Altruism, Cheating, and Anticheater Adaptations in

13 Pages·2002·0.19 MB·English

vol. 160, no. 1 the american naturalist july 2002
Altruism, Cheating, and Anticheater Adaptations
in Cellular Slime Molds
Richard Ellis Hudson,* Juliann Eve Aukema,† Claude Rispe,‡ and Denis Roze§
DepartmentofEcologyandEvolutionaryBiology,Universityof Innature,altruisticindividualsthatsacrificeforthebenefit
Arizona,Tucson,Arizona85721 of other members of their group are common. Sterilein-
dividualswithineusocialinsectcoloniesraisetheprogeny
SubmittedSeptember21,2000;AcceptedDecember13,2001
oftheirqueen,somaticcellswithinmulticellularorganisms
insurethepropagationofthegermline,mitochondriaand
chloroplasts within eukaryoticcellscontributetothesuc-
cess of the nuclear genome, and DNA polymerases rep-
abstract: Cellular slime molds (CSMs) possessa remarkablelife licate genes of other proteins as well as their own(Szath-
cyclethatencompassesanextremeactofaltruism.CSMcellsliveas
ma´ryandMaynardSmith1995).Whencellularslimemold
individual amoebae until starved, then aggregate and ultimately
(CSM) amoebae starve, they join together and eventually
transformthemselvesintoamulticellularfruitingbody.Thisfruiting
transform themselves into a fruiting body (fig. 1). Some
bodyconsistsofstalkcells(altruiststhateventuallydie)andspores
(thebeneficiariesofthissacrifice).Altruisticsystemssuchasthisare amoebae become spores, and others become stalk cells;
vulnerabletocheaters,whichareindividualsunrelatedtothealtruists the former disperse, and the latter die while constructing
that obtain the benefits provided by them without reciprocating. thestalk(WhittinghamandRaper1960).Thus,CSMstalk
Here, we investigate two forces that can maintain CSM altruism
cellsarealtruiststhat,inordertoaidsporedispersal,make
despitecheating:kinselectionandanticheateradaptations.First,we
the greatest sacrifice possible.
present new kinship-based models based on CSM developmental
Altruists can theoretically be eliminated by cheaters,
biologytoevaluatetheefficacyofkinselection.Thesemodelsshow
that stalk-making genotypes canstillbemaintainedwhenaggrega- individuals that obtain the benefits provided by the al-
tionsareinitiatedbymultiple“founder”spores,providedthatspores truists without reciprocating (Frank 1994; Michod
ofstalklessfruitingbodieshavelowratesofdispersalanddispersal 1997). The establishment and persistenceofcheaterge-
success is a concave function of stalk height. Second, we review
notypes require that chimeric groups (groups contain-
proposals that several features of CSM development, such as the
ing both altruists and cheaters) form with sufficient
chemical suppression of the redifferentiation of prestalk cells into
frequency. Because chimeric CSM groups can form by
prespores,actasanticheateradaptations.
theuniqueprocessofcoaggregationofunrelatedamoe-
Keywords:altruism,cellularslimemolds,cheater,Dictyostelium,kin bae, these organismsshouldbeunusuallysusceptibleto
selection,multicellularity.
cheating. In fact, several cheating CSM strains (strains
that have an unusually high propensity to form spores)
have been identified (Filosa 1962; Buss 1982; Hilson et
* Corresponding author. Present address: Department of Biology, Indiana
al. 1994; Ennis et al. 2000; Strassmann et al. 2000; re-
University,Bloomington,Indiana47405;e-mail:[email protected].
viewed in Pa´l and Papp 2000). To assay for cheaters,
† Present address: Pacific Northwest Research Station, U.S. Departmentof
spores of two strains are first sown together into one
Agriculture Forest Service, Olympia, Washington 98512; e-mail:jaukema@
alumni.brown.edu. petri dish. A few days later, spores are randomly har-
vestedfromtheresultingfruitingbodies,andthentheir
‡ Presentaddress:Unite´MixtedeRechercheBiologiedesOrganismesetdes
Populations Applique´e a` la Protection des Plantes, Institut National de la genotypes are determined. In early experiments of this
Recherche Agronomique, Domaine de la Motte, BP35327, 35653 LeRheu, type, the ratio of Dictyostelium mucoroides strain DM-
France;e-mail:[email protected].
4-TYPE to strain DM-4-VAR increased from 1:1
§ Present address: Centre d’Etudes sur le PolymorphismedesMicroorgan- among the spores sown to 5:1 among those harvested,
ismesEquipe:EvolutionTheoriqueetExperimentale,InstitutdeRecherche
indicating that the former was less likely to form stalk
sur le Developpement, 911 avenue Agropolis, BP5045, 34032 Montpellier,
cells (Filosa 1962). In later studies, where harvested
France;e-mail:[email protected].
