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Energy partitioning in the cell cortex

Nature Physics (2024)Cite this article

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Abstract

Living systems are driven far from thermodynamic equilibrium through the continuous consumption of ambient energy. In the cell cortex, this energy is invested in the formation of diverse patterns in chemical and mechanical activities, whose spatial and temporal dynamics determine the cell phenotypes and behaviours. How cells partition internal energy between these activities is unknown. Here we measured the entropy production rate of both chemical and mechanical subsystems of the cell cortex across a variety of patterns as the system is driven further from equilibrium. We do this by manipulating the Rho GTPase pathway, which controls the cortical actin filaments and myosin-II. At lower levels of GTPase-activating protein expression, which produce pulses or choppy Rho and actin filament waves, energy is proportionally partitioned between the two subsystems and is subject to the constraint of Onsager reciprocity. Within the range of reciprocity, the entropy production rate is maximized in choppy waves. As the cortex is driven into labyrinthine or spiral travelling waves, reciprocity is broken, marking an increasingly differential partitioning of energy and an uncoupling of chemical and mechanical activities. We further demonstrate that energy partitioning and reciprocity are determined by the competing timescales between chemical reaction and mechanical relaxation.

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Fig. 1: RGA-3/4 expression drives diverse mechanochemical patterns in the cell cortex.
Fig. 2: Energy partitioning between chemical and mechanical subsystems depends on RGA-3/4.
Fig. 3: Mechanics get decoupled from chemical signalling in the highly activated cell cortex.
Fig. 4: Onsager reciprocity implies energy partitioning among subsystems in a fixed proportion in simulation.
Fig. 5: Competing internal timescales determine a repartition of energy.

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Data availability

Data supporting the findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

Code availability

Custom codes that were used for the data analysis in this manuscript are available via GitHub at http://github.com/shengchen-yale/mech-chem-wave.

References

  1. Niebel, B., Leupold, S. & Heinemann, M. An upper limit on Gibbs energy dissipation governs cellular metabolism. Nat. Metab. 1, 125–132 (2019).

    Article  Google Scholar 

  2. Yang, X. et al. Physical bioenergetics: energy fluxes, budgets, and constraints in cells. Proc. Natl Acad. Sci. USA 118, e2026786118 (2021).

    Article  Google Scholar 

  3. Ariga, T., Tomishige, M. & Mizuno, D. Nonequilibrium energetics of molecular motor kinesin. Phys. Rev. Lett. 121, 218101 (2018).

    Article  ADS  Google Scholar 

  4. Toyabe, S., Watanabe-Nakayama, T., Okamoto, T., Kudo, S. & Muneyuki, E. Thermodynamic efficiency and mechanochemical coupling of F1-ATPase. Proc. Natl Acad. Sci. USA 108, 17951–17956 (2011).

    Article  ADS  Google Scholar 

  5. Lan, G., Sartori, P., Neumann, S., Sourjik, V. & Tu, Y. The energy–speed–accuracy trade-off in sensory adaptation. Nat. Phys. 8, 422–428 (2012).

    Article  Google Scholar 

  6. Tennessen, J. M., Baker, K. D., Lam, G., Evans, J. & Thummel, C. S. The Drosophila estrogen-related receptor directs a metabolic switch that supports developmental growth. Cell Metab. 13, 139–148 (2011).

    Article  Google Scholar 

  7. Jarzynski, C. Diverse phenomena, common themes. Nat. Phys. 11, 105–107 (2015).

    Article  Google Scholar 

  8. Ornes, S. How nonequilibrium thermodynamics speaks to the mystery of life. Proc. Natl Acad. Sci. USA 114, 423–424 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  9. Hong, S. et al. Sub-nanowatt microfluidic single-cell calorimetry. Nat. Commun. 11, 2982 (2020).

    Article  ADS  Google Scholar 

  10. Nagano, Y. & Ode, K. L. Temperature-independent energy expenditure in early development of the African clawed frog Xenopus laevis. Phys. Biol. 11, 046008 (2014).

