Journal of Theoretical Biology 380 (2015) 183–191

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Theoretical lessons for increasing algal biofuel: Evolution of oil accumulation to avert carbon starvation in microalgae Tetsuya Akita a,n,1, Masashi Kamo b a b

Graduate University for Advanced Studies, Hayama, Kanagawa 240-0193, Japan National Institute of Advanced Industrial Science and Technology, 16-1 Onogawa, Tsukuba, Ibaraki 305-8569, Japan

H I G H L I G H T S








Oil accumulation by an alga that utilizes carbon and nitrate is modeled. Allocation ratio for the accumulation at ESS is determined. Direct trigger for accumulation is avoiding an increased death by carbon starvation. Nitrate limitation promotes the accumulation but is not a necessary condition. Strong carbon starvation and moderately limited nitrate will maximize total biofuel.

art ic l e i nf o

a b s t r a c t

Article history: Received 6 October 2014 Received in revised form 21 May 2015 Accepted 22 May 2015 Available online 3 June 2015

Microalgae-derived oil is considered as a feasible alternative to fossil-derived oil. To produce more algal biomass, both algal population size and oil accumulation in algae must be maximized. Most of the previous studies have concentrated on only one of these issues, and relatively little attention has been devoted to considering the tradeoff between them. In this paper, we first theoretically investigated evolutionary reasons for oil accumulation and then by coupling population and evolutionary dynamics, we searched for conditions that may provide better yields. Using our model, we assume that algae allocate assimilated carbon to growth, maintenance, and carbon accumulation as biofuel and that the amount of essential materials (carbon and nitrate) are strongly linked in fixed proportions. Such stoichiometrically explicit models showed that (i) algae with more oil show slower population growth; therefore, the use of such algae results in lower total yields of biofuel and (ii) oil accumulation in algae is caused by carbon and not nitrate starvation. The latter can be interpreted as a strategy for avoiding the risk of increased death rate by carbon starvation. Our model also showed that both strong carbon starvation and moderately limited nitrate will promote total biofuel production. Our results highlight considering the life-history traits for a higher total yields of biofuel, which leads to insight into both establishing a prolonged culture and collection of desired strains from a natural environment. & 2015 Elsevier Ltd. All rights reserved.

Keywords: C/N balance Life-history evolution Seasonal variation Adaptive dynamics Ecological stoichiometry

1. Introduction Proven oil reserves are estimated to be 1652.6 thousand million barrels and daily oil consumption is estimated to be 88,034 thousand barrels (Ruhl, 2012). If oil consumption continues at this rate, world oil reserves will be exhausted in 50 years. New sources of oil are continuously being searched for and found; therefore, it is unlikely that the world supply of fossil oil will be exhausted soon. However, oil n

Corresponding author. E-mail address: [email protected] (T. Akita). 1 Present address: National Research Institute of Far Seas Fisheries, Fisheries Research Agency, 5-7-1 Orido, Shimizu, Shizuoka 424-8633, Japan. http://dx.doi.org/10.1016/j.jtbi.2015.05.027 0022-5193/& 2015 Elsevier Ltd. All rights reserved.

is a finite resource and is certain to be exhausted at some point of time in the future. Thus, it is always important to find renewable oil to enable its continuous consumption by future generations. If renewable oil replaces fossil oil, it will also contribute to reduce the level of carbon dioxide, which is considered the main contributor for global warming (McCarthy, 2001; Solomon, 2007). Producing renewable fuels is a task that requires immediate action in this century. Microalgae came into spotlight as fuel producers during the 1970s when the world faced an oil crisis (Regan and Gartside, 1983; Sheehan et al., 1998). Microalgae assimilate carbon by photosynthesis and accumulate carbon in the form of oils such as triacylglycerides. These accumulated oils yield biodiesel fuels (Georgianna and Mayfield, 2012) (hereafter, we will refer to this

