Rheology of mixed motor ensembles
RRheology of mixed motor ensembles
Justin Grewe and Ulrich S. Schwarz ∗ Institute for Theoretical Physics and Bioquant, Heidelberg University, Heidelberg, Germany
The rheology of biological cells is not only determined by their cytoskeletal networks, but alsoby the molecular motors that crosslink and contract them. Recently it has been found that theassemblies of myosin II molecular motors in non-muscle cells are mixtures of fast and slow motorvariants. Using computer simulations and an analytical mean field theory of a crossbridge modelfor myosin II motors, here we show that such motor ensembles effectively behave as active Maxwellelements. We calculate storage and loss moduli as a function of the model parameters and showthat the rheological properties cross over from viscous to elastic as one increases the ratio of slow tofast motors. This suggests that cells tune their mechanical properties by regulating the compositionof their myosin assemblies.
The rheology of animal cells is essential for many phys-iological functions, including the function of epithelialand endothelial cell layers under continous loading, e.g.in lung, skin, intestines or vasculature [1]. It is also es-sential for single cell processes such as cell migration anddivision, which are characterised by intracellular flowsand deformations [2]. For these reasons, single cells andcell ensembles have been widely studied using rheologi-cal approaches as commonly applied in materials science[3–7]. Cells typically show a wide relaxation spectrumindicating the relevance of different time scales. Oftenpower-law relaxation spectra have been reported [3, 5],but there is also evidence for an upper cut-off at a maxi-mum relaxation time [7]. Despite this complexity of cellrheology, however, for many purposes linear viscoelastic-ity has turned out to be a surprisingly good descriptionof the effective mechanical properties of cells and cellmonolayers [8–14].Cells actively control their mechanical propertiesmainly by changing the assembly status and activity oftheir actomyosin cytoskeleton. Although much is knownabout the effective rheology of these networks from theviewpoint of polymer physics [15, 16], it is not clear howthe microscopic properties of the different types of myosinmotors contribute to cell rheology. Recently it has beenfound that non-muscle cells co-assemble fast and slow iso-forms of myosin II [17, 18]. A very recent computationalstudy showed that the electrostatic interactions betweenthe coiled-coils of the different isoforms leads to a rich en-ergy landscape for mixed assembly and can explain someaspects of their cellular localization [19]. While the fastmyosin II isoform A is mainly found at the front of thecell, where fast assembly and flow is required, the slowmyosin II isoform B is incorporated towards the back,where strong and long-lasting forces are required [17, 18].Here we explore the intriguing possibility that cellscontrol their rheology by differential assembly of theirmyosin minifilaments. We address this important ques-tion theoretically by using a microscopic crossbridgemodel for small ensembles of myosin motors [20–23],which earlier has been applied only to ensembles of oneisoform [24, 25]. By extending this framework to mixed (a)(b) (c)
FIG. 1. Model. (a) Scheme used for rheology simulationsof myosin II ensembles. The central spring has extension z and spring constant k f . A sin ωt is the oscillatory perturba-tion. The myosin crossbridges have motor strains x and linkerstiffness k m . Blue and red myosin crossbridges denote the fastand slow isoforms A and B, respectively. (b) Maxwell elementwith spring constant k and friction coefficient ξ in parallelwith active motor force F m . (c) Crossbridge model showingthe three mechanochemical states (UB = unbound, WB =weakly bound, PPS = post-powerstroke) and the transitionrates between them. ensembles and calculating their complex modulus, weshow that such assemblies operate as active Maxwell ele-ments that can tune their rheology from viscous to elasticby increasing the ratio of slow versus fast motors.Fig. 1a shows a schematic representation of the situa-tion that we analyse here. A central spring, which couldrepresent an optical trap or an elastic substrate, has ex-tension z and spring constant k f . It is pulled from twosides. On the right hand side, we have a mechanicalmotor that pulls with fixed frequency ω and amplitude A . On the left hand side, we have a small ensemble of N myosin II motor heads that walk towards the barbedend of an actin filament. Each motor has a strain x anda spring constant k m . For the myosin II minifilamentsin non-muscle cells, we typically would have N = 15 a r X