Inferring phytoplankton carbon and eco-physiological rates from diel cycles of spectral particulate beam-attenuation coefficient

The diurnal fluctuations in solar irradiance impose a fundamental frequency on ocean biogeochemistry. Observations of the ocean carbon cycle at these frequencies are rare, but could be considerably expanded by measuring and interpreting the inherent optical properties. A method is presented to analyze diel cycles in particulate beam-attenuation coefficient ( cp) measured at multiple wavelengths. The method is based on fitting observations with a size-structured population model coupled to an optical model to infer the particle size distribution and physiologically relevant parameters of the cells responsible for the measured diel cycle incp. Results show that the information related to size and contained in the spectral data can be exploited to independently estimate growth and loss rates during the day and night. In addition, the model can characterize the population of particles affecting the diel variability in cp. Application of this method to spectral cp measured at a station in the oligotrophic Mediterranean Sea suggests that most of Correspondence to: G. Dall’Olmo (gdal@pml.ac.uk) the observed variations in cp can be ascribed to a synchronized population of cells with an equivalent spherical diameter around 4.6 ±1.5 μm. The inferred carbon biomass of these cells was about 5.2–6.0 mg m −3 and accounted for approximately 10 % of the total particulate organic carbon. If successfully validated, this method may improve our in situ estimates of primary productivity.


Introduction
The Earth's climate is profoundly influenced by the ocean biological pump, which absorbs CO 2 from the atmosphere and transfers it to the deep ocean in particulate and dissolved form. Despite its importance, in situ information on this process is scarce and hard to obtain. Ocean particle dynamics can, however, be investigated by means of the inherent optical properties, which have strong potential to improve understanding of the biological pump.
The ocean carbon cycle is also dependent on the Earth's rotation. Indeed, the alternation between day and night G. Dall'Olmo et al.: Diel cycles of spectral c p imposes a fundamental cycle on photosynthesis and the resulting conversion of atmospheric CO 2 into particulate carbon. As a consequence, diel cycles of biogeochemical (Burney et al., 1982;Gasol et al., 1998) and optical (Siegel et al., 1989;Cullen et al., 1992;Marra, 1997) properties have been observed in the sunlit ocean.
The particulate beam-attenuation coefficient, c p , is an inherent optical property that exhibits diurnal variations (e.g., Siegel et al., 1989;Claustre et al., 1999). c p depends, theoretically, on all the particles present in the water column (i.e., autotrophic and heterotrophic micro-organisms, as well as detritus and mineral particles). In practice however, c p is mostly influenced by particles with equivalent spherical diameters (ESD) between 0.5 and 20 µm (Pak et al., 1988). Within this size range, each particle pool is expected to contribute a variable fraction of c p , depending on its concentration, size, refractive index and shape (e.g., Stramski and Kiefer, 1991). In the open ocean, c p is therefore well correlated with particulate organic carbon concentration (POC), but it has been difficult to ascribe variations of c p to any specific particle pool, except when additional measurements are available.
Diel variations of particulate beam-attenuation coefficient are widespread in the surface ocean and have long been investigated as a non-invasive tool for deriving particle growth rates and productivity in situ (Siegel et al., 1989;Cullen et al., 1992;Stramska and Dickey, 1992;Walsh et al., 1995;DuRand and Olson, 1996;Claustre et al., 1999Claustre et al., , 2008Oubelkheir and Sciandra, 2008;Gernez et al., 2011). The diel cycle of c p is generally characterized by an increase during daytime hours and a decrease at night suggesting that photosynthetic production of particles causes the observed c p diel cycles (Siegel et al., 1989). These estimates are considered to be net values, because it is assumed that c p tracks particle concentration, which, in turn, depends on both growth and losses (e.g. Siegel et al., 1989).
Laboratory investigations on phytoplankton cultures have provided significant insights on what controls diel cycles in c p . First, acclimated pico-phytoplankton (<2 µm) populations are characterized by remarkably synchronized growth cycles, with division occurring mostly at night, despite the large genetic diversity of this group (Jacquet et al., 2001). Cell cycle synchronization to light availability has also been observed in larger phytoplankton, such as diatoms (Vaulot et al., 1986;Stramski and Reynolds, 1993), although these larger organisms are also known to be able to divide more than once per day (Chisholm and Costello, 1980). Second, diel variations in c p are more correlated to changes in scattering cross-section (σ b , a measure of the light scattered by a single cell), than to changes in cell concentration (Stramski and Reynolds, 1993;Stramski et al., 1995;DuRand et al., 2002) and σ b appears to vary mostly with changes in cell size DuRand and Olson 1998, but see Stramski and Reynolds, 1993).
Flow cytometric analyses of in situ samples have confirmed and expanded laboratory results. Phytoplankton grow synchronously (i.e., grow during daylight and start dividing around dusk) in different ocean regions (DuRand, 1995;Vaulot and Marie, 1999;Jacquet et al., 2002;Binder and Du-Rand, 2002) and most of the diel variations in c p can be ascribed to variations in the scattering properties of phytoplankton cells (DuRand, 1995;DuRand and Olson, 1996). Phytoplankton, however, seem to contribute only a fraction of the total c p (DuRand and Olson, 1996;Claustre et al., 1999;Grob et al., 2007).
Most studies conducted on c p diel cycles to date employed measurements at a single wavelength (typically 660 nm). The particulate beam attenuation coefficient is, however, wavelength-dependent and its spectral shape can be approximated as a power law (e.g., Boss et al., 2001): where c p (λ 0 ) is c p at a reference wavelength λ 0 and ξ is the spectral slope of c p . Interestingly, a proxy for the "average particle size" can be derived from ξ . If the particle size distribution (PSD) can be represented by a power law (with slope η) that spans all possible sizes and particles are not absorbing, then the spectral slope of c p and the slope of the PSD are approximately related according to ξ η − 3 (Voltz, 1954cited in van de Hulst, 1957Morel, 1973; see also improved relationship presented by Boss et al., 2001). Nevertheless, the practical meaning of "average particle size" remains unclear. Similarly, it is uncertain how the spectral shape of c p changes when deviations from the PSD power law model occur.
While diel cycles in c p are well known, cycles in ξ have only recently been reported in the open ocean. Two field studies showed that ξ decreased during the day and increased at night suggesting that the relative size of the particles affecting c p increased during the day and decreased at night (Oubelkheir and Sciandra, 2008;Slade et al., 2010). Laboratory studies have also shown diel variability in ξ (Claustre et al., 2002;DuRand et al., 2002;Stramski and Reynolds, 1993;Stramski et al., 1995), but only DuRand et al. (2002) clearly reported that the spectral slope of the scattering coefficient decreased during the day and increased at night as recently observed in the field.
No attempt has so far been made to exploit the observed variations in ξ to study diel cycles of phytoplankton. We reasoned that it should be possible to estimate gross growth rates by measuring diel changes in cell size, because the increase in cell volume during the day should be strictly a result of photosynthetic growth in synchronized cell populations (e.g., Sosik et al., 2003). The objective of the current study was thus to develop a methodology to quantitatively interpret the diel variations in spectral c p . Specifically, an existing mathematical model was adapted for simulating the diel variations in size and optical properties of a population of phytoplankton cells. This model was then fitted to in situ measurements of spectral c p to infer physiological parameters of this population.
The population model and approach adopted here is similar to the one employed by Sosik et al. (2003) to infer eco-physiological parameters of a Synechococcus population from a time series of in situ flow cytometry measurements. The main difference is that c p data are related to all particles present in the water column. Thus, the challenge here is to identify the most likely population that is responsible for the diel cycle in c p .

