ABSTRACT (original article): Rapid movement is rare in the plant kingdom, but a prerequisite for ballistic seed dispersal. A particularly dramatic example of rapid motion in plants is the squirting cucumber (Ecballium elaterium) which launches its seeds explosively via a high-pressure jet. Despite intriguing scientists for centuries, the exact mechanism of seed dispersal and its effect on subsequent generations remain poorly understood. Here, through a combination of experimentation, high-speed videography, quantitative image analysis, and mathematical modeling, we develop a full mechanical description of the process. We quantify the turgor pressure driving ballistic ejection, and uncover key mechanical interactions between the fruit and stem both prior to and during seed ejection, including the unique feature that fluid is redistributed from fruit to stem prior to ejection, a developmental event that goes against the paradigm of rapid seed ejection but which is of key importance in successful dispersal for Ecballium. Combining modeling elements, we quantify and simulate the ballistic trajectories of seeds, which are dispersed over distances greater than 2,000 times their length. We demonstrate how together these mechanical features contribute to a nearly uniform distribution of seeds away from the parent plant. Parametric variation of key developmental events in the modeling framework indicates how a suite of adaptive features in combination drives the spatial distribution of offspring over consecutive generations, and suggests that ballistic seed dispersal has a stabilizing effect on population dynamics by reducing intraspecific competition. CITATION (original article): F. Box, D.E. Moulton, D. Vella, Y. Bhagotra, T. Lowe, A. Goriely, C.J. Thorogood (2024), Uncovering the mechanical secrets of the squirting cucumber, Proc. Natl. Acad. Sci. U.S.A. 121 (50) e2410420121. https://doi.org/10.1073/pnas.2410420121
OXFORD: https://wolfr.am/1rxxuQgrH
CNN: https://wolfr.am/1rxxgty8H
NYTimes article: https://wolfr.am/1rwUV5925
BBC: https://wolfr.am/1rxywlZSr
OXFORD: https://wolfr.am/1rxxuQgrH
CNN: https://wolfr.am/1rxxgty8H
NYTimes article: https://wolfr.am/1rwUV5925
BBC: https://wolfr.am/1rxywlZSr
We study the remarkable seed dispersal mechanism of Ecballium elaterium, commonly known as the squirting cucumber, one of the most rapid motions in the plant kingdom. Despite its apparent simplicity, the specifics of the seed ejection process—combining mechanical, hydraulic, and ballistic phenomena—remain largely unexplored. By integrating experiments, high-speed videography, and advanced mathematical modeling, we uncover unique facets of this strategy, including an unusual decrease in fruit volume prior to ejection which stiffens the stem and orients the fruit to an improved angle for dispersal. Our study reveals how the delicate interaction of mechanical components contributes to dispersal efficiency, thereby influencing plant distribution and population dynamics, and offering insights into evolutionary adaptations related to explosive fruit mechanisms.
Main text simulations: Prob success= 3/200, Min distance= 2.5 m
Main text simulations: Prob success= 3/200, Min distance= 2.5 m
This code simulates the full dispersal process of several generations including:
1. Computation of the stem and fruit orientation before and after fluid redistribution,
2. fruit rotation due to stem recoil
3. Seed ejection
4. Seed ballistics for a single fruit
5. Seed ballistics for N=30 fruits distributed around the plant
6. Generational simulation based on chosen values for probability of germination and minimum distance from other plants required (SI simulations may be obtained by changing the parameters)
1. Computation of the stem and fruit orientation before and after fluid redistribution,
2. fruit rotation due to stem recoil
3. Seed ejection
4. Seed ballistics for a single fruit
5. Seed ballistics for N=30 fruits distributed around the plant
6. Generational simulation based on chosen values for probability of germination and minimum distance from other plants required (SI simulations may be obtained by changing the parameters)
The following subsections creates the following figures:
Base case
Base case
Extracting initial velocity
