Uncovering the mechanical secrets of the squirting cucumber

Plant material was acquired from the living collection of the Oxford Botanic Garden (grown under ambient conditions). Fruit were then removed to the laboratory, with their stems intact; the stems were placed in water and the fruit monitored either using a high-speed video camera or time-lapse photography, or subject to experimentation over the course of days prior to seed ejection. Additionally, measurements of the compass orientation of ripe cucumbers was performed on living specimens in Kew Gardens (London, UK) and in the wild (Zaragoza, Spain). Experimental techniques are summarized below, with further detail provided in SI Appendix as indicated.

To complement the experiments described above, we developed a suite of mathematical models at different scales, describing the mechanics of several distinct components: the pressurized fruit, the stem, the rotation of the fruit following detachment, the ballistic trajectories of the seeds, and the spatial distribution of seeds over several plant generations.

These components are described further below. A full description of the models, including underlying assumptions, how they are parameterized, and how different model components are linked, is provided in SI Appendix, section 2.

The fruit of Ecballium is approximately ellipsoidal, with major and minor axes of about 4 and 2 cm respectively (Fig. 2A). It is attached at the end of the plant stem. Typically, a plant produces dozens of fruits which are orientated in different directions. A fruit contains $\sim$50 seeds of (wet) mass $25.0\pm 2$ mg ($N=96$) that roughly take the shape of a flattened ellipsoid of length $l_{\mathit{se}}\approx 5$ mm, width $w_{\mathit{se}}$$\approx 3$ mm and thickness $t_{\mathit{se}}\approx 2$ mm. The seeds are immersed in a mucilaginous fluid matrix (which is comparable in density to water, ${\rho }_{\mathit{fl}}\approx 1,000$ kg m${}^{-3}$) and occupy $\approx$6% of the volume within the fruit (details given in SI Appendix, section 1B). The shell of the fruit is $h=3.20\pm 0.24$ mm ($N=4$) thick and measured Young’s modulus $E_f=0.50\pm 0.09$ M Pa. A Computed Tomography (CT) scan of an intact cucumber, see Fig. 1B (and visualization in SI Appendix), shows that the seeds are distributed in pairs in locules (chambers) along the major axis of the fruit. Individual pairs are connected to four vasculatory pillars (placentae) that are located at 90-degree intervals around the long axis of the fruit. The aperture through which the seeds pass is comparable in diameter to the width of the seeds which, on ejection, forces the seeds to align their long axis in the direction of the liquid jet.

The stored energy in the turgid fruit is converted to kinetic energy of the jet and seeds upon ejection. Our indentation measurements in Fig. 2B show small variations in the indentation stiffness prior to launch, but consistently show a stiffness prior to launch that is around $3.5$ times larger than that after launch, which corresponds to a pressure difference $\mathrm{\Delta }p\approx 1.7\mathrm{bar}$ (see SI Appendix for details). The higher the pressure, the higher the velocity of the seeds (see Section 2.3 below), and thus one might expect that evolution would favor the highest possible pressure that the fruit tissue can withstand. However, the angle at which the seeds are launched is equally important in determining the horizontal distance attained from the mother plant. For example, if the fruit were to launch from a nearly vertical orientation, all seeds would fire straight up, landing very near to the mother plant. Hence, in this scenario, increasing internal pressure would not increase seed dispersal.

A delicate balance between pressure and fruit orientation was observed in a vital developmental sequence in time-lapse videos: In the build-up to launch, the inclination of fruits changes from an approximately vertical position to an angled orientation with a launch angle better suited for increased seed dispersal distances, see Figs. 1C and 2C (and the time-lapse video, Movie S2). The orientation angle of the fruit at launch was observed to be $\varphi \approx 42.7\pm 8.9^{°}$ relative to the horizontal ($N=8$; data from both time-lapse and high-speed images), which is slightly below the optimal launch angle of ${45}^{°}$ usually expected. Accounting for drag decreases the launch optimal angle in general (3, 33); detailed calculations of the optimal launch angle with the drag relevant for Ecballium seeds are given in SI Appendix, section 2.E.1 of the SM and show that the optimal launch angle lies in the range ${37}^{°}\lesssim \varphi \lesssim {44}^{°}$. The reorientation of the fruit is a continuous process occurring over the period of several days prior to ejection. We also observed a qualitative trend in undisturbed fruits growing in the Oxford Botanic Garden: ripe fruits [detectable by a more yellow shade (28)] generally were oriented closer to 45 degrees from the vertical, whereas less ripe fruits (with a more green shade) generally were oriented closer to the vertical, see Fig. 1A.

