Biogeochemistry and maximum entropy production - Semantic Scholar

Earth System Dynamics Discussions

Received: 31 December 2010 – Accepted: 13 January 2011 – Published: 19 January 2011

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Marine Biological Laboratory, Woods Hole, Massachusetts, USA

Correspondence to: J. J. Vallino ([email protected])

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Published by Copernicus Publications on behalf of the European Geosciences Union.

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ESDD 2, 1–44, 2011

Biogeochemistry and maximum entropy production J. J. Vallino

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J. J. Vallino

Discussion Paper

Differences and implications in biogeochemistry from maximizing entropy production locally versus globally

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

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Earth Syst. Dynam. Discuss., 2, 1–44, 2011 www.earth-syst-dynam-discuss.net/2/1/2011/ doi:10.5194/esdd-2-1-2011 © Author(s) 2011. CC Attribution 3.0 License.

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1 Introduction

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There is a long history of research that attempts to understand and model ecosystems 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., ¨ Morowitz, 1968; Schrodinger, 1944; Odum and Pinkerton, 1955; Margalef, 1968; Prigogine 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)

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ESDD 2, 1–44, 2011


Title Page Abstract

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In this manuscript we investigate the use of the maximum entropy production (MEP) principle for modeling biogeochemical processes that are catalyzed by living systems. Because of novelties introduced by the MEP approach, many questions need to be answered and techniques developed in the application of MEP to describe biological systems that are responsible for energy and mass transformations on a planetary scale. In previous work we introduce the importance of integrating entropy production over time to distinguish abiotic from biotic processes under transient conditions. Here we investigate the ramifications of modeling biological systems involving one or more spatial 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 ocean models to illustrate the differences in local versus global entropy maximization. Synthesis and oxidation of biological structure is modeled using two autocatalytic reactions that account for changes in community kinetics using a single parameter each. Our results show that entropy production can be increased if maximized over the system 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.

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ESDD 2, 1–44, 2011


Title Page Abstract

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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 is an inherently thermodynamic-like approach. Thermodynamics appears particularly relevant for understanding biogeochemistry as it is largely under microbial control, or 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 understanding organismal metabolism in an ecosystem context, it is rather surprising that we still lack an agreed upon theory that can explain and quantitatively predict ecosystem biogeochemistry that is independent of the constituent organisms. Perhaps 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 organize in a predictable manner independent of species composition, as the planet’s biogeochemical processes have remained relative stable over hundreds of millions of years but the organisms that comprise ecosystems have changed significantly over this period. Consider primary producers in terrestrial versus marine ecosystems. While there is a definitive morphological discontinuity between marine systems that are composed of planktonic algae and terrestrial ecosystems comprised of trees, shrubs or grasses, 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 freedom, it is likely impossible to predict the nature of the organisms that evolve to 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 biogeochemistry, 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 the expense of details necessary for accurate weather prediction whose accuracy decays rapidly with time. Our objective is to understand biogeochemistry at the level of generic biological structure catalyzing chemical reactions. Understanding energy and

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Title Page Abstract

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mass flow catalyzed by living systems is particularly relevant for understanding how the biosphere will respond to global change, and predicting biogeochemical responses is essential before any geoengineering projects could be responsibly implemented. Many of the thermodynamically inspired ecosystem theories are more similar than 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 systems may organize under nonequilibrium steady state conditions when sufficient degrees of freedom exist. It appears Paltridge (1975) was the first to apply MEP as an objective function for modeling global heat transport, but it was not until the theoretical 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 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, 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 second law of thermodynamics for nonequilibrium systems. In essence, nonequilibrium systems will attempt to approach equilibrium via the fastest possible pathway, which can include the organization of complex structures if they facilitate entropy production. 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 MEP analyses are derived from Jaynes’ (2003) maximum entropy (MaxEnt) formulation that is information based. Furthermore, current MEP theory suggests that maximizing entropy production locally will maximize entropy production over the system domain, at 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 type of spatial optimization affects the solution to a simple, MEP-based biogeochemistry model.

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11 Entropy 11 of Entropy mixingofismixing readilyiscalculated readily calculated between boxes between andboxes boundaries and boundaries based on based flux and on flux and In this manuscript we will assume that biological systems organize to maximize en12 chemical 12 chemical potential potential obtained obtained from concentration from concentration differences differences between boxes between (Meysman boxes (Meysman and and tropy production, which is synonymous with maximizing the dissipation rate of Gibbs ESDD 13 systems Bruers 13 2007, Bruers Kondepudi 2007, Kondepudi and Prigogine and Prigogine 1998). For 1998). our simple For our model, simple wemodel, only permit we only permit free energy for under constant temperature and pressure. Living organisms 2, 1–44, 2011 are viewed here mere diffusive catalyststransport facilitating autocatalytic reactions for dissipa14 as diffusive 14 transport between boxes between andboxes boundaries, and boundaries, so a material so athe flux material of chemical flux of chemical species k species k tion of free energy. We will also continue our main hypothesis (Vallino, 2010), that -2 -1 15 from 15boxfrom or boundary box or boundary [i] to box[i] or to boundary box or boundary [j] is given [j]by is given (mmolby m(mmol d ), m-2 d-1), Biogeochemistry and the acquisition of Shannon information (Shannon, 1948), or more precisely useful inmaximum entropy formation (Adami, 2002; Adami et al., 2000), via evolution facilitates the production of D D 16 over (11) production (11) (kCthe )jβ] i−living )β i , j (k ) , to store information − C(C Fi , j (Ctime. Ckk)[,i ]systems ) k=[ j−]ability k ) =F− i ,It j (C k [ki[]of ,j( entropy when16averaged is   that allows them to out-compete abiotic systems in entropy production under approJ. J. Vallino -5 priate conditions. Here, how information may also facilitate 17 where 17 we where is awill characteristic is a characteristic length scale length for diffusion scale for diffusion (2.5×10 (2.5×10 m), D entropy is-5 the m),diffusion D is the diffusion examine production averaged over space.-4 For system, we will consider a simple 2 -1our -4 model 2 -1 18 where coefficient 18entropy coefficient (1×10 m(1×10 d ) in and m flux 1 ifcompound flux of reactions compound k is allowed k isbetween allowed box between or box or βdi , j)(box kand ) isisβ1igoverned ) is of , if j (k two-box model production each by two that Title Page determined the of (i.e., catalyst) synthesis andand degradation. 19 rate boundary 19biological boundary [i] and structure [j];[i]otherwise and [j]; otherwise it is 0. The it isproduct 0. Theofproduct flux of chemical flux and chemical potential potential differences, differences, | Discussion Paper

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Introduction

i, j 20 divided temperature by temperature gives the gives entropy theproduction entropy production per unit area per unit for mixing area forbetween mixing between ∆µ20 ∆µ ki , j ,by k , divided 2 Model system Conclusions References -1-2 -1 box21[i] tobox [j] for [i] chemical to [j] for chemical species k,species as follows k, as(Jfollows m-2 d-1 (J 21 ºKm ) d ºK-1) Tables Figures To examine how total entropy production depends on either local or global entropy Fi , j Fi , j i, j i, j     ∆ ∆ [ ] [ ] µ µ dS dS C i C i D D production optimization, model k k k k  . biogeo . (C k=[ j−]reminiscent 22 22 wekuse= aFi ,simple (12) (12) ) Ftwo-box − C(Ck [ki[])jR ] −lnCkof [i ]k )ocean R ln j (C k = i , j (C= k )− J I [ ] T in the photic and C dt to capture dt T j C chemistry models developed processes aphotic zones k   k [ j ]  of oceans (Fig. 1) (Ianson and Allen, 2002). Model focus is placed on dissipation of free J I 23 Given 23 Given concentrations of NH CH O,2, NH H2CO H112and CO ,S S with along thetogrowth with the growth energy by synthesis andconcentrations destruction of CH biological and3S )and from 2O,of 3,structures 2O 3, O 3,2(,S 22 1along 2 and chemical constituents (CH2 O, NH3 , O2 and H2 CO3 ) within the two layers. The photic Back 24 efficiencies 24 efficiencies eachε 2box, in each entropy box,production entropy production associatedassociated with reactions with reactions and mixing and mixing Close ε and ε 2ε in 1 and zone is typically energy replete1 but resource limited, while the aphotic zone is its comFull Screen / Esc 25 limited can25 be calculated canresource be calculated from replete. Eqs.from (6-12). Eqs. plement: energy but Our (6-12). model captures these two differing states; however, we replace energy input via photosynthetic active radiation with enPrinter-friendly Version ergy input via the diffusion of chemical potential from a fixed upper boundary condition 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 9 5

