Biogeochemistry and Maximum Entropy Production

Home , Entropy production

icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Earth Syst. Dynam. Discuss., 2, 1–44, 2011 www.earth-syst-dynam-discuss.net/2/1/2011/ Earth System Dynamics ESDD doi:10.5194/esdd-2-1-2011 Discussions 2, 1–44, 2011 © Author(s) 2011. CC Attribution 3.0 License.

Biogeochemistry and This discussion paper is/has been under review for the journal Earth System maximum entropy Dynamics (ESD). Please refer to the corresponding ﬁnal paper in ESD if available. production

J. J. Vallino

Diﬀerences and implications in Title Page biogeochemistry from maximizing Abstract Introduction entropy production locally versus globally Conclusions References Tables Figures J. J. Vallino J I Marine Biological Laboratory, Woods Hole, Massachusetts, USA J I Received: 31 December 2010 – Accepted: 13 January 2011 – Published: 19 January 2011 Back Close Correspondence to: J. J. Vallino ([email protected]) Published by Copernicus Publications on behalf of the European Geosciences Union. Full Screen / Esc

Printer-friendly Version

Interactive Discussion

1 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Abstract ESDD In this manuscript we investigate the use of the maximum entropy production (MEP) principle for modeling biogeochemical processes that are catalyzed by living systems. 2, 1–44, 2011 Because of novelties introduced by the MEP approach, many questions need to be an- 5 swered and techniques developed in the application of MEP to describe biological sys- Biogeochemistry and tems that are responsible for energy and mass transformations on a planetary scale. In maximum entropy previous work we introduce the importance of integrating entropy production over time production to distinguish abiotic from biotic processes under transient conditions. Here we investigate the ramiﬁcations of modeling biological systems involving one or more spatial J. J. Vallino 10 dimensions. When modeling systems with spatial dimensions, entropy production can be maximized either locally at each point in space asynchronously or globally over the system domain synchronously. We use a simple two-box model inspired by two-layer Title Page

ocean models to illustrate the diﬀerences in local versus global entropy maximization. Abstract Introduction Synthesis and oxidation of biological structure is modeled using two autocatalytic re- 15 actions that account for changes in community kinetics using a single parameter each. Conclusions References Our results show that entropy production can be increased if maximized over the sys- Tables Figures tem domain rather than locally, which has important implications regarding how biological systems organize and supports the hypothesis for multiple levels of selection and cooperation in biology for the dissipation of free energy. J I

J I

20 1 Introduction Back Close

There is a long history of research that attempts to understand and model ecosys- Full Screen / Esc tems from a goal-based perspective dating back to at least Lotka (1922), who argued that ecosystems organize to maximize power. Many of the proposed theories (e.g., Printer-friendly Version Morowitz, 1968; Schrodinger,¨ 1944; Odum and Pinkerton, 1955; Margalef, 1968; Pri- Interactive Discussion 25 gogine and Nicolis, 1971; Schneider and Kay, 1994; Bejan, 2007; Jorgensen et al., 2000; Toussaint and Schneider, 1998; Weber et al., 1988; Ulanowicz and Platt, 1985) 2 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion are inspired by or derived from thermodynamics, because ecosystems are comprised by numerous agents, and there is the desire to understand their collective action, which ESDD is an inherently thermodynamic-like approach. Thermodynamics appears particularly 2, 1–44, 2011 relevant for understanding biogeochemistry as it is largely under microbial control, or 5 more aptly molecular machines (Falkowski et al., 2008), that span the realm between pure chemistry and biology. While significant effort and progress has been made in Biogeochemistry and understanding organismal metabolism in an ecosystem context, it is rather surpris- maximum entropy ing that we still lack an agreed upon theory that can explain and quantitatively predict production ecosystem biogeochemistry that is independent of the constituent organisms. Perhaps J. J. Vallino 10 a single theory does not exist, and ecosystem biogeochemistry depends completely or substantially on the nuances of the constituent organisms that comprise an ecosystem. However, at the appropriate scale and perspective it appears likely that systems Title Page organize in a predictable manner independent of species composition, as the planet’s biogeochemical processes have remained relative stable over hundreds of millions of Abstract Introduction 15 years but the organisms that comprise ecosystems have changed significantly over this period. Conclusions References

Consider primary producers in terrestrial versus marine ecosystems. While there is Tables Figures a deﬁnitive morphological discontinuity between marine systems that are composed of planktonic algae and terrestrial ecosystems comprised of trees, shrubs or grasses, J I 20 the distinction and discontinuity is removed when considering solar energy acquisition and carbon dioxide ﬁxation, which both do similarly. Because of the large degrees of J I freedom, it is likely impossible to predict the nature of the organisms that evolve to Back Close facilitate energy and mass transformations in one ecosystem versus the next or how their populations vary over time, but at the level of free energy dissipation and biogeo- Full Screen / Esc 25 chemistry, the problem may be more tractable. The problem is similar to predicting weather versus climate, where the latter can be predicted over long time scales at Printer-friendly Version the expense of details necessary for accurate weather prediction whose accuracy de- cays rapidly with time. Our objective is to understand biogeochemistry at the level of Interactive Discussion generic biological structure catalyzing chemical reactions. Understanding energy and

3 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion mass flow catalyzed by living systems is particularly relevant for understanding how the biosphere will respond to global change, and predicting biogeochemical responses ESDD is essential before any geoengineering projects could be responsibly implemented. 2, 1–44, 2011 Many of the thermodynamically inspired ecosystem theories are more similar than 5 different (Fath et al., 2001; Jorgensen, 1994), but the theory of maximum entropy production (MEP) has made great progress in understanding how biotic and abiotic Biogeochemistry and systems may organize under nonequilibrium steady state conditions when sufficient maximum entropy degrees of freedom exist. It appears Paltridge (1975) was the first to apply MEP as an production objective function for modeling global heat transport, but it was not until the theoret- J. J. Vallino 10 ical support provided by Dewar (2003) that research in MEP theory and applications garnered greater interest, particularly in Earth systems science (Lorenz et al., 2001; Kleidon et al., 2003; Kleidon and Lorenz, 2005a,b; Dyke and Kleidon, 2010). While Title Page Dewar’s initial work has received some criticisms, there have been several other approaches that arrive at the same nonequilibrium steady state result of MEP (see Niven, Abstract Introduction 15 2009 and references therein). The MEP principle states that systems will organize to maximize the rate of entropy production, which is an appealing extension to the sec- Conclusions References

ond law of thermodynamics for nonequilibrium systems. In essence, nonequilibrium Tables Figures systems will attempt to approach equilibrium via the fastest possible pathway, which can include the organization of complex structures if they facilitate entropy production. J I 20 However, none of the current MEP theories incorporate information content of the system, such as that contained within an organism’s genome, even though several of the J I MEP analyses are derived from Jaynes’ (2003) maximum entropy (MaxEnt) formulation Back Close that is information based. Furthermore, current MEP theory suggests that maximizing entropy production locally will maximize entropy production over the system domain, at Full Screen / Esc 25 least for ﬂow-controlled systems (Niven, 2009). Because a MEP-based model can be implemented with either local or global optimization, this manuscript investigates if the Printer-friendly Version type of spatial optimization aﬀects the solution to a simple, MEP-based biogeochemistry model. Interactive Discussion

4 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

11 Entropy of mixing is readily calculated between boxes and | boundaries Paper Discussion | Paper Discussion based | Paper Discussion on | flux Paper Discussion and In this manuscript we will assume that biological systems organize to maximize en- 12 chemical potential obtained from concentration differences between boxes (Meysman and tropy production, which is synonymous with maximizing the dissipation rate of Gibbs ESDD free energy for13 systemsBruers 2007, under Kondepudi constant and temperature Prigogine 1998) and. pressure. For our simple Living model, organisms we only permit 2, 1–44, 2011 are viewed here14 asdiffusive mere transport catalysts between facilitating boxes autocatalytic and boundaries, reactions so a material for theflux dissipa-of chemical species k 5 tion of free energy. We will also continue our main hypothesis (Vallino, 2010), that 15 from box or boundary [i] to box or boundary [j] is given by (mmol m-2 d-1), the acquisition of Shannon information (Shannon, 1948), or more precisely useful in- Biogeochemistry and formation (Adami, 2002; Adami et al.,D 2000), via evolution facilitates the production of maximum entropy entropy when16 averaged overFi, j (C time.k ) = − It is(C thek [ j] ability− Ck [i] of)βi living, j (k) , systems to store information (11) production  that allows them to out-compete abiotic systems in entropy production under appro- J. J. Vallino 10 priate conditions.17 where Here,  we is awill characteristic examine length howinformation scale for diffusion may also(2.5×10 facilitate-5 m), D entropyis the diffusion production averaged over space. For our model system, we will consider a simple × -4 2 -1 β two-box model18 wherecoefficient entropy (1 production10 m d ) inand each i, j ( boxk) is is 1 governedif flux of compound by two reactions k is allowed that between box or Title Page determined the19 rateboundary of biological [i] and structure[j]; otherwise (i.e., it is catalyst) 0. The product synthesis of flux and and degradation. chemical potential differences, Abstract Introduction i, j 20 ∆µk , divided by temperature gives the entropy production per unit area for mixing between 2 Model system Conclusions References 21 box [i] to [j] for chemical species k, as follows (J m-2 d-1 ºK-1)

15 To examine how total entropy production depends on either local or global entropy Tables Figures F dS i, j ∆µ i, j D  C [i]  production optimization, wek use= a simple two-boxk = − model reminiscent−  ofk ocean biogeo- 22 Fi, j (Ck ) (Ck [ j] Ck [i])R ln  . (12) chemistry models developeddt to capture processesT  in the photic and C aphotick [ j] zones of J I oceans (Fig. 1) (Ianson and Allen, 2002). Model focus is placed on dissipation of free J I energy by synthesis23 Given and concentrations destruction of CH biological2O, NH3, structures O2, H2CO3, ( S1 andand S2 2along) from with and the to growth 20 chemical constituents (CH O, NH ,O and H CO ) within the two layers. The photic 24 efficiencies2 ε and3 ε in2 each box,2 entropy3 production associated with reactions and mixingBack Close zone is typically energy replete1 but resource2 limited, while the aphotic zone is its com- plement: energy25 limitedcan be calculated but resource from replete.Eqs. (6-12). Our model captures these two differing Full Screen / Esc states; however, we replace energy input via photosynthetic active radiation with energy input via the diffusion of chemical potential from a fixed upper boundary condition Printer-friendly Version 25 into the surface layer box. Nitrogen in the form of ammonia is allowed to diffuse into the Interactive Discussion bottom box [2] from sediments, which fixes the lower boundary condition [3]. Hence, energy flows into the system via box [1] and resources (nitrogen) flow in via box [2]. 9 5 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

D 16 Fi, j (Ck ) = − (Ck [ j] − Ck [i])βi, j (k) , (11) 