Extracting initial velocity
ClearAll["Global`*"](*stemODEs*)Stem[m0_?NumberQ,E0_]:=Block{sol,s,x,y,θ},sol=NDSolvex'[s]γCos[θ[s]],y'[s]γSin[θ[s]],E0-uhat'[s]+DE0,s-uhat[s]-γCos[θ[s]](ρsg(s-L)-Mfg),x[0]0,y[0]0,θ[0]θ0,θ'[0]m0,{x,y,θ},{s,0,L}[[1]];E0-uhat[s]+Mfgh1Cos[θ[s]]/.sol/.sL;StemSol[m0_,E0_]:=Module{x,y,θ},{x[s],y[s],θ[s]}/.NDSolvex'[s]γCos[θ[s]],y'[s]γSin[θ[s]],E0-uhat'[s]+DE0,s-uhat[s]-γCos[θ[s]](ρsg(s-L)-Mfg),x[0]0,y[0]0,θ[0]θ0,θ'[0]m0,{x[s],y[s],θ[s]},{s,0,L}[[1]];
15π
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15π
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15π
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Clear[c,ds,σ,L,a,b]g=;(*gravity=10m/s^2=cm/ms^2*)ρf=1.02;(*fruitdensity,g*)M=9;(*massoffruitandstemcombined,ing*)ρs=1.03*π*;(*stemdensity(lineardensity)*)uhat=Functions,cExp;(*intrinsiccurvature*)E0=60;(*(effective)Youngsmodulusofstempriortoinflation*)γ=1;(*growthparameter*)L=3.2;(*cmlength*)a=0.17;(*cmradiusatbase*)b=.01;(*tapering,supposeradiusattipdecreasesbyfactor.18/.23*)c=4;(*degreeofintrinsiccurvature*)ds=0.2;(*curvaturedistancefromtip*)σ=0.3;(*definestaperingwidth*)θ0=π/2;(*clampangle*)FruitVol=4/3*Pi*h1*h2^2;(*fruitvolume*)Mf=ρf*FruitVol;(*fruitmass*)(*momentofinertia-formulafromWikipediaforellipsoidwithaxesh1,h2,h2*)Iyy=Mf/5*(h1^2+h2^2);h1=1.75;(*halflengthoffruit,incm*)h2=1.0;(*minoraxisoffruit*)mVal=FindRoot[Stem[m0,E0],{m0,.45}];{x[s_],y[s_],θ[s_]}=StemSol[m0/.mVal,E0];d3=Function[s,{Cos[θ[s]],Sin[θ[s]]}];d1=Function[s,{-Sin[θ[s]],Cos[θ[s]]}];stemBefore=Show[ParametricPlot[{x[L],y[L]}+h1d3[L]+h1Cos[t]d3[L]+h2Sin[t]d1[L],{t,0,2π},PlotStyleGreen],ParametricPlot[{x[s],y[s]},{s,0,L},PlotStyleDashed],ParametricPlot[{x[s],y[s]}+(a-bs)d1[s],{s,0,L},PlotStyleGreen],ParametricPlot[{x[s],y[s]}-(a-bs)d1[s],{s,0,L},PlotStyleGreen],PlotRangeAll,AxesOrigin->{0,0}];(*inflation*)aOld=a;a=aOld;γ=1.45;E0=E0;h1=*h1;h2=*h2;mVal=FindRoot[Stem[m0,E0],{m0,.04}];{x[s_],y[s_],θ[s_]}=StemSol[m0/.mVal,E0];d3=Function[s,{Cos[θ[s]],Sin[θ[s]]}];d1=Function[s,{-Sin[θ[s]],Cos[θ[s]]}];stemAfter=Show[ParametricPlot[{x[L],y[L]}+h1d3[L]+h1Cos[t]d3[L]+h2Sin[t]d1[L],{t,0,2π},PlotStyleGreen],ParametricPlot[{x[s],y[s]},{s,0,L},PlotStyleDashed],ParametricPlot[{x[s],y[s]}+(a-bs)d1[s],{s,0,L},PlotStyleGreen],ParametricPlot[{x[s],y[s]}-(a-bs)d1[s],{s,0,L},PlotStyle->Green],PlotRangeAll,AxesOrigin->{0,0}];Show[stemBefore,stemAfter,PlotRange{{-5,2},{-1.5,6}},AxesOrigin->{0,0},AxesStyleDirective[24],Ticks->{{-4,-2,2},{2,4,6}}]
-3
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-3
10
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(γ(s-(L-ds)))
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σ
.21
.17
500
60
1.65
1.75
.97
1.0
Fluid jet model
Model for initial rotation of the stem during recoil
Seed trajectories
First generation of seeds
First generation of seeds
First we simulate seed trajectories for all fruits around the plant
Selecting some seeds to germinate
Second generation
Second generation
Selecting seeds to germinate
Varying redistribution
Varying redistribution
This code computes seed distribution characteristics for varying degrees of fluid redistribution
Stem fit
Stem fit
This code reproduces the best fit parameter values for the stem before and after fluid redistribution
Note: the images Init.jpg and Inflated.jpg are input into the notebook. This requires having those files in the same directory as this notebook. To obtain these images, they may be saved from below
Note: the images Init.jpg and Inflated.jpg are input into the notebook. This requires having those files in the same directory as this notebook. To obtain these images, they may be saved from below
Init.jpg
Init.jpg
Inflated.jpg
Inflated.jpg
The best fit values
The best fit values
Init
Init
Inflated
Inflated
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Uncovering the mechanical secrets of the squirting cucumber
by Finn Box, Derek E. Moulton, Dominic Vella, Yuvraj Bhagotra, Tristan Lowe, Alain Goriely & Chris J. Thorogood
Wolfram Community, STAFF PICKS, December 5, 2024
https://community.wolfram.com/groups/-/m/t/3334542
by Finn Box, Derek E. Moulton, Dominic Vella, Yuvraj Bhagotra, Tristan Lowe, Alain Goriely & Chris J. Thorogood
Wolfram Community, STAFF PICKS, December 5, 2024
https://community.wolfram.com/groups/-/m/t/3334542