Maximum Entropy Production and their Application to Physics and Biology Roderick C. Dewar Research School of Biological Sciences The Australian National University

Recall Lecture 3 … MaxEnt applied to non-equilibrium systems : Boltzmann MaxEnt applied to non-equilibrium systems : maximum irreversibility steady-state flux selection by MEP macroscopic dynamics Gibbs Shannon Jaynes

Max H : Max I: max H F  0 max I MEP FΩ uV Γ pΓ = 1 normalisation Γ pΓ fΓ = F flux  Ω Γ pΓuΓ = u density  V uΓ /t = –fΓ + QΓ continuity equation Subject to : (Dewar 2003) = entropy production in V = entropy export across Ω Max I: MEP

Part 1: Maximum Entropy (MaxEnt) – an overview Part 2: Applying MaxEnt to ecology Part 3: Maximum Entropy Production (MEP) Part 4: Applying MEP to physics & biology

MEP applications Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001) Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978, 1981; O’Brien & Stephens 1993, 1995) Ocean thermohaline circulation (Shimokawa & Ozawa 2002) Rayleigh-Bénard convection (Malkus 1954) Shear turbulence (Malkus 1956, Busse 1970) Mantle convection (Lorenz 2002) Crystal growth morphology (Hill 1990) Energy dissipation in ecosystems (Schneider & Kay 1994) Photosynthetic free energy transduction (Juretić et al. 2003) Evolutionary optimisation of ATP synthase (Dewar et al. 2006)

MEP applications Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001) Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978, 1981; O’Brien & Stephens 1993, 1995) Ocean thermohaline circulation (Shimokawa & Ozawa 2002) Rayleigh-Bénard convection (Malkus 1954) Shear turbulence (Malkus 1956, Busse 1970) Mantle convection (Lorenz 2002) Crystal growth morphology (Hill 1990) Energy dissipation in ecosystems (Schneider & Kay 1994) Photosynthetic free energy transduction (Juretić et al. 2003) Evolutionary optimisation of ATP synthase (Dewar et al. 2006)

Latitudinal heat transport H = ? Poleward heat transport 170 W m-2 300 W m-2 Latitudinal heat transport H = ? SW T LW

Fsw cT14 cT24 T1 T2 H = DΔT Fsw=cT14+H H=cT24 H = ? Equatorial zone Polar zone H = ? T1 H = DΔT

Trade-off between H and ΔT (Earth) Inter-zonal thermal diffusivity D (W m-2 K-1) EP ΔT H DMEP = 1.4 DEarth ≈ 1.7

MEP works elsewhere EP T0 T1

MEP applications Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001) Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978, 1981; O’Brien & Stephens 1993, 1995) Ocean thermohaline circulation (Shimokawa & Ozawa 2002) Rayleigh-Bénard convection (Malkus 1954) Shear turbulence (Malkus 1956, Busse 1970) Mantle convection (Lorenz 2002) Crystal growth morphology (Hill 1990) Energy dissipation in ecosystems (Schneider & Kay 1994) Photosynthetic free energy transduction (Juretić et al. 2003) Evolutionary optimisation of ATP synthase (Dewar et al. 2006)

Paltridge (1978) : 10-zone climate model EPradiation ? Planetary rotation rate ? LWi SWi N pole Equator S pole θi Fi Ti

Zonal temperature and cloud cover Ti θi

MEP applications Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001) Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978, 1981; O’Brien & Stephens 1993, 1995) Ocean thermohaline circulation (Shimokawa & Ozawa 2002) Rayleigh-Bénard convection (Malkus 1954) Shear turbulence (Malkus 1956, Busse 1970) Mantle convection (Lorenz 2002) Crystal growth morphology (Hill 1990) Energy dissipation in ecosystems (Schneider & Kay 1994) Photosynthetic free energy transduction (Juretić et al. 2003) Evolutionary optimisation of ATP synthase (Dewar et al. 2006)

Raleigh-Bénard convection: MEP = max flux (Ozawa et al 2001, after Malkus 1954) F is maximum when the boundary layer is marginally stable: Cold plate, Tc Hot plate, Th=Tc+ΔT diffusion δ convection d F diffusion δ

Ozawa et al. (2001) after Malkus, Busse (max flux) M: slope = 1/3 (max flux) M: slope = 1

