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Supporting InformationMüller et al. 10.1073/pnas.1313285111SI Text

S1. Experimental MethodsThis section gives additional details about our strains and ex-perimental methods.

S1.1. Strains. The strains used in this work are listed in Table S1.The mutualistic strains LeuFBR Trp– and Leu– TrpFBR are W303strains that are prototrophic for all amino acids except for eitherleucine or tryptophan. They all have the genetic backgroundW303 MATa can1-100 hmlαΔ::BLE leu9Δ::KANMX6 prACT1-yCerulean-tADH1@URA3 and differ only at the LEU4 and TRP2loci, as well as in their fluorescent marker at the HIS3 locus (seebelow). The strains contain the S288C version of BUD4 [stan-dard W303 strains have a mutated BUD4 (1), which causes al-terations in the budding pattern and occasional failures in cellseparation after cytokinesis]. The deletion of the silent mating-type locus HMLα prevents mating-type switching, and thereforemating, because all our strains are MATa at the mating-typelocus. The two strains thus behave essentially as two differentspecies. Leu9 and Leu4 are isozymes for the first step in leucinebiosynthesis (2), which makes the leu9 deletion necessary tomake a leu4 deletion auxotrophic for leucine. With the mCer-ulean gene under the actin promoter prACT1, all strains con-stitutively express a cyan fluorescent protein. We checked byflow cytometry that the cyan fluorescence intensity profiles wereindistinguishable for all strains.The leucine-overproducing strain LeuFBR Trp– is the strain

yMM60, which has the (additional) genetic modifications his3Δ::prACT1-ymCitrine-tADH1:HIS3MX6 LEU4FBR trp2Δ::NATMX4.It constitutively expresses the yellow fluorescent proteinmCitrineand is a tryptophan auxotroph. The LEU4FBR allele differs fromthe wild-type LEU4 by the deletion of the codon 548; it isfunctional in leucine biosynthesis, but feedback resistant (FBR)to inhibition by the end-product leucine. It was identified inref. 3 (named LEU4-1 in ref. 3). This strain is depicted as yellow(in accordance with its fluorescent protein) in the main text.The tryptophan-overproducing strain Leu– TrpFBR is the strainyMM65, with his3Δ::prACT1-ymCherry-tADH1:HIS3MX6 leu4Δ::HPHMX4 TRP2FBR. It constitutively expresses the red fluorescentprotein mCherry and is a leucine auxotroph. The TRP2FBR gene isthe allele TRP2–S76L, which was identified in ref. 4 (designatedL76 in ref. 4) as a feedback-resistant, functional version of TRP2.This strain is depicted as blue in the main text (to enhance itscontrast with the yellow color of the partner strain).To rule out effects of the fluorescent proteins on the fitness

or other phenotypes of the strains, we also used strains with“swapped” fluorescent proteins: yMM61, with his3Δ::prACT1-ymCherry-tADH1:HIS3MX6 LEU4FBR trp2Δ::NATMX4, andyMM64, with his3Δ::prACT1-ymCitrine-tADH1:HIS3MX6 leu4Δ::HPHMX4 TRP2FBR. To compare feedback-resistant and wild-type amino acid production we used the following strains: theFBR strains yMM26 and yMM31, which have the same geneticbackground as yMM60 and yMM65, respectively, except for beingURA3 wild type at the Ura3 locus; and the WT strains yMM29and yMM32, which are LEU4 his3Δ::prACT1-ymCherry-tADH1:HIS3MX6 and TRP2 his3Δ::prACT1-ymCitrine-tADH1:HIS3MX6,respectively.

S1.2. Fitness Costs of Amino Acid Overproduction due to FeedbackResistance. The relative fitness in direct competition in liquidculture was measured by a flow-cytometer–based competition

assay as described in ref. 5. The relative fitness on plates wasdetermined by analyzing the shape of sectors of the two com-peting strains in a colony as described in ref. 6.We first checked that the fluorescent proteins mCitrine and

mCherry were neutral with respect to each other both in the liquidand in the plate competition assays by verifying that the strainsyMM60 and yMM61 as well as yMM63 and yMM64, which differonly in their fluorescent protein, have the same fitness in com-plete synthetic medium (CSM) (for medium definitions, seeMaterials and Methods in the main text).We competed the mutualistic strains LeuFBR Trp– (yMM60) and

Leu– TrpFBR (yMM65) in liquid CSM and found that Leu– TrpFBR

had a 2% fitness advantage compared with LeuFBR Trp–. Thesame was true in a competition of the strains yMM61 andyMM64 with swapped fluorescent proteins. This fitness differenceis entirely due to leucine feedback resistance, as the TrpFBR strainyMM31 and the TrpWT strain yMM32 have the same fitness,whereas the LeuFBR strain yMM26 and the LeuWT strain yMM29also have a 2% fitness difference. Presumably, leucine over-production due to feedback resistance is more costly than tryp-tophan overproduction because leucine is more abundant thantryptophan, with ∼10% of yeast amino acids being leucine andonly ∼1% being tryptophan (7–9).

S1.3. Radial Expansion Velocities.When a yeast colony expands on anagar surface, its radius increases linearly with time after an initialtransient of about 4 d (Fig. S1A). In the linear regime, we charac-terize colony growth by an expansion velocity, defined as the slopeof the radius vs. time. For both mutualistic and nonmutualisticcolonies, the steady-state expansion velocity is independent of itsinitial conditions, such as the inoculation volume and density or theinitial mixing ratio of the blue and yellow strains.Initial transient in radius increase. In contrast, the transient towardthe linear regime can depend on the initial condition, because ittakes longer to reach steady-state growth mode when the initialconditions are farther away from it. As shown in Fig. 3 in the maintext, obligately mutualistic colonies are driven toward an optimalfraction f p = 0:5 of blue cells. Thus, when colonies are inoculatedwith a start fraction f0 close to f p, they approach the steady-stategrowth velocity faster than when inoculated with a start fractionfarther from f p. This effect is particularly noticeable for small f0(orange and red data in Fig. S1A), i.e., when there is a very smallfraction of leucine-requiring cells. In the case of f0 = 0:01, colo-nies never grew on experimental timescales.Growth as a function of the amino acid concentrations. Fig. 2A in themain text shows that, as a function of a required amino acid, thegrowth velocity increases linearly before it saturates at a plateau.These data are reproduced in Fig. S2 A (blue diamonds) and B(yellow squares). In addition, Fig. S2 shows that the growth velocitiesdo not depend on the concentration of the nonrequired amino acid[Fig. S2 A (yellow squares) and B (blue diamonds)]. The maximalgrowth velocities (plateau values) for Leu–TrpFBR andLeuFBR Trp–

are vL = 20:3 μm=h and vT = 19:8 μm=h, respectively. The 2.5%difference in the plateau growth rates agrees, within experi-mental error, with the difference in fitness between the twostrains as measured by the competitive fitness assay describedin SI Text, section S1.2.We also measured the growth rate in well-mixed culture as a

function of the external amino acid concentration. As shown in Fig.S2, the growth rates behave very similarly to the growth velocities. Infact, when scaled by the same factor of bvg = 40 μm, the growthrate plateaus for both Leu– TrpFBR and LeuFBR Trp– overlap with

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the growth velocity plateaus. The growth rates in the linear re-gimes are similar but not identical, because the growth rates inliquid increase slightly faster than the growth velocities on solidmedium. This leads to higher crossover concentrations from thelinear to the plateau regime for plate growth (Fig. S2, blue andyellow vertical lines) than for liquid growth (Fig. S2, red and greenvertical lines). We normalize the leucine and tryptophan con-centrations such that the crossover concentrations are atnEL = ½leu�=½leu�c = 1 and nET = ½trp�=½trp�c = 1 for liquid growth,with ½leu�c = 762 μM and ½trp�c = 97:6 μM, because the liquidconcentrations reflect the concentrations felt by the cells. Theslower growth on plates is due to an amino acid depletion layeraround the colony and leads to crossover concentrations that arelarger by a factor of 1.18 (SI Text, section S2.2).

