Mechanical organization yields degenerate dissipation beyond linear response

17 18 In non-equilibrium (active) systems, increased driving is commonly assumed to amplify energy 19 dissipation. This frames the efficiency of protein -based machines as a fixed or monotonically 20 decreasing function with driving. Using picowatt -sensitive calorimetry and advanced entropy 21 production metrics in reconstituted actomyosin networks, we show that energy dissipation depends 22 non-monotonically on myosin -generated stress (driving). At low driving, dissipation increases 23 proportionally with stress, consistent with near -equilibrium linear response. At high driving, 24 however, dissipation decreases, revealing a far -from-equilibrium regime in which excessive load 25 suppresses motor ATPase activity. This non -monotonicity reflects a transition from spatially 26 localized stress at low driving to delocalized stress at high driving, where force per motor, and thus 27 ATPase suppression, is maximized. Crosslinker mechanics tune this transition as fascin (slip bonds) 28 amplifies stress localization and shifts the dissipation peak to higher driving, whereas α -actinin 29 (catch bonds) stabilizes under load, delocalizes stress, and shifts the peak to lower driving. Thus, 30 enhanced mechanochemical coupling causes additional driving to restructure rather than amplify 31 dissipation, revealing how material system organization (bonding), and not driving alone, governs 32 energy flow far from equilibrium. 33 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint Main Text 34 35

36 37 In non-equilibrium systems, increased driving is commonly assumed to amplify energy 38 dissipation, reflecting the conventional assumption inherited from linear response and irreversible 39 thermodynamics 1-5. This intuition links activity, force generation, and energetic cost , in that 40 stronger driving produces larger fluxes and greater heat loss. It underlies how efficiency is 41 typically understood in systems ranging from molecular machines to active solids, where enhanced 42 activity is often equated with increased dissipation. Whether this assumption remains valid when 43 the system is driven far from thermodynamic equilibrium, however, remains largely unexplored. 44 Linear response theory predicts that fluxes scale proportionally with their conjugate forces 45 and that entropy production increases monotonically with driving 1,6-10. In this regime, dissipation 46 is determined by transport coefficients that are independent of the applied force, and added input 47 is converted directly into heat through Onsager reciprocity6. This framework has been remarkably 48 successful in describing transport 11 and pattern formation in passive matter 12, and is often 49 implicitly extended to active systems13-16. Active materials, however, operate far from equilibrium, 50 where driving does not merely induce motion but impacts the material itself 13,14,16. In the cell 51 cytoskeleton for example , i nternally generated stresses alter connectivity, stiffness, and force 52 transmission, introducing feedback between mechanical organization and energy consumption. 53 Among these feedbacks include load -dependent bonds 17-20, which endow the cytoskeleton with 54 adaptability, in the ability to transition between solid -like and fluid -like states through motor -55 generated stresses transmitted by filamentous networks 21-24. In such systems, dissipation may no 56 longer be set by driving alone, but how the system redistributes or allocates energy across internal 57 degrees of freedom. Directly testing this possibility has been challenging, in part because energy 58 dissipation is rarely measured alongside material organization. As a result, it remains unclear 59 whether dissipation in active matter necessarily increases with driving, or whether it can instead 60 be regulated by the internal organization of the material. 61 Here we show that increased driving does not necessarily amplify dissipation in active 62

but instead reorganizes how energy is dissipated through stress localization and 63 delocalization. To this end, we use the actomyosin cytoskeleton as a model non-equilibrium 64 material, where myosin motors consume chemical energy (ATP) and through the generation of 65 active stress, drive the cytoskeleton beyond the linear regime 22-26. As a function of driving, 66 mechano-sensitive bonds introduce feedback between mechanical force transmission and chemical 67 (ATPase) activity, determining material organization (e.g. crosslinking), stress focusing and a non-68 monotonic dissipation of energy. The impact of the bonds are twofold : catch bonds (𝛼-actinin) 69 strengthen under tensile load 21,27-30, stabilizing stressed networks, whereas slip bonds (fascin) 70 unbind under increased tension 31,32, shifting the peak in dissipation to different driving, and fine 71 tuning stress concentration akin to Anderson Localization in amorphous materials . Thus, b y 72 reconstituting 2D and 3D actomyosin networks with defined bond mechanics, we demonstrate that 73 dissipation does not increase monotonically with driving and directly link non-monotonic behavior 74 to a material transition between stress localization and delocalization 33,34. Combining confocal 75 fluorescence microscopy, entropy production, picowatt-sensitive calorimetry, and rheology, we 76 show how mechanochemical feedback reshapes energy flow in active matter, revealing 77 mechanisms by which driving reorganizes, rather than simply amplifies dissipation in a far from 78 equilibrium regime. 79 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint 80

81 We assemble model actomyosin networks in two configurations 35. First, we assemble a quasi-2D 82 network, amenable for fluorescence imaging and the quantification of filament dynamics 36-38. 83 Second, we assemble a 3D network, more amenable to rheology 39-42 and calorimetry (Methods). 84 Integrating 3D imaging with pico -calorimetry allows direct comparison of network local 85 deformation and heat dissipation, while rheological measurements provide complementary 86 mechanical characterization. The molar ratios of proteins (e.g., crosslinker -to-actin and myosin -87 to-actin) are identical in the 2D and 3D assays, although the use of a crowding agent in the 2D 88 system results in higher effective surface protein concentrations compared to bulk concentrations 89 in 3D networks that are without crowding agent. 90 91 Catch- and slip -bond networks are mechanically indistinguishable under thermal 92 fluctuations but diverge under active stress. 93 We reconstitute a quasi -two-dimensional active actomyosin network by confining pre -94 polymerized actin filaments to a supported lipid bilayer using 0.25% methylcellulose, crosslinking 95 the network with fascin or α-actinin, as fascin is known to be a slip bond, while α-actinin is a catch 96 bond 21(Fig. 1a). We then introduce skeletal muscle myosin II to generate active contractile stress 97 (Fig. 1a). Under high crosslinker density (R c > 0.2), pseudo -2D α-actinin and fascin crosslinked 98 network sheets are morphologically indistinguishable (Fig1 b -d). Both α -actinin and fascin 99 crosslink and bundle F -actin into thick bundles (Fig. d-e). Under thermal driving force, F-actin 100 networks crosslinked by α -actinin and fascin exhibit similar relaxation time, suggestive of 101 comparable rigidity (Fig1. f-g). However, the dynamics of the networks crosslinked by α -actinin 102 and fascin are distinctively different under myosin -induced active stress: f ascin-crosslinked 103 networks contract locally into asters (Fig. 1h, supplementary movie 1 ); i n contrast, α -actinin-104 crosslinked networks behave as cohesive elastic sheets that rupture via crack formation38 (Fig. 1h-105 i, supplementary movie 