*E

J. theor. Biol. (2000) 202, 283}304doi:10.1006/jtbi.1999.1054, available online at http://www.idealibrary.com on

0022

Speci5c and Non-speci5c Defense against Parasitic Attack

STEVEN A. FRANK*

Department of Ecology and Evolutionary Biology, ;niversity of California, Irvine,CA 92697-2525, ;.S.A.

(Received on 29 April 1999, Accepted in revised form on 11 November 1999)

Speci"c defense protects against some parasite genotypes but not others, whereas non-speci"cdefense is e!ective against all genotypes of a parasite. Some empirical studies observe hostswith variability only in non-speci"c defense, other studies "nd only speci"c defense. I analysea model with combined speci"c and non-speci"c defense to determine the conditions that favordetectable variation in each form of defense. High variation in non-speci"c defense is oftenmaintained when resistance increases in an accelerating way with investment, whereas lowvariation tends to occur when resistance increases at a decelerating rate with investment.Variation in speci"c defense rises as the parasite pays a higher cost to attack a broad hostrange (high cost of virulence), as the number of alternative speci"cities declines, and as theaverage level of non-speci"c defense increases. The last condition occurs because greaternon-speci"c protection tends to stabilize the gene frequency dynamics of speci"c defense.Selection favors a negative association between costly components of speci"c and non-speci"cdefense*hosts defended by one component are favored if they have reduced allocation toother costly components. A negative association confounds the measurement of costs ofresistance. Individuals with speci"c defense may have reduced investment in costly non-speci"c defense. This leads to an apparent advantage of speci"cally defended hosts in theabsence of parasites and a measured cost of resistance that is negative.

( 2000 Academic Press

Introduction

Speci"c defense occurs when a host repels attackby some parasitic genotypes but not by others.Speci"city may, for example, be mediated bya biochemical match between host receptors andparasite surface molecules. The match inducesthe host's non-speci"c defenses near the site ofattack. This approach localizes the induced re-sponse and minimizes wasted defensive e!ort.

Non-speci"c defense protects against all para-sitic genotypes of an attacking species. Suchbroad-band defense is sometimes called &&durable

-mail: safrank@uci.edu

}5193/00/040283#22 $35.00/0

resistance'' because it remains e!ective over time,whereas speci"c defense often succumbs to novelparasitic genotypes that evade recognition.Durable resistance has the obvious advantages ofbroad and continuous defense but may some-times require constitutive and costly expressionof defensive systems.

Many models of speci"c and non-speci"c res-istance have studied evolutionary dynamics andmaintenance of polymorphism. Models typicallyfocus on either speci"c or non-speci"c resistance;here I examine the interaction between thesetwo forms of defense. It seems likely that therewould be some interaction*for example, highlysuccessful speci"c and localized defense may

( 2000 Academic Press

284 S. A. FRANK

in#uence the level of constitutive expression.However, interactions between epidemiologyand genetics have sometimes proved to becounterintuitive.

Biological Examples of Speci5c andNon-speci5c Defense

Agricultural studies of plant}pathogen inter-actions provide the best data on speci"c andnon-speci"c defense. Speci"city is de"ned bystatistical description of variation in attack anddefense for a given sample of hosts and parasites(Vanderplank, 1963, 1984). Speci"c resistance isthe variation in host "tness explained by an inter-action between host and parasite genotype. Theremaining variation in host response to parasiticattack is independent of parasite race*this isnon-speci"c resistance.

Resistance in plant}pathogen interactions isoften manifest by a hypersensitive response*theaccumulation of defensive compounds in thetissue surrounding the point of invasion and anassociated con"nement of pathogen proliferation(Gabriel & Rolfe, 1990). Successful pathogens,di!ering by a single mutation from resistant in-teractions (Kearney et al., 1988), do not elicit thehypersensitive response.

Biochemical models suggest that resistanceoccurs only when a pathogen allele produces aparticular gene product (elicitor) that can be rec-ognized by a matching host receptor (Gabriel& Rolfe, 1990). If an elicitor}receptor match oc-curs, then the host induces a non-speci"c hyper-sensitive response and resists attack. If the samepathogen elicitor is present, but the host pro-duces a non-matching receptor, then disease de-velops. Infection also succeeds when a pathogenlacks an elicitor that matches the speci"c hostreceptor.

The mechanisms of durable, non-speci"c res-istance are less clear. Pathogens may containa monomorphic elicitor that is essential for func-tion and cannot be changed. Hosts may havea low level of constitutive expression of the non-speci"c defensive system that provides partialresistance against all pathogen races. Or thehosts may lower the threshold signal at whicha hypersensitive response is induced, leading touniversal resistance against attack but possibly

frequent &&false alarms'' in which defense is trig-gered in response to a harmless invasion orsignal. Finally, mechanical or other biochemicalmechanisms may be continuously expressed ina non-speci"c way. Most non-speci"c mecha-nisms suggest costly expression, although somemay be relatively free of deleterious side e!ects.

The main point here is that mixed speci"cand non-speci"c defensive systems occur. Theplant}pathogen interaction is a particularly well-studied case, but there are many other examplesof induced defense in response to signals ofinvasion or danger (Harvell, 1990; Karban &Baldwin, 1997; Tollrian & Harvell, 1999). Whenthese signals of attack are recognized in a speci"cway, then resistance may be composed of speci"crecognition followed by induction of non-speci"c defense. Such a two-part resistance islikely to be associated with widespread variationin the relative contributions of speci"c and non-speci"c defense toward explaining the observeddistribution of resistance.

In plant}pathogen systems, speci"c variationis commonly observed in both agricultural andnatural populations (reviewed by Burdon, 1987;Frank, 1992, 1997). Non-speci"c defense is widelysought after and sometimes observed in agricul-tural systems (Vanderplank, 1984). The classicexample is the interaction between potato blightand potato. The two potato varieties Maritta andKennebec have the same pro"le of speci"c andcomplete resistance against several pathogenraces but di!er uniformly (non-speci"cally) in theseverity of damage caused by successful infection.Such non-speci"c resistance would, however,have to be reclassi"ed as speci"c if a newly ana-lysed pathogen race overcame what had pre-viously been classi"ed as non-speci"c defense.Thus, classi"cation depends partly on the inten-sity and duration of sampling.

Many studies have analysed variation in hosts'resistance to di!erent parasite isolates and thehost range of those parasite isolates over thesampled hosts. Patterns and generalities remainobscure. I illustrate this literature by two recentexamples.

Fellowes et al. (1998) used laboratory selectionexperiments to study the resistance of Drosophilamelanogaster to the endoparasitoid ¸eptopilinaboulardi. They selected four replicate populations

DEFENSE AGAINST PARASITIC ATTACK 285

of the host for increased resistance to the para-sitoid. Resistance increased from 0.5% to be-tween 40 and 50% in "ve generations. Hosts fromthe selected lines su!ered lower larval survival incomparison with unselected competitors.

Insects resist attack by encapsulating the para-sitoid with defensive cells (Salt, 1970). Resistancehas two important steps: recognition of a foreignobject as non-self and induction of the encapsula-tion process. It is not clear at present whether theincreased resistance observed by Fellowes et al.(1998) resulted from improved recognition ora more vigorous induced response. Genetic vari-ation in insect resistance against endoparasitoidshas been observed in several cases (Kraaijeveldet al., 1998), whereas there are only a few reportsof variation in parasitoid success (summarized bySasaki & Godfray, 1999). If this pattern holds,variation in insect resistance would appear to becontrolled by changes in the threshold and vigorof induced defense rather than by speci"c defense.Firm conclusions are premature, however, be-cause inferring speci"c variation often requiresintensive sampling.

