Single microtubules and small networks become significantly stiffer on short time-scales upon mechanical stimulation

The transfer of mechanical signals through cells is a complex phenomenon. To uncover a new mechanotransduction pathway, we study the frequency-dependent transport of mechanical stimuli by single microtubules and small networks in a bottom-up approach using optically trapped beads as anchor points. We interconnected microtubules to linear and triangular geometries to perform micro-rheology by defined oscillations of the beads relative to each other. We found a substantial stiffening of single filaments above a characteristic transition frequency of 1-30 Hz depending on the filament's molecular composition. Below this frequency, filament elasticity only depends on its contour and persistence length. Interestingly, this elastic behavior is transferable to small networks, where we found the surprising effect that linear two filament connections act as transistor-like, angle dependent momentum filters, whereas triangular networks act as stabilizing elements. These observations implicate that cells can tune mechanical signals by temporal and spatial filtering stronger and more flexibly than expected.


Introduction
Today, we know that cells across all domains are mechanosensitive 1 , and that mechanosensitivity is the base for sensing quite different stimulus qualities including osmotic challenges, gravity, movements or even sound. In addition, mechanosensitivity is used to organize and integrate cells and organs into functional units, e.g., in the course of movements in metazoan organisms or during plant development 2 . Perturbations of mechanotransduction have been implicated in various severe diseases like cancer 3,4 . Remodeling of the cell as a response or adaption to an external, physical stimulus is steered by gene expression in the nucleus 5 . Therefore, the information of the stimulus has to be transported across the cell from the periphery to the center. Common models of cellular mechanotransduction assume the conversion of a physical stimulus to a chemical signal by membrane proteins such as integrins 3 , and the subsequent transport to the nucleus either passively by diffusion or actively by molecular motors, i.e., rather slow processes. However, the direct propagation of a mechanical stimulus by stress waves through stiff cytoskeletal elements connecting the membrane and the nucleus 6 would enable a much faster transport pathway on the microsecond timescale and thus allow almost instantaneous integration of responses across the cell 7 . A model for such a pathway has been proposed by Ingber 8,9 on the basis of a tensegrity model of flexible actin filaments (able to transmit traction forces) connected to the relatively stiff microtubules (able to transmit compression forces). In mammalian cells, microtubules are typically aligned radially inside a cell spanning from the centrosome, located close to the nucleus, to the cell membrane 10 , a set up that would allow for efficient mechanotransduction between cell membrane and nucleus 11 . In fact, mechanical stimulation has been shown recently to induce a perinuclear actin ring, brought about by the activity of actin-microtubule cross-linking formins 12 . MTs are well known as components of mechanosensing in flies 13 as well as in vertebrates 14 and microtubules have also been found to participate in gravity sensing and mechanic integration in plants (reviewed in 15 ).
Remarkably, mutants of Caenorhabditis affected in beta tubulin turned out to be insensitive to mechanic stimulation 16 . Efficiency and specificity of the MT sensory functions, however, depend on their frequency dependent viscoelastic properties, which are characteristic for biological systems.
To address these aspects of microtubule-dependent signaling, we present an approach for the targeted construction of cytoskeletal meshes with defined geometries by using optically trapped beads as anchor points. Existing approaches only demonstrated the construction of small networks without biologically relevant measurements 17 or rely on the stochastic attachment or growth of filaments to optically trapped beads or micro pillars which is less flexible and barely allows control of the number of attached filaments 18,19,20 . We use established micro-rheology techniques 21

Results
To determine the time-dependent viscoelastic properties of single microtubules (MTs) and small networks of MTs, movable Neutravidin coated beads as anchor points were attached to a biotinylated microtubule at defined positions by time-shared optical tweezers (see Methods).
Then, these anchor points were mutually displaced in an oscillatory fashion with defined frequencies and amplitudes along the x-direction as illustrated in Fig. 1. The resulting frequency dependent stretching and buckling behavior of these constructs is measured, which allows determining both the elastic and the viscous properties of the MT constructs in different geometrical arrangements.