1). The distinct contractile behaviors of the two crosslinked networks lead 106 to differences in both the magnitude of strain and the time to material failure. (Fig. 1j-k). We 107 hypothesize the difference in the mechanical dynamics of the two types of networks is due to the 108 difference in bond mechano-sensitivity and mechanics21 (Supplemental Materials). Therefore, due 109 to the opposite mechanism of bond kinetics under load, slip bonds tend to unbind in high -stress 110 regions while catch bonds tend to unbind in low -stress regions. This results in potentially the 111 homogenization of stress in the α-actinin-crosslinked network which enables the active material to 112 endure higher stress. Interestingly, in α-actinin crosslinked networks, the strain rate of deformation 113 exhibits a non-monotonic dependence on myosin concentration , with strain rate peaks at around 114 Rmyo ≈ 0.02 (Fig. 1l). This unique observation of the dynamics and non -monotonicity in the 115 nonlinear/contractile regime is reminiscent of mechanical inhibition of dissipation in active 116 solids43. Therefore, to investigate the origin of the non -monotonicity, we begin to investigate the 117 energy partition and heat dissipation of the networks under active stress. 118 119 Mechano-chemical coupling yields non-monotonic dissipation with driving 120 We reconstitute a three-dimensional active actomyosin network by mixing purified G-actin, 121 skeletal muscle myosin II, and crosslinkers such as α -actinin or fascin with ATP in F -buffer 122 (Methods). This protein mixture was injected into a glass capillary tube mounted inside a custom-123 built picowatt-calorimeter. The calorimeter achieves a lower detection limit of 8 5 pW (Fig. 2a, 124 SFig. 1, Methods) [ Zheng et al., In submission ]. The total heat dissipation results from 125 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint predominantly three mechanisms: 1) ATP hydrolysis of actin polymerization, 2) ATPase kinetics 126 via myosin motors, and 3) ATP regeneration system via pyruvate kinase and lactic dehydrogenase 127 (PK-LDH) along with phosphoenolpyruvate (PEP) (Fig. 2 b, SFig. 1 , Methods ). Upon 128 polymerization, heat dissipation from the F -actin network was captured and exhibited an 129 exponential decay over time, consistent with prior work on actin polymerization kinetics 44 (Fig. 130 2c-d, Supplementary Materials). We directly measure the heat dissipation Δ𝑄, and each data point 131 is collected every 0.5 seconds, results in heat dissipation per unit time (power), 𝑄̇ = Δ𝑄/Δ𝑡. After 132 characterizing the heat dissipation of actin polymerization, we introduce skeletal muscle myosin 133 II dimers into the protein mix, which performs power strokes on F -actin and causes network 134 deformation and contraction45 (Supplementary Materials). We increase ATP concentration while 135 keeping actin and myosin concentration constant, the measured heat dissipation rate 𝑄̇ follows 136 Michaelis–Menten kinetics: 137 𝐽𝐴𝑇𝑃𝑎𝑠𝑒([𝐴𝑇𝑃])= 𝑘𝑐𝑎𝑡 𝐴𝑇𝑃[𝐴𝑇𝑃] 𝐾𝑀 𝐴𝑇𝑃 + [𝐴𝑇𝑃][𝑀]𝑡𝑜𝑡 (1) , where 𝐾𝑀 𝐴𝑇𝑃 = 0.1397 ± 0.09345 µM , of which the difference is within a factor of 2 compared 138 to values from previous studies46 (Fig. 2e), potentially due to the differences in actin concentration 139 and myosin motor type . We then vary the concentration of actin, and the heat dissipation rate 𝑄̇ 140 also follows Michaelis–Menten kinetics: 141 𝐽𝐴𝑇𝑃𝑎𝑠𝑒([𝐴])= 𝑘𝑐𝑎𝑡 [𝐴𝑇𝑃] 𝐾𝐴 + [𝐴𝑇𝑃][𝑀]𝑡𝑜𝑡 (2) , where 𝐾𝐴 = 13.11± 9.6125 µM, agrees with previous work47,48 (Fig. 2f). 142 To further investigate the energetic role of motor activity, we systematically varied the 143 myosin concentration while maintaining a constant actin concentration. The heat dissipation 144 profile of the resulting actomyosin networks exhibit s a non -monotonic trend as myosin 145 concentration increases (Fig. 2g). We hypothesize that above a critical myosin-to-actin molar ratio 146 (Rmyo > 0.02), the network enters a highly connected regime in which active stresses become 147 uniformly distributed due to myosin’s mechanosensitivity. In this regime, myosin thick filaments 148 can behave as catch bonds, and therefore the ATPase activity of myosin is reduced due to increased 149 mechanical load across the network49-52, reflected as a decreased rate of heat dissipation (Fig. 2g). 150 This is reminiscent of the non -monotonicity of dissipation observed in ‘active solids’ as in our 151 contractile 2D-network experiments (Fig. 1) , in which energy dissipation peaks at intermediate 152 stress and is inhibited at large tensile or compressive loads 43. The nuance between the 2D system 153 and 3D system is that the 3D system does not exhibit large scale deformation, potentially due to 154 geometry and the effective protein concentration differences (Fig. 2g, supplementary movie 2). In 155 our case, the heat dissipation primarily comes from the ATPase activity and ATP regeneration 156 system ( SFig. 1, Supplemental Materials ). The heat dissipation is linearly proportional to the 157 concentration (number) of myosin head performing ATPase , 𝑐ℎ𝑒𝑎𝑑 , the myosin ATPase rate , 𝑟 158 (𝑠−1), the volume of the chamber, 𝑉 (80 nL in this case), and Δ𝐺, the free energy , which is ≈159 60 𝑘𝐽/𝑚𝑜𝑙. The heat dissipation rate 𝑄̇ measured then follows: 160 161 . 162 From this, we can calculate an effective myosin ATPase rate ( SFig. 2, Supplemental Materials), 163 which can indirectly infer the level of active stress , 𝜎𝑎, experienced by the motors since myosin 164 ATPase rate 𝑟 is inversely proportional to active stress 𝜎𝑎 49-51,53. 165 𝑄̇ = Δ𝑄/Δ𝑡 = 𝑐ℎ𝑒𝑎𝑑 ∙ 𝑟 ∙ 𝑉 ∙ Δ𝐺 (3) .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint To elucidate how differences in force sensing and mechano -sensitive binding shape 166 energetic output, we examine actin networks crosslinked with either α -actinin (catch bond) or 167 fascin (slip bond) using 3D microscopy and pico -calorimetry. This allows us to directly compare 168 how distinct bond mechanics influence stress generation and heat dissipation under active motor 169 loading. 170 171 Catch bond crosslinked network dissipates more heat under active stress 172 As mentioned previously, t he 3D networks exhibited minimal large -scale deformation 173 under equivalent myosin density , likely due to geometric constraints and effective protein 174 concentrations, as noted in prior studies 26,54 (Fig. 2g). Given the differences in mechanical 175 response between α-actinin and fascin networks under active loading, we next quantif y the total 176 heat dissipation of these systems using picowatt-resolution calorimetry. 177 Under fixed crosslinking and myosin density , increasing the ratio between α-actinin to 178 fascin (Φ𝑎𝑐𝑡𝑖𝑛𝑖𝑛:𝑓𝑎𝑠𝑐𝑖𝑛) results in a n increase in heat dissipation , potentially due to an elevated 179 ATPase rate with higher Φ𝑎𝑐𝑡𝑖𝑛𝑖𝑛:𝑓𝑎𝑠𝑐𝑖𝑛 (Fig. 2h). 180 To further investigate this behavior, we systematically var y the concentration of myosin 181 motors while maintain constant crosslinker density. In both α -actinin and fascin networks, heat 182 dissipation display a non-monotonic trend as a function of motor density (Fig. 2i). However, the 183 dissipation peak occurred at a higher myosin concentration in fascin networks compared to α -184 actinin networks (Fig. 2i). Notably, the dissipation curve for the α -actinin case closely resembles 185 that of a network lacking crosslinkers (Fig. 2i), supporting the hypothesis that myosin exhibits 186 catch bond–like behavior. In contrast, the delayed peak observed in fascin networks is likely due 187 to fascin’s polarity -sorting capacity and slip bond properties, which restrict stress redistribution 188 until the motor density reaches a threshold where catch bond dynamics begin to dominate (Fig. 2i). 189 Having established that bond mechanics influence how stress accumulates and relaxes, we next 190 ask how these differences translate into measurable changes in energy dissipation. 