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How size is controlled is a fundamental question in biology. In this review, we discuss the use of scaling relationships—for example, power-laws of the form yxα—to provide a framework for comparison and interpretation of size measurements. Such analysis can illustrate the biological and physical principles underlying observed trends, as has been proposed for the allometric dependence of metabolic rate or limb structure on organism mass. Techniques for measuring size at smaller length-scales continue to improve, leading to more data on the control of size in cells and organelles. Size scaling of these structures is expected to influence growth patterns, functional capacity and intracellular transport. Furthermore, organelles such as the nucleus, mitochondria and endoplasmic reticulum show widely varying morphologies that affect their scaling properties. We provide brief summaries of these issues for individual organelles, and conclude with a discussion on how to apply this concept to better understand the mechanisms of size control in the cellular environment.
“Everywhere Nature works true to scale, and everything has its proper size accordingly.” With these words,1 D’Arcy Wentworth Thompson elegantly identifies what remains as one of the great mysteries in science: the regulation of the sizes of biological organisms and their substructures. He also suggests the use of scaling relationships—the comparison of measurements in different dimensions (both spatial and in time)—as a powerful tool to address this question. Because cells form the root of all biological structures, in this review we examine scaling relationships from a cellular perspective. To begin, we introduce the concept of scaling and its use in biology; in the next section, we discuss the importance of scaling to various aspects of cell biology; then, we present empirical evidence on size scaling for specific organelles; finally, we conclude with a perspective on using scaling analysis to provide insight on the molecular mechanisms that form the basis of all of biology.
When comparing measurements of size, the dependence of a variable on another can often be described using a power-law relationship where y xα. For example, a sphere’s surface area, A, scales as the radius squared and volume, V, scales as the radius cubed, or A r2 and V ∞ r3. In biology, one of the most commonly reported scalings of this form is the allometric relationship between organism metabolism and body mass, with conflicting evidence based on measurements in homeotherms generally interpreted as supporting one of two widely described values of α = 2/3 (reviewed in ref. 2) or 3/4 (reviewed in refs. 3–5). Continuing exploration of the subject suggests that even the existence of a universal metabolic scaling law is in some doubt, and that the scaling can depend on the subset of organisms examined. 6,7 For instance, heat production measured in protists compared with cell size shows a value of α = 1 (reviewed in ref. 8). Another common subject for scaling in macroscopic organisms is the comparison of the dimensions of support structures (limbs, stems) and the overall mass of the organism.9,10 Greater overall size places greater stress on the support, and the force that can be borne is proportional to the diameter of that support. This constrains the manner in which the size of such structures scales with total mass, which has implications for the maximum size of organisms and modes of locomotion possible with limbs with particular dimensions.
These examples illustrate how scaling can be used to inform our thinking of the underlying biological, chemical and engineering principles of organism design. The driving force behind scaling analysis is typically the assumption that the variable of interest is somehow related to some geometrically-defined parameter for which the scaling can be derived. In the case of metabolism, the 2/3-power scaling has historically been justified as a consequence of maintaining a constant temperature in homeothermic organisms. The exact scaling relationship arises from the relationship between volume and surface area, where the former is estimated to be proportional to body mass, metabolism and heat production and the latter to the rate of heat loss from the body.11 The 3/4-power scaling has been justified by examining the properties of transport networks in larger organisms (which are often fractal-like in nature) and how their geometry scales as they grow.12 Though debate continues on which of these power-law exponents best describes the data, using scaling provides a framework to analyze results as well as a guide to future experiments needed to test these models.
In practice, a common method for determining the scaling relationship between two variables is to transform the data from linear Cartesian coordinates to a logarithmic plot (or semi-logarithmic plot if an exponential dependence is suspected). Mathematically, this method reduces the problem to a linear one whose slope is equal to the power-law exponent α (Fig. 1). Log-log analysis is very powerful in its ability to fit data sets that would otherwise be difficult to interpret. However, the limits of such analysis have been debated, with an emphasis on the ability of data at extremes and outliers to skew the best-fit value of α. Because of the logarithmic transformations involved, α should be viewed as a geometric than arithmetic average of the distribution of possible values.13–15
To date, biological scaling properties have mostly been analyzed for length scales ranging from the level of whole organism to the level of organs, tissues and whole cells. Fewer have been performed at the level of sub-cellular organelles. This is largely because the relevant measurements of size and function are relatively easier to perform on larger organisms. In particular, the diffraction limit is a major obstacle to using light-based microscopy for making size measurements on the micrometer length scale or smaller. These sorts of experimental challenges have limited the ability to collect accurate size data on small and morphologically complex cellular and organellar structures. However, continuing improvements in imaging equipment, techniques and analysis are making such experiments possible, and quantitative size measurements on sub-cellular structures are becoming more routine.16–18 Thus, the time is ripe to examine how scaling pertains to questions of cell biology.
