The fitness value of information

Transcription

1 Oikos 119: , 2010 doi: /j x, # 2009 Th Authors. Journal compilation # 2009 Oikos Subjct Editor: Knnth Schmidt. Accptd 1 Sptmbr 2009 Th fitnss valu of information Matina C. Donaldson-Matasci, Carl T. Brgstrom and Michal Lachmann M. C. Donaldson-Matasci, Dpt of Ecology and Evolutionary Biology, Univ. of Arizona, Tucson, AZ 85721, USA. C. T. Brgstrom, Dpt of Biology, Univ. of Washington, Sattl, WA , USA. M. Lachmann (lachmann@va.mpg.d, Max Planck Inst. for Evol. Anthropology, Dutschr Platz 6, DE04103 Lipzig, Grmany. Communication and information ar cntral concpts in volutionary biology. In fact, it is hard to find an ara of biology whr ths concpts ar not usd. Howvr, quantifying th information transfrrd in biological intractions has bn difficult. How much information is transfrrd whn th first spring rainfall hits a dormant sd, or whn a chick bgs for food from its parnt? On masur that is commonly usd in such cass is fitnss valu: by how much, on avrag, an individual s fitnss would incras if it bhavd optimally with th nw information, compard to its avrag fitnss without th information. Anothr masur, oftn usd to dscrib nural rsponss to snsory stimuli, is th mutual information a masur of rduction in uncrtainty, as introducd by Shannon in communication thory. Howvr, mutual information has gnrally not bn considrd to b an appropriat masur for dscribing dvlopmntal or bhavioral rsponss at th organismal lvl, bcaus it is blind to function; it dos not distinguish btwn rlvant and irrlvant information. In this papr w show that thr is in fact a surprisingly tight connction btwn ths two masurs in th important contxt of volution in an uncrtain nvironmnt. In this cas, a usful masur of fitnss bnfit is th incras in th long-trm growth rat, or th fold incras in numbr of surviving linags. W show that in many cass th fitnss valu of a dvlopmntal cu, whn masurd this way, is xactly qual to th rduction in uncrtainty about th nvironmnt, as dscribd by th mutual information. Information is a cntral organizing concpt in our undrstanding of biological systms at vry scal. Our DNA digitally ncods information about how to crat an organism information that was rfind ovr gnrations through th procss of natural slction (Maynard Smith Snsory systms ar usd to acquir information about th nvironmnt, and th brain procsss and stors that information. A varity of larning mchanisms allow animals to flxibly act upon th information thy rciv. Signals lik th pacock s tail, th honyb waggl danc, and human languag ar usd to convy information about th signalr or th nvironmnt to othr individuals (Maynard Smith and Harpr In th study of human communication and data transfr, information is typically masurd using ntropy and mutual information (Shannon 1948, Winr 1948, Covr and Thomas Entropy is a statistical masur of th amount of uncrtainty about som outcom, lik whthr it will rain tomorrow or not, which has to do with th numbr of diffrnt possibl outcoms and th chanc ach on has to occur. Mutual information masurs th rduction in uncrtainty about that outcom aftr th obsrvation of a cu, lik th prsnc or absnc of clouds in th sky. In som filds of biology, such as nurobiology, information is naturally and usfully masurd with th sam informationthortic quantitis (Borst and Thunissn Howvr, information thortic masurs hav sn substantially lss us in volutionary biology, bhavioral cology and rlatd aras. Why is this? On problm is that masurs of ntropy do not dirctly addrss information quality; thy do not distinguish btwn rlvant and irrlvant information. Whn w think about fitnss consquncs w car vry much about rlvant and irrlvant information. For xampl, from an information-thortic standpoint on has th sam amount of information if on knows th timing of sunris on Mars as on has if on knows th timing of sunris on Earth. Yt individuals of fw if any Earthbound spcis find th timing of sunris on Mars rlvant to thir survival. Information masurs basd on ntropy hav thrfor bn dmd irrlvant to th volutionary cology of information. Instad volutionary biologists and bhavioral cologists tnd to focus on dcision-thortic masurs such as th xpctd valu of prfct information or th xpctd valu of sampl information (Savag 1954, Good 1967, Winklr 1972, Gould 1974, Ramsy 1990, with valu oftn masurd in trms of fitnss consquncs (Stphns and Krbs 1986, Stphns 1989, Lachmann and Brgstrom Th disconnct btwn information-thortic and dcision-thortic masurs is prplxing. Entropy and mutual information appar to masur information quantity whil rflcting nothing about fitnss consquncs; th xpctd valu of information masurs fitnss consquncs but has nothing to do with th actual lngth or 219

2 information quantity of a mssag. But arly work in population gntics (Haldan 1957, Kimura 1961, Flsnstin 1971, 1978 and rcnt analyss of volution in fluctuating nvironmnts (Brgstrom and Lachmann 2004, Kussll and Liblr 2005 hint at a possibl rlation btwn information and fitnss. What is this rlation? Information thorists sinc Klly (1956 hav obsrvd that in spcial circumstancs, information valu and information-thortic masurs may b rlatd. Hr w argu that ths spcial circumstancs ar xactly thos about which biologists should b most concrnd: thy includ th contxt of volution by natural slction in an changing, unprdictabl nvironmnt. Most organisms xprinc som kind of stochasticity in th nvironmnt, but short-livd inhabitants of xtrm habitats ar particularly vulnrabl to its vagaris. A cas in point is dsrt annual plants: onc thy grminat, thy hav just on chanc to rproduc, and in many yars thr simply is not nough rain. Thir adaptiv solution is to somtims dlay grmination for a yar or mor, so that ach plant s sds will grminat ovr a sprad of svral yars, rathr