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dD 18 ) )and is if 1compound (1×10 if fluxlength ofmcompound kdisscale )allowed andfor is box between flux or of compound boxDorisDthek diffusion is allowed between box or βcoefficient (characteristic k ) isof2003). βki ,diffusion (kallowed ) is 1 if ctivities (Alberty here (2.5×10 m), i , j1is j (k ia , j flux jbetween ng alculated between calculated and boundaries boxes and based boundaries on flux based and byboxes temperature gives the entropy production perflux unitand area mixing between µ[readily , between 16 (11) (C∆isk20 ( C ii], )j β Fi , j (Con Ckour Ck [i ])β )Bruers [ jfor ] −simple 13 2007, Kondepudi and Prigogine 1998). For model, 9boundary terms of and instead of activities (Alberty 2003). kj ] ,−divided k [(9) i , j ( k(10) k ) = − (11) j ( k ) , we only permit , [j]; divided gives the entropy production perpotential unit area fori ,mixing between ∆µand [i] it is 0. The product of flux and chemical differences, 2 otherwise -1 by temperature   k -4 oefficient (1×10 m d ) and is 1 if flux of compound k is allowed between box or β ( k ) herwise sial 0. 19 The it boundary product is 0. The of [i] product flux [j]; of otherwise chemical flux and potential it chemical is 0. The differences, potential product differences, of flux and chemical potential differences, i , j -2 -1 -1 ng lated ily is calculated readily between calculated between boxes and boxes between boundaries and boxes boundaries based and on boundaries based flux and on flux based and on flux and omobtained concentration from differences between differences boxes between (Meysman and (Meysman and [1] and [2], but only CH2 O, All constituents arespecies allowed to between freely diffuse between boxes 14concentration diffusive transport boundaries, species k chemical k, as follows (Jboxes mboxes dand ºK i , j [i] to [j] for -2) -1 -1 so a material flux of chemical 0∆µbox ,O divided by temperature gives species the-5entropy per area mixing between box to [j] forare chemical k, asproduction follows (J m dunitºK ) for i , j [i] k21 and H CO permitted across the upper boundary, while only NH can diffuse ESDD -1 eristic length scale diffusion (2.5×10 m), 17 Dmodel, where is the diffusion isonly aand characteristic length scale for diffusion (2.5×10-5 m), D is the diffusion  production 2for 3 3 -2 erature es20 the entropy gives the , production divided entropy by production per temperature area per gives for unit mixing the area entropy for between mixing between perand unit area for mixing between ∆2and µ concentration ed ial from obtained concentration differences concentration differences between differences between boxes (Meysman boxes between (Meysman and boxes (Meysman oundary [i] and [j]; otherwise itunit is 0. The product of flux and chemical potential differences, ondepudi Prigogine 1998). Prigogine For our 1998). simple For model, our simple we only permit we permit kfrom 15 from box or boundary [i] to box or boundary [j] is given by (mmol m d ), ulated between boxes and boundaries based on flux and Fi , j i, j -2 -1 -1 across the lower boundary (Fig. 1).   ∆ [ ] µ dS C i D box [i] to [j] for chemical species k, as follows (J m d ºK ) 2 -1 -4 2 -1 F Entropy of mixing readily boundaries basedβ on(kflux i ,simple j kFor k calculated k-1 , j Prigogine i , jis -1 -2 -2model, -2 2, 1–44, 2011 ogine and 1998). and Prigogine simple For 1998). our simple our only permit we only only permit m1ondepudi dkias and is ifboundaries, of compound 18 kbetween is allowed coefficient between (1×10 box 1 if(12) flux of compound k is allowed between box or β iboxes (1998). kto )simplified ) is and kand area oxes ort between boundaries, and so aflux material flux so akentropy of material flux of chemical species -1m (Cchemical )we .C =our )) model =species −-1we − [iboxes ]permit ln Fdi1,-1 C j ]Dmodel, C k species R(J ∆ µ)model, dS i ]k)and ,)and divided temperature gives the production per mixing µconcentration k, al species follows box k, [i] (J follows m [j] for (J chemical m k, follows m 21 d-1For ºK dunit ºK )orkd[for ,as jby i , jbetween k [as kj (ºK differences boxes our autocatalytic chemical reactions   22 In dt . (12) =k between =(Meysman −we(consider ] − C konly [C ln Fi ,Tj (C k ) system, C k [ jand i ])kR[ two D ]  j    -2 -1 -2 -1 2 chemical potential obtained from concentration differences between boxes (Meysman and F 16 , (11) ( ) = − − β F C C j C i k ( ) [ ] [ ] ( ) ] kspecies T jmaterial iboxes ,box jboundaries, en ort and boxes between and and adt material boundaries, soµby ai , [j] flux of so chemical aand flux material of chemical flux of chemical species -2by -1the i , is j as k constrained kkd ji ] k species undary ox or boundary [i] to [j]so orThe isFboundary given (mmol given m by d19 (mmol m 5boundaries, occur within each box, following k [ jk  Cstoichiometric  -1),C ∆ dS D otherwise itkfor is 0. of flux chemical boundary [i] and [j]; otherwise it is 0.equations: The product of flux and chemical potential differences, gogine For our simple only permit ox [i] [j] chemical species k, as follows (J m dpotential ºK ) ki ,[differences, i ,to j 1998). i ,we j j ]D), k model, i , j product         ( ) . (12) = ( ) = − [ − [ ] ln F C C C i R ∆ [ ] ∆ [ ] [ ] µ µ dS C i C i C i D D -2 -1 -2 -1 -2 -1 i j k k k , k k k k k k Biogeochemistry and 3−rto Bruers 2007, Kondepudi and Prigogine 1998). For our simple model, we only permit         ( ( ) ) ( ) 22 . . (12) (12) . (12) )ndary [ = ] − − [ ] [ ] ln − = [ ] ( ln ) = − [ ] − [ ] ln C j C C i R j C F i R C C j C i R boxboundaries, orconcentrations to boundary box is or [j] by isof (mmol by ism (mmol given by m [ 2j ]along ofd3chemical dt[j] C kS Given CH O, NH , ),O Hd2CO with the growth iS , j 11dand k[i] k given k soboundary kgiven i ,T j [j] k), 3,m k ), 2kflux 2, (mmol        s boundary and a material species k mperature gives the entropy production per 20 unit area for , divided mixing by between temperature gives the entropy production per unit area for mixing between ∆ µ D -5 F ]1)isof [ ] [ ]  T23 r1 :iGiven  217  dt+ where C C T j C j concentrations CH O, NH , O , H CO , S and S along with the growth O − O + ε ρ NH −→ ε + − ε H CO (1) (1 (1 ) i[, jjε , j CH k 2 3 2 2 3 1 2 maximum entropy k k k      2 1 3 1 1 1 2 3 aDcharacteristic lengthsoscale (2.5×10 m), D is kthe diffusion  diffusion ∆ Cak [material i ]for (11) (11) (βCi ,kj[(kj ])=−,transport i ])dS Ck [i ])β i ,between k )k , 4−=C−k [diffusive boxes and boundaries, flux of chemical species j (µ -2 -1   (Centropy (12) (C k )ε 2byin(mmol =-2−box, j ] − Cproduction efficiencies each with reactions and mixing  -1 -2 -1 -1 jand k [ ), k [i ])R ln  associated or boundary [j] εF is1i , given meach d-1 production D  . chemical -4 -1 3box mical species follows (J [i] to [j] is species k, ask follows m din 2122CO d ºKbox ) or efficiencies 24 entropy production associated with reactions and (J mixing εj 1(CH ε 2NH Given of O, ,orO H ,and S21is and along with the growth -2 -1 ]for  3ºK C(S 2),box, ,2m (11) ) ) −]β −O β − βCO i= j)23 krkCH ik,]:box j(δ Ckas i ]and k,Tand []CH ((,concentrations [Given )dt [CO ]i,k, (,concentrations [,H )[i] k [)2jby i(11) 1(11) 18 coefficient (1×10 m d ) if flux of compound is allowed between β k 5fH boundary to box boundary [j] given (mmol m d ), ifrom k) i ,δ , jC , jor , j , O, NH O S , S S along and of CH with S O, along the NH growth with , O , the H CO growth , S and S along with the growth + + − ε O −→ ε + − ε CO + ρ NH (2) ) (1 ) (1 ) (H ). 3  2 2 2 2 3 31 2 1 1 2 32 2 2 1 2 2 32 1 2 2 3 -5 22 2 2 2 3 3 can be calculated from Eqs. (6-12). -5 J. J. Vallino ngth aracteristic scale for length scale (2.5×10 for diffusion m), D (2.5×10 is theproduction diffusion m), D is associated the Fdiffusion efficiencies and each box, entropy with reactions and mixing 25 calculated from Eqs. (6-12). εdiffusion ε 2 in i , j can i, j i, j 1 be O  C k [i ]  entropy , entropy (11) [i2]CO ∆ µflux C2box, dS [ientropy ])β∆iconcentrations )production boundary [i] and otherwise it isS 0. The product of andDchemical potential differences, D19 of efficiencies x,niven 24 each box, production and inassociated with each reactions with and reactions mixing production and mixing associated with and mixing εrassociated ε 2 NH , H , S and with the growth reaction biological structure, , catalyzes its own synthesis from sugar (unit kalong kk k[j]; k reactions j (In ,µ 2-5O, 3,-5 3 1 2 D 1 ,CH -5 1 1 is. the  . 2for-1 22 (12) [=)j−]compound − 1C(diffusion [m), ])RDln = Fi , j ((12) = − (C k [ j ] −(11) C iflux C C k [i ])R ln scale ccan diffusion length for diffusion scale (2.5×10 for (2.5×10 is (2.5×10 m), the is the kC k ) or 6daracteristic ,k diffusion βD C(i C Ciscompound i ][)diffusion [allowed )m), 0-4length is ) =and 1F−if flux is if kof isD allowed boxdtdiffusion or between box βk )im ) scale βbasis) (kkof i , j( kj ) k [ j] − kC calculated Eqs. , be j (k ,from  unit arearole. Tdcarbon j ]i ,j (k between T , j (6-12). and available thereby fulfilling the primary producer Re-  Cbetween kammonia k [ j]  (6-12). kifrom 20 , divided by temperature gives the entropy production per for mixing ∆ µ Eqs. ).ficiencies 25 (6-12). can be calculated Eqs. and in each box, entropy production associated with reactions and mixing ε ε -4 2 -1 -5 flux and m d1action )(diffusion flux ofirepresents ifjcompound of 1 if compound kheterotrophic is chemical allowed ofof kthe is diffusion between allowed k isbetween box allowed orpotential box between or differences, box or βfor kand )1 is (2flux k0.)ofis rβit that consume biological structure syntheTitle Page i , jif ,is 21 scale (2.5×10 m), D iscompound d)e0(k [j]; it)10 isisotherwise 0. The product The flux product and flux potential andorganisms chemical differences, -2 -1 -1 -5 box [i] to [j] for chemical species k, as follows (J m 21 dthe an be calculated from Eqs. (6-12). 7s of CH where characteristic scale forwith diffusion (2.5×10 m),of D CH ismode diffusion  is H2CO 23functions Given thein concentrations growth NH O2, H2CO3, S21 and ,and butSalso a cannibalistic by) 3,consuming as S2 along with the growth sized from (1), 2O, NH 3,aO 2,Eq. 3, S1 1length 2 along 2O,ºK rwise s, jgives [j]; 0. otherwise it is 0. The it of is of product 0. flux The and ofproduct flux chemical and ofchemical flux potential and chemical potential differences, differences, potential differences, isthe 1product if flux compound k isproduction allowed between or (temperature k )The Abstract Introduction edby entropy gives production the entropy per unit area for per mixing unitbox area between for mixing between well. While cannibalism could be replaced by inclusion of a sufficient number of higher Fi ,associated 82 in coefficient (1×10-4 production m2 d-1) and is with 1 ifefficiencies flux kε 2isinallowed between box or each box, entropy reactions and and mixing each box, entropy production associated with reactions and mixing ε 9 i , j of compound j β i , j ( k )24 1 µbetween dSproduction C [i ]  Dbetween trophic levels closure, this would lead degrees of model that are9 -2for -1per -2 -1 -1per∆ k mixing by ature es temperature the entropy the production gives entropy the production area per for mixing area for unit area for between References  k freedom  . ischemical 0. The product ofmas flux and chemical potential differences, (Cmixing 22(Jk, (12) Conclusions =dunit =to−greater F cies k,gives as follows species follows dentropy ºK-1unit )(Jk m i ,ºK j (C k) ) k [ j ] − C k [i ])R ln  (6-12). can calculated from Eqs. (6-12). dt T be typical not important for-1 this study, areproduct more closure techniques in real 9m Eqs. boundary [i] -2 and-1[j]; otherwise it is-1but 0. 25 The offlux and chemical potential  C k [ j ] observed differences, -1 -2 9 k,chemical species as15the follows k,species asi(J follows m k,das(JºK follows m-2)per (J d-1 unit ºKm )area d for ºK-1mixing ) 2000). ves entropy production between Tables Figures j , 9 9 9 environments (Edwards and Yool, To minimize parameterization, we assume  C k [i ] gives the [i ]  C kentropy D temperature k 0 F= −(∆CDµ ki(),Cj∆, µdivided by production per unit area for mixing between     ) ( ) . . (12) (12) [ ] − = − [ ] ln [ ] − [ ] ln j C i R C j C i R 23-2 Given with the(2) growth -1 structure by3, S21 ,and so S that δi in Eq. are i , j, j kthat k biological k k -1 k is degraded 2O, NH3, O2, H2CO 2 along  concentrations  of CH  indiscriminately follows  ∆DµTki , j(J m Dd  ºK 9 ∆k, ] k [ jC] k [i ]   C kk [ij]] µ C k [i)C Dias -2 -1 -1 J I       1Fi ,kj((C box [j] for chemical species k, as follows (J m d ºK ) (CCby, ) ( ) ) . . . (12) (12) (12) − )−j[i] ] −to [ = [ ] − ] − ln [ [ ] ] − ln [ ] ln C=kk[given i j R C C i j R C i R kk k Ck [ j ]   kC ε[1j ]and   εC2 [inj ]each  box, entropy production associated with reactions and mixing  T 24  efficiencies T k k k       C, 3S C and S C [CO i ]2 along D 3of O, ations , −H NH , ln O the along growth with the growth i1  J I 2O, 2CO 2, Hk 2S F[ i , ij ]3)R and µ3.i,, jSi1 ,with (C,δk O − NH [iCH j2]= C for i 2=Eqs. 1, (6-12). 2, (12)  C k [i ]  (3) can be = calculated from kk 25 ∆ dS D k [ ]  C j C,, S +,FO CS 9  . 9 (C2kalong )the 2ations (12) =and (, C )CO3C −the [ the j ] − with Cand R lngrowth CH Hbox, O of ,NH HCH CO O, NH H112CO H21kkalong S ,with S= along S with growth growth with k [i ]mixing T and 3,2O, 2each 3, 2O 32production 3entropy 3i ,, 2j S 2associated 2and and in production associated reactions and mixing ε 22entropy Back Close  reactions dtbox, T 2 1with  Ck [ j]  −3 each entropy production entropy box, entropy production associated with reactions associated with reactions andwith mixing reactions and and mixing ε box, where C production isassociated the concentration (mmol m ) of mixing biological structure i . In both reactions, 2 2in 6-12). d,nd from (6-12). NH ,Eqs. Heach 3, O 2CO 3, Si1 and S2 along with the growth Full Screen / Esc 3 20Given the growth ρ isconcentrations the N:C ratioofofCH biological structure, andSε2i along is thewith growth efficiency for biolog2O, NH3, O 2, H2CO3, S,1 and qs. dx, . from (6-12). Eqs. (6-12). entropy with reactions and mixing icalproduction structure associated synthesis. Note, as growth efficiency approaches zero, both reactions 4 efficiencies ε 1 and ε 2 in each box, entropy production associated with reactions and mixing Printer-friendly Version represent the complete oxidation of reduced carbon to CO2 and H2 O. 2). 9 5 can beTypically, calculatedHolling-type from Eqs. (6-12). 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 paramMax 9 25 eters (e.g., µ , KS ) in Holling-type expressions 9depend on community composition, 9 9 6 9 | Discussion Paper | Discussion Paper | Discussion Paper |

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In Eq. (4), Cj is the concentration (mmol m−3 ) of one or more growth limiting substrates, such as NH3 , εi is the growth efficiency as used in Eqs. (1) and (2), and ν∗ and κ ∗ are universal parameters that remain fixed regardless of community composition. Equation (4) differs from the Monod equation in several important ways. The effective half ∗ 4 saturation “constant” (κ εi ) depends on growth efficiency. The rational for this dependency is that nutrients at low concentration require free energy expenditure to transport them into the cell against a concentration gradient; consequently, organisms that have 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) approximately accounts for thermodynamic constraints on kinetics, because net reaction rate must approach zero as reaction free energy tends toward zero (Jin and Bethke, 2003; Boudart, 1976), which is equivalent to a growth efficiency of 1. The leading ε2i

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µi = εi νi

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and specific growth rate readily follows as (d ),

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so Holling kinetic equations need to be reparameterized as community composition changes. Since an underlying hypothesis of the MEP principle when applied to living systems is that they will evolve and organize to achieve a MEP state (Vallino, 2010), we cannot a prior set parameters in a kinetic model as we do not know the nature of the community composition nor the corresponding kinetic parameter values that will lead to a MEP state. Kinetic parameters need to be dynamic to reflect community composition changes. To achieve this objective, we have developed a novel kinetic expression that can capture reaction kinetics in a manner consistent with community compositional changes. The form of our kinetic expression takes the hyperbolic form of the Monod −1 equation (Monod, 1949), where substrate specific up-take rate is given by (d ), ! Y C j νi = ν∗ ε2i 1 − ε2i (4) Cj + κ ∗ ε4i j =1