i, j 20 ∆µk , divided by temperature gives the entropy production per unit area for mixing between icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 21 box [Alli] to constituents [j] for chemical are species allowed k, as to follows freelydi (Jff muse-2 d between-1 ºK-1) boxes [1] and [2], but only CH2O, O2 and H2CO3 are permitted across the upper boundary, while only NH3 can diffuse ESDD F acrossdS i, j the lower boundary∆µ i, j D (Fig. 1).  C [i]  k = k = − −  k  2, 1–44, 2011 22 In our simplifiedFi, j (Ck ) model system,(Ck [ wej] considerCk [i])R ln only two . autocatalytic chemical(12) reactions dt T   Ck [ j] 5 occur within each box, as constrained by the following stoichiometric equations: Biogeochemistry and 23 Given concentrations of CH2O, NH3, O2, H2CO3, S1 and S2 along with the growth r1 : CH2O + (1 − ε1) O2 + ε1 ρ NH3 −→ ε1 1 + (1 − ε1) H2CO3 (1) maximum entropy 24 efficiencies ε1 and ε 2 in each box, entropy production associated with reactions and mixing production 2 r2 : (δ1 1 + δ2 2) + (1 − ε2) O2 −→ ε2 2 + (1 − ε2)(H2CO3 + ρ NH3). (2) 25 can be calculated from Eqs. (6-12). J. J. Vallino In reaction r1, biological structure, 1, catalyzes its own synthesis from sugar (unit carbon basis) and available ammonia thereby fulfilling the primary producer role. Re- 10 action r2 represents heterotrophic organisms that consume biological structure synthe- Title Page sized from Eq. (1), 1, but also functions in a cannibalistic mode by consuming 2 as well. While cannibalism could be replaced by inclusion of a sufficient number of higher Abstract Introduction 9 trophic levels for closure, this would lead to greater degrees of model freedom that are Conclusions References not important for this study, but are more typical closure techniques observed in real 15 environments (Edwards and Yool, 2000). To minimize parameterization, we assume Tables Figures that biological structure is degraded indiscriminately by 2, so that δi in Eq. (2) are given by, J I C C i i J I δi = = , for i = 1, 2, (3) C + C C T 1 2 Back Close where C is the concentration (mmol m−3) of biological structure i. In both reactions, i Full Screen / Esc 20 ρ is the N:C ratio of biological structure, , and εi is the growth efficiency for biological structure synthesis. Note, as growth efficiency approaches zero, both reactions Printer-friendly Version represent the complete oxidation of reduced carbon to CO2 and H2O. Typically, Holling-type kinetics (Holling, 1965) are used to describe reaction rates for Interactive Discussion Eqs. (1) and (2) based on a limiting substrate or substrates. However, kinetic param- Max 25 eters (e.g., µ ,KS) in Holling-type expressions depend on community composition, 6 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion so Holling kinetic equations need to be reparameterized as community composition changes. Since an underlying hypothesis of the MEP principle when applied to living ESDD systems is that they will evolve and organize to achieve a MEP state (Vallino, 2010), we 2, 1–44, 2011 cannot a prior set parameters in a kinetic model as we do not know the nature of the 5 community composition nor the corresponding kinetic parameter values that will lead to a MEP state. Kinetic parameters need to be dynamic to reflect community composi- Biogeochemistry and tion changes. To achieve this objective, we have developed a novel kinetic expression maximum entropy that can capture reaction kinetics in a manner consistent with community compositional production changes. The form of our kinetic expression takes the hyperbolic form of the Monod −1 J. J. Vallino 10 equation (Monod, 1949), where substrate specific up-take rate is given by (d ), C ! ∗ 2 2Y j νi = ν ε 1 − ε (4) i i ∗ 4 Title Page j=1 Cj + κ εi Abstract Introduction and specific growth rate readily follows as (d−1), Conclusions References µi = εi νi (5) Tables Figures −3 In Eq. (4), Cj is the concentration (mmol m ) of one or more growth limiting substrates, ∗ ∗ 15 such as NH3, εi is the growth efficiency as used in Eqs. (1) and (2), and ν and κ are J I universal parameters that remain fixed regardless of community composition. Equa- tion (4) differs from the Monod equation in several important ways. The effective half J I ∗ 4 saturation “constant” (κ εi ) depends on growth efficiency. The rational for this depen- Back Close dency is that nutrients at low concentration require free energy expenditure to transport 20 them into the cell against a concentration gradient; consequently, organisms that have Full Screen / Esc evolved to grow under low nutrients conditions (i.e., K-selection; Pianka, 1970) will, by 2 thermodynamic necessity, grow at lower efficiencies. The (1 − εi ) term in Eq. (4) ap- Printer-friendly Version proximately accounts for thermodynamic constraints on kinetics, because net reaction rate must approach zero as reaction free energy tends toward zero (Jin and Bethke, Interactive Discussion 2 25 2003; Boudart, 1976), which is equivalent to a growth efficiency of 1. The leading εi 7 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

14 diffusive transport between boxes and boundaries, so a material | Paper Discussion flux | Paper Discussion of | chemical Paper Discussion | Paper Discussion species k term in Eq. (4) is empirically motivated to account for reduced substrate uptake-2 rates-1 at low growth15 effi ciency.from box or boundary [i] to box or boundary [j] is given by (mmol m d ), ESDD The universal parameters have been chosen to qualitatively match observations of D 2, 1–44, 2011 bacterial growth.16 Under oligotrophicFi, j (Ck ) = − conditions(Ck [ j] − Ck [ ofteni])βi, j ( foundk) , in ocean gyres’, bacterial (11) −1 5 specific growth rate is typically 1–2 d , with growth efficiencies of 10–20%, and substrate concentrations in the µM or lower range (Del Giorgio and Cole, 1998;-5 Carlson Biogeochemistry and 17 where  is a characteristic length scale for diffusion (2.5×10 m), D is the diffusion et al., 1999). At the other extreme, bacteria grown under ideal laboratory conditions maximum entropy -4 2 -1 −1 show specific18 growth coefficient rates as (1× fast10 as m 50 d d) and, withβi, j (k growth) is 1 if e fluxfficiencies of compound around k is 50–60%, allowed between box or production provided substrate concentrations are in the mM range (Bailey, 1977; Lendenmann 19 boundary [i] and [j]; otherwise it is 0. The product of flux and chemical potential differences, J. J. Vallino 10 and Egli, 1998). Equations (4) and (5) capture these approximate trophic extremes ∗ i, j −3 ∗ −1 with values of20κ of∆µ 5000k , divided mmol mby temperatureand ν of gives 350 dthe entropy(Fig. 2). pro Theduction usefulness per unit area of thefor mixing between kinetic expressions (Eqs. 4 and 5) is their ability to generate organismal-2 -1 - growth1 kinetics 21 box [i] to [j] for chemical species k, as follows (J m d ºK ) Title Page expected under oligotrophic conditions to extreme eutrophic conditions by varying only

growth efficiency, ε (Fig. 2). Fi, j i, j dS ∆µ D  C [i]  Abstract Introduction k = k = − −  k −3 −1 15 Based on Eq.22 (4), the rate of Eqs.Fi, j ( (1)Ck ) and (2) are expressed(Ck [ j] Ck [i by])R (mmolln m  . d ), (12) dt T   Ck [ j] Conclusions References C ! C ! ∗ 2 2 CH2O NH3 r ν ε − ε C Tables Figures 1 = 1 123 1Given concentrations of CH O, NH , O , H CO , S and S along with the(6) growth C + κ∗ ε4 2 C 3+ κ2∗ ε42 3 1 2 CH2O 1 NH3 1 24 efficiencies ε1 and ε 2 in each box, entropy production associated with reactions and mixing C ! J I ∗ 2 2 T r2 = ν ε 125− εcan be calculated from Eqs.C (6-12). (7) 2 2 C κ∗ ε4 2 J I T + 2 Back Close Equations (6) and (7) allow us to explore reaction rates for different community compositions by varying ε1 and ε2 between 0 and 1. Of course, other mathematical functions Full Screen / Esc 20 could be used in place of Eq. (4), but this is not our primary interest for this manuscript, but it is an interesting topic that warrants further research. For instance, we use the Printer-friendly9 Version ∗ 4 same effective half saturation constant, κ εi , regardless of substrate type in Eqs. (6) or (7), which could likely be improved, as well as accounting for other cellular growth Interactive Discussion complexities (Ferenci, 1999).

8 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 2.1 Free energy and entropy production ESDD For our two box model, entropy production occurs via destruction of chemical potential via Eqs. (1) and (2), and by entropy of mixing associated with relaxing chemical 2, 1–44, 2011 gradients between boxes [1] and [2] and their associated boundaries. For chemical 5 reactions, entropy production per unit area in a given box with depth h (m) is readily Biogeochemistry and determined under constant pressure and temperature from the reaction Gibbs free en- maximum entropy ergy change, ∆Gr, reaction rate, r, and absolute temperature, T , as given by Eu (1992, production 131–141 pp.) (J m−2 d−1 ◦ K−1), J. J. Vallino dSri h = − r ∆ G . (8) dt T i ri

10 While Gibbs free energy of reaction is usually straight forward to calculated for a given Title Page reaction stoichiometry, both Eqs. (1) and (2) require some discussion because both Abstract Introduction involve synthesis or decomposition of living biomass, or more generally, what we refer to as biological structure. Conclusions References Living organisms are often thought to be very low entropy structures because they Tables Figures 15 are highly order systems. However, this common misconception results from the incor- rect association of thermodynamic entropy with order or disorder in a system. In general, system order has little to do with thermodynamic entropy (Kozliak and Lambert, J I 2005; Lambert 1999), as the latter is purely related to dispersion of energy to lower en- J I ergy states, which may or may not be less ordered. Interestingly, both theoretical calcu- 20 lations and empirical data show that bacteria and yeast have higher absolute entropies Back Close (at 298.15 ◦K) than glucose (Battley, 1999a,b, 2003). Biological structure has a higher Full Screen / Esc entropy of formation than “simple” growth substrates, and it can be shown that the standard Gibbs free energy of reaction for synthesizing bacteria or yeast from glucose, Printer-friendly Version ammonia, phosphate and sulfate, without CO2 production, is slightly less than zero, so 25 is a reaction that can occur spontaneously (Vallino, 2010). Hence, it is thermodynam- Interactive Discussion ically possible for Eq. (1) to proceed without additional free energy input with ε1 = 1. However, as mentioned above, reaction rates are also constrained by thermodynamics 9 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

D 16 Fi, j (Ck ) = − (Ck [ j] − Ck [i])βi, j (k) , (11) 

-5 | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 17 where  isas a characteristic Gibbs free energylength scale of reaction for diffusion approaches (2.5×10 m) zero., D is For the adiffusion reaction to proceed at high rate it- must4 2 - proceed1 irreversibly; consequently, organisms whose growth efficiency is 18 coefficient (1×10 m d ) and βi, j (k) is 1 if flux of compound k is allowed between box or ESDD less than 100% can grow faster than those operating at very high efficiencies. The 2, 1–44, 2011 19 boundary [diminishingi] and [j]; otherwise returns it ofis high0. The growth product e ffiofciency flux and is chemical embed inpotential our growth differences kinetics, equation, i, j 5 as plotting Eq. (5) as a function of ε exhibits an optimum, so that maximum specific 20 ∆µk , divided by temperature gives the entropy production per unit area for mixing between d µ q Biogeochemistry and growth rate under nutrient saturation ( = 0) occurs as ε → 3/5 ≈ 0.77. -2 d-1 ε -1 21 box [i] to [j] for chemical species k, as follows (J m d ºKCj →∞) maximum entropy The Gibbs free energy of reaction for Eqs. (1) and (2) can be expressed as a linear production F dScombinationi, j of∆µ biosynthesisi, j D and CH O oxidation, C [i]  as given by (J mmol−1), k = k = − − 2  k  22 Fi, j (Ck ) (Ck [ j] Ck [i])R ln  . (12) J. J. Vallino dt T C [ j] 1−ε1 ε1    k  C C H2CO3 G 1 − ε G◦ ε G◦ RT ln  1  (9) ∆ r1 = ( 1) ∆ Ox + 1 ∆ + 23 Given concentrations of CH2O, NH3, O2, H2CO3, S1 and S2 along with the growth1−ε1 ε1ρ  CCH O C C Title Page 2 O2 NH3 24 efficiencies ε and ε in each box, entropy production associated with reactions and mixing 1 2  1−ε (1−ε )ρ ε  C 2 C 2 C 2 Abstract Introduction H CO NH 25 can be calculated from Eqs. (6-12). ◦ ◦ 2 3 3 2 10 ∆ G = (1 − ε ) ∆ G − (1 − ε ) ∆ G + RT ln  , (10) r2 2 Ox 2  1−ε  Conclusions References C C 2 T O2 Tables Figures −1 ◦ −1 ◦ where R is the ideal gas constant (in J mmol K ), ∆GOx is the standard Gibbs −1 ◦ free energy of CH2O oxidation by O2 to H2CO3 (−498.4 J mmol ) and ∆G is the J I standard Gibbs free energy for biological structure synthesis from CH 9O and NH , 2 3 J I which is set to zero for our study because it is negligibly small (Vallino, 2010). We 15 use the approach of Alberty (2003, 2006) to calculate the standard Gibbs free energy Back Close ◦ of reaction, ∆GOx, which accounts for proton dissociation equilibria between chemical + − Full Screen / Esc species (CO2 + H2O ↔ H2CO3 ↔ H + HCO3 , etc.) at a pH of 8.1 and temperature of ◦ 293.15 K. Ionic strength (IS = 0.7 M) is used to estimate activity coefficients, so that Printer-friendly Version concentrations can be used in the logarithmic correction terms of Eqs. (9) and (10) 20 instead of activities (Alberty, 2003). Interactive Discussion Entropy of mixing is readily calculated between boxes and boundaries based on flux and chemical potential obtained from concentration differences between boxes 10 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