S1.4. Determination of Yeast Colony Growth Parameters. This sub-section summarizes how we determined the model parameterslisted in Table S2.Cellular diffusion constant Ds and genetic diffusion constant Dg. We de-termined the spatial diffusion constant Ds (for cellular diffusiondue to the jostling of cell division) and the genetic diffusionconstant Dg (which characterizes the strength of genetic drift)with the sectoring assay described in ref. 10. When performingthis assay with the noninteracting equally fit strains yJHK111 andyJHK112, we obtained Ds=τg = 15 μm2=h and Dg=τg = 1:3 μm=h,where τg = 1:5 h is the yeast generation time. The value of Ds forcellular diffusion lies in the expected range: In each generation,cells push by a diameter of a= 5 μm, so we expect the diffusionconstant to be of the order of Ds = a2 = 25 μm2 per generation orDs=τg = 17 μm2=h. The value of the genetic diffusion constant Dgallows us to estimate the effective population density ρe ≈ 1=Dg(10). The resulting effective density ρe ≈ 0:5=μm of about 2–3cells in a region of the size of a cell is rather low, especially whenconsidering that yeast colonies pile up to a height of ∼100 cells.This result is similar to even lower effective population densitiesfor the bacteria Escherichia coli and Pseudomonas aeruginosaobserved in ref. 10 and implies that the dynamics of the bound-aries between different genotypes are dominated by the behaviorof the cells at the extreme edge of the growing colonies.Size b of the active layer. To estimate the size b of the active layer,within which cell proliferation drives radial growth, we use theobservation that the maximal colony growth velocity v isbvg = 40 μm times the maximal growth rate g in liquid culture(SI Text, section S1.3). Within a generation time τg ∼ 1=g, thenumber of cells increases by ΔN =Ubc, where U is the colonycircumference and c is the cellular surface carrying capacity. In thesame time τg , the colony increases in area by ΔA=Uvτg =Uv=g,which corresponds to ΔN = cΔA new cells. Thus, v=g= b. Withv=g= bvg = 40 μm, we obtain the reasonable estimate of b= 40 μmfor the size of the active layer (11, 12).Surface carrying capacity c. Colonies grown between 4 and 7 d (toobtain a variety of sizes and to measure in the same time windowused for all other experiments) were imaged with a Zeiss LumarStereoscope to determine the colony area by image analysis (circlefit withMATLAB).Directly after imaging, a small agar padwith thecolony on it was cut out of the plate and suspended in a knownamount of PBS. All cells were washed off the agar by strongvortexing. The cell concentration in the PBS was then determinedby using a Beckman Coulter Counter, from which the total numberof cells in the colony could be calculated. The surface carryingcapacity was determined as the slope of the cell number vs. colonyarea, with the result of 10 cells=μm2. Taking into account thatyeast colonies have a height of 0:5− 1 mm, this surface densitycorresponds to a volume density of about 0:01 cells=μm3, whichis similar to dense spherical packing of yeast cells of diameter5 μm. Because we are interested in the cell density at the colonyboundary, and because yeast colonies are higher in the middlethan at the boundary, we take c= 5 cells=μm2 (11).

Amino acid diffusion constant Da.Diffusion through the agar mediumis similar to diffusion in water, because the agar pore size of about500 Å (13) is much larger than the amino acid hydrodynamicradius of about 3 Å (14). The diffusion constant for the similar-sized leucine and tryptophan molecules is Da = 3 mm2=h (15).Diffusive flux rate d. The derivation of d= 5=h is described in SIText, section S5.Amino acid crossover concentrations [leu]c and [trp]c. The leucineand tryptophan crossover concentrations between the regimewheregrowthdepends linearlyon theaminoacid concentrationsand the regime where growth is saturated are determined inFig. S2 as ½leu�c = 762 μM =̂ 2:3 · 109 molecules=μm2 and ½trp�c =97:6 μM =̂ 2:9 · 108 molecules=μm2. Here, we have used the agarheight of 5 mm to transform the 3D volume concentrations to2D surface concentrations, i.e., to the number of moleculesavailable to cells from the agar directly beneath.Amino acid uptake rates. From the literature, we estimate the aminoacid uptake rates as kL = 105 molecules · s−1 · cell−1 for leucine (8,16, 17) and kT = 1:5 · 104 molecules · s−1 · cell−1 for tryptophan (8,18). These values are in agreement with the value of the pa-rameter κ= kL c=½leu�c=d= kT c=½trp�c=d= 0:18 (determined in-dependently from our growth rate and velocity measurements inFig. S2) and our above estimates of ½leu�c, ½trp�c, c, and d.Amino acid secretion rates. We expect the amino acid leakage ratesto be proportional to the internal cellular amino acid concen-trations. Because yeast cells contain an order of magnitude moreleucine than tryptophan (7–9), we expect the leucine secretionrate to be an order of magnitude larger than the tryptophansecretion rate. From our experimental result that our yeaststrains grow twice as fast under nonmutualistic than under ob-ligate mutualistic conditions (SI Text, section S3.1), we de-termined the parameter ρ= 1:09 from Eq. S38, ρ= 1+ κ=2. Withρ= rL c=½leu�c=d= rT c=½trp�c=d, we obtain the leucine and tryp-tophan secretion rates rL = 7 · 105 molecules · s−1 · cell−1 andrT = 9 · 104 molecules · s−1 · cell−1. These values are of the orderof magnitude of amino acid secretion rates measured in ref. 19, butare on the larger side, presumably because our strains are over-producing leucine and tryptophan due to the engineered feedbackresistance and thus leak large amounts of excess amino acids.

S2. Modeling Growth of the Mutualistic Yeast Strains in thePresence of Amino AcidsIn this section we describe amodel for the growth of ourmutualisticyeast strains in the presence of amino acids in more detail than inthe main text. The model parameters are listed in Table S2.

S2.1. Model for Amino Acid Dynamics. Growth velocities. As shown inSI Text, section S1.3, the growth rates and velocities increaseroughly linearly with the concentration of the required aminoacid until they saturate at a plateau. This motivates us to writethe growth velocities as a function of the rescaled amino acidconcentrations nL = ½leu�=½leu�c and nT = ½trp�=½trp�c,

Leu− Trp FBR expansion velocity : VLðnLÞ= vL TðnLÞ; [S1]

Leu FBR Trp− expansion velocity : VTðnTÞ= vT TðnTÞ; [S2]

where we have defined the threshold function

TðxÞ=�x for x≤ 11 for x> 1: [S3]

Note that the velocities are expressed as a function of concentra-tions nL and nT felt by the cells, which can differ from the ex-ternal amino acid concentrations nEL and nET with which theagar medium was prepared (see below).



Amino acid uptake. Under steady-state growth conditions, the up-take of amino acids that are not produced by the cell must equalthe amount used in the production of cellular biomass. Thus, theamino acid uptake rates must be proportional to the growth rates,

Leu uptake rate of Leu− Trp FBR: KLðnLÞ= kL TðnLÞ; [S4]

Trp uptake rate of Leu FBR Trp− : KTðnTÞ= kT TðnTÞ; [S5]

where kL and kT are the maximal leucine and tryptophan uptakerates, respectively.Amino acid secretion. Because our yeast strains are feedback re-sistant in the production of leucine or tryptophan, they cannotregulate production in response to the available amount of theend product.We therefore assume that they secrete amino acidsat a constant rate,

Leucine secretion rate of Leu FBR Trp−: RLðnLÞ= rL; [S6]

Tryptophan secretion rate of Leu− Trp FBR: RTðnTÞ= rT;[S7]

see also ref. 20.Amino acid dynamics. The dynamics of the leucine and tryptophanconcentrations ½leu� and ½trp� can be described by

∂∂t½leu�= rL cT −KLð½leu�Þ cL +Da Δ½leu�; [S8]

∂∂t½trp�= rT cL −KTð½trp�Þ cT +Da Δ½trp�: [S9]