Am.Nat.2002.Vol.160,pp.31–43.(cid:1)2002byTheUniversityofChicago. spores were used to start subsequent “generations” of
0003-0147/2002/16001-0003$15.00.Allrightsreserved. fruiting bodies, putative cheaters were greatly enriched
32 The American Naturalist
Figure1:Asexuallifecycleandstepsofsimulatedlifecycle
in the final harvest (Buss 1982; Ennis et al.2000).Con- theyareinchimericaggregations(Strassmannetal.2000).
trol experiments have shown that the increase in fre- Our models focus on fixed-allocation cheating because
quency of putative cheaters was not due to a higher variable-allocation cheating should not influence clonal
amoeba-stage replication rate (Buss 1982). Although stalk sizes; in theoretical competitions between variable-
most of these studies employed laboratorymutants,in- allocation strains, the victor is always a strain thatsetsits
vestigations of isolates from soils show that cheating clonalstalksizeatthesamelevelselectedforintheabsence
CSM strains occur naturally (Buss 1982; Strassmann et of cheating (Matsuda and Harada 1990).
al.2000).CheatinghasalsobeendemonstratedforMyx- One might mistakenly conclude that fixed-allocation
ococcus xanthus, a eubacterium with a similar life cycle cheating is unimportant for two reasons: few naturally
(Velicer et al. 2000). occurringcheatersareofthefixed-allocationtype(Strass-
Cellularslimemoldcheateramoebaecomeintwotypes. mann et al. 2000), and fixed-allocation cheaters should
Fixed-allocation cheaters allocate an unusually low pro- always be eliminated by certain variable-allocation com-
portionoftheircellstothestalkinbothchimericandall- petitors (those that form stalk cells in chimeras with the
cheater groups, whereas variable-allocation cheaters allo- same propensity as the former but that produceoptimal-
catealowproportionoftheircellstothestalkonlywhen sized stalks in clonal groups). However, although fixed-
Altruism and Cheating in Slime Molds 33
Table 1: Variablesand parametersused in the text
Notation Description
All models:
a, a Stalk cell to total cell ratio
n Number of founders
k Number of descendantamoebasper founder
N Number of cells in an aggregation/slug/fruitingbody
W Fitness
F Fecundity
D Dispersibility
c Cost of stalklessness
x Exponentthat controlsshapeof dispersibilityfunction
a Stalk to total cell ratio of a given fruitingbody, g
g
Constant-proportionmodel:
a Constantstalkto spore ratio specifiedby genotype j
j
a Stalk to total cell ratio thatmaximizesfitness
m
a∗ Evolutionarilystablestrategyvalue of a
r Kinshipcoefficient
Constant-sizemodel:
a Stalk cell proportion in all-altruistfruitingbody specifiedby genotype j
j
A Stalk cell number in all-altruistfruitingbody specifiedby genotype j
j
N Number of altruisticamoebasin an aggregation
a
allocationcheatersarerelativelylesscommon,theyarenot thesemodelstoanalyzewhetherkinselectionactingalone
rare; they comprised one out of 13 soil isolates in one might be sufficient to maintain altruism in CSMs, a pro-
study(Buss1982)anduptofouroutof12cheatersiden- posal that has been criticized (Atzmony et al. 1997).
tified in another (Strassmann et al. 2000). Furthermore, Kin selection prevents the elimination of altruistic ge-
scarcitydoesnotnecessarilydenoteunimportance;therel- notypesbystrengtheningbetween-groupselectionrelative
ative dearth of fixed-allocation cheaters may reflect the to within-group selection (Sober and Wilson 1998; for
greater efficacy of kin selection against them than against applicationstospecificsystems,seeFrank1994;Rispeand
cheaters of the other type. Regarding the second reason, Moran 2000). Certain group traits, “anticheater” adapta-
these competitively superior variable-allocation strains tions, also contribute to the maintenance of altruism via
probably do not arise as often as their fixed-allocation the same mechanism. We close by describing and evalu-
counterparts because they have a more complex pheno- ating several proposals for such adaptations in CSMs.
type;thesestrainsmustadjusttheirallocationinchimeras
as a function of the allocation patterns of the other
Kin Selection in Cellular Slime Molds:
strain(s) in the group (Matsuda and Harada 1990). Fur-
Two New Models
thermore, the ostensibly invincible variable-allocation
strains postulated by Matsuda and Harada (1990), which Because aggregations descended from two or more CSM
only cheat against strains they recognize as foreign, are sporesareprobablynotrare(WilsonandSober1989),we
still theoretically vulnerable to certain fixed-allocation have used our models to explore the level of stalk for-
cheaters(mutantsarisingfromthemthattheycannotdis- mation that can be maintained when there are multiple
tinguish as alien). founders. Our models simulate competitionsbetweenge-
Inthisarticle,wedescribetwokinship-basedmodelsof notypesthatcodefordifferentlevelsofstalkformationin
competitions between fixed-allocation cheatersandaltru- alifecyclethatconsistsoffourstages:randomassociation
ists.Previouskinship-basedmodelsofCSMslackfeatures ofgerminating“founder”spores,growthandaggregation,
necessary to evaluate the efficacy of kin selectionbecause differentiation, and dispersal (fig. 1; see table 1 for a list
they make unrealistic assumptions: altruists in chimeras of the models’ parameters).First,foundersporesareran-
neverbecomespores(MaynardSmith1964),fruitingbod- domly distributed into groups of size n. In the second
iesofdifferentsizeshavethesamedispersalsuccess(Arm- stage, each founder gives rise to k descendant amoebae,
strong1984),andstalklessfruitingbodieshavezerofitness which then aggregate with the other amoebae of their
(Matapurkar and Watve 1997). Therefore, we have for- foundergrouptoproduceanaggregationofNpnkcells.