- | 2 - Paper 1 Discussion | Paper Discussion | Paper Discussion | Paper Discussion (Meysman15 from and box Bruers, or boundary 2007; [i] Kondepudito box or boundary and Prigogine, [j] is given 1998). by (mmol For m our d simple), model, we only permit diffusive transport between boxes and boundaries, so a material flux D ESDD of16 chemical speciesFi, j (Ck )k= −from(C boxk [ j] − orCk boundary[i])βi, j (k) , [i] to box or boundary [j] is given by (11) −2 −1  2, 1–44, 2011 (mmol m d ), 17 where D is a characteristic length scale for diffusion (2.5×10-5 m), D is the diffusion 5 Fi,j (Ck ) = − (Ck [j] − Ck [i]) βi,j (k), (11) Biogeochemistry and ` × -4 2 -1 β 18 coefficient (1 10 m d ) and i, j (k) is 1 if flux of compound k is allowed between box or maximum entropy −5 where ` is a characteristic length scale for diffusion (2.5 × 10 m), D is the diffusion production 19 boundary [i−] 4and2 [j];− 1otherwise it is 0. The product of flux and chemical potential differences, coefficient (1 × 10 m d ) and βi,j (k) is 1 if flux of compound k is allowed between i, j J. J. Vallino box20 or boundary∆µk , divided [i] and by [temperaturej]; otherwise gives it is the 0. entropy The product production of flux per andunit chemicalarea for mixing potential between i,j differences, ∆ µk , divided by temperature gives the- entropy2 -1 -1 production per unit area 21 box [i] to [j] for chemical species k, as follows (J m d ºK ) −2 −1 ◦ −1 10 for mixing between box [i] to [j] for chemical species k, as follows (J m d K ) Title Page Fi, j i, j Fi,j dS i,j ∆µ D  C [i]  dS k ∆=µ k = − −  k  22k Fki, j (Ck ) D (Ck [ j] Ck [i])R lnCk [i]  . (12) Abstract Introduction = Fi,j (Ckdt) = − T (Ck [j]− Ck [i]) R ln  Ck [ j.] (12) dt T ` Ck [j] Conclusions References Given23 concentrationsGiven concentrations of CH of2 O,CH2 NHO, 3NH,O3,2 O,H2, H2CO2CO33,, S1 andand S2 along2 along with with the growth the growth efficiencies ε and ε in each box, entropy production associated with reactions and Tables Figures 24 efficiencies1 ε 2and ε in each box, entropy production associated with reactions and mixing mixing can be calculated1 from2 Eqs. (6–12). 25 can be calculated from Eqs. (6-12). J I 15 2.2 Transport equations, optimization and numerical routines J I To obtain changes in chemical constituents over time for a given set of boundary condi- Back Close tions, a mass balance model is constructed based on in-and-out fluxes and production rates by reactions r1 and r2 for each constituent k in boxes [1] and [2], which takes the Full Screen / Esc general form, 9 Printer-friendly Version dCk [i] 20 = Fi − 1,i (Ck ) − Fi,i+1 (Ck ) /h[i] + Λk [i] r[i] for i = 1, 2 (13) dt Interactive Discussion where the brackets, [i], following a variable denote the box-boundary location with [0] being the upper boundary and [3] being the lower boundary (Fig. 1). Elements of 11 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

D 16 Fi, j (Ck ) = − (Ck [ j] − Ck [i])βi, j (k) , (11) 

F | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion dS i, j ∆µ i, j D the 1 × 2 vector C [iΛ]  are the stoichiometric coefficients for compound k obtained from k = k = − −  k k 22 Fi, j (Ck ) (Ck [ j] Ck [i])R ln  . (12) T dt T  Eqs. (1) and C (2),k [ j] and r is a 2 × 1 vector of reaction rates, [r1 r2] , given by Eqs. (6) ESDD and (7). The full model equations are detailed in Appendix A for k = CH O, O ,H CO , 2 2 2 3 2, 1–44, 2011 23 Given concentrations of CH2O, NH3, O2, HNH2CO33,, S1 ,and2 Sin2 along the two with boxes, the growth and parameter values used for simulations is given in 5 Table 1. 24 efficiencies ε and ε in each box, entropy production associated with reactions and mixing 1 2 We have intentionally designed our biogeochemistry model to contain only a few Biogeochemistry and 25 can be calculated from Eqs. (6-12). transport related parameters (D, `, h) and effectively only two adjustable biological maximum entropy parameters per box, giving a total of four adjustable parameters for the two box model: production ε [1], ε [1], ε [2], and ε [2]. Because choosing a value for ε [j] effectively selects 1 2 1 2 i J. J. Vallino 10 the kinetic characteristics of the community in box [j], we can examine how entropy production changes as a function of community composition. More importantly for this study, the values of ε [1], ε [1], ε [2], and ε [2] that maximize entropy production 1 2 1 9 2 Title Page can be determined by numerical analysis, and via Eqs. (6) and (7) the biogeochemistry associated with the MEP state is defined. For a zero dimensional system, a variation Abstract Introduction 15 of this approach has been used to determine how biogeochemistry develops over time (Vallino, 2010); however, for a system that has one or more spatial dimensions, a choice Conclusions References needs to be made regarding local versus global optimization of entropy production. For Tables Figures the two-box model, entropy production can be asynchronously maximized locally (i.e., in each box), as given by, J I r r r r Maximize dS 1 [1] dS 2 [1] Maximize dS 1 [2] dS 2 [2] 20 + and + (14) J I ε1 [1], ε2 [1] dt dt ε1 [2], ε2 [2] dt dt Back Close or globally (i.e, across both boxes synchronously) as, Full Screen / Esc r r r r Maximize dS 1 [1] dS 2 [1] dS 1 [2] dS 2 [2] + + + (15) ε1 [1], ε2 [1], ε1 [2], ε2 [2] dt dt dt dt Printer-friendly Version

where both optimizations are bounded within the hypercube, 0 ≤ εi [j] ≤ 1. Of course, Interactive Discussion local maximization, as given by Eq. (14), requires an iterative (i.e., asynchronous) 25 approach, because changing the conditions in one box alters the optimal solution for 12 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion the other. In contrast, global optimization given by Eq. (15) is mathematically well posed, but the parameter space is larger (4-D vs. 2-D in this case). Neither Eq. (14) nor ESDD Eq. (15) include the entropy of mixing contribution given by Eq. (12), as it is uncertain 2, 1–44, 2011 how it should be partitioned, but as will be shown, it is a small contribution compared 5 to entropy production from the reaction terms, so neglecting it in the optimization is acceptable. Biogeochemistry and While optimization of Eqs. (14) and (15) can be conducted for the transient problem maximum entropy (Eq. 13) over a speciﬁed time interval (Vallino, 2010), we are primarily interested in production how spatial optimization alters MEP solutions. Consequently, we are only interested J. J. Vallino 10 in steady state (SS) solutions to Eq. (13). Typically, Newton’s method, or a variation thereof, is used to solve for SS solutions to Eq. (13); however, employing several variations to Newton’s method including a line search method with trust region (Bain, 1993) Title Page was found to fail in some subspaces of 0 ≤ εi [j] ≤ 1. An approach based on interval arithmetic (Kearfott and Novoa, 1990; Kearfott, 1996) was also developed, but proved Abstract Introduction 15 to be too computationally burdensome. The most robust approach found was to integrate the ODE equations (Eq. 13) forward in time using a high precision block implicit Conclusions References

method (Brugnano and Magherini, 2004) from a specified initial condition (Table 2) un- Tables Figures 1 T til 2 dC/dt dC/dt ≤ ζ, where C is a vector of concentrations and ζ was set to −8 −3 −1 6 10 mmol m d . If a steady state solution failed to be obtained after 10 days of in- J I 20 tegration, the point was flagged and removed from the search, but this rarely occurred. Numerical solution to Eq. (15) was obtained by using VTDIRECT (He et al., 2009), J I which employs a parallel version of a Lipschitzian direct search algorithm (Jones et Back Close al., 1993) to find a function’s global optimum without using function derivatives. We also used VTDIRECT to solve Eq. (14), but we implemented an iterative approach, Full Screen / Esc 25 where first box [1] was optimized for given values of ε1 [2] and ε2 [2], then box [2] was Printer-friendly Version optimized given ε1 [1] and ε2 [1] from the previous optimization of box [1]; this iteration proceeded until ε[j]−εˆ[j] ≤10−9 for each box, where is the Euclidian norm and k k k k Interactive Discussion T the vector εˆ (εˆ = [εˆ1 εˆ2] ) is the value of ε from the previous iteration.

13 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

+ 6 proton dissociation equilibria between chemical species (CO2 + H2O ↔ H2CO3 ↔ H + - 7 HCO3 , etc.) at a pH of 8.1 and temperature of 293.15ºK. Ionic strength (IS =0.7 M) is used to 8 estimate activity coefficients, so that concentrations can be used in the logarithmic correction icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 9 terms of (9)3 andResults (10) instead and of discussion activities (Alberty 2003). 10 ESDD 3.1 Model characteristics 2, 1–44, 2011 11 Entropy of mixing is readily calculated between boxes and boundaries based on flux and For any speciﬁed point in εi [j]-space, a solution to Eq. (13) can be found. For instance, 12 chemical potential obtained from concentration differences between boxes (Meysman and Fig. 3 illustrates the transient dynamics to steady state for the point (ε1 [1], ε2 [1], ε1 [2], Biogeochemistry and 5 ε 13 Bruers 2007,2 Kondepudi[2]) = (0.1, and 0.05, Prigogine 0.1, 0.05) 1998) with. For the our parameterssimple model, and we only initial permit and boundary conditions maximum entropy used throughout this study (Tables 1 and 2, respectively). For this particular solu- 14 diffusive transport between boxes and boundaries, so a material flux of chemical species k production tion, SS entropy production dominates in box [1] (575 J m−2 d−1 ◦ K−1), as compared to −2 −1 ◦ −1 -2 -1 15 from box orbox boundary [2] (41.4 [i] to J mbox ord boundaryK ), and[j] is entropiesgiven by (mmol associated m d ) with, mixing between boundary J. J. Vallino [0] and box [1], box [1] and box [2], and box [2] and boundary [3] are 14.6, 0.436 and D−2 −1 ◦ −1 10 16 Fi, j (0.00Ck ) = (J− m (Cdk [ j] −KCk [i),])β respectively.i, j (k) , Due to the signiﬁcant amount of(11) free energy re- lease in oxidation of reduced organic compounds, entropy of mixing is a small fraction Title Page

of the entropy production associated with Eqs.-5 (1) and (2). To demonstrate this more 17 where  is a characteristic length scale for diffusion (2.5×10 m), D is the diffusion Abstract Introduction generally, we randomly chose 100 000 points within the 4-D εi [j]-space and calculated -4 2 -1 18 coefficient the(1×10 steady m d state) and solutionβi, j (k) is to 1 if Eq. flux (13) of compound and associated k is allowed entropy between terms, box whichor shows that Conclusions References 15 entropy of mixing is at most 7.6% of the entropy of reaction, and for most cases much 19 boundary [iless] and than[j]; otherwise that (Fig. it is 4a). 0. The Only product 310 of points flux and out chemical of 100 310potential random differences simulations, did not Tables Figures i, j 6 20 ∆µk , dividedattain by temperature a SS solution gives within the entropy 10 days. production per unit area for mixing between J I An interesting aspect of the model-2 is-1 that-1 the amount of nitrogen extracted from the 21 box [i] to [jsediments,] for chemical or species boundary k, as follows [3], depends (J m d on ºK community) composition, because changing J I 20 Fvalues of ε [j] dramatically affects the total standing SS nitrogen within boxes [1] dS i, j i∆µ i, j D  C [i]  k = k = − − −2  k  Back Close 22 and [2],Fi, j ( calculatedCk ) as (mmol(Ck [ j] N mCk [i]),)R ln  . (12) dt T   Ck [ j] 2 Full Screen / Esc X N = h[i] C [i] + ρ C [i] + C [i] , (16) 23 Given concentrationsTotal of CH2O, NHNH3,3 O2, H2CO3, S1 and S2 along2 with the growth i=1 Printer-friendly Version 24 efficiencies ε and ε in each box, entropy production associated with reactions and mixing which1 is evident2 in the results from the 100 000 random ε [j]-point simulations (Fig. 4b). i Interactive Discussion 25 can be calculatedFor any from given Eqs. reaction(6-12). entropy production defined by Eq. (8), numerous SS solutions 25 can be found that differ in location within εi [j]-space and exhibit differences of up 14

9 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

D 16 Fi, j (Ck ) = − (Ck [ j] − Ck [i])βi, j (k) , (11) 