The first and second terms describe amino acid secretion and up-take, respectively, by Leu– TrpFBR cells (surface concentration cL)and LeuFBR Trp– cells (surface concentration cT). The lastterm describes diffusion of amino acids with diffusion constant Da.Yeast colony growth resembles a chemostat. As described in moredetail in SI Text, section S5, only cells in a small active layernear the colony boundary are actively growing. The growthdynamics within the active layer in the reference frame of theboundary are reminiscent of growth in a chemostat, in whichnutrients are added at a constant rate that is balanced byoutflux of waste (medium and cells) (21). For the active layerof a colony, there is continuous influx of nutrients (e.g., glu-cose) due to diffusive flux toward the colony. Cells are re-moved “behind” the active layer, because cells in the colonyinterior are not growing and therefore effectively “gone”.When the colony grows at constant velocity, nutrient influxand cell “outflux” are balanced, so that the number of activelygrowing cells remains constant. Thus, we can describe celldynamics in terms of the fraction

f =cL

cL + cT[S10]

of Leu– TrpFBR cells of the constant surface carrying capacityc= cL + cT. As explained in SI Text, section S5, chemostat-likegrowth also means that the Laplace operator Δ can be decom-posed (in a frame comoving with the frontier) into angular andradial diffusion in a specific way; e.g., for leucine

Δ½leu�= ∂2

∂x2½leu�− d

�½leu�− ½leu�E�: [S11]

Here, the first term describes angular diffusion along the frontcoordinate x, with ∂x=R∂ϕ. The second term replaces radial

diffusion by effectively describing the radial diffusive flux intoor out of the colony as dilution with rate d into medium withconcentration ½leu�E (the concentration in the agar medium faraway from the colony, i.e., the concentration with which themedium was prepared). With Eqs. S10 and S11, the diffusionEqs. S8 and S9 become

∂½leu�∂t

=Da∂2½leu�∂x2

− d�½leu�− ½leu�E

�+ rL cð1− f Þ−KLð½leu�Þc f

[S12]

∂½trp�∂t

=Da∂2½trp�∂x2

− d�½trp�− ½trp�E

�+ rT c f −KTð½trp�Þcð1− f Þ:

[S13]Nondimensionalization. These equations can be written in a non-dimensionalized form for the rescaled amino acid concentrationsnL = ½leu�=½leu�c and nT = ½trp�=½trp�c by expressing time in unitsof the inverse diffusive rate d and space in units of the diffusionlength scale

ffiffiffiffiffiffiffiffiffiffiffiDa=d

p,

∂nL∂~t

=∂2nL∂~x2

− ðnL − nELÞ+ ρLð1− f Þ− κL TðnLÞf [S14]

∂nT∂~t

=∂2nT∂~x2

− ðnT − nETÞ+ ρTf − κT TðnTÞð1− f Þ; [S15]

with ~t= t d and ~x= x=ffiffiffiffiffiffiffiffiffiffiffiDa=d

p. The secretion and uptake param-

eters are now dimensionless:

ρL ≡rL c

½leu�c d; ρT ≡

rT c½trp�c d

; κL ≡kL c

½leu�c d; κT ≡

kT c½trp�c d

:

[S16]

Intuitively, in the reduced parameter rL c=ð½leu�cdÞ, the “free”secretion rate rL is reduced by the factor d due to diffusive lossof amino acids from the colony. In addition, it is rescaled fromcellular to amino acid concentrations with the “conversion fac-tor” c=½leu�c.Note that for ourmutualistic yeast strains, the tryptophan secretion

rate rT, uptake rate kT, and crossover concentration ½trp�c are alllower than the corresponding quantities for leucine by a factor of 8(SI Text, section S1.4). Because all dimensionless rates depend on theratio of a secretion rate or an uptake rate and the crossover con-centration, this means that the dimensionless secretion and uptakeparameters have the same values for the two strains,

rescaled nutrient uptake rate : κ≡ κL = κT; [S17]

rescaled nutrient secretion rate : ρ≡ ρL = ρT: [S18]

The numerical values of these dimensionless parameters arelisted in Table S2.

S2.2. Steady-State Amino Acid Concentrations. The cell concen-trations cL and cT vary in angular direction due to spatialdemixing into patches or sectors. If only one cell type ispresent, or on length scales large compared with the patch orsector width, the cells in the active layer are essentially wellmixed, and the problem becomes radially symmetric, i.e., in-dependent of the radial angle ϕ. In this case the amino aciddynamics of Eqs. S14 and S15 become

1d∂∂tnL = ρð1− f Þ− κ TðnLÞf − ðnL − nELÞ; [S19]



1d∂nT∂t

= ρ f − κ TðnTÞ ð1− f Þ− ðnT − nETÞ: [S20]

For a calculation that takes into account cellular patches alongthe colony front, see SI Text, section S4.Because the nutrient dynamics are fast compared with colony

growth, we can assume that the cell fraction f is constant onnutrient timescales. The nutrient dynamics equations Eqs. S14and S15 then have the quasi-stationary solution

nLð f Þ=nEL + ρð1− f Þ

1+ κf≡ nmLðf Þ for f ≥ fC1 ðcase nL ≤ 1Þ

nEL + ρð1− f Þ− κf ≡ ndLðf Þ for f ≤ fC1 ðcase nL ≥ 1Þ

8><>:[S21]

nTð f Þ=nET + ρf

1+ κð1− f Þ ≡ nmTðf Þ for f ≤ fC2 ðcase nT ≤ 1Þ

nET + ρf − κð1− f Þ ≡ ndTðf Þ for f ≥ fC2 ðcase nT ≥ 1Þ

8><>:[S22]

with the limiting allele frequencies

fC1 =ρ− 1+ nEL

ρ+ κ=12+2nEL − ð2+ κ− ρÞ

2ðρ+ κÞ ;

fC2 =κ+ 1− nET

ρ+ κ=12−2nET − ð2+ κ− ρÞ

2ðρ+ κÞ :

[S23]

In view of Eqs. S1 and S2, this means that the growth velocities ofthe two strains are given by

VLð f Þ=�vL nmLð f Þ for f ≥ fC1

vL for f ≤ fC1

and

VTð f Þ=�vT nmTð f Þ for f ≤ fC2

vT for f ≥ fC2:[S24]

Crossover concentrations from linear to saturated growth. For thespecial case of single-strain colonies, consisting, e.g., of only Leu–

TrpFBR cells ð f = 1Þ, the steady-state leucine concentration is

nL =

nEL1+ κ

for nEL ≤ 1+ κ;

nEL − κ for nEL ≥ 1+ κ:

8<: [S25]

This means that the colony expansion velocity from Eq. S24 asa function of the external amino acid concentration is

Vcolony =VLðnELÞ=vL

nEL1+ κ

for nEL ≤ 1+ κ;

vL for nEL ≥ 1+ κ:

8<: [S26]

Thus, when measuring the colony velocity as a function of the ex-ternal amino acid concentration, as in SI Text, section S1.3, thevelocity passes from the linear regime (proportional to the aminoacid concentration) to the saturated regime (plateau velocity) atnEL = 1+ κ. This threshold is higher than the crossover value of 1naively expected from Eq. S1 because the amino acid concentra-tion nL felt by the colony is lower than the external concentrationby the factor ð1+ κÞ. Similar equations hold for single-strain

colonies of LeuFBR Trp– cells. In particular, the factor ð1+ κÞis the same for both strains, which we indeed observe experimen-tally (Fig. S2).

S3. Generalized Moran Model for Mutualism at the FrontAllele Frequency Dynamics.The expansion of a circular yeast colonyon an agar surface is a 2D process. However, because only cells ina small active layer near the colony boundary are actively growing(SI Text, section S5), one can approximate the expansion bya one-dimensional process along the front coordinate x (22). Inaddition, because colony front growth is similar to growth ina chemostat (SI Text, section S2.1), the effective population sizeremains approximately constant as long as the inflationary effectof the radius increase is not too large. In the context of pop-ulation genetics, our two different yeast strains can be viewed asrepresenting two different species or two different alleles. Intotal, the selection dynamics for our two “alleles” can be de-scribed within the framework of the Moran model, with theaddition of a frequency-dependent growth rate and spatial dif-fusion (22–24)

∂f∂τ

=Ds∂2f∂x2

+ sð f Þf ð1− f Þ+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifDgð f Þf ð1− f Þ

q Γðx; τÞ; [S27]

where f ðx; τÞ is the fraction of Leu– TrpFBR cells as a function ofthe coordinate x along the front and the time τ measured ingenerations. The first term on the right-hand side of Eq. S27describes spatial diffusion with diffusion constant Ds, whicharises because dividing yeast cells push neighboring cellsaround. The second term describes selection with the selectioncoefficient

sð f Þ≡ VLð f Þ−VTð f ÞfVLð f Þ+ ð1− f ÞVTð f Þ: [S28]

Selection arises due to differences in the growth velocities VL andVT of the two strains, as given by Eq. S24. Since the growthvelocities depend on the “allele frequency” f because of themutualistic interaction, the selection coefficient s is frequencydependent. The last term in Eq. S27 describes genetic drift,i.e., noise due to cell number fluctuations. Γ=Γðx; τÞ is anIt�o delta-correlated Gaussian noise, and the genetic diffusionconstant

fDgð f Þ=Dg VLð f Þ+VTð f Þ

2ð fVLð f Þ+ ð1− f ÞVTð f ÞÞ [S29]

characterizes the strength of the genetic drift. It is frequency de-pendent due to its dependency on the growth velocities. However,its frequency dependence is weak compared with the frequencydependence of sð f Þ so that we use fDgð f Þ≈Dg.

