central region with genetic diversity (two col-ors were mixed together); demixing only occurred at amuch larger colony radius, as displayed in Figs 1(a) and1(b) where the size of the initial inoculum is shown as ablack dashed circle. Genetic domain walls with neutralstrains impinge at right angles to a colony’s front and aredriven by interfacial undulations [7]. Yeast cells grown onthe viscous liquid presented much rougher colony frontsthan on hard agar plates, leading to more irregular do-main walls after the onset of genetic demixing. As theviscosity decreased to η ≈
450 Pa · s, the initially circu-lar colony formed numerous smaller microbial assembliesat its periphery on the media’s surface. The front of theoriginally circular colony became unstable and finger-like structures formed within the first 24h of growth; a largefingering colony spanning an entire Petri dish after 84hours of growth can be seen in Figure 1c) and supple-mental movie 2; a high-magnification picture of a fingeris shown in the bottom panel of Fig. 1c). These fingersform after demixing has occurred, typically leading tomonoclonal aggregates that grow and break up into smallclusters, somewhat reminiscent of a Plateau-Rayleigh in-stability [26, 27]. However, our system is complicated byactive cell divisions and a colony-generated radial veloc-ity field (see Sec. III). Below η = 300 ±
45 Pa · s, the initialcolony fractured into irregular pieces within the first 12hours of expansion, behaving as if they had a shear mod-ulus on our experimental time scales, and formed highlyfragmented colonies as seen in Figure 1d) and supple-mentary movie 3. Colonies in this regime break apartbefore genetic demixing occurred, resulting in geneticallydiverse growing fragments. The regularly interspersedfragments repel each other as they continue to grow, sug-gesting the existence of an underlying repelling flow. Atthe lowest studied viscosity, η = 54 ± · s, these clustersof yeast cells propelled themselves across an entire Petridish within 36 hours, dispersing more than one order ofmagnitude faster than the same yeast strains growing on2% hard agar plates (see Fig. 2 for the radial growth ofour strains on agar and liquid substrates over time). FIG. 2. a) Azimuthally averaged yeast colony radius R ( t )during the first 24h of growth on hard agar, blue circles, andon a liquid substrate with viscosity η = 600 ±
90 Pa · s, greensquares. b) The corresponding colony front velocity extractedfrom R ( t ), the colony exhibits two growth regimes on theliquid substrate: a superlinear regime for < t ∗ and a slowlydecaying phase for t > t ∗ . We found that the colony frontvelocity approaches v ( t ) = 0 . ± .
05 mm/day at long times( t (cid:29) t ∗ ) which is less than the velocity of yeast coloniesgrowing on 2% hard agar plates. c) Consecutive front spatialpositions at equal 40 min intervals during the first 24h ofgrowth on liquid substrate with the same viscosity as in a)and b), overlayed on top of a fluorescent image, top, and onbrightfield image of the colony, bottom. Note that geneticdemixing begins at the edge of the colony after the front hasslowed down. The scale bar corresponds to 1 mm. III. COLONY-GENERATED FLOW
In this Section, we focus, for simplicity, on the high vis-cosity regime 450 (cid:46) η (cid:46)
600 Pa · s where yeast cells forma single, approximately circular colony to investigate thecoupling between its growth and the three-dimensionalfluid flows generated in its vicinity. We imaged yeastcolonies growing during the first 48 hours after inocu-lation and extracted in parallel the fluid velocity nearthe substrate’s surface with particle image velocimetry(PIV). The fluid was seeded with a dilute concentrationof 10 − µ m fluorescent, neutrally buoyant polyethylenebeads, and horizontal slices of the flow were followed atthe desired height by varying the focal plane at which thebeads motion was tracked; see more details in AppendixA. Figure 2 displays the expanding colony average radius R ( t ), velocity v ( t ) and two-dimensional front profile overtime extracted from brightfield images.In contrast to yeast cells growing on hard agar plateswhich expand with approximately constant radial veloc-ity [7, 9, 10, 12], two distinct growth regimes separatedby a characteristic time t ∗ ≈
600 min can be identified on liquid substrates. At early times for t < t ∗ , the colony ra-dius expands superlinearly with time and reaches a maxi-mum horizontal growth velocity of v (cid:39) . ± . / day,while for t > t ∗ the expansion rate gradually slows downto v (cid:39) . ± .
05 mm / day over the rest of the exper-iment as shown in Fig. 2b). This first, approximatelyexponential, growth regime when t < t ∗ suggests thatcells dividing throughout the entire colony contribute toits surface area expansion, in contrast to growth on hardagar where only cells dividing near the front of the colonycontribute to its expansion [7]. A comparison of the ex-pansion rate of the colony with the spatial distributionof the strains reveals that genetically demixed sectorsappear only after the front propagation slowed down to v (cid:46) / day, as shown in Figure 2c), when only thoseregions exhibiting demixing at the edge of the colony aregrowing (see supplemental movie 4).PIV measurements carried out in the same experimentnear the surface of the fluid revealed an outward radialflow centered around the colony which began soon afterthe first cell divisions occurred; two-dimensional snap-shots of the velocity field are displayed in Figs. 3a),3b) and 3c) for t < t ∗ , t > t ∗ and t (cid:29) t ∗ respectively,while Figures 3d), e) and f) display the evolution of theazimuthal average of the velocity field u r ( r, t ) ≡ u ( r, t )over time. The flow is radially symmetric, reflecting thecircular colony shape at high viscosity, and its overallmagnitude increases within 24 hours after inoculation.Two distinct regimes can be identified. At early times,for t < t ∗ , the radial velocity profile exhibits a maximumnear the edge of the growing colony, whose value increasesin time, peaking at u = 6 ± . / day for t (cid:39)
560 minafter inoculation, and rapidly decreases away from thecolony. The similar values and variation exhibited bythe colony front propagation velocity v ( t ) for t < t ∗ , anddisplayed on Fig. 2(b), suggest that the fluid is radiallypushed outwards by the exponentially expanding colonyduring this time period.However, as the expansion slows down after t ∗ , a sec-ondary peak with a smaller amplitude can be observed inFig. 3d) and e). Within 48 hours it approaches a time-independent velocity u = 4 ± . / day, shown in Figs.3c), at about 1.5 colony radii away from the colony centerdespite the fact that the colony expansion velocity hadslowed to v ( t ) (cid:46) . ± .
05 mm / day. These observationssuggest that the expanding edge of the colony pushingthe surrounding fluid is not the unique origin of the ob-served flow and another mechanism is generating the flowin the surrounding media for t (cid:29) t ∗ , an idea we pursuein the next Section. IV. BAROCLINIC INSTABILITY
Plates filled with viscous media and monitored over24 hours under conditions identical to our experimentsshowed no evidence of flow in the absence of growingyeast cells, suggesting that the colony metabolism is re-
FIG. 3. Experimental flow field at the viscous substrate’s surface over the first 48 hours of a compact yeast colony growthfor a substrate viscosity of η = 600 ±
90 Pa · s. The central gray region delineated by red dashed lines indicates the growingcolony’s radius positions masking the fluorescent beads; we could not directly measure the velocity below the colony with thisexperimental setup. The velocity field in the vicinity of the colony was averaged over 3h for t < t ∗ (a), t > t ∗ (b), and t (cid:29) t ∗ (c). The colormap represents the flow velocity amplitude. The azimuthal average of the velocity radial profile is plotted every10 min for t < t ∗ (d), and every 20 min for t > t ∗ (e) and t (cid:29) t ∗ (f). Lighter lines correspond to earlier times. sponsible for the flow observed at t > t ∗ . A wide vari-ety of microbial organisms exploit Marangoni flows [28]to facilitate their horizontal displacement across liquidinterfaces by locally reducing the surface tension [29–31]. Yeast cells secrete a wide variety of molecules intheir vicinity, including ethanol and pheromones, whichcould potentially lower the substrate surface tension inthe colony surrounding. Surfactant-releasing particles,such as camphor boats, can lead to the formation ofmutually-repelling assemblies [32] similar, for example,to the fragmented yeast aggregates we observe under theexperimental conditions shown in Fig. 1d) and 12. Onthe other hand, the yeast cell metabolism could also gen-erate large enough gradients in the surrounding fluid’stemperature or solute concentration, to produce local dif-ferences in density and drive buoyant flows in the pres-ence of a gravitational field [33]. However, as shown inthe work of Benoit et al. [34], temperature gradients canbe ruled out because heat diffuses over 200 times fasterthan small-molecule solutes (such as glucose) in water,minimizing resulting density gradients, and because thecoefficient of thermal expansion is so much smaller thanthe coefficient of solute expansion; large temperature dif-ferences (several degrees Celsius) would be required tocreate the same density difference as a small change insolute concentration (see Appendix C for details).In order to discriminate between these different sources FIG. 4. a) Experimental setup for a yeast colony anchored onthe side wall of a sealed chamber filled with the viscous liquid;no liquid-air interfaces were present, removing the possibilityof Marangoni flows. Gravity points downward, and the fluidwas seeded with fluorescent PIV beads to track fluid motion.b) Fluid flow streamlines over the yeast colony (the dark cir-cular patch) during a time interval of ∆ t ≈ of flow, we conducted a series of experiments where weanchored the colonies on a thin layer of agar to the top,bottom, and sides of sealed chambers filled with our vis-cous media (see Fig. C.1 for details). We found thatcolonies created fluid flows similar in magnitude to ex-periments when the air-liquid interface was present, andregardless of their position in the chamber (even whenplaced at the top of a sealed chamber). The inducedfluid flows always opposed the direction of gravity, suchas the one shown in Fig. 4a), where a colony attached toa vertical wall entirely immersed in the liquid media cre-ated an upwards flow over its surface; one large vortex oneach side of the colony is partially visible in Fig. 4b). Al-though these experiments did not rule out the possibilitythat surface tension gradients affect the flow when a freeinterface is present, they revealed that buoyant forces areprimarily driving the observed flows.The flow was systematically opposing the direction ofgravity regardless of the position of the colony in thesealed chambers, suggesting that the cells’ metabolismaltered the density of the substrate by depleting nutrientsin the surrounding fluid, for instance by taking biomassfrom the solute to create progeny or by converting densersolute molecules into lighter ones (e.g. fermentation con-verts glucose to ethanol and carbon dioxide which areboth less dense than glucose in water). In fact, simi-lar behavior has been observed from E. coli growing insealed chambers filled with liquid media [34]. Measur-ing the initial and final density of the medium after ayeast culture grew to saturation in YPD showed a de-crease in density ∆ ρ = − . ± . / mL, wherethe ± corresponds to the range of density differences wemeasured (see Appendix A for additional details), con-firming that proliferating yeast cells reduce the densityof the surrounding media.However, in contrast to microbes growing at the bot-tom of liquid-filled sealed container that can induce aclassical Rayleigh-Taylor instability [34–36], where theless dense fluid near the colony rises, the cells in our ex-periments grew on the surface of a liquid-air interfaceand could not generate flow with this particular insta-bility. Instead, the yeast produces a localized pocket ofless dense fluid on top of a more dense fluid. In this con-figuration, the resulting density contours’ misalignmentwith the hydrostatic pressure horizontal isobars leads toa thresholdless baroclinic instability. This type of in-stability, common in stratified fluids, generates vorticityand can be observed in atmospheric and oceanic flows[33, 36, 37].The origin of the instability can be understood startingwith the Navier-Stokes equations for the substrate fluid: ∂ u ∂t + ( u · ∇ ) u = − ρ ∇ p + ν ∇ u + g , (1)where u is the fluid velocity, ρ the fluid density, p thepressure, ν = η/ρ the kinematic viscosity of the liquidmedium, and g = − g ˆz the gravitational force. Upontaking the curl of the fluid velocity u we obtain the vor- ticity ω = ∇ × u and find: ∂ ω ∂t + ( u · ∇ ) ω = ( ω · ∇ ) u + 1 ρ ( ∇ ρ × ∇ p ) + ν ∇ ω . (2)In the limit of small flow velocities, second order terms in ω and u (vorticity advection and vortex stretching) canbe neglected, and Eq. (2) simplifies to: ∂ ω ∂t ≈ ρ ( ∇ ρ × ∇ p ) + ν ∇ ω . (3)The viscous term ν ∇ ω simply redistributes the vorticityin the bulk fluid. However, the term ρ ( ∇ ρ × ∇ p ), oftencalled the “baroclinicity” [37], generates vorticity when-ever the contours of constant density ρ and pressure p cross at a finite angle. V. HYDRODYNAMIC SIMULATIONSA. Origin of the Baroclinic Instability
