Bioresource Technology 102 (2011) 111–117

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Bioresource Technology

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Growth and neutral lipid synthesis in green microalgae: A mathematical model

Aaron Packer a,*, Yantao Li b, Tom Andersen c, Qiang Hu b, Yang Kuang a, Milton Sommerfeld b

a School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, United Statesb Department of Applied Sciences, Arizona State University, Mesa, AZ 85212, United Statesc Department of Biology, University of Oslo, P.O. Box 1066, Blindern, 0316 Oslo, Norway

a r t i c l e i n f o

Article history:Received 1 April 2010Received in revised form 28 May 2010Accepted 7 June 2010Available online 8 July 2010

Keywords:Mathematical modelGreen algaeNeutral lipidsPhotosynthesisBiofuel

0960-8524/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.biortech.2010.06.029

Abbreviations: NL, neutral lipid; N, nitrogen; C, cdry weight; ROS, reactive oxygen species; PS I/II, pho

* Corresponding author. Tel.: +1 602 695 5512.E-mail address: aaron.packer@asu.edu (A. Packer).

a b s t r a c t

Many green microalgae significantly increased their cellular neutral lipid content when cultured in nitro-gen limited or high light conditions. Due to their lipid production potential, these algae have been sug-gested as promising feedstocks for biofuel production. However, no models for algal lipid synthesis withrespect to nutrient and light have been developed to predict lipid production and to help improve theproduction process. A mathematical model is derived describing the growth dynamics and neutral lipidproduction of green microalgae grown in batch cultures. The model assumed that as the nitrogen wasdepleted, photosynthesis became uncoupled from growth, resulting in the synthesis and accumulationof neutral lipids. Simulation results were compared with experimental data for the green microalgaePseudochlorococcum sp. For growth media with low nitrogen concentration, the model agreed closelywith the data; however, with high nitrogen concentration the model overestimated the biomass. It islikely that additional limiting factors besides nitrogen could be responsible for this discrepancy.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The realization of industrial-scale production of algal-derivedbiofuel faces many hurdles, and its success may require progressand development in many different scientific and engineering dis-ciplines. An active research problem is to better understand whyand how, under certain environmental conditions, some speciesof algae up-regulate neutral lipids (NL), which can be readily con-verted to biodiesel and other biofuels (Scott et al., 2010; Hu et al.,2008). This article deals specifically with those species of greenmicroalgae such as Pseudochlorococcum sp. that have been ob-served to accumulate extremely high levels of NLs. In the follow-ing, unless specified otherwise, ‘‘algae” will refer to such species.

Cultures suspended in growth media with low nitrogen (low-N)concentration yield biomass with significantly higher lipid contentthan those suspended in high-N media. On the other hand, culturessuspended under high-light tend to yield greater lipid content thanthose suspended under low-light (Hu et al., 2008; Rodolfi et al.,2009). The neutral lipid content can increase from zero to well overfifty percent of dry weight. This trend has been confirmed repeat-edly by laboratory experiments (Hu et al., 2008; Scott et al., 2010;Scragg et al., 2002; Rodolfi et al., 2009).

ll rights reserved.

arbon; Chl, chlorophyll; Dw,tosystem I/II.

Photosynthesis is a highly complex process that is life-sustain-ing; however, excess light energy can be potentially harmful. Algaeabsorb light energy in order to oxidize water, providing electronsthat can face a number of different fates. This process must besafely regulated, for there is an inherent danger to continuouslyexchanging electrons between and within proximity of moleculessuch as singlet oxygen or triplet chlorophyll a, for harmful reactiveoxygen species (ROS) can form. When ROS accumulate and causemore damage than can be reconciled, algae experience photoinhi-bition and oxidative stress. Hence, with increased light there is anincreased susceptibility to photo-oxidative stress (Niyogi, 2000). Ithas been proposed that increased NL synthesis is perhaps the ‘‘de-fault pathway” to defend against photo-oxidative stress that canoccur as a result of too much reducing energy (Hu et al., 2008).Moreover, N-limitation reduces cell growth, a high energy-con-suming process. A lack of electron sinks downstream of photosys-tem I (e.g. carbon fixation) can result in a buildup of electrons inthe electron transport chain and subsequently an increased riskof photoinhibition and ROS production (Niyogi, 2000).