Fellowes et al. (1999) expanded their study ofD. melanogaster resistance. They selected di!er-ent host lines for increased resistance to eitherof the endoparasitoids Asobara tabida or ¸ep-topilina boulardi. They then tested each selectedline against three parasitoids: the two from theselection experiments plus ¸eptopilina hetero-toma. Lines selected against ¸. boulardi had in-creased resistance against all three parasitoidspecies. Lines selected against A. tabida had in-creased resistance against all parasitoids except¸. boulardi. Apart from this last, unexplainedexception, it appears that increased non-speci"cdefense causes the enhanced protection of theselected lines.

Webster & Woolhouse (1998) performed sim-ilar selection experiments on a snail and itsschistosome parasites. They found nearly all thevariation explained by enhanced speci"c defense.They conducted their study with the snail Biom-phalaria glabrata, which is polymorphic in natu-ral populations for resistance to Schistosomamansoni. They selected two strains of the snail forincreased or decreased resistance to two strainsof the schistosome. Third-generation snails weretested for resistance against both parasite strains.

Each snail strain selected for higher or lowerresistance to a particular parasite strain re-sponded accordingly to that strain, but had nochange in resistance to the other parasite strain.Thus variation in defense appears to be entirelyspeci"c. There was no trade-o! in resistance, thatis, increased resistance to one parasite strain hadno association with resistance to the other para-site strain.

I draw four conclusions from the limited dataavailable. First, protection against internal para-sites often depends on recognition of invasionfollowed by induction of defense. Either phase,recognition or induced defense, may be speci"cor non-speci"c for a given sample of hosts andparasites.

Second, one cannot infer recognition or non-speci"c defense when expressed in a constant wayby all individuals of a given sample, even if suchcomponents of defense are critical aspects ofa multistep process of resistance. Thus observedresistance always means variable resistance. Thequestion is: Which factors favor variable speci"cdefense and which factors favor variable non-speci"c defense?

Third, the biochemical and physiologicalmechanisms of recognition and induced defenselikely in#uence variability in resistance. Forexample, if successful recognition of an invadingparasite can be turned to failure by small bio-chemical changes in elicitors and receptors, thenspeci"c recognition is likely to be highly variable.By contrast, if recognition depends on constantbiochemical features of the parasites, then defensewill appear as non-speci"c.

Fourth, given that much of the variation acrosshost}parasite systems may depend on the pres-ently unknown biochemistry and physiology ofthe interaction, the role of general models isnecessarily limited. General models can clarifythe broad categories of speci"c and non-speci"cdefense that may occur, the conditions thatmaintain variable and thus observable resistance,and the potential interactions between di!erentcomponents of a defensive cascade.

Model

I use a system of Lotka}Volterra equations tomodel the host}parasite dynamics (e.g. May,

286 S. A. FRANK

1974). This allows joint study of #uctuatingabundances caused by ecological dynamics (epi-demiology) and #uctuating gene frequenciescaused by host}parasite selection. I use a formthat also includes competition among hosts forlimited resources, imposing a carrying capacityon the hosts in the absence of parasites andavoiding the peculiar, neutrally stable limit cyclesthat occur in the absence of host competition.

The system is given by

Dhij"h

ijArcij!r+x

+y

cxy

hxy

/K!m +k

dijk

pkBDt,

(1a)

Dpk"p

kA!s#bk+i

+j

dijk

hijBDt. (1b)

The abundance of hosts with genotype ij is givenby h

ij. The host genotypes have varying aspects

of speci"c and non-speci"c defense. For speci"cdefense, each host carries a speci"c recognitionallele, i"0, 1,2, n. For non-speci"c defense,each host carries an allele j"0, 1/N,2, N/N,where the level of defense varies from 0 to 1.A host that carries no defenses (i"j"0) has anintrinsic rate of increase of r. Defenses have costs

cij"G

1!aj, i"0,

(1!a) (1!aj), i'0.(2)

which are composed of two parts. Non-speci"cdefense reduces the rate of increase by 1!aj,where j is the level of constitutive expressionand a is a cost parameter. Speci"c defense reducesthe rate of increase by 1!a where a is a costparameter. The allele i"0 provides no speci"cdefense and has no cost.

The middle term of eqn (1a) describes hostcompetition*the pressure on a host's rate ofincrease caused by other hosts' tendency toincrease, scaled by the carrying capacity ofhosts, K.

The "nal term is the morbidity and mortalityof parasitic attack, m, weighted by the success ofattack over all parasites. Each parasite genotype

k succeeds against host genotype ij according to

dijk"G

0, i"kO0,

1!jz otherwise,

where speci"c resistance prevents attack whenhost and parasite have matching speci"cities,i"k, and the host is not carrying the null allele,i"0. When speci"c resistance fails, hosts resistaccording to their constitutive level of non-speci-"c defense, jz, where z(1 causes diminishingbene"ts with investment in constitutive defense,and z'1 causes accelerating bene"ts.

The time-scale of the interaction is set by thediscrete time step Dt. As DtP0, the interactionbecomes continuous (di!erential) in time.

The abundance of parasites with genotypek"0, 1,2, n is given by p

k. The death rate of

parasites in the absence of hosts is s. The growthof parasites on successfully infected hosts is b

k,

where bk"b for k'0 and b

0"(1!v)b. The

zero allele has cost of virulence, v, and gains theadvantage of universal host range because thereis no matching host type that can provide speci"cresistance against this parasite genotype.

Non-dimensional analysis simpli"es the sys-tem by reducing the number of parameters andclarifying the scaling relations among processes(Segel, 1972; Murray, 1989). I use the followingsubstitutions:

q"rt, hK "h/K, pL "(m/r)p,

sL"s/r, bK "bK/r, (3)

where the new variables are all non-dimensionalquantities. The new time-scaling, q, expressestime in terms of the host population's potentialrate of increase. For example, in a continuous-time model, the doubling time of the host popula-tion in the absence of competition or parasitismis ln(2)+0.7 non-dimensional time units. Thistransformed time-scaling holds for any set ofparameters and provides easy comparisonamong systems that have di!erent natural time-scales.

The host abundances, hK , are fractions of thecarrying capacity, which range from 0 to 1.Again, this allows comparison on a common

DEFENSE AGAINST PARASITIC ATTACK 287

scale for models that assume di!erent spatialdistributions of resources and host competition.When I apply the substitution for parasiteabundance, pL , below, the parameter m disappearsfrom the system. From this we can immediatelydeduce that morbidity and mortality per infec-tion has no in#uence on ecological or evolution-ary dynamics, but rather a lower value of msimply translates into a proportionately higherabundance of parasites, that is, parasite pressureremains constant.

The "nal interesting point concerns the scalingfor parasite growth rate. On the non-dimensionalscale, bK depends on the host's carrying capacity,K, showing the formal equivalence of increasedparasite growth and increased host density on theincrease of the parasite population.

I make these non-dimensional substitutionsinto eqns (1), expressing the system only in non-dimensional quantities. For convenience, the hatsare dropped from the non-dimensional para-meters, and from this point all parameters areassumed to be non-dimensional quantities unlessotherwise noted. The substitutions yield

Dhij"h

ijAcij!+x

+y

cxy

hxy!+

k

dijk

pkBDq,

(4a)

Dpk"p

kA!s#bk+i

+j

dijk

hijBDq, (4b)

where the parameters r, K, and m have beenabsorbed by the scaling relations.