Methods
The data analyzed in this study were collected during the summer of 2008 in the framework of the "Biogeochemistry from the Oligotrophic to the Ultra-oligotrophic Mediterranean" (BOUM) cruise. Specifically, we report on measurements collected during a three-day intensive sampling station that took place in the center of a gyre located in the Algero-Provençal basin (approximately at 39°N and 7°E; Moutin et al., 2011). Water samples were collected continuously from a custom-made clean underway system including a Teflon diaphragm pump to draw water from just under the ship's hull (about 9 m). The average chlorophyll and POC concentrations during this long duration station were approximately 0.05 mg m −3 and 50 mg m −3 , respectively . Because of the large computational cost of the optimization procedure described below, the inversion scheme was tested on a single day of measurements during which cloud cover was almost absent, as verified by incident PAR measurements.

In situ optical measurements
Continuous c p measurements were conducted with by a Wet-Labs ACs hyperspectral absorption and attenuation meter (400-750 nm, every 5 nm) and two WetLabs C-star transmissometers (nominally centered at 532 and 660 nm). Data were processed following protocols described elsewhere Slade et al., 2010). Briefly, the particulate optical signals were calculated by subtracting from the bulk raw measurements the optical signals generated by 0.2 µm-filtered seawater. Such signals were automatically recorded every hour for ten minutes Slade et al., 2010). All instruments had 25-cm pathlengths. The suitability of the pumping system for conducting these measurements was verified by comparing C-star based c p (660) values measured on the flow-through system with surface (average between 5 and 15 m) c p values derived from another C-star transmissometer (nominally, 660 nm, 25-cm pathlength) that was mounted on the CTD. CTD values of c p were derived by subtracting the minimum c p value measured at each station from the upcast profiles. This comparison yielded a mean value for the ratio c flow p : c CTD p of 0.998 and a standard deviation of 0.071 (N = 42). Photosynthetically active radiation (PAR) was determined with a calibrated LI-COR Li-190 quantum sensor mounted on the front mast of the ship.

Flow cytometric measurements
2 ml samples were collected every hour in cryo-vials from the pump-system described above, fixed using 1% (final) formaldehyde, flash frozen in liquid nitrogen and preserved at −80°C (Troussellier et al., 1995). Samples were later thawed at room temperature and analyzed with a FAC-SCan flow Cytometer (BD-Biosciences), equipped with an air-cooled argon laser (488 nm, 15 mW). Phytoplanktonic cells were enumerated according to right-angle light scatter properties (SSC, roughly related to cell size), and the orange (585/42 nm) and red (> 650 nm) fluorescence emissions from phycoerythrin and chlorophyll pigments, respectively. 1 µm beads (Polysciences) were added to all samples as internal standard. Data were acquired through the CellQuest software (BD-Biosciences). Four cell types were determined: "Syn", for Synechococcus sp., "euk" for small eukaryotes, and two nanoplankton groups ("nano-1" and "nano-2") according to their increasing SSC and red fluorescence properties. Approximate size ranges for the eukaryotic groups are 1-3 µm for "euk", 3-5 µm for "nano-1" and 5-8 µm for "nano-2". Bacterial counts were obtained on the same sample, after SyBRGreen-I staining (Marie et al., 1997) allowing measurements of their side scattering and green fluorescence (530/30 nm).