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microalgal oil as biofuel). Algae possess numerous advantages over higher plants as being a source of biofuels, particularly for reducing the environmental load and are a competition for conventional agriculture (Chisti, 2008; Smith, 2010). However, in the 1970s, the prices of oil produced by algae were much higher than those of fossil-derived oils; therefore, the effort was abandoned (Haag, 2007). Thereafter, with the continuous rise in oil prices and the increasing global warming, “algae bloomed again” (Haag, 2007) and vast research and development resources are now being invested in studies seeking algae with high oil production efficiency (Georgianna and Mayfield, 2012). Researchers looked for conditions leading to high biofuel accumulation within cells and found that deficiencies of nutrients such as nitrogen, phosphorus, and silicon were triggers for oil accumulation (e. g., Sheehan et al., 1998; Rodolfi et al., 2008; Griffiths and Harrison, 2009; Přibyl et al., 2014). Although the physiological/biochemical processes responsible for oil accumulation induced by nutrient deficiency are unclear (Přibyl et al., 2014), the underlying principle is that insufficient nutrients for protein synthesis necessary for growth channels would transfer the excess carbon produced during photosynthesis into storage molecules such as triacylglycerides (Rodolfi et al., 2008; Scott et al., 2010; Přibyl et al., 2014). It was also found that nutrient deficiencies inhibited cell division, resulting in low population growth (Rodolfi et al., 2008; Scott et al., 2010). Because total biofuel yield depends both on the amount of biofuel in cells and the number of algae in a given population, maximizing the oil within cells does not necessarily maximize its total yield (Chisti, 2007; Rodolfi et al., 2008; Smith, 2010; Shurin et al., 2013). With the aim of overcoming the tradeoff between growth rate and fuel accumulation by identifying the metabolic processes involved in fuel accumulation with the aid of genetic engineering, the mainstream of this research area has shifted to physiology and biochemistry (Sheehan et al., 1998; Shurin et al., 2013). Despite the lack of knowledge pertaining to the metabolic processes associated with nutrient deficiencies, considerable effort has been made to identify genes involved in higher growth rates and/or increased biofuel accumulation (Hu, 2008; Beer et al., 2009; Miller et al., 2010; Radakovits et al., 2010, 2012; Rismani-Yazdi et al., 2012; Tanaka et al., 2015). If genes with higher performance are identified, would it then be possible that a strain with a high growth rate and high biofuel accumulation in cells could be used to establish a culture system for high biofuel yield? This task may not be so easy. Cell division and fuel accumulation require certain resources. In microalgae, these resources assimilate carbon by photosynthesis and nutrient uptake. Because available resources are always limited, it is natural that microalgae will allocate these resources in a manner to maximize their fitness (Maynard-Smith and Price, 1973). Thus, although “good” genes are incorporated into a microalga, these genes may not function well unless the resources are allocated for gene expression. Together with these considerations, to achieve higher biofuel yield from microalgal culture, it is important to consider the adaptation underlying the microalgal allocation of resources (Bull and Collins, 2012; Shurin et al., 2013). On a time scale much longer than the average lifetime of algae, biofuel accumulation may be acquired through an evolutionary process. What are the function and the adaptive advantages of oil accumulation in algae? It is natural to consider that its main function is the storage of energy and carbon (but see Solovchenko, 2012). Intuitively, because nutrient deprivation inhibits cell division, carbon assimilated by photosynthesis and not used for cell division (the portion of carbon not used for cell growth is termed “surplus carbon” throughout this paper) is expected to be translocated to organelles for storage as oils (e.g., Roessler, 1990; Rodolfi et al., 2008; Scott et al., 2010; Přibyl et al., 2014). This verbal argument is commonly accepted as conventional wisdom by the microalgal biochemistry community and leads to the following

speculation: the adaptive purpose of oil accumulation is that surplus carbon resulting from nutrient deprivation would be used for storage (Solovchenko, 2012). However, this speculation is not so straightforward, given that this surplus carbon can be used for other purposes such as maintaining life by ATP synthesis. Algae that do not accumulate these fuels probably allocate this surplus carbon to maintain life. Thus, oil accumulation is an adaptive strategy for a specific environment, wherein these algae have higher fitness. Accordingly, it is important to identify the types of selection pressures directed towards the evolution of accumulation depending on environmental conditions. Previous studies based on theoretical evolutionary ecology have shown the possibility that food deprivation promoted the evolution of energy storage. For example, Shertzer and Ellner (2002) analyzed algarotifer (prey-predator) dynamics, and showed that energy storage of rotifers would evolve to overcome periodically or chaotically fluctuating algal density. Kooi and Troost (2006) analyzed the similar dynamic but assumed an external periodic supply of algae that are non-vital. However, these studies considered binary allocation: to store or to consume immediately. Alga uses carbon and nitrate as essential resources. It is considered that the amount of these two resources are dependently interacted each other since alga is composed of carbon and nitrate brought together in a nonaribitrary proportion (C/N ratio). Therefore, ecological-stoichiometry model that explicitly incorporates the fixed proportion and the conservation of mass in chemical reactions (Elser et al., 2012) is suited for describing dynamics in alga. With incorporating the usage of two resources into population dynamics, optimal life-history trait responsible for accumulation is little known. Although, using such stoichiometric models, several studies have investigated how organisms use multiple resources to maximize their growth rates (e.g., Klausmeier et al., 2007; Litchman et al., 2007; Bonachela et al., 2013), energy storage as a risk-hedging strategy to avert future food deprivation in the presence of multiple resources has not been considered. In this study, we considered the allocation of two resources (carbon and nitrate) and sought an evolutionary optimum. We also aimed to identify a condition leading to the maximum cultivation of oil from algae. For this purpose, not only the amount of stored oil within an alga but also the number of algal individuals must be considered because the total yield of oils is multiple of these two. By artificially strengthening selection pressure, we can generally select algae with increased accumulations of biofuel. Although we cannot predict the direction in which this evolution will occur, it is natural to expect that it will occur in the direction of rapid growth if there are sufficient resources in the culture environment (Bull and Collins, 2012), i.e., natural selection favors individuals that allocate far more resources to cell division than to oil storage. Establishing a culture system that prevents microalgal strains from evolving in this undesired direction would be required. Here we theoretically model physiological responses in microalgae and investigate conditions that can lead to the evolution of increased biofuel accumulation. Our model explicitly includes a tradeoff for a resource allocated for growth, fuel accumulation, and the maintenance of life. Based on the analyses of population dynamics, we investigated the optimum balance of allocations for both growth and accumulation such that accumulation could be maximized. We proposed a hypothesis for the adaptive reasons that explained the evolution of biofuel accumulation.