Special Cases

To understand the interaction between non-speci"c and speci"c defense one must "rst havean understanding of each process acting alone.The general model in eqns (4) subsumes fourspecial cases.

NON-SPECIFIC RESISTANCE

Assume that hosts can defend non-speci"callybut lack the matching speci"cities of host}parasite recognition. In eqns (4) the matchingspeci"cities of host and parasite are set toi"k"0; in e!ect, the universal host-range

allele of the parasite, k"0, has no cost, v"0,and thus goes to "xation. Hosts can vary at thenon-speci"c locus indexed by j"0, 1/N,

2,N/N, thus j varies between 0 and 1 andmeasures the fraction of maximal investment indefense.

The dynamics of non-speci"c defense are de-scribed by

Dhj"h

jC(1!aj)!+k

(1!ak)hk!p(1!j z)DDq,

Dp"pC!s#b +j

(1!j z)hjDDq.

Let H be the total abundance of hosts. Thenhj/H is the frequency of the j-th allele and

+j(h

j/H) j"jM is the average level of expression in

the population. Similarly, +j(h

j/H) j z"jM z is the

average level of bene"t provided by non-speci"cresistance.

With these de"nitions, the system can bewritten as

Dhj"h

j[(1!aj)!H(1!ajM )!p(1!j z)]Dq,

(5a)

Dp"p[!s#bH (1!jM z)]Dq, (5b)

and the equilibrium total abundance of hosts andparasites is given by

H*"s

b(1!jM z),

P*"(1!ajM )[b(1!jM z)!s]

b(1!jM z )2.

The distribution of genotypes must be ana-lysed in two separate cases, for diminishing bene-"ts with investment in non-speci"c resistance,z(1, and for accelerating bene"ts, z'1 [seeBoots & Haraguchi (1999) for a related analysis].

I "rst examine the case z(1 at equilibrium.Note that the dynamical equations for the di!er-ent host genotypes have the form Dh

j"h

jw

jDq,

where wjis the rate of increase ("tness) of each

type. Suppose that, for a given distribution of

288 S. A. FRANK

genotypes, we study the "tness of a single indi-vidual in a large population, considering how its"tness changes as a function of its genotype, j.The change in "tness with a small change ingenotype is given by the derivative

dwj

d j"!a#z j z~1p.

With z(1, the second derivative is negative,thus there may exist a single genotype, 0(j(1,that has maximum "tness and is the mono-morphic equilibrium. This candidate winner isfound by solving dw

j/dj"0 at j"j* yielding the

equation

zj z~1(1!aj)[(1!j z)!s/b]!a (1!j z)2"0.

(6)

This must be solved numerically given the para-meters z, a, and s/b. I made a brief exploration ofthe parameter space and found that the equilib-rium level of investment in non-speci"c defense,j*, follows simple trends with respect to theparameters. In particular, as s/b declines, j*,increases*greater parasitic pressure impliesa higher level of defense; as a increases, j* de-clines*greater cost of defense lowers the levelof defense; and as z increases, j* increases*asthe marginal gains for defense decline more slow-ly with investment, the allocation to defense rises.

Interesting problems arise when consideringhow to measure and de"ne resistance. One choiceis to measure the "tness of an attacked individualdivided by the "tness that individual would haveif not attacked. If attack has no e!ect, then resist-ance is one; if attack kills the host, then resistanceis zero. Although this measure appears reason-able at "rst glance, it is rarely used. Consider, forexample, a population in which hosts alwayshave their "tness reduced by one-half fromattack. One would rarely describe the hosts ashaving 50% resistance in this case. Many para-sites damage but do not kill their hosts, and thisinteraction by itself is not considered an expres-sion of resistance.

Variation in resistance is often a more interest-ing quantity. For example, a population respondsto selection for increased resistance in proportion

to its variance in resistance. Thus, a selectionexperiment that found a response mainly ofnon-speci"c defense suggests greater variabilityin non-speci"c than in speci"c defense for thesample of hosts and parasites studied. By con-trast, individuals may invest heavily in costlynon-speci"c defense but lack variation in thatinvestment. In this case, di!erent populations ex-posed to various levels of parasite pressure mayhave di!erent levels of constant expression tonon-speci"c defense.

The important point is that both the averageand the variability of investment in non-speci"cdefense are important quantities. The variation isrelatively easy to measure within populations,whereas the average investment can only be infer-red by comparison among populations.

The condition in eqn (6) describes the forcesacting on average investment, j*. What aboutvariability? Numerical studies with mutationshow that the stabilizing force of diminishingreturns, with z(1, typically maintains a co-e$cient of variation about the equilibrium of lessthan 5%. In addition, non-speci"c defense reduc-es parasite growth, which in turn reduces thetendency for non-equilibrium #uctuations inabundance. This ecological stability helps tomaintain low levels of genetic variability in resist-ance.

The second case concerns accelerating bene"tsof resistance per unit investment, z'1. Thesecond derivative of "tness with respect tophenotype, d2w

j/dj2"z(z!1)j z~2p, is positive

in this case implying disruptive selection. Thus,candidate equilibria include mixtures of min-imum and maximum expression, j"0 and 1,respectively, or "xation of either type. Non-speci-"c resistance appears as a qualitative factor eventhough there is a potential for quantitative vari-ation.

The dynamics of the two host types and theparasite are given by

Dh0"h

0[1!(h

0#(1!a)h

1)!p]Dq,

Dh1"h

1[1!a!(h

0#(1!a)h

1)]Dq,

Dp"p[!s#bh0]Dq.

FIG. 1. The various e!ects caused by cost of non-speci"cresistance, a. An increase in a directly reduces h

1. The e!ect

of increasing a on parasite abundance, p, is obtained bymultiplying the e!ects through the pathway: negative on h

1,

negative on h0

by density-dependent competition amonghosts, and positive on p, yielding a net positive e!ect. Inter-estingly, increasing a has no net e!ect on the abundance ofthe susceptible host, h

0. This lack of e!ect arises from two

opposing forces: the e!ect on h0

is positive via reduction incompetition with h

1(the product of two negative paths from

a to h0), and the e!ect on h

0is negative via an increase in

parasite abundance, p.

DEFENSE AGAINST PARASITIC ATTACK 289

The equilibrium is a mixture of minimum andmaximum expression when the cost of non-speci"c resistance is bounded such that a(1!s/b,in which case the equilibrium is given by

h*0"s/b,

h*1"

1!a!s/b1!a

,

p*"a,

with frequency of resistance, h*1/(h*

0#h*

1), as

f *"1!a!s/b

1!a!a(s/b). (7)

The equilibrium abundances present an interest-ing pattern. The abundance of the susceptiblehost, h

0, is independent of cost, a. By contrast,

the parasite, which can only attack the suceptiblehost, has equilibrium abundance equal to a. Thediagram in Fig. 1 shows how density-dependentcompetition among hosts causes this pattern ofequilibrium abundances.

High parasite growth rates, b, and long timelags in interaction, Dq, destabilize the ecologicalinteractions and cause boom and bust cyclesof parasite epidemics. These epidemics tend tomaintain high levels of resistance because an out-break drives the frequency of resistance near"xation. The parasites may die o! as resistancerises, followed by a decline in the frequency ofresistance. This decline in resistance sets up an-other round of parasite boom and bust.

