Differential growth regulates asymmetric size partitioning in Caulobacter crescentus

Differential growth corrects errors in size partitioning

To study the control of asymmetric size partitioning in C. crescentus cells, we first analyzed cell growth dynamics during the division cycle with various size discrepancies between the stalked and swarmer compartments. C. crescentus cells exhibit a pronounced invagination near the cell center, which is identifiable at the beginning of the division cycle, even before the onset of constriction (17). The location of the invagination, which ultimately becomes the division plane, divides the cell into two compartments—the stalked compartment, with length $L^{S t}$ , and the swarmer compartment with length $L^{S w}$ (Fig 1B). As previously reported (17), C. crescentus cells exhibit a tight regulation of the division size ratio. The ratio of the stalked-to-swarmer compartment lengths at division is ≈1.2, with a coefficient of variation (CV) ≈ 0.095. Similar observations for high precision in division plane positioning have been reported for symmetrically dividing Escherichia coli and Bacillus subtilis with CV of 0.05 and 0.08, respectively (27). However, the ratio of stalked-to-swarmer compartment lengths $(L^{S t} / L^{S w})$ at the beginning of the division cycle is a noisy parameter with CV = 0.209 (Fig 1C), which may be attributed in part to the shallowness of the minimum width at early times, making it difficult to precisely determine the location of primary invagination. However, as the cell cycle advances, the invagination becomes more pronounced, facilitating a more accurate determination of $(L^{S t} / L^{S w})$ . We asked how C. crescentus cells achieve a high precision in $(L^{S t} / L^{S w})$ in predivisional cells.

From a previously published dataset of 2,448 individual cell generations at 24°C (17), we extracted a subset where the stalked compartment size is substantially larger than the swarmer compartment size at birth (L^St/L^Sw > 1.4, n = 403), and conversely a subset (L^St/L^Sw < 1.0, n = 347) where the swarmer compartment is larger. Surprisingly, at 60% of the division cycle period τ, both the subsets reach the average value $〈 L^{S t} / L^{S w} 〉 \approx 1.2$ (Figs 1D and S1A–D). This time point coincides with a previously reported crossover time (t_c = 0.63τ) between lateral cell-wall growth and septal growth in C. crescentus (18), which marks the onset of rapid cell-wall constriction. This suggests that the cell corrects deviations from the average value for $L^{S t} / L^{S w}$ before the onset of cell-wall constriction.

Figure S1. Representative trajectories and intergenerational variability in the location of division plane.

(A) Three representative trajectories of L^St/L^Sw over a single generation with time t normalized by generation time τ. Red line corresponds to initial length ratio L^St/L^Sw > 1.4, blue line is for L^St/L^Sw < 1.0, and black line is for $L^{S t} / L^{S w} \sim 〈 L^{S t} / L^{S w} 〉$ . (B, C, D) Intergenerational variability in L^St/L^Sw for initial length ratio $〈 L^{S t} / L^{S w} 〉$ < 1.0 (B), $〈 L^{S t} / L^{S w} 〉$ > 1.4 (C) and all cells (D). Shaded regions represent ±1 SD. Data are taken at 24°C.

