Norman Y. Yao^{§*}, Yi-Chia Lin^{*}, Chase P. Broedersz^{?}, Karen E. Kasza^{‡}, Frederick C. MacKintosh^{?}, and David A. Weitz^{*‡¶}

^{§}Harvard College 2008; ^{*}Department of Physics, Harvard University, Cambridge, MA 02138, USA;^{ †}Department of Physics and Astronomy, Vrije Universiteit, 1081HV Amsterdam, The Netherlands;^{ ‡}School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.

**Neurofilaments are found in abundance in the cytoskeleton of neurons, where they act as an intracellular framework protecting the neuron from external stresses. To elucidate the nature of the mechanical properties that provide this protection, we measure the linear and nonlinear viscoelastic properties of networks of neurofilaments. These networks are soft solids that exhibit dramatic strain stiffening above critical strains of 30–70%. Surprisingly, divalent ions, such as Mg ^{2+}, Ca^{2+}, and Zn^{2+} act as effective cross-linkers for neurofilament networks, controlling their solid-like elastic response. This behavior is comparable to that of actin-binding proteins in reconstituted filamentous actin. We show that the elasticity of neurofilament networks is entropic in origin and is consistent with a model for cross-linked semiflexible networks, which we use to quantify the cross-linking by divalent ions.**

## Introduction

The mechanical and functional properties of cells depend largely on their cytoskeleton, which is comprised of networks of biopolymers; these include microtubules, actin, and intermediate filaments. A complex interplay of the mechanics of these networks provides cytoskeletal structure with the relative importance of the individual networks depending strongly on the type of cell [1]. The complexity of the intermingled structure and the mechanical behavior of these networks *in vivo* has led to extensive *in vitro* studies of networks of individual biopolymers. Many of these studies have focused on reconstituted networks of filamentous actin (F-actin) which dominates the mechanics of the cytoskeleton of many cells [2-7]. However, intermediate filaments also form an important network in the cytoskeleton of many cells; moreover, in some cells they form the most important network. For example, in mature axons, neurofilaments, a type IV intermediate filament, are the most abundant cytoskeletal element overwhelming the amount of actin and outnumbering microtubules by more than an order of magnitude [8]. Neurofilaments (NF) are assembled from three polypeptide sub-units NF-Light (NF-L), NF-Medium (NF-M), and NF-Heavy (NF-H), with molecular masses of 68* *kDa, 150* *kDa and 200* *kDa, respectively [8]. They have a diameter *d ~ *10* *nm, a persistence length believed to be of order *l _{p }~ *0.2

*µm and an*

*in vitro*contour length

*L ~*5

*µm. They share a conserved sequence with all other intermediate filaments, which is responsible for the formation of coiled dimers that eventually assemble into tetramers and finally into filaments. Unlike other intermediate filaments such as vimentin and desmin, neurofilaments have long carboxy terminal extensions that protrude from the filament backbone [9]. These highly charged “side-arms” lead to significant interactions among individual filaments as well as between filaments and ions [10]. Although the interaction of divalent ions and rigid polymers has been previously examined, little is known about the electrostatic cross-linking mechanism [11]. Networks of neurofilaments are weakly elastic; however, these networks are able to withstand large strains and exhibit pronounced stiffening with increasing strain [12, 13]. An understanding of the underlying origin of this elastic behavior remains elusive; in particular, even the nature of the cross-linkers, which must be present in such a network, is not known. Further, recent findings have shown that NF aggregation and increased network stiffness are common in patients with amyotrophic lateral sclerosis (ALS) and Parkinson’s. Thus, an understanding of the fundamental mechanical properties of these networks of neurofilaments is an essential first step in elucidating the role of neurofilaments in a multitude of diseases [14]. However, the elastic behavior of these networks has not as yet been systematically studied.*

Here, we report the linear and nonlinear viscoelastic properties of networks of neurofilaments. We show that these networks form cross-linked gels; the cross-linking is governed by divalent ions such as Mg^{2+} at millimolar concentrations. To explain the origins of the network’s elasticity, we apply a semiflexible polymer model, which ascribes the network elasticity to the stretching of thermal fluctuations; this quantitatively accounts for the linear and nonlinear elasticity of neurofilament networks, and ultimately, even allows us to extract microstructural network parameters such as the persistence length and the average distance between cross-links directly from bulk rheology.

