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Conformational control of mechanical networks

Nature Physics volume 15, pages 714–720 (2019)Cite this article

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Abstract

Understanding conformational change is crucial for programming and controlling the function of many mechanobiological and mechanical systems such as robots, enzymes and tunable metamaterials. These systems are often modelled as constituent nodes (for example, joints or amino acids) whose motion is restricted by edges (for example, limbs or bonds) to yield functionally useful coordinated motions (for example, walking or allosteric regulation). However, the design of desired functions is made difficult by the complex dependence of these coordinated motions on the connectivity of edges. Here, we develop simple mathematical principles to design mechanical systems that achieve any desired infinitesimal or finite coordinated motion. We specifically study mechanical networks of two- and three-dimensional frames composed of nodes connected by freely rotating rods and springs. We first develop simple principles that govern all networks with an arbitrarily specified motion as the sole zero-energy mode. We then extend these principles to characterize networks that yield multiple specified zero modes, generate pre-stress stability and display branched motions. By coupling individual modules, we design networks with negative Poisson’s ratio and allosteric response. Finally, we extend our framework to networks with arbitrarily specifiable initial and final positions to design energy minima at desired geometric configurations, and create networks demonstrating tristability and cooperativity.

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Fig. 1: Graphical representations of Maxwell frames.
Fig. 2: Solution space of unspecified nodes is determined by the specified nodes.
Fig. 3: Construction and control of frames with specified outward motion.
Fig. 4: Intersections of solution spaces for multiple non-rigid motions.
Fig. 5: Combining network motions by merging nodes and adding edges.
Fig. 6: Designing finite motions and bistable networks with cooperativity.

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Data availability

There are no data with mandated deposition used in the manuscript or Supplementary Information. All analyses and figures were created in MATLAB, and can be publicly accessed on GitHub at https://github.com/jk6294/Mechanical-Networks.git with a test script that will exactly replicate most of the key figures and results in this manuscript.

References

  1. Papadopoulos, L., Porter, M. A., Daniels, K. E. & Bassett, D. S. Network analysis of particles and grains. J. Complex Netw. 6, 485–565 (2018).

    Article  MathSciNet  Google Scholar 

  2. Picu, R. C. Mechanics of random fiber networks—a review. Soft Matter 7, 6768–6785 (2011).

    Article  ADS  Google Scholar 

  3. Vermeulen, M. F. J., Bose, A., Storm, C. & Ellenbroek, W. G. Geometry and the onset of rigidity in a disordered network. Phys. Rev. E 96, 053003 (2017).

    Article  ADS  Google Scholar 

  4. Bassett, D. S., Owens, E. T., Daniels, K. E. & Porter, M. A. Influence of network topology on sound propagation in granular materials. Phys. Rev. E 86, 041306 (2012).

    Article  ADS  Google Scholar 

  5. Shi, F., Wang, S., Forest, M. G. & Mucha, P. J. Network-based assessments of percolation-induced current distributions in sheared rod macromolecular dispersions. Multiscale Model. Simul. 12, 249–264 (2014).

    Article  MathSciNet  Google Scholar 

  6. Detweiler, C., Vona, M., Yoon, Y., Yun, Seung-Kook & Rus, D. Self-assembling mobile linkages. IEEE Robotics Autom. Mag. 14, 45–55 (2007).

    Article  Google Scholar 

  7. Patek, S. N., Nowroozi, B. N., Baio, J. E., Caldwell, R. L. & Summers, A. P. Linkage mechanics and power amplification of the mantis shrimp’s strike. J. Exp. Biol. 210, 3677–3688 (2007).

    Article  Google Scholar 

  8. Crapo, H. Structural rigidity. Struct. Topol. 73, 26–45 (1979).

    MathSciNet  MATH  Google Scholar 

  9. Maxwell, J. C. On the calculation of the equilibrium and stiffness of frames. Phil. Maga. Ser. 4 27, 294–299 (1864).

    Article  Google Scholar 

  10. Grimm, H. & Dorner, B. On the mechanism of the α-β phase transformation of quartz. J. Phys. Chem. Solids 36, 407–413 (1975).

    Article  ADS  Google Scholar 

  11. Bertoldi, K., Vitelli, V., Christensen, J. & van Hecke, M. Flexible mechanical metamaterials. Nat. Rev. Mater. 2, 17066 (2017).

    Article  ADS  Google Scholar 

  12. Guo, J. & Zhou, H. X. Protein allostery and conformational dynamics. Chem. Rev. 116, 6503–6515 (2016).

    Article  Google Scholar 

  13. Kempe, A. B. On a general method of describing plane curves of the nth degree by linkwork. Proc. Lond. Math. Soc. s1–7, 213–216 (1875).

    Article  MathSciNet  Google Scholar 

  14. Goodrich, C. P., Liu, A. J. & Nagel, S. R. The principle of independent bond-level response: tuning by pruning to exploit disorder for global behavior. Phys. Rev. Lett. 114, 225501 (2015).