Metabolic pathways downstream of the electron transportchain may serve as important defenses against ROS productionduring nutrient limitation. During N-stress, cell growth is halted,but carbon fixation may continue at rates exceeding the needs ofthe cell. It may be that the ‘‘uncoupling of photosynthesis andbiomass production” as reviewed in Berman-Frank and Dubinsky(1999) is the mechanism that allows adequate electron sinks tofunction when cell growth is hindered as a result of nutrient

112 A. Packer et al. / Bioresource Technology 102 (2011) 111–117

limitation. Some species up-regulate nitrogen-free pigments orsimply excrete excess photosynthate during stressful growth con-ditions–the many mechanisms with which phytoplankton handleexcess carbon reduction is an important research area (Dubinskyand Berman-Frank, 2001; Hessen and Anderson, 2008).

Up-regulation of NL synthesis may be a means by which energycan be spent during stressed conditions, helping to maintain a safeturnover rate of the ATP and reductant pools sustained by the lightreactions. Fatty acid production is expensive in terms of ATP andreducant requirements (Xiong et al., 2010). NLs store significantlymore energy than carbohydrates do: 37 kJ/g versus 17 kJ/g, respec-tively; and, on a per-mass basis, NL synthesis requires twice thereducing energy (NADPH) than that of carbohydrate or proteinsynthesis (Hu et al., 2008). NL synthesis is an effective energy sink.It may be that certain species maintain a relatively high rate ofphotosynthesis during N-stress, but compensate by synthesizingNLs. It has been suggested that newly fixed carbon is used for NLsynthesis (Scott et al., 2010), particularly in instances when theNL dry weight of a suspension exceeds its initial dry weight (Rodol-fi et al., 2009). Oleaginous species of algae use excess carbon andenergy to synthesize storage lipids under N-stress, whereas non-oleaginous species synthesize carbohydrates or halt growth (Ro-dolfi et al., 2009).

Since N-limitation appears to be a key catalyst for excessive NLaccumulation, an immediate question is whether or not ecologicalmodels of phytoplankton–nutrient interactions can be extended tothis phenomenon. Ecological stoichiometry (Sterner and Elser,2002) in particular provides a useful foundation for mathematicalmodels by considering the relationship between the elementalcompositions of organisms and their environment. Given thatNLs serve as C storage in N-limited environments, the N:C ratioof an algal suspension may provide the means of modeling TAGaccumulation using plausible ecological models. The N:C ratio de-creases with decreasing N-availability and increasing irradiance,both of which have been observed to increase NL synthesis. CanNL synthesis be simplified ecologically and mathematically as thecause (or effect) of a low N:C? This question is important, as eco-logical stoichiometry may be applicable to other bioengineeringprocesses, see e.g. (Mauzerall, 2008).

As of yet, there have been no published mathematical models ofalgal lipid production with respect to nutrient and light conditions.Such models are important not only for gaining insight into andtesting theories of oleaginous NL synthesis, but also for optimizingbiodiesel production. Devising a mathematical model of phyto-plankton NL production poses several challenges. The mechanismsbehind NL synthesis are not fully understood (Hu et al., 2008), andmodeling NL production entails modeling non-NL dynamics suchas photosynthesis and nutrient assimilation. While there are exist-ing models of phytoplankton growth with respect to both light andnutrient limitation (Geider et al., 1998; Flynn, 2001; Loladze et al.,2000), it remains an important research area with need for moremodels (Klausmeier et al., 2008). The model to be formulated be-low, while comprehensive, is largely based on existing but uncou-pled modeling efforts (Geider et al., 1998; Huisman, 1999; Loladzeet al., 2000). Existing frameworks are combined in order to modelboth biomass and NL production subject to nutrient and lightlimitation.

