Source-linked AI summary
Designing allostery-inspired response in mechanical networks
Jason W. Rocks, Nidhi Pashine, Irmgard Bischofberger, Carl P. Goodrich, Andrea J. Liu, Sidney R. Nagel
TL;DR
The paper addresses whether disordered spring networks can be engineered to produce targeted allosteric responses rather than only tuned global properties. It introduces an efficient bond-pruning framework and demonstrates that networks can achieve precise single, multiple, and independent source–target responses with minimal pruning. The authors also reproduce these responses in physical two- and three-dimensional systems and identify thermal effects and lower-connectivity regimes as scope boundaries.
Problem
The paper asks whether disordered networks can be tuned to develop specific allosteric structural responses, a relatively unexplored problem distinct from controlling pre-existing allosteric properties.
Method
The authors introduce a computationally efficient formalism that evaluates bond contributions and prunes bonds to tune strain responses between arbitrary source and target node pairs.
Results
|η| ∼1 responses were achieved with almost 100% success, alongside simultaneous independent source/target responses in theoretical and physical networks.
Takeaways & Limitations
The ease of creating targeted allosteric responses suggests a possible explanation for the prevalence of allostery in large biological molecules.
Takeaways & Limitations
The networks are nonthermal, whereas proteins are thermal, so thermal effects on designing desired responses remain to be investigated.
Abstract
from arXiv · showhide
Recent advances in designing meta-materials have demonstrated that global mechanical properties of disordered spring networks can be tuned by selectively modifying only a small subset of bonds. Here, using a computationally-efficient approach, we extend this idea in order to tune more general properties of networks. With nearly complete success, we are able to produce a strain between any pair of target nodes in a network in response to an applied source strain on any other pair of nodes by removing only ~1% of the bonds. We are also able to control multiple pairs of target nodes, each with a different individual response, from a single source, and to tune multiple independent source/target responses simultaneously into a network. We have fabricated physical networks in macroscopic two- and three-dimensional systems that exhibit these responses. This targeted behavior is reminiscent of the long-range coupled conformational changes that often occur during allostery in proteins. The ease with which we create these responses may give insight into why allostery is a common means for the regulation of activity in biological molecules.
INTRODUCTION
The paper asks whether disordered mechanical networks can acquire specific allosteric responses through bond pruning, extending prior control of global properties to targeted local responses. It introduces an efficient framework for programming source–target strain relationships and demonstrates multiple allosteric functions in physical networks.
- Motivation: Disordered networks generally lack allosteric behavior, motivating bond pruning to create specific structural responses between distant node pairs.Allostery involves local binding affecting activity at a distant site and often couples conformational changes between sites.
- Method: The approach calculates each bond’s contribution to network-wide mechanical response under arbitrary source strain, then sequentially removes bonds to reach a target response.The method targets the strain between an arbitrary pair of target nodes in response to strain applied between an arbitrary source pair.
- Scope: The framework extends mechanical-network design from tuning global properties to controlling local responses and multiple source–target relationships.The authors target multiple pairs of targets from one source and independent responses to different local strains.
- Physical realization: Theoretical networks can be converted into macroscopic two- and three-dimensional physical systems that reproduce the designed responses.The demonstrations use laser-cut planar sheets and 3D-printed structures.
- Implication: The central result is that precise allosteric conformational responses can be created with minimal changes to network structure.The authors view this ease of construction as a possible first step toward understanding why allostery is common in biological molecules.
- Biological relevance: The construction of new allosteric functions is presented as relevant to broader efforts to understand and control biological allostery, including potential drug-design applications.The paper distinguishes introducing new allosteric functions from work focused on pre-existing properties.
COMPUTATIONAL RESULTS
The study tests whether bond pruning can program targeted source–target responses in disordered spring networks. Its computational approach achieves precise responses with few removals, including multiplexed and independent controls, while the authors identify lower-connectivity and large-response regimes as boundaries.
- Single-response demonstrations: η = +1 and η = −1 were achieved in one two-dimensional network by removing 6 of 407 bonds, using different six-bond sets.Some removed bonds overlap between the two designs.
- Efficiency and reliability: ~1% of bonds in two dimensions and ~0.5% in three dimensions were removed on average to reach η = ±1.The averages were about 5 of 400 bonds in two dimensions and about 4 of 740 bonds in three dimensions.
- Efficiency and reliability: The failure rate was below 2% in two dimensions and below 1% in three dimensions for strain ratios up to |η| = 1.The algorithm therefore achieved precise control in the vast majority of tested networks.
- Connectivity dependence: At ΔZ = 1.0, pruning narrows into a thin region connecting source and target, whereas more complicated responses generally fail more often at lower ΔZ in two dimensions.At very small ΔZ, few bonds can be removed without compromising rigidity.
