Triggered contraction of self-assembled micron-scale DNA nanotube rings

Contractile rings are formed from cytoskeletal filaments during cell division. Ring formation is induced by specific crosslinkers, while contraction is typically associated with motor protein activity. Here, we engineer DNA nanotubes and peptide-functionalized starPEG constructs as synthetic crosslinkers to mimic this process. The crosslinker induces bundling of ten to hundred DNA nanotubes into closed micron-scale rings in a one-pot self-assembly process yielding several thousand rings per microliter. Molecular dynamics simulations reproduce the detailed architectural properties of the DNA rings observed in electron microscopy. Theory and simulations predict DNA ring contraction – without motor proteins – providing mechanistic insights into the parameter space relevant for efficient nanotube sliding. In agreement between simulation and experiment, we obtain ring contraction to less than half of the initial ring diameter. DNA-based contractile rings hold promise for an artificial division machinery or contractile muscle-like materials.

We present additional details of the theory of bundle contraction.
We consider a toroidal DNA nanotube bundle with radius D and a roughly circular cross section containing N DNA nanotubes resulting in a total length L tot = πN D. The total length is fixed as we assume a fixed number of DNA nanotube monomers throughout contraction.This length can change by depolymerization or polymerization processes of DNA nanotubes.The bundle has a bending rigidity κ b (N ), which is related to the bending energy κ of individual DNA nanotubes via κ b (N ) = κN α , with α = 1 for decoupled sliding DNA nanotubes and α = 2 if crosslinking resists shear [4].We assume an adhesion energy g per length for the DNA nanotubes inside the bundle.For roughly circular cross sections, there should be N s = a s N 1/2 out of N DNA nanotubes at the bundle surface with a geometric factor a s of order unity (in principle, it is possible to consider a wider class of bundle cross section geometries along the lines of Schnurr et al. [10]).This results in an effective DNA nanotube length at the interior of the DNA nanotube bundle, which is the length that is fully accessible to the attraction of strength g.Contraction of toroidal bundles can then be described by the total free energy which is the sum of bending energy, adhesion energy at the interior and the entropic free energy F c of the crosslinker gas at the interior of the bundle, which will arise if crosslinkers are mobile.
We neglect the entropy of DNA nanotube shape fluctuations because we focus on the bundled state; this entropy is relevant, however, for the process of bundle formation [5].
In the bundled state, the entropy of the quasi-one-dimensional gas of N c crosslinkers distributed over the accessible bundle length L i becomes relevant.We approximate F c by the (entropic) free energy of a Tonks gas of N c non-overlapping particles of crosslinker size b c in a one-dimensional volume of length L i with a line density of 1/b c of possible binding sites, Minimizing F with respect to D gives the equilibrium diameter D of the toroidal DNA nanotube bundle, We minimize at fixed L tot , i.e., using N (D) = L tot /πD; the effective interior bundle length to use in minimization becomes The bending energy will favor large D Adhesion energy and crosslinker entropy favor ring contraction via increasing the overlapping interior length L i of the bundle.The crosslinker gas increases L i by exerting an ideal gas pressure onto the boundaries of the overlapping region, Neglecting prefactors of order unity, the minimization condition (5) leads to the equilibrium diameter (Eq.( 3) in the main text) where