191 Therefore, we explore how heat dissipation depends on crosslinking density. For fascin, 192 we observe a sharp drop in dissipation around Rc ≈ 0.001, near the expected bundling threshold 193 and percolation critical point for 3D networks41 (Fig. 2j). In contrast, heat dissipation in α-actinin-194 crosslinked networks remain relatively stable with only a modest decline as Rc increased (Fig. 2j). 195 Across all crosslinker densities, α -actinin networks exhibit higher heat dissipation than fascin 196 networks. 197 We hypothesize that this is due to α-actinin’s catch bond nature, which allows crosslinkers 198 in low-stress regions, where unbinding rates are higher, to relocate to regions of higher stress. This 199 dynamic redistribution promotes stress homogenization across the network 55. Because myosin 200 ATPase activity is inversely correlated with the mechanical load experienced by motors49-51,53, the 201 more evenly distributed stress in α -actinin networks may result in higher overall ATP turnover 202 compared to fascin networks, as also shown from the effective ATPase rate calculation (SFig. 2). 203 To directly probe the mechanical stress landscape in these systems and validate this hypothesis, 204 we next perform rheological measurements to characterize the viscoelastic properties of the two 205 network types (Fig. 3a). 206 207 Catch bond crosslinked network homogenizes and lowers stress 208 To characterize the material properties of crosslinked F-actin networks under active stress 209 generated by myosin motors , we use a stress -controlled rheometer (Fig. 3a, Supplementary 210 Materials). Previous studies have suggested that the mechanical response of actively stressed 211 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint cytoskeletal networks is analogous to that of passive networks driven into the nonlinear regime via 212 prestress23,25,56. Here, we indeed observe that the viscoelastic moduli (G′ and G″) of the network 213 increase with higher Rmyo (Fig. 3b). This is not due to crosslinking by myosin dimers or thick 214 filaments, as our ATP concentration is within the saturating regime in which myosin dimers have 215 an binding/unbinding time of ~5 ms23. 216 In linear frequency sweep experiments, the crossover frequency ( 𝜔𝑐𝑜), the frequency at 217 which the storage modulus G′ equals the loss modulus G″ , serves as a metric for the timescale at 218 which the network transitions from solid -like to fluid-like behavior. More adaptive and transient 219 networks exhibit lower 𝜔𝑐𝑜 values. At equivalent crosslinking densities, α -actinin–crosslinked 220 networks display significantly lower 𝜔𝑐𝑜 than fascin-crosslinked ones, indicating more fluid -like 221 and dynamic behavior (Fig. 3c-d). As a control, we also examine networks crosslinked via biotin–222 streptavidin, a permanent and non -dynamic linkage. Biotin–streptavidin-crosslinked networks 223 display a crossover at very high frequency or almost no crossover, behaving as elastic solids over 224 a wide frequency range (Fig. 3c-d). 225 As myosin density increases , both G′ and G″ increase monotonically for α-actinin–226 crosslinked networks, consistent with stress -induced stiffening observed in prior work 23,56 (Fig. 227 3e-f). In contrast, fascin -crosslinked networks exhibit a non -monotonic response: both moduli 228 initially increased with motor density but later decreased (Fig. 3e-f). This behavior may reflect 229 structural reorganization of the fascin -actin bundles or a mechanical transition from slip bond –230 dominated to catch bond–dominated dynamics. 231 To quantify the internal stress exerted by myosin motors, we perform nonlinear rheology 232 on passive networks of matched crosslinking density (Methods). We first appl y a step shear 233 prestress 𝜎0 and allow the sample to reach a quasi -steady state. We then superimpose a small 234 sinusoidal stress 𝛿𝜎(𝑡)= |𝛿𝜎|𝑒𝑖𝜔𝑡 with amplitude kept within the linear response of the 235 prestressed state (| 𝛿𝜎| ≤ 𝜎0/10) so that linear response theory holds 57-59. The resulting strain 236 response 𝛿𝛾(𝑡) defines the complex differential modulus as the local slope of the stress –strain 237 relation about 𝜎0: 238 𝐾∗(𝜔,𝜎0)= 𝛿𝜎(𝜔) 𝛿𝛾(𝜔)|𝜎0 (4) , with 𝐾′(𝜔,𝜎0)= 𝑅𝑒(𝐾∗)and 𝐾"(𝜔,𝜎0)= 𝐼𝑚(𝐾∗). In the limit of zero prestress, this reduces to 239 the conventional linear viscoelastic moduli, 𝐾′(𝜔,0)= 𝐺′(𝜔) and 𝐾"(𝜔,0)= 𝐺"(𝜔). We fit the 240 differential elastic modulus with: 241 𝐾′ 𝑝𝑎𝑠𝑠(𝜎0,𝜔) = 𝐾′ 0 + 𝐴 𝜎0 𝛼 (5) By extracting the differential elastic modulus K′ as a function of applied prestress, we map the 242 linear moduli (G′, G″) of the active networks onto the stress–stiffening curves of passive ones, thus 243 estimating the active stress imparted by motors (Fig. 3g-h): 244 𝜎𝑎 ≈ ((𝐺′ 𝑎𝑐𝑡 − 𝐾′ 0) 𝐴 ) 1 𝛼 (6) . At low myosin concentrations, active stress in fascin -crosslinked networks exceed that of α -245 actinin networks (Fig. 3h). However, above Rmyo ≈ 0.05, the active stress in fascin networks 246 decreases. This trend mirrors the calorimetric data (Fig. 2i), in which fascin networks begin to 247 dissipate more heat than α-actinin networks at high motor densities. 248 We also use velocity information calculated from PIV on 3D network data to extract the 249 mechanical dissipation rate ⟨𝑝𝑚𝑒𝑐ℎ⟩: 250 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint ⟨𝑝𝑚𝑒𝑐ℎ⟩ = 2 𝛽 ∫ 𝜂′(2𝜋 𝑓) [ 𝑆𝑥𝑥(𝑓) + 𝑆𝑦𝑦(𝑓) + 2 𝑆𝑥𝑦(𝑓) ] 𝑑𝑓 𝑓𝑚𝑎𝑥 0 (7) , in which 𝛽 is the inverse thermal energy of the bath, S(𝑓) is the power spectral density function 251 for the strain rate (divergence of velocity), and dynamic viscosity 𝜂′(2𝜋 𝑓) = G"(𝑓) 2𝜋 𝑓 252 (Supplemental Materials). We then calculate the mechanical efficiency 𝜂 = ⟨𝑝𝑚𝑒𝑐ℎ⟩ ⟨𝑄̇⟩ , for both slip 253 bond- (fascin) and catch bond- (α-actinin) crosslinked networks. 𝜂 lies between 5 × 10−5~1.5×254 10−4, with slip bond -crosslinked network showing a higher mechanical efficiency than catch 255 bond-crosslinked network (SFig. 3). 256 257 Monotonic increase in driving leads to degeneracy in dissipation 258 Finally, to elucidate further on the non-monotonic behavior in dissipation due to chemical-259 mechanical coupling, we perform actin network velocity correlation and entropy production rate 260 analysis33,34,60 using the actin and myosin fluorescence channel (Methods). The entropy production 261 rate (EPR) provides a lower bound for the dissipation of the system, and further, the 262 thermodynamic molecular details of the system due to mechano-chemical feedbacks. We observe 263 that the EPR exhibit non -monotonic trend as a function of myosin motor concentration, which 264 shows degeneracy at low (Rmyo = 0.002) and high (Rmyo = 0.1) driving (Fig. 4 a, d). 265 Despite their similar dissipation rates, these regimes differ fundamentally in how 266 mechanical stress propagates through the network. Velocity correlation analysis shows that at low 267 myosin activity, stress transmission is short-ranged and localized, whereas at high myosin activity 268 it becomes more delocalized, giving rise to coherent flows across the network (Fig. 4b,c). Thus, 269 comparable entropy production can emerge from distinct stress-propagation modes, demonstrating 270 a degeneracy between dissipation and mechanical organization in active cytoskeletal 271 materials.(Rmyo = 0.002) 272 273