2. Methods

2.1. Mathematical model

The model describes four state variables:

A(t) = algal biomass concentration, excluding neutral lipids(g dw m�3),

L(t) = neutral lipid concentration (g NL �m�3),H(t) = chl a content of A (g chl g�1 dw),N(t) = extracellular nitrogen concentration (g N m�3),

The phytoplankton mass is divided into two compartments:non-NL biomass A and neutral lipids L. Therefore the total algaedensity is the sum of the two compartments, A + L. The model isderived based upon four major assumptions:

A1. The growth rate of A, defined as l :¼ 1A

dAdt , is either N- or light

limited. N-limited growth takes the form of the well-estab-lished (Droop) cell-quota model. An increase in A, the non-lipiddry weight, requires a fixed proportion c (g C g�1 dw) of accu-mulated carbon.A2. The net carbon fixation rate is governed by the standard sin-gle-hit Poisson model of photosynthesis, normalized to the chl acontent of A.A3. Following Geider et al. (1998), chlorophyll a synthesis iscoupled with nitrogen uptake. The proportion of nitrogendevoted to chl a synthesis is regulated by the utilization touptake ratio of carbon. N-uptake is regulated by Q.A4. NL synthesis results from an excess of C-fixation relative tothe C requirements for growth. Therefore, when Q = q, allincreases in total biomass are due to de novo NL synthesis.

Based upon these assumptions, a system of four ODEs is pro-posed to describe algal growth, N-assimilation, and chlorophylland NL synthesis of a batch culture:

dAðtÞdt¼ l A; L;H;Nð ÞAðtÞ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

cell growth

; ð1Þ

dLðtÞdt¼ p A; L;H;Nð Þ � cl A; L;H;Nð Þ½ �AðtÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

NL synthesis

; ð2Þ

dHðtÞdt¼ c

lp

A; L;H;Nð Þqv A;Nð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}N uptake devoted to chl a synthesis

� HðtÞl|fflffl{zfflffl}growth dilution

; ð3Þ

dNðtÞdt¼ �v A;Nð ÞA|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

N uptake

; ð4Þ

where

QðtÞ ¼ Að0ÞQ0 þ Nð0Þ � NðtÞAðtÞ ; ð5Þ

l A; L;H;Nð Þ ¼ min lm 1� qQðtÞ

� �;p A; L;H;Nð Þ

c

� �; ð6Þ

p A; L;H;Nð Þ ¼ HðtÞpm A; L;Nð Þ 1� exp�aUI A;Hð Þpm A; L;Nð Þ

� �� �; ð7Þ

pm A; L;Nð Þ ¼ ðAQÞ2ðtÞp0

ðAQÞ2ðtÞ þ q2 AðtÞ þ LðtÞð Þ2; ð8Þ

I A;Hð Þ ¼ I0

aHðtÞAðtÞz 1� exp �aHðtÞAðtÞzð Þð Þ; ð9Þ

v A;Nð Þ ¼ qM � QðtÞqM � q

vmNðtÞNðtÞ þ vh

� �: ð10Þ

A full list of parameter descriptions is given in Table 1.

2.1.1. Formulation of (6), l(A,L,H,N)There are two aspects of growth considered in the model. The

first, referred to as simply ‘‘growth,” is exclusive to A, the non-NLbiomass. The other aspect is NL synthesis. NLs are considered aform of biomass that exists transiently under stressed conditions.When growth of A ceases, excess carbon and energy is used to

Table 1A list of the model parameter descriptions and units.