Stiffening of single filaments at high oscillation frequencies
Upon force generation, the beads are displaced from their equilibrium position with a straight microtubule as depicted in Fig. 1. The displacements x B1 (t) -x L1 (t) and x B2 (t) -x L2 (t) of bead 1 (actor) and bead 2 (sensor) relative to the laser trap positions x L1 and x L2 , are shown exemplarily in Fig. 2a,b for two different actor displacement frequencies, f a = 0.1 Hz and f a = 100 Hz, at a displacement amplitude A a = 500 nm. Due to high tensile and small buckling forces, the sensor bead is pulled out of the trap center by up to x B2 ≈ 80nm and pushed only slightly by less than x B2 ≈ 10 nm during each half period. This situation changes significantly at high frequencies f a = 100Hz. While the maximum displacements x B2 during microtubule stretching were approximately the same at f a = 100 Hz and f a = 0.1 Hz, the displacement increased by an order of magnitude at high frequencies during buckling, i.e., x B2 (f a = 100 Hz) ≈ 10⋅x B2 (f a = 0.1 Hz). Therefore, only the compression and buckling of single filaments will be analyzed in this study. The complete frequency dependence of the filament -bead construct is expressed by the average maximum distance change between both beads ∆L x = A a -x max1 -x max2 as shown in Fig. 2c

Excitation and relaxation of higher MT deformation modes
As introduced above, the oscillatory driving force counteracts against the viscous and the elastic forces of both the MT and the two beads. The behavior of the semi-flexible MT of length L is described by the hydrodynamic beam equation, which predicts that induced MT deformations can be described by a superposition of sine waves with wave numbers n n L q π ⋅ = and a characteristic relaxation time proportional to 1/q 4 ~ L 4

(see Methods and Supplementary
Results). Hence, higher deformation modes n > 1 can only be excited at higher driving frequencies ω = 2π⋅ f a , leading to the effect of MT stiffening. The stiffening can be described by the frequency dependent complex shear modulus G(ω) = G′(ω) + i⋅G′′(ω) (see Methods Section), where a representation of all forces in frequency space allows to extract the elastic component G′(ω) and the viscous component G′′(ω).
The elastic modulus G′(ω) shown in Fig. 3  We checked whether the measured frequency response and apparent stiffening of single filaments indeed results from the excitation of higher deformation modes as described by equation (1). Therefore, we analyzed the dynamics of the trapped anchor points with two particle active micro-rheology techniques (see 22,23 and Supplementary Methods for details) as described in the following.
The frequency dependent elastic response of the single microtubule was analyzed in terms of which increases with frequency because of a successive excitation of higher modes at It can be seen in Fig. 3d that the persistence length depends sensitively on the contour length L of the MT. We find l p (0) = (0.33 ± 0.05) mm for L = 5 µm and all stabilizations. For L = 15 µm we find l p (0) = (4.06 ± 0.26) mm stabilized with 10 µM Taxol, l p (0) = (5.80 ± 0.39) mm for 100 µM Taxol, and l p (0) = (12.10 ± 0.66) mm for GMPCPP. The estimates based on equation (2) agree well with the published dependency of l p on the filament contour length 38 and stabilization 39 , as further elucidated in the discussion.
Transition frequency. Beyond a characteristic frequency, a visible increase of G'(ω) is manifested due to the excitation of higher deformation modes. We define this transition by the We find that the transition frequencyω t = 2πf t scales by a factor of 3 with the ground mode.
We obtain f t = (5.3 ± 1.2) Hz and f t = (4.2 ± 1.7) Hz for short filaments (L = 5µm) stabilized with 10 µM or 100 µM Taxol, respectively. For long filaments (L = 15µm), we find f t = (0. shows the result for a bead oscillation lateral to the axes of long (L = 15µm) GMPCPP stabilized filaments. Again, we observe a plateau value for frequencies ω < ω t and a power law rise for ω > ω t with p = 1.76 ± 0.02, i.e., a 40% larger stiffening exponent than the predicted value p = 1.25 for a longitudinal oscillation. In contrast, the plateau (0) G ⊥ ′ = 0.54 mPa is approximately one order of magnitude smaller than G || '(0) in axial direction. The transition frequency f t = 3 Hz obtained from ( ) 1.
is approximately the same as in axial direction.