274 Near thermodynamic equilibrium, linear response theory predicts that increasing driving 275 forces lead to a monotonic increase in dissipation: energy fluxes scale linearly with their conjugate 276 forces, and additional input is converted directly into heat through Onsager reciprocity 1,2,7,10,61. 277 While not an explicit assumption, t his expectation is often implicitly extended to active matter, 278 where stronger motor activity is assumed to amplify dissipation 62,63. Our results demonstrate a 279 fundamental breakdown of this intuition. Instead of monotonically increasing, energy dissipation 280 peaks at intermediate driving and is suppressed at high stress. This non -monotonicity reflects a 281 far-from-equilibrium regime in which added energy is not dissipated more strongly, but reallocated 282 through feedback between mechanical stress, bond kinetics, and motor chemistry. Such behavior 283 is inaccessible to linear response and reveals that, in active materials, driving can restructure 284 pathways of energy flow rather than simply increase entropy production. These results may be 285 potentially a general feature of active systems. While previous work could not statistically 286 distinguish between saturation and modest decrease 64, the non -monotonicity may arise robustly 287 from mechanochemical feedback between stress organization and motor activity. 288 The physical manifestation of this energetic reallocation is a transition in how mechanical 289 stress is organized in space analogous to Anderson localization in disordered media 65,66. Rather 290 than remaining uniformly distributed, active stress either concentrates into localized domains or is 291 delocalized across the network, depending on bond mechano-sensitivity (Fig. 4e). Stress focusing 292 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint amplifies local force per motor, suppressing ATP turnover and limiting dissipation, whereas stress 293 defocusing distributes load over many motors, sustaining higher chemical flux and elevated heat 294 production. This stress-organization transition provides a concrete, mesoscale mechanism linking 295 microscopic mechanochemistry to macroscopic thermodynamics: dissipation is regulated not by 296 the magnitude of driving alone, but by how that driving is partitioned across space. In this sense, 297 stress localization acts as a control variable for energy flow in active matter . The impact of stress 298 localization is not only to alter the propensity for mechanical propagation, but also in the induction 299 of other active processes, such as transforming mechanical structure into a regulator of metabolic 300 cost. 301 Non-monotonic behaviors in system variables are ubiquitous and diverse 67,68. These 302 behaviors can arise from feedback between system variables, such as reactant and product 303 concentration, or through mechanical effects, such as contractions that alter the flow and 304 concentration of chemical species 69. However, the mechanisms do not necessarily impact 305 thermodynamic forces, such as the ab initio injection of energy. Here, we demonstrate that 306 chemical energy is not a bath, but a subsystem. 307 Collective organization is known to qualitatively reshape dissipation in condensed -matter 308 systems70-72. Calorimetry reveals that networks crosslinked with α-actinin dissipate more heat than 309 those crosslinked with fascin under comparable connectivity and active stress, indicating that 310 stabilizing a stressed cytoskeleton can be more energetically demanding than contracting it. Unlike 311 in superconductors where coherence suppresses dissipation 71,73, the actomyosin network 312 represents an inverted dissipative Bardeen -Cooper-Schrieffer (BCS) system: collective stress 313 homogenization increases the density of active mechanochemical cycles, amplifying energy flux 314 until feedback through bond stabilization suppresses it at high stress. 315 316 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint Figures 317 318 Fig. 1 Activity reveals divergent mechanical responses of catch - and slip-bond crosslinked 319 networks. (a) Schematic of pseudo-2D actomyosin network crowded by Methylcellulose on the 320 lipid bilayer. (b) Confocal fluorescence microscopy image of pseudo-2D actomyosin network 321 crosslinked by fascin (top) and α -actinin (bottom) on a supported lipid bilayer. Left is before 322 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint crosslinking, and right is post crosslinker addition. Scalebar is 5 µm. (c) Kymograph of the yellow 323 dashed line in b), scalebars are 5µm (horizontal) and 10s ( vertical). (d) Exemplary linescan 324 intensity (normalized) for conditions in e. (e) Bundling metric 𝜆 of fascin and α-actinin crosslinked 325 networks as well as F-actin network without crosslinkers. N = 3 for each condition. (f) Exemplary 326 autocorrelation function of actin network fluctuations as function of lag time Δ𝑡. (g) Characteristic 327 time 𝜏 for different conditions. N = 5,3,3,3,3,3 respectively. (h) Confocal fluorescence microscopy 328 image of 2D actomyosin network crosslinked by fascin (top) and α-actinin (bottom) deformed and 329 contracted under myosin active stress over time. Scalebars are 20 µm. Heatmaps show 330 accumulative strain of the final frame. Quiver plot overlay shows the instantaneous velocity. ( i) 331 Kymograph of α-actinin network rupture (dashed red line in h). Scalebars are 10µm and 10 s. (j) 332 Mean strain of the network during deformation caused by myosin induced active stress over 333 time. The slope of the strain curve is the strain rate 𝑑𝜀 𝑑𝑡. (k) Maximum mean strain 𝑚𝑎𝑥 of 334 fascin and α-actinin crosslinked networks at Rc = 0.2. N = 3 for each condition. 𝑝𝑓𝑎𝑠−𝑎𝑎 = 0.0101. 335 (l) Strain rate 𝑑𝜀 𝑑𝑡 of the network crosslinked by α -actinin at R c = 0.1 at various myosin 336 concentrations. N = 2,3,3,3,2,2,2 from low to high concentration respectively. 337 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint 338 339 Fig. 2. Catch- and slip-bond crosslinkers tune non-monotonic heat dissipation generated by 340 the actomyosin crossbridge . (a) Schematic of calorimeter and the zoomed -in circle shows the 341 actomyosin sample details in the capillary tube that contains the sample. (b) Schematic of ATPase 342 activities from 1. actin polymerization ; 2. myosin ATPase in the network ; and 3. the chemical 343 reaction of ATP regeneration system via Pyruvate assay. (c) Heat dissipation rate (𝑄̇ ) of actin 344 polymerization at various actin concentrations over time. (d) Initial (black circles) and steady state 345 (blue diamonds) heat dissipation rate (𝑄̇ ) at various actin concentrations. Gray line is a Michaelis-346 Menten curve fit. (e) Heat dissipation rate of actomyosin network at various ATP concentrations 347 (Cactin = 37.5µM, R myo = 0.02.) Black line is the fit of Michaelis-Menten curve to data. (f) Heat 348 dissipation rate of actomyosin network at various actin concentrations (Rmyo = 0.02.) Black line is 349 the fit of Michaelis-Menten curve to data. (g) Heat dissipation rate (𝑄̇ )of actomyosin network at 350 various myosin-to-actin ratios (Cactin = 37.5µM, CATP = 6.25mM.) (h) Heat dissipation rate (𝑄̇ ) of 351 actomyosin network under same crosslinking ratio R CL = 0.001 but at different α -actinin:fascin 352 ratios (Φ𝑎𝑐𝑡𝑖𝑛:𝑓𝑎𝑠𝑐𝑖𝑛). Gray dashed line is a linear fit to the data. (i) Heat dissipation rate (𝑄̇ ) of 353 actomyosin network under same crosslinking ratio R CL = 0.001and C actin = 37.5µM, at various 354 Rmyo. (j) Heat dissipation rate (𝑄̇ ) of actomyosin network at R myo = 0.02, C actin = 37.5µM, and 355 various crosslinking ratio RCL. 356 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint 357 358 Fig. 3 Rheological mapping reveals delocalized active stress in catch -bonded networks. (a) 359 Schematic of crosslinked network inside a cone -and-plate rheometer for (1) under active stress 360 (top right) and (2) under rheometer-induced prestress ( bottom right). (b) Frequency sweep (G’, 361 elastic modulus, and G”, viscous modulus) of the α -actinin crosslinked network ( RCL = 0.001) 362 under active stress induced by myosin (Rmyo = 0.02, maroon, and 0.1, blue). (c) Frequency sweep 363 (G’, elastic modulus, and G”, viscous modulus) of the α -actinin (maroon hexagons), fascin (blue 364 triangles), and biotin -streptavidin (black diamonds) crosslinked networks. Vertical dashed lines 365 indicate the G’ and G” crossover frequency co for three cases. (d) Crossover frequency co for 366 the three conditions in (c). (e) Elastic modulus G’ of the three types of network with various Rmyo. 367 (f) Viscous modulus G” of the three types of network with various R myo. (g) Differential elastic 368 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint modulus K’ for the three types of networks as a function of prestress 0. Black line is indicative 369 of slope of 1. (h) Indicated active stress a as a function of Rmyo for the three types of networks. 