    Article  ADS  Google Scholar 

  11. Auberson, L. C., Kanbier, T. & von Stockar, U. Monitoring synchronized oscillating yeast cultures by calorimetry. J. Biotechnol. 29, 205–215 (1993).

    Article  Google Scholar 

  12. Hur, S., Mittapally, R., Yadlapalli, S., Reddy, P. & Meyhofer, E. Sub-nanowatt resolution direct calorimetry for probing real-time metabolic activity of individual C. elegans worms. Nat. Commun. 11, 2983 (2020).

    Article  ADS  Google Scholar 

  13. Imamura, H. et al. Visualization of ATP levels inside single living cells with fluorescence resonance energy transfer-based genetically encoded indicators. Proc. Natl Acad. Sci. USA 106, 15651–15656 (2009).

    Article  ADS  Google Scholar 

  14. Yang, X., Ha, G. & Needleman, D. J. A coarse-grained NADH redox model enables inference of subcellular metabolic fluxes from fluorescence lifetime imaging. eLife 10, e73808 (2021).

    Article  Google Scholar 

  15. Wiechert, W. 13C metabolic flux analysis. Metab. Eng. 3, 195–206 (2001).

    Article  Google Scholar 

  16. Park, J. S. et al. Mechanical regulation of glycolysis via cytoskeleton architecture. Nature 578, 621–626 (2020).

    Article  ADS  Google Scholar 

  17. Sakamoto, R. & Murrell, M. P. F-actin architecture determines the conversion of chemical energy into mechanical work. Nat. Commun. 15, 3444 (2024).

    Article  ADS  Google Scholar 

  18. Serra-Picamal, X. et al. Mechanical waves during tissue expansion. Nat. Phys. 8, 628–634 (2012).

    Article  Google Scholar 

  19. Banerjee, S., Utuje, K. J. & Marchetti, M. C. Propagating stress waves during epithelial expansion. Phys. Rev. Lett. 114, 228101 (2015).

    Article  ADS  Google Scholar 

  20. Alert, R. & Trepat, X. Physical models of collective cell migration. Annu. Rev. Condens. Matter Phys. 11, 77–101 (2020).

    Article  ADS  Google Scholar 

  21. Flemming, S., Font, F., Alonso, S. & Beta, C. How cortical waves drive fission of motile cells. Proc. Natl Acad. Sci. USA 117, 6330–6338 (2020).

    Article  ADS  Google Scholar 

  22. Boocock, D., Hino, N., Ruzickova, N., Hirashima, T. & Hannezo, E. Theory of mechanochemical patterning and optimal migration in cell monolayers. Nat. Phys. 17, 267–274 (2021).

    Article  Google Scholar 

  23. Munro, E., Nance, J. & Priess, J. R. Cortical flows powered by asymmetrical contraction transport PAR proteins to establish and maintain anterior-posterior polarity in the early C. elegans embryo. Dev. Cell 7, 413–424 (2004).

    Article  Google Scholar 

  24. Solon, J., Kaya-Copur, A., Colombelli, J. & Brunner, D. Pulsed forces timed by a ratchet-like mechanism drive directed tissue movement during dorsal closure. Cell 137, 1331–1342 (2009).

    Article  Google Scholar 

  25. Martin, A. C., Kaschube, M. & Wieschaus, E. F. Pulsed contractions of an actin–myosin network drive apical constriction. Nature 457, 495–499 (2009).

    Article  ADS  Google Scholar 

  26. Munjal, A., Philippe, J.-M., Munro, E. & Lecuit, T. A self-organized biomechanical network drives shape changes during tissue morphogenesis. Nature 524, 351–355 (2015).

    Article  ADS  Google Scholar 

  27. Bailles, A. et al. Genetic induction and mechanochemical propagation of a morphogenetic wave. Nature 572, 467–473 (2019).

