Cost theory and the cost of substitution a clarification

Cost theory and the cost of substtuton a clarfcaton Walter J. ReMne The cost of substtuton has been wdely msnterpreted, whch has lmted ts utlty. Ths paper clarfes the cost concept and re-establshes ts vtal role for nvestgatng populaton phenomena. Many factors that tradtonally caused confuson are dentfed and dsmssed, ncludng genetc death, genetc load, the envronment, and extncton, whch are not essental to the cost of substtuton. Instead, the cost concept s here defned as the reproducton rate requred by a scenaro. Unlke the tradtonal vew, ths cost concept s general and apples under the wdest crcumstances, ncludng extreme fluctuatons n populaton sze and selectve value, and for dscrete- or contnuous-generaton models. Yet under the smplfed crcumstances tradtonally assumed, ths concept reduces exactly to the tradtonal formulas. For sngle substtutons, proofs show that the mnmum total cost of substtuton occurs when the cost remans constant throughout the substtuton. A clarfed bass for a generalpurpose cost theory s offered. J.B.S. Haldane 1 frst descrbed the cost of substtuton and ts lmtaton on the speed of evoluton. That gave rse to a problem (see, for example, Dodson 2 ), known today as Haldane s Dlemma. The problem s more severe n organsms wth low reproducton rate and long generaton tme, such as the hgher vertebrates: elephants, whales, apes and humans, etc. Evolutonary genetcsts saw ths as a compellng ssue. Maynard-Smth 3 and Kmura 4 each cted t as the man reason for ther revolutonary new vews of evolutonary process. Yet the fundamental cost concept fell nto long-lved confuson, whch lmted ts deployment. Today, most commentators say the problem s solved, but exhbt lttle agreement as to why. One modern authorty, George C. Wllams, 5 asserted, the [Haldane s Dlemma] problem was never solved, by Wallace or anyone else. Ths paper wll show the problem cannot be solved so long as confuson prevals over the fundamentals. For a layman s perspectve see Haldane s dlemma has not been solved, pp. 20 21 Ths paper clarfes the cost concept and re-establshes t as a vtal tool for nvestgatng populaton phenomena and our attempts to explan or descrbe them (heren called scenaros). A cost, as here defned, s smply the reproducton rate requred by a gven scenaro. If the gven speces cannot supply that reproducton rate, then the scenaro s not plausble. In more concse wordng, the speces cannot pay the cost. At ts core, a cost argument s that smple. A cost and ts payment are both reproducton rates. They are defned dentcally, except that a cost s requred by a gven scenaro, whereas a payment s actually produced wthn the gven speces. Hstorcally, much confuson was created from overemphaszng secondary matters such as genetc death, the prevous organsms that are elmnated, the envronment, ftness and genetcs. So keep n mnd, a cost s about the requred reproducton rate, and all else s secondary. Indeed, the unt of cost s reproducton rate (.e. offsprng per ndvdual per generaton). Every cost s specfed n terms of reproducton rate. Ths also means every cost s specfed as a cost per generaton, because per generaton s the understood manner for specfyng reproducton rate. As s customary, all reproducton rates are normalzed by the number of producers. For example, f a male and a female produce sx progeny, ths s a reproducton rate of three, not sx. Ths consstent focus on reproducton rate clarfes the cost concept. There are many types of cost, each named after what t models. The smplest cost, called the cost of contnuty (C C ), s the reproducton rate necessary to contnue the reproducers from generaton to generaton. Ths cost s 1.0; that s, a reproducton rate of 1.0 s necessary solely to sustan the reproducers over the long term. All other costs requre reproductve excess a reproducton rate n excess of mere contnuty. These other costs wll tabulate n addton to the cost of contnuty. Such costs nclude: the cost of random loss (C R ), the cost of elmnatng harmful mutatons, also known as the cost of mutaton (C M ), the cost of segregaton (C X ), and several others. Ths paper focuses on a specfc cost: the cost of Note: Ths paper was submtted prevously to the journal Theoretcal Populaton Bology, where renowned evolutonary genetcsts Warren J. Ewens and James F. Crow revewed t, along wth Alexey Kondrashov and John Sanford. They all acknowledged ths paper s essentally correct n all matters of substance. However, Ewens and Crow rejected t from publcaton on the grounds that t s not suffcently new or dfferent from what was known by themselves and some of ther colleagues n the 1970s. However, they never communcated ths knowledge to the greater scentfc communty, nor to the publc at large. There were rare correct nsghts scattered sparsely n the lterature, but those were ncomplete, overwhelmed by confuson, and never communcated together n a coherent manner. Ths has all been very unfortunate, as there contnues to be wdespread msunderstandng wthn the scentfc communty regardng these mportant matters, even among those who have studed the cost lterature for years. It s hoped that the clarfcatons presented n ths paper, whch are sound, wll eventually reach the greater scentfc communty. Walter J. ReMne. TJ 19(1) 2005 113

Cost theory and the cost of substtuton a clarfcaton ReMne substtuton (C S ). In ths paper, a trat s a hertable bologcal characterstc. The adjectve old refers to the prevously predomnant trat that s beng replaced by somethng new (also called the substtutng trat). The new trat begns as a unque mutaton and s substtuted nto the populaton. The old trat s elmnated. Indvduals that have the old trat (new trat) are called the old-type ( new-type ). Evoluton requres the substtuton of trats nto a populaton. The new trat goes from few n number to many n number. The new trat ncreases and grows. In ths paper, ncrease and growth refer to the actual number of copes of the trat (not the frequency of the trat, and not growth of the trat n the embryo or phenotype). For example, a new trat grows from one copy to one mllon copes. Ths growth requres reproductve excess a cost. The cost of substtuton (C S ) s the extra reproducton rate requred to ncrease a specfc genotype at the rate gven by an evolutonary scenaro. Ths defnton s devod of genetc detal because the cost of substtuton s not prmarly about genetcs; rather, t s about the ncrease of a trat or trats (va the ncrease of genotypes) and the reproducton rate requred for achevng t. The genetc detals can be fleshed-n, when needed n specfc case studes. For smplcty, ths paper deals only wth ncreases acheved by excess reproducton rate. (Increases or decreases due to mgraton and recurrent mutaton are mnor phenomena, to be dealt wth n another paper.) Also, ths paper focuses on sngle substtutons (non-overlappng n tme). Multple substtutons (overlappng n tme) are an advanced topc to be covered n another paper. Untl the bascs are covered, we need to leave out genetc complextes, lke sexual reproducton and dplody. As a smple example, assume that, averaged over the long-term, the populaton has a unform reproducton rate. Nonetheless, a new trat can substtute