370 371 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint 372 Fig. 4. Degeneracy in dissipation emerges from stress localization and delocalization . (a) Heat 373 map of azimuthally averaged Entropy Production Factor (EPF) as a function of frequency (𝜔) and 374 wave vector (𝑞𝑟). Rmyo = 0.02. (b) Heatmap shows accumulative strain (colorbar) after 5 minutes 375 of completion of polymerization. Quiver plot overlay shows the mean velocity within the 5 minutes. 376 Arrow color indicates the degree of alignment of the velocity with its neighbors. Red -white-blue 377 indicates ‘highly aligned’ - ‘not aligned’ – ‘anti-aligned’. R myo = 0.1. (c) Azimuthally averaged 378 velocity spatial correlation (𝐶𝑣𝑣) as a function of radius 𝑅. N = 3 for both cases. Errorbars are the 379 s.t.d. of the mean. Gray dashed lines are the exponential fit. (d) Entropy Production Rate (EPR, 380 maroon, left y-axis) and characteristic length (𝜆𝑣, blue, right y-axis) as a function of R myo. N = 3 381 for all conditions. Errorbars are the s.t.d. of the mean. (e) Schematic of how degeneracy in 382 dissipation can arise from the two ends of the driving due to stress de/localization. 383 384 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint Acknowledgments 385 386 We are grateful for insightful discussions with Dr. Daniel Needleman and Dr. Peter Foster. Z.G.S. 387 acknowledges fundings and support from Yale PEB, his family, and friends during his exchange 388 period at Harvard. This work was supported by funding ARO MURI W911NF-14-1-0403, 389 the National Institutes of Health (NIH) R01 GR130179, Sloan Matter-to-Life G-2025-79182 , and 390 Human Frontiers Science Program (HFSP) grant number RGP012/2025 to M.P.M. 391 392 Author Contributions 393 394 Z.G.S. & M.P.M designed and conceived the work. Z.G.S. & M.P.M. drafted the paper. Z.G.S, 395 M.P.M, & J.J.V. edited the paper. Z.G.S. and A.P.T. performed experiments. Z.G.S. analyzed the 396 data. J.Z. instructed on calorimeter operation. Z.G.S, M.P.M, J.J.V ., & J.Z. participated in the 397

of the work. M.P.M. & J.J.V. supervised the work. 398 399 400 Competing Interests 401 A patent, entitled as “Microcalorimetry for high-throughput screening of bioenergetics”, has been 402 filed for the devices by Harvard University (inventors: J. Zheng, J.J. Vlassak, D.J. Needleman). 403 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint

404 405 1 Onsager, L. Reciprocal relations in irreversible processes. II. Physical Review 38, 2265-406 2279 (1931). https://doi.org:DOI 10.1103/PhysRev.38.2265 407 2 Onsager, L. Reciprocal relations in irreversible processes. I. Physical Review 37, 405-426 408 (1931). https://doi.org:DOI 10.1103/PhysRev.37.405 409 3 Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep 410 Prog Phys 75, 126001 (2012). https://doi.org:10.1088/0034-4885/75/12/126001 411 4 Battle, C. et al. Broken detailed balance at mesoscopic scales in active biological systems. 412 Science 352, 604-607 (2016). https://doi.org:10.1126/science.aac8167 413 5 Mizuno, D., Tardin, C., Schmidt, C. F. & Mackintosh, F. C. Nonequilibrium mechanics of 414 active cytoskeletal networks. Science 315, 370-373 (2007). https://doi.org:315/5810/370 415 [pii] 416 10.1126/science.1134404 417 6 Onsager, L. Reciprocal relations in irreversible processes. I. Physical review 37, 405 (1931). 418 7 De Groot, S. R. & Mazur, P. Non-equilibrium thermodynamics. (Courier Corporation, 419 2013). 420 8 Prigogine, I. & Van Rysselberghe, P. Introduction to thermodynamics of irreversible 421 processes. Journal of The Electrochemical Society 110, 97C (1963). 422 9 Chaikin, P. M., Lubensky, T. C. & Witten, T. A. Principles of condensed matter physics . 423 Vol. 10 (Cambridge university press Cambridge, 1995). 424 10 Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. 425 Reports on progress in physics 75, 126001 (2012). 426 11 Kubo, R. Statistical-Mechanical Linear Response Theory of Irreversible Processes. P Jpn 427 Acad 45, R11-& (1969). 428 12 Cross, M. C. & Hohenberg, P. C. Pattern-Formation Outside of Equilibrium. Rev Mod Phys 429 65, 851-1112 (1993). https://doi.org:DOI 10.1103/RevModPhys.65.851 430 13 Julicher, F., Kruse, K., Prost, J. & Joanny, J. F. Active behavior of the cytoskeleton. Physics 431 Reports-Review Section of Physics Letters 449, 3 -28 (2007). 432 https://doi.org:10.1016/j.physrep.2007.02.018 433 14 Kruse, K., Joanny, J. F., Julicher, F., Prost, J. & Sekimoto, K. Asters, vortices, and rotating 434 spirals in active gels of polar filaments. Phys Rev Lett 92 (2004). https://doi.org:ARTN 435 078101 436 10.1103/PhysRevLett.92.078101 437 15 Prost, J. Physics of active gels. Abstr Pap Am Chem S 230, U3582-U3582 (2005). 438 16 Prost, J., Julicher, F. & Joanny, J. F. Active gel physics. Nat Phys 11, 111-117 (2015). 439 https://doi.org:10.1038/Nphys3224 440 17 Tabatabai, A. P. et al. Detailed Balance Broken by Catch Bond Kinetics Enables 441 Mechanical-Adaptation in Active Materials. Advanced Functional Materials n/a, 2006745 442 https://doi.org:https://doi.org/10.1002/adfm.202006745 443 18 Marshall, B. T. et al. Direct observation of catch bonds involving cell-adhesion molecules. 444 Nature 423, 190-193 (2003). https://doi.org:10.1038/nature01605 445 19 Thomas, W. E., Vogel, V. & Sokurenko, E. Biophysics of catch bonds. Annu Rev Biophys 446 37, 399-416 (2008). https://doi.org:10.1146/annurev.biophys.37.032807.125804 447 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint 20 Kong, F., García, A. J., Mould, A. P., Humphries, M. J. & Zhu, C. Demonstration of catch 448 bonds between an integrin and its ligand. J Cell Biol 185, 1275 -1284 (2009). 449 https://doi.org:10.1083/jcb.200810002 450 21 Mulla, Y. et al. Weak catch bonds make strong networks. Nature materials 21, 1019-1023 451 (2022). 452 22 Murrell, M. P. & Gardel, M. L. F -actin buckling coordinates contractility and severing in 453 a biomimetic actomyosin cortex. Proceedings of the National Academy of Sciences 109, 454 20820-20825 (2012). 455 23 Koenderink, G. H. et al. An active biopolymer network controlled by molecular motors. 456 Proceedings of the National Academy of Sciences 106, 15192-15197 (2009). 457 24 Humphrey, D., Duggan, C., Saha, D., Smith, D. & Käs, J. Active fluidization of polymer 458 networks through molecular motors. Nature 416, 413-416 (2002). 459 25 Mizuno, D., Tardin, C., Schmidt, C. F. & MacKintosh, F. C. Nonequilibrium mechanics of 460 active cytoskeletal networks. Science 315, 370-373 (2007). 461 26 Bendix, P. M. et al. A quantitative analysis of contractility in active cytoskeletal protein 462 networks. Biophysical journal 94, 3126-3136 (2008). 463 27 Sokurenko, E. V., Vogel, V. & Thomas, W. E. Catch-bond mechanism of force-enhanced 464 adhesion: counterintuitive, elusive, but… widespread? Cell host & microbe 4, 314 -323 465 (2008). 466 28 Xu, X.-P. et al. Structural basis of αE-catenin–F-actin catch bond behavior. Elife 9, e60878 467 (2020). 468 29 Wang, A., Dunn, A. R. & Weis, W. I. Mechanism of the cadherin –catenin F-actin catch 469 bond interaction. Elife 11, e80130 (2022). 470 30 Thomas, W. E., Vogel, V. & Sokurenko, E. Biophysics of catch bonds. Annu. Rev. Biophys. 471 37, 399-416 (2008). 472 31 Evans, E. & Ritchie, K. Dynamic strength of molecular adhesion bonds. Biophysical 473 journal 72, 1541-1555 (1997). 474 32 Merkel, R., Nassoy, P., Leung, A., Ritchie, K. & Evans, E. Energy landscapes of receptor–475 ligand bonds explored with dynamic force spectroscopy. Nature 397, 50-53 (1999). 476 33 Seara, D. S. et al. Entropy production rate is maximized in non -contractile actomyosin. 477 Nature communications 9, 4948 (2018). 478 34 Chen, S. et al. Energy partitioning in the cell cortex. Nature Physics 20, 1824-1832 (2024). 479 35 Murrell, M., Thoresen, T. & Gardel, M. Reconstitution of contractile actomyosin arrays. 480

