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# Generalized size scaling of metabolic rates based on single-cell measurements with freshwater phytoplankton

Contributed by Andrea Rinaldo, July 1, 2019 (sent for review April 30, 2019; reviewed by Paul G. Falkowski and Pablo A. Marquet)

## Significance

Empirical laws predicting metabolic rates of a species by its average body mass neglect intraspecies variability arising from the range of physiologically feasible rates and phenotypic heterogeneity. We describe an exploratory experimental test of a theory that explicitly accounts for such variations. We show, by single-cell joint measurements of mass and uptake rates, how marginal distributions of mass and rates collapse onto a common master distribution for species spanning 3 orders of magnitude in cell volume. These results demonstrate the potential of a generalized scaling theory that goes beyond population averages to incorporate within-species variation.

## Abstract

Kleiber’s law describes the scaling of metabolic rate with body size across several orders of magnitude in size and across taxa and is widely regarded as a fundamental law in biology. The physiological origins of Kleiber’s law are still debated and generalizations of the law accounting for deviations from the scaling behavior have been proposed. Most theoretical and experimental studies of Kleiber’s law, however, have focused on the relationship between the average body size of a species and its mean metabolic rate, neglecting intraspecific variation of these 2 traits. Here, we propose a theoretical characterization of such variation and report on proof-of-concept experiments with freshwater phytoplankton supporting such framework. We performed joint measurements at the single-cell level of cell volume and nitrogen/carbon uptake rates, as proxies of metabolic rates, of 3 phytoplankton species using nanoscale secondary ion mass spectrometry (NanoSIMS) and stable isotope labeling. Common scaling features of the distribution of nutrient uptake rates and cell volume are found to hold across 3 orders of magnitude in cell size. Once individual measurements of cell volume and nutrient uptake rate within a species are appropriately rescaled by a function of the average cell volume within each species, we find that intraspecific distributions of cell volume and metabolic rates collapse onto a universal curve. Based on the experimental results, this work provides the building blocks for a generalized form of Kleiber’s law incorporating intraspecific, correlated variations of nutrient-uptake rates and body sizes.

Research across diverse systems has revealed remarkable regularities in the distributions of species, their abundances, and metabolic requirements (1⇓⇓–4), providing a foundation to predict how ecological communities assemble and respond to environmental change. Kleiber’s law (1) is widely regarded as one of the most important of these regularities. It states that, across many orders of magnitude, the average metabolic rate B of an ensemble of organisms of a species scales with its average body mass, M, according to the power law

Most studies investigating Kleiber’s law have measured average metabolic rates and body masses within individual species, so that each species is described by one point in

Community effects of variability in individual traits, including size and metabolic rates, have been predicted theoretically (22). Experimentally, the variability of metabolic rates among individuals has been examined within single species or classes (14, 23, 26⇓–28), often looking at species-specific molecular pathways as well as temporal fluctuations of individual rates (29). However, no systematic measurements of common patterns of intra- and interspecific metabolic rate heterogeneity and their covariation with body size seem to have been performed to date across a significant range of body sizes. Here, we aim at characterizing how the variability of body sizes and of metabolic rates within a species affects Kleiber’s law and whether one can formulate a more general law that takes the intraspecific heterogeneity of physiological traits explicitly into account. At the community level, the metabolic theory of ecology can be used to link individual metabolism to the global carbon cycle (5, 30). The characterization of intraspecific variability of carbon uptake rates is a prerequisite for understanding how it affects the global carbon cycle—an interesting and timely avenue for future research.

## Results

Previous work (31) has shown that intraspecific body size distributions of microorganisms display universal features such that the average body volume V of a species is sufficient to completely characterize such distributions. In mathematical terms, the intraspecific body size distribution *SI Appendix*). Existing data pose a constraint on the functional form of *Materials and Methods*), which encapsulates the covariation of v and b, and of which

Our experiments are aimed at demonstrating the universality of the intraspecific variability of v and b. Reconstructing the joint distribution *SI Appendix* for the properties that F and G must satisfy). This hypothesis can be shown to imply the canonical form of Kleiber’s law with exponent α for the mean metabolic rate B and the mean volume V (i.e., **2** has been proposed previously (14) and shown to be compatible with experimental data on metabolic rate fluctuations for small mammals. We discuss the relationship between Eq. **2** of this work and equation 2 of ref. 14 in *Materials and Methods*. In our experiments, we measured 2 nutrient uptake rates (**2** with 2 different exponents (