Momentum transport along a linear chain of connected MTs
An important question is whether the findings for single filaments can be used to predict the momentum transport through small networks of filaments -in analogy to Kirchhoff's circuit laws for the connection of currents in network nodes. However, for connected microtubules, i.e. for different networks, the compression of one filament usually results in a stretching of another filament and vice versa, such that a separation of compression and stretching is not possible anymore. Therefore, the complete oscillation period of the actor and sensor beads will be analyzed in the following.
In a first step, we constructed a linear network consisting of three optically trapped beads and two microtubule filaments as shown in Fig. 4A. This construct was probed such that trap 1 was oscillated sinusoidally at varying frequency and amplitude, while trap 2 and 3 remained stationary. In this way, we investigated the momentum transfer along the first microtubule, while attached to a second microtubule, through (1,2) for the two step connection is larger than for the single step. As shown in Fig Lateral chain oscillation: As shown in Fig. 4b . Beyond the transition frequency, the frequency dependent elasticity

Momentum transport in an equilateral triangle
We used GMPCPP filaments to construct equilateral triangles of 15 µm side length as depicted in Fig. 5a. The trap 1 is again oscillated in x or y, resulting in a trap movement radial or tangential to the connection between bead 1 and the center of the triangle. An overlay of brightfield and fluorescence images of one radial oscillation period at f a = 0.1 Hz (T = 1 / f a = 10 s) and A a = 600 nm along x is shown in Fig. 5b. In contrast to single filaments and the linear chain, here, in total two filaments are always buckled or tense, while at the same time, the third one behaves in the opposite manner, i.e., is tense or buckled.

Triangles are stiffer than single filaments and have a similar high frequency response
Due to the symmetric configuration of the equilateral triangle, the elastic modulus for both connections 1→2 and 1→3 should be identical, except for different oscillation directions.
This is indeed the case as shown in Fig. 6a,b for an exemplary construct, where the radial and tangential elastic responses, (i, j) ( ) According to equation (3), the larger static elasticities should result in an increase of the transition frequency ω t by a factor 25, i.e., f t = (810 ± 308) Hz and f t = (75 ± 13) Hz for both oscillation directions. 810 Hz is much larger than the measured maximum frequency, so that we cannot observe a power law rise for an oscillation along x. However, the extrapolated intersection of the single filament response (fit with free exponent according to equation (1)) with the plateau of the triangle can be estimated to f t ≈ 800 Hz, which is in good agreement with the theoretical estimate of equation (3). For the tangential oscillation direction (y) displayed in Fig. 6b, the network stiffens already at a transition frequency f t ≈ (100 ± 10) Hz

Discussion
Microtubule stiffness depends on the contour length. We have analyzed the elastic behavior of single and inter-connected MTs by means of the elastic modulus G'(ω), which can be described by a low frequency plateau (0)~(0) p G′  , and a rise at high frequencies above a characteristic transition frequency ω t , defined by a 50% increase of G'(ω).
Varying the molecular composition of the filaments, by stabilization agents had no visible effect on the static elasticity of short MTs (5 µm length). Interestingly, this was different for which is approximately quadratic and results in a 9-fold higher plateau for 3-fold shorter MTs, we find a reasonable match with our measurements shown in Fig. 3. From the two MT lengths, we also find that our results for (0, ) p L  agree well to those reported previously 34, 38 .
Frequency dependent persistence length and stiffness. The novelty of our observations is the increase of the persistence length, or correspondingly the elastic modulus G'(ω), of a single microtubule with the displacement frequency ω (Fig. 3). In the Methods section, we show that this is caused by the excitation of higher deformation modes, which means that filaments become stiffer on shorter timescales, such that filament buckling is suppressed. In other words, molecular relaxation processes as a consequence of internal stress along the MT cannot follow on too short timescales. The timescale of molecular relaxation is approximated by the transition frequency ω t ≈ 3⋅ω n=1 , which we indicated in all plots of G'(ω). Beyond this frequency, the second deformation mode (n = 2) renders the filament about 1-4 times stiffer, beyond ω = 20⋅ω n=1 the third deformation mode (n = 3) stiffens the filament 4-10 times relative to ω = 0 as explained in the Supplementary Results (Fig. S4). Our measurements confirm the general, theoretically predicted trend of a smaller transition frequency for longer MTs. According to ( ) ( ) For short MTs we measured a stiffness increase according to 3/4 ( )G ω ω ′ at high frequencies (50 Hz < ω < 100 Hz), whereas for longer MTs we found In biological and other noisy systems, the signal energy stored in various degrees of freedom (translation, oscillation, etc.) is significantly less pronounced at higher frequencies (e.g., a 1/ω² decay for thermal motion). In this way, microtubules should act as transmission amplifiers or high pass filters for mechanical signals, based on our observations that mechanical stimuli are transferred much more efficiently at higher frequencies. Interestingly, this situation resembles an (electronic) transistor, where a small input signal (here, a mechanical stimulus) controls a strong current (here, the momentum transport from bead 1→3). It will be interesting to perform further experimental and theoretical investigations to explain the elastic behavior, where momentum transport between two network nodes can be steered by an intermediate node.