To better understand how yeast colonies living at a liq-uid interface can trigger a baroclinic instability, we firstassume a fluid at rest and numerically investigate howbaroclinicity is created as the cells deplete the surround-ing nutrient field by examining the resulting density andpressure contours. We assume the fluid has a density ρ which depends on the local concentration field c ( r , t ) of adiffusing nutrient solute such as glucose. The solute con-centration is depleted near the metabolizing yeast cellssuch that the mass density of the fluid, given by ρ ( r , t ) = ρ + δρ ( r , t ) = ρ [1 + βc ( r , t )] , (4)locally decreases, where ρ is the fluid density withoutnutrient solute, β = ρ (cid:16) ∂ρ∂c (cid:17) is the solute expansioncoefficient, and δρ ( r , t ) = ρ βc ( r , t ) gives the local in-crease in density due to the presence of nutrients [34].Let c be the initial reference nutrient concentration be-fore any metabolic depletion occurs, such that, close tothe metabolizing colony, there is a reduction in ρ ( r , t )and c ( r , t ) < c . In the absence of a flow, the momentumequation (1) simplifies to a hydrostatic pressure balancecoupled to nutrient diffusion in the substrate fluid andbecomes: − ∇ p + ρ g = 0 (5) ∂c∂t = D ∇ c, (6)where D is the diffusion constant of the nutrient solutemolecules.We account for the colony nutrient absorption by im-posing a nutrient mass flux normal to the colony’s sur-face j col = ac ˆn , where a is the mass flux rate into thecolony per unit nutrient concentration and ˆn is the unit TABLE II. Model parameters and their experimentally measured values, where appropriate. For additional details, see theAppendix D. Unless otherwise indicated, the error bars correspond to the standard deviation.
Parameter Value Units Description ν − / s Kinematic viscosity; varies with polymer concentration D . ± . × − cm / s Diffusion coefficient of small nutrient molecules ρ . ± .
005 g / mL Density of the viscous substrate with nutrients βc . ± .
001 None Product of the expansion coefficient β and c ac ± / ( µ m h) Product of the mass flux into the yeast colony a and c H −
10 mm Fluid height in the Petri dish ( h ≈ r petri ± | g | .
81 m / s Gravitational acceleration R − (cid:96) ≡ ρ βD/a . ± . FIG. 5. Baroclinic vorticity generation rate ∂ ω /∂t ≈ ρ ( ∇ ρ × ∇ p ) normal to a radial cross-section before flow isinitiated by a yeast colony fixed at the surface of the vis-cous fluid in a radially symmetric Petri dish. The colonyposition is indicated by the thick orange line, the pressureisobars in blue, and the density contours in green. The iso-bars are near-horizontal due to small density differences orig-inating from nutrient depletion (in this simulation, ∆ ρ max ∼− .
003 g / mL). Whenever the pressure and density contourscross at an angle, vorticity is generated via the baroclinic termin Eq. (2). normal vector to the interface, such that larger nutri-ent concentrations lead to a larger nutrient absorptionrate [38]. In contrast, no-nutrient-flux boundary condi-tions are applied elsewhere, on the walls of the domainaway from the colony, D ∇ c · ˆn = 0. The mass flux dueto transport and diffusion in the bulk fluid is given by j fluid = ρ β ( u c − D ∇ c ). We assume that u = 0 for now,and upon applying continuity on the solute flux acrossthe colony boundary ( j colony = j fluid ) (cid:12)(cid:12) colony , the bound-ary condition can be rewritten as:( ∇ c · ˆn ) (cid:12)(cid:12)(cid:12)(cid:12) colony = c(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) colony (7)where (cid:96) = ρ βD/a = 1 . ± . (cid:96) is different thanthe nutrient screening length inside the yeast colony [38],as discussed in Appendix D. Note that our yeast cells donot absorb the concentration field fast enough to warrantsetting c = 0 at the interface between the colony and thefluid substrate as indicated by the dimensionless numbersdiscussed in Appendix E.The actual colony expansion is neglected for simplic-ity, so we consider a colony of fixed radius R at the sur-face of the viscous fluid, in a radially symmetric Petridish as shown in Figure 5; the yeast colony is repre-sented by the thick orange line. We use OpenFOAM5.0 [39] to simulate equations (5)-(7) using the program diffusionPressureFoam [40] and the measured param-eters from Table II; additional details about the numeri-cal scheme appear in Appendix F. Figure 5 displays theresulting density contours and isobars. Once the cellsstart absorbing nutrient mass from the fluid, a curveddensity gradient that conforms to the finite size of thecolony is created in its vicinity; supplementary Fig. F.1shows an example of a corresponding simulated concen-tration field. The pressure contours, on the other hand,remain nearly horizontal over the entire domain as thedensity differences due to solute depletion are so small.The finite crossing angle of the pressure and density con-tours leads to vorticity generation via the baroclinic term ρ ( ∇ ρ × ∇ p ) in Eq. (3) below the edge of the yeastcolony, where the gradient of density is large and nearlyperpendicular to the pressure gradient. As long as theyeast cells deplete the surrounding nutrients, the createddensity difference will generate vorticity via this thresh-oldless baroclinic instability. B. Comparison with Experiment
We now determine the flow produced by the baroclinicinstability in the liquid substrate by simulating the hy-drodynamic flow equations, and compare our simulationswith the experimental flow velocities. The diffusing so-
FIG. 6. a) Snapshot of the simulated flow field below theyeast colony (brown bar) after flow is initiated, for t (cid:29) t ∗ .The simulated flow field qualitatively matches our experi-ments with a vortex ring produced around the colony. b)Azimuthal average of the numerical flow field using the mea-sured parameters in Table II plotted every 12 hours at thesubstrate fluid surface. Black circles, experimental flow radialprofile measured for similar flow parameters after 24 hours ofgrowth and an initial η = 600 ±
90 Pa · s. c) Simulated andexperimental peak radial velocity, determined from PIV mea-surements, as a function of fluid height below the colony. Theblue line with circles corresponds to the simulated values us-ing the parameters in Table II, the black shaded region isthe standard deviation of the simulated points, and the blackcircles corresponds to experimental data. lute field is now coupled with the incompressible Navier-Stokes equations, and we now must solve the full set ofequations, ∂c∂t + u · ∇ c = D ∇ c, (8) ∂ u ∂t + ( u · ∇ ) u = − ρ ∇ p + ν ∇ u + g , (9) ∇ · u = 0 . (10)In the limit of small local density variations, δρ ( r ) /ρ (cid:28)
1, we can apply the Boussinesq approximation [33, 34],such that equation (1) becomes: ∂ u ∂t + ( u · ∇ ) u = − ρ ∇ p (cid:48) + ν ∇ u + βc ( r , t ) g , (11)where the pressure p (cid:48) = p − ρ gz is the pressure measuredrelative to the hydrostatic pressure at constant density ρ . We now introduce rescaled variables for space ˜ r = r /H , time ˜ t = tD/H , velocity ˜ u = u H/D , pressure˜ p = pH /Dη and nutrient concentration relative to itsvalue c in the absence of the colony ˜ c = c/c , where H is the depth of the substrate fluid. In the creeping flowregime, appropriate to our experiments, inertial terms on the left-hand side of the equation (11) can be neglected,and the governing equations then become (see AppendixE for details): ∂c∂t + u · ∇ c = ∇ c, (12) ∇ u − ∇ p − Ra c ˆz = , (13) ∇ · u = 0 , (14)where the tildes have been dropped for convenience and r , t , u , p and c now denote nondimensional variables.In Eq. (13), the Rayleigh number Ra = h βc g/Dν ,compares the buoyant forces to the stabilizing effect ofthe viscous forces.We can consider again a colony with fixed radius R ,provided the characteristic eddy turnover time τ eddy ∼ τ growth = R ( t ) / ( dR/dt ) ∼
10 days for t (cid:29) t ∗ . The colony expan-sion rate is slower than the induced flow velocity, andstarts behaving like a solid in this regime, so we applya no-slip boundary condition just below the colony. Wealso apply a no-slip boundary condition to the walls ofthe Petri dish and a free boundary condition to the air-substrate interface such that there is no normal velocity, v z = 0, and negligible shear stress, ∂v r /∂z = 0. Weapply the same nutrient absorption boundary conditionbelow the yeast colony because the normal componentof the fluid velocity at the boundary with the colonyvanishes, and no-flux boundary conditions on both thewalls of the petri dish and the fluid surface to the dif-fusing nutrient field. We use OpenFOAM 5.0 [39] tosolve the governing Equations (12)-(14) with the bound-ary conditions given by Eq. (7), using the program stokesBuoyantSoluteFoam [40] (available on GitHub)with the experimentally measured parameters in TableII; see Appendix F for additional numerical details.The baroclinic effect leads to an intense vortex ringbeneath the outer edge of the colony, as revealed by thetransverse section shown in Fig. 6 a). The flow geometryand intensity on the surface of the fluid resemble theexperimental flow field shown in Fig. 3 around the colony.As shown in Figure 6b), the corresponding radial velocityprofile at the fluid interface is in good agreement withthe experimental profile, with a strong peak at about1.5 times the colony radius. Figure 6 c) compares themaximum radial velocity measured in the stationary flowregime, reached after 48 hours in the experiments, withsimulations as a function of the substrate fluid height H .Our minimal buoyant flow model tracks the experimentalpeak velocities, supporting the hypothesis of a buoyancy-driven flow produced by a baroclinic instability in ourexperiments. FIG. 7. Morphologies of yeast colonies growing on a liquid media substrate over time at a variety of viscosities. Quotedsubstrate viscosities are accurate to about 10% (see Table I). The figure shows merged brightfield and fluorescent images.White: transmitted brightfield, red: YFP strains, and cyan: mCherry strains. All images have the same scale and the scalebar at the lower right corresponds to 10 mm. The left column is an enlargement on the colonies after 12h of growth and itsscale bar corresponds to 5 mm.