To test this hypothesis, we measured experimentally nutrient uptake rates and cell volumes of individual freshwater phytoplankton cells of the 3 species *Synechococcus* sp., *Scenedesmus obliquus*, and *Cryptomonas ovata*, together covering 3 orders of magnitude in cell volume (*Materials and Methods*) in combination with stable isotope labeling to simultaneously measure cell size and rates of nutrient (carbon and nitrogen) uptake—the latter a measure of metabolic rate of particular relevance for phytoplanktonic communities—although aware of the difficulties of constraining measures of heat production and photosynthetic energy conversion in nature and in cultures (33, 34). Specifically, we targeted nitrogen and carbon uptake because phytoplankton photosynthesis has been routinely measured via C uptake for decades (16, 32) and because C and N uptakes tend to be coupled during photoautotrophic exponential growth (7, 35). The cell size was measured as the volume v of individual cells, rather than their mass, based on density being nearly constant within species. Monocultures of the 3 study species were exposed to a medium enriched in 2 rare isotopes of carbon and nitrogen, *Materials and Methods*). By measuring the uptakes of *Materials and Methods* and Fig. 1). The total cell content of *SI Appendix*) as explained in *Materials and Methods*. The uptake was then obtained by subtracting the natural content of

The experiments thus provide a dataset of N joint measurements of single-cell uptake rates *A* and *B*, for each of the 3 phytoplankton species employed here. Each point represents a single cell and the species averages V, *A* and *B* is based on the assumption that the interspecific scaling of carbon and nitrogen content holds also at the intraspecific level (*Materials and Methods*, *Computation of Size and Uptake Rates*), an assumption which could not be verified experimentally due to the impossibility of measuring absolute C and N levels in cells to the accuracy required with the available experimental methods. On the other hand, unit carbon and nitrogen uptake rates *C* and *D*, are not affected by this assumption. Within each species, the correlations of unit carbon and nitrogen uptake rates with cell volume are weak, implying that the variabilities of cell size and unit uptake rates for cells of the same species are approximately independent. The correlations of total uptake rates *Synechococcus* cells, possibly affected by errors during manipulation or NanoSIMS analysis, were excluded from the analysis (*SI Appendix*). Carbon-specific and nitrogen-specific uptake rates show a strong positive correlation (Pearson coefficient 0.83, Fig. 2*E*). Note that we consider nutrient-specific rates to assess the existence of correlations because the multiplication by cell volume (which is necessary to obtain total uptake rates) introduces a strong spurious effect.

The marginal distributions *A* and 4 *A* and *C*, respectively. The validity of the scaling forms in Eqs. **1** and **2** can be tested via data collapse as follows. If Eq. **1** is verified, plotting the quantity **2** holds, the curves obtained by plotting *B* for *B* and *D* for **1** and **2** hold true. We estimated the exponents *SI Appendix*). Note that by fixing the threshold for the acceptance of the collapse at *SI Appendix*). Furthermore, if Eq. **2** holds true, the probability distribution for the variable **1**, where the rescaled variable used for the statistical testing is *P* value = 0.17), supporting the hypothesis that cell size distributions have the scaling form given by Eq. **1**.

Similarly, for both carbon and nitrogen uptake rates, the k-sample Anderson–Darling tests on the rescaled samples (*Materials and Methods*) do not reject the hypothesis that the 3 samples come from the same distribution at the 5% confidence level (with *P* values of 0.09 and 0.07, respectively), supporting the rescaling framework hypothesized in Eq. **2**. We note that when testing for the universality of scaling probability distributions, we find it remarkable that the statistical test does not rule out the null hypothesis. In fact, any small correction (e.g., logarithmic) to the scaling form in Eq. **2** could make the statistical test fail, although it would have little effect on the consequences of Eq. **2**. In biological or ecological contexts, logarithmic corrections may be more important than in classical statistical physics, given that the sizes of the systems under investigation here are much farther from the thermodynamic limit than, say, a gas of *SI Appendix*). Intrinsic variability, as given for example by phenotypic heterogeneity, is therefore a crucial contributor to the manifestation of biological variability and should be accounted for in any generalized scaling theory.

