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The cusp of an apple

Abstract

Singularities are common in diverse physical systems1 and lead to universal structures2,3. This universality suggests that they should also naturally arise in biological systems, where active growth, autonomous motion, kinesis and taxis focus deformations in spacetime, as exemplified in the morphogenetic processes determining biological size and shape4. A familiar example of a morphogenetic singularity is seen in the humble apple, which forms in the neighbourhood of the stalk as the apple grows. Here we study the geometry and morphogenesis of the cusp of an apple by combining observations of fruit growth with a simple theory, finite element simulations and controlled swelling experiments using a physical gel simulacrum. Our observations show that the axisymmetric cusp develops into a self-similar form, which can be understood in terms of a mechanical theory for the inhomogeneous growth of a soft sphere. Physical experiments using local inhibition in swelling gels corroborate our theoretical predictions. These experiments further show that axisymmetric cusps can lose stability and become lobed. We use simulations to show that the number of cuspidal lobes depends on the ratio of the size of the stalk to the size of the sphere, as well as the amplitude and periodicity of perturbations that mimic the role of fruit anatomy, consistent with observations of multi-cusped fruits.

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Fig. 1: Experimental observations of the growth and form of the cusp of an apple.
Fig. 2: Numerical simulations of a growing solid and physical experiments with swelling gels recapitulate axisymmetric cusp formation.
Fig. 3: Instability of an axisymmetric cusp can lead to multi-cusped forms.

Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

Source codes to plot the graphs are provided with this paper.

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Acknowledgements

We acknowledge support from the Swiss National Science Foundation (T.C.T.M.); Peterhouse, Cambridge (T.C.T.M.); the China Scholarship Council (file no. 201906210341); NSF DMR-2011754 (L.M.); NSF DMR-1922321 (L.M.); NSF DMS 1764269 (L.M.); and the Henri Seydoux Fund (L.M.).

Author information

Authors and Affiliations

Authors

Contributions

L.M. conceived of the problem and designed the study and approaches. A.C. developed and performed the swelling experiments of the physical gel models. T.C.T.M. and L.M. developed the theoretical models. T.C.T.M. performed the apple cross-section measurements. S.Y. performed the finite element simulations of growth. E.S. numerically solved the eikonal equation. A.C., T.C.T.M. and L.M. wrote the paper. L.M. supervised the study.

Corresponding author

Correspondence to L. Mahadevan.

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Competing interests

The authors declare no competing interests.

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Peer review informationNature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Raw and collapsed cusp profiles for apples, simulations and physical gel models.

(A) In vivo: Experimental measurements of profiles near the cusp of apple cross-sections at different stages of growth. 2 pickings of the apples were made in June 2018. At the time of picking, the fruits were cut through the middle at the stalk, (0,0), and images of the cross section were taken. The apple profile was traced from the images of 100 apples at different stages of growth. The colors indicate time point (t) during growth, where t represents the normalized time scale for cusp growth in apples. The inset shows an apple cusp at t = 1. (B) In silico: Simulation profiles as a function of time. The inset shows a cusp at t = 0.5. (C) In vitro: Gel profiles of the cusp obtained from laser sheet experiments as a function of time, going from 0 min to 130 min, where t represents the normalized time with respect to the total time of the experiment, that is 130 min. The inset shows a cusp at t = 0.8.

Source data

Extended Data Fig. 2 Description of FEM model and growth profiles.