Parameter Description Units

I0 Incident irradiance mol photons m�2 d�1

z Light path ma Optical cross section of chl a m2 g�1 chlU Quantum efficiency g C (mol photons) �1

q Minimum/subsistence N quota g N g�1 dwqM Maximum N quota g N g�1 dwc C subsistence quota g C g�1 dwvm Maximum uptake rate of nitrogen g N g dw�1 d�1

vh Half-saturation coefficient g N m�3

q Maximum chl:N g chl a g�1 Nlm Maximum N-limited growth rate d�1

p0 Maximum photosynthesis rate g C g�1 chl d�1

A. Packer et al. / Bioresource Technology 102 (2011) 111–117 113

create NLs, consistent with observations in green microalgae thatNLs are synthesized with newly fixed carbon (Rodolfi et al., 2009).

Growth is either N- or light-limited. N-limited growth is mod-eled using the cell quota model lmð1�

qQÞ, where lm is the theoret-

ical maximum growth rate (d�1) and q the minimum cell quota(g N g�1 dw) (Leadbeater, 2006). To incorporate light, it is assumedthat the C-content of A is the constant c (g C g�1 dw). Therefore, themaximum growth rate possible is p/c (d�1), since p (g C g�1 dw d�1)is the influx of C. Applying Liebig’s Law of the Minimum yields theexpression (6) for the specific growth rate of A (Loladze et al., 2004;Kuang et al., 2004; Loladze et al., 2000).

2.1.2. Formulation of (7), p(A,L,H,N)The Poisson single-hit model of photosynthesis is a commonly

used and studied model of phtyoplankton photosynthesis (MacIn-tyre et al., 2002; Behrenfeld et al., 2004). Its general form ispðIÞ ¼ pm 1� exp �aIU

pm

� �h i, where pm is the light-saturated photo-

synthesis rate, and �aIU is the light-limited rate. The ratioIk = pm(aU)�1 is considered the ‘‘light saturation index”, wherebyphotosynthesis is light limited for I < Ik and light saturated forI > Ik. The photosynthesis rate is often normalized to chl a (g C(g chl a d)�1), which is important for a model that includes lightconditions since the chl a content of green algae is highly variableand can change significantly on the order of only a few days. Mul-tiplying the chl a normalized rate by H yields the photosynthesisrate normalized to non-NL biomass (7).

2.1.3. Formulation of (8), pm(A,L,N)The maximum photosynthesis rate pm is generally considered to

be nutrient dependent. In Geider et al. (1998), pm is modeled as alinear function of the N quota of algal biomass. In the proposedmodel, since Q is the N quota of non-lipid biomass, the N quotaof the total biomass is defined and calculated as follows:eQ ðtÞ :¼ AðtÞQðtÞ

AðtÞþLðtÞ. The functional response f ðeQ Þ ¼ pm is adopted as anon-linear function of eQ for several reasons.

Theoretically, pm can be considered a function of the number offunctional PSII and the turnover rate of electrons from H2O to CO2

(Behrenfeld et al., 2004). Under N-replete conditions, the dark reac-tions are generally the rate-limiting factors for pm, as electron turn-over depends on the availability of substrate downstream of PSI.Therefore, when Q > q, the functional response f ðeQ Þ should be sim-ilar to that of the N-limited growth, with f0 > 0, f00 < 0. Under N-stressed conditions, significant changes are induced in the func-tioning of PSII, and there is an observed down-regulation of theCalvin Cycle coupled with accumulation of inactive PSII duringN-limitation and/or arrestment of cellular division (Behrenfeldet al., 2004; MacIntyre et al., 2002). Although f should remainstrictly positive for Q = q, it should satisfy f00 > 0 once eQ decreasesbelow a certain value less than q. The proposed function (8)

resembles the cell-quota model dynamics for Q ; eQ > q, but is posi-tive for eQ 6 q and decreases rapidly as eQ ! 0.

2.1.4. Formulation of (2), ddt LðtÞ

As Q ? q and l decreases, carbon fixation becomes uncoupledfrom growth, potentially increasing susceptibility to photo-oxida-tive stress. Using the influx of carbon to synthesize NLs yieldstwo advantages. First, since NL synthesis requires twice the NADPHon a per-weight basis than carbohydrates or proteins, it helps com-pensate for the decrease in NADPH utilization. Second, it utilizescarbon from the Calvin cycle, helping to compensate for the de-crease in the turnover rate of electrons. The rate of carbon fixationin excess of growth is p � cl. Since it is assumed that the dryweight of the NL pool is roughly equal to its dry weight of carbon,multiplying by A gives L0.

