A hydraulic instability drives the cell death decision in the nematode germline

Oocytes are large cells that develop into an embryo upon fertilization1. As interconnected germ cells mature into oocytes, some of them grow—typically at the expense of others that undergo cell death2–4. We present evidence that in the nematode Caenorhabditis elegans, this cell-fate decision is mechanical and related to tissue hydraulics. An analysis of germ cell volumes and material fluxes identifies a hydraulic instability that amplifies volume differences and causes some germ cells to grow and others to shrink, a phenomenon that is related to the two-balloon instability5. Shrinking germ cells are extruded and they die, as we demonstrate by artificially reducing germ cell volumes via thermoviscous pumping6. Our work reveals a hydraulic symmetry-breaking transition central to the decision between life and death in the nematode germline.

Oocytes are large cells that develop into an embryo upon fertilization 1 . As interconnected germ cells mature into oocytes, some of them grow-typically at the expense of others that undergo cell death [2][3][4] . We present evidence that in the nematode Caenorhabditis elegans, this cell-fate decision is mechanical and related to tissue hydraulics. An analysis of germ cell volumes and material fluxes identifies a hydraulic instability that amplifies volume differences and causes some germ cells to grow and others to shrink, a phenomenon that is related to the two-balloon instability 5 . Shrinking germ cells are extruded and they die, as we demonstrate by artificially reducing germ cell volumes via thermoviscous pumping 6

. Our work reveals a hydraulic symmetry-breaking transition central to the decision between life and death in the nematode germline.
The germline of the adult Caenorhabditis elegans hermaphrodite captures all the essential features to identify the mechanisms by which germ cells are selected to live or die. The nematode gonad is a tubular syncytium consisting of germ cells that surround a central cytoplasmic compartment called rachis, to which all the germ cells are connected via openings called rachis bridges 3,7-10 ( Fig. 1a). Germ cells originate in a mitotic zone from a pool of stem cells residing in the distal tip of each gonad arm, and undergo meiotic maturation as they move towards the proximal turn 3,11 . During this progression, some of the germ cells grow to become oocytes, while the rest shrink and die by physiological apoptosis 12 (Fig. 1a). Although the core apoptotic machinery was shown to drive the final steps of cell death, the mechanisms that select and initiate apoptosis in individual germ cells are still unclear 13,14 .
Oocyte growth in C. elegans has been shown to rely on long-range cytoplasmic streaming 15,16 , but how and why germ cells shrink remains elusive. To identify a potential relationship between germ cell growth, shrinkage and apoptosis, we first set out to both quantify where germ cells grow along the gonad and characterize how this growth proceeds (Fig. 1b). Confocal imaging followed by three-dimensional (3D)-membrane-based segmentation of adult germlines expressing the membrane marker mCherry::PH(PLC1delta1) allowed us to measure individual germ cell volumes along the distal to proximal axis until the turn region (Supplementary Video 1). We find that germ cells near the distal tip have a volume of approximately 100 fl. As germ cells mature along the gonad, they first collectively grow in volume to approximately 150 fl ( Supplementary Fig. 1a). Before the proximal turn, the variation in germ cell volumes increases drastically, and germ cells range from very small (~65 fl) to very large (~1,200 fl) sizes. While the distribution of germ cell volumes is unimodal in the distal region, it becomes bimodal close to the turn ( Fig. 1b and Supplementary Fig. 1b). This suggests a transition from a homogeneous to a heterogeneous growth mode of germ cells along the gonad. To identify the precise location where this transition occurs, we investigated the average and standard deviation of germ cell volumes in different regions along the gonad (Fig. 1b, bottom inset). Both quantities appear to be linearly related but the associated slope changes sharply, which can be used to locate the transition zone to 65% ± 3.75% germline length. Two alternative methods for investigating the unimodality of distributions to identify the transition zone gave a similar result (Supplementary Information). Note that physiological apoptosis occurs proximal to this transition point, from about 70% to 90% germline length ( Supplementary Fig. 1c).
Both homogeneous and heterogeneous modes of germ cell growth must rely on the addition of cytosolic volume. Germ cells can either grow by receiving material from the rachis inside or from the surrounding tissue outside, such as the intestine 17 . We set out to identify the two regions of growth from where the corresponding cytosolic volume is available. For this, we made use of the fact that cytosol is incompressible 18 and determined the volume flux of cytoplasmic material through the rachis along the gonad Q r (x), where x denotes the position along the distal-proximal axis of the germline (Supplementary Information). In this one-dimensional (1D) representation, the flux balance at the steady state can be expressed as where J(x) denotes the germ-cell-to-rachis current. Hence, an increase in rachis flux Q r along the gonad implies that germ cells contribute material to the rachis, while a decrease means that germ cells receive material from the rachis. We performed particle image velocimetry (PIV) 19 on mid-plane confocal sections of the germline expressing LifeAct::mKate to determine a cytoplasmic velocity field inside the rachis, which we then used to infer the steady-state rachis flux (Fig. 1c Supplementary Fig. 1a) despite losing cytoplasm to the rachis. This implies that they must be receiving material from the outside. We inferred the profile of material uptake S(x) (Fig. 1d) from the total flux balance at the steady state: where Q c (x) denotes the volume flux associated with germ cells moving from distal to proximal 11 (Supplementary Information). We find that in the distal region and up to approximately 60% gonad length, material uptake S is positive and germ cells grow by receiving material from the outside. Material uptake becomes negative beyond 60%, indicating a loss of material to the outside, possibly via removal of apoptotic cells. We conclude that material associated with the homogeneous growth of germ cells comes from the outside, while the heterogeneous growth mode is associated with germ cells receiving cytoplasm from the rachis.