S3.1. The Effect of Mutualistic Selection.We first examine the effectof the mutualistic interaction that is incorporated in the selectioncoefficient sð f Þ. If the selection term dominates in Eq. S27, forexample in a well-mixed culture with high cell numbers (soDg ≈ 0), the dynamics become

∂f∂τ

= sð f Þf ð1− f Þ: [S30]

As shown above, amino acid secretion and uptake are symmetricfor our two strains because of κ= κL = κT and ρ= ρL = ρT (Eqs.S17 and S18). For simplicity, we neglect here the 2% differencein growth velocities and use v= vL = vT. (We consider the effectof the small growth velocity difference later in SI Text, sectionS3.2.) With these assumptions, the selection coefficient can be



obtained from the steady-state amino acid concentrations (21,22) via the growth velocities (24). Its form depends on whichof the limiting allele frequencies fC1 and fC2 given in Eq. S23is larger,

fC1 ≤ fC2 ⇔ nEL + nET ≤ 2 + κ− ρ: [S31]

In the case fC1 ≤ fC2, the selection coefficient is

and in the opposite case fC1 ≤ fC2,

In these equations, we have labeled the frequency region inwhich both strains are amino acid limited ðnL; nT < 1Þ as M for“mutualistic”, because in this regime both strains benefit from theadditional amino acids due to the mutualistic interaction. Thiscontrasts with the “neutral” N region in which both strains aresaturated for their required amino acids, nL; nT > 1, so that thereis no benefit due to mutualism. The “commensal” regions CT andCL denote the cases ðnL > 1; nT < 1Þ and ðnL < 1; nT > 1Þ, respec-tively, in which one strain is amino acid limited but the other isnot. Some of these regions may vanish, depending on whether thelimiting fractions fC1 and fC2 are in the relevant interval ð0; 1Þ:

0≤ fC1 ≤ 1 ⇔ 1− ρ ≤ nEL ≤ 1+ κ; [S34]

1≥ fC2 ≥ 0 ⇔ 1− ρ≤ nET ≤ 1+ κ: [S35]

“Phase diagram”. In total, the selection coefficient sð f Þ depends onthe parameters κ and ρ and the external amino acid concen-trations nEL and nET. Depending on the values of the allelefrequencies fC1 and fC2 (that mark the transitions between thecommensal and the mutualistic or neutral regimes of the selec-tion coefficient) in Eqs. S31, S34, and S35, sð f Þ could be in 1 of10 possible “phases,” M, CTM, MCL, CTMCL, CTNCL, CTN,NCL, CT, CL, and N. Here, for example, the notation CTMmeans that sð f Þ has the functional form of the CT regime(tryptophan limiting, leucine not) in Eq. S31 for 0< f < fC1 andthe functional form of the M regime (tryptophan and leucineboth limiting) in Eq. S31 for fC1 < f < 1. As we will see below, foryeast parameters, only 5 of the 10 possible phases appear; theyare shown in the ðnEL; nETÞ plane in Fig. S3A. To interpret the

graphs of the selection term sð f Þf ð1− f Þ as a function of theallele frequency f, note that positive values of the selection termmean that selection acts to increase f, whereas negative valuesdrive toward smaller f. A selection term of zero, sð f Þf ð1− f Þ= 0,defines a fixed point of the selection dynamics. The fixed point isstable if the selection term is positive to the left and negative tothe right of it. Thus, in the CTMCL phase, selection drives thesystem toward a stable fixed point within the M regime (unless

f = 0 or f = 1, which are absorbing boundaries of the system). Inthe CTNCL, CTN, and NCL phases, selection drives the systemfrom the commensal regimes CT or CL toward the neutral Nregime where the selective force is zero. If genetic drift is added,it will dominate in this neutral regime, causing random fluctua-tions of the allele frequency f.The phase diagram in Fig. S3A shows that for all amino acid con-

centrations below the crossover concentration, nEL; nET < 1+ κ, se-lection acts toward the interior of the frequency interval ð0; 1Þ andthus promotes mixing (pink area in Fig. S3A). For very small aminoacid concentrations (dark pink triangle), selection drives the systemtoward a well-defined stable frequency f p, whereas for intermediateconcentrations (light pink area), selection drives toward a neu-tral region only within the interval ð0; 1Þ. For larger concen-trations, nEL; nET > 1+ κ, there is no barrier to fixation at f = 0 orf = 1. If genetic drift is added, random fluctuations in fwill lead tofixation at one of these absorbing boundaries. The change in theselection term sðf Þf ð1− f Þ along the “diagonal” nE = nEL = nET isshown in Fig. S3B.Mutualism with a fixed point. We first take a closer look at cases inwhich sð f Þ has a mutualistic M region, where both amino acidconcentrations are below the crossover concentration required formaximum growth; i.e., nEL + nET ≤ 2+ κ− ρ, and nEL; nET < 1+ κ.This mutualistic regime is characterized by a uniquely determinedfixed point f p for which sð f pÞ= 0:

f p=12+12

κ+ 22ρð1+ κÞ+ κðnEL + nETÞ ðnEL − nETÞ: [S36]

Because of the symmetry of the mutualistic interaction parame-ters (κ= κL = κT, ρ= ρL = ρT), the stationary allele frequency f p

sð f Þ=

1− nmTð f Þf + ð1− f ÞnmTð f Þ for 0 ≤ f ≤ fC1 ðregime CT with nL > 1; nT < 1Þ

nmLð f Þ− nmTð f ÞfnmLð f Þ+ ð1− f ÞnmTð f Þ for fC1 ≤ f ≤ fC2 ðregime M with nL < 1; nT < 1Þ

nmLð f Þ− 1fnmLð f Þ+ ð1− f ÞnmT

for fC2 ≤ f ≤ 1 ðregime CL with nL < 1; nT > 1Þ

8>>>>>>>>><>>>>>>>>>:[S32]

sð f Þ=

1− nmTðf Þf + ð1− f ÞnmTðf Þ for 0 ≤ f ≤ fC2 ðregime CT with nL > 1; nT < 1Þ

0 for fC2 ≤ f ≤ fC1 ðregime N with nL > 1; nT > 1ÞnmLðf Þ− 1

fnmLð f Þ+ ð1− f ÞnmTfor fC1 ≤ f ≤ 1 ðregime CL with nL < 1; nT > 1Þ:

8>>>>>>><>>>>>>>:[S33]



equals 1/2 unless driven away from it by unequal external nutri-ent concentrations nEL ≠ nET. Vanishing selection sðf Þ= 0 in themutualistic regime means that the amino acid concentrationsnLð f pÞ= nTð f pÞ are equal (Eq. S32):

nL�f *�= nT

�f *�=nEL + nET + ρ

2+ κ: [S37]

As expected, the amino acid concentrations increase with the ex-ternal concentrations nEL, nET and the cellular secretion rate ρ,but decrease with the uptake rate κ.Obligate mutualism. In the obligate case, nEL = nET = 0, the steady-state allele frequency is f p = 1=2 due to the symmetry, as observedexperimentally. The steady-state amino acid concentrations arenLð f pÞ= nTð f pÞ= ρ=ð2+ κÞ, so that the expansion velocity ac-cording to Eq. S24 is v nLð f pÞ= v nTð f pÞ= v ρ=ð2+ κÞ. In ourexperiment, obligate mutualists expand with a velocity v=2 that ishalf the maximum velocity (Fig. 4F in the main text). Thus, we have