    Article  ADS  Google Scholar 

  28. Wigbers, M. C. et al. A hierarchy of protein patterns robustly decodes cell shape information. Nat. Phys. 17, 578–584 (2021).

    Article  Google Scholar 

  29. Raskin, D. M. & De Boer, P. A. Rapid pole-to-pole oscillation of a protein required for directing division to the middle of Escherichia coli. Proc. Natl Acad. Sci. USA 96, 4971–4976 (1999).

    Article  ADS  Google Scholar 

  30. Sedzinski, J. et al. Polar actomyosin contractility destabilizes the position of the cytokinetic furrow. Nature 476, 462–466 (2011).

    Article  ADS  Google Scholar 

  31. Bement, W. M. et al. Activator–inhibitor coupling between Rho signalling and actin assembly makes the cell cortex an excitable medium. Nat. Cell Biol. 17, 1471–1483 (2015).

    Article  Google Scholar 

  32. Ramm, B. et al. A diffusiophoretic mechanism for ATP-driven transport without motor proteins. Nat. Phys. 17, 850–858 (2021).

    Article  Google Scholar 

  33. Murrell, M., Oakes, P. W., Lenz, M. & Gardel, M. L. Forcing cells into shape: the mechanics of actomyosin contractility. Nat. Rev. Mol. Cell Biol. 16, 486–498 (2015).

    Article  Google Scholar 

  34. Ajeti, V. et al. Wound healing coordinates actin architectures to regulate mechanical work. Nat. Phys. 15, 696–705 (2019).

    Article  Google Scholar 

  35. Banerjee, S., Gardel, M. L. & Schwarz, U. S. The actin cytoskeleton as an active adaptive material. Annu. Rev. Condens. Matter Phys. 11, 421–439 (2020).

    Article  ADS  Google Scholar 

  36. Wu, M. & Liu, J. Mechanobiology in cortical waves and oscillations. Curr. Opin. Cell Biol. 68, 45–54 (2021).

    Article  Google Scholar 

  37. Yan, V. T., Narayanan, A., Wiegand, T., Jülicher, F. & Grill, S. W. A condensate dynamic instability orchestrates actomyosin cortex activation. Nature 609, 597–604 (2022).

    Article  ADS  Google Scholar 

  38. Bement, W. M., Goryachev, A. B., Miller, A. L. & von Dassow, G. Patterning of the cell cortex by Rho GTPases. Nat. Rev. Mol. Cell Biol. 25, 290–308 (2024).

    Article  Google Scholar 

  39. Michaud, A. et al. A versatile cortical pattern-forming circuit based on Rho, F-actin, Ect2, and RGA-3/4. J. Cell Biol. 221, e202203017 (2022).

    Article  Google Scholar 

  40. Benink, H. A., Mandato, C. A. & Bement, W. M. Analysis of cortical flow models in vivo. Mol. Biol. Cell 11, 2553–2563 (2000).

    Article  Google Scholar 

  41. Markovich, T., Fodor, É., Tjhung, E. & Cates, M. E. Thermodynamics of active field theories: energetic cost of coupling to reservoirs. Phys. Rev. 11, 021057 (2021).

    Article  Google Scholar 

  42. Seara, D. S., Machta, B. B. & Murrell, M. P. Irreversibility in dynamical phases and transitions. Nat. Commun. 12, 392 (2021).

    Article  ADS  Google Scholar 

  43. Zhang, D., Cao, Y., Ouyang, Q. & Tu, Y. The energy cost and optimal design for synchronization of coupled molecular oscillators. Nat. Phys. 16, 95–100 (2020).

    Article  Google Scholar 

  44. Lynn, C. W., Holmes, C. M., Bialek, W. & Schwab, D. J. Decomposing the local arrow of time in interacting systems. Phys. Rev. Lett. 129, 118101 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  45. Cao, Y., Wang, H., Ouyang, Q. & Tu, Y. The free-energy cost of accurate biochemical oscillations. Nat. Phys. 11, 772–778 (2015).