nto the populaton. Our example scenaro clams the new trat ncreases by 114 New-type New-type One generaton later Fgure 1. A smple scenaro showng the cost of substtuton. Four elephants are the ntal new-type. One generaton later, fve elephants are the new-type. For that to be plausble, a reproducton rate of at least 1.25 (= 5/4) s requred n that generaton. twenty-fve-percent n one generaton (as n fgure 1). The speces s therefore requred to supply a reproducton rate of 1.25; n other words a cost of contnuty (C C ) of 1, plus a cost of substtuton (C S ) of 0.25. Ths example dsplays the cost of substtuton at ts smplest and most essental. If the gven speces cannot actually supply a reproducton rate of 1.25 (whch would requre 2.5 brths per female n speces wth a 1:1 male/female sex rato), then the speces cannot pay the cost and the scenaro s not plausble. That s how a cost argument operates. The cost amount cannot be reduced by further detals (concernng the envronment, populaton sze, the fate of old-type ndvduals, the manner of selecton process, nor by fluctuatons n any of those). For example, envronmental change and soft-selecton (no matter how those are defned) cannot reduce that cost. The cost s smple, mechancal and unavodable. When the scenaro clamed an ncrease of twenty-fve percent n one generaton, at the moment the scenaro made that smple clam, a cost of substtuton of 0.25 was requred. In the prevous scenaro, assgnng the old trat a lower reproducton rate does not reduce the cost. Instead, t affects who must pay the cost. In the frst scenaro, the reproducton rate s unform throughout the populaton, so that requres all genotypes to be produced at a reproducton rate of at least 1.25, and they all ncur the same cost of substtuton (= 0.25). In the second scenaro, only the substtutng genotype ncurs that cost. The phrases Genotype X ncurs a cost C and The cost of genotype X s C are shorthand for The effectve producers of genotype X progeny are requred to produce genotype X progeny at a reproducton rate of C or greater, devoted to the task dentfed by the type of cost. For greatest physcal understandng, focus on the genotype wth the largest growth n a gven generaton (.e. focus on the substtutng genotype), because that sets the crtcal amount of the cost of substtuton, and that genotype always ncurs that cost. Moreover, that amount remans fxed regardless of whether the reproducton rate s unform. Call ths the cost equvalence prncple. Snce ths cost amount s unaffected by varaton n ndvdual reproducton rates, we can safely presume a unform reproducton rate. (Haldane lkewse used that smplfcaton n all hs tutoral examples, though he dd not explan t well.) We can now gnore the reproducers of the old-type ndvduals, and focus only on the reproducers of the substtutng trat. These reproducers are requred to supply a reproducton rate suffcent to cover all the costs of the scenaro. These reproducers must supply replacements for themselves (whch ncurs cost C C ), plus supply those ndvduals elmnated through random losses (whch ncurs cost C R ), plus supply those ndvduals elmnated because they possess harmful mutaton (whch ncurs cost C M ), plus supply an ncrease of the substtutng trat (whch ncurs cost C S ). And so forth through the other costs. The reproducers must supply ths full reproducton rate; otherwse the scenaro s not plausble. In other words, the full requrement the cost of evoluton sums all these costs. The costs all have TJ 19(1) 2005

Cost theory and the cost of substtuton a clarfcaton ReMne Papers the same unts (they are all reproducton rates), so they add together convenently. Cost of evoluton = C C + C R + C M + C X + C S +... (1) By focusng on the new trat (not the old trat, as was tradtonally done), one gets a clear physcal understandng of the reproductve requrements and why they are mandatory. Let R be the actual (realzed) reproducton rate for the new-type progeny. R s called the payment, and s an observable feature of natural populatons. A scenaro s mplausble f the payment, R, s less than the cost of evoluton. R < Cost of evoluton (2) In other words, the scenaro s mplausble f the actual reproducton rate (for new-type progeny) s less than the requred reproducton rate (for new-type progeny). Ths plausblty crteron can be appled repeatedly, to each generaton n turn. It can also be appled over the longterm by usng long-term averages of costs and payments. Ths framework wll seem foregn to students schooled n the tradtonal vew, whch ntroduces confuson through a varety of other approaches. Amd the varous approaches, ths paper derves from the followng narrow lneage. The cost concept orgnated wth Haldane. 1 Its math was mproved by Crow, 6 wheren he also explctly brought n the term reproductve excess. Ne 7 ntroduced a concept called accumulated fertlty excess necessary for a gene substtuton, 8 whch he defned for a constant populaton sze and small selecton coeffcent. Merrell 9 best summarzed the md-term comprehenson. Varous cost ssues were clarfed by ReMne. 10 The present paper draws on those sources, further clarfes cost theory, and elmnates tradtonal sources of confuson, whle progressvely addng depth. The result s a clarfed cost concept, harmonous wth Haldane s orgnal ntenton and calculatons. Calculatng the cost of substtuton Let X be any hertable trat, where the trat s cyto-genetc dentty s unque, ndvsble, and consstent throughout the substtuton. (The trat s not multfactoral, where varous mxes of dfferent DNA appear to cause the same phenotypc outcome.) Therefore, the trat can be dentfed generatonby-generaton and we may dscuss ts growth. Also, the trat s nherted, or not, as a complete unt (.e. no partal nhertance of the trat). One may thnk of the trat as a block of DNA of some arbtrary length, or a cytoplasmc character. We should not overfocus on the trat and ts nature. The ssue here s not the trat, but rather the growth of the trat, and the reproducton rate necessary to acheve t. Let reproducton occur shortly after the start of each generaton. In generaton, let P be the number of effectve producers of type-x progeny; call ths the effectve startng count. As the cycle of that generaton comes to a close, let P be the endng count of type-x adults who produce progeny n the succeedng generaton. In other words, we are countng partcular groups of reproducers at the generaton s start and end. The frst group s the sole producer of the second group, and that places a reproductve requrement on the frst group. (Note: throughout ths paper, a prme, such as P, denotes a quantty as the generaton comes to a close.) The ncrease s P = P P. Then n generaton, the cost of substtuton for type-x s: P P Cost = = 1, f > 0; P P otherwse, Cost = 0 (3) In smple cases (such as asexual haplods), the effectve startng count equals the prevous generaton s endng count. In other words, P and P count the same group of parents, one generaton apart. Snce the latter group s produced solely by the former group, the requred reproducton rate s P /P. Subtract the cost of contnuty (= 1) to obtan the cost of substtuton. Unlke the tradtonal vew, ths cost concept does not assume a constant populaton sze. Indeed, the populaton sze