17 where  is a characteristic length scale for diffusion (2.5×10-5 m), D is the diffusion -4 2 -1 18 coefficient (1×10 m d ) and βi, j (k) is 1 if flux of compound k is allowed between box or 19 boundary [i] and [j]; otherwise it is 0. The product of flux and chemical potential differences, i, j 20 ∆µk , divided by temperature gives the entropy production per unit area for mixing between icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion -2 -1 -1 21 box [i] to [j] for chemicalto four species orders k, ofas follows magnitude (J m in d N ºKTotal) (Fig. 4b). While the vast majority of solutions result in an N less than 104 mmol m−2, 33 solutions produced high to extremely ESDD F Total dS i, j ∆µ i, j D  C [i]  k = high NTotalk =values.− Examination−  of communityk  kinetics that lead to high NTotal values 22 Fi, j (Ck ) (Ck [ j] Ck [i])R ln  . (12) 2, 1–44, 2011 dt revealsT an apparent necessary condition Ck [ j] that the growth efficiency of Eq. (2), ε2 [j], 5 in both boxes [1] and [2] must be less than 0.02 (Fig. 5). Inspection of the transient 23 Given concentrationssolution of CH2O, reveals NH3, O that2, H2 highCO3, S1 andconcentration S2 along with develops the growth due to extremely slow growth Biogeochemistry and rates of 2 caused by small ε2 [j] values (data not shown). Over time, 2 develops maximum entropy 24 efficiencies ε1 and ε 2 in each box, entropy production associated with reactions and mixing sufficient concentration to place “top down” control on 1, but this leads to elevated production 25 can be calculated fromSS Eqs.concentrations (6-12). of both and . 1 2 J. J. Vallino 10 Inspection of total entropy production by Eqs. (1) and (2) in the random 100 000 εi [j]-point simulations reveals that the top 17 SS solutions have total reaction entropy −2 −1 ◦ −1 production distributed over a narrow range from 620.11 to 621.15 J m d K ; how- Title Page ever, the corresponding εi [j] points have a much broader distribution over εi [j]-space (Table 3), which produce significant differences in transient dynamics and SS standing Abstract Introduction 9 15 stocks (Fig. 6). Most significant are differences in SS standing stocks for biological −2 Conclusions References structures 1 and 2 that can range from 26 to 29 600 and from 2.2 to 340 mmol m in box [1], respectively. These state variations that have nearly identical entropy pro- Tables Figures ductions illustrate, at least qualitatively, an important concept in MEP theory (Dewar,

2005, 2009), which is there should be many micropath solutions that produce the same J I 20 macropath (or MEP) state. Since MEP theory rests on probabilities (Dewar, 2009; Lorenz, 2003), slight variations in entropy production exhibited by the top SS entropy J I

producing solutions (Table 3) can be viewed as interchangeable, since any real system Back Close is subject to noise that would make the small diﬀerences in dSr/dt indistinguishable. For our model system, there are clearly multiple solutions that give rise to what is Full Screen / Esc 25 eﬀectively the same MEP state. Printer-friendly Version

Interactive Discussion

15 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 3.2 Global versus local MEP solutions ESDD The global MEP solution obtained by maximizing reaction associated entropy produc- 2, 1–44, 2011 tion, Eq. (8), summed over boxes [1] and [2] by simultaneously varying ε1 [1], ε2 [1], −2 −1 ◦ −1 ε1 [2] and ε2 [2] as given by Eq. (15) is 621.5 J m d K (Table 4). Because of Biogeochemistry and 5 the small 4-D parameter space of our model, this solution was almost located in the maximum entropy random 100 000 point simulations (Table 3). To examine the solution in the vicinity of r production the global maximum, we plot dS /dt in box [1] for any given value of ε1 [1] and ε2 [1] while holding ε1 [2] and ε2 [2] ﬁxed at their global optimum values, and vice versa for J. J. Vallino box [2] (Fig. 7). It is clear from this plot that dSr/dt in box [1] of 619.6 J m−2 d−1 ◦ K−1 r 10 is near the dS /dt maximum in box [1] given ε1 [2] and ε2 [2] (Fig. 7a, light gray point); however, it is also evident that dSr/dt in box [2] that contributes 1.92 J m−2 d−1 ◦ K−1 Title Page r (Fig. 7b, light gray point) to the global solution is far removed from the maximum dS /dt Abstract Introduction possible in box [2] that could be attained given ε1 [1] and ε2 [1], which is approximately 236 J m−2 d−1 ◦ K−1. This result indicates that the global optimum does not correspond Conclusions References

15 to local optimums. Entropy productions associated with mixing, Eq. (12), summed Tables Figures over all relevant state variables for the globally-optimized solution are 18.6, 0.06 and −2 −1 ◦ −1 0.0 J m d K for ﬂuxes F0,1, F1,2 and F2,3, respectively. While entropy production J I from mixing does contribute to total entropy production, it is clear that it is small relative to the entropy of reaction terms. J I 20 If reaction associated entropy production is maximized locally as given by Eq. (14), Back Close then the total entropy production summed over both boxes is only 429.4 J m−2 d−1 ◦ K−1, −2 −1 ◦ −1 with 224.4 and 204.9 J m d K being produced in boxes [1] and [2], respectively Full Screen / Esc (Table 4). Examination of entropy production in boxes [1] and [2] in the vicinity of the locally-optimized solutions (Fig. 8) illustrates that the iterative procedure used to Printer-friendly Version 25 solve Eq. (14) has indeed obtained a solution that simultaneously maximizes entropy production in both boxes, because dSr/dt in box [1] cannot be improved with the given Interactive Discussion r values of ε1 [2] and ε2 [2], nor can dS /dt in box [2] be improved given ε1 [1] and ε2 [1]; 16 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

D 16 Fi, j (Ck ) = − (Ck [ j] − Ck [i])βi, j (k) , (11) 

i, j | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 20 ∆µk , divided by temperature gives the entropy production per unit area for mixing between the solution is stable. The corresponding entropy of mixing terms for fluxes F0,1, F1,2 −2 −1 ◦ −1 -2 -1 -1 and F2,3 are 3.21,21 2.60box [i and] to [ 0.0j] for J mchemicald speciesK , respectively. k, as follows (J m d ºK ) ESDD Comparing steady state solutions from the global versus the local optimizations F 2, 1–44, 2011 dS i, j ∆µ i, j D  C [i]  shows that CH2O concentrationk = in boxesk [1]= − and [2] is− near zero ink the globally- 22 Fi, j (Ck ) (Ck [ j] Ck [i])R ln  . (12) 5 optimized solution, but onlydt partially depletedT in the locally-optimize solution Ck [ j] (Table 5). Biogeochemistry and The inability to effectively oxidize CH2O in box [1] in the locally-optimized solution re- maximum entropy sults from an23 insu ffiGivencient concentrations development of of CH biological2O, NH3, O structure2, H2CO3, ( S1 andand S2 2along), as with it is the ev- growth ident in Fig. 4b that a lower boundary on N exists for a given entropy production production 24 efficiencies ε and ε in eachTotal box, entropy production associated with reactions and mixing rate. Biological reaction rates1 are ultimately2 limited by the quantity of catalyst avail- J. J. Vallino 10 able that in turn25 iscan limited be calculated by elemental from Eqs. resources. (6-12). The spatial model configuration is such that state variables in box [1] control energy acquisition across boundary [0], while box [2] controls resource input across boundary [3]. Limitations in either flux will Title Page result in lower entropy production rates. In the local optimization given by Eq. (14), maximizing entropy production in box [2] ultimately results in less N transported into Abstract Introduction 15 the system from boundary [3]. In the globally optimize solution, net synthesis of 1[2] Conclusions9 References is critical in N acquisition, because when reaction r1 exceeds r2 in box [2] there is a net transport of NH3 into the system that accumulates in 1 [2]. In the locally optimize Tables Figures solution, entropy production can be increased by a large loop flow between reactions r and r , but this limits N acquisition and entropy production in box [1]. These simula- 1 2 J I 20 tions clearly demonstrate that optimizing local entropy production limits whole system resource and energy acquisition, which results in lower entropy production compared J I to the globally optimized solution that more effectively utilizes available resources to Back Close dissipate free energy. We also investigated globally and locally optimized solutions with different values of Full Screen / Esc 25 the transport parameters (namely D, `, h[1] and h[2]), but in each case the general con- clusion was supported; entropy production is greater when maximized globally rather Printer-friendly Version than locally. A second variation of the model was also investigated that included two ∗ ∗ Interactive Discussion additional state variables, 1 [j] and 2 [j], in each box, where the * superscript indicates inactive biological structure that does not catalyze reactions r1 or r2. The rational 17 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

D 16 Fi, j (Ck ) = − (Ck [ j] − Ck [i])βi, j (k) , (11) 

F dS i, j ∆µ i, j D  C [i]  k = k = − −  k  22 Fi, j (Ck ) (Ck [ j] Ck [i])R ln  . (12) dt T   Ck [ j] icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 23 Given concentrations offor CH this2O, modelNH3, O2 was, H2CO that3, Si1 optimizedand S2 along in with box [thej] wouldgrowth be poorly adapted to conditions in box [k] and would be outcompeted, but could serve as substrate in reaction r . In this 24 efficiencies ε and ε in each box, entropy production associated with reactions and mixing 2 ESDD 1 2 alternate model, diﬀusion across boxes occurred between active and inactive variants ∗ ∗ ∗ ∗ 2, 1–44, 2011 25 can be calculated from Eqs.of (6i ,- as12). given by 1 [1]↔ 1 [2], 2 [1]↔ 2 [2], 1 [1]↔ 1 [2] and 2 [1]↔ 2 [2]. 5 This modiﬁed version of the model also produced similar results, where global optimization resulted in total entropy production of 620.9 J m−2 d−1 ◦ K−1, but local optimization Biogeochemistry and generated entropy at a rate of only 415.0 J m−2 d−1 ◦ K−1. Next, we examine how our maximum entropy approach and conclusions may be extrapolated to a more ecologically relevant context. production

9 J. J. Vallino 4 Broader ecological context

10 The model we have developed is obviously chosen to illustrate various aspects in the Title Page application of MEP in spatially explicit situations, and does not represent any natural system per say. The ideas we explore here, based on the development and results of Abstract Introduction

our modeling exercise, are intended to examine the usefulness of the MEP principle Conclusions References for understanding biogeochemical processes. The two main questions the MEP com- 15 munity must address are (1) do biological systems organize towards a MEP state and Tables Figures (2) if so, is this knowledge useful for understanding and modeling biogeochemistry? This manuscript does not address question (1), as we implicitly assume it to be true; J I instead, we focus on question (2). Ecosystem processes and the resulting biogeochemistry are often viewed from an J I 20 organismal perspective, because organisms are the autonomous parts that comprise Back Close an ecosystem. Furthermore, since macroscopic ecosystem constituents appear relatively stable over timescales of human interest, it is common practice to characterize Full Screen / Esc and model ecosystems with a relatively static species composition. However, the fossil record clearly shows that ecosystem communities exhibit great change over time (Gai- Printer-friendly Version 25 dos et al., 2007), which is particularly evident in microbial systems whose characteristic Interactive Discussion timescales are short (Falkowski and Oliver, 2007; Fernandez et al., 1999, 2000). The MEP perspective indicates there should be many diﬀerent means of attaining the same 18 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

15 from box or boundary [i] to box or boundary [j] is given by (mmol m d ), | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion MEP state under a given set of constraints (Dewar, 2003). Consequently, ecosystem biogeochemistryD is not constrained to one particular set of species, but can be attained ESDD 16 Fi, j (Ck ) = − (Ck [ j] − Ck [i])βi, j (k) , (11) by an infinite number of complementary species sets, with the sets being interchange- 2, 1–44, 2011 able. By complementary we mean the community must be comprised of organisms -5 17 where5 capable is a characteristic of energy length acquisition scale for anddiffusion element (2.5× recycling,10 m), D is so the that diffusion at steady state only free energy is-4 dissipated.2 -1 The Earth’s biosphere obviously operates in this mode, as there Biogeochemistry and 18 coefficient (1×10 m d ) and βi, j (k) is 1 if flux of compound k is allowed between box or is no net accumulation or loss of biomass over time, but the conversion of electro- maximum entropy 19 boundarymagnetic [i] and [j]; radiation otherwise intoit is longer0. The product wavelengths of flux producesand chemical entropy. potential We differences have attempted, to production i, j embody the idea of species composition fluidity in kinetic expressions Eqs. (4) and (5), 20 ∆µk , divided by temperature gives the entropy production per unit area for mixing between J. J. Vallino 10 where the choice of εi defines the kinetics of the community. To implement MEP, dif- 21 box [i] toferent [j] for modelingchemical species approaches k, as follows need (J tom- be2 d-1 developed ºK-1) that are independent of species composition, which Eq. (4) is but one example. F dS i, j ∆µ i, j D  C [i]  Title Page kThe= MEP approachk = − also highlights− the importancek  of resource acquisition for the 22 Fi, j (Ck ) (Ck [ j] Ck [i])R ln  . (12) constructiondt of biologicalT  structures needed Ck to[ j] dissipate free energy. Different com- Abstract Introduction 15 munity parameterizations lead to different levels of N acquisition and biological struc- Conclusions References 23 Given concentrationsture concentrations of CH2O, with NH3 a, O minimum2, H2CO3, ST1 andconcentration S2 along with required the growth to achieve a given rate of entropy production (Fig. 4b). This perspective differs from the current paradigm 24 efficiencies ε and ε in each box, entropy production associated with reactions and mixing Tables Figures of Liebig’s1 Law2 of the Minimum, which focuses solely on elemental limitations at the 25 can be calculatedorganismal from level. Eqs. (6 Because-12). organism stoichiometry changes to match resource avail- J I 20 ability over evolutionary time scales (Elser et al., 2000, 2007), Liebig’s law becomes a weak constraint. From the MEP perspective, resources will be extracted based on J I catalyst compositions that achieve the highest rate of free energy dissipation as con- Back Close strained by organic chemistry. To quote Lineweaver and Egan (2008) “This represents a paradigm shift from “we eat food” to “food has produced us to eat it”.” Full Screen / Esc 9 25 The 17 highest entropy producing solutions in the 100 000 random-point simulations illustrate that very different “food web” configurations, as defined by εi [j], can never- Printer-friendly Version theless dissipate free energy at statistically similar rates (Table 3). Each of these MEP solutions can be considered as meta-stable or as alternate stable states, because Interactive Discussion a perturbation of sufficient magnitude, or alteration of initial conditions, can lead the