VI. MODEL COUPLING GROWTH WITHDILATIONAL FLOW
In this Section, we investigate how substrate viscos-ity influences colony morphology and describe a simplephenomenological model for colony growth, expansion,and thinning in the spirit of the so-called lubrication ap-proximation [41]. Figure 7 displays five characteristiccolony morphologies over time growing on liquid mediafor the entire range of studied viscosities (Table I), from η = 54 ± · s to η = 600 ±
90 Pa · s. Our measurements offlow velocity shown in Figure 3 reveal that metabolicallydriven buoyant flows become apparent as early as 2 hoursafter inoculation, suggesting that yeast cells can depleteenough mass to induce a flow even at this initial stage ofgrowth. The first column to the left on Figure 7 shows anenlargement of the colonies 12h after inoculation. Theirshape already shows a strong dependence to the substrateviscosity, suggesting that the future morphology of thecolony is determined during the early growth.When viscosity decreases, the amplitude of the toroidal0flow field beneath the colony increases and eventually ap-plies enough force to alter the initial circular morphologyof the colony. For instance, experiments performed at η = 54 ± · s indicate that the flow velocity can reachmagnitudes up to 20 mm/day and apply non-negligiblestresses on the colony. Once the cell division rate fallsbehind the colony’s advancing front at t ≈ t ∗ , the bulk ofthe colony ceases to behave as a liquid with internal mo-tion due to cell division and begins to behave more likea viscoelastic material. Given the much faster fluid sub-strate velocities outside the colony relative to the colonyexpansion speed, the colony starts experiencing radialshear stresses imposed by the flow. One possible expla-nation for the especially intriguing colony morphologydisplaying multiple elongated fingers around the colonyedge, close to η = 450 ±
70 Pa · s, could be a mechanismsimilar to viscous fingering instabilities. Under these con-ditions, competition between relaxation forces, due tothe attractive interaction between cells and an outwardpulling force produced by the radial flow could drive aninstability resembling those that arise in rotating oil films[42], suitably modified to allow for colony growth and thediscreteness of the underlying cells. However, when theviscosity drops below η (cid:46) ±
45 Pa · s, the radial expan-sion imposed by the vortex ring under the colony startsto outcompete the colony expansion due to cell divisions,such that growth cannot accommodate the dilational flowduring the initial stage. This results in a rapid separationof the cells and holes start opening up within the centerof the colony.A complete understanding of the complex experimen-tal behaviors described here (exponential stretching priorto genetic demixing, a fingering instability with fingersthat break into droplet-like clusters and fragmentation;see Fig. 1) would require a detailed theory of the fluid dy-namics of the substrate fluid coupled to the visco-elasticbehavior of a colony of approximately 5-micron-sized cellswith both excluded volume and attractive interactions,all while cells are actively dividing, as well as interactingwith the substrate fluid during the range expansion. Wehope that the results described here will encourage suchtheoretical investigations, which might also need to ac-count for the discreteness of the cells in the colony andassess the impact of the fluid mechanics on the geneticdemixing observed in our experiments.Here, we propose, instead, a simple phenomenologicalmodel that provides insight into the exponential stretch-ing and colony thinning during the early stages of therange expansion when the colony maintains its circu-lar symmetry and behaves approximately like a two-dimensional liquid. In analogy with treatments of colonyexpansions on hard agar plates [43], we describe thedynamics of the colony height by a generalized Fisherpopulation dynamics equation [44] for the colony height h ( r , t ), namely: ∂h ( r , t ) ∂t + ∇ · [ h ( r , t ) v ( r )]= D h ∇ h ( r , t ) + µh ( r , t ) (cid:20) − h ( r , t ) h (cid:21) , (15)where v ( r ) is the advecting hydrodynamic flow velocitythat acts on the colony and µ is an effective colony verti-cal growth rate when its height is small. The quantity h is the steady state colony thickness in the absence of flowand spatial gradients of the height field, which we expectwill depend on quantities such as nutrient penetrationdepth inside the colony [38] and strength of, e.g. , theVan der Waals and gravitational forces that attract thecells to the liquid substrate. The parameter D h is a dif-fusion constant that promotes an approximately uniformcolony height – a similar term appears in, e.g., the hy-drodynamic equations that describe capillary wave-likeexcitations in thin helium films [45].One source of the radially outward flows we observenear the surface during the early stages of our range ex-pansions on liquid substrates is the outward pushing bythe growing quasi-two-dimensional yeast colony. To de-termine the form of this contribution to the substrateflow, we assume that, at least during the early stages ofthe expansion, the colony behaves like a two-dimensionalliquid where all the cells in the colony receive enough nu-trients to actively divide. We further assume that thetwo-dimensional colony viscosity can be neglected com-pared to the overdamped frictional coupling to the liquidsubstrate. We can then apply a simple hydrodynamicmodel, [46–49], which leads to: ∇ p d = − γ ∇ · v = − γα (16)where p d is an effective two-dimensional pressure fieldinside the colony [49]. Here, α arises from cell divi-sions that, as shown below, will give rise to a horizontalradial velocity field within the quasi-two-dimensional liq-uid colony averaged over the thickness of the colony. Thequantity γ is a frictional coefficient due to the motion ofthe colony relative to the liquid substrate. If the liquidsubstrate has a dynamical viscosity η s and depth H , inthe limit of colony radius larger than H , we then expect γ ≈ η s /hH [50], where h is the thickness of the colony.We can now exploit an electrostatic analogy, such thatthe two-dimensional pressure field inside the colony sat-isfies a Poisson equation, and where the height-averagedgrowth rate α determines a 2d “charge density”. Thecolony velocity field (like the 2d electric field inside acharged disk in two dimensions) that solves Eq. (16) hasthe radially symmetric form: v ( x, y ) = 12 α r ˆr , r = (cid:112) x + y . (17)When coupled to an underlying viscous substrate fluid,this dilational flow field within the colony will act to in-1 FIG. 8. Magnification of demixing patterns formed by twodifferent yeast strains growing on a substrate with a viscosity η = 600 ±
90 Pa · s at different time points. The first threeimages have the same scale represented by the white bar onthe upper right of the images; the scale bar corresponds to100 µ m. The final picture to the right shows the same featureat the larger colony scale; the scale bar now corresponds to500 µ m.FIG. 9. a) Snapshot of the simulated flow field below theyeast colony (brown bar) after flow is initiated when t (cid:29) t ∗ .The simulated flow field is very similar to the one displayedin Fig. 6a) except with free boundary condition beneath theyeast colony. b) Azimuthal average of the numerical flowfield using the measured parameters in Table II plotted every12 hours at the substrate fluid surface with free boundarycondition beneath the yeast colony. duce flows in the underlying liquid, in qualitative agree-ment with our PIV measurements near the surface shownin Fig. 3(a). The development of expanding genetic pat-terns during the approximately exponential growth for t < t ∗ is shown in Fig. 8. The figure highlights one par-ticular feature inside the black dashed square which onlyundergoes a dilatation when expanding over time, as ifthe genetic patterns were painted on the surface of aninflating balloon, which is also consistent with Eq. (17).Estimates of this dilational expansion velocity for t < t ∗ gives values of the order of 4 − t > t ∗ , and becomesdominant at later growth stages for high substrate vis-cosity, and at increasingly earlier times with decreasingsubstrate viscosity. Triggered by the metabolic uptake ofnutrients, this additional flow is potentially responsiblefor the fingering and fragmentation instabilities observedwhen the substrate viscosity decreases and flow ampli-tude becomes larger. If we express the flow produced bycell divisions occurring throughout a circular colony inthe form v ( r ) = α r ˆ r , we expect then another contri-bution to this velocity of the form v ( r ) = α r ˆ r oncethe baroclinic instability establishes a vortex ring in thesubstrate fluid beneath the colony with a size of order 1.5times the colony radius (Figure 3). A simple model of avortex ring submerged in substrate fluid with an imagevortex ring with opposite circulation above the colonysatisfies the requisite boundary conditions beneath thecolony (the resulting velocity field resembles the mag-netic field from a pair of anti-Helmholtz coils). Thisansatz leads to a radial velocity field at the colony whichvanishes linearly in r for small r , and falls off roughlylike 1 /r for r large compared to the colony radius. Tocheck these ideas for the substrate-induced velocity fieldacting on the colony, we have repeated the simulationsof Sec.V.B under identical conditions with, however, freeinstead of no-slip boundary conditions at the interfacebetween the colony and the substrate fluid. We thusassume that active cell divisions throughout a circularcolony cause it to behave like a two-dimensional liquid,with a contribution to the in-plane colony velocity fieldimposed directly by the substrate fluid. The resultingflow snapshot for the substrate fluid velocity field be-low the colony, displayed in Fig. 9a), is qualitativelysimilar to Fig. 6a) indicating a submerged vortex ring.Now, however, the absence of a no-slip boundary condi-tion leads to a velocity field right at the colony-substrateinterface. The azimuthal average of our numerical flowfield is shown in Fig. 9b), again at 12 hour time inter-vals. The results are similar to Fig. 6b), except thatthey clearly show a linear behavior of the velocity fieldunderneath the colony, consistent with the ideas in thepreceding paragraph.With these motivations, it seems reasonable to assumethat the advecting velocity field in Eq. (15) takes theform: v ( r ) = 12 αr ˆ r (18)where α is an effective dilational flow parameter thatincludes the effect of the baroclinic instability as wellpushing generated by dividing cells within the colony. Weexpect α to increase with decreasing substrate viscosity,reflecting a stronger baroclinic instability.2With these assumptions, Eq. (15) takes the form: ∂h ( r , t ) ∂t + 12 αr ˆr · ∇ h ( r , t )= D h ∇ h ( r , t ) + ( µ − α ) h ( r , t ) − µh ( r , t ) h . (19)In regions where the colony height is spatially uniform,we have for the height h ( t ), ∂h ( r ,t ) ∂t = ( µ − α ) h ( r , t ) − µh ( r , t ) /h , and thus: h ( t ) = h (0) e ( µ − α ) t µh (0) /h µ − α (cid:0) e ( µ − α ) t − (cid:1) . (20)We can now look for a radially symmetric solution withan interpolating step-like function Θ( x ) = 1 , x (cid:28) , Θ( x ) = 0 , x (cid:29) h ( r, t ) = h ( t )Θ [( R ( t ) − r ) /δ ] , (21)where R ( t ) defines a colony radius smeared out over aninterfacial width δ . It is easy to see from Eq.