The pre-ejection reorientation of the fruit may be understood in terms of observed changes to the stem during the days prior to ejection. The time-lapse videos show that the stem becomes significantly straighter and longer, increasing in radius by approximately 20%, and increasing in length by approximately 45%. The stem forms a thick-walled hollow tube, and this expansion suggests an increase in internal pressure. During the same time period, the fruit itself undergoes an active contraction, decreasing in volume; see images in Fig. 2 A and D, where we provide approximate volumetric measurements based on the ellipsoidal form of the fruit. Simultaneously, the stem increases in volume; see Fig. 2E, where we provide volumetric measurements assuming the stem takes the form of a conical frustum. However, in monitoring the combined mass of fruit plus stem over several days prior to launch, we detect no appreciable change; see Fig. 2F. This suggests that reorientation of the fruit is driven by a redistribution of fluid from the fruit, which contracts, to the stem, which expands and elongates. Accordingly, pressure in the fruit decreases prelaunch to gain a better orientation. Indeed, the geometric changes in the stem have the effect of straightening it from its initial configuration; the stiffer stem is less bent over by the weight of the fruit (Fig. 2A). To quantify this effect, we model the stem as an elastic rod with tapered cross-section (details given in SI Appendix, section 2A). When the fruit is attached to the stem, it provides a couple (a combined force and moment) to the tip of the stem. For given properties of the fruit (mass and length), the shape of the deformed stem supporting the fruit may be computed from the balances of linear and angular momenta. Stem dimensions were extracted from laboratory images, and an effective Young’s modulus $E_s$ was estimated both before and after fluid redistribution by fitting the deformed shape to lab images (Fig. 2A). An excellent fit was produced with the single fitting parameter and shows an approximately five fold increase in $E_s$ over the time period from 4 d prior to launch. The stiffening of a stem under increased pressure, lengthening, or tissue-tension is well known in other plants and due to the anisotropic stiffening of the fiber-reinforced outer layers of epithelial cell (34, 35). We hypothesize that a similar mechanism is at play in Ecballium, though investigation of such properties is beyond the current work.

Upon dehiscence, the fluid and seeds stream through the fruit aperture. From images of seed ejection obtained at high frame rate, we measured the initial speed of individual seeds and the time of their ejection after the onset of abscission, see Fig. 3A. These data show that the speed of ejection decreases with time after abscission, and further, that this decrease is approximately linear. This observation is consistent with the intuitive picture that the high pressure within the fruit (which drives the fluid flow) is caused by the fruit containing excess liquid volume, $\mathrm{\Delta }V$, which decreases as liquid is expelled: As liquid is expelled, the excess volume liquid decreases, along with the driving pressure and so the speed at which fluid is ejected therefore also decreases. More quantitatively, linear elasticity theory for a shell (36, 37) shows that the driving pressure difference, $\mathrm{\Delta }p(t)\propto \mathrm{\Delta }V(t)$, which can also be related to the fluid ejection speed $U(t)$ using Bernoulli’s principle (38) so that $U(t)={(2\mathrm{\Delta }p/{\rho }_{\mathit{fl}})}^{1/2}\propto \mathrm{\Delta }V{(t)}^{1/2}$. Finally, we note that, by conservation of volume, the rate of change of the excess liquid volume $\dot{\mathrm{\Delta }V}\propto U\propto {(\mathrm{\Delta }V)}^{1/2}$ so that ${(\mathrm{\Delta }V)}^{1/2}\propto t_{\ast }-t$ for some constant $t_{\ast }$—this immediately gives that the speed of ejection $U(t)\propto t_{\ast }-t$, as observed experimentally (Fig. 3A). (More details of this calculation, including estimates of the relevant prefactors, are given in SI Appendix, section 2D.)