In summary, diminishing returns on defensiveinvestment lead to limited quantitative polymor-phisms about an intermediate level of expression.By contrast, accelerating bene"ts favor a poly-morphism of high and low expression (Boots& Haraguchi, 1999). Conditions favoring slowgrowth of parasites (low b) maintain a high fre-quency of non-expressing hosts with a low fre-quency of resistant hosts, whereas high hostdensity and fast parasite growth (high b) favora high frequency of resistance.

In the Discussion, I will consider these newresults for non-speci"c resistance in light of theexperiments by Fellowes et al. (1998, 1999). The

following three sections review earlier models forspeci"c defense. This review is useful because itbrings together in a common framework di!erentassumptions and methods of study. This back-ground is also required to understand the inter-actions between speci"c and non-speci"c defense,which I describe after summarizing the models ofspeci"c defense.

SPECIFIC DEFENSE: HAPLOID GENE-FOR-GENE

Flor's (1971) classical gene-for-gene modelstates that for each locus in the host there isa matching locus in the parasite that conditionssuccess of attack. The typical patterns of domi-nance and phenotypic interaction at each condi-tioning locus yield a 2]2 matrix such that a hosthas either a susceptible or a resistant phenotype

290 S. A. FRANK

and the pathogen has either a virulent or anavirulent phenotype. A resistant reaction for thisgene pair only occurs with a match between thehost resistance phenotype and the parasite arivu-lence phenotype. Any of the other three pheno-typic combinations allows parasitic attack.

This pairwise gene-for-gene interaction occursover all the matching gene pairs of the host andparasite. If the interaction leads to host resistanceat any single matching gene pair, the net e!ect isresistance. A parasite succeeds only when all itsavirulence phenotypes are matched by host sus-ceptibility phenotypes. Note that plant pathol-ogists use the word &&virulence'' to mean hostrange, whereas in other "elds &&virulence'' meansthe aggressiveness of the parasite.

My model is haploid and there is only a singlematching locus between host and parasite. Weobtain the simplest form of a gene-for-genematching speci"city from eqns (4) by turning o!non-speci"c defense, j"0, and limiting the num-ber of matching speci"cities to n"1. Droppingthe j subscript for hosts, the full system is

Dh0"h

0[1![h

0#(1!a)h

1]

!(p0#p

1)]Dq, (8a)

Dh1"h

1[1!a![h

0#(1!a)h

1]!p

0]Dq, (8b)

Dp0"p

0[!s#b (1!v)(h

0#h

1)]Dq, (8c)

Dp1"p

1[!s#bh

0]Dq. (8d)

The two top equations describe host susceptibil-ity (h

0) and resistance (h

1); the two bottom

equations describe parasite virulence (p0) and

arivulence (p1). From earlier de"nitions, the cost

of resistance is a; the cost of virulence is v, wherevirulence is more aptly called &&universal hostrange''. This is a special case of the multilocusmodel in Frank (1993a), according to which atequilibrium the relative abundances of resistance(R) to susceptibility (S) and virulence (<) to aviru-lence (A) are

R :S"v : 1!v,

< :A" 1!a!H*(1!av) : a,

where H*"s/[b(1!v)] is the total equilib-rium abundance of hosts. Note that there is aminor error in eqn (6) of Frank (1993a), inwhich (1!a)N(1!H*) should be (1!a)N!H*(1!av)N. This correction applies also to eqn(12) of Frank (1993a) and eqn (5) of Frank(1993b). This change does not a!ect in any waythe conclusions from those earlier papers.

These equations emphasize the three key pro-cesses that in#uence the equilibrium frequenciesof resistance and virulence. First, the frequency ofhost resistance rises with the cost of virulence, v,and is independent of the cost of resistance. Sec-ond, the frequency of virulence declines with thecost of resistance, a, which can be seen by rewrit-ing the < :A ratio as 1!a(1!vH*)!H* : aand noting that 1!vH*'0 when all types arepresent. Third, the cost of virulence in#uences thefrequency of virulence only through the e!ectof density-dependent competition among hosts,which can be seen by rewriting the < :A ratio as1!a![h*

0#(1!a)h*

1] : a and noting that the

term [h*0#(1!a)h*

1] derives from host com-

petition in the solution of p*0

from eqn (8b). In theabsence of host competition, the < : A ratio is1!a : a. Figure 2 provides an intuitive explana-tion for these e!ects. (Many other gene-for-genemodels have been published. See the references inLeonard & Czochor, 1980; Burdon, 1987; Frank,1992, 1993a).

SPECIFIC DEFENSE: MATCHING ALLELE

The classical gene-for-gene model requirescosts of resistance and virulence to maintainpolymorphism. Without a cost of virulence, theuniversal host-range allele (V) of the parasite goesto "xation; without a cost of resistance, the hosts'resistance allele (R) goes to "xation. Alterna-tively, there may be a series of matching allelicspeci"cities at each locus, which allows poly-morphism to be maintained by frequencydependence.

I developed a formal model of matching speci-"cities, which I called the matching-allele model(Frank, 1993b). The model is obtained from eqns(4) by assuming no non-speci"c defense and anabsence of the special 0 alleles of the matchingspeci"cities such that the matching alleles vary asi, k"1,2, n. These modi"cations are equivalent

FIG. 2. The feedbacks in a gene-for-gene model of speci-"c defense. A rise in the cost of resistance, a, directly lowersthe abundance of resistant hosts, h

1, but indirectly raises h

1through a feedback e!ect. The feedback is calculated bymultiplying the individual e!ects of the three steps in thefeedback loop: a negatively in#uences, h

1, which in turn

positively in#uences the virulent phenotype of the parasite,p0, which negatively in#uences h

1. The net e!ect of a rise in

a is a decrease in p0and no change in h

1. Similar calculations

show that an increase in v raises h1

but has no net e!ect onp0

in this diagram. The e!ect of host competition is notshown here. The only e!ect of more intense host competitionis to reduce the abundance of the virulent parasite pheno-type because, by weakening the hosts, the parasites' abilityto maintain a costly phenotype is reduced. This occursbecause the parasites e!ectively graze excess host energy.Diagram modi"ed from Frank (1992).

DEFENSE AGAINST PARASITIC ATTACK 291

to assuming a very large cost of non-speci"cdefense, aPR, no cost for host resistance alleles,a"0, and zero "tness for the parasite allelelabeled as 0 that has universal host range, that is,the cost of virulence is v"1. With these assump-tions, we have the system

Dhi"h

i[1!H!(P!p

i)]Dq, (9a)

Dpk"p

k[!s#b(H!h

k)]Dq, (9b)

where H and P are the total abundances of hostsand parasites, and the genotypes range overi, k"1,2, n. Frank (1993b) provided a full analy-sis. I summarize a few details as background forthe extended model studied below.

The dynamics of the system are controlled bythe equilibrium with all hosts and parasites pres-ent, which occurs at h*"s/[b(n!1)] and

p*"(1!H*)/(n!1), where H*"nh* and, bythe symmetry of the system, h*

i"h* and p*

k"p*

for all i and k. This equilibrium point is unstablewhen there are discrete time lags (Dq'0) in thecompetitive e!ects among hosts and in the inter-actions between host and parasite. This equilib-rium is neutrally stable when interactions occurin continuous time (DqP0).

