Microrheology of DNA hydrogel gelling and melting on cooling

Mechanical response of biocompatible all-DNA hydrogels with tuneable properties.

shows the g 1 (t) autocorrelation function measured at T = 20°C and wave vector q = 0.0186 nm −1 for three different values of φ. All curves are rather well described by a single exponential decay (black solid lines). A proper passive microrheology experiments requires that the probe particles do not interact, such that the measured scattering function is dominated by the self component. It is normally assumed that hard sphere particles do not interact when the distance between them is larger than 20 times their diameter 1 . This factor 20 leads to a maximum φ ∼ 6·10 −3 . Indeed, with the aim to minimise particle interactions and to guarantee the validity of this approximation we select a volume fraction one order of magnitude lower (φ = 3.4 · 10 −4 ). Under this condition, g 1 (τ ) = exp(−q 2 < r 2 (τ ) > /6), where < r 2 (τ ) > is the mean square displacement of the probe particle. In addition, at this value, the scattering intensity from the colloidal particles is observed to be more than twenty times larger than the one of a DNA hydrogel sample. As a further check we evaluated the viscosity (η) of water at different T s ranging from 15°C up to 52°C. The DLS results and the corresponding viscosities are shown in Fig. 2 together with the theoretical values from the NIST database 2 . Finally, from the single exponential fittings of the autocorrelation functions presented in Fig. 2(a) it is possible to estimate the hydrodynamic radius of the colloidal tracers. The average of the measurements, using as reference η of water at the corresponding T s, it is estimated in ∼ 268 ± 5 nm.

II. DNA SEQUENCES
The tetrafunctional DNA particles of the first hydrogels system were prepared by diluting equimolar concentrations of the sequences NS 1 -NS 4 where sequences with the same colour indicate complementary strands.
In the reentrant-gel system, four sequences (A 1 − A 4 ) are needed to form the tetrafunctional nanostar structures and two single stranded sequences (B 1 − B 2 ) to form the so-called competitors: Here the nucleotides involved in the formation of the arms of the tetravalent nanostar are reported in blue, followed by a 12-long sticky-end sequence. Competitor particles are designed to be complementary to different parts of the nanostar sticky overhangs, being able to cap the nanostar sticky-ends at low T . The breaking of the inter nanostar bonds results in the melting on cooling of the gel network. The characteristic re-entrant behaviour introduced by the competition between the configurations stabilised by energy and those stabilised by entropy is discussed in more detail in Ref. 3.

III. EVALUATION OF THE VISCOSITY FROM THE MSD USING PARTICLE TRACKING MICROSCOPY (PTM)
Calling (x a , y a ) and (x b , y b ) the discretized 2D trajectory of two tracked PS microspheres (whose relative distance is always larger than 7 µm) we compute the MSDs along the x and y axis as: The sum runs over all the possible time origins for which t j + ∆t i does not exceed the observation time. To properly cover the diffusive part of the trajectory we use a frame rate between 10 and 50 frames per second, with a exposure time T e of 10 ms. This T e value allows us to properly measure η but it prevents us from exploring the viscoelastic behaviour of the medium. Figures 3(a) and 3(b) show the measured MSD x and MSD y at two different temperatures, 21°C and 39°C, respectively. Given the large distance between the two colloids, hydrodynamic interactions can be neglected. We thus assume the microspheres' Brownian displacements to be uncorrelated. In this case, eq. 1, for both x and y directions, results in a MSD growing at long times as (D a + D b )t where D a and D b are the diffusion coefficients of the two microspheres. Finally, considering an equal size of the microspheres' (within a 5% uncertainty coming from colloids' radii polydispersity), we identify D a = D b = D. Having access to two independent measurements of the MSD, respectively along x and y directions, the precision of the fitting procedure can be estimated. Defining D x and D y as the values corresponding to the fit of M SD x and M SD y , we estimate the relative error as the average value of 2|D x − D y |/(D x + D y ) for all the investigated temperatures, resulting in a relative error of ∼ 10%. When computing η via the Stokes-Einstein relation, we also take into account the 5% uncertainty on the microspheres' radii, obtaining a final 12% precision.
We note that with the selected experimental set-up, the value of q does not provide a measure of the squared size of the typical cage confining the probe particle. Indeed, the value of q is affected by instrumental tracking artefacts due to the finite camera exposure time, since the measured position of a colloid corresponds to the temporal average of the real trajectory during T e . Following Ref. 4, the experimentally observed mean square displacement MSD e at time ∆t is related to the real MSD by an averaging procedure over the exposure time T e (that we assume larger than the time the particle spends in the cage T cage ) A detailed computation of MSD e is described in Ref. 4. Here, we will provide a back of the envelope evaluation of MSD e for the case in which ∆t is larger than the caging time (∆t >> T e > T cage ). In Equation 3 we approximate the (one-dimensional) MSD with the linear function Correspondingly, we substitute MSD(∆t ± t ) = 2D(∆t ± t ) + ξ 2 and MSD(t ) ≈ ξ 2 where we assumed a relaxation time in the cage T cage shorter than T e . After integration we find MSD e (∆t) = 2D∆t leading to q ≈ 0. Therefore, a finite exposure time T e > T cage , leads to an error in the intercept of the order of the of the squared cage size. Additionally, by taking into account the tracking error associated to the noise and pixelation of the camera (∼ 3 nm) the experimental value q for the intercept becomes unusable. For this reason, the fitted intercept value from the MSD curves displayed in Fig. 7 of the main text has been removed.
Errors on the MSD shown in Fig. 7 have been estimated by dividing the 10 minutes long trajectories into N sub subtrajectories of length T sub = 100 s. For each subtrajectory we compute the MSD along a coordinate (x or y) that we indicate with MSD i (∆t), where i = 1, 2, . . . , 6 accounts for the subset of trajectories. For each time ∆t we compute the mean MSD = N sub In a similar way, in order to verify the stationarity of the diffusion process, again for each temperature, we plot the MSD i (∆t) at a fixed ∆t (see Fig. 3). MSD i fluctuates around its mean value within the expected standard deviation, confirming the stationarity of the diffusion processes.

