# Phytoplankton Temporal Strategies Increase Entropy Production in a Marine Food Web Model

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## Abstract

**:**

^{−1}, the optimization-guided model selects for phytoplankton ecotypes that exhibit complementary for winter versus summer environmental conditions to increase entropy production. We also present a new type of trait-based modeling where trait values are determined by maximizing entropy production rather than by random selection.

## 1. Introduction

^{9}or more individuals, thousands to perhaps tens of thousands of different species [22], all of which are subject to predation and viral attack; soils and sediments are approximately 1000 times more complex. While it may not be possible to model all the details of these communities and the associated chemistry they catalyze, MEP provides an opportunity for prediction, assuming living systems evolve, organize, and function to dissipate free energy. If MEP theory does not explain microbial systems, there seems little expectation that it would be useful in describing biogeochemistry of higher trophic levels where the theory of large numbers is even less applicable [23], so microbial systems are a good place to test MEP-based hypotheses.

## 2. Model Description

#### 2.1. Entropy Production

#### 2.2. Metabolic Reactions

_{2}producing O

_{2}; bacteria, ${\mathrm{\mathbb{S}}}_{B}$, that consume labile organic carbon, ${C}_{L}$, and decompose detrital organic carbon, ${C}_{D}$, and nitrogen, ${N}_{D}$, into labile constituents; consumers, ${\mathrm{\mathbb{S}}}_{C}$, that prey on phytoplankton and bacteria, as well as themselves and produce detrital organic carbon and nitrogen (Figure 1). Unlike our previous models that contained just one state variable for each functional group, in the trait-based approach, there are ${n}_{P}$, ${n}_{B}$, and ${n}_{C}$ instances or ecotypes of ${\mathrm{\mathbb{S}}}_{P\left\{i\right\}}$, ${\mathrm{\mathbb{S}}}_{B\left\{i\right\}}$, and ${\mathrm{\mathbb{S}}}_{C\left\{i\right\}}$, respectively, where a particular ecotype is distinguished using braces nomenclature, such as $P\left\{i\right\}$. The symbol $\mathrm{\mathbb{S}}$ represents biological Structure to emphasize its action as a reaction catalyst as opposed to the organismal centric view typically pursued in biology, and for simplicity the elemental composition for all $\mathrm{\mathbb{S}}$ is assumed to be the same, as given by $C{H}_{{\alpha}_{\mathrm{\mathbb{S}}}}{O}_{{\beta}_{\mathrm{\mathbb{S}}}}{N}_{{\gamma}_{\mathrm{\mathbb{S}}}}{P}_{{\delta}_{\mathrm{\mathbb{S}}}}$. The generalized reactions each functional group catalyzes are listed in Table 1, and the stoichiometrically balanced reactions are provided in Section S2.2 of the Supplementary Material. Two different kinetics expressions govern growth of phototrophs, ${\mathrm{\mathbb{S}}}_{P}$, one of which is also used for heterotrophs, ${\mathrm{\mathbb{S}}}_{B}$, ${\mathrm{\mathbb{S}}}_{C}$, as described in Section 2.3 below.

_{2}) into sugar,$\text{}{C}_{P\left\{i\right\}}$, and ${\mathrm{O}}_{2}$ (anabolic reaction). The term above the arrow, ${\mathsf{\Omega}}_{1.P\left\{i\right\}}{\mathrm{\mathbb{S}}}_{P\left\{i\right\}}\text{}$, indicates the reaction is catalyzed by phytoplankton, ${\mathrm{\mathbb{S}}}_{P\left\{i\right\}}$, but only by the faction of phytoplankton biomass that is allocated to photosynthesis specified by the variable ${\mathsf{\Omega}}_{1.P\left\{i\right\}}$. The stoichiometric coefficient ${n}_{1,P\left\{i\right\}}$ is the moles of photons needed to reversibly fix one mole of carbon dioxide and is calculated from the free energy of photons and the free energy of reaction for CO

_{2}fixation accounting for the reaction quotient under the prevailing environmental conditions (Equations (S4)–(S6)). Consequently, when ${\epsilon}_{P\left\{i\right\}}$ equals 1, free energy is conserved, no entropy is produced, and the Gibbs free energy of reaction for ${R}_{1,P\left\{i\right\}}$, defined as ${\Delta}_{r}{G}_{1,P\left\{i\right\}}$, equals 0. That is, the reaction is at equilibrium so the net reaction rate, defined by ${r}_{1,P\left\{i\right\}}$, is also 0 (however, see Supplementary Materials S2.2.1 on photon free energy dissipation by reaction versus particles). At the other extreme, when ${\epsilon}_{P\left\{i\right\}}$ equals 0, all the photon free energy is dissipated as heat, no CO

_{2}is fixed, and entropy production based on Equation (1) is maximized, where the rate of reaction, ${r}_{1,P\left\{i\right\}}$, is based on the rate of photon interception described in Section 2.3 below.