mulatedmodelsthatrelaxtheseassumptions.Wethenuse (We do not model the movement of foraging amoebae
34 The American Naturalist
because this process should be less important than spore
dispersal in generating the spatial distribution of cells.)
During the third stage, cells differentiate into stalk cells
and spores in accordance with their genotypes; each cell
of genotype j has an a (model 1) or a (model 2) prob-
j j
ability of becoming a stalk cell. Finally, spores disperse,
with their success depending on the height of their stalk,
which is a function of the values of a (or a) of the ge-
j j
notypes that make up each fruiting body.
The fitness of a genotype in a given fruiting body is a
function of both the number of its spores and the size of
the stalk:
W(a)pF(a)D(a ). (1)
j j g
The variables W(a) and F(a) represent, respectively, the
j j
fitnessandfecundity(numberofspores)ofgenotypejper
founder cell of that genotype, and D(a) stands for the
g
dispersibility of a fruiting body, g, where the proportion
of cells that are stalk cells equals a; the value of D(a)is
g g
anincreasingfunctionofa,andF(a)pk(1(cid:1)a).Given
g j j
i different genotypes, a is the weighted average of the i
g
valuesofa.HavingDincreasewitha meansthathaving
j g
a larger stalk bestows a greater benefit, such as enhanced
dispersal (Bonner 1982) or enhanced displacement of
spores away from harmful agents in the soil (Gadagkar
and Bonner 1994). A D that increases with a is also ap-
g
propriate if prestalk cells provide benefits that increase
withtheirfrequency,suchasanimprovedabilitytolocate
and move to better dispersal sites (Inouye and Takeuchi
1980;MiuraandSiegert2000).Althoughdispersibilityin-
creases with a, having too many stalk cells can be detri-
mentalbecauseagenotypethatincreasesitsDbyincreas-
ing its a also decreases its fecundity.
j
OurformulaforDcontainstwoparameters:c,thecost
of stalklessness, and x, which determines the shape of D:
D(a ,c,x)p1(cid:1)c(cid:2)cax. (2)
g g
Theparametercpermitstheassignmentofanonzerodis-
persibilitytostalklessfruitingbodies.Sporesofsuchfruit-
ingbodiesmayoccasionallyhavesome“dispersal”success;
they may be dispersed at a rate lower than that of spores
borne on stalks, or they may germinate without being
dispersed. Thus, we have set D (a p0) equal to 1(cid:1)c;
g
c then represents the cost of having no stalk.
Values of x greater than, equal to, and less than one
generate convex, linear, and concave D versus a curves,
respectively. (Figure 2 shows the effects of x on fitness.)
The shape of the D versus a curve describes where the
Figure 2: Fitness asa function of a, c,and x. Within eachgraph,the
greatestgainsinthedispersibilitycomponentoffitnessare curves(fromtoptobottom)areforcp0,0.2,0.4,0.6,0.8,and1.This
made; if, for example, dispersal success increases at a formulationappliestobothmodels.
Altruism and Cheating in Slime Molds 35
higher rate per unit length for taller stalks, thenaconvex
Dismostappropriate.Smallvaluesofxaremostsuitable
if there is a maximum dispersibility asymptotically ap-
proached by taller and taller fruiting bodies.
The Constant-ProportionModel
We have formulated two models that represent different
ways that stalk-making genotypes allocate their cells in
chimeric groups. In our first model, each genotypecodes
for a constant proportion of stalk cells, whichisexpected
whenthepropensitytobecomeaprestalkcelldependson
Figure3:Thevalueofathatmaximizesfitness,a ,asafunctionofc
acell-autonomousmechanism.Inbothpurefruitingbod- m
and x. The value of D is concave forx!1 and convex for x11. The
iesandchimeras,thefractionofallcellsofgenotypejthat valueofa is0forxp1,c!0.5andforxp2,c!0.75.
m
are stalk cells always equals a (i.e., the presence of cells
j
with other genotypes has no effect on genotype j’s cell
allocation).
