Differences and implications in biogeochemistry from maximizing

Earth Syst. Dynam., 2, 69–85, 2011 www.earth-syst-dynam.net/2/69/2011/ doi:10.5194/esd-2-69-2011 © Author(s) 2011. CC Attribution 3.0 License.

Earth System Dynamics

Differences and implications in biogeochemistry from maximizing entropy production locally versus globally J. J. Vallino Marine Biological Laboratory, Woods Hole, Massachusetts, USA Received: 31 December 2010 – Published in Earth Syst. Dynam. Discuss.: 19 January 2011 Revised: 25 May 2011 – Accepted: 27 May 2011 – Published: 17 June 2011

Abstract. 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 over space, 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.

1

Introduction

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

organize to maximize power. Many of the proposed theories (e.g., Morowitz, 1968; Schr¨odinger, 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) 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. Being that ecosystems are open, energetic systems, thermodynamics is particularly relevant for understanding biogeochemistry, which 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 (West et al., 1997), 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

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

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In92this manuscript In this manuscript we willwe assume will assume that biological that biological systemssystems organizeorganize to maximize to maximize entropyentropy production, 93 production, which iswhich synonymous is synonymous with maximizing with maximizing the dissipation the dissipation rate of Gibbs rate offree Gibbs energy free energy for 94systems for systems under constant under constant temperature temperature and pressure. and pressure. Living organisms Living organisms are viewed are viewed here as here as 70

J. J. Vallino: Differences and implications in biogeochemistry

mere 95 catalysts mere catalysts facilitating facilitating autocatalytic autocatalytic reactions reactions for the dissipation for the dissipation of free energy. of free energy. We willWe will the the organisms that evolve to facilitate ing reactions for the dissipation of free energy. also 96 continue alsonature continue our of main ourhypothesis main hypothesis (Vallino (Vallino 2010), that 2010), theenergy that acquisition the acquisition of autocatalytic Shannon of Shannon information information and mass transformations in one ecosystem versus the next

We will also continue our main hypothesis (Vallino, 2010),

(Shannon 97 or (Shannon 1948), or 1948), more orprecisely more vary precisely useful useful information (Adami 2002, that Adami 2002, Adami et al. 2000), et al. 2000), via how their populations overinformation time, but at(Adami the level the acquisition of via Shannon information (Shannon, 1948), free energy dissipation via of biogeochemistry, the problem or more useful information (Adami, 2002; Adami evolution 98 of evolution facilitates facilitates the production the production entropy of entropy when averaged when averaged over time. over It time. is precisely theItability is the of ability of may be more tractable. The problem is similar to predict-

et al., 2000), via evolution facilitates the production of en-

living 99 systems living systems to store to information store information that allows them allows out-compete to out-compete abiotic abiotic systems systems in in time. It is the ability of living ing weather versus climate, where thethat latter can to bethem predicted tropy when averaged over longproduction time scales at theappropriate expense of details necessary forwe systems store that allows them to out-compete 100 entropyover entropy production under appropriate under conditions. conditions. Here, we Here, will examine will examine howtoinformation howinformation information weather prediction whose accuracy decays rapidly with time.

abiotic systems in entropy production under appropriate con-

101 may also may facilitate also facilitate entropy entropy production production averaged averaged over space. For space. ourFor system, our system, we will we consider will consider Our objective is to understand biogeochemistry at over the level ditions. Here, we will examine how information may also faof generic biological structure catalyzing chemical reactions. cilitate entropy production averaged over space. For our sysUnderstanding energy and mass flow catalyzed by living system, we will consider a simple two-box model where entropy 103 reactions reactions thatisdetermined that determined the ratethe of rate of biological structure structure (i.e., catalyst) (i.e., production catalyst) synthesis synthesis andis largely governed by two reactions tems particularly relevant forbiological understanding how the bioinand each box sphere will respond to global change, and predicting biogeothat determined the rate of biological structure (i.e., catalyst) 104 degradation. degradation. chemical responses is essential before any geoengineering synthesis and degradation. projects could be responsibly implemented. 105 Many of the thermodynamically inspired ecosystem the2 Model system ories are more similar than different (Fath et al., 2001; Jor106 2 Model 2 Model System System gensen, 1994), but the theory of maximum entropy producTo examine how internal entropy production depends on eition (MEP) has greatentropy progress in depends understanding how Inmade this manuscript we will assume that systems organize to maximize entropy 107 To examine To examine how92internal how internal entropy production production depends onbiological either on local either orlocal global oror entropy global entropy ther local global entropy production 92 In this manuscript we will assume that biological systems organize to maximize entropy optimization, we use biotic and abiotic systems may organize under nonequilib92 In this manuscript we will assume that biological systems organize to entropy 93 production, which is synonymous with maximizing the dissipation rate of Gibbs free energy a simple two-box model reminiscent of maximize ocean biogeochem108 production production optimization, optimization, we use we a simple use a two-box simple two-box model reminiscent model reminiscent of ocean of ocean rium steady state when sufficient degrees of free93 conditions production, which is synonymous with maximizing the dissipation rate of Gibbs free energy models developed to capture the photic and 94 It appears for systems constantwas temperature pressure.istry Living organisms arethe viewed hereprocesses as rate of in dom exist. Paltridge (1975) the firstistoand apply 93under which with maximizing dissipation Gibbs free energy 109 biogeochemistry biogeochemistry models developed developed toproduction, capture to capture processes processes insynonymous the and photic in pressure. theaphotic and photic aphotic and aphotic zones ofzones of 94 models for systems under constant temperature Living organisms are viewed here and as Allen, zones of oceans (Fig. 1) (Ianson 2002). As MEP as an objective function forwe modeling global trans- systems In this manuscript will autocatalytic assume thatheat biological organize to maximize entropyWe will 9592 mere catalysts facilitating reactions for the dissipation of to free energy. 94 for systems under constant temperature and pressure. Living organisms are viewed here as customary in oceanography, we model a water column where In 92 this manuscript In this manuscript we will 92 we assume will In this assume that manuscript biological that biological we systems will systems assume organize that organize to biological maximize to maximize systems entropy entropy organize maximize entropy 95 mere catalysts facilitating autocatalytic reactions for the dissipation of free energy. We will 110 oceans port, (Fig. oceans 1)(Fig. (Ianson 1)not (Ianson and Allen and2002). Allen As 2002). customary As customary in oceanography, in oceanography, we model we amodel water a water but it was until the theoretical support provided by 93 also production, which is synonymous with maximizing the dissipation rateboundaries ofof Gibbs freehave energy all boxes and unit surface area, so that ex96 continue our main hypothesis (Vallino 2010), that the acquisition Shannon information production, 93 Dewar production, which iswhich synonymous 93is synonymous production, with with which maximizing is synonymous thehypothesis dissipation the dissipation with ratemaximizing ofautocatalytic Gibbs rate offree Gibbs theenergy dissipation free energy rate Gibbs free energy (2003) that research inmaximizing MEP theory andfacilitating applications 95 mere catalysts reactions forof the dissipation of free energy. We will 96 also continue our main (Vallino 2010), the acquisition of Shannon information 111 columncolumn where all where boxes all and boxes boundaries and boundaries have unit have surface unitand surface area, soarea, that extensive sothat that extensive variables variables tensive variables only depend on box depth, h, and will be 94 for systems under constant temperature pressure. Living organisms are viewed here as garnered greater interest, particularly in Earth systems sci(Shannon 1948), or more precisely useful information (Adami Adamiare al. 2000), via for 94systems for systems under97 constant under 94constant temperature for systems temperature and under pressure. and constant pressure. Living temperature organisms Livinguseful organisms and arepressure. viewed are viewed here Living as2002, here organisms as viewed here as 2 et 97 (Shannon 1948), or more precisely (Adami 2002, Adami et al. 2000), via 2 information 2 (Vallino calculated on a per m basis. Model focus is placed on dissi96 also continue our main hypothesis 2010), that the acquisition of Shannon information ence et al., al., 2003; Kleidon and 112 only depend only (Lorenz depend on box on depth, box2001; h, depth, andKleidon will h, and beetwill calculated be calculated on areactions peronm a for basis. perthe m dissipation Model basis. focus Model isfocus placed isWe placed mere catalysts facilitating autocatalytic of free energy. will 9895 facilitating evolution facilitates the production of dissipation entropy when averaged over time. is ability mere 95 catalysts mere catalysts facilitating 95autocatalytic mere autocatalytic catalysts reactions facilitating reactions for the dissipation for autocatalytic the of reactions free energy. of free forenergy. We the dissipation will We will of Itfree energy. Weof will destruction of biologipation of free energy bythe synthesis and 98b; Dyke evolution facilitates the1948), production of entropy when averaged over time. It is the 2002, ability of Lorenz, 2005a, and Kleidon, 2010). While Dewar’s 97 (Shannon or more precisely useful information (Adami Adami et al. 2000), via 96 alsofree continue ourby main hypothesis (Vallino 2010), that the acquisition of Shannon information 113 on dissipation on dissipation of free energy of energy by synthesis synthesis and destruction and destruction of biological of biological structures structures (S and (S S ) and S ) and to chemical constituents cal structures ( abiotic 2 1 and 212) from 99 living systems to (Vallino store that allows them out-compete systems in also 96 continue also continue our main our hypothesis 96 main also hypothesis (Vallino continue 2010), our information main that 2010), hypothesis the that acquisition the (Vallino acquisition of Shannon 2010), ofto Shannon that information the acquisition information of Shannon information initial work has received some criticisms, there have been 99(Shannon living systems to store information that allows (Adami them toentropy out-compete (CHof NH3Adami , Owhen H22000), COsystems within the two but en-of 97 approaches 1948), or more precisely useful information 2002, etabiotic al. 98 evolution facilitates the production averaged overintime. It islayers, the ability 2 O, 2 and 3 ) via several other that(CH arrive the same nonequilib(Shannon 97 and (Shannon 1948), or 1948), 97 orprecisely more (Shannon precisely useful 1948), information useful orNH information more Adami 2002, Adami et al.3)2000), et(Adami al. via 2000), 2002, viaAdami et but al. 2000), via 114 from from to chemical and tomore chemical constituents constituents O,at (CH ,(Adami Oprecisely NH ,2002, O Huseful and Hwithin the within two the layers, two but layers, 100 entropy production appropriate conditions. Here, we will examine how information 2under 23O, 2 and 3(Adami 2 CO 3)information 2CO tropy of mixing between boxes and boundaries is accounted 100 entropy production under appropriate conditions. Here, we will examine how information 98 state evolution facilitates the production of store entropy when averaged over time.them It is the ability of rium steady result 99 of MEP (see Niven, 2009 andinformation refliving systems to that allows towe out-compete in evolution 98 entropy evolution facilitates the production 98 the evolution production of entropy facilitates of entropy when the averaged when production averaged over oftime. entropy over It time. is when the Itability is averaged the of ability over of time. It is the of abiotic 101 may also facilitate entropy production averaged over space. For our system, willability consider for. The photic zone is typically energy repletesystems but resource 115 entropy offacilitates mixing of mixing between between boxes and boxes boundaries and boundaries isproduction accounted is accounted for. The for. photic The zone photic is zone is erences therein). The MEP principle states that systems may also facilitate entropy averaged over space. abiotic For oursystems system,inwe will consider 99 101living systems to store information that allowswill them to limited, out-compete while theabiotic aphotic isexamine its complement: energy living 99 systems living systems to102 store to information store information living that systems allows thattothem allows store toinformation them out-compete to out-compete that abiotic allows abiotic systems them systems toinout-compete systems in 100 entropy production under appropriate conditions. Here, wezone will how information a99 simple two-box model where entropy production in each box isinlargely governed by two organize to maximize rate of entropy production, which is 116 typically typically energy energy replete replete buta the resource buttwo-box resource limited, limited, while the while aphotic the production aphotic zone zone itsin complement: is its complement: energy energy 102entropy simple where entropy each box ishow largely governed by two captures these two 100 production undermodel appropriate conditions. Here,is we will examine information limited but resource replete. Our model appealing extension to theconditions. second law of thermodynamics 100 entropyan entropy production production under 100 appropriate under entropy appropriate production conditions. Here, under we appropriate Here, will we examine will conditions. examine how information how Here, information we will examine how information 101 may also facilitate entropy production averaged over space. For ourenergy system, we via willphotoconsider 103 reactions that determined the rate of biological structure (i.e., catalyst) synthesis and differing states; however, we replace input 117 limited for limited butnonequilibrium resource but replete. replete. Our model Our captures model captures these two these two states; states; however, wewe will we 103 reactions that determined the rate ofdiffering biological structure (i.e., catalyst) synthesis and 101resource maysystems. also facilitate entropy production averaged overdiffering space. Forhowever, our system, consider In essence, nonequilibrium sys101 may also may facilitate also104 facilitate entropy 101 entropy production mayproduction alsoaveraged facilitate averaged over entropy space. over production For space. ourFor system, averaged our system, weover willwe space. consider willactive For consider ourradiation system, we willenergy consider synthetic with input governed via the diffusion degradation. 102 amodel simple two-box model where entropy in each box is largely by two tems willenergy attempt tophotosynthetic approach equilibrium via the fastest pos102 ainput simple two-box where entropy production in each box isproduction largely governed 104 degradation. 118 replace replace input via via photosynthetic active radiation active radiation with energy with input energy via input the diffusion via the diffusion ofbyatwo oftwo of chemical potential from fixed upper boundary condition 102 a simplesible aenergy two-box simple two-box model 102 where model a entropy where simple entropy two-box production production model in each where in box each entropy is largely box production is largely governed governed in by each two box by two is largely governed by pathway, which can include the organization of com103 reactions that determined the rateinto of biological structure catalyst) and 103 reactions thatfixed determined the rate of biological structure (i.e., catalyst) synthesis and(i.e., 105 surface layer box. Nitrogen in thesynthesis form of ammonia 119 chemical chemical potential potential from athey fixed from upper boundary condition condition into theinto surface thethe surface layer box. layer box. 103 reactions reactions that determined that determined 103 rate reactions the of aupper biological ratethat of boundary biological determined structure structure the (i.e.,rate catalyst) (i.e., of biological catalyst) synthesis structure synthesis and (i.e., and catalyst) synthesis and 105 plex structures ifthe facilitate entropy production. Howis allowed to diffuse into the bottom box [2] from sediments, 104 degradation. 104System degradation. ever, none the current theories incorporate informa104 degradation. degradation. 104 degradation. 106 2 form Model 120 Nitrogen Nitrogen in the form inofthe of ammonia of MEP ammonia is allowed is allowed to diffuse to diffuse into theinto bottom thewhich bottom box [2] box from from fixes the[2] lower boundary condition [3]. Hence, energy 106 2 Model tion content of the system, suchSystem as that contained within an 105 flows into the system box [1] and resources (nitrogen) 105 121 sediments, sediments, which which fixes lower the boundary lower boundary condition condition [3]. analyHence, [3]. Hence, energy flows energy into flows theglobal intovia the 105 organism’s 105 genome, even though several of the MEP 107fixes Tothe examine how internal entropy production depends on either local or entropy 107 To examine how internal entropy production depends onvia either local orAll global entropy are allowed to freely flow in box [2]. constituents ses derived from Jaynes (2003) maximum entropy 106box 2[1] Model System 122 system system via are box via [1] and resources and resources (nitrogen) (nitrogen) flow via box in(Maxvia [2].box Allmodel [2]. constituents All constituents are of allowed are[1] allowed 108 production optimization, we useinaflow simple two-box reminiscent ocean 106 2 Model 2 Model System System 106 2 Model System diffuse between boxes and 106 2optimization, Model 108 that production weFurthermore, use a simple two-box model reminiscent of ocean [2], but only CH2 O, O2 and Ent) formulation is information based.System H2either CO are permitted acrosszones the upper 3 local 109 biogeochemistry models developed to capture processes in the photic and aphotic of boundary, while only 107 theory To examine howthat internal entropy production depends on or global entropy current suggests maximizing entropy pro107 To examine To examine howMEP internal how 107 internal entropy Toentropy production examine production how depends internal depends onentropy either onlocal production either local global depends or entropy globalonentropy either local or global entropyzones of 109 biogeochemistry models developed toor capture processes in the photic and aphotic NH can diffuse across the lower boundary (Fig. 1). 107optimization, To examine internal entropy depends on either local or global entropy duction 110 locally will maximize entropy over 108 oceans production weproduction usehow a simple two-box modelproduction reminiscent of ocean (Fig. 1) (Ianson and Allen 2002). Asofthe customary in3 oceanography, weocean model a water 108 production production optimization, optimization, 108 we use production we a simple use a two-box simple optimization, two-box model wereminiscent model use a reminiscent simple two-box ocean of ocean model reminiscent ofmodel 110 oceans (Fig. 1) (Ianson and Allen 2002). As customary in oceanography, we model a water In our simplified system, we consider only two ausystem domain, at least for flow-controlled systems (Niven, 108 allproduction optimization, we use ainsimple two-box model reminiscent of ocean 4variables 4 within 109 column biogeochemistry models developed to capture processes the photic and extensive aphotic zones of 111 where boxes and boundaries unit surface area, so that tocatalytic chemical reactions occur each box, as con109 biogeochemistry biogeochemistry models 109 models developed biogeochemistry developed to where capture to capture processes models developed in the photic in have to thecapture and photic aphotic processes and aphotic zones inofzones the photic ofso that and aphotic zones of 2009). Because an MEP-based model canprocesses be implemented 111 column all boxes and boundaries have unit surface area, extensive variables 2 the following strained by stoichiometric equations, 110 oceans (Fig. 1) (Ianson and Allen 2002). As customary in oceanography, we model a water 109 biogeochemistry models developed to processes in the photic and aphotic zones of 112 local only depend on box depth, h, and will calculated onwe a aper m Model focus isaplaced either orAllen global optimization, this in110 oceans with (Fig. oceans 1) (Fig. (Ianson 1)112 (Ianson 110 and oceans and 2002). Allen (Fig. As 2002). 1) customary (Ianson As depth, customary and in manuscript oceanography, Allen inbe 2002). oceanography, Aswe customary model model water incapture abasis. water model water only depend on box h, and will be calculated on aoceanography, per m2 basis. we Model focus is placed 111 column where all boxes and boundaries have unit surface area, so that extensive variables vestigates if the type of spatial optimization affects the solu113 on dissipation ofhave freeall energy by synthesis and destruction of biological structures (S 110boundaries oceans (Fig. 1) (Ianson and Allen 2002). As so customary in oceanography, we model a water 11 and S2) 111 columncolumn where all where boxes all 111 and boxes boundaries column and where unit have boxes surface unit and surface area, boundaries area, that extensive so have that unit extensive variables surface variables area, that extensive variables 113 on dissipation of free energy byso synthesis and destruction ofεbiological structures and Sε2)1 )H2 CO3 CH2 O +2 (1 − (1 − (1) 1 )O2 + ε1 ρNH 3 −→ ε1(S1 + tion to a simple, MEP-based 112 only depend onbiogeochemistry box depth, h, and model. will 2be calculated on a per m basis. Model focus is placed 2 2 from and to chemical constituents (CH O,mNH and His2CO within the two layers, but 112 only depend only on box on depth, 112 box h, depth, only will h, depend and beassume will calculated onbe box calculated depth, onbiological a per h,all on and maboxes basis. per will be Model basis. calculated Model on focus aplaced per is placed basis. Model focus is placed 2systems 3, O 2 focus 3) m 111 column where and boundaries have unit surface area, so that extensive variables In depend this114 manuscript we will that 114on dissipation from andoftofree chemical constituents (CH O, NH3, Oof 2 and H2CO3) within the two layers, but 113 energy by synthesis and 2destruction biological structures (S21 and S 2) organize to maximize production, which is synony113 on dissipation on dissipation of115 free energy of113 free energy byonentropy synthesis dissipation by synthesis and of destruction free and energy destruction of by biological synthesis of biological structures and destruction structures (S and S biological ) 2structures and Sε2) )(H2 CO3 focus entropy of mixing between boxes and boundaries is accounted for. The zone ism212+ (δ δof + (1S −photic )O −→ (1 − + ρNHis 2ε 1 1+ 2(S 212))and 3 ) (2) 112 of mixing only depend onboxes box depth, h, and will be calculated on2photic a perε2(S basis. placed 115 entropy between and boundaries is accounted for. The zone is 2 Model 114 from andthe to chemical constituents (CH2O,free NHenmous with maximizing dissipation rate of Gibbs 3, O2 and H2CO3) within the two layers, but 114 from and from to chemical and116 to chemical constituents 114 constituents from (CH and2O, toreplete (CH NH chemical , Obut NH constituents H2 CO and3)limited, Hwithin CO32)O, the within NH two the O layers, and two H but layers, within the two layers,energy but typically energy resource while zone is its complement: 23O, 2 and 3, O 2(CH 3,the 2 aphotic 2CO 3)but 116entropy typically energy replete butand resource limited, while thefor. aphotic zone iszone its complement: energy 113 onbetween dissipation of free energy synthesis and destruction structures where the stoichiometry biological structure, (Si1, and is S2) ergy for 115 systems under constant temperature and pressure. of mixing boxes boundaries isby accounted The photic isof biological 115 entropyLiving entropy of mixing of mixing between 115 between boxes entropy boxes boundaries ofhere and mixing boundaries between is accounted isboxes accounted for. and The boundaries for. photic The zone photic istwo accounted isO(NH zone is)for. The photic zone is of biological structure and 117 limited butand resource replete. Our model captures these differing states; however, we CH , ρ is the N:C ratio organisms are viewed as mere catalysts facilitatρ 2 limited butreplete resource Our model captures these two differing we 116 117typically energy butreplete. resource limited, while the aphotic is its energy 114 from and to chemical constituents (CH O, 3NH O2 states; and H2however, CO 2zone 3, complement: 3) within the two layers, but 116 typically typically energy118 energy replete116 replete but resource typically but resource limited, energy limited, while replete the while but aphotic resource the aphotic zone limited, is zone itsradiation complement: is while its complement: the aphotic energy zone energy is its energy replace energy input via photosynthetic active with energy input viacomplement: the diffusion of replace energy inputof via photosynthetic radiation withstates; energy input via diffusion of 117 118limited but resource replete. Our modelbetween capturesactive these two differing however, wethefor. 115 entropy mixing boxes and boundaries is accounted The photic zone is 117 limited Earth limited but resource but resource replete. 117 replete. Our limited model Our but captures model resource captures these replete. two these differing Our two model differing states; captures however, states; these however, we two differing we states; however, we Syst. 2, 69–85, 2011 www.earth-syst-dynam.net/2/69/2011/ 119 Dynam., chemical potential from a fixed upper boundary condition into the surface layer box. chemical potential from a fixed upper condition thevia surface layer box. 118 119replace energy input via photosynthetic activeboundary radiation with energyinto input the diffusion of 118 replace replace energy energy input input 118 photosynthetic viareplace photosynthetic active input radiation active via radiation photosynthetic withisenergy with input energy active via input radiation the into diffusion via limited, the with diffusion of energy of input the diffusion 116 typically energy replete but resource while the zone of is its complement: energy 120 via Nitrogen in theenergy form of ammonia allowed to diffuse the bottom box [2]viaaphotic from 119 120chemical potential from a fixed upper boundary condition into the layer box. Nitrogen in the form of ammonia is allowed to diffuse intosurface the bottom box [2] from 119 chemical chemical potential potential from119 a fixed fromchemical aupper fixedboundary upper potential boundary condition from aresource condition fixed into upper theinto surface boundary the surface layer box. layer box. into the surface layer box. 117 replete. Ourcondition model captures these states; however, we 121 sediments, whichlimited fixes thebut lower boundary condition [3]. Hence, energy flowstwo intodiffering the 120 121Nitrogen in the form of ammonia allowed to diffuse into the[3]. bottom box energy [2] fromflows into the sediments, which fixes the is lower boundary condition Hence,