may fluctuate wldly, and any cost assocated wth that wll automatcally be ncluded wthn the above defnton, wth no specal handlng. Ths makes sense because populaton growth and substtuton are vrtually the same thng as far as the trat s concerned they both ncrease the trat. The total cost of substtuton ( total cost ) merely sums the cost of substtuton over an entre substtuton cycle, from begnnng to end. Total cost of substtuton = Σ Cost (4) For a gven populaton sze, the total cost often remans approxmately constant over wde ranges of other parameters. Ths consstency makes the total cost a useful fgure for characterzng substtutons n a gven populaton, and s generally used for that purpose. Equaton 4 represents what the lterature loosely has called the cost of substtuton. But what s the physcal meanng of a reproducton rate summed over many generatons? It has no obvous physcal nterpretaton, and was a source of confuson. Equaton 4 s actually an ntermedate step n fndng C S, the substtutonal cost per generaton, whch s the real focus of our concern. Therefore, total cost s calculated n equaton 4, then averaged over all the substtuton cycles of the scenaro, and later dvded by the average number of generatons per substtuton cycle, to obtan the average cost per generaton, whch s used for C S. We could just as well obtan C S by averagng equaton 3 over the long-term. In the same way, equaton 3 drectly gves C S for any sngle generaton we choose to analyze. In any case, C S s the quantty we seek, and equaton 3 s what TJ 19(1) 2005 115

Cost theory and the cost of substtuton a clarfcaton ReMne ultmately supples knowledge of t by narrowng the term the cost of substtuton to these quanttes, confuson can be reduced. Cost averagng can produce smple, powerful arguments. For example, Haldane 1 estmated the total cost of substtuton s 30 (on average); therefore, f there s one substtuton per 300 generatons (on average), then the cost of substtuton, C S, s 30/300 = 0.1 (on average). Indeed, that was the knd of averagng argument that Haldane made. Equaton 3 contans an f-clause. It models the fact that decrements (decreases) of a substtutng trat n a gven generaton have no physcal means to reduce the total cost or the average cost. On the contrary, such decrements wll ncrease total costs, and the f-clause precsely ncorporates that fact. Obvously, when the substtutng trat decreases, t does not requre extra reproducton rate, so there s zero cost, as defned heren. In ths way, costs are nherently non-negatve. Cost averagng s most useful over tme ntervals where the cost each generaton, Cost, remans reasonably constant; otherwse the serousness of cost problems can be underestmated or obscured. For example, take a scenaro one mllon generatons n length, where Cost s one mllon for one generaton, and zero thereafter. The average cost s 1. But that average cost fgure conceals the extremely hgh cost n one generaton. It obscures the fact that no speces on Earth can pay enough n that one generaton to make the scenaro plausble. In other words, average cost s a helpful statstc so long as the maxmum cost does not make the scenaro mplausble (see fgure 2). Some scenaros may requre partal (or pece-wse) averagng to obtan dfferent average cost values for dfferent tme-perods. Very complex scenaros may requre abandonng the averagng technques altogether, and nstead requre comparson of costs and payments generaton-by-generaton. 116 Cost versus speed A correct framng of the cost concept places unavodable concrete lmtatons on scenaros. Indeed, cost theory s so unavodable t even apples to computer smulatons of evoluton. As a smple example, set asde all the genetc complextes (dplody, etc.) and examne a smple speces possessng the one essental attrbute reproducton. Suppose a sngle substtuton requres a new trat to start n one adult, and ncrease to one mllon adults. Say a gven scenaro allows only one generaton. Ths would requre a reproducton rate of at least one mllon progeny per adult. That equals a cost of contnuty (C C ) of 1, plus an extra reproducton rate of 999,999 whch s the cost of substtuton (C S ). It s mpossble to get a lower cost for ths scenaro. Because of the hgh cost, ths scenaro would be mpossble for all speces. The only thng that can pay the cost s excess reproducton rate. The benefcal-ness of the trat cannot pay the cost the trat cannot pay for tself. Only genune reproductve excess can pay the cost of substtuton. The cost can be reduced, by lowerng the substtuton rate. Suppose the substtuton takes three generatons. I arbtrarly assgn numbers (500 and 20,000) for the ntermedate generatons, and show the costs n Table 1. Table 1. A three-generaton substtuton n a populaton of 1 mllon No. of ndvduals wth trat Cost per generaton Total cost of substtuton Tme (n generatons) 0 1 2 3 1 500 20,000 1,000,000 499 39 49 587 Suppose a scenaro clams the substtuton happens exactly as shown n Table 1. Ths would requre the speces to pay costs of 499, 39 and 49 n the frst, second and thrd generatons, respectvely. If the speces cannot plausbly come up wth those payments (n precsely those generatons), then the scenaro s not plausble. In ths way the argument apples tghtly, even on a generaton-by-generaton bass. My arbtrarly chosen numbers happen to gve a total cost of 587. Yet no matter how you adjust the fgures for the ntermedate generatons, the total cost wll always be at least 297. It s mpossble to get lower. An excess reproducton rate of 99 for each of three generatons s just barely suffcent to satsfy the scenaro. Other arrangements always total more than 297, but never less. Thus, even f a scenaro does not specfy the ntermedate generatons, we can stll set a lower Requred reproducton rate 10 9 8 7 6 5 4 3 2 1 0 Tme (n generatons) Maxmum Average Actual Fgure 2. Cost vs Tme, showng how the average cost may not adequately represent the maxmum cost. In cases lke ths, where the cost spkes hgh, the maxmum cost must be used as the more strngent test of the scenaro. In cases wthout such spkng, the average cost s often a more convenent test of the scenaro. TJ 19(1) 2005

Cost theory and the cost of substtuton a clarfcaton ReMne Papers bound on the total cost. So our framework stll has force, even when scenaros are not specfc. For a substtuton of a gven duraton, a smple proof shows that the lowest total cost s acheved when the costper-generaton remans constant (see Appendx). Table 2 summarzes ths pont. Table 2. Mnmum total cost for ncreasng a trat by a factor of one mllon Substtuton duraton (n generatons) Mnmum Total Cost 1 999,999.0000 2 1,998.0000 3 297.0000 4 122.4911 5 74.2447 6 54.0000 7 43.3780 8 36.9873 9 32.7743 10 29.8107 30 17.5468 100 14.8154 300 14.1386 1,000 13.9114 13.8155 Table 2 s gven by the followng formula. Let P S and P E be the number of copes of the new trat at the start (S) and end (E) of the substtuton cycle. Defne the substtuton growth factor K = P E /P S. Let N be the number of generatons for the substtuton. The mnmum total cost s gven by: N Total Cost of substtuton = N K 1 (5) These optmally low costs requre the trat to have a constant growth rate throughout all generatons. Snce nature does not provde ths constancy, real cases wll always have hgher costs. Also, f the new trat decreases even momentarly, then the total cost ncreases, because some costs wll be ncurred more than once. Some theorsts beleve cost problems can be solved by a non-constant growth rate such as frequency- or denstydependent ftness, as employed n soft selecton. 