ρ= 1+κ

2: [S38]

We use this condition to determine the value of the rescaled se-cretion rate ρ from the value of κ. Eq. S38 shows that for ourmutualistic strains ρ> 1, so that the conditions nEL; nET > 1− ρin Eqs. S34 and S35 are always satisfied. In consequence, onlythe phases CTMCL, CTNCL, CTN, NCL, and N are relevant forour system, as shown in Fig. S3.Absorbing boundaries for obligate mutualism. In the obligate casenEL = nET = 0, the selection coefficient in the mutualistic M re-gime in Eq. S32,

sðf Þ= −12

ð1+ κÞ�f −

12

ð2+ κÞf ð1− f Þ ; [S39]

diverges when the fraction f of leucine-requiring cells approaches0 or 1. The reason is, for example in the case f → 0, that the growthvelocity VL of the rare Leu– TrpFBR cells approaches a constant,whereas the velocity VT of the abundant LeuFBR Trp– cells van-ishes linearly with f. Similarly, the average population velocityfVL + ð1− f ÞVT approaches zero linearly with f. Thus, as f → 0,population growth ceases and thereby prevents the system fromreaching the boundary f = 0 within a finite time. A population withf > 0 will therefore never fix at f = 0. Note that the selection termsðf Þf ð1− f Þ in Eq. S27 remains finite for all f ∈ ½0; 1�.Asymmetric mutualism. If, e.g., more leucine than tryptophan issupplied in the medium, nEL − nET > 0, mutualism is asymmetric,favoring leucine consumers (Fig. S4A). Compared with symmetricmutualism with a stable fraction f p = 0:5, the stable fraction iscloser to f = 1, and the barrier toward fixation at f = 1 (the maxi-mum value of the absolute selection term in the interval ½ f p; 1�) issmaller. Both effects make fixation of leucine consumers at f = 1more likely. Likewise, tryptophan consumers are more likely to fixif there is more tryptophan than leucine in the medium.Mutualism with a neutral region. The phase diagram in Fig. S3 exhibitsan unusual phase CTNCL in which the selection coefficient sðf Þ= 0for a finite region in the interior of the interval ð0; 1Þ. In this phase,selection in the commensal CT and CL regimes drives the systemtoward the neutral N regime, but dynamics within this regime areneutral. The existence of the neutral N regime is due to the pla-teaus in the velocity curves in Fig. S2: If both amino acid con-centrations are nL; nT ≥ 1, the precise concentration values do notmatter because the colony always grows at maximum velocity.No mutualism. For nEL; nET > 1+ κ, the system does not havea barrier toward fixation at one or both of the absorbing fre-

quency boundaries f = 0; = 1 (green regions in Fig. S3). In theNCL and CTN phases, selection acts toward the neutral N regimein which s= 0. Because the neutral regions connect to f = 0 orf = 1, number fluctuations can push the system toward an ab-sorbing boundary, thus leading to demixing. In the N phase,genetic drift is the only acting force, leading to eventual fixationat f = 0 or f = 1.

S3.2. The Effect of Small Fitness Differences. So far we have neglectedthe small fitness advantage sd = 0:02 of the Leu– TrpFBR strainover the LeuFBR Trp– strain (SI Text, section S1.2). For low ex-ternal amino acid concentrations, mutualism is a strong selectiveforce with selection coefficients large compared with sd (Fig. S3),and it is justified to neglect this fitness difference. Experimentally,this approximation results in the observed symmetric steady-statecell fraction f p = 0:5“despite” the fitness difference. However, forhigher amino acid concentrations, the fitness difference becomesrelevant as mutualistic selection becomes smaller.No mutualism. For no mutualism, the fitness difference is the onlyselective force so that the dynamics of the fraction f of leucine-requiring cells become

∂f∂τ

= sd f ð1− f Þ: [S40]

Thus, the frequency changes from the start frequency f0 as

f ðτÞ= f0 e sdτ

1+ f0ðe sdτ − 1Þ [S41]

and approches fixation of Leu– TrpFBR; i.e., f = 1, for long times.This equation has been used to predict the cell fraction dynamicsin Fig. 3B in the main text and Fig. S4C.Weak mutualism. The fitness difference is also important for weakmutualism, in particular when the selection coefficient has neutralregimes, as in the N, CTN, NCL, and CTNCL phases: It “tilts” flatneutral regimes toward f = 1. This results in selection toward fC2in the CTNCL phase, increases the likelihood for fixation at f = 1in the N and CTN phases, and decreases the probability of fixa-tion at f = 0 in the NCL phase. This bias has been taken intoaccount in the theoretical phase diagrams shown in Fig. 4B in themain text and explains part of the asymmetries in the experi-mental phase diagram in Fig. 4 A and B in the main text.

S3.3. Transient Approach to Steady-State Growth. Fig. S4 B and Cshows the dynamics of the boundary fraction f of blue cells asa function of the radial distance from the homeland, which in ourspatial expansion is analogous to time. In the case of obligatemutualism, the experimental boundary fraction converges to thesteady-state value f p = 0:5, independent of the inoculation frac-tion f0. The approach happens rapidly, within less than 1 mmdistance from the homeland with a small overshoot towardhigher f, a feature expected for cross-feeding interactions (25).Our simple mutualism model, designed for steady-state colonyexpansion, fails to produce this overshoot (although it doescapture the timescale of the approach to steady state) (solid linesin Fig. S4B), as well as asymmetric time lags to initiate growth[colonies with f0 = 0:15 are smaller because they take longer tostart growing (SI Text, section S1.3) and colonies with f0 = 0:01do not grow at all].In contrast to obligate mutualism, selection for no mutualism is

not frequency dependent, and the colony boundary fraction fdepends on the start fraction f0 (Fig. S4C). The fraction f of bluecells increases during expansion because the blue tryptophanproducers have a 2% fitness advantage over the yellow leucineproducers under these conditions (SI Text, section S1.2).



S3.4. Antagonism Between Selection and Genetic Drift. As shown inSI Text, section S3.1, the mutualistic selection coefficient selectsfor “mixing,” i.e., for cell fractions f in the interior of the intervalð0; 1Þ, as long as the amino acid concentrations are below thecrossover concentrations nEL; nET = 1+ κ= 1:18. However, ex-perimentally local demixing into domains with f = 0 or f = 1 oc-curs for much lower concentrations nEL; nET ≥ 0:25 (Fig. 4 in themain text). This is due to the genetic drift term in Eq. S27. Tocompare the strengths of mutualism and genetic drift, we non-dimensionalized Eq. S27 by measuring distance in units of Ds=Dgand time in units of sc ≡D2

g=Ds to obtain (22, 24)

∂f∂~τ

=∂2f∂~x2

+sð f Þsc

f ð1− f Þ+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ð1− f Þ

p Γ�~x;~τ�

[S42]

with the dimensionless distance ~x= x=ðDs=DgÞ and time~τ= t=ðDs=D2

gÞ. As shown in ref. 24, the strength of mutualismand genetic drift become comparable if mutualistic selectionbecomes comparable to the critical selection coefficient

sc ≡D2

g

Ds[S43]

that describes the strength of local demixing due to genetic drift.The “mutualistic barrier” to demixing, i.e., fixation at f = 0 orf = 1 when the system is close to the stable fraction f p, can beestimated as the maximum of the selection term jsðf Þf ð1− f Þj(Fig. S3B). One can see in Fig. S3B and in Fig. 4D in the maintext that for symmetric mutualism with nE ≡ nEL = nET this bar-rier becomes comparable to sc at about nE = 0:25. Thus, weexpect that mutualism is dominant for nE < 0:25, leading to ex-pansion in a mixed pattern, whereas genetic drift dominates fornE > 0:25, leading to demixed expansion. This is indeed foundexperimentally (Fig. 4 in the main text).For asymmetric mutualism, local fixation of the favored strain

becomes more likely because the stable cell fraction f p movescloser to a fixation boundary f = 0 or f = 1 and because themutualistic barrier to fixation at that boundary becomes smaller(Fig. S4). A more sophisticated analysis of the critical selectioncoefficient sc for asymmetric mutualism is the topic of a forth-coming theoretical paper (26).