than all togthr a stratgy known as risksprading or bt-hdging (Cohn 1966, Coopr and Kaplan 1982, Sgr and Brockmann This stratgy, though it allows a linag to prsist through drought yars, is somwhat wastful; all th sds that do happn to grminat in a drought yar di with no chanc of rproducing. What if, instad, sds wr snsitiv to nvironmntal cus that could hlp prdict th chanc of a drought in th coming yar? Th bt-hdging stratgy could b improvd, by adjusting th probability of grmination in rspons to that cu, according to th conditional probability of a drought (Cohn 1967, Haccou and Iwasa How dos this improvd stratgy translat into incrasd fitnss, and how dos that rlat to th amount of information th cu convys about th nvironmnt? W prsnt a simpl modl of volution in an uncrtain nvironmnt, and calculat th incras in Darwinian fitnss that is mad possibl by rsponding to a cu convying information about th nvironmntal stat. W show that in crtain cass this fitnss valu of information is xactly qual to th mutual information btwn th cu and th nvironmnt. Mor gnrally, w find that this mutual information, which smingly fails to tak anything about organismal fitnss into account, nonthlss imposs an uppr bound on th fitnss valu of information. Two masurs of information Environmntal cus can hlp organisms living in an uncrtain nvironmnt prdict th futur stat of th nvironmnt, and thrby can allow thm to choos an appropriat phnotyp for th conditions that thy will fac. W will considr a population of annual organisms living in a variabl nvironmnt. Th nvironmntal stat E and th nvironmntal cu C ar corrlatd random variabls indpndntly drawn vry yar; both ar common to all individuals, so that all individuals ncountr th sam conditions in a givn yar. W might masur th information convyd by th nvironmntal cu C in two diffrnt ways. Th typical approach in statistical physics, communication nginring, nurobiology, and rlatd filds is to us an information-thortic masur such as th mutual information. Th mutual information dscribs th xtnt to which a cu rducs th uncrtainty about th nvironmnt, masurd in trms of ntropy. Following Covr and Thomas (1991, w dfin th ntropy of th random variabl E rprsnting th nvironmntal stat as H(E X p( log p( whr p( is th probability of obsrving th stat. Th mor diffrnt stats of th nvironmnt that ar possibl, and th closr thos stats ar to qually likly, th highr th uncrtainty about which stat will actually occur. Aftr th organism obsrvs a cu, th chancs of ach nvironmntal stat may chang. W dfin th conditional ntropy of th nvironmnt E, onc th random variabl C rprsnting th cu has bn obsrvd, as H(E½C X p(c X p(½c log p(½c c whr p(c is th probability of obsrving cu c, and p(jc is th conditional probability that th nvironmnt is in stat, givn that cu c has bn obsrvd. This is a masur of th rmaining uncrtainty about th stat of th nvironmnt, onc a cu has bn obsrvd. Dfinition Th mutual information btwn a cu C and a random nvironmntal stat E masurs how much th cu rducs th uncrtainty about th stat E: thus I(E; C H(EH(E½C: If th cu is compltly unrlatd to th stat of th nvironmnt, thn th uncrtainty about th nvironmnt rmains th sam aftr th cu has bn obsrvd, and th mutual information is zro. Howvr, if thr is som rlationship btwn thm, thn th cu rducs th uncrtainty about th nvironmnt, so th mutual information is positiv. At bst, a prfctly informativ cu would xactly rval th stat of th nvironmnt; th rmaining uncrtainty would b zro and th mutual information btwn th cu and th nvironmnt would b xactly th amount of uncrtainty about th nvironmnt. Th ntropy masur is most familiar to cologists as th Shannon indx of spcis divrsity, which taks into account th numbr of diffrnt spcis and th frquncy of ach (Shannon Th mor diffrnt spcis ar prsnt, and th closr thy ar to qually frqunt, th highr th spcis divrsity. Considr a fild cologist obsrving random individuals in a particular habitat, and writing down th spcis of ach individual as it is obsrvd. Th mor diffrnt spcis ar prsnt in th habitat, th mor diffrnt squncs of spcis ar possibl. Howvr, squncs in which a rar spcis is obsrvd many tims and a common spcis is obsrvd just a fw tims ar quit unlikly. Th numbr of squncs that ar likly to actually occur thus dpnds also on th frquncy of ach spcis. 220

3 For xampl, if thr ar just two spcis that ar qually frqunt, th most likly squncs of tn obsrvations will hav fiv individuals of on spcis and fiv of th othr; thr ar (10!/5!5! 252 such squncs. In contrast, if thr ar just two spcis, but on is nin tims mor frqunt than th othr, th most likly squncs of tn obsrvations will hav just on individual of th rar spcis; thr ar only 10!/(9!1!10 such squncs. If w considr vry long squncs of obsrvations, th numbr of likly squncs is clos to 2 HN, whr H is th divrsity indx and N is th numbr of individuals obsrvd. With ach nw obsrvation, th numbr of possibl squncs is multiplid by th numbr of spcis, but th numbr of likly squncs incrass by a factor of 2 H. Thus th divrsity indx H can b intrprtd as th fold incras in likly outcoms with ach additional obsrvation a masur of th uncrtainty about th nxt spcis to b obsrvd. Whn w ar obsrving individuals that liv in diffrnt habitats, w can ithr ignor th habitat and masur th divrsity of all individuals poold togthr, or w can masur th divrsity within ach habitat. If w masur within-habitat divrsity, and thn avrag across habitats according to th numbr of individuals obsrvd in ach habitat, w will usually find a lowr divrsity than w would by pooling across habitats. Th only cas whr th avrag within-habitat divrsity will b as high as th ovrall divrsity is whn habitat plays no rol th frquncy of ach spcis is th sam in th diffrnt habitats. Th diffrnc btwn th ovrall divrsity and th avrag within-habitat divrsity