(19) that,provided r (cid:29) δ and r (cid:29) (cid:112) D h /α , the colony radiusgrows exponentially in time: R ( t ) = R (0) e αt . (22)Figure 10 shows the numerical solution of equation(15), assuming radial symmetry for the colony height h ( r , t ) = h ( r, t ), at different values of α/µ using theprogram forcedThinFilmFoam [51]; see Appendix F formore details. In the absence of an advecting velocityfield, α = 0 in Fig. 10a), Eq. (19) has the usual Fisherwave solution of an outwardly expanding colony frontcircumference with constant velocity v F = 2 √ D h µ when-ever the colony radius is much greater than the interfacialwidth l F = (cid:112) D h /µ [44]. However, for nonzero α suchthat µ − α > exponentially fast advance of the wave: if the shoulder of the popula-tion wave in this case occurs at x when t = 0, then theposition of the shoulder at time t is as x exp [(1 / αt ],with a width δ of order (cid:112) D h / ( µ − α ), consistent withour early time observations in Fig. 2. In this regime, thecolony advances but is thinned down to a height givenby the long time limit of Eq(20): h ∗ = h (cid:18) − αµ (cid:19) . (23)Thus, with increasing α , the flow becomes strongerand the exponential advance of the colony is fasterbut the colony becomes progressively thinner. Inter-estingly, when α = µ , equation (20) becomes h ( t ) = h (0) / (cid:16) h (0) h µt (cid:17) , and approaches zero as h ( t ) ≈ h / ( µt ) for large times. In fact, when time is substi-tuted with R using equation (22), we find that at largetimes the height at the midpoint of the shoulder behaves FIG. 10. Numerical solution of Eq. (15) for h ( r , t ) /h at dif-ferent values of µ − α and equal time intervals. The radialcoordinate r is measured in units of (cid:112) D h /µ , the width of theFisher wave in the absence of a dilational flow. The coloreddots correspond to the prediction of h as a function of timefrom equation (20); it is clear that there is good agreement be-tween the theoretical prediction and simulation. a) for α = 0the colony height increases to h/h = 1 and the front prop-agates radially with a constant velocity v F = 2 √ D h µ with µ = 1. b) When α < µ the colony front propagation veloc-ity increases exponentially with time and the colony heightdecreases to h ∗ = h (1 − α/µ ) < h . c) When α = µ thedilational flow is strong enough to decrease the colony heightbelow one cell size and h ( t ) goes to zero logarithmically withradius. d) For α > µ , the colony thins exponentially fast,potentially signalling that holes open during its early expo-nential growth; these holes may be responsible for the highlyfragmented colonies at later times. according to h s [ R ( t )] ∼ h ln (cid:104) R ( t ) R (0) (cid:105) (24)such that h decreases logarithmically with radius, leadingto the formation of a wide plateau due to the extremellyslow decay of h over time, as can be seen in Figure 10c).For sufficiently strong flows such that α > µ , there is a“thinning catastrophe”, see Figure 10d), such that thecolony population collapses at long times. In this limit,of course, the discrete nature of the cells making up thecolony, neglected in Eqs (15) and (19), becomes impor-tant.Finally, we check the qualitative agreement betweenthis simplified model and the experiments by determiningthe colony expansion rate during the superlinear growthregime ( t < t ∗ ) as a function of substrate viscosity. A de-tailed measurement of the radial expansion coefficient’sviscosity-dependence, α = α ( η ), would provide a morequantitative test. Here, we explore this idea further by3 FIG. 11. a) Colony radius as function of time during thefirst day of growth for two different viscosities; blue circles, η = 600 ±
90 Pa · s, green squares: η = 300 ±
45 Pa · s andblack line: exponential fit realized for t < t ∗ . The shorttime behavior is consistent with an exponential growth of thecolony radius in both cases, but the growth is much faster atlower viscosity. b) same as in a) with experiments realized forcolonies growing on a 1 mm thin substrate liquid film on thetop of a nutrient rich gel layer. We found that the exponentialfit realized for t < t ∗ exhibit a similar expansion rate for bothviscosities. reproducing the same experiments as the ones describedin Sec. II for two different substrate viscosities: by re-lating the above model predictions to the early colonymorphologies, one may be able to estimate a critical vis-cosity below which the flow becomes strong enough tocause a “thinning catastrophe”. As can be seen in thefirst column of Fig. 7, for η (cid:46) ±
45 Pa · s, holes startopening up in the center of the colonies during the earlyexpansion, indicating that the flow dilation rate is largerthan the colony height growth rate and corresponds tothe height profile regime described by α > µ . Assuming η (cid:39) ±
45 Pa · s is the highest viscosity at which we canobserve a catastrophic thinning of the colony height inthe early growth regime, our model suggests that α ≈ µ in this experiment.Figure 11a) displays the colony radius R ( t ) over timegrowing on two different liquid substrates with viscosity η = 600 ±
90 Pa · s and η = 300 ±
45 Pa · s. The colonyexpansion rate described by Eq. (22) can then be esti-mated from an exponential fit of R ( t ) for t < t ∗ , andgives α = 4 . ± . − for the higher viscosity, and alarger rate α = 7 . ± . − for the lower viscosity.Assuming that the critical value of α , for which we have α ≈ µ , is close to the colony expansion rate measuredfor η = 300 ±
45 Pa · s, we estimate µ ≈ . ± . − ,which gives a characteristic division time of τ ≈
140 minin the vertical direction of the colony, in approximateagreement with yeast colony growth rates on hard agarplates [49].The dilational coefficient α in Eq. (18) is presumablya combination of the α and α contributions discussedabove. Although it is difficult to determine the value of α for t < t ∗ , as the metabolic velocity field is weaker atshort times, we were able to isolate the constant α , re-lated to the flow contribution coming from cell-divisionsat a liquid interface but without the enhanced dilational velocity due to the metabolic flow. To do this, the sameexperiments were repeated on a much thinner 1 mm thicklayer of liquid substrate deposited on the top of a regular,nutrient rich gel plate. This geometry allowed us to dampout the baroclinic instability in the thin liquid layer, andrevealed a nearly identical expansion rate this time, with α = 4 . ± . − for both η = 300 ±
45 Pa · s and η = 600 ±
90 Pa · s, suggesting that α is independent ofsubstrate viscosity for 300 ≤ η ≤
600 Pa · s. Note that themeasured value of α is similar to the expansion rate α wefound for thicker substrates at higher viscosity, while itis significantly less than the measured α for the substratewith lower viscosity. This suggest that the metabolic flowdoesn’t contribute significantly to the colony expansionfor η = 600 ±
90 Pa · s, while it considerably increasesthe colony dilation rate for η = 300 ±
45 Pa · s even atearly times for t < t ∗ . Although further experimentswould be required to fully map out the colony dynamicsas a function of substrate thickness and viscosity, our ex-perimental results suggest a qualitative agreement withequations (15) and (19). VII. DISCUSSION
We investigated the growth of yeast range expansionson the surface of an extremely viscous nutrient-rich liq-uid substrate. Capillary forces keep our yeast cells at thesurface for many days, and the extreme viscosity of thefluid insures that cell clumps that break the surface ofthe air-liquid interface settle slowly. The large viscosityalso prevented thermal convection from mixing the me-dia. Previous experiments of range expansions on solidagar media featured a thin layer of proliferating cells atthe frontier of radially expanding circular colonies [7]. Wefound that colonies grown on a liquid medium, where thesubstrate can flow and friction between the cells and themedium is much lower, behave very differently.In the early stages of these range expansions, for t < t ∗ ,colony radii grew in a superlinear, approximately expo-nential fashion and the growth was dominated by activecell divisions throughout the colony. However, for t > t ∗ ,yeast metabolism generated fluid flows in the surround-ing media many times larger than their basal expansionvelocity. This flow dramatically altered the colony mor-phology, depending on the surrounding substrate viscos-ity.Compact circular colonies grew for η ≈ ±
90 Pa · s(3.0% polymer), the largest viscosity we tested, featur-ing a regime of roughly exponential stretching and thin-ning where strains remained mixed together, and latera period of slow, linear expansion where strains geneti-cally demixed and resembled expansions on agar plates[7] with more wiggly domain walls. The expansion likelyslowed because of nutrient depletion.As the viscosity of the medium decreased, hydrody-namic forces acting on the colony were eventually suffi-cient to produce fingering and fragmentation instabilities4and led to two additional morphologies. At intermediateviscosities between η = 450 ±
70 Pa · s and η = 300 ± · s (2.8% – 2.6% polymer), compact colonies developed“fingers”, an instability that allowed thin streams of cellsto be ripped away from colonies resembling dendriticcrystal growth in the presence of a solute-driven buoyantflow [52] or fingering instabilities in spinning drops [42]and Marangoni flow [53]. We attribute this liquid-like be-havior at the colony perimeter to the lubricating effect ofactive cell divisions. The filaments then broke into clus-ters via a process reminiscent of capillary forces in theRaleigh-Plateau instability [54, 55], with, however, differ-ences due to actively dividing, discrete cells. The compe-tition between the self-induced flow, diffusion of nutrientsand the attractive forces between the cells might triggera selection for a characteristic finger width.For viscosities lower than η = 300 ±
45 Pa · s, growingcolonies exhibited solid-like behavior in the interior; theyfractured into many irregularly shaped repelling island-like fragments. These repelling fragments could colo-nize an entire Petri dish within 36 hours, presumablybecause each fragment metabolically generated its ownsubmerged vortex ring. This conjecture about a vortexring under each solid-like colony fragment is consistentwith the image shown in Fig. 12, taken under experimen-tal conditions similar to Fig. 1d), but with a shallowersubstrate fluid. As opposed to the nearly monoclonalfingers separating from the initial colony after demixing,island-like fragments tended to be genetically diverse asthe entire colony broke apart.Our experiments and simulations provide strong evi-dence that yeast metabolism creates fluid flow in the sur-rounding media via a baroclinic instability: yeast createda pocket of less dense fluid on top of a more dense onethat generated vorticity near the colony edge when theisobars and isoclines of the underlying fluid crossed eachother at an angle. Minimal buoyant fluid flow simula-tions calibrated to experiments with independently mea-sured parameters capture our experimentally observedflow fields. Interestingly, as discussed in AppendixD,these calibrations allowed us to measure the mass fluxrate into the yeast colony in rich nutrient conditions as ac = 5 ± / ( µ m hour); the authors are unaware ofother literature measuring this quantity. Furthermore,this mass flux rate is consistent with a nutrient screeninglength of about 50 µ m inside yeast colonies (AppendixD), consistent with that measured in prior work [38].Furthermore, colonies always generated fluid flowsagainst the direction of gravity, regardless of their po-sition in a sealed chamber, and we found that yeastcells grown to saturation in overnight culture decreasethe surrounding media’s density by ∆ ρ = − . ± . / mL. We believe that surface tension gradients(the Marangoni effect) played only a minor role in gen-erating the observed flows, because yeast attached to thesurface of a sealed chamber generated fluid flow com-parable in magnitude, and because the above argumentssuggest that buoyancy alone sufficiently explains the phe- FIG. 12. Low viscosity ( η = 300 ±