11 But that does not reduce the problem, at least not for sngle substtutons, as shown above. Rather, constancy s requred to mnmze the problem, as t allows the lowest possble total cost for a substtuton of any gven duraton. The total cost s extremely hgh for fast substtutons, and decreases for slower substtutons. The absolute lowest total cost occurs under condtons never met n nature. It occurs only when the trat ncreases monotoncally, at a constant growth rate, and when that rate s nfntely slow. In that case, equaton 4 becomes: Total cost of substtuton = PE dp P E = loge = log e K (6) P P PS S (Note: When N goes to nfnty, equaton 5 smplfes exactly to equaton 6, as expected.) For a sngle trat to ncrease by a gven substtuton growth factor K, equaton 6 gves the mnmum total cost under deal condtons whch are never met n nature. Ths has also been the most commonly used equaton for the total cost of substtuton. Tradtonal sources of confuson The cost of substtuton depends foremost on the growth rate clamed n a scenaro. It s not prmarly about the envronment, the old-type organsms that are replaced, or ther lfe hstores. These had prevously been suggested as ways to reduce or elmnate the cost, but they cannot possbly reduce the cost lower than the mnmums gven above. Ther predomnant effect s to rase the costs and/or lower the payments, and thereby aggravate cost problems. Many commentators (e.g. Van Valen, 12 Felsensten, 13 Hartl, 14 Merrell 9 and Grant 15 ) ndcate that accordng to the cost of substtuton, f a populaton undergoes too many substtutons too rapdly, the cost wll be too great and the speces wll become extnct. That nterpretaton s faulty. The cost of substtuton s not a theory of extncton, and cost, by tself, does not cause extncton. Rather, the cost of substtuton supples a crteron of plausblty, whch compels us to reject some scenaros as mplausble. Then extncton may, or may not, result for varous addtonal reasons. For example, f a scenaro assumes a certan substtuton rate s necessary to fend off extncton, then ths assumpton tself forges the lnk to extncton, not cost theory. The target of cost theory s the plausblty, or otherwse, of a gven scenaro. Some commentators (e.g. Brues 16 and Merrell 9 ) suggest the cost of not evolvng s greater than the cost of evolvng. That reckonng uses the word cost nformally, as though not evolvng leads to extncton, and extncton has a hgh cost. But that nterpretaton does not correspond to any populaton genetc defnton of cost. Used correctly, the cost of not evolvng the cost of no substtutons s zero. Other thngs beng equal, the cost of adaptve evoluton s always greater than the cost of not evolvng. The cost of substtuton was tradtonally taught usng the concept of genetc death. The dea was that a substtuton requres the genetc death, or elmnaton, of the old-type ndvduals. That created confuson by emphaszng the wrong thng: the death of the old-type, ther lfe hstory, how they de, and so forth. Fundamentally, the cost of substtuton s not about that. It s about the ncrease the excess brth of addtonal copes of the new-type. The tradtonal approaches TJ 19(1) 2005 117

Cost theory and the cost of substtuton a clarfcaton ReMne assumed a constant populaton sze, where, n each generaton, the actual reducton of the old-type precsely equals the excess brths of the new-type. So the concept had some ntutve appeal, and n the smplest cases yelded a correct result. Ths paper dsmsses the concept of genetc death along wth the requrement for constant populaton sze. The above dscusson dd not menton whether the substtutng trat s benefcal, neutral or harmful because t does not matter. Whenever the trat ncreases through reproductve means, there s a cost. Ths contradcts almost all the tradtonal lterature, whch holds that neutral substtutons have no cost (e.g. Merrell 9 ). I clarfy the dscrepancy by notng that neutral substtutons ndeed have a hgh cost (because they fluctuate up and down, repeatedly ncurrng addtonal costs), but ther overall substtuton rate s not cost-lmted. 17 Neutral and slghtly harmful mutatons ndvdually substtute very slowly, but they have specal mechansms that allow hgh overall rates of substtuton unlmted by cost (but nstead lmted by the mutaton rate). 1. Hgher payments There are a few specal mechansms for supplyng larger reproductve payments. These mechansms employ randomness n the payment a stochastc reproductve excess at the level of the ndvdual, plus, n sexual speces, also at the level of the gene. [Note: n a sexual speces, n each generaton, each gene locus experences a doublng (at fertlzaton) and halvng (at meoss). From the gene s-eye-vew, the doublng s a source of reproductve excess, though of a stochastc or random nature. The doublng-andhalvng gves an average gene-level reproducton rate of 1.0, but randomly fluctuates above and below that, to provde stochastc reproductve excess. Crossng-over s an addtonal source. These sources at gene-level are suffcent to pay for substtutons, even f there were no reproductve excess at the level of the ndvdual.] Vrtually all of a speces reproducton s random wth respect to a gven neutral substtuton, and the stochastc component of all that reproducton can be large. Ths stochastc reproductve excess s what pays for genetc drft and the substtutons of neutral and harmful trats. These mechansms rely on randomness, and are overwhelmngly non-benefcal or harmful n outcome. Wthn these mechansms, benefcal evoluton becomes a moot pont. 2. Lower costs There are also specal mechansms for reducng the average cost of substtuton when many mutatons occur near each other n tme, lnked near each other on a chromosome, then they substtute together. In ths way many are substtuted for the cost of one substtuton. However, to have much effect, ths mechansm requres a super-abundance of the new mutatons to be substtuted. For example, wth unlmted hgh rates of neutral or harmful mutaton, unlmted hgh rates of these substtutons are achevable unlmted by cost. Those same mechansms, however, are not avalable to benefcal substtutons. Long before 118 those mechansms can ad benefcal substtutons, the populaton wll be overwhelmed (and destroyed) by harmful mutaton. (For example, the desred benefcal mutatons would far more often be lnked to harmful mutatons so these would travel together, usually to elmnaton, wth no net beneft to the speces.) In ths case, the cost of elmnatng harmful mutaton (C M ) would be hgh thereby aggravatng cost problems. Lastly, the populaton would lkely be n error catastrophe, where harmful mutatons would chroncally accumulate faster than they could be elmnated. All ths happens because harmful mutatons vastly outnumber benefcal mutatons, n quantty and effect. Long before there s the necessary super-abundance of benefcal mutatons, the scenaro would suffer severely from hgh rates of harmful mutaton. In such