102 a simplea two-box simple two-box model where modelentropy where entropy production production in each in box each is largely box is largely governed governed by two by two

108 production optimization, 2 106 Model 2System Model Systemwe use a simple two-box model reminiscent of ocean 109 models developed to capture processes in the photic and aphotic zones of Model 2106biogeochemistry Model System 2 Model SystemSystem To examine 107 To how examine internalhow entropy internal production entropy production depends ondepends either local on either or global localentropy or global entropy 110 oceans (Fig. 1) (Ianson and Allen 2002). As customary in oceanography, we model a water J. J. Vallino: Differences and implications in biogeochemistry 71 Model System examine To 107 how To internal examine how internal entropy how entropy production entropy production depends depends on either depends on local either on orlocal global either local entropy globalorof entropy global production 108examine production optimization, optimization, we internal use a simple we use two-box aproduction simple model two-box reminiscent model reminiscent of or ocean oceanentropy 111 column where all boxes and boundaries have unit surface area, so that extensive variables duction production 108 production optimization, we optimization, usemodels we a simple use awe simple two-box use depends a two-box simple model two-box reminiscent model reminiscent model ofor reminiscent ocean of ocean of ocean 109 optimization, biogeochemistry models developed to developed capture processes to capture inprocesses the photic in and theecosystems aphotic photic and zones aphotic of zones of 2000). To minimize parameobiogeochemistry examine how internal entropy production on either local global entropy Yool, 112 only depend6 on box depth, h, and willQ be calculated on a per m2 basis. Model(Edwards focus is and placed  terization, we assume that biological structure is degraded ingeochemistry biogeochemistry 109 biogeochemistry models developed models capture developed totwo-box capture processes to model capture processes in in the processes photic in the and photic the aphotic and photic aphotic zones and of aphotic zones ofzones of oceans 110(Fig. oceans 1)models (Ianson (Fig. and 1) (Ianson Allen 2002). and Allen As customary 2002). As customary oceanography, inin we model awe water model a water roduction optimization, we reminiscent ofoceanography, ocean (kuse )developed Sˆaktosimple 113 on dissipation F offree energy by synthesis and destruction of biological structures S2)the stoichiometric coefficients, discriminately by(S21 ,and so that ans oceans 110 (Fig. 1)(Fig. oceans (Ianson 1)kboxes (Ianson anddeveloped 1) Allen (Ianson and 2002). Allen and 2002). As Allen customary As 2002). customary in Asoceanography, customary in photic oceanography, inand oceanography, we model we amodel water weof amodel water a water Tprocesses column 111 where column all and all boundaries boxes and boundaries have unit surface have unit area, surface so that area, extensive variables extensive (Fig. 1where iogeochemistry models to capture in the aphotic zones δi ,so inthat reaction (2) arevariables given by (mole compound (mole reac114 from and to chemical constituents (CH2O, NH3, O2 and 2H2CO3) within the two layers, but −1 2 tion extent) ),placed umn column 111 where column all where boxes all where and boxes boundaries all and boxes boundaries and have boundaries unit have surface unit have surface area, unit so surface area, that so extensive area, that extensive so variables that extensive variables variables only 112 depend only on box depend depth, on h, box and depth, will be h, and calculated will be on calculated a per m on basis. a per Model m basis. focus Model is focus is placed ceans (Fig. 1) (Ianson and Allen 2002). As customary in oceanography, we model a water CH O[0], O [0], H CO [0] 115 entropy of mixing between boxes and boundaries is accounted for. The photic zone is 2 2 2 2 of 2destruction Cfocus C) is and yondepend only 112 depend ononly box on depth, box h, depth, on and box will h,depth, and be2will calculated h, and be calculated will on bea3calculated per on m a per basis. m aModel basis. per m Model focus basis. is Model placed isfocus placed placed dissipation 113where on ofdepend free energy free by synthesis energy by and synthesis and destruction of biological ofstructures biological (S structures S) i1 and S2(S i1 olumn alldissipation boxes and boundaries have unit surface area, soon that extensive variables , for i2 = 1,2, δ = = (3) i 116 typically energy replete but resource limited, while the aphotic zone isCits complement: energy + C C 2of)and dissipation on 113 dissipation of on free dissipation of energy free energy by of free synthesis by energy synthesis and by destruction synthesis and destruction and of biological destruction biological structures of biological structures (S and (S S ) (S S ) and S ) F ( k ) 2 2 T1 11 structures 21 2and from 114 and to from chemical and to constituents chemical constituents (CH O, NH (CH , O O, and NH H CO , O within H CO the ) two within layers, the two but layers, but 2 3 2 2 3 2 3 2 3 nly depend on box depth, h, and will be calculated on a per m basis. Model focus is placed 0 , 1 systems will assume organize that biological to maximize systems entropy organize to maximize entropy 117 limited but resource replete. Our model captures these two differing states; however, we m and from 114 to chemical and from to chemical and constituents to chemical constituents (CH constituents O,(CH NH , destruction O (CH NH O H2NH and H )Owithin and within the two the within layers, two layers, but but layers, where C(Sthe thezone concentration (mmol m−3 ) of biological entropy 115 of mixing entropy between of mixing boxes between and boundaries boxes and is accounted for. accounted The photic zone The photic isistwo is but 2 23O, 2 and 3boundaries 2,O, 2CO 2is 2CO 3)H 2CO 3)for. nynonymous of free energy bythe synthesis and of3,3free biological structures g dissipation the dissipation with rate maximizing of Gibbs freedissipation energy rate of Gibbs energy i1 and S2) 118 replace energy input CH via 2photosynthetic active radiation with energy input via the diffusion of O[1], Oboundaries structure 2[1], ropy 115 of mixing entropy of mixing between ofconstituents between mixing boxes between and boxes boundaries and boxes and isOlimited, accounted boundaries isviewed accounted for. is accounted Thefor. photic The for. zone photic The isi.zone photic is zone is energy typically 116 typically replete energy but resource replete but limited, resource while the aphotic while the zone aphotic isthe its two complement: zone is its complement: energy ure. stant Living temperature organisms and pressure. are viewed Living here as organisms are here as om entropy and toenergy chemical (CH O, NH , and H CO ) within layers, but 2 3 2 2 3 Typically, Holling-type kinetics (Holling, 1965) are used h [ 1 ] 119 chemical potential from a fixed upper boundary condition into the surface layer box. H CO [1], NH [1] 2 3 3 assume that biological systems organize to maximize entropy ically typically 116 energy typically energy replete replete energy but resource but replete resource limited, but resource limited, while the limited, while aphotic the while aphotic zone the is aphotic zone its complement: is its zone complement: is its energy complement: energy energy to describe reaction rates for reactions (1) and (2) based on limited 117 but limited resource but replete. resource Our replete. model captures Our model these captures two differing these two states; differing however, states; we however, we for tingthe autocatalytic dissipation reactions of free energy. for theand We dissipation will of is free energy. We ntropy of mixing between boxes boundaries accounted for. will The photic zone is S [1], S [1] 120 Nitrogen in the form of ammonia is allowed to diffuse into the bottom box [2] from a limiting substrate or substrates. However, kinetic parame1 captures 2 captures ymous with maximizing the dissipation rate of Gibbs free energy limited 117 but resource but limited resource replete. but resource replete. Our model replete. Our model Our model these captures two these differing two these differing states; two differing however, states; however, states; we however, we we of replace 118 energy replace input energy via photosynthetic input via photosynthetic active radiation active with radiation energy input energy via the input diffusion via theofdiffusion max n,ited that hypothesis theenergy acquisition (Vallino of 2010), Shannon that information thelimited, acquisition ofthe Shannon information ypically replete but resource while aphotic zone iswith its complement: energy ters (e.g., µ ,K ) in Holling-type expressions depend on 121 sediments, which fixes the organisms lower boundary condition [3]. Hence, energy flows intoM the temperature and pressure. Living are viewed here as lace replace 118 energy replace energy input via input energy photosynthetic via input photosynthetic via photosynthetic active radiation active radiation active with energy radiation with energy input with via input energy the via diffusion input the diffusion via of the diffusion of of community composition, so Holling kinetic equations need F1upper kcondition )boundary chemical 119 chemical from potential a fixed upper a fixed boundary into condition the surface layer surface box.welayer box. ore tion (Adami 2002, useful Adami information etOur al.from 2000), (Adami via 2002, et al. 2000), viaintohowever, mited but potential resource replete. model captures these two differing states; , 2to(Adami willprecisely assume that biological systems organize maximize entropy 122 system via box [1] and resources (nitrogen) flow in via box [2]. All constituents are allowed to be reparameterized as community composition changes. utocatalytic reactions for the dissipation ofboundary free energy. We will mical chemical 119 potential potential from potential aof from fixed aupper fixed from upper aallowed fixed upper condition boundary condition condition into surface the into surface layer the box. surface layer box. layer box. Nitrogen 120 inchemical Nitrogen the form in the ammonia form ofisboundary ammonia to is to diffuse allowed into to diffuse the bottom into box the bottom [2] from box [2] of from we will assume that biological systems organize maximize entropy he en production averaged over of entropy time. It when is the averaged ability of over time. It is the ability of eplace energy input via photosynthetic active radiation with energy input via the diffusion Since an underlying hypothesis of the MEP principle when ynonymous with maximizing theO[2], dissipation rate of Gibbs free energy CH O [2], othesis (Vallino 2010), that the acquisition of Shannon information 2 2 applied to living systems is that they will evolve and organize rogen Nitrogen 120 in the Nitrogen form in the of form ammonia in the of form ammonia is of allowed ammonia is allowed to diffuse is allowed to diffuse into the to into diffuse bottom the into bottom box the [2] box bottom from [2] box from [2] from sediments, which sediments, fixes which lower fixes boundary the lowercondition boundary [3]. condition Hence, [3]. energy Hence, flows energy into the flows into the withfrom maximizing dissipation rate of systems Gibbs free eshemical msynonymous information to121 out-compete that allows abiotic them systems tothe out-compete in abiotic in energy potential athe fixed upper boundary condition into the surface layer box. stant temperature and pressure. Living organisms are viewed here as to achieve an MEP state (Vallino, 2010), we cannot a prior set hvia [fixes 2which ]resources H CO [2], NH [2] recisely useful information (Adami 2002, Adami et al. 2000), via 2fixes 3the 3(nitrogen) iments, sediments, 121 which sediments, fixes which the lower the boundary lower boundary lower condition boundary condition condition Hence, [3]. Hence, energy [3]. energy Hence, flows into flows energy the flows the into the system 122 via box system [1] and box [1] and (nitrogen) resources flow in[3]. via flow box [2]. inbottom via All box constituents [2]. All constituents areinto allowed aremodel allowed onstant temperature and pressure. Living organisms are viewed here as 4as we do not know the nature der Here, appropriate we will examine conditions. how Here, information we will examine how information Nitrogen in the form of ammonia is allowed to diffuse into the box [2] from parameters in a kinetic ting autocatalytic reactions forS the[2], dissipation of free energy. We will S 1 resources 2[2] duction of entropy when averaged over time. It is the of of the community composition nor the corresponding kinetic tem system 122 via box via system [1] box and via [1] resources box and [1] resources (nitrogen) and (nitrogen) flow (nitrogen) in flow via box inability flow via [2]. box in All via [2]. constituents box All[2]. constituents All are constituents allowed are allowed tating autocatalytic reactions for the dissipation of free energy. We will ropy space. production For(Vallino ourfixes averaged system, over wethat will space. consider For condition our system, we will consider ediments, which the lower boundary [3]. Hence, energy flows into theare allowed nver hypothesis 2010), the acquisition of Shannon information parameter values that will lead to an MEP state. Kinetic parmation that allows them2010), to out-compete abiotic systems in information ain hypothesis (Vallino that the acquisition of Shannon F ( k ) del nystem in where each box entropy is largely production governed in each by two box is largely governed by two via box [1] and resources (nitrogen) flow in via box [2]. All constituents are allowed rameters need to be dynamic to reflect community composi2 , 3 Adami et al. 2000), via ore precisely useful information (Adami 2002, 4 this objective, 4 ppropriate conditions. Here, we will examine how information tion changes. To achieve we have developed a more precisely usefulsynthesis information (Adami 2002, synthesis Adami et and al. 2000), via ned ucture the (i.e., rate of catalyst) biological structure and (i.e., catalyst) he production of entropy when averaged over time. It is the ability of novel kinetic expression that can capture reaction kinetics in 4 4 4 NHour production averaged over space. For system, we will 3[3] the production of entropy when averaged over time. It is consider the ability of a manner consistent with community compositional changes. e information that allows them to out-compete abiotic systems in 4 expression takes the familiar hyperhereinformation entropyFig. production in each box isbox largely governed bya two The form of our kinetic 1. Schematic of theto two model representing waterin colore that allows them out-compete abiotic systems nder appropriate conditions. Here, we will examine how information bolic shape of the Monod equation (Monod, 1949), where umn with structure unit cross (i.e., sectional area with associated state variables he rateappropriate of biological catalyst) synthesis and under conditions. Here, we will examine how information substrate specific up-take rate is given by (d−1 ), and boundary conditions. All fluxes boxeswe andwill boundaries ropy production averaged over space. For between our system, consider ! are governed by diffustion (see Eq. (12)) only CH O2consider and ntropy production averaged over space. For ourand system, we2 O, will Y Cj del where entropy each box is largely by two ∗ 2 2 H2 CO3 production are exchangedinaccross boundary [0] andgoverned box [1]. Similarly, νi = ν εi (1 − εi ) (4) nal ndsentropy on either production local depends entropy either or global entropy only NH3or isglobal allowed across boxbox [2]local and boundary [3]. Boundary odel where entropy production inoneach is largely governed by two Cj + κ ∗ εi4 j =1 ned the rateconcentrations of biologicalare structure (i.e., catalyst) synthesis and held at fix values throughout the simulation and xon, model wethe use reminiscent arepresent simple two-box of external ocean model reminiscent of ocean mined rate of biological structure (i.e., catalyst) synthesis the environmental conditions (see Table and 2). Interand specific growth rate readily follows from Eq. (4) as



−1

entropy production for the whole system, σ˙ I aphotic , upper box, σ˙ I [1], (d ), els cesses developed in thenalphotic to capture and aphotic processes zones in the of photic and zones of and lower box, σ ˙ [2], are conducted around the red, blue and green I ntropy production depends on either local or global entropy µi = εi νi (5) nmary and in Allen oceanography, 2002).boundary As customary weboxes, modelresepctively. in a water oceanography, we model a water dashed e use a simple two-box model reminiscent of ocean es surface and boundaries area, so that have extensive unit surface variables area, so that extensive variables In Eqs. (4) and (5), ν ∗ εi2 (1 − εi2 ) is the maximum specific 2 ∗ 2 mveloped to capture processes in the photic2 and aphotic zones of 2 ed epth, h, a per andmwill basis. be calculated Model focus onon aisper placed m local basis.orModel focus is placed uptake rate of substrate, εi ν εi (1 − εi ) is equivalent to the nalonentropy production depends either global entropy ε is the growth efficiency for biological structure synthesis maximum specific growth rate term in the Monod equaAllenentropy 2002).i production As customary in oceanography, weor model a water −3 ) of one or more ernal depends on either local global entropy energy truction by of synthesis biological and structures destruction (S of and biological S ) structures (S and S ) from either CH O or represents moles of protion, C is the concentration (mmol m 2 2 i1 j 2 modeli1 reminiscent on, we use a simple two-box of ocean dtion, boundaries have unit surface area, so that extensive variables duced per mole of substrate consumed, so it is dimensionless. growth limiting substrates, such as NH 3 , εi is the growth efwe use 3(CH a) simple two-box model reminiscent of ocean Oels constituents H2CO within the ,two O2 layers, and Hin2CO but withinand the two layers, butof 2 and 2O, NH 3processes 3)photic developed to capture aphotic zones 2 the Note, as growth efficiency approaches zero, both reactions ficiency (dimensionless) as used in reactions (1) and (2), and h, anddeveloped will be calculated onprocesses a per m in basis. Modeland focus is placed ∗ (d−1 ) and κ ∗ (mmol m−3 ) are universal parameters that odels to capture the photic aphotic zones of represent the complete oxidation of reduced carbon to CO ν ween accounted boxes for.boundaries The iszone accounted is oceanography, for. The photic zone is 2 nisand Allen and 2002). Asphotic customary in we model a water y byand synthesis and destruction of(1), biological structures (Smodel S ) and2002). H2 O. In reaction biological structure, catalyzes remain fixed regardless of community composition. Equa2water 1 ,and son Allen As customary in oceanography, we a te e the but resource limited, is itswhile complement: thesugar aphotic energy zone is extensive its complement: energy tion (5) differs from the Monod equation in several impores andaphotic boundaries have unit surface area, socarbon that itszone own synthesis from (unit basis) andvariables available ituents (CH O2 and Hsurface thethat twoextensive layers, but 2O, NH3,have 2CO3) within oxes boundaries unit area, so variables ammonia thereby fulfilling the producer role. Retant ways. The effective Monod “constant”, KM , is repreeplete. heseand two differing model states; however, these wediffering states; however, epth, h,Our and will becaptures calculated ontwo a per m2primary basis. Model focus iswe placed 2 action (2) represents heterotrophic organisms that consume boxes and boundaries is accounted The photic Model zone isfocus is placedsented by κ ∗ εi4 , so depends on growth efficiency. The radepth, h, and will active be calculated onwith afor. perenergy mof basis. iation ia photosynthetic with energy input via radiation the diffusion input via the of) tional for this dependency is that nutrients at low concentraenergy by synthesis destruction of biological structures (S1 ,and biologicaland structure synthesized from reaction (1),diffusion but S 2 resource limited, while the aphotic zone is its complement: energy e energy by synthesis andlayer of biological (S21 and ) require free energy expenditure to transport them into as S2tion also functions indestruction a cannibalistic mode by structures consuming ondition m a fixed into upper the boundary surface condition box. into the surface layer box. constituents (CH2O, NH3, O2 and H2CO3) within the two layers, but well. While cannibalism could be replaced by inclusion of the cell against a concentration gradient; consequently, orOur model captures these two and differing states; however, we al.fuse constituents (CH2O,box NH Hthe the[2]two layers, but 3, O 2 into 2CO 3) within of ammonia into the is bottom allowed to [2] diffuse from bottom box from a sufficient number of higher trophic levels for closure, this ganisms that have evolved to grow under low nutrients conween boxes and boundaries is accounted for. The photic zone is otosynthetic active radiation with energy input via the diffusion of would lead to greater degrees of model freedom that are not ditions (i.e., K-selected, Pianka, 1970) will, by thermodyetween boxes and boundaries is accounted for. The flows photicinto zonethe is sion the [3]. lower Hence, boundary energy condition flows the into [3]. theHence, energy te but resource limited, while aphotic zone is its complement: energy namic necessity, grow at lower efficiencies. The (1 − ε2 ) important for this study, but would be more typical of natural ixed upper boundary condition into the surface layer i lete resource while thebox aphotic zone is itsbox. complement: energy deplete. nresources viabut box (nitrogen) Alllimited, constituents flow in via are allowed [2]. All constituents are allowed Our[2]. model captures these two differing states; however, we monia is Our allowed to captures diffuse into thetwo bottom box [2] fromhowever, we replete. model these differing states; www.earth-syst-dynam.net/2/69/2011/ Earth Syst. Dynam., 2, 69–85, 2011 via photosynthetic active radiation with energy input via the diffusion of lower boundary condition Hence, energy flows into via photosynthetic active [3]. radiation with energy input viathe the diffusion of m a fixed upper boundary condition into the surface layer box. 4 4 urcesa fixed (nitrogen) in via condition box [2]. All allowed rom upperflow boundary intoconstituents the surface are layer box. of ammonia is allowed to diffuse into the bottom box [2] from