Several scaling relationships are important in cell structure. One is dimensional scaling—how the volume, surface area, and length of cells and its substructures relate to one another as the cell grows. These relationships depend both on the cell’s shape and the manner in which the cell grows. Consider the case where a cell doubles in volume before division then divides to form two daughters of similar shape (Fig. 2). In the ideal case of isotropic, three-dimensional growth of spherical cell (Fig. 2A), a doubling of volume requires a roughly 60% increase in surface area and a 25% increase in diameter. There is a disparity as volume has increased two-fold but surface area has not. To achieve similarity of shape, dividing spherical cells must therefore provide more membrane surface area19,20 and/or remove volume.21 Different scaling relationships between cell dimensions can be achieved during the cell division cycle for cells with non-spherical cell shapes or polarized growth. For the roughly cylindrical fission yeast Schizosaccharomyces pombe, growth of the cell occurs by elongation along the cylindrical axis while preserving the cross-sectional area (Fig. 2B).22 In effect, the geometry has been reduced from three dimensions to one, and so volume increases are linearly proportional to both surface area and axial length. Thus, cell division requires no dramatic shape changes. The situation is more complicated in budding yeast, Saccharomyces cerevisiae, where the mother cell volume remains largely constant while the bud size shows a combination of polarized (or apical) and isotropic growth (Fig. 2C).23 Here, the shape of the mother-bud pair just before division has roughly double the original volume and surface area, similar to the case in fission yeast. However, the increases in volume and surface area during bud growth are less straightforward to correlate with one another.
Even the most basic question of how cell growth scales with time or progression along the cell cycle has also been a matter of intense study and debate. The two most prevalent models for cell growth are linear and exponential which are familiar as the solution to zeroth- and first-order rate equations. Both linear24 and exponential growth25,26 have been reported in various cell types, though these possibilities are often difficult to distinguish,27,28 and more complex growth patterns have also been reported.26,29,30 In the former, the rate at which a cell increases in size (typically measured by volume) is independent of its current size, and is hypothesized to be a result of a growth-limiting factor such as nutrient import which holds constant regardless of cell size.24 Exponential growth occurs when there is a linear relationship between growth rate and size, suggesting that the larger cell has proportionately more capacity for metabolism and growth.28
At a simple level, cells are membrane-bound structures that include smaller sub-structures or organelles. Each of these organelles performs a specific function and shows characteristic morphologies, though these vary from organism to organism. As the cell grows, generally organelles do as well to accommodate the typically greater need for their functions.31 How organelle size scales with cell size is a question garnering more attention as advances in microscopy and other imaging techniques allow for better quantification of cell and organelle size. A simple model to consider is that functional need for organelles increases with cell size, and that correspondingly organelle size increases with a straightforward linear scaling relationship. However, even in this overly basic framework, several questions arise. What is the relevant measure of size, both for the cell and for the organelle? Does the relevant cell size measure differ among the organelles? What is the relationship between organelle size and function, and how can this be measured?
Morphology can impact function in several ways. In terms of capacity, intuitively, as an organelle gets bigger, it will be able to perform more of the functions for which it is responsible, including metabolism, signaling, storage and homeostasis. Most organelles have membrane and lumenal environments to perform these functions, and these parameters are characterized by surface area and volume, respectively. The balance between two- and three-dimensional size determines the possible shapes of the organelle.