To investigate how individual compartments dynamically regulate their size, we computed the instantaneous growth rate κ for the whole cell using the formula, $κ (t) = L {(t)}^{- 1} d L (t) / d t = d \log L (t) / d t$ , and the instantaneous growth rates for the individual cell compartments as $κ^{S t} (t) = d \log L^{S t} (t) / d t$ and $κ^{S w} (t) = d \log L^{S w} (t) / d t$ . With these definitions, the gradients of the curves in Fig 1E (log[L(t)/L(0)] versus t) indicate the instantaneous growth rates for the whole cell and its two compartments. At the beginning of the cell division cycle, the swarmer compartment in the “high” L^St/L^Sw subset (L^St/L^Sw > 1.4 at birth) has a higher growth rate than the stalked compartment, whereas, the swarmer compartment grows slower than the stalked compartment in the “low” L^St/L^Sw subset (L^St/L^Sw < 1.0 at birth). Towards the end of the division cycle, both compartments achieve similar growth rates, regardless of the initial size discrepancy (Fig 1E and G). This implies that at the beginning of the division cycle (before t_c), the stalked and the swarmer compartments grow at different rates to correct the deviations from the average value for L^St/L^Sw (Fig 1F). The difference in their growth rates (pre-t_c) also increases as $L_{b}^{St} / L_{b}^{Sw}$ deviates from the range 1.1–1.2, where most cells fall into (Fig 1F). Interestingly, after t_c, both growth rates reach similar values, with that of the swarmer compartment being slightly higher, regardless of $L_{b}^{St} / L_{b}^{Sw}$ (Fig 1G). These correlations also hold at other temperature conditions where cells grew at different rates (Fig S2A–D). Generally, the cell compartments grow faster after t_c, leading to a higher whole-cell growth rate, consistent with previous measurements (18). Whereas it is known how C. crescentus cells globally control cell size (18), the existing models for cell size control are agnostic about how individual cell compartments regulate their size. The differential growth of the stalked and swarmer compartments implies the existence of a negative feedback between compartment size and their respective growth rates. This begs the question of how each cell compartment regulates its size during the division cycle.

Figure S2. Differential growth for C. crescentus cells growing at different temperatures.

(A, B, C) Mean growth rate before t_c and after t_c of stalked, swarmer and the whole cell, binned by $L_{b}^{St} / L_{b}^{Sw}$ . Cells were grown at 17°C, 31°C and 34°C in (A, B, C) respectively. Error bars represent ±1 SEM.

Asymmetric size control ensures asymmetric size partitioning at division

Cell size control in C. crescentus has recently been studied for individual stalked and swarmer cells (7, 12, 18, 25, 26). Analysis of single-cell growth and morphological data revealed that C. crescentus stalked cells follow the mixer model for cell size homeostasis that combines an adder and a timer component (18): $L_{d} = a \cdot L_{b} + Δ$ , where L_b is the cell length at birth (i.e., at the beginning of the stalked cell cycle), L_d is the cell length at division (Fig 2C), and the parameters a and Δ depend on the growth conditions. This model, however, does not reveal how individual cell compartments regulate their size to ensure proper size asymmetry between the daughter cells.

Figure 2. Size regulation of stalked and swarmer cell compartments.

(A) Added length in the stalked compartment (ΔL^St) versus added length in the swarmer compartment (ΔL^Sw) during one division cycle. The “Mid” subset consists of points with $1.0 \leq L_{b}^{St} / L_{b}^{Sw} \leq 1.4$ . Dashed line corresponds to the fit of the adder model ( $Δ L^{S t} + Δ L^{S w} = 〈 Δ L 〉$ , R² = −0.1976). Red solid line represents a least square linear fit to all data points (n = 2,448, R² = 0.1442). (B) Correlation coefficient matrix between length variables in all cells in the sample (n = 2,448). (C) Scatter plots of various length variables at division versus L_b. Red lines represent least square fits of the data corresponding to the bin means, which are taken from 10 equally spaced bins between L_b = 2.4 μm and L_b = 2.8 μm. Black line displays the expected correlation according to Equations (3), (4), and (5). (inset) Difference between the stalked and the swarmer compartment lengths at division is independent of the temperature.

At any time, $L (t) = L^{S t} (t) + L^{S w} (t)$ . We find that the amount of length added in the stalked compartment during the division cycle, ΔL^St, is negatively correlated with the length added in the swarmer compartment ΔL^Sw (Fig 2A). With an adder model for size control (a = 1), ΔL = ΔL^St + ΔL^Sw is a constant. Thus, we would predict a slope −1 for the correlation between ΔL^St and ΔL^Sw. However, the data significantly deviate from the adder model predictions (Fig 2A). The division lengths of stalked and swarmer compartments ( $L_{d}^{St}$ and $L_{d}^{Sw}$ ) are moderately correlated to the birth lengths of the respective compartments ( $L_{b}^{Sw}$ and $L_{b}^{St}$ ), whereas they are strongly correlated to the total cell length at birth (Fig 2B). From multivariate least square model fit, we found

L_{d}^{St} = 0.93 + 0.63 L_{b}^{St} + 0.59 L_{b}^{Sw},

(1)

L_{d}^{Sw} = 0.57 + 0.61 L_{b}^{St} + 0.63 L_{b}^{Sw} .