-1

Specific Growth Rate, μ (d )

iogeochemistry models developed 109 biogeochemistry to capture processes models in the developed photic and to capture aphoticprocesses zones of in the photic and aphotic zones of 109110 biogeochemistry developed capture in in theoceanography, photic and aphotic zonesaof oceans (Fig. 1)models (Ianson and Allento2002). Asprocesses customary we model water ceans (Fig. 1) (Ianson and 110 Allen 2002). oceansAs (Fig. customary 1) (Ianson in oceanography, and Allen 2002). weAs model customary a waterin oceanography, we model a water 110111 oceans (Fig.where 1) (Ianson and and Allen 2002). Ashave customary in oceanography, model avariables water column all boxes boundaries unit surface area, so thatwe extensive olumn where boundaries column have where unit all surface boxesarea, and so boundaries that extensive have unit variables surface area, so that extensive variables 72all boxes and111 J. J. Vallino: Differences and implications in biogeochemistry 2 111112 column and boundaries unit surface extensive variables only where dependallonboxes box depth, h, and willhave be calculated onarea, a perso mthat basis. Model focus is placed 2 2 ! is placed nly depend on box depth, h, 112 and will onlybedepend calculated on box on adepth, per mh, basis. and will Model be calculated focus is placed on a per m basis. CModel focus 11250 on box h, andbywill be calculated on a per mof2 biological basis. placed 113 onlyondepend dissipation of depth, free energy synthesis and destruction structures ∗ 2Model 2 focus is(S T1 and S2) C . r2 = ν2 C ν S ε2 (1 ε2 ) (7) n dissipation of free energy113 by synthesis on dissipation and destruction of free energy of0.7 biological by synthesis structures and destruction (S21 = and ) −biological 2of C(S structures + κ ∗S ε24 ) (S21 and S2) 113114 on dissipation of free energy by synthesis and destruction of biological structures and from and to chemical constituents (CH2O, NH3, O2 and H2CO3) within the two layers, but2 T1 om and to chemical constituents 114 (CH from2O, and NH to3,chemical O2 and Hconstituents (CH the2O, twoNH layers, and H2CO3) within the two layers, but 0.63) within 2CO 3, O2 but 11440 and toof chemical (CH NH3, O2 and H2CO3)(6) within the allow two layers, but 115 from entropy mixing constituents between boxes and is accounted for. photic is Equations and The (7) uszone to explore reaction rates for 2O,boundaries ntropy of mixing between 115 boxes and entropy boundaries of0.8mixing is accounted between boxes for. The andphotic boundaries zonecommunity isis accounted for. The photic zoneonly is ε1 and ε2 different compositions by varying 115116 entropy of mixing and boundaries is accounted for. zone The photic zone is typically energybetween replete boxes but resource limited, while the aphotic is its complement: energy between 0 and 1. Of course, other mathematical ypically energy replete but 116 resource typically limited,energy while the replete aphotic but resource zone is itslimited, complement: while the energy aphotic zone is its complement: functions energy 0.5 11630 replete replete. but resource limited, while the aphotic zoneinisplace its complement: could be used of Eq. (4), butenergy this is not our primary 117 typically limitedenergy but resource Our model captures these two differing states; however, we mited but resource replete.117 Our model limited captures but resource these two replete. differing Our model states;interest captures however, we two differing wetopic that forthese this manuscript, butstates; it is anhowever, interesting 117118 limited but energy resource replete. Our model captures two differing states; however, we replace input via photosynthetic activethese radiation with energy input via the diffusion of warrants further research. For instance, we use the same 0.9radiation eplace energy input via photosynthetic 118 replace active energy input via withphotosynthetic energy input via active the radiation diffusion with of energy input via the diffusion of ef∗ ε 4 , regardless of substrate type in fective Monod constant, κ 11820 energy input via photosynthetic active radiation with energy input via the diffusion of 119 replace chemical potential from a fixed upper boundary condition into the surface layer box. i 0.4 hemical potential from a fixed 119 upper chemical boundary potential condition frominto a fixed the surface upper boundary layer(6) box. condition into the surface box. nor do we Eqs. or (7), which could likely belayer improved, 119120 chemical potential upperisboundary condition surface box layer[2] box. Nitrogen in the from form aoffixed ammonia allowed to diffuse intofor theother bottom from account complexities of cellular growth (Ferenci, Nitrogen in the form of ammonia 120 isNitrogen allowed in to the diffuse forminto of ammonia the bottomis box allowed [2] from to diffuse into the bottom box [2] from 1999; Smith et al., 2009). 12010 in thewhich form of ammonia is allowed to diffuse into the bottom box [2] from 121 Nitrogen sediments, fixes the lower boundary condition [3]. Hence, energy flows into the 0.3 ediments, which fixes the lower 121 boundary sediments, condition which fixes [3]. Hence, the lower energy boundary flowscondition into the [3]. Hence, energy flows into the 121122 sediments, which the resources lower boundary condition [3]. energy flows into balance the system via boxfixes [1] and (nitrogen) flow in via Hence, box [2]. Alland constituents are allowed 2.1 Free energy entropy equations 0.15 ystem via box [1]0 and resources 122 (nitrogen) system via flow boxin[1] viaand boxresources [2]. All (nitrogen) constituents flow are in allowed via box [2]. All constituents are allowed 122 system via box [1] and 1.0 resources (nitrogen) flow in via box [2]. All constituents are allowed 0

1000

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 parameters values of 350 d−1 and 5000 mmol m−3 for ν* and κ*, respectively.

A standard entropy balance around the control volume (Fig. 1, red dashed boundary) takes the general form, 6 4 Q˙ dS X = F (k)Sˆk + + σ˙ I dt k=1 T

4 4

4 (8)

where S is system entropy per unit area (J m−2 ◦ K−1 ), Sˆk is the molar entropy (J mmol−1 ◦ K−1 ) of constituent k trans˙ ported across the boundary by flux F (mmol m−2 d−1 ), Q −2 d−1 ), T is temperais the heat flux into the system (J m term in Eq. (4) approximately accounts for thermodynamic ture (◦ K) and σ˙ I is the internal entropy production rate per on biological kinetics, systems becauseorganize net reaction rate must apmanuscript ganize to maximize we constraints will assume entropythat to maximize entropy unit area of water column (J m−2 d−1 ◦ K−1 ). Under steady proach zero as reaction free energy tends toward zero (Jin  pation on, which rate is ofsynonymous Gibbs free energy with maximizing the dissipation rate of Gibbs free energy and Bethke, 2003; Boudart, 1976), which is equivalent to a state conditions, dS dt = 0 and internal entropy production ms organisms under constant are viewed temperature here as andofpressure. Living organisms areEq. viewed as growth efficiency 1. The leading εi2 term in (4) ishere emequals the entropy transported into the control volume via ipation alysts facilitating of freepirically energy. autocatalytic We will reactions for the dissipation of free energy. uptake We will motivated to account for reduced substrate mass and heat flux. In this manuscript, we are solely conrates atinformation low(Vallino growth 2010), efficiency. tinue quisition our main of Shannon hypothesis that the acquisition of Shannon information cerned with internal entropy production, σ˙ I , and we will asuniversal ν ∗ and2002, κ ∗ , Adami have been chosenvia sume homogeneity and isothermal conditions prevail in each mi n 1948), 2002, Adami or moreetThe precisely al. 2000), useful via parameters, information (Adami et al. 2000), box. Unlike other analyses (Meysman and Bruers, 2007), to qualitatively match observations of bacterial growth. Undn over facilitates time. the It isproduction the ability of entropy when averaged over time. It is the ability of we include entropy of mixing within the system boundary der oligotrophic conditions often found in ocean gyres’, bacystems ompetetoabiotic storeterial information systems in thatgrowth allows them to typically out-compete abiotic in (Fig. 1), as biological systems can significantly alter transspecific rate is 1–2 d−1 , systems with growth port characteristics (Erwin, 2008; Jones et al., 1994); howefficiencies ofconditions. 10–20 %, Here, and substrate concentrations in the will production examineunder how information appropriate we will examine how information ever, we will not consider direct alteration of transport coefµM range (Delover Giorgio Cole, 1998; et oFor facilitate our system, entropy we or production willlower consider averaged space. and For our system, weCarlson will consider ficients in our analysis. al., 1999). At the other extreme, bacteria grown under ideal ox two-box is largely model governed whereby entropy two production in each box is largely governed by two For our two box model, internal entropy production occurs laboratory conditions show specific growth rates as fast as −1 via destruction of chemical potential via reactions (1) and (2), ,s catalyst) that determined synthesis rate of biological structure (i.e.,around catalyst) synthesis and 50 dtheand , with growth efficiencies 50–60 %, provided σ ˙ , ˙ F , associated with relaxing r and by entropy of mixing, σ substrate concentrations are in the mM range (Bailey, 1977; ion. chemical potential gradients between boxes [1] and [2] and Lendenmann and Egli, 1998). Equations (4) and (5) cap∗ their associated boundaries. For chemical reactions, entropy ture these approximate trophic extremes with values of κ −3 ∗ −1 production per unit area in a given box with depth h (m) is of 5000 mmol m and ν of 350 d (Fig. 2). The usefuldel System readily determined under constant pressure and temperature ness of the kinetic expressions (Eqs. 4 and 5) is their ability from the Gibbs free energy of reaction, 1G , reaction rate, r, r mine her local howor internal global entropy entropyproduction depends on either local or globalunder entropy to generate organismal growth kinetics expected oligand absolute temperature, T , as given by (Eu, 1992 pp. 131– otrophic to extreme miniscent on optimization, of ocean we useconditions a simple two-box model eutrophic reminiscentconditions of ocean by vary−2 d−1 ◦ K−1 ), 141) (J m ing only growth efficiency, ε (Fig. 2). he hemistry photic models and aphotic developed zones of to capture processes in the photic and aphotic zones of Based on Eq. (4), the rate of reactions (1) and (2) are exh[j ] eanography, Fig. 1) (Ianson weand model Allen a water 2002). As customary in oceanography, we model a water σ˙ ri [j ] = − ri [j ]1Gri [j ], (9) pressed by (mmol m−3 d−1 ), T

where a, so that all boxes extensive and variables boundaries have unit surface area, so that extensive variables

where [j ] corresponds to either box [1] or [2] and i to either reactions (1) or (2). While Gibbs free energy of reaction CCH2 O CNH3 r1 = ν1 Cby = ν ∗S ε12 (1 − ε2 ) C(S and (6)S2) pation biological of free structures energy (S11synthesis and of biological structures 4 4 ∗ ∗ 2) and1destruction 1 1 is usually straight forward to calculate for a given reaction CCH2 O + κ ε1 CNH3 + κ ε1

! is placed pend m2 basis. on boxModel depth,focus h, and is will placed be calculated on a per m2!basis. Model focus

O d 3to) within chemical theconstituents two layers, but (CH2O, NH3, O2 and H2CO3) within the two layers, but

of d for. mixing The between photic zone boxes is and boundaries is accounted Earth Syst. Dynam., 2, 69–85, 2011 for. The photic zone is

yicenergy zone isreplete its complement: but resource energy limited, while the aphotic zone is its complement: energy

but ffering resource states; replete. however, Our we model captures these two differing states; however, we

www.earth-syst-dynam.net/2/69/2011/

96 alsoconditions. continue our main (Vallino 2010), that the acquisition of Shannon information 100 entropy production under appropriate Here, wehypothesis will examine how information may101 also facilitate production averaged over space. Foror our system, wesystem, willuseful consider may alsoentropy facilitate entropy production averaged over space. For our weinformation will consider(Adami 2002, Adami et al. 2000), via 105 97 (Shannon 1948), more precisely