2.1.5. Formulation of (10), v(A,N)The model for N uptake and assimilation is adopted from Geider

et al. (1998) but without temperature dependence. Uptake of N is aMichaelis–Menten function of the external N-concentration, butwith negative feedback linearly dependent on Q. The maximumuptake rate is a decreasing, linear function of Q, scaled to be vm

for Q = q and zero for Q = qM (Geider et al., 1998). The reason is thatnitrate reduction is expensive in that it requires a significantamount of energy, and so algae tend not to accumulate such highlevels of N as they do with the luxury uptake of P, for example(Flynn, 2008). The parameters vm (g N g�1 dw d�1) and vh (g N m�3)are the maximum uptake rate and half-saturation constant.

2.1.6. Formulation of (3), ddt HðtÞ

The chl a content of the algae is important for a model of bothlight and N limited growth. It not only affects the photosynthesisrate, but also is affected by the availability of N. As in Geideret al. (1998), chl a synthesis is coupled with N uptake and assimi-lation. The amount of chl a synthesized per unit N assimilated isregulated by the ratio of C utilization to net C fixation, cl

p . This ratiois the fraction of q, the maximum chl a:N ratio, that determines theproportion of v devoted to chl a synthesis. Since H is the chl a rel-ative to non-NL biomass, (3) is found by simply expandingðAHÞ0 ¼ cl

p qvA.

2.1.7. Formulation of (9), I(A,H)The Lambert–Beer law is used in order to incorporate the depth

of the reactor and self-shading. It is assumed that as light travelsthrough the reactor, its rate of absorption by phytoplankton isanalogous to attenuation described by the Lambert–Beer model(Huisman, 1999). At a depth of x m, the irradiance is related tothe incident (surface) irradiance I0 by the functionI(x) = I0exp(�aHAx), where HA is the chlorophyll density (g chla m�3) and a the absorption coefficient of the algae normalizedto chl a density (m2 g�1 chl a). This model assumes that the bio-mass density is homogenous due to mixing, and (9) is simply theaverage irradiance in a bioreactor of depth z.

2.2. Experimental

An experiment was designed to investigate the effect of nitrogenon the biomass and NL yield of this green microalgae. It was per-formed in batch culture using 60 � 10 � 3 cm cuboid-shape, flat-panel photobioreactors containing 1 L of growth medium with con-stant irradiance of 600 lmol photon m2 s�1 (51.8 mol pho-ton m2 d�1) through the 60 cm side. BG-11 medium (Stanier et al.,1971) was modified by adjusting the NaNO3 concentration to100%, 25%, or 0% of the original level (0, 0.06, and 0.240 g N L�1,respectively). All cultures were agitated by aeration with 1.5% (v/v) CO2, and in our preliminary experiments, the air flow rate

Table 2Simulation parameter values 2a and simulation errors 2b.

Parameter Value Units

(a) Parameter valuesI0 51.8 mol photon m�2 d�1

z 0.03 ma 4.82 m2 g�1 chlc 0.610 g C g�1 dwq 0.0278 g N g�1 dwqM 0.0935 g N g�1 dwvm 5.96 � 10�1 g N g�1 dw d�1

vh 1.03 � 10�5 g N m�3

q 0.283 g chl g�1 Nlm 3.26 d�1

p0 90.1 g C g�1 chl d�1

U 9.84 � 10�2 mol C mol�1 photon

0% 25% 100%

(b) Residual sum of square errorA(t) + L(t) 0.085 0.434 17.6L(t) 0.0332 0.584 8.53N(t) 0 6 � 10�4 0.0678

114 A. Packer et al. / Bioresource Technology 102 (2011) 111–117

(160 mL/s) has been optimized so that full mixing is achieved. Addi-tionally, the CO2 level has been optimized so that maximum bio-mass and lipid yield can be achieved while CO2 is not a limitingfactor. The biomass yield ofPseudochlorococcum sp. culture grownunder 1.5% CO2 was similar to that grown under 10% CO2.