is th mutual information btwn habitat and spcis: how much uncrtainty about which spcis will b ncountrd nxt is rmovd, if w know th habitat in which th ncountr taks plac? Anothr approach, common in dcision thory, conomics, bhavioral cology, and rlatd filds, is to look at th xpctd valu of information: how information improvs th xpctd payoff to a dcision-makr (Gould W writ th maximal fitnss obtainabl without a signal as F(Emax x a p(f(x; ; whr x is a stratgy, and f(x, is th fitnss of that stratgy whn th nvironmntal stat occurs. Similarly, w writ th optimal fitnss attainabl with a signal F(E½C X p(cmax x X p(½cf(x c ; whr x is a stratgy usd in rspons to th cu c and dpnds on that cu. Dfinition Th dcision-thortic valu of information that a cu C provids about th stat of th world E is dfind as th diffrnc btwn th maximum xpctd payoff or fitnss that a dcision-makr can obtain by conditioning on C, and th maximum xpctd payoff that could b obtaind without conditioning on C. This is writtn as DF(E; C F(E½CF(E: An illustrativ xampl To illustrat th diffrnc btwn th two masurs of information dscribd abov, w start with a simplifid modl. Th nvironmnt has two possibl stats, such as wt and dry yars, and ach organism has two possibl phnotyps into which it can dvlop. Fitnss dpnds on th match of phnotyp to nvironmnt, as follows: Phnotyp 8 1 Phnotyp 8 2 Environmnt Environmnt Givn that th probability of nvironmntal stat 1 is p, and of stat 2 is (1p, what should ths individuals do in th absnc of information about th condition of th nvironmnt? Th organism maximizs its singl-gnration xpctd fitnss by dvloping into phnotyp 8 1 if p1/3, and into phnotyp 8 2 othrwis. This is optimal bcaus 5 p1(1 p 1 p3(1 p whn p 1/3. Th payoff arnd with this stratgy would b F(E max[5p(1p; p3(1p]: 8 >< 5p(1p for p] 1 3 F(E >: p3(1p for pb 1 3 Now w suppos that thr is a prfctly informativ nvironmntal cu C which accuratly rvals th stat E of th nvironmnt. How do w masur th information providd by this cu? Within an information-thortic framwork, w masur th amount of information in th cu by calculating mutual information btwn th cu and th nvironmnt; this masurs how much th cu tlls us about th nvironmnt. Sinc th cu is prfctly informativ, th conditional uncrtainty about th nvironmnt onc th cu is obsrvd is zro: H(EjC 0. Th mutual information is thrfor I(E; C H(EH(E½C H(E p log p(1p log (1p (2 Figur 1A plots th mutual information btwn cu and nvironmnt as a function of th probability p that nvironmnt 1 occurs. Within a dcision thortic framwork, w masur th valu of th information in th cu by calculating how much th ability to dtct th cu improvs th xpctd fitnss of th organism. Whn an organism rcivs th cu, it can always dvlop th appropriat phnotyp for th nvironmnt, and thus obtain a singl-gnration xpctd fitnss of F(EjC 5p3(1p 32p. Th dcisionthortic valu of information in this cas is thrfor DF(E; C F(E½CF(E 8 >< 22p for p] 1 3 >: 4p for pb 1 3 By using th cu, th organism incrass its singlgnration xpctd fitnss by th dcision-thortic (1 (3 221

4 (A 1.0 (B 2.0 Mutual information I(E; C Valu of information F(E; C Probability of nvironmnt 1, p( 1 Probability of nvironmnt 1, p( Figur 1. Two diffrnt masurs of th information in a cu ar commonly usd. (A Th mutual information btwn th cu and th nvironmnt masurs th rduction in nvironmntal uncrtainty onc th cu has bn obsrvd. (B Th dcision-thortic valu of information masurs th chang in xpctd fitnss that is mad possibl by using th cu. masur of th valu of information. This quantity, illustratd in Fig. 1B, diffrs considrably from th information-thortic masur of mutual information shown in Fig. 1A. Not only do th graphs tak on diffrnt forms, but thir units of masurmnt diffr. Mutual information is masurd in bits, whras th valu of information is masurd in fitnss units. Th fitnss valu of information In th prvious sction, w masurd th valu of information by its ffct on xpctd fitnss ovr a singl gnration. But as many authors hav shown (Dmpstr 1955, Haldan and Jayakar 1963, Cohn 1966, Lwontin and Cohn 1969, Gillspi 1973, Yoshimura and Jansn 1996, organisms will not always b slctd to us a stratgy that maximizs thir fitnss in a singl gnration. Instad, a bttr proxy for th likly outcom of volution is to think of organisms as maximizing th long-trm growth rat of thir linag. This distinction is critical whn th nvironmnt changs from on gnration to th nxt, and affcts all individuals within on gnration in th sam way as, for xampl, with drought or abundant spring rains. Undr ths circumstancs, maximizing th long-trm growth rat ovr a vry larg numbr of gnrations is quivalnt to maximizing th xpctd valu of th logarithm of th fitnss in a singl gnration. From an volutionary prspctiv, it thrfor maks sns to dfin th valu of a cu not in trms of singlgnration fitnss consquncs, but rathr in trms of th incras in long-trm growth rat it maks possibl. Lt g(xs p( log f(x, b th xpctd long-trm growth rat of a stratgy x. W writ th maximum long-trm growth rat obtainabl without a cu as G(Emax x S p( log f(x,. Similarly, w writ th maximum longtrm growth rat attainabl with a cu as G(EjCS c p(c max xc S p(jc log f(x c,. Dfinition Th fitnss valu of information DG(E; C associatd with a cu or signal is th gratst fitnss dcrmnt or cost that would b favord by natural slction in xchang for th ability to dtct and rspond to this cu: DG(E; C G(EjCG(E. Proportional btting To xplor th connction btwn nvironmntal uncrtainty and long-trm growth rat, w will first look at an vn simplr xampl, whr th organism survivs only if it matchs its phnotyp to th nvironmnt prfctly. Phnotyp 8 1 Phnotyp 8 2 Environmnt Environmnt 2 0 7/2 In th short run, individuals maximiz xpctd fitnss by mploying th highst-payoff phnotyp only. But w can immdiatly