- Pruning statistics and localization: Required pruning distributions broaden and shift upward as η increases, but collapse when normalized by the average number of removed bonds ⟨Nr⟩.The resulting target response is localized: other node pairs are essentially unaffected by the source strain.
- Multiple responses: One source can control three targets with distinct strain ratios, while two source–target pairs can be tuned simultaneously to respond independently.For each independent pair, its target responds strongly while the other target does not respond.
EXPERIMENTAL RESULTS
Physical two- and three-dimensional networks reproduced the designed strain responses, with experiments closely matching simulations and retaining responses beyond the linear regime. Quantitative tests included strain ratios and bond-level agreement across realizations.
- η = 1.00 ± 0.01 matches the theoretical prediction for the two-dimensional network’s target-to-source strain response.
- The three-dimensional 3D-printed network was designed with a strain ratio of η = −5.
- The experiments measured strain on every bond to quantitatively compare physical and simulated networks.
- C = 0.98 ± 0.02 across 4 experimental realizations indicates highly accurate agreement between experiments and theoretical models.
- Experimental networks differ from theoretical models through bond-bending forces, possible out-of-plane buckling, and nonlinear finite-strain effects.
- Responses predicted by linear response theory survived into the nonlinear regime in a significant fraction of studied realizations.
DISCUSSION
The approach makes it easy to program localized, long-range mechanical responses by modifying a small subset of bonds, including multiple simultaneous source/target behaviors. These responses are experimentally robust and suggest connections to allostery and other programmable network designs, while important limits and physical effects remain to be characterized.
- Programmable responses: Removing only a small fraction of bonds can tune arbitrary spring networks to produce targeted allosteric deformation responses.The approach performs a discrete optimization of network response and is described as computationally efficient.
- Programmable responses: The strain ratio can be tuned to large positive or negative values, with values of order |η| ∼1 achieved with almost 100% success.The same network can be tuned to have large negative or positive strain ratios.
- Programmable responses: Multiple target pairs can be controlled from the same source, and multiple independent source/target responses can be tuned simultaneously into one network.The authors also report similarly excellent results in periodically continued systems.
- Optimization and generalization: A continuous bond-stiffness optimization is equally successful but less efficient, while offering the advantage of tuning nonlinear behavior and permitting other bond manipulations.The approach can be generalized to introducing new bonds.
- Biological connection: The responses resemble localized, long-range-correlated deformations in proteins and may help explain common allostery and multifunctional behavior.The authors suggest extensions toward identifying protein interactions that could be modified to create new allosteric functions.
- Open problems: Further work is needed to explain why specific bond removals achieve the desired response, although stress-basis analysis indicates that stress states are fundamental to this understanding.The limits on the number of controllable targets and independent responses are not yet known.
- Open problems: The theory’s scope is limited by unresolved dependence on network size and connectivity and by omitted temperature, dynamics, pre-stress, bending, and finite-strain nonlinearities.Protein contact networks include pre-stressed bonds and bond-bending and twisting constraints absent from the theoretical networks.
Computed Networks and Choice of Source and Target Nodes
The networks are constructed computationally and analyzed through bond extensions, tensions, and equilibrium-matrix bases. Bonds are removed sequentially to reduce the difference between actual and desired source–target strain responses while avoiding zero modes.
- Computed networks: Finite networks are generated by cutting bonds crossing a boundary, then removing nodes associated with zero-energy modes until rigidity is restored.Zero modes are identified spectrally from the dynamical matrix, and the node with the largest displacement amplitude is removed for each mode.
- Node selection: Source nodes are placed on one exposed surface and target nodes on the opposing pole, with directly bonded pairs excluded.This avoids surface bonds whose tensions do not couple to the rest of the network.
- Response calculation: The response ratio η = εT/εS is computed from bond extensions and tensions under applied source-bond tensions.In the linear regime, explicitly applying a strain is unnecessary because only the target-to-source strain ratio is required.
- Sequential tuning: The algorithm evaluates each candidate bond removal, selects the bond producing the largest decrease in the cost function, and recomputes the bases after removal.Bonds that would introduce zero modes are excluded.
- Computational method: A computationally efficient basis rotation reduces the matrix inversion to the small subspace associated with modified bonds.The approach avoids the naive O(NbN^3) cost of testing each bond through a full matrix inversion.
- Response calculation: Singular-value decomposition of the equilibrium matrix separates states of self-stress from states of compatible stress, providing complete bond-space bases.The basis dimensions satisfy Nc + Ns = Nb.
Experimental Networks
The designed networks are fabricated as macroscopic two- and three-dimensional physical systems using flexible struts connecting nodes. Fabrication choices reduce out-of-plane buckling, central strut buckling, and bond-bending effects.