Dissolved organic carbon measurements
Discrete samples were collected from the outlet of the flowthrough system approximately every three hours during the long duration station. The method, standards and accuracy assessment of the dissolved organic carbon (DOC) determination is described in detail in Pujo-Pay et al. (2011).

Size-dependent model of cell growth
A discrete time-and size-structured model of cell growth was adapted for the purpose of this study (Gage et al., 1984;Smith, 1996;Arino et al., 2002). The details of the model are reported in the given references, but we describe briefly its main characteristics and the modifications implemented.
The model simulates the biovolume dynamics of a population of phytoplankton cells that is subdivided into n discrete classes each with a given average size. The cell classes are further divided into two sub-categories: "new born" and "mature". Both new born and mature cells can grow in size, provided light is available, but only mature cells can divide. When a mature cell divides, two new born cells are generated, each with volume equal to half the volume of the mature

Parameter Definition Units
index for all size classes j index for mature size classes m real part of relative refractive index n total number of size classes n b number of "new-born" size classes n m number of "mature" size classes t iteration number iter v t vector of cell biovolumes at iteration t µm 3 cm −3 A t projection matrix at iteration t -D i mean cellular diameter for size class i µm M fractional volume increment from one size class to the following -PAR above-surface photosynthetically active radiation µmoles m −2 s −1 Q c,i,t (λ) spectral efficiency factor for attenuation for class i and iteration t -Q ACs c,i,t (λ) ACs spectral efficiency factor for attenuation for class i and iteration t -T iteration period days/iter β i phase function for size class i sr −1 η PSD slope ϑ scattering angle degrees ξ spectral slope of c p t division efficiency at iteration t mother cell (see Tables 1 and 2 for symbols definitions and units).
During the illuminated part of the 24 hours, all cells grow and mature cells can divide. At night cell growth is assumed to be negligible, but division can continue in mature cells. All cells are subject to a loss rate ζ that is assumed to be constant, but with different rates between day and night. The nighttime part of the 24 h was defined by the changes in trend of c p (690) and, in practice, corresponded to the part of the 24-h period during which PAR was approximately smaller than 70 µmoles m −2 s −1 .
Since modeled cells grow exponentially in discrete time steps, the number of size classes is dependent on the cell growth rate and the length of the model iteration (Arino et al., 2002). Here, the iteration period was fixed to 30 min, the maximum growth rate was set to 2 day −1 and the resulting total number of classes was 12. These classes were equally sub-divided between new born and mature cells (i.e., n = n b +n m , where n b and n m are the numbers of "new-born" and "mature" cells, respectively). Although the number of size classes is known, the average diameter of each size class is determined by the model parameters and cannot be specified a priori.
The model is formally expressed by the following system of difference equations: where v t is the vector with the biovolume of each class at iteration t, ζ is the loss rate and A t is the Leslie or projection matrix at iteration t (Caswell, 2001). The expression for A t is presented in Eq.
(3) for a simplified case (i.e., n b = n m = 3): Here γ t and δ j,t are the fractions of biomass in a given class that grow enough to pass to the next class and that divide in the j mature class, respectively; and M = 2 1 n b is the fractional volume increment for a cell growing from one size class to the following (for more details see Arino et al., 2002).
The main diagonal of A t (red elements) specifies the fraction of biomass that remains in the same class. The lower diagonal (blues elements) of A t , instead, describes the growth from one class to the following. Finally, the upper diagonal biovolume of class n b at t = t 0 10 10 5 µm 3 cm −3 D min minimum cellular diameter 0.2 20 µm (black non-zero elements) of A t represents cell division or the transfer of biomass from mature to new-born cells.
Each row i of the matrix A t describes the interactions of the cells in size class i with the cells in the other size classes during a given model iteration. For example, the third row of A t informs us that the total biovolume of cells belonging to class 3 increases due to the growth of cells in class 2 (the Mγ t term in the second column, third row) and decreases because part of the biovolume of class 3 grows enough to pass to class 4 (the 1−γ t term in the third column, third row). Similarly, the biovolume of class 4 in A t decreases due to growth (the 1−γ t term in the fourth column, fourth row) and, as a consequence, the biovolumes of class 1 and 5 increase (the Mγ t δ 1,t and Mγ t (1 − δ 1,t ) terms in the fourth column, first and fifth rows, respectively).
γ t was assumed to be constant for each size class  and parametrized as a function of PAR: where µ max and k PAR are optimization parameters (see below) and the factor T /(M − 1) is used to convert the growth rate into the fraction of population that grows to the next class at each iteration (Arino et al., 2002). The subscript t indicates quantities that are dependent on iteration number (i.e., time). Similarly, loss rates were independent of size class (Eq. 2). δ j,t was assumed to be a linear function of size class and its magnitude during the course of the day was modulated by an inverse function of light according to: where t = 1 − PAR t /max(PAR t ) and δ max is the value of δ in the last mature size class (index n m ) when t = 1. Here, t imposes a linear decrease in division efficiency with increasing PAR, in accord with the hypothesis that natural populations minimize DNA damage by avoiding division when UV radiation is high (Vaulot et al., 1995;Vaulot and Marie, 1999). t forces division to zero when PAR reaches its maximum. This parametrization was selected because it allowed us to modulate division rates without introducing excessive complexity in the model.
The solution of Eq. (2) is the distribution of biovolumes in each size class as a function of time. This distribution is then converted into a number distribution (see below) and used to simulate the optical properties of the cell population.