_{2}and water is the catabolic reaction that drives biosynthesis, which is included in Equation (4), where again ${\epsilon}_{P\left\{i\right\}}$ provides coupling between the two sub-reactions. When ${\epsilon}_{P\left\{i\right\}}$ equals 1, ${R}_{2,P\left\{i\right\}}$ is formulated to be at equilibrium (see Equation (S16) for details), while when ${\epsilon}_{P\left\{i\right\}}$ equals 0, ${C}_{P\left\{i\right\}}$ is completely combusted and all free energy is dissipated as heat resulting in maximum entropy production. The variable ${\mathsf{\Omega}}_{2.P\left\{i\right\}}$ is the fraction of phytoplankton biomass allocated to catalyzing the biosynthesis reaction, ${R}_{2,P\left\{i\right\}}$, but since ${\mathsf{\Omega}}_{1.P\left\{i\right\}}{\mathrm{\mathbb{S}}}_{P\left\{i\right\}}+{\mathsf{\Omega}}_{2.P\left\{i\right\}}{\mathrm{\mathbb{S}}}_{P\left\{i\right\}}$ must sum to ${\mathrm{\mathbb{S}}}_{P\left\{i\right\}}$, increasing the rate of biosynthesis by allocating more catalyst to ${R}_{2,P\left\{i\right\}}$ results in a decreased allocation of catalyst to CO

_{2}fixation, ${R}_{1,P\left\{i\right\}}$, and vice versa for increasing catalyst allocation to ${R}_{1,P\left\{i\right\}}$. Note, the complete stoichiometry for reaction ${R}_{2,P\left\{i\right\}}$, Equation (S16), is also balanced so that when ${\epsilon}_{P\left\{i\right\}}$ equals 1, ${\Delta}_{r}{G}_{2,P\left\{i\right\}}$ equals 0.

#### 2.3. Reaction Kinetics

_{2}under thermodynamic reversibility (see Equation (S6)), so the maximum rate of reaction ${R}_{1,P\left\{i\right\}}$ is given by $\frac{\Delta {I}_{P\left\{i\right\}}}{{n}_{1,P\left\{i\right\}}}$. However, the reaction rate can also be limited by CO

_{2}plus HCO

_{3}

^{-}concentration, so the overall rate expression for photon driven reaction, ${R}_{1,P\left\{i\right\}}$, is given by,

^{−1}and 5000 mmol m

^{−3}, respectively, for all functional groups and reactions, except detritus decomposition, there are no adjustable parameters other than the two control variables ${\epsilon}_{\chi \left\{i\right\}}$ and ${\mathsf{\Omega}}_{j,\chi \left\{i\right\}}$ governing reaction rates and stoichiometry. For decomposition of detrital organic matter given by reactions ${R}_{2,B\left\{i\right\}}$ and ${R}_{3,B\left\{i\right\}}$, ${\nu}_{D}^{\ast}$ replaces ${\nu}^{\ast}$ to reflect the slower kinetics associated with detritus utilization, where ${\nu}_{D}^{\ast}$ is set to 175 d

^{−1}.

#### 2.4. Model Domain and Simulation Details

_{2}and CO

_{2}with the atmosphere. The pond is modeled as a well-mixed system (0D) with equal input and output flows, $F\u27e6{m}^{3}{d}^{-1}\u27e7$, and a fixed volume, $V\u27e6{m}^{3}\u27e7$, which defines a dilution rate given by $D=\frac{F}{V}\u27e6{d}^{-1}\u27e7$. A governing set of ordinary differential equations (ODEs, Equation (S1)) is derived from mass balances (Supplementary Materials S2.3) around the six constituents $\left({H}_{2}C{O}_{3},\text{}{O}_{2},\text{}N{H}_{3},\text{}{C}_{L},\text{}{C}_{D},\text{}{N}_{D}\right)$ and ${n}_{P}$ phytoplankton, ${n}_{P}$ phytoplankton carbon stores, ${n}_{B}$ bacteria, and ${n}_{C}$ consumers. An additional three ODEs (Equations (S70)–(S72)) integrate irreversible entropy production, $\dot{\sigma}$, to obtain total entropy production, ${\sigma}^{T}$, with contributions from reactions, ${\sigma}^{R}$, particles, ${\sigma}^{P}$, and water, ${\sigma}^{W}$. All simulations are run for two years with constant inputs at a dilution rate of 0.2 d

^{−1}unless otherwise specified.

#### 2.5. Optimize Trait-Based Model

#### 2.6. Temporal Strategies

_{2}fixation reaction, ${R}_{1,P\left\{i\right\}}$) and ${\mathsf{\Omega}}_{2,P\left\{i\right\}}$ (allocation to phytoplankton biosynthesis reaction, ${R}_{2,P\left\{i\right\}}$), remain constant for the duration of the simulation. In circadian allocation, the resource allocation trait, ${\mathsf{\Omega}}_{1,P\left\{i\right\}}$, can vary with time. Initial studies used a sinusoid function for ${\mathsf{\Omega}}_{1,P\left\{i\right\}}\left(t\right)$, Equation (S75), where frequency, ${f}_{P\left\{i\right\}}$, and phase, ${\phi}_{P\left\{i\right\}}$, were used as traits and determined by EP maximization along with all other traits. While this approach worked, the global optimum was always found with ${f}_{P\left\{i\right\}}=1\text{}{d}^{-1}$, but this global solution was computationally difficult to locate due to the narrowness of the optimum (Figure S1). To increase computational speed, we choose a square-wave function for ${\mathsf{\Omega}}_{1,P\left\{i\right\}}\left(t\right)$, Equation (S76), that varies on a diel cycle, where three trait parameters, ${t}_{On\left\{i\right\}}$, ${t}_{Off\left\{i\right\}}$, and ${\Omega}_{amp\left\{i\right\}}$ are used to set the time of step-up, step-down, and amplitude of the square wave, respectively. Setting ${t}_{On\left\{i\right\}}$ and ${t}_{Off\left\{i\right\}}$ to 0 and 1 d

^{−1}, respectively, produces the same results as the passive storage strategy.

