The size of a cell’s internal parts are scaled to its overall size. This size-scaling behavior has been demonstrated for organelles as well as large macromolecular assemblies, illustrating the broad importance of adapting the size of internal structures to the geometric dimensions of the cell (Rafelski et al., 2012; Levy and Heald, 2010; Hazel et al., 2013; Good et al., 2013; Weber and Brangwynne, 2015; Greenan et al., 2010; Jorgensen et al., 2007; Decker et al., 2011; Neumann and Nurse, 2007; Lacroix et al., 2018). A popular model of cellular scaling is the limiting pool mechanism, wherein maintaining a constant concentration of molecular components enables the subcellular structure to increase in size proportionally with cell volume (Goehring and Hyman, 2012; de Godoy et al., 2008) This allows larger cells to assemble larger structures, since the total number of molecular building blocks increases proportionally with cell volume. Additionally, this mechanism is biochemically simple because it does not require active processes that dynamically tune concentrations or activity levels of proteins involved in the construction . Indeed, cells appear to use a limiting pool mechanism to scale the size of their nucleoli, centrosomes, and mitotic spindles (Hazel et al., 2013; Good et al., 2013; Weber and Brangwynne, 2015; Greenan et al., 2010; Decker et al., 2011; Lacroix et al., 2018). However, limiting pool models cannot explain how the size of a linear subcellular structure scales with the linear dimensions of a cell, rather than its volume. Namely, these mechanisms predict that a two fold increase in the radius of a spherical cell will increase the length of a linear structure eight fold, following the eight fold increase in cell volume. This suggests that other mechanisms must account for how some subcellular structures are scaled with the linear dimensions of a cell.

Polarized actin cables in S. cerevisiae are an example of a linear structure that appear to grow to match the linear dimensions of the cell in order to effectively deliver secretory vesicles (Moseley and Goode, 2006). These cables are linear bundles of crosslinked actin filaments assembled by formins, which extend along the cell cortex and serve as tracks for intracellular transport of cargo from the mother cell to the growing bud, or daughter cell. Complementary sets of cables are assembled by two formins, Bni1 at the bud tip and Bnr1 at the bud neck (Pruyne et al., 2004). Throughout the cell cycle, cables are continuously polymerized, turn over at high rates, and appear to grow until they reach the back of the mother cell (Yu et al., 2011; Yang and Pon, 2002; Eskin et al., 2016). This prompted us to more rigorously investigate the relationship between cable length and cell size.

We started by comparing cable lengths to the lengths of the mother cells in which they grew. Cables were imaged in fixed wild-type haploid cells using super-resolution microscopy. Cable lengths were measured from their site of assembly (the bud neck) to their distal tip in the mother cell (note that mother cell and cell are synonymous and used interchangeably from this point on) (Figure 1—figure supplement 1A). Average cable length and average cell length were remarkably similar (4.5 ± 0.3 µm and 4.5 ± 0.2 µm, respectively), suggesting a scaling relationship. However, we note that there was a wider range in cable lengths (2.0–8.7 µm) compared to cell lengths (3.7–5.5 µm), presumably because cables in fixed cells are at different stages of growth. Further, because cables grow along the cortex of an ellipsoid shaped cell, their length can exceed the length of the cell while not growing past the back of the cell. Therefore, a cable that grows from the bud neck to the back of the cell is expected to be slightly longer than the direct distance between these two points.

The observations above led us to ask whether the relationship between cable length and cell length is maintained as cell size increases. To address this, we compared cable lengths in haploid and diploid cells, and cdc28-13^ts temperature-sensitive mutants that grow abnormally large. Diploid mother cells had an ~2-fold increase in volume compared to haploid mother cells (81.8 ± 6.3 µm³ and 44.9 ± 4.7 µm³, respectively) (Figure 1A,B and E), consistent with previous studies (Jorgensen et al., 2002). The cdc28-13^ts strain exhibited a normal haploid mother cell size at the permissive temperature. However, this strain displayed a ~ 5-fold increase in volume (198.3 ± 5.5 µm³ versus 40.9 ± 2.3 µm³) after growth at the restrictive temperature (37°C) for 8 hr, followed by 1 hr of growth at the permissive temperature (25°C) to allow cell polarization and bud growth (Figure 1C,D and E; and Figure 1—figure supplement 1B and C; Allard et al., 2018). Accordingly, cell length increased with cell volume (Figure 1F). Cable length was greater in diploids (6.3 ± 0.7 µm) compared to haploids (4.5 ± 0.3 µm), and greater in induced (8.2 ± 0.4 µm) compared to uninduced (4.3 ± 0.1 µm) cdc28-13^ts cells (Figure 1G). However, the distribution of cable lengths for all strains collapsed when we divided the lengths of cables by the lengths of the cells in which they grew (Figure 1H and I). These results strongly suggest that cables grow to a length that matches cell length.