    Article  Google Scholar 

  46. Oberreiter, L., Seifert, U. & Barato, A. C. Universal minimal cost of coherent biochemical oscillations. Phys. Rev. E 106, 014106 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  47. Marsland, R., Cui, W. & Horowitz, J. M. The thermodynamic uncertainty relation in biochemical oscillations. J. R. Soc. Interface 16, 20190098 (2019).

    Article  Google Scholar 

  48. Onsager, L. Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405 (1931).

    Article  ADS  Google Scholar 

  49. Ohga, N., Ito, S. & Kolchinsky, A. Thermodynamic bound on the asymmetry of cross-correlations. Phys. Rev. Lett. 131, 077101 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  50. Gingrich, T. R., Rotskoff, G. M. & Horowitz, J. M. Inferring dissipation from current fluctuations. J. Phys. A: Math. Theor. 50, 184004 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  51. Qiu, Y., Nguyen, M., Hocky, G. M., Dinner, A. R. & Vaikuntanathan, S. A strong nonequilibrium bound for sorting of cross-linkers on growing biopolymers. Proc. Natl Acad. Sci. USA 118, e2102881118 (2021).

    Article  Google Scholar 

  52. Chun, H.-M., Gao, Q. & Horowitz, J. M. Nonequilibrium Green-Kubo relations for hydrodynamic transport from an equilibrium-like fluctuation-response equality. Phys. Rev. Research 3, 043172 (2021).

    Article  ADS  Google Scholar 

  53. Luo, R., Benenti, G., Casati, G. & Wang, J. Onsager reciprocal relations with broken time-reversal symmetry. Phys. Rev. Res. 2, 022009 (2020).

    Article  Google Scholar 

  54. Wang, X., Dobnikar, J. & Frenkel, D. Numerical test of the Onsager relations in a driven system. Phys. Rev. Lett. 129, 238002 (2022).

    Article  ADS  Google Scholar 

  55. De Corato, M. & Pagonabarraga, I. Onsager reciprocal relations and chemo-mechanical coupling for chemically active colloids. J. Chem. Phys. 157, 084901 (2022).

    Article  ADS  Google Scholar 

  56. Banerjee, D., Souslov, A., Abanov, A. G. & Vitelli, V. Odd viscosity in chiral active fluids. Nat. Commun. 8, 1573 (2017).

    Article  ADS  Google Scholar 

  57. Swider, Z. T. et al. Cell cycle and developmental control of cortical excitability in Xenopus laevis. Mol. Biol. Cell 33, ar73 (2022).

    Article  Google Scholar 

  58. Westendorf, C. et al. Actin cytoskeleton of chemotactic amoebae operates close to the onset of oscillations. Proc. Natl Acad. Sci. USA 110, 3853–3858 (2013).

    Article  ADS  Google Scholar 

  59. Cunnane, S. C. et al. Brain energy rescue: an emerging therapeutic concept for neurodegenerative disorders of ageing. Nat. Rev. Drug Discov. 19, 609–633 (2020).

    Article  Google Scholar 

  60. Kistamas, K. et al. Calcium handling defects and cardiac arrhythmia syndromes. Front. Pharmacol. 11, 72 (2020).

    Article  Google Scholar 

Download references

Acknowledgements

We acknowledge funding through ARO MURI W911NF‐14‐1‐0403 (to M.P.M.), National Institutes of Health RO1 GM126256 (to M.P.M.), NIH U54 CA209992 (to M.P.M.), Human Frontier Science Program (HFSP) grant no. RGY0073/2018 (to M.P.M.), National Institutes of Health grant RO1GM052932 (to W.M.B.) and NSF grant 2132606 (to W.M.B.). Any opinion, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NIH, NSF, ARO or HFSP. We acknowledge helpful discussions with B. Machta, H. Levine, S. Amiri and J. Ke.