S4. Model for Patches of Obligate MutualistsIn colonies of obligate mutualists, the two mutualistic strains forma pattern of blue and yellow “patches” (Fig. 2 in the main text).Because of the effective symmetry of our mutualistic interaction,yellow and blue patches have the same width (Fig. S5A). After aninitial transient of 1–4 d, the patch pattern becomes stationary; i.e.,although patches may form and disappear, the overall pattern andpatch widths remain statistically the same during the expansion.In this section, we derive an approximation for the charac-

teristic width of the patches during steady-state growth by con-sidering the amino acid dynamics. Basically, the width of a patchof leucine-consuming cells is limited by how far leucine candiffuse into it from a neighboring leucine-producing patch beforecellular uptake makes its concentration too low for cells in thepatch interior to grow well. Although the patches are probably notcompletely demixed, here we first assume, for simplicity, thata yellow patch consists entirely of yellow LeuFBR Trp– cells anda blue patch consists entirely of blue Leu– TrpFBR cells. Wedenote the average width of blue and yellow patches parallel tothe front as LL and LT, respectively (Fig. S5B).

Diffusion with Sources and Sinks. We first consider leucine, whosedynamics in the active layer of a colony are described by Eq. S14.In a leucine “production patch” of LeuFBR Trp– cells, we havef = 0 and thus

Production patch : 1d∂∂tnL = ρ− ðnL − nELÞ+Da

d∂2

∂x2nL: [S44]

Similarly, the leucine dynamics in a leucine “consumption patch”with f = 1 are given by

Consumption patch : 1d∂∂tnL = − κnL − ðnL − nELÞ+Da

d∂2

∂x2nL:

[S45]

Here, the uptake rate in the leucine-limited regime ðnL < 1Þ hasbeen used; for higher amino acid concentrations, the sink termwould equal −κL.As shown in Fig. S5B, a consumption patch is surrounded on

both sides by production patches. Because we assume all con-sumption and production patches to have the same widths LL

and LT, respectively, the symmetry of the problem dictates that,in steady state, there is no amino acid current through the middleof the patches. Using the coordinate system shown in Fig. S5B,this leads to the boundary conditions

∂∂x

nL

x=0

= 0 and ∂∂x

nL

x=± LL+LT

2

= 0: [S46]

In addition, the amino acid concentration and current have to becontinuous at the patch boundaries:

nL and ∂∂x

nL continuous at x= ±LL

2: [S47]

Solution to the Diffusion Problem. Because we are interested indescribing the pattern of the mutualistic strains during steady-state growth, we solve the diffusion Eqs. S44 and S45 for thestationary state in which the amino acid profiles do not changeover time. The diffusion problem [S44, S45] with the bound-ary conditions [S46, S47] can be solved analytically in terms ofhyperbolic functions. The minimal and maximal leucine con-centrations within the consumption patch are attained in themiddle x= 0 and at the boundary x=LL=2 of the consumptionpatch, respectively. For obligate mutualism with nEL = 0, thesevalues can be obtained from the full analytic solution

nL;min = nLð0Þ= ρ

Σsinh

LT

2ξp

![S48]

nL;max = nLðLL=2Þ= ρ

Σsinh

LT

2ξp

!cosh

�LL

2la

; [S49]

where

ξp =

ffiffiffiffiffiffiDa

d

r and la =

ffiffiffiffiffiffiffiffiffiffiffiDa=dκ+ 1

r[S50]

are the length scales characterizing the “piling up” of leucine inthe production patch and the decay of the leucine concentra-tion in the consumption patch, respectively. We also used theabbreviation

Σ= cosh�LL

2la

sinh

LT

2ξp

!+ξplasinh

�LL

2la

cosh

LT

2ξp

!: [S51]



Velocity Gradients Within Consumption Patches. In the amino acid-limited regime, the growth velocity is proportional to the concen-tration of the required amino acid (Eqs. S1 and S2). This means thatthe growth velocity varies within a consumption patch: It is smallestin the center and largest at the boundary. If cells grew only in theexpansion direction, these velocity differences along the colonyfront would lead to nonuniform front propagation and thus to anunstable, undulated colony front. However, experiments show thatthe obligate mutualists expand together with a smooth colony front(Fig. 2 in the main text). In addition, the velocity differences gen-erate a selection force perpendicular to the expansion direction. Ina leucine consumption patch, this perpendicular velocity gradientcan be estimated by (6, 27)

V⊥L = vL ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2L;max − n2L;min

q: [S52]

There must be a force that counteracts the selective force gener-ated by the velocity differences within a path.

Patch Boundary Diffusion. Colony growth of nonmotile yeast cellsproceeds due to cell division, which results in pushing andshuffling of cells. This leads to a diffusion-like process that occursparallel to as well as perpendicular to the expansion direction.Experimentally, this diffusive mixing can be observed as thediffusion of sector boundaries with diffusion constant Ds=τg,where τg is the generation time at the patch boundary (27, 28).Because the growth velocity at the boundary equals v= vLnL;max,and because the growth rate at the boundary is g= 1=τg = v=b (SIText, section S1.3), we can estimate the diffusion “velocity” re-quired to diffuse on a length scale of half a patch of width LL as

VDL =4�Ds=τg

�LL

=4Ds vLnL;max

b LL=2lb vLnL;max

LL; [S53]

where we have introduced the boundary diffusion length scalelb = 2Ds=b.

Balance Condition Determines Patch Width. This diffusive processcan counteract the imbalances created by the velocity differencesif both velocities are of the same order of magnitude,

V⊥L =VDL: [S54]

An analogous equation holds for the width LT of tryptophan con-sumption patches. In the case of obligate mutualism, i.e., no exter-nal amino acids nEL = nET = 0, the mutualistic interaction is fullysymmetric so that LL =LT =L. In this case, Eq. S54 yields

L= 2 lb coth�L2la

; [S55]

which can be solved approximately for small patch widths to yieldEq. 5 in the main text,

L= 2 ffiffiffiffiffiffiffiffila lb

p: [S56]

S5. Radial Flux of Amino Acids into and out of a ColonyIn this section, we derive an approximate description for the diffusiveflux of amino acids into and out of the actively growing boundarylayer of a colony. We show that, on the experimental timescales ofmany hours to days, the amino acid diffusion dynamics can be ap-proximately described by a chemostat-like process as used in Eq. 1 inthe main text, and we derive an expression for the effective influx/outflux rate d in terms of the colony growth parameters.

S5.1. Nutrient Diffusion and Colony Growth. Diffusion timescales. Thecells in a growing yeast colony obtain their nutrients, e.g., glucose andamino acids, from the agar medium on which they grow. Nutrientsreach cells via diffusion with a diffusion constantDnut of a few squaremillimeters per hour (15, 29). This parameter value means that dif-fusion can sample the 5-mmheight of the agar in the Petri dishwithina few generations (generation time τg ≈ 1:5 h). Thus, on experi-mental timescales of several days, we integrate out the diffusionperpendicular to the agar surface and consider the 2D diffusion ofnutrients only in the plane of the colony (Fig. S6). In addition, dif-fusion is fast compared with the colony expansion velocity ofv≈ 20 μm=h:A colonywill outrun nutrient diffusion only after aboutDnut=v2 ≈ 1 y.We can therefore ignore colony expansion for nutrientdynamics on experimental timescales of several days.The active layer. We first consider the diffusion and uptake of anessential nutrient that is not produced by the cells, such as glucose.After an initial transient of 1–2 d, in which the colony depletes theglucose beneath it, new glucose reaches the colony by radial dif-fusion toward the colony. This glucose allows only the cells in theactive layer of size b= 40 μm near the colony boundary to grow(11, 12). Consistent with the chemostat-like nature of the frontierregion, these cells in the colony boundary consume all of theglucose, so that cells in the colony interior cannot grow. This ge-ometry is shown in Fig. S6A.

