Cooperative effect of local active stresses on the macroscopic contractility of elastic fiber networks

Abstract

The collective action of actively contractile units embedded in elastic biopolymer networks plays a crucial role in regulating the network's macroscopic mechanical response. Here, we investigate how the macroscopic boundary stress in model elastic fiber networks depends on the number and nature of embedded contractile units, each exerting an isotropic force dipole, as well as on the bending stiffness of fibers. We find that the macroscopic stress increases nonlinearly with the number of dipoles due to mutual stiffening of initially soft, bending-dominated networks. Using effective medium theory, we relate this enhanced contractility to an increase in the effective average network coordination number due to constraints imposed by the force dipoles. By comparing three distinct force dipole models that differ in their local structures, we demonstrate that the specific manner in which an active unit constrains the network strongly influences the onset and nature of the stiffening transition. Our results highlight that not only the quantity but also the local geometry of force-generating units critically determines the macroscopic mechanical behavior. This framework provides a physical basis for understanding how biological systems-such as molecular motors in the cytoskeleton, or adherent cells in the extracellular matrix-can modulate network-scale nonlinear elastic properties through local tuning of active force-generating units.

Figures (8)

• Figure 1: Model setup of elastic fiber network with embedded force dipoles. (a) A representative network configuration with a circular outer boundary ($R_{2} = 25$), with fixed boundary nodes (colored in magenta). Isotropic force-producing units (nodes colored in green) are randomly placed in an inner circular region ($R_{1} = 12$), marked with darker colored nodes. For each dipole number, a total of 10-40 random dipole placements are generated in the inner region. The network configuration in the annular outer region ($R_{1} < r < R_{2}$), is varied over 10-25 different realizations for each random dipole placement. (b) A representative force-producing unit is depicted. It comprises a central node and six outer nodes, all shown in green. All six radial bonds coming out of the central node are required to be present during network construction. These bonds are given a rest-length of 0.9, less than the initial length of all bonds, which is set to 1.0. Therefore, these bonds are in a state of prestress and tend to contract towards their rest lengths as the network reaches its force balanced mechanical equilibrium configuration. The net effect is to produce an isotropic, contractile, local force dipole on the network. (c) The individual bonds in the network can stretch and compress, and are modeled as linear springs. Additionally, co-linear bonds can also bend by changing their relative angle with a bending energy cost, typically much smaller than the stretching/compression energy cost. (d) The final network configuration at mechanical equilibrium, i.e. after the network shown in (a) is relaxed to its energy minimum state using the conjugate gradient method. Here, highly stretched(compressed) bonds are colored red(blue) for visualization purposes, if the magnitude of bond strain exceeds a threshold value, $\epsilon_0 = 10^{-6}$.
• Figure 2: Scaling of average far-field force dipole moment with bending modulus and number of force dipoles in bending-dominated networks with $\boldsymbol{p=0.55}$. (a) The average far-field dipole moment, $\langle D_{far} \rangle$, a measure of force transmission to the network boundary, increases linearly with dipole number $N_d$ for low $N_d$, but shows positive (upward) curvature at higher $N_d$. The straight lines show a linear scaling of $D_{far}$ with $N_d$. $\langle D_{far} \rangle$ increases approximately linearly with the reduced fiber bending modulus $\widetilde{\kappa}$, but the onset of nonlinearity is at a similar value, $N_d \gtrapprox 15$, for all three representative $\widetilde{\kappa} \ll 1$ cases simulated. The insets show two representative simulated network configurations with $N_d = 5$ dipoles each, but different bending moduli: $\widetilde{\kappa} = 10^{-5}$ (top left) and $\widetilde{\kappa} = 10^{-6}$ (bottom right). Blue (red) colors indicate highly stretched (compressed) bonds above a strain magnitude threshold, $\epsilon_0 = 10^{-6}$. The softer network ($\widetilde{\kappa} = 10^{-6}$; bottom right) has visibly fewer highly tensed or compressed bonds ("force chains") than the stiffer network ($\widetilde{\kappa} = 10^{-5}$; top left). (b) Average Far-field dipole moments, $\langle D_{far} \rangle$, scaled by the corresponding average value for one dipole unit $N_{d} = 1$, at the same $p$ and $\widetilde{\kappa}$ values. $\langle D_{far} \rangle$ values for bond-diluted ($p=0.55$) networks at different bending moduli, $\widetilde{\kappa}$, collapse on the same master curve, indicating universal nonlinear stiffening of bending-dominated networks with increasing dipole number. The $p=1$ networks, by contrast, show linear increase of $D_{far}$ with $N_d$, characteristic of linear elastic media. [Inset] Local dipole moment scales linearly with the number of dipoles, for all networks with $p=0.55$ with different bending moduli. Thus, our simulation procedure ensures an approximately identical local force dipole is exerted by each active unit. Each plotted value of $\langle D_{far} \rangle$ is an average over all bending-dominated networks obtained from a total of $100$ simulations ($10$ dipole configurations $\times 10$ network configurations), except for the one dipole case ($N_d=1$), where averaging is over 1000 simulations. Error bars indicate 95 $\%$ confidence intervals on the mean value obtained from bootstrapping.