2. Modeling We used a simple resource allocation model whose simplest case has been well illustrated by Gurney and Nisbet (1998). We modeled (1) algal population dynamics and (2) carbon dynamics within an alga. These two models were coupled and followed by the

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investigation of the total yield of biofuels. We considered a situation in which a microalgal population lived within a finite space (such as open ponds or microtubes). These spaces were assumed to be well mixed and nutrients were uniformly distributed within them. The carbon available to an alga for a given unit of time was the amount of carbon assimilated by photosynthesis plus carbon that had accumulated as oils (called biofuel throughout this paper). We consider carbon cycle within a cell between carbon stored as oil and the net amount of carbon available for allocation to strategize an optimal life. The net amount of carbon ðC sum Þ available for an alga at a given time is a sum of carbon assimilated by photosynthesis (Ci) and carbon retranslocated from that previously stored as oil (Cr). Cr is the carbon accumulated as biofuel with which we are concerned here. C sum is allocated to three optimal life strategies (growth, maintenance of life, and storage). Fig. 1A outlines this carbon flow. Once carbon is allocated for storage, it is retranslocated at a rate e to carbon available for allocation. The value of e may be flexibly changed depending on the quality of environment (e.g., e may be increased under carbonlimited conditions); however, for the sake of simplicity we assume here that the value is fixed. When the carbon stored as oil is retranslocated, energy loss is assumed to occur at a rate 1 k ð0 r k r 1Þ. Therefore, the amount of carbon retranslocated in a unit of time is the product of the amount of carbon, e, and k. Thus, the net amount of carbon is C sum ¼ C i þkeC r . C sum is allocated to (1) accumulation in a fraction ua, (2) maintenance of life in a fraction um, and (3) growth (or cell division) in a fraction ug. By definition, these fractions must sum up to 1 ðua þ um þ ug ¼ 1Þ. Because the total amount of available carbon is C sum , the amount of carbon available for accumulation (Ca), maintenance, (Cm) and growth (Cg) are ua C sum , um C sum , and ug C sum , respectively. Cm is used for maintenance, including defense against pathogens and not for oil accumulation or growth. It should be noted that Ca is distinct from Cr: Cr is the amount of carbon in oil form that is reserved in algal organelles and Ca is the amount of carbon for accumulation that will be translocated into carbon in oil form, as explained later. We define that carbon that is not allocated for growth (i.e., cell division) is surplus carbon. Thus, the amount of surplus carbon is written by C sum  C g . At a glance, such surplus carbon is used for the maintenance of life, and there seems to be no reason to reserve it as oil form. This seems to be contradict to the observation that many algal strains reserves biofuels as noted in Introduction, and we seek an evolutionary condition which leads to the accumulation of oil. The death rate of an alga is a function of Cm and is defined as d¼

d0 ; Cm

ð1Þ

where d0 is the baseline death rate. The growth rate is a function of Cg; however, in this case this rate is affected by the amount of nitrate available to an alga. Although there are exceptions (e.g. Martiny et al., 2013), the amount of carbon and nitrate in a body system are finely balanced and each algal species has a specific C/N ratio (Redfield, 1958). We assume that the carbon/nitrate ratio is α/1 and nitrate uptake for growth occurs based on this ratio. When the amount of available nitrate for an alga is Ni, the amount of carbon for optimal growth is αN i . This relationship is modeled as ( r 0 C g =α ðC g =α r N i ; carbon limitationÞ r¼ ð2Þ ðC g =α 4 N i ; nitrate limitationÞ r0 Ni where r0 is the baseline growth rate. This type of function is known as “essential resources,” in which both carbon and nitrate are required for algal survival, leading to colimitation (Tilman, 1982; Abrams, 1987). Fig. 1B and C illustrates these relationships. For simple explanations, we introduce the amount of carbon that is scheduled to be allocated for growth but not used due to nitrate limitation ðC dis Þ. Based on the C/

185

Fig. 1. (A) An illustration of carbon flow in an individual microalga. (B and C) An illustration of the C/N balance theory that governs an algal growth process (Eq. (2)). For carbon limitation ðC g o αN i Þ only N 0i is used for growth such that C g ¼ αN 0i (B). For nitrate limitation ðC g 4 αN i Þ only C 0g is used for growth such that C 0g ¼ αN i (C).

N ratio, C dis corresponds to C g  αNi if the condition of nitratelimitation is satisfied (otherwise zero). For the sake of simplicity, we assume that C dis is abandoned and not retranslated to carbon available for allocation, although C dis would be reduced by decreasing of ug. Although the growth rate may be affected by other nutrients such as phosphorus, here for simplicity we focus only on nitrate as a representative nutrient. By combining these definitions, we have the dynamics of carbon accumulated in an alga at a population density (denoted by x). The rate of change in the amount of carbon accumulated in an alga as oil can be described by the accumulated carbon (Ca) minus biofuel returned to available carbon, such that dC r ¼ C a  eC r : dt

ð3Þ

As noted above, once assimilated carbon is translocated into reserved carbon in oil form, the reserved carbon will be returned to the available carbon (at rate e) and can be used for three purposes (accumulation, maintenance, and growth) (Fig. 1A). The net growth of the population is determined by the growth and death rates of the algae; therefore, dx ¼ ðr dÞx: dt

ð4Þ

Finally, we define the available carbon (Ci) and nitrate (Ni) within an alga. If there is only one alga, it can use all of the sunlight and nitrates in the system, although as the population grows, resource limitation occurs because of self-shading (Radakovits et al., 2012) and competition for resources. Thus, Ci and Ni are assumed to be described by decreasing functions of population size (x) and defined by Ci ¼

aS ; 1þx

ð5Þ

bN ; 1þx

ð6Þ

and Ni ¼

where a; S; b; and N are constants representing, carbon assimilation efficiency, surface area of the living space, coefficient of nitrate intake, and total nitrate in this space, respectively. Both Ci and Ni are the