S5.2. Radial Amino Acid Dynamics. We now consider the radialdiffusion dynamics of leucine (tryptophan works analogously),which is produced or taken up by cells in the active layer ofa colony consisting of one or both of the mutualistic strains. Here,we are interested in length scales larger than the angular width ofpatches or sectors. We therefore consider the cells in the activelayer to be essentially well mixed, so that they can be described bythe r-independent concentrations cL and cT of Leu– TrpFBR andLeuFBR Trp– cells, respectively. The 2D dynamics of the radialleucine concentration ½leu�r≡½leu�ðr; tÞ in the active layer can thenbe described by the radially symmetric diffusion equation,

for rl < r< rc : ∂∂t½leu�r =Da

1r∂∂r

�r∂∂r

½leu�r

−kL

½leu�ccL ½leu�r + rLcT; [S57]

where rc is the colony radius and rc − rl = b is the size of the activelayer. Outside of the colony, leucine diffuses freely,

for r≥ rc : 1Da

∂∂t½leu�r =

1r∂∂r

�r∂∂r

½leu�r: [S58]

This diffusion causes a diffusion zone around the colony, withinwhich the leucine concentration differs from its concentration nELin the agarmedium far away from the colony (the concentration withwhich the medium was prepared). On the experimentally relevanttimes of 4–7 d, the size of the diffusion layer, rd − rc ∼

ffiffiffiffiffiffiffiffiffiffi4Dat

p, in-

creases very slowly from34 mm to 44 mm.For distances larger thanthis, the leucine concentration equals its value far away fromthe colony,

½leu�rðr≥ rdÞ= ½leu�E: [S59]

In a quasi-stationary approximation, we can solve the diffusionEqs. S57 and S58 for their stationary states with the boundarycondition [S59]. In addition to the boundary condition [S59],the amino acid concentration and current have to be contin-uous at the connection between the active layer and the dif-fusion zone:



½leu�r; ∂½leu�r∂r

continuous at r= rc: [S60]

Furthermore, because cells in the colony interior are inactive,amino acids diffuse freely for r< rl, with a zero-current conditionin the colony center at r= 0 because of symmetry. This translatesinto a zero-current condition at the boundary between the activelayer and the colony interior:

∂½leu�r∂r

ðr= rlÞ= 0: [S61]

Solution to the diffusion problem. The diffusion problem [S57, S58]with the boundary conditions [S59–S61] can be solved analyti-cally. The concentration within the active layer is a superpositionof the modified Bessel functions of the first and second kind,whereas the concentration decays logarithmically in the diffusionzone. It is plotted for yeast parameters in Fig. S6B.Flux into and out of the active layer.As can be seen in Fig. S6B, radialdiffusion happens on length scales large compared with the ra-dial size of the active layer b= 40 μm. Thus, we can approximatethe leucine concentration within the active layer, nL, by theconcentration at the colony boundary, ½leu�≡ ½leu�rðrcÞ. The fluxat the colony boundary can be calculated from the exact solutionto the diffusion problem and has the simple form

jL ≡ jLðrcÞ=Da∂½leu�r∂r

r=rc

= −D

rc ln�rdrc

�½leu�− ½leu�E�: [S62]

Due to the boundary flux jL, the number of leucine molecules inan area element b Δx of the active layer changes in a time in-crement Δt according to

½½leu�ðt+ΔtÞ− ½leu�ðtÞ� b Δx= jL Δx Δt; [S63]

or

∂∂t½leu�= jL

b= − d

�½leu�− ½leu�E�

[S64]

with the diffusive flux constant

d=Da

b rc ln�rdrc

=Da

b rc ln�1+

ffiffiffiffiffiffiffiffiffiffi4Dat

prc

: [S65]

This diffusive flux is balanced by leucine secretion and uptake inthe active layer, leading to an effective dynamic equation for thenutrient concentration nL in the active layer,

∂∂t½leu�= − kLcL½leu�+ rLcT − d

�½leu�− ½leu�E�: [S66]

A similar equation holds for tryptophan dynamics. With the aminoacid diffusion constant Da = 3 mm2=h, the size b= 40 μm of theactive layer, a colony radius of rc = 3 − 5 mm, and the sizeffiffiffiffiffiffiffiffiffiffi4Dat

p≈ 34 − 44 mm of the diffusion zone, the diffusive flux

constant is of the order of d≈ 5=h.

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19. Siegel SJ, Swenson PA (1964) Loss of nucleotide and amino acid pool componentsfrom yeast cells following exposure to ultraviolet and photoreactivating radiations. JCell Physiol 63:253–260.

20. Swenson PA, Dott DH (1961) Amino acid leakage and amino acid pool levels ofultraviolet-irradiated yeast cells. J Cell Comp Physiol 58:217–232.

21. Koster DA, Mayo A, Bren A, Alon U (2012) Surface growth of a motile bacterialpopulation resembles growth in a chemostat. J Mol Biol 424(3–4):180–191.

22. Korolev KS, Avlund M, Hallatschek O, Nelson DR (2010) Genetic demixing andevolution in linear stepping stone models. Rev Mod Phys 82(2):1691–1718.

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25. Holland JN, DeAngelis DL (2010) A consumer-resource approach to the density-dependent population dynamics of mutualism. Ecology 91(5):1286–1295.

26. Lavrentovich MO, Nelson DR (2013) Asymmetric mutualism in two and three di-mensions. arXiv:1309.0273v1.

27. Hallatschek O, Nelson DR (2010) Life at the front of an expanding population.Evolution 64(1):193–206.

28. Hallatschek O, Nelson DR (2008) Gene surfing in expanding populations. Theor PopulBiol 73(1):158–170.

29. Stecchini ML, et al. (1998) Influence of structural properties and kinetic constraints onBacillus cereus growth. Appl Environ Microbiol 64(3):1075–1078.



0 50 100 1500

2

4

0 0.5 1

10

15

20

obligate mutualism

0.99

0.500.30

0.85

0.15

0.70

0.01

f

Steady−state expansion velocity

colo

ny r

adiu

s [m

m]

Colony expansion history

time [h] start fraction

no mutualism

BA

vgr

owth

vel

ocity

[

m/h

]

f0

no mutualism

obligate mut.

0

Fig. S1. Colony expansion velocities. (A) Radius increase over time for individual colonies of the mutualistic strains Leu– TrpFBR and LeuFBR Trp–, inoculated atdifferent start fractions f0 of blue cells, under conditions of no (red circles) and obligate mutualism (green circles). In the no mutualism case, all colonies exhibitsimilar radius increases and reach a velocity of 20.7 ± 0.4 μm/h for times larger than 90 h. In contrast, obligately mutualistic colonies start expanding at differentpoints in time: Colonies with start fraction f0 = 0:15 start expanding at later times (solid purple line through green circles), and colonies with f0 = 0:01 do notexpand at all (solid red line through green circles). Expanding colonies reach a velocity of 10.7 ± 0.6 μm/h for times larger than 90 h, indicating that themutualistic strains have reached steady-state expansion. (B) Growth velocities for times larger than 90 h are independent of the start fraction f0 both fornonmutualistic (red circles) and for obligately mutualistic (green circles) colonies. Solid lines indicate the average velocities, which are the slopes of the cor-responding red and green dashed lines in A.

0 100 200 300 6500

5

10

15

20

grow

th v

eloc

ity [

m/h

]

0 1000 2000 45000

5

10

15

20

grow

th v

eloc

ity [

m/h

]

Leu Trp , liquidLeu Trp , liquid

Leu Trp , plateLeu Trp , plate −

−

A

−

−0.25

B

−

−

−

−

0

grow

th r

ate

[1/h

]

E

0.5

[leu] [ M]

0

grow

th r

ate

[1/h

]

0.5

0.25

FBR

Leu Trp , liquid

FBR

Leu Trp , plate

FBR

FBR

[trp] [ M]E

FBR

FBR

Leu Trp , liquid

FBR

FBR

Leu Trp , plate

Fig. S2. Comparison of growth velocities on plates and growth rates in liquid of LeuFBR Trp– (yellow squares for plates and green circles for liquid) and Leu–

TrpFBR (blue diamonds for plates and red stars for liquid) for (A) varying leucine and (B) varying tryptophan concentrations. Growth velocities and rates areindependent of nonrequired amino acids, but increase linearly with a required amino acid concentration before they saturate at a plateau. When the liquidgrowth rates (red stars for Leu– TrpFBR and green circles for LeuFBR Trp–) are scaled by a factor of bvg = 40 μm, they are very similar to the growth velocities onplates (blue diamonds for Leu– TrpFBR and yellow squares for LeuFBR Trp–). However, crossovers from linear to plateau regimes for liquid growth occur atconcentrations ½leu�c = 762 μM (red vertical line) and ½trp�c =97:6 μM (green vertical line) that are lower by a factor of 1+ κ= 1:18 than the correspondingcrossover concentrations (blue and yellow vertical lines) for the plate growth velocities, as explained in SI Text, section S2.2.