## Materials and Methods

#### Materials

Neurofilaments are purified from bovine spinal cords using a standard procedure [9, 15, 16]. The fresh tissue is homogenized in the presence of buffer A (Mes 0.1 M, MgCl_{2} 1 mM, EGTA 1 mM, pH 6.8) and then centrifuged at a K-factor of 298.8 (Beckman 70 Ti). The crude neurofilament pellet is purified overnight on a discontinuous sucrose g radient with 0.8 M sucrose (5.9 ml), 1.5 M sucrose (1.3 ml) and 2.0 M sucrose (1.0 ml). After overnight sedimentation, the concentration of the purified neurofilament is determined with a B radford Assay using bovine serum albumen (BSA) as a standard. The purified neurofilament is dialyzed against buffer A containing 0.8 M sucrose for 76 hours and then 120 μl aliquots are flash frozen in liquid nitrogen and stored at -80 °C.

#### Bulk Rheology

The mechanical response of the cross-linked neurofilament networks is measured with a stress-controlled rheometer (HR Nano, Bohlin Instruments) using a 20 mm diameter 2 degree stainless steel cone plate geometry and a gap size of 50 μm. Before rheological testing, the neurofilament samples are thawed on ice, after which they are quickly pipetted onto the stainless steel bottom plate of the rheometer in the presence of varying concentrations of Mg^{2+}. We utilize a solvent trap to prevent our networks from drying. To measure the linear viscoelastic moduli, we apply an oscillatory stress of the form *σ*(*t*) = *A* sin(*ωt*), where A is the amplitude of the stress and ω is the frequency. The resulting strain is of the form *γ*(*t*)* = B *sin(*ωt + φ*)* *and yields the storage modulus and the loss modulus . To determine the frequency dependence of the linear moduli, G'(ω) and G”(ω) are sampled over a range of frequencies from 0.006–25 rad/s. In addition, we probe the stress dependence of the network response by measuring G'(ω) and G”(ω) at a single frequency varying the amplitude of the oscillatory stress. To probe nonlinear behavior, we utilize a differential measurement, an effective probe of the tangent elastic modulus, which for a viscoelastic solid such as neurofilaments provides consistent nonlinear measurements of elasticity in comparison to other nonlinear methods [17‑19]. A small oscillatory stress is superimposed on a steady pre-stress, σ, resulting in a total stress of the form *σ*(*t*) = *σ* + |*δσ*| sin(*ωt*). The resultant strain is *γ*(*t*) = *γ* + |*δγ*| sin(*ωt + φ*), yielding a differential elastic modulus and a differential viscous modulus [2].

#### Scaling Parameters

To compare the experiments with theory, we collapse the differential measurements onto a single master curve by scaling the stiffness K’ and stress σ by two free parameters for each data set. According to theory, the stiffness versus stress should have a single, universal form apart from these two scale factors. We determine the scale factors by cubic-spline fitting the data sets to piecewise polynomials; these polynomials are then scaled onto the predicted stiffening curve using a least squares regression.

## Results and Discussion

To quantify the mechanical properties of neurofilaments, we probe the linear viscoelastic moduli of the network during gelation, which takes approximately one hour; we characterize this by continuously measuring the linear viscoelastic moduli at a single frequency, ω = 0.6 rad/s. Gelation of these networks is initiated by the addition of millimolar amounts of Mg^{2+} and during this process we find that the linear viscoelastic moduli increase rapidly before reaching a plateau value. We measure the frequency dependence of the linear viscoelastic moduli over a range of neurofilament and Mg^{2+} concentrations. To ensure that we are probing the linear response, we maintain a maximum applied stress amplitude below 0.01 Pa, corresponding to strains less than approximately 5%; we find that the linear moduli are frequency independent for all tested frequencies, 0.006–25 rad/s. Additionally, neurofilament networks behave as a viscoelastic solid for all ranges of Mg^{2+} concentrations tested and the linear storage modulus is always at least an order of magnitude greater than the linear loss modulus, as shown in Fig. 1. This is indicative of a cross-linked gel and allows us to define a plateau elastic modulus *G _{0}*

_{ }[20].

The elasticity of neurofilament networks is highly nonlinear; above critical strains *γ _{c}* of 30–70%, the networks show stiffening up to strains of 300% [21], as shown in Fig. 2. This marked strain-stiffening occurs for a wide variety of Mg

^{2+}and neurofilament concentrations. In addition, by varying the neurofilament concentration

*c*and the Mg

_{NF}^{2+}concentration

*c*, we can finely tune the linear storage modulus

_{Mg}*G*

_{0}over a wide range of values, as seen in Fig. 3. The strong dependence of

*G*

_{0}on Mg

^{2+}concentration is reminiscent of actin networks cross-linked with the incompliant cross-linkers such as scruin [2, 22, 23]; this suggests that in the case of neurofilaments, Mg