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amount of intake for an alga for a given unit of time. Table 1 summarizes all symbols used in the models. The allocation ratio is assumed to be a phenotypic trait that is subject to evolutionary processes. After long-term evolution, the ratio in the population is expected to approach an evolutionarily stable strategy (ESS; Maynard-Smith and Price, 1973) at which the ratio can prevent invasions by any other mutants. Such an ESS ratio can be determined within an adaptive dynamics framework (Geritz et al., 1997). Because evolutionary pressure depends on environmental quality, the ESS ratio will differ depending on the environment in which the algae live. This ESS is achieved to maximize algal fitness and will not necessarily be desirable for the maximization of biofuel yield by human intervention. For example, no biofuel accumulation may evolve or the population size may be very small under the ESS condition. We here investigate an environmental condition for the ESS maximizing a value of x  C r . For the first step in studying the evolutionary implications of oil accumulation in algae, our interests are to construct a simple toy model describing algal evolution and to investigate qualitative but not quantitative properties of this system (see Bees and Croze, 2014, concerning summarizing models for quantitative purposes). Thus, the units of parameters are arbitrary in our paper.

3. Results 3.1. Allocation for carbon accumulation does not evolve in a stable environment ESS allocation ratios may be analytically derived as   8 αbN > > 40:5 ð0; 0:5; 0:5Þ if > < aS  ðuna ; unm ; ung Þ ¼  α bN α bN > > > ðotherwiseÞ : 0; 1  aS ; aS

ð7Þ

where at ESS, the conditions αbN=aS 4 0:5 and αbN=aS o 0:5 correspond to algal carbon- and nitrate-limited environments, respectively (see details in Appendix). Hereafter, “n” indicates ESS values. As shown in Fig. 2A (carbon-limited) and 2(B) (nitrate-limited), these ESS ratios were numerically confirmed. In either environment, allocation for carbon accumulation within an alga does not evolve (i.e., una ¼ 0), meaning that the amount of biofuel is always zero. Under conditions of nitrate limitation, as approaching to ESS, allocation for growth (Cg) decreases, and C ndis (i.e., C g  αN i at ESS) is equal to zero (see details in Appendix), meaning that no carbon is abandoned. This ESS condition is true under both carbon- and nitratelimited conditions. In the carbon-limited environment, this finding is obvious, given that allocation to biofuel is of no benefit to algae because of the energy loss from the accumulated carbon. The carbon may be used immediately for growth and maintenance. In the nitrate-limited environment, the reason for no allocation for carbon accumulation is not straightforward. Because there is a C/N balance (Fig. 1C) and decreasing of allocation for growth is favored, there is a larger amount of surplus carbon (i.e., C sum  C g ) than under carbon-limited conditions. One may think that the surplus carbon would be allocated for accumulation; however, this is not true. The allocation of carbon for accumulation is an “investment” for the future. If a carbon-limited condition is expected in the future, the investment can be used; however, in a stable environment under nitrate-limited conditions, carbon limitation will never occur. Owing to the tradeoff among allocations, allocation for accumulation leads to reduced growth and/or increased death; therefore, an algal strain that allocates carbon to accumulation cannot win over algal strains that allocate carbon to growth and maintenance. Our conclusion so far is that algae never accumulate carbon in a stable environment. The C/N balance itself cannot be a factor in the evolution of biofuel accumulation. There must be other factors that promote carbon accumulation. We focus next on the role of seasonal variation in resources.

Table 1 List of mathematical symbols. Allocation ratios ua um ug

Allocation ratio for accumulation Allocation ratio for maintenance Allocation ratio for growth

State variables x Cr Ci Ni C sum Ca Cm Cg C dis r d

Algal population size Amount of oil-formed carbon that is reserved in algal organelle Carbon assimilation rate Nitrate uptake rate ðkeC r þ C i Þ: Net amount of carbon ðua C sum Þ: Amount of allocated carbon available for accumulation ðum C sum Þ: Amount of allocated carbon available for maintenance ðug C sum Þ: Amount of allocated carbon available for growth ðmaxfC g  αN i ; 0gÞ: amount of carbon that is scheduled to be allocated for growth but not used due to nitrate limitation Algal growth rate Algal death rate

Parameters α e k a S b N δC δN

Carbon/nitrate ratio Coefficient for conversion efficiency and returned rate between available carbon and oil-formed carbon Extant rate of returned carbon Coefficient of carbon assimilation Surface area of living space for algae Coefficient for nitrate intake Total amount of nitrate in living space Strength of a seasonal variation in carbon assimilation Strength of a seasonal variation in available nitrate

T. Akita, M. Kamo / Journal of Theoretical Biology 380 (2015) 183–191

ua

ug

ua

um

ug

ua

ug

um ua

um

ug

um

Fig. 2. The transitions of allocation ratios to evolutionarily stable states (open stars) beginning from some initial states ðua ; ug ; um Þ ¼ ð0:8; 0:1; 0:1Þ (black dots), ð0:1; 0:8; 0:1Þ (open-circles), and ð0:1; 0:1; 0:8Þ (gray dots). (A) N ¼10 and δC ¼ 0, (B) N ¼0.1 and δC ¼ 0, (C) N¼ 10 and δC ¼ 1, and (D) N ¼0.1 and δC ¼ 1. The mutation rate is 0.001 and the interval for the ratios is 0.02. Other parameters: a¼ 0.3, S ¼100, b¼ 1, α ¼ 10, r 0 ¼ 3, d0 ¼ 1, e¼ 2, k ¼ 0.3, and δN ¼ 0.