s that th long-run fitnss of linag is not maximizd in th sam way: playing only on stratgy will invitably lad to a yar with zro fitnss and consqunt xtinction for th linag. Thus organisms will b slctd to hdg thir bts, randomizing which of th two phnotyps thy adopt (Coopr and Kaplan 1982, Sgr and Brockmann If nvironmnt 1 occurs with probability p and nvironmnt 2 occurs with probability (1p, with what probability should an individual adopt ach phnotyp? As w considr a largr and largr span of gnrations, a largr and largr fraction of th probability is takn up by typical squncs of nvironmnts, in which nvironmnt 1 occurs around Np tims, and nvironmnt 2 occurs around N(1p tims (Covr and Thomas A stratgy that maximizs th growth rat ovr ths typical squncs will, with vry high probability, b th on that is obsrvd as th rsult of natural slction (Robson t al To find this gnotyp, lt us assum a gnotyp that dvlops with probability x into phnotyp 8 1 and with probability (1x into phnotyp 8 2 ; th population growth ovr a typical squnc of N vnts will b (7x Np (7=2 /(1x N(1p : Maximizing th population growth is quivalnt to maximizing th pr-gnration xponnt of growth, or th log of th xprssion dividd by N: 222

5 g(x plog(7x(1p log (7=2(1x p log (x(1p log (1xp log (7 (4 (1p log (7=2 Th only part of this quation that dpnds on x is p log (x(1p log (1x; so any dpndnc on th fitnsss whn th organism proprly matchs th nvironmnts (i.. on th valus 7 and 7/2 has droppd out. Th maximum occurs whn xp, a stratgy calld proportional btting. With this stratgy, organisms dvlop into th two phnotyps in proportion to th probabilitis of th two nvironmnts and ths optimal proportions do not dpnd on th fitnss bnfits (Covr and Thomas Th optimal growth rat is thus G(Ep log (p(1p log (1p p log 7(1p log (7=2 (5 p log 7(1p log (7=2H(E Uncrtainty and optimal growth To undrstand th connction btwn nvironmntal uncrtainty and optimal growth mor fully, it is instructiv to gnraliz th simpl modl abov to includ svral diffrnt nvironmnts and phnotyps. Lt us assum that thr ar n nvironmnts, and that for ach nvironmnt thr is on optimal phnotyp. Th payoff for phnotyp 8 in nvironmnt is d, and th payoff for any othr phnotyp in nvironmnt is 0. Lt th probability of nvironmnt occurring b p(, and th probability of dvloping into phnotyp 8 b x(. Thn th growth rat of th linag ovr a typical squnc of nvironmnts, in which nvironmnt occurs approximatly Np( tims, is P (d x( Np( : Again, instad of maximizing th abov xprssion, w can maximiz its log, dividd by N: g(x X p( log (d x( X p( log d X (6 p( log x( Th lft part dos not dpnd on th stratgy x, so w just nd to maximiz th right part, which as bfor is indpndnt of th fitnss valus d in th diffrnt nvironmnts. But instad of simply giving th solution, lt us rwrit th abov xprssion as g(x X p( log d X p( log p( X p( log p( X p( log x( X p( log d X (7 p( log p( X p( log p( x( X p( log d H(ED KL (p½½x Th trm D KL (p½½xa p( log (p(=x( is th KullbackLiblr divrgnc (KL divrgnc btwn th distribution of nvironmnts and th distribution of phnotyps. Th K-L divrgnc, also known as th rlativ ntropy, quantifis how gratly a givn distribution x( varis from a rfrnc distribution p(. To illustrat its maning, w rturn to th xampl of a fild cologist rcording th spcis of ach individual as it is obsrvd. If th tru frquncis of ach spcis ar givn by th distribution p(s, thn th most likly squncs of obsrvations ar th ons whr ach spcis s is obsrvd in proportion to its frquncy, n(s/n:p(s. Th obsrvd frquncy of ach spcis is thus th maximum liklihood stimat for th tru frquncy of ach spcis. Howvr, othr typs of squncs ar possibl, whr th obsrvd frquncy of ach spcis q(s n(s/n dos not match th tru frquncy lading th cologist to an incorrct stimat. How oftn dos this happn? As long as th total numbr of obsrvations N is larg, th probability of obsrving a squnc with spcis frquncis q whn th tru spcis frquncis in th habitat ar p is approximatly 2 D(qjjpN. As th numbr of obsrvations grows, dviations from p bcom lss and lss likly. Th KullbackLiblr divrgnc is a masur of th dviation of a distribution p from anothr distribution q, which rflcts how unlikly it is that w would obsrv spcis frquncis q whn th tru frquncis ar p. Th rprsntation of th growth rat givn in Eq. 7 allows us to gnraliz and intrprt th rsults of th prvious sction. W not that th trm D KL (pjjx is th only part of th xprssion for g(x that dpnds on th organism s choic of stratgy; this is zro whn x( p(. Thrfor th optimal assignmnt of phnotyps will b a proportional btting stratgy, and and will achiv a growth rat of G(E X p( log d H(E (8 similar to Eq. 5 abov. Whn th distribution of phnotyps is x( instad of th optimal p(, thn th growth rat pr gnration will b rducd by D KL (pjjx. Furthrmor, if th organism knw xactly what th nvironmnt would b at vry gnration, it would choos th optimal phnotyp vry tim. Th growth rat would thn b P (d Np( ; corrsponding to an avrag log growth rat of a p( log d : Comparing this quantity to Eq. 8, w s that th growth rat associatd with th proportional btting stratgy is qual to th growth rat that could b achivd with full information, minus th ntropy of th unknown nvironmntal stat. At last for this spcial cas, nvironmntal uncrtainty rducs th log growth rat by an amount qual to th ntropy of th nvironmnt! Th illustrativ. xampl, rvisitd In th prvious two sctions, w assumd that organisms can only surviv whn thy choos xactly th right phnotyp for th nvironmnt. In gnral, howvr, choosing th wrong phnotyp nd not b fatal. How dos this affct th optimal growth rat, and th fitnss valu of information? Lt us rturn to th first xampl prsntd in sction An illustrativ xampl ; in this xampl th wrong 223