Author information

Author notes

  1. These authors contributed equally: William M. Bement, Michael P. Murrell.

Authors and Affiliations

  1. Department of Biomedical Engineering, Yale University, New Haven, CT, USA

    Sheng Chen & Michael P. Murrell

  2. Systems Biology Institute, Yale University, West Haven, CT, USA

    Sheng Chen, Daniel S. Seara & Michael P. Murrell

  3. Department of Physics, Yale University, New Haven, CT, USA

    Daniel S. Seara & Michael P. Murrell

  4. James Franck Institute, University of Chicago, Chicago, IL, USA

    Daniel S. Seara

  5. Cellular and Molecular Biology Graduate Program, University of Wisconsin–Madison, Madison, WI, USA

    Ani Michaud, Songeun Kim & William M. Bement

  6. Center for Quantitative Cell Imaging, University of Wisconsin–Madison, Madison, WI, USA

    Ani Michaud, Songeun Kim & William M. Bement

  7. Department of Integrative Biology, University of Wisconsin–Madison, Madison, WI, USA

    William M. Bement

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Contributions

M.P.M. and W.M.B. designed and conceived the project. W.M.B. designed and conceived the experimental work. M.P.M., S.C. and D.S.S. designed and conceived the analytical, computational and theoretical work. W.M.B., A.M. and S.K. performed the experiments. S.C. and D.S.S. analysed the experimental data. S.C. implemented the model and performed and analysed the simulations. S.C. conducted the theoretical analysis. D.S.S. performed the theoretical analysis of the Stuart–Landau equation. S.C., D.S.S., W.M.B. and M.P.M. wrote the manuscript.

Corresponding authors

Correspondence to William M. Bement or Michael P. Murrell.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–16, Notes I–III, Table 1, captions for videos and references.

Reporting Summary

Supplementary Video 1

Pattern manipulations of Rho/F-actin chemical waves in the activated cell cortex. Prototypical examples of different wave phenotypes with [RGA-3/4] = 0, 33, 333 and 1,000 ng µl–1 are shown in sequence. From left to right, the panels show Rho waves (left), F-actin waves (middle) and a merged image with Rho and F-actin shown in cyan and red (right), respectively. The video corresponds to the data shown in Fig. 1c.

Supplementary Video 2

Mechanical dynamics of the activated cell cortex. The panel on the left shows an example of the F-actin wave in the activated cortex of the Xenopus egg. The arrows in the right panel show the corresponding deformation rates of the cortex interpolated from tracking pigment granules’ velocities.

Supplementary Video 3

Pattern manipulations of Rho/F-actin chemical waves in the simulation. Prototypical examples of different wave phenotypes with the RGA-3/4-regulated Rho-GTP hydrolysis rate kRGA = 0, 0.3 and 0.6 (s–1) are shown in sequence. From left to right, panels show Rho waves (left), F-actin waves (middle) and a merged image with Rho and F-actin shown in cyan and red (right), respectively. The video corresponds to the data shown in Supplementary Fig. 1.

Supplementary Video 4

Experimental perturbations using the F-actin-severing protein, cofilin. Normal oocyte cells (as control) and cells with the overexpression of cofilin were microinjected with 200 ng µl–1 Ect2 (the standard dose) and 166 ng µl–1 RGA-3/4. The characteristic wave phenotypes of the control and cofilin experiments are shown in sequence. From left to right, panels show Rho waves (left), F-actin waves (middle) and a merged image with Rho and F-actin shown in cyan and red (right), respectively. The video corresponds to the data shown in Fig. 5.

Source data

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Statistical source data.

Source Data Fig. 2

Statistical source data.

Source Data Fig. 3

Phase portrait source data.

Source Data Fig. 4

Statistical source data.

Source Data Fig. 5

Statistical source data.

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Chen, S., Seara, D.S., Michaud, A. et al. Energy partitioning in the cell cortex. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02626-6

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