^{2+}is effectively acting as a cross-linker leading to the formation of a viscoelastic network. Thus, the neurofilaments are cross-linked ionically on length scales comparable to their persistence length; hence, they should behave as semiflexible biopolymer networks. We therefore hypothesize that the network elasticity is due to the stretching out of thermal fluctuations. These thermally driven transverse fluctuations reduce neurofilament extension resulting in an entropic spring. To consider the entropic effects we can model the Mg

^{2+}-cross-linked network as a collection of thermally fluctuating semiflexible segments of length

*l*, where

_{c}*l*is the average distance between Mg

_{c}^{2+}cross-links. A convincing test of the hypothesis of entropic elasticity is the nonlinear behavior of the network. When the thermal fluctuations are pulled out by increasing strain, the elastic modulus of the network exhibits a pronounced increase.

To probe this nonlinear elasticity of neurofilament networks, we measure the differential or tangent elastic modulus *K’*(*σ*) at a constant frequency ω = 0.6* *rad/s for a variety of neurofilament and Mg^{2+} concentrations. If the network elasticity is indeed entropic in origin, this can provide a natural explanation for the nonlinear behavior in terms of the nonlinear elastic force-extension response of individual filaments that deform affinely. Here, the force required to extend a single filament diverges as the length approaches the full extension *l _{c}*, since [24-26]. Provided the network deformation is affine, its macroscopic shear stress is primarily due to the stretching and compression of the individual elements of the network. The expected divergence of the single-filament tension leads to a scaling of ; we therefore expect a scaling of network stiffness with stress of the form

*K’*(

*σ*) ~

*σ*

^{3/2}in the highly nonlinear regime [2]. Indeed, ionically cross-linked neurofilament networks show remarkable consistency with this affine thermal model for a wide range of neurofilament and cross-link concentrations, as shown in Fig. 4. This consistency provides convincing evidence for the entropic nature of the network’s nonlinear elasticity [2, 25].

The affine thermal model also suggests that the functional form of the data should be identical for all values of *c _{Mg}* and

*c*. To test this, we scale all the data sets for

_{NF}*K’*(

*σ*) onto a single master curve. This is accomplished by scaling the modulus by a factor

*G’*and the stress by a factor

*σ’*. Consistent with the theoretical prediction, all the data from various neurofilament and Mg

^{2+}concentrations can indeed be scaled onto a universal curve, as shown in Fig. 5. The scale factor for the modulus is the linear shear modulus

*G’*=

*G*

_{0}, while the scale factor for the stress is a measure of the critical stress

*σ*at which the network begins to stiffen. This provides additional evidence that the nonlinear elasticity of the Mg

_{c }^{2+}-cross-linked neurofilament networks is due to the entropy associated with single filament stretching.

To explore the generality of this ionic cross-linking behavior, we use other divalent ions including Ca^{2+} and Zn^{2+}. We find that the effects of both of these ions are nearly identical to those of Mg^{2+}; they also cross-link neurofilament networks into weak elastic gels. This lack of dependence on the specific ionic cross-link lends evidence that the interaction between filaments and ions is electrostatic in nature. This electrostatic interaction would imply that the various ions are acting as salt-bridges, thereby cross-linking filaments into low energy conformations.

The ability to scale all data sets of K'(σ) onto a single universal curve also provides a means to convincingly confirm that the linear elasticity is entropic in origin. To accomplish this, we derive an expression that relates the two scaling parameters to each other. For small extensions δl of the entropic spring, the force required can be derived from the wormlike chain model giving . Assuming an affine deformation, whereby the macroscopic sample strain can be translated into local microscopic deformations, and accounting for an isotropic distribution of filaments, the full expression for the linear elastic modulus of the network is given by

(1)

where κ = *k _{B}Tl_{p} *is the bending rigidity of neurofilaments,

*k*is the thermal energy, and

_{B}T*ρ*is the filament-length density [2, 25, 27, 28]. The density

*ρ*is also proportional to the mass density

*c*, and is related to the mesh size ζ of the network by [29]. Furthermore, the model predicts a characteristic filament tension proportional to , and a characteristic stress

_{NF}(2)

[2, 22, 25]. Thus, if the network’s linear elasticity is dominated by entropy, we expect the scaling *c _{NF}^{1/2}G*

_{0 }~

*σ*, where the pre-factor should depend only on

_{c}^{3/2}*k*and

_{B}T*l*; although the pre-factor will differ for different types of filaments it should be the same for different networks composed of the same filament type and at the same temperature, such as ours. Thus, plotting