45 Pa · s) range expan-sion on a liquid substrate in the fragmentation regime. Thisimage was taken for t (cid:29) t ∗ in a single experiment under con-ditions similar to those in Fig.1d), except that the substratefluid height was H = 4mm instead of 7mm. The more iso-lated cell fragments clearly collect on the mid-planes separat-ing the larger “continents”, consistent with the down-wellingsassociated with a vortex ring underneath each continent, assuggested by the sketch on the top. The scale bar correspondsto 10 mm. nomenon. To the best of our knowledge, this unusualbaroclinic instability has not been previously investigatedin a biological context.The work described here suggests a number of intrigu-ing avenues for future work: for example, can othermicroorganisms growing on or near the surface of liq-uids generate buoyant flows similarly to our experiments?Preliminary experiments with immotile E. coli colonieshave exhibited similar flows when growing on the surfaceof liquid substrates with comparable viscosity, and havealso exhibited fascinating colony morphologies [20]. Itis intriguing to speculate that similar instabilities mightoccur at much higher Reynolds numbers in the oceans,beneath plankton blooms confined to, say, the first 50meters of depth. It would also be interesting to experi-mentally test if microbial colonies that generate buoyantflows have a selective advantage relative to those that donot. Induced fluid flows clearly allow more efficient redis-tribution of nutrients and provide a mechanism for themore rapid dispersal of colony fragments. Preliminarynumerical investigations when viscosity is lowered from5infinity (i.e. modeling hard agar substrates), increasingthe Rayleigh number from 0 to 10 in our Petri dishgeometry, increased the nutrient absorption rate of theyeast colony by a factor of about 1.5, suggesting thatcolonies generating stronger buoyant flows could indeedhave a selective advantage (see Appendix G for details).Although yeast colonies might develop fluid-mechanics-like instabilities reminiscent of classicalones in the presence of flow [27, 42, 53, 54], they differin two key ways: 1) Dividing cells cause growth overtime, stressing the need for further theoretical work tounderstand instabilities arising from the competitionbetween flow and growth; and 2) the discreteness ofthe dividing cells may play an important role nearthe “thinning catastrophe” discussed for a simpletheoretical model in section VI. The transition froman approximately exponential to a slower expansionrate, corresponding to the transition from liquid-like tosolid-like behavior of the yeast colony, could also benefitfrom a fluid-mechanical perspective to model the yeastfingering instability, assuming a liquid-like behavior dueto agitation by cell-divisions at the frontier.The origin of the quantitative differences between yeastcolony growth on the highest viscosity substrates and onhard agar plates, such as the more wiggly genetic domainboundaries has yet to be understood. Systematic inves-tigations of how colony morphology and genetic patternsvary with nutrient concentration (glucose) in addition toviscosity, similar to the pioneering work of Wakita et al.[43], would also be of interest. Furthermore, it is worthnoting that we modeled the rheology of the liquid sub-strate as a Newtonian fluid despite the shear-thinningproperties measured in the media at very large polymerconcentrations as discussed in Appendix B; future workshould investigate how more pronounced non-Newtonianeffects could impact the fluid flows induced by the yeast,in the context of microbial populations growing in mucusfor instance [56].Lastly, the fluid used in this paper is viscous enoughthat it can be advected at a velocity as low as 1 mm / day[20], matching the expansion rates of E. coli and thebaker’s yeast
S. cerevisiae on agar [9, 11], over an entire10 cm Petri dish over many days of growth [20]. Theextreme viscosity of the fluid allows for the impositionof slow, controlled fluid flows at a macroscopic scale thatcan advect microbial colonies and provides an alternativeto working with microfluidic devices where complicationsarise when microbes stick to the walls of their enclosure[20, 57]. Using syringe pumps, one could impose well-defined flows on microbial colonies and systematically re-peat previous experiments with microbial range expan-sions on hard agar plates [9–15] on viscous liquid sub-strates like those studied here but with additional typesof advection. Investigating the evolutionary dynamics ofcolonies composed of complementary strains that secretepublic goods such as leucine and tryptophan [12] couldbe especially relevant because the secretions would betransported by the fluid flow. In conclusion, our results suggest that microbial rangeexpansions on the surface of a highly viscous fluid pro-vide a versatile laboratory system to explore the interplaybetween advection and spatial population genetics.
ACKNOWLEDGMENTS
We would like to thank all members of A.W. Mur-ray’s group for their indispensable and generous helpthroughout this project. Joanna Aizenberg’s lab kindlyallowing us to use their Kr¨uss tensiometer to measurethe surface tension of our fluid; we would especially liketo thank Daniel Daniel and Michael Kreder for theirtime and helpful input. We would also like to thankJennifer Lewis’s and Dave Weitz’s labs for allowing usto use their rheometers, and would like to particularlythank Sean Wei and Liangliang Qu for helping us op-timize our measurements. S.A. and B.T.W. would liketo thank Andrea Giometto for interesting discusions andhis useful suggestions. We would also like to acknowl-edge conversations with Michael P. Brenner. B.T.W.would like to thank Maxim Lavrentovich and Steven We-instein for their helpful comments and guidance. Workby B.T.W. was supported by the Department of EnergyOffice of Science Graduate Fellowship Program (DOESCGF), made possible in part by the American Recoveryand Reinvestment Act of 2009, administered by ORISE-ORAU under contract number DE-AC05-06OR23100, bythe US Department of Energy (DOE) under Grant No.DE-FG02-87ER40328. The Harvard MRSEC (DMR-1608501) helped to fund our usage of the Anton Paarrheometer. B.T.W., S.A., and D.R.N. benefitted fromthe National Science Foundation through grants DMR-1608501 and via the Harvard Materials Science and En-gineering Center through grant DMR-1435999. S.A. andB.T.W. also benefitted from the National Science Foun-dation through DMS 1406870.
Appendix A: MATERIALS AND METHODS1. Liquid substrate preparation
To produce our highly viscous medium, standard richgrowth medium for yeast (YPD), consisting of 1% Bac-toYeast extract, 2% BactoPeptone, and 2% anhydrousdextrose (glucose), was mixed in autoclaved water andfiltered into a sterile glass bottle using a Zapcap (MaineManufacturing item number 10443430) to remove con-taminants. We then systematically increased the sub-strate viscosity by adding 2-Hydroxyethyl cellulose, anextremely long-chain polymer with a viscosity-averagedmolecular weight of 1 . × (Sigma-Aldrich productnumber 434981), at concentrations ranging from 2.0%to 3.0% w/v into 300 mL aliquots of the media, as shownin Table I. We used a strong magnetic mixer (IKA RCTbasic magnetic stirrer) to rapidly stir the media with a6sterile magnetic bar until it became homogenously vis-cous over the course of three hours. We found that themodel of the magnetic mixer was important; the mixerneeded to be able to deliver enough torque to the stirbarso that it would continue spinning as the media becamevery viscous. Furthermore, if we used too much me-dia in the mixing flask (typically volumes greater than300 mL), the polymer would not mix evenly. The fi-nal mixture was sterilized to avoid contaminants broughtin from the polymer. Because the extreme viscosity ofour fluid prevented it from being filtered, we sterilized itby microwaving it for three minutes (with a “Panasonicmodel number NN-SN9735” microwave). In contrast tomicrowaving, sterilization via autoclaving produced in-consistent viscosities between replicates. We found thatit was essential to let the media cool to room temper-ature in the bottle before pouring it into Petri dishes;yeast colony morphologies were not reproducible wheninoculated onto substrates prepared with different heat-ing protocols. As discussed in Appendix B, the fluid’sviscosity dropped almost 20% over the first 24 hours andthen slowly decreased as a function of time. The cellswere consequently always inoculated 24 hours after pour-ing the media. Future work should investigate how tomake the fluid viscosity more stable.
2. Strains
We used the prototrophic (capable of synthesizingall required amino acids) yeast strains yJHK041 andyJHK042 which were derived from the W303 back-ground. The two strains were virtually identical anddiffered only by the expression of different fluores-cent proteins under the control of an ACT1 promoter.yJHK041 expressed mCitrine and was colored red inour figures while yJHK042 expressed mCherry andwas colored cyan for visual clarity. yJHK041 hadthe genotype can1-100 bud4 prACT1-ymCitrine-tADH1-His3MX6:prACT1-ACT1 and yJHK042 had the samegenotype except with ymCitrine replaced with the ym-Cherry . The two strains had identical growth rates inliquid culture and expanded at the same rate when de-posited separately on agar plates.
3. Standard experimental setup
To prepare the saturated yeast cultures that we inocu-lated on our viscous media, we followed a similar proce-dure used for bacteria by Weinstein et al. [11]. We tooka single colony of yeast growing on an agar plate and in-oculated it in 10 mL of YPD media in a glass tube. Thetube was then shaken overnight for roughly 16 hours at30 ◦ C as the yeast grew to saturation. The next morn-ing, we used optical density measurements to place equalproportions of yJHK041 and yJHK042 in an Eppendorftube with a final volume of 1 mL. After vortexing the Ep-
FIG. A.1. Upon deposition on the highly viscous substratefluid, yeast cells first spread uniformly in the circular inocu-lant region usually called the “homeland” [7], and then clumptogether via a coarsening process. a) Distribution of cells20 minutes after inoculation on the viscous substrate, andb) zoom in immediately after inoculation (left) and after 20minutes (right). The initially uniform distribution of yeastsegregates into large clumps in a phase-separation process,suggestive of attractive interactions. The bottom scale barcorresponds to 100 µ m. pendorf tube, 2 µ L of saturated culture was taken fromthe tube and was inoculated on the surface of 40 mL ofviscous fluid in a 94 ×
16 mm Petri-dish (Greiner Bio-Oneitem number 633181), leading to an average fluid heightof H = 7 ± . ◦ C.