Angular momentum filtering in a linear
The MT triangle -a uni-directional stable network. Displacement of the actor bead in either radial x-or tangential y-direction as illustrated in Fig. 5 results in a very direct and efficient transport of momentum in direction towards the one or the other sensor bead.
Remarkably, the measured elasticity behavior described by the modulus G'(ω) is the same as in the single filament case. It consists of a static elasticity G'(0), and a strong rise of G'(ω) when higher deformation modes are excited beyond the transition frequency ω t . This strong rise is clearly visible at f t = 100 Hz for a tangential oscillation, but could not be resolved for a radial oscillation. This is probably due to a much faster rise (larger exponent) of G' in a direction lateral to the filament axis, as we observed this phenomenon for single filaments and the linear MT chain as well. However, based on our observations for single filaments, we could estimate the transition frequency for a radial oscillation of the triangle to be f t ≈ 800 Hz.
In the static case, the triangle is about 25 times stiffer than a single filament. This can be explained by the fact that every radial or tangential displacement of the actor bead results in a compression and stretching of another MT at the same time. Since MTs are hardly stretchable 32 , this results in static elasticities of G'(0) ≈ 20-50 mPa. Comparing the estimates for the transition frequency ω t obtained from G'(0) and equation (3), these extrapolated values come close to the frequency where G'(ω) ≈ 1.5⋅G'(0). Again, we interpret the increased transition frequency as a result of the intermolecular relaxations of or between two tubulin heterodimers, which cannot follow on timescales below 2π/ω t < 10 ms. A stiffening beyond a transition frequency of f t ≈ 200 Hz could also be observed in cross-linked actin networks 50 .
Whereas the optically trapped anchor points could rotate and act as hinges in the previous configurations, the anchor points of the triangle can hardly rotate, and therefore rather resemble a movable support only. This triangular situation is relevant for the radial MT arrays that form around the nuclei of many cells by a mechanism where microtubule-nucleation factors are directionally transported by dynein motors 51 . In addition, the forces conveyed to the nucleus by this network would act, via links of the cytoskeleton to the nuclear lamina, on structure and dynamics of the chromatin 52 , providing a mechanism how mechanic signals can modulate gene activity in the network's center.

Summary and conclusions
Motivated by the capability of individual microtubules and inter-connected microtubule networks to transduce a mechanical stimulus over a long distance within short times, we clearly identified substantial differences in response for different network topologies and at different stimulation frequencies ω 35 . This has a couple of interesting implications for biology: The rather low stiffness at frequencies below the characteristic transition frequency, ω < ω t , of single filaments or the linear network is expected to dampen the transmission of mechanical signals, while the rise at ω > ω t would allow for an enhanced transmission of signals that typically show a reduced amplitude in noise driven systems such as living cells.
Interestingly, this transition frequency is in a physiologically relevant range (1-10 Hz). For instance, the mammalian heartbeat ranges between 1 Hz in humans up to 18 Hz in mice 53 , and muscles undergo an innate oscillation of around 20 Hz 54 .
A second aspect of the strong influence of network topology is the comparatively high stiffness at low frequencies of triangular networks. This displays a stiff, load bearing scaffold, which could be used to reinforce the cell against external pressure in densely packed tissues, or enable the contraction of large scale MT networks 55 . The specific mechanical properties of triangular networks are relevant for nuclear positioning, since the nucleus is tethered and positioned by radial arrays that are stabilized by cross-connection in many organisms integrated into cell polarity 56 . A third implication of our findings is linked with the "mechanic transistor" function of microtubule networks, where small mechanical forces can control a large amount of momentum transport.
Microtubule crosslinkers have recently been reported to be able to generate entropic forces on the pN range 57 , which could lead to passive changes of network elasticity over time by prestretching individual filaments of a network. This would provide a mechanism how cells can control the directionality of mechanic signaling, which is relevant for mechanic integration of cells into organs, or of organs into organisms 2 . These implications show that our bottom-up approach to analyze the transmission of mechanic forces in networks of increasing complexity is relevant to understand, how mechanic signals can shape biology.