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of mixing is readily calculated between and boundaries based 1998). on flux For andour simple model, we only permit 13 Bruers 2007,boxes Kondepudi and Prigogine 1 Entropy of mixing is readily calculated between boxes and boundaries based on flux and potential obtained from differences between boxes 14 concentration diffusive transport between boxes and boundaries, so a material flux of chemical species k xing is readily calculated between boxes and boundaries based on (Meysman flux and and 2 chemical from concentration differences boxes substrate (Meysman uptake and term inpotential Eq. (4)obtained is empirically motivated to accountbetween for reduced rates 007, andconcentration Prigogine 1998). For ourbetween simple model, weboundary only permit 15 from box or boundary [i] to box or [j] is given by (mmol m-2 d-1), ntialKondepudi obtained from differences boxes (Meysman and at low growth efficiency. ESDD 3 Bruers 2007, Kondepudi and Prigogine 1998). For our simple model, we only permit transport The between boxes and boundaries, so a been material flux of chemical speciesmatch k parameters have chosen to qualitatively observations of Kondepudi anduniversal Prigogine 1998). For our simple model, we only permit D 4 diffusive transport boxes flux chemical 16between (11)2, 1–44, 2011 ((mmol = − conditions Fi , j and Ck [ jso i ]-1)β i , j (found k ) , of in (Ck )boundaries, ] −aCmaterial -2 bacterial Under oligotrophic ocean species gyres’,k bacterial k [often or boundary togrowth. box boundary [j] by m d ), sport between [i] boxes andorboundaries, soisa given material flux of chemical species k  5 5from box orgrowth boundary [i] to or boundary [j], is given by (mmol m-2 d-1), of 10–20%, and subspecific rate is box typically 1–2 d−1 with growth efficiencies -2 -1 oundarystrate [i]Dto concentrations box or boundary [j] given by d ), -5 in is the or(mmol lower m range and (2.5×10 Cole, 1998; a characteristic length(Del scaleGiorgio for diffusion m), DCarlson is the diffusion Biogeochemistry and  is µM (11) Fi , j (Ck ) = − (Ck [ j ] − 17 Ck [Di ])βwhere i , j (k ) , maximum entropy et al., 1999). At the other extreme, bacteria grown under ideal laboratory conditions 6 (11) − (Ck [ j ] − Ck [i ])β i , j (-4k ) , 2 -1 −1 D Fi , j (Ck ) =18 coefficient (1×10 m 50 d d ) and flux of compound k is50–60%, allowed between box or production β i , j (kgrowth ) is 1 ifefficiencies  show specific growth rates as fast as , with around , (11) ( ) = − − β C j C i k ) [ ] [ ] ( ) k k k i, j -5 provided substrate is a characteristic length scaleconcentrations for diffusion (2.5×10 is therange diffusion are inm), theDitmM (Bailey, 1977; 19 boundary [i] and [j]; otherwise is 0.-5 The flux andLendenmann chemical potential differences, J. J. Vallino 7 10where is a characteristic length scale for diffusion (2.5×10 m), product D is the of diffusion Egli, and -5 (5) capture these approximate trophic extremes -4 2 -1 1998). Equations (4) and characteristic m), Dkisisthe diffusion nt (1×10 m length d ) andscale isi ,1j if flux(2.5×10 of compound allowed box or β i , j∗2for (k )-1diffusion −1between -4 κ by−3temperature entropy production per unit area for mixing between with values m and ν∗ ofgives 350 dthe (Fig. 2). The usefulness of the k , divided 8 coefficient (1×10of20 m of d∆)µ5000 and βmmol i , j ( k ) is 1 if flux of compound k is allowed between box or -4 2 -1 ×10 m d ) and is 1 if flux of compound k is allowed between box or β i , j (kit) is 0. (Eqs. expressions 4 and of 5) flux is their to potential generatedifferences, organismal growth kinetics y [i] andkinetic [j]; otherwise The and ability chemical box [i]product to [j]0.forThe chemical species as follows (J potential m-2 d-1 ºK-1) 21 Title Page 9 boundary [i] and [j]; oligotrophic otherwise it is product of fluxk,and chemical expected under conditions to extreme eutrophic conditionsdifferences, by varying only and [j];by otherwise it is 0. The product of flux and chemical potential differences, vided temperature gives the entropy production per unit area for mixing between i, j F efficiency, ε (Fig. 2). Abstract Introduction 0 entropy ∆ production ∆µgrowth  C [between µ ki , j dS k i , j the i ]  −1 Dper unit area for mixing k , divided by temperature gives -2 F-1per -1) area  . d ), (C k [ jbetween 22 (12) ((1) =forare − mixing ] − C k [iby ])R(mmol ln k m−3 Cunit by15for temperature thek,(4), entropy production Based gives on Eq. the rate Eqs. expressed k )and (2) od [j] chemical species as follows (Jof m= d i , j ºK Conclusions References  Ck [ j]  box [i] to [j] for chemical speciesdt k, as follows ! (J mT-2 d-1 ºK-1) 1 ! -2 -1 -1 C C orFchemical species k, as follows (J m d ºK ) NH3 CH2 O 2µ i , j  C [i ]  ∆ dS k i , j r = ν∗ Fε D2 k 1 − ε Tables Figures (6)growth C and S2 along with the j]concentrations i, j 23 (C∆µk [ki ,jC = F1i , j (CdS = − 1Given − C kD [i ])R ln∗  4ofk CH2.O, NH 3, [O 2CO3, S11(12)  4 ]2∗, H C k )k 1 k iκ + κ ε C + ε i , j T= F (C ) [ ] C j   CH O NH ( ) 2 dt . (12) = − [ ] − [ ] ln C j C i R 1 1 2 k [i ] k k  C ∆µ k Di , j k T 3C [ j ]  in. each box, (12) associated with reactions and mixing J = Fi , j (C k ) dt = − 24(C k [efficiencies j ] − C k [i ])R εln1 andk ε 2! k  entropy  production I T Ck [ j]    C ncentrations of∗ CH NH3,2can O2,be H2calculated CO3, ST1 and S along with the growth 2 2O,25 2 from Eqs. (6-12). r2 =concentrations ν ε2 1 − ε (7) 3 Given of2 CH2O, NH3, O2∗, H42CO3C, S21 and S2 along with the growth J I C + κ ε trations of εCH NHbox, H2CO3,production ST1 and S2associated along withwith the growth ies each reactions and mixing ε 1 and 2 2O, 3, O2,entropy 2 in 4 efficiencies ε 1 and ε 2 in each box, entropy production associated with reactions and mixing Back Close and εEquations in each box, entropy reactions andformixing (6) and (7)production allow us associated to explorewith reaction rates different community compoEqs. (6-12). 1lculated 2 from 5 cansitions be calculated from εEqs. (6-12). Full Screen / Esc by varying 1 and ε2 between 0 and 1. Of course, other mathematical functions ated from Eqs. (6-12). 20 could be used in place of Eq. (4), but this is not our primary interest for this manuscript, 9 but it is an interesting topic that warrants further research. For instance, we use the Printer-friendly Version ∗ 4 same effective half saturation constant, κ εi , regardless of substrate type in Eqs. (6) Interactive Discussion or (7), which could likely be improved, as well as accounting for other cellular growth 9 complexities (Ferenci, 1999). 9 9 8 | Discussion Paper | Discussion Paper | Discussion Paper |

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h d S ri = − ri ∆ Gri . (8) dt T While Gibbs free energy of reaction is usually straight forward to calculated for a given reaction stoichiometry, both Eqs. (1) and (2) require some discussion because both involve synthesis or decomposition of living biomass, or more generally, what we refer to as biological structure. Living organisms are often thought to be very low entropy structures because they are highly order systems. However, this common misconception results from the incorrect 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, 2005; Lambert 1999), as the latter is purely related to dispersion of energy to lower energy states, which may or may not be less ordered. Interestingly, both theoretical calculations and empirical data show that bacteria and yeast have higher absolute entropies ◦ (at 298.15 K) than glucose (Battley, 1999a,b, 2003). Biological structure has a higher 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, ammonia, phosphate and sulfate, without CO 2 production, is slightly less than zero, so is a reaction that can occur spontaneously (Vallino, 2010). Hence, it is thermodynamically 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

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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 gradients between boxes [1] and [2] and their associated boundaries. For chemical reactions, entropy production per unit area in a given box with depth h (m) is readily determined under constant pressure and temperature from the reaction Gibbs free energy change, ∆Gr , reaction rate, r, and absolute temperature, T , as given by Eu (1992, 131–141 pp.) (J m−2 d−1 ◦ K−1 ),

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 15 from box or boundary [i] to box or boundary [j]-5is given bychemical (mmol mspecies d ), k 14 where diffusive between boxesscale and boundaries, so(2.5×10 a material flux of 17 a characteristic length for diffusion m), D is the diffusion  istransport -2 -1 12 chemical potential obtained from concentration differences between boxes (Meysman and 5 from box or boundary [i] to box or boundary [j] is given by (mmol m d ), -5 -2 -1 ere  isas a characteristic length scale for diffusion D is the diffusion -4 2(2.5×10 -1 to boxm), 15 from box or boundary [i] or boundary [j] is given by (mmol m d ), D Gibbs free18energy of 16 reaction approaches reaction to proceed at highbetween box or coefficient (1×10 m Fid, j (2007, )Cand is ifaCflux k is allowed β (C) kFor −zero. β i , compound i ])of [ j1] −and i , j (k 13 Bruers k ) = Kondepudi k [Prigogine j ( k ) ,1998). For our simple model, we only permit(11) -4 2 -1 D  rate it must proceed irreversibly; consequently, organisms whose growth efficiency is ESDD k is allowed between box or 6fficient (1×10 (11) )and βflux Fi , j (m Ck )d= )−and(Cβk i[, jj(]k−) Cisk [1i14 kdiffusive ]if )of, Dcompound i , j ([j]; transport between boxes and boundaries, so a The material flux of chemical species k 19 boundary otherwise itC is[i0. The product of flux and chemical potential differences, 16 can , (11) ( ) = − − β C C j k ( ) [ ] ] ( ) less than 100% growFi[i] faster than those operating at very high efficiencies. k k k i, j ,j -5 2, 1–44, 2011  isflux where a characteristic length scale for diffusion (2.5×10 D is m the-2 diffusion of undary [i] and [j]; otherwise it isi , high 0. 17 The product and chemical potential diminishing returns growth efficiency isgives embed in growth kinetics equation, 15 box or boundary [i] toour box ordifferences, boundary [j] is given bym), (mmol d-1), 20 byfrom temperature the ∆of µ k j , divided -5 entropy production per unit area for mixing between 7 i , j 5where scale for diffusion (2.5×10 m), D is so the that diffusion  is a characteristic -4 2 optimum, as plotting Eq. (5)gives aslength athe function of ε exhibits -5 of compound 18 coefficient (1×10 m d-1area ) and 1-1 maximum if flux k is allowed between box or βmixing entropy production peran unit for between i , j ( k ) -2is(2.5×10 q 17 where is a characteristic length scale for diffusion m), Dspecific is the diffusion  -1 k , divided by temperature [i] to [j] for chemical species k, as follows (J m 21 2 box d ºK ) D -4 -1 Biogeochemistry(11) and d µ of compound k is allowed between box or 8 coefficient (1×10 m d ) and is 1 if flux β ( k ) (Ck [ itj ] is−as )β j (k3/5 Ck [εi ]→ ) , ≈ 0.77. isaturation , j 16 growth rate under nutrient ( -1-1εFand =−0) occurs i , j -1(C k) = boundary 0. Thei ,product of flux and chemical potential x [i] to [j] for chemical18 species k, as19 follows (J-4mm-22ddd[i] )[j]; coefficient (1×10 )ºK and or differences, β i ,otherwise Cj →∞ entropy j ( k ) is 1 if flux of compound k is allowed between boxmaximum Fi , j i, j C ∆µof D chemical 9 boundary and [j]; otherwise itdS 0. The and potential differences, The [i] Gibbs free energy20 ofiskreaction for) Eqs. (2) can expressed i , j product k flux k [i ] as a linear production  −1  . ( , divided by(1) gives entropy production per unit for(12) mixing between 22 = ( =temperature −isand − C kbe [ithe ])of ln F C CThe R ∆ µ -5 area Fi , j i , j otherwise k k [ j ]product i , j boundary [i] andk [j]; 19 it 0. flux and chemical potential differences, 17 where is a characteristic length scale for diffusion (2.5×10 m), D is the diffusion    ∆ [ ] µ dScombination C i D [ ]  dt T C j of biosynthesis and CH O oxidation, as given by (J mmol ), ki , j k k k   2   . per unit area for mixing 0 between = Fi , j (Cby = − (gives − Centropy C k [ j ]the ∆µ k , divided -2 -1 -1 k ) temperature k [i ])R lnproduction (12) J. J. Vallino [i]temperature to [j]for chemical as (J munit dofarea ºK ) mixing -4 the 2species -1 1 k, production 1−ε ε1 follows ] dt C (1×10 20T ∆µ ki ,j , 21 divided bycoefficient entropy perflux for k [ jgives  18box m d ) and is 1 if compound k is between allowed between box or β ( k ) i , j C C -2 -1 -1 box [i] to [j] for chemical species k, as follows m 2O, 1 d NH ºK3, )O2, HH2 2CO 23 Given CO S2-1along with the growth 3 3, S11 and  ◦ concentrations ◦ of(JFCH -2 -1 i, j i , j species ∆ Gr1 = (1 −21ε1 ) ∆ GOx + [j] ε ∆ chemical G +R T[i]ln (9)  chemical potential differences, box [i] to for k, as∆ follows (J m d ºK )   1 C µ dS D 19 boundary and [j]; otherwise it is 0. The product of flux 1−ε1 ε1 ρ ven concentrations of 24 CH2O,efficiencies NH3, O growth k S2 along with the k k [i ]and 2, H2CO3, S1 and  ( ) 22 (12) = ( ) = − [ ] − [ ] ln F C C j C i R and in each box, entropy production associated with reactions ε ε C C C Fi , j i, j k k 1 2 Title Page  C [ j ]  . and mixing  Ckk [CH ∆µ ki , j dS i ] T2O O2  NH 3 i , j dt FD k   i , j lnby i, j 20 , divided temperature gives the entropy production per unit area for mixing between ∆ µ   ( ) 2ciencies ε 1 andk ε 2=inFieach . (12) ( ) = − [ ] − [ ] C C j C i R associated with reactions and mixing  C [i ]  k dS k production D  kbox, entropy k k ∆µ k ,j 25 Eqs. (6-12). ] k [ j ] 1−ε  = Ffrom dt 22 canTbe calculated (12) =−C k [ (jC − C2k [i ])R(1−ε ln2 )ρ k ε2 . Abstract Introduction i , j (C k ) C C C [Cj ] (J1 and H box [i] to◦ T [j] for chemical species k,2NH m-2 d-1 ºK-1) be calculated from Eqs. (6-12). ◦ 23 dt21Given concentrations of CH NH ,asH3follows 2 CO 2O, 3,3 O 2COk 3, S 2  S2 along with the growth 10 ∆ Gr2 = (1 − ε2 ) ∆ GOx − (1 − ε2 ) ∆ G + R T ln   , (10) Conclusions References 1−ε2 3 Given concentrations of CH2O,24NH3,efficiencies O2, H2CO3, S1and and S2inalong with the growth each box, entropy production associated with reactions and mixing ε ε Fi , j C C i j , 1 2 23 Given concentrations of CH 3, ST1Dand O2S2 along with the  Cgrowth [i ]  dS k2O, NH3, O2, H∆2µCO Tables Figures (12) = Fi , j (C k ) withk reactions = − (Cand − C k [i ])R ln k  . 4 efficiencies ε 1 and ε 2 in each box,22 entropy production associated k [ j ]mixing 25 εcanand be εcalculated from Eqs.  associated with reactions dt box, T ◦ −1 ◦entropy −1 (6-12). 24 ideal efficiencies production  C k [ j ] and  mixing 1 2 inJeach is the standard Gibbs where R is the gas constant (in mmol K ), ∆G Ox 5 can be calculated from Eqs. (6-12). −1 J I 9 free energy of CH O oxidation by O to H CO (−498.4 J mmol ) and ∆G ◦ is the 25 can be calculated from (6-12). 2 2 Eqs. 2 3 23 Given concentrations of CH2O, NH3, O2, H2CO3, S1 and S2 along with the growth standard Gibbs free energy for biological structure synthesis from CH29O and NH3 , J I andnegligibly box, entropy production with reactions and mixing ε 2 in eachsmall which is set to zero for our 24 studyefficiencies because εit1 is (Vallino, 2010). associated We Back Close 15 use the approach of Alberty 25 (2003, to calculate the standard can2006) be calculated from Eqs. (6-12). Gibbs free energy ◦ 9 of reaction, ∆GOx , which accounts for proton dissociation equilibria between chemical Full Screen / Esc + − 9 species (CO2 + H2 O ↔ H2 CO3 ↔ H + HCO3 , etc.) at a pH of 8.1 and temperature of 9 ◦ 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 9 Entropy of mixing is readily calculated between boxes and boundaries based on flux and chemical potential obtained from concentration differences between boxes 10 | Discussion Paper | Discussion Paper | Discussion Paper |