3.2. Allocation for carbon accumulation can evolve under seasonal variation In nature, all algal populations inhabit environments that are subject to variation. Their resources oscillate during a year, month, or week, meaning that these algae have been subjected to selective pressure resulting from variation in their resources. We here consider whether such variations can give rise to allocation for carbon accumulation via adaptive evolution. In the present study, the unit of oscillation in time can be treated arbitrarily. If the rate of a phase change is not much faster than that of evolution, the ESS ratios will change in association with a phase change depending on the algal mutation rate. At present, we do not consider such a situation. However, for simple explanations we use seasonal variation with a 1-year cycle. Although in a massculture system, the resource conditions for algae that produce biofuel may be constant, it is natural to consider that these strains have previously adapted to any variations in resources and have maintained the phenotypic trait of carbon accumulation as they have been collected from a natural environment that exhibited seasonal variations. We incorporate seasonal variation into our resource allocation model with other parameters held constant. First, seasonal variation in carbon assimilation is modeled as Ci ¼

aS ð1 þ δC sin ð2π tÞÞ: 1þx

when it was low (Fig. 2D). Irrespective of the nitrate level, because carbon starvation leads to a sudden increase in algal death rate and decreased growth rate, an alga without biofuel would not survive throughout the seasons. In this case, the accumulation of carbon as a biofuel would be advantageous. This situation can be interpreted as deployment of biofuel production as a hedging strategy against the risk of carbon starvation. This evolutionary consequence holds qualitatively when we use death rate functions such that the death rate does not increase to infinity when Cm ¼0 (not shown). We use the term “carbon starvation” to indicate a low level of Ci associated with seasonal variation such as the survival of algae would be impossible without the use of reserved carbon. We also assume that before the starvation period, a higher level of Ci was attained during the oscillation; otherwise there is not enough carbon for accumulation in algae. Fig. 3A and B shows the changes in growth rate (r) and death rate (d) under the ESS during seasons. With a large amount of nitrate, a carbon-limited phase dominates during the seasons; thus, both growth and death rates change for maximizing fitness only on carbon availability. Eventually, a high growth rate is achieved during a carbon-rich period and a high death rate is achieved during a carbon starvation period (Fig. 3A). With a smaller amount of nitrate, the situation is more complex because the dynamic behaviors of the growth and death rates are affected by both carbon and nitrate. In this case, most seasons are under nitrate limitation except for an extreme carbon starvation period, and the investment for growth is limited (growth rates are flat during most seasons, as shown in Fig. 3B). This situation generates a larger amount of surplus carbon (i.e., C sum  C g ), and this surplus is allocated for maintenance and accumulation, resulting in lower growth and death rates than when there is a higher amount of nitrate (Fig. 3B). We found a tradeoff relationship between algal population density and the amount of accumulated carbon in an algal cell under the ESS that was associated with a change in the amount of nitrate. Fig. 3C and D shows the dynamic behaviors of population density (x), accumulated carbon (Cr), and the factor for seasonal variation in carbon assimilation ð1 þ δC sin ð2π tÞÞ. With a smaller amount of nitrate (Fig. 3D), population density is smaller owing to a lower investment for growth; however, the amount of accumulated carbon is greater than that with a larger amount of nitrate (Fig. 3C). This result can be explained as the accumulation of surplus carbon because of nitrate limitation increasing investment in maintenance and accumulation by decreasing the investment in growth; thus, the population density decreases. It should be noted that nitrate limitation qualitatively promotes the evolution of algal carbon accumulation but is not a necessary condition. The direct trigger for accumulation is avoiding an increased death rate during the period of carbon starvation. For the next case, seasonal variation in the supply of nitrate and corresponding variation in available nitrate are modeled as

ð8Þ

We use a sinusoidal function to represent a seasonal variation in carbon assimilation and δC ð0 r δC r1Þ is the strength of this seasonal variation. When δC ¼ 1, the value of Ci periodically changes from zero to 2aS=ð1 þxÞ during a 1-year cycle; therefore, the algae undergo a period of complete carbon starvation (Ci ¼0). We assume that the rate of phase change is much faster than that of evolution; therefore, allocation ratios are constant under the ESS. A complete analysis of models with sinusoidal functions is very difficult. We accordingly searched numerically for ESS allocation ratios. Under seasonal variation in carbon assimilation, we found that carbon accumulation of biofuel evolved ðuna ⪢0Þ in both cases: when the total amount of nitrate was relatively high (Fig. 2C) and

187

Ni ¼

bN ð1 þ δN sin ð2π tÞÞ; 1þx

ð9Þ

where δN ð0 r δN r 1Þ represents the strength of the seasonal variation in available nitrate similar to the variation in carbon assimilation. When δN ¼ 1, the value of Ni periodically changes from zero to 2bN=ð1 þ xÞ during a 1-year cycle, meaning that algae undergo a period of complete nitrate starvation (i.e., Ni ¼0). It should be noted that in contrast to carbon starvation, a period of nitrate starvation does not affect the death rate. Thus, an algal “risk hedge” as a compensation strategy for maintenance to avoid a sudden increase in death rate will not work. Under seasonal variations in available nitrate, the evolution of carbon accumulation does not occur. This consequence holds when the baseline nitrate intake rate is relatively low (not shown).