Enzymol 540, 265 -282 (2014). https://doi.org:10.1016/B978-0-12-397924-481 7.00015-7 482 36 Murrell, M. P. & Gardel, M. L. F -actin buckling coordinates contractility and severing in 483 a biomimetic actomyosin cortex. Proc Natl Acad Sci U S A 109, 20820 -20825 (2012). 484 https://doi.org:10.1073/pnas.1214753109 485 37 Murrell, M. & Gardel, M. L. Actomyosin Sliding is Attenuated in Contractile Biomimetic 486 Cortices. Mol Biol Cell (2014). https://doi.org:10.1091/mbc.E13-08-0450 487 38 Sun, Z. G. et al. Cofilin-mediated actin filament network flexibility facilitates 2D to 3D 488 actomyosin shape change. European journal of cell biology 103, 151379 (2024). 489 39 Sun, Z. G. & Murrell, M. Cofilin -Mediated Filament Softening and Crosslinking 490 Counterbalance to Enhance Actin Network Flexibility. Physical Review Letters 133, 491 218402 (2024). 492 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint 40 Luan, Y., Lieleg, O., Wagner, B. & Bausch, A. R. Micro-and macrorheological properties 493 of isotropically cross-linked actin networks. Biophysical journal 94, 688-693 (2008). 494 41 Lieleg, O., Claessens, M. M., Heussinger, C., Frey, E. & Bausch, A. R. Mechanics of 495 bundled semiflexible polymer networks. Physical review letters 99, 088102 (2007). 496 42 Lieleg, O., Claessens, M. M. & Bausch, A. R. Structure and dynamics of cross-linked actin 497 networks. Soft Matter 6, 218-225 (2010). 498 43 Cocconi, L., Chatzittofi, M. & Golestanian, R. Mechanical inhibition of dissipation in a 499 thermodynamically consistent active solid. arXiv preprint arXiv:2506.18000 (2025). 500 44 Pollard, T. D. Rate constants for the reactions of ATP-and ADP-actin with the ends of actin 501 filaments. The Journal of cell biology 103, 2747-2754 (1986). 502 45 Baker, J. E., Brosseau, C., Joel, P. B. & Warshaw, D. M. The biochemical kinetics 503 underlying actin movement generated by one and many skeletal muscle myosin molecules. 504 Biophysical journal 82, 2134-2147 (2002). 505 46 Stewart, T. J., Murthy, V., Dugan, S. P. & Baker, J. E. Velocity of myosin -based actin 506 sliding depends on attachment and detachment kinetics and reaches a maximum when 507 myosin-binding sites on actin saturate. Journal of Biological Chemistry 297 (2021). 508 47 Resnicow, D. I., Deacon, J. C., Warrick, H. M., Spudich, J. A. & Leinwand, L. A. 509 Functional diversity among a family of human skeletal muscle myosin motors. 510 Proceedings of the National Academy of Sciences 107, 1053-1058 (2010). 511 48 Haldeman, B. D., Brizendine, R. K., Facemyer, K. C., Baker, J. E. & Cremo, C. R. The 512 kinetics underlying the velocity of smooth muscle myosin filament sliding on actin 513 filaments in vitro. Journal of Biological Chemistry 289, 21055-21070 (2014). 514 49 Finer, J. T., Simmons, R. M. & Spudich, J. A. Single myosin molecule mechanics: 515 piconewton forces and nanometre steps. nature 368, 113-119 (1994). 516 50 Sweeney, H. L. & Houdusse, A. Structural and functional insights into the myosin motor 517 mechanism. Annual review of biophysics 39, 539-557 (2010). 518 51 Johnson, C. A. et al. The ATPase cycle of human muscle myosin II isoforms: Adaptation 519 of a single mechanochemical cycle for different physiological roles. Journal of Biological 520 Chemistry 294, 14267-14278 (2019). 521 52 Geeves, M. A. & Holmes, K. C. The molecular mechanism of muscle contraction. 522 Advances in protein chemistry 71, 161-193 (2005). 523 53 Kovács, M., Thirumurugan, K., Knight, P. J. & Sellers, J. R. Load -dependent mechanism 524 of nonmuscle myosin 2. Proceedings of the National Academy of Sciences 104, 9994-9999 525 (2007). 526 54 Soares e Silva, M. et al. Active multistage coarsening of actin networks driven by myosin 527 motors. Proceedings of the National Academy of Sciences 108, 9408-9413 (2011). 528 55 Ruiz-Franco, J., Tauber, J. & van der Gucht, J. Cross -linker mobility governs fracture 529 behavior of catch-bonded networks. Physical Review Letters 130, 118203 (2023). 530 56 Broedersz, C. & MacKintosh, F. Molecular motors stiffen non-affine semiflexible polymer 531 networks. Soft Matter 7, 3186-3191 (2011). 532 57 Shin, J. H., Gardel, M., Mahadevan, L., Matsudaira, P. & Weitz, D. Relating microstructure 533 to rheology of a bundled and cross -linked F-actin network in vitro. Proceedings of the 534 National Academy of Sciences 101, 9636-9641 (2004). 535 58 Gardel, M. L. et al. Elastic behavior of cross -linked and bundled actin networks. Science 536 304, 1301-1305 (2004). 537 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint 59 Gardel, M. L. et al. Prestressed F-actin networks cross-linked by hinged filamins replicate 538 mechanical properties of cells. Proceedings of the National Academy of Sciences 103, 539 1762-1767 (2006). 540 60 Seara, D. S., Machta, B. B. & Murrell, M. P. Irreversibility in dynamical phases and 541 transitions. Nature communications 12, 392 (2021). 542 61 Callen, H. B. Thermodynamics and an Introduction to Thermostatistics. John Wiley& Sons 543 2 (1993). 544 62 Marchetti, M. C. et al. Hydrodynamics of soft active matter. Reviews of modern physics 545 85, 1143-1189 (2013). 546 63 Jülicher, F., Grill, S. W. & Salbreux, G. Hydrodynamic theory of active matter. Reports on 547 Progress in Physics 81, 076601 (2018). 548 64 Foster, P. J. et al. Dissipation and energy propagation across scales in an active cytoskeletal 549 material. Proceedings of the National Academy of Sciences 120, e2207662120 (2023). 550 65 Anderson, P. W. Absence of diffusion in certain random lattices. Physical review 109, 1492 551 (1958). 552 66 Sun, Z. G. et al. Feedback between F-actin organization and active stress governs criticality 553 and energy localization in the cell cytoskeleton. Nature Physics, 1-13 (2025). 554 67 Strogatz, S. H. Nonlinear dynamics and chaos: with applications to physics, biology, 555 chemistry, and engineering. (Chapman and Hall/CRC, 2024). 556 68 J Evans, D. & P Morriss, G. Statistical mechanics of nonequilbrium liquids. (ANU Press, 557 2007). 558 69 Gnesotto, F. S., Mura, F., Gladrow, J. & Broedersz, C. P. Broken detailed balance and non-559 equilibrium dynamics in living systems: a review. Reports on Progress in Physics 81, 560 066601 (2018). 561 70 De Gennes, P.-G. Superconductivity of metals and alloys. (CRC press, 2018). 562 71 Tinkham, M. Introduction to superconductivity. (Courier Corporation, 2004). 563 72 Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Reviews of 564 modern physics 65, 851 (1993). 565 73 Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Theory of superconductivity. Physical review 566 108, 1175 (1957). 567 568 569 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint 570