#### 2.7. Optimization and Computational Approach

## 3. Results

#### 3.1. Phytoplankton Temporal Strategies and Entropy Production

^{−1}with input concentrations given in Table 2. Under these input conditions, aerobic oxidation of all the supplied organic carbon $\left({C}_{L}+{C}_{D}\right)$ in a 1 m deep pond with 1 m

^{2}surface over a two-year period produces 0.025 MJ K

^{−1}of entropy from chemical energy dissipation, while the dissipation of solar radiation input from a latitude of 42° over the 1 m

^{2}surface produces 27.1 MJ K

^{−1}of entropy over the same two year period. Consequently, energy input and entropy production from electromagnetic radiation is more than 1000 times greater than that from chemical free energy in the nominal simulations. In Section 3.3, results from simulations where energy inputs are similar will be examined, but in this section and the next, nominal conditions (Table 2) will be used that vastly favor dissipation of electromagnetic free energy.

^{−1}, and phytoplankton reached a maximum concentration of approximately 35 mmol m

^{−3}with various diel and seasonal dynamics (Figure 2). The random solutions serve as a null model that the optimum solutions can be compared to with respect to entropy production. They also reveal a complex relationship between phytoplankton dynamics (Figure 2a) and entropy production (Figure 2b). For instance, the solution with the highest phytoplankton concentration only produces entropy at an intermediate level (Figure 2, blue lines). Furthermore, if the growth efficiencies for phytoplankton, bacteria, and consumers are set to zero so that no growth occurs, the total entropy produced is 0.3077 MJ K

^{−1}due to particle absorption of solar radiation by biomass in the input (Table 2).

^{−3}(Figure 3a) and all reduced ammonium concentrations from the input of 5 mmol m

^{−3}to 0.5 to 2 mmol m

^{−3}(Figure 3d). Similarly, detrital nitrogen, ${N}_{D}$, was drawn down from 7 mmol m

^{−3}to approximately 1.5, 1.2, and 0.7 mmol m

^{−3}in the balanced growth, passive, and circadian simulations, respectively (not shown). In all three simulations, optimal solutions excluded consumers from growing by selecting consumer growth efficiencies, ${\epsilon}_{C\left\{1\right\}}$, near 0 or 1 (Table 3). Bacteria concentrations were highest in the circadian strategy, at 12 mmol m

^{−3}, and lowest in the passive strategy, at 2.5 mmol m

^{−3}(Figure 3b). Both the circadian and passive strategies accumulated high concentrations of phytoplankton internal carbon, ${C}_{P\left\{1\right\}}$, with a strong seasonal signal, while ${C}_{P\left\{1\right\}}$ in the balanced growth strategy never exceeded 70 mmol m

^{−3}(Figure 3c). The high ${C}_{P\left\{1\right\}}$ concentrations in the passive and circadian strategies produced phytoplankton C:N ratios that varied from 115 to 230, and from 140 to 270, respectively, while the phytoplankton C:N ratio in the balanced growth solution was held fixed at 14 (data not shown).

_{2}fixation reaction rate, ${r}_{1,P\left\{i\right\}}$, is not constrained by either the thermodynamic, ${F}_{T}$, or kinetic, ${F}_{K}$, drivers, then the free energy released by ${R}_{1,P\left\{i\right\}}$ contributes entirely to ${\sigma}^{R}$. However, if ${r}_{1,P\left\{i\right\}}$ is reduced by ${F}_{T}$ (which occurs as ${\epsilon}_{P\left\{i\right\}}$ approaches 1) or by ${F}_{K}$ (which occurs as the concentration of CO

_{2}+ HCO

_{3}

^{−}becomes low relative to ${\kappa}^{\ast}{\epsilon}_{P\left\{i\right\}}^{4}$) then some of the photon free energy dissipated contributes to ${\sigma}^{P}$ as defined by Equations (S13)–(S15), because the phytoplankton photosynthetic machinery is behaving like a particle then. This accounting does not change the model simulation results, just how entropy production results are tabulated.

^{−1}, followed closely by the passive strategy at 18.95 MJ K

^{−1}, which were both much higher than the balanced growth solution, which only produced 3.984 MJ K

^{−1}(Table 3). The source of entropy production for all three strategies is largely due to light attenuation due to particles in the water column, with ${\sigma}^{P}$ accounting for 86%, 94%, and 94% for the balanced growth, passive storage, and circadian strategies, while entropy production from reactions, ${\sigma}^{R}$, only accounts for 7.1%, 5.2%, and 5.0% of ${\sigma}^{T}$, respectively (Table 3). These results are not surprising given the amount of free energy entering the system from light versus chemical potential. For aquatic systems, dissipating electromagnetic energy is mostly about synthesizing particles to intercept high frequency photons and dissipate their energy as heat [17]. Light attenuation by water, ${\sigma}^{W}$, contributes 6.9%, 0.89%, and 0.82% to entropy production for balanced, passive, and circadian strategies, respectively.