(2)

Because, L_d = 1.49 + 1.24 L_b for the same set of cells (Fig 2C, top left), the regression coefficients for $L_{b}^{St}$ and $L_{b}^{Sw}$ in Equations (1) and (2) are approximately half of the regression coefficient of L_b. This suggests a simple mathematical model for asymmetric size control:

L_{d}^{St} \approx \frac{a}{2} L_{b} + δ,

(3)

L_{d}^{Sw} \approx \frac{a}{2} L_{b} + Δ - δ,

(4)

where δ = 0.93 μm, Δ = 1.49 μm, a = 1.24. The above relations are in excellent quantitative agreement with the least square linear fit to the mean trend in the experimental data $L_{d}^{S t}$ versus $L_{b}$ and $L_{d}^{S w}$ versus $L_{b}$ (Fig 2C). This model also predicts that the difference in lengths between the stalked and the swarmer compartments at division is constant and uncorrelated to cell length at birth:

L_{d}^{St} - L_{d}^{Sw} \approx 2 δ - Δ .

(5)

Indeed, experimental data show that there is no correlation between $L_{d}^{St} - L_{d}^{Sw}$ and L_b (Fig 2C, bottom right). Furthermore, the mean of the binned data is in excellent agreement with the predicted value 2δ − Δ = 0.37 μm, at all temperatures. Thus, the asymmetric size control models given by Equations (3) and (4) lead to a constant size difference between the stalked and the swarmer compartments at the time of cell division.

Size-independent partitioning of growth regulators ensures robust asymmetric size partitioning

Experimental data (Fig 1F) suggest a negative feedback between the size of individual compartments and their respective growth rates, such that the larger compartment grows at a slower rate to correct for initial size discrepancies. This differential rate of change in compartment length could arise from differential elongation rates in each compartment, or active movement of the division plane relative to the cell. In the absence of experimental evidence for the latter, we constructed a mathematical model for differential growth. In this model, we assume that the growth rate of the cell is proportional to the concentration of a regulatory molecule (which we call a growth regulator), whose abundance at time t is given by ɛ(t). At the beginning of the stalked cell cycle, these growth regulators can be partitioned between the stalked and swarmer compartments in two possible ways: partitioning by size and partitioning by amount (Fig 3A). In the first model, the growth regulators are partitioned such that their abundance in each compartment is proportional to the size of that compartment, resulting in equal concentration in each compartment. The growth rate, being proportional to the concentration, is the same in each compartment and thus independent of compartment size. Partitioning by size, therefore, cannot account for size-dependent growth such that the bigger compartment grows slower to achieve the correct size partitioning ratio at division (Fig 1F).

Figure 3. Cell size correction requires size-independent partitioning of molecules regulating growth.

(A) Schematic of model for size-based partitioning (top) and amount-based partitioning (bottom) of growth regulators. Amount-based partitioning model ensures differential concentration of growth regulators, thereby predicting a negative correlation between growth rates and lengths of each compartment. Solid lines near the mid-cell do not represent a physical barrier but the location of the minimum cell width separating the swarmer and stalked compartments. (B) Schematic of differential growth model for typical asymmetric and highly asymmetric size ratios. (C) Amount-based partitioning model predicts negative correlation between individual compartment growth rates and birth lengths (solid lines). Scattered points are experimental data, black points are binned data, and lines are model fit to (8). (D) The model predicts negative correlation between growth rate ratios and birth length ratios, given by Equation (14). Best fit to binned data, $κ^{St} / κ^{Sw} = (1.26 \pm 0.03) L_{b}^{Sw} / L_{b}^{St}$ .

By contrast, if the growth regulators are partitioned in fixed amounts between the two compartments, then the regulator concentration is lower in the bigger compartment compared to the smaller compartment (Fig 3A). As a result, we would expect a negative correlation between growth rate and the compartment size, consistent with experimental data.