6 phnotyp had a fitnss of 1 instad of 0. Now th xpctd log growth rat in th absnc of a cu is g(x p log (4x1(1p log (32x and hr, organisms will not follow a strict proportional-btting stratgy. Instad, th optimal stratgy x* will b to always dvlop into a singl phnotyp whn p is nar 0 or 1, and to hdg bts whn p taks on an intrmdiat valu: 8 0 for p5 1 always phnotyp >< 7p 1 1 x*(p for 4 7 BpB5 bt thdging 7 5 >: 1 for 7 5p always phnotyp 8 1 This yilds a growth rat without th cu of 8 (1p log 3 for p5 1 7 >< G(E p log 7(1p log 7 2 H(E for 1 7 BpB5 7 >: p log 5 for p] 5 7 (10 If organisms could sns a cu that prfctly rvald th stat of th nvironmnt, all individuals could dvlop as th phnotyp which matchs th nvironmnt, and w would s a log growth rat of G(E½Cp log (5 (1p log (3: Th diffrnc in xpctd log fitnss and thus th fitnss valu of information is givn by: 8 p log 5 for p5 1 7 >< 1 H(Ep log 5=7 for DG(E; C 7 BpB5 7 (1p log 6=7 >: (1p log 3 for p] 5 7 (9 (11 1 In th cntral rgion, whr 7 BpB5, th fitnss 7 valu of information is qual to th ntropy of th nvironmnt plus a linar function of th probability of ach nvironmnt: p log 5=7(1p log 6=7: Outsid th rang, whn th optimal stratgy invsts in only on of th phnotyps, th valu of th cu dpnds linarly on th probability of ach nvironmnt: p log 5 or (1p log 3. Sinc in this rgion th stratgy usd is idntical to th stratgy optimizing fitnss in a singl gnration, th valu of th cu is just lik th diffrnc in long trm growth rat whn optimizing ovr only a singl gnration. Calculus rvals that th function is continuous and onc continuously diffrntiabl vrywhr. Th fitnss valu of information for th cu appars to b rlatd to th mutual information btwn th cu and nvironmnt. In th nxt sctions, w prsnt a gnral modl that will allow us to quantify and intrprt that rlationship. Effctiv proportional btting In ordr to xplor th connction btwn fitnss valu and information, w will dvlop a gnral modl basd on th on prsntd in sction Uncrtainty and optimal growth, but which rlaxs th assumption that dvloping th wrong phnotyp is always fatal. Lt us again assum that an organism has to mak a dvlopmntal dcision btwn n possibl phnotyps, ach of which is a bst match to on of n nvironmnts. Each nvironmntal stat occurs with probability p(, and th fitnss of phnotyp 8 in nvironmnt is f(8,. Th bst match for nvironmnt is phnotyp 8 ; that is, max 8 [f(8; ]f(8 ; : Th stratgy x dfins th probability x(8 that an individual will dvlop into phnotyp 8. W want to find th stratgy that maximizs th xpctd long-trm growth rat, g(xa p(loga 8 f(8; x(8: In a prvious papr (Donaldson-Matasci t al. 2008, w introducd a mthod for calculating optimal bt-hdging stratgis that will prov usful in th prsnt analysis. What follows is a vry brif outlin of th mthod; full dtails ar providd in th prvious papr. W first dfin a st of hypothtical xtrmist phnotyps which fit th modl of sction Uncrtainty and optimal growth, so that ach hypothtical phnotyp 8? is idally adaptd to on nvironmnt, whr it has fitnss d, but fails to surviv in any othr nvironmnt. W nxt aim to dscrib ach actual phnotyp 8 as a bt-hdging stratgy combining th hypothtical phnotyps 8?. That is, w would lik to find fitnsss d for th xtrmist phnotyps and mixing stratgis s(j8 across xtrmist phnotyps such that f(8; s(½8d (12 for all nvironmnts and phnotyps 8. For ach phnotyp 8, th stratgy s(j8 dscribs th bt-hdging proportions of hypothtical phnotyps 8? that would produc th sam fitnss, masurd sparatly for ach nvironmnt. This problm is quivalnt to dfining th fitnss matrix F, with ntris F 8 f(8,, as a product of two unknown matrics: S, with ntris S 8 s(j8, and D, with diagonal ntris D d and 0 lswhr. Solving for ths two matrics is straightforward and can almost always b don uniquly (Donaldson-Matasci t al Th advantag of this approach is that it will allow asy comparison to th simplifid modl prsntd in sction Uncrtainty and optimal growth, which highlights th connction btwn growth rat and uncrtainty. W can xprss a stratgy x as a row vctor x; with ach lmnt x/ x f x(f rprsnting th probability of dvloping phnotyp f. To dscrib th stratgy s fitnss in any particular nvironmnt, w nd simply look at th -th lmnt of th vctor xf: f(x; [ xsd] [ xs] d y(d (13 This mans that th stratgy x, which producs ach phnotyp 8 with probability x(8, is xactly quivalnt to a stratgy y that producs ach hypothtical phnotyp 8? with probability y([ xs] : W can writ down th long-trm growth rat for a linag that uss stratgy x by calculating th growth rat for th quivalnt stratgy y: 224

7 (B f(ϕ' 1, 1 f(ϕ 1, 1 cost of uncrtainty H(E cost of constraint D KL (p s i f(ϕ' 2, 2 f(ϕ 2, 2 Long trm growth rat (log scal (A rgion of bt-hdging Optimal growth rats: with prfct cu, using phnotyps ϕ' with no cu, using phnotyps ϕ' with prfct cu, using phnotyps ϕ with no cu, using phnotyps ϕ using phnotyp ϕ 1 using phnotyp ϕ 2 f(ϕ 1, 2 f(ϕ 2, s( 1 ϕ 2 s( 1 ϕ Probability of nvironmnt 1, p( 1 Figur 2. Maximal growth rat whn choosing th wrong phnotyp is not fatal. Th rd solid lin indicats th maximal growth rat using th two phnotyps 8 1 and 8 2. Th black solid lin is th growth rat that could b achivd with an unconstraind stratgy, using th hypothtical phnotyps 8? 1 and 8? 