Fig. S3. “Phases” of mutualistic selection for yeast parameters as listed in Table S2. (A) Deterministic “phase diagram” in the plane of external amino acidconcentrations ðnEL,nETÞ. Insets show the selection term sðfÞfð1− fÞ in Eq. S27 on a vertical scale of −0:5 to 0.5 as a function of the fraction f of leucine-requiringcells in the interval 0≤ f ≤1. A positive selection term, sðfÞfð1− fÞ> 0, acts to increase f, whereas negative values drive toward smaller f. The selective forcevanishes for a selection term of zero, sðfÞfð1− fÞ= 0. The selection term can be in one of five possible phases designated as CTMCL, CTNCL, CTN, NCL, and N. Inthe phase names, M denotes the mutualistic regime in which both strains are amino acid limited, nL, nT < 1, so that they benefit from their mutualistic in-teraction. N represents the neutral regime with nonlimiting amino acid concentrations nL, nT > 1 and therefore no mutualism. CT denotes a commensal regimein which the leucine-requiring strain is saturated, nL > 1, whereas the tryptophan-requiring strain is still limited, nT < 0. CL describes the inverse situation withnL < 1, nT > 1. The vertical red and green lines (Insets) mark the transition frequencies fC1 and fC2 between the different regimes. The pink-shaded phases CTMCL

and CTNCL represent mutualistic phases, in which selection drives the system toward a stable frequency f*∈ ð0,1Þ within the M regime (dark pink) or towarda neutral N region in the interior of the interval f ∈ ð0,1Þ (light pink). In the green-shaded, nonmutualistic phases CTN, N, and NCL, the system has no barrier tofixation at f = 0 and/or f = 1. (B) Selection term sðfÞfð1− fÞ along the diagonal nEL =nET≡nE with the same notation as in A. The black horizontal lines at ± sc,with sc =D2

g=Ds = 0:18, become comparable to the selection term around nE = 0:25.



Fig. S4. (A) The frequency-dependent selection term sðfÞfð1− fÞ of Eq. S27 for different amino acid asymmetries nEL −nET, with fixed total amino acid amountnEL +nET = 0:6. For increasing asymmetry nEL −nET, fixation of leucine consumers at f =1 becomes more likely, because (i) the stable fraction f* (indicated byvertical lines) of leucine consumers moves closer to f = 1 and (ii) the barrier toward fixation at f = 1 (the absolute value of the minimum of the selection term)decreases. (B and C) Dynamics of the colony boundary fraction f, as a function of radial distance R−R0 from the homeland of radius R0, for colonies withdifferent inoculation fractions f0. (B) For obligate mutualism, the boundary fraction reaches the steady-state fraction f*= 0:5 within a 0.5-mm radial expansionfrom the homeland with weak transient oscillations. (C) For no mutualism, the boundary fraction increases during expansion because blue Leu– TrpFBR cells are2% more fit than yellow LeuFBR Trp– cells. Data points in B and C are from flow cytometry of cells harvested from colony boundaries. Solid lines are solutions toselection dynamics with selection coefficient sðfÞ of Eq. S28 for obligate and s= 0:02 for no mutualism.

0 0.5 1 1.50

25

50

75

radial distance R−R [mm]

patc

h w

idth

L [

m] yellow patch width

LLLT LT

2LL L T

2 2 +L L

A

0

productionconsumption

patch

productionpatchpatch

Ln

leuc

ine

conc

entr

atio

n

0front coordinate x

blue patch widthB

width of all patches

Fig. S5. Determination of patch widths along the front coordinate x. (A) Average patch widths (parallel to the front) as a function of the radial distance R−R0

from the homeland with radius R0, determined via different methods for the same colony of obligate mutualists. Yellow patch width (yellow data) and bluepatch width (blue data) are the average angular local maximum-to-minimum and minimum-to-maximum distances, respectively, of the derivative of the yellowfluorescence intensity. The width of all patches (green data) is reproduced from Fig. 2D in the main text. Because it is is calculated as the circumference dividedby the number of local maxima of the yellow fluorescence intensity, it does not equal the average of the yellow and blue patch widths. All patch widths areidentical within error bars and saturate at L= 52 μm (black horizontal line) for radial distances larger than 1 mm. (B) The amino acid leucine is secreted byLeuFBR Trp– in yellow production patches of width LT and consumed by Leu– TrpFBR cells in blue consumption patches of size LL. The leucine concentration nL

thus “piles up” in the production patches and decreases in the consumption patches. The width of the consumption patch is limited by this decrease, asexplained in the main text.

Fig. S6. (A) Geometry to calculate amino acid diffusion within and out of a colony. In a colony of radius rc, only cells in an active layer of size b= rc − rl near thecolony boundary grow, whereas cells in the colony interior r ≤ rl do not grow due to depletion of essential nutrients (e.g., glucose). In the active layer, aminoacids diffuse and are produced and taken up by cells. Outside the colony, amino acids diffuse freely and create a diffusion zone of radial size rd − rc ∼

ffiffiffiffiffiffiffiffiffiffiffi4Dat

p. (B)

Radial amino acid concentration profile nLðrÞ= ½leu�r=½leu�c for yeast parameters (Table S2) for external nutrient concentrations nEL = ½leu�=½leu�c = 0 (red) andnEL = 0:5 (blue).



Table S1. Strains used in this work

Strain name Strain ID Genotype at LEU4 and TRP2 loci Fluorescent proteins

LeuFBR Trp– yMM60 LEU4FBR trp2Δ::NATMX4 ymCitrine, yCeruleanLeu– TrpFBR yMM65 leu4Δ::HPHMX4 TRP2FBR ymCherry, yCerulean

yMM61 LEU4FBR trp2Δ::NATMX4 ymCherry, yCeruleanyMM64 leu4Δ::HPHMX4 TRP2FBR ymCitrine, yCeruleanyMM26 LEU4FBR trp2Δ::NATMX4 ymCitrineyMM31 leu4Δ::HPHMX4 TRP2FBR ymCherryyMM29 LEU4 trp2Δ::NATMX4 ymCherryyMM32 leu4Δ::HPHMX4 TRP2 ymCitrine

Noninteracting strain yJHK111 LEU4 TRP2 ymCitrineNoninteracting strain yJHK112 LEU4 TRP2 ymCherry

All strains have the genetic background W303 MATa can1-100. All yMM strains have in addition hmlαΔ::BLEleu9Δ::KANMX6. The fluorescent markers are incorporated as follows: ymCitrine, his3Δ::prACT1-ymCitrine-tADH1:HIS3MX6; ymCherry, his3Δ::prACT1-ymCherry-tADH1:HIS3MX6; and yCerulean, prACT1-yCerulean-tAD-H1@URA3.

Table S2. Parameters used in the theoretical calculations of SI Text, sections S3 and S4

Symbol Definition Value Description

κ kL½leu�c

cd=

kT½trp�c

cd 0.18 Reduced amino acid uptake rate

ρ rL½leu�c

cd=

rT½trp�c

cd 1.09 Reduced amino acid secretion rate

laffiffiffiffiffiffiffiffiDa=d1+ κ

q710 μm Length scale of amino acid depletion

lb2Dsb 0.83 μm Length scale of patch boundary diffusion

sc D2g=Ds 0.18 Critical selection strength to overcome genetic drift

As given by the definitions, these “reduced” parameters are compounds of directly biologically relevantparameters: the leucine and tryptophan uptake rates kL and kT, the leucine secretion rates rL and rT, the leucineand tryptophan crossover concentrations ½leu�c and ½trp�c, the cellular surface carrying capacity c, the amino aciddiffusive flux rate d, the amino acid diffusion constant Da, the size b of the actively growing layer, the cellulardiffusion constant Ds, and the genetic diffusion constant Dg. The determination of the parameter values isdescribed in SI Text, section S1.4.


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