19 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion system to a different MEP basin of attraction where different ecosystem processes are exhibited for the same boundary conditions, such as occurs in shifts between macroal- ESDD gae and coral reefs (Mumby, 2009), in microbial communities (Price and Morin, 2004) 2, 1–44, 2011 and in many other systems (Schroder¨ et al., 2005). Likewise, altering system drivers 5 or boundary conditions will change the ecosystem configuration necessary to maintain a MEP state, which would likely lead to regime shifts (Brock and Carpenter, 2010). All Biogeochemistry and of the solutions in Table 3 are stable states, as they were obtained via integration of maximum entropy the ODEs (Appendix A) to steady state. Unlike most analyses based on a discrete production selection of food web components with fixed parameter values, our approach based on J. J. Vallino 10 Eqs. (4) and (5) selects from a continuum of possible food web configurations that could arise over long periods of evolution, but does not produce a statistically unique solution (Table 3). While we have not investigated stability aspects of the MEP solutions in this Title Page manuscript, the MEP perspective may be useful in such analyses. The optimization criteria implicit in MEP allows parameter values to be determined Abstract Introduction 15 for a given set of environmental conditions and drivers instead of being fit to experimental data (e.g., Table 4). This improves model robustness as parameter values can Conclusions References

be determined for conditions where experimental data are lacking, or where model Tables Figures resolution is coarse compared to the underlying physics, such as in modeling global heat transport (Paltridge, 1975; Kleidon et al., 2003). The optimization approach also J I 20 replaces knowledge on information content in a system. As stated in the introduction, genomic information allows biological systems to out-compete abiotic systems J I under some situations when averaged over time and/or space. However, it is not just Back Close Shannon information, but useful information (Adami et al., 2000) that is relevant; e.g., genes that allow sulfate to serve as an electron acceptor are useless if sulfate is not Full Screen / Esc 25 present in the environment. This context dependency of genomic information com- plicates decoding an ecosystem’s metagenome. While great strides are being made Printer-friendly Version at reading and decoding environmental genomic data (DeLong, 2009; Gianoulis et al., 2009), we are still far from using that information to predict ecosystem biogeochemistry Interactive Discussion (Keller, 2005; Frazier et al., 2003). By assuming all known metabolic capabilities are

20 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion present in a system, MEP-based optimization can determine which pathways may be expressed and when (Vallino, 2010); furthermore, genomic information can be incor- ESDD porated into the optimization as constraints when genomic surveys indicate the lack of 2, 1–44, 2011 certain metabolic capabilities. Hence, MEP theory can be used to link genomic content 5 to free energy dissipation. An interesting result from our MEP-based modeling exercise is that entropy produc- Biogeochemistry and tion can be increased when optimized globally versus locally. However, to achieve high maximum entropy entropy production requires that the system organize such that certain locations within production the model domain function “sub-optimally”, as evident in Fig. 7b. That is, free energy J. J. Vallino 10 dissipation in box [2] is lower than possible, but this allows greater entropy production in box [1], which more than oﬀsets the lower entropy production in box [2]. This “altruistic behavior” is inconsistent with standard Darwinian evolution, which places op- Title Page timization at the level of the individual; however, our results are consistent with ideas on evolution of cooperation and multilevel selection theory (Bastolla et al., 2009; Clutton- Abstract Introduction 15 Brock, 2009; Goodnight, 1990; Hillesland and Stahl, 2010; Nowak, 2006; Traulsen and Nowak, 2006; Wilson and Wilson, 2008). For instance, bacterial colonies are large Conclusions References

spatial structures (∼cm) compared to the bacteria (∼µm) that comprise them, yet it is Tables Figures well established that bacteria produce quorum sensing compounds that cause bacteria on the periphery of a colony to express diﬀerent genes than those in the colony’s J I 20 center (Camilli and Bassler, 2006; Keller and Surette, 2006; Shapiro, 1998). Com- munication between plants via emission of volatile organic compounds has also been J I demonstrated to improve plant defenses over spatial scales greater than the individual Back Close (Heil and Karban, 2010). Below ground, a single fungus can attain immense size and occupy up to 1000 ha with hyphae (Ferguson et al., 2003). Given that mycorrhizae Full Screen / Esc 25 can integrate with plant roots (Maherali and Klironomos, 2007; Vandenkoornhuyse et al., 2007; Whitﬁeld, 2007) suggests some level of below ground communication occurs Printer-friendly Version across entire forests. From a physics perspective, even an individual multicellular organism represents a large spatial structure that is 103 to 106 times larger than the cells Interactive Discussion that comprise it, and surely those cells are spatially coordinated. Consequently, it is

21 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion clear that information exchange over large spatial scales exists in nature, so it is con- ceivable that ecosystems could coordinate over space to maximize energy acquisition ESDD and dissipation. 2, 1–44, 2011 If ecosystem information exchange occurs over large spatial scales, and if this com- 5 munication facilitates increased free energy dissipation, then the domain of a model needs to be chosen judiciously. Specifically, the model domain is defined by the spa- Biogeochemistry and tial extent of information exchange, largely dictated by chemical signaling. In certain maximum entropy systems, such as microbial mats (van Gemerden, 1993), it seems reasonable to model production these systems as spatially connected systems, while other systems, such as the sur- J. J. Vallino 10 face and deep ocean, it is uncertain to what extent, if any, they are informationally connected over space. Furthermore, it is possible that entropy production from local optimization is statistically similar to that from global optimization, so there may be little Title Page gained from spatial communication, and local optimization (e.g., Follows et al., 2007) would be sufficient to describe system dynamics. We do not know the answer to this Abstract Introduction 15 question, as all current biogeochemical models depend solely on local conditions and are Markovian in nature; more research is needed. Conclusions References

Tables Figures 5 Conclusions J I The maximum entropy production principle (Paltridge, 1975; Dewar, 2003) proposes that nonequilibrium abiotic or biotic systems with many degrees of freedom will or- J I 20 ganize towards a state of maximum entropy production, which is synonymous with Back Close maximizing free energy dissipation for chemical systems. The usefulness of MEP theory for understanding and modeling biogeochemistry is still under debate, and nu- Full Screen / Esc merous applied and theoretical challenges need to be address before MEP becomes a common tool for solving problems in biogeochemistry. In previous work we have Printer-friendly Version 25 demonstrate how the MEP principle can be applied to biological systems under tran- Interactive Discussion sient conditions provided entropy production is integrated over time (Vallino, 2010). In this manuscript we have examined how biogeochemical predictions diﬀer if entropy 22 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion production is maximized locally versus globally for systems involving spatial dimensions. Biological systems explore biogeochemical reaction space primarily via changes ESDD in community composition and gene expression; consequently, we have developed a 2, 1–44, 2011 simple kinetic expression, given by Eq. (4), to capture changes in community com- 5 position that govern biogeochemistry by using a single parameter, εi , that varies between 0 and 1. Investigating MEP in a simple two-box biogeochemistry model involving Biogeochemistry and two chemical reactions has revealed that globally optimized solutions generate higher maximum entropy entropy production rates than locally optimized solutions (Table 4), and there exists production many alternate steady state solutions that produce statistically similar rates of entropy J. J. Vallino 10 (Table 3). Results from globally optimized MEP solutions support hypotheses regarding the evolution of cooperation in biological systems and the beneﬁt of exchanging information over space to extract and dissipate all available free energy. Title Page

Abstract Introduction Appendix A Conclusions References Model equations Tables Figures The complete expansion of Eq. (12) for the 6 constituents in the two boxes with associated boundaries shown in Fig. 1 are given below. For box [1] constituents we have, J I

dCCH O[1] J I 2 = F C − F C /h[1] − r [1] (A1) dt 0,1 CH2O 1,2 CH2O 1 Back Close

dCO [1] 2 = F C −F C /h[1]−(1 − ε [1]) r [1] − (1−ε [1]) r [1] (A2) Full Screen / Esc dt 0,1 O2 1,2 O2 1 1 2 2 dC [1] Printer-friendly Version H2CO3 = F0,1 CH CO − F1,2 CH CO /h[1] (A3) dt 2 3 2 3 Interactive Discussion

+ (1 − ε1[1]) r1[1] + (1 − ε2[1]) r2[1] 23 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

D 16 Fi, j (Ck ) = − (Ck [ j] − Ck [i])βi, j (k) , (11) 

F | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion dS i, j ∆µ i, j D  C [i]  k = k = − −dCNH [1] k  22 Fi, j (Ck ) (Ck [ j] Ck [i])R3 ln  . (12) dt T =C [−j]F , C /h[1] − ρ ε [1] r [1] + ρ (1 − ε [1]) r [1] (A4)  dt  k 1 2 NH3 1 1 2 2 ESDD

23 Given concentrations of CH O, NH , O , H COdC, S [1]and S along with the growth 2, 1–44, 2011 2 3 2 2 3 1 = 2−F C /h[1] + ε [1] r [1] − δ [1] r [1] (A5) 1,2 1 1 1 1 2 24 efficiencies ε and ε in each box, entropy productiondt associated with reactions and mixing 1 2 Biogeochemistry and dC [1] 25 can be calculated from Eqs. (6-12). 2 maximum entropy = −F1,2 C /h[1] + (ε2[1] − δ2[1]) r2[1] (A6) dt 2 production For box [2] constituents, the equations are, J. J. Vallino dCCH O[2] 2 = F C /h[2] − r [2] (A7) dt 1,2 CH2O 1 9 Title Page dCO [2] 2 = F C /h[2] − (1 − ε [2]) r [2] − (1 − ε [2]) r [2] (A8) dt 1,2 O2 1 1 2 2 Abstract Introduction

dCH CO [2] Conclusions References 2 3 = F C /h[2] + (1 − ε [2]) r [2] + (1 − ε [2]) r (A9) dt 1,2 H2CO3 1 1 2 2 Tables Figures dC [2] NH3 = F1,2 CNH − F2,3 CNH /h[2] − ρ ε1[2] r1[2] (A10) dt 3 3 J I

+ ρ (1 − ε2[2]) r2[2] J I dC [2] 1 Back Close = F1,2 C /h[2] + ε1[2] r1[2] − δ1[2] r2[2] (A11) dt 1 Full Screen / Esc dC [2] 2 = F1,2 C /h[2] + (ε2[2] − δ2[2]) r2[2] (A12) dt 2 Printer-friendly Version

Fluxes, Fi,j (Ck ), where βi,j (k) in Eq. (11) equals zero have been removed from the Interactive Discussion above expressions. Parameter values, initial conditions plus boundary conditions used for all simulations are given in Tables 1 and 2, respectively. 24 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Acknowledgements. I thank Axel Kleidon for organizing the 2010 workshop on Non-Equilibrium Thermodynamics and MEP in the Earth system that lead to this manuscript. The work pre- ESDD sented here was funded by the NSF Advancing Theory in Biology Program (NSF EF-0928742), the NSF PIE-LTER program (NSF OCE-0423565) as well as from NSF CBET-0756562 and 2, 1–44, 2011 5 OCE-0852263. Biogeochemistry and References maximum entropy production Adami, C.: Sequence complexity in Darwinian evolution, Complexity, 8(2), 49–56, doi:10.1002/cplx.10071, 2002. J. J. Vallino Adami, C., Ofria, C., and Collier, T. C.: Evolution of biological complexity, P. Natl. Acad. Sci. 10 USA, 97(9), 4463–4468, 2000. Alberty, R. A.: Thermodynamics of biochemical reactions, Hoboken, NJ, Wiley & Sons, 397 pp., Title Page 2003. Alberty, R. A.: Biochemical thermodynamics: Applications of Mathematica, Hoboken, NJ, Wi- Abstract Introduction ley & Sons, 464 pp., 2006. 15 Bailey, J. E.: Biochemical engineering fundamentals, New York, NY, McGraw-Hill, 753 pp., Conclusions References 1977. Bain, R. S.: Solution of nonlinear algebraic equation systems; and, Single and multire- Tables Figures sponse nonlinear parameter estimation problems, Ph.D. Thesis/Dissertation, University of Wisconsin-Madison, 1993. J I 20 Bastolla, U., Fortuna, M. A., Pascual-Garcia, A., Ferrera, A., Luque, B., and Bascompte, J.: The architecture of mutualistic networks minimizes competition and increases biodiversity, J I Nature, 458, 1018–1020, doi:10.1038/nature07950, 2009. Battley, E.: Absorbed heat and heat of formation of dried microbial biomass: Studies Back Close on the thermodynamics of microbial growth, J. Therm. Anal. Calorim., 74(3), 709–721, Full Screen / Esc 25 doi:10.1023/B:JTAN.0000011003.43875.0d, 2003. Battley, E. H.: An empirical method for estimating the entropy of formation and the absolute entropy of dried microbial biomass for use in studies on the thermodynamics of microbial Printer-friendly Version growth, Thermochim. Acta, 326(1–2) 7–15, 1999a. Interactive Discussion Battley, E. H.: On entropy and absorbed thermal energy in biomass; a biologist’s perspective, 30 Thermochim. Acta, 331(1), 1–12, 1999b.

