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

2. 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

3. 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

4. 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

5. 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)

6. 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)

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

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

9. 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

10. MEP works elsewhere EP T0 T1

11. 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)

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

13. Zonal temperature and cloud coverTi θi

14. 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)

15. 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

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

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

18. 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

19. 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)

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

21. 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

22. 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)

23. 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: % power transfer efficiency
optimal quantum yield = 94.6% power transfer efficiency = 87.8%

24. 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)

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

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

27. 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

28. 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

29. 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)

30. 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 ( ), Split (2006), Jena ( ) ….
Jaynes