i v : . [ q - b i o . S C ] J u l [19, 26, 27]. From these N motor heads, N a are assumedto be of the fast isoform A. Then N b = N − N a are ofthe slow isoform B.We first show that in this setup, both the total mechan-ical system as well as the motor ensemble itself shouldbehave effectively like an active Maxwell element as de-picted in Fig. 1b. The Maxwell element is the simplestpossible viscoelastic model and features a spring withspring constant k and a dashpot with friction coefficient ξ in series; in an active Maxwell model, there is a con-stant motor force F m operating in parallel. We assumethat the motor ensemble depicted in Fig. 1a has a well-defined force-velocity relation v ( F ), with a free velocity v at F = 0 and vanishing velocity at the stall force F = F s . With all motors having the same crossbridgespring constant k m , the motor ensemble has an effectiveensemble spring constant k e = nk m , with n being thetypical number of bound motors. Force balance leads toa differential equation for the extension z ˙ z = k e k e + k f [ v ( k f z ) + Aω cos( ωt )] . (1)After expanding the force-velocity relation around thestall force F s with a slope v (cid:48) ( F s ) = − /ξ , we easily cansolve this equation: z ( t ) = F s k f + C exp (cid:18) − k t ξ t (cid:19) + ωk t Ak f (cid:112) ω + ( k t /ξ ) sin( ωt − δ )(2)with the total spring constant k t = k e k f / ( k e + k f ) andtan δ = k t /ωξ . We obtain three terms, each with a clearphysical meaning. The first term is the constant pullof the active Maxwell element, thus F m = F s as for alinear force-velocity relation [28]. The second term isinitial relaxation with a constant C determined by theinitial conditions. The third term is our most impor-tant result: the system response is oscillatory with thesame frequency as the external perturbation, but with aloss angle δ that depends on the parameters of the mo-tor ensemble. δ increases from 0 to π/ k t and decreases with increasing friction ξ . The calculatedoscillations correspond to a complex modulus G ∗ = k t ω ξ k t + ω ξ + i k t ωξk t + ω ξ (3)which is exactly the result for a Maxwell model with theeffective spring constant k = k t and a friction coefficient ξ . This result applies to the system that includes theexternal stiffness k f . If we restrict ourselves to the motorensemble, we also obtain an active Maxwell model, withthe same friction coefficient ξ and motor force F m , butwith the spring constant k = k e rather than k t . This isequivalent to assuming an infinitely stiff environment, as k t → k e as k f → ∞ . F [pN] v [ n m / s ] (a) N a = 0 N a = 5 N a = 10 N a = 15 0 50 100 t [s] F [ p N ] (b) FIG. 2. Computer simulations of the crossbridge model formixed ensembles with N = 15 motor heads. (a) Force-velocityrelations. Zero crossings define the stall force F s and the lin-ear slope around this point the friction coefficient ξ . Through-out the paper we color code the fraction of NM IIA motors asa gradient from dark blue for N a = 0 to light red for N a = N .(b) Force on central spring as a function of time. The col-ored areas denote the region of one standard deviation. Thesolid lines denote fits with sine functions and the vertical ticsindicate where the new period starts. In order to validate our prediction that motor en-sembles should effectively behave as active Maxwell sys-tems, we conducted computer simulations of a micro-scopic crossbridge model for myosin II as shown inFig. 1c. In our model, each of the N crossbridges ofthe ensemble is in one of three mechanochemical statesthat are connected by force dependent transition rates[21, 22, 25]. The transition from the unbound (UB) tothe weakly bound (WB) state occurs with rate k =0 . − . The strain-dependent reverse rate is k ( x ) = k exp( k m x + /f s ), where k = 0 .
004 s − is the rate atzero strain, k m = 0 . x + = max(0 , x ) is the positive part of the cross-bridge strain x and f s = 10 .
55 pN/nm the internal forcescale. From the WB-state, the motor can perform a pow-erstroke of distance d = 8 nm, which also determineshow far the motors can advance. The rate for transi-tion into the post-powerstroke (PPS) states is governedby the difference in elastic energy stored in the cross-bridges, ∆ E el , and the free energy of ATP-hydrolysis,∆ G = −
60 pN nm. We use k / = k ps exp( ± β (∆ E el +∆ G ) /
2) with k ps = 1000 s − . Finally, the unbindingfrom the PPS-state is modeled as a catch-slip bond, i.e. k ( x ) = k a/b (∆ c exp( − k m x + /f c ) + ∆ s exp( k m x + /f s )).Here ∆ c = 0 .
92 is the fraction following the catch-path at zero force with force scale f c = 1 .
66 pN, ∆ s = 0 .
08 isthe fraction following the slip-path at zero force. For thefaster NM IIA motors we use k = 1 .