There are several transport factors that can depend on morphology. The first is transport between the organelle and other points in the cell. A centralized organelle like the nucleus will obviously sample a much more restricted region of the cell than a distributed network, and this localization can affect how long it takes to deliver cargo to and from that organelle. Further, cellular transport occurs by many mechanisms, and we will discuss two of these: diffusion and active transport using motor proteins traveling along the cytoskeleton. These modes of transport show different dependencies of average transport time vs. distance. Many signaling processes involve influx and subsequent diffusion of Ca2+-ions or other species that are generally found at varying concentrations.32 The effective distance, d, over which the signaling species propagates scales as the square root of time (d t1/2), and therefore diffusion is more efficient at shorter length-scales. Motor protein-mediated transport requires input of ATP and allows the cargo to be targeted to particular destinations. Distance traveled scales linearly with time (d t), which is more efficient than diffusion at longer length-scales. Then, there is transport of material from inside to outside the organelle (or vice versa), which is achieved by a number of passive or active mechanisms. The rate of these processes is likely limited in part by the amount of membrane or surface area in the organelle, and the surface area-to-volume ratio may be regulated to achieve the appropriate amount of transport back-and-forth between the organelle interior and cytoplasm. Such regulation would result in changes in membrane or lumen amounts and can occur somewhat independently for the two parameters, and this would be reflected in the overall shape of the organelle.
Just as cells show different growth patterns, there are several ways by which organelle size can increase: (I) Isotropic, three-dimensional expansion is generally found for larger, round organelles such as the nucleus. (II) Distributed networks—such as found for mitochondria and tubular endoplasmic reticulum (ER) in certain cells—are propagated by increasing the linear length of the network, keeping the cross-sectional dimensions roughly constant. (III) Organelles existing in multiple copies can simply be increased in number, as is the case for peroxisomes. Combinations of these scaling behaviors are also possible, as for example in the fungal vacuole, which can be found in a single, round morphology (I) as well as a fragmented collection of smaller vesicles (III).
Furthermore, each of these cases has different scaling properties as organelle and cell size increase (Fig. 3). Linear volume scaling between an organelle showing isotropic growth and the cell would give a constant proportion between organelle and cell dimensions (Fig. 3, purple sphere). With only one such organelle, it is trivial to optimize the transport distances to other places in the cell by placing it in the center. However, having two or more organelles necessarily breaks this symmetry, leaving at least one further away from some areas of the cell. This effect is well illustrated in electron tomography micrographs of the smallest known eukaryote, Ostreococcus tauri. O. tauri cells contain only single copies of several organelles, which are tightly packed into a relatively small cell volume, with many organelles located at the cell periphery.33 Because of this organism’s size, all organelles are still within a short distance of all other points in the cell volume. However, these distances will grow longer with larger cells, and this may be a reason why in other organisms, only a small number of organelles tend to have this kind of morphology. Thus, organelle shape can affect cellular organization.
Organelles with network morphology (Fig. 3 and black lines) or multiple copies (Fig. 3, red spheres) can be more readily distributed around the cell as needed. An interesting scenario arises for linear networks that are localized to the periphery of an isotropically growing cell, as in the case of mitochondria in S. cerevisiae.34 Assuming linear scaling between cell volume and organelle length, as the cell grows its surface area increases as Vcell2/3, meaning there will be proportionately less area per unit network length, and the area density of the organelle will increase. Network properties such as branching and spacing can adjust to accommodate this density scaling to a point. There is an upper limit on how much packing can be achieved, indicating a transition point at which the network is forced to enter into the cell volume to access more available space. With these general considerations in mind, we next consider the scaling and size regulation of particular organelles.
