The fitness value of information in an uncertain environment

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1 Mark Dimmit T. Beth Kinsey Clauss & Venable (2000) fireflyforest.com The fitness value of information in an uncertain environment Matina Donaldson- Matasci Harvey Mudd College NIMBioS Information Theory Workshop April 8-10

2 How do organisms deal with all this uncertainty?

Developmental plasticity a heritable mechanism that

switch Acorn barnacles respond to the presence of

3 Developmental plasticity a heritable mechanism that generates predictive phenotypic diversity Acorn barnacle (Cthalamus anisopoma) C. Lively (1986) Evolution conditional developmental switch Acorn barnacles respond to the presence of snails by developing a predator- resistant bent shell shape. C. Lively (1986) Am Nat

$Nat stochastic developmental switch Desert annual plants can delay germination. Each year, only a fraction of seeds germinate, hedging bets against drought. D. Cohen (1966) J Theor Biol$

4 Bet- hedging a heritable mechanism that generates random phenotypic diversity 50% dry years 50% wet years 50% 50% Desert Indianwheat (Plantago insularis) Clauss & Venable (2000) Am Nat stochastic developmental switch Desert annual plants can delay germination. Each year, only a fraction of seeds germinate, hedging bets against drought. D. Cohen (1966) J Theor Biol

5 Conditional bet- hedging a heritable mechanism that generates partially predictive, partially random phenotypic diversity Early spring rain Late spring rain 25% 75% wet years 75% dry years 25% 25% 75% 75% 25% Desert Indianwheat (Plantago insularis) Clauss & Venable (2000) Am Nat conditional stochastic developmental switch Germination is more likely in years with early spring rains, because favorable growing conditions are more likely to follow D. Cohen (1967) J Theor Biol

6 How does the fitness value of a developmental cue relate to the amount of information it conveys?

7 The amount of information in a cue entropy entropy conditional entropy mutual information frequency of environment A Shannon (1948) Bell Syst Tech J

The amount of information in a cue entropy total variation information uncertainty measures the total uncertainty about an event, when no cue has been received conditional entropy

8 The amount of information in a cue entropy total variation information uncertainty measures the total uncertainty about an event, when no cue has been received conditional entropy measures the remaining uncertainty, once a cue has been received mutual information measures the reduction in uncertainty caused by receiving the cue Shannon (1948) Bell Syst Tech J

9 The fitness value of a cue The difference between the optimal fitness with the cue and the optimal fitness without the cue F c = f(ĝ c ) f(ĝ) How we measure fitness depends how risk is distributed

10 Individual- level risk every year Under individual- level risk, natural selection favors genotypes with a high mean fitness.

11 Population- level risk 4 in 5 years

12 Population- level risk 1 in 5 years Under population- level risk, natural selection favors genotypes with a high mean log fitness. Dempster (1955) CSH Symposia Quant Biol

13 The fitness value of a cue The difference between the optimal fitness with the cue and the optimal fitness without the cue F c = f(ĝ c ) How we measure fitness depends how risk is distributed i.i.d. between individuals in a generation f = X e f(ĝ) Pr(e)f(g, e) i.i.d. from one generation to the next r = X e Pr(e) log f(g, e)

Modeling developmental strategies in an uncertain environment per generation per individual Pr(e) environment E g(x) phenotype X The long- term fitness of a strategy g: r(g) = X e If each

14 Modeling developmental strategies in an uncertain environment per generation per individual Pr(e) environment E g(x) phenotype X The long- term fitness of a strategy g: r(g) = X e If each phenotype only survives in the right environment, the optimal strategy is proportional betting: ĝ(x e )=Pr(e) Pr(e) log X x g(x)f(x, e) Donaldson- Matasci, Bergstrom & Lachmann, Oikos (2010)

15 Optimal fitness with uncertainty long- term growth rate (generations with environment A) (generations with environment B) (on average) dry years wet years 50% dry years 50% wet years Donaldson- Matasci, Bergstrom & Lachmann (2010) Oikos

16 Optimal fitness with no uncertainty long- term growth rate (generations with environment A) (generations with environment B) (on average) dry years wet years 50% dry years 50% wet years Donaldson- Matasci, Bergstrom & Lachmann (2010) Oikos

The cost of environmental uncertainty growth rate with no uncertainty growth rate with uncertainty long- term growth rate (with no uncertainty)

17 The cost of environmental uncertainty growth rate with no uncertainty growth rate with uncertainty long- term growth rate (with no uncertainty) (with uncertainty) long- term growth rate cost of uncertainty is entropy frequency of environment A Donaldson- Matasci, Bergstrom & Lachmann (2010) Oikos

18 Modeling developmental strategies in an uncertain environment per generation per individual environment E Pr(c e) cue C g(x c) phenotype X The long- term fitness of a strategy: r(g c )= X e Pr(e) X c If each phenotype only survives in the right environment, the optimal strategy is conditional proportional betting: ĝ c (x e c) =Pr(e c) Pr(c e) log X x g(x c)f(x, e) Donaldson- Matasci, Bergstrom & Lachmann, Oikos (2010)

19 Optimal fitness with a partially informative cue cost of uncertainty (with cue 1) dry years wet years 50% dry 75% years dry years 50% wet years 25% Donaldson- Matasci, Bergstrom & Lachmann (2010) Oikos