71 s − , while forthe slower NM IIB we use k = 0 .
35 s − [25].By mixing N a and N b motors, we now can explorehow the minifilament composition determines its effec-tive rheology. Fig. 2a shows that indeed our microscopicmodel leads to a well-defined force-velocity relation foreach value of N a , which then defines both the stall force F s and the effective friction coefficient ξ . We find that asthe number of fast motors N a is increased, both F s and ξ decrease. Fig. 2b shows that as predicted by Eq. (2),the system response is oscillatory and can be fit well witha sine wave F ( t ) = F s + ∆ F sin( ωt − δ ) with an ampli-tude ∆ F , loss angle δ and force offset F s that depend onmodel parameters. Here we have identified the constantoffset force with the stall force as suggested by Eq. (2) for A = C = 0. We see that the loss angle δ increases whilethe amplitude and the constant offset force F s decreasewith increasing NM IIA content. This suggests that thesystem crosses over from viscous to elastic as the fastmyosin IIA motors are replaced by the slow myosin IIBmotors.In order to achieve a deeper understanding and anaccurate mapping between the microscopic motor ratesand the effective Maxwell rheology, we next developed aself-consistent mean field treatment of our crossbridgemodel for motor ensembles [21, 22]. In steady statebinding and unbinding from the track is balanced. As-suming the powerstroke is performed immediately afterbinding and approximating the stall force of the ensem-ble as the sum of the single motor stall forces, we have F s = nk m d and n = k N/ ( k ( d ) + k ). Using theformerly derived relation between speed and the numberof bound motors when all of them are in the PPS-state[22], v ( F ) = ( N − n ) k ( d − F/ ( nk m )) / (( n + 1)), we canderive the total spring constant k t and the friction coef-ficient ξ as functions of the mechanochemical rates, theenvironmental stiffness k f and the ensemble size N : k t = k m N k ( d ) k + k m Nk f , (4) ξ = k m n ( n + 1)( N − n ) k . (5)To approximate these quantities for motor ensembleswith heterogeneous composition, we reason that the rel-evant factor is the time spent by each motor bound tothe filaments. For the rates, this implies that we shouldtake the harmonic mean k ( N a , N b ) = N a + N b N a /k a + N a /k b , (6)with N a , k a , N b , k b the total number and the off-ratesof NM IIA and B heads, respectively.In Fig. 3(a) we show that the results from the com-puter simulations (symbols, obtained via calculating G ∗ = ∆ F (cos δ + i sin δ ) /A ) and from the analyticalmean field theory (interrupted lines, obtained by combin-ing Eqs. (3)-(6)) are in very good agreement with eachother. In addition we plot fits of Eq. (3) to the resultsof the computer simulations (solid lines). The mean fieldtheory does not perform perfectly for N a = N because − f [Hz] . . . . G ∗ [ p N / n m ] (a) N a . . . . k t [ p N / n m ] (b) simulationmean field N a ξ [ p N s / n m ] (c) N a F s [ p N ] (d) FIG. 3. Mechanical response as a function of ensemble iso-form content for ensemble size N = 15 and external stiffness k f = 1 pN/nm. (a) The dynamic modulus G ∗ of a ensembleswith N a ∈ { , , } (dark blue, light violet, light red). Thediamonds and crosses denote the storage and loss as deter-mined from the simulation, respectively, while the black linesare Maxwell model fits. The interrupted lines are the resultof the analytical mean field theory. (b, c, d) Spring constant k t , friction coefficient ξ and stall force F s as a function of NMIIA content N a . Blue diamonds and black crosses denote theresults obtained from the simulation and the analytical meanfield theory, respectively. in this case of only fast motors, it can happen that allmotors are unbound at the same time, while the meanfield theory assumes that n always has a finite value. InFigs. 3(b) - (d) we show that the mean field theory alsoperforms well for predicting total spring constant k t , fric-tion coefficient ξ and stall force F s as a function of thenumber of fast motors. For (b) and (c), we obtainedthese values through fits of Eq. (3) to the results of thecomputer simulations. One sees that the effective springconstant decreases slightly with increasing NM IIA con-tent, which is consistent with the lower duty ratio of asingle NM IIA motor compared to NM IIB. The frictioncoefficient decreases markedly with increasing NM IIAstarting from ξ ≈