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Figure 1—source data 1

Complete results from statistical tests performed in this study.

The p-value is indicated for student’s t-test used to compare the data presented in Figure 1 and Figure 1—figure supplement 1.

https://cdn.elifesciences.org/articles/68424/elife-68424-fig1-data1-v2.xlsx

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Next, we used a power law analysis to rigorously test the scaling relationships of cable length with cell length and volume (Figure 1J and K). Generally, scaling relations can be described by the power law $y=A{x}^a$, where $a$ is the scaling exponent that reflects the relationship between the two measured quantities, $x$ and $y$ (Reber and Goehring, 2015). This analysis revealed isometric scaling (${\displaystyle a_L=0.91\pm 0.03,R^2=0.50}$) between cable length and cell length (Figure 1J), whereas scaling between cable length and cell volume was hypoallometric (${\displaystyle a_V=0.36\pm 0.01,R^2=0.46}$) (Figure 1K).

To uncouple cell length from cell volume, we compared the length of cables in cells of different morphology. We computed the aspect ratio (the ratio of cell length to cell width) for the same cells analyzed above. This revealed that while some cells had nearly spherical morphologies, others had highly elongated morphologies (Figure 1—figure supplement 2A and B). Despite these differences in cell shape, the ratio of cable length to cell length, and the scaling exponents were similar for all cells (Figure 1—figure supplement 2C–G). Therefore, in cells of vastly different size and shape, the cable length directly scales with cell length, rather than with other dimensions such as cell surface area or volume.

We considered two distinct models to explain the control of cable length. In both models, the length of a cable is determined by competing rates of actin assembly $\left(k_{+}\right)$ at the barbed ends of cables and disassembly $\left(k_{-}\right)$ at the pointed ends of cables (Figure 2A and B). Therefore, at any given time, the extension rate of a cable is determined by the difference in its assembly and disassembly rates (Figure 2B). In the boundary-sensing model, the assembly rate is greater than the disassembly rate until the extending cable physically encounters the rear of the cell, causing one or both rates to abruptly change (Figure 2C, and Figure 2—figure supplement 1A; Reber and Goehring, 2015). This model predicts that the cable extension rate will be constant until the cable tip encounters the back of the cell. In contrast, the balance-point model requires that either the assembly rate, the disassembly rate, or both rates are length-dependent, and defines steady state cable length as the point at which these two rates are balanced (Figure 2D, and Figure 2—figure supplement 1B; Mohapatra et al., 2016). In clear contrast to the boundary-sensing model, this model predicts that the cable extension rate will steadily decrease as the cable lengthens.

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To directly test the predictions of the two models, we used live imaging to track the tips of cables as they grew from the bud neck into the mother cell (Figure 2E, Video 1, and Figure 2—figure supplement 2A–C). Initially cables extended at 0.36 ± 0.02 µm s⁻¹, and as they grew longer their extension rates steadily decreased (Figure 2F, Figure 2—figure supplement 2D). Accordingly, we observed greater changes in cable length during earlier phases of cable growth (Figure 2G, Figure 2—figure supplement 2E). Thus, as cables lengthen their growth rate decelerates.

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Note that we detected cables that were very short (<2 µm) by live imaging, which were not seen in our analysis of fixed cells. We expect that this is because shorter cables extend at a faster rate compared to longer cables and are therefore less prevalent in fixed cell populations.

Our experimental observations above support a balance-point model in which steady state cable length is reached when the assembly and disassembly rates are balanced. In this model, the rate of cable extension at any given time is given by the difference between the assembly and disassembly rates, which we call the feedback function, $f=k_{+}-k_{-}.$ To account for the observed scaling of cable length with cell length (Figure 1H and I), we assume that $f$ depends on the cable length ($L_{cable}$) scaled by the cell length ($L_{cell}$), that is $f\left(L_{cable}{,L}_{cell}\right)=f(L_{cable}/L_{cell})$. The steady state cable length $\left(L_{cable}^{*}\right)$ is reached when the feedback function equals zero, $f(L_{cable}^{*}/L_{cell})=0$. Therefore, the scale-invariant feedback function leads to the scaling of $L_{cable}^{*}$ with $L_{cell}$ seen in Figure 1J. (Further mathematical details in Materials and methods.)