25 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Bejan, A.: Constructal theory of pattern formation, Hydrol. Earth Syst. Sci., 11, 753–768, doi:10.5194/hess-11-753-2007, 2007. ESDD Boudart, M.: Consistency between kinetics and thermodynamics, J. Phys. Chem., 80(26), 2869–2870, 1976. 2, 1–44, 2011 5 Brock, W. A. and Carpenter, S. R.: Interacting regime shifts in ecosystems: implication for early warnings, Ecol. Monogr., 80(3), 353–367, 2010. Brugnano, L. and Magherini, C.: The BiM code for the numerical solution of ODEs, J. Comput. Biogeochemistry and Appl. Math., 164–165, 145–158, doi:10.1016/j.cam.2003.09.004, 2004. maximum entropy Camilli, A. and Bassler, B. L.: Bacterial Small-Molecule Signaling Pathways, Science, production 10 311(5764), 1113–1116, 2006. Carlson, C. A., Bates, N. R., Ducklow, H. W., and Hansell, D. A.: Estimation of bacterial respira- J. J. Vallino tion and growth efficiency in the Ross Sea, Antarctica, Aquat. Microb. Ecol., 19(3), 229–244, doi:10.3354/ame019229, 1999. Clutton-Brock, T.: Cooperation between non-kin in animal societies, Nature, 462(7269), 51–57, Title Page 15 doi:10.1038/nature08366, 2009. Del Giorgio, P. A. and Cole, J. J.: Bacterial growth efficiency in natural aquatic systems, Annu. Abstract Introduction Rev. Ecol. Syst., 29, 503–541, 1998. DeLong, E. F.: The microbial ocean from genomes to biomes, Nature, 459(7244), 200–206, Conclusions References doi:10.1038/nature08059, 2009. Tables Figures 20 Dewar, R.: Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states, J. Phys. A, 36, 631– 641, 2003. J I Dewar, R.: Maximum Entropy Production as an Inference Algorithm that Translates Physical Assumptions into Macroscopic Predictions: Don’t Shoot the Messenger, Entropy, 11(4), 931– J I 25 944, doi:10.3390/e11040931, 2009. Back Close Dewar, R. C.: Maximum entropy production and the fluctuation theorem, J. Phys. A, 38(21), L371–L381, doi:10.1088/0305-4470/38/21/L01, 2005. Full Screen / Esc Dyke, J. and Kleidon, A.: The Maximum Entropy Production Principle: Its Theoretical Founda- tions and Applications to the Earth System, Entropy, 12(3), 613–630, 2010. Printer-friendly Version 30 Edwards, A. M. and Yool, A.: The role of higher predation in plankton population models, J. Plankton Res., 22(6), 1085–1112, doi:10.1093/plankt/22.6.1085, 2000. Interactive Discussion

26 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Elser, J. J., Sterner, R. W., Gorokhova, E., Fagan, W. F., Markow, T. A., Cotner, J. B., Harrison, J. F., Hobbie, J. E., Odell, G. M., and Weider, L. W.: Biological stoichiometry from genes to ESDD ecosystems, Ecol. Lett., 3(6), 540–550, 2000. Elser, J. J., Bracken, M. E. S., Cleland, E. E., Gruner, D. S., Harpole, W. S., Hillebrand, H., 2, 1–44, 2011 5 Ngai, J. T., Seabloom, E. W., Shurin, J. B., and Smith, J. E.: Global analysis of nitrogen and phosphorus limitation of primary producers in freshwater, marine and terrestrial ecosystems, Ecol. Lett., 10(12), 1135–1142, doi:10.1111/j.1461-0248.2007.01113.x, 2007. Biogeochemistry and Eu, B. C.: Kinetic theory and irreversible thermodynamics, Montreal, John Wiley & Sons maximum entropy Canada, Ltd., 752 pp., 1992. production 10 Falkowski, P. G. and Oliver, M. J.: Mix and match: how climate selects phytoplankton, Nat. Rev. Micro., 5(10), 813–819, doi:10.1038/nrmicro1751, 2007. J. J. Vallino Falkowski, P. G., Fenchel, T., and DeLong, E. F.: The Microbial Engines That Drive Earth’s Bio- geochemical Cycles, Science, 320(5879), 1034–1039, doi:10.1126/science.1153213, 2008. Fath, B. D., Patten, B. C., and Choi, J. S.: Complementarity of Ecological Goal Functions, J. Title Page 15 Theor. Biol., 208(4), 493–506, 2001. Ferenci, T.: “Growth of bacterial cultures” 50 years on: towards an uncertainty princi- Abstract Introduction ple instead of constants in bacterial growth kinetics, Res. Microbiol., 150(7), 431–438, doi:10.1016/S0923-2508(99)00114-X., 1999. Conclusions References Ferguson, B. A., Dreisbach, T. A., Parks, C. G., Filip, G. M., and Schmitt, C. L.: Coarse-scale Tables Figures 20 population structure of pathogenic Armillaria species in a mixed-conifer forest in the Blue Mountains of northeast Oregon, Can. J. Forest Res., 33(4), 612–623, doi:10.1139/X02-165, 2003. J I Fernandez, A., Huang, S., Seston, S., Xing, J., Hickey, R., Criddle, C., and Tiedje, J.: How Stable Is Stable? Function versus Community Composition, Appl. Environ. Microbiol., 65(8), J I 25 3697–3704, 1999. Back Close Fernandez, A. S., Hashsham, S. A., Dollhopf, S. L., Raskin, L., Glagoleva, O., Dazzo, F. B., Hickey, R. F., Criddle, C. S., and Tiedje, J. M.: Flexible community structure correlates with Full Screen / Esc stable community function in methanogenic bioreactor communities perturbed by glucose, Appl. Microbiol. Biotechnol., 66(9), 4058–4067, 2000. Printer-friendly Version 30 Follows, M. J., Dutkiewicz, S., Grant, S., and Chisholm, S. W.: Emergent Biogeog- raphy of Microbial Communities in a Model Ocean, Science, 315(5820), 1843–1846, Interactive Discussion doi:10.1126/science.1138544, 2007.

27 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Frazier, M. E., Johnson, G. M., Thomassen, D. G., Oliver, C. E., and Patrinos, A.: Realizing the Potential of the Genome Revolution: The Genomes to Life Program, Science, 300(5617), ESDD 290–293, 2003. Gaidos, E., Dubuc, T., Dunford, M., McAndrew, P., Padilla-Gamino, J., Studer, B., Weersing, 2, 1–44, 2011 5 K., and Stanley, S.: The Precambrian emergence of animal life: a geobiological perspective, Geobiology, 5(4), 351–373, doi:10.1111/j.1472-4669.2007.00125.x, 2007. Gianoulis, T. A., Raes, J., Patel, P.V., Bjornson, R., Korbel, J. O., Letunic, I., Yamada, T., Pacca- Biogeochemistry and naro, A., Jensen, L. J., Snyder, M., Bork, P., and Gerstein, M. B.: Quantifying environmental maximum entropy adaptation of metabolic pathways in metagenomics, P. Natl. Acad. Sci., 106(5), 1374–1379, production 10 doi:10.1073/pnas.0808022106, 2009. Goodnight, C. J.: Experimental studies of community evolution II: The ecological basis of the J. J. Vallino response to community selection, Evolution, 44(6), 1625–1636, 1990. He, J., Watson, L. T., and Sosonkina, M.: Algorithm 897: VTDIRECT95: Serial and Paral- lel Codes for the Global Optimization Algorithm DIRECT, Am. T. Math. Softw., 36(3), 17, Title Page 15 doi:10.1145/1527286.1527291, 2009. Heil, M. and Karban, R.: Explaining evolution of plant communication by airborne signals, Abstract Introduction Trends Ecol. Evol., 25(3), 137–144, doi:10.1016/j.tree.2009.09.010, 2010. Hillesland, K. L. and Stahl, D. A.: Rapid evolution of stability and productivity at the origin of a Conclusions References microbial mutualism, P. Natl. Acad. Sci., 107(5), 2124–2129, doi:10.1073/pnas.0908456107, Tables Figures 20 2010. Holling, C. S.: The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entom. Soc. Can., 45, 1–60, 1965. J I Ianson, D. and Allen, S. E.: A two-dimensional nitrogen and carbon ﬂux model in a coastal upwelling region, Global Biogeochem. Cy., 16(1), 1011, doi:10.1029/2001GB001451, 2002. J I 25 Jaynes, E. T.: Probability theory: The logic of science, Cambridge, Cambridge University Press, Back Close 758 pp., 2003. Jin, Q. and Bethke, C. M.: A New Rate Law Describing Microbial Respiration, Appl. Environ. Full Screen / Esc Microbiol., 69(4), 2340–2348, 2003. Jones, D. R., Perttunen, C. D., and Stuckman, B. E.: Lipschitzian optimization without the Printer-friendly Version 30 Lipschitz constant, J. Optimiz. Theory App., 79(1), 157–181, 1993. Jorgensen, S. E.: Review and comparison of goal functions in system ecology, Vie Milieu, Interactive Discussion 44(1), 11–20, 1994.

28 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Jorgensen, S. E., Patten, B. C., and Straskraba, M.: Ecosystems emerging: 4. growth, Ecol. Modell., 126(2–3), 249–284, 2000. ESDD Kearfott, R. B.: Algorithm 763: Interval Arithmetic: A Fortran 90 Module for An Interval Data Type, Am. T. Math. Softw., 22(4), 385–392, 1996. 2, 1–44, 2011 5 Kearfott, R. B. and Novoa, M.: Algorithm 681: INTBIS, A Portable Interval Newton Bisection Package, Am. T. Math. Softw., 16(2), 152–157, 1990. Keller, E. F.: The century beyond the gene, J. Biosci., 30(1), 3–10, 2005. Biogeochemistry and Keller, L. and Surette, M. G.: Communication in bacteria: an ecological and evolutionary per- maximum entropy spective, Nat. Rev. Micro., 4(4), 249–258, doi:10.1038/nrmicro1383, 2006. production 10 Kleidon, A. and Lorenz, R. D.: Non-equilibrium thermodynamics and the production of entropy, Springer-Verlag, Berlin, 260 pp., 2005a. J. J. Vallino Kleidon, A. and Lorenz, R.: Entropy production by earth system processes, in: Non-equilibrium thermodynamics and the production of entropy: life, earth and beyond, edited by: Kleidon, A. and Lorenz, R. D., Springer-Verlag, Berlin, 1–20, 2005b. Title Page 15 Kleidon, A., Fraedrich, K., Kunz, T., and Lunkeit, F.: The atmospheric circula- tion and states of maximum entropy production, Geophys. Res. Lett., 30(23), 1–4, Abstract Introduction doi:10.1029/2003GL018363, 2003. Kondepudi, D. and Prigogine, I.: Modern thermodynamics: From heat engines to dissipative Conclusions References structures, New York, Wiley & Sons, 486 pp., 1998. Tables Figures 20 Kozliak, E. I. and Lambert, F. L.: “Order-to-Disorder” for entropy change? Consider the num- bers!, Chem. Educator, 10, 24–25, 2005. Lambert, F. L.: Shuﬄed cards, messy desks, and disorderly dorm rooms – Examples of entropy J I increase? Nonsense!, J. Chem. Educ., 76, 1385–1387, 1999. Lendenmann, U. and Egli, T.: Kinetic models for the growth of Escherichia coli with J I 25 mixtures of sugars under carbon-limited conditions, Biotechnol. Bioeng., 59(1, 99–107, Back Close doi:10.1002/(SICI)1097-0290(19980705)59:1<99::AID-BIT13>3.0.CO;2-Y, 1998. Lineweaver, C. H. and Egan, C. A.: Life, gravity and the second law of thermodynamics, Phys. Full Screen / Esc Life Rev., 5,(4) 225–242, doi:10.1016/j.plrev.2008.08.002, 2008. Lorenz, R.: Computational mathematics: Full Steam Ahead-Probably, Science, 299(5608), Printer-friendly Version 30 837–838, 2003. Lorenz, R. D., Lunine, J. I., and Withers, P. G.: Titan, Mars and Earth: Entropy production by Interactive Discussion latitudinal heat transport, Geophys. Res. Lett., 28(3), 415–418, 2001.