4. Imaging
The microbial colonies were imaged with an incubatedZeiss Lumar.V12 Stereoscope held at 30 ◦ C with bothfluorescent (eYFP and mCherry) and brightfield chan-nels. In order to image large fields of view (i.e. an en-tire Petri dish), we stitched many images together andblended their overlapping regions using Axiovision 4.8.2software. Our fluid was viscous enough that panning themicroscope stage did not adversely shake the fluid andmicrobes. Fluid flows were imaged by adding fluorescentgreen polyethylene microspheres between 10 and 20 µ min diameter (Cospheric item number “UVPMS-BG-1.02510 −
5. Density measurements
To test if yeast colonies depleted the density of thesurrounding substrate as they metabolized, we comparedthe density of YPD media before and after the cells grewto saturation in it with an Anton Paar DMA 38 densitymeter. To conduct this experiment, we placed a controltest tube of YPD and another tube inoculated with ourstrains of yeast on a shaker overnight in a 30 ◦ C room;the yeast culture grew to saturation. The next day, wecentrifuged both tubes, depositing the yeast on the bot-tom of the second tube, and measured the supernatantdensity of each. We repeated this experiment three timesand found that the average density of our control tubewas ρ YPD = 1 . ± . / mL and that the den-sity of the supernatant where the yeast had grown was ρ saturated = 1 . ± . / mL, leading to a changein density of ∆ ρ = − . ± . / mL where the ± corresponds to range of densities that we measured. Appendix B: LIQUID SUBSTRATE RHEOLOGY
The substrate rheology was characterized with anAnton-Paar MCR 501 rheometer in a 50 mm disk ge-ometry with a 1 mm gap. Fig. B.1a) displays steady-state flow tests for various polymer concentrations re-alized with logarithmic sweeps of the shear rate rangingfrom 10 to 10 − ◦ C (the yeast incubation temperature), andthe measurements were performed a day after the viscousmedia was microwaved, corresponding to the time thatstrains were inoculated on it.Our viscous substrate exhibited a clear shear-thinningbehavior, i.e. the viscosity decreased with increasingshear rate larger than ˙ γ (cid:38) − s − but presented aplateau for smaller shear rates. At cellulose concentra-tions higher than 2%, the viscosity continued to decreasewith shear rate for ˙ γ (cid:46) − / s and we found an ap-proximate power law relation between the shear stress τ and shear rate ˙ γ ; Figure B.1b) shows a fit to τ = m ˙ γ n , inaccord with the “Power-Law” model of Ostwald and deWaele [58–60], where the amplitude m is the flow consis-tency index and the exponent n corresponds to the flowbehavior index. The effective “Newtonian” viscosity ofour fluid can then be expressed as η ( ˙ γ ) = m ˙ γ n − [58],where n = 1 describes Newtonian fluids and n < FIG. B.1. (a) Shear viscosity and (b) corresponding shearstress for different polymer concentrations measured viasteady-state flow tests. The fluid was weakly shear thinningfor ˙ γ (cid:46) − s − and reached a Newtonian plateau at lessthan or equal to 2% polymer. For other concentrations, apower law of τ = m ˙ γ n described the shear stress for shearrates of ˙ γ (cid:46) − s − . (c) and (d) plot m and n as a func-tion of polymer concentration; the fluid became more non-newtonian (shear thinning) as more polymer was added. dicates shear thinning behavior. We determined m and n as a function of polymer concentration by fitting thepower law behavior at shear rates lower than 10 − / sas shown in Figure B.1. Our liquid substrate exhibitsincreasing shear-thinning behavior (decreasing n ) withlarger polymer concentration; we found n = 0 . ± . n = 0 . ± .
05 at 3%, suggesting asmall, but measurable departure from Newtonian behav-ior across all polymer concentrations in this regime.The typical shear rate in our experiments was on theorder of 10 − ≤ ˙ γ ≤ − / s, estimated from themeasured surface flow velocity generated by the yeastcolonies, 1 ≤ u ≤
20 mm/day, and with ˙ γ = u/H fora fluid with a typical height H ≈ γ = 10 − / s (the lowest shear rate atwhich the rheometer give reproducible results); the cor-responding values as we varied the polymer concentrationare shown in Table I. The media rheology was monitoredover one week and presented a slow decrease in viscosityas a function of time after being microwaved (less than10% per day), and was neglected within the 3-5 day timescale of our experiments. Although we did not inves-tigate closely, the viscosity of yeast complete syntheticmedia (CSM) appeared to be more stable as a functionof time; future work should investigate this phenomenon.8 FIG. C.1. Fluid flow in sealed chambers with various yeastcolonies’ position relative to gravity. a) Schematic of a colonygrowing on a thin layer of agar on the bottom, top or side ofa sealed container filled with our viscous media. Fluid flownear the colony, regardless of its position, is always gener-ated opposing the direction of gravity. The fluid circulationwas consistent in all cases with the vorticity direction pre-dicted by baroclinic instability embodied in Eq. (2). b) samefigure as in Fig. 4b) in the main text. Fluid flow stream-lines corresponding to the yeast colony attached to the side.The streamlines were obtained by taking a maximum inten-sity projection of green fluorescent bead motion over severalhours. The scale bar is 2 mm, and the red arrow indicatesthe direction of the flow for streamlines near the top of thecolony.
Appendix C: OTHER COLONYCONFIGURATIONS
We explored other experimental geometries, summa-rized in Fig. C.1, to determine if other mechanisms (e.g. Marangoni flows [28]) might account for the flowgenerated in the substrate fluids by the colony. How-ever, experiments conducted where we anchored yeastcolonies on a thin layer of agar to the top, bottom, andsides of sealed chambers filled with our viscous mediaFig. C.1b) cast doubt on the Marangoni flow hypoth-esis. We found that colonies created fluid flows in thesurrounding media similar in magnitude to experimentswhen the air-liquid interface was present regardless oftheir position in the chamber (even when placed at thetop of the chamber), and also found that the inducedfluid flows always opposed the direction of gravity. Al-though these experiments did not rule out the possibilitythat surface tension gradients drove flow when the freeinterface was present, they do suggest that metabolically-induced buoyant forces opposing the direction of gravitycould be completely responsible for our observed flow.Buoyant flows result from differences in density in thepresence of a gravitational field [33] and, in our experi-ments, could originate from gradients in fluid tempera-ture and solute concentration. One possibility is that en-vironmental temperature gradients (i.e. in the chamberwhere the yeast were imaged) drove fluid flows. As men-tioned earlier, the very high viscosity of our liquid me-dia substrates coupled with estimates of critical Rayleighnumbers strongly suggest that stray thermal gradientswould be insufficient to produce convection in our ex-periments [34]. In fact, plates filled with viscous media, monitored over 24 hours, showed no evidence of a flowin the absence of yeast cells. The yeast colonies them-selves must have induced buoyant flows by generatinglocal gradients in the surrounding fluid’s temperature orsolute concentration. Similar to the work of Benoit etal. [34], temperature gradients can be ruled out becauseheat diffusivity D heat is much larger than the moleculardiffusivity D glucose of glucose in water, minimizing result-ing density gradients caused by thermal gradients. Thisis can be estimated via the Lewis number of our media: L = D heat /D glucose
300 indicative of an isothermal fluid.In addition, the coefficient of thermal expansion is muchsmaller than the coefficient of solute expansion; largetemperature differences (several degrees Celsius) wouldbe required to create the same density difference froma small change in solute concentration [34]. Estimatesof the yeast cell metabolic heat production seem insuf-ficient to produce the requisite thermal gradient. Forinstance, comparing the density change induced only bythe cells glucose uptake ∆ ρ G , with the density decreasedue to the fluid thermal expansion caused by the heatproduced during yeast glucose fermentation ∆ ρ T , givesan estimate of ∆ ρ G / ∆ ρ T ≈ Appendix D: CALIBRATING SIMULATION TOEXPERIMENTS
Table II in the main text shows the values used tofit our model to experiment (i.e. Figure 6), and the re-mainder of this appendix discusses how we obtained thesevalues.
1. Viscous media density: ρ As discussed in section A 5, we found that the densityof YPD media without adding the cellulose polymer was ρ YPD = 1 . ± . / mL. Mixing hydroxyethyl cel-lulose with water within the range of the concentrationwe used in our experiments, i. e. between 2% and 3%,did not significantly affect the solution density [23]. Ad-ditional density measurements of the polymer solutionswhen mixed with YPD solutions [20] also didn’t showa significant change in density of the substrate withinexperimental error.Yeast colonies deplete the density of the surroundingmedia in order to create more biomass. In the modelused by Benoit et al. [34]), cells can absorb moleculeswith a variety of sizes and with correspondingly differ-ent concentration fields and diffusion constants. Here,the change in density we observed in overnight culture∆ ρ = − . ± . / mL was consistent with approx-imatelly all of the glucose (originally 2%) in the mediabeing depleted within a factor of two [61]. For simplic-9 FIG. D.1. Fit to the fluorescein diffusion constant D inour viscous fluid. The blue line corresponds to the measuredinitial radial profile, and the green line is measured the finalprofile. Black dotted line represents the profile predicted byequation (D.1) with the best fit value of D ity, we consequently used a single concentration field c tomodel the diffusion and absorption of glucose only.
2. Solute Expansion Coefficient: β The solute expansion coefficient β only enters in ourdimensionless simulations via the combination βc in theRayleigh number, Ra = h βc gDν , where c is the initial concentration of solute in the sys-tem. In our experiments, c is also the maximum con-centration, since uptake of nutrients and excretion of lessdense waste products leads to a net depletion of the ef-fective concentration field. Hence, to estimate βc , wesimply note that the density change when all solute isdepleted is ∆ ρ = − . ± . / mL from our ex-periments measuring the density of yeast overnight cul-ture (Section A 5). Because the density of our media is ρ = ρ (1 + βc ), and after a day of growth in well mixedculture, the glucose is completely depleted as the yeastcan no longer reproduce, we estimate ∆ ρ ≈ − ρ βc , im-plying that βc = − ∆ ρρ . After including appropriatesources of error, we thus find βc = 0 . ± .