a simple model where entropy production in each boxiniseach largely governed by two 102 two-box a simple two-box model where entropy production is largelyof governed two averaged over time. It is the ability of 98implications evolution thebox production entropy by when 2 Differences Model System J. 106 J. Vallino: and infacilitates biogeochemistry 73 reactions that determined the rate of biological structure (i.e., catalyst) synthesis and 103 reactions that determined the rate of biological structure (i.e., catalyst) synthesis and 99 living systems to store information that allows them to out-compete abiotic systems in 107 To examine how internal entropy production either local or global entropy degradation. stoichiometry, both reactions (1) and (2) require somedepends dis- onTable 104 degradation. 1. Model fixed parameter values. 100 entropy production under appropriate conditions. Here, we will examine how information cussion bothoptimization, involve synthesis oradecomposition 108 because production we use simple two-boxofmodel reminiscent of ocean living biomass, or more generally, what wefacilitate refer to as bio- production averaged 101 may also entropy over space. For our system,Value we will 105 (units)consider 109 biogeochemistry models developed to capture processes inParameter the photic Description and aphotic zones of logical structure. 102 and a simple two-box model where production in each box is largely governed by) two 2 106 Model2System ν∗ Universal kinetic 350 (d−1 110 oceans (Fig.are 1) (Ianson Allen customary in entropy oceanography, we model aparameter water Model System Living organisms often thought to 2002). be veryAslow en−3 ) κ* Universal kinetic parameter 5000 (mmol m tropy structures because they are highly orderthat systems. How-surface 103 reactions determined the rate of biological structure (i.e., catalyst) synthesis and 111 column where all boxes and boundaries have area, so thatDiffusion extensive variables To examine how internal entropy production depends ondepends either local or global entropy coefficient 1 × 10−4 (m2 d−1 ) 107 To examine how internal entropy production onunit either ever, statistical thermodynamic calculations show that en- localDor global entropy ` m2 basis.Characteristic length 2.5 × 10−5 (m) 104 degradation. 112 contributions only depend box depth, h, and willreminiscent besignificant calculated on a per Model focus is placed production optimization, we usedue aon simple two-box model of ocean tropy to patterns only become 108 production optimization, we use a simple two-box model reminiscent of ocean h[1], h[2] Depth of boxes [1] and [2] 10 (m) compared todissipation the entropyof offree theenergy substrate the in pattern is writ113 on by synthesis and destruction of biological structures (S and S ) biogeochemistry models developed to capture processes the photic and aphotic zones of ρ N to C ratio of 1/6 (atomic) 1 2 i 105 109 biogeochemistry to capture processes in the photic and aphotic zones of ten in (i.e., DNA,models protein)developed when ordering occurs at the atomic T Temperature 293.15 (◦ K) oceans (Fig. 1) (Ianson and Allen 2002). As customary in oceanography, we H model water 114 from and to chemical constituents (CH O, NH Oof2 and CO3a)we within layers, but 2 3 2pH scale (Morrison, 1964). on information content pHthea two 8.1 110 oceans (Fig. 1) (Ianson andBased Allen 2002). As customary in, oceanography, model water 1061955;2 Johnson, Model1970), System ISvariables Ionic Strength 0.7 (M) a 115 bacterial cell (Morowitz, entropy column where all boxes and boundaries have unit surface area, so that extensive entropy of mixing between boxes and boundaries is accounted for. The photic zone is 111 column where all boxes and boundaries have unit surface area, so that extensive variables ◦ 1G Standard Gibbs free energy −498.4 (J mmol−1 ) reduction due to molecular order is trivial compared to enOx logical systems organize to maximize entropy 2 107 To examine how internal entropy production depends on either local or global entropy 2 aphotic only112 depend on box depth, h, and will be calculated on a per m basis. Model focus is placed 116 typically energy replete but resource limited, while the zone is its complement: energy of CH O oxidation to H CO 2 2 3 only on box depth, h, andsynthesis will be calculated on1964). a per m basis. Model focus is placed tropydepend contributions of substrate (Morrison, ◦ 1G Standard Gibbs free energy 0 (J mmol−1 ) aximizing the dissipation Gibbs energy me that biological systems organize toa free maximize entropy Hence the totalrate entropy of random sequence polymer108 production optimization, we two use differing a (S simple two-box model reminiscent of ocean limited but resource Our model captures we on dissipation of free energy byofsynthesis and destruction ofof biological structures S ) however, 1 andstates; 113 on117 dissipation of free energy byreplete. synthesis and destruction ofthese biological structures and S2) from of2(S1 synthesis CH2 O ized aminothe acids is only slightly a energy specific amino dfrom pressure. Living are(CH viewed here asthan and NH ous with dissipation rate ofgreater Gibbs freeH 118 organisms replace energy input photosynthetic active radiation with energy via3 the diffusion of andmaximizing tofrom chemical NH and )models within the two layers, butinput 109 developed to capture processes in the photic and aphotic zones of 2O,via 3, O2O, 114 toconstituents chemical constituents (CH NH32,CO O23and HFuracid and sequence that gives rise to abiogeochemistry functioning protein. 2 2CO3) within the two layers, but actions for the dissipation of free energy. We will 119 potential fromare a fixed upper boundary condition into the surface perature pressure. Living organisms viewed here entropy mixing between boxes and110 boundaries is accounted for. The photic zone is thermore, both theoretical calculations and empirical data oceans (Fig. 1) (Ianson and Allen 2002). As customary 115 ofand entropy ofchemical mixing between boxes and boundaries isas accounted for. The photic zone islayer box.in oceanography, we model a water at biological systems organize to maximize entropy show that bacteria and yeast have higher standard specific otypically 2010), that the acquisition of information 120replete Nitrogen in Shannon the form of ammonia allowed to diffuse into the bottom [2]surface from area, so that extensive variables energy butdissipation resource limited, while the is aphotic zone is its energy catalytic for the of free energy. We will 111 column all boxes andcomplement: boundaries havebox unit 116 reactions typically energy replete but resource limited, while the aphotic zone is itsRcomplement: than glucose 1999a, b, where 2003). Biological th maximizing dissipation rate(Battley, of Gibbs free energy where is the ideal energy gas constant (8.3145 × 10−3 J mmol−1 will assumeentropies thatthe biological systems organize to maximize entropy 121 but sediments, which fixes the lower boundary condition [3]. energy into information (Adami 2002, Adami et112 al. 2000), via ◦ is flows ◦ KHence, −1 ), we limited butlimited resource replete. Our model captures these two differing states; however, esis (Vallino 2010), that acquisition of Shannon information structure has athe higher specific entropy than “simple” growth the free energy CH2isOplaced only depend on box depth, h, and will1G be calculated on athe perGibbs m2 basis. Model of focus 117 resource replete. Our model captures these two differing states; however, we standard Ox ure and pressure. Living organisms are viewed here as ynonymous with maximizing the dissipation rate of Gibbs free energy e that biological systems organize to maximize entropy −1 ) and 1G◦ is substrates, and it can be shown that the standard Gibbs free 122 system via box [1] and resources (nitrogen) flow in via box [2]. All constituents are allowed oxidation by O to H CO (−498.4 J mmol replace energy input via photosynthetic active radiation with energy input via the diffusion of2diffusion opy when averaged over time. via It2002, isphotosynthetic the ability of 2 3 sely useful information (Adami al. 2000), viafreewith 118 replace energy input active radiation energy via the of 113Adami on et dissipation energy by input synthesis and destruction of biological structures (S1 and S2) energy of reaction for synthesizing bacteria or of yeast from ticwith reactions for the dissipation ofLiving free energy. We will temperature and pressure. organisms are viewed here as sstant maximizing the dissipation rate of Gibbs free energy the standard Gibbs free energy for biological structure synchemical potential from a fixed upper boundary condition into the surface layer box. ows them to out-compete abiotic systems in glucose, ammonia, phosphate and sulfate, without tion119 of entropy when averaged over time. Itfrom is the ability ofCO2 prochemical potential from a fixed upper boundary condition into the thesis surface layer box.O 114 and to chemical constituents (CH O, NH , O and H CO from CH and NH , which is set the to two zero layers, for ourbut 2 3 2 2 3) within 2 3 Vallino 2010), thatreactions the acquisition Shannon information ing autocatalytic the dissipation of free energy. We will erature and pressure. Living organisms areless viewed here as duction (i.e., ε =for 1), isisof slightly than zero, so is a reacNitrogen inNitrogen thewe form ammonia allowed to diffuse into the bottom boxbottom [2]study from ditions. will examine how information because it is negligibly small (Vallino, 2010). We use 120 inofto the form of ammonia is allowed to diffuse into the box [2] from tion thatHere, allows them out-compete abiotic systems in 115 et al. entropy ofvia mixing between boxes and boundaries is accounted for. 4 The photic zone is tion that can occur spontaneously (Vallino, 2010). Although seful information (Adami 2002, Adami 2000), ntalytic hypothesis (Vallino 2010), that theofacquisition ofWe Shannon information reactions for the dissipation freecondition energy. will theinto approach of Alberty (2003, 2006) to calculate the stansediments, which fixes the lower boundary [3]. Hence, energy flows the raged space. ourwe system, we will boundary consider 121over sediments, which fixes lower condition [3].sysHence, energy flows into the the aphotic zone ◦is its complement: energy poorly For appreciated, thethe “order” ascribed to biological opriate conditions. Here, will examine how information 116 typically energy replete but resource limited, dard Gibbs free while energy of reaction, 1GOx , which accounts fssystem entropy when averaged over time. It is the ability of ore precisely useful information (Adami 2002, Adami et al. 2000), via (Vallino 2010), that the acquisition of Shannon information tems[1] simply notand thermodynamically box and (nitrogen) flow in important via flow box [2]. constituents are allowed 122 via via box [1] resources (nitrogen) incompared viaAll box [2]. All are allowed forconstituents proton dissociation equilibria between chemical species oduction insystem each isbox isresources largely governed by two duction averaged over space. For our system, we will consider 117 limited but resource replete. Our model captures these two differing states; however, we to of their bulk composition. Hence, it in is thermodynamically hat allowsinformation them to out-compete abiotic systems e production entropy when 2002, averaged overettime. It is the (CO2 + H2 O ↔ H2 CO3 ↔ H+ + HCO− ly useful (Adami Adami al. 2000), viaability of 3 , etc.) at a pH of gical structure (i.e., catalyst) synthesis and governed to proceed without freephotosynthetic possible for reaction (1) ◦ K.energy entropy production in each box is largely byadditional twoinput via 118 replace energy active radiation with input via diffusion of 8.1 and temperature of 293.15 Ionic strength (ISthe = 0.7 M) e information conditions. Here, we will examine that allows them out-compete abiotic systems in on of entropy when averaged over time. It information is the ability of energy input with εto 1. how However, as mentioned above, 1= is used to estimate activity coefficients, so that concentrations ate of biological structure (i.e.,also catalyst) synthesis andpotential fromasa fixed upper boundary 4 condition 119 we chemical the surface layer box. reaction ratesFor are constrained thermodynamics 4 into n averaged over space. our system, willbyconsider can be used in the logarithmic correction terms in Eqs. (10) der appropriate conditions. Here, we will examine how information on that allows them to out-compete abiotic systems in Gibbs free energy of reaction approaches zero. For a reand is (11) insteadtoofdiffuse activities as the described Alberty (2003), 120 Nitrogen the form of ammonia allowed into bottombybox [2] from py production inaveraged each box is largely byproceed twoinwe ropy production space. Forhow system, will consider action to proceed at high governed rate itour must irreversibly; riate conditions. Here, weover will examine information and the 106 terms convert µmolar (or mmol m−3 ) to standard 121 growth sediments, which fixes the lower boundary condition [3]. Hence, energy flows into the consequently, organisms whose efficiency is less than biological structure (i.e., catalyst) synthesis andwill molar concentrations. el where entropy production in our each box is we largely governed by two ction averaged over space. For system, consider 100 % can grow faster than those operating at very high effiof mixing readily between boxes 122 system via box [1] and resourcesEntropy (nitrogen) flow inis via boxcalculated [2]. All constituents areand allowed diminishing returns of high efficiency ned the rate of structure (i.e., catalyst) synthesis ntropy production in The each box is largely governed bygrowth two and on dependsciencies. onbiological either local or global entropy boundaries based on flux and chemical potential obtained is embed in our growth kinetics equation, as plotting Eq. (5) from concentration differences between boxes (Meysman of biological (i.e., catalyst) andthat maximum spetwo-box model reminiscent of ocean as astructure function of εeither exhibits ansynthesis optimum, py production depends on local or global so entropy and Bruers, 2007; Kondepudi and Prigogine, 1998). For our cificingrowth ratereminiscent under nutrient saturation ( dµ simple model, we only permit diffusive transport between 4 ture processes themodel photic and aphotic of dε C →∞ = 0) e a simple two-box ofzones ocean j √ boxes and boundaries, so a material flux of chemical species occurs as ε → in3/5 0.77. As customary in processes oceanography, we model water zones of oped to capture the≈ photic anda aphotic k from box or boundary [i] to box or boundary [j ] is given oduction depends either local or global entropy The on Gibbs free energy change for reactions (1) and (2) by (mmol m−2 d−1 ), ave unit surface area, so that extensive variables en 2002). As customary in oceanography, we model a water can be expressed as a linear combination of biosynthesis and

mple two-box model 2reminiscent oceanlocal −1 nal entropy production depends onofeither or ),global entropy CH O oxidation, asModel given by (Jextensive mmol calculatedhave on a 2unit per m basis.area, focus is placed oundaries surface so that variables oproduction capture inon photic and aphotic zones of ◦local ◦ on, we useprocesses a1G simple two-box model reminiscent of ocean depends or global entropy −the εeither + ε 1G (10) r1 = (1 1 )1G 1 2 Ox andwill destruction of biological S2) is placed 1 andfocus nd be calculated on aperstructures m basis. (S Model  02). As customary in oceanography, we model a water 1−εand ε1 1 els developed to capture in of theocean photic aphotic zones of a simple two-box model processes reminiscent Cbut H2 CO3 C(S and S ) the two layers, 3, O2 and 2CO3) within yNH synthesis andHdestruction of6(1−ε biological structures 2 (1−ρ)) 11 1   +RT lnin the 10 that ies have unit2002). surface so extensive variables ned and Allen Asarea, customary in oceanography, we a water to capture processes photic and aphotic zones ρ 1−ε 1of 1 εmodel C C C CH O ndaries is accounted for. The photic zone is 2 NHbut O nts (CH2O, NH3, O2 and H22CO3) within the two layers, 3 2 ll2002). be onhave a perinunit moceanography, basis. focus placed andcalculated boundaries surfaceModel area,we so thatisextensive nes As customary model a water variables ◦ zone is ed, and while the aphotic is its◦ complement: xes boundaries accounted for.(1The 1G (1zone − ε2 )1G − ε2photic )1Genergy (11) r2 = is Ox −structures hesis h, and destruction of biological (Svariables and S2focus ) 1Model epth, and will calculated on a per m basis. isplaced ndaries have unitbe surface area, so that extensive  (1−ε )ρ ε2 ptureslimited, these two differing states;zone however, we 2 ource while the aphotic is its complement: CH1−ε CNH3 2 energy C 2 Hwill NH , O2 and H the two 2 COstructures 3but 2O, be 3synthesis 2CO 3) within energy bycalculated and destruction of Model biological and S2) on a per m6(1−ε basis. focus is placed(S21  −ρ(1−ε ))layers, 2 2  +RT ln 10 , tive radiation with energy input via the diffusion of ur model captures these two differing states; however, we 1−ε2 C dynthesis boundaries is accounted for. The photic zone isCtheT1 two O2 layers, constituents O and(CH destruction biological structures and S2) but 2O, NH3,of 2 and H2CO 3) within (S ndary condition into the surface layer input box. via the diffusion of synthetic active radiation with energy while the aphotic zone is its complement: energy ween and boundaries is3)accounted photic slimited, (CHboxes within thefor. twoThe layers, butzone is 2O, NH 3, O 2 and H2CO to diffuse into the bottominto boxthe [2]surface from layer box. dedupper boundary condition www.earth-syst-dynam.net/2/69/2011/ captures these differing states; however, eand but resource limited, while the aphotic zonezone is we itsiscomplement: energy sdel boundaries istwo accounted for. The photic y condition [3].to Hence, into the[2] from nia is allowed diffuse energy into theflows bottom box tic active with energy input via the diffusion of plete. Ourradiation model these twoisdiffering states; however, we urce limited, whilecaptures the aphotic zone its complement: energy n) in viacondition box [2]. All are flows allowed erflow boundary [3]. constituents Hence, energy into the

D (12) (Ck [j ] − Ck [i])βi,j (k), ` where ` is a characteristic length scale for diffusion (2.5 × 10−5 m), D is the diffusion coefficient (1 × 10−4 m2 d−1 ) and βi,j (k) is 1 if flux of compound k is allowed between box or boundary [i] and [j ]; otherwise it is 0. Technically, Eq. (12) should be based on chemical potential gradient; however, this can be reasonably approximated by the concentration gradient in this case. The product of flux and chemical potential differences, 1µi,j (k), divided by temperature gives the entropy production per unit area for mixing Fi,j (k) = −

Earth Syst. Dynam., 2, 69–85, 2011

107 107To examine To examine howhow internal internal entropy entropy production production depends depends on either on either locallocal or global or global entropy entropy 108 108production production optimization, optimization, we use we ause simple a simple two-box two-box model model reminiscent reminiscent of ocean of ocean

74

J. J. Vallino: Differences and implications in biogeochemistry

109 109biogeochemistry biogeochemistry models models developed developed to capture to capture processes processes in the inphotic the photic and aphotic and aphotic zones zones of of being the lower boundary (Fig. 1). Elements between box oceans [i]entropy to [j ] entropy for species k, 2002). as As follows sdepends on eitheronlocal either or global or global 110 110local oceans (Fig. (Fig. 1) (Ianson 1)chemical (Ianson and Allen and Allen 2002). customary As customary in oceanography, in oceanography, we model we model a water a water of the 1×2 vec(J m−2 d−1 ◦ K−1 )

tor 3k are the stoichiometric coefficients for compound k

o-box odel reminiscent model111 reminiscent of column ocean of where ocean 111 column where all boxes all boxes and boundaries and boundaries havehave unit unit surface surface area,area, so that so that extensive extensive obtained from reactions (1) variables and variables (2), and r is a 2 × 1 vec1µi,j (k) T , given by Eqs. (6) and (7). 2 2 tor of reaction rates, [r r ] σ˙in = F (k) (13) 1 2 eses processes in the photic the and photic aphotic and zones aphotic of zones of i,jonly Fi,j (k) 112 112only depend depend on box on box depth, depth, h, and h, will and will be calculated be calculated on a on pera m perbasis. m basis. Model Model focusfocus is placed is placed T



The full model equations are detailed in Appendix A for



Cby customary ry in oceanography, in113 oceanography, weD model weamodel water a water k [i] 113on dissipation on dissipation of free of free energy energy by synthesis synthesis and destruction and destruction of 2biological of Obiological structures and21 S and ) Stwo k = CH O, NH3 ,(S1 ,(S in 2the 2) boxes, and 2 , H2 CO 3 ,structures =−

(Ck [j ] − Ck [i])βi,j (k)R ln

` extensive Ck [j ] parameter values used for all simulations are given in Table 1. rface unit area, surface so114 area, that114 extensive so that from from andvariables to andchemical to variables chemical constituents constituents (CH2(CH O, NH , O23,and O2 H and H32)CO within the two the two layers, layers, but but 2O,3NH 2CO 3) within We have intentionally designed our biogeochemistry

2 again activities been approximated by concenon culated a peron m2awhere basis. per m Model basis. focus Model isofhave focus placed is placed model to contain a photic few transport 115 115 entropy entropy of mixing mixing between between boxes boxes and boundaries and boundaries is accounted is accounted for. The for.only The photic zone zone is isrelated parameters (D,

trations. Given concentrations of the six constituents, CH2 O,

`, h) and effectively only two adjustable biological paramction d destruction of biological structures (S1 energy and (S S221replete )and ) resource andreplete corresponding to k =limited, 1,. .while . 6, while O116 , Hbiological NHstructures 2but 2of 2 CO 3 , typically 3 ,energy 116 typically butSresource limited, the aphotic the aphotic zonezone is itsiscomplement: its complement: energy energy along with the growth efficiencies ε and ε in each box,

eters per box, giving a total of four control variables for

1 2 H nd3,HO22CO and within within two layers, the two layers, butreplete. 3) H 2CO 3)the the two box model: ε1 [1], εhowever, ε1 [2], 117 117 limited limited but resource butbut resource replete. Our Our model model captures two two differing differing states; states; however, weand ε2 [2]. Because 2 [1], we internal entropy production per unit water-column areacaptures as- thesethese

choosing a value for εi [j ] effectively selects the kinetic charsociated with reactions and mixing canphotosynthetic be calculated from accounted aries is accounted for. The for. photic The zone photic isenergy zone isvia photosynthetic 118 118 replace replace energy input input via active active radiation radiation withwith energy energy inputinput via the viadiffusion the diffusion of of acteristics of the community in box [j ], we can examine

Eqs. (6–13) and is given by (J m−2 d−1◦ K−1 ),

hewhile aphotic the zone aphotic its zone complement: is its potential complement: energy energy how entropy changes as a function of commu119 is119 chemical chemical potential fromfrom a fixed a fixed upper upper boundary boundary condition condition into into the production surface the surface layer layer box.box. 6 X

nity composition, which has a collective respiration given by
Pinto ures e two these differing two differing states; however, states; however, we we 2 into 120 Nitrogen the the σ˙form of ammonia of+ammonia is allowed is allowed to diffuse to diffuse the bottom the bottom box box [2] from [2] from σ˙120 [1] = σ˙Nitrogen [1] + σ˙ in [1] +inform σ˙ (14) I

r1

r2

F0,1 (k)

i=1 (1 − εi [j ])ri [j ].

F1,2 (k)

More importantly for this study, the

k=1 eonradiation with energy with energy viasediments, input the diffusion via the diffusion of fixes of values of Hence, ε1 [1], εenergy ε1flows [2], [2] that maximize en2 [1], 121input 121 sediments, which which fixes the lower the lower boundary boundary condition condition [3]. [3]. Hence, energy flows intoand into theε2the

tropy production 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 of this approach has been used to determine how biogeochemistry develops over time (Vallino, 2010); how4 4 dimensions, ever, for a system that has one or more spatial 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), with the cost functions defined by Eqs. (14) and (15) as given by,

6 dition ary condition into the122 into surface the layer surface box. layer box.
(nitrogen) 122 system system via box via X box [1] and [1] resources and resources (nitrogen) flowflow in via inbox via box [2]. [2]. All constituents All constituents are allowed are allowed

σ˙ I [2] = σ˙ r1 [2] + σ˙ r2 [2] +

to e into diffuse the into bottom the box bottom [2] from box [2] from k=1

σ˙ F1,2 (k) + σ˙ F2,3 (k)

(15)

ondition [3]. Hence, [3]. energy Hence,flows energy into flows the into the

σ˙All σ˙ r1 [1] + constituents σ˙ r2 [1] +σ ˙ r1 [2] +σ ˙ r2 [2] I =[2]. low ia box in via [2]. box constituents All are allowed are allowed +

6 X

(16)


σ˙ F0,1 (k) + σ˙ F1,2 (k) + σ˙ F2,3 (k) ,

k=1

4 associated 4 where σ˙ I [1] is entropy production with box [1] (blue dashed box, Fig. 1), σ˙ I [2] is entropy production associated with box [2] (green dashed box, Fig. 1) and σ˙ I is total internal entropy production associated with both boxes (red dashed box, Fig. 1). We note that σ˙ I [1] and σ˙ I [2] will server merely as numerical objective functions for local optimization and could be defined differently, especially with respect to how entropy of mixing flux between boxes, σ˙ F1,2 (k) , could be partitioned. We have chosen to include σ˙ F1,2 (k) in both σ˙ I [1] and σ˙ I [2] because state changes in either box directly alters the F1,2 (k) flux; consequently, σ˙ I = P σ˙ I [1]+ σ˙ I [2]− 6k=1 σ˙ F1,2 (k) . For comparing internal entropy production between local versus global optimization below, only σ˙ I defined by Eq. (16) is relevant. 2.2

Transport equations, optimization and numerical routines

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 (1) and (2) for each constituent k in boxes [1] and [2], which takes the general form,
dCk [i] = Fi−1,i (k) − Fi,i+1 (k) / h[i] + 3k [i]r[i] for i = 1,2 (17) dt

where the brackets, [i], following a variable denote the boxboundary location with [0] being the upper boundary and [3] Earth Syst. Dynam., 2, 69–85, 2011

Maximize ε1 [1],ε2 [1]

σ˙ I [1]

and

Maximize ε1 [2],ε2 [2]

σ˙ I [2].

(18)

Globally (i.e, across both boxes synchronously), the optimization is defined by the cost function given by Eq. (16), as given by, Maximize ε1 [1],ε2 [1],ε1 [2],ε2 [2]

σ˙ I ,

(19)

where both optimizations are bounded within the tesseract, 0 ≤ εi [j ] ≤ 1. Of course, local maximization, as given by Eq. (18), requires an iterative (i.e., asynchronous) approach, because changing the conditions in one box alters the optimal solution for the other. In contrast, global optimization given by Eq. (19) is mathematically well posed, but the parameter space is larger (4-D vs. 2-D in this case). For local optimization, Eq. (18), we have chosen to associate the entropy of mixing terms that directly contribute to the box being optimized. While other partitions of mixing entropy can be envisioned, we will demonstrate below that the entropy of mixing terms represent very minor contributions to internal entropy production, and simulations with mixing entropies completely removed had insignificant impact on the results and conclusions. www.earth-syst-dynam.net/2/69/2011/

107

To107 examine To how examine internal howentropy internalproduction entropy production depends on depends either local on either or global local or entropy global entropy

108

production 108 production optimization, optimization, weand useimplications a simple we use two-box a in simple model two-box reminiscent model reminiscent of ocean of ocean J. J. Vallino: Differences biogeochemistry

109

biogeochemistry 109 biogeochemistry models developed models developed to capturetoprocesses capture processes in the photic in the andphotic aphotic andzones aphotic of zones of

110

Table 2. Model initial1) conditions (for boxes [1]2002). and [2])As andcustomary Dirichlet boundary conditions (for and [3]) used for all model simulations. oceans 110 (Fig. oceans 1) (Ianson (Fig. and (Ianson Allen and 2002). Allen As customary in oceanography, in oceanography, we model we a[0] water model a water

111

column 111 where column all where boxes all andboxes boundaries have unit have surface unitarea, surface so that area, extensive so that extensive variables variables Box orand boundaries

112

2 2 (mmol m ) Variable only 112depend onlyondepend box depth, on box h,boundary and depth, willh,be and calculated will be calculated on a perConcentrations mon basis. a per mModel basis.focus Model is placed focus is placed

113

CCH [i] Cand [i] C C(S [i] C(S) [i] NH3 [i] structures O2 [i]destruction 2 O destruction 2 CO3 of biological on113 dissipation on dissipation of free energy of free byenergy synthesis by synthesis and ofCHbiological structures 2 21 and S2) 11 and S

114

from 114and from to chemical and to chemical constituents (CH2O,100 NH (CH O, NH3H, 2OCO H2COthe within two layers, the0.1 two butlayers, 3, 2O 2 and 2 and 3) within 3) 1.0 [1] constituents 290 2000 0.1 but

115

entropy 115 ofentropy mixingofbetween mixing boxes between andboxes boundaries and boundaries is accounted is accounted for. The photic for. The zone photic is zone is [3] –* – – 1.0 – –

116

typically 116 energy typically replete energy butreplete resource but limited, resourcewhile limited, the while aphotic thezone aphotic is itszone complement: is its complement: energy energy

117

limited 117 but limited resource but replete. resourceOur replete. model Our captures model these captures twothese differing two states; differing however, states; however, we we

75

−3

[0] [2]

100 100

290 290

2000 2000

– 1.0

– 0.1

– 0.1

* Indicates variable is not allowed to cross boundary (i.e., βi,j (k) = 0 in Eq. (12)).