29 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Lotka, A. J.: Contribution to the Energetics of Evolution, P. Natl. Acad. Sci., 8(6), 147–151, 1922. ESDD Maherali, H. and Klironomos, J. N.: Influence of Phylogeny on Fungal Com- munity Assembly and Ecosystem Functioning, Science, 316(5832), 1746–1748, 2, 1–44, 2011 5 doi:10.1126/science.1143082, 2007. Margalef, R.: Perspectives in ecological theory, Chicago, University of Chicago Press, 112 pp., 1968. Biogeochemistry and Meysman, F. J. R. and Bruers, S.: A thermodynamic perspective on food webs: Quantify- maximum entropy ing entropy production within detrital-based ecosystems, J. Theor. Biol., 249(1), 124–139, production 10 doi:10.1016/j.jtbi.2007.07.015, 2007. Monod, J.: The growth of bacterial cultures, Annu. Rev. Microbiol., 3, 371–394, 1949. J. J. Vallino Morowitz, H. J.: Energy flow in biology: biological organization as a problem in thermal physics, New York, Academic Press, 179 pp., 1968. Mumby, P. J.: Phase shifts and the stability of macroalgal communities on Caribbean coral Title Page 15 reefs, Coral Reefs, 28(3), 761–773, 2009. Niven, R. K.: Steady state of a dissipative flow-controlled system and the maximum entropy pro- Abstract Introduction duction principle, Phys. Rev. E, 80(2), 021113–021115, doi:10.1103/PhysRevE.80.021113, 2009. Conclusions References Nowak, M. A.: Five Rules for the Evolution of Cooperation, Science, 314(5805), 1560–1563, Tables Figures 20 doi:10.1126/science.1133755, 2006. Odum, H. T. and Pinkerton, R. C.: Time’s speed regulator: the optimum efficiency for maximum power output in physical and biological systems, Am. Sci., 43, 321–343, 1955. J I Paltridge, G. W.: Global dynamics and climate-a system of minimum entropy exchange, Q. J. Roy. Meteor. Soc., 104, 927–945, 1975. J I 25 Pianka, E. R.: R-Selection and K-Selection, Am. Nat., 104(940), 592–597, 1970. Back Close Price, J. E. and Morin, P. J.: Colonization history determines alternate community states in a food web of intraguild predators, Ecology, 85(4) 1017–1028, doi:10.1890/03-0157, 2004. Full Screen / Esc Prigogine, I. and Nicolis, G.: Biological order, structure and instabilities, Q. Rev. Biophys., 4, 107–148, 1971. Printer-friendly Version 30 Schneider, E. D. and Kay, J. J.: Complexity and thermodynamics: towards a new ecology, Futures, 26(6), 626–647, 1994. Interactive Discussion Schroder,¨ A., Persson, L., and De Roos, A. M.: Direct experimental evidence for alternative stable states: a review, Oikos, 110(1), 3–19, doi:10.1111/j.0030-1299.2005.13962.x, 2005. 30 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Schrodinger,¨ E.: What is life?, Cambridge, UK, Cambridge University Press, 1944. Shannon, C. E.: A mathematical theory of communication, Bell System Technical Journal, 27, ESDD 379-423-623-656, 1948. Shapiro, J. A.: Thinking about bacterial populations as multicellular organisms, Annu. Rev. 2, 1–44, 2011 5 Microbiol., 52, 81–104, 1998. Toussaint, O. and Schneider, E. D.: The thermodynamics and evolution of complexity in biological systems, Comp. Biochem. Physiol. A, 120(1), 3–9, 1998. Biogeochemistry and Traulsen, A. and Nowak, M. A.: Evolution of cooperation by multilevel selection, P. Natl. Acad. maximum entropy Sci, 103(29), 10952–10955, doi:10.1073/pnas.0602530103, 2006. production 10 Ulanowicz, R. E. and Platt, T.: Ecosystem theory for biological oceanography, Ottawa, Can. Bull. Fish. Aquat. Sci., 213, 260 pp., 1985. J. J. Vallino Vallino, J. J.: Ecosystem biogeochemistry considered as a distributed metabolic network ordered by maximum entropy production. Philos. T. Roy. Soc. B, 365(1545), 1417–1427, doi:10.1098/rstb.2009.0272, 2010. Title Page 15 van Gemerden, H.: Microbial mats: A joint venture, Mar. Geol., 113(1–2), 3–25, doi:10.1016/0025-3227(93)90146-M, 1993. Abstract Introduction Vandenkoornhuyse, P., Mahe, S., Ineson, P., Staddon, P., Ostle, N., Cliquet, J. B., Francez, A. J., Fitter, A. H., and Young, J. P.: Active root-inhabiting microbes identified by rapid in- Conclusions References corporation of plant-derived carbon into RNA, P. Natl. Acad. Sci., 104(43), 16970–16975, Tables Figures 20 doi:10.1073/pnas.0705902104, 2007. Weber, B. H., Depew, D. J., and Smith, J. D.: Entropy, information, and evolution: New perspective on physical and biological evolution, MIT Press, Cambridge, MA, 390 pp., 1988. J I Whitfield, J.: Fungal roles in soil ecology: Underground networking, Nature, 449(7159), 136– 138, doi:10.1038/449136a, 2007. J I 25 Wilson, D. S. and Wilson, E. O.: Evolution “for the Good of the Group”, Am. Sci., 96(5), 380– Back Close 389, doi:10.1511/2008.74.1, 2008. Full Screen / Esc

Printer-friendly Version

Interactive Discussion

31 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

11 Entropy of mixing is readily calculated between boxes and boundaries based on flux and 12 chemical potential obtained from concentration differences between boxes (Meysman and 13 Bruers 2007, Kondepudi and Prigogine 1998). For our simple model, we only permit icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 14 diffusive transport between boxes and boundaries, so a material flux of chemical species k 15 from box or boundary [i] to box or boundary [j] is given by (mmol m-2 d-1), ESDD 2, 1–44, 2011 D 16 Fi, j (Ck ) = − (Ck [ j] − Ck [i])βi, j (k) , (11)  Biogeochemistry and × -5 17 where  is a characteristic length scale for diffusion (2.5 10 m), D is the diffusion maximum entropy -4 2 -1 18 coefficient (1×10 m d ) and βi, j (k) is 1 if flux of compound k is allowed between box or production Table 1. Model fixed parameter values. 19 boundary [i] and [j]; otherwise it is 0. The product of flux and chemical potential differences, J. J. Vallino 20 ∆µ i, j , divided by temperature gives the entropy production per unit area for mixing between k Parameter Description Value (units) -2 -1 -1 21 box [i] to [j] for chemical∗ species k, as follows (J m d ºK ) −1 ν Universal kinetic parameter 350. (d ) Title Page κ∗ Universal kinetic parameter 5000. (mmol m−3) Fi, j i, j ∆µ   −4 2 −1 dS k D k DiDffusion coefficientCk [i] 1 × 10 (m d ) Abstract Introduction 22 = Fi, j (Ck ) = − (Ck [ j] − Ck [i])R ln  . (12)   −5 dt ` T Characteristic length Ck [ j] 2.5 × 10 (m) h[1], h[2] Depth of boxes [1] and [2] 10. (m) Conclusions References 23 Given concentrationsρ of CH2O, NHN3, toO2 C, H ratio2CO of3, Si1 and S2 along with the growth 1/6 (atomic) T Temperature 293.15 (◦K) Tables Figures ε ε 24 efficiencies 1 and pH2 in each box, pH entropy production associated with reactions and mixing 8.1 IS Ionic Strength 0.7 (M) 25 can be calculated from Eqs.◦ (6-12). −1 J I ∆GOx Standard Gibbs free energy of CH2O oxidation to H2CO3 −498.4 (J mmol ) G◦ −1 ∆ Standard Gibbs free energy of S synthesis from CH2O and NH3 0. (J mmol ) J I

Back Close

9 Full Screen / Esc

Printer-friendly Version

Interactive Discussion

32 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

17 where  is a characteristic length scale for diffusion (2.5×10-5 m), D is the diffusion 2, 1–44, 2011 -4 2 -1 18 coefficient (1×10 m d ) and βi, j (k) is 1 if flux of compound k is allowed between box or Biogeochemistry and 19 boundary [i] and [j]; otherwise it is 0. The product of flux and chemical potential differencesmaximum, entropy

i, j production 20 ∆µk , divided by temperature gives the entropy production per unit area for mixing between 21 box [i] to [j] for chemical species k, as follows (J m-2 d-1 ºK-1) J. J. Vallino Table 2. Model initial conditions (for boxes [1] and [2]) and Dirichlet boundary conditions (for [0] and [3]) used for all model simulations.F dS i, j ∆µ i, j D  C [i]  k = k = − −  k  22 Fi, j (Ck ) (Ck [ j] Ck [i])R ln  . (12) Title Page dt T  −3 C [ j] Box or Variable Concentrations (mmol m )  k  Abstract Introduction Boundary CCH O [i] CO [i] CH CO [i] CNH [i] C [i] C [i] 23 Given concentrations2 2 of CH22O,3 NH3, O2, 3H2CO3, S1 and S2 along2 with the growth Conclusions References [0] 100. 290. 2000. – – – 24 efficiencies ε1 and ε 2 in each box, entropy production associated with reactions and mixing [1] 100. 290. 2000. 1.0 0.1 0.1 Tables Figures [2]25 can be 100. calculated 290. from Eqs. 2000. (6-12). 1.0 0.1 0.1 [3] –* – – 1.0 – – J I

* Indicates variable is not allowed to cross boundary (i.e., βi,j (k) = 0 in Eq. 11). J I

Back Close 9 Full Screen / Esc

Printer-friendly Version

Interactive Discussion

ESDD 2, 1–44, 2011

Table 3. Values in 4-D εi [j]-space that correspond to the highest entropy production −2 −1 ◦ −1 Biogeochemistry and (J m d K ) by reactions (r ) and (r ) summed over boxes [1] and [2] from the 100 000 ran- 1 2 maximum entropy dom εi [j]-point simulations. production ε ε ε ε r 1 [1] 2 [1] 1 [2] 2 [2] dS /dt J. J. Vallino 0.14046 0.019324 0.21967 0.99543 621.15 0.12696 0.020367 0.80546 0.98968 620.93 0.18904 0.040613 0.88768 0.99995 620.86 Title Page 0.12239 0.063907 0.99076 0.013594 620.85 0.10072 0.043317 0.7629 0.98881 620.84 Abstract Introduction 0.10663 0.059295 0.64977 0.038126 620.73 0.088998 0.029534 0.35951 0.9788 620.72 Conclusions References 0.1354 0.034642 0.7582 0.99138 620.70 0.10056 0.061161 0.59806 0.9968 620.69 Tables Figures 0.10912 0.05965 0.3372 0.99783 620.64 0.1313 0.054689 0.62598 0.011278 620.62 J I 0.38034 0.011359 0.58493 0.0005084 620.51 0.096701 0.074526 0.20127 0.0053228 620.48 J I 0.10842 0.071069 0.5166 0.024946 620.45 Back Close 0.23553 0.029255 0.90925 0.0095011 620.36 0.09848 0.02539 0.11833 0.97672 620.28 Full Screen / Esc 0.082633 0.059068 0.35479 0.0277 620.11

Printer-friendly Version

Interactive Discussion

ESDD 2, 1–44, 2011

Biogeochemistry and maximum entropy production

J. J. Vallino Table 4. Global and local MEP solutions (J m−2 d−1 ◦ K−1) based on solution to Eqs. (14) and (15), respectively.