Theoretical description of viscoelastic behavior
This section introduces the relevant forces acting on a single filament and its resulting deformations as well as the relative bead displacements during an oscillation longitudinal to the MT. Through a representation of all forces in frequency space, the elastic and viscous components of the filament can be extracted using the frequency dependent complex shear modulus G(ω).
To separate the viscoelastic contributions of the filament and the trapped beads, we analyzed the data by means of active two particle micro-rheology in frequency space. Here, is the elastic optical force and ( )  (4) can then be given explicitly: Hence, equation (5) represents a set of m coupled differential equations, where m is the number of beads. These are solved pair wise using relative and collective coordinates Since the contribution of the filament acts in opposite directions for each bead (points away from the MT ends, ± in equation (5)), this effect cancels out in the The buckling of the filament contour u(x B1 , x B2 , x, t) is a function of the compression given by the bead positions x B1 and x B2 and is assumed to be deformed in lateral direction y only with small angles to the x-axis. The deformation amplitude can be written as a superposition of sinusoidal modes with wavenumber q n = n⋅π/L (n≥1) 59, 60, 61 : The amplitudes The filament is axially compressed by  , equation (7) reads in frequency space: The spectral forces acting on the beads with relative position ( ) R x ω  are known and can be subtracted, such that the following response equation holds: deforms the MT at different temporal and spatial frequencies.

Theoretical estimate for MT stiffening on short timescales
The question is how well our observations can be explained on the basis of an equation of forces, as introduced in equation (4) Using this fit function, the transition frequency ω t ≈ 3ω 1 could be extracted from the experimental data. ω t was interpreted as the frequency at which molecular relaxations cannot follow the external filament deformation. The frequency independent stiffness at low frequencies and the sudden increase in G'(ω) on a double-logarithmic scale could be well observed in single filaments as well as in the linear and triangular MT arrangements. From these observations, we conclude that the description of forces chosen in equations (4) and (5) to quantify our mechanistic model is reasonable. However, the stronger stiffening at high frequencies with p > 5/4 needs a more thorough theoretical investigation. In addition, the theoretical approach has to be extended in the future, to also integrate the porous molecular structure, especially to explain the dependence of the transition frequency on chemical stabilization of the microtubule (see Supplementary Results).

Stiffness estimate for a linear MT chain
The two-step elastic modulus (1,3) G′ , resulting in a fivefold stiffening in longitudinal direction and fivefold softening in lateral direction compared to the one-step modulus (1,2) G′ , can be modelled as serial connection of two springs (two filaments, 2fl) with MT length L / 2 or wave number 2q, such that . Reciprocal addition of two single filament results in a two filament sum elasticity, 2 (0, 2 ) , which is two times softer than that of a single filament. Alternatively, the two-step modulus (1,3) G′ can be identified with a single filament of length L, or wave number q, such that (1,3) . However, this results in a fivefold decrease of the elasticity relative to that of a single filament with length L / 2, according to 1 The factor 1/3 arises from the length dependence of l p (q).
Hence, an additional coupling term cpl G′ is required to explain the elastic behavior of the linear construct, such that ( ) 0, cpl q G′ must be positive for longitudinal momentum transport, and negative for lateral momentum transport.

Experimental setup with optically trapped beads as actor and sensor
A single biotinylated microtubule was attached laterally to two Neutravidin coated beads displaced by x B1 and x B2 relative to the trap centers. During both half periods of an oscillation, the distance between the beads was first increased and then decreased resulting in tensile and compressive forces acting on the microtubule, respectively. Since microtubules are practically inextensible, they are bent locally at the point of attachment to beads (Fig. 1a) during the first half period 32 and buckled during the second half period due to their high compliance to compression forces 63 . The buckling amplitude along the filament is denoted by u(x, t) as illustrated in Fig. 1c.
In the experiments, we used a lateral stiffness of κ opt ≈ 25pN/µm per trap. The actor trap was typically oscillating at frequencies 0. usually resulted in filament breaking close to one bead. In such cases, the measurements were excluded from further analysis. Also, we did not observe significant differences for repetitive measurements on the same filament indicating that microtubules were not structurally damaged during oscillation. In some cases, one of the filaments was detached of a bead.
These experiments have also been excluded from analysis. Elastic effects of the biotin linker can be neglected, since the effective length of this linker is in the Ångstrom range 64 and its spring constant 65, 66 is much larger than that of microtubules, both for buckling and stretching 32 .

