## Abstract

The rod-like viruses show anomalously rapid diffusion in bio-tissue networks originated from the rotation-facilitated transportation; however, the experimental investigation of the correlation of the rotational and translational dynamics is still in blank. Herein, typical rod-like and spherical gold nanoparticles (NPs) are dispersed in the classical Tetra-PEG gels, respectively, as model systems for light scattering studies. The contributions from translational and rotational diffusive dynamics, and network fluctuation dynamics can be well-resolved and the stretch exponent of rotational dynamics at 0.25 is proven to be the fingerprint for the coupled rotational and translational dynamics of nanorods. The rotation facilitated re-orientation finally leads to the fast transportation of nanorods. The discoveries are confirmed to be valid for rod-like biomacromolecule systems by studying the diffusive dynamics of Tobacco mosaic virus in gels. The work can be inspiring for the development of protocols to prevent infection of microorganism and regulate the transportation of nano-medicines.

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## Introduction

The transportation of biological and artificial nanoparticles (NPs) in various tissues represents the critical process for the infection of microorganisms and the targeted delivery of nano-medicines and diagnostic probes^{1,2,3,4}. Generally, the underlying physics can be described as the diffusion of NPs confined in networks with different viscoelasticity. Interestingly, in comparison to isotropic NPs, the nanorods show anomalously rapid diffusion in networks from theoretical studies^{5,6}. Experimentally, the rod-like biomacromolecules and micro-organisms are observed to possess high mobility in tissues^{1,7,8,9,10}. The Stokes-Einstein relation is invalid to describe the dynamics of nanorods due to the breakdown of continuum model^{11}. The increased mobility of nanorods has been attributed to the additional freedom degrees of rotation, while the rotational dynamics allow nanorods to enter adjacent mesh more easily compared to isotropic NPs with similar sizes^{7,12}. In previous research, Jain et al. reported the faster tumor penetration speed of rod-like NPs in comparison to spherical NPs by using the multiphoton microscopy technique^{1}. Gao et al. observed a similar phenomenon in stimulated emission of depletion microscopy studies, and further simulation was applied to confirm the enhancement of translational diffusion by rotational dynamics^{7}. Yan et al. focused on the fast sliding dynamics of thick rod in a network both in experiment and simulation and revealed the physical origin based on the calculation of the free energy landscape^{13}. Composto et al. applied single particle tracking to study the rod dynamics during the gelation process and elucidated the potential mechanisms for the caging to mobile behavior of nanorods^{8}. The non-Gaussian displacement distributions of nanorods is also uncovered, which is in consistence with the results obtained by Conrad et al.^{14}. Although the explanations inferred yet are reasonable, the quantitative relationship between rotational and translational dynamics and their correlation with nanorod size and network mesh size still hides in the mist.

Quasi-elastic scattering can be applied to probe relaxation dynamics of soft matter at selected spatiotemporal ranges, and the measurements contain both information on relaxation time and the corresponding spatial scale for certain relaxation processes^{15,16,17}. When the spatio-temporal window is broad enough, the contribution of hierarchical relaxation processes can be quantitatively determined, and their structural origin can be ascribed^{18}. Moreover, due to their sensitivity to the rotation motion of anisotropic particles, the depolarized mode of quasi-elastic scattering can be used to separately measure rotational relaxation dynamics^{19,20,21}. Herein, depolarized dynamic light scattering (DDLS) together with dynamic light scattering (DLS) is applied to study the relaxation dynamics of rod and spherical gold nanoparticles (NPs) confined in classical tetra-poly (ethylene glycol) (tetra-PEG) hydrogel. For anisotropic particles, the off-diagonal elements of the polarizability tensor are not zero, which leads to the changing of the polarization. Therefore, optically anisotropic particles will contribute a signal of depolarized scattering, and the orientational correlation function can be measurable^{19}. As a result, the translational and rotational dynamics are quantitively and separately determined, and the correlation can be deciphered. With rigorous correction of the intrinsic nonergodicity of hydrogel system, we report a rotation re-orientation induced translation dynamical behavior of nanorod whose translation is highly coupled with its rotation process, and the same behavior can also be extended to biological systems, e.g., tobacco mosaic virus (TMV).