2. Sinc th hypothtical phnotyps ar fatal in th wrong nvironmnt, th optimal unconstraind stratgy always uss proportional btting. In contrast, th optimal constraind stratgy bt-hdgs only in an intrmdiat rang of nvironmntal frquncis, labld th rgion of bt-hdging. Outsid that rgion, it uss th singl phnotyp that dos bst on avrag. (A Within th rgion of bt-hdging, th constraind stratgy dos just as wll as th unconstraind stratgy. Compard to an unconstraind stratgy that can prfctly prdict th nvironmnt, both stratgis incur a cost of uncrtainty qual to th ntropy of th nvironmnt H(E (Eq. 8, 15. (B Outsid th rgion of bt-hdging, th constraind stratgy dos wors than th unconstraind stratgy, sinc it cannot bt-hdg anymor. It always uss phnotyp 8 1 on th right of th rgion, whr nvironmnt 1 is mor common, and 8 2 on th lft. Th growth rat achivd is thrfor xactly as with th dcision-thortic stratgy, optimizing fitnss in just on gnration. In this cas, th constraind stratgy pays not only th cost of uncrtainty, H(E, but also a cost of constraint that ariss from th inability to bt strongly nough on th most common nvironmnt. This constraint furthr rducs th growth rat by th Kullback-Liblr divrgnc D KL (pjjs i, which gts largr as w gt farthr from th boundary of th rgion of bt-hdging (Eq. 14. g(x X p( log f(x; X p( log y(d X (14 p( log d H(ED KL (p½½y This quation is vry similar to Eq. 7, xcpt that instad of masuring th Kullback-Liblr divrgnc of th stratgy x from th nvironmntal distribution, w masur th divrgnc of th ffctiv stratgy y from th nvironmntal distribution. Th maximum growth rat that can b achivd occurs whn y([ xs] p(; in which cas D KL (p½½y0 (Fig. 2A. If thr is no stratgy x that can achiv this, thn th stratgy that minimizs th Kullback- Liblr divrgnc is optimal (Fig. 2B. If w think of th ffctiv stratgy y as rprsnting th ffctiv bts th stratgy is placing on ach nvironmnt, thn w s that th optimal stratgy ffctivly dos proportional btting or as clos as it can gt. Th optimal growth rat for a stratgy that ffctivly dos proportional btting is thrfor G(E X p( log d H(E (15 just as it was for a diagonal fitnss matrix (Eq. 8. Thus, vn whn choosing th wrong phnotyp is not fatal, th optimal growth rat is limitd by th ntropy of th nvironmnt. Howvr, th first trm is no longr th growth rat that could b achivd if individuals could prdict th nvironmnt prfctly. Instad, it is th growth rat that could b achivd if individuals could prdict ach nvironmnt prfctly, and instad of using th actual phnotyp 8 with fitnss f(8,, thy could us th highr-fitnss hypothtical phnotyp 8? with fitnss d. Th valu of prfct information is thrfor th rduction in ntropy it facilitats, H(E, plus a ngativ trm that rflcts th fitnss cost du to th fact that individuals ar in practic rstrictd to th actual phnotyps rathr than th hypothtical ons, S p( log f(8,/d (.g. Eq Information and fitnss valu Until now, w hav considrd only cus that allow individuals to prdict th stat of th nvironmnt prfctly. W would now lik to calculat th valu of a partially informativ cu. 225

8 f(ϕ' 2, 2 f(ϕ 2, 2 Long trm growth rat (log scal (B (A (C rgion of bt-hdging f(ϕ' 1, 1 cost of full uncrtainty H(E f(ϕ 1, 1 Optimal growth rats: cost of partial uncrtainty H(E C valu of information I(E; C with prfct cu, using phnotyps ϕ' with no cu, using phnotyps ϕ' with prfct cu, using phnotyps ϕ with no cu, using phnotyps ϕ with prdictiv cu c, using phnotyps ϕ f(ϕ 1, 2 f(ϕ 2, p( 1 c 2 p( 1 p( 1 c Probability of nvironmnt 1, p( 1 Figur 3. Whn th optimal stratgy is to bt-hdg, both with and without a cu, th fitnss valu of information is qual to th mutual information btwn th cu and th nvironmnt. W calculat th valu of a partially informativ cu by looking at th rduction in growth rat, as in Fig. 2, rlativ to a prfctly informd, unconstraind stratgy. (A With no cu at all, th cost of uncrtainty is qual to th ntropy of th nvironmnt H(E. (B Onc a particular cu c i has bn obsrvd thn th rduction in growth rat is just th cost of uncrtainty, H(Ejc i. Avraging across th diffrnt cus, th rduction in growth rat for a stratgy using a partially informativ cu is simply th conditional ntropy H(EjC. (C Th fitnss valu of information is, in this cas, th amount by which th cu rducs uncrtainty about th nvironmnt that is, xactly th mutual information btwn th cu and th nvironmnt (Eq. 18. All individuals within a gnration obsrv th sam nvironmntal cu c, which occurs with probability p(c. Onc that cu has bn obsrvd, th probability of ach nvironmntal stat is givn by th conditional probability distribution, p(jc. A conditional stratgy x spcifis th probability of dvloping into ach phnotyp 8, aftr obsrving th cu c: x(8jc. This can b rprsntd as a matrix X, with ntris X c8 x(8jc. To dscrib th stratgy s fitnss in a particular nvironmnt, aftr a cu c has bn obsrvd, w can look at th c-th row and th -th column of th matrix XF: f(x; [XSD] c [XS] c d y(½cd (16 This shows that a conditional stratgy y which producs th hypothtical phnotyp 8? with probability y(jc[xs] c, conditional on obsrving th cu c, is xactly quivalnt to th conditional stratgy x. Th growth rat of th stratgy x can thrfor b writtn as: g(x X X p c p(½clog X x(8½cf(8; c 8 X X p c p(½clog y(½cd c X (17 p(log d H(E½CD KL (p(½c½½y(½c which is lik a conditional vrsion of Eq. 14. Instad of th uncrtainty of th nvironmnt H(E, w hav th conditional uncrtainty aftr obsrving a cu, H(EjC. Instad of th rlativ ntropy D KL (pjjy, w masur th conditional rlativ ntropy D KL (p(jcjjy(jc, which rflcts th diffrnc btwn th bts th stratgy ffctivly placs on nvironmnts and th nvironmntal probabilitis, conditional on which cu is obsrvd. As usual, th bst stratgy is ffctiv proportional btting, conditional on th cu, but this may not always b possibl. What is th fitnss valu of th cu C? First of all, considr th situation whr a bt-hdging stratgy can ffctivly do proportional btting, both without th cu and with ach possibl cu. Thn th Kullback-Liblr divrgnc trms in Eq. 14 and 17 ar always zro. W can thrfor writ: DG(E; C G(E½CG(E X p(½c log d H(E½C X p( log d H(E H(EH(E½C I(E; C (18 Th valu of rciving a cu whn ffctiv proportional btting is possibl is xactly th mutual information btwn th cu and th nvironmnt (Fig. 3. Now lt us considr th mor gnral situation, whr ffctiv proportional btting may not b possibl. Lt y* b th bst possibl ffctiv btting stratgy whn no cu is availabl, and lt y* C b th bst possibl ffctiv btting stratgy whn th cu C is availabl. Th fitnss valu of information is thn 226