Within each chimeric group, the genotype with the
No Chimeras, Nonlinear Dispersability (np1, x(1).
lower value of a is cheating on the other genotype, and
j When x can vary, the value of a that maximizes fitness
its frequency increases within groups. Consider two ge- m
is the solution of the following equation (see also fig. 3;
notypeswherea1!a2.Supposethattheratioofgenotype when the solution is negative, a p0):
1 amoebae to genotype 2 amoebae is initially 1:1; the m
ratio of their spores in the resulting fruiting body will be
x 1(cid:1)c
(1(cid:1)a):(1(cid:1)a )11. ax (cid:1) ax(cid:1)1(cid:2) p0,for0!c≤1. (5)
1 2 m x(cid:2)1 m c(x(cid:2)1)
To analyze the constant-proportion model, we deter-
mined the evolutionarily stable strategy (ESS; Maynard
Varyingxaffectstheoutcomesasfollows:Stalk-makingge-
Smith and Price 1973), which is the strategy that resists
notypes have the highest fitness only when x is below a
displacement by all other strategies. The ESS is the value
of a∗ that satisfies the following equation: threshold value because only then is the dispersibility of
short-stalked fruiting bodies significantly higher than that
F ofstalklessfruitingbodies.ThismeansthatconcaveDfunc-
(cid:1)Wa,a∗ p0, (3) tions generallyfavoraltruistsandconvexDfunctionsgen-
(cid:1)a apa∗ erallyfavorstalklessgenotypes.Butthisdoesnotmeanthat
more extreme concavity always favors higher levels of al-
whereW representsthefitnessofafounderofgenotype truism; in many cases, increasing concavity (by decreasing
a,a∗
ainagroupwherealltheotherfoundersareofgenotype x) actually decreases a (fig. 3). In such cases, decreas-
m
a∗. ing x decreases the dispersal advantage of intermediate-
sizedstalksrelativetosmallonesandthusselectsforsmaller
No Chimeras, Linear Dispersability (np1, xp1). In stalks.
these conditions, the genotype that maximizes fruiting The parameters c and x interact as follows: As c in-
body fitness, m, drives all others to extinction: creases, the threshold value of x increases, making it less
likelythatthestalklessgenotypehasthemaximumfitness.
{ 1 As c increases for a given x, the stalk size that maximizes
0 forc≤
fitness increases; in other words, larger values of c favor
a∗pamp 2c(cid:1)1 12. (4) greater degrees of altruism.
forc1
2c 2
Chimeras Form, Linear Dispersability (n11, xp1). To
Equation (4) shows that when c, the cost of stalklessness, solveequation(3)fortheESS,wehavesetW ,thefitness
a,a∗
islow,thestalklessformhasthemaximumfitness.Asthe ofagenotype-afounderinagroupwhereallotherfoun-
costofstalklessnessincreasesaboveone-half,thestalksize dersareofgenotypea∗,equaltothegenotype’sfecundity,
that maximizes fitness increases. k (1–a), multiplied by the group’s dispersibility. When
36 The American Naturalist
xp1andcp1,thegroup’sdispersibilityequalsitsstalk
cell proportion, a p(1/n)a(cid:2)[(n(cid:1)1)/n]a∗, giving
g
(1 n(cid:1)1 )
Wa,a∗pk(1(cid:1)a)na(cid:2) n a∗ . (6)
Solving for the ESS gives
1
a∗p . (7)
1(cid:2)n
The ESS can also be expressed in terms of the kinship
coefficient, r (Hamilton 1964, 1970; Frank 1994), which
equalsda/da.Todoso,wetakethederivativeofequation
g
(6):
F
(cid:1)W [ d ]
a,a∗ pk(cid:1)a (cid:2)(1(cid:1)a) a
(cid:1)a apa∗ g da g
pk[(cid:1)a∗(cid:2)(1(cid:1)a∗)r]. (8)
Setting (cid:1)W/(cid:1)ap0 and solving for a∗ in terms of r gives
Figure4:Evolutionarilystablestrategystalksizeasafunctionof(A)c
andx(whennp2)and(B)nandc(whencp0.9).
r
a∗p . (9)
1(cid:2)r
{ n
0 forc≤
When each aggregation is founded by only one spore a∗p 1(cid:1)(1(cid:1)c)(1(cid:2)n) 1(cid:2)nn. (11)
(np1),theamoebaeinonegroupareallrelated,kinship forc1
(1(cid:2)n)(cid:1)(1(cid:1)c)(1(cid:2)n) 1(cid:2)n
is at its maximum (rp1), and the ESS is the same as
a , the value of a that maximizes group fitness. As n
m Equation(11)showsthatwhenthecostofstalklessness
increasesandkinshipdecreases,a∗decreases,althougha
m is low or when the number of founders is high, the
remainsunchanged.Thedifferencebetweena∗anda can
m ESS is to make no stalk (a∗p0). When c exceeds the
be thought of as the amount of potential fitness not
n/(1(cid:2)n) threshold, the ESS increases as c increases but
achieved; this difference grows larger as n increases.
decreases as n increases.