Simulation of optical properties
The particulate beam-attenuation coefficient was assumed to be composed of two spectrally dependent parts: a background component, c p0 (λ), that was constant throughout the simulation and a time-varying component, c p1,t (λ), dependent on the dynamics of the cell population (this is similar to the assumption of Cullen and Lewis, 1995). The cell concentration of each class, N i , was computed by dividing the biovolume in each class by its mean cellular volume v i = M i−1 v min , where v min = π/6D 3 min (i.e., cells were assumed to be spherical; Arino et al., 2002). The refractive index of the cell population relative to that of seawater (m) was assumed to be a real number (i.e., absorption was neglected) and constant (in practice the values 1.04, 1.06 and 1.08 were used, Aas 1996). The former assumption is supported by the negligible (i.e., ≤ 2.5 %) contribution of particulate absorption to c p above 550 nm in our data set (data not shown). The latter is in agreement with laboratory results DuRand and Olson, 1998), although it may not apply in all cases (Stramski and Reynolds, 1993).
The spectral efficiency factors for attenuation for each class and iteration, Q ACs c,i,t (λ), were computed using a standard Mie code (Bohren and Huffman, 1983), accounting for the acceptance angle of the ACs transmissometer (i.e., ϑ 0 = 0.93 G. Dall'Olmo et al.: Diel cycles of spectral c p degrees; Boss et al. 2009): where i is the size-class index, Q c,i,t (λ) andβ i,t (λ) are the efficiency factor for attenuation and the spectral phase function, respectively, for size class i and ϑ is the scattering angle. In practice, a look-up table was employed to speed up computations.
The spectral beam-attenuation coefficient of the population of cells undergoing diel cycles and for the background were then computed as: Both c p0 (λ 0 ) and ξ 0 are optimization parameters (see below). In this study, λ 0 = 500 nm, but the results are not affected by this choice. Finally, the simulated spectral particulate beamattenuation coefficient was computed as:

Model inversion
The solutions to Eqs. (2) and (9) were used to simulate a time series of c p values that was compared to in situ observations. To obtain a model solution, first the system (Eq. 2) was brought to steady state (the system was solved for 100 iterations, i.e., about 3 months). The resulting biovolume distribution was normalized by the biovolume of size class n b and multiplied by the parameter v nb0 to obtain the initial conditions for the following simulation. The system was then solved at each iteration for the entire period and the timevarying spectral c p was computed.
The model parameters were estimated by minimizing the following cost function: where σ cp is the uncertainty in c p which was conservatively estimated as 0.1 × c obs p,t (λ). The global optimization algorithm Adaptive Simulated Annealing (http://www.ingber.com/ASA-README. html) was employed to derive robust estimates of the model parameters, which were constrained within realistic ranges (see Table 2). To better constrain the fit at the extremes of the simulation period, the data set used for the fit was composed of one day of data repeated twice and in succession. Thus, the data to be fitted consisted of two identical days of data. To improve the ability of the cost function to detect the global minimum, simulated c p values were compared to data smoothed by a median filter (window size of 2 h). As traditional simulated annealing, the ASA algorithm introduces random variations in the search for the global optimum and is rather insensitive to initial values of the parameters, given enough time for the optimization. Nevertheless, to test the robustness of our results, the initial values of the parameters were selected randomly from uniformly distributed ranges (Table 2). In addition, the optimization was repeated multiple times for different values of the initial parameters and random seeds. The optimal parameters and their confidence intervals were finally determined from the regions of the solution space where the cost function was minimal and where the inferred phytoplankton carbon (see below) was smaller than 70 % of the measured POC.

Carbon estimates
The size distribution of phytoplankton cells retrieved by the inversion scheme affords us with the ability to estimate the carbon biomass, C φ , of the population of cells that drives the diel c p cycle. Two methods were adopted.
First, the biovolume distribution was converted into phytoplankton carbon by means of published relationships between biovolume and carbon (Montagnes et al., 1994;Menden-Deuer and Lessard, 2000). Of these two studies, the former is considered more relevant to the oligotrophic conditions sampled in the current study, because it was based on data from cells ranging in ESD down to about 2 µm. In contrast, the relationship presented by Menden-Deuer and Lessard (2000) focuses mostly on larger cells (minimum ESD about 7 µm) and thus its application here depends on the conserved nature of the carbon-density vs. biovolume relationship found for large cells.
The second method employed to compute C φ is based on the linear correlation between the real part of the relative refractive index, m, and the intracellular carbon concentration, c i (Stramski, 1999), which has been confirmed for a variety of species spanning from small cyanobacteria to larger eukaryotes : c i = 3946 m − 3922, where c i is expressed in kg m −3 . Phytoplankton carbon was computed as the product of the estimated biovolume distribution and c i .
Flow cytometry data were used for preliminary validation of the inverted C φ estimates. Specifically, the carbon biomass of different populations was computed from published values of carbon densities, assumed ranges of cell diameters and measured cellular abundances. Different values were adopted for eukaryotes (0.22 pgC µm −3 ; Booth, 1988) and prokaryotes (0.35 pgC µm −3 ; Bjornsen, 1986). For comparison, the carbon biomass of heterotrophic bacteria was also estimated.