NAturE PhySicS
The observed inversion of current J around 60% germline length implies an inversion of the pressure difference across the rachis bridges separating the germ cells from the rachis. To shed light on the underlying force balances, we construct a 1D physical model that relates pressure profiles to flows of germ cells and rachis cytoplasm as well as material exchange between germ cells and rachis ( Fig. 2a and Supplementary Information). The germ-cell-to-rachis current is driven by differences in germ cell pressure P c (x) and rachis pressure P r (x), and it can be expressed as J = α(P c -P r ). Here α denotes an effective hydraulic conductivity of rachis bridges, which depends on rachis bridge radii. Using the profile of material uptake (Fig. 1d), this theory recapitulates rachis and germ cell fluxes (indicated by solid lines in Fig. 1c and Supplementary Fig. 1e) and predicts a profile of germ-cell-to-rachis current J that matches the experimental estimates (Fig. 2b). Consistent with the observation that the rachis flux peaks around 60% gonad length, this current changes sign at the same location (Fig. 2b). Because pressure differences drive the germ-cell-to-rachis current J, the pressure difference between the cells and rachis, P c − P r , also changes sign at this location. Next, we investigated if this inversion of the pressure difference might be the key to understanding the transition from homogeneous to heterogeneous mode of germ cell growth. We note that the change in unimodal to bimodal volume distribution is indicative of instability in the germ cell volumes during growth. Similar instability arises when simultaneously blowing into two rubber balloons in an attempt to inflate them both. Here only one balloon inflates. Because the larger balloon can be inflated at lower pressures than the smaller one, the situation where both simultaneously inflate is mechanically unstable 20 .
Could such an instability also arise in the germline [21][22][23][24][25][26][27] ? We consider the mechanics of a simplified configuration of two germ cells with volumes V 1 and V 2 that surround a common rachis to which they are connected by rachis bridges (Fig. 2c). We take into x (% length)

Fig. 2 | A theoretical model of germ cell and rachis fluxes reveals a hydraulic instability. a,
Schematic of a 1D hydrodynamic model for pressures, material fluxes and volume exchange in the C. elegans gonad. P c and P r denote the pressure field in germ cells and rachis, respectively; S, profile of material uptake from the outside; J, germ-cell-to-rachis current associated with flows through the rachis bridges; Q r , rachis flux; and v c , germ cell velocities. b, Green open circles, estimated germ-cell-to-rachis current J along the gonad length; vertical dashed line and grey bar denote the region of transition between the growth modes of germ cells (Fig. 1b). Solid line, best parameter theory fit given the profile of material uptake S shown in Fig. 1d. c, Schematic of a germ cell doublet and two connected balloons depicting how a difference in volumes leads to difference in pressures. d, Evolution of small volume differences between coupled germ cells with time. The symmetric state (ν = 0) of equal germ cell volumes is unstable when the pressure in the rachis is higher than in germ cells (right).
Here the initially larger cell grows at the expense of the smaller one. e, Relative volume difference ν in a cell doublet bifurcates around 60% germline length where material uptake S vanishes and pressure difference P c - P r changes sign. Insets illustrate cell configurations and corresponding effective potentials W as a function of ν normalized by its value at ν = 0 (Supplementary Information). Error bars indicate the error of the mean at 95% confidence.