Supplementary Note 2: MD simulations Equilibration
We perform coarse-grained Molecular Dynamics (MD) simulations using LAMMPS [11] with a Langevin dynamics thermostat.DNA nanotubes are represented as bead-spring polymers with bending rigidity.
Beads interact by an attractive Lennard-Jones (LJ) interactions with a length scale σ, which represents both the range of attraction and the size of the hard core.
Equilibration is impeded by the tendency of neighboring bead-spring polymers in a bundle to "lock-in" by attraction to two adjacent beads on neighboring polymers, see Supplementary Fig. 12.
Supplementary Fig. 12: Sketch of a bundle of two mutually attractive bead-spring polymers.The transparent spheres represent the range of the LJ attraction.The beads assume a configuration where each bead has two attractive overlaps with beads from the neighboring polymer.This leads to a "locked-in" configuration, where sliding motion of beads requires activation over a barrier, where a bead has only one attractive interaction.
Bead motion along the neighboring polymer then requires activation over energy barriers, which impedes polymer sliding in simulations as the essential process for the toroidal ring to find its equilibrium diameter.
Therefore, we employ an annealing protocol to facilitate faster equilibration, where we increase the temperature to T 2 ∼ 2-8 T 1 for short time intervals for a simulation at temperature T 1 (measurements are always performed after switching back to temperature T 1 and after a certain waiting time at T 1 ).This allows the bundle to "loosen" at T 2 (because ε/k B T 2 is reduced) such that sliding becomes possible.The value of T 2 is adapted depending on the simulated potential strength ε to prevent complete unbundling of the ring structure during the interval at T 2 .For small potentials 0.2 < ε ≤ 0.5 k B T , an annealing temperature T 2 = 2T 1 is sufficient to equilibrate the rings within achievable simulation times.The annealing temperature is gradually increased with the potential up to T 2 = 8T 1 for ε = 2.0 k B T because annealing at T 2 = 2T 1 turns out to be insufficient to "loosen" the bundle effectively for these larger values of ε.For very small ε/k B T 1 ≤ 0.2, we do not use simulated annealing.
We can test equilibration by starting from different initial ring diameters for the same system parameters and monitoring how long the MD simulation requires to contract rings to their common equilibrium diameter, see Supplementary Fig. 13.Equilibration in the MD simulation at a fixed temperature T 1 is fairly slow for all interaction strengths ε/k B T 1 ≳ 0.5 well above the bundling threshold.The comparison of the different simulations in Supplementary Fig. 13 demonstrates a much more efficient equilibration with simulated annealing.For ε/k B T 1 = 1 equilibration of DNA nanotube rings with parameters typical for experiments is practically arrested without annealing.For small ε/k B T 1 ≲ 0.2, however, simulated annealing should not be employed as it results in an artificial widening of the ring.
We equilibrate rings over ∼ 10 9 MD steps and measure ring diameters in a fixed time interval between 8-10 × 10 8 MD time steps.lowest curves in the Supplementary Fig. 14b show the resulting bundle contraction during equilibration in this regime.
Extended simulation runs up to 2 × 10 9 MD simulation steps actually show that the lowest three simulations are not fully equilibrated and the ring diameter is still decreasing at the end of the simulations.This explains the relatively large deviation of the first three light yellow crosses in Fig. 5c in the main text from the scaling law (the time interval for ring diameter measurements is shifted to 18 − 20 × 10 8 ∆t for these simulations).The annealing procedure is less effective for strong attraction and small bending rigidity because small bending rigidities reduce the bending moments expanding the ring after annealing has effectively weakened the attraction by thermal shape fluctuations.This considerably slows down "loosening" of the bundle during annealing.We argue that this will correspond to equilibration times if we could simulate with the high viscosity of water.For high viscosities such that τ γ (η) < τ LJ , the simulation is overdamped already on the Lennard-Jones time scale such that inertia can be neglected, and we essentially have a Brownian dynamics.For the viscosity of water, we have τ γ (η Water ) = 10 −3 τ LJ and are deep in the overdamped limit (as is typical for all biological systems on the micron scale).Within the overdamped Brownian regime, the equilibration time t scales linearly with viscosity t ∼ η such that t Water ∼ (η Water /η)t because all motion is slowed down proportional to friction.This can be shown strictly by non-dimensionalizing time in the Brownian equation of motion in units of τ γ , which completely removes viscosity from the equation of motion.For lower viscosities such that τ γ (η) > τ LJ , on the other hand, the simulation is underdamped and the equilibration time t becomes essentially independent of viscosity.Combining the findings for overdamped and underdamped regimes leads to the above overall result (11) for the equilibration time at different viscosities.To further support this argument we performed MD simulations of a faster process than ring contraction, namely the pulled sliding of only two bundled polymers with a fixed external pulling force F , see Fig. 15a, where we consider the desorption time t des to separate both polymers.This process is faster, such we can explore a wider range of viscosities in MD simulations.According to our arguments leading to Eq. ( 11), the total desorption time t des should scale linearly with viscosity in the overdamped regime τ γ (η) < τ LJ and become essentially independent of viscosity in the underdamped regime τ γ (η) > τ LJ , Extrapolating desorption times from a lower viscosity η to the high viscosity of water with τ γ (η Water ) < τ LJ gives again in agreement with Eq. (11).