30 pN s/nm without NM IIA and end-ing at ξ ≈ F s goes from ∼
20 pN/nm for ensemblesof pure NM IIB ensembles to ∼
10 pN/nm for pure NMIIA ensembles.Non-muscle myosin II assemblies are very dynamic and − f [Hz] . . . . G ∗ [ p N / n m ] (a) N k e [ p N / n m ] (b) N ξ [ p N s / n m ] (c) N F s [ p N ] (d) FIG. 4. Mechanical response as a function of ensemble size.(a) Dynamic modulus for NM IIA ensembles of size N = N a ∈ { , , } (light to dark red) with k f = 1 pN/nm (forsymbol description see Fig. 3(a)). (b, c, d) motor ensemblespring constant k e , friction coefficient ξ and ensemble stallforce F s as a function of ensemble size N for N a ∈ { , N/ , N } (dark blue, violet, light red). Symbols and lines represent thecomputer simulations and the analytical mean field theory,respectively. often change their size in the cellular context. We there-fore next investigated the size-dependence of the mechan-ical response, as shown in Fig. 4. Ensemble spring con-stant k e = nk m , friction coefficient ξ and stall force F s rise linearly with size. This suggests, that one can thinkof a single motor of the ensemble as an active Maxwellelement which in the context of the ensemble is in par-allel to others. The linear relationships also motivatecalculating the effective Young’s modulus E = k e l/πr ,viscosity η = ξl/πr and the active stress σ m = F s /πr generated by one half-minifilament of length l ≈
150 nmwith a typical crosssectional radius of r ≈
20 nm, i.e.the distance the heads typically splay outward from thecenter of the filament with N = 15 motors [26]. Wefind E = 160 −
460 kPa, η = 0 . − . σ a = 2 − E and η are higherthan typical cellular values because they describe onlythe condensed situation in the myosin assemblies, the ac-tive stress σ a is exactly the order of magnitude measurede.g. with monolayer stress microscopy [29].The viscoelastic relaxation time of the ensemble followsas τ = ξ/k e = η/E ≈ s , which is exactly the orderof magnitude observed in laser cutting experiments [30,31]. Our results suggest that the exact relaxation time − − f [Hz] . . . G ∗ [ p N / n m ] (a) − − k f [pN/nm] . . . k t / k f (b) − − k f [pN/nm] τ t [ s ] (c) − − k f [pN/nm] F s [ p N ] (d) FIG. 5. Mechanical response as a function of external stiffness k f . (a) Dynamic modulus for NM IIA and NM IIB ensemblesof size N = 15 at k f = 0 .
01 pN/nm (dark blue, light red)(for symbol description see Fig. 3(a)). (b, c, d) Normalizedstiffness k t /k f , total relaxation time τ t and stall force F s as afunction of external stiffness k f for N a ∈ { , , , } (darkblue, dark violet, light violet, light red). Symbols and linesrepresent the simulations and the analytical mean field theory,respectively. should depend on the mix of NM IIA and B motors in thestress fiber, as indeed reported experimentally [32, 33].In particular, the relaxation time of mature stress fiberscan be reduced by suppressing NM IIB gene expression,as predicted here [34].Cells respond very sensitively to the stiffness of theirenvironment and we therefore also investigated the roleof the external stiffness k f as shown in Fig. 5. For suf-ficiently high external stiffness, the normalized stiffness k t /k f and the total relaxation time τ t = ξ/k t both de-crease with increasing environment stiffness, while thestall force F s does not depend much on the external stiff-ness. Interestingly, the mean field approximation failsdramatically for pure NM IIA ensembles of size N = 15for environments less stiff than k f = 0 . ∗ [email protected][1] X. Trepat and E. Sahai, Nature Physics , 671 (2018).