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Fig. 3. The dynamic behaviors of state variables during one period under carbon oscillation with an ESS. (A and B) Growth rates (r, indicated by thin black lines) and death rates (d, indicated by bold gray lines). (C and D) Population density (x, indicated by bold gray lines), accumulated carbon (Cr, indicated by bold black lines) and the factor for seasonal variations in carbon assimilation with δC ¼ 1 (1 þ sin ð2πtÞ, indicated by thin black lines). Results in A and C correspond to the parameters and ESS values (Fig. 2C). Results in B and D correspond to the parameters and ESS values (Fig. 2D).

It may be assumed that a strategy by which surplus carbon generated during the nitrate-limited period is allocated for growth during the nitrate-rich period would be favored, but this strategy is not beneficial because of a lower allocation rate for maintenance during the nitrate-limited period. This lower rate can be attributed to the constant allocation rates under the ESS owing to a lower rate of adaptation than that for the seasonal variation, meaning that an alga does not adjust these ratios in accordance with the prevailing nitrate condition. Furthermore, when nitrate variation is added to the situation with carbon variation, the ESS allocation ratios for carbon accumulation do not increase (not shown). Thus, the major factor promoting the evolution of accumulation for “risk hedging” is carbon starvation throughout these seasonal variations. We next describe the evolutionary response of the system to changes in the severity of carbon starvation and the amount of nitrate. 3.3. Severe carbon starvation and moderately limited nitrate maximize the total amount of biofuel Provided that an alga can adapt to carbon starvation by accumulation, under what conditions does an alga evolve to perform high carbon accumulation? Practically, we are interested in maximizing total biofuel yields and not the accumulation of biofuel per individual. We now investigate the evolutionary consequence of the total amount of biofuel corresponding to the product of population density and the amount of biofuel per alga (denoted by x  C r ). First, we investigate the sensitivity of population density and the amount of biofuel per alga to environmental conditions (δC and N) after which we investigate the total amount of biofuels. In the following, let x and Cr be observed values at t such that sin ð2π tÞ ¼ 0 as both of these fluctuate with seasonal variation. Fig. 4 shows the sensitivities of population density (x), amount of biofuel per alga (Cr) and the total amount of biofuel ðx  C r Þ to the total amount of nitrate (N) and the degree of carbon starvation (δC)

at ESS. The population density increases with the amount of nitrate as long as the phase of nitrate limitation holds (N o 1; Fig. 4A); otherwise the population density does not change with the amount of nitrate. This relationship is obvious, given that an increasing nitrate contributes directly to the growth rate under nitrate limitation (Eq. (2)). As the degree of carbon starvation increases, the population density decreases because much more carbon is required in a carbon-starvation period and its allocation for growth is relatively small (Fig. 4A). The overall trend of biofuel per alga decreases with the amount of nitrate (Fig. 4B). This is because when the amount of nitrate is relatively small (N o 1; Fig. 4B), the amount of carbon for growth is limited and accordingly, far better accumulation can be achieved using the surplus carbon, as noted earlier (Fig. 3A and B). In addition, when the amount of nitrate exceeds a certain value (N 42:5; Fig. 4B), then the phase is changed such that the dynamics always depends only on the condition of carbon assimilation (carbon limitation). In this case, biofuel per alga does not depend on the amount of nitrate. As shown in Fig. 4B, biofuel per alga increases with the degree of carbon starvation, and when this degree is lessened, biofuel per alga is nearly zero (e.g., δC ¼ 0:8), indicating that strong variation is needed for the evolution of accumulation. Finally, we investigate the conditions for maximizing the total amount of biofuel. As noted earlier, we found a tradeoff relationship between population density and biofuel per alga under the ESS through a change in the strength of carbon starvation and the nitrate intake rate (higher N leads to higher x but smaller Cr and higher δC leads to smaller x but higher Cr and vice versa). With our parameter settings, we found that strong carbon starvation and moderate nitrate limitation maximized the total amount of biofuel, as shown Fig. 4C. For a given N, the value of Cr always increased with δC as long as the population did not become extinct ðx 4 0Þ, meaning that with our settings, an increase in Cr associated with δC canceled out a decrease in x. Thus, the greater the degree of carbon starvation, the greater was the amount of total biofuel obtained.

Population density, x

T. Akita, M. Kamo / Journal of Theoretical Biology 380 (2015) 183–191

5.0 4.0 3.0 2.0 1.0 0.01

0.1

1

10

Amount of individual biofuel, Cr

Total amount of nitrate, N 5.0 4.0 3.0 2.0 1.0 0.01

0.1

1

10

Total amount of nitrate, N

Amount of total biofuel, xCr

3.0 2.5 2.0 1.5 1.0 0.5 0.01

0.1

1

10

Total amount of nitrate, N Fig. 4. Relationship between the total amount of nitrate and (A) population density, (B) amount of biofuel per alga, (C) total amount of biofuel under an ESS. Responses to seasonal variation in carbon assimilation for different values (filled circles: δC ¼ 1; open triangles: δC ¼ 0:95; filled rectangles: δC ¼ 0:9; open circles: δC ¼ 0:8). Other parameters are the same as those of Fig. 2.