The most important aspect of dynamics con-cerns a qualitative change from limit cycles tocolonization-extinction dynamics as the numberof speci"cities, n, increases (Frank, 1993b, 1997).Systems with low n maintain all speci"cities with-in the local population and follow nonlinear #uc-tuations governed by eqns (9). As the number ofspeci"cities rises, the average abundance per typedeclines. Lower average abundance increases theprobability that one or more types become ex-tinct locally.

Such extinctions make local populations proneto invasion and temporary dominance by colon-ists. For example, suppose that a particular para-site speci"city was lost. Then, the matching hosttype will be driven to local extinction because itdoes not resist any parasite with which it inter-acts and has lower "tness than hosts resistingsome of the locally present parasites. Loss of thishost type sets up an opportunity for the matchingparasite to invade by colonization; this parasitetype will increase rapidly because no local hostsresist it. The colonizing parasite favors the rein-troduction of the matching host type. The nete!ect is boom and bust cycles for matching types,with common local extinctions followed event-ually by recolonizations.

SPECIFIC DEFENSE: HYBRID MODEL

Perhaps the most realistic model of speci"citycontains a mixture of gene-for-gene and match-ing-allele components (Frank, 1993b). On thegene-for-gene side, novel mutations to universalhost range (virulence) are sometimes deletions(Flor, 1971), suggesting that virulence is the lossof a speci"c elicitor. Such deletions likely havea cost relative to various forms of the allele thatact as elicitors, otherwise the deletions wouldquickly go to "xation. On the matching-alleleside, it seems likely that small biochemicalchanges can alter a pathogen elicitor from a state

292 S. A. FRANK

that induces host defense to a state that avoidsrecognition and produces a successful infection.Small changes in the pathogen allele could becountered by matching host alleles, leading toa series of matching speci"cities.

A hybrid model with costly universal virulenceand a series of cost-free matching speci"cities hasnot been studied. Within the present framework,such a model has the following form:

Dh0"h

0[1!((1!a)H#ah

0)!P]Dq,

Dhi"h

i[1!a!((1!a)H#ah

0)!(P!p

i)]Dq,

Dp0"p

0[!s#b(1!v)H]Dq,

Dpk"p

k[!s#b(H!h

k)]Dq,

where H and P are the total abundances of hostsand parasites, respectively. The host has a 0 allelethat matches none of the parasites, the parasitehas a 0 allele that escapes recognition by all hosts,and the matching speci"cities i, k"1,2, n causeresistance when i"k. The hosts' matching speci-"cities have a cost a relative to the null allele; theparasites' virulence allele has a cost v relative tothe matching speci"cities.

A candidate equilibrium with all types presentoccurs at

h*0"

sb(1!v)

(1!nv),

h*i"

svb(1!v)

,

H*"s

b(1!v),

p*0"P*!na,

p*i"a,

P*"1![(1!a)H*#ah*0],

where this equilibrium exists when v(1/n anda(P*/n(1/n and P*'0. For the last condi-tion, b(1!v)'s is su$cient and converges to

the necessary condition as n increases. A fullanalysis of this model would be useful, but myfocus here is on the interaction between speci"cand non-speci"c defense.

Interaction between Speci5c and Non-speci5cDefense

I focus on the three problems. First, whatparameters favor qualitative polymorphism innon-speci"c defense and little polymorphismfor speci"c defense? This matches the data ofFellowes et al. (1998, 1999) in which hostssucceed or fail qualitatively and non-speci"callyagainst various parasites.

Second, what parameters favor polymorphismof speci"c defense but little variation in non-speci"c defense. This matches the data ofWebster & Woolhouse (1998) in which hostswere polymorphic for highly speci"c defense todi!erent parasite genotypes but apparentlylacked variation in non-speci"c defense.

Third, how does the interaction between speci"cand non-speci"c defense in#uence the measure-ment of the costs of resistance? The role of costsand the measurement of costs have been widelydebated (Simms, 1992). Analysis of this modelclari"es some of the di$culties involved.

To address these questions, I "rst introduce mymethods of analysis. I then analyse the interac-tions between non-speci"c defense and thevarious forms of speci"c defense outlined in theprevious section.

PARAMETERS AND VARIABLES

The model requires computer analysis. I de-scribe the parameters, variables, and additionalassumptions in this section.

The non-dimensional parameters in eqns (4)were described earlier; I summarize them here.The epidemiology is controlled by three para-meters: s and b are the non-dimensionallyscaled parasites' death and birth rates, and Dqis the time step over which interactions occur.Recall that the non-dimensional parameter bincreases linearly with host density (carryingcapacity), K, as described in eqn (3).

The cost of non-speci"c defense is a. The bene-"t of non-speci"c defense is a power function of

DEFENSE AGAINST PARASITIC ATTACK 293

the amount of investment, j z, where j is invest-ment and z is an exponent that determines thebene"t. The di!erent levels of non-speci"c de-fense are coded by alternative alleles at a singlelocus. I use N"127, with non-speci"c defensevarying over j"0, 1/N,2, N/N.

The number of matching host and parasitespeci"cities is n. Both host and parasite have anadditional, null allele. For the host, this null allelematches none of the parasites (universal suscepti-bility). For the parasite, the null allele is matchedby none of the hosts (universal host range). Thehosts' cost of speci"c defense (for alleles otherthan the null allele) is a. The parasites' cost forthe null allele that confers universal host range(virulence) is v.

The host has two haploid loci, one for speci"cdefense and the other for non-speci"c defense.There are n#1 alleles for speci"c resistance andN#1 alleles for non-speci"c resistance, yielding(N#1)(n#1) possible genotypes. The parasiteshave n#1 alternative alleles for speci"city attheir single locus. Thus, the system in eqns (4) has(N#1)(n#1)2 equations. For all analyses I iter-ated these equations over discrete time steps ofsize Dq.

I assumed that any genotype that had anabundance below 10~8 was locally extinct. Lo-cally extinct genotypes could be introduced byeither immigration or mutation. For immigra-tion, after each non-dimensional time unit hadpassed (1/Dq iterations), I chose a random num-ber on the interval [0, 1]. If that number was lessthan the immigration rate, then a random hostgenotype was introduced with abundance 10~4.I then chose a second random number and, if itwas less than the immigration rate, introduceda random parasite genotype. I "xed the immigra-tion rate at 10~1 for all runs.

I also performed stepwise mutation on thehosts' quantitative trait for non-speci"cresistance. Each trait allele with value j mutatedto the next higher trait value, a step of 1/N,with probability k/2, and to the next lower traitvalue with the same probability. For all runs, I setthe mutation rate to 10~5 per non-dimensionaltime unit and performed mutation every 1/Dqiterates.

In some runs I allowed recombination betweenthe two host loci. Recombination was accomp-

lished in the standard way by random matingbetween all haploid genotypes. The diploid mat-ing pairs generated haploid progeny; loci wererecombined with probability o and remained inparental coupling with probability 1!o. I per-formed recombination in every iterate of Dq timeunits.

The mathematical formulation of the modelprovides no structure for de"ning generationsand degree of overlap in survival of parents ando!spring. The only intrinsic condition is that thehost population doubles every ln(2)+0.7 non-dimensional time units in the absence of para-sitism and host competition. I typically usedDq"0.2, which implies approximately 3.5 re-combination events per host doubling time. Thiscauses much more recombination than is likelyfor most host populations. Thus, a recombina-tion rate of o"0.05 probably approximatesa typical amount of recombination betweenfreely segregating loci. I shall return to this pointlater.