The parameter k is the number of amoebae descended
from each founding spore. Because it does not differ
among genotypes, the value of k does notaffectthevalue Chimeras Form, Nonlinear DispersabilityFunctions(n11,
of the ESS nor does it affect the outcomes of the second x(1). When x can vary, the fitness ofstrategya∗(when
model. rare) becomes
Whenxp1butc(1,thefitnessofstrategya(when
rare) becomes [ ]
(1 n(cid:1)1 )x
Wa,a∗pk(1(cid:1)a)1(cid:1)c(cid:2)c na(cid:2) n a∗ . (12)
[ (1 n(cid:1)1 )]
Wa,a∗pk(1(cid:1)a)1(cid:1)c(cid:2)c na(cid:2) n a∗ . (10) The ESS is the solution of the following equation:
TheESSforthisfitnessfunctionisasfollows(seealsofig. ax(cid:1) x ax(cid:1)1(cid:2)(1(cid:1)c)n p0. (13)
4): x(cid:2)n c(x(cid:2)n)
Altruism and Cheating in Slime Molds 37
Table 2: Numberof differentcell typesin chimericfruitingbodiesof constant-sizemodel
Total altruistcells Total stalkcells Total spores Cheater-genotypesporesa Altruist-genotypespores
N (!A) N N (cid:1) N N (cid:1) N 0
a j a a a
N (1A) A N (cid:1) A N (cid:1) N N (cid:1) A
a j j j a a j
a Allcheatersbecomespores.
Varying x has the following effects: when x is low,a∗ is genotypeamoebaebecomespores.Whenthereisonlyone
always 1 0; when x is high, a∗p0 for much of the pa- altruist-genotype founder,N pk andA paNpank,
a j j j
rameter space (fig. 4). As before, threshold values of x which gives the following:
exist,abovewhicha∗p0;thesethresholdvaluesarelower
than those when np1 but behave similarly otherwise. {0 fork≤ank
F(lonealtruist)p j . (15)
k(cid:1)ank fork1ank
j j
The Constant-Size Model
The fitness of the altruistic founder, whenan!1 is then
j
It is possible that in chimeras, altruists, being unable to
distinguish between themselves and others, increase their WpFDp(k(cid:1)kna)(1(cid:1)c(cid:2)cax). (16)
j j
probability of becoming stalk cells to compensate for the
deficit of cheater-genotype stalk cells. Allocating cells in In the all-cheaters group, the fecundity of one cheater-
this way is expected when the propensity of analtruistto genotypefounderisk,andthegroup’sdispersibilityisthat
becomeaprestalkcelldependsoncertaindiffusiblefactors of a stalkless fruitingbody,1(cid:1)c.Thus,thefitnessofone
asdescribedin“Discussion.”Oursecondmodelsimulates cheater-genotype founder is
competitions between altruists employing such an allo-
cationmechanismandastalklesscheatergenotype.Inthis Wpk(1(cid:1)c). (17)
model, each altruistic genotype is characterized by a and
j
A, the proportion and number, respectively, of its cells The altruistic genotype spreads when rare only when the
j
that become stalk cells in all-altruist fruiting bodies: quantitygivenbyequation(16)isgreaterthanthequantity
given by equation (17). This inequality simplifies to
stalkcellsinpurefruitingbody A
a p p j. (14)
j totalcellsinpurefruitingbody N (1)( a n ) c
j ! . (18)
ax 1(cid:1)an 1(cid:1)c
j j
Inchimericgroups,ifthenumberofaltruisticcells(N)
a Analogous reasoning shows that the fitness of a lone
is insufficient to form a stalk of A cells, then all altruists
j cheater-genotype founder in a group of altruists always
become stalk cells, and if N exceeds A, then A altruists
a j j exceeds the fitness of an altruist-genotype founder in an
become stalk cells and the rest become spores (table 2).
all-altruists group, except for the implausible case where
The constant-size model cannot be analyzed using the a is high (1n/[n(cid:1)1]) and the dispersibility function is
ESS approach because in this model,W is not defined j
a,a∗ extremely convex:
for a∗(0. Instead, we analyzed pairwise competitions
when either cheaters or altruists were rare, with the fol-
lowing possible outcomes: first, cheaters could prevail in ( D(n/(n(cid:1)1)))
D(a)1 . (19)
both conditions, thus going to fixation; second, altruists j 1(cid:1)a
j
could prevail in both conditions; third, both genotypes
could increase in frequency when rare,leadingtoastable Therefore,cheatersalmostalwaysexistatequilibrium;fur-
polymorphic equilibrium; and fourth, both genotypes thermore, they drive altruists to extinction whenever in-
could increase in frequency whencommon,indicatingan equality (18) is false.