118

replace 118 energy replace input energy via input photosynthetic via photosynthetic active radiation active radiation with energy with input energy via input the diffusion via the diffusion of of 92 In this 92 manuscript In this manuscript we(18) will and assume we(19) willthat assume biological that biological systems organize systems organize maximize to maximize entropy entropy While optimization of Eqs. can be con3 Results andto discussion 119 chemical 119 potential from potential a fixed from upper a fixed upper boundary condition condition into the surface into thelayer surface box.layer box. ductedchemical for the transient problem, Eq.boundary (17), over a specified 93 production, 93 production, which is synonymous which is synonymous with maximizing with maximizing the the characteristics dissipation rate of Gibbs rate free of Gibbs energy free energy 3.1 dissipation Model time interval (Vallino, 2010), we are primarily interested 120 Nitrogen 120 Nitrogen in the form in of theammonia form of ammonia is allowedistoallowed diffuseto into diffuse the bottom into thebox bottom [2] from box [2] from in how optimization alters MEP solutions. Conse94 spatial for 94 systems forunder systems constant under temperature constant temperature and pressure. and pressure. Living organisms Living organisms are viewed arehere viewed as here as For any specified point εi [jthe ]-space, a solution to Eq. (17) 121 sediments, 121 sediments, which which the lower fixes the boundary boundary condition condition [3]. Hence, [3]. energy Hence, flows energy into flows the ininto quently, we arefixes only interested inlower steady state (SS) solu95 mere 95catalysts mere facilitating catalysts facilitating autocatalytic autocatalytic reactions reactions for can the dissipation the dissipation of free energy. of free We energy. We will befor found. For instance, Fig. 3 will illustrates the transient tions to Eq. (17). Typically, Newton’s method, or a variation 122 system 122 viasystem box [1] viaand boxresources [1] and resources (nitrogen)(nitrogen) flow in via flow boxin[2]. via All box constituents [2]. All constituents are state allowed arethe allowed dynamics to steady for point (ε [1], ε 1 2 [1], ε 1 [2], thereof, used to solve for main SS solutions Eq. (17); how96 isalso 96continue also our continue our hypothesis maintohypothesis (Vallino 2010), (Vallino that 2010), the acquisition that the acquisition of Shannon of information Shannon information ε [2]) = (0.1, 0.05, 0.1, 0.05) with the parameters and ini2 ever, employing severalthat variations to Newton’s method inn this manuscript will97assume organize to maximize 97 we (Shannon (Shannon 1948), biological or 1948), more precisely orsystems more precisely useful information useful information 2002, (Adami Adami 2002, et Adami al. 2000), et al. via2000), tial(Adami andentropy boundary conditions used throughout thisvia study (Tacluding a line search technique with trust region (Bain, 1993)

bles 1 free and 2, respectively). For this particular solution, SS roduction, which is synonymous with subspaces maximizing the dissipation rate of Gibbs energy was98found to98 fail inevolution some of the 0 ≤production εiof [j ]entropy ≤ 1. An evolution facilitates facilitates the production of when entropy averaged when averaged over time.over It istime. the 4ability It is the of4ability of internal entropy production dominates in box [1] (590 J m−2 approach basedtemperature on interval arithmetic (Kearfott andorganisms Novoa, are−1 ◦ −1 or systems under constant and pressure. Living viewed here as dto out-compete K to, out-compete blueabiotic bounding box in Fig. 1),inas compared to 99 Kearfott, living 99 1996) systems living storedeveloped, information to store but information that allows allows them systems abiotic insystems 1990; wastosystems also proved tothat bethem −2 d−1 ◦ K−1 , green bounding box in Fig. box [2] (41.8 J m mere catalysts facilitating autocatalytic reactions for therobust dissipation ofconditions. free energy. Weexamine will too computationally burdensome. Theappropriate most approach 100 entropy 100 production entropy production under under appropriate conditions. Here, we Here, will we will how examine information how information

1), and entropies associated with mixing between boundary found was to integrate the ODEs, Eq. (17), forward in time using a high precision block implicit method (Brugnano and aryet[3] are 14.6,via 0.436 and 0.00 (J m−2 d−1 ◦ K−1 ), respecShannon 1948), precisely information (Adami 2002, al.box 2000), Magherini, 2004) from auseful specified initial condition (Table 2)Adami 102ormore a simple 102 two-box a simple model two-box where model entropy where production entropy production in each in each is largely box isgoverned largely governed by two by two tively. Due to the significant amount of free energy release in  T dC  1 dC until ≤ ζ , where C is a vector of convolution facilitates production of entropy when averaged over Itoxidation is the ability of synthesis dt of reduced organic compounds, entropy of mixing, 1032 the reactions 103 reactions thatdt determined that determined the rate ofthe biological rate oftime. biological structure (i.e., structure catalyst) (i.e., catalyst) synthesis and and −8 mmol m−3 d−1 . If a steady σ ˙ , is a small fraction of the entropy production associated centrations and ζ was set to 10 ving systems 104 to storedegradation. information that allows them to out-compete abiotic Fsystems in 104 degradation. with reactions, σ˙ r . To demonstrate this more generally, we state solution failed to be obtained after 106 days of integrantropy production under appropriate conditions. Here,from we the willsearch, examine how information randomly chose 100 000 points within the 4-D εi [j ]-space tion time, the point was flagged and removed 105 105 and calculated the steady state solution to Eq. (17) and assobut this rarely occurred. may also facilitate entropy production averaged over space. For our system, we will consider ciated entropy terms, which shows that entropy of mixing is Numerical solution to Eq. (19) was obtained by using VT106model 2 106Model 2 System Model System simple two-box production in aeach box version is largely governed by%two at most 7.2 of total entropy production and for most cases DIRECT (Hewhere et al., entropy 2009), which employs parallel much less than that (Fig. 4a). Only 310 points out of 100 310 of a Lipschitzian direct search algorithm (Jones et al., 1993) eactions that determined the rate biological (i.e., catalyst) synthesis and on 107 To107 examine Toof how examine internal howstructure entropy internalproduction entropy production depends on depends either local either or global local or entropy global entropy Monte Carlo simulations did not attain a SS solution within to find a function’s global optimum without using function 6 10 days. egradation. derivatives. 108 production 108 production optimization, optimization, we use simple we use a(18), simple model two-box reminiscent model reminiscent of ocean of ocean We also used VTDIRECT to asolve Eq.two-box An interesting aspect of the model is that the amount of but we implemented an iterative approach, where first box 109 biogeochemistry 109 biogeochemistry models developed models developed to capturetoprocesses capture processes in the photic in the andphotic aphotic andzones aphotic of zones of nitrogen extracted from the sediments, or boundary [3], de[1] was optimized for given values of ε1 [2] and ε2 [2], then on community composition, because box optimized given ε1 [1] andAllen ε2 [1]and from the pre110[2] was oceans 110 (Fig. oceans 1) (Ianson (Fig. 1)and (Ianson 2002). Allen As2002). customary Aspends customary in oceanography, in oceanography, we modelwe a water model achanging water values Model System of ε [j ] dramatically affects the total standing SS nitrogen vious optimization of box [1]; this iteration proceeded until i

111

column 111 where column all where boxes all andboxes boundaries and boundaries have unit have surface unitarea, surface so[1] that area, extensive so that extensive variables variables

ε[j ] − εˆ [j within boxes and [2], calculated as (mmol N m−2 ), ] ≤ 10−9 for each box, where kk is the Euclid2 2 o examine how internal entropy production either local 112 only 112 depend only ondepend depth, on depth, h,value be and calculated will beglobal calculated on a entropy per mon a per mModel basis.focus Model is placed focus is placed ian norm and the vector εˆbox (ˆε = [ˆ εdepends εˆ 2h, ]T and ) isonwill the of εor 1 box 2 basis. X from the previous iteration. Wetwo-box note thatmodel the local optimiza- of ocean NTotal = biological h[i](C +C [i])), (20) roduction optimization, we use simple reminiscent NH 3 [i] + ρ(C 113 on113 dissipation ona dissipation of free energy of free byenergy synthesis by synthesis and destruction and destruction of of biological structures structures (S11[i] and S (S 2)21 and S2) tion search converged to the same solution regardless of the i=1 iogeochemistry developed to capture processes in theO,photic and aphotic zones initial choice ofand ε1 [2] εand 114models from 114 from toor chemical constituents constituents (CH NH (CH O, NH , 2OCO and Hof two layers, the two butlayers, but 2 [2].to chemical 2 3, 2O 2 and 3H 3) within 2COthe 3) within which is2 evident in the results from the 100 000 Monte Carlo ceans (Fig. 1)115 (Iansonentropy and Allen 2002). As customary in oceanography, we model a water simulations (Fig. 4b). ForThe any given 115 ofentropy mixingofbetween mixing boxes between andboxes boundaries and boundaries is accounted is accounted for. The photic for. zone photic is entropy zone is production defined by Eq. (16), numerous SS solutions can be found that olumn where 116 all boxes and boundaries have unit surface so that extensive variables typically 116 energy typically replete energy but replete resource butarea, limited, resource while limited, the while aphotic thezone aphotic iswithin itszone complement: is ]-space its complement: energy energy differ in location ε [j and exhibit differences i 2 nly depend on117 box depth, h, and will be calculated on a per m basis. Model focus is placed of up to four orders of magnitude in N (Fig. 4b). Varialimited 117 but limited resource but replete. resourceOur replete. model Our captures model these captures twothese differing two states; differing however, states; however, we Total we tion in C in the vicinity of MEP may explain how biomass n dissipation of free energy by synthesis and destruction of biological structures (S and S )

so continue our hypothesis (Vallino 2010),production that the production acquisition of Shannon information [0] and box [1], and box[2], and boxconsider [2] and bound101mainmay 101also may facilitate also entropy facilitate entropy averaged averaged over space. over For space. our box system, For[1] our we system, will consider we will

2energy T1 with 118 replace 118 energy replace input energy via input photosynthetic via photosynthetic active radiation active radiation with energy input via input the diffusion via the diffusion of of om and to chemical (CH2O,from NH3a, fixed O2from and withinboundary the two layers, butsurface 2CO 3)upper 119 constituents chemical 119 chemical potential potential upper aHfixed boundary condition condition into the into thelayer surface box.layer box.

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Earth Syst. Dynam., 2, 69–85, 2011

ntropy of mixing andform boundaries is accounted for.isto The photic zone 120 between Nitrogen 120boxes Nitrogen in the in of theammonia form of ammonia is allowed allowed diffuse tointo diffuse theisbottom into thebox bottom [2] from box [2] from

ypically energy replete but resource limited, while the the aphotic is its complement: 121 sediments, 121 sediments, which fixes which the lower fixes boundary lowerzone boundary condition condition [3]. Hence, [3].energy energy Hence, flows energy intoflows the into the

76

J. J. Vallino: Differences and implications in biogeochemistry -3

100

CH2O (mmol m )

(a)

300

-3

O2 (mmol m )

-3

H2CO3 (mmol m ) (c)

(b)

280 2100

80 260 60

240

40

220

20

200 180

0 0

92 93

2

2050

250

500

2000 0

250

500

0

250

500

(d) manuscript Time (d)that biological In92 this Time manuscript In this we will we assume will assume that biological systemssystems organizeTime organize to (d) maximize to maximize entropy entropy

NH (mmol m-3)

120

(mmol m-3)

(mmol m-3)

(d) (e)the 4dissipation 1 production, 93 3 production, which iswhich synonymous is synonymous with maximizing with maximizing the 2dissipation rate of Gibbs rate (f) offree Gibbs energy free energy 100

anize to maximize entropy941.5 for 94systems for systems under constant under80constant temperature temperature and pressure. and pressure. Living organisms Living organisms are viewed are viewed here as here as 3 ms organize to maximize entropy ation rate of Gibbs free energy 95 entropy mere 95 catalysts mere catalysts facilitating facilitating autocatalytic autocatalytic reactionsreactions for the dissipation for the dissipation of free energy. of free energy. We willWe will 60 stems organize to maximize 2 dissipation rate of Gibbs free 1energy gical systems to maximize organisms are organize viewed here as also 96 96entropy continue also continue our mainour hypothesis main hypothesis (Vallino(Vallino 2010), that 2010), the that acquisition the acquisition of Shannon of Shannon information information 40 the dissipation rate of Gibbs free energy systems logical systems organizeorganize to maximize to 0.5 maximize entropy iving organisms are viewed here as entropy 1 mizing dissipation ratewill of Gibbs energy ation ofthe free energy. We 97 (Shannon 97free 1948), or 1948), moreor precisely more precisely useful information useful information (Adami (Adami 2002, Adami 2002, et Adami al. 2000), et al.via 2000), via 20 e. Living organisms are viewed here as (Shannon ng aximizing the dissipation the dissipation rate of Gibbs rate of free Gibbs energy free energy dissipation of free energy. We will pressure. organisms as facilitates uisition ofLiving Shannon information 98are evolution 98will here evolution facilitates the production of entropy of entropy when averaged when averaged over time. over It time. is the Itability is theof ability of 0 viewed 0 0 the production r the dissipation of free energy. 0Weviewed 500 0 250 500 0 250 500 ure. dhepressure. Living organisms Living organisms are viewed are here Time as250 here as acquisition of Shannon information (d) Time (d) Time (d) ions forAdami the dissipation of 99 free We will 2002, et al. 2000), via energy. living 99 systems living systems to store to information store information that allows thatthem allows to them out-compete to out-compete abiotic systems abiotic systems in in hat the acquisition of Shannon 700 information-2 -1 30 -2 -1 actions for the dissipation for theAdami dissipation of et free of free energy. Weσr will Adami 2002, al.energy. 2000), via (J mWe d will °K) σF (J m d °K) (g) (h) 2010), that the of600 Shannon information over time. It isacquisition the ability100 entropy 100 via entropy production production under appropriate under appropriate conditions. conditions. Here, weHere, will we examine will examine how information how information on (Adami 2002, Adami et al. 2000), o, that 2010), the that acquisition the acquisition Shannon of Shannon information information eraged over time. It isofthe ability of formation (Adami 2002, al. also 2000), via mpete abiotic systems in Adami 500 et may 101 facilitate also facilitate entropy20entropy production production averagedaveraged over space. overFor space. our For system, our system, we will we consider will consider averaged over time. It is101 the ability of may ation information (Adami(Adami 2002, Adami 2002, et Adami al. 2000), et al. via 2000), via out-compete abiotic systems in y when averaged over time. It isa102 the abilitysimple of two-box ill examine how information 102400 model where modelentropy where entropy production production in each box in each is largely box is governed largely governed by two by two to out-compete abiotic systems insimplea two-box en opy whenexamine averaged over time. over It time. is 300 the Itability is theof ability of weaveraged will how information ws them to out-compete abiotic systems or our system, we will consider reactions 103 in reactions that determined that determined the rate the of biological rate of biological structurestructure (i.e., catalyst) (i.e., catalyst) synthesis synthesis and and 10 ere, we will examine how103 information 200 em ows toFor them out-compete to system, out-compete abiotic systems abiotic systems in in ace. our we will consider Here,governed we will examine how104 information xions. is largely by two 104100 r space. For our system, we will degradation. considerdegradation. ditions. Here, will we examine will examine howby information how information ach boxwe isHere, largely governed two 0 ged over synthesis space. Forand our system, we will consider catalyst) n each box is largely governed two 250 105 0 0by 105 500 0 250 500 raged space. overFor space. our For system, ourand system, we will we consider will consider ever (i.e., catalyst) synthesis Time (d) Time (d) uction in each box is largely governed by two ture (i.e., catalyst) synthesis and noduction in each in box each is largely box is governed largely by two 106 governed 2106 Model 2by two Model SystemSystem cal structureFig. (i.e.,3. catalyst) synthesis Concentration of stateand variables (a–f) and entropy production from reactions (g) and mixing (h) for the point (ε 1 [1], ε 2 [1], ε 1 [2], = (0.1, 0.05, 0.1, 0.05) based Eq. (17). In (a–g), green line represents variables in box [1], while red line is for variables in box [2]. gical ucturestructure (i.e.,εcatalyst) (i.e., catalyst) synthesis and and 2 [2]) 107 synthesis To 107examine Toonexamine how internal how internal entropy entropy production production dependsdepends on eitheronlocal either or local globalorentropy global entropy 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. 108 production 108 production optimization, optimization, we use awe simple use a two-box simple two-box model reminiscent model reminiscent of oceanof ocean