3. Diffusion Constant: D The glucose concentration field in our liquid substrateis difficult to track. In order to estimate the diffusionconstant in our medium, we instead tracked the diffu-sion of fluorescein molecules as a proxy for glucose inour substrate over the course of several days (see [20]for additional details). A circular droplet of approxi- mately 7 mm in diameter was deposited on the surfaceof a thin, 2 . ◦ C). The radial density profile of a dif-fusing concentration c ( (cid:126)r, t ) can be related to its originalprofile c t = c ( r, t = 0) via an integral representationthat depends on the diffusion constant D [62]: c ( r, t ) = 12 Dt (cid:90) ∞ ds s c t I (cid:16) rs Dt (cid:17) e − ( r s ) Dt (D.1)where I is the modified Bessel function of the first kindand we take the limit of the plate radius to infinity forsimplicity. We ignored diffusion in the third dimension(towards the bottom of the plate) as the fluid layer wassmall relative to the droplet diameter. We fit the fluores-cein diffusion constant D in equation (D.1) by insertingour experimentally measured initial concentration field at c ( r, t = 0), numerically evaluating the integral, and com-paring the predicted concentration field at later times toour experimental measurements. We adjusted the valueof D using least-squares to find the best-fit to our exper-imental measurement.Figure D.1 displays the original radial profile of the flu-orescein, the final profile, and the predicted fit from equa-tion (D.1) with the best value of D . We repeated this ex-periment three times on media with 2.0% and 3.0% poly-mer concentration and found identical diffusion constantswithin experimental error, D = 2 . ± . − cm / s.These results are consistent with the assumption thatfluorescein diffusion is dominated by motion throughthe gaps between the long chains of hydroxyethyl cel-lulose polymer. Noting that the diffusion constants offluorescein and dextrose (glucose) are similar in waterat 25 ◦ C: D fluorescein = 4 . ± .
01 10 − cm / s and D dextrose = 5 . − cm / s [63] , for simplicity, we as-sumed that the nutrient diffusion constant in the sub-strate is similar to D = 2 . ± . − cm / s in oursimulations of the substrate fluid.
4. Mass flux rate into the yeast colony in richnutrient conditions: ac We fit ac , the mass flux rate into the yeast colonyin rich nutrient conditions, by calibrating our simulationto experiments in a situation which negated the effect ofsurface tension: A yeast colony anchored on a thin agarsheet on the bottom of a sealed petri dish filled with ourviscous nutrient-containing fluid at η = 54 ± · s (seeFigure C.1a); upper left). Under these conditions, thesimulated yeast colony nutrient uptake created a buoy-0 FIG. D.2. Simulated average flow velocity as a function ofmass flux rate j colony = ac into a submerged yeast colonyin rich nutrient conditions. The velocity field is determinedabove the center of the colony, in the rising plume of fluidfrom the bottom to the top of the domain. ant plume in the direction opposing gravity, and the fluidflow reached a maximum, stable magnitude after abouta day of growth. Note from the lower left side of Fig-ure C.1a) that the induced flow in this case opposes theoutward growth-induced expansion velocity of the colony.We adjusted the product ac until the simulated averageflow velocity in the plume above the colony, as shown inFigure D.2, matched the average experimental velocity oftracer beads moving in the rising fluid from the bottomto the top of the container above the colony. The bestmatch for v experimental = 30 ±
10 mm / day resulted in avalue of ac = 5 ± / ( µ m hour), where the ± is thestandard deviation.We now argue that this mass flux rate is consistentwith a simple, order of magnitude estimate and also showthat it predicts a nutrient screening length inside yeastcolonies in agreement with earlier investigations [38]. a. Order of magnitude estimate for ac A singleyeast cell consumes about N ∼ glucose moleculesper cell division when fermenting at high glucose con-centrations [64], and glucose has a molar mass of M = 180 .
156 g / mol. Yeast divide roughly every τ g ≈
90 minutes in rich media, have a radius of approximately r yeast ≈ . µ m, and are approximately spherical whennot actively dividing; they consequently have an area of A yeast = 4 πr . Therefore, the glucose mass flux intoa spherical yeast cell must be on the order of j cell = a yeast c ∼ M NA yeast τ g ∼ . µ m hour . (D.2)In rich nutrient conditions, we assume that the concen-tration field is at its maximum value of c = c just outsidethe yeast cell walls, implying that j cell = a yeast c . Ourorder of magnitude estimate of j cell allows us then to esti-mate that a yeast c ∼ . / ( µ m hour), which is in thesame order of magnitude as ac , the nutrient flux into the colony. b. Consistency with nutrient screening length insidea yeast colony In the main text, we used our measuredvalue of ac , the mass flux into the colony, to calculatethe nutrient screening length in the fluid (cid:96) = ( ρ βD ) /a =5 ± ac toestimate the nutrient screening length inside the yeastcolony given in [38]: ζ = (cid:115) Dρ solute ˙ ρ (D.3)where ρ solute = ρ βc is the characteristic density of so-lute and ˙ ρ is the rate at which the solute is depleted.With the volume of a yeast cell V yeast = (4 / πr andthe packing fraction of spherical cells in a colony N ∼ . ρ can then be estimated as:˙ ρ ∼ N j col A yeast V yeast = 3 N ac r yeast (D.4)implying that the nutrient screaning length inside thecolony is ζ = (cid:114) Dρ βr yeast N a ∼ µ m , (D.5)in approximate agreement with the work of Lavrentovichet al. [38]. Appendix E: NONDIMENSIONALIZING THE SETOF EQUATIONS
As discussed in Sec. VI of the main text, after cou-pling the Navier-Stokes equations with the diffusing so-lute field, applying the Boussinesq approximation as thelocal density variations are small in our experiments( δρ/ρ (cid:28) ∂c∂t + u · ∇ c = D ∇ c (E.1) ∂ u ∂t + u · ∇ u = − ∇ pρ + ν ∇ u + βc g (E.2) ∇ · u = 0 (E.3)( ∇ c · ˆn ) (cid:12)(cid:12)(cid:12)(cid:12) colony = (cid:16) c(cid:96) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) colony , (E.4)where c is the nutrient concentration field, u the fluid ve-locity, D the nutrient diffusion contant in the substrate, ν the fluid’s kinematic viscosity, ρ the substrate densitywithout nutrient, p the fluid’s pressure, g the downwardacceleration due to gravity, β the solute expansion co-efficient, ˆn the normal unit vector to the interface, and (cid:96) = ρ βD/a the characteristic nutrient depletion lengthin the substrate fluid.1To better understand the dynamics of our model, wenon-dimensionalize equations (E.1)-(E.4) by choosing acharacteristic length scale L = H , the height of the fluidin the Petri dish, a time scale T = H /D , (the timeit takes solute to diffuse from the bottom to the top ofthe fluid in the petri dish), and the initial, maximum glucose concentration c (the initial concentration has themaximum value before the yeast cells deplete nutrients).The non-dimensionalized equations become: ∂ ˜ c∂ ˜ t + ˜u · ∇ ˜ c = ∇ ˜ c (E.5)1Sc (cid:20) ∂ ˜u ∂ ˜ t + ˜u · ∇ ˜u (cid:21) = − ∇ ˜ p + ∇ ˜u + Ra ˜ c ˆg (E.6) ∇ · ˜u = 0 (E.7)where the dimensionless concentration field is givenby ˜ c = c/c , the dimensionless velocity is ˜ u = u/ ( L/T ) = u/ ( D/H ) and the dimensionless pressureis ˜ p = p/ ( Dρ ν/H ). The non-dimensional Navier-Stokes equation reveals two key dimensionless param-eters: the Schmidt number, Sc = ν/D , the ratio ofthe momentum diffusion to solute diffusion, and theRayleigh number, Ra = ( H βc g ) / ( Dν ) which quanti-fies the strength of the dimensionless buoyant force [36].Non-dimensionalizing the flux boundary condition for theconcentration field at the yeast colony’s border reveals afinal key parameter; the boundary condition becomes( ∇ ˜ c · ˆn ) (cid:12)(cid:12) colony = ( G ˜ c ) (cid:12)(cid:12) colony (E.8)where the “mass flux number,” G = ( Ha ) / ( ρ βD ) ≡ H/(cid:96) is the dimensionless ratio of the fluid height H to thenutrient depletion length in the fluid (cid:96) .The interplay between the Rayleigh, Schmidt, andmass flux numbers in our simulated geometry control thedynamics of our model. However, the large Schmidt num-ber Sc = ν/D ∼ − (using the parameter valuesin Table II) allows us to set the inertial terms in equa-tion (E.6) to zero; this simplification corresponds to theStokes regime. Thus, we need only consider the inter-play between the Rayleigh and mass flux numbers. Forthe standard fluid height used in our experiments (40mL of fluid in a standard 94 mm diameter Petri dish, or H = 7 ± . to 10 as we vary the fluid viscosity from η = 54 ± · sto η = 600 ±