_{2}fixation (Table 3; ${\mathsf{\Omega}}_{amp\left\{1\right\}}=1$) when the fractional time of day, ${t}_{D}$, falls within the interval $0.2389\text{}d\le {t}_{D}\le 0.7799\text{}d$, and redirected to biosynthesis outside the interval (Figure 4a, black lines). In both the balanced growth and passive storage strategies, the optimal solutions locate a compromise between allocation of biomass to ${R}_{1,P\left\{1\right\}}$ versus ${R}_{2,P\left\{1\right\}}$, where 67.5% and 80.4% of biomass is allocated to carbon fixation, ${R}_{1,P\left\{1\right\}}$, at all times for the balanced growth and passive storage optimal solutions, respectively (${\mathsf{\Omega}}_{amp\left\{1\right\}}$, Table 3; Figure 4a, blue and red lines). These different allocation strategies significantly impact the rates of the two reactions associated with phytoplankton (${r}_{1,P\left\{1\right\}}$ and ${r}_{2,P\left\{1\right\}}$).

_{3}available, also limits CO

_{2}fixation rate. For instance, considering just two days in the two year simulation (Figure 4b), ${r}_{1,P\left\{1\right\}}$ and ${r}_{2,P\left\{1\right\}}$ in the balanced growth strategy (Figure 4b, blue lines), both equal 0 at night, and the decrease in both CO

_{2}fixation (blue solid line) and biosynthesis (blue dashed line) during the day is due to NH

_{3}limitation occurring (Figure 4b). The passive storage strategy avoids the reaction coupling limitation, so that biosynthesis can occur at night (Figure 4b, dashed red line), but because resource allocation is fixed in the passive strategy, all cellular resources cannot be allocated to growth at night, nor CO

_{2}fixation in the day. The circadian strategy relaxes this problem by allocating resources dynamically, so that growth at night can be maximized (Figure 4b, black dashed line), yet still be able to fix CO

_{2}during daylight at maximum rate as well (Figure 4b, black solid line). However, since free energy dissipated by chemical reactions contribute little to entropy production in all strategies (Table 3, ${\sigma}^{R}$), the differences in total entropy production lies elsewhere.

^{−3}, the balanced growth solution increases ${\sigma}^{T}$ to 15.33 MJ K

^{−1}and maintains a phytoplankton concentration of ~300 mmol m

^{−3}.

#### 3.2. Entropy Production and Food Web Complexity

^{−1}, produced nearly the same amount of entropy as the $1P1B1C$ solution (Table 4), and nutrient and organism dynamics were very similar to the $1P1B1C$ solutions as well, with only minor or duplicate contributions from the additional ecotypes (data not shown). For instance, in the $3P3B3C$ food web using passive storage strategy, two phytoplankton exhibited nearly identical dynamics and each attained a steady state concentration of ~30 mmol m

^{−3}, so when summed together they were equivalent to the $1P1B1C$ solution (Figure 3a, red line). As in the $1P1B1C$ solutions, consumers were nearly absent. The additional ecotypes in the more complex food web were effectively superfluous as far as the entropy maximization is concerned. However, the complexity of the food web became more important as dilution rate was increased, as well as the circadian strategy compared to the passive strategy.

^{−1}, the added food web complexity and the circadian strategy showed enhanced entropy production relative to the other simulations (Table 5). There is approximately a 7% to 12% increase in ${\sigma}^{T}$ associated with the increase in food web complexity from $1P1B1C$ to $2P2B2C$ or from $2P2B2C$ to $3P3B3C$ regardless of the temporal strategy employed, but a much greater increase in ${\sigma}^{T}$ occurred as temporal strategies were changed. There is approximately a 320% increase in ${\sigma}^{T}$ as the strategy was changed from balanced growth to passive storage. In fact, phytoplankton in the solutions using the balanced growth strategy were near washout conditions at a dilution rate of 1.5 d

^{−1}, as their concentrations only attain ~1 mmol m

^{−3}for a short period during the peak of summer. When the temporal strategy was switched from passive to circadian, there was approximately a 130% increase in ${\sigma}^{T}$, which indicates the usefulness of an explicit clock in improving entropy production over the passive solution. Furthermore, optimal solutions at high dilution rates exhibited complementary when more complex food webs were used.

^{−1}or more, phytoplankton (as well as the other functional groups to a lesser extent) exhibited complementary in solutions using the $2P2B2C$ or $3P3B3C$ food webs with either the passive storage or circadian strategies (Figure 5). For instance, at a dilution rate of 1.5 d

^{−1}, the best circadian solution selects for phytoplankton with traits that are complementary with respect to winter versus summer (Figure 5a, red versus black lines). Note, simulations did not investigate seasonal fluctuations in temperature, just solar radiation. In the circadian solution, ${\epsilon}_{P\left\{1\right\}}$ and ${\epsilon}_{P\left\{2\right\}}$ equal 0.341 and 0.401, respectively, which allows ${\mathrm{\mathbb{S}}}_{P\left\{2\right\}}$ to grow slightly more efficiently than ${\mathrm{\mathbb{S}}}_{P\left\{1\right\}}$ giving the former an advantage during winter when light intensity is lower and NH

_{3}is ~2 mmol m

^{−3}higher. The advantage of the circadian strategy over the passive strategy is evident in phytoplankton concentrations between the two simulations. At the same dilution rate of 1.5 d

^{−1}, the passive strategy has lower summer time phytoplankton concentration, ($P\left\{2\right\}$, Figure 5b, black line), and the winter ecotype, $P\left\{1\right\}$, is closer to being washed out of the system (Figure 5b, red line), which results in less entropy production compared to the circadian strategy (Table 5). At a dilution rate of 2.0 d

^{−1}, the $2P2B2C$ food web using the circadian strategy has an entropy production of 1.5042 MJ K

^{−1}and looks very similar to Figure 5b, which implies the circadian strategy has approximately a 0.5 d

^{−1}specific growth rate advantage over the passive strategy, which only produces 0.5943 MJ K

^{−1}at the 2.0 d

^{−1}dilution rate.