unstable polymorphic equilibrium. Inequality (18) and figure 5 show that for altruists to
The frequency of an altruistic genotype increaseswhen exist, four conditions must be met. First, n must be rel-
rare only if the fitness of a single altruistic founder in a ativelylow;smallervaluesofnfavoraltruistsbecausethen
group of cheaters exceeds the fitness of a single cheating fewerchimericgroupsform.Second,cmustexceedacer-
founder in a group of cheaters. To derive the fitness for- tain critical value that can be determined by rearranging
mulafortheformer,recallthatnoaltruist-genotypeamoe- inequality (18). Third, x must not be too high. As with
bae become spores unless N 1A, then N (cid:1)A altruist- the other model, the stalkless genotype wins when x ex-
a j a j
38 The American Naturalist
cells (Nanjundiah and Bhogle 1995). Since prestalk cells
are lost during slug migration and then replaced, 14%
represents a minimum value for the initial prestalk cell
fraction.Therefore,wewillassumethattheinitialprestalk
fraction (a , a∗, or a, depending on the model used) is
m j
approximately 20%.
For kin selection alone to produce a value for a /a∗/a
m j
of0.2,cmustberelativelyhigh,xmustbeincertainranges,
and n must be relatively low. For the case when no chi-
meras form, a never equals 0.2 when x!0.25 or x1
m
1.5. In the former case, a is always smaller; in the latter
m
case, a is either 0 or 10.2. For the stalk cell fraction a
m m
to reach 0.2, it is necessary, but not sufficient, that c≥
0.6. In the constant-proportion model, a∗ can reach 0.2
when there are more than one founders, if c is high and
if x is intermediate. For example, when xp1/2 and
cp0.9, an ESS of 0.2 is achieved when np1.6; lower
valuesofnproducelargerESSs.Similarly,intheconstant-
size model, altruists with a p0.2 can exist when n11,
j
provided that c is large and x is not too large (table 3).
Theoutcomesfortheconstant-proportionandconstant-
size models differ when x is low; in the former, altruists
with a stalk cell proportion of 0.2 are always eliminated
byshorter-stalkedaltruists;inthelatter,suchaltruistscan
still persist. This difference arises because genotypes that
make stalks of different sizes can compete against one
another in the constant-proportion model but cannot in
the constant-size model; in this model, one of the com-
Figure5:Conditionsforthestableexistenceofaltruistsintheconstant-
sizemodelasafunctionof(A)c,a,andx(np2)and(B)n,a,and peting strains is always the stalkless strain. Thus, in the
x(cp0.9).Forxp1/4,1/2,and1,jaltruistsstablycoexistwithchejaters constant-size model when x is low, altruists with ajp
in the parameter space below the curves (in both graphs), and only 0.2 are not eliminated by strains forming shorter stalks
cheatersstablypersistabovetheselines.Forxp2,altruistsandcheaters because they do not face them in direct competition.
coexistinthestripedareas,andonlycheaterspersistoutsidetheseareas.
ceeds a certain threshold, unless n is low and c is very
Discussion
high. Fourth, a must not be too high (or, in some cases,
j
too low). The variable a can be thought of as the cost of
j Our models show that competitions involving fixed-
altruism; it is the minimum proportion of altruists that
allocationcheatersselectforsuboptimalstalks,incontrast
become stalk cells and therefore die. As with a∗, the de-
to theoretical competitions involving variable-allocation
parture of the maximum value of a from a represents
j m strains, where clonal stalks of suboptimal size are not se-
theamountofpotentialfitnessnotachieved,andthisdif-
lected for (Matsuda andHarada 1990). TheratioofCSM
ference increases as n increases and as c decreases. The
value of a never exceeds 1/n and has a minimum value
j
when x is high; in these circumstances, the advantage of Table 3: Maximum values of n that allow altruists with a p
j
having a very short stalk is too small to be worth it. 0.2 to coexistwith cheatersat equilibrium
x c Maximumn
Sample Calculations .25 .5 2.0
.25 .9 4.3
Our models can be used to determine what values of c, .5 .5 1.5
x, and n are required for kin selection to select for the .5 .9 4.0
observed proportion of prestalk cells in clonal groups. In 1.0 .5 NAa
normal Dictyostelium discoideum fruiting bodies, approx- 1.0 .9 3.2
imately14%ofthecellsarestalkcellsandothernonspore a Noaltruistsatequilibrium.