    Article  ADS  Google Scholar 

  15. Bolker, E. & Roth, B. When is a bipartite graph a rigid framework? Pac. J. Math. 90, 27–44 (1980).

    Article  MathSciNet  Google Scholar 

  16. Guest, S. The stiffness of prestressed frameworks: a unifying approach. Int. J. Solids Struct. 43, 842–854 (2006).

    Article  Google Scholar 

  17. Asimow, L. & Roth, B. The rigidity of graphs. Trans. Am. Math. Soc. 245, 279–289 (1978).

    Article  MathSciNet  Google Scholar 

  18. Changeux, J.-P. & Edelstein, S. J. Allosteric receptors after 30 years. Neuron 21, 959–980 (1998).

    Article  Google Scholar 

  19. Allewell, N. M. Escherichia coli aspartate transcarbamoylase: structure, energetics, and catalytic and regulatory mechanisms. Annu. Rev. Biophys. Biophys. Chem. 18, 71–92 (1989).

    Article  Google Scholar 

  20. Macol, C. P., Tsuruta, H., Stec, B. & Kantrowitz, E. R. Direct structural evidence for a concerted allosteric transition in Escherichia coli aspartate transcarbamoylase. Nat. Struct. Biol. 8, 423–426 (2001).

    Article  Google Scholar 

  21. Cockrell, G. M. et al. New paradigm for allosteric regulation of Escherichia coli aspartate transcarbamoylase. Biochemistry 52, 8036–8047 (2013).

    Article  Google Scholar 

  22. Rocks, J. W. et al. Designing allostery-inspired response in mechanical networks. Proc. Natl Acad. Sci. USA 114, 2520–2525 (2017).

    Article  Google Scholar 

  23. Whiteley, W. Infinitesimal motions of a bipartite framework. Pac. J. Math. 110, 233–255 (1984).

    Article  MathSciNet  Google Scholar 

  24. Silverberg, J. L. et al. Using origami design principles to fold reprogrammable mechanical metamaterials. Science 345, 647–650 (2014).

    Article  ADS  Google Scholar 

  25. Paulose, J., Chen, B. G.-g & Vitelli, V. Topological modes bound to dislocations in mechanical metamaterials. Nat. Phys. 11, 153–156 (2015).

    Article  Google Scholar 

  26. Körner, C. & Liebold-Ribeiro, Y. A systematic approach to identify cellular auxetic materials. Smart Mater. Struct. 24, 025013 (2015).

    Article  ADS  Google Scholar 

  27. Lukin, J. A. & Ho, C. The structure function relationship of hemoglobin in solution at atomic resolution. Chem. Rev. 104, 1219–1230 (2004).

    Article  Google Scholar 

  28. Lee, J.-H., Singer, J. P. & Thomas, E. L. Micro-/nanostructured mechanical metamaterials. Adv. Mater. 24, 4782–4810 (2012).

    Article  Google Scholar 

  29. Dagdelen, J., Montoya, J., de Jong, M. & Persson, K. Computational prediction of new auxetic materials. Nat. Commun. 8, 323 (2017).

    Article  ADS  Google Scholar 

  30. Yu, Z. & Qian, W.-H. Dynamic force distribution in multifingered grasping by decomposition and positive combination. IEEE Trans. Robotics 21, 718–726 (2005).

    Article  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge conversations with B. Chen, A. E. Sizemore, E. J. Cornblath, E. Teich and M. X. Lim. J.Z.K. acknowledges support from the NIH T32-EB020087, PD: F. W. Wehrli, and the National Science Foundation Graduate Research Fellowship no. DGE-1321851. S.H.S. acknowledges support from the United States NSF grant nos. DMS-1513179 and CCF-1522054. D.S.B. acknowledges support from the John D. and Catherine T. MacArthur Foundation, the ISI Foundation, the Alfred P. Sloan Foundation, an NSF CAREER award PHY-1554488, and from the NSF through the University of Pennsylvania Materials Research Science and Engineering Center (MRSEC) DMR-1720530.

Author information

Authors and Affiliations

  1. Department of Bioengineering, University of Pennsylvania, Philadelphia, PA, USA

    Jason Z. Kim, Zhixin Lu & Danielle S. Bassett

  2. Center for Applied Mathematics, Cornell University, Ithaca, NY, USA

    Steven H. Strogatz

  3. Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, PA, USA

    Danielle S. Bassett

  4. Department of Electrical & Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA

    Danielle S. Bassett

  5. Department of Neurology, University of Pennsylvania, Philadelphia, PA, USA

    Danielle S. Bassett

  6. Department of Psychiatry, University of Pennsylvania, Philadelphia, PA, USA

    Danielle S. Bassett

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  1. Jason Z. Kim
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Contributions

J.Z.K. and D.S.B. wrote the manuscript, with feedback from Z.L. and S.H.S. J.Z.K. conceived the idea, formalized the math, and performed the simulations and analyses with feedback from Z.L., S.H.S. and D.S.B.

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Correspondence to Danielle S. Bassett.

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Kim, J.Z., Lu, Z., Strogatz, S.H. et al. Conformational control of mechanical networks. Nat. Phys. 15, 714–720 (2019). https://doi.org/10.1038/s41567-019-0475-y

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