Ring formation
In a different set of simulations we aim to follow the process of ring formation starting from a solution of polymers.To observe the process of ring formation on computationally accessible time scales, we employ a reduced bending stiffness because the bending energy enters the time scale for ring closure in an Arrhenius-type exponential dependence.We use κ = 200k B T σ as compared to the realistic stiffness of κ = 600k B T σ for DNA nanotubes.Fig. 4c in the main text shows a sequence of snapshots of a ring formation simulation.We find that ring formation in simulations is optimal for an initial small bundle containing few DNA nanotubes such that its total contour length is ∼ 3L p .For all MD simulations of ring contraction (Fig. 5 in the main text), we enhance initial ring formation by initially positioning DNA nanotubes in a ring-like structure ´and employ realistic bending stiffnesses.

Measuring ring diameters
In order to calculate the ring diameters of DNA nanotube bundles in simulations we compute the 3x3 gyration tensor by averaging over all bead positions ⃗ r (i) .The tensor is symmetric and has three eigenvalues X 2 , Y 2 and Z 2 .For a homogeneous torus with major axis b and minor axis a, we can calculate the eigenvalues of the gyration tensor as Therefore, we calculate the eigenvalues X 2 , Y 2 , and Z 2 of the measured tensor S in a toroidal ring configuration numerically and identify as ring diameter D.

for physical system
Temperature dependence of parameters According to the theoretical result (9) the ring diameter is controlled by several parameters that can be controlled experimentally and in simulations and that are possibly temperature dependent: • The bending rigidity κ appears via the corresponding persistence length L p = κ/k B T (which measures κ in units of k B T ).The bending rigidity κ is assumed to be temperature independent (both in experiments and simulations) such that the persistence length L p ∝ 1/T by definition.
• The (free) adhesion energy g per length for the DNA nanotubes inside the bundle appears as g/k B T , which is an inverse length measuring g in units of k B T .The free energy of adhesion g is temperature-dependent in general.
Close to the bundling threshold it will contain entropic contributions from shape fluctuations of the DNA nanotubes [5].This complication is absent sufficiently deep in the bundled phase; this is the limit we focus on in experiments and simulations.Additional temperature dependencies can arise depending on the nature of the interaction.
Crosslinker-mediated adhesion can be assumed to lead to a largely temperature independent g.The only temperature dependence stems from the unbinding and rebinding equilibrium of crosslinking molecules [5].For a fixed number N c of firmly bound crosslinkers, this temperature-dependence is absent.Then g/k B T ∝ 1/T .This is the relevant situation for crosslinked DNA nanotube bundles.
Depletion attraction by additional crowding agents is an entropic interaction with a linear temperature-dependence g ∝ k B T such that g/k B T becomes temperature independent.This is relevant for the additional depletion attraction triggering ring contraction.
In simulations, we employ a temperature independent fixed g, which is given by a temperature independent parameter ε via g ≈ ε/σ.
• The total contour length L tot is assumed to be temperature independent, since we neglect the possibility of melting.Therefore, also the average crosslinker distance L tot /N c is temperature independent (for fixed number N c of crosslinkers).
• Geometric parameters such as the crosslinker size b c are temperature independent.