While our model does not explicitly identify the growth regulators, we can narrow down the potential candidates based on the physical model. To maintain differential concentration, it is necessary that these growth regulators do not significantly diffuse through the cytoplasm since diffusion would tend to equalize concentration. Furthermore, diffusion barrier in C. crescentus is set just before cell division (28). Therefore, one possible hypothesis is that the growth regulators are immobile by being bound to the genome, which is partitioned evenly between the stalked and the swarmer cells and DNA replication begins early in the S-phase of the cell cycle (29). Possible candidates for these growth regulators include RNA polymerases, mRNAs and actively translating ribosomes, which are known to regulate bacterial growth rate (30, 31, 32, 33), and display limited mobility owing to their localization at the sites of transcription (34). Experimental observations indicate minimal mobility of active ribosomes in C. crescentus, characterized by a low-micrometer-scale diffusion coefficient (34, 35). This limited diffusivity can be attributed to the localization translating ribosomes at mRNA transcription sites, anchored to their corresponding genes (34, 36). Thus, the chromosome can serve as an internal template for amount-based partitioning of growth regulators into daughter cells, much like chromatin-based partitioning of cell size regulators in budding yeast and plant cells (20, 37). Aside from ribosomes, differential recruitment of MreB and PBP2 could also play a role in differential growth of cell compartments.

Mathematical model for asymmetric growth control

We first start with a mathematical model for whole-cell growth, where the rate of cell elongation is proportional to the abundance of the regulators ɛ,

\frac{d L}{d t} = α \int_{0}^{L} λ (l) d l = α ε (t),

(6)

where α is a constant that depends on cell geometric parameters and the speed of peptidoglycan insertion, and λ is the number density of the regulatory molecules that coordinate cell growth. If λ has a specific spatial profile such that the growth is localized to a fixed region on the cell surface (2, 19, 38, 39, 40, 41), then $ε = \int_{0}^{L} λ (l) d l$ is constant and independent of L. This results in a negative feedback between cell length and growth rate κ = αɛ/L, consistent with data for M. smegmatis cells where growth is localized to the cell’s poles (42) (Fig S3A). By contrast, E. coli cells exhibit uniform lateral growth along the cell, such that ɛ ∝ L. As a result, there is negligible correlation between growth rate and cell length (Fig S3B). On the other hand, C. crescentus cells grow by inserting peptidoglycan both laterally and at the septal plane (18, 38, 44). C. crescentus cell growth is uniform before the beginning of the constriction phase (t < t_c), whereas growth is localized to the septum after t > t_c (18). Our model then predicts no correlation between cell length at birth and the growth rate before t_c, and a negative correlation between cell length at t_c and growth rate after t_c, which is consistent with experimental data (Fig S3C).

Figure S3. Growth patterns in different organisms predict correlation between growth rate and cell length.

Area of localised growth is shown in green. (A) Growth rate versus length at birth for C. crescentus. (B) In M. smegmatis growth is localised at the old end before New End Takes Off (NETO) when new cell pole starts growing (43). Data obtained from Robertson and Shahrezaei lab (42). (C) In E. coli, growth occurs throughout the cell length resulting in non-negative correlation between growth rate and cell length. Single-cell mother machine data obtained from Suckjoon Jun laboratory (9).

We apply the whole-cell growth model to each compartment of the cell, assuming that the rate of change in length of the stalked and the swarmer compartments are given by

\frac{d L^{i}}{d t} = α ε^{i} (t),

(7)

where i = {St, Sw} and ɛⁱ is the abundance of the growth regulators in compartment i. Instantaneous growth rate of each compartment is then given by

κ^{i} (t) = \frac{1}{L^{i} (t)} \frac{d L^{i}}{d t} = \frac{1}{L^{i} (t)} α ε^{i} (t) .