## Results

### Hydrogel characterization

The mesh size \(\xi\) of the host hydrogel is measured using ensemble-averaged static light scattering intensity (\({\left\langle I\left(q\right)\right\rangle }_{E}\)) upon the fitting using the Ornstein-Zernike equation^{15,22,23}. As shown in Supplementary Fig. 1 (a), a plot of \(1/{\left\langle I\left(q\right)\right\rangle }_{E}\) versus \({q}^{2}\) is linear, and the respective mesh size is 20 nm, and this mesh size is also consistent with the results of rheology (Supplementary Fig. 2a, b, and Supplementary Note 3). The scattering intensity at different sub-ensembles is consistent, verifying the near-perfect homogeneity of the selected model gel system (Supplementary Fig. 3a). For the pure host gel, two relaxation modes can be observed in DLS measurements (Supplementary Fig. 1b), and the dominant mode contributes about 93% of the correlation function, whose decay rate is proportional to \({q}^{2}\) (Supplementary Fig. 1c) with a diffusion coefficient of \({D}_{{gel}}=\left(5.44\pm 0.13\right)\times {10}^{-11}{\,m}^{2}/s\). For gels, the electric field correlation function \({g}_{1}(q,t)\) is proportional to the correlation function of the longitudinal displacement \({\mu }_{l}(q,\, t)\) of strands, as expressed:

where \({D}_{{gel}}=\frac{{M}_{l}}{f}\) (\({M}_{l}\) is the longitudinal modulus of the gel, \(f\) is the friction coefficient). \({D}_{{gel}}\) is related to the gel elasticity as it is proportional to the longitudinal modulus of gel^{24}.

As shown in Supplementary Fig. 1d, the decay rate of the other mode is not linear with \({q}^{2}\), which represents non-diffusive dynamics^{25,26}. The minor mode is related to the tiny amount of unavoidable chemical inhomogeneities, and it has negligible effect on the following studies of NP diffusive dynamics since the ratio is low. The homogeneity has also been verified by scattering intensity at different sub-ensembles (Supplementary Fig. 3a)^{27}.

### Dynamics of gold nanorods in hydrogels

We have studied two gold nanorods named NR50 and NR85 with sizes 50 nm * 20 nm and 85 nm * 17 nm, respectively, and the size of TMV used is 300 nm * 18 nm, while the gold nanosphere named NS32 has a diameter of 32 nm. These particles are loaded within the gels without affecting the structure of networks or exhibiting strong interaction with gel strands, as supported by the rheology results (Supplementary Fig. 2). The structure parameters of these particles have been confirmed from small angle X-ray scattering (SAXS) (Supplementary Fig. 3b–e), DLS studies (Supplementary Table 1) and TEM (transmission electron microscopy) (Supplementary Fig. 4). The normalized first-order correlation function \({g}_{1,F,{VH}}(q,t)\) s of DDLS of NR50 in hydrogel matrix at different *Q*s together with the fittings are shown in Fig. 1a, and the respective best-fitting function can be expressed as a sum of two exponential decays and a stretched exponential decay:

where \(\Gamma\) is the decay rate, \(A\) is the weighting factor, and \(\beta\) is the stretched exponent. The linear fitting of the obtained decay rate on Q affords the translational and rotational diffusion coefficients from the slope and intercept respectively. As shown in Fig. 1b, the Q dependence of the decay rate of the first mode (\({\Gamma }_{1}\)) of NR50 exhibits a slope of zero, suggesting the absence of the translational motion of the particles. The first mode \({\Gamma }_{1}\) can be attributed to the locally partial rotation of the nanorod within meshes since the \({D}_{r}\) is identical to that of the nanorod in water after viscosity correction. The apparent \({D}_{r}\, \left(\right.7194\) \(\left.{s}^{-1}\right)\) in gels is about 25% of \({D}_{r}\, \left(\right.27874\) \(\left.{s}^{-1}\right)\) of nanorods in water, while the viscosity of tetra-PEG solution with the same concentration as the hydrogel matrix is about four times of pure water (Supplementary Fig. 3f). Similarly, the second mode \({\Gamma }_{2}\) between \({\Gamma }_{1}\) and \({\Gamma }_{s}\), only exports a rotational diffusion coefficient of \(381\) \({s}^{-1}\) without obvious translational contributions, and we consider this mode as another local rotation process. The stretched mode \({\Gamma }_{s}\) is actually an anomalous rotation process since its deviation from exponential decay^{28}, and the averaged \(\beta\) at different angles (Supplementary Fig. 5a) is:

The stretched exponential decay represents a distribution of relaxation times of rods taking different positions and orientations inside the gel meshes, whereas the elastic deformation energy of gel strands confining a nanorod varies depending on the specific orientations of the rods. The thermal fluctuation could possibly drive the rotation of the rod to achieve the hopping of rod orientation, which finally results in the rotational diffusion within adjacent meshes. The rotational mode of a re-orientation process can contribute to the translational diffusion of rods, and the translation is highly coupled to the rotation.

Theoretical analysis regarding the rotation process is also conducted for the deep interpretation of experimental results. The rotation and translation of the rod will be confined once the rod length is larger than the network mesh size and therefore, the confined rods can have different orientations within meshes (Fig. 1e–h)^{12}. Each orientation corresponds to a certain possible rotatable angle \(\psi\), which is of order:

where \(d\) is the rod diameter, \(\xi\), and \(l\) is the length of rod inside the gel meshes. As shown in Fig. 1e–h, the rod is sliced by several meshes, and the meshes are disturbed and stretched by the rod. The parameter \(l\) is used to describe the length of the subregions of rods inside one gel mesh. Here, \(\frac{\xi }{l}\) is the Taylor expansion of \(\arcsin (\frac{\xi }{l})\), which is the main part of the description of the rotatable angle. However, the above term holds assuming a very small diameter, which is not valid in our system, and a correction term \(\frac{\xi }{d}\) is introduced as the rotatable angle becomes smaller for a larger diameter. Here, the parameter \(\psi\) doesn’t equal to the practical value of the rotatable angle but only plays as a scaling argument for the measure of the rotatable extent of the rod. Therefore, the confinement free energy, \({F}_{{conf}}\), of the strands can be derived by considering the elastic deformation energy of gel strands under the interference of the confined rods with specific orientation^{29,30} (Supplementary Note 1):

where \({A}_{l}\) is the pre-factor (\({A}_{l} \sim \frac{1}{\xi }\)), \({k}_{B}\) is the Boltzmann constant and \(T\) is the absolute temperature. The corresponding probability of the existence of a specific orientation can be described by the Gibbs distribution function as^{31}:

The rotation process is actually a re-orientation process during a short time interval (\(\Delta t\)), and the relaxation rate is

the corresponding displacement during \(\Delta t\) is in connection with \(\xi\) as we assume that the re-orientation induces the entrance into adjacent meshes, and such consideration is reasonable as suggested from the similar trend of TMSD (translational mean-square displacements) and RMSD (rotational mean-square displacements) of rods (Fig. 2). The time interval follows \(\Delta t \sim \frac{{\xi }^{2}{\eta }_{{local}}l}{{k}_{B}T}\) based on the diffusion equation consideration^{32}, where the \({\eta }_{{local}}\) is the local viscosity as the rod doesn’t experience the bulk viscosity, and the expression for \({D}_{r}\) covers the contribution from the parameter of \(d\) and \(\xi\). Then, the relaxation rate can be expressed as:

The experimentally measured \({g}_{1}(t)\) is the superposition of rods taking different orientations considering their respective probability (\(P\left(\psi \right) \sim P(l)\)):

Substituting Eqs. (5) and (7) into Eq. (8), and computing the integral via saddle point approximation (Supplementary Note 2) considering the feature of fast decay of exponential function, we get the expression^{25,33}:

The obtained stretched exponential decay originating from the linear superposition of relaxations with different characteristic times is different from the ordinary exponential decay suggested by the Stokes-Einstein-Debye relation. As a result, the proposed rotation dynamics for the re-orientation of rods at different orientations in the gel meshes lead to the theoretical derivation of \(\beta \, \approx \frac{1}{4}\), which is consistent with the above experimental observations. Actually, the exponent of 0.25 is directly related to the relaxation time distribution while such theoretical assumption can also be well verified by a recent MD simulation work by Yan et al.^{12}. The performed integral considers not only the cooperative results of the subregions of one rod but also the rods with various orientations. It’s actually the ensemble-averaged results that we measure in the scattering experiment. The coincidence confirms that the re-orientation process helps the fast translation of the rod within networks, as the re-orientation of the rod raises the probability of moving into other meshes (Fig. 1h). This should be a reliable explanation for the speculation about the rotation-facilitated translation of nanorods by previous work^{7,8}.

Compared to NR50, NR85 exhibits similar dynamical behaviors both in DLS and DDLS results, but an extra exponential decay is required to fulfill best-fitting (Fig. 1c, d). The extra dynamical mode with a ratio of 9% corresponds to the rotation coupled to network fluctuation, as the relaxation time (\({\Gamma }_{1} \sim 174600\) \({s}^{-1}\)) is similar to the characteristic fluctuation time of the network (\({\Gamma }_{{gel}}=\frac{{D}_{{gel}}}{{\xi }^{2}} \sim 135000\) \({s}^{-1}\)). This extra rotation coupled to the network is due to the large rod length compared to the mesh size, and the coupling is weak. Considering the low ratio of coupling in the NR85 system, the coupling effect of NR50 may be too weak to be observed in our experiments. Apart from the extra rotation process, the other three dynamical modes are the same as NR50, and the characteristic rotation-induced translation dynamic still holds as the averaged exponent at different angles is *β* = 0.27 ± 0.015 (Supplementary Fig. 5 (b)). This consistency of the obtained dynamics with theoretical analysis and the gel’s intrinsic properties further verifies the credibility of the fitting function.

### The coupling of translational and rotational diffusive dynamics

The diffusive dynamics of spherical NPs are studied as a control to understand the coupling of translational and rotational diffusive dynamics of nanorods, and NS32 is selected by considering that its translational diffusion coefficient is similar to NR85 and NR50 in water. As for nanospheres with a smaller size comparable to mesh size, free diffusion of nanosphere slower than the prediction of the Stokes-Einstein relation is observed, and the detailed analysis is shown in Supplementary Note 4. The DDLS and DLS studies of NS32 in gels give different results (Fig. 3a), where a mode at micro-seconds scale can be found absent in DDLS compared to DLS obviously. This absent mode is related to translational fluctuations, which are filtered out by the polarizer in DDLS^{34}. The best fit for DLS can be accomplished by \({g}_{1,F,{VV}}(q,\, t)={A}_{1}\exp (-{\Gamma }_{1}t)+{A}_{2}\exp (-{({\Gamma }_{s1}t)}^{\alpha })+(1-{A}_{1}-{A}_{2})\exp (-{({\Gamma }_{s2}t)}^{\beta })\), while DDLS only needs two stretched exponential decays to fulfill the best fit in comparison to DLS, and these two modes correspond to rotational motions. Actually, the non-perfect surface of the nanosphere with relatively small sizes contributes to its anisotropy and the so-resulting signals of DDLS^{35}.