Conclusion for the physical system from theoretical predictions
The theoretical result (Eq.9) predicts several experimentally testable scenarios under which ring contraction could occur given the temperature dependence of the parameters: In experiments, a decrease in persistence length (i) could be achieved by increasing the temperature, as temperature is known to decrease the persistence length of semiflexible polymers and in particular DNA [2].As an experimental control, the persistence length of the DNA nanotube bundle can be increased by depletion when first adding a molecular crowder, like dextran, and second adding the crosslinker starPEG-(KA7) 4 in order to form larger rings.In the simulation experiments, the bending stiffness κ/k B T σ can be altered directly.
(ii) Rings contract if the total DNA nanotube length is decreased, for example, by depolymerization, D ∝ L In addition, experiments have shown that in absence of free tiles, the depolymerization rate measures around 0.3 layers per second at 40 • C [3].Moreover, the gradual disassembly of DNA nanotubes confined in water-in-oil droplets has been shown for temperatures above room temperature [13].
At the interior of the bundle, the concentration of free tiles is higher (due to recent depolymerization and depletion effects), leading to an effective increase of the melting temperature.A moderate increase in temperature will thus likely lead to depolymerization of the loose ends of the DNA nanotubes that were seen experimentally in STED (Fig. 3b, Supplementary Fig. 7) and electron microscopy (Fig. 4e) as well as in simulation experiments (Fig. 4c, Fig. 4d, Supplementary Movies 2,3) .DNA nanotube depolymerization can thus enhance ring contraction further.However, depolymerization is not included in the simulations.For simplicity L tot is kept constant.
(iii) If crosslinker entropy can be neglected, the ring diameter will contract with increasing g according to D ∝ (κ/g) 1/(3/2+α) .This means that if g is an entropic depletion attraction with g ∝ T ring contraction occurs for increasing temperature or decreasing persistence For a purely crosslinked system (only starPEG-(KA7) 4 , no dextran) the DNA nanotube colocalization assay indicates that nanotube-nanotube interaction relies on specific crosslinking (Fig. 2a,b).Moreover, increasing starPEG-(KA7) 4 concentration alone did not contract the ring size (Fig. 3c), indicating that assumption (iii.b) is not valid on its own.Experimentally, g can be increased by an additional depletion force (iii.a).The addition of dextran could shed light on the contraction for increasing attraction strength g and, furthermore, on the temperature dependence of the contractile system.
(iv) For dominant crosslinker entropy, rings will contract with increasing temperature or decreasing persistence length, D ∝ T −1/(3/2+α) ∝ L In our experimental system (iv) would indicate that at low crosslinker concentrations temperature increase would not cause crosslinker entropy to constrict the ring size.However, since this setting cannot be completely isolated experimentally from crosstalk effects, we can only make sure that we are in the parameter range of a dense crosslinker gas to obtain efficient constriction.Then (iv.a) predicts ring contraction for increasing temperature and further ring contraction upon increasing the depletion attraction, for example, by increasing the depletant concentration or the molecular size of depletants.
In summary, the key parameters for ring constriction are L p and g.These can be controlled in simulations as well as experiments with DNA nanotube rings, making the theoretical predictions testable.
the average distance between crosslinkers (while b c is their minimal possible distance) and L p = κ/k B T is the persistence length of individual DNA nanotubes.
