Motor crosslinking augments elasticity in active nematics

In active materials, uncoordinated internal stresses lead to emergent long-range flows. An understanding of how the behavior of active materials depends on mesoscopic (hydrodynamic) parameters is developing, but there remains a gap in knowledge concerning how hydrodynamic parameters depend on the properties of microscopic elements. In this work, we combine experiments and multiscale modeling to relate the structure and dynamics of active nematics composed of biopolymer filaments and molecular motors to their microscopic properties, in particular motor processivity, speed, and valency. We show that crosslinking of filaments by both motors and passive crosslinkers not only augments the contributions to nematic elasticity from excluded volume effects but dominates them. By altering motor kinetics we show that a competition between motor speed and crosslinking results in a nonmonotonic dependence of nematic flow on motor speed. By modulating passive filament crosslinking we show that energy transfer into nematic flow is in large part dictated by crosslinking. Thus motor proteins both generate activity and contribute to nematic elasticity. Our results provide new insights for rationally engineering active materials.


SUPPLEMENTARY TEXT
In the main text, we estimated the characteristic velocity v and characteristic length ℓ using quantities (ε and P cℓ ) derived from the kinetic simulations.Here, we consider an alternative model in which the motor velocity v m and crosslinking probability P cl are assumed to follow simple Michaelis-Menten-like dependences on [ATP].Following [1], we assume that the motor velocity v m takes the form in which v m∞ is the maximum velocity (corresponding to [ATP] → ∞, which we take to be 1 for simplicity) and C 1 is a constant with units of concentration that represents the value of [ATP] at which v m = v m∞ /2.We then assume that the probability of motor attachment P cl takes the form in which P cl0 is the maximum motor attachment probability (corresponding to [ATP] = 0, which we also take to be 1) and C 2 is a constant with units of concentration that represents the value of [ATP] at which P cl = P cl0 /2.The experimental observations and simulations both suggest that In Fig. S6A,B, we plot the normalized motor velocity v m /v m∞ and normalized motor crosslinking probability P cl /P cl0 for C 1 = 80 µM and C 2 = 400 µM.
The extension rate ε is assumed to be proportional to the product of the motor speed v m and the motor attachment probability P cl : As in the main text, we assume that α ∼ ε β and K = K 0 + κc e , where c e = c p + c m P cl , and K 0 , κ, and c m are positive constants.In the absence of passive crosslinker (c p = 0), we can estimate the characteristic active length scale discussed in the main text as and the characteristic velocity scale as For nonzero K 0 , the limiting behavior in the small concentration ( and the limiting behavior in the large concentration (   [ATP] (μM) v (norm.) [ATP] (μM) [ATP] (μM)

Figure S1 :Figure S2 :
FigureS1: Details of engineered myosin constructs.(A) Block diagrams and schematics of the engineered myosin XI trimer, tetramer, and octamer.Constructs consist of a Chara corallina myosin XI motor domain (MXI MD), a lever arm containing two spectrin-like repeats from Dictyostelium α-actinin (2R), a flexible "slack" element containing one spectrin-like repeat flanked by glycine-serine-glycine linkers (∼1R∼), a multimerization domain (TRI, TET or OCT for the trimer, tetramer, and octamer, respectively), and a Halotag (HT).(B) Amino acid sequences of the engineered myosin constructs.Sequences are shown at the junctions between the different protein fragments within each construct.

Figure S3 :
Figure S3: ℓ vort robustly captures nematic length scale.Comparison of ℓ vort as calculated in [2] and the traditional velocity correlation length at which C vv = 1/e for nematics driven by (A) 120 pM tetramers and (B) 50 pM octamers.Error bars are averages over five separate frames for C vv and five 10 s segments for ℓ vort .

Figure S4 :
Figure S4: Nematics driven by tetrameric motors exhibit nonmonotonic speed with increasing [ATP].(A) Average nematic speed (v RMS ) over time for samples driven by tetrameric motors with varying [ATP].(B) Histograms of nematic speed for individual frames in (A) sampled at 20 s intervals.

Figure S12 :
Figure S12: Peak shift is robust across replicates and motor concentrations.[ATP] at which peak speed occurred as a function of motor valency.Colors indicate cluster concentration and separate points indicate independent replicates.

Figure S13 :
Figure S13: Nematic elasticity increases energy in the nematic.Elastic (black) and kinetic (red) energy for nematics in lattice Boltzmann simulations with constant α = 0.01 across a range of K.