571 Preparation of small unilamellar vesicles (SUVs) 572 Lipids were combined by mixing 100 µL of egg phosphatidylcholine (Egg PC, 25 mg mL⁻¹; Avanti 573 Polar Lipids , 840051C) with 10 µL of 1,2 -dihexadecanoyl-sn-glycero-3-phosphoethanolamine 574 (DHPE, 0.5 mg mL⁻¹), optionally conjugated to Oregon Green 488 ( Invitrogen). All lipids were 575 dissolved in chloroform and transferred to a clean glass vial pre -equilibrated with argon to 576 minimize oxidative degradation. The solvent was removed under a gentle stream of argon, yielding 577 a uniform lipid film coating the bottom of the vial. The dried lipid film was rehydrated with 5 mL 578 of vesicle buffer (100 mM NaCl, 20 mM HEPES, pH 7.5) and vortexed until the suspension 579 became turbid. The dispersion was then sonicated in a bath sonicator for approximately 1 h, or 580 until the solution turned optically clear, indicating the formation of small unilamellar vesicles 581 suitable for bilayer formation. 582 Fluorescent labeling of skeletal muscle myosin 583 Skeletal muscle myosin (Heavy Meromyosin from rabbit, Cytoskeleton Inc.) is fluorescently 584 labeled with Alexa Fluor 647 C2 Maleimide under reducing conditions. Initially, myosin is 585 reduced in a labeling buffer containing 50 mM HEPES, 0.5 M KCl, 1 mM EDTA, and 10 mM 586 DTT at pH 7.6. Following reduction, the sample is dialyzed overnight against the same buffer 587 without DTT. After dialysis, the solution is centrifuged to eliminate any insoluble components. 588 The resulting supernatant is reacted with Alexa Fluor C2 Maleimide at a 5:1 molar ratio of dye to 589 myosin. Labeling is carried out at 4°C for one hour, after which the reaction is quenched by adding 590 1 mM DTT. The labeled protein is purified using a desalting column (Pierce, 5K MWCO, 5 mL). 591 Absorbance readings at 280 nm and 647 nm are then used to calculate the degree of labeling. This 592 protocol is adapted from Verkohovsky and Borisy1. 593 594 2D actomyosin contraction experiments 595 Each chamber is of cylindrical shape, with diameter of 12mm. The top and bottom piece are 596 magnetically locked, with a rubber piece sealing the middle and a glass slide sandwiched in 597 between. The glass slide is first washed with 50% ethanol to clean any residues left on the surface. 598 The slide is then exposed under UV light for 5 min to induce hydrophilicity to the surface. 300 μL 599 of SUV solution is added to the chamber. Once the surface of the glass is coated with lipid bilayer, 600 we take 100 μL solution out and wash the chamber with 400 μL of 1x F-buffer (10 mM imidazole, 601 1 mM MgCl2, 50 mM KCl, 2 mM EGTA, 0.5 mM ATP, pH = 7.5) . Dark G-actin (Cytoskeleton) 602 is mixed with rhodamine labelled F -actin (20% fluorescent, Cytoskeleton) to a final molar 603 concentration of 1.4 µM, and is stabilized with 1 μM phalloidin (Cytoskeleton) and crowded to the 604 surface of a 97% Egg Phosphatidyl Choline (Avanti Polar Lipids)/3% FITC -DHPE (Molecular 605 Probes) phospholipid bilayer, using 0.2% 14,000 MW methyl -cellulose (Sigma, 15 cP) as a 606 depletion agent (Fig. 1A). The actin mixed soup is placed in an Eppendorf tube on ice for 1 hour 607 to reach full polymerization (50 μL protein mix in total). Once F-actin is polymerized, it is added 608 to the chamber, along with methyl-cellulose. Once the F-actin network is crowded onto the surface, 609 we add crosslinkers 𝛼-actinin or fascin of various concentrations. Then, Skeletal muscle myosin 610 II (various concentrations), labelled with Alexa Flour 647nm C2 Maleimide (Molecular Probes) is 611 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint added in solution in dimeric form, which polymerizes into thick filaments onto the F -actin. This 612

in a two-dimensional actomyosin network. 613 614 Removing non-catalytically active skeletal muscle myosin 615 To isolate only catalytically active myosin dimers for experimental use, myosin is subjected to a 616 selective centrifugation process in the presence of polymerized actin. First, actin is polymerized 617 for one hour at 4°C in a high -salt environment (1X F -buffer supplemented with 4 M KCl) and 618 stabilized using phalloidin. After polymerization, ATP is added to reach a final concentration of 1 619 mM, followed by the addition of freshly thawed myosin. This actin -myosin mixture is incubated 620 at 4°C for 10 minutes, then centrifuged at 128,360 × g for 30 minutes. During centrifugation, 621 enzymatically inactive myosin remains bound to the F-actin network and sediments, while active 622 myosin motors detach and remain in the supernatant, which is then collected. The concentration 623 of active myosin in the supernatant is quantified by measuring absorbance at 647 nm, based on a 624 pre-determined labeling efficiency. Myosin is freshly prepared for each experiment and used 625 within 24 hours. 626 627 Microscopy 628 The image stack data are collected using Leica DMi8 inverted microscope equipped with a 63×, 629 40×, or 20×, and 1.4, 1.3, and 0.75 -NA oil immersion lens respectively (Leica Microsystems), a 630 spinning-disk confocal (CSU22; Yokagawa), and sCMOS camera (Zyla; Andor Technology) 631 controlled by Andor iQ3 (Andor Technology). Image time series stack data are taken with time 632 interval of 0.5-10 seconds. 633 634 Picocalorimetry Measurements 635 Metabolic heat was quantified using a custom micromachined capillary -based picocalorimetry 636 system, as previously described (Zheng et al.). Three borosilicate glass capillaries (400 µm × 400 637 µm outer dimension, 100 µm wall thickness) were bonded onto gold -coated regions of a silicon-638 nitride membrane integrating two Nichrome/Constantan thermopiles and a tungsten micro-heater. 639 One capillary was loaded with the biological sample, and two identical capillaries containing pure 640 water served as thermal references. Heat generated by the sample produced a temperature 641 differential relative to the references, which was converted into a voltage by the thermopiles 642 through the Seebeck effect. The monitored sensing volume within the sample capillary was ~80 643 nL. 644 The calorimeter assembly was mounted on a custom PCB and placed inside a thermally insulated 645 vacuum chamber evacuated to ~10⁻⁵ Torr to suppress convective heat loss and reduce electrical 646 noise. Measurements were performed at 2 5 °C. Thermopile voltages were acquired using two 647 Keithley 2182A nanovoltmeters operated at 6 PLC with power-line synchronization and auto-zero 648 enabled. Data were sampled at 2.31 Hz and processed using a moving-average filter. Under these 649 measurement conditions, the system exhibited an effective noise floor corresponding to a power 650 sensitivity of ~85 pW. 651 652 Experimental setup and procedures for rheology experiments 653 Rheological measurements of actin networks were performed using a stress -controlled rheometer 654 (Anton Paar M502). Actin was first prepared following the composition detailed in Table 1, and 655 the final sample mixture (Table 2) was assembled to a total volume of 1000 µL. After gentle 656 mixing, the solution was loaded onto the preheated bottom plate maintained at 25 °C. Both the 657 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint cone and plate surfaces were sandblasted to prevent sample slippage. Following equilibration, 658 several drops of silicone oil were added around the sample edge to minimize evaporation, and a 659 solvent trap was placed over the geometry for additional protection. Gelation kinetics were 660 monitored under the parameters described in the main text. After gelation reached a steady plateau 661 in both elastic (G′) and viscous (G″) moduli (typically after ~2 h), measurements in the linear and 662 nonlinear regimes were performed. After each experiment, the setup was thoroughly cleaned with 663 sequential rinses of 70% ethanol, water, ethanol, and water to ensure complete removal of residual 664 protein and prevent cross-contamination between runs. 665 666 Experimental setup and procedures for microscopy experiments 667 Unlabeled rabbit skeletal muscle actin (>99% purity; Cytoskeleton, Inc.)