Letters
NAturE PhySicS account force balances due to cortical tension and the fact that material uptake from the outside of each cell is proportional to the basal surface area exposed to the outside. Note that one germ cell can increase in size at the expense of the other. Furthermore, large germ cells tend to have larger rachis bridges than small germ cells ( Supplementary Fig. 1f). With these components, we obtain an equation of motion for the relative volume difference ν = ( where A c is the combined cross-sectional area of the two germ cells; λ is the relative difference in the basal surface area of the germ cells, which depends on ν; and α 0 is the hydraulic conductivity. The average pressure between the two germ cells P (1) c and P (2) c can be expressed as Pc = (P (1) c + P (2) c )/2. Pressure in the two germ cells can differ by ΔP = P (1) c − P (2) c . This pressure difference ΔP depends on the relative volume difference ν and is similar to the pressure difference between two balloons of different sizes (Fig. 2c). For two germ cells, ΔP ≈ Tν/R, where T is an effective cortical tension 9,15 and R is the radius of the gonad (Supplementary Information). This pressure difference, therefore, tends to destabilize the symmetric configuration with equal germ cell volumes. Two additional contributions in equation (3) can stabilize the symmetric state. First, the effects of material uptake S > 0 are generally stabilizing. Second, the contributions from the difference in germ cell and rachis pressure become stabilizing when P c > P r . Hence, material uptake in the distal region (S > 0) leads to a situation with P c > P r , both of which stabilize the symmetric state of equal germ cell volume in this region. At around 60% gonad length, material uptake vanishes (S ≈ 0) leading to inversion of the pressure difference P c − P r at a point slightly proximal to the point at which the uptake vanishes (Figs. 1d and 2b). Beyond this point, the stabilizing effects are absent and the symmetric state is, therefore, unstable. As a result, small differences in germ cell volumes increase, leading to the growth of the larger germ cell at the expense of the smaller one until the small cell loses its cytoplasm

NAturE PhySicS
at ν = ±1 (Fig. 2d). Equation (3) can be expressed in terms of an effective potential dν/dt = -dW/dν (Fig. 2e, inset). This effective potential W either has a single minimum at ν = 0 when the symmetric state with ν = 0 is stable or exhibits two minima at ν = ±1 corresponding to the two completely asymmetric configurations, while the symmetric state at ν = 0 is at a maximum. Hence, germ cells undergo homogeneous growth in the distal region before 60%, beyond which they undergo heterogeneous growth as is observed in this regime (beyond 65% gonad length; Fig. 1b). The transition between these two regimes is associated with a hydraulic instability, which is triggered by the loss of material uptake and the associated inversion of the pressure difference P c − P r . This hydraulic instability presents a possible mechanism by which germ cells become fated to die: generating a few large cells at the expense of smaller shrinking cells in a coarsening process. Apoptosis is then triggered in shrinking cells, which leads to their removal. However, an alternative scenario is that unknown molecular signals first induce apoptosis, which subsequently leads to the shrinkage of those cells fated to die. To test this alternative possibility, we inhibited the apoptosis of germ cells by RNA interference (RNAi) 28 targeted against the caspase CED-3 (refs. 12,13 ) and evaluated if germ cells still shrink. We find that in the absence of apoptosis, germ cells are no longer removed 12 ; however, we still observe that some cells in the proximal region shrink (blue dots in Fig. 3a). Similar to unperturbed conditions, ced-3(RNAi) gonads show a transition from a homogeneous to a heterogeneous mode of growth (grey bar in Fig. 3a); further, the position of this transition is close but a bit proximal to the location where both germ-cell-to-rachis current and pressure difference between the germ cells and rachis change sign (Fig. 3a), and where the rachis flux peaks (Fig. 3b). Together, this eliminates apoptosis as the cause of germ cell shrinkage and supports the idea that germ cell fate is determined by a hydraulic instability.
Our results suggest that a hydraulic instability generates large and small germ cells; the latter are eliminated by physiological apoptosis. This implies that increasing the number of small germ cells should lead to an increase in apoptotic germ cells. To test this, we depleted the small anillin isoform ANI-2 by RNAi, which is reported to affect the germline architecture and result in smaller germ cells 29 . Indeed, treatment with ani-2(RNAi) led to a higher fraction of proximal germ cells smaller than ~150 fl ( Supplementary  Fig. 2f,g) and a concomitant increase in the number of apoptotic cells 29,30 (Supplementary Fig. 2h). Next, we set out to test if we can bias the outcome of this hydraulic decision-making process via direct mechanical manipulation. In particular, artificially reducing the volume of individual germ cells should increase their likelihood to undergo apoptosis. We tested this prediction by unidirectional thermoviscous pumping (focused-light-induced cytoplasmic streaming (FLUCS)) 6 for 15-20 min to pump germ cell cytoplasm out of individual germ cells through their rachis bridge, and monitoring the subsequent fate of the manipulated cells for 3 h (Fig. 3c). As a control, we performed bidirectional FLUCS by rapidly switching between pumping cytoplasm into and out of individual germ cells, with an overall similar dosage of laser light but without inducing a net flow (Supplementary Video 5). In the control scenario, 14.3% germ cells (3 out of 21) commence apoptosis within the following 3 h (Fig. 3d), as judged by a characteristic rounding up of apoptotic germ cells 12 . This number favourably compares with the rates of apoptosis in the unperturbed situation, since an apoptotic rate of 4-6% per hour ( Supplementary Fig. 1c) leads to 11.5-17% germ cells commencing apoptosis within 3 h (Methods; note that we arrive at similar numbers when analysing neighbouring cells not subjected to FLUCS (Fig. 3d)). In contrast, 52.6% germ cells (10 out of 19) commence apoptosis within the three hours following unidirectional FLUCS (Fig. 3c,d and Supplementary Video 6). We conclude that a hydraulic manipulation to reduce the volume of individual germ cells results in an increased likelihood of commencing apoptosis. Together, this lends credence to the statement that the life and death decision in the gonad is of mechanical nature.
The hydraulic instability that we have discovered amplifies small differences in germ cell volumes and redistributes material from smaller to larger cells. This is consistent with observations in which larger oocytes continued to grow at the expense of smaller neighbours even after the blockage of rachis with an oil drop 15 . Because germ cell fate is determined by size, the mechanism selects for larger-and perhaps-fitter cells while making use of the resources of the dying ones 29,31 . The mechanism we have discovered here bases a cell-fate decision on a hydraulic instability, presenting a robust alternative to biochemical switches usually invoked in cellular decision-making processes [32][33][34] .