_{p}*c*

_{NF}^{1/2}G_{0 }as a function of

*σ*for different neurofilament networks at the same temperature should result in collapse of the data onto a single curve characterized a 3/2 power law; this even includes systems with different divalent ions or different ionic concentrations. For a variety of divalent ions, we find that

_{c}*c*

_{NF}^{1/2}G_{0 }~

*σ*, where

_{c}^{z}*z =*1.54

*±*

*0.14 in excellent agreement with this model, as shown in Fig. 6. It is essential to note that the 3/2 exponent found here is not a direct consequence of the 3/2 exponent obtained in Fig. 5, which characterizes the highly nonlinear regime. Instead, the plot of*

*c*

_{NF}^{1/2}G_{0}

*as a function of*

_{ }*σ*probes the underlying mechanism and extent of the linear elastic regime.

_{c}For a fixed ratio of cross-links *R = c _{Mg}/c_{NF}*, we expect cross-linking to occur on the scale of the entanglement length, yielding

*l*~

_{c }*c*

_{NF}

^{‑}*[25, 27, 30]. Thus, we expect the linear storage modulus to scale with neurofilament concentration as*

^{2/5 }*G*

_{0 }~

*c*[25]. For

_{NF}^{11/5 }*R*= 1000, we find an approximate scaling of

*G*

_{0 }~

*c*, consistent with the predicted power law, as shown in the inset of Fig. 6. Interestingly, the stronger concentration dependence of

_{NF}^{25}*G*

_{0 }may be a consequence of the dense cross-linking that we observe. Specifically, for densely cross-linked networks, corresponding to a minimum

*l*on the order of the typical spacing between filaments as we observe here, the model in Eq. (1) predicts

_{c }*G*

_{0 }~

*c*[25]. The agreement with the affine thermal model in both the linear and nonlinear regimes confirms the existence of an ionically cross-linked neurofilament gel whose elasticity is due to the pulling out of thermal fluctuations.

_{NF}^{25}The ability of the affine thermal model to explain the elasticity of the neurofilament network also suggests that we should be able to quantitatively extract network parameters from the bulk rheology. The model predicts that

(3)

and

(4)

where *ρ _{ }≈* 2.1×10

^{13}

*m*

^{-2 }for neurofilament networks at a concentration of 1

*mg/mL. This yields a persistence length*

*l*0.2

_{p }≈*µm which is in excellent agreement with previous measurements [31]. In addition, we find that*

*l*0.3

_{c }≈*µm which is close to the theoretical mesh size ≈ 0.26 μm; surprisingly, this is far below the mesh size of 4 µm inferred from tracer particle motion [1]. Such particle tracking only provides an indirect measure: in weakly cross-linked networks, for instance, even particles that are larger than the average inter-filament spacing will tend to diffuse slowly.*

To further elucidate the cross-linking behavior of Mg^{2+}, we explore the dependence of *l _{c}* on both

*c*and

_{Mg}*c*, based on Eq. (3-4). Based on the form of

_{NF}*G*

_{0}and

*σ*, we expect that . Assuming that Mg

_{c}^{2+}is acting as the cross-linker and that

*l*is also the typical distance between binary collisions of filament chains we would expect that , where

_{c}*l*is the entanglement length. Thus, for a given concentration of neurofilaments

_{e}(5)

[25]. This yields

(6)

where *X *is the exponent of the Mg^{2+} concentration. Naively, we would expect that *X ≈ *1* *which would imply that doubling the concentration of Mg^{2+} would halve the average distance between cross-links. Empirically we find a much weaker dependence on *c _{Mg}*, where . This weaker dependence suggests that mM concentrations of Mg

^{2+}actually saturate our networks. This is consistent with a calculation of the percentage of Mg

^{2+}ions, which actually act as cross-links. The number of cross-linking ions per cubic meter is ; the number density of ions,

*N*in a standard 5

*mM Mg*

^{2+}concentration is

*N ≈*30

*×*

*10*

^{23}. Thus, there is an excess of Mg

^{2+}ions available to act as cross-linkers; this may account for the weak cross-link dependence. A similarly weak dependence has been seen previously with actin networks in the presence of the molecular motor heavy meromysin where

*X*was found to be 0.4 and thus, , where

*c*is the actin concentration and

_{A}*c*is the heavy meromysin concentration [32]. Utilizing our empirical power law for

_{HMM}*c*, we are able to collapse the curves such that which is in excellent agreement with the predicted exponent 2/5, as shown in Fig. 7. The fact that the cross-linking distance

_{Mg}*l*scales directly with

_{c }*c*further confirms the role of Mg

_{Mg}^{2+}as the effective ionic cross-linker of the neurofilament networks. Thus, our findings demonstrate both the entropic origin of neurofilament network’s elasticity as well as the role of Mg

^{2+}as an effective ionic cross-linker.