Generally speaking, if a cell contains a nucleus it will contain only one, typically shaped as a single spheroidal object positioned somewhat centrally. The primary function of the nucleus is storage and maintenance of genetic material, which entails DNA replication and regulation of mRNA transcription. Ribosome assembly also occurs in a sub-organelle, the nucleolus. The nucleus’s importance and relatively simple geometry have made it perhaps the best studied organelle in terms of scaling.35
The size of the nucleus would at first-approximation be related to the amount of DNA in the cell, and its volume would therefore be the relevant size parameter to determine capacity. Nucleus size does show a correlation with genome size36,37 and ploidy.38,39 Since chromosomal DNA is replicated during S-phase in the cell cycle, nuclear size might therefore be expected to show a dependence on cell cycle. Such a trend was observed in HeLa cells, where nucleus volume calculated from microscopy imaging was observed to roughly double by the end of S-phase.40
The ratio between nuclear and cytoplasmic volume, or the karyoplasmic ratio, has long been observed to be maintained at a constant value,41,42 which suggests that total cell or cytoplasmic volume is another possible regulator of nucleus size. The adjustment of nucleus size upon transplantation between cells of varying size supports this idea,43 as does the eventual decrease in macronuclei in hypernucleated Stentor cells.44 More recent studies on S. cerevisiae45 and S. pombe46 yeast have shown general linear scaling relationship between the extrapolated volumes of the nucleus and the cell. The scaling seems to be independent of ploidy, with diploids having both larger nuclei and larger cell sizes than haploids at roughly similar proportions.46,47 Interestingly, nuclear size in S. cerevisiae and S. pombe was not found to increase dramatically at the onset of S-phase (in contrast to HeLa cells) which would be expected if DNA content were the primary determinant.45,46 Furthermore, experiments on multi-nucleated S. pombe show a relationship between nucleus size and the volume of the neighboring cytoplasm, providing more evidence in yeast for a mechanism to maintain a constant karyoplasmic ratio.46
Often, the mitochondria in a eukaryotic cell will be organized in a network with branches containing single units connected tip-to-tip,48 though other morphologies such as isolated or aggregated units can also be found.49 The individuals remain distinct but are constantly involved in fusion and fission events with are necessary to maintain the overall morphology.50,51 Mitochondrial DNA (mtDNA) copy number has been found to be greater for larger cells,52 and though mtDNA has not been definably linked to mitochondrial amount,53 the general implication is that larger cells contain more mitochondria. In HeLa cells, the ratios of mitochondrial number, outer-membrane area, and volume with respect to cytoplasmic volume are constant throughout the cell cycle.54 Linear dimensional scaling is due to the fact that mitochondrial unit shape is roughly constant, and extension of the mitochondrial network increases the length while preserving outer surface area and volume per unit length.
Mitochondria have an outer membrane that defines the canonical pill-shape exterior of the organelle. The final reactions leading to ATP synthesis occur across the inner membrane, which has many cristae perhaps in part to increase the organelle’s capacity for these reactions. Outer and inner membrane topology must therefore be balanced to three separate volumes (cytoplasmic, inter-membrane space, inner matrix). Synthesized ATP is transported across the organelle membranes then diffuses throughout the cell to provide the chemical driving force for other functions. The network morphology of mitochondria helps to optimize these transport problems because it has increased surface area-to-volume ratio in comparison to a spherical organelle and allows for distribution throughout the cell.12,55 Such networks may be designed to ensure a minimal distance to other parts of the cell, which would result in a dependence between mitochondrial and cell morphologies.
The ER is perhaps morphologically the most complex organelle and can be classified into several distinct types. Rough ER (rER) is decorated with ribosomes and is largely responsible for protein synthesis and translocation. Functions performed by the smooth ER (sER) range from lipid metabolism in the membrane to calcium storage in the lumen. Other types of ER include transitional ER (tER)—sites of protein delivery into budding vesicles to enter the secretory pathway—and the nuclear envelope—site of transport in and out of the nucleus. These classes can show various morphologies ranging from networks of tubules to larger sheets to the double membrane shell of the nuclear envelope. The different curvatures of the membrane in the rER and sER has been proposed as way to sort proteins and keep their functions in physically separated regions.56
Measurement of the total amount of ER across all these distinct and complex structures is typically difficult, and recent development of techniques to measure network properties promise to provide novel insights into ER morphology and size scaling.57 Such quantification will allow analysis on whether the branching and density of ER is held constant with respect to cell size. ER size has been shown to depend on functional demand, with rER and sER proliferating when greater amounts of secretion and detoxification are needed, respectively58–60 One interesting pathway which affects ER proliferation is the unfolded protein response (UPR) during which the accumulation of misfolded proteins in the ER lumen triggers a series of responses including upregulation of lumenal proteins and lipid synthesis.61 Thus, the UPR induces changes in both volume and surface amounts as has been observed for rER and has been suggested as a major role in ER biogenesis.62
The ER is also closely associated or even continuous with many other organelles including the mitochondria63–66 and plasma membrane.67,68 These connections allow for fast mechanisms of transfer of ions and newly synthesized lipids, and they also place constraints on the distribution of the ER. Thus, how the ER size scales with cell size is likely to show some feedback with the growth of other organelles.