#### 3.3. Dissipation of Chemical versus Electromagnetic Free Energy

^{−3}to 12 mol m

^{−3}. With these changes, maximum possible entropy produced over a two-year period from electromagnetic radiation reduces to 2.7068 MJ K

^{−1}and that from chemical free energy increases to 2.8061 MJ K

^{−1}. Simulations are run at a dilution rate of 0.2 d

^{−1}with a $1P1B1C$ food web configuration using only the circadian temporal strategy, and we only consider a single optimization run using the standard 90 initial conditions in trait space based on Latin hypercube sampling.

^{−1}(Figure 6a, red lines), and all these solutions invest mainly in bacterial growth (Figure 6b, red lines) to oxidize ${C}_{L}$ from the initial 12.00 down to 11.232 mol m

^{−3}but leave ${C}_{D}$ unused. No phytoplankton are produced (Figure 6c, red lines), and for most solutions, consumers remain at low concentrations (Figure 6d, red line). The next 16 solutions locate an entropy maximum that is a little higher at an average of 0.4583 MJ K

^{−1}(Figure 6a, blue lines). These solutions do not invest in phytoplankton either and still produce entropy by bacterial oxidation of ${C}_{L}$, but these solutions lower the concentration of ${C}_{L}$ further to 10.485 mol m

^{−3}by investing some N resources in consumers, which results in lower bacteria concentrations (Figure 6b,d, blue lines). By investing in consumers, which remineralize N in bacteria as NH

_{3}and ${N}_{D}$ by grazing, the strategy reduces the N limitation on bacterial growth by rapid recycling of N that allows bacteria to consume and oxidize more ${C}_{L}$ and produce more entropy than solutions without consumers, even though bacteria biomass is lower, bacterial production is higher.

^{−1}and imparting smooth oscillations in cumulative entropy production due to the seasonal nature of solar radiation over the two year period (Figure 6a, grey and black lines). There appears to be either several local optimum in these solutions, or the optimization routine may have had difficulty locating the true global optima because the entropy production from the 38 solutions span a range from a minimum of 0.4846 to the maximum of 0.9041 MJ K

^{−1}(Figure 6a, grey and black lines). These solutions invest minimally in bacteria (Figure 6b), which are used primarily to remineralize ${N}_{D}$ to NH

_{3}, which is evident in values of ${\mathsf{\Omega}}_{1,B\left\{1\right\}}$, ${\mathsf{\Omega}}_{2,B\left\{1\right\}}$, and ${\mathsf{\Omega}}_{3,B\left\{1\right\}}$ traits. In the first 52 of 90 solutions discussed above, the reaction for bacterial growth, ${R}_{1,B\left\{1\right\}}$, is heavily favored with ${\mathsf{\Omega}}_{1,B\left\{1\right\}}\cong 0.95$, with the remainder of the bacterial catalyst allocated to ${N}_{D}$ decomposition by ${R}_{3,B\left\{1\right\}}$ with ${\mathsf{\Omega}}_{3,B\left\{1\right\}}$ set to 0.05. In the phytoplankton-based strategy, the weighting of bacterial catalyst to reactions is more variable, but solutions are in the neighborhood defined by ${\mathsf{\Omega}}_{1,B\left\{1\right\}}\cong 0.5$ and ${\mathsf{\Omega}}_{3,B\left\{1\right\}}\cong 0.5$. In all solutions, ${\mathsf{\Omega}}_{2,B\left\{1\right\}}\cong 0$, so that ${C}_{D}$ remains unused.

## 4. Discussion

^{−1}before washout occurs, which is near observed maximum phytoplankton growth rates at 20 °C [68]. These results are encouraging in that our formulation does not include a parameter for maximum specific growth rate like standard kinetic models, but only uses photon interception rate combined with an MEP-determined trait on growth efficiency, ${\epsilon}_{P\left\{i\right\}}$, to set the specific growth rate. We have not included temperature dependency in the MEP model, so we cannot compare our results to the full Eppley curve [69], but adding temperature dependency is a next step in model development. The addition of the phytoplankton carbon storage improves growth rate for both the passive and circadian strategies over the balanced growth strategy, which is consistent with observation on improving competitive advantage in fluctuating environments [67,70].