9 f(ϕ' 1, 1 f(ϕ' 2, 2 f(ϕ 2, 2 Long trm growth rat (log scal (A rgion of bt-hdging (B f(ϕ 1, 1 Fitnss valu of information G(E; C Optimal growth rats: Mutual information I(E; C with prfct cu, using phnotyps ϕ' with no cu, using phnotyps ϕ' with prfct cu, using phnotyps ϕ with no cu, using phnotyps ϕ f(ϕ 1, Probability of nvironmnt 1, p( 1 Figur 4. Th mutual information btwn a cu and th nvironmnt is an uppr bound on th fitnss valu of that information. For an unconstraind stratgy, using th xtrmist phnotyps 8?, th valu of a cu is xactly qual to th information it convys. W illustrat two cass of a constraind stratgy whr th valu of information is strictly lss than th amount of information. (A For xampl, say that th optimal stratgy without a cu would b to bt-hdg; th optimal rspons to a prfctly informativ cu would b to choos th singl bst phnotyp 8 1 or 8 2. For an unconstraind stratgy, th valu of this prfct cu would b qual to th mutual information. Th fitnss valu for th constraind stratgy is lowr, bcaus although it can achiv just th sam growth rat as th unconstraind stratgy without information, onc information is availabl th unconstraind stratgy can do bttr. (B If thr is no bt-hdging vn without a cu, thn th constraind stratgy dos wors both with and without th cu. Th valu of th cu using a constraind stratgy is thus not dirctly comparabl to th valu of th cu whn using an unconstraind stratgy. Howvr, w prov in th txt that th diffrnc in growth rats for th constraind stratgy cannot xcd th diffrnc in growth rats for th unconstraind stratgy (Eq. 22. f(ϕ 2, 1 DG(E; C G(E½CG(E I(E; C(D KL (p(½c½½y + C (½cD KL (p½½y+ (19 W would lik to show that th mutual information I(E; C is an uppr bound for th fitnss valu of information DG(E; C. That mans w nd to show that th right-hand trm is nvr ngativ: th cost of constraining th unconditional stratgy cannot b gratr than th cost of constraining th conditional stratgy. W ll do this in two stps. First of all, w dfin th unconditional stratgy y + C (a p(cy + C (½c as th stratgy an obsrvr would s, watching somon play y + C (½c in rspons to th cus c, but without obsrving th cus. Th first stp is to show that this marginal stratgy can b no farthr from th marginal distribution of nvironmnts than th conditional stratgy is from th conditional distribution of nvironmnts. W can writ th KullbackLiblr divrgnc btwn th two joint distributions ovr cus and nvironmnts in two diffrnt ways: D KL (p(; c½½y + C (; cd KL (p(c½½½y+ C (c½ D KL (p(½½y + C ( D KL (p(½c½½y + C (½cD KL (p(c½½y+ C (c (20 Howvr, th marginal distribution ovr cus is th sam for th two distributions, bcaus y + C is dfind in trms of th way it rsponds to cus gnratd according to th distribution p. This mans th last trm is zro, so D KL (p(; c½½y + C (; cd KL (p(c½½½y+ C (c½ D KL (p(½½y + C ( D KL (p(½c½½y + C (½cD KL (p(½½y+ C ( D KL (p(c½½½y + C (c½]0 (21 as dsird. Finally, w not that sinc y* is dfind as th optimal unconditional stratgy for th nvironmntal distribution p; D KL (p(½½y + (5D KL (p(½½y* C (: This shows that th fitnss valu of a cu cannot xcd th mutual information btwn that cu and th nvironmnt: DG(E; CI(E; C(D KL (p(½c½½y + C (½cD KL (p½½y+ 5I(E; C(D KL (p(½c½½y + C (½cD KL (p½½y+ C 5I(E; C (22 Figur 3 illustrats a cas whr th fitnss valu of th cu is xactly qual to th information it convys; Fig. 4 illustrats two cass in which th fitnss valu of th cu is strictly lss than th information it contains. 227

10 Discussion Many organisms living in a fluctuating nvironmnt show rmarkabl plasticity in lif history traits. Dsrt annual plants oftn fail to grminat in thir first yar, on th chanc that futur conditions will b bttr (Philippi 1993, Clauss and Vnabl Similarly, som inscts and crustacans can ntr diapaus to wait out unfavorabl conditions (Danforth 1999, Philippi t al For amphibians and fish, thr is a tradoff btwn producing a fw larg ggs or many small ons; if th smallst ggs can only surviv undr th bst conditions, this can provid an incntiv to mak ggs of variabl siz (Crump 1981, Koops t al Furthrmor, som amphibians that brd in tmporary pools show xtrmly variabl tim to mtamorphosis, bcaus th pools somtims dry up bfor th tadpols matur (Lan and Mahony 2002, Mory and Rznick Aphids and som plants can switch btwn sxual and asxual mods of rproduction dpnding on nvironmntal conditions and uncrtainty (Brg 1998, Halktt t al In all ths cass, th obsrvd variation in lif historis is thought to b an adaptation to nvironmntal variability; th bst studis show a quantitativ agrmnt btwn th amount of obsrvd plasticity and what is prdictd to b optimal (Vnabl 2007, Simons Howvr, it is oftn difficult to tll mpirically whthr th lif history variation is producd randomly, as in bt-hdging, or in rspons to prdictiv nvironmntal cus (Philippi 1993, Clauss and Vnabl 2000, Adondakis and Vnabl 2004, Mory and Rznick 2004; in som cass, it may actually b a combination of both mchanisms (Richtr Boix t al In this papr, w xamind th adaptiv valu of rsponding to prdictiv cus in th contxt of nvironmntal uncrtainty. W hav shown that th fitnss valu of using information about th nvironmnt gaind from prdictiv cus is intimatly rlatd to th amount of information th cus carry about th nvironmnt. Undr appropriat circumstancs, th fitnss bnfit of bing abl to dtct and rspond to a cu is xactly qual to th mutual information btwn th cu and th nvironmnt. Mor gnrally, th mutual information provids an uppr bound on th fitnss valu of rsponding to th cu. Ths rsults ar surprising, in that th mutual information masur smingly taks into account nothing about th fitnss structur of th nvironmnt. Why do w obsrv this connction btwn th fitnss