Suitability of experimental approach
The use of optically trapped beads as anchor points for simple microtubule networks turned out to be a very suitable approach. Potential phototoxic effects such as bleaching and filament

SI Methods and Material
The elastic relaxation forces along the filament ∫dF κ,MT are reduced by filament friction ∫dF γ,MT and act in lateral y direction. Due to the homolog constraint of the filament of constant length L and its connection to the optically trapped beads, the resulting elastic forces of the microtubule F κ,MT push the beads outwards in x-direction and are counteracted by the optical forces F opt . Whereas the friction force F γ,B on the bead in the sensor trap (blue) is negligible small, the viscous force on the oscillating actor bead (red) counteracts the driving force F drive . The tension free equation of motion for relaxation (F drive in positive x-direction) reads 1 1 1 1 , the left bead and   2   1 1 for the right bead.

Micro-rheology analysis
The measured, frequency dependent displacements x Bi , y Bi of bead i as a response to an applied actuation force ( ) In active micro-rheology as it is used here, the driving force F = κ·x L (t) is generate by a sinusoidal oscillation x L (t) = A a sin(ω a t) of one optical trap with stiffness κ, amplitude A a and driving frequency ω a . To obtain the complete spectrum A(ω), the experiment has to be repeated several times for different actuation frequency ω a and evaluated according to Eq. (S1) for each frequency ω a . As explained in the main paper, the measured bead displacements x B , y B are a superposition of the elastic trapping force, the viscous drag of beads and the wanted viscoelastic properties of the material under investigation, i.e., of the microtubule filaments in our case. Hence, the response function A is a superposition of these contributions as well. As explained in (1, 2) given by Eq. (S2), this can be separated to obtain the pure viscoelastic response function G MT of the filament: Here, κ (i) and κ (j) are the trap stiffnesses of the corresponding traps which have to be determined independently by calibration (3,4). Different pre-factors 4πL and 8πL for different directions take care of the hydrodynamic coupling To ensure correct results we tested the software implementation and measurement procedure for simple beads in water, where the viscoelastic response is known theoretically and measured experimentally (1). Since the motivation of the paper is to study the transport of mechanical stimuli, only the elastic components ( , ) i j G′ (real part of G) are shown and discussed in the main paper. Viscous components (imaginary part of G) are shown in the SI Results (see below).

The molecular architecture of differently polymerized and stabilized filaments
We choose filaments polymerized in the presence of a non or slowly hydrolysable GTP analog (GMPCPP) in addition to filaments assembled with GTP since both filament types have significantly different mechanical properties (5) ultimately governed by different molecular configurations as illustrated in Fig. S2. After polymerization, GTP molecules in the microtubule lattice hydrolyze stochastically to GDP. While GTP and GTP-analog tubulins adopt a straight conformation, the hydrolysis at the β-tubulin leads to a kink of the GDP tubulin dimer resulting in an intrinsic strain in the microtubule lattice (6,7). This conformational change is slowed down by Taxol (8) which binds on the inside of the hollow tube (9, 10) and has been used to theoretically recapitulate the tip structure and rates of assembly/disassembly of microtubules (11), the occurrence of long-lived arcs and rings in kinesin-driven gliding assays (12) and to transform MTs into inverted tubules facing their inside out by a specifically induced conformational change using spermine, a polyamine present in eukaryotic cells (13). Further, microtubules polymerized in the presence of slowly or non-hydrolyzable GTP analogs such as GMPCPP or γ-S-GTP have additional lateral interprotofilament contacts between β-tubulins compared to GTP/GDP microtubules (5,14). Assuming that the connection between individual αβtubulin dimers can be approximated by damped harmonic springs (15,16), the damping of the intermolecular connections should affect the temporal response upon exertion of mechanical stimuli and thereby the transition frequency ω t .

Lateral forces are negligible
In addition to the data shown in Fig. 2 of the main paper, we here compare the bead displacements along the x and y direction during a single filament rheology experiment at two oscillation frequencies f = 0.1Hz and f = 100Hz. As Fig. S 3 shows, the total contributions in lateral y-direction are negligibly small. Here, the lateral elastic MT buckling force F κMT,y (x,x Bj ) is increased (reduced) by the MT drag force F γMT,y (x, x Bj ) for deformation (relaxation). Both MT forces are equilibrated by the strong optical forces F opt,y (x Bj ) and the weak viscous drag forces of the beads F γ,y (x Bj ) in lateral direction. The sum of these forces is zero for all oscillation frequencies and phasings, i.e., F opt,y (x Bj ) + F γ,y (x Bj ) + F κMT,y (x, x Bj ) + F γMT,y (x, x Bj ) ≈ 0. This situation is revealed in Fig. S