(8)

If the growth regulators are partitioned in proportion to compartment size then ɛⁱ ∝ Lⁱ, leading to a constant size-independent growth rate in each compartment, inconsistent with experimental data. By contrast, if the growth regulators are partitioned by amounts independent of size, then ɛⁱ does not dependent on Lⁱ. It then follows from Equation (8) that κⁱ ∝ 1/Lⁱ. To quantify the relationship between growth rate and compartment length, we note that the cell compartments grow at equal rates if the stalked-to-swarmer length ratio at birth is equal to its average value (Figs 1F and G and 3A). Therefore, $κ_{b}^{St} = κ_{b}^{Sw}$ if $L_{b}^{St} = 〈 L_{b}^{St} 〉$ and $L_{b}^{Sw} = 〈 L_{b}^{Sw} 〉$ , where the subscript (b) refers to values at birth at the beginning of the stalked cell cycle and the angular brackets denote average across all cells. This results in the constraint

\frac{ε_{b}^{St}}{ε_{b}^{Sw}} = \frac{〈 L_{b}^{St} 〉}{〈 L_{b}^{Sw} 〉} = \frac{0.55}{0.45} = γ^{*},

(9)

where γ* is defined as the average size ratio between stalked and swarmer cell compartments at birth. The numerical value for γ* is determined by the size control parameters a, b, δ and Δ, as defined in Equations (3) and (4).

When the septal invagination is formed at distance x with respect to the average septum location (Fig 3B), such that $L_{b}^{St} = 〈 L_{b}^{St} 〉 + x$ and $L_{b}^{Sw} = 〈 L_{b}^{Sw} 〉 - x$ , we then have

κ_{b}^{St} = \frac{1}{L_{b}^{St}} \frac{d L^{St}}{d t} = \frac{1}{〈 L_{b}^{St} 〉 + x} α ε_{b}^{St},

(10)

κ_{b}^{Sw} = \frac{1}{L_{b}^{Sw}} \frac{d L^{Sw}}{d t} = \frac{1}{〈 L_{b}^{Sw} 〉 - x} α ε_{b}^{Sw} .

(11)

The above equations suggest a negative correlation between the growth rates of individual compartments and their respective birth lengths, which fit very well to experimental data (Fig 3C). Differential growth rate of daughter cells upon asymmetric division has been recently reported in E. coli (45 Preprint), where negative correlation between cell size and growth rate arises from equipartitioning of ribosomes that localize near the cell poles.

Size-independent synthesis of growth regulators

Next, we prescribe the dynamics of ɛ^St(t) and ɛ^Sw(t) to predict how the cell dynamically corrects deviations in stalked-to-swarmer size ratio from their homeostatic values. If the growth regulators are synthesized in proportion to cell size such that dɛⁱ/dt (i = {St, Sw}) is proportional to Lⁱ, then such a model would accelerate the growth of the bigger compartment relative to the smaller compartment, unlike what is observed in data. We therefore considered a model of size-independent synthesis such that

\frac{d ε^{i}}{d t} = k ε^{i},

(12)

where k is a constant rate of synthesis. Using Equations (7) and (12) we derive the time-dependence of the stalked and swarmer compartment lengths, given by,

L^{i} (t) = L_{b}^{i} + α ε_{b}^{i} (e^{k t} - 1) / k .

(13)

The above equation predicts that the compartments elongate at different rates unless γ(t) = L^St(t)/L^Sw(t) is equal to the homeostatic value $γ^{*} = ε_{b}^{St} / ε_{b}^{Sw}$ . In particular, the model leads to the relation

\frac{κ^{St} (t)}{κ^{Sw} (t)} = \frac{γ^{*}}{γ (t)},

(14)

which predicts a negative correlation between the growth rate ratio and the ratio between the stalked and swarmer compartment lengths, which is in excellent agreement with experimental data (Fig 3D). As γ(t) approaches γ* for t > k⁻¹, both the compartments grow exponentially at equal growth rates.

Combining (Equation (6)) for whole-cell elongation with Equation (12) for size-independent synthesis of growth regulators, we can derive the time evolution of cell length as $L (t) = L_{b} + α ε_{b} k^{- 1} (e^{k t} - 1)$ . This predicts super-exponential growth of the whole cell, such that the instantaneous growth rate $κ (t) = L {(t)}^{- 1} d L (t) / d t$ increases with time, in agreement with single-cell data for C. crescentus and E. coli (46, 47). Our model is thus relevant for other cell types that exhibit super-exponential growth, with appropriate modifications in the patterns of growth and cell size partitioning ratio.