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diffusive 14 transport diffusive between transportboxes between andboxes boundaries, and boundaries, so a material so aflux material of chemical flux of chemical species k species k

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15 from 15box from or boundary box or boundary [i] to box [i] or to boundary box or boundary [j] is given [j]by is (mmol given by m-2(mmol d-1), m-2 d-1), (Meysman and Bruers, 2007; Kondepudi and Prigogine, 1998). For our simple model, we only permit diffusive transport D Dbetween boxes and boundaries, so a material flux ( = − − Fi , j (Ck )k=F− C C j C(kC j ]i ,−j (Ckk)[,i ][i)β] i ,to ( ) [ ] [ik][)β ) , or boundary [j ] is given by (11) of 16 chemical16species from box or boundary i, j k k j ( k box   −2 −1 (mmol m d ),

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17 where 17 D is where a characteristic length scale length for diffusion scale for diffusion (2.5×10-5 (2.5×10 m), D is-5the m),diffusion D is the diffusion  is a characteristic Fi ,j (Ck ) = − (Ck [j ] − Ck [i ]) βi ,j (k), (11) Biogeochemistry and 2 -1 -4 18 coefficient 18 ` coefficient (1×10-4 m(1×10 d ) and m2βdi ,-1j )(kand 1 compound if flux of compound k is allowed k isbetween allowedbox between or box ormaximum entropy ) is β1i ,ifj (flux k ) isof where ` is a characteristic length scale for diffusion (2.5 × 10−5 m), D is the diffusion 19 boundary 19 boundary [i] and [j];[i]otherwise and [j]; otherwise it is 0. The it is product 0. Theofproduct flux and ofchemical flux and chemical potential differences, potential differences, production coefficient (1 × 10−4 m2 d−1 ) and βi ,j (k) is 1 if flux of compound k is allowed between i, j ,j J. J. Vallino box [j ]; otherwise it isthe 0.gives The the product of production flux potential 20or boundary by , divided temperature by temperature gives entropy production entropy perand unit chemical area per for unitmixing area forbetween mixing between ∆µ20 ∆[iµ] kiand k , divided i ,j differences, ∆ µk , divided by temperature gives the -2entropy production per unit area -1 -2 box21 [i] to box [j] for [i] chemical to [j] for chemical species k,species as follows k, as(Jfollows m d-1 (J 21 ºKm ) d-1 ºK-1)−2 −1 ◦ −1 for mixing between box [i ] to [j ] for chemical species k, as follows (J m d K ) Title Page Fi , j Fi , j i, j i, j Fi ,j  C k ∆µ k ∆Dµ k [i ]   C k [i ]  dS k dS k D i ,j  . d S22 ∆=µFk i , j (C k )= F 22 (12) (12) ])jR] −ln CCk k[i[i]])R ln. Di , j (C=k )− (C k=[ j−] − C(C k [ki [ Abstract Introduction k dt= − T (Ck [j ]T− Ck [i ]) R ln = Fi ,j (Ckdt) (12)  C k [ j.]   C k [ j ]  dt T ` Ck [j ] Discussion Paper

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d Ck [i ] = Fi − 1,i (Ck ) − Fi ,i +1 (Ck ) /h[i ] + Λk [i ] r [i ] for i = 1, 2 (13) dt 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

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To obtain changes in chemical constituents over time for a given set of boundary conditions, 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 general form, 20

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23 concentrations Given 23 concentrations Given concentrations of2 O, CH2NH O,of NH , NH H , H11 2and CO ,S S with the growth withgrowth the growth Given of CH , CH O ,O, and3S along with the 3,22O 2H 2CO 3, 3O 3,,2S 2 along 1 and 2 along 3 2 CO 2 efficiencies ε1 and ε2 in each box, entropy production associated with reactions and 24 efficiencies 24 efficiencies eachε box, in each entropy box,production entropy production associatedassociated with reactions with reactions and mixing and mixing ε 1 and ε 2 εin 1 and mixing can be calculated from Eqs.2 (6–12). 25 can25 be calculated can be calculated from Eqs.from (6-12). Eqs. (6-12). 2.2 Transport equations, optimization and numerical routines

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es as k, follows as follows (J m (Jdm ºKd )ºK ) Discussion Paper | Discussion Paper Discussion Paper |

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where both optimizations are bounded within the hypercube, 0 ≤ εi [j ] ≤ 1. Of course, local maximization, as given by Eq. (14), requires an iterative (i.e., asynchronous) approach, because changing the conditions in one box alters the optimal solution for

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or globally (i.e, across both boxes synchronously) as, r r r r d S 1 [1] d S 2 [1] d S 1 [2] d S 2 [2] Maximize + + + ε1 [1], ε2 [1], ε1 [2], ε2 [2] dt dt dt dt

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 C k [iΛ ]C [are i ]  the stoichiometric coefficients for compound k obtained from D D the 1 × 2 vector  kk.  . )Rk [ln (12) (12) = (−C k [ (jC ] −k [Cj k] [−i ]C i ])R ln T ]and   Eqs. (1) and C(2), k [jC k [ j ] r is a 2 × 1 vector of reaction rates, [r1 r2 ] , given by Eqs. (6) and (7). The full model equations are detailed in Appendix A for k = CH2 O, O2 , H2 CO3 , ,NH O2,3,HONH S113,,and S21S and S2 along withboxes, with the growth theand growth in2 along the two parameter values used for simulations is given in 2CO 2, H 33,2,CO 5 Table 1. box, entropy entropy production production associated associated with with reactions reactions and mixing and mixing We have intentionally designed our biogeochemistry model to contain only a few 12). transport related parameters (D, `, h) and effectively only two adjustable biological parameters per box, giving a total of four adjustable parameters for the two box model: ε1 [1], ε2 [1], ε1 [2], and ε2 [2]. Because choosing a value for εi [j ] effectively selects 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 ε1 [1], ε2 [1], ε1 [2], and ε2 [2] that maximize entropy production 9 9 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 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 needs to be made regarding local versus global optimization of entropy production. For the two-box model, entropy production can be asynchronously maximized locally (i.e., in each box), as given by, d S r2 [1] d S r1 [2] d S r2 [2] d S r1 [1] Maximize Maximize 20 + and + (14) ε1 [2], ε2 [2] ε1 [1], ε2 [1] dt dt dt dt

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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 Eq. (15) include the entropy of mixing contribution given by Eq. (12), as it is uncertain how it should be partitioned, but as will be shown, it is a small contribution compared to entropy production from the reaction terms, so neglecting it in the optimization is acceptable. While optimization of Eqs. (14) and (15) can be conducted for the transient problem (Eq. 13) over a specified time interval (Vallino, 2010), we are primarily interested in how spatial optimization alters MEP solutions. Consequently, we are only interested 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) 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 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 method (Brugnano and Magherini, 2004) from a specified initial condition (Table 2) unT til 12 d C/d t d C/d t ≤ ζ , where C is a vector of concentrations and ζ was set to 10−8 mmol m−3 d−1 . If a steady state solution failed to be obtained after 106 days of integration, 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), which employs a parallel version of a Lipschitzian direct search algorithm (Jones et 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, where first box [1] was optimized for given values of ε1 [2] and ε2 [2], then box [2] was optimized given ε1 [1] and ε2 [1] from the previous optimization of box [1]; this iteration ˆ ]k≤10−9 for each box, where k k is the Euclidian norm and proceeded until kε[j ] − ε[j T the vector εˆ (εˆ = [εˆ 1 εˆ 2 ] ) is the value of ε from the previous iteration.

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mate 8 activity estimate coefficients, activity coefficients, so that concentrations so that concentrations can be usedcan in the be used logarithmic in the logarithmic correction correction

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i =1

iencies 24 efficiencies and box, each box, production entropy production associated with associated reactions withand reactions mixingand mixing ε 1 and ε 2 inε 1each ε 2 inentropy which is evident in the results from the 100 000 random εi [j ]-point simulations (Fig. 4b). be 25calculated can from calculated Eqs. reaction (6-12). from Eqs.entropy (6-12). production defined by Eq. (8), numerous SS solutions Forbe any given 25 can be found that differ in location within εi [j ]-space and exhibit differences of up