Secretory vesicles containing newly synthesized proteins from the ER are delivered to the Golgi. Here, the proteins are post-translationally modified, then sorted for delivery via the secretory pathway to the relevant end-destinations. The Golgi is a set of membrane cisternae which can be distinct and distributed as is found in budding yeast, or arranged in a stack as found in animal cells. The stack structure has a polarity with a cis and a trans face often orientated towards the ER and the nucleus (or plasma membrane), respectively.69 To our knowledge, only limited measurements have been directly made on Golgi size, and its scaling with cell size has not yet been measured.70,71 It seems likely that the overall size would be determined by a dynamic equilibrium between the vesicles delivered to and budded from the organelle.
The morphology of fungal vacuoles is highly variable and dynamic. In yeast, the vacuole can exist as anywhere from a single, round structure to a collection of smaller structures.72 It becomes more or less fragmented through fusion and fission mechanisms,73–75 and transitions to more complex structures during cell division.76 The morphology of the organelle is highly dependent on environment and its dynamics are implicated in response to various cues. In S. cerevisiae, a correlation is found between larger vacuole size and nutrient depletion, possibly indicating a greater demand for the vacuole’s recycling functions to continue synthesizing needed proteins when presented with limited resources. The vacuole also plays a large role in response to osmotic stress.77,78 Measurements in S. pombe have shown that vacuole fusion and swelling comprise one component in the cell’s response to hypo-osmotic conditions.79 This increases the possible range of scaling of vacuole volume per unit surface area and increases the capacity of the vacuole to take on excess water and buffer the cell’s osmolyte concentrations. A large number of mutations have been identified which are classified by their effects on vacuole morphology, and these will be useful for measuring the relationship between size scaling and function.80–83 Plant vacuoles show a similar variety of possible structures, though sometimes differ in that they often take up over half of the available cell volume. Thus, in addition to the functions listed above for yeast, they also maintain turgor pressure for structural support to the cell.84
Lyosomes in animal cells are typically not centralized organelles, but rather exist as a smaller round structures present in large copy numbers85 or as a tubular network.86 They are transported along microtubules,87 and this could allow them to sample more of the cellular volume for elements that need to be degraded. Recent studies have shed light on genetic mechanisms coupling lysosome biogenesis with cellular need for its function.88
The cell contains many organelles, which typically exist in multiple discrete copies. Peroxisomes, for example, are relatively small, round vesicle-like structures responsible for peroxide detoxification and fatty acid oxidation. With multiple copies, they can be distributed around the cell as needed. They are able to fuse,89 they divide during the cell cycle, and they proliferate and are degraded in response to environmental cues.90 Variability in peroxisome number and size could make correlation to overall cell size difficult to discern. Centrioles, on the other hand, help to organize the mitotic spindle and can also act to anchor cilia to the cell membrane. They are typically present in the cell at a defined number, and this implies strict regulation of replication and segregation during the cell division cycle. In ciliates, single-celled organisms can contain hundreds or thousands of centrioles and cilia, and it has been shown that the number of centrioles scales linearly with cell length, suggesting coordination between centriole production and cell growth.91
As an organelle that protrudes from the surface, the cilium (flagellum) is a special case for scaling as it does not experience the same constraints to size as other internal cellular structures. Communication with the main body of the cell is limited to the attachment point at the membrane. Both the number of cilia and their lengths vary greatly among different types of cells, as is expected for cells with different functions and motility requirements. Cilia geometry is primarily measured by its length, which affects the types of beating possible to create movement. A scaling analysis of the force exerted by a cilium shows that the resulting fluid velocity goes as the inverse square of cilium length.92 Theoretical models based on these sorts of scaling insights can provide insight into possible microscopic mechanisms by which the observed beating behavior is achieved.93 Cilia length appears to scale linearly with the diameter of cells (Marshall WF, unpublished data) and mutants have been reported that simultaneously alter both cell size and ciliary length.94 This scaling relation may reflect the fact that cell size determines the rate of synthesis of ciliary proteins, which in turn affects the growth rate of the cilia.95
Due to the relatively simple geometry, mechanisms affecting ciliary length have been widely studied which has provided many insights into basic principles of organelle size control.96 As proposed in the balance point model, the length of this dynamic, pseudo-one dimensional structure depends on a competition between growth and disassembly.97,98 While the rate of disassembly is relatively constant, the rate of growth depends on several factors, including flagellar length, frequency of injection of building material from the cell, amount of building material per injection, and speed at which this material is delivered to the tip of the flagellum.99 The mechanisms affecting some of these variables are still largely unknown, but the transport speed of various cargo protein complexes has been measured in many different organisms.100,101
Interpreting data in the context of scaling faces several challenges. First, which are the relevant parameters of organelle size to compare? For three dimensional objects, there are generally three parameters to be measured—length, surface area and volume. This gives at first glance nine possible comparisons of size to make between two structures, and the number rises rapidly with each different object or variable (time, density, sub-populations) to be included. Some subset of these scaling relationships will often end up being redundant and therefore do not provide additional information. For example, in the case where two structures exhibit the same pattern of growth (i.e., isotropic, elongation, etc.,), length-length, area-area and volume-volume comparisons will all show similar scaling trends (Fig. 3, purple object). Or, in the case of certain geometries (i.e., for a sphere) where length, surface area and/or volume area are easily related to one another, the different scalings of those parameters will be predictable. Hypothesis-driven assumptions and prior knowledge can of course also be used to help limit the parameters for comparison. The remaining scaling relationships can be analyzed individually using the mathematical methods presented earlier. Once found, certain scaling trends tend to be simpler to justify with existing models, as was discussed with linear vs. exponential growth rates or power-law relationships that can be derived from basic kinetics or geometric principles.
