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Edited by: Pia H. Moisander, University of Massachusetts Dartmouth, United States

Reviewed by: Ralf Steuer, Humboldt University of Berlin, Germany; Juan A. Bonachela, Rutgers, The State University of New Jersey, United States

This article was submitted to Aquatic Microbiology, a section of the journal Frontiers in Microbiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We present a model of the growth rate and elemental stoichiometry of phytoplankton as a function of resource allocation between and within broad macromolecular pools under a variety of resource supply conditions. The model is based on four, empirically-supported, cornerstone assumptions: that there is a saturating relationship between light and photosynthesis, a linear relationship between RNA/protein and growth rate, a linear relationship between biosynthetic proteins and growth rate, and a constant macromolecular composition of the light-harvesting machinery. We combine these assumptions with statements of conservation of carbon, nitrogen, phosphorus, and energy. The model can be solved algebraically for steady state conditions and constrained with data on elemental stoichiometry from published laboratory chemostat studies. It interprets the relationships between macromolecular and elemental stoichiometry and also provides quantitative predictions of the maximum growth rate at given light intensity and nutrient supply rates. The model is compatible with data sets from several laboratory studies characterizing both prokaryotic and eukaryotic phytoplankton from marine and freshwater environments. It is conceptually simple, yet mechanistic and quantitative. Here, the model is constrained only by elemental stoichiometry, but makes predictions about allocation to measurable macromolecular pools, which could be tested in the laboratory.

Phytoplankton are responsible for the majority of photosynthesis in the ocean (Field et al.,

The elemental stoichiometry and growth rate of phytoplankton are not independent. Robust qualitative relationships between growth rate, elemental stoichiometry, and resource availability are evident in controlled laboratory cultures spanning wide taxonomic and allometric ranges. We illustrate this in ^{2} values in ^{2} values in

Compiled laboratory data of growth rate and light dependence of chlorophyll and elemental stoichiometry, and light dependence of ^{−2} s^{−1}.

Here we define

Illustration of general trends in laboratory data in chemostat culture studies. Growth rate (μ) and light dependence of

The common patterns in

Approximate elemental stoichiometry of key macromolecular pools.

Chlorophyll | 55:4:0 | Chlorophyll A |

Protein | 4.49:1:0 | Average value from Brown ( |

RNA | 10.7:3.8:1 | Based on GC = 0.563: ^{*} |

DNA | 11.1:3.8:1 | Based on GC = 0.563: ^{*} |

P lipid | 40:0:1 | Phosphatidylglycerol with C16 fatty acids |

C store | 1:0:0 | Carbohydrate and non-phospholipid |

N store | 2:1:0 | Cyanophycin |

P store | 0:0:1 | Polyphosphate |

Models of phytoplankton physiology have sought to relate growth rate (related to fitness) and elemental stoichiometry (related to biogeochemical impacts) to external resource availability (Riley,

In

Schematic of the two different views of the model: CFM-Phyto.

In the following sections, we outline an idealized, allocation-based model of phytoplankton physiology and growth rate under a range of resource conditions (N, P, light). We show that the observed relationships between Chl:C, N:C, P:C growth rate and light (

Laboratory studies have shown that almost all the cellular carbon in phytoplankton is accounted for by the major macromolecular pools: proteins (_{Pro}), chlorophyll and other pigments (_{Chl}), nucleic acids (_{Nuc}), carbohydrates (_{Carb}), and lipids (_{Lip}) (Anderson, _{Cell} (mol C cell^{−1}) is thus defined as the sum of these components along with carbon associated with nitrogen storage molecules, _{Nsto}:

While chlorophyll is a relatively minor contribution in this regard, it provides a constraint on the light harvesting capacity of the cells and is routinely measured. Here we neglect the contribution from intra-cellular, dissolved metabolites which are typically minor [e.g., ~4% of cellular dry weight in