20 Optimal fitness with a partially informative cue cost of uncertainty (with cue 1) (with cue 2) (on average) dry years wet years 25% 75% wet years Donaldson- Matasci, Bergstrom & Lachmann (2010) Oikos

21 The cost of remaining uncertainty growth rate with no uncertainty growth rate with uncertainty cost of remaining uncertainty (with cue 1) (with cue 2) long- term growth rate (on average) conditional entropy frequency of environment A Donaldson- Matasci, Bergstrom & Lachmann (2010) Oikos

22 The value of information growth rate with no uncertainty growth rate with uncertainty cost of uncertainty (with no cue) (with a cue) long- term growth rate value of information frequency of environment A Donaldson- Matasci, Bergstrom & Lachmann (2010) Oikos

23 Information theory links ecology and value of information environmental variation adaptive strategy fitness amount of information amount I(E;C) of uncertainty H(E) remaining uncertainty H(E C) amount of plasticity amount I(X;C) of bet- hedging H(X) amount of bet- hedging H(X C) value of information cost of uncertainty cost of remaining uncertainty The amount of information in a cue determines its value Donaldson- Matasci, Bergstrom & Lachmann, Oikos (2010)

24 When does the fitness value of a developmental cue equal the amount of information it conveys?

25 Three assumptions 1. Phenotypes only survive in the right environment Donaldson- Matasci, Lachmann & Bergstrom, EER (2008) Donaldson- Matasci, Bergstrom & Lachmann, Oikos (2010) 2. Environments are distributed i.i.d. across generations, but shared within generations Kussell & Leibler, Science (2005) Donaldson- Matasci, Lachmann & Bergstrom, EER (2008) 3. Cues are distributed i.i.d. across generations, but shared within generations Rivoire & Leibler, J Stat Phys (2011) Donaldson- Matasci, Bergstrom & Lachmann, Am Nat (2012)

26 What if phenotypes survive in many environments? f(ϕ' 2,e 2 ) f(ϕ 2,e 2 ) Long term growth rate (log scale) (A) (B) (C) region of bet-hedging f(ϕ' 1,e 1 ) f(ϕ 1,e 2 ) f(ϕ 2,e 1 ) 0.0 p(e 1 c 2 ) p(e 1 ) p(e 1 c 1 ) 1.0 Probability of environment 1, p(e 1 ) cost of full uncertainty H(E) f(ϕ 1,e 1 ) Optimal growth rates: cost of partial uncertainty H(E C) value of information I(E; C) The amount of information with perfect cue, using in a cue phenotypes is an ϕ' upper bound on its fitness value with no cue, using phenotypes ϕ' with perfect cue, using phenotypes ϕ R c apple I(E; C) with no cue, using phenotypes ϕ with predictive cue c, using phenotypes ϕ Donaldson- Matasci, Bergstrom & Lachmann, Oikos (2010)

Valente, Swarthmore College Photo: Mark Yokoyama Pupal

27 What if individuals receive different information? warm- weather morphs cold- weather morphs Photo: Tom Valente, Swarthmore College Photo: Mark Yokoyama Pupal temperature is influenced by weather patterns and microclimate

e) log X e q c Pr(c q) X x g(x c)f(x, e) If each phenotype only survives in the right environment, the optimal strategy is

28 Modeling developmental strategies when individuals receive different information environment E per generation Pr(q e) predictor Q Pr(c q) per individual cue C g(x c) phenotype X The long- term fitness of a strategy: r(g c )= X Pr(e) X Pr(q e) log X e q c Pr(c q) X x g(x c)f(x, e) If each phenotype only survives in the right environment, the optimal strategy is effectively conditional proportional betting: X Pr(c q)ĝ(x e c) =Pr(e q) c Donaldson- Matasci, Bergstrom & Lachmann, Am Nat (2012)

29 When all individuals receive the same information environmental variation adaptive strategy fitness amount of information I(E;Q) amount of plasticity I(X;Q) remaining uncertainty H(E Q) amount of bet- hedging H(X Q) Donaldson- Matasci, Bergstrom & Lachmann, Am Nat (2012)

30 When individuals receive different information environmental variation adaptive strategy fitness amount of information I(E;C) amount of plasticity I(X;C) remaining uncertainty H(E C) bet- hedging H(X C) Uncertainty increases, but bet- hedging decreases Donaldson- Matasci, Bergstrom & Lachmann, Am Nat (2012)

31 Information theory links ecology and value of information environmental variation adaptive strategy fitness population- level information I(E;Q) predictive plasticity I(X;Q) value of information shared uncertainty H(E Q) bet- hedging H(X Q) cost of remaining uncertainty The reduction in shared uncertainty determines a cue s value Donaldson- Matasci, Bergstrom & Lachmann, Am Nat (2012)

32 In many situations, the fitness value of a developmental cue is bounded by the amount of information it conveys

33 Thanks to Michael Lachmann (MPI Evol. Anth.) Santa Fe Institute Carl Bergstrom Univ. of Washington For comments and suggestions: Ben Kerr, Joe Felsenstein, Ray Huey, Nils Bertschinger, Eckehard Olbricht, Arthur Robson, Dean Foster, Martin Rosvall, Ola Olsson, Ken Schmidt, Naim Matasci and you! For more, follow

The fitness value of information

The fitness value of information Matina C. Donaldson-Matasci Department of Ecology & Evolutionary Biology University of Arizona Tucson, AZ 85721 email: matina@email.arizona.edu Carl T. Bergstrom Department