## Discussion

When confronting the diversity of phytoplankton form and function, 2 broadly divergent approaches exist: one emphasizing the existence of master traits, such as cell size, that underlie much of the diversity, and the other emphasizing the importance of phylogenetic variability. The latter implies that taxonomic differences would be crucial to explain functional differences. Phytoplankton species are particularly relevant for the general study of size-dependent vital rates. On the one hand, in fact, changes in phytoplankton community structure as a consequence of global changes in ocean chemistry and circulation, and in light and nutrient regimes, are expected to have major cascading effects on primary production, food web dynamics, the structure of the marine food web, and biogeochemical cycles (37). On the other hand, cell size has been shown to be a key determinant of phytoplankton metabolism and community structure (17, 38⇓⇓–41).

Our main result is that species that differ widely in their phylogenetic affiliation show patterns of intraspecific variability of metabolic rates and cell sizes that are identical, when appropriately rescaled according to the average species’ mean cell size, as suggested by the collapses shown in Fig. 4 *B* and *D*. As a consequence, size predicts not only the average metabolic rate, as per Kleiber’s law, but also the variability that is expected around that average. The scaling in Eq. **2** suggested by our results implies that such variability scales with a power of size, supporting size as a fundamental trait to determine the structure of microbial communities (17), and consequently their functioning and response to environmental fluctuations (22).

Eq. **2** predicts that the maximum and the minimum metabolic rates *SI Appendix*). This suggests that discrepancies in the measured value of the scaling exponent α in the literature might not be explained merely by discrepancies of measurements (say, targeting basal, field, or maximum metabolic rates), but rather represent real shifts in the scaling exponents.

In this work, we characterized physiological variations of body size and metabolic rate within clonal populations at a constant temperature. In natural populations, different sources of variability will also be important, most notably genetic and environmental ones. An environmental variable that is well known to affect metabolic rate (42) and body size is temperature (31, 43), which enters the classical Kleiber’s law through an Arrhenius-like term as

The limited number of cells that can be analyzed with the experimental approach adopted here precludes a detailed identification of the joint probability distribution of mass and metabolic rate, *Materials and Methods*), or the identification of the effects of cell cycle stage or nutrient limitations (44). Because we posit that v and b are correlated random variables, their joint probability distribution cannot simply be obtained by the product of marginal; i.e., *Materials and Methods*.

Experimental circumstances may have limited the generality of our results, but nonetheless support the need for broader investigations along these lines. First, cultures were grown at likely subsaturating irradiance, which may have affected the observed size-scaling exponents (0.69 for carbon and 0.59 for nitrogen), which were lower than the exponents estimated in previous studies (in the range 0.8 to 0.9) obtained by measuring bulk rates of nutrient uptake under light-saturating conditions (7, 15, 35). Low light is known to induce a shallower slope in the scaling between metabolic rate and cell size, possibly because larger cells suffer from “package” effects (a reduction in chlorophyll-specific light absorption) that become progressively more important as light levels decrease (low light induces higher pigment content and therefore a lower absorption efficiency) (15). Second, determining the carbon and nitrogen content at the single-cell level would be a challenging major advance. In fact, our results rest on the assumption that individual elemental content can be predicted by cell volume by Eq. **4**, although their relationship at the intraspecific level has not been tested to date. Combining electron-probe X-ray microanalysis (XRMA) for measurement of single-cell elemental content with NanoSIMS analyses is a possible solution (45). Relatedly, experimental observations (16) have shown that, at the interspecific level, elemental content is a better predictor of metabolic rates than cell volume in phytoplankton, possibly because it is a better proxy for body mass. Therefore, relating metabolic rate to elemental content rather than volume would be a promising development.