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Germ cell volume quantification. Confocal imaging was performed on UM208 young adult worms and 100 different z planes spaced 0.5 μm apart were acquired in 3 to 4 fields before being stitched together using Fiji's pairwise stitching plugin. The 3D segmentation on the membrane marker was performed using Imaris Cell plugin (version 9.2).

Rachis flow quantification.
Confocal imaging was performed on SWG007 young adult worms and five different z planes spaced 2 μm apart were acquired every 10 s. To capture full gonads, 3 to 4 fields were acquired and stitched together using Fiji's pairwise stitching plugin. The z stack corresponding to the central-most plane of the rachis was kept for analysis. Flow velocities were obtained by PIV tracking of LifeAct::mKate timelapse acquisitions 19 . PIV template size was 16 pixels corresponding to 1.695 μm. The x axis is defined as the distal to proximal axis. Due to geometric irregularities in the distal region of the rachis for ced-3(RNAi) germlines, we are only able to reliably identify steady velocity fields beyond 16% gonad length in the distal-proximal axis of these samples. The estimation of rachis flux Q r from the velocity field is based on erosion-based segmentation to find the centreline and is described in Supplementary Information. Note that erosion-based segmentation fails near the proximal turn due to high curvatures. As a result, at the very proximal end and at 100% gonad length, the rachis flow into the turn is captured by the germ-cell-to-rachis current J rather than Q r . In our analysis, this results in the rachis flux Q r to drop to nearly zero at 100% gonad length (Fig. 1c), slightly before the rachis flux actually drops to zero at the last oocyte 15 .

Rachis opening perimeter.
Confocal imaging was performed on UM208 young adult worms and 100 different z planes spaced 0.5 μm apart were acquired in 3 to 4 fields before stitching them together using Fiji's pairwise stitching plugin. After maximal projections of the ANI-2::GFP slices corresponding to the upper half of the germline, a 2σ Gaussian blur filter was applied and the perimeter of each opening was quantified semi-automatically using Fiji's wand tool. A conservative estimate for uncertainty is provided by the z spacing, which is 0.5 μm in this case.