a stuaton, benefcal evoluton becomes a moot pont. Evoluton must operate wthn relatvely low rates of benefcal mutaton, and ths precludes ths mechansm for reducng the average cost. In short, whle all substtutons have a cost, ths fact only lmts the benefcal substtuton rate. Examples from nature showng rapd rates of neutral or harmful substtuton do not contradct cost theory or the cost of substtuton, nor do they explan the bologcal desgns that evoluton s called on to explan. Some commentators (e.g. Felsensten 13,18 ) have clamed the substtuton of favorable mutants n the absence of envronmental change does not mpose any cost. That s mstaken. The cost of substtuton embodes the unyeldng fact that growth requres reproductve excess. The equatons gven n ths paper represent optmal stuatons, ndependent of the envronment. On average, envronmental change can only ncrease costs and/or decrease payments, thereby ntensfyng cost problems. Substtutons when populaton sze fluctuates Some nvestgators have argued as follows: suppose the populaton comprses one mllon old-type ndvduals and one new-type ndvdual. If the one mllon old-type ndvduals de wthout hers, then fxaton occurs extremely rapdly, n one generaton. Thus, elmnaton of the old-type (whch they equated to the cost of substtuton ) places no lmtaton on the substtuton rate. However, that scenaro s ncomplete, snce t does not represent a complete cycle of substtuton. It does not represent how evoluton happens over the long term. Ths scenaro reduced the populaton sze down to one ndvdual, where countless generatons would pass before the next benefcal mutaton would occur, allowng the cycle of substtuton to begn anew. (For example, a populaton of one wll receve benefcal mutatons one mllon tmes slower than a populaton of one mllon.) Ths mode of evoluton would be exceedngly slow. To speed thngs up, the scenaro can clam the populaton grows (perhaps returnng to ts prevous sze), but that requres reproductve excess a TJ 19(1) 2005

Cost theory and the cost of substtuton a clarfcaton ReMne Papers cost of substtuton, as defned above. A speed lmt occurs because the populaton growth rate s lmted to the speces avalable excess reproducton rate. Once agan, the focus on elmnaton of the old-type created confuson by focusng on the wrong thng. The ssue s not elmnaton of the oldtype, but rather the growth of the new-type. Ths paper takes a broader vew of substtuton. Under the tradtonal vew, a substtuton s defned by changes n allele frequency, and a substtuton ends at the moment of fxaton thereby excludng any perods where allele frequences reman constant, and gnorng populaton growth as rrelevant. That vew s nadequate for studyng substtuton rates over the long term. For ths purpose a substtuton cycle can be usefully defned as an nterval of tme begnnng wth the ntroducton of a mutaton that reaches fxaton, then contnues after fxaton untl the ntroducton of the next mutaton to reach fxaton. Ths nterval wll be at least as long as, and usually longer than, one entre substtuton (and may nvolve substtutons overlappng n tme). Under ths defnton, there s a contguous set of substtuton cycles, wth no overlap, or omssons n tme. The goal s always to tally the reproductve requrements of the scenaro exactly once, wth no tme perods omtted or counted twce. The above defnton allows calculaton of the total cost even when populaton sze fluctuates (whch the tradtonal cost concept dd not allow). Such fluctuatng scenaros are dverse and awkward to generalze. Ther detals are an advanced topc, outsde ths paper. The assumpton of constant populaton sze s here taken solely for smplfcaton, so further ssues can be clarfed. Haplods, clonal, or self-fertlzng organsms, or maternally nherted cytoplasmc characters For sake of clarty, the cost concept (defned above) was stated n ts smplest, non-genetc form. When appled to specfc case studes, the defnton can take on the tradtonal termnology of populaton genetcs where the substtutng trat could be a pont mutaton, nserton, deleton, allele, nverson, duplcaton, the relatve order of genes on a chromosome, or smlar. To smplfy dscusson, the tradtonal term allele wll be used as an exemplar for all these. In a haplod, durng a sngle substtuton, let there be P ndvduals wth allele A, and one generaton later a scenaro clams t ncreases by P. The cost s gven drectly by equaton 3. Notce ths does not depend on the old-type allele, ts characterstcs, how t des, or the envronment. The cost smply depends on the growth of the substtutng allele. Let N e be the effectve breedng populaton sze. It s typcally the number of adults who breed. (Its precse defnton s not crtcal to our calculatons, because ts mpact s solely to measure the populaton growth or decay. It drops out and has no effect on calculatons when t remans constant.) Let the populaton growth factor G = N e /N e. When N e remans constant, G = 1. The concept of effectve producer of progeny allows generalzaton of our notons about reproducton. It allows the converson of sexual reproducers nto reproductvely equvalent asexual reproducers. To see how ths works, adopt the usual conventons of populaton genetcs. Let the new and old alleles be A and a, wth parental frequences p and q, where p + q = 1. Multply that equaton by N e to obtan N e p + N e q = N e. The terms of that equaton dentfy the effectve producers of the two genotypes. In effect there are N e p producers of new-type progeny, and N e q producers of old-type progeny. It s as though all the reproducton of N e p adults goes to producng genotype-a progeny, and all the reproducton of N e q adults goes to producng genotype-a progeny. Those fgures are exact for asexual speces. For sexual speces, those fgures are average values. In any case, we have subdvded the populaton s reproductve capacty nto portons responsble for producng each genotype. A genotype s effectve startng count s the number of effectve producers of that genotype. For the substtutng allele, that fgure s P = N e p. By defnton, the endng count s P = N e p. Then restate equaton 3: Cost = N e p 1= G p 1, f > 0; N p p e otherwse, Cost = 0 (7) Equaton 7 s vald under wde condtons, such as erratc fluctuatons n both the populaton sze and growth rate of genotype A. Next, assume a constant populaton sze (G = 1), and utlze the fact that genotype growth s usually specfed n terms of selectve values. Let genotypes A and a have selectve values 1 and 1-s, respectvely, where s > 0. Genotype: A a Selectve value: 1 1 s Frequency: p + q = 1 (8) Apply Selecton: p + (1 s)q = 1 sq (9) Normalze: p 1 sq (1 s)q + = 1 (10) 1 sq The left-hand term s the new allele frequency p, whch, by equaton 7, gves: sq Cost = 1 sq (11) Equaton 11 gves the mnmum requred excess reproducton rate. It s accurate regardless of the envronment (changng or unchangng), the selecton coeffcent (small or large), or the type of selecton (soft or hard). For example, n a populaton of one mllon adults, where one has the new allele A (therefore, p = 10 6 ), suppose s = 1 (whch means the substtuton occurs n one generaton), then Cost = 999,999. That reconfrms the frst scenaro n the above secton, cost versus speed. TJ 19(1) 2005 119