Altruism and Cheating in Slime Molds 39
prestalk cells to prespores, 1:5, is very low; this suggests selection is sufficient to maintain the observed level of
that natural selection has acted to reduce CSM stalk size. stalk formation even if there are multiple founders, pro-
Although the effects of n and c on expected stalk size vided that n is low, x is small, andc is large.Thenumber
arestraightforward—increasingtheformeranddecreasing offounders,n,shouldbelowwhensporesareverywidely
thelatterreducesthestalksizeselectedfor—theeffectsof dispersed, when amoebae descendedfromdifferentfoun-
the shape of dispersibility curve are complicated. High ders avoid coaggregation (Buss 1982), or whengerminat-
valuesofx,encodingconvexdispersibilityfunctions,gen- ing spores inhibit the germination of their neighbors, as
erally make it harder to select for a stalk. However,when hasbeendemonstratedforyoungspores(RussellandBon-
stalks are selected for and when x is below a threshold ner1960;CohenandCeccarini1967).Determiningwhich
value, higher values of x tend to favor larger stalks, as values of x are most realistic awaits the identification of
indicatedbytheincreasesina∗andintheminimumvalue the mechanisms of spore dispersal, but concave dispersi-
for a. bility functions are not implausible. The cost of stalkless-
j
Each of our models can be related to specific mecha- ness,c,shouldbelargeifsporesonshortstalksareunlikely
nisms that control differentiation into stalk cells and to be dispersed and if survival rates at the (potentially
spores.Theinitialdifferentiationofamoebaeisinfluenced exhausted) site of the parentfruitingbodyarelowerthan
by an amoeba’s cell-cycle position; amoebae that at the those elsewhere.
onset of starvation are in mid– and late–growth-2 (G2)
stage normally become prespores, and amoebae in other
Anticheater Adaptations
stages normally become prestalk cells (Zimmerman and
Weijer 1993; Gomer and Amman 1996; Araki andMaeda Asnotedabove,theevolutionarymaintenanceofaltruism
1998; for a review, see Brown and Firtel 1999). Consider is facilitated in theory by phenomena that diminish the
a hypothetical mutant with an extended G2 stage but a amountofselectionwithingroupsrelativetothatbetween
normal-length cell cycle, which would produce a short- groups.Theputativeanticheateradaptationsdescribedbe-
stalked clonal fruiting body. In chimeras, cells of mutant lowdothisinoneoftwoways:byreducingthefrequency
andnonmutantstrainsshoulddifferentiatewiththesame of chimeric groups or by reducing the fitness advantage
propensities as they would have in pure fruiting bodies, of cheating within chimeras.
as in our constant-proportion model, because the differ-
entiation of each cell is controlled autonomously, by its
Reducing the Probability of Forming Chimeras
cell-cycle position and its genotype.
Cheating on another regulatory mechanism should CSM strains with somatic-compatibility systems, the first
lead to the dynamics of the constant-size model. Pre- of three chimera-avoidance mechanisms, avoid aggregat-
stalk cells have the ability to redifferentiate into pre- ing with cells of other genotypes (Buss 1982). Doing so
spores, and vice versa (Bonner 1982), apparentlyunder shouldbeespeciallyeffectiveagainstcheatingwhencheat-
the control of a secreted factor that controls gene ex- ing strains are preferentially avoided, as was the case in
pression, the differentiation-inducing factor (DIF-1; Buss’s (1982) coculture experiments. Models by Arm-
Gross 1994). The concentration of this factor most likely strong (1984) and Matapurkar and Watve (1997) show
is regulated by a feedback mechanism arising from its how a second mechanism could work: the advantage of
simultaneousproductionanddegradation,respectively,by altruism is enhanced when the distance that amoebae
prespores and prestalk cells (Brookman et al. 1987; Kay travel as they forage is reduced. Third, reducing the size
et al. 1993). When the concentration of DIF-1 rises, the of aggregation territories (the territories within which all
prestalk cellpopulationshouldincrease,andtheprespore amoebae coaggregate to form one fruiting body; Bonner
population should decrease, thus increasing DIF-1 deg- and Dodd 1962) should decrease the probability that de-
radation capacity and decreasing DIF-1 production ca- scendants from different founders coaggregate. Aggrega-
pacity (Insall et al. 1992; Loomis 1993). Suppose that a tion territory size is negatively correlated with the con-
mutant that only forms prespores arises and that its pre- centrationofasecretedfactor,countin(BrockandGomer
sporesproduceDIF-1attheusualrate.Inchimerasofthis 1999), and thus is under genetic control.
mutant and a wild-type strain, the number of wild-type
cells that become prestalk cells should equal that num-
Reducing the Fitness Advantage of Cheating
ber in all-wild-type fruiting bodies, as postulated by our
within Chimeric Groups
constant-sizemodel,becausetheratesofDIF-1production
and degradation are the same as in the wild-type slug. RandomizingWhichCellsBecomePrespores.Thethreepu-
Giventhatfixed-allocationcheatingispossible,whydo tative anticheater adaptations in this category (using cell
CSMs make any stalks at all? Our analyses show that kin quality to determine cell fate, coercing low-energy cells
40 The American Naturalist
into differentiating, and usingcell-cyclepositionasasec- differentcell-cyclepositionsmostlikelyunderliethiseffect
ondary determinant of cell fate) act against two types of (Gadagkar and Bonner 1994).