valu of information and th mutual information? To answr that qustion, it hlps tak a closr look at th information-thortic dfinition of information: information is th rduction of uncrtainty, whr uncrtainty masurs th numbr of stats a systm might b in. Thus mutual information btwn th world and a cu is th fold rduction in uncrtainty about th world aftr th cu is rcivd. For xampl, if a systm could b in any of six quiprobabl stats, and a cu srvs to narrow th ralm of possibility to just thr of ths, th cu provids a twofold rduction in uncrtainty. For rasons of convninc, information is masurd as th logarithm of th fold rduction in uncrtainty. Logarithmic units nsur that th masur is additiv, so that for xampl w can add th information rcivd by two succssiv cus to calculat th total information gaind (Nyquist 1924, Hartly 1928, Shannon Thus whil information concpts ar oftn thought to b linkd with th famous sum S p log (p, th fundamntal concpt is not a particular mathmatical formula. Rathr, it is th notion that information masurs th fold rduction in uncrtainty about th possibl stats of th world. With this viw, it is asir to s why information bars a clos rlation to biological fitnss. For simplicity, considr an xtrm xampl in which individuals surviv only if thir phnotyp matchs th nvironmnt xactly, and suppos that thr ar tn possibl nvironmnts that occur with qual probability. In th absnc of any cu about th nvironmnt, th bst th organism can do is randomly choos on of th tn possibl phnotyps with qual probability. Only on tnth of th individuals will thn surviv, sinc only a tnth will match th nvironmnt with thir phnotyp. If a cu convys 1 bit of information and thus rducs th uncrtainty about th nvironmnt twofold, th nvironmnt can b only in on of fiv possibl stats. Th organism will now choos randomly on of fiv possibl phnotyps, and now a fifth of th population will surviv a twofold incras in fitnss, or a gain of 1 bit in th log of th growth rat. What happns whn th nvironmnts ar not quiprobabl? In this cas w can undrstand th connction btwn information and fitnss by looking to long squncs of nvironmnts and th thory of typical squncs. Th thory tlls us that almost surly on of th typical squncs thos squncs in which th nvironmnts occur in thir xpctd frquncis * will occur (Covr and Thomas Morovr, all typical squncs occur with qual probability. Thus a linag is slctd to divid its mmbrs qually among all typical squncs. Sinc any on mistak in phnotyp is lthal, only a fraction of ths linags, namly thos that hav just th right squnc of phnotyps, will surviv. Th numbr of typical squncs in this cas is xactly 2 NH(E whr N is th numbr of gnrations in th squnc and H(E is th ntropy of th nvironmnt. Corrspondingly, th fraction of surviving linags will b 2 NH(E If a cu C is rcivd that rducs th uncrtainty of th nvironmnts by I(E;C, thn th fraction of surviving linags can b incrasd by xactly 2 NI(E;C This is analogous to th situation in communication: if w nd to ncod a string of symbols that ar not quiprobabl, w turn to a long squnc of such symbols. Our cod thn nds only to b fficint for rprsnting typical squncs of symbols, and thos typical squncs occur with qual probability. Th numbr of such squncs is 2 NH, whr N is th lngth and H is th ntropy of th symbols. If th mssag rcipint also obtains sid information rlatd to th mssag itslf, thn th mutual information I btwn th mssag and th sid information masurs th rduction in th numbr of possibl mssags that nd to b ncodd by th transmittr. This numbr of mssags is rducd xactly 2 NI -fold by th prsnc of th sid information. Finally, what happns whn having th wrong phnotyp is not lthal, but simply dcrass fitnss? In this cas, w can no longr simply count th numbr of linags that hav th corrct squnc of phnotyps to dtrmin th 228

11 fraction that surviv. Howvr, w can transform th systm into on whr this is possibl, by constructing an altrnat st of hypothtical phnotyps that surviv in just on nvironmnt, and xprssing vrything in trms of thos phnotyps. W imagin that, instad of an individual dvloping a singl phnotyp, it dvlops a crtain combination of th altrnat phnotyps; instad of following linags of individuals, w follow linags of ths altrnat phnotyps. Th fraction that surviv without information is, at bst, 2 NH(E, whil th fraction that surviv with information is, at bst, 2 NH(EjC. Th mutual information I(E;C placs an uppr limit on th fold incras of linags that surviv whn a cu is availabl. W can now s why th concpt of information is th sam across diffrnt disciplins. In communication thory, th transmission of information is th rduction of uncrtainty about what signals will com through a channl, from an initial st of all possibl signals down to th post hoc st of signals actually rcivd. In thrmodynamics, a dcras in ntropy rfrs to th fold rduction in th numbr of stats that a systm can b in. In volutionary biology, th fitnss valu of a cu about an uncrtain nvironmnt rfrs to th fold incras in th numbr of surviving linags mad possibl by rsponding to th cu. Acknowldgmnts This work was supportd in part by a UW RRF award to CTB and was initiatd during a visit to th H. R. Whitly Cntr in Friday Harbor, WA. Th authors thank Sidny Frankl, Arthur Robson and Martin Rosvall for thir hlpful discussions. ML compltd th arly stags of this papr whil at th Max Planck Institut for Mathmatics, which providd a fruitful nvironmnt to work on such qustions. Rfrncs Adondakis, S. and Vnabl, D Dormancy and grmination in a guild of Sonoran Dsrt annuals. Ecology 85: Brg, H. and Rdbo-Torstnsson, P Clistogamy as a bt-hdging stratgy in Oxalis actoslla, a prnnial hrb. J. Ecol. 86: Brgstrom, C. T. and Lachmann, M Shannon information and biological fitnss. 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