Maximum entropy & maximum entropy production in biological systems: survival of the likeliest?

Transcription

1 Maxmum entropy & maxmum entropy producton n bologcal systems: survval of the lkelest? Roderck Dewar Research School of Bology The Australan Natonal Unversty, Canberra Informaton and Entropy n Bologcal Systems 8 10 Aprl 2015, NIMBIOS

2 Maxmum Entropy (MaxEnt)

3 Maxmum (relatve) entropy: Maxmse H p ln p p q C w.r.t. p subject to qe q e x x p p x 1 X constrants (C) What s the ratonale for MaxEnt? p C nformaton theory (least based ) p C combnatorcs of sample frequences (most lkely )

4 MaxEnt: the combnatoral ratonale N ndependent observatons M possble outcomes for each observaton, = 1.M q = pror probablty of outcome Outcome observed n tmes frequency dstrbuton lm N 1 ln Pr Pr n N 1 n N! M p ln p q M n q 1 n! p n N = relatve entropy of p and q p The MaxEnt dstrbuton C s by far the most lkely long-term frequency dstrbuton of outcomes, among all those dstrbutons consstent wth gven constrants C (transparent connecton to observatons)

5 The predcton challenge n bology: bologcal systems are complex, open, non-equlbrum energy n matter n energy out matter out many nternal degrees of freedom Statstcal mechancs: Some (many?) detals of the underlyng dynamcs are rrelevant for makng predctons at larger scales

6 statstcal mechancs Shannon Boltzmann Gbbs maxmum (relatve) entropy detaled dynamcs x(t) Jaynes most lkely p(xkey dynamcal constrants )

7 Maxmum (relatve) entropy: Maxmse H p ln p p q C w.r.t. p subject to qe q e x x p p x 1 X constrants (C) What do the constrants represent? nformaton theory: C = what we know statstcal mechancs: C = the relevant dynamcs (key resource constrants, steady-state balance )

8 The role of MaxEnt n statstcal mechancs MaxEnt as a statstcal selecton prncple Known constrants C most lkely p(xc) MaxEnt as a tool for dentfyng the relevant dynamcs (C) Guess constrants C most lkely p(xc) observed p(x) fracton of tme system s n state x

9 Combnng mechansm and drft* n ecology Bertram & Dewar (2015) Relevant dynamcs C + MaxEnt most lkely p(xc) mechansm drft* * n the bologcal sense!

10 MaxEnt: a non-neutral, resource-based approach Dewar & Porté (2008), Bertram & Dewar (2013, 2015) mechansm drft Speces per capta resource use r Smax Speces abundances n Smax MaxEnt p(n 1, n 2. n Smax R) ( Bose-Ensten) r 2 n 2 r 1 Area constrant (hard) : Smax 1 n 1 Mean annual resource balance : (mean use = mean supply) n a Smax 1 A n r R unfes dfferent macroecologcal patterns: SAD s energy equvalence dversty-productvty dversty-stablty

11 The statstcal mechancs of savannas 3 1 f r Bertram & Dewar (2013) r tree > r grass > r bare 95% 5% R For gven R what s most lkely p( f tree, f grass )? mean H 2 O supply, R (mm yr -1 )

12 Maxmum Entropy Producton (MEP)

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14 Paltrdge (1978): 10-zone energy balance model N pole SW LW S pole T Maxmze EP 10 LW 1 SW T 9 X 1 1 T 1 1 T entropy producton (t s not!) subject only to steady-state energy balance

15 Merdonal heat transport (PW) Paltrdge (1978): 1D model obs. MaxEP Herbert & Pallard (2013): 2D model MEP IPSL-CM4 lattude

16 MEP applcatons across physcs & bology Paltrdge (1978) Malkus (2003) Man & Naylor (2008) some ntrgung successes, but what does t mean? Dewar et al (2006) Juretć et al (2003) Dewar (2010) Martyushev et al (2000)

17 Some potentally msleadng statements about Maxmum Entropy Producton MEP s a corollary to the second law (ds unverse /dt 0) whch states that S unverse ncreases as fast as possble MEP means that, over tme, EP approaches a maxmum n the steady-state

18 MEP as a statstcal stablty crteron for non-equlbrum statonary states R D p p ln 0 p R wth equalty ff p p R D s a generc measure of rreversblty / tme-reversal symmetry breakng / dstance from equlbrum d Fluctuaton theorem (follows from defnton of d ): p( d) p( d) d e If p(d) = N(D,σ 2 ): 2 2D CV D 1 D s mnmal when D s maxmal Use MaxEnt to construct p : D depends on the constrants C

19 Chemcal entropy producton (kg m -2 yr -1 ) Tree Physol. 32, (2012) MEP mmcs tradtonal maxmum ftness models ; β = carbon-use effcency Fne-root bomass, R (kg m -2 )

20 Evolutonary optmsaton of Rubsco: Earth s most abundant proten Enzyme state Enzyme knetcs Knetc desgn k = (k 1, k 2 k 10 ) Apply MaxEnt to p(, k) p( k) p( k) CO 2 carboxylaton, v C oxygenaton, v O O 2 p( k) e H ( k) p( k)ln p( k) EP(k) v C v ln v C C H ( k ) v O v ln v O O Maxmse smultaneously w.r.t. k 6 & k 3

21 EP and H Evolutonary optmsaton of Rubsco: Earth s most abundant proten Enzyme knetcs MaxEnt & MEP CO 2 carboxylaton, v C 10 k 3 = 6 mm -1 s -1 H oxygenaton, v O O 2 5 EP k 6 (mm -1 s -1 )

22 k cat,c / K C (mm -1 s -1 ) k cat,o / K O (mm -1 s -1 ) K C (μm) k cat,c (s -1 ) Rubsco adaptaton to dfferent kcatc (s-1) v s S CO 2 /O 2 envronments hgh CO 2 /O low CO 2 /O Specfcty, S C/O kcatc/kc (mm-1 s-1) v s S Specfcty, S C/O MaxEnt & MEP predctons default parameters k 7 1/10 k 4 1/ Dewar et al. (n prep.) Specfcty, S C/O Specfcty, S C/O

23 J net (s -1 ) F 0 F 1 -ATP synthase : Nature s smallest rotary motor Net ATP synthess vs. proton drvng force MaxEnt & MEP Transthylakod pmf, Δμ H+ (ph unts) Dewar, Juretc & Zupanovc (2006)

24 Evoluton of ATP-synthase knetcs MaxEnt & MEP predct optmal angular poston for ATP synthess close to observed (0.6) optmal gearng rato (H + /ATP) 1 / pmf J net maxmally senstve to pmf hgh free-energy converson effcency (69%) wthn the expermental range (50 80%) observed knetc desgn of ATP synthase consstent wth most lkely desgn

25 K C (μm) Optmal trat adaptaton to changng constrants (C 1 C 2 ): survval of the lkelest MaxEnt: C 1 most lkely p(xc 1 ) 100 most lkely p(xc 2 ) C MEP: max stablty? (cf. P-V-T gas laws) Specfcty, S C/O

26 Where next? Theory Bass of MEP Non-statonary behavour Applcatons Ecologcal nteractons: r r j Food webs Plant adaptatons (e.g. leaf stomatal responses)