^{−1}), but they did produce more entropy at higher specific growth rates (1.5 and 2.0 d

^{−1}) compared to the $1P1B1C$ food web configuration. At the higher specific growth rates, the more complex models discovered complementarity [74,75], where trait values for one phytoplankton specialized in high light intensity during the summer and another ecotype had trait values that performed better under winter conditions. Complementarity is also exhibited in trait-based models [76] provided the initial population contains ecotypes with diverse parameterizations. In our approach, this is not necessary as the optimization sets the trait values and will select for complementary ecotypes when the strategy increases entropy production.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Food web structure used in the marine plankton model consisting of three function groups representing phytoplankton, ${\mathrm{\mathbb{S}}}_{P}$, with their internal CH

_{2}O storage, ${C}_{P}$, bacteria, ${\mathrm{\mathbb{S}}}_{B}$, and consumers, ${\mathrm{\mathbb{S}}}_{C}$. Colored lines correspond to reactions a functional group is capable of catalyzing. The thermodynamic properties of glucose (unit carbon basis) is used to represent labile and detrital carbon, ${\left(C{H}_{2}O\right)}_{L}$ and ${\left(C{H}_{2}O\right)}_{D}$, respectively, while ammonium is used to represent detrital nitrogen, ${\left(N{H}_{3}\right)}_{D}$.

**Figure 2.**(

**a**) Phytoplankton concentration and (

**b**) cumulative total entropy production, ${\sigma}^{T}$, over two years for 90 simulations with random selection of trait values for a $1P1B1C$ food web configuration using nominal input concentrations (Table 2). Three of the 90 solutions are highlighted by color in (

**a**) and (

**b**) corresponding to high (red), intermediate (blue), and low (orange) entropy production.

**Figure 3.**Variations in (

**a**) phytoplankton, (

**b**), bacteria, (

**c**) phytoplankton carbon storage ${C}_{P}$, and (

**d**) ammonium concentrations (mmol m

^{−3}) over the two-year simulations under the three different temporal strategies: Blue, balanced growth; red, passive storage; black, circadian allocation.

**Figure 4.**(

**a**) How phytoplanton resource allocation, ${\mathsf{\Omega}}_{1,P\left\{1\right\}}$ (solid lines) and ${\mathsf{\Omega}}_{2,P\left\{1\right\}}$ (dashed lines) and (

**b**) reactions for CO

_{2}fixation, ${r}_{1,P\left\{1\right\}}$, (solid lines) and biosynthesis, ${r}_{2,P\left\{1\right\}}$ (dashed lines), vary over a two-day period in the two-year simulations associated with balanced growth (blue lines), passive storage (red lines), and circadian resource allocation (black lines).

**Figure 5.**Phytoplankton concentration for a $2P2B2C$ food web configuration at a dilution rate of 1.5 d

^{−1}using (

**a**) the circadian allocation strategy versus (

**b**) the passive storage strategy.

**Figure 6.**All 90 solutions from simulations using a $1P1B1C$ food web configuration with phytoplankton circadian allocation strategy where the input ${C}_{L}$ concentration has been increased to 12 mole m

^{−3}and the solar radiation has been decreased by a factor of 10 so that electromagnetic and chemical free energy inputs are nearly equal. (

**a**) Cumulative total entropy production, (

**b**) bacteria, (

**c**) phytoplankton, and (

**d**) consumer concentrations over the two year simulation, where the line colors highlight solutions grouped around the three different optimum attractors corresponding to bacteria only (red), bacteria plus consumers (blue), and phytoplankton (grey and black).

**Table 1.**Reactions associated with the three functional groups (phytoplankton $\left(P\right)$, bacteria $\left(B\right)$, and consumers $\left(C\right)$ ), where ${R}_{j,\chi \left\{i\right\}}$ represents sub-reaction $j$ of biological catalyst ${\mathrm{\mathbb{S}}}_{\chi \left\{i\right\}}$, and $\chi \left\{i\right\}$ is ecotype $\left\{i\right\}$ of $P$, $B$, or $C$, while the rate of reaction ${R}_{j,\chi \left\{i\right\}}$ is symbolized ${r}_{j,\chi \left\{i\right\}}$. For consumers, $i$ spans all $P\left\{i\right\}$, $B\left\{i\right\}$ and $C\left\{i\right\}$; consequently, among the three functional groups there are a total of $2{n}_{P}+3{n}_{B}+\left({n}_{P}+{n}_{B}+{n}_{C}\right){n}_{C}$ reactions. Reactions are shown here to emphasize function only, and the third column shows which functional group catalyzes the reaction. Complete reaction stoichiometries are provided in Section S2.2 of the Supplementary Material and ${\mathrm{H}}_{3}{\mathrm{PO}}_{4}$ and ${P}_{D}$ (detrital ${\mathrm{H}}_{3}{\mathrm{PO}}_{4}$ ) are only used in thermodynamic calculations but are not state variables in the model. Definitions: ${H}_{2}C{O}_{3}$, dissolved inorganic carbon; ${\gamma}_{H}$, high-energy photons; ${C}_{P\left\{i\right\}}$, internal carbon storage of phytoplankton; ${C}_{L}$, labile organic carbon (glucose on unit carbon basis); ${C}_{D}$, detrital carbon (glucose on unit carbon basis); ${N}_{D}$, detrital nitrogen (composition as ${\mathrm{NH}}_{3}$ ).