Fig. 14a shows all measured equilibration curves of ring diameters for weak attraction ε/k B T = 0.3 and Supplementary Fig. 14b for strong attraction ε/k B T = 1.0.For strong attraction and smallest bending rigidity resulting in the smallest diameters, equilibration is most difficult because of the lock-in of neighboring polymers.The Supplementary Fig. 13: Comparison of equilibration dynamics by monitoring ring contraction from three different initial ring diameter to the final identical equilibrium diameter.All plots show ring diameter as a function of simulation time.Top row: Standard MD simulations for a fixed temperature k B T 1 for different interactions ε/k B T 1 = 0.2, 0.3, 1.0 (from left to right).Equilibration is only reached for small ε/k B T 1 = 0.2, 0.3.Bottom row: MD simulations employing simulated annealing, where we increase the temperature to T 2 ∼ 2T 1 (for ε/k B T 1 = 0.2, 0.3) or T 2 ∼ 4T 1 (for ε/k B T 1 = 1.0) for short time intervals.This allows us to achieve equilibration also for larger ε/k B T 1 = 1.0.For small ε/k B T 1 = 0.2, however, simulated annealing results in an artificial widening of the ring.
Supplementary Fig. 14: Bundle contraction during equilibration (ring diameter as a function of MD simulation time) for different single DNA nanotube rigidities κ for a, ε/k B T = 0.3 (showing data for more κ-values as compared to Fig. 5a in the main text) and b, ε/k B T = 1.0.Time scales of MD simulationThe natural time scale of the LAMMPS MD simulation is the Lennard-Jones times scale τ LJ = mσ 2 /ε.This is the time scale over which the inertial force⃗ F i = m ⃗ r i ofa bead becomes comparable to the typical Lennard-Jones force between beads.The mass m of a DNA nanotube piece of length σ ≈ 12nm is m ≈ 240kDa using a mass density λ ≈ 20kDa/nm [8].For ε = 1k B T , this gives a Lennard-Jones time scale τ LJ ≈ 3.8 ns.The actual LAMMPS MD simulation time step ∆t is chosen much smaller than τ LJ to assure accurate results, ∆t = 5 × 10 −3 τ LJ ≈ 0.02 ns.Using the Langevin thermostat we introduce an additional friction force ⃗ F f = −γm ⃗ r with a friction coefficient γ = 3πση/m for each bead and a corresponding thermal noise satisfying the Einstein relation.Here, η is the viscosity of the surrounding medium.The friction force sets a frictional time scale τ γ = γ −1 ∝ η −1 , over which the inertial force ⃗ F i of a bead becomes comparable to the friction force such that for times t > τ γ the motion becomes overdamped.The equilibration process during which rings contract to their final equilibrium diameter takes up to ∼ 10 9 MD steps for the smallest ring diameters.This corresponds to a simulation time t = 10 9 ∆t = 20 ms.To achieve such a fast equilibration we employ, however, viscosities η much lower than the realistic viscosity of water, η ≈ 10 −5 η Water or τ γ = 100τ LJ ≈ 10 5 τ γ,Water .
Supplementary Fig.15: a, MD simulation of two bundled polymers sliding apart by pulling with a force F .Polymers have 580 beads, the pulling force is F = 1k B T /σ, and the interaction energy is ε/k B T = 0.5.b, Desorption time t des in units of τ LJ as a function of the damping time scale τ γ (η)/τ LJ ∝ 1/η.

( 1 /
2+α)/(3/2+α) tot .its single stranded state.The critical melting temperature for DNA nanotubes can be calculated as a function of the enthalpy of disassembly, the entropy, the number of sticky end bonds and base pairs per bond and the concentration of free tiles[3,12].Decreasing free tile concentrations lower the corresponding melting temperature of a DNA nanotube in solution.Assuming 50 nM final tile concentration (of assembled DNA nanotubes, they are pre-formed at 5 µM and then diluted 1:100), one could gently estimate the concentration of free tiles to be below 5 nM.This would result in a melting temperature of 37.2 • C (compare to 50 nM free tiles resulting in 42.0 • C).

.
The crosslinker entropy is dominant for gd c < k B T /(1 − b c /d c ), where gd c is the average adhesion energy per crosslinker; in particular, it becomes dominant for a dense crosslinker gas with b c /d c ≤ 1 if the average adhesion per crosslinker is of the order of several k B T .An additional entropic depletion attraction with g ∝ T will further contract rings under these conditions (iv.a).