For dry weight measurement, a 10 mL culture sample was fil-tered through pre-weighed Whatman GF/B filter paper (WhatmanInternational Ltd., Maidstone, UK). The filter paper was dried over-night in oven at 100 �C. The difference between the final weight

0 2 4 6 8 10 120

1

2

3

4

5

6

7

Days

g/L 25% (Model)

0% (Model)25% (Data)0% (Data)

(a) Total dry weight

0 2 4 60

1

2

3

4

5

6

7

8

9

g dw

/L

Da

100%25%0%

(c) Total dry

Fig. 1. Simulation results versus data for the total dry wei

and the weight before filtration was the dry weight of the sample.Neutral lipid content was analyzed by an improved Nile red fluo-rescence method (Chen et al., 2009). Nitrate analysis was per-formed with a Lachat QC8500 (Lachat Instruments, Milwaukee,WI, USA) according to the manufacturer instructions. Modelingthe NL yield in such a setting is a critical first step for modeling lar-ger-scale and more complicated photobioreactors.

2.3. Parameter constraints and initial estimates

The model was initially parameterized using values and rangestaken from the literature. When using arbitrary but realisticparameter values, the model agrees relatively well with the data.

2.3.1. PhotosynthesisThe photosynthesis-irradiance function requires three parame-

ters: a, U, and pm. For models of steady state growth dynamics,these parameters are often considered constants. The theoreticalupper bound for U can be calculated directly from the mass bal-ance equations for photosynthesis. In terms of oxygen evolution,this value is eight photons per molecule of O2, or 0.125 mol O2

(mol photons)�1. However, values of U for O2 evolution are typi-cally lower, ranging from 0.077 to 0.12 mol O2 (mol photons)�1

(MacIntyre et al., 2002). Processes such as nonphotochemicalquenching are responsible for this decrease from the maximum.In terms of net CO2 fixation, U is even less, with values reportedin the range of 0.001 to 0.10 mol C (mol photons)�1 (MacIntyreet al., 2002; Sakshaug et al., 1997; Behrenfeld et al., 2004). The rea-son is that there are many pathways for electrons downstream of

0 2 4 6 8 10 120

0.5

1

1.5

2

g dw

/L/d

ay

Days

25% (Model)0% (Model)25% (Data)0% (Data)

(b) Total dry weight production rates

8 10 12ys

weight data

ght density (A + L) and daily biomass production rate.

0 2 4 6 8 10 120

1

2

3

4

Days

g N

L / L

25% (Model)0% (Model)25% (Data)0% (Data)

(a) NL density

0 2 4 6 8 10 120

20

40

60

80

Days

%

25% (Model)0% (Model)25% (Data)0% (Data)

(b) NL percentage of total dry weight

0 2 4 6 8 10 120

10

20

30

40

50

60

%

Days

100%25%0%

(c) NL percentage data

Fig. 2. Simulation results versus data for NL density and NL percentage of biomass.

A. Packer et al. / Bioresource Technology 102 (2011) 111–117 115

PSII other than CO2. We express U, the net quantum yield(g C mol�1 photons) of photosynthesis, as a constant parameterwith units chosen for consistency with experimental data. To con-vert from mol C (mol photons)�1 to g C (mol photons)�1, the con-version factor 12 g C mol photons (mol C mol photons)�1 is used.

The maximum rate of photosynthesis is often expressed interms of either net carbon fixation or net oxygen evolution, nor-malized to a measure of biomass such as chl a or C. When normal-ized to chl a, pm has been measured to be as low as 0.5 and as highas 29.3 g C (g chl h)�1 (Behrenfeld et al., 2004; MacIntyre et al.,2002). Our initial chl-specific maximum photosynthesis rate, p0,is consistent with the values regularly found in the literature.