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3.1 Model characteristics opy 11 of Entropy mixing isofreadily mixingcalculated is readily between calculated boxes between and boundaries boxes and boundaries based on flux based andon flux and For any specified point in εi [j ]-space, a solution to Eq. (13) can be found. For instance, mical 12 potential chemical obtained potentialfrom obtained concentration fromdynamics concentration differences differences between boxes between and Fig. 3 illustrates the transient to steady state for(Meysman theboxes point (Meysman (ε1 [1], ε2 and [1], ε1 [2], 5 Bruers ε2Kondepudi [2])2007, = (0.1, 0.05, 0.1, with and initialpermit and boundary ers 132007, Kondepudi and Prigogine and0.05) 1998). Prigogine Forthe 1998). our parameters simple For our model, simple we only model, we only permit conditions used throughout this study (Tables 1 and 2, respectively). For this particular soluusive 14 transport diffusivebetween transport boxes between and boundaries, boxes and boundaries, so a material so flux a material of chemical flux ofspecies chemical k species k tion, SS entropy production dominates in box [1] (575 J m−2 d−1 ◦ K−1 ), as compared to -2 -1 -2 -1 −2 −1 ◦ −1 m15 box orfrom boundary box(41.4 or[i] boundary to box or toKbox),orand [j] boundary isentropies given [j] by is(mmol givenm by (mmol d ),withmmixing d ), between boundary box [2] Jm d[i]boundary associated [0] and box [1], box [1] and box [2], and box [2] and boundary [3] are 14.6, 0.436 and D−2 −1 ◦ D−1 d) j=] −K free energy 1610Fi , j (0.00 (11) re−Fm Ck ) =(J Ck [(C i),]k)β[respectively. ) ,k [i ])β i , j (kDue ) , to the significant amount of(11) i , j ((C k [ ij,]j (−kC lease in oxidation of reduced organic compounds, entropy of mixing is a small fraction of the entropy production associated with Eqs. (1) and -5 -5 (2). To demonstrate this more re 17 is where a characteristic length scalelength for diffusion scale for(2.5×10 diffusion m), (2.5×10 D is the m), diffusion D is the diffusion  is a characteristic generally, we randomly chose 100 000 points within the 4-D εi [j ]-space and calculated -4 -4 ficient 18 coefficient (1×10 m2 (1×10 d-1state ) and m isto1βif is 1compound if flux of compound k is allowed k between is allowed box between or box or that β2i ,dj (-1k))and (k(13) ) of the steady solution Eq. and associated entropy terms, which shows i , jflux 15 entropy of mixing is at most 7.6% of the entropy of reaction, and for most cases much ndary 19 [i] boundary and [j]; otherwise [i] and [j];itotherwise is 0. Theitproduct is 0. The of flux product and of chemical flux andpotential chemicaldifferences, potential differences, less than that (Fig. 4a). Only 310 points out of 100 310 random simulations did not 6 j ,j attain a SS solution within 10 20 , divided , divided temperature by temperature gives the entropy givesdays. the production entropy production per unit area perforunit mixing area for between mixing between ∆µ kiby An interesting aspect of the model that the amount of nitrogen extracted from the -2 is [i] 21to [j] box for[i] chemical to [j] forspecies chemical k, asspecies follows k, (J as m follows d-1 ºK(J-1m ) -2 d-1 ºK-1) sediments, or boundary [3], depends on community composition, because changing 20 Fvalues of F ε [j ] dramatically affects the total standing SS nitrogen within boxes [1]     dS i , j dS k i , j i∆µ ki , j D∆µ ki , j D (C k [ =j ]N− m ([Ci]),k)[Rj ]ln−CCk k[[ii]])R .ln C k [i]  . 22 kand= [2], (12) (12) −k ) (mmol Fi , j (calculated C k ) = Fi , j (=Cas C k−2  T  dt dt T  Ck [ j]   Ck [ j]  2 X N = h[i ] C [i ] + ρ C [i ] + C [i ] , 2 along (16) Total concentrations en 23concentrations Given of CH2O, NH ofNH3CH ,3 O22O, , HNH and S3,2 S along S the growth with the growth 2CO 3,3,OS 2,11H 2CO 21 andwith

Discussion Paper

s 9of (9)terms and (10) of (9) instead and (10) of activities instead of(Alberty activities 2003). (Alberty 2003). 3 Results and discussion

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tween xes boxes boundaries, ak )material material chemical fluxofofdifferences, chemical species k species k itbetween -4The 2 product 2ofso is 0.and and chemical 19 potential boundary [i] [j]; otherwise is 0.or The product of flux and chemical potential differences, 1×10 fficient m(1×10 d-1and )-4and mboundaries, isβentropy 1so isproduction of 1ofif compound flux compound kunit isand allowed is between allowed box box or βd-1i ,flux (akflux )flux ed by temperature gives the per areakfor mixing between j)(and i , jif -2 otherwise -1 i , j -2 -1it is 0. The product of flux and chemical potential differences, 19 boundary [i] and [j]; yves x[i] orthe to boundary box or boundary [j] is given [j]by is given (mmol by m(mmol d∆µ ),k m, -1divided d between ), by temperature gives the entropy production per unit area for mixing between entropy production per unit20 area -2for -1mixing ndary and [j]; [i] and otherwise it isk,0.of it isproduct 0. (J The product and of chemical flux and chemical potential differences, differences, for chemical species as follows mof ) (Fig. tootherwise four[j]; orders magnitude indflux NºK 4b). Whilepotential the vast majority of solutions i , j The Total 20 -2 ∆-1µ k -1, divided by temperature gives the entropy production per unit area-2 for mixing between 4 [i] to [j] −2 (J man dNTotal box for, chemical speciesproduced k, as follows (J m ºK )less than 21 10 d-1 ºK-1) i ,k, j as follows result in mmol m 33 solutions high to extremely ESDD ed ,by divided temperature temperature entropy theproduction entropy production per unit area per(11) unit for mixing area(11) forbetween between βii,, jj gives ] i−, j C (kkby )[i,∆])µ (k ) , thegives j(C -2mixing k,C k [ki[])jβ ofCspecies boxD[i] toExamination [j] for chemical (J m 21 d-1lead ºK-1to ) high NTotal values high N values. community kinetics that k k [i ]  k, as follows Total (12) = Fi , j (C k ) = − (C k [ j ] − C k [i ]-2)R -1ln -1-2 -1Fi , .j -1 2, 1–44, 2011 asCfollows εC k [j necessary ∆µ ki , j efficiency [i ]], D for [i]Dto chemical [j] for chemical species k,species as(Jfollows m d (J mk)[dS ºK dj ]k ºKthat ) the growth T[an C apparent condition of Eq. (2), k [i ]k,  2     ) ( ) . 22 (12) . (12) − (Creveals [ ] − ] ln = ( ) = − [ ] − [ ] ln j C i R F C C j C i R -5Fi , j j k k i, j k [1]  C [ j]   C kk [i ]  ofk the transient µ dS gth ristic for scale diffusion for diffusion (2.5×10 (2.5×10 D is-5 the m), diffusion D∆less iski ,the diffusion D 0.02 [ ] scale  C j dt T 5 length in both boxes and [2] must be than (Fig. 5). Inspection km), k k     . 22 j (12) = Fi , j (C k ) = − (C k [ j ] − C k [i ])R ln Fi , j j j S  growthdue to extremely ∆µ ki2,O, ∆D ] the µ3,ki ,O C k2T[along i ]   Cwith D Biogeochemistry and 2 -1 dSsolution dt C k [ j ]  (12) slow growth entrations of CH NH H CO S reveals that concentration k[idevelops 2,of 2high 3,allowed 1 and 1 1 if flux is of 1 if compound flux compound k is k is between allowed box between or box or βd=i , j)F (and ki ,)jk(is ( k )     ( ( ) ) . . (12) = ) ( = ) − = [ − ] − [ [ ] ] − ln [ ] ln Cβ F C C j C C i j R C i R k i, j i, j k k kk k C] concentrations   C(data  of maximum entropy [ ] [ ]   T T j j NH3, Odt , H CO , S and S along with 23 the Given growth CH O, NH , O , H CO , S and S along with the growth rates of caused by small ε [j values not shown). Over time, develops 3 1 2 2 3 2 2 3 1 2 k  k   with  2 2 box, entropy production2 associated reactions and mixing ε 1 and2 ε 2 2in each production otherwise it is 0. sufficient The it isproduct 0. of product fluxconcentrations and of chemical flux chemical potential potential differences, 23 The Given of“top CH2O, NH3, control O differences, , H CO withtotheelevated growth concentration to and place down” on3, S11 ,and butSthis leads 2 along efficiencies x, entropy production associated with 24 reactions and mixing box, entropy production associated with reactions and mixing ε 1 and 2ε 2 in2 each lated from ntrations en concentrations CH2(6-12). O,ofNH CH O,2,NH H2both CO , O32, ,S H112CO and S2 21along the growth with the growth SSofEqs. concentrations of and .and Swith 3,2O 3,S 2 along mperature gives the gives entropy theproduction entropy production per3 εunit area perε 2unit for mixing area box, forbetween mixing efficiencies 24 in each entropybetween production associated with reactions and mixing J. J. Vallino 1 and 2). 10 25 can be calculated from (6-12). Inspection of total entropy production by Eqs. Eqs. (1) and (2) in the random 100 000 ciencies each in-2each entropy box, production entropy production associated associated with reactions with reactions and mixing and mixing ε 1 and ε 2ε 1inand ε 2box, can Eqs.the (6-12). ies icalk,species asεfollows k, 25 as(Jfollows msimulations (JºKcalculated m-1-2) d-1reveals d-1be ºK-1from ) that top 17 SS solutions have total reaction entropy i [j ]-point −2 −1 ◦ −1 lated be calculated from Eqs. from (6-12). Eqs. (6-12). production distributed over a narrow range from 620.11 to 621.15 J m d K ; howTitle Page i, j  C k [i ]  ε C[jk []ipoints  ∆D ] µ kever, the D corresponding have a much broader distribution over ε i [j ]-space  . (Ck [ki[])jR] −lnCk [i])R ln . i (12) (12) C=k )− (C k=[ −j ] − C Abstract Introduction T (Table 3), which produce Ck [ j]  differences in transient dynamics and SS standing  C k [ j ]  significant 9 15 stocks (Fig. 6). Most significant are differences in SS standing stocks for biological 9 9 Conclusions References ,ofNH CH O O,2,NH H2CO H112CO and S2 21along and with the growth withfrom the growth and thatS can range 26 to 29 600 and from 2.2 to 340 mmol m−2 3, 2structures 3, O32, ,S 3,S 2 along 9 in box [1], respectively. These state variations that have nearly identical entropy proTables Figures 9 9 box, in each entropy box,production entropy production associatedassociated with reactions with reactions and mixing and mixing ductions illustrate, at least qualitatively, an important concept in MEP theory (Dewar, m -12). Eqs. (6-12). 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; J I Lorenz, 2003), slight variations in entropy production exhibited by the top SS entropy producing solutions (Table 3) can be viewed as interchangeable, since any real system Back Close r is subject to noise that would make the small differences in d S /d t indistinguishable. Full Screen / Esc For our model system, there are clearly multiple solutions that give rise to what is 9 9 25 effectively the same MEP state. | Discussion Paper | Discussion Paper | Discussion Paper |

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The global MEP solution obtained by maximizing reaction associated entropy production, 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 the small 4-D parameter space of our model, this solution was almost located in the random 100 000 point simulations (Table 3). To examine the solution in the vicinity of the global maximum, we plot d S r /d t 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 box [2] (Fig. 7). It is clear from this plot that d S r /d t in box [1] of 619.6 J m−2 d−1 ◦ K−1 r is near the d S /d t maximum in box [1] given ε1 [2] and ε2 [2] (Fig. 7a, light gray point); however, it is also evident that d S r /d t in box [2] that contributes 1.92 J m−2 d−1 ◦ K−1 r (Fig. 7b, light gray point) to the global solution is far removed from the maximum d S /d t possible in box [2] that could be attained given ε1 [1] and ε2 [1], which is approximately −2 −1 ◦ −1 236 J m d K . This result indicates that the global optimum does not correspond to local optimums. Entropy productions associated with mixing, Eq. (12), summed over all relevant state variables for the globally-optimized solution are 18.6, 0.06 and 0.0 J m−2 d−1 ◦ K−1 for fluxes F0,1 , F1,2 and F2,3 , respectively. While entropy production from mixing does contribute to total entropy production, it is clear that it is small relative to the entropy of reaction terms. If reaction associated entropy production is maximized locally as given by Eq. (14), then the total entropy production summed over both boxes is only 429.4 J m−2 d−1 ◦ K−1 , with 224.4 and 204.9 J m−2 d−1 ◦ K−1 being produced in boxes [1] and [2], respectively (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 solve Eq. (14) has indeed obtained a solution that simultaneously maximizes entropy production in both boxes, because d S r /d t in box [1] cannot be improved with the given values of ε1 [2] and ε2 [2], nor can d S r /d t in box [2] be improved given ε1 [1] and ε2 [1];

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3.2 Global versus local MEP solutions