When considering the sensitivity of the total amount of biofuel to the amount of nitrate, we can limit the situation of nitrate limitation (N o 1, Fig. 4C), given that in a carbon-limited situation, the amount of nitrate does not affect the evolutionary consequences. In our setting, although the population density increases linearly with the amount of nitrate, biofuel per alga displays a convex decrease. Thus, a moderate amount of nitrate maximizes the total amount of biofuel. In the cases we have investigated numerically, this result is true for other functional forms, such that Ci is sigmoidal-decreasing with x and death rate is exponentialdecreasing with Cm (nor shown). In summary, an algal strain under (i) strong carbon starvation and (ii) nitrate-limited conditions, although moderate, may potentially produce higher amount of algal biofuel. It should be noted that, if nitrate limitation is very severe, carbon starvation promotes the extinction of the algal population.

4. Discussion In this paper, we have discussed evolutionary scenarios for algae to accumulate carbon as biofuel. Understanding the evolutionary reasons is important for designing an improved biofuel

189

production system and this evolution has been a neglected topic (but see Bull and Collins, 2012). We assume that algae allocate assimilated carbon to growth (cell division), maintenance (ATP synthesis), and carbon accumulation as biofuel and that the amount of essential materials (carbon and nitrate) are strongly linked in fixed proportions (CN balance theory, Redfield, 1958). Using an adaptive dynamics technique, we have analyzed evolutionarily stable allocation ratios. Carbon and nitrates are essential for algal growth. Algae can assimilate carbon through photosynthesis, but cannot produce nitrates. On an evolutionary time scale, algae (or any other photosynthetic organisms) adapted to carbon-rich but nitratelimited conditions. These nitrate-limited conditions led to a surplus of carbon (that is assimilated by photosynthesis and not used for cell division). It is generally believed that carbon accumulation as biofuel is promoted by the storage of this surplus carbon (e.g. Roessler, 1990; Rodolfi et al., 2008; Scott et al., 2010; Přibyl et al., 2014). By analyzing our model, we first found that carbon accumulation never evolves in a stable environment, even if the amount of nitrate that algae can use is very small. This finding indicates that nitrate starvation conditions (and thus, surplus carbon) cannot be a trigger for the evolution of carbon accumulation. Then, we analyzed our model by assuming seasonal variation in the amount of available nitrate and carbon and found that variation in nitrate played no role, whereas that in carbon did play a role. This result indicates that encountering carbon starvation may be the trigger for carbon accumulation. We also found that nitrate limitation qualitatively promotes the evolution of algal carbon accumulation at ESS ratios but is not a necessary condition. One of our main results is that starvation is necessary for carbon accumulation, which is consistent with the experimental results of Duong et al. (2012) who showed that high oil-accumulating microalgae were more likely to be isolated under conditions in which environmental variations were large. Our results also suggest that prolonged culture of algae in stable environments will eventually cause artificial selection toward no oil accumulation (Bull and Collins, 2012). Therefore, adding environmental fluctuation or the periodic replacement of algal strains may be desirable. Our results show that accumulation of oil as biofuel is a strategy for avoiding the risk of death by carbon starvation. As shown in Eq. (1), we assumed that the death rate of algae sharply increases as the amount of carbon allocated for maintenance is reduced. At the lowest level of seasonal variation (Eq. (8)), the total amount of carbon that algae can use becomes extremely small. Algae store carbon to avoid a high death rate under such a carbonstarvation condition. Accumulated carbon is a nest egg. If there is no plan to use it in the future, there is no reason to accumulate carbon even if there is sufficient surplus carbon. The only reason to accumulate carbon can be to avert death by carbon starvation in the future. The population growth of algae adapted to a fluctuating environment is slow (Fig. 4A). Intuitively, using such a strain for biomass production would provide no benefits. However, fluctuation is the only driving force for carbon accumulation in our model, and a strain that is adapted to a fluctuating environment will accumulate more carbon (Fig. 4B). Thus, there is a tradeoff between algal growth and the amount of carbon accumulation. The total biomass is the product of total population density and carbon accumulated in an individual algal cell. Previous studies considered only one resource for accumulation (e.g., Shertzer and Ellner, 2002; Kooi and Troost, 2006) or multiple resources only for maximizing growth rate (e.g., Klausmeier et al., 2007; Litchman et al., 2007; Bonachela et al., 2013). We considered the allocation of two resources: carbon is allocated for growth, maintenance, and accumulation; nitrate is used only for growth but determines the

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T. Akita, M. Kamo / Journal of Theoretical Biology 380 (2015) 183–191

amount of surplus carbon (that is not used for growth) due to C/N balance. Usage of the two resources is explicitly modeled so that we can consider the maximization of total biofuel production at ESS ratios given the amount of the two resources. Our results suggest that using algal strains occupying an environment with strong carbon starvation and moderately limited nitrate will maximize total biofuel production (Fig. 4C). Evolutionary biologists have mainly discussed only the outcomes of individual fitness and population dynamics have sometimes been ignored. Even if we can find a set of genes for high carbon accumulation, an undesired result will occur if the growth rate of the strain is low. In addition, increasing the amount of nitrate to achieve rapid growth results in lower amount of biomass (Fig. 4C). The important task is to determine the rates of growth and allocation that afford maximum biomass yield.