Measured values of a has been reported to range from <1 to>40 m2 (g chl a)�1 (Falkowski et al., 1985; Behrenfeld et al.,2004). When normalized to chl a, this parameter assumes thatthe absorption capacity of a culture is directly proportional to itschl a density. In reality, the proportion of photoprotective pig-ments relative to chl a can fluctuate with irradiance and nutrientconditions, resulting in non-constant values for a. On the otherhand, a has also been found to remain constant despite growthconditions (MacIntyre et al., 2002).

2.3.2. ChlorophyllFor green microalgae, the chl a proportion of biomass (or equiv-

alently the chl a:C ratio) is highly variable with respect to bothsteady state and transient growth conditions. In batch cultures,the chl a content can increase from <1% to >3% of dry weight overthe course of a few days (see e.g. (Li et al., 2008; Geider et al., 1998).Unfortunately chl a was not measured in thePseudochlorococcum

sp. experiment; however, when using realistic values between0.005 and 0.015 g chl a (g dw)�1 for the initial chl a content, themodel agrees relatively well with the data.

In the model of Geider et al. (1998), values for q range fromapproximately 0.3 to 0.4 g chl (g N)�1. The value of q calculatedfor our model is consistent with these values, and yields expecteddynamics for the chl a content of a batch culture.

2.3.3. Growth rateThe Droop model requires two parameter values (q, lm) that are

widely investigated and discussed in the literature. In models ofphytoplankton growth, the cell quota is often treated as the N:C ra-tio (g N g�1 C) of biomass rather than the N-proportion of dryweight (g N g�1 dw). The N:C ratio also associates well with ecolog-ical stoichiometry theory, which can serve as a useful foundationfor phytoplankton models. Nonetheless, only the dry weight den-sity and external NaNO3 concentration is available for thePseudo-chlorococcum sp. experiment. If we assume that the C-content(g C g�1 dw) remains relatively constant before NL accumulation,then using the N-content of biomass is equivalent to using theN:C. Possible values for q range from 0.02 g N to 0.07 (g�1 C) (Geid-er et al., 1998; Ho et al., 2003, which can be translated to N-content(g N g�1 dw) by assuming a certain C-content of dry weight (e.g.0.45 g C g�1 dw).

2.3.4. Nitrate uptakeSince the nitrate uptake model is directly taken from Geider et

al. (1998), it was parameterized similarly (Geider et al., 1998) (withunits of g C converted to g dw). It was found that even if the same

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

g N

/L

Days

100% (Model)25% (Model)0% (Model)100% (Data)25% (Data)0% (Data)

(a) N (t)

0 2 4 6 8 10 120

0.02

0.04

0.06

0.08

g N

/g d

w

Days

100% (Model)25% (Model)0% (Model)

(b) Q(t)

0 2 4 6 8 10 120

0.005

0.01

0.015

0.02

0.025

g ch

l a /

g dw

Days

100% (Model)25% (Model)0% (Model)

(c) H(t)

Fig. 3. Simulation results for N(t), Q(t), and H(t). Data for N(t) is also shown.

116 A. Packer et al. / Bioresource Technology 102 (2011) 111–117

values were used, then the model agreed with data for the extra-cellular N content to a large degree.

3. Results and discussion

In order to determine parameter values for a best fit of the data,the MATLABfminsearch routine was used with the objective func-tion as the residual sum of squres for total dry weight, NL content,and medium N concentration. This algorithm utilizes the simplexsearch algorithm described in Lagarias et al. (1998). The valuesdetermined from this algorithm are entirely consistent with the lit-erature. Refer to Table 2a for a list of parameter values.

Since the model assumes and is therefore only valid for micro-algal cultures limited by N, the data for the 100% N cultures wasnot used to determine a best fit. The 100% N culture medium hadfour times the N concentration of the 25% culture medium, yet onlyyielded approximately 36% more biomass despite the algae havinguptaken all of the N from the medium. This result in addition to thefact that BG-11 is very rich in N very strongly implies that the 100%culture’s growth was limited by another nutrient.