Rxn. | Abbreviated Stoichiometry | Cat. |
---|---|---|

${R}_{1,P\left\{i\right\}}$ | ${\mathrm{H}}_{2}{\mathrm{CO}}_{3}+{\gamma}_{H}\to {C}_{P\left\{i\right\}}+{\mathrm{O}}_{2}$ | ${\mathrm{\mathbb{S}}}_{P\left\{i\right\}}$ |

${R}_{2,P\left\{i\right\}}$ | ${C}_{P\left\{i\right\}}+{\mathrm{NH}}_{3}+{\mathrm{H}}_{3}{\mathrm{PO}}_{4}+{\mathrm{O}}_{2}\to {\mathrm{\mathbb{S}}}_{P\left\{i\right\}}+{\mathrm{H}}_{2}{\mathrm{CO}}_{3}$ | ${\mathrm{\mathbb{S}}}_{P\left\{i\right\}}$ |

${R}_{1,B\left\{i\right\}}$ | ${C}_{L}+{\mathrm{NH}}_{3}+{\mathrm{H}}_{3}{\mathrm{PO}}_{4}+{\mathrm{O}}_{2}\to {\mathrm{\mathbb{S}}}_{B\left\{i\right\}}+{\mathrm{H}}_{2}{\mathrm{CO}}_{3}$ | ${\mathrm{\mathbb{S}}}_{B\left\{i\right\}}$ |

${R}_{2,B\left\{i\right\}}$ | ${C}_{D}\to {C}_{L}$ | ${\mathrm{\mathbb{S}}}_{B\left\{i\right\}}$ |

${R}_{3,B\left\{i\right\}}$ | ${N}_{D}\to {\mathrm{NH}}_{3}$ | ${\mathrm{\mathbb{S}}}_{B\left\{i\right\}}$ |

${R}_{\chi \left\{j\right\},C\left\{i\right\}}$ | ${\mathrm{\mathbb{S}}}_{\chi \left\{j\right\}}+{C}_{\chi \left\{j\right\}}+{\mathrm{O}}_{2}\to {\mathrm{\mathbb{S}}}_{C\left\{i\right\}}+{\mathrm{H}}_{2}{\mathrm{CO}}_{3}+{C}_{D}+{\mathrm{NH}}_{3}+{N}_{D}+{\mathrm{H}}_{3}{\mathrm{PO}}_{4}+{P}_{D}$ | ${\mathrm{\mathbb{S}}}_{C\left\{i\right\}}$ |

**Table 2.**Concentrations of state variables in the feed, as well as environmental conditions for the nominal simulations, where ${I}_{s}$ is ionic strength and ${I}_{0}^{M}$ is the maximum surface solar radiation at 0° latitude.

Input | Value | Input | Value |
---|---|---|---|

${\mathit{I}}_{\mathbf{0}}^{\mathit{M}}\mathbf{\u27e6}\mathit{m}\mathit{m}\mathit{o}\mathit{l}\mathbf{-}\mathit{\gamma}\text{}{\mathit{m}}^{\mathbf{-}\mathbf{2}}\text{}{\mathit{d}}^{\mathbf{-}\mathbf{1}}\mathbf{\u27e7}$ | 406,000 | $\left[{C}_{L}\right]\u27e6mmol\text{}{m}^{-3}\u27e7$ | 10 |

$\mathit{T}\mathbf{\u27e6}\mathit{K}\mathbf{\u27e7}$ | 293 | $\left[{C}_{D}\right]\u27e6mmol\text{}{m}^{-3}\u27e7$ | 100 |

$\mathit{p}\mathit{H}$ | 8.1 | $\left[{N}_{D}\right]\u27e6mmol\text{}{m}^{-3}\u27e7$ | 7 |

${\mathit{I}}_{\mathit{s}}\mathbf{\u27e6}\mathit{M}\mathbf{\u27e7}$ | 0.72 | $\left[{\mathrm{\mathbb{S}}}_{P\left\{i\right\}}\right]\u27e6mmol\text{}{m}^{-3}\u27e7$ | 0.1 |

$\mathbf{\left[}{\mathit{H}}_{\mathbf{2}}\mathit{C}{\mathit{O}}_{\mathbf{3}}\mathbf{\right]}\mathbf{\u27e6}\mathit{m}\mathit{m}\mathit{o}\mathit{l}\text{}{\mathit{m}}^{\mathbf{-}\mathbf{3}}\mathbf{\u27e7}$ | 2000 | $\left[{C}_{P\left\{i\right\}}\right]\u27e6mmol\text{}{m}^{-3}\u27e7$ | 0.1 |

$\mathbf{\left[}{\mathit{O}}_{\mathbf{2}}\mathbf{\right]}\mathbf{\u27e6}\mathit{m}\mathit{m}\mathit{o}\mathit{l}\text{}{\mathit{m}}^{\mathbf{-}\mathbf{3}}\mathbf{\u27e7}$ | 225 | $\left[{\mathrm{\mathbb{S}}}_{B\left\{i\right\}}\right]\u27e6mmol\text{}{m}^{-3}\u27e7$ | 0.1 |

$\mathbf{\left[}\mathit{N}{\mathit{H}}_{\mathbf{3}}\mathbf{\right]}\mathbf{\u27e6}\mathit{m}\mathit{m}\mathit{o}\mathit{l}\text{}{\mathit{m}}^{\mathbf{-}\mathbf{3}}\mathbf{\u27e7}$ | 5 | $\left[{\mathrm{\mathbb{S}}}_{C\left\{i\right\}}\right]\u27e6mmol\text{}{m}^{-3}\u27e7$ | 0.1 |

**Table 3.**Optimal trait values obtained from maximizing total entropy production, ${\sigma}^{T}$, in a $1P1B1C$ food web model over a two-year period for the three different temporal strategies under nominal conditions (Table 2) at a dilution rate of 0.2 d

^{−1}.