Frequency dependent bead displacements
The displacements x Bi of the beads are governed by the elastic optical trapping force, the viscous drag force of the beads as well as the viscoelastic force from the microtubule filament according to Eq. (2) of the main article. In Fig. S4, we show the frequency dependence of the maximum actor and sensor bead displacement |x Bi -x Li | during filament buckling and filament stretching. While an increase of the maximum amplitude of bead displacements of approximately one order of magnitude can be observed during buckling, bead displacements stay approximately constant and are proportional to the actor amplitude A a during filament stretching. Already here, the connection between the constant low frequency plateau of G' and its power law rise above f t ≈ 2Hz to filament buckling can be anticipated. The viscoelastic contribution of the trapped beads alone moving in the purely viscous buffer medium is much smaller than the effect observed here and is dominated by the corner frequency ω c = κ / 6πR B η ≈ 2500Hz of the position power spectral density |x B (ω)|² of the bead motion, which is much larger than the transition frequencies estimated for our MT constructs.

Estimation of the total viscous force
To distinguish the absolute role of the different forces introduced in Eqs. (1) and (2) of the main paper, we determine these contributions in the following. The results shown here are the basis of Fig. 2D of the main paper.
The most obvious force is the viscous drag ,tran B F γ of bead translation. The actor bead approximately follows the sinusoidal movement of its trapping focus and is much larger than that of the sensor bead, which is neglected for this reason. The movement of the actor trap is x L1 (t) = A a sin(2πf a t) resulting in the velocity v L1 (t) = ∂/∂t x L1 (t) = 2πf a A a cos(2πf a t). Only the maximal force components are considered in the following. Hence, the translational viscous drag force is given by Eq. (S3) and shown by the yellow line in Fig. S5B.
The buckling filament causes both beads to rotate resulting in a rotational drag force ,rot For comparison, we also plotted the experimentally obtained frequency dependence of the total force of a bead alone (yellow markers) and of a bead / MT construct (red markers) together with the total force of all contributions estimated above (red line) in Fig. S5B   For the experimental situation with MT filament, the total force increases continuously but still intersects with the sum of all estimates of the viscous contributions made above at approximately f = 100Hz. This indicates a strong additional contribution from the MT, very likely of elastic nature as we obtained by the micro-rheology analysis for G'.

Contributions of deformation modes to G'(ω)
As stated in the main paper, we estimate to excite N = 3 deformation modes at oscillation frequencies up to f a = 100Hz. The contributions of each additional mode are illustrated in Fig.   S6 where we plotted the theoretical slope of ( ) indicate the influence of a single mode and at which frequency the next mode kicks in.
The first mode, causing a constant plateau, is dominant for low frequencies up to ω ≈ ω 1 .

Viscous components of single filaments
The viscous modulus G''(ω) of a simple bead in a ideally viscous solution such as water is A comparison to the theoretical prediction shown in Fig. S7B reveals that high deformation modes n ≥ 2 only slightly change this linear relationship and only for relatively high frequencies ω > 10ω 1 , which is roughly the maximum frequency we resolve in our experiments (ω 1 ≈ 4Hz typically, see main paper).

Viscous components of a linear connection of MTs
Similarly to the viscous components of single MT filaments, the viscous components

Pre-stress in triangular networks
Free floating filaments are subject to Brownian forces and sometimes bend heavily. This can cause pre-stress during the construction of a network, i.e., the subsequent attachment of a filament to optically trapped beads. This effect becomes more prominent if the number of filaments of a network increases. Fig. S9 shows the elastic modulus G' of three different equilateral triangles with a side length of 15µm. For both oscillation directions, the plateau of G' is larger for the connection of the first filament 1→2 and smaller for the connection of the second filament 1→3 for the first two triangles, compared to the third triangle where the elastic moduli for both connections are approximately equal. This clearly indicates a prestress of the first MT compared to the second filament. (1) (1)

Viscous components of triangular networks
Again, we observe a linear relation between the viscous components G''(ω) of filaments in a triangular networks and the frequency ω as shown in Fig. S10 together with power law fits with free exponent p ≈ 1. For the tangential oscillation along y, the viscous component (1,2) G′′ of the first filament deviates strongly from the expected linear response for the first two triangle constructs, indicating that maybe the connection to either of the beads (1) or (2) was not perfect. A B (1) (2)