- Fabrication: Macroscopic two- and three-dimensional realizations are fabricated from simulated node and strut positions.Two-dimensional networks are laser-cut from silicone rubber sheets, while three-dimensional networks are produced by 3D printing.
- Mechanical design: Struts are narrowed near their ends so they deform primarily near nodes rather than buckling at their centers.This geometry is used to minimize forces due to bond-bending.
- Three-dimensional networks: Three-dimensional networks use a rubber–rigid-plastic mixture with Shore value A85 and struts whose dimensions have an approximately 1:1:11 ratio.The material simulates styrene-based thermoplastic elastomers and ABS.
Ghost bonds
Ghost bonds provide a convenient way to apply tensions and measure extensions between source or target nodes without changing the network’s calculated response. Node pairs connected by uncoupled surface bonds are excluded.
- Uncoupled surface bonds: Surface nodes with exactly d bonds in d dimensions can carry bonds that are uncoupled from the rest of the network.Tensions on such bonds do not communicate tensions or extensions to the rest of the network to linear order.
- Node-pair selection: Source and target pairs are chosen not to share a bond, preventing selection of uncoupled bonds.This restriction avoids pairs whose direct bond would not transmit the intended response through the network.
- Ghost-bond construction: A zero-stiffness ghost bond is introduced between each source or target pair so tensions can be applied and extensions measured conveniently.Because calculations involve bonds, the ghost bonds provide formal source and target bonds without affecting results.
Creating identical bond stiffnesses
The analysis maps networks with heterogeneous bond stiffnesses to an equivalent system with identical default stiffnesses, then handles modified bonds through a reduced compatible-stress basis. This makes repeated response calculations efficient while retaining the relevant stiffness changes.
- Stiffness mapping: Experimental bonds can have non-identical stiffnesses because equal material moduli combine with different equilibrium lengths.For bond i, the stiffness is ki = λi/li; in the experiments λi = λ while li varies.
- Stiffness mapping: A flexibility matrix and scaled equilibrium matrix map heterogeneous stiffnesses to an equivalent system with identical default bond stiffnesses.The transformation uses ¯Q = QF^-1 and cannot scale out zero stiffnesses.
- Modified-bond set: The modified-bond set B contains bonds whose stiffness differs from the default, including ghost bonds and bonds tested for removal.The typical set includes the source and target ghost bonds plus the candidate removal bond.
- Reduced basis: Rotating the compatible-stress basis isolates the modified bonds into a special subset V, while the remaining basis vectors have zero projection onto them.Only the vectors in V are needed for the solution.
- Reduced basis: The full matrix inversion is reduced to inversion of the submatrix ˜K, making calculations fast when B is small.The submatrix is defined over V, and its inverse determines the extension changes caused by stiffness modifications.
Avoiding the introduction of zero modes
The method removes only bonds contributing to the states of self stress, preserving the network’s zero-mode count while updating its bases efficiently.
- Avoiding the introduction of zero modes: Bonds are removed only when they contribute to the SSS sub-basis, so each removal eliminates a unique state of self stress without creating a zero mode.This follows from the Maxwell–Calladine counting relation and the selected-bond construction.
- Avoiding the introduction of zero modes: The unique SSS associated with a removable bond is calculated analogously to the basis construction and then removed from the SSS sub-basis.
- Avoiding the introduction of zero modes: After bond removal, the method subtracts projections, deletes the bond entries from the SSS and SCS bases, and reorthonormalizes both bases.A modified Graham–Schmidt algorithm performs the reorthonormalization.
- Avoiding the introduction of zero modes: O(NbN^3) is the cost of repeatedly solving Eq. (4), whereas the basis-update procedure is reported as significantly faster.The supplied passage identifies repeated equation solving as the slower comparison.
Animations of nonlinear response
The paper evaluates tuned networks beyond their linear design response by minimizing nonlinear configurational energy under progressively larger source strains. At source strains up to 40%, the nonlinear strain ratio remains within a factor of two of the desired linear result.
- Animations of nonlinear response: The supplementary animations show tuned expansion and contraction responses, including networks designed for η = +1 in the linear regime.The animations use oscillatory source strains with amplitude 40% and nonlinear energy minimization.
- Animations of nonlinear response: 40% source strain leaves the resulting nonlinear strain ratio within a factor of two of the desired linear result.
- Animations of nonlinear response: The nonlinear response is calculated from a tuned network by minimizing configurational energy as source strain increases from 0% to 40%.The calculation uses increments of the imposed source strain.
- Animations of nonlinear response: The energy is minimized over node displacements while the source strain εS is held at a specified value.Bond vectors and extensions determine the configurational energy being minimized.
- Animations of nonlinear response: The tuned networks’ responses are compared with untuned responses across source and target pairs, with other pairs essentially unaffected.The target strain ratio is marked by a vertical dashed line, and distributions include contracting and expanding responses.