In summary, for all the results below I "xed thefollowing parameters: Dq"0.2, N"127, s"0.2,a mutation rate of 10~5, an immigration rate of10~1, an abundance of immigrants of 10~4, anda truncation level for local extinction of 10~8.I varied the following parameters: the number ofspeci"c types, n; parasite growth rate (includeshost density), b; the cost of non-speci"c defense, a;the cost of speci"c defense, a; the cost of viru-lence, v; the exponent determining rate of returnon investment in non-speci"c defense, z, and therecombination rate, o.

The following sections focus on measurementsof "ve variables. I assume, for each variable, thatspeci"c and non-speci"c components can be sep-arately measured. This may be done by di!erentappearance, response, or molecular indicators ofgenotype or physiology.

The "rst variable is variation in non-speci"cdefense, which I measure simply as the standarddeviation in the level of investment, j. The e!ec-tiveness of the defense is j z, but measuring vari-ation in investment provides a good indicator ofpolymorphism in non-speci"c defense.

The second variable is the frequency of speci"cdefense. This is the probability that, for all pairsof hosts and parasites, an interaction leads tospeci"c resistance.

294 S. A. FRANK

The third variable measures the cost of non-speci"c defense. I calculate that as the regressionof cost on non-speci"c defense. I de"ne the costexpressed by an individual host as its "tnessdeviation relative to average "tness in theabsence of parasites. For host ij, this observedcost is

cost (ij)"1!cij

E(cij),

where cij

was given in eqn (2) as

cij"G

1!aj, i"0,

(1!a)(1!aj) i'0,

and E is the expectation (average) over the fre-quency of host genotypes. Quantitative resistanceis simply j z. From these de"nitions, when averageexpression, j, is high, then E (c

ij) is small. Thus,

small changes in j cause small changes in non-speci"c resistance but large changes in the e!ec-tive cost. Note also that an association betweenthe cost of speci"c resistance, a, and non-speci"cexpression, j, can in#uence the observed cost ofnon-speci"c defense. This association will bemeasured by the "fth variable, given below.

The fourth variable is the cost of speci"c resist-ance. I measure this e!ect as the regression of coston the presence or absence of an allele that con-fers speci"c resistance. An allele confers speci"cresistance if it is e!ective against more than 1%but fewer than 99% of parasites. Presence isassigned a value of one and absence is assigneda value of zero. Cost is de"ned as in the previousparagraph.

The "nal variable is the correlation betweenspeci"c and non-speci"c defense. I coded speci"cdefense as a zero or one for absence or presenceas de"ned in the previous paragraph. Non-speci-"c resistance is j z.

For each parameter combination, I begana run by setting the abundance of a randomlychosen host genotype to the carrying capacity.All other hosts and parasites were initially absentand were subsequently introduced by mutationor immigration. I completed initialization by iter-ating over 20 000 non-dimensional time units,that is, 20 000/Dq iterations. During the following

2000 non-dimensional time units, I collected dataon the "ve variables plus other measures. I reportbelow on the median value of each variable overthe 2000 time units for which I collected data.

GENE-FOR-GENE SPECIFICITY PLUS NON-SPECIFIC

DEFENSE

I analysed this model with a factorial designover the following parameter combinations:b"1, 3, 9; a"0.2, 0.4, 0.8; z"0.4, 0.8, 1.6;a"0.05, 0.1, 0.2; v"0.05, 0.1, 0.2; o"0, 0.005,0.05, 0.5. This design has 4 ) 35"972 parametercombinations. I discuss the results with respect tothe three questions given in the introduction tothis section.

The "rst two questions can be taken as a pair:What parameters favor qualitative polymor-phism in non-speci"c defense and little polymor-phism for speci"c defense? What parametersfavor polymorphism of speci"c defense but littlevariation in non-speci"c defense?

The answers to these two questions are consis-tent with the special cases discussed earlier. Non-speci"c defense varies qualitatively when z'1and quantitatively when z(1. For z'1, poly-morphism of non-speci"c defense varies in accordwith eqn (7). For z(1, quantitative variation innon-speci"c defense tends to increase as z in-creases, which corresponds to a decline in thestrength of stabilizing selection. Figure 3 showsvariation in non-speci"c defense for a few para-meter combinations.

Variation in speci"c defense is conrolled main-ly by the parameters v and z. An increase inv causes a rise in speci"c resistance [Fig. 4(a)], asexpected from the gene-for-gene models de-scribed earlier. An increase in z causes a decline inspeci"c resistance [Fig. 4(b)]. This probably oc-curs because a rise in non-speci"c defense reducesthe bene"t of speci"c resistance.

The third question is: How does the interactionbetween speci"c and non-speci"c defense in#u-ence the measurement of the cost of resistance?The brief answer is that selection can create verystrong negative correlations between the expres-sions of speci"c and non-speci"c resistance. Forexample, an individual with speci"c resistancemay have relatively low expression of costly non-speci"c resistance. This can lead to a measured

FIG. 3. Variation in non-speci"c defense in a model with both gene-for-gene speci"city and non-speci"c resistance. Theplots show the standard deviation in j, the investment in non-speci"c resistance. For each of the following plots, each point isthe median value of the response variable over the 2000 time units of a run with a particular set of parameters. The observedvalues #uctuate over time, particularly as the tendency for epidemics (high b) increases. As explained in the text, six parameterswere varied over 972 combinations. The parameters shown provide the best explanation for the observed variation.

FIG. 4. Frequency of speci"c resistance in a model with both gene-for-gene speci"city and non-speci"c resistance. See thelegend of Fig. 3 for further details.

DEFENSE AGAINST PARASITIC ATTACK 295

cost of speci"c resistance that is negative because,when parasites are absent, individuals with speci-"c resistance have higher "tness than those thatlack speci"c resistance.

Figures 5 and 6 show the measured cost ofspeci"c resistance for various parameter combi-nations. There are several points to be noted.Observed costs are frequently negative. The

FIG. 5. Observed cost of speci"c resistance in a model with both gene-for-gene speci"city and non-speci"c resistance.Values less than !0.5 were truncated to !0.5. A value of zero was assigned when there was no variation in speci"cresistance. Recombination is measured by a scaled ratio of recombination to selection intensity (see text). See the legend ofFig. 3 for further details.

296 S. A. FRANK

physiological cost of speci"c resistance, a, is notshown as an explanatory parameter because itexplains little of the observed variation. An in-crease in b is associated with greater non-equilib-rium #uctuations (epidemics), more frequentnegative values for observed costs, and a widerrange of measured values. Higher costs of non-speci"c resistance, a, cause more frequent nega-tive values and a wider spread. Increasing recom-bination moves observed costs toward positivevalues and reduces the range of values*the roleof recombination is discussed further below.

Figure 7 shows the measured cost of non-speci"c defense. The striking observation here isthe very high values. These high values occurbecause average expression of non-speci"c de-fense can be high, causing lowered average "tnessand therefore greater variation in relative "tnessfor variations in expression (see above for thede"nition of non-speci"c cost). Another interest-ing observation is that negative values only occurin the lower-left panel, for high intrinsic costs, a,

and for low recombination rates (analysis of re-combination not shown). When intrinsic costsare moderate, a"0.2, observed costs vary mod-erately between 0 and 0.5.

Figure 8 shows the correlation between thepresence or absence of speci"c defense and thelevel of non-speci"c defense (see the de"nitionabove). The correlations vary widely and areoften strongly negative. Recombination explainsmuch of the variation. When recombination islow, selection builds negative associations*indi-viduals with strong non-speci"c defense are nat-urally favored if they do not waste resources onspeci"c defense; those with speci"c defense arefavored if they do not waste resources on non-speci"c defense. Recombination breaks downthese associations created by selection.