r local or global entropy 109 biogeochemistry 109 biogeochemistry models models developed developed to capture to capture processes processes in the photic in theand photic aphotic and aphotic zones ofzones of n either local or global entropy over a in narrow range fromwe 642.22 toamodel 643.07 m−2 specific entropy production change with1)ecosystem maniscent of ocean 110 oceans 110 can (Fig. oceans 1) (Fig. (Ianson (Ianson and Allen and 2002). Allentributed As 2002). customary As customary oceanography, in oceanography, model we water aJ water −1 ◦ −1 ds on either turity local (Aoki, or global entropy d K ; however, the corresponding εi [j ] points have a of solutions reel reminiscent of ocean 2008). While the4 vast majority −2 depends onsult either local or entropy photic and aphotic zones ofglobal 111 column 111 column where all where boxes all and boxes boundaries and boundaries have unit have surface unit surface area, so area, that extensive extensive variables much broader distribution over εso ]-space (Table variables 3), which in an N less than 10 mmol m , 33 solutions proi [jthat Total model reminiscent of ocean ends on photic either on local either or local global or entropy global entropy son in depends the and aphotic zones of 2 2 transient dynamics and SS produce significant differences in duced high to extremely high N values. Examination of Total o-box model of ocean anography, wereminiscent model a water 112 only 112 of depend only depend on box depth, on boxh,depth, and will h, and be calculated will be calculated on a peron m abasis. per m Model basis. focus Modelis focus placedis placed sses in the photic and aphotic zones standing stocks (Fig. 6). Most significant are differences in community kinetics that lead to high NTotal values reveals xtwo-box reminiscent model reminiscent oceanof ocean inmodel oceanography, weofmodel a water re processes in apparent the photic and aphotic zones of ofthefree , so that extensive variables 113 113 dissipation on dissipation energy of free by energy synthesis by synthesis and destruction andstocks destruction offorbiological of biological structures structures (S1 and(S S21)and SS standing biological structures that S2) an condition that growth efficiency ary in oceanography, wenecessary model aon water −2 ture cesses processes in the photic in the and photic aphotic and aphotic zones of zones of and [2] must be less ce area, so that extensive variables 2 can range from 52 to 19 020 and from 2.2 to 174 mmol m of reaction (2), ε [j], in both boxes [1] 2 in oceanography, we from model afrom water mcustomary basis. Model focus is placed 114 variables 114 and to chemical and to chemical constituents constituents (CH2O, (CH NH32,O, O2NH and3, H O22CO and3)Hwithin within two the layers, twobut layers, but 2CO3)the urface area, so that extensive in box [1], respectively. These state variations that have than 0.02 (Fig. Inspection of the transient solution reveals As customary in2 oceanography, in Model oceanography, we5).model weamodel water a water amary per m basis. focus is placed eiological unit surface area, so that extensive variables structures (S and S 2that nearly identical entropy productions illustrate, least isqualihigh concentration develops due to extremely slow 115 115 entropy of mixing of between mixing between boxes and boxes boundaries and boundaries is accounted is accounted for. Thefor. photic The zone photic isatzone 2) entropy 1 on a per m basis. Model focus is placed ave surface unit surface area, sostructures area, that extensive so that extensive variables variables on of biological (S S 2rates tatively, an important concept in MEP theory (Dewar, 2005, growth of caused small ε [j] values (data not 1 and 2)by 2 2 culated onthe a per basis.but Modeltypically focus is placed replete twomlayers, 116 116 typically energy energy but replete resource but resource limited, limited, while the while aphotic theshould aphotic zone isbeits zone complement: is its complement: energy energy 3) within uction of biological structures (S S 2 2 develops 2009), which is there many micropath solutions shown). Over time, 1 and 2) sufficient concentration to 2 calculated ed a3)per on m a basis. per m Model basis. focus Model isfocus placedis placed H CO within the two layers, but 2on dfor. destruction of biological structures Sbut place “top down” control but this to elevated SS Ourcaptures that produce thetwo same macropath (or MEP) The photic zone is 117 2)leads limited 117on(Sbut limited resource resource replete. replete. Our model model captures these these differing two differing states; however, states;state. however, weSince MEP we 1 ,and and 2CO3) within the two layers, but truction and H destruction of biological of biological structures (S1 and(S S212.)and S2) concentrations of both theory rests on probabilities (Dewar, 2009; Lorenz, 2003), ounted for. The photic zone isstructures and the two layers, but input cHzone its H complement: energy 118 replace 118 replace energy energy viainput photosynthetic via photosynthetic active active radiation with energy withproduction input energy viainput the diffusion via thebydiffusion of top of 3, O2is 2CO3) within slight radiation variations in entropy exhibited the accounted for. The photic zone is Oaphotic NH and , H O CO and ) H within CO ) the within two the layers, two but layers, but by reactions (1) and Inspection of total entropy production zone energy 2 3 2 3is its 2 complement: 3 SS entropy producing (Table can be box. viewed as aries isstates; accounted for. The photicchemical zone ischemical fering however, we 119 119 potentialreveals potential from athat fixed from atop fixedboundary upper boundary condition condition intosolutions the into surface the surface layer3)box. layer Monte Carlo simulations theupper the aphotic (2) zonefrom is itsthe complement: energy interchangeable, since any real system is subject to noise that ndaries is accounted is accounted for. The for. photic Thezone photic is zone is wo differing states; however, we 20 have total internal entropy production dis,nergy whileinput the aphotic zone is120 its complement: viaSSthesolutions diffusion of Nitrogen 120 Nitrogen inenergy the form in the of form ammonia of ammonia is allowed is allowed to diffuse to into diffuse the into bottom the box bottom [2] box from[2] from se two differing states; however, we ed, ewith the while aphotic the aphotic zone is zone its complement: is its complement: energy energy energy input via the diffusion of ures two differing however, wewhich fixes the these surface layer box. states; sediments, 121 of sediments, which thefixes lower theboundary lower boundary condition condition [3]. Hence, [3]. energy Hence, flows energyinto flows the into the tion with energy input Dynam., via121 the diffusion Earth 2, however, 69–85, www.earth-syst-dynam.net/2/69/2011/ hese ptures two these two Syst. differing states; however, states; we 2011 we on into thediffering surface layer box. ve radiation energy input viasystem the diffusion e bottom boxwith [2] from 122box. 122 via system boxof via [1] box and [1] resources and resources (nitrogen) (nitrogen) flow in via flowbox in via [2].box All[2]. constituents All constituents are allowed are allowed dition into the surface layer iation tivethe radiation with energy with energy via input the diffusion via the diffusion of of nto bottom boxinput [2] from ary condition intointo the the surface layer box. nce, energy flows se into the bottom box [2] from ondition ndary condition into theinto surface theinto surface layerthebox. layer box. ]. Hence, energy flows

J. J. Vallino: Differences and implications in biogeochemistry

77

Table 3. Values in 4-D ε i [j ]-space that correspond to the 20 highest internal entropy production rates, σ˙ I Eq. (16), (J m−2 d−1 ◦ K−1 )  6  P from the 100 000 Monte Carlo simulations. Here σ˙ r = σ˙ r1 [1] + σ˙ r2 [1] + σ˙ r1 [2] + σ˙ r2 [2] and σ˙ F = σ˙ F0,1 (k) + σ˙ F1,2 (k) + σ˙ F2,3 (k) are k=1

total internal entropy productions from reactions and mixing, respectively. ε1 [1]

ε2 [1]

ε1 [2]

ε2 [2]

σ˙ r

σ˙ F

σ˙ I

0.115840 0.081216 0.040506 0.034556 0.037706 0.012111 0.022522 0.125490 0.071504 0.029651 0.075518 0.048038 0.071063 0.118860 0.107290 0.120840 0.257120 0.114720 0.095344 0.057887

0.008848 0.008126 0.004444 0.013810 0.010242 0.003315 0.018299 0.018242 0.023712 0.004722 0.035487 0.037938 0.024790 0.030821 0.035433 0.032660 0.013008 0.036114 0.030462 0.044734

0.19774 0.14751 0.45347 0.63265 0.38227 0.57708 0.08346 0.30648 0.82380 0.03499 0.86522 0.78268 0.23527 0.90396 0.54022 0.10968 0.75947 0.09940 0.18539 0.98099

0.001492 0.004026 0.008376 0.005105 0.018611 0.009786 0.008856 0.002544 0.024850 0.028105 0.014358 0.016527 0.032890 0.007209 0.006473 0.000561 0.001172 0.021301 0.996260 0.033569

602.46 599.40 596.74 601.00 604.91 596.47 605.33 610.46 612.19 613.23 613.77 614.59 614.71 615.21 615.57 615.69 617.01 617.89 618.96 618.67

40.607 43.66 46.319 42.057 38.142 46.576 37.708 32.566 30.783 29.699 29.151 28.273 28.149 27.641 27.255 27.126 25.638 24.576 23.323 23.554

643.067 643.060 643.059 643.057 643.052 643.046 643.038 643.026 642.973 642.929 642.921 642.863 642.859 642.851 642.825 642.816 642.648 642.466 642.283 642.224

would make the small differences in σ˙ I indistinguishable. For our model system, there are clearly multiple solutions that give rise to what is effectively the same MEP state.

Table 4. Global and local MEP solutions (J m−2 d−1 ◦ K−1 ) based on solutions to Eqs. (19) and (18), respectively. Variable

3.2

Global versus Local MEP solutions

The global MEP solution obtained by maximizing total internal entropy production, σ˙ I Eq. (16), over the whole domain (red dashed box, Fig. 1) by simultaneously varying ε1 [1], ε2 [1], ε1 [2] and ε2 [2] as given by Eq. (19) is 643.10 J m−2 d−1 ◦ K−1 (Table 4). Because of the small 4-D parameter space of our model, this solution was almost located by the Monte Carlo simulations (Table 3). To examine the solution in the vicinity of the global maximum, we plot σ˙ I [1], Eq. (14), as a function of ε1 [1] and ε2 [1] while holding ε1 [2] and ε2 [2] fixed at their global optimum values, and vice versa for σ˙ I [2] (Fig. 7). It is clear from this plot that σ˙ I [1] of 642.96 J m−2 d−1 ◦ K−1 is near the σ˙ I [1] maximum for box [1] given ε1 [2] and ε2 [2] (Fig. 7a, light gray point); however, it is also evident that σ˙ I [2] that contributes 0.142 J m−2 d−1 ◦ K−1 (Fig. 7b, light gray point) to the global solution is far removed from the maximum σ˙ I [2] possible in box [2] that could be attained given ε1 [1] and ε2 [1], which is approximately 215 J m−2 d−1 ◦ K−1 . This result indicates that the global optimum does not correspond to local optimums. Entropy productions associated with mixing, Eq. (13), summed www.earth-syst-dynam.net/2/69/2011/

ε 1 [1] ε 2 [1] ε 1 [2] ε 2 [2] σ˙ r [1] σ˙ r [2] P6 σ˙ F0,1 (k) Pk=1 6 σ˙ F1,2 (k) Pk=1 6 k=1 σ˙ F2,3 (k) σ˙ I [1] σ˙ I [2] σ˙ I

Global, Eq. (19)

Local, Eq. (18)

0.0754 0.0123 0.2222 0.9998 611.64 0.139 31.32 0.0027 0.0 642.96 0.142 643.10

0.0991 0.0352 0.0997 0.1100 225.92 204.97 3.25 2.67 0.0 231.84 207.64 436.81

over all relevant state variables for the globally-optimized solution are 31.32, 0.0027 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 (Table 4).

Earth Syst. Dynam., 2, 69–85, 2011

78

J. J. Vallino: Differences and implications in biogeochemistry

0.02

ε2[1] or ε2[2]

0.015

0.01

Total N -2 (mmol m ) 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

0.005

92

In92 this manuscript In this manuscript we will we assume will assume that biological that biological systemssystems organizeorganize to maximize to maximize entropy entropy

0 93 production, 93 production, which iswhich synonymous is synonymous with maximizing with maximizing the dissipation the dissipation rate of Gibbs rate offree Gibbs energy free energy o maximize entropy 0 0.2 0.4 0.6 0.8 1 o maximize entropy 94 for 94systems for systems under constant under constant temperature temperature and pressure. and pressure. Livingε1organisms Living are viewed are viewed here as here as [1] or ε1[2]organisms te of Gibbs free energy te of Gibbs free energy 95 mere 95 catalysts mere catalysts facilitating facilitating autocatalytic autocatalytic reactionsreactions for the dissipation for the dissipation of free energy. of free energy. We willWe will ms are viewed here as Fig. 5. Points (large triangles and squares) in 4-D ε i [j ]-space ms are viewed here as that resulted total steady state standing stock (N 96 also 96 continue also continue our mainour hypothesis main hypothesis (Vallino (Vallino 2010),inthat 2010), the that acquisition the N acquisition of Shannon of Shannon information information Total ) greater of free energy. We will than 104 mmol m−2 . Triangles mark locations and magnitude of N of free energy. We will 97 (Shannon 97 (Shannon 1948), or 1948), moreor precisely more precisely useful information useful (Adami 2002, Adami 2002, et Adami al. 2000), et al.via 2000), via standing stockinformation for ε(Adami n of Shannon information 1 and ε 2 in box [1], while squares mark locaand N standing stock for ε 1 and ε 2 in box [2]. Also shown as n of Shannon information 98 evolution 98 evolution facilitates facilitates the production the production of tions entropy of entropy when averaged when averaged over time. over It time. is the Itability is theof ability of Adami et al. 2000), via small points are a subset of the ε 1 and ε 2 in box [1] (green) and Adami et al. 2000), via [2]that (red) from 100,000 Monte Carlosystems simulations 99 living 99 systems living systems to store to information store information thatbox allows them allows tothe them out-compete to out-compete abiotic abiotic systems in that have in me. It is the ability of NTotal < 104 mmol m−2 . Note, the y-axis range from 0 to 0.021 me. It is the ability of 100 entropy 100 entropy production production under appropriate under appropriate conditions. conditions. Here,that weHere, willthe we examine will examine how information how information abiotic systems in captures all points meet minimum NTotal requirement of abiotic systems in 104 mmol m−2 . mine how information 101 may 101 alsomay facilitate also facilitate entropy entropy production production averagedaveraged over space. overFor space. our For system, our system, we will we consider will consider mine how information system, we will consider 102 a102 simplea two-box simple two-box model where modelentropy where entropy production production in each box in each is largely box is governed largely governed by two by two system, we will consider gely governed by two 103 from reactions 103 thatMonte determined that determined the rate the of biological rate ofproduction biological structurein structure (i.e., (i.e., synthesis synthesis and tropy boxescatalyst) [1] andcatalyst) [2] in the vicinityand of the Fig. 4. Extracted results the 100reactions 000 Carlo simulations gely governed by two (a) Comparison of entropy of mixing, t) synthesis with and randomly selected locally-optimized solutions (Fig. 8) illustrates that the itera104εi [j ].degradation. 104 degradation. t) synthesis Eq. and (13), as a function of total internal entropy production from tive procedure used to solve Eq. (18) has indeed obtained a Eq. (16). (b) Total 105 steady state solution that simultaneously maximizes entropy production 105N standing stock (NTotal ) given by Eq. (20) as a function of total internal entropy production. in both boxes, because σ˙ I [1] in box [1] cannot be improved with the given values of ε1 [2] and ε2 [2], nor can σ˙ I [2] in 106 2106 Model 2 Model SystemSystem box [2] be improved given ε1 [1] and ε2 [1]; the solution is Table 5. Steady state (mmol m−3internal ) ofhow state internal variables stable. The corresponding entropy mixing terms for fluxes 107concentration To 107examine To examine how entropy entropy production production depends depends on eitheron local either oroflocal global orentropy global entropy at maximum internal entropy production associated with local and F0,1 , F1,2 and F2,3 are 3.25, 2.67 and 0.0 J m−2 d−1 ◦ K−1 , or global entropy production 108 production optimization, optimization, we use awe simple use a two-box simple two-box model reminiscent model reminiscent of oceanof ocean global solutions. 108 respectively. or global entropy t of ocean 109 biogeochemistry 109 biogeochemistry models models developed developed to capture to capture processes processes instate the photic in theand photic aphotic and zones zones of Comparing steady solutions from the aphotic global of versus t of ocean Box [1] Box [2] c and aphotic zones of the As local optimizations shows that CH concentration 110 oceans 110 (Fig. oceans 1) (Fig. (Ianson 1) (Ianson andLocal Allen and2002). Allen 2002). customary As customary in oceanography, in oceanography, we2 O model weamodel water a in water c and aphotic zones of Variable Global Local Global boxes [1] and [2] is near zero in the globally-optimized soluphy, we model a water 111 32.085 column 111 column where where boxes boundaries and boundaries have have surface unit surface area, so area, that so that extensive variablesvariables tion, butunit only partially depleted in extensive the locally-optimize soluphy, we modelCH a water 0.013016 1.6025all × boxes 10−4 alland 2.6750 2O at extensive variables 2 to effectively 2 O2 190.01 222.09 190.00 189.58 tion (Table 5). The inability oxidize CH 2 Oisinplaced 112 only 112 depend only depend on box depth, on boxh,depth, and will h, and be calculated will be calculated on a peron m abasis. per m Model basis. focus Modelis focus placed at extensive variables H CO 2100.0 2067.9 2100.0 2100.4 box [1] in the locally-optimized solution results from an in3 s. Model focus 2is placed NH 1.0004 0.48473 1.0000 1.0000 3 113 on 113 dissipation on dissipation of free energy of free by energy synthesis by synthesis and destruction and destruction of biological of biological structures structures (S and (S S and S2) sufficient development of biological structure ( s. Model focus is placed 1 21)), al structures (S1 and S2) 315.25 9.1628 314.17 6.0450 as it is evident in Fig. 4b that illustrates a lower boundary 114 0.69131 from 114 and from to chemical and58.360 to chemical constituents constituents (CH2O, (CH NH ,O, O2NH and3, H O CO and3)Hwithin within two the layers, twobut layers, but al structures (S21 and S2) 57.280 0.71748 2CO3)the on NTotal32exists for a 22given entropy production rate. Biologin the two layers, but ical reactions ultimately limited byThe thezone quantity of catain the two layers, but 115 entropy 115 entropy of mixing of between mixing between boxes and boxes boundaries and boundaries isare accounted is accounted for. Thefor. photic photic is zone is he photic zone is lyst available that in turn is limited by elemental resources. he photic zone is 116 typically 116 typically energy replete energy but replete resource but resource limited, limited, while the while aphotic the aphotic zone isisits zone complement: isthat its state complement: energy energy The spatial model configuration such variables is its complement: energy is its complement: energy entropy If internal is resource maximized locally in captures box [1]captures control energy acquisition 117 production limited 117 but limited but resource replete. replete. Ourasmodel Our model these two these differing two differing states; across however, states;boundary however, we [0], we states; however, we given by Eq. (18), then the total entropy production, σ˙ I while box [2] controls resource input across boundary [3]. states; however, we 118 J m−2 replace 118d−1 replace energy energy input photosynthetic via active radiation active radiation with energy withresult input energy viainput the diffusion via the diffusion of of ◦ K−1 , input input via theEq. diffusion (16), of is 436.81 with via 231.84 andphotosynthetic Limitations in either flux will in lower entropy pro−2 −1 ◦ −1 input via the207.64 diffusion J m of d 119 K chemical beingchemical produced by σ˙aI [1] and In the localthe optimization given by Eq. (18), 119 potentialpotential from fixed from upper a fixedduction boundary upper rates. boundary condition condition into into surface the surface layer box. layer box. rface layer box. σ ˙ [2], respectively (Table 4). Examination of internal enmaximizing entropy production in box [2] ultimately results I rface layer box. 120 Nitrogen 120 Nitrogen in the form in the of form ammonia of ammonia is allowed is allowed to diffuse to into diffuse the into bottom the box bottom [2] box from[2] from m box [2] from m box [2] from 1212, 69–85, sediments, 121 2011 sediments, which fixes which thefixes lower theboundary lower boundary condition condition [3]. Hence, [3]. energy Hence, flows energyinto flows the into the ergy flows into theSyst. Dynam., Earth www.earth-syst-dynam.net/2/69/2011/ ergy flows into the 122 system 122 via system box via [1] box and resources [1] and resources (nitrogen) (nitrogen) flow in via flowbox in via [2].box All[2]. constituents All constituents are allowed are allowed constituents are allowed constituents are allowed

J. J. Vallino: Differences and implications in biogeochemistry

(a)

σI (J m-2 d-1 °K-1)

800

.