Growth rate estimates
The average population volume at a given time, τ , is: where the sums are over all the size classes. The phytoplankton growth rate based on cellular volume, µ V , was then computed between sunrise, τ 1 , and sunset, τ 2 , as: where f d is the illuminated fraction of the day and τ is measured in days. The important property of µ V is that it depends only on the relative distribution of biovolume across the different size classes (Eq. 11) and not on its absolute value. µ V (and γ t ) can thus be determined independently of losses (i.e., ζ d ), since the model is able to estimate the relative size of the cells.

Results
The c p time series displayed typical diel cycles that were spectrally dependent (Fig. 1a). As a consequence, distinct trends between day and night were observed in the spectral slope of c p , with increasingly flatter slopes during the day and a steepening in the spectra at night (Fig. 1b). Other quantitative spectral differences were also evident: the peakto-peak variations of c p (550) were smaller than those of c p (750) and the maximum value of c p (550) was broader and lasted about three hours longer than that at c p (750). c p spectra were, in general, smooth decreasing functions of wavelengths, as previously observed (inset in Figure 1b; Boss et al., 2001). Figure 2 presents the time series of flow cytometric abundances (left column) and relative side scattering (right column). To quantitatively identify significant periods in these time series, Lomb-Scargle normalised periodograms (Press et al., 1992) were computed (solid grey lines in Fig. 2). Results were different for the abundance and scatter data. The only cell population that showed a highly significant period of 1 day in abundance was "euk". "syn" was characterised by two periods (0.8 and 1.3 days) and all the other groups did not have significant periods near 1 day. On the other hand, the relative side scatter data from "syn", "euk" and "nano-1" all showed a highly significant period of 1 day. Thus, cycles with a period of 1 day (i.e., diel cycles) were more evident in side scatter measurements than in abundances. These diel patterns are typical in oceanic regions (e.g., Vaulot and Marie, 1999) and have been previously observed in the surface Mediterranean Sea (Jacquet et al., 2002;Oubelkheir and Sciandra, 2008). Modeled c p values were in agreement with measured c p with relative residuals (c mod p /c obs p − 1) spanning ±2 % (Fig. 3). Estimates of model parameters for three different values of the refractive index are presented in Table 3. The relative uncertainties of the parameters were typically below 24 %. Most parameters were independent of the refractive index, which suggests that they can be estimated accurately, provided the model assumptions are correct. On the other hand, the estimates of D min and v nb0 depended on the value of m selected for the optimization (Table 3), as did the population time-and size-average diameter and total number of cells (Table 4). These parameters therefore cannot be estimated independently of m (see also Discussion section).
Cellular promotion rate (γ t ) did not reach full saturation and volume-average division rates peaked at the end of the afternoon (Fig. 4). As in Sosik et al. (2003), γ t is a two parameter function of light, but it displayed considerably lower values in our study. This could be due to the very different environments where the two studies have been conducted (i.e., productive coastal waters vs. oligotrophic open ocean) or to the different phytoplankton groups under analysis (Synechococcus vs. all phytoplankton and most likely nanoeukaryotes). Instantaneous division rates peaked at dusk (Fig. 4b) (Bjornsen, 1986); in M94 and MDL00 algal carbon densities were taken from the relationships published by Montagnes et al. (1994) and Menden-Deuer and Lessard (2000), respectively. HB carbon density was 0.148 pgC µm −3 (exponentially growing organisms, Vrede et al., 2002) in M94 and 0.039 pgC µm −3 in MDL00 (phosphorous limited, Vrede et al., 2002). presented in Sosik et al. (2003) and other studies (e.g., Binder and DuRand, 2002).
Volume-based cellular growth rate (µ V ) was relatively high, with a mean daily value of 0.65 day −1 (Table 4). Cell losses were negligible during the day, but high at night and compensated daytime growth. These results are consistent with previous studies which indicated moderate to high growth rates in oligotrophic regions and near balance between day growth and night losses (Siegel et al., 1989;Lessard and Murrell, 1998;Claustre et al., 2008). The background value of c p at 500 nm contributed about 75 % of the measured c p , in agreement with the conclusions of other studies (e.g., DuRand and Olson, 1996;Claustre et al., 1999). Figure 5 presents the size distribution of the cell population as a function of time for the median set of parameters and m = 1.06. The cell population gradually increased in size during the illuminated part of the day as a consequence of photosynthetic growth. At dusk (20:24) the population was characterized by a relatively larger number of dividing cells (red empty circles in Fig. 5). As night started, cells divided, new born cells increased in number and the average size of the population decreased. These dynamics are further demonstrated in Fig. 6, where the total number of cells and their average scattering cross-section are plotted. From this figure it is evident that the model ascribes the observed increase in c p during the illuminated part of the day to an increase in cellular scattering, rather than cell abundance (compare Fig. 6a and b), as suggested by previous studies (e.g., Stramski et al., 1995;DuRand et al., 2002). The c p dynamics at night are instead related to both variations in cell abundance and scattering cross-section (Fig. 6).
In general, our results are in qualitative agreement with flow cytometry studies that show increases in cell size from dawn to dusk and opposite patterns at night, with negligible changes in cell abundances Jacquet et al., 2002). The specific shape of the modeled size distribution, however, is slightly different from those of observed phytoplankton distributions which are typically log-normal (e.g., Fig. 2a in Stramski et al., 1995or Fig. 4 in Sosik et al., 2003. This difference is likely caused by the simplified model formulation (e.g., it may require a more complex formulation for δ or implementation of asymmetric cell division). In addition, σ b decreases faster than flow-cytometric side scattering (compare right column of Figs. 2 and 6b), which is likely related to the assumed function describing the probability of division for each mature cell (δ, Eq. 5). Finally, it is worth mentioning that the total number of cells retrieved by the inversion scheme is in relatively good agreement with the the concentrations of eukaryotic cells determined by flow cytometry (compare Figs. 6a and 2).
Phytoplankton carbon estimates based on flow cytometric cell abundances were characterized by large uncertainties due to uncertainties in cell size and carbon densities (Table 5). Nonetheless, nano-eukaryotes ("nano-2" and "nano-1") contributed the majority of the phytoplankton carbon, while pico-eukaryotes ("euk") and Synechococcus ("syn") contributed a considerably smaller fraction of C φ . Depending on the parametrization, C φ contributed from 5 to 48% of the measured POC (i.e., 50 mg m −3 , Pujo-Pay et al., 2011). The model inferred phytoplankton carbon biomass, C m φ , varied by about 15 % when m was employed in the calculation and accounted for about 10 % of the observed POC (Table 6). Instead, variations of up to a factor 2 in the inferred C φ were observed when only the PSD and published carbon densities were employed in the calculation, due to the observed covariation between m andD avg .