Comparison of transition frequencies
As described in the main paper, we found that he transition frequency ω t , separating the constant plateau value of G' for low frequencies from the high frequency rise approximately proportional to ω 1.25 , depends on filament length, stabilization, polymerization and especially on the geometry of the network. This is summarized in Fig. S11A. The transition frequency is the highest for the triangular network, which is also the stiffest. The difference of filament stabilization is clearly visible for long filaments. There is also a clear difference visible for different oscillation directions of all geometries, where the transition frequency is much smaller for an oscillation lateral to the filament axis, indicating much faster stiffening in this direction.
In order to analyze how well the experimental transition frequencies match the theoretical predictions according to Eq. 3 of the main paper, we plotted the transition frequency as a function of MT contour length as shown in Fig. S11B. Here, we included the length dependence of the MT persistence length according to Pampaloni (18). For the persistence length p l ∞ of MTs much longer than a critical length l c = 21µm, we used the values for L = 15µm long MTs obtained in this study to reflect the different stabilizations.

Geometric effects of beads
MT filaments are attached laterally to the anchor beads with a diameter d = 1062nm. This results in a torque on the beads during filament buckling because the optical trapping force acts on their geometric centers. This causes the point of attachment of the filament to a bead to rotate. Hence, the precisely measured distance ∆x L + x B1 -x B2 between both beads does not coincide with the actual projected length p MT of the buckled filament as illustrated in Fig.   S12A. For convenience, we assume the symmetric case with x B1 = x B2 = x B and ∆ 1 = ∆ 2 = ∆ B = R B ·sin(ϕ) ≈ R B ·ϕ in the following. Neither the actual compression δ L given by Eq. (S7) nor the rotation angle ϕ² = 4δ L / L MT can be measured directly.
( ) However, both unknowns depend on each other leading to the quadratic relation ( ) This can be substituted in Eq. (S7) to calculate the actual compression δ L and to plot force compression curves, i.e., the buckling force F = κ 1 ·x B1 + κ 2 ·x B2 versus the compression δ L as we show for two filaments with different length in Fig. S12C+D. The data shown here were obtained in a quasi-equilibrium where we moved trap 1 in discrete steps of ∆x L1 = 50nm every ∆ t = 100ms.
In the ideal case, i.e., a perfectly axial application of force on the filament, the MT should not buckle, i.e., δ L (F < F crit ) = 0, until a finite critical buckling force F crit = π²EI / L² MT is reached, above which the filament behaves like a spring with spring constant κ MT , i.e., δ L (F > F crit ) = (F -F crit ) / κ MT (17). Here, we observe a nearly exponential dependence of the force on the compression for small δ L < 400nm and a linear dependence for δ L > 400nm. This is due to the imperfect, lateral application of the force on the filament. We indicated this behavior in However, we did not consider this geometric effect in our rheology experiments. We expect this to have only a minor effect on the measured viscoelastic properties G MT of the filaments, but this has to be tested and included into the theory in the future.

Microtubule buckling amplitude
For the integration step in equation (5)  Since this square root cannot be solved analytically, we investigate the buckling amplitude u qn as a function of arc length n L′ for single bending modes n, where the ground mode n=1 allows to estimate the maximum possible deflection u q1 > u(x).
Substituting ϕ = q n x leads to However, the sum of all deformations in a filament is always smaller than the ground mode buckling, i.e. u(x) <u q1 .

Linear system theory
Our theoretical description as well as our analysis assume a linear relationship between the microtubule buckling amplitude u qn and the driving force F D , such that ( ) ( ) ( ) In order to test whether the buckling responses of single MTs are linear with force, we analyzed the force dependency F(δ L ) on a stepwise MT compression by δ L . Fig. S14 shows that F(u qn ) increases indeed roughly linearly for not too large buckling amplitudes u qn (δ L ), in accordance with to Eq. S13.
Linearity: By analyzing the normalized χ² value as a function of the number of data points included in a linear fit to the data, we find an approximately linear response up to u qn ≤ 300 nm for short microtubules (L = 5µm) and u qn ≤ 1.4µm for long filaments (L = 15µm). This is equivalent to an oscillation amplitude of the laser trap x L < 500nm for short and x L < 1200 nm for long microtubules, respectively. In our rheology experiments, we usually analyze the microtubule response for three different oscillation amplitudes A a = 200 nm, A a = 400 nm, and A a = 600 nm. Hence, linear response is well fulfilled for long microtubules and at least for the two smaller oscillation amplitudes for short filaments. In addition, we always compare the results for different oscillation amplitudes to each other and never observe a significant difference.

Local forces acting along the filament
The curve of a filament deformed in the mode q n can be described by the vector We like to point out the strong dependence of the local bending force on the third power of the mode number n, meaning that the highest present order always dominates the buckling force.