IV. THEORETICAL BACKGROUND: DLS MICRORHEOLOGY
The g 1 (t) is related to the mean square displacement (MSD) of the colloidal tracer particle < ∆r 2 (τ ) > by where the wave vector q is defined as and n stands for the refractive index of the medium, λ is the laser wavelength and θ is the scattering angle. DLS microrheology relies on the thermal energy k B T associated to the tracer colloids. According to the generalized Stokes-Einstein (GSER) equation 5 , defined in a similar way to the Stokes-Einstein equation but as a function of Laplace transformed quantities, it is possible to relate the diffusion coefficient (D) to the viscosity (η) of the material asD whereD(s) andη(s) are the diffusion coefficient and the viscosity as a function of the Laplace frequency s and a is the tracer particle radius. From eq. 7, and assuming that inertial effects are negligible at the experimentally accessible ω, the Laplace transformed of the complex modulusG(s) can be expressed as 6 where < ∆r 2 (s) > represents the Laplace transform of the measured MSD. Then, the linear viscoelastic moduli as a function of the frequency ω, G (ω) and G (ω), can be obtained according to the frequency transformation s = iω and to the relation G * (ω) = G (ω) + iG (ω) [6][7][8] . Here, G (ω) and G (ω) represent the real and the imaginary part of the complex modulus G * (ω) which are related to the elasticity and the viscosity of the material, respectively. As a result where F{< ∆r 2 (τ ) >} is the one side Fourier transform of the MSD. A convenient method to avoid spurious fluctuations in the Laplace and Fourier transforms has been developed in Ref. [ 7,4,9 ] and often applied to soft-matter systems 10 . < ∆r 2 (τ ) > is expanded locally around τ = 1/ω assuming where α(ω) is estimated as the local slope of the MSD logarithmic time derivative In purely viscous media α(ω) takes a value of 1, while if the particle is completely arrested in an elastic medium the MSD slope would be 0. Therefore, in viscoelastic materials, α(ω) is expected to range within these two cases, 0 < α(ω) < 1. From eq. 10 and eq. 11, the GSER in the Fourier space can be expressed as where |G * (ω)| = k B T πa < ∆r 2 (1/ω) > Γ [1 + α(ω)] being Γ the gamma function. Two limits are worth discussing. If the MSD is time-independent (complete caging, < ∆r 2 (τ ) >= ξ 2 0 ) then α(ω) = 0 and  Hence G and G are parallel power-law in ω. If β = 0.5 G (ω) = G (ω).