Smy1 is a factor implicated in cable length control, and therefore we considered whether it might be required for cable deceleration. It has been reported that cables are longer in smy1∆ compared to wildtype cells, and that Smy1 directly inhibits Bnr1-mediated actin assembly (Eskin et al., 2016; Chesarone-Cataldo et al., 2011). Further, Smy1 is transported by myosin along cables to the bud neck where Bnr1 is anchored. Based on these observations, an ‘antenna mechanism’ has been proposed in which longer cables deliver more Smy1 to slow cable extension and limit cable length (Mohapatra et al., 2015). We confirmed the increase in cable length in smy1∆ cells (Figure 3A, and Figure 3—figure supplement 1A and B; Eskin et al., 2016), but found that cables continued to decelerate in the absence of Smy1 (Figure 3B and C). Furthermore, we observed an increase in the initial cable extension rate in smy1Δ (0.42 ± 0.04 µm s⁻¹) compared to wild-type cells (0.35 ± 0.02 µm s⁻¹) (Figure 3D and E). Interestingly, the initial extension rate in smy1Δ cells increased by the same magnitude (1.2-fold ± 0.2) as the measured increase in cable length (1.2-fold ± 0.1). Thus, Smy1 affects cable length by limiting the initial cable growth rate (Figure 3F) but does not provide the feedback that results in cable deceleration. Importantly, this does not rule out the possibility of other cellular factors acting through an antenna mechanism to control cable growth in a length-dependent manner.

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Our model makes an interesting quantitative prediction for cables that have abnormally fast initial extension rates, such as those measured in smy1Δ cells above. Specifically, our model predicts that this increase in initial extension rate will lead to a proportional increase in the initial deceleration (see Equation 7 in Materials and methods). Thus, the measured 1.2-fold increase in initial extension rate seen in smy1∆ cells is expected to lead to a 1.2-fold increase in initial deceleration of cables, shortly after they emerge from the bud neck. Indeed, linear fits to the cable extension rate, as a function of time over the first 10 s (i.e. the first few microns of cable extension), yield, $d_o^{smy1\mathrm{\Delta }}=-0.018\pm 0.010\mu m/s^2$ and $d_o^{wt}=-0.015\pm 0.005\mu m/s^2$, for smy1∆ and wild-type cells, respectively (Figure 3B and C). The ratio of these two, $d_0^{smy1\mathrm{\Delta }}/d_0^{wt}=1.2\pm 0.7$, matches the ratio of the initial extension rates, $f{\left(0\right)}^{smy1\mathrm{\Delta }}/f{\left(0\right)}^{wt}=1.2\pm 0.2$. Therefore, these data lend additional quantitative support for our model.

A key prediction of our balance-point model is that cable extension rates should depend on cell length, that is a cable of a given length should grow faster (or slow down more gradually) in longer cells compared to shorter cells (Figure 4A, top). Further, it predicts that the cable extension rate profiles from cells of different lengths will collapse when cable length is normalized to cell length (Figure 4A, bottom; predictions of model derived in Materials and methods). To test these predictions, we compared cable extension dynamics in uninduced and induced cdc28-13^ts cells (Figure 4B and C, Figure 4—figure supplement 1A and B, and Videos 2, 3, 4). When cables began to grow, they extended at similar rates in shorter and longer cells (Figure 4—figure supplement 1C). However, as the cables grew longer, they decelerated more gradually in the longer cells (Figure 4D–F). This led to longer cables in longer cells (Figure 4—figure supplement 1D). Linear regression analysis revealed that there is a nearly 2-fold greater initial deceleration in the shorter, uninduced cdc28-13^ts cells ($d_0^{uninduced}=-0.019\pm 0.005\mu m/s^2$) compared to the longer, induced cdc28-13^ts cells ($d_0^{induced}=-0.010\pm 0.003\mu m/s^2$). To determine how deceleration changes with respect to cell length, we compared the ratio of initial deceleration and cell length in induced ${\displaystyle \left(L_{induced}=8.2\pm 0.4\mu \mathrm{m}\right)}$ and uninduced ${\displaystyle \left(L_{uninduced}=4.3\pm 0.1\mu \mathrm{m}\right)}$ cdc28-13^ts cells. We found that the initial deceleration rate is inversely proportional to cell length ($d_0^{uninduced}/d_0^{induced}=2\pm 1$; ${(L_{uninduced}/L_{induced})}^{-1}=1.9\pm 0.1$), consistent with the predictions of our balance-point model (Figure 4D and E and Equation 7 in Materials and methods). Further, once cable length was normalized to cell length, cables extended with similar dynamics (Figure 4G), as predicted by our model.