Discussion
Similarly to our study, other investigators have previously employed a size-structured population model to infer ecophysiological parameters of a Synechococcus population from continuous in situ flow-cytometry data . The novelty of the present work is that the dynamics of both the magnitude and the spectral shape of c p are exploited to characterize the phytoplankton population that most likely is responsible for the diel variations in c p . In addition, for the first time to our knowledge, our model exploits empirical relationships on optical parameters (i.e., refractive index and size distribution) to infer phytoplankton carbon from in situ diel cycles of spectral c p .

Partitioning c p
It has long been recognized that the background component of c p can cause significant underestimation of growth rates calculated from diel cycles of c p (e.g., Siegel et al., 1989;Cullen et al., 1992). Here, a simple model was employed to partition the measured c p into background and time-varying components. In doing so, the model infers the size distribution of the cell population that is responsible for the diel variations in c p , as well as other eco-physiological parameters. This is important because the estimated physiological rates are no longer dependent on the background c p value and the population of particles that most likely affects c p1 can now be (partially) characterized. Some of the model parameters (D min and v nb0 ), however, were found to be correlated with the refractive index, suggesting that their estimates are unreliable. The correlation between m and D min (or, in general, D) is expected, because these variables appear as a product in the expression for the phase shift parameter, ρ = 2πDλ −1 (m − 1), that approximately controls Q c (van de Hulst, 1957). Thus, (m−1)D is a first independent model parameter. On the other hand, D min and v nb0 (and similarlyD avg andN tot ) are inversely related because c p1 can be safely approximated as c p1 ≈N totḠQc ∝ (N totD 2 )Q c , since the size distribution of the population that generates the diel component of c p is rather narrow (van de Hulst, 1957).N totD 2 , is thus a second parameter that partially overlaps with (m − 1)D.