79

600

400

200

92

In this manuscript we will assume that biological systems organize to maximize entropy

93

production, which is synonymous with maximizing the dissipation rate of Gibbs free energy

92

In this manuscript assume thatconstant biological systems organize to maximize 94 we for will systems under temperature and pressure. Living entropy organisms are viewed here as

93

0 production, which synonymous with maximizing the dissipation free energy 95 is mere catalysts facilitating autocatalytic reactionsrate for of theGibbs dissipation of free energy. We will 0

1000

2000

3000

(d) our 94 for systems temperature andhypothesis pressure. (Vallino Living organisms here as ume that biological systems under organize toTime maximize entropy 96 constant also continue main 2010), thatare theviewed acquisition of Shannon information

(b) 95 1600 mere catalysts autocatalytic reactions for the dissipation of free(Adami energy.2002, We will ous with maximizing the dissipation rate of Gibbs energy 97facilitating (Shannon 1948),free or more precisely useful information Adami et al. 2000), via C 1[1] (mmol m-3)

also continue our main hypothesis (Vallino 2010), thatof theentropy acquisition Shannonover information mperature 96 and pressure. Living organisms arefacilitates viewed here as 1400 98 evolution the production whenofaveraged time. It is the ability of

1200 (Shannon more precisely (Adami 2002, Adami et al. 2000),abiotic via systems in ocatalytic 97 reactions for the 1948), dissipation of free energy. We will 99 orliving systems touseful store information that allows them to out-compete

1000 98 2010), evolution facilitates the production entropy when averaged over time. It is ability of how information hesis (Vallino that the acquisition of production Shannonofinformation 100 entropy under appropriate conditions. Here, wethe will examine 800

99information living systems store allows them to out-compete abiotic systems in system, we will consider isely useful (Adami Adami et al.that 2000), viaproduction 101to 2002, mayinformation also facilitate entropy averaged over space. For our 600

100 entropy production conditions. weproduction will examine how box information ction of entropy when averaged time.appropriate Ittwo-box is the ability ofwhere Here, 102 over aunder simple model entropy in each is largely governed by two 400

also facilitate production averaged space. For our system,(i.e., we will consider ation that101 allowsmay them to out-compete abiotic systems in 103 entropy reactions that determined the over rate of biological structure catalyst) synthesis and 200

102 a simple where production in each box is largely governed by two ropriate conditions. we examine howentropy information 104willmodel degradation. 0 Here,two-box 0

1000

2000

3000

107 108

CNH3[1] (mmol m-3)

Time reactions that determined the(d) rate biological oduction 103 averaged over space. For our system, we of will consider structure (i.e., catalyst) synthesis and 105 degradation. e entropy104 production (c) 4 in each box is largely governed by two 106 2 Model System rate of biological structure (i.e., catalyst) synthesis and 105 Fig. 7. Internal production (a) boxentropy [1], Eq. (14) as a 3 107 To examine how internal entropy production dependsentropy on either local orin global function ε and ε in box [1] for a fixed value of ε 1 and ε 2 in box 1 2 106 2 Model System 108 production optimization, we use a simple model reminiscent ocean [2] oftwo-box 0.2222 and 0.9998, respectively,ofand (b) in box [2], Eq. (15), 2

as a function ε and ε in box [2] for a fixed value of ε and ε 2 in box [1] of 0.0754 and 0.0123, respecitvely. White point in (a) and (b) dentotes theofposition of the global MEP solutionainwater boxes [1] production optimization, we use a simple two-box model reminiscent ocean 110 oceans (Fig. 1) (Ianson and Allen 2002). As customary in oceanography, we model 1 and [2] based on Eq. (19) (Table 4). 1 global 2 entropy 1 To examine109 how internal entropy production depends on local or biogeochemistry models developed to either capture processes in the photic and aphotic zones of

109 depends biogeochemistry capture in have the photic and aphotic opy production on 111 eithermodels local ordeveloped global column whereentropy alltoboxes andprocesses boundaries unit surface area, zones so thatofextensive variables

110two-box oceans (Fig.112 1) (Ianson Allen As customary inbe oceanography, model a waterModel focus is placed 0 model se a simple reminiscent of ocean only and depend on2002). box depth, h, and will calculated onwe a per m2 basis. 0

1000

2000

3000

in less N transported into the system from boundary [3]. In

where allphotic boxes and aphotic boundaries area, sodestruction thatoptimize extensive variables loped to 111 capturecolumn processes in the and of unit 113 on dissipation of zones free have energy bysurface synthesis and of biological structuresof(S1 [2] andisS Time (d) the globally solution, net synthesis crit2)

ical m in2Nbasis. acquisition, because reaction (1) exceeds re112As customary only depend on boxfrom depth, will abewater calculated on(CH a per Model focus iswhen placed llen 2002). in114 oceanography, we model andh, toand chemical constituents H2CO the two layers, but 2O, NH 3, O2 and 3) within Fig. 6. Transient data for the top 20 entropy producing solutions

action (2) in box [2] there is a net transport of NH3 into the

from unit the dissipation 100,000 Carlo simulations. (a) Total internal en- and system oundaries have surfaceMonte area, soentropy that extensive variables 113 on of free energy synthesis and destruction of biological structuresin(S andInS the sothat accumulates 115 ofby mixing between boxes boundaries is accounted for.1 [2]. The photic zone optimize is 2) locally tropy production from both boxes [1] and [2] whoes steady state

lution, entropy production in box [2] occurs via a large loop flow between reactions (1) and (2), but this limits N acqui, and (c) NH centration of biological structure in box [1], C 3 sition and entropy production box [1]. These simulations by synthesis destruction of biological (Sboundaries and S2)Our 115and entropy of mixing between boxes and is model accounted for. The photic zone is instates; 117 limitedstructures but resource captures these two differing however, we 11replete. over the first 3000 days of simulation. Values of ε i [j ] for the above clearly demonstrate that optimizing local entropy production energy resource limited, while the aphotic zone issystem its complement: energy ents (CH116 NHtypically andgiven H118 ) within the two layers, butphotosynthetic replace energy input via active radiation with energy input via acquisition, the diffusion of simulations in Table 3. but 2O, 3, O2 are 2CO 3replete limits whole resource and energy which results in lower entropy production compared to the globally 117 limited but 119 resource replete. Our zone model these two differing states; however, xes and boundaries is accounted for. The photic is captures chemical potential from a fixed upper boundary condition into the we surface layer box. optimized solution that more effectively utilizes available re118 while replace viais photosynthetic radiation with energy inputinto via the diffusion of [2] from source limited, theenergy aphotic zone its complement: 120input Nitrogen in the form active ofenergy ammonia is allowed totodiffuse theenergy. bottom box sources dissipate free

and will 114 be values calculated on atoper m2 to basis. Model placed −2 focus ◦is −1 from chemical constituents (CH O, NH , ConO2 and H theaphotic two layers, 116 typically energy replete but limited, while the zonebut is its complement: energy 2K 3resource 2CO3) within rangeand from 642.22 643.07 Jm d−1 . (b)

chemical potential from a fixed upperfixes boundary condition into condition the surface[3]. layer box. energy flows into the Our model119 captures these two differing states; however, we the lower 121 sediments, which boundary Hence,

www.earth-syst-dynam.net/2/69/2011/ Earth Dynam., 2, 2011 Nitrogen the energy form ofinput ammonia is[1] allowed toofdiffuse into the bottom [2] from osynthetic120 active radiationinwith the diffusion 122 system viavia box and resources (nitrogen) flow in box via box [2]. AllSyst. constituents are69–85, allowed

121 sediments, fixes the lower condition [3]. Hence, energy flows into the d upper boundary conditionwhich into the surface layerboundary box.

122 tosystem box and resources (nitrogen) flow in via box [2]. All constituents are allowed nia is allowed diffusevia into the[1] bottom box [2] from

108 production n optimization, production 108optimization, we production optimization, 108 use a we simple production optimization, 108 use we two-box a simple use production optimization, a we model simple two-box useoptimization, reminiscent two-box a model simple we use reminiscent model two-box aof we simple ocean reminiscent usemodel two-box aof simple ocean reminiscent of model two-box oceanreminiscent of model oceanreminiscent of ocean of ocean 109 biogeochemistry 109 biogeochemistry 109 modelsbiogeochemistry developed models to capture developed models processes developed to capture in the toprocesses capture photic and processes in aphotic the photic inzones the and photic of aphotic andzones aphotic of zones of 109 biogeochemistry emistrybiogeochemistry models 109 developed biogeochemistry models 109 models developed to biogeochemistry 109 capture developed models to biogeochemistry processes capture developed tomodels capture processes in thedeveloped tomodels processes photic capture in the and developed processes to photic inaphotic capture the and photic tozones in processes aphotic capture the and of photic aphotic zones processes in and theofzones photic aphotic in the ofand zones photic aphotic ofandzones aphotic of zones of 110 oceans (Fig. 110 1) oceans (Ianson 110 (Fig. and oceans Allen 1) (Ianson (Fig. 2002). 1)and (Ianson AsAllen customary and 2002). Allen inAsoceanography, 2002). customary As customary in oceanography, we model in oceanography, a water we modelwe a water model a water 110 oceans ig. 1) (Ianson oceans (Fig. 110 1)and (Fig. (Ianson oceans Allen 110 1) (Ianson and (Fig. 2002). oceans Allen 110 1)and (Ianson As2002). (Fig. Allen customary oceans 1) and As 2002). (Ianson (Fig. Allen customary inAs 1) oceanography, 2002). and (Ianson customary Allen inAs oceanography, and customary 2002). inAllen we oceanography, model As2002). customary inwe oceanography, a water model As we customary inamodel oceanography, water we ainwater model oceanography, awe water modelwe a water model a water 80 column 111 J.have J.area, Vallino: Differences and implications invariables biogeochemistry 111 where all column 111 boxeswhere column and boundaries allwhere boxesall and have boxes boundaries unitand surface boundaries unit sohave surface that unit extensive area, surface so variables that area, extensive so that extensive variables 111 column here allcolumn where 111 boxes all where and column boxes 111 boundaries alland where boxes column 111 boundaries have all andboxes where boundaries column unithave and surface allwhere boxes boundaries unit have area, surface all and unit boxes soboundaries have that surface area, and extensive unit soboundaries area, that surface have so extensive variables unit that area, have surface extensive so unit that area, surface extensive variables so that area, extensive so that variables variables 2variables 2 variables 2 extensive 112 only depend 112 ononly box 112depend depth, only h,on depend and boxwill depth, onbebox calculated h, depth, and will h,onbe and a calculated per willmbebasis. calculated on aModel per m onfocus basis. a perismModel placed basis.focus Model is placed focus is placed 2 2 model, diffusion 2 alternate boxes occurred ac112 only nd ondepend box only 112 depth, depend on box only h,112 on depth, and depend box will only h, depth, 112 on be and depend calculated box will h,only depth, and beonwill depend calculated box on h, be and a depth, per calculated onwill m box on2h, be basis. a depth, and per calculated onm will Model a2h,per basis. be and m calculated on focus will Model basis. a per be is m placed calculated focus Model onbasis. a per isfocus placed Model m on2 basis. aisper placed focus macross Model basis. is placed focus Model is placed focusbetween is placed 113 on dissipation 113 of on113 free dissipation energy on dissipation by of synthesis free energy of free and byenergy destruction synthesis by tive synthesis and ofand biological destruction and destruction structures of biological of i1biological structures S2) by structures (S1 [1] and↔S (S2)∗11 [2], and S2) inactive variants of(S , and as given ∗ ∗ and 113 ation on dissipation ofon free 113 dissipation energy ofon free dissipation 113 byof energy synthesis free on113 energy by dissipation ofsynthesis and free on by destruction energy dissipation synthesis of and free by destruction synthesis energy of and ofbiological free destruction byenergy of and synthesis biological structures destruction of by biological synthesis and structures (S destruction of21 [1] and biological and structures (S destruction and structures biological (S S∗112[1] and )of↔ biological S (S structures andand S structures (S (S and This S2) ↔S ↔S [2], 2)21of 2)1 [2] 2) 21 [1] 2)21 [2]. 114 from and 114 to chemical from 114 and constituents from to chemical and to (CH chemical constituents H2NH CO (CH O, within NH3H ,the O and ) layers, within H2CO3the but ) within two layers, the twobut layers, but 2O, NHconstituents 3, O(CH 2 and 2O, 33,)2O 2 and 2CO 2two modified version of the3 model also produced similar results, 114 to rom chemical and from 114 to chemical and constituents from to chemical 114 and constituents (CH tofrom chemical 114 constituents NH (CH from to O O,2and (CH and NHto O, , chemical O CO NH and O H O,22constituents CO and NH(CH , within O and )layers, (CH the within H32, CO O O,2but and NH )layers, within two ,O but the and two Hbut layers, ) within two but layers, the two butlayers, but 2O,and 3,constituents 2chemical 23H 2constituents 2(CH 33), 2within 3the 3)H 2CO 2two 2O, 3NH 2two 3the 3H 2CO 2layers, 3) within 2CO 3the where global optimization total entropy produc115 entropy115 of mixing entropy 115 between of entropy mixing boxesofbetween and mixing boundaries boxes between and is boxes accounted boundaries and boundaries for. is The accounted photic is accounted for. zoneresulted The is photic for.inThe zone photic is zone is −2 d−1 ◦ K−1 , but local optimization genertion 643.1 Jm 115 entropy f mixing entropy of 115 between mixing ofentropy mixing boxes between 115 of and between entropy mixing boxes 115 boundaries and boxes between ofentropy mixing boundaries isand accounted boxes ofboundaries between mixing isand accounted for. boxes boundaries between isThe accounted and photic for. boxes boundaries isThe accounted zone for. and photic The boundaries is isof zone photic accounted for. The is zone is photic accounted for. is The zonephotic for. is The zone photic is zone is 116 typically116 energytypically 116 replete typically but energy resource replete energy limited, but replete resource while butthe resource limited, aphotic limited, zonetheis aphotic its complement: the aphotic itszone energy complement: complement: energy energy ◦K −1 . Next, ated while entropy atwhile a rate ofzone onlyis419.8 J mis−2itsd−1 116 energy ypically typically replete 116 energybut typically energy replete resource 116 replete but energy typically limited, resource 116but replete resource typically while energy limited, butthe replete limited, resource while energy aphotic but the while replete limited, zone resource aphotic the isbut while its aphotic zone limited, resource complement: the is zone its aphotic while limited, complement: is its energy the zone complement: while aphotic is its energy the zone complement: aphotic energy is its zone complement: energy is its complement: energy energy we examine how our approach and however, conclusions ex117 limited 117 but resource limited 117 replete. but limited resource Ourbut model resource replete. captures Our replete. model these Our two captures model differing these captures states; twothese differing however, two differing states; we states; however, wemay bewe trapolated tostates; adiffering more ecologically relevanthowever, context. we 117 utimited resource limited but 117replete. resource but limited resource 117 Our replete. model butlimited replete. resource 117 Our captures model but Our limited replete. resource these captures model but Our two captures replete. resource these model differing two Our these captures replete. differing states; model twoOur these however, differing captures states; model two however, differing states; these we captures two however, these we two however, we differing states;we however, states; we 118 replace 118 energy replace input 118 viaenergy replace photosynthetic input energy via input photosynthetic activevia radiation photosynthetic active with energy radiation active input radiation with viaenergy thewith diffusion input energy viaofinput the diffusion via the diffusion of of 118 nergy eplaceinput replace energy 118viaenergy input photosynthetic replace 118 via input energy photosynthetic replace via 118 active input photosynthetic energy replace via radiation active photosynthetic input energy radiation with active via input photosynthetic energy radiation with active viainput photosynthetic energy radiation with via active energy input the with diffusion radiation via input active energy the via diffusion of radiation with input theenergy diffusion via ofwith the input energy diffusion of via input theofdiffusion via the diffusion of of 119 chemical 119 potential chemical 119 from chemical apotential fixed upper potential fromboundary a fixed fromupper condition a fixed boundary upper intoboundary the condition surface condition into layer thebox. surface into thelayer surface box.layer box. 119 chemical potential chemical 119 potential from achemical potential fixed from 119upper a potential from fixed chemical 119 boundary aupper fixed from chemical potential boundary upper condition a fixed boundary potential from upper condition into a fixed the boundary from condition surface into upper a fixed the condition boundary layer into surface upper the box. surface boundary layer into condition the box. layer surface condition into box. the layer surface into box. thelayer surface box.layer box. 4 isthe Broader ecological context 120 Nitrogen120 in theNitrogen form 120 of ammonia Nitrogen in the form in is allowed the of ammonia formto ofdiffuse ammonia is allowed into to allowed diffuse bottom tointo box diffuse the [2] bottom from into thebox bottom [2] from box [2] from 120 Nitrogen in the form Nitrogen 120 in the of ammonia form Nitrogen in the 120 ofform ammonia isinNitrogen allowed the of 120 ammonia form isto Nitrogen allowed inof diffuse the ammonia isform allowed toin into diffuse the ofis the ammonia form to allowed bottom diffuse into ofthe ammonia is to box into bottom allowed diffuse [2] thefrom is bottom box into to allowed diffuse [2] thebox from bottom tointo [2] diffuse from the boxbottom into [2] from thebox bottom [2] from box [2] from 121 sediments, 121which sediments, 121 fixes the sediments, which lower fixes boundary which the fixes lower condition the boundary lower [3]. boundary Hence, condition energy condition [3]. flows Hence, [3]. into energy Hence, the flows energy intoflows the into the The we have developed is obviously chosen 121 s, sediments, whichsediments, 121 fixes which thesediments, fixes lower which 121 the boundary fixes sediments, lower which 121 theboundary fixes lower condition sediments, which the boundary lower condition [3]. fixes which Hence, boundary the condition [3]. fixes lower energy Hence, the condition boundary [3].lower flows Hence, energy boundary [3]. condition intoenergy flows Hence, themodel into condition [3]. flows energy the Hence, into [3]. flows the energy Hence, into flows the energy intoflows the into theto illus122 system via 122box system [1] 122andvia resources system box [1] via (nitrogen) and boxresources [1] and flowresources (nitrogen) in via box (nitrogen) flow [2]. in Allvia flow constituents boxin[2]. via box All are constituents [2]. allowed All constituents allowed are allowed trate various aspects in the application of are MEP in spatially that biological organize systems organize maximize tovia entropy maximize entropy 122 system aical boxsystems [1] system via 122 and box resources via [1] system box and 122to [1] resources via (nitrogen) and system box 122 resources [1] (nitrogen) flow and system box resources in (nitrogen) [1] via via flow and box box in (nitrogen) resources [2]. flow [1] via and box All in via constituents resources [2]. flow (nitrogen) box All in[2]. via constituents (nitrogen) flow box All are allowed constituents in [2].via flow All are box allowed constituents in[2]. via are box All allowed [2]. are All allowed are allowed allowed explicit situations, and constituents does not constituents represent anyarenatural sys-

mizing with maximizing the dissipation the dissipation rate of Gibbs rate free of energy Gibbs free energy

alressure. systems organize to maximize entropy ature andLiving pressure. organisms Living are organisms viewed are hereviewed as here as

ing for the dissipation ratedissipation Gibbs free energy alytic ons reactions the dissipation for the ofoffree energy. of free Weenergy. will We will

ssure. Living organisms viewed here as 010), (Vallino that 2010), the acquisition that the are acquisition of Shannon of information Shannon information

s useful for theinformation dissipation of free energy. We willetvia yormation (Adami 2002, (Adami Adami 2002, et al.Adami 2000), al. 2000), via

the acquisition of Shannon information yn0), when ofthat entropy averaged whenover averaged time. Itover is the time. ability It isof the ability of

mation (Adami 2002, Adami al. abiotic 2000), ns that themallows to out-compete them to out-compete abioticetsystems in via systems in

when averaged time. ishow the abilityhow of information ons. ate conditions. Here, we over will Here, examine weItwill examine information

hem to out-compete abiotic systems in consider ed tionover averaged space.over For our space. system, For our we system, will we will consider

s. Here, we will how tropy ction production in each boxexamine iniseach largely box governed isinformation largelybygoverned two by two

over space.(i.e., For our system, we willand consider al of structure biological structure catalyst) (i.e., synthesis catalyst) synthesis and

Fig. is 8. largely Interal entropy production on in each box governed by two in (a) box [1], Eq. (14), as a

function ε1 and ε 2 in box [1] for a fixed value of ε 1 and ε 2 in box

tructure (i.e., [2] catalyst) of 0.0997 synthesis and 0.1100,and respectively, and (b) in box [2], Eq. (15), as a function ε 1 and ε 2 in box [2] for a fixed value of ε 1 and ε2 in box [1] of 0.0991 and 0.0352, respecitvely. White point in (a) and (b) dentotes the position of the local MEP solution in boxes [1] and [2] based on Eq. (18) (Table 4).

production depends ondepends either local on either or global localentropy or global entropy

-box simple model two-box reminiscent model reminiscent of ocean of ocean

on either entropy epends dprocesses to capture inprocesses thelocal photic inglobal and the aphotic photic and zones aphotic of zones of We alsoorinvestigated globally and locally optimized solutions with different values of the transport ox model ofoceanography, ocean 2002). ustomary Asreminiscent in customary oceanography, in we model awe water model a water parameters

(namely D, `, h[1] and h[2]), but in each case the gen-

rocesses in eral the photic and aphotic zones of daries unit surface have unit area, surface so that area, extensive that variables extensive variables conclusion wassosupported; entropy production is greater

2 2 when than locally. A second varitomary in aoceanography, model arather water will ulated be on calculated per mmaximized on basis. a perwe Model mglobally basis. focus Model is placed focus is placed

ation of the model was also investigated that included two

∗ [j itdestruction surface area, so thatstate extensive variables ynthesis and destruction of biological ofstructures biological (S structures S2(S ) ∗21 [j and S2each ) additional variables, ] and ], in box, 11 and

2 the * superscript indicates inactive biological strucwhere ated aNH per focus isthe placed H (CH and H32m ,CO O2basis. within HModel the3)two within layers, two but layers, but 3, Oon 2O, 3)and 2CO

ture that does not catalyze reactions (1) or (2). The rational

estruction offorbiological structures (Si1photic and this model was thatThe optimized and ries boundaries is accounted isfor. accounted The photic for. zone is Szone 2) inisbox [j ] would be poorly adapted to conditions in box [k] and would be out-

O2limited, andtheH2aphotic CO theitstwo layers, while ce while the zone aphotic is complement: zone is as itsbut complement: energyin reaction energy 3) within competed, but could serve substrate (2). In this

tem per say. The ideas we explore here, based on the development and results of our modeling exercise, are intended 4 to examine the usefulness of the MEP4principle for under4 4 processes. 4 4 two main4 questions4 standing biogeochemical The the MEP community must address are (1) do biological systems organize towards an 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., 2000, 1999). The MEP perspective indicates there should be many different means of attaining the same MEP state under a given set of constraints (Dewar, 2003). Consequently, ecosystem biogeochemistry is not constrained to one particular set of species, but can be attained by an infinite number of complementary species sets, with the sets being interchangeable. By complementary we mean the community must be comprised of organisms capable of energy acquisition and element recycling, so that at steady state only free energy is dissipated. The Earth’s biosphere obviously operates in this mode, as there is no significant net accumulation or loss of biomass over time, but the conversion of electromagnetic radiation into longer wavelengths produces entropy. We have attempted to embody the idea of species composition fluidity in the kinetic expressions, Eqs. (4) and (5), where the choice of εi defines the kinetics of the community. To implement MEP, different modeling approaches

es accounted for. The photic zonestates; is we model resisthese captures two differing these two states; differing however, however, we

Earth Syst.iswith Dynam., 69–85, 2011 hile theactive aphotic zone its complement: energy ethetic radiation with radiation energy input energy via2,the input diffusion via theofdiffusion of

sry these two differing welayer box. per condition boundary into condition thestates; surface intohowever, the layer surface box.

adiation with energy inputthe viabottom thefrom diffusion is o diffuse allowed into to diffuse the bottom into box [2] box [2]offrom

www.earth-syst-dynam.net/2/69/2011/

4

global entropy J. J. Vallino: Differences and implications in biogeochemistry

f ocean

need to be developed that are independent of species compo-

nd aphotic zones sition,of which Eq. (4) is but one example.