FLUCS.
Young adult worms were paralysed and mounted between a 3% agarose pad and a 18 mm × 18 mm coverslip (0.17 mm thickness). The sample was placed on a sapphire microscope slide equipped with Peltier cooling elements, sealed with dental silicone (Picodent twinsil, Picodent) and mounted on the FLUCS microscope stage 6 . The samples were imaged using  Fig. 2f).
Mitotic and apoptotic rates in the germline. SWG59 (for mitosis) or SWG16 (for apoptosis) young adult worms were paralysed in 0.1% tetramisole (Sigma-Aldrich T1512) for 3 min on a cover slip precoated with 0.1% poly-l-lysine (Sigma-Aldrich P8920) and mounted on 2% agarose pads. Images were acquired with a spinning-disk confocal microscopy (Zeiss C-Apochromat, ×63/1.2NA, Yokogawa CSU-X1 scan head and Hamamatsu ORCA-Flash4.0 camera) on 41 different z planes spaced 1 μm apart, recording one full stack every 30 s. After maximal projections of individual movies in Fiji, the number of metaphase spindles or each new engulfment event-together with their position relative to the distal tip or proximal turn-were manually measured. The frequency of mitotic or apoptotic events along the total amount of imaging time was then binned to obtain a probability of event per unit length of the germline (Supplementary Fig. 1c). From these rates, an estimate of cumulative probability can be deduced. For example, at position x of the gonad, considering that the rate of apoptosis per hour is p a , then the associated total probability of a cell to survive a period of 3 h is ps = (1 − pa) 3 . Hence, the estimated total cell death should be ~100 × (1 − p s )%.
Cell number density in the germline. UM208 young adult worms (24 h post L4) were paralysed in 0.1% tetramisole (Sigma-Aldrich T1512) for 3 min on a cover slip precoated with 0.1% poly-l-lysine (Sigma-Aldrich P8920) and mounted on 2% agarose pads. Images were acquired with spinning-disk confocal microscopy (Zeiss C-Apochromat, ×63/1.2NA, Yokogawa CSU-X1 scan head and Hamamatsu ORCA-Flash4.0 camera) on 100 different z planes spaced 0.5 μm apart. To capture the full gonads, 3 to 4 fields were acquired and stitched together using Fiji's pairwise stitching plugin. Curved gonads were straightened using a 300 pixels (32 μm) wide line on the x-y stacks using Fiji's straighten tool and resliced along the z axis to get the cross-sections along the distal to proximal axis (semi-automatic using 'sideviews' Fiji macro). Cell numbers around the rachis were counted in 30 slices along the distal proximal axis and the average cell length along the distal proximal axis was estimated.
Estimating interface curvature. UM208 young adult worms (24 h post L4) were paralysed in 0.1% tetramisole (Sigma-Aldrich T1512) for 3 min on a cover slip precoated with 0.1% poly-l-lysine (Sigma-Aldrich P8920) and mounted on 2% agarose pads. Images were acquired with spinning-disk confocal microscopy (Zeiss C-Apochromat, ×63/1.2NA, Yokogawa CSU-X1 scan head and Hamamatsu ORCA-Flash4.0 camera) on 100 different z planes spaced 0.5 μm apart. To capture the full gonads, 3 to 4 fields were acquired and stitched together using Fiji's pairwise stitching plugin. Curved gonads were straightened using a 300 pixels (32 μm) wide line on the x-y stacks using Fiji's straighten tool and resliced along the z axis to get the cross-sections along the distal to proximal axis (semi-automatic using 'sideviews' Fiji macro). For 1/100th of the cross-section slices, the curvature along all the cell-cell interfaces were calculated by fitting a 5 point B-spline curve using the Kappa plugin in Fiji. The average curvature (μm −1 ) at each cross-section was then plotted along the distal to proximal axis. A smooth curved surface of a cell of 10.0 μm height and 0.5 μm maximal deflection would correspond to a curvature of ~0.04 μm −1 , larger than the estimated curvatures shown in Supplementary Fig. 4b.
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All data generated or analysed in this study are available from the corresponding authors upon reasonable request.

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Codes to analyse the data and perform numeric calculations are available from the corresponding authors upon reasonable request.  Extended Data Fig. 1 | Volumes and fluxes in nematode germline. A, Average volume of germ cells along the distal to proximal axis (18 germlines, bin size 2.5 % gonad length). Vertical dashed line and grey region indicate the transition region between growth modes (Fig. 1b). B, Transition from a unimodal to a bimodal distribution of germ cell volumes (left: blue dots, individual germ cell volumes as in Fig. 1b). Histograms of germ cell volumes in five different regions along the gonad (right). The distribution of germ cell volumes is unimodal (Gaussian fits in (i)-(iv)) in the distal region and bimodal in the proximal region (superposition of two Gaussian's in (v)). C, Rates of mitotic (positive) and apoptotic events (negative) along germline length, analysed from 6 gonads for 30

Corresponding author(s): Stephan Grill
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