As mentioned above, a recombination rate of0.5 may be much higher than free segregation.I performed complete random mating and recom-bination after every iterate of length Dq"0.2.This roughly corresponds to a generation

FIG. 6. Observed cost of speci"c resistance in a model with both gene-for-gene speci"city and non-speci"c resistance.These plots highlight z along the rows rather than a as in Fig. 5. Recombination is measured by a scaled ratio ofrecombination to selection intensity (see text). See the legend of Fig. 3 for further details.

FIG. 7. Observed cost of non-speci"c resistance in a model with both gene-for-gene speci"city and non-speci"c resistance.See the legend of Fig. 3 for further details.

DEFENSE AGAINST PARASITIC ATTACK 297

FIG. 8. Correlation of speci"c and non-speci"c resistance in a model with both gene-for-gene speci"city and non-speci-"c resistance. A value of zero was assigned when either speci"c or non-speci"c resistance lacked variation. Recombinationis measured by a scaled ratio of recombination to selection intensity (see text). See the legend of Fig. 3 for furtherdetails.

298 S. A. FRANK

time for hosts that is equivalent to the timerequired for the host population to increase bye0.2!1+0.22 (22%) in the absence of parasitesand density-dependent competition. If, by con-trast, a generation of hosts corresponds to a pot-netial increase of x, then generation time isln(1#x), where generation time is the expectedtime between recombination events. For exam-ple, if a tree had a potential increase of 104 pergeneration, then the expected time between mat-ing and recombination events would beln(1#104)+9.2 non-dimensional time units, or46 times less often than imposed in my analyses.Put another way, high fecundity imposes strongselection in relation to recombination (seeDiscussion).

In summary, physical linkage or high fecundityof hosts per lifetime can create a strong negativeassociation between speci"c and non-speci"c de-fense. Such association may lead to measurementof negative costs for speci"c resistance.

MATCHING-ALLELE SPECIFICITY PLUS

NON-SPECIFIC DEFENSE

I analysed this model with a factorial de-sign over the following parameter combina-tions: n"3, 7, 15; b"1, 3, 9; a"0.2, 0.4,0.8; z"0.4, 0.8, 1.6; a"0; v"1; o"0, 0.005,0.05, 0.5. This design has 4 ) 34"324 parametercombinations. The combination of a"0 andv"1 excludes the host susceptibility allele (be-cause the cost of resistance is zero) and the para-site virulence allele (because the cost of virulenceis one).

This matching-allele model typically main-tained maximum diversity of speci"c resistance.Figure 9 shows that the frequency of resistance isapproximately 1/n for most parameters. Thecases that deviate from this equilibrium occuronly when there is disruptive selection on thenonspeci"c defense trait, z"1.6, and there is norecombination, o"0.

DEFENSE AGAINST PARASITIC ATTACK 299

In the absence of non-speci"c defense, thematching-allele model has widely #uctuatingallele frequencies (Frank, 1993b). Those #uctu-ations tend to increase as the parasites' capacityfor increase, b, rises. Here the e!ective growthrate of the parasites is kept relatively low byan increase in non-speci"c defense with a rise

FIG. 10. Level of non-speci"c defense in a model with both min non-speci"c defense is di$cult to measure directly; one can onindividuals. The high levels of average investment shown here e

FIG. 9. Frequency of speci"c resistance in a model withboth matching alleles and non-speci"c defense.

in b (Fig. 10). This led to an endemic parasitewith nearly stable allele frequencies for matchingspeci"cities.

There was no cost for speci"c resistance in thismodel, a"0. Figure 11 shows the measured cost,which tends to be either close to zero or negative.In the gene-for-gene model with costly speci"cresistance (Figs 5 and 6), the observed costs weresometimes strongly negative. By contrast, thevalues here are only weakly negative. Thus, posit-ive costs were more likely to lead to stronglynegative observed values for the cost of resistancethan were no costs.

What causes the contrast in measured costsbetween models? A cost of speci"c resistancefavors the absence of such resistance in a hostthat also has strong non-speci"c resistance. Thiscreates the appearance of high costs for the ab-sence of speci"c resistance via the associationwith high levels of non-speci"c defense. Whenspeci"c resistance has no cost, there is no pres-sure to associate the presence of non-speci"c de-fense with the absence of speci"c resistance.Consequently, the correlation between speci"c

atching alleles and non-speci"c defense. Average investmently compare di!erences in defense and apparent costs betweenxplain the genetic stability near equilibrium shown in Fig. 9.

FIG. 11. Measured cost of speci"c resistance in a model with both matching alleles and non-speci"c defense. Recombina-tion is measured by a scaled ratio of recombination to selection intensity (see text).

300 S. A. FRANK

and non-speci"c defense is weak in this model(data not shown).

Finally, the measured cost of non-speci"c de-fense had a pattern similar to that shown inFig. 7. In this case, negative values never occur-red because the association between speci"c andnon-speci"c defense was weak.

MIXED SPECIFICITY PLUS NON-SPECIFIC DEFENSE

The "nal model examines a hybrid of the gene-for-gene and matching-allele assumptions forspeci"city plus the usual assumptions for non-speci"c defense. The design repeats the prior sec-tion, but with a cost of resistance and a cost ofvirulence of a"v"0.1. The interesting contrastsfrom the prior models concern the frequency ofspeci"c resistance and the measured cost of speci-"c resistance.

Figure 12 shows the frequency of speci"c de-fense, which tends to be low for all parametercombinations. There is a small amount of speci"cresistance maintained when the cost of non-speci-"c defense is high and there are relatively few

speci"cities (lower-left panel). Otherwise, thevalue of speci"c resistance is reduced by the si-multaneous presence of non-speci"c defense tothe point where the bene"ts do not outweigh thecosts. Non-speci"c defense tends to be highlyexpressed in a pattern similar to that shown inFig. 10.

Figure 13 shows the measured cost of speci"cdefense. As in Fig. 5, stronger parasite pressure(high b) reduces the observed cost of resistance bycausing #uctuating selection and a negative asso-ciation between speci"c and non-speci"c defense.Higher costs of non-speci"c defense, a, magnifythese e!ects. Interestingly, weaker parasite pres-sure and high costs of non-speci"c defense oftencause observed costs to exceed the actual costs of0.1 (lower-right panel).

Discussion

SPECIFIC VS. NON-SPECIFIC DEFENSE

The speci"c defense observed by Webster& Woolhouse (1998) and the non-speci"c defense

FIG. 12. Frequency of speci"c resistance in a model with multiple matching alleles, costly speci"c resistance and virulence,and non-speci"c defense.

FIG. 13. Measured cost of speci"c resistance in a model with multiple matching alleles, costly speci"c resistance andvirulence, and non-speci"c defense. Recombination is measured by a scaled ratio of recombination to selection intensity (seetext).

DEFENSE AGAINST PARASITIC ATTACK 301

302 S. A. FRANK

observed by Fellowes et al. (1998) provide aninteresting contrast. Recall that Webster &Woolhouse found all of the variation amongtheir snail hosts to be speci"c against particularstrains of schistosome parasites. By contrast,Fellowes et al. found variation only in non-speci-"c defense of Drosophila melanogaster againstvarious species of parasitoid. In addition, thenon-speci"c defense appeared to be qualitative*either succeeding or failing rather than graded.