—hereafter referred to as 668 dark actin—was reconstituted in 1× G -buffer (see Table 3) to a concentration of 10 mg mL⁻¹. 669 Rhodamine-labeled actin (Cytoskeleton, Inc.) was prepared using the same procedure. Both 670 preparations were depolymerized for over 24 h at 4 °C in the dark (see Table 3). The two actin 671 species were then mixed at a 9:1 ratio (dark:rhodamine) to yield G -actin. Polymerization was 672 initiated by combining the protein mixture with polymerization buffer (Table 4). Glucose oxidase, 673 catalase (GOC), and glucose were added as an oxygen scavenging system to minimize 674 photobleaching during imaging. 675 Samples were loaded into a custom four -well round chamber, each well comprising a cylindrical 676 cavity sealed by a 12 mm glass coverslip and a rubber spacer to prevent leakage. Coverslips were 677 sequentially cleaned with 70% ethanol, dried, and exposed to UV light for 5 min to render the 678 surface hydrophilic. To reduce actin adsorption, surfaces were coated with small unilamellar 679 vesicles (SUVs; protocol available upon request). After gentle mixing (30 s), 200 µL of the protein 680 mixture was injected into each well. Samples were imaged using a Leica confocal microscope (see 681 Microscopy section). 682 683 ATP-regeneration system 684 ATP regeneration system was used in all experiments, containing ATP at the indicated 685 concentrations (0.625–12.5 mM), 40 mM phosphoenolpyruvate (PEP), and a coupled pyruvate 686 kinase/lactate dehydrogenase (PK/LDH) enzyme mixture (Sigma-Aldrich). The PK/LDH enzymes 687 were mixed at a 1:1 ratio (1 mg mL⁻¹ stock concentration), and 2.5 µL of the enzyme mixture was 688 added per reaction volume of 800 µL. 689 690 Particle Image Velocimetry (PIV) 691 Particle image velocimetry (PIV) is applied on the fluorescent actin images in MATLAB (mPIV, 692 https://www.mn.uio.no/math/english/people/aca/jks/matpiv/). The extent of contraction is 693 calculated by defining a mean strain: 694 = (1) , as the divergence of the displacement field. The data is analyzed with PIV window size 32 and 695 overlap 0.5. Window size of 16 and 64 have also been used to generate the data, and eventually 696 determined that 32 is the best parameter value because of the high signal-to-noise ratio. 697 698 2D image thresholding method for calculating bundling parameter 699 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint For each microscopy image of F-actin network, a line scan of approximately 20 m is done to the 700 image and the fluorescence intensity is extracted from the line scan. The bundling parameter 𝜆 is 701 calculated using the equation: 702 𝜆 = 𝜈/𝜇 (2) 703 , in which 𝜈 is the standard deviation of the fluorescence intensity of the line scan, and 𝜇 the mean 704 of the fluorescence intensity of the same line scan. 705 706 Autocorrelation analysis 707 After the F-actin network reached a steady state, time-lapse fluorescence image sequences were 708 acquired with randomly chosen starting times. To quantify the temporal dynamics of the network 709 at the global level, we computed the traditional temporal autocorrelation of the fluorescence 710 intensity using custom MATLAB scripts. 711 For each frame, the fluorescence intensity was spatially averaged over all pixels in the image, 712 yielding a single intensity time series 𝐼(𝑡). The mean intensity ⟨𝐼⟩was then subtracted to isolate 713 temporal fluctuations, 𝛿𝐼(𝑡)= 𝐼(𝑡)− ⟨𝐼⟩. The normalized temporal autocorrelation function was 714 calculated according to 715 𝐶(Δ𝑡)= ⟨𝛿𝐼(𝑡) 𝛿𝐼(𝑡 + Δ𝑡)⟩𝑡 ⟨𝛿𝐼(𝑡)2⟩𝑡 (3) where ⟨⋅⟩𝑡denotes an average over all valid time points separated by a delay Δ𝑡. This 716 normalization ensures 𝐶(0)= 1. Because the number of statistically independent pairs decreases 717 with increasing delay time, the autocorrelation at large Δ𝑡 is increasingly affected by finite-718 sampling noise and was not interpreted quantitatively. 719 The experimentally measured autocorrelation functions were fit using a double-exponential 720 decay model, 721 𝐶(Δ𝑡)= 𝑎 e−Δ𝑡/𝜏1 + (1− 𝑎) e−Δ𝑡/𝜏2 (4) 722 𝜏 = 𝑎𝜏1 + (1− 𝑎) 𝜏2 (5) where 𝑎 and 1− 𝑎 are the relative amplitudes of the two relaxation modes, and 𝜏1and 𝜏2are their 723 associated characteristic times. Fitting was restricted to the initial portion of the correlation curve 724 (typically the first 30% of the available delay times), as long-lag data points are dominated by 725 statistical uncertainty arising from finite acquisition length. 726 727 Mechanical power calculation using PIV velocity field 728 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint Defining 𝜂′(𝜔)= 𝐺′′(𝜔) 𝜔 , this is: 729 ⟨𝑝⟩ = ∫ 𝜂′(2𝜋𝑓) ∞ 0  𝑆𝛾̇(𝑓) 𝑑𝑓 (6) 730 For a 2D/3D velocity field, using the strain-rate tensor 𝐷𝑖𝑗(𝑟⃑,𝑡) and its PSDs: 731 ⟨𝑝⟩ ≈ 2𝛽 ∫ 𝜂′(2𝜋𝑓) 𝑓𝑚𝑎𝑥 0  [𝑆𝑥𝑥(𝑓)+ 𝑆𝑦𝑦(𝑓)+ 2𝑆𝑥𝑦(𝑓)] 𝑑𝑓 (7) For more details, please refer to Supplemental Materials. 732 733 Velocity spatial correlation analysis 734 Equal-time spatial velocity correlations were computed from particle image velocimetry (PIV) 735 data using a custom MATLAB pipeline. The velocity correlation function was defined as 736 𝐶𝑣𝑣(𝑟)= ⟨𝐯(𝐫′,𝑡)⋅ 𝐯(𝐫′ + 𝐫,𝑡)⟩𝐫′ ⟨∣ 𝐯(𝐫′,𝑡)∣⟩𝐫′ 2 , (8) where 𝐯(𝐫′,𝑡)denotes the local velocity fluctuation at position 𝐫′and time 𝑡, and ⟨⋅⟩𝐫′indicates 737 spatial averaging over all interrogation windows. 738 Velocity fields in the 𝑥and 𝑦directions (𝑣𝑥,𝑣𝑦) were first preprocessed to handle missing values 739 arising from PIV failures. NaN entries were replaced using MATLAB’s fillmissing function with 740 linear interpolation to ensure spatial continuity of the velocity field. 741 To compute the correlation efficiently and without directional bias, the calculation was 742 performed in Fourier space. For each time point, the mean velocity was subtracted from the raw 743 velocity field 𝐯0(𝐫′,𝑡)to obtain velocity fluctuations, 744 𝐯(𝐫′,𝑡)= 𝐯0(𝐫′,𝑡)− ⟨𝐯0(𝐫′,𝑡)⟩𝐫′. (9) The resulting field was then normalized by its spatial root-mean-square magnitude, 745 𝐯norm(𝐫′,𝑡)= 𝐯(𝐫′,𝑡)/√∑ ∣ 𝐯( 𝐫′ 𝐫′,𝑡)∣2. (10) A two-dimensional fast Fourier transform (FFT) was applied to the normalized velocity field, 746 and the autocorrelation was obtained via inverse FFT of the power spectrum: 747 𝐶𝑣𝑣(𝐫)= Re[fftshift(ifft2(𝐯̂norm(𝐤,𝑡) 𝐯̂norm ∗ (𝐤,𝑡)))], (11) where 𝐯̂norm(𝐤,𝑡)denotes the Fourier-transformed velocity field and ∗ indicates complex 748 conjugation. The fftshift operation was used to center the zero-frequency component. 749 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint The resulting two-dimensional correlation map was radially averaged using a custom radialavg 750 routine to obtain 𝐶𝑣𝑣(𝑟). The decay of the correlation function was quantified by fitting to a 751 single-exponential form, 752 𝐶𝑣𝑣(𝑟)= 𝑏1exp (−𝑟/𝜆𝑣)+ 𝑏2, (12) where 𝜆𝑣 defines the characteristic velocity correlation length. 753 754 Entropy production calculation 755 The entropy production factor (EPF) and entropy production rate (EPR) are: 756 757 𝐸𝑃𝐹 (𝒒,𝜔)= 1 2[𝐶−1(𝒒,−𝜔)− 𝐶−1(𝒒,𝜔)]𝑖𝑗𝐶𝑗𝑖(𝒒,𝜔) (13) 𝐸𝑃𝑅 = ∫𝑑𝜔 2𝜋 𝑑2𝒒 (2𝜋)2 𝐸𝑃𝐹 (𝒒,𝜔) (14) , where 𝐶𝑖𝑗(𝒒,𝜔) is the dynamic structure factor (for more derivations, please refer to Seara, 758 Machta, and Murrell, 2021). The entropy calculation and analyses are carried out using customized 759 Python script utilizing frequent and freqentn package developed by Dr. Daniel S. Seara 760 (https://github.com/lab-of-living-matter/freqent/tree/epf_paper/freqent). 761 762 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint Data availability 763 Source data are provided with this paper. Raw data supporting the findings of this manuscript are 764 available from the corresponding authors upon reasonable request. A reporting summary for this 765 Article is available as a Supplementary Information file. 766 767 Code availability 768 Code supporting the findings of this manuscript are available from the corresponding authors upon 769 reasonable request. A reporting summary for this Article is available as a Supplementary 770 Information file. 771 772 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint