A Science Friction Story: Molecular Interactions in Semiflexible Polymer Networks

Established model theories, developed to capture the mechanical behavior of soft, complex materials composed of semiflexible polymers, assume that entropic interactions between filaments are primarily responsible for determining the mechanical response. In recent studies, the generally accepted tube model has been challenged in terms of this basic assumption about filament–filament interactions, but also because of its predictions regarding the frequency dependence of the elastic modulus in the intermediate frequency regime. A central question is how molecular interactions and friction between network constituents influence the rheological response of isotropic entangled networks of semiflexible polymers. It has been previously shown that friction forces between aligned pairs of actin filaments are not negligible. Here, the influence of friction forces and attractive interactions on network rheology is systematically investigated by means of targeted surface modification. It is shown that these forces have a qualitative and quantitative influence on the viscoelastic properties of semiflexible polymer networks and contribute to their response to nonlinear deformations. By comparing two polymer model systems with respect to their surface compositions, a possible explanation is given about the origin of acting forces on a molecular level.


Introduction
Semiflexible polymers are intensively studied materials due to their special mechanical properties and their significant role in complex soft matter.Biological systems rely on semiflexible polymers as building blocks of intracellular scaffolds and extracellular matrices.As essential components for cell-shape regulation and force generation to promote cell migration and division, they take on a special role within the intracellular skeleton. [2][12] The characterization of individual filaments is theoretically embedded within the generally accepted wormlike chain (WLC) model. [5,13]Based on the WLC, the tube concept is constructed to capture the network properties of entangled polymer solutions.It exclusively accounts for entropic interactions between filaments while neglecting any attractive or repulsive effects caused by the molecular structure.Based on these assumptions, fluctuation modes are suppressed due to surrounding polymers establishing a confinement in a tube-like region. [11,14]Some important signatures in the frequency dependent rheology are qualitatively reflected by the model.However, experimental studies on filamentous actin (F-actin) and DNA-based (Deoxyribonucleic acid) semiflexible polymers questioned the model's central scaling prediction with respect to persistence length. [15,16]Additionally, recent work emphasized the importance of attractive interfilamentous interactions in networks caused by hydrophobicity or electrostatic interactions, which are not considered by the tube model. [17,18]Despite representing some crucial effects in the viscoelastic network behavior, the tube model predicts a flat plateau of G′ in the intermediate frequency range in a double logarithmic representation.[19][20][21] Employing optical tweezers, Ward et al. showed surprisingly high friction forces between a pair of aligned actin filaments. [22]In multifilament bundles, contractile forces that can be attributed to pairwise filament Figure 1.Modification of semiflexible filaments with PEG molecules.a) Actin monomers have primary amines like those contained in lysine residues exposed on their surface, which serve as targets for a modification via NHS Ester reaction chemistry (polymerization relevant domains are labeled green [1] ).Semiflexible polymers can be artificially recreated via DNA nanotechnology.DX5 nanotubes are a DNA-based reference system, composed of five different, partial complementary, single-stranded DNA oligonucleotides, which hybridize in b) unit elements that build up to elongated tubes and can be modified via copper-free click chemistry.c) PEG-modified unit elements/monomers hybridize/polymerize into filaments with PEGylated surfaces.d) PEG molecules act as spacers to screen filament-filament interactions over three controlled distances determined by the length of the respective attached PEG molecules.e) The viscoelastic properties of reconstituted networks are experimentally accessible via bulk shear rheometry.
interactions determine the relaxation behavior of the bundles. [23]he mechanical response of actively excited actin bundles formed by depletion forces suggests that interactions and crosslinking are difficult to distinguish. [23,24]On a network level, recent studies found that attractive filament-filament interactions have a qualitative and quantitative impact on the rheological behavior of isotropic networks of semiflexible polymers. [17,18,21,25]A theory that accounts for attractive sticky interactions in networks and explains the experimentally observed power law behavior was proposed by Kroy and Glaser. [26]The so-called glassy wormlike chain (GWLC) extends the wormlike chain by incorporating interactions between individual filaments in a network by presuming a glassy surrounding. [26]This is mathematically described by the stretching of the WLC's long-wavelengths Eigen modes with an Arrhenius-like exponential factor.The model has been successfully employed in experimental research, providing a quantitative characterization of non-specific interactions between network components.Previous work suggested by comparing different model systems within the framework of the model that rheological signatures depend on interactions and are not determined by persistence length alone, as previously thought.However, no solid statements can be made from these comparisons, which make it difficult to isolate effects due to inherently different material parameters. [17,18,21]However, the interactions that are investigated and quantified with the model are not specified, and their origins remain speculative.Ward et al. established an approach to reduce the friction between actin filaments by a surface modification with polyethylene glycol (PEG) molecules that act as entropic springs to prevent friction interactions in an anisotropic arranged two-filament system. [22]Here, we exploit this strategy to systematically modify the prerequisites for networks comprised of two different polymer model systems, thus, for the first time, enabling an examination of the role friction plays in bulk behaviors.By comparing their rheological properties, we isolated the impact of friction and/or attractive forces on network mechanics.We compared F-actin to a DNA-based polymer model system with a double-crossover (DX) motif built according to the construction plans of DNA nanofabrication. [27]The particular DNA-based nanotube (DX5 nanotubes) design in this study, introduced by Rothemund et al., [28] exhibits similar mechanical properties as F-actin.With comparable contour lengths and bending rigidities of respective individual filaments, the DX5 nanotubes differ from actin in their surface composition with respect to topography, hydrophobicity, and electric charges, each of which is plausible to contribute to friction and/or attractive filament-filament interactions.Previous studies compared different polymer model systems with regard to their "stickiness" to understand their network architectures and their role within the cytoskeleton. [17,18,21]Instead, here we systematically investigated the influence of surface-surface interactions between filaments in isotropic networks of preserved geometry, pioneering the exploration of molecular interactions in multi-filament systems and their intrinsic impact on higher ordered structures.

Linear Network Rheology
Networks consisting of each filament type were compared to those comprised of modified components (Figure 1).We synthesized three different modifications of actin as well as DX5 nanotubes by decorating the filaments' surfaces with PEG molecules of three different lengths (see Supporting Information).Gel electrophoresis indicated successful PEG binding to the respective surfaces as well as a more precise functionalization for the DX5 tubes, reflected in narrower bands (Figure S1, Supporting Information).PEG modification is an established method to reduce the nonspecific binding of quantum dots in live cell assays. [29]The screening effect of PEG was shown to depend on molecule length. [30]Here, the surface coating prevents attractive filament-filament interactions on length scales depending on the end-to-end distance of the PEG chain, which is ≈6.35 nm per kDa in outstretched configuration. [31]With this approach, we were able to systematically alter mutual inter-filament interactions.
Viscoelastic properties of suspended networks of semiflexible polymers are experimentally accessible via bulk shear rheology, yielding the complex shear modulus G*(f) = G′(f) + i G″(f).The results of linear rheology are summarized in Figure 2.For both F-actin and DX5 nanotubes, elastic moduli as well as loss moduli decreased with increasing lengths of PEG chains (see Figure S2, Supporting Information).The loss factor tan , quantifying the relation between viscosity and elasticity as tan  = G″/G′, shows a general increase upon PEG modification for the DX5 system (Figure 2c).This reflects a gradual transition from elasticity-to more viscosity-dominated networks.We suggest that the lack of a clear trend in the behavior of the loss factor for actin is due to the less controllable nature of the surface modification of the filaments compared to DX5 tubes.This is also reflected in broad bands in the gel electrophoresis (see Supporting Information).By fitting the power law behavior of the elastic modulus in the double logarithmic representation to a linear function f fit = a × x + b over frequencies ranging from 1 to 10 Hz, we determined the slopes of the elastic plateau for both systems (Figure 2d).With 0.22, the slope value for unmodified F-actin networks is in agreement with previous findings. [21]The value for unmodified DX5 tube networks was calculated to be 0.26 and is thus comparable to F-actin networks.Both systems displayed slopes that incrementally increased up to values of 0.49 for actin-PEG-5 kDa and 0.61 for DX5-PEG-5 kDa.These findings confirm the results of previous experimental studies pointing to higher values for loss factor and the slope of plateau, which describes the slope of the elastic modulus G′ in the intermediate frequency regime (in a log-log plot), for presumably less interactive polymer systems. [17,18,21,32]

Glassy Wormlike Chain
The impact of the surface modification on the frequency dependent viscoelastic properties of the investigated networks can be explained within the frame of the GWLC model. [26]The filaments' mode relaxation times   >  Λ of all Eigen modes of (half) wavelength  longer than a characteristic interaction length Λ are stretched by the factor exp (N).Here, N ≡ /Λ − 1 gives the interactions per wavelength  and corresponds to the entanglement length. [26]Attractive "sticky" interactions between filaments are assumed to cause a retardation of relaxation dynamics, in particular for long wavelength modes, which are thought to determine the points of contact between filament contours.This is governed by the stretching parameter .We attribute this slowdown of polymer dynamics to nonspecific filament-filament interactions and consequently interpret  as a quantification of the interaction strength. [17,18,21,26]A more detailed description of this model is presented in the Supporting Information.
The linear rheology data was analyzed by fitting the storage modulus G′(f) to the mean curves of the measured data with where  = 2f and () is the micro-rheological response function of the GWLC to a point force at its ends in the high frequency limit, which describes the motion of a filament segment as a sum of modes. [33]The fitting routine was implemented in a selfwritten Python script.Contour lengths and persistence lengths for actin as well as DX5 nanotubes were measured via fluorescence microscopy and used accordingly for the evaluation (Section SIV, Supporting Information).As we compare networks of the same molarities, we emanate from similar mesh sizes, leaving us with the characteristic interaction length Λ and the stretching parameter  as the only non-fixed fitting parameters.Assuming a cubic network of rigid rods with mass per length m L and the protein concentration c, the mesh size  of a semiflexible polymer network can be estimated via With m L = 2.66 × 10 −11 g/m for actin and m L = 2.49 × 10 −11 g/m for DX5 nanotubes, [28] the employed concentration of c = 1 g/L for both investigated systems yields  Actin = 0.28 μm and  DX5 = 0.26 μm.With a contour length of roughly 6.35 nm per kDa, the end-to-end distance of the longest applied PEG molecules in solution can be expected to be in the order of 10 nm.It can be assumed that the effect of the PEGylation on the mesh size of the modified networks is negligible.The drag coefficient was set to  ⊥ = 2 mPas, a representative value for water.We found  = 3.33 for untreated actin and  = 1.47 for DX5 nanotubes, respectively.We interpret the stretching parameter  as the strength of attractive interactions between individual filaments in a network.The magnitude of this parameter is sensitive to small deviations in the preset of fixed parameters such as mesh size or contour lengths of filaments.Hence, we did not only consider absolute values but also accounted for the gradual decline of the strength of interactions upon surface modification.
We examined the stretching parameter  for both polymer systems and their respective modifications.Distributions differed significantly for all combinations within the F-actin derivatives, except for actin-PEG-1 kDa compared to actin-PEG-2 kDa.For the DX5 nanotube system, only DX5-PEG-2 kDa and DX5-PEG-5 kDa were recorded as directly neighbored distributions with no significant deviation.Interestingly, the reduction of the interaction strength upon a gradual increase in the distance over which interactions are shielded is not the same for actin and DX5 tube networks.The decrease in the interaction strength  can be de-scribed with an exponential function for both systems (Figure 3c).With an exponent of  = −0.27for actin, the overall reduction is clearer than for the DX5 system, where we found an exponent of  = −0.12.Conjectures about the principles of filament-filament interactions can be derived from the step-wise reduction, illustrated in Figure 3d.This reveals a more pronounced interaction decrement for F-actin networks of filaments covered with 1 kDa PEG. Ward et al. showed that friction forces between aligned actin filaments depend on their orientation of polarity with respect to each other, concluding that the surface topography itself has a strong impact on the inter-filament friction. [22]Actin filaments, in contrast to DX5 nanotubes, have a surface structure that promotes friction.In this context, the drastic reduction can be attributed to the compensation of steric forces that originate from surface-surface friction.With a diameter of ≈7 nm, [34] the roughness of the actin filaments' surface is compensated by 1 kDa PEG molecules, with a contour length of roughly 6.35 nm.With a further increase in PEG molecule lengths from 1 to 2 kDa,  reduces more for the DNA-based system.Hydrophobic interactions between molecules reportedly decay exponentially like van der Waals dispersion forces over a range of ≈10 nm and are an order of magnitude stronger. [35]Monomeric actin consists of 375 amino acids, some of which are hydrophobic. [36]owever, during protein folding, most of these are inherently turned to the monomer's interior to prevent contact with its aqueous surrounding. [37,38]Consequently, the reduction due to the elimination of hydrophobic effects is presumably stronger for the DX5 nanotubes, wherein adenine and thymine in particular promote hydrophobic-driven interactions. [39]Electrostatic interactions can be orders of magnitude stronger than van der Waals interactions. [40]Being relevant for DNA-based materials as well as amino acids, [41,42] they presumably contribute to the filamentfilament interactions of both systems in a comparable manner.

Nonlinear Rheology
We tested the mechanical behavior of F-actin and DX5 nanotube networks under high deformations by applying a constant strain rate and measuring the stress responses.We exposed the samples to strains in the nonlinear regime prior to calculating the differential modulus K, defined as the local derivative of stress  over strain  as described by Semmrich et al. [32] Consistent with previous studies, we found a weak stiffening behavior for F-actin networks. [17,18,21,32]This is reflected in a slight increase of K as illustrated in Figure 4a.For modified F-actin networks as well as for networks of both unmodified and modified DX5 nanotubes (Figure 4b), we found no stiffening behavior under nonlinear deformations.This is in line with the  values calculated from the linear data.The yield stresses, which denote the transition from elastic to plastic behavior, shifted to lower values for decreasing associated  values obtained from the model for the linear data (Figure 3).Network stiffening as a response to large deformations has been shown for various intermediate filaments and, in particular, for keratins, which are known to provide structure and protection against large deformations to epithelial cells. [21,43,44]][47] Previous studies showed that systems consisting of filaments that are assigned high  values express drastic stiffening upon shear deformations. [17,18,21]Here we show that stiffening behavior for F-actin networks is eliminated while softening behavior is generally increased by screening interactions at distances of a few nanometers.From this, we conclude that actin filaments in entangled networks interact strongly enough to influence the nonlinear network behavior.

Statistical Analysis
The entire data processing and presentation were done with self elaborated Python scripts.The investigation of linear rheology is based on a data set of 12 measurements per sample.Evaluated parameters illustrated in Figures 2 and 3c are presented in a box plot representation, showing the mean values, interquartile ranges, as well as whiskers.For statistical analysis, a Mann-Whitney-Utest was used to determine p values.Significance levels are indicated by * symbols with significance levels *p < 0.05, **p < 0.01, and * * *p < 0.001.Elastic moduli, presented in Figure 3, are plotted as mean curves with respective standard deviations indicated by shaded areas in the respective colors.Nonlinear rheology presented in Figure 4 shows the mean curves, based on four measurements per sample.Apparent noise in the low stress regime was compensated by the application of a spline smoothing function.Noise dominated data was excluded from further evaluation.Each curve was set to a starting value of 1 by division by K lin , which was defined at the first non-negative stress value.

Discussion
Previous in vitro studies investigated the influence of crosslinkers of various kinds on the mechanics of isotropic networks of semiflexible polymers. [3,8,33,45,46]Here, we show that eliminating attractive interactions and friction forces by means of filament surface modification leads to faster stress relaxation.Through a systematic screening of surface-surface interactions, we determined rheological signatures associated with friction or attractive interactions.The introduced modification approaches are based on copper free click chemistry in the case of DX5 nanotubes and on amide linkage between NHS ester activated compounds and primary surface amines in the case of actin.In general, the presented surface functionalization can be transferred to any system that is based on proteins, amino molecules, peptides, or DNA.However, our study also suggests that the modification of complex biomolecules, in contrast to simple systems such as the DNA-based nano tubes, leads to less significant results.
We found a progressive network softening for increasing spacer molecule lengths between filaments.This is accompanied by a significant general increase in the slopes of the elastic moduli over frequency and a transition to less elastic networks.
The apparent increase in loss factor values for the DX5 tube system is not visible for the actin networks, which show no significant change upon modification.We attribute this to the less targeted modification for actin resulting in less significant data, which complicates the identification of small changes in viscoelastic properties.
The evaluation with the GWLC model yielded gradually rising relaxation dynamics for long wavelength Eigen modes, indicating a decrease of the impact of attractive interactions in the overall network rheology.From this we conclude that there is a causal relationship between the apparent power law behavior as well as the network softening with the dominance of mutual interactions of semiflexible polymers in an entangled network, confirming the assumptions of earlier studies. [17,18,21]In contrast to transient crosslinkers, which stretch the relaxation times of long wavelength Eigen modes depending on their binding affinity, [48] the reduction of interactions leads to shorter relaxation times.These results are not explainable in the frame of the tube model, which does not consider attractive interactions on a molecular level.The increase in power law behavior upon reduction of interactions contradicts the theory, which predicts a flat plateau for the elastic modulus while not considering attractions between network constituents.
Mathematically explaining the slowdown of filament dynamics with unspecified interactions, the GWLC model does not provide an explanation about their fundamental nature.The decomposition of acting forces is impeded by the fact that range alone is not a sufficient criterion to be able to differentiate explicitly between different types of interactions.However, we can provide a rough estimate of the various forces.Hydrogen bonding constitutes a crucial mechanism for the stabilization of monomeric actin as well as its assembly to F-actin. [3]In DNA, it drives the formation of base pairs by hybridization of complementary nucleotides. [27]Being a relevant force on the sub-nanometer range for intramolecular processes as well as for nucleotide base pairing, we do not consider hydrogen bonding as a main contribution to filament-filament interactions, due to its short range and the occupation of binding partners as a result of assembly or hybridization processes.Ward et al., measured surprisingly high friction forces for a pair of actin filaments moved against each other. [22]In compliance with this, we assume that steric hindrance due to filament surface texture plays a similar role in entangled network rheology.On intermediate distances, our results show a stronger reduction of attraction for the DNA based system, suggesting that hydrophobicity causes attractive forces.This is in line with recent findings that fibrils of -synuclein stiffen significantly due to temperature dependent interactions between hydrophobic patches on the fibril surface. [25]The described interactions and friction forces influence the rheological behavior of networks under large deformations in the nonlinear regime.Strain stiffening and strain softening are critical properties that determine a system's mechanical integrity.Keratin filaments intrinsically form cross-linked networks. [49]The stiffening/softening behavior in vimentin networks was shown to be determined by crosslinker unbinding rates. [50]Our results suggest that interactions alone can induce a weak strain stiffening behavior in F-actin networks, which play a crucial role in cellular dynamics. [12,33,45,46,51]Mutual interactions between filaments have an impact on the mechanics of higher-ordered structures.
On a cellular level, hydrophobic forces between cytoskeletal filaments due to the presence of deuterium oxide are suspected to slow down cellular dynamics like proliferation and migration to a remarkable degree. [9,52]This highlights the relevance of interactions for cellular systems that mechanically rely on networks of semiflexible polymers.

Figure 2 .
Figure 2. a) The loss factor tan  = G″/G′ exhibits a general increase upon PEG modification for DX5 nanotubes, indicating a tendency toward viscous behavior.A decreasing slope of the elastic modulus is associated with a transition to networks of highly inter-attractive networks and cross-linked systems, where the frequency dependence is less pronounced.b) In contrast, we found increasing slopes for increasing spacers between interacting filaments, reflecting the successive elimination of the contribution of attractive interactions.Significant different distributions, according to a Mann-Whitney-U-Test, are indicated by * symbols with significance levels *p < 0.05, **p < 0.01, and * * *p < 0.001.

Figure 3 .
Figure 3. Attractive interactions between individual filaments in networks of entangled semiflexible polymers decrease in strength for increasing length of PEG chains covering the filaments' surface.This is deduced from the frequency dependent elastic modulus for a) actin and b) DX5 nanotubes.Solid lines show the mean curves of measured G′(f), smoothed with a univariate spline fit.Dashed lines represent the fits according to the GWLC model.Each curve represents the mean of 12 measurements.c) The  values resulting from fitting to the GWLC model are listed in the legend and plotted with error bars representing respective standard deviations.Dashed lines represent exponential fits according to f(x) = exp (x).The resulting  values indicate a more drastic decrease for the F-actin networks.The relative gradual change upon stepwise extension of PEG molecules is illustrated in (d).Significant different distributions, according to a Mann-Whitney-U-Test, are indicated by * symbols with significance levels *p < 0.05, **p < 0.01, and * * *p < 0.001.

Figure 4 .
Figure 4. Attractive interactions affect the mechanic behavior under nonlinear network deformations.a) Stiffening is weak for actin networks and vanishes for networks of PEG modified actin filaments, manifesting in a flattening of the differential shear modulus.b) The DX5 nanotube networks exhibit no stiffening behavior.The yield stress is shifted to lower stress values for increasing lengths of PEG molecules in both investigated systems.