• Figure 3: Scaling of $\boldsymbol{\langle D_{far} \rangle}$ with stretching and bending energies for bending- dominated networks at $\boldsymbol{p=0.55}$ and $\boldsymbol{\widetilde{\kappa}=10^{-6}}$. (a) $\langle D_{far} \rangle$ values scale as the square root of the average stretching energy as dipole number increases. This is the expected stress-energy relationship from usual elasticity. (b) $\langle D_{far} \rangle$ scales linearly with the average bending energy as dipole number increases. This is a departure from the usual elastic stress-energy relationship, and can be explained based on how bending contributes to bond tension (see main text). Each plotted value of $\langle D_{far} \rangle$ is an average over all bending-dominated networks obtained from a total of $100$ simulations ($10$ dipole configurations $\times 10$ network configurations), except for the one dipole case, where averaging is over 1000 simulations. Error bars indicate 95 $\%$ confidence intervals on the mean value obtained from bootstrapping.
• Figure 4: Dipoles impose additional constraints and increase effective network connectivity in bending-dominated networks. (a) Ratio of $\langle D_{far} \rangle$ for networks at $p = 0.55$ and $p = 1$. This is a measure of the fraction of forces applied by local dipoles, that is transmitted to the boundary, and therefore of effective network stiffness. This ratio increases with increasing dipole number and bending modulus. (b) The effective bond probability, $p_{eff}$, of a corresponding passive network with equivalent stiffness, $\mu_m$, to our local dipole-stressed networks, is obtained using effective medium theory (EMT) (see main text for details). The linear dependence of $p_{eff}$ on $N_d$ shows that each dipole applies a fixed number of constraints, corresponding to Eq. \ref{['eq:peff']} in the text. The value of the slope of linear fit (shown here for $\widetilde{\kappa}$ = $10^{-6}$) corresponds to $n^{d}_{c} = 17.2 \pm 0.1$ constraints per dipole. The increase in $p_{eff}$ with $N_d$ is however very similar for the three different bending modulus $\widetilde{\kappa} \ll 1$ values simulated. This indicates that a similar number of floppy modes are "pulled out" for networks in this bending-dominated regime. Each plotted value of $\langle D_{far} \rangle$ is an average over all bending-dominated networks obtained from a total of $100$ simulations ($10$ dipole configurations $\times 10$ network configurations), except for the one dipole case, where averaging is over 1000 simulations. Error bars indicate 95 $\%$ confidence intervals on the mean value obtained from bootstrapping.
• Figure 5: Comparison of three local force dipole models showing significantly different distributions of network deformation energy. In all three dipole models, the central dipole node is fully coordinated, i.e. all six radial bonds are present. Contraction is applied by reducing their rest lengths relative to initial value. (a) Model 1: All six transverse bonds between six outer dipole nodes are removed. Representative force-balanced network configuration shows very few strained bonds (red/blue). (b) Model 2: Transverse bonds connecting the outer dipole are present with a probability $p$. Representative force-balanced network configuration shows more strained bonds (red/blue) than model 1. (c) Model 3: All six transverse bonds connecting the outer dipole nodes are present. Representative force-balanced network configuration shows more strained bonds (red/blue) than models 1 and 2. The distribution of ratios of network stretching and bending energies shown in (d,e,f) are obtained from $1000$ representative simulations for a five-dipole configuration in $p=0.55$ networks with $\widetilde{\kappa}=10^{-6}$ ($40$ random dipole placements, and $25$ random outer network configurations for each dipole placement, leading to $1000$ total simulations). All three histograms (d,e,f) are normalized by maximum bin height. (d) Model 1: All networks simulated are bending-dominated ($E_{st}/E_{bend} < 1$), as expected for small deformations of sub-isostatic networks. (e) Model 2: A fraction of the random network realizations are stretching-dominated ($E_{st}/E_{bend} > 1$) . (f) Model 3: A majority of networks turn stretching-dominated despite being at $p=0.55$. This suggests that progressively more pre-stress and thus effective constraints are applied by the force dipoles as we go from model 1 to 2 to 3.