## 4 Discussion
Furthermore, the fluorescent PSDs used in the ChIA calculations are not always easily reconciled with a Junge-like size distribution, and thus cannot be extended beyond the FC functional size range in a straightforward manner (Figure 3c). This can easily lead to unobserved particle fractions (especially for particle sizes above \(\sim\)10 \(\upmu\)m) and thus to underestimation of the actual total fluorescence. This underestimation has a direct effect on the parameter values of the optimized refractive index-based models, which were found to be very large compared to those found in the literature (Equations (4)-(7), (18), and (19)). However, it should be noted that while the large value of the parameter \(a_{chl}\) can be explained by the issues mentioned above, the large value of the parameter \(a_{C}\) may instead be explained by the very different context of the model's application. The original \(C_{i}\) model was developed from the analysis of a small number of cultured phytoplankton species, which compared tothe varied composition and extended size range of the He442 samples are essentially monotypic and probably log-normally distributed in their size. For both parameters, the good agreement obtained here (Figure 11) further confirms that fractal models of particle volume are appropriate and essential for successful closure between the results provided by linear formulations of biogeochemical parameters and their corresponding bulk measurements when natural mixed particle populations are concerned. Given the results obtained for SPM, ISM, and OSM and for refractive index-derived values of POC and ChLA using a fractal model of particle volume, at a first glance the application of POC and ChLA volume scaling models to simple spherical volumes appears incongruuous. However, close inspection of Equations (10)-(12) reveals that the POC and ChLA empirical volume scaling relationships hide proportionalities which are very close to those of the SPM, ISM, and OSM models. Specifically, from Equations (10) and (12) and for a single particle  
\[ \begin{split} m&=r_{o}^{3-F(r)}\rho\Big{(}\tfrac{4 \pi}{3}\Big{)}^{1-\frac{F(r)}{3}}\Big{(}\tfrac{4}{3}\pi r^{3}\Big{)}^{\frac{F(r) }{3}}=\\ &\qquad=u(r)V^{\frac{F(r)}{3}}\propto y=kV^{h}.\end{split}  \tag{20}\]  
Of particular interest then is the comparison between the exponents \(h\) and \(F(r)/3\). Over the 0.05-2000 \(\upmu\)m size range considered here, the value of \(F(r)\) was found to vary between 3 and \(\sim\)2, producing values of \(F(r)/3\) between 1 and 2/3. These not only cover the range of values of exponent \(h\) in literature-derived and UKCW optimized volume scaling models for both POC and ChLA, but also echo general results found in literature for the volume scaling coefficients of chlorophyll and organic carbon. A physical interpretation of these results is that intracellular carbon is not uniformly distributed within cells, and this non-uniform distribution of intracellular carbon causes the observed non-linear relationship with cell volume. Based on the analysis above, we have found that the observed exponent in the cell volume model reflects the non-uniform distribution of carbon within the cell and can be used to predict the subsequent impact on the fractal dimensionality of the particle. As shown earlier in the paper, the fractal dimension of the particles controls the relationship between particle size distribution and observed mass concentrations (Figure 5). Non-uniform distribution of intracellular carbon is also closely linked with cellular metabolism. Metabolic rate under optimal growth conditions is seen to scale with volume following a 3/4 exponent for a large number of organisms in what is known as the 3/4 rule or Kleiber's rule [37], and in phytoplankton this relationship is directly tied to the photosynthetic rate, and ultimately to the intracellular chlorophyll-a concentration. In general, phytoplankton cells regulate their pigment concentration in response to environmental irradiance changes [37].