Growth and loss rate estimates
The biomass-specific growth rate estimates presented in Table 4 and based on Eq. (12) implicitly incorporate losses due to phytoplankton respiration and, therefore, are not gross growth rates sensu strictu. Nevertheless, these estimates are growth rates independent of losses such as grazing and sinking. The ability of the model to decompose the spectral optical measurements and simultaneously follow biovolume and cell size is the reason why growth and losses can be independently differentiated. Indeed, while biovolume depends both on growth and losses, the average population cell size (Eq. 12) should be independent of losses, if these are not size specific.
For comparison, the "diurnal rate of variations" (Eq. (6) in Gernez et al., 2011), scaled by f d and computed from c p at 450 and 730 nm were 0.07 and 0.14 d −1 , respectively, and significantly lower than the values (0.65 d −1 ) presented in Table 4. Hence, estimates of growth rate based on singlewavelength c p measurements, beside being significantly underestimated due to the background c p0 , appear to be wavelength dependent and should be used with caution.
Reliable loss estimates during the day and night are fundamental for deriving dependable eco-physiological parameters (Cullen and Lewis, 1995). Flow cytometry data did not show large decreases in cell abundances during the day (with the exception of "euk") and thus is, in general, consistent with negligible losses during the day. In addition, the maximum side scattering values achieved at the end of the daylight period ( Figure 2) are consistent with cells dividing at the beginning of the night (e.g., Vaulot and Marie, 1999;Binder and DuRand, 2002) and should correspond to an increase in cell numbers. Such increase is not observed (Fig. 2), however, indicating that cell losses at night are approximately balancing cell division.
Micro-zooplankton (i.e., <200 µm) are the main predators of phytoplankton in oligotrophic regions (Calbet, 2008, and references therein). Although data are still scarce, some evidence suggests that even these smaller organisms display diel vertical migrations (Denis et al., 2000;Perez et al., 2000) and cycles in grazing activity (Tsuda et al., 1989), possibly as a consequence of the light cycle (Economou et al., 2008). Interestingly, diel cycles in grazing activity cannot be resolved by the dilution technique (one of the main experimental tools for determining microzooplankton grazing rates), because samples are incubated for 24 h (Landry and Hassett, 1982), as verified for open-ocean tropical/subtropical waters (representative of oligotrophic conditions) in the references cited by Calbet and Landry (2004). In addition, if vertical migrations of micro-zooplankton, as observed in the cited studies for the North-western Mediterranean Sea, are widespread in the open ocean, then bottle incubation may introduce further biases in the estimation of grazing rates as they would effectively prevent any migrating organism to enter (or leave) the incubated water sample.
An alternative or complementary hypothesis could be that the large loss rates inferred by our model at night are caused by large, vertically migrating, filter feeders (e.g., tunicates). These organisms can be centimeters in size, but appear to be able to filter sub-micron particles by means of their fine mucous mesh (Sutherland et al., 2010) and could thus be responsible for the significant night time losses inferred by our model. The available Acoustic Doppler Current Profiler (ADCP) measurements partially support this hypothesis by showing the presence near the surface of relatively large organisms (∼2 cm) at night and their absence during the day (Fig. 7). Future investigations should nevertheless focus on validating the current inversion scheme by collecting, among others, in situ measurements of loss rates as well as groupspecific growth rates.

Phytoplankton carbon estimates
A novel output of our inversion scheme is the estimated carbon biomass of the phytoplankton population that most likely causes the measured diel cycle in c p . Estimates based on the conversion of biovolume into carbon biomass were associated with large uncertainties due to the discussed covariation between refractive index and diameter. On the other hand, phytoplankton carbon estimates based on biovolume and the carbon density derived from m (i.e., C m φ )  195 195.5 196 196.5 197 197.5 198  were more self-consistent and varied approximately between 5.2-6.0 mgC m −3 for the different values of m used. These estimates can be used to compute time-and size-averages of c * p1 = c p1 : C m φ . At 670 nm, the mean (st. dev.) c * p1 was equal to 3.4(0.1), 3.7(0.1) and 4.0(0.1) m 2 gC −1 for m=1.04, 1.06 and 1.08 respectively, in agreement with laboratory-based estimates measured on cells spanning diameters between 2 and 5 µm .
The inversion scheme, however, retrieves C m φ of a phytoplankton population distributed over a limited range of diameters (about a factor of 1.5 between minimum and maximum values, Fig. 5) and likely underestimates the total C φ . Thus, the inferred relative contribution of C m φ to POC (∼10%) is not inconsistent with previous estimates indicating that the C φ :POC ratio is typically around 20-40% in the upper oligotrophic ocean (e.g. DuRand et al., 2001).

Productivity estimates
Net primary productivity (NPP) was computed from the model output and compared to single-band c p estimates. The difference in phytoplankton carbon biomass between dusk and dawn yielded a mean (st. dev.) NPP value of 2.5(0.2) mgC m −3 d −1 for m varying from 1.04 to 1.08. On the other hand, the rate of POC increase between dusk and dawn was calculated from measurements of c p (670) and yielded 3.4 and 4.7 mgC m −3 d −1 , using a conversion factor between POC and c p (670) of 2.47 m 2 gC −1 (Loisel et al., 2011) and 1.78 m 2 gC −1 (Oubelkheir and Sciandra, 2008), respectively.
The c * p values typically adopted to compute productivity from c p cycles are derived empirically using bulk c p and POC measurements. These c * p are therefore dependent on the background components of both POC and c p and assume that the same carbon-to-attenuation ratios exist for background and time-varying components of the particle pool. The differences found between c * p values taken from the literature and c * p1 derived from our model suggest that the conversion factors for c p1 and c p0 might have been significantly different in the current study. These differences can also explain the discrepancies in productivity highlighted above.
Particulate primary production based on 14 C additions was approximately 2.5 mgC m −3 d −1 at a depth of about 12 m (López-Sandoval et al., 2011) and thus in agreement with the estimate based on diurnal C m φ variations, but lower (44% and 88 %, respectively) than those estimated using POC variations. However, the C m φ -based NPP estimate is assigned by the model to a single population of cells, while NPP based on 14 C refers to the entire phytoplankton community. Thus, our model may force the dynamic population to be responsible for the entire NPP, because of the assumption that a single cell population drives the diel cycle of c p . Alternatively, the 14 C determinations could be underestimating the real NPP, for example because of artifacts due to bottle incubations (Claustre et al., 2008;Quay et al., 2010).
An independent, but nevertheless 14 C-based, measurement of size-fractionated NPP at 12 m yielded a total value of about 3.1 mgC m −3 d −1 , half of which (47 %) was due to cells smaller than 2 µm (Lagaria, unpublished data). Assuming that this relative contribution of NPP is less affected by the above bottle artifacts than its absolute value, c p -based estimates of primary production could be missing a significant part of the total particulate production, because they appear to be most sensitive to cells larger than 2 µm. This conclusion is also supported by theoretical calculations that demonstrate that volume-normalized c p of phytoplankton-like particles (i.e. m =1.05) is approximately a bell-shaped function with a maximum around 5 µm ( Fig. 6 in Boss et al., 2001). In other words, c p is more sensitive to the biomass of phytoplankton with sizes around 5 µm than to smaller or larger cells.
Finally, diurnal variations of DOC were also evident and of larger magnitude than the c p -inferred variations of POC (Figure 8). If the observed increase in DOC during the day was due to passive or active exudation of organic carbon from phytoplankton, then primary production estimates based on particle dynamics may be further underestimated.