## Conclusion

We measure the linear and nonlinear viscoelastic properties of cross-linked neurofilament solutions over a wide range Mg^{2+} and neurofilament concentrations. Neurofilaments are interesting intermediate filament networks whose nonlinear elasticity has not been studied systematically. We show that the neurofilament networks form densely cross-linked gels, whose elasticity can be well understood within an affine entropic framework. We provide direct quantitative calculations of *l _{p}* and

*l*from bulk rheology using this model. Furthermore, our data provides evidence that Mg

_{c}^{2+}acts as the effective ionic cross-linker in the neurofilament networks. The weaker than expected dependence that we observe suggests that Mg

^{2+}may be near saturation in our networks. Future experimental work with other multivalent ions is required to better understand the electrostatic interaction between filaments and cross-links; this would lead to a better microscopic understanding of the effects of electrostatic interactions in the cross-linking of neurofilament networks. Moreover, the effect of divalent ions on the cross-linking of networks of other intermediate filaments would also be very interesting to explore.

## Acknowledgments

This work was supported in part by the NSF (DMR-0602684 and CTS-0505929), the Harvard MRSEC (DMR-0213805), and the Stichting voor Fundamenteel Onderzoek der Materie (FOM/NWO).

## References

- S. Rammensee, P. A. Janmey, and A. R. Bausch, Eur. Biophys. J. Biophy. 36, 661 (2007).
- M. L. Gardel
*et al.*, Science 304, 1301 (2004). - B. Hinner
*et al.*, Phys. Rev. Lett. 81, 2614 (1998). - R. Tharmann, M. Claessens, and A. R. Bausch, Biophys. J. 90, 2622 (2006).
- C. Storm
*et al.*, Nature 435, 191 (2005). - J. Y. Xu
*et al.*, Biophys. J. 74, 2731 (1998). - J. Kas
*et al.*, Biophys. J. 70, 609 (1996). - P. C. Wong
*et al.*, J. Cell Biol. 130, 1413 (1995). - J. F. Leterrier
*et al.*, J. Biol. Chem. 271, 15687 (1996). - S. Kumar, and J. H. Hoh, Biochem. Biophy. Res. Co. 324, 489 (2004).
- G. C. L. Wong, Curr. Opin. Colloid In. 11, 310 (2006).
- O. I. Wagner
*et al.*, Exp. Cell Res. 313, 2228 (2007). - L. Kreplak
*et al.*, J. Mol. Biol. 354, 569 (2005). - S. Kumar
*et al.*, Biophys. J. 82, 2360 (2002). - A. Delacourte
*et al.*, Biochem. J. 191, 543 (1980). - J. F. Leterrier, and J. Eyer, Biochem. J. 245, 93 (1987).
- N. Y. Yao, R. Larsen, and D. A. Weitz, J. Rheol. 52, 13 (2008).
- C. Baravian, G. Benbelkacem, and F. Caton, Rheol. Acta 46, 577 (2007).
- C. Baravian, and D. Quemada, Rheol. Acta 37, 223 (1998).
- M. Rubenstein, and R. Colby, Polymer Physics (Oxford University Press, Oxford, 2004).
- D. A. Weitz, and P. A. Janmey, P. Natl. Acad. Sci. USA 105, 1105 (2008).
- M. L. Gardel
*et al.*, Phys Rev Lett 93 (2004). - A. R. Bausch, and K. Kroy, Nat. Phys. 2, 231 (2006).
- C. Bustamante
*et al.*, Science 265, 1599 (1994). - F. C. Mackintosh, J. Kas, and P. A. Janmey, Phys. Rev. Lett. 75, 4425 (1995).
- M. Fixman, and J. Kovac, J. Chem. Phys. 58, 1564 (1973).
- A. N. Semenov, J. Chem. Soc. Faraday T. Ii 82, 317 (1986).
- F. Gittes, and F. C. MacKintosh, Phys. Rev. E 58, R1241 (1998).
- C. F. Schmidt
*et al.*, Macromolecules 22, 3638 (1989). - T. Odijk, Macromolecules 16, 1340 (1983).
- Z. Dogic
*et al.*, Phys. Rev. Lett. 92 (2004). - R. Tharmann, M. Claessens, and A. R. Bausch, Phys. Rev. Lett. 98 (2007).