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+0. Theof + and 19 chemical boundary 19 species boundary [i] and [j];[i] otherwise [j]; itHis 0. The it↔H isproduct product flux flux chemical potential potential differences, 15 from box or [i] or [j]and is given by (mmol m-2differences, d-1), ciation proton dissociation equilibria between equilibria between chemical (CO (CO +and H2O ↔otherwise +CO H23O ↔ H + CObox Hboundary + of chemical 2species 2 2boundary 2to 3↔ 14 diffusive transport between boxes and boundaries, so a material flux of chemical species k i, j 208.1 and , divided divided temperature by temperature gives theM) gives entropy the production entropy production pertounit area per unit for mixing area forbetween mixing between ∆µ20 ∆µ ki , j ,by k 293.15ºK. at a pH etc.) of 8.1 atand a pHtemperature ofis temperature of of Ionic 293.15ºK. strength Ionic (IS =0.7 strength (ISused =0.7 toM) for is used HCO 3 , the solution stable. The corresponding of [j] mixing terms fluxes Disis 15 from box or boundary orentropy boundary given by (mmol mk -2 d-1F),0,1 , F1,2 16[i]−2to box , (11) ( ) = − − β F C C j C i ( ) [ ] [ ] ( ) -1 j k i-1 , j -2 -1 −1 ◦chemical −1i , used estimate vity coefficients, activity so that so21[i] that can in can the in logarithmic box toconcentrations box [j] for [i] chemical toused [j]dfor species k,k species as the follows k, as (Jfollows m-2 kdcorrection (J 21 concentrations ºKm ) d ºK-1) correction ESDD and F2,3 coefficients, are 3.21, 2.60 and 0.0 Jbe m Kbelogarithmic , respectively. D Comparing steady state2003). solutions from the global versus the local optimizations and terms (10) of (9) instead and16 (10) of activities instead (Alberty ofFiactivities (Alberty 2003). -5 (11) 2, 1–44, 2011 ,j ij, ] j − C k [ii ,]j)β i , j ( k ) ,i , j k [Fwhere , j (C k ) =Fi− 17(C is aand characteristic length scale  Cinkfor   globally∆µk [1] ∆D [ithe ] diffusion µ k [2] isDnear dS dS C k [i ] (2.5×10 m), D is the diffusion  shows that CH O concentration in boxes zero k k 2     22 22 . (12) (12) = Fi , j (C k = ) Fi , j (C=k )− (C k=[ j−] − C(Ck [ki[])jR ] −lnCk [i ])R ln  . C [ j ] 5). T -4  ) and β (solution T in(1×10 5 optimized solution, but onlydtpartially the locally-optimize k  C k [ j ]  (Table  18 dtdepleted coefficient m2 d-1 i ,-5 j k ) is 1 if flux of compound k is allowed between box or 17 to where scale for diffusion (2.5×10 m), D is the diffusion  is a characteristic Biogeochemistry and The inability effectively oxidize CHlength 2 O in box [1] in the locally-optimized solution remixing Entropy issults of readily mixing calculated is readily between calculated boxes between and boundaries boxes and based boundaries on flux based and on flux and -4 2 -1 19 boundary [i] and [j]; otherwise it is 0. The product of flux and chemical potentialentropy differences, 23 Given 23 concentrations Given concentrations of CH O, of NH CH , O O, , NH H CO , O , , S H and CO S , S along and S with along the growth with the growth from insufficient development ), as it2 between is ev- box or maximum 2 3 2structure 2 2 3 3 2( 1 12 and 3k is 22 1allowed 18 ancoefficient (1×10 m d ) andofβbiological i , j ( k ) is 1 if flux of compound production i, j N ident infrom Fig. 4b that a lower boundary on exists for and a(Meysman given entropy production chemical ential obtained potential obtained concentration differences differences between boxes andentropy , boxes divided by temperature gives the per unit area mixing between efficiencies efficiencies 24 from 24concentration andbetween in each box, in (Meysman each entropy box, production entropy production associated associated withproduction reactions with reactions and mixing and for mixing ε 20 ε 2∆εµ εTotal 1 kand 19 boundary [i] and [j];1 are otherwise it is 0.2 limited The product of quantity flux and chemical potential differences, rate. Biological reaction rates ultimately by the of catalyst availBruers Kondepudi 2007,and Kondepudi Prigogine and 1998). Prigogine For our 1998). simple Formodel, our simple we only model, permit we only permit J. J. Vallino box [i] to [j]Eqs. for chemical speciesmodel k, as follows (J m-2 d-1 ºK-1) 21calculated 25 i , jiscan 25 be calculated can be from Eqs. from (6-12). (6-12). 10 able that limited bytemperature elemental resources. The spatial 20in turn by gives the entropy production per unit configuration area for mixing between ∆µ k , divided nsport diffusive between between and boundaries, boxes and so boundaries, material so flux a material of chemical fluxacquisition species of chemical k species is transport suchboxes that state variables inabox [1] control energy across kboundary [0], Fi , j i, j -2 -1 -1 -1follows -2 -1 ∆d dS to resource [j] chemical species as (Jdm 21 [2] while controls input across boundary [3]. Limitations will  C k [i ]  . k ), from boundary box or [i] boundary tobox box or box [i] boundary to[i]box or [j]for boundary is given [j](mmol is given mk,-2 by d(mmol ), µ k ºK= −) D in (C keither 22by (12) = Fm [ j ] − Cflux Title Page i , j (C k ) k [i ])R ln   dt local optimization T result in lower entropyFproduction rates. In the given by Eq. (14),  C k [ j ]   C k [N ∆µ i , j dS k i , j i ] transported into D D D Abstract Introduction in results less   . (11) (12) =i ,Fj (i ,kj ()C − (C k [ j ] − C k [i ])Rinln , k [i ])β , k )box k [2]= ultimately (11) )βk i[, jj(]k−)production =− −Ck [entropy Ck ) = − maximizing Fi ,(jC(C i(]C C k [k j)]22  optimize solution, dt  system from boundary 23 theT Given concentrations of CH2O, NH and S2 along with the growth 15 the [3]. In globally net 3],O2, H2COof 3, S1 1 [2] k [ jsynthesis C 9 9 Conclusions References is critical in N acquisition, because -5when reaction r exceeds r in box [2] there is a 1 2 -5 efficiencies 24 in each box, entropy production associated with reactions and mixing εm), 1 and 2 the where characteristic is atransport characteristic length for length diffusion scale for (2.5×10 diffusion m), (2.5×10 D D εis diffusion net 23 scale Given concentrations of CHthat NHis3,the O2,diffusion H2CO along with the growth of NH in [2]. S In2 the locally optimize 2O, accumulates 3, S1 1 and Tables Figures 3 into the system -4 solution, 2 -1 -4entropy 2 -1 production can 25 can be calculated from Eqs. (6-12). be increased by a large loop flow between reactions 1×10 coefficient m d(1×10 ) and and 1 ifβflux iscompound 1εif in flux ofk compound is allowed kbetween isproduction allowed boxbetween or boxwith or reactions and mixing (dk )) is (k )of i, jε 24 βmi , j efficiencies and each box, entropy associated 2 r1 and r2 , but this limits N1 acquisition and entropy production in box [1]. These simulaJ I boundary and20[j];tions otherwise [i] and it is otherwise 0. The product it is 0.that The offrom flux product and (6-12). chemical of flux potential chemical differences, potential differences, 25[j]; can be calculated Eqs. clearly demonstrate optimizing localand entropy production limits whole system J I i , j resource and energy acquisition, which results in lower entropy production compared ed , divided by temperature gives the entropy gives production the entropyper production unit areaper for unit mixing areabetween for mixing between ∆µby k temperature to the globally optimized solution that more effectively utilizes available resources to Back Close -2 -1 box for chemical [i] dissipate to [j] for species chemical as species followsk,(Jas mfollows d ºK-1(J) m-2 d-1 ºK-1) freek, energy. 9 Full Screen / Esc We also investigated globally and locally optimized solutions with different values of F j  C k [`,  C [i ]  but in each case the general conµ ki , j dS i , ∆ i]  D ∆µ ki , j D  . (12) (12) =25Fi , j the (C kk)transport = Fi ,=j (C −parameters [i(]C)Rk [ln C k(namely j ]−D, C k [i ])h[1] R. lnandk h[2]), 9 k ) (C k [ j ]=−−  T  dt T was supported; Ck [ j]  clusion entropy when maximized globally rather Printer-friendly Version k [ j]  production isC greater than locally. A second variation of the model was also investigated that included two Interactive Discussion Given ntrations concentrations of CH2O, state NHof3,CH O H2CO NH33,, S O∗112,and H]2CO S23,along S∗21 and along growth with the growth additional variables, [j and [jwith ], S in2the each box, where the * superscript indi2,2O, cates inactive biological structure that does not catalyze reactions r1 or r2 . The rational box, entropy each box, production entropyassociated productionwith associated reactions with andreactions mixing and mixing εefficiencies ε 2 in 1 and ε 2 in εeach 1 and 17 can atedbefrom calculated Eqs. (6-12). from Eqs. (6-12).

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mical x]ollows [i] forto chemical species box [j] (J21 m for species as box [j] follows for to chemical species as box [j] (Jfollows for m[i]k,21 chemical to as[j] (J follows for box m)k,21 chemical species as (Jto follows mbox [j]k, species as [i](J follows chemical tom[j] as(JºK follows chemical species m ) d k, (J as m)follows m d (J 21 dto ºK 21 ) k, dspecies ºK ºK ) dfor ºK )k,dfor ºKspecies d k, ºKas(J)follows ºKm) d ºK ) [i] T[i]k,chemical Cd[i] k [ j] 

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i , ji , j i , ji , j i , ji , j j ,j i, j ,j i, j i, j dS  µCkibe  C k [i ] in C k [i ]  H  ∆D  D ∆dS ∆D [i∆]2was ]µ ]µki, j C ]D ∆ µfor µmodel µ µ kii1k, jioptimized dS dS dS C k [∆iS Cin iD dS Cµkki[, ij adapted C [i ]to conditions D2ki , along tions of NH ,D ,D S and with the growth this that box [j kkCH kk C kk 2CO k [∆ k][k iwould k [i ]poorly 2O, 3,k O 3∆            ) ( ( ) ( ) ( ) ( ) ( ) )R]k(12) 22 22 22 . 22 22 . (12) . . (12) . .] −lnC k(12) (12) (12) [ ) = ] − ( [ = ] = − ) ln ( = [ ) = − ] − ( [ = ] = [ − ) ] − ln ( [ = [ ) ] = − ] − ln ( [ = ] [ − ) ] − ln = [ = [ ] ( − ] − ln ) = [ ] [ ( ] = − ln − ) ]C =[lnj.−]− Ck(C [ki ][)jR [i ])R ln. (12)  . (12) CC j F C C i R F C C j F C C C i j R F C C C i j R F C C C i j R C C F i j C R C C F i j R C C kk i, j k k i , j  kk i , j k kk i , j  kkk i, j  k kk kki , j k  k ki , j k  k [i( k(12)    [ j ]Tdt be outcompeted, [ j]Tcould [ jserve ]  C [ j ]substrate [ j ]this Tdt C k [ j ]T  Cdt Tk [ j ]  in reaction Tdt dt C k would C kdt T kas C C Ck [ j]  box [k]Tand r2C. kIn ESDD kbut k [ j]          nd ε 2 in each box, entropy production associated with reactions and mixing alternate model, diffusion across boxes occurred between active and inactive variants 2, 1–44, 2011 ∗ ∗ ∗ dven from Eqs. (6-12). 23 ,of HCH CO Given O, S of NH concentrations and Given ,O S of , 2H NH along CH concentrations ,by O S ,NH H11CH 23 concentrations CO ,O S ,Given S NH CH along CO and ,2O, O concentrations of S ,NH Given along the ,O concentrations ,,with S NH along of and the , CH O growth S , 22H O, CO of the NH CH growth ,3with ,S along the H NH S with growth ,2S the , H∗21 2and growth CO with ,S the growth with the growth with the growth [1]↔ [1]↔ given [1]↔ [1]↔ [2] and [2]. S 2entrations 2concentrations 2of 3,23 3as 2O, 223 2CO 3Given 2O, 3,with 2of 2and 3the 2O, 3growth 2,of 2H 1223 3[2], 3,S 2with 2H 1CH 2and 3CO 2O, 3S 2growth 2H 12CO 3[2], 3,S 2with 1along 2and 3S 2O 2O, 12,and 2CO 3,2O 3along 3S 2 along 1 and 2 along i1,CH 2 1 2 1 2 1 5 This modified version of the model also produced similar results, where global optimizaiciencies efficiencies efficiencies efficiencies efficiencies opy production each and each entropy associated in each production entropy and 24 reactions each production entropy associated in each production 24 and entropy associated mixing box, in reactions production entropy and with box, reactions entropy in and each with mixing box, production and in associated with entropy reactions box, and associated with production mixing entropy reactions andwith production mixing associated reactions and mixing associated with andreactions mixing with reactions and mixing and mixing εin ε24 εbox, ε 224 εbox, εwith εbox, ε 2 εbox, εwith εeach ε 2production εand εreactions −2 −1each ◦mixing −1 224 1 and 2 1in 1 and 2 1in 1 and 2 efficiencies 1 associated 1 associated Biogeochemistry and tion resulted in total entropy production of 620.9 J m 2d K , but local optimization −2 −1 ◦ −1 nulated m25 be Eqs. calculated from can (6-12). 25be Eqs. calculated from can (6-12). 25 be Eqs. calculated from can (6-12). be 25 calculated (6-12). can Eqs. be from (6-12). calculated can Eqs. be(6-12). calculated (6-12). (6-12). maximum entropy generated entropy atEqs. afrom rate of25 only 415.0 Jfrom m Eqs. d from K Eqs. . Next, we examine how our production approach and conclusions may be extrapolated to a more ecologically relevant context.