Regarding the relaxation dynamics of NS32, the first dynamical mode is diffusive since the angle dependence of \({\Gamma }_{1}\) follows \(\Gamma=D{q}^{2}\) (Fig. 3b). The diffusion coefficient obtained is \(D=5.7\times {10}^{-11}\) \({m}^{2}/s\), which is even faster than the diffusion of the nanosphere in water (Supplementary Table 1). Therefore, this diffusive mode is definitely not the free diffusion of NS32 within gel networks. Actually, the diffusion coefficient is close to the intrinsic fluctuation dynamics of gels (\({D}_{{gel}}=5.4\times {10}^{-11}\) \({m}^{2}/s\)). Therefore, the diffusive mode is highly coupled to the fluctuation of the hydrogel. The diameter of NS32 is about 1.5 times the network mesh size, and the sphere will be forced to fluctuate together with the network (Fig. 3c), and the mean squared displacement of NS32 will be immobilized within the mesh as shown in the plateau in Fig. 2b. In fact, similar observation is also reported, where gold nanospheres are used as tracers to monitor the gelation process, and the spheres exhibit a diffusion coefficient faster than in pure solvents^{36}. The other two stretched exponential decays are from the rotation coupled to chain dynamics, and they are consistent in DLS and DDLS measurements (Fig. 3d). The stretched exponent \(\alpha\) is close to 0.5 (Supplementary Fig. 5c), and the same phenomenon is always observed in polymer rubbery states^{37}. The stretched exponent \(\beta\) is close to 0.4 (Supplementary Fig. 5c), which is consistent with the dynamical feature of Zimm dynamics originating from the crosslinked polymer networks^{33}. In short, the NS32 is highly confined in gel mesh and its relaxation dynamics is on the same tune with the gel network, which is the same as the principle of nano-rheology^{38,39,40}.

Translational (TMSD, \(\left\langle \delta {r}^{2}(t)\right\rangle\)) and rotational mean-square (RMSD, \(\left\langle \delta {\theta}^{2}(t)\right\rangle\)) displacements are extracted from the correlation function of DDLS studies for direct comparison of the diffusive behavior of spherical NPs and nanorods. According to Eqs. (16) and (17) in methods part, we can obtain the TMSD and RMSD of the NPs confined in gels. As shown in Fig. 2a, the diffusion of NR50 is much faster than NR85 and NS32 within the hydrogel matrix, even if these three particles have similar diffusion coefficients in water (Supplementary Table 1). However, the dynamics of NR50 within the experimental time scale don’t exhibit Brownian diffusion feature. We attribute the difference between NR85 and NR50 to the stronger confinement of gel on NR85, and that’s also the reason for the observable weak coupling between NR85 and gel network (mode \({\Gamma }_{1}\) with a low ratio of 9%) (Fig. 1d). This difference doesn’t affect the confirmation of the characteristic rotation re-orientation dynamics. Within the experimental scale, the diffusion of NR85 is slower than NS32, but the NR85 tends to move to broader space compared to NS32, as the TMSD of NS32 exhibits a plateau (Fig. 2b). Such plateau-like regime at the long time scale of NS32 is consistent with the fitting results, the dynamics of spherical NPs is on the same tune with gel networks due to the strong confinement effect. For NR85, the slope of 0.1 of TMSD at a long time scale represents the sub-diffusion feature, which is also consistent with the fitting analysis. It confirms that the characteristic rotation re-orientation dynamics do contribute to the diffusion of rods. The same trend can be observed in RMSD and TMSD for both NR85 and NR50, especially at longer time scales, which further verifies the coupling between rotation and translation, and the translation induced by rotation re-orientation.

### Extension to biological system

The understanding of the dynamics of nanorods is important for biomedical applications, and herein, rod-like biomacromolecule TMV, is studied in gels for their unique diffusive dynamics^{41}. The dynamics of TMV in pure water were first analyzed as a control, the rotational and translational diffusion coefficients were obtained (Supplementary Table 1), which are the same as in previous literature. Once the TMV is confined in the gel network, the dynamics of TMV are severely restricted, as suggested by our light scattering experiments. The fastest relaxation mode observed in DLS is absent in DDLS, and the respective correlation functions are shown in Fig. 4a. The invisibility of this mode in DDLS demonstrates its translational nature. We consider this process as the motion of TMV coupled to the network fluctuation, as the fastest dynamical mode \({\Gamma }_{1}\) follows \(\Gamma=D{q}^{2}\) and the diffusion coefficient is \(\left(5.2\pm 0.6\right)\times {10}^{-11}\) \({\,m}^{2}/s\), which is close to the \({D}_{{gel}}\) (Fig. 4b).