Variable | Balanced Growth | Passive Storage | Circadian Allocation |
---|---|---|---|

${\mathit{\epsilon}}_{\mathit{P}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}$ | 0.2536 | 0.3452 | 0.3788 |

${\mathit{t}}_{\mathit{O}\mathit{n}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}\text{}\mathbf{\left(}\mathit{d}\mathbf{\right)}$ | 0.0000 * | 0.0000 * | 0.2389 |

${\mathit{t}}_{\mathit{O}\mathit{f}\mathit{f}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}\text{}\mathbf{\left(}\mathit{d}\mathbf{\right)}$ | 1.0000 * | 1.0000 * | 0.7799 |

${\mathit{\Omega}}_{\mathit{a}\mathit{m}\mathit{p}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}$ | 0.6746 | 0.8036 | 1.0000 |

${\mathit{\epsilon}}_{\mathit{B}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}$ | 0.1686 | 0.1618 | 0.1628 |

${\mathit{\Omega}}_{\mathbf{1}\mathbf{,}\mathit{B}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}$ | 0.3351 | 0.3852 | 0.4349 |

${\mathit{\Omega}}_{\mathbf{2}\mathbf{,}\mathit{B}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}$ | 0.1997 | 0.1609 | 0.2869 |

${\mathit{\Omega}}_{\mathbf{3}\mathbf{,}\mathit{B}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}$ | 0.4651 | 0.4539 | 0.2782 |

${\mathit{\epsilon}}_{\mathit{C}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}$ | 0.0001 | 0.9971 | 0.0001 |

${\mathit{\Omega}}_{\mathit{P}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}\mathbf{,}\mathit{C}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}$ | 0.7288 | 0.0000 | 0.6547 |

${\mathit{\Omega}}_{\mathit{B}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}\mathbf{,}\mathit{C}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}$ | 0.0729 | 0.3888 | 0.5809 |

${\mathit{\Omega}}_{\mathit{C}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}\mathbf{,}\mathit{C}\mathbf{\left\{}\mathbf{1}\mathbf{\right\}}}$ | 0.4396 | 0.5868 | 0.0963 |

${\mathit{\sigma}}^{\mathit{R}}\text{}\mathbf{\left(}\mathit{M}\mathit{J}\text{}{\mathit{K}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)}$ | 0.2826 | 0.9901 | 0.9787 |

${\mathit{\sigma}}^{\mathit{W}}\text{}\mathbf{\left(}\mathit{M}\mathit{J}\text{}{\mathit{K}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)}$ | 0.2751 | 1.687 | 1.616 |

${\mathit{\sigma}}^{\mathit{P}}\text{}\mathbf{\left(}\mathit{M}\mathit{J}\text{}{\mathit{K}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)}$ | 3.426 | 17.79 | 18.60 |

${\mathit{\sigma}}^{\mathit{T}}\text{}\mathbf{\left(}\mathit{M}\mathit{J}\text{}{\mathit{K}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)}$ | 3.984 | 18.95 | 19.74 |

**Table 4.**Total entropy production, ${\sigma}^{T}$, from three different food web configurations over a two-year period with the three different temporal strategies under nominal conditions (Table 2) at a dilution rate of 0.2 d

^{−1}.

Strategy | $1\mathit{P}1\mathit{B}1\mathit{C}$ | $2\mathit{P}2\mathit{B}2\mathit{C}$ | $3\mathit{P}3\mathit{B}3\mathit{C}$ |
---|---|---|---|

Balanced | 3.9837 | 4.034 | 4.0811 |

Passive | 18.9511 | 18.9976 | 19.0152 |

Circadian | 19.7359 | 19.7844 | 19.7987 |

**Table 5.**Total entropy production, ${\sigma}^{T}$, from three different food web configurations over a two-year period, each run using the three different temporal strategies under nominal inputs concentrations (Table 2) but at a dilution rate of 1.5 d

^{−1}.

Strategy | $1\mathit{P}1\mathit{B}1\mathit{C}$ | $2\mathit{P}2\mathit{B}2\mathit{C}$ | $3\mathit{P}3\mathit{B}3\mathit{C}$ |
---|---|---|---|

Balanced | 0.3226 | 0.3455 | 0.3684 |

Passive | 1.3374 | 1.4995 | 1.6432 |

Circadian | 3.0891 | 3.4000 | 3.6242 |

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Vallino, J.J.; Tsakalakis, I. Phytoplankton Temporal Strategies Increase Entropy Production in a Marine Food Web Model. *Entropy* **2020**, *22*, 1249.
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Vallino JJ, Tsakalakis I. Phytoplankton Temporal Strategies Increase Entropy Production in a Marine Food Web Model. *Entropy*. 2020; 22(11):1249.
https://doi.org/10.3390/e22111249

**Chicago/Turabian Style**

Vallino, Joseph J., and Ioannis Tsakalakis. 2020. "Phytoplankton Temporal Strategies Increase Entropy Production in a Marine Food Web Model" *Entropy* 22, no. 11: 1249.
https://doi.org/10.3390/e22111249