(A) Three dimensional model illustration for the FEM simulations. (B) For the cusp formation in 2D FEM simulations, the growth profile along radius, f(R), is assumed in the form of a sigmoid function. (Right): Growth distribution for single cusp formation in asymmetric geometry for FEM simulations. The swelling and non-swelling zone are fixed throughout the process. To model the isotropic inhomogeneous growth we use the following profile: G = g(x, y, t)I, where the growth rate is given by g(x, y, t) = 1 + f(R)h(d)t, with \(f(R)={\left\{1+\exp \left[-{k}_{R}(R-{R}_{{{{\rm{cr}}}}})/{R}_{0}\right]\right\}}^{-1}\), and \(h(d)={\left\{1+\exp \left[-{k}_{d}(d-{d}_{{{{\rm{cr}}}}})/{d}_{0}\right]\right\}}^{-1}\). Here \(R=\sqrt{{x}^{2}+{({R}_{0}+y)}^{2}}\) and \(d=\sqrt{{x}^{2}/d_{a}^{2}+{y}^{2}/d_{b}^{2}}\). This growth profile allows g to vary smoothly from g = 1 in the core and the neighborhood of the stalk to g > 1 in the growing cortex. (C) The growth profile, f(R), moves inwards as a function of time in 2D simulations for the formation of a single cusp at times t1 − t3. (Right): Growth profile for single cusp formation with a moving swelling zone for each time interval t1 − t3. (D) For the cusp formation in 3D FEM simulations, the growth profile, f(R), moves inwards as a function of time at time points t1 − t4. (Right): Growth profile used in 3D FEM simulations (\({D}_{{{{\rm{stalk}}}}}/{D}_{{{{\rm{sphere}}}}}=0.3\)) for each corresponding time intervals.

Source data

Extended Data Fig. 3 Stress distribution in FEM simulations and gel models for multicusp formation.

(A) Plan view and cross-section from FEM simulations with n = 8 cusps showing temporal evolution of the compressive stresses around the central stalk region, which are responsible for the formation of the multiple cusps. Respective color legends shows the magnitude and sign of the Hoop stress. (B) Two examples of gel models, one with a small non-swelling stalk (left, \({D}_{{{{\rm{stalk}}}}}/{D}_{{{{\rm{sphere}}}}}=0.05\)) and a large stalk (right, \({D}_{{{{\rm{stalk}}}}}/{D}_{{{{\rm{sphere}}}}}=0.5\)) denoted by arrows. At t = 0, both these gel models were cut through their mid-plane to show their cross-section before swelling. Both the halves remain flat after cutting showing no pre-stresses in the samples. A replicate of the model for each case is swelled in hexane until it develops the cusp. At t = 1 (75 min of swelling reaching the final cusped state), when the swollen gel spheres are cut into halves, each of these fold inwards demonstrating the presence of compressive hoop stresses in the outer region. These hoop stresses are responsible for formation of multiple cusps when \({D}_{{{{\rm{stalk}}}}}/{D}_{{{{\rm{sphere}}}}}\) is large. All scale bars denote 2 mm. The white dashed lines are guide to the eyes at locations where the cusps are observed before longitudinally slicing them.

Supplementary information

Supplementary Information

Supplementary text and Figs. 1–5.

Reporting Summary

Source data

Source Data Fig. 1

Statistical source data: individual Excel files for each time point t for apple profiles (x,y) and a MATLAB code to plot Fig. 1d (collapsed universal self-similar curve).

Source Data Fig. 2

Statistical source data for Fig. 2b: individual .rpt files (open with .txt) for each time point t for simulations (x,y) and a MATLAB code to plot Fig. 2b (collapsed universal self-similar curve). Statistical source data for Fig. 2d: individual Excel files (open with .txt) for each time point t for gel swelling (x,y) and a MATLAB code to plot Fig. 2d (collapsed universal self-similar curve).

Source Data Fig. 3

Statistical source data: Excel files for the data points for Fig. 3a,c,e.

Source Data Extended Data Fig. 1

Statistical source data: each folder contains the individual Excel files for each time point t for apple profiles (Extended Data Fig. 1a), simulations (Extended Data Fig. 1b) and gel swelling (Extended Data Fig. 1c).

Source Data Extended Data Fig. 2

Codes for generating the growth front graphs in Extended Data Fig. 2c,d.

Source Data Supplementary Fig. 2

Statistical source data: data for the eikonal equation.

Source Data Supplementary Fig. 3

Statistical source data: data from Fig. 8 in ref. 7 and the re-analysed data.

Source Data Supplementary Fig. 4

Statistical source data: data and code for generating the plots in Supplementary Fig. 4b.

Source Data Supplementary Fig. 5

Statistical source data.

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Chakrabarti, A., Michaels, T.C.T., Yin, S. et al. The cusp of an apple. Nat. Phys. 17, 1125–1129 (2021). https://doi.org/10.1038/s41567-021-01335-8

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