Kleiber’s law is particularly valuable for its use in theoretical and computational models of community dynamics, because it allows theoretical ecologists to account for the dependence of metabolic rates of different species on their typical body size (16, 17, 46) via a simple power-law relationship that bypasses the need to explicitly account for each species’ physiology (4). This dependence has been used, for example, to predict the height distribution of trees in tropical forests (47), to explore the covariation of macroecological scaling laws (4), and to study the fluctuations of the number of species inhabiting islands of different areas (48). In all these applications, Kleiber’s law provides the link between the resources available in a given ecosystem and the consumption rate of individuals. Our characterization of the scaling of intraspecific variability of metabolic rates with body size is a first step toward understanding the intraspecific correlation of metabolic rate with body size and being able to account for intraspecific variability in theoretical and computational models of community dynamics. An outstanding question in this field is whether and how the correlated fluctuations of body size and metabolic rates at the individual level affect size-related ecological patterns such as the interspecific body size distribution and the species–area relationship at the community level. To get there, we first need to close the circle and identify how the joint distribution depends on v, b, and V.

In conclusion, common scaling features of body sizes and uptake rates are shown to hold for species of freshwater phytoplankton across 3 orders of magnitude in cell volume. Such features imply the collapse of distinct experimental distributions of body size and uptake rates onto a universal master curve once suitably rescaled. The foundations are thus provided for a unified framework for metabolic rate–organismic body size relations embedding fluctuations and operating across taxa and scales.

## Materials and Methods

### Labeling Experiment.

Monocultures of the cyanobacterium *Synechococcus* sp., the chlorophyte *S. obliquus*, and the cryptophyte *C. ovata* obtained from the Culture Collection of Algae and Protozoa (CCAP) were grown in culture medium for freshwater diatoms (WC medium) at *SI Appendix* for details) and then fixed in 1% paraformaldehyde. A comparison of cell volumes between fixed and unfixed cultures showed that fixation did not cause cell shrinkage.

### NanoSIMS Measurements.

NanoSIMS is an ion microprobe that performs mass spectrometry on secondary ions sputtered from the top few atomic monolayers of a solid target by the impact of a primary beam of charged particles (49). The high spatial resolution of the ion beam (∼100 nm) allows the creation of an ion image of the sample through a raster of the primary beam on the sample surface. The color of a pixel of the ion image corresponds to the counts of that ion obtained from the sputtering of that pixel. The ratios *SI Appendix*). Cells were bombarded with a beam of C^{12}^{12}^{12}^{12}^{13}*SI Appendix*. The average C and N content of cells, necessary to compute the total C and N uptake during the incubation period, was measured by gas chromatography (*SI Appendix*).

### Computation of Size and Uptake Rates.

Regions of interest (ROIs) were defined for each imaged cell, following its contour (Fig. 1). A total of, respectively, 37, 54, and 35 cells were imaged for each species. Individual cell volumes were inferred from cell cross-sections by measuring the axes and assuming a spherical cell shape for *Synechococcus* and an ellipsoidal one for *Scenedesmus* and *Cryptomonas*, as in ref. 50. Average isotope ratios *F*). By assuming that the scaling holds at the individual cell level within a species, *SI Appendix*).

### Scaling Functional Form of p ( v ∣ V ) .

We find that the function F from Eq. **1** obtained from the data collapse of Fig. 3*B* is fitted well by a parabola in log-log space (*B*). Note that this is a 1-parameter fit because a log-normal distribution has the scaling form of Eq. **1** only if *A*. Note that even though **1**.

### Elements for a Further Generalization of Kleiber’s Law.

Experimental results allowed the characterization of the marginal distribution of metabolic rates (2). b and v being 2 correlated variables, their joint distribution is not simply given by the product of the 2 marginals, but rather by **2**), but the data do not allow distinguishing between the 2 for 2 reasons. First, the number of data points is not sufficient to verify the scaling form of *A* and *B* is affected by our assumption that the C and N content of cells scales with volume at the intraspecific level with the same exponent of the interspecific scaling. This assumption needs to be verified experimentally before drawing conclusions regarding the intraspecific scaling of metabolic rates

#### Case 1.

In the case in which the intraspecific scaling of b has the same exponent of the interspecific scaling, a reasonable further assumption is*SI Appendix*). Note that Eq. **5** (but not our Eq. **2**, in which **5** and equation 2 of ref. 14 coincide, one needs to multiply and divide Eq. **5** by *SI Appendix*); therefore α is the exponent of the intraspecific size scaling of b. The joint distribution would then have the form**2**, for which