Title Page Variable Global Eq. (15) Local Eq. (14) Abstract Introduction ε1 [1] 0.1312 0.0992 ε2 [1] 0.0123 0.0356 Conclusions References ε1 [2] 0.3621 0.1005 ε2 [2] 0.9957 0.1106 Tables Figures dSr [1]/dt 619.6 224.4 dSr [2]/dt 1.92 204.9 J I dSr [1]/dt + dSr [2]/dt 621.5 429.7 J I

Back Close

Full Screen / Esc

Printer-friendly Version

Interactive Discussion

35 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

11 Entropy of mixing is readily calculated between boxes and boundaries based on flux and 12 chemical potential obtained from concentration differences between boxes (Meysman and

13 Bruers 2007, Kondepudi and Prigogine 1998). For our simple model, we only permit | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 14 diffusive transport between boxes and boundaries, so a material flux of chemical species k ESDD -2 -1 15 from box or boundary [i] to box or boundary [j] is given by (mmol m d ), 2, 1–44, 2011

D 16 Fi, j (Ck ) = − (Ck [ j] − Ck [i])βi, j (k) , (11)  Biogeochemistry and maximum entropy 17 where  is a characteristic length scale for diffusion (2.5×10-5 m), D is the diffusion production -4 2 -1 18 coefficient (1×10 m d ) and βi, j (k) is 1 if flux of compound k is allowed between box or J. J. Vallino Table 5. Steady state concentration (mmol m−3) of state variables at maximum entropy produc- 19 boundary [i] and [j]; otherwisetion associated it is 0. The with product local of and flux global and solutions.chemical potential differences,

20 ∆µ i, j , divided by temperature gives the entropy production per unit area for mixing between Title Page k Box [1] Box [2] -2 -1 -1 21 box [i] to [j] for chemical species k, as follows (J mVariable d ºK ) Global Local Global Local Abstract Introduction

Fi, j i, j CH O 0.568 32.3 0.336 2.92 Conclusions References dS ∆µ D 2  C [i]  k = k = − − O  k 190.6 222.3 190.3 189.9 22 Fi, j (Ck ) (Ck [ j] Ck [i])2R ln  . (12) dt T  C [ j] Tables Figures H2CO3 k 2099. 2068. 2100. 2100. NH3 0.996 0.489 1.00 1.00 S S 60.2 9.05 52.1 5.97 23 Given concentrations of CH2O, NH3, O2, H2CO3, 1 and 2 along with the growth J I 98.7 0.686 106.7 0.712 24 efficiencies ε and ε in each box, entropy production2 associated with reactions and mixing 1 2 J I 25 can be calculated from Eqs. (6-12). Back Close

Full Screen / Esc

Printer-friendly Version

9 Interactive Discussion

CH2O[0], O2[0], H2CO3[0] ESDD 2, 1–44, 2011 F0,1(Ck ) Biogeochemistry and maximum entropy CH2O[1], O2[1], production H2CO3[1], NH3[1] J. J. Vallino S1[1], S2[1]

Title Page F1,2 (Ck ) Abstract Introduction

Conclusions References CH2O[2], O2[2], Tables Figures H2CO3[2], NH3[2] S [2], S [2] 1 2 J I F (C ) J I 2,3 k Back Close

NH3[3] Full Screen / Esc

Fig. 1. Schematic of the two box model with associated state variables and boundary condi- Printer-friendly Version tions. All fluxes between boxes and boundaries are governed by diffustion (see Eq. 11) and Interactive Discussion only CH2O, O2 and H2CO3 are exchanged accross boundary [0] and box [1]. Similarly, only NH3 is allowed across box [2] and boundary [3]. Boundary concentrations are held at fix values throughout the simulation (see Table 2). 37 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

50 ESDD 0.7 2, 1–44, 2011

0.6

) 40 Biogeochemistry and -1 maximum entropy 0.8 (d production μ J. J. Vallino 30 0.5

Title Page 0.9 20 Abstract Introduction 0.4 Conclusions References

Tables Figures 10

Specific Growth Rate, 0.3 J I

J I 0.15 0 1.0 Back Close

0 1000 2000 3000 4000 5000 Full Screen / Esc Substrate Concentration (mmol m-3) Printer-friendly Version Fig. 2. Speciﬁc growth rate as a function of substrate concentrations based on Eqs. (4) and (5) for diﬀerent values for ε (0.15, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.8, 1.) using the universal kinetic Interactive Discussion parameters values of 350 d−1 and 5000 mmol m−3 for ν∗ and κ∗, respectively.

-3 300 -3 -3 100 CH2O (mmol m ) (a) O2 (mmol m ) (b) H2CO3 (mmol m ) (c) 280 ESDD 80 2100 260 2, 1–44, 2011 60 240

40 220 2050 Biogeochemistry and 20 200 maximum entropy 180 0 2000 0 250 500 0 250 500 0 250 500 production Time (d) Time (d) Time (d) 120 NH (mmol m-3) S (mmol m-3) S (mmol m-3) 2 3 (d) 1 (e) 4 2 (f) J. J. Vallino 100

1.5 80 3

60 1 2 Title Page 40 0.5 1 Abstract Introduction 20

0 0 0 Conclusions References 0 250 500 0 250 500 0 250 500 Time (d) Time (d) Time (d) 700 30 dSr/dt (J m-2 d-1 °K) (g) dSF/dt (J m-2 d-1 °K) (h) Tables Figures 600

500 20 J I 400 300 10 J I 200 100 Back Close 0 0 0 250 500 0 250 500 Time (d) Time (d) Full Screen / Esc

Fig. 3. Concentration of state variables (a–f) and entropy production from reactions (g) and Printer-friendly Version mixing (h) for the point (ε1 [1], ε2 [1], ε1 [2], ε2 [2]) = (0.1, 0.05, 0.1, 0.05) based on Eq. (13). In (a–g), green line represents variables in box [1], while red line is for variables in box [2]. Interactive Discussion The green, blue and red lines in (h) correspond to entropy of mixing between boundary [0] and box [1], box [1] and box [2], and box [2] and boundary [3], respectively. 39 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

ESDD 2, 1–44, 2011

Biogeochemistry and maximum entropy production

J. J. Vallino

Title Page

Abstract Introduction

Conclusions References

Tables Figures

J I

Back Close

Full Screen / Esc

Printer-friendly Version

Fig. 4. Extracted results from 100 000 simulations with randomly selected εi [j]. (a) Com- Interactive Discussion parison of total entropy of mixing from Eq. (12) as a function of total entropy of reaction from Eq. (8). (b) Total steady state N standing stock (NTotal) given by Eq. (16) as a function of entropy production from reactions. 40 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Total N -2 0.02 (mmol m ) ESDD 1.2E+06 2, 1–44, 2011 8.0E+05 4.0E+05 2.0E+05 Biogeochemistry and 0.015 1.0E+05 8.0E+04 maximum entropy 5.0E+04 production 4.0E+04

[2] 3.0E+04

2 J. J. Vallino

ε 2.0E+04 0.01 1.5E+04 1.0E+04

[1] or Title Page 2 ε Abstract Introduction 0.005 Conclusions References

Tables Figures

0 J I

0 0.20.4 0.6 0.8 1 J I ε ε 1[1] or 1[2] Back Close

Fig. 5. Points (triangles and squares) in 4-D εi [j]-space that resulted in total steady state N Full Screen / Esc 4 −2 standing stock (NTotal) greater than 10 mmol m . Triangles mark locations and magnitude of N standing stock for ε and ε in box [1], while squares mark locations and N standing stock 1 2 Printer-friendly Version for ε1 and ε2 in box [2]. Also shown as small points are a subset of the ε1 and ε2 in box [1] (green) and box [2] (red) from the 100 000 randomly selected εi [j]-point simulations that have Interactive Discussion 4 −2 NTotal < 10 mmol m . Note, the y-axis range from 0 to 0.021 captures all points that meet the 4 −2 minimum NTotal requirement of 10 mmol m . 41 -1 -1 ° 1 where R is the ideal gas constant (in J mmol ºK ), ∆GCH 2Ox is the standard Gibbs free

-1 ° 2 energy of CH2O oxidation by O2 to H2CO3 (-498.4 J mmol ) and ∆GS is the standard Gibbs

3 free energy for biological structure synthesis from CH2O and NH3, which is set to zero for our 4 study because it is negligibly small (Vallino 2010). We use the approach of Alberty (2003,

° 5 2006) to calculate the standard Gibbs free energy of reaction, ∆GCH 2Ox , which accounts for

+ | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 6 proton dissociation equilibria between chemical species (CO2 + H2O ↔ H2CO3 ↔ H + (a) - 800 7 HCO3 , etc.) at a pH of 8.1 and temperature of 293.15ºK. Ionic strength (IS =0.7 M) is used to 700 ESDD

) 600 -1 K

8 estimate activity coefficients, so that concentrations can be ° used in the logarithmic correction 500 -1 2, 1–44, 2011 d 9 terms of (9) and (10) instead of activities (Alberty 2003). -2 400 300 /dt (J m /dt r 200 10 dS Biogeochemistry and 100 maximum entropy 0 0 100 200 300 400 500 11 Entropy of mixing is readily calculated between boxes and boundaries basedTime on (d) flux and production

(b) 12 chemical potential obtained from concentration differences between300 boxes (Meysman and J. J. Vallino 13 Bruers 2007, Kondepudi and Prigogine 1998). For our simple250 model, we only permit ) -3 200 14 diffusive transport between boxes and boundaries, so a material flux of chemical species k 150 Title Page -2 -1

15 from box or boundary [i] to box or boundary [j] is given by[1] (mmol m (mmol100 m d ), S1

C Abstract Introduction 50 D = − ( − )β 0 Conclusions References 16 Fi, j (Ck ) Ck [ j] Ck [i] i, j (k) , 0 100 200 300 400 (11)500  Time (d) Tables Figures 5 (c) 17 where  is a characteristic length scale for diffusion (2.5×10-5 m), D is the diffusion 4

-4 2 -1 ) -3 18 coefficient (1×10 m d ) and βi, j (k) is 1 if flux of compound k is allowed between box or J I 3

19 boundary [i] and [j]; otherwise it is 0. The product of flux and2 chemical potential differences, J I [1] m (mmol NH3 i, j C 1 Back Close 20 ∆µk , divided by temperature gives the entropy production per unit area for mixing between -2 -1 -1 0 21 box [i] to [j] for chemical species k, as follows (J m d ºK ) 0 100 200 300 400 500 Full Screen / Esc Time (d)

Fi, j i, j dS ∆µFig. 6. DTransient data for the C top[i] 17 entropy producing solutions from the 100 000 random- Printer-friendly Version 22 k = F (C ) k = − (C [ j] − C [i])R ln k  . (12) i, j k point simulations.k (a)k Total reaction entropy production from both boxes [1] and [2] whose dt T   Ck [ j] steady state values range from 620.11 to 621.15 J m−1 d−1 ◦ K−1. (b) Concentration of biological Interactive Discussion structure in box [1], C , and (c) NH3 over the ﬁrst 500 days of simulation. Values of εi [j] for 23 Given concentrations of CH2O, NH3, O2, H2CO3, S1 and S2 along with the growth the above simulations are given in Table 3. 24 efficiencies ε1 and ε 2 in each box, entropy production associated with reactions42 and mixing 25 can be calculated from Eqs. (6-12).

ESDD 2, 1–44, 2011

Biogeochemistry and maximum entropy production

J. J. Vallino

Title Page

Abstract Introduction

Conclusions References

Tables Figures

J I

Back Close

Full Screen / Esc

Printer-friendly Version Fig. 7. Reaction associated entropy production in (a) box [1] as a function ε1 and ε2 in box [1] for a ﬁxed value of ε and ε in box [2] of 0.3621 and 0.9957, respectively, and (b) in box [2] 1 2 Interactive Discussion as a function ε1 and ε2 in box [2] for a ﬁxed value of ε1 and ε2 in box [1] of 0.1312 and 0.0123, respecitvely. Light gray points in (a) and (b) dentote the position of the global MEP solution in boxes [1] and [2] based on Eq. (15). 43 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

ESDD 2, 1–44, 2011

Biogeochemistry and maximum entropy production

J. J. Vallino

Title Page

Abstract Introduction

Conclusions References

Tables Figures

J I

Back Close

Full Screen / Esc

Printer-friendly Version Fig. 8. Reaction associated entropy production in (a) box [1] as a function ε1 and ε2 in box [1] for a ﬁxed value of ε andε in box [2] of 0.1005 and 0.1106, respectively, and (b) in box [2] as 1 2 Interactive Discussion a function ε1 and ε2 in box [2] for a ﬁxed value of ε1 and ε2 in box [1] of 0.0992 and 0.0356, respecitvely. Light gray points in (a) and (b) dentote the position of the local MEP solutions in boxes [1] and [2] based on Eq. (14). 44