Cost theory and the cost of substtuton a clarfcaton ReMne 120 Genetc death Equaton 11 s dentcal to Crow s cost formula, and (under Haldane s addtonal assumpton of s << 1) t smplfes exactly to Haldane s formula. Ths s remarkable because ths equaton derves from dfferent physcal reasonng (though wth a smlar goal n mnd). Haldane reasoned that the populaton begns wth a normalzed sze of 1 (on the rght sde of equaton 8). After selecton, that quantty s reduced by an amount seen at the rght sde of equaton 9 (sq, n ths case). Haldane dentfed ths reducton as the genetc deaths dvded by the adult populaton sze (so t has unts of genetc deaths per adult), and defned t as the cost. Haldane 1 used ths same mathematcal method (reflexvely, and wthout physcal justfcaton) n all hs varous case studes. Later, Crow s formula covered the full range of s-values, but based on the same physcal reasonng as Haldane. Crow defned the cost as the rato of those elmnated (sq), to those not elmnated (1 sq). That defnton twce uses Haldane s concept of genetc death. Ther formulas gve mathematcally correct predctons, but based on dubous physcal nterpretaton. As ponted out by Feller, 19 Moran, 20 Hoyle 21 and Wallace, 22 Haldane s defnton of genetc death does not make sense physcally, and they gve compellng examples of ts falure. I here dentfy the reason. Equatons 9 and 10 are separate mathematcal steps that represent a sngle ndvsble physcal process. In nature, there s no step of applyng selecton, followed by a separate step of normalzng. Rather, ths all happens n one swoop. Haldane s genetc death concept exsts only n the mddle of these equatons (at equaton 9). It s a mathematcal phantom that does not exst n physcal realty. The soluton s to jettson hs concept of genetc death, and buld upon frmer ground. The physcal meanng of a genetc death has been a contnual obstacle. Examne the smplest case [a sngle generaton sometme durng a sngle substtuton n haplods, wth constant populaton sze, and unform reproducton rate (followed by selecton through juvenle death), and s << 1]. In ths smplest case, and usng the genetc death concept, we can construct a correct argument that lmts the substtuton rate. But extreme care s requred because the logc s ndrect and complcated. Frstly, the deaths had to be converted (somewhere, somehow) nto a requrement on reproducton rate. Tradtonally, ths converson was done mplctly (not explctly), so t was not well understood. Secondly, n ths smplest case, genetc death was nterpreted as a death of old-type ndvduals caused by the presence of the new allele or, equvalently, the deaths would not have occurred f the new allele had been absent. So, dentfyng genetc death seemed to requre that the scenaro be examned under two crcumstances wth and wthout the substtutng allele. But what possble relevance could the second crcumstance have? Why would the second crcumstance where substtutons are absent have any bearng whatever on lmtng the substtuton rate? Ths mystery was cause for confuson. Some researchers nterpreted the mystery as follows. If the envronment s deteroratng, and the new allele s benefcal relatve to the deteroratng envronment, then the same deaths would occur even f the new allele had been absent. Lkewse, the presence of the new benefcal allele would not ncrease the deaths. Therefore, the deaths are not caused by the new allele, but are entrely caused by the envronmental deteroraton. On such a bass, some researchers (e.g. Felsensten 13,18 ) concluded that n a non-deteroratng envronment, benefcal substtutons would have zero cost. That faulty concluson, though stll common today, was due to genetc death and the confuson t creates. In short, even n the smplest cases, many nvestgators struggled to understand genetc death (of the old allele) and how t could lmt the substtuton rate (of the new allele). But the stuaton quckly gets worse. When the scenaro moves away from a unform reproducton rate (and allows selectve elmnatons due to lowered fertlty), then some, or all, of the genetc deaths are vrtual, or magnary (not actual deaths). Haldane sad to count these vrtual deaths as equvalent to real deaths, but he dd not explan why. Snce these ndvduals are never even conceved, ther physcal nterpretaton s a source of confuson. For example, why would vrtual (or magnary) deaths of the old allele lmt the substtuton rate of the new allele? The answer can now be realzed. Haldane s vrtual death concept s merely an artfcal tabulaton devce that helps us calculate the correct answer. Tabulate all selectve elmnatons (even those due to lowered reproducton rate) as though they are actual deaths that really means all scenaros are calculated as though they have a unform reproducton rate. (In effect, Haldane was usng the cost equvalence prncple.) Usng that artfcal tabulaton devce (and when the selecton coeffcent approaches zero), the number of selectve deaths of old-type ndvduals dvded by the adult populaton sze (whch s Haldane s cost concept) happens to equal the requred excess reproducton rate for the entre populaton. In other words, there are a certan number of selectve deaths of the old-type, and the populaton s requred to produce an excess reproducton rate suffcent to supply those old-type progeny. However, under a unform reproducton rate, ths same rate s also requred for producng new-type progeny, and ths latter requrement remans unaffected even f the reproducton rate s not unform (ths agan uses the cost equvalence prncple). In ths way, we can translate Haldane s cost concept (genetc death) nto my cost concept, and provde legtmate physcal ratonale for Haldane s cost argument. That ratonale, however, s also awkward and convoluted, whch has kept Haldane s argument under a cloud of confuson. Larger selecton coeffcents add another layer of complcaton (and confuson). Moreover, when a scenaro ncludes multple substtutons (overlappng n tme), or dplods wth non-domnant substtutons, a correct physcal nterpretaton of Haldane s argument appears ntractable. TJ 19(1) 2005

Cost theory and the cost of substtuton a clarfcaton ReMne Papers Genetc death quckly becomes a mere mathematcal equaton, the correct physcal nterpretaton of whch s obscure at best. Nonetheless, many nvestgators advanced the genetc death concept and the accompanyng confuson fostered varous mstaken solutons to Haldane s Dlemma. They faled to realze that Haldane was not focused on death, but on somethng else. Haldane, a consummate mathematcan, probably frst dscovered hs cost concept as t lay exposed and beckonng wthn hs math. Haldane s math accurately predcts somethng mportant, but he was graspng at how to explan t. Crow broadened the math somewhat, but based t on the same faulty physcal reasonng. Defnng cost n terms of reproducton rather than genetc death clarfes the stuaton by tyng the math wth physcal realty. Ths s supported by the followng fact: n all the varous case studes, my substtuton cost for a gven generaton reduces exactly to Crow s (by assumng a constant populaton sze), and then exactly to Haldane s (under hs addtonal assumpton of s << 1). Yet ths cost concept s more general and has a concrete physcal nterpretaton. Haldane s 1 paper repeatedly speaks of reproductve capacty ; t was clearly a key focus of hs thnkng. Hs paper dd not explctly use the term reproductve excess, though later commentators, such as Merrell, 9 understandably attrbuted that essental concept to hm. Crow and Merrell advanced ths clearer wordng (though stll heavly ntermngled wth the concept of genetc death). Ther usage of the term reproductve excess further suggests that the cost concept they were graspng for s clarfed n ths paper. Unfortunately, ther proper focus on reproductve excess was largely brushed asde by the rse of a new concept genetc load. Genetc load The substtutonal load s defned as the percentage decrease of average ftness wthn a gven generaton, caused by the substtuton of benefcal mutaton. Let w O be the ftness of the optmal genotype, and let w A be the average ftness of the populaton. The substtutonal load s (w O w A )/w A. In the haplod case, w O = 1, and w A = 1 sq, so the load s sq/(1 sq). In other words, the load and cost happen to be equal. (Note: some researchers defne load as (w O w A )/w O = w O w A = sq, whch makes load equal to cost so long as s << 1.) A smlar equvalence s also found for dplod cases. Kmura popularzed ths equvalence, and load soon became the predomnant way of dscussng the cost of substtuton. That s unfortunate, because the load concept caused much confuson. Frst, cost and load became vewed as dentcal concepts, when they are dfferent physcally. They merely happen to have a weak mathematcal equvalence, and then only under specal crcumstances (such as constant populaton sze, s << 1, etc.). Load obscures the physcal processes (especally n dplods), and defocuses the concept of reproducton rate (whch s rghtly the central focus of cost theory). Second, substtutonal load has a strong counterntutve flavour. Kmura notes that One popular crtcsm s that the substtutonal load of a more advantageous allele for a less advantageous one cannot be consdered a load, snce the ftness of the speces s thereby ncreased. 23 The load argument clams the substtuton of a new benefcal mutaton nto the populaton temporarly decreases the average ftness of the populaton, so a benefcal mutaton causes a ftness depresson. Whle that made some sense to mathematcans, ts physcal nterpretaton has been a major source of confuson. Thrd, load strongly emphaszes the concept of ftness, whch creates confuson: Ftness values are re-relatvzed frequently as varous mutatons enter, ext, or reach fxaton, and the tmng of whch s generally unknown. The contnual conveyorbelt of re-relatvzed ftness values creates confuson, much lke dentfyng the tenth man on an escalator. The load calculaton depends on w O, the ftness of the theoretcally optmal ndvdual whose dentty (and ftness) s ever changng. Moreover, n scenaros nvolvng substtutons at many loc, a theoretcally optmal ndvdual typcally never exsts wthn the populaton. Load calculatons have confuson surroundng the optmal ftness (w O ) and how to handle t. (And the current defnton of load does not clarfy t.) A gven ftness value can be realzed wth dramatcally dfferent tradeoffs between reproducton rate and vablty rate. These dfferences were often vewed as rrelevant, because load focuses so ntensely on ftness. Ths led to ndscrmnant use of, and vacllaton between, these ftness components, and confuson was the result. (Cost theory focuses on reproducton rate, and thus keeps a clear eye on ths dstncton.) Ths confuson ncreases many-fold n stuatons nvolvng numerous substtutons overlappng n tme. Fourth, load furthered the mstaken dea that neutral substtutons have no cost. Ths occurred because nearly neutral substtutons have very low load; 24 and neutral substtutons have a load (or ftness depresson) of zero, by defnton and load was casually equated wth cost. 25 Moreover, neutral mutatons (and ther substtuton) cause zero genetc deaths, and ths seemed confrmed by the cost equatons: snce s = 0 for neutrals, ther cost ( sq) seems to be zero. However, that s a faulty nterpretaton, because these cost equatons are derved usng s-values to specfy the growth of the allele. When s = 0 there s no growth, and hence no cost but no substtuton ether. In effect, t calculates the cost of not substtutng a neutral mutaton zero. Ths agan shows a real dfference between the load and cost concepts. The erroneous noton that neutral substtutons have no cost caused confuson and encouraged varous clams that benefcal substtutons lkewse have no cost. Ffth, load s a sngle value for the entre populaton (calculated usng the entre populaton s average ftness, w A, TJ 19(1) 2005 121

Cost theory and the cost of substtuton a clarfcaton ReMne and optmal ftness, w O ). There s no generally acknowledged method for calculatng the load for each genotype, and, even f there were, there s no f-clause to rectfy negatve loads and provde consstent physcal meanng sutable for testng scenaros. By contrast, the cost of substtuton has a dstnct physcally meanngful value for each genotype (though usually only the largest costs are calculated for testng a scenaro), and these gve the reproductve requrements for each genotype. In other words, load theory s a blunt nstrument, whle cost theory provdes fner detal and a more transparent connecton to the underlyng physcal processes. Cost and load are dfferent lnes of physcal reasonng. These terms ought to be used more carefully and no longer nterchangeably. The cost concept has a drect, clearer physcal meanng, whle the load concept, n my vew, s prone to confuson. In hs book Ffty Years of Genetc Load, Bruce Wallace revews hs eventual dsenchantment wth genetc load theory. 26 An expert user of load arguments, Ewens often sensed a frustraton among bologsts (ncludng Mayr) who felt the [prevalent substtutonal and segregatonal load] concepts were msguded but who could not see ther way through the mathematcal dervatons and so fnd the errors n the load arguments. 27 Load theory brought confuson, and a clarfed cost theory can renvgorate the feld. 122 Contnuous-generaton models The dscrete-generaton model (from equatons 3 and 4) can be generalzed nto a contnuous-generaton model. At tme t, let P(t) and dp(t)/dt be the effectve number of ndvduals who produce type A progeny, and ts tme rate of change, respectvely. Let T be the effectve generaton length ths s approxmately the parents ages when they gve each brth, averaged over all brths that reach md-parenthood. T s also equvalent to the tme-unt from the correspondng dscrete-generaton model. Ths s what lnks the two models together. The nstantaneous cost per unt tme s: 1 P(t) Cost_densty(t) = d, f > 0; P(t T) dt otherwse, Cost_densty(t) = 0 (12) Ths equaton ncludes the fact that, at any nstant, the ncrease n P s due to the value of P from a tme one generaton prevous. That s, the T-parameter models the tme-delay between when progeny are born and when, on average, the next generaton of progeny s produced. Wth ths n mnd, the analogy wth equaton 3 s exact. The dfference s that equaton 3 gves a cost per whole generaton, whereas equaton 12 gves a cost per nfntesmal tme-slce (not per whole generaton). Other than that, they both represent the same thng the requred excess reproducton rate. As before, the total cost merely sums (.e. ntegrates) over a substtuton cycle. Total cost of substtuton = Cost_densty(t) dt (13) If the substtuton s very slow, compared to the tmedelay T, then the total cost wll asymptotcally approach the classc formula (= log e K, my equaton 6). However, the total cost ncreases rapdly for faster substtuton speeds (as shown n Table 2). Ths contnuous model s drectly applcable to haplods, and can be extended to dplods usng methods gven n the next secton. We can contrast that wth Flake and Grant s contnuousgeneraton haplod model. They defne the cost densty as the rato of the nstantaneous rate of loss of [the old trat] to the nstantaneous populaton sze. 28 Ther dervaton then arrves at equatons dentcal n form to equatons 12 and 13, absent the T-parameter. Ther fnal result then gves a total cost dentcal to the classc formula (= log e K, equaton 6), ndependently of substtuton speed. Ths result s naccurate as no T-parameter was ncluded to model the above-descrbed tme-delay between brth and md-parenthood. Ther T s zero (therefore all substtutons are very slow compared to ther T), and ths s why ther total cost s ndependent of substtuton speed. Ths mstake arose because of a focus on genetc death and loss of the old trat where a tme-delay would not suggest tself. 29 Once agan, genetc death was a source of confuson. Dplods Snce we are nterested n requred reproducton rates, we must refer to some dentfable reproductve ndvduals. Ths paper vews the ndvdual n the ordnary sense, as a body. 30 Therefore, we focus on genotypes, because genotypes correspond to ndvduals bodes capable of gvng progeny. For example, genotype AA corresponds to a well-defned reproductve body, but allele A does not (as t ambguously refers to AA or Aa). Alleles substtute, but genotypes are the vehcles for the ncreases. Therefore, we always calculate the costs of specfc genotypes (not alleles). Ths dstncton s nvsble for sngle substtutons n haplods (where the allele and genotype have a one-to-one correspondence), but ths dstncton s essental n all other cases. In dplody, ndvduals correspond to the three genotypes: AA, Aa and aa. At the start of generaton, let the startng count of breedng adults be denoted as: P AA + P Aa + P aa = N e. (The endng count of one generaton becomes the startng count of the next.) Dvdng by N e gves the startng genotype frequences, denoted by f AA + f Aa + f aa = 1. The parental allele frequences are then easly calculated: p = f AA + ½f Aa, and q = 1 p. Mendelan segregaton, n combnaton wth varous matng schemes (such as random matng and nbreedng), alter genotype frequences between parents and progeny, whle tendng to leave allele frequences unaffected. By ths means, some genotypes can decrease, whle others ncrease. Ths TJ 19(1) 2005

Cost theory and the cost of substtuton a clarfcaton ReMne Papers predctable change s due solely to the passve remxng of alleles at the gene level, and does not requre reproductve excess of whole bodes. Therefore, we do not tally t nto costs that whole bodes must pay. So we next assess ts effect and dsallow t from our tally of costs. That s accomplshed wth the concept of effectve producers. I wll use random matng to exemplfy how all cases are handled. Random matng produces genotype frequences gven by p 2 + 2pq + q 2 = 1. (For other matng schemes, that equaton wll be dfferent, but the followng steps wll usually reman the same.) Multply by the effectve breedng sze of the populaton to obtan: N e p 2 + N e 2pq + N e q 2 = N e. That equaton gves the effectve producers of the three genotypes. (An effectve producer can be modelled as an abstract asexual adult that, n effect, produces progeny of a gven genotype at the same reproducton rate requred of ts real sexual counterparts.) Ths parcels the breedng populaton nto three portons, each porton gong toward the producton of a specfc genotype (see fgure 3, for example.) It s as though all the reproducton of N e p 2 (asexual) adults goes solely toward producng AA genotypes, whle all the reproducton of N e 2pq (asexual) adults goes solely toward producng Aa genotypes, and so forth. That equaton shows how the populaton s reproductve capacty s redstrbuted. A genotype s effectve startng count s the number of effectve producers of that genotype, and s gven by the terms of the prevous equaton, * * * here labelled as P AA + P Aa + P aa = N e. 31 As the cycle of one generaton completes, call the number of breedng adults the endng count (labelled wth a prme) as P AA + P Aa + P aa = N e = N e (f AA + f Aa + f aa ). In generaton, each genotype has a cost, gven by equaton 3: PAA Cost G f AA _ AA = 1 = 1, f > 0; 2 P * p AA otherwse, Cost_ AA = 0 (14) Cost _ Aa PAa G f Aa = 1 = 1, f > 0; P * 2pq Aa otherwse, Cost_ Aa = 0 (15) Cost _ aa Paa G f aa = 1 = 1, f > 0; 2 P * q aa otherwse, Cost_ aa = 0 (16) Those equatons have a straghtforward nterpretaton. Take a case where populaton sze s constant (G = 1), and focus on the substtutng genotype AA. Its effectve producers have a frequency p 2, and ths genotype ends the generaton at a greater frequency, f AA. That requres a reproducton rate f AA /p 2, whch s a cost of contnuty of 1, plus a cost of substtuton of (f AA /p 2 ) 1. Wthn a gven generaton, the three costs Cost_AA, Cost_Aa and Cost_aa do not stack onto each other. They are three separate ppelnes ncurred, and pad, n parallel. The payments that go toward one cannot go toward payng the others. 32 Those three costs are the mnmum concevable cost of substtuton ncurred by each genotype. Each of those s a mechancal lmt that cannot go lower, regardless of further detals about the selecton process. For decreasng genotypes (such as Aa durng the second half of the substtuton, and aa), ths lmt s zero. But ther cost can, and usually does, go substantally hgher. For pure vablty selecton, all subgroupngs of the populaton have the same reproducton rate (then some unfavoured genotypes are elmnated by premature death); therefore, all genotypes ncur the same cost of substtuton: Cost_AA. On the other hand, for pure reproductve selecton, genotypes dffer n the rates at whch they are produced; therefore, ther costs may approach these mechancal mnmums. In any case, the cost of the substtutng genotype Cost_AA s unaffected by those crcumstances, and remans the crtcal focus for testng the plausblty of a scenaro. It s usually suffcent to focus on the greatest of the three costs, as ths cost (and ts payment) almost always forms the most strngent test of the scenaro: thus: Cost = Maxmum (Cost_AA, Cost_Aa, Cost_aa ) (17) For a well-behaved substtuton, Cost_AA always domnates, therefore (by equaton 4): Total cost of substtuton = Σ Cost_AA (18) The above method apples under the wdest of crcumstances. Under the same model assumptons used by Crow and Haldane, the above equatons reduce to thers, ncludng Haldane s (1957) equatons for all hs substtuton case stud- aa 36% Fgure 3. Reproductve capacty s redstrbuted by Mendelan segregaton n dplods. Ths example shows the percentage of the populaton s conceptons that produce progeny of each genotype. In ths example, where p = 0.4, only 16% of the conceptons go toward reproducng progeny of genotype AA. Aa 48% AA 16% TJ 19(1) 2005 123