#### Case 2.

If the intraspecific scaling of b has a different scaling exponent than the interspecific one, we can hypothesize*SI Appendix* for the properties of **7** implies**2** for the marginal distribution

## Acknowledgments

Funds from the Swiss National Science Foundation, through Projects 200021_157174, SINERGIA 2019 CRSII5_186422/1, and IZSEZ0_177302 are gratefully acknowledged. A.G. thanks the Swiss National Science Foundation for funding through projects P2ELP2_168498 and P400PB_180823. This work was partly supported by a grant from the Simons Foundation (542395, to R.S.), as part of the Principles of Microbial Ecosystems Collaborative. A. Maritan thanks the Cariparo 2018 Excellence Projects for funding. We thank Frank Schreiber for suggestions on sample preparation and Aline Reynaud and Marta Reyes for precious help in the execution of the experiments.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: andrea.rinaldo{at}epfl.ch.

Author contributions: A. Maritan and A.R. designed research; S.Z., A.G., E.M., and A. Meibom performed research; S.Z., E.M., S.E., A. Meibom, and R.S. contributed new reagents/analytic tools; S.Z., A.G., E.M., S.E., A.A., R.S., A. Maritan, and A.R. analyzed data; and S.Z., A.G., E.M., A.A., R.S., A. Maritan, and A.R. wrote the paper.

Reviewers: P.G.F., Rutgers University; and P.A.M., Pontificia Universidad del Chile.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1906762116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

## References

- ↵
- ↵
- P. A. Marquet et al.

- ↵
- I. A. Hatton et al.

- ↵
- S. Zaoli,
- A. Giometto,
- A. Maritan,
- A. Rinaldo

- ↵
- ↵
- A. Ahluwalia

- ↵
- B. A. Ward,
- E. Marañón,
- B. Sauterey,
- J. Rault,
- D. Claessen

- ↵
- P. McMahon,
- J. T. Bonner

- ↵
- R. H. Peters

- ↵
- V. M. Savage et al.

- ↵
- G. B. West,
- J. H. Brown,
- B. J. Enquist

- ↵
- ↵
- J. H. Brown

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- J. R. Banavar,
- T. J. Cooke,
- A. Rinaldo,
- A. Maritan

- ↵
- ↵
- ↵
- F. Schreiber et al.

- ↵
- M. E. Malerba,
- M. M. Palacios,
- Y. M. Palacios Delgado,
- J. Beardall,
- D. J. Marshall

- ↵
- A. K. Pettersen,
- D. J. Marshall,
- C. R. White

- ↵
- E. Kussell,
- S. Leibler

- ↵
- ↵
- N. Welkenhuysen et al.

- ↵
- F. A. Labra,
- P. A. Marquet,
- F. Bozinovic

- ↵
- A. P. Allen,
- J. F. Gillooly,
- J. H. Brown

- ↵
- A. Giometto,
- F. Altermatt,
- F. Carrara,
- A. Maritan,
- A. Rinaldo

- ↵
- ↵
- M. D. Johnson et al.

- ↵
- P. G. Falkowski,
- H. Lin,
- M. Y. Gurbunov

- ↵
- M. Olofsson et al.

*skeletonema*. Environ. Microbiol. 21, 12941–12945 (2019). - ↵
- S. M. Bhattacharjee,
- F. Seno

- ↵
- Z. V. Finkel et al.

- ↵
- P. G. Falkowski,
- A. D. Woodhead,
- K. Vivirito

- S. W. Chisholm

- ↵
- ↵
- ↵
- ↵
- J. F. Gillooly,
- J. H. Brown,
- G. B. West,
- V. M. Savage,
- E. L. Charnov

- ↵
- M. Daufresne,
- K. Lengfellner,
- U. Sommer

- ↵
- K. Banse

- ↵
- M. Segura-Noguera,
- D. Blasco,
- J. M. Fortuño

- ↵
- E. Marañón,
- P. Cermeño,
- M. Latasa,
- R. D. Tadonléké

- ↵
- T. Anfodillo et al.

- ↵
- S. Zaoli,
- A. Giometto,
- J. Giezendanner,
- A. Maritan,
- A. Rinaldo

- ↵
- ↵
- I. Olenina et al.

- ↵
- V. Gosselain,
- P. B. Hamilton,
- J. P. Descy

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