The significance of our studies on the rotation re-orientation process relies on the extension in biological systems, and we demonstrate its existence via the stretch exponent \(\beta=0.26\pm 0.03\) of the stretched exponential decay (Supplementary Fig. 5d). However, the ratio of this process is just about 10%, which is much lower than in NR50 (~45%) and NR85 (~40%). Although the confinement is strong considering the large length, the dynamics of TMV are not completely frozen. It still exhibits the same fingerprint of coupled rotation and translation as observed in gold nanorod systems, even if the ratio is low. The other mode \({\Gamma }_{2}\) corresponds to rotation coupled to network fluctuation at a larger scale, since \(\frac{{D}_{{gel}}}{{L}_{{TMV}}^{2}}=600\) \({s}^{-1}\), which is close to \({\Gamma }_{2}\) measured (Fig. 4c). In short, the other two relaxation process apart from the rotation re-orientation process are dynamics coupled to network fluctuation. To further study the strong confinement exerted on the TMV, the gel mesh sizes have been varied to study the confined dynamics of TMV systematically. A PEGDA hydrogel with mesh size ~40 nm is prepared with TMV loaded (Supplementary Fig. 6d and Supplementary Note 3, 5). The dynamics of TMV within the PEGDA gel are similar to NR50 and NR85, and the normalized first-order correlation function \({g}_{1,F,{VH}}(q,\, t)\) s of DDLS of TMV in PEGDA hydrogel at different Qs can be well fitted by a sum of two exponential decays and a stretched exponential decay (Supplementary Fig. 5e):

The decay rates of these three relaxations exhibit the same Q dependence of \(\Gamma \, \sim \, {q}^{0}\) (Fig. 4d). Similar to NR85 and TMV in TPEG gels, the first mode \({\Gamma }_{1}\) corresponds to the rotation coupled to network fluctuation, since \(\frac{{D}_{{gel}}}{{\left(90{nm}\right)}^{2}} \sim 7037\) \({s}^{-1}\), and this length scale is smaller than the coupling scale of TMV in TPEG hydrogel while larger than the coupling scale of NR85. Taking the lengths of nanoparticles and mesh sizes into consideration, this phenomenon should originate from different extents of confinement.

The second mode \({\Gamma }_{2}\) is the locally partial rotation of TMV within meshes, as the \({D}_{r}\) is the same that of TMV in water after viscosity correction. The apparent \({D}_{r}\, \left(\right.62\) \(\left.{s}^{-1}\right)\) TMV in PEGDA gels is about 33% of \({D}_{r}\, \left(\right.191\) \(\left.{s}^{-1}\right)\) of TMV in water, while the viscosity of PEGDA solution with the same concentration as the gel matrix is about three times of pure water (Supplementary Fig. 3f). The third mode of stretched exponential decay exhibits stretched exponent of \(\beta=0.26\pm 0.011\) (Supplementary Fig. 5f), which demonstrates the existence of the characteristic rotation re-orientation process in TMV-PEGDA gel system. In comparison to TMV in TPEG gel, the ratio of such relaxation is raised to about 28 % due to weaker confinement. Besides, the \({D}_{r}\) of this process is faster compared with TMV within TPEG gel where the mesh size is smaller, suggesting the potential effect of the term \(\frac{\xi }{d}\), which we have introduced in Eq. (3), on the rotation process. The TMSD of the TMV-TPEG system exhibits a plateau-like regime at a long time scale (Supplementary Fig. 7), which is consistent with the fitting analysis. The ratio of the characteristic rotation re-orientation mode is only 10%, and its contribution to the translational diffusion is limited. On the other hand, both the TMSD and data fitting of the TMV-PEGDA system are similar to that of the NR85-PEG system. The diffusion of TMV in the PEGDA matrix is even faster than NR85 in the PEG matrix at a long time scale, which may have originated from the effect of large mesh size.

The extension to the biological system not only proves the universality of our method and theory but also indicates the existence of the typical range for the application of the dynamics model. With the increasing of the ratio \(\frac{L}{\xi }\), the ratio of the characteristic rotation re-orientation process decreases. There should be a critical value of \(\frac{L}{\xi }\) above which the proposed dynamics model doesn’t hold, as the physical picture of the system will transform to rod reptation, and this value should be close to \(\frac{L}{\xi } \sim 15\), as suggested by our experiments of the TMV-TPEG system.