Ozawa et al. (2001) after Malkus, Busse (max flux) M: slope = 1/3 (max flux) M: slope = 1

Global entropy production (mW m-2 s-1) Tuning GCM parameters using MEP (Kleidon et al. 2006) total Global entropy production (mW m-2 s-1) vertical horizontal k = 0.4 von Karman parameter, k

MEP applications Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001) Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978, 1981; O’Brien & Stephens 1993, 1995) Ocean thermohaline circulation (Shimokawa & Ozawa 2002) Rayleigh-Bénard convection (Malkus 1954) Shear turbulence (Malkus 1956, Busse 1970) Mantle convection (Lorenz 2002) Crystal growth morphology (Hill 1990) Energy dissipation in ecosystems (Schneider & Kay 1994) Photosynthetic free energy transduction (Juretić et al. 2003) Evolutionary optimisation of ATP synthase (Dewar et al. 2006)

Different growth morphologies are labelled by their Miller indices <111>, <110> …... ….. the 3D orientations of the different crystal faces that are growing : <111> <110>

Hill (1990) : crystallization of NH4Cl EP = F  X EP<111> EP<110> flux F F = L  (X - X0) XMEP = 0.21 Xobs = 0.216 force X = liq - solid

MEP applications Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001) Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978, 1981; O’Brien & Stephens 1993, 1995) Ocean thermohaline circulation (Shimokawa & Ozawa 2002) Rayleigh-Bénard convection (Malkus 1954) Shear turbulence (Malkus 1956, Busse 1970) Mantle convection (Lorenz 2002) Crystal growth morphology (Hill 1990) Energy dissipation in ecosystems (Schneider & Kay 1994) Photosynthetic free energy transduction (Juretić et al. 2003) Evolutionary optimisation of ATP synthase (Dewar et al. 2006)

2-state chlorophyll model: Juretić & Županović (2003) optimal quantum yield = 97% power transfer efficiency = 91% High efficiency ( 90%) is due to non-linear flux-force relation 5-state chlorophyll model: cf. linear flux-force relations: 50% power transfer efficiency optimal quantum yield = 94.6% power transfer efficiency = 87.8%

MEP applications Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001) Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978, 1981; O’Brien & Stephens 1993, 1995) Ocean thermohaline circulation (Shimokawa & Ozawa 2002) Rayleigh-Bénard convection (Malkus 1954) Shear turbulence (Malkus 1956, Busse 1970) Mantle convection (Lorenz 2002) Crystal growth morphology (Hill 1990) Energy dissipation in ecosystems (Schneider & Kay 1994) Photosynthetic free energy transduction (Juretić et al. 2003) Evolutionary optimisation of ATP synthase (Dewar et al. 2006)

F0F1-ATP synthase : Nature’s smallest rotary motor pmf-driven H+ transport  γ stalk torsion  ATP synthesis

Key functional parameter : κ = angular position of γ at which ADP+Pi  ATP (motor timing)

Transition rates between the 5 open (O) states of F1 were calculated using the kinetic model of Panke & Rumberg (1999) O: O:ATP JATP O:ADP O:ADP+Pi O:Pi

Transition rates between the 5 open (O) states of F1 were calculated using the kinetic model of Panke & Rumberg (1999) O: O:ATP JATP O:ADP O:ADP+Pi O:Pi

MaxEnt predicts observed kinetic design of ATPase Sstate and EPATP : simultaneous maxima at κ = 0.598 JATP (s-1) XATP (102 J mol-1) (κempirical fit  0.6) Sstate (102) EPATP (10 J K-1 mol-1 s-1) Relative angular position of γ at which ADP+Pi  ATP (κ)

MaxEnt and MEP …what next? Theory MaxEnt basis of MEP : info theory vs. max probability (N ) Boltzmann Applications in Global Change Science and beyond …. Climate and climate change : EPwater, cloud & water vapour feedbacks Plant and ecosystem responses to climate change : MEP = plant optimisation (“survival of the likeliest”) Climate-biosphere feedbacks : MEP = Gaia for grown-ups Other complex, non-equilibrium systems : e.g. plasmas, economies, networks Gibbs Shannon Annual MEP Workshops Bordeaux (2003-05), Split (2006), Jena (2007-09) …. Jaynes