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The model we have developed is obviously various aspects in9 the 9 chosen to illustrate 9 9 9 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 our modeling exercise, are intended to examine the usefulness of the MEP principle for understanding biogeochemical processes. The two main questions the MEP community must address are (1) do biological systems organize towards a MEP state and (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; instead, we focus on question (2). Ecosystem processes and the resulting biogeochemistry are often viewed from an organismal perspective, because organisms are the autonomous parts that comprise an ecosystem. Furthermore, since macroscopic ecosystem constituents appear relatively stable over timescales of human interest, it is common practice to characterize and model ecosystems with a relatively static species composition. However, the fossil record clearly shows that ecosystem communities exhibit great change over time (Gaidos et al., 2007), which is particularly evident in microbial systems whose characteristic 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

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4 Broader ecological context

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om box or boundary [i] to box or boundary [j] is given by (mmol m-2 d-1), MEP state under a given set of constraints (Dewar, 2003). Consequently, ecosystem D to one particular set of species, but can (11)be attained (Ck [ jis Fbiogeochemistry Ck [iconstrained ] −not ])β i , j (k ) , i , j (C k ) = −  number of complementary species sets, with the sets being interchangeby an infinite able. By complementary we mean the community must be comprised of organisms -5 where5  capable is a characteristic length scale for and diffusion (2.5×10 m), D issothe diffusion of energy acquisition element recycling, that at steady state only free -4 2 -1 energy dissipated. Earth’s biosphere obviously operates in this mode, as there oefficient (1×10is m d ) and β iThe , 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 electrooundarymagnetic [i] and [j];radiation otherwiseinto it is longer 0. The product of flux produces and chemical potentialWe differences, wavelengths entropy. have attempted to embody idea of species fluidity kinetic expressions Eqs. (4) and (5), by the temperature gives the composition entropy production perinunit area for mixing between ∆µ ki , j , divided 10 where the choice of εi defines the kinetics of the community. To implement MEP, difox [i] toferent [j] for modeling chemical species k, as follows m-2be d-1developed ºK-1) approaches need(Jto that are independent of species composition, which Eq. (4) is but one example. Fi , j  ∆µ i , j dS kThe D highlights the importance of resource acquisition for the (C k [ j ] − C k [i])R ln C k [i]  . (12) = MEP ) k = − also Fi , j (C kapproach [ j ]dissipate  structures needed dt T construction of biological free energy. Different com C kto  15 munity parameterizations lead to different levels of N acquisition and biological strucGiven concentrations of CH2O,with NH3a , Ominimum S2 along with the growth ture concentrations concentration required to achieve a given rate 2, H2CO3, ST1 and of entropy production (Fig. 4b). This perspective differs from the current paradigm fficiencies ε 1 and ε 2 in each box, entropy production associated with reactions and mixing of Liebig’s Law of the Minimum, which focuses solely on elemental limitations at the an be calculated fromlevel. Eqs. (6-12). organismal Because organism stoichiometry changes to match resource avail20 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 catalyst compositions that achieve the highest rate of free energy dissipation as constrained 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”.” 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 nevertheless 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 a perturbation of sufficient magnitude, or alteration of initial conditions, can lead the

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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 macroalgae and coral reefs (Mumby, 2009), in microbial communities (Price and Morin, 2004) ¨ and in many other systems (Schroder et al., 2005). Likewise, altering system drivers 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 of the solutions in Table 3 are stable states, as they were obtained via integration of the ODEs (Appendix A) to steady state. Unlike most analyses based on a discrete selection of food web components with fixed parameter values, our approach based on 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 manuscript, the MEP perspective may be useful in such analyses. The optimization criteria implicit in MEP allows parameter values to be determined 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 be determined for conditions where experimental data are lacking, or where model 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 replaces knowledge on information content in a system. As stated in the introduction, genomic information allows biological systems to out-compete abiotic systems under some situations when averaged over time and/or space. However, it is not just 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 present in the environment. This context dependency of genomic information complicates decoding an ecosystem’s metagenome. While great strides are being made 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 (Keller, 2005; Frazier et al., 2003). By assuming all known metabolic capabilities are

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present in a system, MEP-based optimization can determine which pathways may be expressed and when (Vallino, 2010); furthermore, genomic information can be incorporated into the optimization as constraints when genomic surveys indicate the lack of certain metabolic capabilities. Hence, MEP theory can be used to link genomic content to free energy dissipation. An interesting result from our MEP-based modeling exercise is that entropy production can be increased when optimized globally versus locally. However, to achieve high entropy production requires that the system organize such that certain locations within the model domain function “sub-optimally”, as evident in Fig. 7b. That is, free energy 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 optimization 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; CluttonBrock, 2009; Goodnight, 1990; Hillesland and Stahl, 2010; Nowak, 2006; Traulsen and Nowak, 2006; Wilson and Wilson, 2008). For instance, bacterial colonies are large spatial structures (∼cm) compared to the bacteria (∼µm) that comprise them, yet it is 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 center (Camilli and Bassler, 2006; Keller and Surette, 2006; Shapiro, 1998). Communication between plants via emission of volatile organic compounds has also been demonstrated to improve plant defenses over spatial scales greater than the individual (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 can integrate with plant roots (Maherali and Klironomos, 2007; Vandenkoornhuyse et al., 2007; Whitfield, 2007) suggests some level of below ground communication occurs across entire forests. From a physics perspective, even an individual multicellular or3 6 ganism represents a large spatial structure that is 10 to 10 times larger than the cells that comprise it, and surely those cells are spatially coordinated. Consequently, it is

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The maximum entropy production principle (Paltridge, 1975; Dewar, 2003) proposes that nonequilibrium abiotic or biotic systems with many degrees of freedom will organize towards a state of maximum entropy production, which is synonymous with maximizing free energy dissipation for chemical systems. The usefulness of MEP theory for understanding and modeling biogeochemistry is still under debate, and numerous applied and theoretical challenges need to be address before MEP becomes a common tool for solving problems in biogeochemistry. In previous work we have demonstrate how the MEP principle can be applied to biological systems under transient conditions provided entropy production is integrated over time (Vallino, 2010). In this manuscript we have examined how biogeochemical predictions differ if entropy

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clear that information exchange over large spatial scales exists in nature, so it is conceivable that ecosystems could coordinate over space to maximize energy acquisition and dissipation. If ecosystem information exchange occurs over large spatial scales, and if this communication facilitates increased free energy dissipation, then the domain of a model needs to be chosen judiciously. Specifically, the model domain is defined by the spatial extent of information exchange, largely dictated by chemical signaling. In certain systems, such as microbial mats (van Gemerden, 1993), it seems reasonable to model these systems as spatially connected systems, while other systems, such as the surface 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 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 question, as all current biogeochemical models depend solely on local conditions and are Markovian in nature; more research is needed.

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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,

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production is maximized locally versus globally for systems involving spatial dimensions. Biological systems explore biogeochemical reaction space primarily via changes in community composition and gene expression; consequently, we have developed a simple kinetic expression, given by Eq. (4), to capture changes in community composition 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 two chemical reactions has revealed that globally optimized solutions generate higher entropy production rates than locally optimized solutions (Table 4), and there exists many alternate steady state solutions that produce statistically similar rates of entropy (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.

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oncentrations in the logarithmic -2 flux -1 -1 correction j]; otherwise itcan is be 0.as used The product of and chemical potential differences, hemical follows m dbe ºK efficients, entropyspecies production per unit (J area mixing so thatk,concentrations canfor used) inbetween the logarithmic correction es (Alberty 2003).   [ ] C i temperature gives the entropy production per unit area for mixing between k d Cboxes -1 -1(Alberty iNH j]-2 ,iactivities ºKboundaries )R3 d[1] ln C C(J between and based on flux and (12) )follows instead of k [ j] −∆ k [m )− F . 2003).   [ ] µ C i D k k = C /h[1] − ρ ε + and ρ (1 −(12) ε2 [1]) r2 [1] (A4) C(C -1 boundaries -1 1 [1] ron 1 [1] k [k j[]j]1,2 )Rand − boxes ]-2 C(Jk [m iNH shemical calculated flux 3 ln i , readily j (C k ) species follows )  .andbased d t= −k,asbetween ºK entration differences between boxesd(Meysman [ ] T C j  k   C k [i ]  differences btaineddfrom concentration between boxes (Meysman and C [1] OC2between ,1998). H CO , S and S along with the growth i j ,   . ] −∆ [i ])our CFor and based and (12) 3kboxes 2 model, simple we onlyon permit 11R ln k [ j2  [+ µ Cflux i ] ε =Cboundaries  D k − F C /h[1] r1only [1]on − δ1 [1] (A5) ons of)CH O,t=NH O S and S2kmodel, along with the growth  .1 [1] 2]CO (Ck2[k, j[H]j1,2 − 3, between −For lnboundaries C3k,[our i ])11Rand spudi calculated boxes based flux andr2 [1](12) 2d Prigogine 1998). simple we permit i , readily j (C kand entration differences boxes (Meysman and boundaries, so a associated material flux of reactions chemical species k ropy production with and mixing [ ]  between T C j  k  btained from concentration differences between boxes (Meysman boxes andentropy boundaries, so a-1material flux with of chemical species k detween in d each box, production associated reactions andand mixing ε -2 C [1] 2 Ondary , H2CO ,S S2(mmol along with growth 21998). 3is our simple model, we only permit [j]For given by m dthe ), 21 and -2 -1 = − F C /h[1] + [1] − δ [1]) r [1] (A6) (ε ons CH O2, H1,2 ,our S21 simple and S2model, along the 2growth 2(6-12). 3, 1998). 2CO pudi Prigogine For we only permit2 ry [i]ofand to box or boundary [j] is 3given by (mmol m2with d ), from Eqs. dO, t NH boundaries, so a associated material flux of reactions chemical and species k ropy production with mixing boxesbox, andentropy boundaries, so a the material flux with of(11) chemical species k detween production associated reactions and mixing ε in each equations are, j (k2) , For box [2] constituents, D ndary [j] is given by (mmol m-2 d-1), (11) ( Ck [ j ] − Ck [i ])β i , j (k ) , ry or boundary [j] is given by (mmol m-2 d-1), dbox C(6-12). from Eqs.  [i] to CH2 O [2] (A7) =-5 Fm), /h[2] − r1 [2] 1,2 D C for diffusion (2.5×10 isCH the 2 Odiffusion , (11) (k ) d t -5 j D eristic length scale for diffusion (2.5×10 m), D is the diffusion (11) (Ck [ jd] −CCofk [[2] i ])β i , j (k ) , kis allowed is compound between box or 9 2 1-1if flux O2 9 m d ) and β i , j (k )=is F1-51,2 if flux compound k is−allowed CDOof /h[2] − (1 ε1 [2]) between r1 [2] − box (A8) (1 −orε2 [2]) r2 [2] for diffusion (2.5×10 m), is 2 the diffusion d tof flux and chemical potential The product -5 differences, eristic length scale for diffusion (2.5×10 m), D is the 9diffusion it 0. The product and of fluxbetween chemical isotherwise 1 if flux ofiscompound k is allowed box orpotential differences, d Cproduction H2 CO3 [2] per unit area for mixing between e 9 2entropy -1 m d ) and β i , j (k ) is 1=ifFflux ofCcompound k is + allowed /h[2] − between εmixing [2] or + (1 − ε2 [2]) r2 (A9) (1 for 1,2 production H2 CO3 per 1 [2]) rbetween 1box mperature gives the entropy unit area d-2t flux -1 chemical potential differences, The product follows (J mof d-1 ºKand ) -2 flux -1 and -1 chemical otherwise it is[2] 0.as The of potential differences, product d Cproduction mical species follows (Jm ) NHk, 3 e entropy per unit areadforºKmixing between F CNH3 − F2,3 CNH3 /h[2] − ρ ε1 [2] r1 [2] (A10) = [i ] 1,2 Centropy mperature gives the production per unit area for mixing between d )tR ln -1  -1k i , j [i ]-2  . (12) Cfollows k [ j ]∆−µC k (J m dD ºK )   C k [i ]  k [j ]j ](1 C -1  r -1 )2R[2]) (12) (C k ) species = −k, as(Cfollows −C(J− iε]-2 +k k[ρ k [m mical dln )  . 2 [2] ºK  T  Ck [ j]  C k [i ]  [2]ln S OC2k, [Hj 2]∆CO j, S and  . withthegrowth −µd Ci ,3C 2 along k [i ])11R D C k [+ i ] ε [2] r(12)  k = F C,[S /h[2] [2] growth − δ1 [2] r2 [2](12) (A11) [ ] C j 1,2 CH2O,d=NH , O , CO and S along   .1 with1the k[H   ( ) (sCof − ] − ] ln C j C i R 3 2 2 3 1 2 1 k) k k t associated with reactions  C [ jand ropy production T  k ]  mixing entropy production associated with reactions and mixing ε in each withthe d3C,box, O22, H2CO S21 [2] and S2 along growth =2, FH1,2 C, S1 and /h[2] + (ε2with [2] −theδ2growth [2]) r2 [2] (A12) s ofEqs. CH2(6-12). O,dNH , O CO S2 along 3 2 3 2 om t associated with reactions ropy production and mixing box, entropy associated with reactions and mixing ε 2 in each Fluxes, Fi ,j (Ckproduction ), where β i ,j (k) in Eq. (11) equals zero have been removed from the om Eqs.above (6-12).expressions. Parameter values, initial conditions plus boundary conditions used for all simulations are given in Tables 1 and 2, respectively. 924 9

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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 presented 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 OCE-0852263.