90 Pa · s, and the mass flux number remainsconstant at G ∼ .
4. The yeast do not deplete nutrientsquickly enough to allow us to set c = 0 on the bottomof the colony, corresponding to the G → ∞ limit. Bothquantities consequently play a role in our experiments. Appendix F: SIMULATION METHODS
In this appendix, we discuss how we utilized Open-FOAM 5.0 [39] to simulate the buoyant fluid flow createdby our yeast colonies and the early stages of yeast colony growth. Specifically, we discuss the particular programsthat we created, how we prepared meshes and geome-tries for use in OpenFOAM, and how we analyzed andvisualized simulation output. diffusionPressureFoam The program diffusionPressureFoam (available onGitHub [65]) simulates a yeast colony that absorbs a dif-fusing concentration field and calculates the resulting hy-drostatic pressure. We used diffusionPressureFoam toshow how the baroclinic instability began before advec-tion began to dominate, as seen in Fig. 5.To create diffusionPressureFoam , we modifiedthe standard solver packaged with OpenFOAM called laplacianFoam , which simulates a diffusing scalar field.At each timestep, we let the concentration field diffuseand possibly be absorbed by the yeast. We utilized swak4foam [66], an extension of OpenFOAM, to imposethe absorption boundary condition that( ∇ c · ˆn) (cid:12)(cid:12) colony = (cid:18) acρ βD (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) colony . (F.1)The hydrostatic pressure inside a fluid is given by [36] ∇ p = ρ (1 + βc ) g . (F.2)However, to calculate the hydrostatic pressure numeri-cally, we took the divergence of equation (F.2) and solved ∇ p = ∇ · [ ρ (1 + βc ) g ] . (F.3)At the free interface, we imposed the boundary conditionthat p = p atmospheric while on other walls, we imposed thecondition (again using swak4foam [66]) that[ ∇ p · ˆn] (cid:12)(cid:12) walls = [ ρ (1 + βc ) g · ˆn] (cid:12)(cid:12) walls . (F.4)We always assumed radial symmetry when simulat-ing yeast colonies on the surface of a viscous nutrient-containing liquid or at the bottom of a sealed petri dish.To create our radially symmetric geometry, we used gmsh . ◦ which is the appropriate setup forradially symmetric simulations in OpenFOAM. We sim-ulated our experiments at a resolution such that 20 simu-lation cells spanned the yeast colony radius. We wrappedthe gmsh geometry creation script in python scripts thatcould automatically generate geometries, change simula-tion parameters, and quickly analyze simulation output.After running a simulation, we used Paraview [68], anopen-source tool to visualize large geometrical datasets,to visualize the results and create figures such as the con-centration field showed in Fig. F.1. To quickly analyzedata from many simulations, we used automated Python2 FIG. F.1. Radially symmetric simulation of the concentrationfield c/c below a yeast colony (the thick brown bar) on thesurface of our viscous liquid after 48 hours and at η = 400 ±
50 Pa · s. Equally spaced contours of constant concentrationare shown. scripts to extract relevant data such as the velocity on thefluid’s surface and the total amount of solute present in apetri dish. To create the baroclinicity field ρ ( ∇ ρ × ∇ p )seen in Fig. 5, we utilized the funkySetFields utility, apart of swak4foam [66], which can algebraically manipu-late the output of OpenFOAM simulations. stokesBuoyantSoluteFoam The program stokesBuoyantSoluteFoam simulatedhow yeast depleted the density of the surrounding fluidand calculated the resulting fluid flow. We used thisprogram to generate the quantitative agreement be-tween experiment and simulation in Figure 6. Specifi-cally, stokesBuoyantSoluteFoam solves the dimension-less equations (E.5)-(E.7) and the dimensionless massflux boundary condition below the yeast colony in equa-tion (E.8). It assumes that the Schmidt number S c = ν/D is infinite (as discussed above) and consequentlysolves ∂ ˜ c∂ ˜ t + ˜u · ∇ ˜ c = ∇ ˜ c (F.5)0 = − ∇ ˜ p + ∇ ˜u + Ra ˜ c ˆg (F.6) ∇ · ˜u = 0 (F.7)( ∇ ˜ c · ˆn ) (cid:12)(cid:12) colony = ( G ˜ c ) (cid:12)(cid:12) colonys (F.8)at each timestep.We again utilized swak4foam [66] to implement theconcentration boundary condition at the yeast colonyboundary (equation F.8). At each timestep, the so-lute diffused and was absorbed by the yeast. Af-ter diffusing, stokesBuoyantSoluteFoam calculated thesteady-state velocity field using the same techniqueas buoyantBoussinesqSimpleFoam (the SIMPLE algo-rithm [69]) which is packaged with OpenFOAM. The ve-locity from the previous timestep was used as an initialguess for the velocity field in the next time step to im-prove its convergence speed. To avoid stability problems FIG. F.2. A polyhedral dual mesh (blue cells) and cor-responding height field h used by the forcedThinFilmFoam program [51]. resulting from a high Courant number [69], we adap-tively changed the timestep to ensure that the maximumCourant number in the simulation remained below 0.5and also used the implicit Crank-Nicolson technique [69]to evolve the concentration field. Geometry preparationand postprocessing for stokesBuoyantSoluteFoam wasthe same as that for diffusionPressureFoam . forcedThinFilmFoam forcedThinFilmFoam (available on GitHub [51]) solvesequation (15) in the main text, or ∂h ( r , t ) ∂t + ∇ · [ h ( r , t ) v ( r )]= D h ∇ h ( r , t ) + µh ( r , t ) (cid:20) − h ( r , t ) h (cid:21) . (F.9)and leads to the radial height profiles shown in Fig. 10.Although we could simulate arbitrary velocity fields, weused the radially symmetric field of v ( r ) = (1 / αr ˆ r ,matching equation (18).We used gmsh dramat-ically impacted simulation performance; using a regularcartesian grid led to pronounced lattice artifacts, likelybecause of the autocatatalytic growth term on the rightside of equation (F.9). We obtained the best results whenwe converted a Delaunay triangular mesh to its dual poly-hedral mesh using OpenFOAM’s polyDualMesh utility,3 FIG. G.1. Total number of nutrient molecules absorbed bythe yeast relative to the original number of nutrient moleculesin the fluid
N/N as the fluid viscosity is varied. The height ofthe fluid is H = 7 mm and the rest of the simulation parame-ters are set using the values in Table II. The Rayleigh number,Ra = ( H βc g ) / ( Dν ), varies from 0 to approximately 10 due to the changing viscosity ν , and the mass flux number, G = ( Ha ) / ( ρ βD ) ≡ H/(cid:96) , is fixed at G ≈ .
4. Note that thestronger advection of the substrate fluid at lower viscositiesleads to an enhenced uptake of nutrient molecules. similar to other work simulating fluid flow in radial ge-ometries [70].To allow for advection-dominated simulations, we useda flux-limiting Super bee scheme when calculating thedivergence term, or ∇ · [ h ( r , t ) v ( r )]. To prevent sta- bility problems, we ensured that the maximum Courantnumber was less than 0.1 and used the implicit Crank-Nicolson technique [69] to evolve the height field. Weagain utilized Python scripts to analyze the data coupledwith OpenFOAM’s postprocessing singleGraph tool. Appendix G: Simulated Nutrient Absorption vs.Flow Rate
To investigate if microbial colonies generating buoyantflows absorb more nutrients than those that do not, wesimulated a yeast colony on the surface of our fluid (againwith a fixed colony radius for simplicity) and varied thesubstrate viscosity, from 10 Pa · s to 100 Pa · s, allowing usto control the magnitude of the buoyant flow. We alsosimulated a substrate with infinite viscosity where no flowwas allowed. We kept the rest of the simulation param-eters fixed to the values in Table II with H = 7 mm andrecorded the nutrient uptake by the colony over time.The Rayleigh number, Ra = ( H βc g ) / ( Dν ), of thesesimulations ranged between 0 and 10 and the mass fluxnumber, G = ( Ha ) / ( ρ βD ) ≡ H/(cid:96) , was fixed at G ≈ . · s was about 1.5 times largerthan at infinite viscosity. It is possible that microbesgrowing on less viscous fluids could induce more intenseflows, enhancing this effect even further. It thus seemsplausible that colonies generating stronger buoyant flowscould indeed have a selective advantage. [1] T. T´el, A. de Moura, C. Grebogi, and G. K´arolyi,Physics Reports , 91 (2005).[2] E. R. Abraham, Nature , 577 (1998).[3] E. R. Abraham, C. S. Law, P. W. Boyd, S. J. Lavender,M. T. Maldonado, and A. R. Bowie, Nature , 727(2000).[4] F. D’Ovidio, S. De Monte, S. Alvain, Y. Dandonneau,and M. Levy, Proceedings of the National Academy ofSciences , 18366 (2010).[5] Z. I. Johnson, Science , 1737 (2006).[6] M. J. Follows, S. Dutkiewicz, S. Grant, and S. W.Chisholm, Science , 1843 (2007).[7] O. Hallatschek, P. Hersen, S. Ramanathan, and D. R.Nelson, Proceedings of the National Academy of Sciences , 19926 (2007).[8] K. S. Korolev, M. Avlund, O. Hallatschek, and D. R.Nelson, Reviews of Modern Physics , 1691 (2010).[9] K. S. Korolev, M. J. I. M¨uller, N. Karahan, A. W. Mur-ray, O. Hallatschek, and D. R. Nelson, Physical Biology , 026008 (2012).[10] M. Gralka, F. Stiewe, F. Farrell, W. M¨obius, B. Waclaw,and O. Hallatschek, Ecology Letters , 889 (2016).[11] B. T. Weinstein, M. O. Lavrentovich, W. M¨obius, A. W.Murray, and D. R. Nelson, PLOS Computational Biology , e1005866 (2017).[12] M. J. I. Muller, B. I. Neugeboren, D. R. Nelson, andA. W. Murray, Proceedings of the National Academy ofSciences , 1037 (2014).[13] M. F. Weber, G. Poxleitner, E. Hebisch, E. Frey, andM. Opitz, Journal of The Royal Society Interface ,20140172 (2014).[14] L. McNally, E. Bernardy, J. Thomas, A. Kalziqi, J. Pentz,S. P. Brown, B. K. Hammer, P. J. Yunker, and W. C.Ratcliff, Nature Communications , 14371 (2017).[15] M. O. Lavrentovich, M. E. Wahl, D. R. Nelson, andA. W. Murray, Biophysical Journal , 2800 (2016).[16] S. Pigolotti, R. Benzi, P. Perlekar, M. H. Jensen,F. Toschi, and D. R. Nelson, Theoretical population bi-ology , 72 (2013).[17] S. Pigolotti, R. Benzi, M. H. Jensen, and D. R. Nelson,Physical Review Letters , 57 (2012).[18] P. Perlekar, R. Benzi, D. R. Nelson, and F. Toschi, Phys-ical Review Letters , 3 (2010).[19] T. Chotibut, D. R. Nelson, and S. Succi, Physica A: Sta-tistical Mechanics and its Applications , 500 (2017).[20] See also B. T. Weinstein, Microbial Evolutionary Dynam-ics and Transport on Solid and Liquid Substrates , Ph.D.thesis, Harvard University (2018). [21] M. B. Short, C. A. Solari, S. Ganguly, T. R. Powers,J. O. Kessler, and R. E. Goldstein, Proceedings of theNational Academy of Sciences , 8315 (2006).[22] A. J. T. M. Mathijssen, F. Guzm´an-Lastra, A. Kaiser,and H. L¨owen, Phys. Rev. Lett. , 248101 (2018).[23] D. Boutelier, A. Cruden, and B. Saumur, Journal ofStructural Geology , 153 (2016).[24] D. Vella and L. Mahadevan, American Journal of Physics , 817 (2005).[25] J. S. Langer, Annals of Physics , 53 (1971).[26] L. Rayleigh, Proceedings of the London mathematicalsociety , 4 (1878).[27] J. Eggers, Reviews of Modern Physics , 865 (1997).[28] L. E. Scriven and C. V. Sternling, Nature , 186(1960).[29] T. E. Angelini, M. Roper, R. Kolter, D. A. Weitz, andM. P. Brenner, Proceedings of the National Academy ofSciences , 18109 (2009).[30] A. Yang, W. S. Tang, T. Si, and J. X. Tang, BiophysicalJournal , 1462 (2017).[31] D. B. Kearns, Nature Reviews Microbiology , 634(2010).[32] S. Soh, K. J. M. Bishop, and B. A. Grzybowski, Journalof Physical Chemistry B , 10848 (2008).[33] J. S. Turner, Buoyancy Effects in Fluids , 1st ed. (Cam-bridge University Press, Cambridge, 1973).[34] M. R. Benoit, R. B. Brown, P. Todd, E. S. Nelson, andD. M. Klaus, Physical Biology , 046007 (2008).[35] D. Sharp, Physica D: Nonlinear Phenomena , 3 (1984).[36] E. Guyon, J.-P. Hulin, L. Petit, and C. D. Mitescu, Phys-ical Hydrodynamics , 2nd ed. (Oxford University Press,New York, NY, 2015).[37] J. Marshall and A. R. Plumb,
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