There are two ways in which to consider thespeci"city of the Webster & Woolhouse snailsvs. the non-speci"c defense of the Fellowes et al.,Drosophila. On the one hand, the snails maysimply lack the biochemical potential to producenon-speci"c defense against the various schisto-some strains, and D. melanogaster may lackthe biochemical potential to produce speci"cresponses to di!erent parasitoids. On the otherhand, both systems may contain the potential forspeci"c and non-speci"c defense, but the geneticand epidemiological parameters may favor speci-"city in the case of snails and variable non-speci-"c defense in the case of Drosophila.

One cannot choose between the biochemicalpotential and the di+erent parameter alternativesbased solely on theoretical analysis. But thewidespread distribution of both speci"c and non-speci"c defense suggests that biochemical poten-tial is often not the limiting factor. Thus, it isuseful to consider what parameters would favorone type of defense over the other.

What parameters favor the observation of spe-ci"c defense, as in the case of the snails? Whenbene"ts of non-speci"c defense increase in a di-minishing way with expression, z(1, then vari-ation tends to be limited to small, quantitativeincrements (Fig. 3). If speci"c variation is alsopresent, these small levels of quantitative vari-ation in non-speci"c defense may be overlookedduring selection experiments that emphasizelarge, qualitative e!ects. Variation in speci"cdefense is maintained unless: parasites can pro-duce a universal host-range allele with very lowcost of virulence (vP0); or there are several alter-native matching speci"cities and the hosts paysa signi"cant cost of resistance, a. These condi-tions can be roughly summarized as z(1, v'0,and a;1/n, where n is the number of matchingspeci"cities.

What parameters favor the observation of rare,qualitative, non-speci"c defense, as in the case ofD. melanogaster and its parasitoids? When bene-"ts of non-speci"c defense increase in an acceler-ating way with expression, z'1, then variationin non-speci"c defense is qualitative and in-#uenced in frequency by the conditions ineqn (7). In particular, non-speci"c defense de-clines as the cost, a, rises and as the potential rateof increase of the parasite, b/s, declines, that is,costly defense and endemic parasites with a rela-tively low rate of potential increase. These condi-tions appear to match the situation for the D.melanogaster}parasitoid interaction: Felloweset al. (1998) found that defense was indeed costly,and parasitoids have relatively low rates of in-crease compared to other kinds of parasites. Theconditions for limited variation in speci"c defenseare, inverting from the previous paragraph, vP0or a not much less than 1/n.

I suggested above that biochemical potentialprobably does not, by itself, exclude either speci-"c or non-speci"c defense. But the biochemistryof the host}parasite interaction certainly mustin#uence some of the key parameters, such as thenumber of matching speci"cities and the costs ofresistance and virulence. Thus, full knowledge ofpopulation genetics and epidemiology dependson biochemical and mechanistic knowledge;likewise, the biochemistry and mechanisms ofhost}parasite interaction only have meaningwhen embedded within the dynamics of popula-tion genetics and epidemiology (Frank, 1994).This mutual dependence between the physicalmechanisms of interaction and the populationbiology is further illustrated by the costs of resist-ance.

MEASURING THE COSTS OF RESISTANCE

The theory emphasizes costs of resistance indetermining evolutionary outcome. Much e!ortat measurement and discussion of signi"cancehave been attached to this subject (Simms, 1992).The analyses here make clear that even perfectmeasurement of natural populations can be verymisleading about the intrinsic (or mechanistic)level of costs.

The key di$culty is that selection powerfullybuilds negative associations between speci"c and

DEFENSE AGAINST PARASITIC ATTACK 303

non-speci"c defense or, more generally, betweenthe many di!erent components of the defensivearsenal. Individuals that are particularly well de-fended by any single defensive component dobest when they have strongly reduced expressionof other costly components of defense. In themodels presented here, speci"c defense with anintrinsic (mechanistic) cost often appeared tohave a negative cost*that is, speci"c defense wasassociated with higher "tness in the absence ofparasites (Figs 5, 6 and 13). Negative associationsbetween speci"c and non-speci"c defense are il-lustrated in Fig. 8.

What conclusions can be drawn? Variable in-trinsic costs provide comparative predictionsabout changing pattern; comparative patternsobserved in natural populations suggest hy-potheses about variable intrinsic costs. But costsare di$cult to measure by population analyses;we ultimately depend on future physiological andbiochemical studies. These future studies, withestimated costs, must then be fed back into popu-lation-level analyses such as those presented here.

What should be done before mechanistic esti-mates for costs are available? Probably the mostfruitful line of research will be comparisonsamong populations and among systems. I set upthe Drosophila vs. snail comparison to showa strong contrast between very di!erent systems.Studies of di!erent snails and di!erent Drosophilawill begin to "ll in a richer comparative dataset.The population outcomes will then suggesthypotheses about intrinsic costs, which can ulti-mately be tested by physiological and bio-chemical studies.

ASSOCIATIONS BETWEEN SPECIFIC AND NON-SPECIFIC

DEFENSE

I have emphasized that selection creates anegative association between costly componentsof a defensive system. An individual, successful inone component of defense, gains by reducingcostly expression of other forms of resistance.

The state of the population depends on thebalance between the rate at which selectioncreates associations and the rate at which recom-bination breaks up those associations. Recombi-nation is usually expressed as one minus theprobability that a pair of loci in a parent will be

transmitted together to an o!spring*the rate atwhich loci are separated per generation. Freerecombination corresponds to independent as-sortment of loci, that is, a recombination prob-ability of 0.5 per generation.

The model I used scaled time according to thehosts' intrinsic rate of increase, r, rather than togeneration time. This e!ectively scales time andthe strength of selection to the hosts' potentialdoubling rate rather than to the tempo of mating,birth, and death. As explained in the text, eachtime step at which I applied recombination corre-sponded to the time required for the host popula-tion to increase by approximately 22% in theabsence of competition and parasitism. If a hostpopulation could increase by only 22% per gen-eration, then the recombination rate applied cor-responds to the per-generation recombinationrate. By contrast, if a population had the poten-tial to increase by a larger factor per generation,then the strength of selection rises and the e!ec-tive recombination}selection ratio declines.

The recombination parameter shown inthe "gures is the recombination}selection ratio(Figs 5, 6, 8, 11 and 13). For example, a value of0.05 corresponds roughly to free recombination(0.5) in each generation for a population thatcould potentially increase by a factor of 6.4 ineach generation (see calculations in the sectionGene-for-gene speci,city plus non-speci,c de-fense). The model is not designed for thesenumbers to be exact, but one can extract theapproximate trends for the recombination}selec-tion scaling. The point is that strong selection cancreate negative associations even when recombi-nation is high.

In this model, separate loci encode the twoindependently expressed components of speci"cand non-speci"c defense. Associations betweenlevels of expression can only arise passively bystatistical processes of selection and recombina-tion. Clearly, hosts can gain an advantage bycoupling di!erent components into a commonregulatory network. For example, costly non-speci"c expression may follow only after a triggerby recognition of invasion. The model presentedhere shows how the early stages of associationamong components may have arisen by statist-ical coupling of independent components. Thissets the stage for modi"ers of regulatory control

304 S. A. FRANK

that can enforce coupling by controlled responsesto particular stimuli. It may be that inducibledefense and other complex defensive cascadesarose in this way.

This research is supported by National ScienceFoundation grant DEB-9627259.

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