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Collectively, our observations demonstrate that cables grow until their length matches the length of the cell, and that this is achieved by length-dependent deceleration of cable extension. The precise mechanism providing the feedback to enable cable deceleration is not yet clear. One possibility is that it is controlled by a gradient of actin disassembly-promoting activity that is highest at the rear of the cell. Such a gradient could be established by the retrograde transport of disassembly factors on cables, leading to their release at the rear of the cell. This would produce a higher concentration of disassembly factors, and greater disassembly rate for cables, at the back of the cell. An alternative possibility is a reaction-diffusion mechanism, achieved by anchoring an activator of disassembly factors (such as a kinase) at the rear of the cell while having an inhibitor (such as a phosphatase) in the cytosol. This would be similar conceptually to how Ran GTPase gradients form around chromatin (Kalab et al., 2002), although it would require additional features to produce the scaling that we observe (Ben-Zvi et al., 2011). Either of these two mechanisms (retrograde transport or modified reaction-diffusion) has the potential to create a gradient that is shallower in longer cells compared to shorter cells, accounting for the cell-length-sensitive cable deceleration (Figure 2—figure supplement 1D). This mechanism also would allow cables to sense the rear of the cell without requiring physical interactions with that boundary. A third possibility, which is not mutually exclusive with either mechanism above, would be length-dependent inhibition of cable assembly, that is an antenna mechanism, albeit one that is dependent on cellular factors other than Smy1 (Mohapatra et al., 2015).

It has recently been shown for other subcellular structures (e.g. nucleus, spindle, centrosome, and nucleolus) that their sizes scale with cell volume, and this scaling is explained by limiting pool models (Hazel et al., 2013; Good et al., 2013; Weber and Brangwynne, 2015; Decker et al., 2011; Neumann and Nurse, 2007; Lacroix et al., 2018). However, we found that polarized actin cables scale with cell length rather than volume. This length control cannot be explained by a limiting pool mechanism, and instead is explained, both theoretically and experimentally, by a balance-point model. These results reveal a new strategy by which cells solve engineering challenges, enabling them to scale internal structures with the linear dimensions of the cell (Kirschner et al., 2000). Similar principles may underlie the length control of other polarized, linear actin structures, such as filopodia and stereocilia. Further, related strategies may be used to control the growth of radial microtubule arrays that reach the cell periphery (Lacroix et al., 2016; Wühr et al., 2010), and may explain the scaling relationships observed between flagellar length and cell length (Bauer et al., 2021) and between contractile ring diameter and cell diameter (Kukhtevich et al., 2020). Ultimately, the model of size control that we have presented here expands our understanding of the mechanisms used by cells to sense specific aspects of their geometry, including length, surface area, and volume, to assemble structures that scale with these different dimensions (Rieckhoff et al., 2020; Brownlee and Heald, 2019).

All data points are shown in the main figures and figure supplements, and all cell images and source code are archived at Zenodo.

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Author details

1. Shane G McInally

   1. Department of Biology, Brandeis University, Waltham, United States
   2. Department of Physics, Brandeis University, Waltham, United States

   Contribution

   Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Writing - original draft, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-6145-4581

2. Jane Kondev

   Department of Physics, Brandeis University, Waltham, United States

   Contribution

   Conceptualization, Supervision, Funding acquisition, Methodology, Writing - original draft, Project administration, Writing - review and editing

   For correspondence

   kondev@brandeis.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-7522-7144

3. Bruce L Goode

   Department of Biology, Brandeis University, Waltham, United States

   Contribution

   Conceptualization, Supervision, Funding acquisition, Writing - original draft, Project administration, Writing - review and editing

   For correspondence

   goode@brandeis.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-6443-5893

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