Conformational control of mechanical networks

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.

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.

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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.

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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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The authors declare no competing interests.

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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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