[2] P. Kollmannsberger and B. Fabry, Annual Review of Ma-terials Research , 75 (2011).[3] B. Fabry, G. N. Maksym, J. P. Butler, M. Glogauer,D. Navajas, and J. J. Fredberg, Physical Review Let-ters , 148102 (2001).[4] A. Micoulet, J. P. Spatz, and A. Ott, ChemPhysChem , 663 (2005).[5] M. Balland, N. Desprat, D. Icard, S. F´er´eol, A. Asnacios,J. Browaeys, S. H´enon, and F. Gallet, Physical ReviewE - Statistical, Nonlinear, and Soft Matter Physics (2006), 10.1103/PhysRevE.74.021911.[6] P. Fern´andez and A. Ott, Physical Review Let-ters (2008), 10.1103/PhysRevLett.100.238102,arXiv:0706.3883.[7] E. Fischer-Friedrich, Y. Toyoda, C. J. Cattin, D. J.M¨uller, A. A. Hyman, and F. J¨ulicher, Biophysical Jour-nal , 589 (2016).[8] M. Mayer, M. Depken, J. S. Bois, F. J¨ulicher, and S. W.Grill, Nature , 617 (2010).[9] X. Serra-Picamal, V. Conte, R. Vincent, E. Anon, D. T.Tambe, E. Bazellieres, J. P. Butler, J. J. Fredberg, andX. Trepat, Nature Physics , 628 (2012).[10] R. Vincent, E. Bazelli`eres, C. P´erez-Gonz´alez, M. Uroz,X. Serra-Picamal, and X. Trepat, Physical Review Let-ters , 1 (2015).[11] P. W. Oakes, E. Wagner, C. A. Brand, D. Probst,M. Linke, U. S. Schwarz, M. Glotzer, and M. L. Gardel, Nature Communications (2017),10.1038/ncomms15817.[12] A. Saha, M. Nishikawa, M. Behrndt, C. P. Heisenberg,F. J¨ulicher, and S. W. Grill, Biophysical Journal ,1421 (2016), arXiv:1507.00511.[13] M. Vishwakarma, J. Di Russo, D. Probst, U. S. Schwarz,T. Das, and J. P. Spatz, Nature Communications (2018), 10.1038/s41467-018-05927-6.[14] T. P. Wyatt, J. Fouchard, A. Lisica, N. Khalilgharibi,B. Baum, P. Recho, A. J. Kabla, and G. T. Charras,Nature Materials , 109 (2020).[15] D. Mizuno, C. Tardin, C. F. Schmidt, and F. C. MacK-intosh, Science , 370 (2007).[16] C. P. Broedersz and F. C. MacKintosh, Reviews of Mod-ern Physics , 995 (2014).[17] J. R. Beach, L. Shao, K. Remmert, D. Li, E. Betzig,and J. A. Hammer, Current Biology , 1160 (2014),arXiv:NIHMS150003.[18] J. R. Beach and J. A. Hammer, Experimental Cell Re-search , 2 (2015).[19] T. L. Kaufmann and U. S. Schwarz, PLOS Computa-tional Biology , e1007801 (2020), publisher: PublicLibrary of Science.[20] T. A. J. Duke, Proceedings of the National Academy ofSciences , 2770 (1999).[21] T. Erdmann and U. S. Schwarz, Physical Review Letters , 188101 (2012), arXiv:1202.3038.[22] T. Erdmann, P. J. Albert, and U. S. Schwarz,The Journal of chemical physics , 175104 (2013),arXiv:1307.6510.[23] L. Hilbert, S. Cumarasamy, N. Zitouni, M. Mackey, andA.-M. Lauzon, Biophysical Journal , 1466 (2013).[24] S. Stam, J. Alberts, M. Gardel, and E. Munro, Biophys-ical Journal , 1997 (2015).[25] T. Erdmann, K. Bartelheimer, and U. S. Schwarz, Phys-ical Review E , 052403 (2016).[26] N. Billington, A. Wang, J. Mao, R. S. Adelstein, andJ. R. Sellers, Journal of Biological Chemistry , 33398(2013).[27] J. Grewe and U. S. Schwarz, Phys. Rev. E , 022402(2020).[28] A. Besser and U. S. Schwarz, New Journal of Physics ,1 (2007).[29] X. Trepat, M. R. Wasserman, T. E. Angelini, E. Millet,D. A. Weitz, J. P. Butler, and J. J. Fredberg, NaturePhysics , 426 (2009).[30] J. Colombelli, A. Besser, H. Kress, E. G. Reynaud, P. Gi-rard, E. Caussinus, U. Haselmann, J. V. Small, U. S.Schwarz, and E. H. Stelzer, Journal of Cell Science ,1665 (2009).[31] E. Kassianidou, C. A. Brand, U. S. Schwarz, and S. Ku-mar, Proceedings of the National Academy of Sciences ofthe United States of America , 2622 (2017).[32] K. Tanner, A. Boudreau, M. J. Bissell, and S. Kumar,Biophysical Journal , 2775 (2010).[33] S. Lee, E. Kassianidou, and S. Kumar, Molecular Biologyof the Cell , 1992 (2018).[34] C. W. Chang and S. Kumar, Scientific Reports5