Second, what does scaling tell us, specifically in the case of organelles and cell biology? Size scaling implies a mechanism for both sensing and control. This could be achieved both directly as the primary function of some regulatory pathway, or indirectly as the consequence of the actions of other processes. In either case, the nature of the scaling can provide information on what parameters are important in determining size. As discussed, possible regulatory elements for nucleus size include total DNA amount and maintaining a constant proportion to cytoplasmic volume. Functional necessity varies between tissue types and influences mitochondria size and density, both of which tend to be greater in muscle cells relative to those of other organs.102 A macroscopic-microscopic functional connection is also found between human fitness level and mitochondrial abundance.103
Size scaling analysis can also provide insight into how organelle morphology is dependent on the way in which is it constructed (and vice versa). Changes in membrane and lumen amount must be coordinated for organelles to achieve their proper shape.104 For organelles such as the fungal vacuole, lipid is delivered through vesicles in the secretory or endosomal trafficking pathways, which generally have a much higher membrane-lumen ratio than the final structure. Then to maintain the overall morphology of the organelle requires influx of water, and this further necessitates osmolyte transport to maintain osmotic balance. These functions are regulated by concentration gradients and membrane channels, which are responsible for diffusion and active transport across the membrane. Another example is the size control of tER protein patches on the ER. In the model proposed by Glick, the tER site grows by protein aggregation and shrinks by protein removal in budding vesicles. The rates of these processes scale with patch circumference and area, respectively, and the competition between them defines a steady-state patch size.105 Vesicle budding and fusion from the ER in turn influence the formation and maturation of Golgi cisternae.
The principle of dynamic size control (which was described in the balance-point model for ciliary length control) applies to other organelles as well, all of which have mechanisms for increasing and decreasing size. Mitochondria can divide through fission. The Golgi, vacuole and plasma membrane increase in lipid amount via vesicle fusion. As the site of lipid synthesis, the ER can expand based on diffusion of lipids within its membrane, and this process may be relevant for membrane growth in other organelles continuous with the ER. Reduction in organelle size can occur by vesicle budding, endocytosis, partitioning to daughter cells, and autophagy. The relative contributions of these competing processes leads to growth or shrinkage, and size scaling and dynamics provide a framework with which to interpret these changes.
Two recent studies on cell division processes show the importance of scaling to understanding the dynamics of subcellular structures.106 Hara and Kimura find that the mitotic spindle and its elongation rate both scale with cell size,107 and Carvalho et al. find that the time for the contractile ring to close during cytokinesis is independent of cell size.107 Both studies utilize scaling to inform the development of models to explain the observed behavior based on the proteins and forces involved, and thus they illustrate how in combination with genetics and biochemistry, size scaling analysis leads to insight on the molecular mechanisms, which ultimately govern all cellular processes. As experimental techniques for measuring size at sub-cellular length scales improves, we will better understand the principles of how cell and organelle sizes are controlled.
We would like to thank Will Ludington and Susanne Rafelski for helpful comments. Y.-H.M.C. acknowledges the support of the Herbert Boyer Postdoctoral Fellowship. W.F.M. acknowledges the support of the Searle Scholars Program and the W.M. Keck Foundation.
Previously published online: www.landesbioscience.com/journals/organogenesis/article/11464