Limitations
An important limitation of the current implementation of the technique is its large computational cost, which depends mostly on the need to repeat the optimization many times to ensure the resulting optimal parameters are robust. Despite each set of global optimizations being distributed to several parallel processors, relatively long running times (about 12 h with over 40 CPUs) were required to identify the global optima for each m. Nevertheless, we expect that improved hardware and smarter algorithms will become available, making the current technique (or adaptation thereof) more affordable.
Limitations from the eco-physiological and optical point of view are stated in the model assumptions. Important issues include the assumption that only one population of cells is responsible for the observed cycle in c p and the assumptions of constant loss rates and refractive index during the two simulated phases of the cell cycle. Other complexities of microbial food webs, such as mixotrophy (Talarmin et al., 2011), may further invalidate our hypothesis of absence of growth during the night.
Modeled and observed c p values are in agreement, however, it is clear that the fit is not perfect and the largest deviations occur at the transition between day and night (i.e., Fig. 3). Different species are known to start dividing at different times Binder and DuRand, 2002) and assuming a single species is responsible for all of the diel variations in c p is likely the cause of the mismatch between model and observations. The time-discrete model potentially introduces additional discrepancies, especially at the transition between day an night when the model allows the loss rate to change. Furthermore, we assumed (as in Sosik et al. 2003) a growth rate that only depends on the intensity of light, which may also be questionable as it has been shown that synchronized cells may alter their growth rates depending on the stage of the cell cycle (Harding et al., 1981;Bruyant et al., 2005). It is, however, worth noticing that this assumption does not imply that nutrient limitation is a priori negligible, but that it has a constant intensity during the diel cycle. Further, it is possible that the time-varying population generates a time-varying detrital component that is ignored by our model, potentially skewing the interpretation of the inferred eco-physiological and morphological parameters. Finally, our analysis is based on the assumption that diurnal variations in vertical mixing do not influence the diel cycles in spectral c p , because these spectral measurements were only available from the flow-through system.
Nevertheless, these approximations represent a trade off between model complexity and information content of the data. It is unlikely that more parameters could be estimated from the available c p data, but inclusion of additional independent pieces of information (e.g., angularly resolved scattering data) may permit greater complexity to be resolved and may yield more robust results. Future work, however, should first implement and validate the method under different environments, for example by exploiting carefully calibrated flow cytometry data (e.g., Green et al., 2003).

Conclusions
A model was employed to invert measurements of spectral particulate beam attenuation collected over a diel cycle. The model partitions c p into constant and time-varying components and outputs novel, independent estimates of growth and loss rates, as well as the carbon biomass of the population of cells responsible for the measured variability in c p .
This technique is considerably different from previous work based on measurements at a single wavelength. Specifically, no assumptions are made regarding the background component of c p and the carbon-to-c p conversion factor. The derived growth and net diurnal productivity rates can be, as a consequence, significantly different from those estimated from single-band c p cycles.
The method presented retains the appeal of previous work on c p diel cycles as it only requires measurements of particulate beam-attenuation at multiple wavelengths. If successfully validated, it could provide a powerful tool to interpret diel cycles of c p and assess the eco-physiological status of phytoplankton populations from in situ measurements, thus enhancing our understanding of the ocean carbon cycle.
Acknowledgements. The authors would like to thank the captain, chief engineer, electricians and crew of the N/O L'Atalante for their support during the cruise. S. Sakatahe is thanked for making available the "asamin" interface (http://faculty.arts.ubc.ca/ssakata/public html/software/) which was employed to run the ASA algorithm within GNU Octave. S. Bonnet is thanked for providing the Teflon pump used for the pumping system. M. Grant at PML is acknowledged for his help with distributed computing. S. Ciavatta is thanked for helpful discussions during model development and inversion. An anonymous reviewer is acknowledged for providing constructive criticisms. A thorough review by J. Cullen significantly improved an initial draft of this paper. G. D. O. and E. B. were supported through NASA grant NNX09AK30G and the UK National Centre for Earth Observation.