Differential growth maintains division size asymmetry in cell population

Having developed a quantitative model for asymmetric size partitioning in single cells, we asked if differential growth-mediated size correction is sufficient to achieve tight regulation of cell division ratio at the population level. To this end, we performed stochastic single-cell simulations of growth and division for a population of asynchronous C. crescentus cells. Briefly, we simulated a collection of n = 10⁴ cells where each cell consists of a stalked and a swarmer compartment that can grow at differential growth rates as given by Equation (8). At the beginning of the division cycle, we chose the position of the pre-cleavage furrow from a Gaussian distribution such that $L_{b}^{St} / L_{b}^{Sw} = 1.21 \pm 0.25$ as experimentally observed, with the constraint $L_{b}^{St} + L_{b}^{Sw} = L_{b}$ (Fig 1B). Initial Gaussian distribution is the only source of noise in our simulations. Based on their chosen birth lengths, the stalked and the swarmer compartments grew at size-dependent rates as deduced from experimental data (Fig 4A) and our mathematical model (Fig 4B). In our simulations, individual compartments grew with size-dependent growth rates till the crossover time t_c = 0.6 τ (Figs 4B and S2A–D), after which they grew exponentially at constant rates independent of their size (Figs 4C and S4A–D). Interdivision time τ is computed using the formula, $τ = κ^{- 1} \log (a + \frac{Δ}{L_{b}})$ . Once the simulations reached steady-state, we collected the data for division ratios $L_{d}^{S t} / L_{d}^{S w}$ and compared them with the experimentally obtained distribution for division ratios (Fig 4D).

Figure 4. Simulations of bacterial population with differential growth control predict cell length distributions in agreement with experiments.

(A) Probability distribution of initial growth rates of stalked and swarmer compartments for: (top) $L_{b}^{St} > L_{b}^{Sw}$ , (middle) $L_{b}^{St} = γ^{*} L_{b}^{Sw}$ , and (bottom) $L_{b}^{St} < L_{b}^{Sw}$ . (B) Differential growth rates before t_c are fitted to (14), yielding κ^St(h ^{− 1}) = 0.186 (1 + 1/γ) and κ^Sw(h ^{− 1}) = 0.166 (1 + γ), where $γ \equiv L_{b}^{St} / L_{b}^{Sw}$ . (C) Growth rates after t_c. (D) Simulation predictions of division ratio statistics for different growth models. (B, C) When differential growth model was used (data from panel (B, C) as input), prediction of division ratio distribution (orange solid line) matched with experimental data (green solid line). CVs were 0.095, 0.068, 0.206, 0.211, and sample sizes were 2,448, 90,713, 40,310, 89,404 for experimental data, differential growth model, “same” (equal growth rate model), and random growth rate model, respectively.

Figure S4. Differential growth for C. crescentus cells growing at different temperatures.

(A, B, C, D) Mean growth rate before t_c of stalked, swarmer and the whole cell, binned by $L_{b}^{St} - L_{b}^{Sw}$ . Cells were grown at 14°C, 17°C, 31°C, and 37°C in (A, B, C, D) respectively. Lines represent model fit, see Mathematical model for asymmetric growth control section of the main text. Error bars represent ±1 SEM.

Our simulation results were contrasted with two other models for growth control, where both compartments either grew at the same rate or randomly chosen growth rates independent of their initial lengths (Fig 4D). As expected, high discrepancies from experimental data were observed for non-differential growth models (same growth rates or random growth rates), whereas the empirically observed differential growth model quantitatively matched the experimentally observed distribution for cell division ratio quite well. Interestingly, the precision of cleavage positioning for differential growth model (CV = 0.06) was slightly smaller than for experimental data (CV = 0.09), suggesting additional sources of noise for division septum positioning. Taken together, the stochastic simulations based on the differential growth model show that size-dependent regulation of stalked and swarmer compartment growth is sufficient to quantitatively explain asymmetric division control and tight regulation of daughter cell size ratios in C. crescentus cells.

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Differential growth regulates asymmetric size partitioning in Caulobacter crescentus

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