A key step in the above calculation is the use of Bernoulli’s principle to write the excess pressure $\mathrm{\Delta }p(t)=\frac{1}{2}{\rho }_{\mathit{fl}}U{(t)}^2$. Such an assumption is justified by the inertial nature of the flow (the jet Reynolds number $\mathrm{Re}\approx 20\mathrm{m}/\mathrm{s}\times 2.5\mathrm{mm}/{10}^{-6}\text{m}{\text{s}}^{-2}=5\times {10}^4\gg 1$). Nevertheless, an independent test of this assumption can be gained from comparing the initial jet-ejection speed, $U_0\approx 16.4$ ms${}^{-1}$, with that expected based on the measured excess pressure prior to dehiscence, $\mathrm{\Delta }p(0)\approx 1.7\mathrm{bar}$ (see SI Appendix for details), i.e. $U_0^{\mathrm{pred}}\approx {(3.4\times {10}^5\mathrm{Pa}/1,000\text{k}\text{g}{\text{m}}^{-3})}^{1/2}\approx 18.5\text{m}{\text{s}}^{-1}$. These estimates of the initial velocity of ejection agree to within $\approx$10%, validating a key step in our modeling.

The high-speed images of the fruit immediately following dehiscence also reveal that as it falls, the fruit rotates away from the stem at a nearly constant rate (Fig. 3B). Moreover, this rotation is in a direction that increases the launch angle of the seeds (relative to the horizontal) and occurs on the same timescale as the ejection of the seeds—the launch angle of each seed varies with its position in the launch sequence. The direction and rate of rotation were consistent among the fruit launches we observed, suggesting that this rotation is a robust feature. A variable launch angle leads to a wider distribution of landing spot of the seeds, which likely imparts an evolutionary advantage. To understand the mechanism behind this rotation, we first note that the fruit itself is approximately symmetric about the major axis, and the aperture is also symmetric. Moreover, the seeds are distributed uniformly and symmetrically within the fruit, as can be observed from the CT scan (Fig. 1B). Therefore, the reaction force on the fruit due to the ejection of the jet and seeds cannot impart a consistent and robust torque on the fruit itself. Rather, we postulate that the rotation derives from the interaction between the fruit and the stem during the dehiscence event at the very start of the ejection. Once the fruit has broken free from the stem, the stem recoils away from the fruit (due to breaking free from supporting the mass of the fruit). During the first couple hundred microseconds of ejection, while the fruit and stem are breaking apart but still in contact, the tip of the stem rotates away from the fruit, and the balance of angular momentum implies that the fruit should rotate in the opposite direction. To provide an estimate of the rotation, we develop an idealized model (SI Appendix, section 2C) of the fruit/stem interaction during these first instants of ejection, introducing inertia to the elastic rod framework and assuming that the moment applied by the fruit to the end of the stem continuously decreases to zero over the dehiscence time scale; this analysis predicts a rotation rate that is consistent with our experimental observations (see the dashed line in Fig. 3B).

The ballistic trajectories $\mathbf{r}(t)$ of individual seeds of mass $m$ satisfy the kinematic relation

where $g$ is the gravitational acceleration, acting in the vertical direction ${\mathbf{e}}_z$, and $F_d$ is the drag force (SI Appendix, section 2E). For given initial seed velocities and launch angles, obtained either via mathematical modeling or empirical best fits to data, we integrate numerically the trajectories, from which we compute the horizontal distance from the mother plant at which each seed lands. We first compute the landing distance for the seeds of a single fruit. Our results indicate that the seeds are projected over a horizontal distance between $\sim$4 and $\sim$12 m. This is in agreement with previous measurements of a 5 to 6 m range (28) and more recent reports of a dispersal distance up to 10.2 m, (see ref. 39 and references therein).

F.B., D.E.M., D.V., A.G., and C.T. designed research; F.B., D.E.M., D.V., Y.B., A.G., and C.J.T. performed research; T.L. contributed new reagents/analytic tools; F.B., D.E.M., and D.V. analyzed data; and F.B., D.E.M., D.V., A.G., and C.J.T. wrote the paper.

The authors declare no competing interest.