## Discussion

In summary, we have combined theoretical analysis, DLS, and DDLS experiments to investigate the dynamics of nanorods under confinement. The rotation process is highly coupled to the translational diffusion of nanorods. Our experiments demonstrate that the faster translation of the rod compared to the sphere within the networks originates from the characteristic rotation re-orientation process of the rod, and this process can be extended to biological systems, e.g., rod-like TMV, projecting the applications of our discovery in biomedical engineering. As the key discovery, the combination of theoretical analysis and light scattering studies shows that this rotation behavior follows a stretched exponential decay with exponent \(\beta=0.25\), while the dependence of \({D}_{r}\) on mesh size, rod length, and rod diameter is also verified via scaling argument. Our work provides direct evidence for the rotation-facilitated translation of nanorods under nanoconfinement. The conclusions provide feasible approaches to tune the rotation and translation dynamics of synthetic or biological NPs confined in bio-tissues for practical applications.

## Methods

### Materials

The tetra-PEG macromers, tetra-PEG-amine (TPEG-A) 40 k, and tetra-PEG-succinimidyl glutarate (TPEG-S) 40k are purchased from JenKem. The PEGDA macromer is purchased from Peking PEG. Citric acid, dibasic sodium phosphate, sodium phosphate dibasic heptahydrate, and sodium phosphate monobasic dihydrate are purchased from Aladin. The gold nanoparticles NR50, NR85, NS32, and NS16 are purchased from LMnano, and these gold nanoparticles are grafted with CTAB (Hexadecyltrimethylammonium bromide). TMV is extracted from infected *Nicotiana cbenthamiana* leaves^{42,43}.

### Sample preparation

TPEG-A and TPEG-S are dissolved in phosphate buffer (50 mM, pH 7.4) and phosphate-citrate buffer (50 mM, pH 5.8), respectively, by vortexing the solutions for 30 s. All studies are conducted at a TPEG concentration of 40 mg/mL. For samples loaded with nanoparticles, the nanoparticle solutions are directly added to the TPEG solution. All of the solutions are filtered using a membrane of 450 nm meshed. The filtered solutions are injected into the sample cell, and the solutions will become gel in 30 min. PEGDA is dissolved in water at a concentration of 25 mg/mL, the initiator 2959 was used to initiate the reaction under UV light, and the concentration of initiator is 2 mg/mL. The method loading of TMV within PEGDA hydrogel is the same as in TPEG system.

### SAXS

The SAXS experiments were conducted at the BL10U1 beamline at the Shanghai Synchrotron Radiation Facility.

### TEM

The TEM experiments were conducted at the JEM-1400 flash built by JEOL. The sample is prepared by soaking the holey carbon film on the copper grid in the solution sample for 1 s and dried in the air.

### Rheology

Rheological investigations were performed at Anton Paar MCR302 rheometer.

### Light scattering and the respective correction of non-ergodicity

The light scattering spectrometer used is a 3D-DLS Spectrometer (LS instruments, Switzerland), and the wavelength is 660 nm.

Dynamic light scattering (DLS) probes the dynamics of target objects at spatial scales of the q vector:

Where \(n\) is the refractive index of the solvent, \(\lambda\) is the wavelength of the incident laser, and \(\theta\) is the scattering angle^{19}. The temporal window of dynamics measurements is limited by the sensitivity of the photon detector and the processing capability of digital correlators, and generally, the window is from 10 ns to 1 s, covering the typical rotation and translation dynamics of NPs and network fluctuation dynamics.

The formed tetra-PEG hydrogels and the hydrogels with NPs incorporated systems are nonergodic since the spatial constraints from the gel networks inhibit colloidal objects to decorrelate from their initial position. As a result, the time-averaged quantities could not be consistent with the properties of ensemble-averaged quantities. Pusey and co-workers developed a method to acquire the ensemble-averaged first-order correlation function \({g}_{1,E}\left(q,\, t\right)\) based on time-averaged correlation functions of a sub-ensemble^{44}. The specific expression is shown below:

where \(Y={\left\langle I\left(q\right)\right\rangle }_{E}/{\left\langle I\left(q\right)\right\rangle }_{T}\). \({\left\langle I\left(q\right)\right\rangle }_{E}\) is the ensemble-averaged scattering intensity measured from the time-averaged scattering intensity while rotating the sample (different sub-ensembles), and \({\left\langle I\left(q\right)\right\rangle }_{T}\) is the time-averaged scattered intensity measured similar to \({\left\langle I\left(q\right)\right\rangle }_{E}\) but keeping the sample at a fixed position. \({g}_{2}(q,t)\) is the second-order correlation function obtained from experiments. \({\sigma }_{T}^{2}\) is the mean-square intensity fluctuation (\({\sigma }_{T}^{2}=\frac{{\left\langle {I}^{2}\left(q\right)\right\rangle }_{T}}{{\left\langle I\left(q\right)\right\rangle }_{T}^{2}}-1\)), which is the same as the intercept of \({g}_{2}\left(q,\, t\right)-1\). The first-order correlation function can be divided into two parts, \({g}_{1,F}(q,\, t)\) (dynamic component) and \({g}_{1,s}\left(q\right)\) (static component), as follows:

On the other hand, the DDLS technique is applied by integrating a polarizer in front of the photon detector, which is perpendicular (horizontal) to the polarization direction of the incident laser (vertical). Therefore, only the depolarized signal (VH) reaches the detector, and in our systems, only the dynamics information of nanorod samples is visible under DDLS mode since only anisotropic particles can change the polarization direction of the scattered light^{35}. Both the translational and rotational processes contribute to the measured correlation function, and they are mixed as follows:

or

where \(W\left(t\right)\), \(\Omega \left(t\right)\), \({D}_{t}\) and \({D}_{r}\) denote translational process, rotational process, and translational and rotational diffusion coefficients, respectively. Translational (TMSD, \(\left\langle \delta {r}^{2}(t)\right\rangle\)) and rotational mean-square (RMSD, \(\left\langle \delta {\theta }^{2}(t)\right\rangle\)) displacements can also be extracted as follows^{20}:

In short, DDLS can extract the dynamical processes and the comparison between DDLS and DLS is crucial to the attribution of hierarchical dynamics of particles confined in networks, especially for the identification of rotational processes.

### Data fitting of the stretched exponential functions

Both of multiple modes fitting and CONTIN method are used to analyze the correlation function, where CONTIN is used to help the confirmation of the multiple modes. The correlation function after the non-ergodicity correction was fitted by a sum of exponential decays and stretched exponential decays in order to facilitate comparison to the theory proposed. The correlation function was fitted using ORIGIN software by minimizing the error between the fitted curve and the data, and Iterations were performed until the best-fitting curve was obtained. Besides, the residuals are randomly distributed about zero, and no systematic fluctuations in their mean can be observed.

### Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

## Data availability

All data that support the findings of this work are available within the paper and the Supplementary Information files. All data are available from the corresponding author upon request.

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## Acknowledgements

We would like to thank the support from BL16B1, BL19U2, and BL10U1 of Shanghai Synchrotron Radiation Facility for the access to the synchrotron-based SAXS. The helpful discussion with Prof. Litang Yan is acknowledged. Y.T. thanks the National Natural Science Foundation of China (31971303 and 52273160) for financial support. P.Y. acknowledges the financial support from the National Natural Science Foundation of China (Nos. 22241501 and 92261117), the Guangdong-Hong Kong-Macao Joint Laboratory for Neutron Scattering Science and Technology, the Open Fund of the China Spallation Neutron Source Songshan Lake Science City (DG24313519) and TCL science and technology innovation fund (No. 20222056).

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P. C. Y. designed the research; B. H. X. and Y. L. prepared samples and performed research; Y. T. prepared TMV samples; B. H. X. analyzed data, and B. H. X. and P. C. Y. wrote the paper.

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Xue, B., Liu, Y., Tian, Y. *et al.* The coupling of rotational and translational dynamics for rapid diffusion of nanorods in macromolecular networks.
*Nat Commun* **15**, 6502 (2024). https://doi.org/10.1038/s41467-024-50859-z

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DOI: https://doi.org/10.1038/s41467-024-50859-z

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