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Architecture and Co-Evolution of Allosteric Materials

Le Yan, Riccardo Ravasio, Carolina Brito, Matthieu Wyart

arXiv:1609.03951v1physics.bio-phcond-mat.mtrl-scicond-mat.soft

TL;DR

Specific mechanical responses at a distance are difficult to explain because ordinary elastic networks respond non-specifically and their signals decay rapidly. This paper evolves elastic networks for an allosteric task and finds a trumpet-like architecture whose soft edge modes and evolution-generated correlations reveal aspects of its design.

  • Problem

    In ordinary networks, responses are generally non-specific and decay rapidly, while the physical mechanisms of protein allostery over distance remain elusive.

    Source: p. 1

  • Method

    The authors evolve elastic networks under a fitness function requiring a stimulus at one site to produce a specified displacement at a distant active site.

    Source: p. 1, Description of the evolution model · p. 2, Numerical solutions · p. 3

  • Results

    Allosteric networks use soft edge modes near the active site, and correlations generated during evolution select links belonging to the functional trumpet.

    Source: p. 1, Physical processes underlying allostery · p. 4, Conservation and co-evolution · p. 5

  • Takeaways & Limitations

    The scheme can characterize solution architectures and correlations, and the results support using sequence correlations to reveal compact regions central to function.

    Source: Conclusion · p. 5

    The model omits protein-chain folding and stability constraints, and its asymmetric structures have about four times larger thermally induced motion near the active site than at other surface nodes.

    Source: Conclusion · p. 5

Abstract

from arXiv · show

We introduce a numerical scheme to evolve functional materials that can accomplish a specified mechanical task. In this scheme, the number of solutions, their spatial architectures and the correlations among them can be computed. As an example, we consider an "allosteric" task, which requires the material to respond specifically to a stimulus at a distant active site. We find that functioning materials evolve a less-constrained trumpet-shaped region connecting the stimulus and active sites and that the amplitude of the elastic response varies non-monotonically along the trumpet. As previously shown for some proteins, we find that correlations appearing during evolution alone are sufficient to identify key aspects of this design. Finally, we show that the success of this architecture stems from the emergence of soft edge modes recently found to appear near the surface of marginally connected materials. Overall, our in silico evolution experiment offers a new window to study the relationship between structure, function, and correlations emerging during evolution.

f and N ′

The constraint-counting algorithm builds a compact region by iteratively updating a subnetwork, with its result depending on how external connections are counted.

f and N ′

  • The algorithm updates the subnetwork by adding the nearest node with minimal N′_c, then checks nodes whose neighbors are all already included.It repeats the inclusion procedure until its stopping step.Source: f and N ′ · p. 6
  • The identified compact region depends on q, the parameter that counts external connections.Source: f and N ′ · p. 6
  • The reported Fig. 4.C result uses q = 0.5; q = 1 corresponds to the pebble game.The authors describe q = 0.5 as appropriate in the SI.Source: f and N ′ · p. 6

A. Distortion of triangular lattice

The triangular lattice is distorted to remove straight lines that can create unphysical localized floppy modes, without introducing frozen disorder.

A. Distortion of triangular lattice

B. “Trumpet” structure in floppy networks

In floppy networks, evolution forms a trumpet-shaped structure connecting the allosteric and active sites below the transition temperature.

B. “Trumpet” structure in floppy networks

C. Thermal noise

The thermal-noise analysis uses B-factors to characterize motion and finds enhanced motion in the trumpet, especially near the active site.

C. Thermal noise

  • The B-factor quantifies thermally induced motion and is related to the stiffness matrix.The calculation uses temperature T = 1.Source: C. Thermal noise · p. 8
  • Fig. S3 maps B-factor spatially for Te = 0.30 and Te = 0.05, with a logarithmic color scale and z = 5.0.Source: Figure S3 · p. 8
  • B-factors in the trumpet near the active site are about 20 times larger than at other system-boundary nodes, corresponding to about four times larger motion amplitude.The comparison is reported for configurations at high and low evolution temperature.Source: C. Thermal noise · p. 8

D. Counting of the external constraints

The procedure for detecting the nearly isostatic region near target nodes depends on how external constraints are counted. For the mosaic network, counting each external constraint as half a constraint works better numerically than counting it as zero or one.

D. Counting of the external constraints

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