Programming ultrasensitive threshold response through chemomechanical instability

The ultrasensitive threshold response is ubiquitous in biochemical systems. In contrast, achieving ultrasensitivity in synthetic molecular structures in a controllable way is challenging. Here, we propose a chemomechanical approach inspired by Michell’s instability to realize it. A sudden reconfiguration of topologically constrained rings results when the torsional stress inside reaches a critical value. We use DNA origami to construct molecular rings and then DNA intercalators to induce torsional stress. Michell’s instability is achieved successfully when the critical concentration of intercalators is applied. Both the critical point and sensitivity of this ultrasensitive threshold reconfiguration can be controlled by rationally designing the cross-sectional shape and mechanical properties of DNA rings.


Supplementary Note 1. Double-stranded DNA (dsDNA) ring analysis
To understand the mechanics of Michell's instability, a finite element analysis was performed for the idealized dsDNA ring which consists of 336 base-pairs (BPs) using SNUPI (Structured NUcleic-acids Programming Interface) 1 . The configuration of each BP was abstracted into a node and triad, and the connection of two successive BPs was substituted by Euler-Bernoulli beam finite element. In the model, the dsDNA was assumed to be intrinsically straight and have the regular geometry of B-form DNA (axial rise of 0.34 nm and helicity of 10.5 BP per turn) and mechanical properties (stretching rigidity (S) of 1100 pN, bending rigidity (B) of 230 pN nm 2 , and torsional rigidity (C) of 460 pN nm 2 ). Here, the coupling coefficients were ignored. The bending moment was applied at both ends of a stress-free straight dsDNA to construct the initial configuration of the dsDNA ring structure. After fixing one end, the torsional displacement (θ) was applied at the other end while its other degree of freedom was also fixed. As the θ is increased, the configuration of the dsDNA ring structure was calculated through nonlinear static analysis with only geometrical nonlinearity until self-contact of the structure occurs. In each incremental step, the stretching (πS), bending (πB), and torsional (πC) strain energies of a finite element were calculated as πS = SΔ 2 /2L, πB = Bφ 2 /2L, and πC = Cω 2 /2L, where L, Δ, φ, and ω indicate the axial length, the length change, the bending angle, and the twist angle, respectively. Its total strain energy was obtained by summing these energies.

Supplementary Note 2. Finite element analysis of DNA nanostructures
We carried out a finite element (FE) simulation for DNA nanostructures to predict their equilibrium shapes using SNUPI (Structured Nucleic acids Programming Interface), which would be provided in our recent work 1 . From caDNAno design files, 2 SNUPI obtains information on connectivity and mechanical perturbation among BPs, and constructs FE model, where a BP was abstracted into a node and two successive BPs were connected by a beam finite element. In the FE model, the DNA duplex was assumed to have the regular B-form DNA geometry (diameter of 2.25 nm, axial rise of 0.32 nm and helicity of 10.5 BP per turn) with S of 1100 pN, B of 230 pN nm 2 , C of 460 pN nm 2 and torsional-stretch coupling (g) of -180 pNnm. Note that we did not consider any sequence-dependent properties and assumed a nicked DNA duplex to be the same with a regular DNA duplex in terms of its geometrical and mechanical properties. Crossovers were modeled by a beam finite element that connects two BPs that belong to two adjacent helices. Their mechanical stiffness was defined by multiplying a scale factor (SF) to those of DNA as 1 for S, 0.12 for B, and 0.1 for C, respectively. To simulate the unwinding effect induced by EtBr binding on structural shape, in addition, the equilibrated configuration of the structure was obtained through the nonlinear static analysis with increasing the helicity of dsDNA (BP/turn). In each incremental step, the total strain energy (π) was calculated by summing the strain energy of each finite element as π = Σπ (e) with π = S(Δ (e) ) 2 /2L (e) + B(φ (e) ) 2 /2L (e) + C(ω (e) ) 2 /2L (e) + gΔ (e) ω (e) /2L (e) .

Supplementary Note 3. Calculation of mechanical properties of DNA nanostructures
Normal mode analysis (NMA) was performed at the straight configuration to compute the lowest 40 normal modes of DNA nanostructures. Given the FE model for a DNA nanostructure under free boundary condition, a generalized eigenvalue problem is considered as follows.
( 1) where K is the global stiffness matrix, M is the global mass matrix and λ represents the eigenvalue as the square-root of natural frequency (λ = ! ). Among the eigenvalues obtained, only the two eigenvalues for the first bending modes were selected to calculate bending rigidity (EI).
From the Euler-Lagrange equation for a beam representing effectively DNA nanostructures, we could obtain the following free vibration equation. (2) where w describes the lateral deflection of the beam, x represents the axial position, and µ is the mass per unit length of the beam. Solving analytically the above equation, we can obtain the natural frequencies of the first bending vibration (ω " ) as where m is the total mass of DNA nanostructures, and λ " is the eigenvalues for the first bending mode. Since the L # is defined as where k " is the Boltzmann constant and T is temperature, assumed to be 300 K. Then, L # of DNA nanostructures becomes (6) In a three-dimensional structure, there are always two first bending modes. Therefore, we defined the first bending mode with a smaller eigenvalue as a major bending mode (L #,% ) and the other with a larger eigenvalue as a minor bending mode (L #,! ). Using both bending modes, also, we defined an effective bending persistence length (L #,& ), which we compared with the measured one from AFM image analysis.
Also, introducing the continuum assumption provides the torsional rigidity of the DNA nanostructure from the eigenvalues by NMA. The governing equation was given as where β is the wavenumber. In free-free end boundary condition, the wavenumber satisfies a relation with the lowest mode number as (10, 11) Accordingly, the torsional persistence length (L ' ) was derived as follows. (12) where the ratio of the polar moment of inertia to the area (J ! /A) can be pre-determined by the cross-sectional geometry of the DNA nanostructure.