and

8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ua α d0 > > > > < e r 0 ug um sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C r;eq ¼ > > ua aSd0 > > > : e bNr 0 um ð1  kua Þ

  ug αbN 4 if aS 1  kua

;

ð14Þ

ðotherwiseÞ

respectively. If xeq r 0, the population should go extinct. Assuming that a rare mutant invades the resident population at equilibrium, the dynamics of a rare mutant for the population size and the amount of biofuel are 0

dx 0 ¼ ðr 0  d Þx0 ; dt

ð15Þ

and 0

dC r ¼ u0a C 0sum  eC 0r ; dt Acknowledgments

ð16Þ 0

The authors very much thank the two anonymous reviewers for their useful and kind comments that improved the manuscript a lot and also thank Mayumi Seto for helpful comments and valuable discussion. T.A. is a Research Fellow of the Japan Society for the Promotion of Science (JSPS).

respectively, where C 0sum ¼ aS=ð1 þ xeq Þ þ keC r . Because the change in C 0r is much faster than that in x0 associated with the appearance of mutants and their fixation, we can assume a quasi-equilibrium 0 for C 0r . From dC r =dt ¼ 0 (Eq. (16)), we obtain C 0r;eq ¼

aSu0a : 0 eð1  kua Þð1 þ xeq Þ

ð17Þ

By removing C 0r;eq , the total amount of carbon of a rare mutant at equilibrium will be Analytical treatment of the coupled dynamics at ESS when there is stable supply of essential resources for algae Once the growth rate (r) has reached an upper limit associated with increasing x, x should approach an equilibrium at which the growth rate equals the death rate (d), denoted by xeq (hereafter, the subscripted eq indicates the equilibrium state). Given xeq , from dC r =dt ¼ 0 (Eq. (3)), the equilibrium for the amount of biofuel is uniquely determined by

C 0sum;eq ¼

0

u0g u0m 0

aS

4

aS : ð1  kua Þð1 þ xeq Þ

ð11Þ

ug : 1  kua

ð12Þ

When Eq. (12) is satisfied, the population dynamics balance depends on the carbon fixation rate; otherwise, it depends on the nitrate fixation rate. At equilibrium, from r ¼ d (Eq. (4)) and substituting C sum;eq into Eqs. (1) and (2), after some algebra, we obtain explicit solutions for the population size and the amount of biofuel as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 aS r 0 ug um > > > 1þ > 1  kua α d0 < sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xeq ¼ > aSbNr u 0 m > > 1þ > : d0 ð1  kua Þ

ug um ð1  kua Þ2

;

ð19Þ

and

Substituting C sum;eq into the condition of carbon limitation ðC g =α o N i Þ, we obtain the following condition at equilibrium:

αbN

4

ð10Þ

From the definition that C sum ¼ C i þ keC r , the total amount of carbon at equilibrium is C sum;eq ¼

ð18Þ

From Eq. (15), r 0 4 d is required for the mutant to invade the population. Using Eqs. (13) and (18), after some algebra, we can obtain the invasion conditions under carbon limitation and under nitrate limitation, as follows:

ð1  kua Þ2

aSua : C r;eq ¼ eð1  kua Þð1 þ xeq Þ

aS : 0 ð1 kua Þð1 þ xeq Þ

  αbN ug 4 if aS 1  kua ðotherwiseÞ

;

ð13Þ

u0m um ; 0 4 1  kua 1  kua

ð20Þ

respectively. These inequalities indicate that evolution occurs so as to maximize ug um =ð1  kua Þ2 or um =ð1  kua Þ corresponding to the C/N ratio. Although the condition under nitrate limitation (Eq. (20) indicates the evolution of a large value for um, a decrease in ug with an increase in um would result in a carbon limitation condition for algae (Eq. (12)); thus, both ung and unm would show moderate values. Obviously, under both situations, ua would evolve to zero, which indicates that the evolution of biofuel accumulation does not occur. This is because, due to the energy loss at reflux (i.e., k r 1), using the assimilated carbon for biofuel accumulation is no longer advantageous for an alga relative to its growth or maintenance. The optimal values of the allocation ratio can be analytically obtained, but are not very simple because the C/N condition for an alga changes with the evolution of this ratio. In the carbonlimited case, the allocation ratio that maximizes ug um =ð1  kua Þ2 is (ua ; um ; ug )¼ð0; 0:5; 0:5Þ, and the condition of carbon limitation at ESS is given by αbN=aS 4 0:5 (substituting the optimal ratio on right hand in Eq. 12). If this inequality is not satisfied, the optimal ratio should deviate from ð0; 0:5; 0:5Þ. In this case, um increases unless ug o αbN=aS (note that ug ¼ 1  um , see also Eq. ((12)).

T. Akita, M. Kamo / Journal of Theoretical Biology 380 (2015) 183–191

Taken together, the optimal ratio is given by   8 αbN > > 4 0:5 ð0; 0:5; 0:5Þ if > < aS  ðuna ; unm ; ung Þ ¼  αbN αbN > > > 0; 1  ; ðotherwiseÞ : aS aS

ð21Þ

This result was numerically confirmed as shown in Fig. 2A and B. When αbN=aS o 0:5, it is found that C ng and αN ni are balanced, meaning that the amount of carbon that is scheduled to be allocated for growth but not used due to nitrate limitation ðC dis Þ is equal to zero at ESS (i.e., C ng  αN ni ¼ 0; this can be confirmed by substituting Eq. (21) into Eqs. (13), (11) and (6)).

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