cheaters: those that manipulate the fate-determination One would expect that a cheating mutant that arrests
mechanisms to produce more prespores and those that its development during the late G2 phase (andthuslocks
cheat by foregoing these mechanisms. in a high level of energy reserves) would insure that it
An important determinant of an amoeba’s fate is its would become a prespore. However, this typeofcheating
levelofenergyreserves;amoebaewithmoremitochondria would be penalized, if starvation were not imminent,be-
or with higher levels of stored carbohydrates are more cause wild-type cells would continue to divide while the
likely to become prespores (Noce and Takeuchi 1985; cheaters remained idle. A cheatingmutantthatdecouples
Shaulsky and Loomis 1995). Cheating by strains that ad- its fate from its cell-cycle stage should suffer the same
heretothissystemissuppressedbecausethedetermination penalty as one that decouples its fate from its energy-
of cell fate acts at random with respect to genotype for reserve level because cell-stage and reserve levels are
strainsthatproducecellswithcomparableenergyreserves correlated.
(GadagkarandBonner1984).Cheaters(andaltruists)that
donotusereservelevelstodeterminecellfateshouldalso Dividing Asymmetrically to Insure That Some AltruistsBe-
sufferapenalty,iftheadvantageofallocatinghigh-energy come Spores. Sister Dictyostelium discoideum amoebae,
cells to the spore fraction (increased spore survival) out- upon starvation, differentiate into different types of cells,
weighs the structural disadvantage of using low-energy andamoebaethatbecomeprestalkcellsalmostalwayshave
cells for the stalk. sisters that eventually become spores (Gomer and Firtel
Ithasbeenproposedthathigh-qualitycells(thosewith 1987). If differentiation of altruist-genotype sisters oper-
the greatest energy reserves) secrete DIF-1, a toxin, to ates in the same way in chimeric fruiting bodies, then it
coerce low-qualitycellsintobecomingprestalkcells(Atz- shoulddecreasethefitnessadvantageofalltypesofcheat-
monyetal.1997).Secretionoccursuponstarvation,when ers by increasing the probability that at least one of each
theonlyalternativesavailabletoamoebaearetoaggregate pair of altruist-strain sister amoebae will become a
(and differentiate) or to remain solitary and face certain prespore.
death. The high-quality cells would be able to simulta- In Gomer and Firtel’s (1987) study, divisions thatpro-
neously detoxify this compound and differentiate into duced daughters with different predispositions occurred
prespores, but low-quality cells would not and wouldbe- notonlyduringtheaggregationphasebutalsoduringthe
come prestalk cells by default. Cheaters abstaining from vegetative phase, prior to starvation. This suggests that
toxin degradation to free up resources for differentiating eventhissolitaryphaseoftheCSMlifecycleencompasses
into prespores wouldbe penalizedbecausetheywouldbe a feature that is an anticheater adaptation.
more susceptible to the toxin.
Although DIF-1 appears too late to initiate differenti- Policing to Prevent Prestalk Cells from Abandoning Their
ation (Shaulsky and Loomis 1996), as required by this Fate. Incipient prestalk cells unrelated to most prespores
scenario, another molecule, cyclic adenosine monophos- in their group should increase their inclusive fitness by
phate(cAMP),mayfulfilltheroleproposedforthistoxin. redifferentiating into prespores. In theory, policing for
Cyclic AMP is the molecule responsible for initiatingdif- such transitions (i.e., detecting and eliminating them) is
ferentiation (Firtel 1995) and is initially secreted only by advantageous at the group level and can be selected for
the amoebae most likely to become prespores (Huang et (Frank 1995; Michod 1996). In CSM mounds and slugs,
al. 1997). In addition, although cAMP is not a toxin, it DIF-1 inhibits the redifferentiation of prestalk cells into
doescoercelow-qualitycellstorespondtoitbypenalizing prespores (Firtel 1995); therefore, this compoundmaybe
them if they do not; amoebae that disregard this signal considered a policing agent. This interpretation can ex-
must also forego aggregation (Firtel 1995). Thus, cAMP, plainseveralphenomena:thecellsthatwouldrequirepo-
likethetoxininthescenarioabove,isinitiallysecretedby licing (prestalk cells) are the ones most sensitive to DIF-
high-quality cells, initiates differentiation, induces low- 1; DIF-1 is not secreted until after differentiation has
qualitycellstobecomeprestalkcells,andpenalizescheat- started, which is when the need for a policing agent first
ers that ignore it. arises; and the cells that would benefit most from the
As noted previously, a cell’s fate is largely determined policing agent (prespores) are the ones that produce it.
byitscell-cyclepositionattheonsetofstarvation.Among
amoebae with similar life cycle durations, this system for
Conclusions
allocating cells acts at random with respect to genotype
andthereforeshouldsuppresscheating(WilsonandSober Kin-selection and anticheater adaptations may seem like
1989). Differences in energy-reserve levels among cells in alternative explanations for the maintenance of CSM al-