MEP approach also highlights the importance of rey, we model aThe water

source acquisition for the construction of biological struc-

extensive variables tures needed to dissipate free energy. Different commu-

parameterizations lead to different levels of N acquisiModel focusnity is placed tion and biological structure concentrations with a minimum

structures (ST1 and S2) concentration required to achieve a given rate of entropy

production (Fig. 4b). This perspective differs from the current paradigm of Liebig’s Law of the Minimum, which fophotic zonecuses is solely on elemental limitations at the organismal level. Because organism stoichiometry changes to match resource ts complement: energyover evolutionary time scales (Elser et al., 2000, availability 2007), tes; however, we Liebig’s law becomes a weak constraint. From the MEP perspective, resources will be extracted based on cataput via the diffusion of lyst compositions that achieve the highest rate of free energy dissipation as constrained by organic chemistry. To quote ace layer box. Lineweaver and Egan (2008) “This represents a paradigm box [2] fromshift from “we eat food” to “food has produced us to eat it”. The 20 highest entropy producing solutions in the Monte y flows into the Carlo simulations illustrate that very different “food web” nstituents are allowed 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,4or alteration of initial conditions, can lead the 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 (Schr¨oder et al., 2005). Likewise, altering system drivers or boundary conditions will change the ecosystem configuration necessary to maintain an 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 intro-

the two layers, but

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81 duction, 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 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 “suboptimally”, 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 loss of entropy production in box [2]. This “altruistic behavior” is inconsistent with standard Darwinian evolution, which places optimization soley at the level of the individual; however, our results are consistent with ideas on evolution of cooperation and multilevel selection theory (Bastolla et al., 2009; Clutton-Brock, 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 organism represents a large spatial structure that is 103 to 106 times larger than the cells that comprise it, and surely those cells are spatially coordinated. Consequently, it is clear that information Earth Syst. Dynam., 2, 69–85, 2011

manuscript we will assume that biological systems organize to maximize entropy 92 In this manuscript we will assume that biological systems organize to maximize entropy ction, which is synonymous with maximizing the dissipation rate of Gibbs free energy 93 production, which is synonymous with maximizing the dissipation rate of Gibbs free energy 82 J. J. Vallino: Differences and implications in biogeochemistry tems under constant temperature and pressure. Living organisms are viewed here as manuscript we will assume that biological systems to Living maximize entropy 94 for systems under constant temperature andorganize pressure. organisms are viewed here as 92 In exchange this manuscript we will assume that biological systems organize to maximize of cooperation inentropy biological systems and the benefit of exover large spatial scales exists in nature, so it is atalysts facilitating autocatalytic reactions for the dissipation of free energy. We will ction, which is catalysts synonymous with maximizing thereactions dissipation rate of free 95 mere facilitating autocatalytic forspace thedissipation dissipation ofenergy free energy. We willspace for resource acquisition to over conceivable that ecosystems couldwith coordinate over toGibbschanging 93 production, which is synonymous maximizing the rate ofinformation Gibbs free energy ontinue our maximize main hypothesis (Vallino thatLiving the acquisition ofare Shannon information maximize dissipation of available free energy. energy acquisition and dissipation. tems under constant and 2010), pressure. organisms viewed here as 96 also continuetemperature our main hypothesis (Vallino thatLiving the acquisition ofare Shannon 94 for systems under constant temperature and 2010), pressure. organisms viewedinformation here as If ecosystem information exchange occurs over large spanon 1948), or more precisely useful information (Adami 2002,ofAdami et al. 2000), via atalysts facilitating autocatalytic for the dissipation free energy. will 97 (Shannon 1948), or precisely useful information (Adami 2002,We Adami et al. 2000), via tial catalysts scales, and if more this reactions communication facilitates increased 95 mere facilitating autocatalytic reactions for the dissipation of free energy. We will Appendix A ion facilitates the production of entropy when averaged over time. It is the ability of free energy dissipation, then the domain of a model needs to ontinue our main hypothesis (Vallino 2010), that the acquisition of Shannon 98 evolution facilitates thehypothesis production of entropy whenthat averaged over information time. It is the ability of 96 also our main 2010), becontinue chosen judiciously. Specifically,(Vallino the model domain isthe de-acquisition of Shannon information systems to store information that allows them to out-compete abiotic systems in equations Model andinparameter values non99 1948), or more precisely useful information (Adami 2002, et al. 2000), via living systems to or store information thatinformation allows them toAdami out-compete fined by 1948), the spatial extent of mass and exchange, 97 (Shannon more precisely useful information (Adami 2002,abiotic Adamisystems et al. 2000), via y production under appropriate conditions. Here, we will examine information where the latter isunder largely achieved by chemical signaling. ion facilitates the production production of entropy when averaged over time.we Ithow is the ability of 100 entropy appropriate conditions. Here, will examine how information The complete expansion of 98 evolution facilitates production of entropy when averaged over time. It is the ability of Eq. (17) for the 6 constituents in In certain systems,the such as microbial mats (van Gemerden, boxes with associated boundaries shown in Fig. 1 are lso facilitate entropy production averaged over space. For our system, wethe willtwo consider systems to store information that allows toaveraged out-compete abiotic systems in 101 may also entropy production overas our system, we will in consider 1993), itfacilitate seems reasonable to them model systems spa- For 99 living systems to store information thatthese allows them tospace. out-compete givenabiotic below.systems For box [1] constituents we have, tially connected systems, while other systems, such as the le two-box model where entropy production in each box is largely governed by two y 102 production undertwo-box appropriate conditions. Here,production we will examine how information aentropy simple model entropy in each largely governed by two 100 underwhere Here, webox willisdC examine how information
surfaceproduction and deep ocean, itappropriate is uncertainconditions. to what extent, if any, CH2 O [1] = F0,1 (CH2 O) − F1,2 (CH2 O) (A1) ons that determined the rate of biological structure (i.e.,For catalyst) synthesis and consider lso103 facilitate entropy production averaged over space. our system, we will reactions that determined the rate of biological structure (i.e., catalyst) synthesis and they are informationally connected over space. Furthermore, dt 101 may also facilitate entropy production averaged over space. For our system, we will consider itmodel is possible entropy production frombox local dation. le104 two-box where that entropy production in each is optimizalargely governed by two / h[1] − r1 [1] degradation. 102 a simple model where entropy in each box is largely governed by two tion is two-box statistically similar to that from production global optimization, ons that determined the rate of biological structure (i.e., catalyst) synthesisdC and so there may be little gained from spatial communication, O2 [1] 103 reactions that determined the rate of biological structure (i.e., catalyst) synthesis and) − F (O )
/ h[1] 105 = F0,1 (O (A2) 2 1,2 2 and local optimization (e.g., Follows et al., 2007) would be dt dation. 104 System degradation. sufficient to describe system dynamics. We do not know the odel −(1 − ε1 [1])r1 [1] − (1 − ε2 [1])r2 [1] 106 2 Model System answer to this question, as all current biogeochemical mod-

manuscript wedepend willentropy assume that biological systems organize entropy 105 how amine internal production depends on either localtoormaximize global entropy els solely on local conditions and are Markovian in
92 this manuscript we will assume that biological systems toHmaximize entropy dC 107 InTo examine how internal entropy production depends onorganize either local COglobal 2or 3 [1] nature; more research ismaximizing needed. ction, which is synonymous with the dissipation rateofofocean Gibbs freedtenergy= F0,1 (H2 CO3 ) − F1,2 (H2 CO3 ) / h[1] (A3) odel System ction optimization, we use a simple two-box model reminiscent 93 which is synonymous maximizing dissipation rate of 106 2 Model System 108 production, production optimization, we use with a simple two-boxthe model reminiscent of Gibbs ocean free energy tems undermodels constant temperature and pressure. Living are viewed here of as +(1 − ε1 [1])r1 [1] + (1 − ε2 [1])r2 [1] ochemistry to capture processes theorganisms photic and aphotic zones amine how internal developed entropy production depends onin either localLiving or global entropy 94 forbiogeochemistry systems under constant temperature and pressure. organisms are aphotic viewed zones here asof 109 models developed to capture processes in the photic and 107 To5examine how internal entropy production depends on either local or global entropy Conclusions atalysts facilitating autocatalytic reactions for the dissipation of free energy. We will sction (Fig. 1) (Ianson and Allen 2002). As customary in oceanography, we model a water dC 3 [1]energy. optimization, we1) use a simple two-box2002). model reminiscent 95 catalysts facilitating autocatalytic reactions for model the of dissipation ofNHfree Wea will 110 oceans (Fig. (Ianson and As customary inocean oceanography, we model water = −F (A4) 1,2 (NH3 )/ h[1] − ρε1 [1]r1 [1] 108 mere production optimization, weAllen use a simple two-box reminiscent of ocean dt The maximum entropy production principle (Paltridge, 1975; ontinue our main hypothesis (Vallino 2010), that thearea, acquisition of Shannon information n where all boxes and boundaries have unit surface so that extensive variables ochemistry models developed tohypothesis capture processes in the photic andacquisition aphotic zones of+ρ(1 − 96 also continue ourall main (Vallinohave 2010), that the ofextensive Shannon information 111 column where boxes and boundaries unit surface so that variables ε2 [1])rof 2 [1] 109 biogeochemistry models developed to capture processes inarea, the photic and aphotic zones Dewar, 2003) proposes that nonequilibrium abiotic or biotic 2 non 1948), or more precisely useful information (Adami 2002, Adami et al. 2000), via epend on box depth, h, and will be calculated on a per m basis. Model 2focus is placed s 97 (Fig. 1) only (Ianson and Allen 2002). As in oceanography, wem model a water systems with degrees ofcustomary freedom will organize towards 1948), or more precisely useful information (Adami 2002, Adami et model al. 2000), via 112 depend onmany box depth, h,Allen and will be calculated on ainper basis. Model focus placed 110 (Shannon oceans (Fig. 1) (Ianson and 2002). As customary oceanography, we aiswater [1] a state of maximum production, which is synonyion facilitates the production of entropy entropy when averaged over time.structures It is thedC ability of sipation of free energy by synthesis and destruction of biological (S and S ) 2 ability of n98 whereevolution all boxes and boundaries haveby unit area, so averaged that extensive variables facilitates the energy production ofsurface entropy when time.11 Itstructures is=the 113 on dissipation of synthesis and destruction of over biological (S and S+ h[1] −F ( 1 )/ 2) ε1 [1]r1 [1] − δ1 [1]r2 [1] (A5) 1,2 111 column where allfree boxes and have unit area, so that extensive variables mous with maximizing freeboundaries energy dissipation forsurface chemical dt systems to store information that allows them to out-compete abiotic systems in 2) within the two layers, but nd to chemical constituents (CH O, NH , O and H CO 2 3 2 2 3 systems. The usefulness of MEPthat theory for understanding epend onfrom box depth, will be calculated on a per m basis. Model focus is placed 99 toand store information allows toon out-compete abiotic inis but 114 and to h, chemical constituents (CH NHthem within thesystems twofocus layers, 2O,calculated 3, O2 and 2CO 112 living only systems depend on box depth, h, and will be aHper m32) basis. Model placed and modeling biogeochemistry is still under debate, and nuyy of production under appropriate conditions. Here, we will examine how information dC [1] mixing between boxes and boundaries isconditions. accounted for. Thewe photic zone is1 and 100 entropy production under appropriate Here, will examine how information sipation of free energy by synthesis and destruction of biological structures (S S ) 2 photic2 zone is 115 entropy ofapplied mixing between and boundaries is accounted for. The and theoretical needdestruction to be address 113 onmerous dissipation of free energyboxes bychallenges synthesis and of biological structures and S+ = −F1,2 (S ( 21 )/ h[1] (A6) 2) (ε2 [1] − δ2 [1])r2 [1] lso facilitate entropy production averaged over space. For our system, we will consider dt lly energy replete but resource limited, while the aphotic zone is its complement: energy before MEP becomes common toolaveraged for H solving problems inFor 101 may alsoconstituents facilitate entropy production over space. system, we will consider nd to chemical (CHa2but O, NH the twoour layers, but 116 typically energy replete resource limited, while the aphotic zone is its complement: energy 3, O2 and 2CO 3) within 114 from and to chemical constituents (CH H2CO3) For within the layers, but 2O, NH 3, O2 and how biogeochemistry. In previous work we have demonstrate box [2] two constituents, the equations are, lebut two-box model where entropy production in each box is largely governed by two d102 resource replete. Our model captures these two differing states; however, we a simple two-box model where entropy production in each box is largely governed by two y 117 of mixing between boxes and istoaccounted Thetwo photic zone states; is limited but resource replete. Our model capturesfor. these differing however, the MEP principle canboundaries be boxes applied biological systems un115 entropy of mixing between and boundaries is accounted for. The photic zone iswe dC ons that determined the rate of biological structure (i.e., catalyst) synthesis and CH2 O [2] of e energy input photosynthetic active radiation with energy via the diffusion der via transient conditions provided production inte103 reactions that determined the rate ofentropy biological structure catalyst) synthesis and lly energy replete but resource limited, while the aphotic zone isinput is(i.e., its complement: energy = Fthe h[2] (A7) 118 replace energy input via photosynthetic active radiation with energy input via diffusion of − r1 [2] 1,2 (CH 2 O)/ 116 typically energy replete but resource limited, while the aphotic zone is its complement: energy dt grated over time (Vallino, 2010). In this manuscript we have dation. cal potential from a fixed boundary condition the surface layer box. we degradation. d104 but resource replete. Ourupper model these twointo differing states;into however, 119 chemical potential from acaptures fixed the surface box. we examined how biogeochemical predictions differcondition if entropy 117 limited but resource replete. Ourupper modelboundary captures these two differing states;layer however, en in the form of ammonia is allowed to diffuse into the bottom box [2] from production is maximized locally versus globally for systems dCdiffusion e 120 energy input via photosynthetic active radiation withtoenergy via the of from O2 [2] Nitrogen in the input form via of ammonia is allowed diffuseinput into bottom box [2] 105 118 replace energy photosynthetic active radiation withtheenergy input =via F1,2the (Odiffusion (1 − ε1 [2])r1 [2] (A8) 2 )/ h[2] − of involving spatial dimensions. Biological systems explore dtthe ents, which fixes the lowerupper boundary condition [3]. Hence, energy flows into cal potential from a fixed boundary condition into the surface layer box. 121 sediments, which fixes the lower boundary condition [3]. Hence, energy flows into the biogeochemical reaction space over short time scales primarodel 119 System potential from a fixed upper boundary condition into the surface−(1 layer − εbox. 2 [2])r2 [2] 2 chemical Model System m106 viainbox [1] and resources flow in viainto boxand [2].gene Allexpresconstituents are allowed ily via changes in(nitrogen) community composition en the form of ammonia is allowed to diffuse the bottom box [2] from 122 system viainbox and resources (nitrogen) flow in viainto the [2]. All constituents are allowed 120 how Nitrogen the [1] form of is allowed to diffuse box [2] from sion; consequently, weammonia have developed simple kinetic amine internal entropy production depends ona either local or exglobal bottom entropy dC [2] 107 To examine how internal entropy production depends on either local or global entropy H CO ents, which pression, fixes the given lowerby boundary condition [3]. Hence, energy flows into 2the3 = F1,2 (H2 CO3 )/ h[2] to capture changes in community Eq.the (4),lower (A9) 121optimization, sediments,we which boundary condition [3]. Hence, energy flows into the ction use afixes simple two-box model reminiscent of ocean dtocean composition that govern biogeochemistry by using a single 108 production optimization, we use a simple two-box model reminiscent of m via box [1] and resources (nitrogen) flow in via box [2]. All constituents are allowed+(1 − ε1 [2])r1 [2] + (1 − ε2 [2])r2 [2] 122 system viadeveloped box and resources (nitrogen) flow in viaand boxaphotic [2]. Allzones constituents parameter, εi ,[1] that between 0 andin 1. Investigating ochemistry models tovaries capture processes the photic of 4 are allowed 4 109 biogeochemistry models developed to capturemodel processes in the photic and aphotic zones of MEP in a simple two-box biogeochemistry involving
s (Fig. 1) (Ianson and Allen 2002).has As revealed customary in oceanography, we model a 3water dCNH [2] two(Fig. chemical reactions that optimized 110 oceans 1) (Ianson and Allen 2002). Asglobally customary in oceanography, we=model a water F1,2 (NH (A10) 3 ) − F2,3 (NH3 ) / h[2] dt higher entropy production thanextensive lon where all solutions boxes andgenerate boundaries have unit surface area,rates so that variables 4 111 column all boxes and (Table boundaries surface −ρεvariables 1 [2]r1 [2] + ρ(14− ε2 [2])r2 [2] callywhere optimized solutions 4), andhave thereunit exists manyarea, al- so that extensive 2 epend on box depth, h, and will be calculated on a statistically per m basis. Model 2 focus is placed ternate steady state solutions that produce similar 112 only depend on box depth, h, and will be calculated on a per m basis. Model focus is placed dC(S [2] ratesenergy of entropy (Table 3).and Results from globally optimized sipation of free by synthesis destruction of biological structures and S2) 113 on dissipation of free energyhypotheses by synthesis and destruction of biological 11structures and S+ = F1,2 (S ( 1 )/ h[2] 2) ε1 [2]r1 [2] − δ1 [2]r2 [2] (A11) MEP solutions support regarding the evolution dt nd to chemical constituents (CH2O, NH3, O2 and H2CO3) within the two layers, but 114 from and to chemical constituents (CH2O, NH3, O2 and H2CO3) within the two layers, but y of mixingEarth between boxes and2,boundaries is accounted for. The photic zone is Dynam., 69–85, 115 entropy ofSyst. mixing between boxes2011 and boundaries is accounted for. The photic zone is www.earth-syst-dynam.net/2/69/2011/ lly energy replete but resource limited, while the aphotic zone is its complement: energy 116 typically energy replete but resource limited, while the aphotic zone is its complement: energy d but resource replete. Our model captures these two differing states; however, we

es in the photic and aphotic zones of aphy, we model a water y in oceanography, we model a water at extensive variables ace area, so that extensive variables J. J. Vallino: Differences and implications in biogeochemistry s. Model focus is placed a per m2 basis. Model focus is placed dC(S [2] cal structures and S2) ion of biological 21structures and S+ h[2] (A12) = F1,2 (S ( 21 )/ 2) (ε2 [2] − δ2 [2])r2 [2] dt hin the two layers, but d H2CO3) within the two layers, but Fluxes, Fi,j (k), where βi,j (k) in Eq. (12) equals zero have The photic zone is fromis the above expressions. Parameter valcounted for.been The removed photic zone ues, initial conditions plus boundary conditions used for all is its complement: energy aphotic zone is its complement: simulations are given inenergy Tables 1 and 2, respectively. states; however, we two differing states; however, we Acknowledgements. I thank Axel Kleidon for organizing the 2010 input via the diffusion of n with energy input via the diffusion ofThermodynamics and MEP in the workshop on Non-Equilibrium urface layerEarth box.system that lead to this manuscript. The work presented here ion into the surface layer box. was funded by the NSF Advancing Theory in Biology Program m box [2] from (NSF box EF-0928742), into the bottom [2] from the NSF PIE-LTER program (NSF OCEnergy flows 0423565) into the as well as from NSF CBET-0756562 and OCE-0852263. 3]. Hence, energy flows into the by: V. Lucarin constituentsEdited are allowed a box [2]. All constituents are allowed References

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