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ivities (Alberty 2003). 07, Kondepudi and Prigogine 1998). For our simple model, we only permit

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transport between boxes and boundaries, so a material flux of chemical species k ated between[i] boxes andorboundaries based on flux or boundary to box boundary [j] is given by and (mmol m-2 d-1),

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oncentration differences between boxes (Meysman and D (11) (Ck [our j ] −simple Ck [i ])βmodel, ) = − For gine i , j ( k ) , we only permit , j (C k1998).  and boundaries, so a material flux of chemical species k is a characteristic length scale for diffusion (2.5×10-5 m), D is the diffusion boundary [j] is given by (mmol m-2 d-1), nt (1×10-4 m2 d-1) and β i , j (k ) is 1 if flux of compound k is allowed between box or

vided by temperature gives-5the entropy production per unit area for mixing between Parameter cale for diffusion (2.5×10Description m), D is the diffusion [j] for chemical species k, as follows (J m-2parameter d-1 ºK-1) ∗ ν of compound Universal kineticbetween k is allowed box or (k ) is 1 if flux

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κ Universal kinetic parameter ,j µ kiflux S0. The product∆of C [i ]  D chemical potential differences, and D (C k [ j ] coefficient (12) = Fi , j (C k ) = −Diffusion − C k [i ])R ln k  .  dt T C k [ j]   between ` production Characteristic s the entropy per unit area forlength mixing h[1], h[2] Depth of boxes [1] and [2] -2 -1 -1 , as followsρ(J ncentrations of m CH NH O2C, H N)3,to ratio 2dO,ºK 2COof 3, Si1 and S2 along with the growth T Temperature es ε 1 and ε pH entropy production associated with reactions and mixing 2 in each box,pH  C [i ]  D ( (12) C k [ j ] − IC k [i ])R ln kIonic .Strength lculated fromS Eqs. (6-12).  Ck [ j]   ◦ ∆GOx Standard Gibbs free energy of CH2 O oxidation to H2 CO3 Standard Gibbs free energy of S synthesis from CH2 O and NH3 ∆G ◦ Fi , j k

350. (d ) 5000. (mmol m−3 ) 1 × 10−4 (m2 d−1 ) 2.5 × 10−5 (m) 10. (m) 1/6 (atomic) 293.15 (◦ K) 8.1 0.7 (M) −498.4 (J mmol−1 ) 0. (J mmol−1 )

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, (11) ])β i , j (k )Table 1. Model values. [i] and [j]; otherwise it is fixed 0. Theparameter product of flux and chemical potential differences,

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15

D 16 Fi , j (Ck ) = − (Ck k)[=j ]−−DCk(C[ik])[βji], j−(kC)k,[i])β i , j (k ) , Fi , j (C  

Discussion Paper

16

from15box or from boundary box or [i] boundary to box or [i] boundary to box or [j] boundary is given[j]byis(mmol given by m (mmol d ), m d ), (11)

(11)

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-5 where 17  iswhere a characteristic length scale length for diffusion scale for (2.5×10 diffusion m), (2.5×10 D is-5the m),diffusion D is the diffusion2, 1–44, 2011  is a characteristic

18

20

-4 coefficient 18 coefficient (1×10-4 m2(1×10 d-1) and m2β di , j-1()kand if flux of compound k is allowed k isbetween allowedbox between or box or ) is 1β if k ) isof1compound i , j (flux Biogeochemistry and boundary 19 [i] boundary and [j];[i] otherwise and [j]; otherwise it is 0. Theit product is 0. The ofproduct flux andofchemical flux andpotential chemicaldifferences, potential differences, maximum entropy i, j i, j production , divided temperature by temperature gives the entropy gives the production entropy production per unit area perfor unit mixing area for between mixing between ∆µ 20, divided ∆µ by

21

box 21 [i] to [j] boxfor [i]chemical to [j] forspecies chemical k, as species follows k, as (J follows m-2 d-1 ºK (J -1m) -2 d-1 ºK-1)

19

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17

k

Table 2. Model initial conditions (for boxes [1] and [2]) and Dirichlet boundary conditions (for F F [0] and [3]) used for all model simulations.  C k [i ]   C k [i ]  ∆µ ki , j dS k i , j dS k i , j ∆µ ki , j D D

22

22

dt

Box or

 . .  = Fi , j (C k ) = Fi , j (C = k−) (C k [=j ]−− C(kC[ik][)Rj ] ln − C k [i ])Rln T  −3 dt T  Ck [ j]   Ck [ j]  Variable Concentrations (mmol m )

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k

J. J. Vallino

(12)

Title(12) Page

Abstract

Introduction

Conclusions

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J

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2

22

2 3 3 2 11 2

3 2 21

2

290. 2000. – – production – associated efficiencies 24100.efficiencies and entropy box, production entropy associated with reactions with and reactions mixingand mixing ε 1 and ε 2 εin1 each εbox, 2 in each 100.

290.

2000.

1.0

–*

–

–

1.0

can 25 be100. calculated can be290. calculated from Eqs.2000. from (6-12). Eqs. (6-12). 1.0

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

0.1 0.1 –

0.1 0.1 –

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[0] 24 [1] [2] 25 [3]

3

|

CH2 CO [i ] CH [iCO ] ,O C Boundary C [i ] CO2 [i ]of CH ] S C [i ] with NH 23 Given 23CHconcentrations O,3 NH of ,C OO, , 3HNH ,S , H[i and CO , Salong and S along the growth with the growth 2 O Given concentrations


33

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0.019324 0.020367 0.040613 0.063907 0.043317 0.059295 0.029534 0.034642 0.061161 0.05965 0.054689 0.011359 0.074526 0.071069 0.029255 0.02539 0.059068

0.21967 0.80546 0.88768 0.99076 0.7629 0.64977 0.35951 0.7582 0.59806 0.3372 0.62598 0.58493 0.20127 0.5166 0.90925 0.11833 0.35479

0.99543 0.98968 0.99995 0.013594 0.98881 0.038126 0.9788 0.99138 0.9968 0.99783 0.011278 0.0005084 0.0053228 0.024946 0.0095011 0.97672 0.0277

621.15 620.93 620.86 620.85 620.84 620.73 620.72 620.70 620.69 620.64 620.62 620.51 620.48 620.45 620.36 620.28 620.11

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0.14046 0.12696 0.18904 0.12239 0.10072 0.10663 0.088998 0.1354 0.10056 0.10912 0.1313 0.38034 0.096701 0.10842 0.23553 0.09848 0.082633

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dS r /dt

Discussion Paper

ε2 [2]

2, 1–44, 2011


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ε1 [2]

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ε2 [1]

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ε1 [1]

Discussion Paper

Table 3. Values in 4-D εi [j ]-space that correspond to the highest entropy production (J m−2 d−1 ◦ K−1 ) by reactions (r1 ) and (r2 ) summed over boxes [1] and [2] from the 100 000 random εi [j ]-point simulations.

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Discussion Paper |

Local Eq. (14)

ε1 [1] ε2 [1] ε1 [2] ε2 [2] dS r [1]/dt r dS [2]/dt r r dS [1]/dt + dS [2]/dt

0.1312 0.0123 0.3621 0.9957 619.6 1.92 621.5

0.0992 0.0356 0.1005 0.1106 224.4 204.9 429.7

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Global Eq. (15)


35

2, 1–44, 2011


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Variable

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

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9 9

Discussion Paper

l obtained from concentration differences between boxes (Meysman and l obtained from concentration differences between boxes (Meysman and depudi and Prigogine 1998). For our simple model, we only permit depudi and Prigogine 1998). For our simple model, we only permit t between boxes and boundaries, so a material flux of chemical species k t between boxes and boundaries, so a material flux of chemical species k dary [i] to box or boundary [j] is given by (mmol m-2 d-1), dary [i] to box or boundary [j] is given by (mmol m-2 d-1), D (11) − (Ck [ j ] − Ck [i ])β i , j (k ) , D (11) −  (Ck [ j ] − Ck [i ])β i , j (k ) ,  acteristic length scale for diffusion (2.5×10-5 m), D is the diffusion acteristic length scale for diffusion (2.5×10-5 m), D is the diffusion 4 m2 d-1) and β i , j (k ) is 1 if flux of compound k is allowed between box or 4 m2 d-1Table ) and 5. if flux of compound(mmol k is allowed box or at maximum entropy producconcentration m−3 ) ofbetween state variables β i , jSteady (k ) is 1state j]; otherwise it is 0. The product flux and solutions. chemical potential differences, tion associated with local of and global j]; otherwise it is 0. The product of flux and chemical potential differences, temperature gives the entropy production per unit area for mixing between Boxarea [1] for mixing between Box [2] temperature gives the entropy production per unit -2 -1 -1 hemical species k, as follows (J m d ºK ) Variable Global Local Global Local hemical species k, as follows (J m-2 d-1 ºK-1) CH2 O  C [0.568 32.3 0.336 2.92 ∆µ ki , j D k i]    ( ) . (12) ( ) = − [ ] − [ ] ln C C j C i R O 190.6 222.3 190.3 189.9 i j , 2 i, j k ∆µ k k  ij]]   (C [ j ] − C [H Tk = − D  . )RCOln3 CCkk[[2099. 2068. 2100. (12) 2100. i , j (C k ) k k i ]2  T j ]  NH3  C k [0.996 0.489 1.00 1.00 ons of CH2O, NH3, O2, H2CO3, S11 and S2 60.2 along with9.05 the growth52.1 5.97 ons of CH2O, NH3, O2, H2CO3, S21 and S2 98.7 along with0.686 the growth106.7 0.712 d ε 2 in each box, entropy production associated with reactions and mixing d ε 2 in each box, entropy production associated with reactions and mixing from Eqs. (6-12). from Eqs. (6-12).

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F0,1 (Ck )

Discussion Paper

F2,3 (Ck )

| Fig. 1. Schematic of the two box model with associated state variables and boundary conditions. All fluxes between boxes and boundaries are governed by diffustion (see Eq. 11) and only CH2 O, O2 and H2 CO3 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).

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Discussion Paper

NH3[3]

ESDD 2, 1–44, 2011


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CH2O[2], O2[2], H2CO3[2], NH3[2] S1[2], S2[2]

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F1, 2 (Ck )

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CH2O[1], O2[1], H2CO3[1], NH3[1] S1[1], S2[1]

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CH2O[0], O2[0], H2CO3[0]

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50 0.7

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0.8

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0.5

30

0.9

20

0.4

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10

0.3

-1

0.15

0 2000

3000

4000

5000

-3

Substrate Concentration (mmol m ) 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 −1 −3 ∗ ∗ parameters values of 350 d and 5000 mmol m for ν and κ , respectively.

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1000

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1.0

0

2, 1–44, 2011


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0.6

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(a)

300

O2 (mmol m-3)

H2CO3 (mmol m-3) (c)

(b)

280 2100

80 260 60

240

40

220

20

200

2050

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CH2O (mmol m-3)

100

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180

0 250

Time (d)

NH3 (mmol m-3)

2

500

(d)

2000 0

120

250

Time (d)

S1 (mmol m-3)

500

(e)

0

250

Time (d)

S2 (mmol m-3)

4

500

(f)

100 1.5

3

80 60

1

2

40 0.5

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0

1

20

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0

0

0

700

250

Time (d)

500

dSr/dt (J m-2 d-1 °K) (g)

0 30

250

Time (d)

500

0

250

Time (d)

500

dSF/dt (J m-2 d-1 °K) (h)

600 500

20

400 300 10 200 0 0

250

Time (d)

500

0

250

Time (d)

500

Fig. 3. Concentration of state variables (a–f) and entropy production from reactions (g) and 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]. 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.

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Discussion Paper | Discussion Paper |

Fig. 4. Extracted results from 100 000 simulations with randomly selected εi [j ]. (a) Comparison 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

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0.01

Discussion Paper

ε2[1] or ε2[2]

0.015

|

1.2E+06 8.0E+05 4.0E+05 2.0E+05 1.0E+05 8.0E+04 5.0E+04 4.0E+04 3.0E+04 2.0E+04 1.5E+04 1.0E+04

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Total N -2 (mmol m )

0.02

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0.005

0

0.2

0.4

0.6

0.8

1

ε1[1] or ε1[2]

Discussion Paper

0

| Fig. 5. Points (triangles and squares) in 4-D εi [j ]-space that resulted in total steady state N standing stock (NTotal ) greater than 104 mmol m−2 . Triangles mark locations and magnitude of N standing stock for ε1 and ε2 in box [1], while squares mark locations and N standing stock 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 4 −2 NTotal < 10 mmol m . Note, the y-axis range from 0 to 0.021 captures all points that meet the minimum NTotal requirement of 104 mmol m−2 .

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CH 2 Ox

(a)

of 8.1 and temperature of 293.15ºK. Ionic strength (IS =0.7 M) is used to 700 800

500 400 300

r

) instead of activities (Alberty 2003).

-2

-1

-1

dS /dt (J m d °K )

600 fficients, so that concentrations can be used in the logarithmic correction

200

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equilibria between chemical species (CO2 + H2O ↔ H2CO3 ↔ H+ +

100

|

0

500

btained from concentration differences between boxes (Meysman and 300

(b)

pudi and Prigogine 1998). For our simple model, we only permit -3

CS1[1] (mmol m )

250 200

etween boxes and boundaries, so a material flux of chemical species k 150

ry [i] to box or boundary [j] is given by (mmol m-2 d-1), 100

2, 1–44, 2011


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D (Ck [ j ] − Ck [i])β i , j (k ) , 

0 100

200

300

400

(11) 500

Time (d)

(c)

5

4

-3

CNH3[1] (mmol m )

m2 d-1) and β i , j (k ) is 1 if flux of compound k is allowed between box or 3

otherwise it is 0. The product of flux and2 chemical potential differences,

0

100

200

300

400

500

Time (d)

i, j  Ctop ∆µFig. [i ]  entropy producing solutions from the 100 000 randomD k  k 17  . (C k [ j ] − data (12) =6.− Transient C k [i ])for R lnthe point simulations. (a) Total reaction  T  C k [ j ]  entropy production from both boxes [1] and [2] whose

−1

−1 ◦

−1

ε 2 in each box, entropy production associated with reactions 42 and mixing

|

steady state values range from 620.11 to 621.15 J m d K . (b) Concentration of biological in box [1], C ,and andS(c)along NH3 with over the the growth first 500 days of simulation. Values of εi [j ] for s of CHstructure O, NH , O , H CO , S 2 3 2 2 3 2 11 the above simulations are given in Table 3.

Discussion Paper

mical species k, as follows (J m-2 d-1 ºK-1)

0

|

1 mperature gives the entropy production per unit area for mixing between

Discussion Paper

0

eristic length scale for diffusion (2.5×10-5 m), D is the diffusion

(C k )

Discussion Paper

0 100 200 s readily calculated between boxes and boundaries based on300flux400and Time (d)

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Fig. 7. Reaction associated entropy production in (a) box [1] as a function ε1 and ε2 in box [1] for a fixed value of ε1 and ε2 in box [2] of 0.3621 and 0.9957, respectively, and (b) in box [2] 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).

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Fig. 8. Reaction associated entropy production in (a) box [1] as a function ε1 and ε2 in box [1] for a fixed value of ε1 andε2 in box [2] of 0.1005 and 0.1106, respectively, and (b) in box [2] as 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).

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