Elemental variations referenced to carbon (C:C, N:C, P:C) present clearer relationships with μ and light than cellular quotas (Caperon and Meyer, _{Cell}:

where ^{−1}). Proteins account for a large fraction of the carbon and nitrogen in a phytoplankton cell (Anderson,

Recent quantitative proteomics studies revealed a coarse-grained reorganization of the proteome of

We also resolve a fixed-size pool of “essential” proteins,

Nucleic acids include contributions from DNA and RNA:

where the contribution from RNA is significantly more variable and related to growth rate; discussed in more detail below. Intracellular dissolved pools are not resolved since they generally represent <5% of the total cellular mass (Lengeler et al.,

The lipid pool can be separated into three components. A large fraction of thylakoid membrane is lipid (~30%) (Kirchhoff,

We represent the total cellular carbohydrate pool as the sum of two contributions: a flexible component,

Cellular nitrogen (_{Cell}) is mostly associated with protein (_{Pro}) (Anderson, _{RNA}), DNA (_{DNA}), chlorophyll (_{Chl}), and nitrogen storage (_{Sto}):

Since N:C presents clearer relationships with μ and light than _{Cell} (Caperon and Meyer, _{Cell}:

where ^{−1}). Each of the macromolecular pools has a distinct elemental stoichiometry (see ^{−1}) are constructed accordingly. For example, the total nitrogen content of cellular protein,

Nucleic acids (_{RNA} and _{DNA}), phospholipids in the thylakoid membrane (_{Thy}), and storage compounds including polyphosphate (_{Sto}) are observed to account for most of the cellular phosphorus in phytoplankton and bacteria (Anderson, _{Other0}. We also account for the flexible part of non-Thylakoid P-lipid in _{Sto}. Here we resolve phosphorus allocation to these distinct pools:

We note that a full accounting for the phosphorus in phytoplankton has not been experimentally characterized to date (Moreno and Martiny, _{Cell} and _{Cell}, we divide both sides of Equation (9) and obtain P:C:

where ^{−1}). As for N:C,

In addition to these statements of mass conservation and allocation, we must connect macromolecular allocation to rates. We do this assuming four mathematical representations of which three are well-supported by laboratory observations:

_{I} ^{−1} ^{−1}^{−2} ^{−1}

Here _{I} is a coefficient characterizing the absorption cross-section and turnover time of the photosynthetic unit (Cullen,

and

where _{Pho} and

This is consistent with the observed linear increase in the investment in ribosomal proteins with growth rate in multiple cultures of

where ^{−1}), which occurs at zero growth rate. This relationship says that cells need more RNA to divide faster and/or to reproduce a higher cellular protein quota.

Relationships in (i), (iii), and (iv) above are directly supported by empirical data in the associated citations. The relationship between components of the light harvesting and photosynthesis machinery in (ii) is logical and simple, but unconfirmed by direct empirical data to our knowledge.

Using the above statements of mass conservation (Equations 12, 13) and representations of key relationships between fluxes and pools (Equations 14, 15), we model the observed dependencies of cellular stoichiometry (i.e., Chl:C, N:C, and P:C) on growth rate, light intensity and limiting factor, as well as the variation of maximum growth rate,

Consider the rate of change of the cellular carbon quota, which is increased by photosynthesis, and reduced by division with population growth rate μ (d^{−1}) and maintenance respiration rate ^{−1}) (e.g., Geider et al.,

where ^{−1}], and _{I} is the per chlorophyll rate of photosynthesis as defined in Equation (11). _{5}H_{7}O_{2}N_{1}P_{1/30} using nitrate as the nitrogen source with energy transfer efficiency of 0.6 (Rittmann and McCarty, _{Cell} in Equation 16).

In steady-state, the solution of Equation (16) anticipates the observed linear relationship between the cellular chlorophyll to carbon ratio (

where _{Chl}(_{I}(_{Chl}(_{I}(_{I} were optimized in the model-data fit, with single values for _{I} for all light levels.

Equation (17), indicates that the cell must maintain pigments to sustain maintenance respiration even at a net zero growth rate (y-intercept). It predicts a linear relationship between Chl:C and μ, as well as an increase in both the slope and intercept of the Chl:C ratio with decreasing irradiance (

Model-data comparison of chlorophyll per C of ^{−1}). Legend values are light intensities (μmol m^{−2} s^{−1}).

Consider the case of nitrogen limitation, when allocation to nitrogen storage is small. In this case, to a first approximation, the cellular quota of nitrogen is dominated by that of protein (Liefer et al.,

Nucleic acids account for <7% of cellular dry weight in phytoplankton (Parsons et al.,

Here nitrogen and carbon are linked by the constant elemental ratio for protein,

Equation (19) predicts a linear relationship between N:C and growth rate which has decreasing slope and intercept as with photon flux, qualitatively consistent with the observed data in

Model-data comparison of N:C and model prediction of macromolecular allocation of ^{−2} s^{−1}) are in the legend. Curves are model results and points are data (Healey, ^{−2} s^{−1}). Black points are data for total values under the same light intensity (Healey,

The model suggests that N:C increases with growth rate at a fixed light intensity because there is a linear increase in the investment in both biosynthetic protein (Equation 14) and photosynthetic protein, latter being the associated with the linear increase in Chl:C with growth rate at fixed light intensity (Equations 12, 17). Likewise, a reduction of light intensity at a fixed growth rate also demands a higher investment in both chlorophyll and photosynthetic proteins, hence the slope of N:C increases with decreasing light intensity.

Equation (19) can be fit to the data on _{I} (and thus _{Chl} and _{Chl}) were solved by fitting Equation (17) above so the N:C vs. μ data provide constraints on _{Pho} and _{Bio} (the parameters which scale photosynthetic protein to chlorophyll and biosynthetic protein to growth rate) as well as the fixed pool of “essential” protein,

The model result shows a similar relative increase in the investment in photosynthetic and biosynthetic protein at moderate to high light levels (

A significant fraction of cellular phosphorus is present in nucleic acids, lipid membranes, and storage compounds such as polyphosphate. Consider the case for P-limited culture in which luxury storage is small and the cellular quota of phosphorus is approximated by the sum of the three pools:

where

Here we invoke two of the fundamental relationships discussed earlier: the investment in thylakoid phospholipid,

where

Equation (21) predicts a quadratic relationship of P:C with growth rate, μ, is qualitatively consistent with the non-linear relationship in the P-limited cultures of

Model-data comparison of P:C and model prediction of macromolecular allocation in P for ^{−2} s^{−1}) are in the legend. Curves are model results and points are data (Healey, ^{−2} s^{−1}). Black points are data for total values under the same light intensity (Healey,

Expanding the cellular carbon quota in terms of the macromolecular components as described by Equations (2)–(6), and accounting for only the most quantitatively influential molecules in order to provide an analytic solution (the full, un-approximated model is described in Methods), we describe carbon allocation in the cell as

Using Equation (3), we can further resolve the proteomic contributions into photosynthetic, biosynthetic, structural/other:

where

This equation indicates that as the investment in carbon storage

This inference is logical: the maximum growth rate for a given light intensity,

Simulated carbon allocation and ^{−2} s^{−1}). See

The limit of the model, which occurs when _{I} saturates. When

Simulated light dependence of maximum growth rate (_{I}) under N limitation for three different phytoplankton. ^{−2} s^{−1}) for

Growth rates and photosynthesis are often used interchangeably and the relationships for photosynthesis and light have often been applied to growth rates in ecosystem models (Moore et al., _{.} Growth rate has a non-zero intercept on the light axis, which represents the minimum light intensity required for cellular maintenance. Notably, _{I}. This can be seen in _{I}. This highlights the high cost of the photosynthetic apparatus.

The N:P ratio of plankton has been a topic of interest going back to Redfield (

Using the comprehensive data set of Healey (

Model-data comparison of N:P. ^{−2} s^{−1}).

The model qualitatively captures the variations in N:P with growth rate under both N and P limitation. Under N-limitation, N:P (

We have used laboratory data on elemental stoichiometry to constrain a model which resolves macromolecular allocation. As such, the model makes testable predictions. Our estimated photosynthetic parameters (_{I}) sit within the range of observation (Platt et al.,

Model-data comparison of macromolecular allocation. ^{−2} s^{−1} under N limitation. ^{−2} s^{−1} under N limitation (Felcmanová et al.,

The model presented above provides a conceptually simple, yet quantitative description of the relationship between the elemental stoichiometry of phytoplankton, their growth rate, resource availability, and macromolecular allocation (

How macromolecular allocation of phytoplankton responds to light and nutrient, constraining the growth rate based on CFM-Phyto.

The framework of the model is conceptually simple and steady-state solutions can be solved algebraically and parameters optimized to match empirical data. We have used it to model and interpret laboratory data relating the elemental stoichiometry of

By explicitly resolving the macromolecular allocation, the model captures and provides a simple interpretation for the contrast between cellular nitrogen quota (or N:C ratio) which varies linearly with growth rate under N-limitation, and the phosphorus quota (or P:C) which varies non-linearly with growth rate under P-limitation. At fixed light intensity and under nitrogen limitation, cellular protein increases linearly with growth rate (Felcmanová et al.,

We suggest that the explicit resolution of macromolecular reservoirs provides an important advantage over more idealized frameworks. It allows the exploitation of key observed relationships (e.g., RNA:protein vs. μ) and it explicitly couples the dynamics of N, P and C providing a comprehensive framework (c.f. N:C only in the case of Geider et al.,

We suggest that the explicit macromolecular representation also has some interpretive and predictive advantages. It has the potential to be directly compared with direct proteomic and macromolecular observations (McKew et al.,

The model illustrates the relationship between the maximum growth rate at a given light intensity and storage. In order to increase the growth rate, cells invest in protein at the expense of storage compounds. The maximum growth rate for a given light intensity occurs when storage is minimized and functional allocation is maximized. In some circumstances, maximizing growth rate will be the best measure of fitness, but in others storage is likely to be advantageous. For example, if we consider phytoplankton-bloom conditions, maximizing growth rate may be more important since in such situations, phytoplankton with faster growth can outcompete others and dominate the region (Dutkiewicz et al.,

While we have focused our development and discussion around the data set of Healey (

The model presented here represents an attempt to provide a minimal, transparent and biologically meaningful framework which relates allocation between and within the major macromolecular pools to elemental ratios and growth rates under diverse environmental conditions. It is framed so that the internal allocation is, in principle, in terms of measurable quantities (though not all were available in the data sets studied here). These measurable pools can be mapped into categories which are not directly measurable, but which are grouped by function (

As with any quantitative model, there is a trade-off between realism, data constraints and insight, and we have not resolved a number of potentially important factors. We have assumed a fixed composition of thylakoid membranes which may vary in reality. In particular, the fraction of light harvesting machinery might change relative to other components, which would alter the chlorophyll to protein or lipid ratio. The model could be improved with an additional layer of detail, separating the light harvesting and other components. To constrain the model, combined measurements of chlorophyll and proteomics (e.g., McKew et al., ^{−1}% of total spectral counts in a recent proteomic study; McKew et al.,

We have also neglected the substitution of non-P-lipids for P-lipids under low P concentrations (Van Mooy et al.,

Despite the limitations of the study listed above, we have shown that a conceptually simple model rooted in mass balance and a few basic, empirically sound representations can capture the relationships between growth rate and elemental stoichiometry under a variety of environmental conditions accurately. We suggest that the explicit representation of measurable macromolecular pools allows an advantage over more abstracted forms rooted in elemental quotas. It allows the exploitation of key physiological observations such as the changes in RNA:protein with μ, as well as testable predictions regarding macromolecular allocation. Parameters controlling rates and allocation can be calibrated with laboratory data, either inverted from stoichiometric data as we have done here, or directly measured (McKew et al.,

Physiological models of “intermediate complexity” such as this have a role to play in ecological and biogeochemical studies. While Monod (

Here we provide a complete version of the model: CFM-Phyto (

Equations (2)–(6) lead to the following accounting for total cellular carbon in various macromolecular pools:

where ^{−1} C). The cellular components can be re-arranged and gathered into four functional classes, as depicted in

where

We first obtain

where

To compute biosynthetic apparatus, we first compute

where

where

Here, the positive solution for μ

N:C is represented by the sum of N from N-containing molecules normalized by cellular C quota:

Here, we define

Then, by substituting Equations (15, 3, 14, 12, 17) (in this order) into Equation (32), we obtain

where

When the nitrogen content of RNA is accounted for, we predict a quadratic relationship between N:C and growth rate. However, since the contribution from RNA is small, that from protein dominates and the linear approximation of Equation (19) works well (as seen in the data and un-approximated solution shown in _{N} as N:C and

P:C is represented by the sum of N from N-containing molecules normalized by cellular C quota:

We define

By substituting Equations (15, 3, 14, 13, 12, 17) (in this order) into Equation (36), we obtain

where

We define _{P} as P:C and

Since RNA is the dominant contribution to cellular phosphorus, the relationship between P:C and growth rate is non-linear (

Once we obtain N:C and P:C, N:P can be obtained as follows:

There are three types of storage: C in carbohydrates and lipids, N storage assumed to be cyanophycin, and P storage assumed to be polyphosphate. Only C and N storage affect the carbon budget. To compute N and P storage, which we assume accumulate only when each element _{Cell}]_{i} (mol C m^{−3}), where _{Cell}]_{i}, is by definition the product of the cellular carbon quota, _{Cell} (mol C cell^{−1}), and the cell density, _{i} (cell m^{−3}):

Under N limitation, the time variation of dissolved inorganic N (or NO^{−3}) in the culture is based on the balance between dilution and uptake:

where ^{−1}), [_{in} (mol N m^{−3}) is the concentration of dissolved inorganic N (or NO_{N} is the N uptake rate per cellular C (mol N mol C^{−1} d^{−1}), _{N} (cell m^{−3}) is the cell density in the culture under N limitation. We also consider the time variation of _{N}:

At steady state (i.e., d[N]/dt = 0), Equation (41) suggests that

where [_{Cell}]_{N} (mol C m^{−3}) (= _{Cell}_{N}) is the carbon biomass in the culture under N limitation. The steady state of Equation (42) leads to the following well-known relation for a chemostat at steady state:

To relate _{N} to known parameters, we further consider the balance of _{N} :

The steady state of this equation leads to a simple relation between the uptake and consumption of N:

as _{N} = _{in} >> [_{Cell}]_{N}:

We follow the same procedures above (Equations 41–47) by replacing N with P to obtain an expression for the carbon biomass in the culture under P limitation [_{CCell]P} (mol C m^{−3}):

Here [_{in} (mol m^{−3}) is the concentration of dissolved inorganic P (PO

We assume that the limiting resource is that which gives the smallest cellular C concentration in the culture. For example, when [_{Cell}]_{N} < [_{Cell}]_{P}, the culture is limited by N since this relationship with Equations (47) and (48) leads to the following equation:

showing that the input N:P (left hand side) is lower than required N:P (right hand side). On the other hand, when [_{Cell}]_{N} > [_{Cell}]_{P}, input N:P is higher than required N:P indicating excess N and thus P limitation. Once the nutrient limitation is determined, we define the actual cellular C concentration in the culture [_{Cell}] (cell^{−1} m^{−3}):

With this equation, we have simulated the relationship between biomass, [_{Cell}], and growth rate, μ, which reveals a decreasing trend with dilution rates, capturing the observation (Healey,

If the culture is N limited, there is an excess of P, which could be stored in the cell. To determine the potential level of cellular P based on the P availability (^{−1}), we follow the same procedure as when determining [_{Cell}]_{N} from Equations (41)–(47), by using P instead of N, except for using _{N} and [_{Cell}]_{N} obtained previously:

Then we compare this potential quota with the maximum capacity of cellular P (^{−1}) (mol P mol C^{−1}) and define _{P}:

and by rearranging Equation (38) we determine

Under P limitation, N storage is accumulated since excess N is available. By following the same steps as above with reversed N and P and by rearranging Equation (34), we obtain the following relations:

where ^{−1}) is the potential cellular quota of N normalized by C based on the N availability and the cellular C determined by P limitation and ^{−1}) is the maximum cellular capacity of N normalized by cellular C. We note that in most cases, _{in}/[_{in}), the excess nutrient is relatively small and ^{−2} s^{−1} in

The data and model together reveal that N and P “storage” work differently (^{−1}) and maximum N storage per carbon (^{−1}); thus

This model of the storage pools is simple and logical, yet still somewhat

N storage has an associated C (e.g., cyanophycin), which can be obtained from a given elemental ratio of N storage:

The difference between the total cellular C and computed sum of macromolecular C is assumed to be C storage:

Elemental stoichiometry of each molecule and some parameters are assumed based on available information (_{I}, _{Pho}, _{Bio}, _{I}). We estimate these independently of the other parameters, which do not influence chlorophyll. Next, we estimate nitrogen related parameters (_{Pho}, _{Bio},

For _{I}, and N:C (_{Pho}, _{Bio},

The model is first solved with an initial set of parameters, which are determined by manually tuning values until model solutions are reasonably consistent with the data. The algorithm then proceeds in a series of steps (a “chain” of steps) that introduce random perturbations to the parameter values. It finds a set of values that provide a good fit to the data by keeping new parameter values that fit the data well (the new parameters become the “current” state of the parameters), and usually discarding others. That is, if the new parameters fit the data much more poorly than the “current” state, there is a high probability they will be rejected. In aggregate, the algorithm evaluates many combinations of parameters in the search for globally optimal solutions.

For a given set of parameters (

where _{j}, and σ_{k} is an estimate of the measurement error of the _{k} and add them up to obtain the error covering all the datasets:

This normalization by _{k} is intended to give similar weight to data sets with different numbers of observations and resulted in slightly improved model-data fit for

At each step in the chain, we generate a new parameter set with small random perturbations of the previous set (in this study, within the range of ±20%). In _{j} we only accept positive values and chlorophyll related parameters less than certain values (mostly ~5 times of the estimated values), since values outside of these ranges are less likely. We then compare the fit of the

Once we obtain the likelihood ratio, we generate a uniform random number (_{j}. If _{j} > _{j} is smaller than _{Current} (i.e., the model with _{j} fits better than the current model), the current parameters are updated to _{j}. However, if _{Current} is smaller than _{j}, the ^{6} steps), we identified the parameter set that gives the smallest errors between model and data.

The model code for this study can be found in GitHub

KI, JB, and MF designed the study. KI, AO, DT, and MF gathered data. KI developed a model and led the project. KI parameterized the model with help of AO, DT, and JB. KI, CD, and MF acquired funding. CD supervised KI. MF advised KI, AO, and DT. All the authors wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Rogier Braakman, Stephanie Dutkiewicz, Takako Masuda, Jodi N. Young, and Sallie W. Chisholm for useful discussions. We are grateful to Hedy Kling and Christine Sherratt for sharing the information about

The Supplementary Material for this article can be found online at:

_{12}and marine ecology. IV. The kinetics of uptake, growth and inhibition in