The simulation results are shown in Figs. 1–3. The simulationsfor 0% and 25% NaNO3 agree with the data to a great degree(Fig. 1). For error values see Table 2b. The model still agrees qual-itatively with the data for the 100% NaNO3 treatment. The poor re-sults for the 100% NaNO3 reveal that the N-quota explanation forthe transition to NL accumulation can only be valid for N-limitedcultures. Starvation of other nutrients, such as iron and phospho-rous, have also been observed to result in NL accumulation. Though

the specific nutrients related to NL synthesis may be species spe-cific, the model suggests that if nitrogen alone is to explain NL syn-thesis, then the N quota may not be a sufficient explanation. Thereason is that the growth of the high-N culture was likely limitedby an element other than N. The possibility remains that it is thestoichiometric ratio of whichever nutrient is limiting that drivesthe NL dynamics.

It is important to note that growth, with respect tototal biomass(A + L), continues for Q = q. This assumption breaks away from thetraditional treatment of the cell-quota model. The minimum cellquota q in the Droop model is often considered as the threshold va-lue for not only growth but also subsistence. It is important toremember, however, that the relationship discovered betweenthe growth rate and Q was for steady state, cell-specific growthrates of chemostat cultures (Leadbeater, 2006). When the cell quo-ta is not measured as g N (cell)�1, it is relevant to consider theimplications of phytoplankton plasticity. If biomass is as measuredas total dry weight, then dry weight-specific growth may be non-zero even when the cell-specific growth is zero. This notion is rel-evant to NL synthesis because the experimental data reveal thatafter a certain time, nearly all increases in dry weight result fromincreases in NL density. The model is based upon the notion thatthis phenomenon can be understood through the cell quota: inN-limited cultures, the minimum quota q represents the thresholdthat activates NL synthesis.

A readily apparent problem with the model is the over-calcula-tion of NL synthesis beginning between days 6 and 8 for the 25%NaNO3 culture (Fig. 2). It appears as though the NL synthesis rate(i.e. net photosynthesis rate) does not saturate as quickly as it

A. Packer et al. / Bioresource Technology 102 (2011) 111–117 117

should. In fact, for some T > 0, we have that for all t > T, Q(t) = q, andso L0(t) > 0. The reason is that if Q = q then L0 > 0 and A0 = 0. Othercontributors are that U is constant and the derived model for pm

may disregard significant factors other than the N-status of a cellthat contribute to regulation of pm. Following Geider et al.(1998), in the proposed model a and U are constants. The param-eter a describes the absorption efficiency of the algae normalizedto the density of chl a, and can vary with the chl a content of a cul-ture. In particular, regulation of U can occur on the order of min-utes to days (MacIntyre et al., 2002). Furthermore, during N-stress, U decreases in addition to pm, resulting in possible covaria-tions of these two parameters. Such parallel changes in U and pm

may result in behavior that is significantly different than that pre-dicted by models in which only pm decreases (Behrenfeld et al.,2004).

4. Conclusion

The model demonstrates that NL production may be simplifiedwithin the framework of ecological stoichiometry. In addition, thedecoupling of photosynthesis from cellular growth is a possibleexplanation for excessive NL synthesis in oleaginous green micro-algae. Future experiments designed to measure information suchas the N:C of biomass can help determine if there is a thresholdN quota for excessive NL production. Measuring cell count in addi-tion to dry weight will help determine the role that imbalancedgrowth has in NL production. Lastly, data from other oleaginousspecies need to be analyzed and used to test the model.

Acknowledgements

The authors would like to thank the anonymous reviewers fortheir valuable comments and suggestions. The research of A.P.and Y.K. was partially supported by DMS-0436341 and DMS-0920744. The research of T.A. was partially supported by grantNFR 187162/S30 ‘‘Optimizing Lipid Production by PlanktonicAlgae” from the Norwegian Research Council.

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