{"paper_id":"2abe58a4-ded4-473f-aee6-35b054e48573","body_text":"Mechanical organization yields degenerate dissipation beyond linear 1 \nresponse 2 \nZachary Gao Sun 1,2,3,4, Juanjuan Zheng 4, A. Pasha Tabatabai 2,5, Joost J. Vlassak 4, and Michael 3 \nMurrell1,2,3,5, * 4 \n 5 \n1 Department of Physics, Yale University, 217 Prospect Street, New Haven, Connecticut 06511, 6 \nUSA 7 \n2 Systems Biology Institute, Yale University, 850 West Campus Drive, West Haven, Connecticut, 8 \n06516, USA 9 \n3 Integrated Graduate Program in Physical and Engineering Biology, Yale University, New Haven, 10 \nConnecticut 06520, USA 11 \n4 John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, 12 \nMA, USA 13 \n5 Department of Biomedical Engineering, Yale University, 55 Prospect Street, New Haven, 14 \nConnecticut 06511, USA 15 \n*Corresponding author. Email address: michael.murrell@yale.edu .  16 \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nAbstract 17 \n 18 \nIn non-equilibrium (active) systems, increased driving is commonly assumed to amplify energy 19 \ndissipation. This frames the efficiency of protein -based machines as a fixed or monotonically 20 \ndecreasing function with driving. Using picowatt -sensitive calorimetry and advanced entropy 21 \nproduction metrics in reconstituted actomyosin networks, we show that energy dissipation depends 22 \nnon-monotonically on myosin -generated stress (driving). At low driving, dissipation increases 23 \nproportionally with stress, consistent with near -equilibrium linear response. At high driving, 24 \nhowever, dissipation decreases, revealing a far -from-equilibrium regime in which excessive load 25 \nsuppresses motor ATPase activity. This non -monotonicity reflects a transition from spatially 26 \nlocalized stress at low driving to delocalized stress at high driving, where force per motor, and thus 27 \nATPase suppression, is maximized. Crosslinker mechanics tune this transition as fascin (slip bonds) 28 \namplifies stress localization and shifts the dissipation peak to higher driving, whereas α -actinin 29 \n(catch bonds) stabilizes under load, delocalizes stress, and shifts the peak to lower driving. Thus, 30 \nenhanced mechanochemical coupling causes additional driving to restructure rather than amplify 31 \ndissipation, revealing how material system organization (bonding), and not driving alone, governs 32 \nenergy flow far from equilibrium.  33 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nMain Text 34 \n 35 \nIntroduction 36 \n 37 \nIn non-equilibrium systems, increased driving is commonly assumed to amplify energy 38 \ndissipation, reflecting the conventional assumption inherited from linear response and irreversible 39 \nthermodynamics 1-5. This intuition links activity, force generation, and energetic cost , in that  40 \nstronger driving produces larger fluxes  and greater heat loss. It underlies how efficiency is 41 \ntypically understood in systems ranging from molecular machines to active solids, where enhanced 42 \nactivity is often equated with increased dissipation. Whether this assumption remains valid when 43 \nthe system is driven far from thermodynamic equilibrium, however, remains largely unexplored. 44 \nLinear response theory predicts that fluxes scale proportionally with their conjugate forces 45 \nand that entropy production increases monotonically with driving 1,6-10. In this regime, dissipation 46 \nis determined by transport coefficients that are independent of the applied force, and added input 47 \nis converted directly into heat through Onsager reciprocity6. This framework has been remarkably 48 \nsuccessful in describing transport 11 and pattern formation in passive matter 12, and is often 49 \nimplicitly extended to active systems13-16. Active materials, however, operate far from equilibrium, 50 \nwhere driving does not merely induce motion but impacts the material itself 13,14,16. In the cell 51 \ncytoskeleton for example , i nternally generated stresses alter connectivity, stiffness, and force 52 \ntransmission, introducing feedback between mechanical organization and energy consumption. 53 \nAmong these feedbacks include load -dependent bonds 17-20, which endow the cytoskeleton with 54 \nadaptability, in the ability to transition between solid -like and fluid -like states through motor -55 \ngenerated stresses transmitted by filamentous networks 21-24. In such systems, dissipation may no 56 \nlonger be set by driving alone, but how the system redistributes or allocates energy across internal 57 \ndegrees of freedom. Directly testing this possibility has been challenging, in part because energy 58 \ndissipation is rarely measured alongside material organization. As a result, it remains unclear 59 \nwhether dissipation in active matter necessarily increases with driving, or whether it can instead 60 \nbe regulated by the internal organization of the material. 61 \nHere we show that increased driving does not necessarily amplify dissipation in active 62 \nmaterials but  instead reorganizes how energy is dissipated through stress localization and 63 \ndelocalization. To this end, we  use the actomyosin cytoskeleton as a model non-equilibrium 64 \nmaterial, where myosin motors consume chemical energy (ATP) and through the generation of 65 \nactive stress, drive the cytoskeleton  beyond the linear regime 22-26. As a function of driving, 66 \nmechano-sensitive bonds introduce feedback between mechanical force transmission and chemical 67 \n(ATPase) activity, determining material organization (e.g. crosslinking), stress focusing and a non-68 \nmonotonic dissipation of energy. The impact of the bonds are twofold : catch bonds (𝛼-actinin) 69 \nstrengthen under tensile load 21,27-30, stabilizing stressed networks, whereas slip bonds (fascin) 70 \nunbind under increased tension 31,32, shifting the peak in dissipation to different driving, and fine 71 \ntuning stress concentration akin to Anderson Localization in amorphous materials . Thus, b y 72 \nreconstituting 2D and 3D actomyosin networks with defined bond mechanics, we demonstrate that 73 \ndissipation does not increase monotonically with driving and directly link non-monotonic behavior 74 \nto a material transition between stress localization and delocalization 33,34. Combining confocal 75 \nfluorescence microscopy, entropy production, picowatt-sensitive calorimetry, and rheology, we 76 \nshow how mechanochemical feedback reshapes energy flow in active matter, revealing 77 \nmechanisms by which driving reorganizes, rather than simply amplifies dissipation  in a far from 78 \nequilibrium regime. 79 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\n 80 \nResults 81 \nWe assemble model actomyosin networks in two configurations 35. First, we assemble a quasi-2D 82 \nnetwork, amenable for fluorescence imaging and the quantification of filament dynamics  36-38. 83 \nSecond, we assemble a 3D network, more amenable to rheology 39-42 and calorimetry (Methods).  84 \nIntegrating 3D imaging with pico -calorimetry allows direct comparison of network local 85 \ndeformation and heat dissipation, while rheological measurements provide complementary 86 \nmechanical characterization. The molar ratios of proteins (e.g., crosslinker -to-actin and myosin -87 \nto-actin) are identical in the 2D and 3D assays, although the use of a crowding agent in the 2D 88 \nsystem results in higher effective surface protein concentrations compared to bulk concentrations 89 \nin 3D networks that are without crowding agent. 90 \n 91 \nCatch- and slip -bond networks are mechanically indistinguishable under thermal 92 \nfluctuations but diverge under active stress. 93 \nWe reconstitute a quasi -two-dimensional active actomyosin network by confining pre -94 \npolymerized actin filaments to a supported lipid bilayer using 0.25% methylcellulose, crosslinking 95 \nthe network with fascin or α-actinin, as fascin is known to be a slip bond, while α-actinin is a catch 96 \nbond 21(Fig. 1a). We then introduce skeletal muscle myosin II to generate active contractile stress 97 \n(Fig. 1a). Under high crosslinker density (R c > 0.2), pseudo -2D α-actinin and fascin crosslinked 98 \nnetwork sheets are morphologically indistinguishable (Fig1 b -d). Both α -actinin and fascin 99 \ncrosslink and bundle F -actin into thick bundles (Fig. d-e). Under thermal driving force, F-actin 100 \nnetworks crosslinked by α -actinin and fascin exhibit similar relaxation time, suggestive of 101 \ncomparable rigidity (Fig1. f-g). However, the dynamics of the networks crosslinked by α -actinin 102 \nand fascin are distinctively different under myosin -induced active stress: f ascin-crosslinked 103 \nnetworks contract locally into asters (Fig. 1h, supplementary movie 1 ); i n contrast, α -actinin-104 \ncrosslinked networks behave as cohesive elastic sheets that rupture via crack formation38 (Fig. 1h-105 \ni, supplementary movie 1). The distinct contractile behaviors of the two crosslinked networks lead 106 \nto differences in both the magnitude of strain and the time to material failure.  (Fig. 1j-k). We 107 \nhypothesize the difference in the mechanical dynamics of the two types of networks is due to the 108 \ndifference in bond mechano-sensitivity and mechanics21 (Supplemental Materials). Therefore, due 109 \nto the opposite mechanism of bond kinetics under load, slip bonds tend to unbind in high -stress 110 \nregions while catch bonds tend to unbind in low -stress regions. This results in potentially the 111 \nhomogenization of stress in the α-actinin-crosslinked network which enables the active material to 112 \nendure higher stress. Interestingly, in α-actinin crosslinked networks, the strain rate of deformation 113 \nexhibits a non-monotonic dependence on myosin concentration , with strain rate peaks at around 114 \nRmyo ≈ 0.02  (Fig. 1l). This unique observation of the dynamics and non -monotonicity in the 115 \nnonlinear/contractile regime is reminiscent of mechanical inhibition of dissipation in active 116 \nsolids43. Therefore, to investigate the origin of the non -monotonicity, we begin to investigate the 117 \nenergy partition and heat dissipation of the networks under active stress.  118 \n 119 \nMechano-chemical coupling yields non-monotonic dissipation with driving 120 \nWe reconstitute a three-dimensional active actomyosin network by mixing purified G-actin, 121 \nskeletal muscle myosin II, and crosslinkers such as α -actinin or fascin with ATP in F -buffer 122 \n(Methods). This protein mixture was injected into a glass capillary tube mounted inside a custom-123 \nbuilt picowatt-calorimeter. The calorimeter achieves a lower detection limit of 8 5 pW (Fig. 2a,  124 \nSFig. 1,  Methods) [ Zheng et al., In submission ]. The total heat dissipation  results from 125 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\npredominantly three mechanisms: 1) ATP hydrolysis of actin polymerization, 2) ATPase kinetics 126 \nvia myosin motors, and 3) ATP regeneration system via pyruvate kinase and lactic dehydrogenase 127 \n(PK-LDH) along with phosphoenolpyruvate (PEP) (Fig. 2 b, SFig. 1 , Methods ). Upon 128 \npolymerization, heat dissipation from the F -actin network was captured and exhibited an 129 \nexponential decay over time, consistent with prior work on actin polymerization kinetics 44 (Fig. 130 \n2c-d, Supplementary Materials). We directly measure the heat dissipation Δ𝑄, and each data point 131 \nis collected every 0.5 seconds, results in heat dissipation per unit time (power), 𝑄̇ =  Δ𝑄/Δ𝑡. After 132 \ncharacterizing the heat dissipation of actin polymerization, we introduce skeletal muscle myosin 133 \nII dimers into the protein mix, which performs power strokes on F -actin and causes network 134 \ndeformation and contraction45 (Supplementary Materials). We increase ATP concentration while 135 \nkeeping actin and myosin concentration constant, the measured heat dissipation rate 𝑄̇  follows 136 \nMichaelis–Menten kinetics:  137 \n𝐽𝐴𝑇𝑃𝑎𝑠𝑒([𝐴𝑇𝑃])= 𝑘𝑐𝑎𝑡\n𝐴𝑇𝑃[𝐴𝑇𝑃]\n𝐾𝑀\n𝐴𝑇𝑃 + [𝐴𝑇𝑃][𝑀]𝑡𝑜𝑡 (1) \n, where 𝐾𝑀\n𝐴𝑇𝑃 = 0.1397 ± 0.09345  µM , of which the difference is within a factor of 2 compared 138 \nto values from previous studies46 (Fig. 2e), potentially due to the differences in actin concentration 139 \nand myosin motor type . We then vary the concentration of actin, and the heat dissipation rate 𝑄̇  140 \nalso follows Michaelis–Menten kinetics: 141 \n𝐽𝐴𝑇𝑃𝑎𝑠𝑒([𝐴])= 𝑘𝑐𝑎𝑡\n [𝐴𝑇𝑃]\n𝐾𝐴\n + [𝐴𝑇𝑃][𝑀]𝑡𝑜𝑡 (2) \n, where 𝐾𝐴\n = 13.11± 9.6125 µM, agrees with previous work47,48 (Fig. 2f). 142 \nTo further investigate the energetic role of motor activity, we systematically varied the 143 \nmyosin concentration while maintaining a constant actin concentration. The heat dissipation 144 \nprofile of the resulting actomyosin networks exhibit s a non -monotonic trend as myosin 145 \nconcentration increases (Fig. 2g). We hypothesize that above a critical myosin-to-actin molar ratio 146 \n(Rmyo > 0.02), the network enters a highly connected regime in which  active stresses become 147 \nuniformly distributed due to myosin’s mechanosensitivity. In this regime, myosin thick filaments 148 \ncan behave as catch bonds, and therefore the ATPase activity of myosin is reduced due to increased 149 \nmechanical load across the network49-52, reflected as a decreased rate of heat dissipation (Fig. 2g). 150 \nThis is reminiscent of the non -monotonicity of dissipation observed in ‘active solids’  as in our 151 \ncontractile 2D-network experiments (Fig. 1) , in which energy dissipation peaks at intermediate 152 \nstress and is inhibited at large tensile or compressive loads 43. The nuance between the 2D system 153 \nand 3D system is that the 3D system does not exhibit large scale deformation, potentially due to 154 \ngeometry and the effective protein concentration differences (Fig. 2g, supplementary movie 2). In 155 \nour case, the heat dissipation primarily comes from the ATPase activity and ATP regeneration 156 \nsystem ( SFig. 1, Supplemental Materials ). The heat dissipation is linearly proportional to  the 157 \nconcentration (number) of myosin head performing ATPase , 𝑐ℎ𝑒𝑎𝑑 , the myosin ATPase rate , 𝑟 158 \n(𝑠−1), the volume of the chamber, 𝑉 (80 nL in this case), and Δ𝐺, the free energy , which is ≈159 \n60 𝑘𝐽/𝑚𝑜𝑙. The heat dissipation rate  𝑄̇  measured then follows:  160 \n 161 \n. 162 \nFrom this, we can calculate an effective myosin ATPase rate ( SFig. 2, Supplemental Materials), 163 \nwhich can indirectly infer the level of active stress , 𝜎𝑎, experienced by the motors  since myosin 164 \nATPase rate 𝑟 is inversely proportional to active stress 𝜎𝑎\n49-51,53.  165 \n𝑄̇ =  Δ𝑄/Δ𝑡 = 𝑐ℎ𝑒𝑎𝑑 ∙ 𝑟 ∙ 𝑉 ∙ Δ𝐺 (3) \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nTo elucidate how differences in force sensing and mechano -sensitive binding shape 166 \nenergetic output, we examine actin networks crosslinked with either α -actinin (catch bond) or 167 \nfascin (slip bond) using 3D microscopy and pico -calorimetry. This allows us to directly compare 168 \nhow distinct bond mechanics influence stress generation and heat dissipation under active motor 169 \nloading. 170 \n 171 \nCatch bond crosslinked network dissipates more heat under active stress 172 \nAs mentioned previously, t he 3D networks exhibited minimal large -scale deformation 173 \nunder equivalent myosin density , likely due to geometric constraints  and effective protein 174 \nconcentrations, as noted in prior studies 26,54 (Fig. 2g). Given the differences in mechanical 175 \nresponse between α-actinin and fascin networks under active loading, we next quantif y the total 176 \nheat dissipation of these systems using picowatt-resolution calorimetry. 177 \nUnder fixed crosslinking and  myosin density , increasing the ratio between α-actinin to 178 \nfascin (Φ𝑎𝑐𝑡𝑖𝑛𝑖𝑛:𝑓𝑎𝑠𝑐𝑖𝑛) results in a n increase in heat dissipation , potentially due to an elevated 179 \nATPase rate with higher Φ𝑎𝑐𝑡𝑖𝑛𝑖𝑛:𝑓𝑎𝑠𝑐𝑖𝑛 (Fig. 2h).  180 \nTo further investigate this behavior, we systematically var y the concentration of myosin 181 \nmotors while maintain constant crosslinker density. In both α -actinin and fascin networks, heat 182 \ndissipation display a non-monotonic trend as a function of motor density  (Fig. 2i). However, the 183 \ndissipation peak occurred at a higher myosin concentration in fascin networks compared to α -184 \nactinin networks (Fig. 2i). Notably, the dissipation curve for the α -actinin case closely resembles 185 \nthat of a network lacking crosslinkers  (Fig. 2i), supporting the hypothesis that myosin exhibits 186 \ncatch bond–like behavior. In contrast, the delayed peak observed in fascin networks is likely due 187 \nto fascin’s polarity -sorting capacity and slip bond properties, which restrict stress redistribution 188 \nuntil the motor density reaches a threshold where catch bond dynamics begin to dominate (Fig. 2i). 189 \nHaving established that bond mechanics influence how stress accumulates and relaxes, we next 190 \nask how these differences translate into measurable changes in energy dissipation. 191 \nTherefore, we explore how heat dissipation depends on crosslinking density. For fascin, 192 \nwe observe a sharp drop in dissipation around Rc ≈ 0.001, near the expected bundling threshold 193 \nand percolation critical point for 3D networks41 (Fig. 2j). In contrast, heat dissipation in α-actinin-194 \ncrosslinked networks remain relatively stable with only a modest decline as Rc increased (Fig. 2j). 195 \nAcross all crosslinker densities, α -actinin networks  exhibit higher heat  dissipation than  fascin 196 \nnetworks. 197 \nWe hypothesize that this is due to α-actinin’s catch bond nature, which allows crosslinkers 198 \nin low-stress regions, where unbinding rates are higher, to relocate to regions of higher stress. This 199 \ndynamic redistribution promotes stress homogenization across the network 55. Because myosin 200 \nATPase activity is inversely correlated with the mechanical load experienced by motors49-51,53, the 201 \nmore evenly distributed stress in α -actinin networks may result in higher overall ATP turnover 202 \ncompared to fascin networks, as also shown from the effective ATPase rate calculation (SFig. 2). 203 \nTo directly probe the mechanical stress landscape in these systems and validate this hypothesis, 204 \nwe next perform rheological measurements to characterize the viscoelastic properties of the two 205 \nnetwork types (Fig. 3a). 206 \n 207 \nCatch bond crosslinked network homogenizes and lowers stress  208 \nTo characterize the material properties of crosslinked F-actin networks under active stress 209 \ngenerated by myosin motors , we use a stress -controlled rheometer (Fig. 3a, Supplementary 210 \nMaterials). Previous studies have suggested that the mechanical response of actively stressed 211 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\ncytoskeletal networks is analogous to that of passive networks driven into the nonlinear regime via 212 \nprestress23,25,56. Here, we indeed observe that the viscoelastic moduli (G′ and G″) of the network 213 \nincrease with higher  Rmyo (Fig. 3b). This is not due to crosslinking by myosin dimers or thick 214 \nfilaments, as our ATP concentration is within the saturating regime in which myosin dimers have 215 \nan binding/unbinding time of ~5 ms23.  216 \nIn linear frequency sweep experiments, the crossover frequency ( 𝜔𝑐𝑜), the frequency at 217 \nwhich the storage modulus G′ equals the loss modulus G″ , serves as a metric for the timescale at 218 \nwhich the network transitions from solid -like to fluid-like behavior. More adaptive and transient 219 \nnetworks exhibit lower 𝜔𝑐𝑜 values. At equivalent crosslinking densities, α -actinin–crosslinked 220 \nnetworks display significantly lower 𝜔𝑐𝑜 than fascin-crosslinked ones, indicating more fluid -like 221 \nand dynamic behavior (Fig. 3c-d). As a control, we also examine networks crosslinked via biotin–222 \nstreptavidin, a permanent and non -dynamic linkage. Biotin–streptavidin-crosslinked networks 223 \ndisplay a crossover at very high frequency or almost no crossover, behaving as elastic solids over 224 \na wide frequency range (Fig. 3c-d). 225 \nAs myosin density increases , both G′ and G″ increase monotonically  for α-actinin–226 \ncrosslinked networks, consistent with stress -induced stiffening observed in prior work 23,56 (Fig. 227 \n3e-f). In contrast, fascin -crosslinked networks exhibit a non -monotonic response: both moduli 228 \ninitially increased with motor density but later decreased  (Fig. 3e-f). This behavior may reflect 229 \nstructural reorganization of the fascin -actin bundles or a mechanical transition from slip bond –230 \ndominated to catch bond–dominated dynamics. 231 \nTo quantify the internal stress exerted by myosin motors, we perform nonlinear rheology 232 \non passive networks of matched crosslinking density  (Methods). We first appl y a step shear 233 \nprestress 𝜎0 and allow the sample to reach a quasi -steady state. We then superimpose a small 234 \nsinusoidal stress 𝛿𝜎(𝑡)= |𝛿𝜎|𝑒𝑖𝜔𝑡  with amplitude kept within the linear response of the 235 \nprestressed state (| 𝛿𝜎| ≤ 𝜎0/10) so that linear response theory holds 57-59. The resulting strain 236 \nresponse 𝛿𝛾(𝑡) defines the complex differential modulus  as the local slope of the stress –strain 237 \nrelation about 𝜎0: 238 \n𝐾∗(𝜔,𝜎0)= 𝛿𝜎(𝜔)\n𝛿𝛾(𝜔)|𝜎0 (4) \n, with 𝐾′(𝜔,𝜎0)= 𝑅𝑒(𝐾∗)and 𝐾\"(𝜔,𝜎0)= 𝐼𝑚(𝐾∗). In the limit of zero prestress, this reduces to 239 \nthe conventional linear viscoelastic moduli, 𝐾′(𝜔,0)= 𝐺′(𝜔) and 𝐾\"(𝜔,0)= 𝐺\"(𝜔). We fit the 240 \ndifferential elastic modulus with: 241 \n𝐾′\n𝑝𝑎𝑠𝑠(𝜎0,𝜔) = 𝐾′\n0 +  𝐴 𝜎0\n𝛼 \n \n(5) \nBy extracting the differential elastic modulus K′ as a function of applied prestress, we map the 242 \nlinear moduli (G′, G″) of the active networks onto the stress–stiffening curves of passive ones, thus 243 \nestimating the active stress imparted by motors (Fig. 3g-h):  244 \n𝜎𝑎 ≈ ((𝐺′\n𝑎𝑐𝑡 − 𝐾′\n0)\n𝐴 )\n1\n𝛼\n \n(6) \n. At low myosin concentrations, active stress in fascin -crosslinked networks exceed that of α -245 \nactinin networks  (Fig. 3h). However, above Rmyo ≈ 0.05, the active stress in fascin networks 246 \ndecreases. This trend mirrors the calorimetric data (Fig. 2i), in which fascin networks begin to 247 \ndissipate more heat than α-actinin networks at high motor densities.  248 \nWe also use velocity information calculated from PIV on 3D network data to extract the 249 \nmechanical dissipation rate ⟨𝑝𝑚𝑒𝑐ℎ⟩: 250 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\n⟨𝑝𝑚𝑒𝑐ℎ⟩ =  2 𝛽 ∫ 𝜂′(2𝜋 𝑓) [ 𝑆𝑥𝑥(𝑓) + 𝑆𝑦𝑦(𝑓) +  2 𝑆𝑥𝑦(𝑓) ] 𝑑𝑓\n𝑓𝑚𝑎𝑥\n0\n \n(7) \n, in which 𝛽 is the inverse thermal energy of the bath,  S(𝑓) is the power spectral density function 251 \nfor the strain rate (divergence of velocity), and dynamic viscosity 𝜂′(2𝜋 𝑓) = \nG\"(𝑓)\n2𝜋 𝑓  252 \n(Supplemental Materials). We then calculate the mechanical efficiency 𝜂 = \n⟨𝑝𝑚𝑒𝑐ℎ⟩\n⟨𝑄̇⟩ , for both slip 253 \nbond- (fascin) and catch bond- (α-actinin) crosslinked networks. 𝜂 lies between 5 × 10−5~1.5×254 \n10−4, with slip bond -crosslinked network showing a higher mechanical efficiency than catch 255 \nbond-crosslinked network (SFig. 3).  256 \n 257 \nMonotonic increase in driving leads to degeneracy in dissipation 258 \n Finally, to elucidate further on the non-monotonic behavior in dissipation due to chemical-259 \nmechanical coupling, we perform actin network velocity correlation and entropy production rate 260 \nanalysis33,34,60 using the actin and myosin fluorescence channel (Methods). The entropy production 261 \nrate (EPR) provides a lower bound for the dissipation of the system, and further, the 262 \nthermodynamic molecular details of the system due to mechano-chemical feedbacks. We observe 263 \nthat the EPR exhibit non -monotonic trend as a function of myosin motor concentration, which 264 \nshows degeneracy at low (Rmyo = 0.002) and high (Rmyo = 0.1) driving (Fig. 4 a, d).  265 \nDespite their similar dissipation rates, these regimes differ fundamentally in how 266 \nmechanical stress propagates through the network. Velocity correlation analysis shows that at low 267 \nmyosin activity, stress transmission is short-ranged and localized, whereas at high myosin activity 268 \nit becomes more delocalized, giving rise to coherent flows across the network (Fig. 4b,c). Thus, 269 \ncomparable entropy production can emerge from distinct stress-propagation modes, demonstrating 270 \na degeneracy between dissipation and mechanical organization in active cytoskeletal 271 \nmaterials.(Rmyo = 0.002)  272 \n 273 \nDiscussion 274 \nNear thermodynamic equilibrium, linear response theory predicts that increasing driving 275 \nforces lead to a monotonic increase in dissipation: energy fluxes scale linearly with their conjugate 276 \nforces, and additional input is converted directly into heat through Onsager reciprocity 1,2,7,10,61. 277 \nWhile not an explicit assumption, t his expectation is often implicitly extended to active matter, 278 \nwhere stronger motor activity is assumed to amplify dissipation 62,63. Our results demonstrate a 279 \nfundamental breakdown of this intuition. Instead of monotonically increasing, energy dissipation 280 \npeaks at intermediate driving and is suppressed at high stress. This non -monotonicity reflects a 281 \nfar-from-equilibrium regime in which added energy is not dissipated more strongly, but reallocated 282 \nthrough feedback between mechanical stress, bond kinetics, and motor chemistry. Such behavior 283 \nis inaccessible to linear response and reveals that, in active materials, driving can restructure 284 \npathways of energy flow rather than simply increase entropy production.  These results may be 285 \npotentially a general feature of active systems. While previous work could not statistically 286 \ndistinguish between saturation and modest decrease 64, the non -monotonicity may arise robustly 287 \nfrom  mechanochemical feedback between stress organization and motor activity. 288 \nThe physical manifestation of this energetic reallocation is a transition in how mechanical 289 \nstress is organized in space analogous to Anderson localization in disordered media 65,66. Rather 290 \nthan remaining uniformly distributed, active stress either concentrates into localized domains or is 291 \ndelocalized across the network, depending on bond mechano-sensitivity (Fig. 4e). Stress focusing 292 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\namplifies local force per motor, suppressing ATP turnover and limiting dissipation, whereas stress 293 \ndefocusing distributes load over many motors, sustaining higher chemical flux and elevated heat 294 \nproduction. This stress-organization transition provides a concrete, mesoscale mechanism linking 295 \nmicroscopic mechanochemistry to macroscopic thermodynamics: dissipation is regulated not by 296 \nthe magnitude of driving alone, but by how that driving is partitioned across space. In this sense, 297 \nstress localization acts as a control variable for energy flow in active matter . The impact of stress 298 \nlocalization is not only to alter the propensity for mechanical propagation, but also in the induction 299 \nof other active processes, such as transforming mechanical structure into a regulator of metabolic 300 \ncost.  301 \nNon-monotonic behaviors in system variables are ubiquitous and diverse 67,68. These 302 \nbehaviors can arise from feedback between system variables, such as reactant and product 303 \nconcentration, or through mechanical effects, such as contractions that alter the flow and 304 \nconcentration of chemical species 69. However, the mechanisms do not necessarily impact 305 \nthermodynamic forces, such as the ab initio injection of energy. Here, we demonstrate that 306 \nchemical energy is not a bath, but a subsystem. 307 \nCollective organization is known to qualitatively reshape dissipation in condensed -matter 308 \nsystems70-72. Calorimetry reveals that networks crosslinked with α-actinin dissipate more heat than 309 \nthose crosslinked with fascin under comparable connectivity and active stress, indicating that 310 \nstabilizing a stressed cytoskeleton can be more energetically demanding than contracting it. Unlike 311 \nin superconductors where coherence suppresses dissipation 71,73, the actomyosin network 312 \nrepresents an inverted dissipative Bardeen -Cooper-Schrieffer (BCS) system: collective stress 313 \nhomogenization increases the density of active mechanochemical cycles, amplifying energy flux 314 \nuntil feedback through bond stabilization suppresses it at high stress. 315 \n  316 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nFigures 317 \n 318 \nFig. 1 Activity reveals divergent mechanical responses of catch - and slip-bond crosslinked 319 \nnetworks. (a) Schematic of pseudo-2D actomyosin network crowded by Methylcellulose on the 320 \nlipid bilayer. (b) Confocal fluorescence microscopy image of  pseudo-2D actomyosin network 321 \ncrosslinked by fascin (top) and α -actinin (bottom) on a supported lipid bilayer.  Left is before 322 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\ncrosslinking, and right is post crosslinker addition. Scalebar is 5 µm. (c) Kymograph of the yellow 323 \ndashed line in b), scalebars are 5µm (horizontal) and 10s ( vertical). (d) Exemplary linescan 324 \nintensity (normalized) for conditions in e. (e) Bundling metric 𝜆 of fascin and α-actinin crosslinked 325 \nnetworks as well as F-actin network without crosslinkers. N = 3 for each condition. (f) Exemplary 326 \nautocorrelation function of actin network fluctuations as function of lag time Δ𝑡. (g) Characteristic 327 \ntime 𝜏 for different conditions. N = 5,3,3,3,3,3 respectively. (h) Confocal fluorescence microscopy 328 \nimage of  2D actomyosin network crosslinked by fascin (top) and α-actinin (bottom) deformed and 329 \ncontracted under myosin active stress over time. Scalebars are 20 µm. Heatmaps show 330 \naccumulative strain of the final frame. Quiver plot overlay shows the instantaneous velocity. ( i) 331 \nKymograph of α-actinin network rupture (dashed red line in h). Scalebars are 10µm and 10 s. (j) 332 \nMean strain < 𝜀 > of the network during deformation caused by myosin induced active stress over 333 \ntime. The slope of the strain curve is the strain rate  \n𝑑𝜀\n𝑑𝑡. (k) Maximum mean strain < 𝜀 >𝑚𝑎𝑥 of 334 \nfascin and α-actinin crosslinked networks at Rc = 0.2. N = 3 for each condition.  𝑝𝑓𝑎𝑠−𝑎𝑎 = 0.0101. 335 \n(l) Strain rate \n𝑑𝜀\n𝑑𝑡  of the network crosslinked by α -actinin at R c = 0.1 at various myosin 336 \nconcentrations. N = 2,3,3,3,2,2,2 from low to high concentration respectively.  337 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\n 338 \n 339 \nFig. 2. Catch- and slip-bond crosslinkers tune non-monotonic heat dissipation generated by 340 \nthe actomyosin crossbridge . (a) Schematic of calorimeter and the zoomed -in circle shows the 341 \nactomyosin sample details in the capillary tube that contains the sample. (b) Schematic of ATPase 342 \nactivities from 1. actin polymerization ; 2. myosin ATPase in the network ; and 3. the chemical 343 \nreaction of ATP regeneration system via Pyruvate assay. (c) Heat dissipation rate (𝑄̇ ) of actin 344 \npolymerization at various actin concentrations over time. (d) Initial (black circles) and steady state 345 \n(blue diamonds) heat dissipation rate (𝑄̇ ) at various actin concentrations. Gray line is a Michaelis-346 \nMenten curve fit. (e) Heat dissipation rate of actomyosin network at various ATP concentrations 347 \n(Cactin = 37.5µM, R myo = 0.02.) Black line is the fit of Michaelis-Menten curve to data. (f) Heat 348 \ndissipation rate of actomyosin network at various actin concentrations (Rmyo = 0.02.) Black line is 349 \nthe fit of Michaelis-Menten curve to data. (g) Heat dissipation rate (𝑄̇ )of actomyosin network at 350 \nvarious myosin-to-actin ratios (Cactin = 37.5µM, CATP = 6.25mM.) (h) Heat dissipation rate (𝑄̇ ) of 351 \nactomyosin network under same crosslinking ratio R CL = 0.001 but at different α -actinin:fascin 352 \nratios (Φ𝑎𝑐𝑡𝑖𝑛:𝑓𝑎𝑠𝑐𝑖𝑛). Gray dashed line is a linear fit to the data. (i) Heat dissipation rate  (𝑄̇ )  of 353 \nactomyosin network under same crosslinking ratio R CL = 0.001and C actin = 37.5µM, at various 354 \nRmyo. (j) Heat dissipation rate (𝑄̇ ) of actomyosin network at R myo = 0.02, C actin = 37.5µM, and 355 \nvarious crosslinking ratio RCL.  356 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\n 357 \n 358 \nFig. 3 Rheological mapping reveals delocalized active stress in catch -bonded networks. (a) 359 \nSchematic of crosslinked network inside a cone -and-plate rheometer for (1)  under active stress 360 \n(top right) and (2) under rheometer-induced prestress ( bottom right). (b) Frequency sweep (G’, 361 \nelastic modulus, and G”, viscous modulus) of the α -actinin crosslinked network ( RCL = 0.001) 362 \nunder active stress induced by myosin (Rmyo  = 0.02, maroon, and 0.1, blue). (c) Frequency sweep 363 \n(G’, elastic modulus, and G”, viscous modulus) of the α -actinin (maroon hexagons), fascin (blue 364 \ntriangles), and biotin -streptavidin (black diamonds) crosslinked networks. Vertical dashed lines 365 \nindicate the G’ and G” crossover frequency co for three cases. (d) Crossover frequency co for 366 \nthe three conditions in (c). (e) Elastic modulus G’ of the three types of network with various Rmyo. 367 \n(f) Viscous modulus G” of the three types of network with various R myo. (g) Differential elastic 368 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nmodulus K’ for the three types of networks as a function of prestress 0. Black line is indicative 369 \nof slope of 1. (h) Indicated active stress a as a function of Rmyo for the three types of networks. 370 \n  371 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\n 372 \nFig. 4. Degeneracy in dissipation emerges from stress localization and delocalization . (a) Heat 373 \nmap of azimuthally averaged Entropy Production Factor (EPF) as a function of frequency (𝜔) and 374 \nwave vector (𝑞𝑟). Rmyo = 0.02. (b)  Heatmap shows accumulative strain (colorbar) after 5 minutes 375 \nof completion of polymerization. Quiver plot overlay shows the mean velocity within the 5 minutes. 376 \nArrow color indicates the degree of alignment of the velocity with its neighbors. Red -white-blue 377 \nindicates ‘highly aligned’ - ‘not aligned’ – ‘anti-aligned’. R myo = 0.1.  (c) Azimuthally averaged 378 \nvelocity spatial correlation (𝐶𝑣𝑣) as a function of radius 𝑅. N = 3 for both cases. Errorbars are the 379 \ns.t.d. of the mean. Gray dashed lines are the exponential fit. (d) Entropy Production Rate (EPR, 380 \nmaroon, left y-axis) and characteristic length (𝜆𝑣, blue, right y-axis) as a function of R myo. N = 3 381 \nfor all conditions. Errorbars are the s.t.d. of the mean. (e) Schematic of how degeneracy in 382 \ndissipation can arise from the two ends of the driving due to stress de/localization. 383 \n  384 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nAcknowledgments 385 \n 386 \nWe are grateful for insightful discussions with Dr. Daniel Needleman and Dr. Peter Foster. Z.G.S. 387 \nacknowledges fundings and support from Yale PEB, his family, and friends during his exchange 388 \nperiod at Harvard. This work was supported by funding  ARO MURI  W911NF-14-1-0403, 389 \nthe National Institutes of Health (NIH) R01 GR130179, Sloan Matter-to-Life G-2025-79182 , and 390 \nHuman Frontiers Science Program (HFSP) grant number RGP012/2025 to M.P.M. 391 \n 392 \nAuthor Contributions 393 \n 394 \nZ.G.S. & M.P.M designed and conceived the work. Z.G.S. & M.P.M. drafted the paper.  Z.G.S, 395 \nM.P.M, & J.J.V. edited the paper. Z.G.S. and A.P.T. performed experiments. Z.G.S. analyzed the 396 \ndata. J.Z. instructed on calorimeter operation. Z.G.S, M.P.M, J.J.V ., & J.Z. participated in the 397 \ndiscussion of the work. M.P.M. & J.J.V. supervised the work. 398 \n 399 \n 400 \nCompeting Interests  401 \nA patent, entitled as “Microcalorimetry for high-throughput screening of bioenergetics”, has been 402 \nfiled for the devices by Harvard University (inventors: J. Zheng, J.J. Vlassak, D.J. 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S., Mura, F., Gladrow, J. & Broedersz, C. P. Broken detailed balance and non-559 \nequilibrium dynamics in living systems: a review. Reports on Progress in Physics  81, 560 \n066601 (2018).  561 \n70 De Gennes, P.-G. Superconductivity of metals and alloys.  (CRC press, 2018). 562 \n71 Tinkham, M. Introduction to superconductivity.  (Courier Corporation, 2004). 563 \n72 Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Reviews of 564 \nmodern physics 65, 851 (1993).  565 \n73 Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Theory of superconductivity. Physical review 566 \n108, 1175 (1957).  567 \n 568 \n  569 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\n 570 \nMethods 571 \nPreparation of small unilamellar vesicles (SUVs) 572 \nLipids were combined by mixing 100 µL of egg phosphatidylcholine (Egg PC, 25 mg mL⁻¹; Avanti 573 \nPolar Lipids , 840051C) with 10 µL of 1,2 -dihexadecanoyl-sn-glycero-3-phosphoethanolamine 574 \n(DHPE, 0.5 mg mL⁻¹), optionally conjugated to Oregon Green 488 ( Invitrogen). All lipids were 575 \ndissolved in chloroform and transferred to a clean glass vial pre -equilibrated with argon to 576 \nminimize oxidative degradation. The solvent was removed under a gentle stream of argon, yielding 577 \na uniform lipid film coating the bottom of the vial. The dried lipid film was rehydrated with 5 mL 578 \nof vesicle buffer (100 mM NaCl, 20 mM HEPES, pH 7.5) and vortexed until the suspension 579 \nbecame turbid. The dispersion was then sonicated in a bath sonicator for approximately 1 h, or 580 \nuntil the solution turned optically clear, indicating the formation of small unilamellar vesicles 581 \nsuitable for bilayer formation. 582 \nFluorescent labeling of skeletal muscle myosin  583 \nSkeletal muscle myosin (Heavy Meromyosin from rabbit, Cytoskeleton Inc.) is fluorescently 584 \nlabeled with Alexa Fluor 647 C2 Maleimide under reducing conditions. Initially, myosin is 585 \nreduced in a labeling buffer containing 50 mM HEPES, 0.5 M KCl, 1 mM EDTA, and 10 mM 586 \nDTT at pH 7.6. Following reduction, the sample is dialyzed overnight against the same buffer 587 \nwithout DTT. After dialysis, the solution is centrifuged to eliminate any insoluble components. 588 \nThe resulting supernatant is reacted with Alexa Fluor C2 Maleimide at a 5:1 molar ratio of dye to 589 \nmyosin. Labeling is carried out at 4°C for one hour, after which the reaction is quenched by adding 590 \n1 mM DTT. The labeled protein is purified using a desalting column (Pierce, 5K MWCO, 5 mL). 591 \nAbsorbance readings at 280 nm and 647 nm are then used to calculate the degree of labeling. This 592 \nprotocol is adapted from Verkohovsky and Borisy1. 593 \n 594 \n2D actomyosin contraction experiments 595 \nEach chamber is of cylindrical shape, with diameter of 12mm. The top and bottom piece are 596 \nmagnetically locked, with a rubber piece sealing the middle and a glass slide sandwiched in 597 \nbetween. The glass slide is first washed with 50% ethanol to clean any residues left on the surface. 598 \nThe slide is then exposed under UV light for 5 min to induce hydrophilicity to the surface. 300 μL 599 \nof SUV solution is added to the chamber. Once the surface of the glass is coated with lipid bilayer, 600 \nwe take 100 μL solution out and wash the chamber with 400 μL of 1x F-buffer (10 mM imidazole, 601 \n1 mM MgCl2, 50 mM KCl, 2 mM EGTA, 0.5 mM ATP, pH = 7.5) . Dark G-actin (Cytoskeleton) 602 \nis mixed with rhodamine labelled F -actin (20% fluorescent, Cytoskeleton) to a final molar 603 \nconcentration of 1.4 µM, and is stabilized with 1 μM phalloidin (Cytoskeleton) and crowded to the 604 \nsurface of a 97% Egg Phosphatidyl Choline (Avanti Polar Lipids)/3% FITC -DHPE (Molecular 605 \nProbes) phospholipid bilayer, using 0.2% 14,000 MW methyl -cellulose (Sigma, 15 cP) as a 606 \ndepletion agent (Fig. 1A). The actin mixed soup is placed in an Eppendorf tube on ice for 1 hour 607 \nto reach full polymerization (50 μL protein mix in total). Once F-actin is polymerized, it is added 608 \nto the chamber, along with methyl-cellulose. Once the F-actin network is crowded onto the surface, 609 \nwe add crosslinkers 𝛼-actinin or fascin of various concentrations. Then, Skeletal muscle myosin 610 \nII (various concentrations), labelled with Alexa Flour 647nm C2 Maleimide (Molecular Probes) is 611 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nadded in solution in dimeric form, which polymerizes into thick filaments onto the F -actin. This 612 \nresults in a two-dimensional actomyosin network.  613 \n 614 \nRemoving non-catalytically active skeletal muscle myosin 615 \nTo isolate only catalytically active myosin dimers for experimental use, myosin is subjected to a 616 \nselective centrifugation process in the presence of polymerized actin. First, actin is polymerized 617 \nfor one hour at 4°C in a high -salt environment (1X F -buffer supplemented with 4 M KCl) and 618 \nstabilized using phalloidin. After polymerization, ATP is added to reach a final concentration of 1 619 \nmM, followed by the addition of freshly thawed myosin. This actin -myosin mixture is incubated 620 \nat 4°C for 10 minutes, then centrifuged at 128,360 × g for 30 minutes. During centrifugation, 621 \nenzymatically inactive myosin remains bound to the F-actin network and sediments, while active 622 \nmyosin motors detach and remain in the supernatant, which is then collected. The concentration 623 \nof active myosin in the supernatant is quantified by measuring absorbance at 647 nm, based on a 624 \npre-determined labeling efficiency. Myosin is freshly prepared for each experiment and used 625 \nwithin 24 hours. 626 \n 627 \nMicroscopy 628 \nThe image stack data are collected using Leica DMi8 inverted microscope equipped with a 63×, 629 \n40×, or 20×, and 1.4, 1.3, and 0.75 -NA oil immersion lens respectively (Leica Microsystems), a 630 \nspinning-disk confocal (CSU22; Yokagawa), and sCMOS camera (Zyla; Andor Technology) 631 \ncontrolled by Andor iQ3 (Andor Technology). Image time series stack data are taken with time 632 \ninterval of 0.5-10 seconds.  633 \n 634 \nPicocalorimetry Measurements 635 \nMetabolic heat was quantified using a custom micromachined capillary -based picocalorimetry 636 \nsystem, as previously described (Zheng et al.). Three borosilicate glass capillaries (400 µm × 400 637 \nµm outer dimension, 100 µm wall thickness) were bonded onto gold -coated regions of a silicon-638 \nnitride membrane integrating two Nichrome/Constantan thermopiles and a tungsten micro-heater. 639 \nOne capillary was loaded with the biological sample, and two identical capillaries containing pure 640 \nwater served as thermal references. Heat generated by the sample produced a temperature 641 \ndifferential relative to the references, which was converted into a voltage by the thermopiles 642 \nthrough the Seebeck effect. The monitored sensing volume within the sample capillary was ~80 643 \nnL. 644 \nThe calorimeter assembly was mounted on a custom PCB and placed inside a thermally insulated 645 \nvacuum chamber evacuated to ~10⁻⁵ Torr to suppress convective heat loss and reduce electrical 646 \nnoise. Measurements were performed at 2 5 °C. Thermopile voltages were acquired using two 647 \nKeithley 2182A nanovoltmeters operated at 6 PLC with power-line synchronization and auto-zero 648 \nenabled. Data were sampled at 2.31 Hz and processed using a moving-average filter. Under these 649 \nmeasurement conditions, the system exhibited an effective noise floor corresponding to a power 650 \nsensitivity of ~85 pW. 651 \n 652 \nExperimental setup and procedures for rheology experiments 653 \nRheological measurements of actin networks were performed using a stress -controlled rheometer 654 \n(Anton Paar M502). Actin was first prepared following the composition detailed in Table 1, and 655 \nthe final sample mixture (Table 2) was assembled to a total volume of 1000 µL. After gentle 656 \nmixing, the solution was loaded onto the preheated bottom plate maintained at 25 °C. Both the 657 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\ncone and plate surfaces were sandblasted to prevent sample slippage. Following equilibration, 658 \nseveral drops of silicone oil were added around the sample edge to minimize evaporation, and a 659 \nsolvent trap was placed over the geometry for additional protection. Gelation kinetics were 660 \nmonitored under the parameters described in the main text. After gelation reached a steady plateau 661 \nin both elastic (G′) and viscous (G″) moduli (typically after ~2 h), measurements in the linear and 662 \nnonlinear regimes were performed. After each experiment, the setup was thoroughly cleaned with 663 \nsequential rinses of 70% ethanol, water, ethanol, and water to ensure complete removal of residual 664 \nprotein and prevent cross-contamination between runs. 665 \n 666 \nExperimental setup and procedures for microscopy experiments 667 \nUnlabeled rabbit skeletal muscle actin (>99% purity; Cytoskeleton, Inc.)—hereafter referred to as 668 \ndark actin—was reconstituted in 1× G -buffer (see Table 3) to a concentration of 10 mg mL⁻¹. 669 \nRhodamine-labeled actin (Cytoskeleton, Inc.) was prepared using the same procedure. Both 670 \npreparations were depolymerized for over 24 h at 4 °C in the dark (see Table 3). The two actin 671 \nspecies were then mixed at a 9:1 ratio (dark:rhodamine) to yield G -actin. Polymerization was 672 \ninitiated by combining the protein mixture with polymerization buffer (Table 4). Glucose oxidase, 673 \ncatalase (GOC), and glucose were added as an oxygen scavenging system to minimize 674 \nphotobleaching during imaging. 675 \nSamples were loaded into a custom four -well round chamber, each well comprising a cylindrical 676 \ncavity sealed by a 12 mm glass coverslip and a rubber spacer to prevent leakage. Coverslips were 677 \nsequentially cleaned with 70% ethanol, dried, and exposed to UV light for 5 min to render the 678 \nsurface hydrophilic. To reduce actin adsorption, surfaces were coated with small unilamellar 679 \nvesicles (SUVs; protocol available upon request). After gentle mixing (30 s), 200 µL of the protein 680 \nmixture was injected into each well. Samples were imaged using a Leica confocal microscope (see 681 \nMicroscopy section). 682 \n 683 \nATP-regeneration system 684 \nATP regeneration system was used in all experiments, containing ATP at the indicated 685 \nconcentrations (0.625–12.5 mM), 40 mM phosphoenolpyruvate (PEP), and a coupled pyruvate 686 \nkinase/lactate dehydrogenase (PK/LDH) enzyme mixture (Sigma-Aldrich). The PK/LDH enzymes 687 \nwere mixed at a 1:1 ratio (1 mg mL⁻¹ stock concentration), and 2.5 µL of the enzyme mixture was 688 \nadded per reaction volume of 800 µL. 689 \n 690 \nParticle Image Velocimetry (PIV) 691 \nParticle image velocimetry (PIV) is applied on the fluorescent actin images in MATLAB (mPIV, 692 \nhttps://www.mn.uio.no/math/english/people/aca/jks/matpiv/). The extent of contraction is 693 \ncalculated by defining a mean strain: 694 \n< ε >=< 𝛁⃗⃗ ∙ 𝐱⃗ 𝐜 > \n \n(1) \n, as the divergence of the displacement field. The data is analyzed with PIV window size 32 and 695 \noverlap 0.5. Window size of 16 and 64 have also been used to generate the data, and eventually 696 \ndetermined that 32 is the best parameter value because of the high signal-to-noise ratio.  697 \n 698 \n2D image thresholding method for calculating bundling parameter 699 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nFor each microscopy image of F-actin network, a line scan of approximately 20 m is done to the 700 \nimage and the fluorescence intensity is extracted from the line scan. The bundling parameter 𝜆 is 701 \ncalculated using the equation: 702 \n𝜆 = 𝜈/𝜇 (2) \n 703 \n, in which 𝜈 is the standard deviation of the fluorescence intensity of the line scan, and 𝜇 the mean 704 \nof the fluorescence intensity of the same line scan. 705 \n 706 \nAutocorrelation analysis 707 \nAfter the F-actin network reached a steady state, time-lapse fluorescence image sequences were 708 \nacquired with randomly chosen starting times. To quantify the temporal dynamics of the network 709 \nat the global level, we computed the traditional temporal autocorrelation of the fluorescence 710 \nintensity using custom MATLAB scripts. 711 \nFor each frame, the fluorescence intensity was spatially averaged over all pixels in the image, 712 \nyielding a single intensity time series 𝐼(𝑡). The mean intensity ⟨𝐼⟩was then subtracted to isolate 713 \ntemporal fluctuations, 𝛿𝐼(𝑡)= 𝐼(𝑡)− ⟨𝐼⟩. The normalized temporal autocorrelation function was 714 \ncalculated according to 715 \n𝐶(Δ𝑡)= ⟨𝛿𝐼(𝑡) 𝛿𝐼(𝑡 + Δ𝑡)⟩𝑡\n⟨𝛿𝐼(𝑡)2⟩𝑡\n \n \n(3) \nwhere ⟨⋅⟩𝑡denotes an average over all valid time points separated by a delay Δ𝑡. This 716 \nnormalization ensures 𝐶(0)= 1. Because the number of statistically independent pairs decreases 717 \nwith increasing delay time, the autocorrelation at large Δ𝑡 is increasingly affected by finite-718 \nsampling noise and was not interpreted quantitatively. 719 \nThe experimentally measured autocorrelation functions were fit using a double-exponential 720 \ndecay model, 721 \n𝐶(Δ𝑡)= 𝑎 e−Δ𝑡/𝜏1 + (1− 𝑎) e−Δ𝑡/𝜏2 \n \n(4) \n 722 \n𝜏 = 𝑎𝜏1 + (1− 𝑎) 𝜏2 \n \n(5) \nwhere 𝑎 and 1− 𝑎 are the relative amplitudes of the two relaxation modes, and 𝜏1and 𝜏2are their 723 \nassociated characteristic times. Fitting was restricted to the initial portion of the correlation curve 724 \n(typically the first 30% of the available delay times), as long-lag data points are dominated by 725 \nstatistical uncertainty arising from finite acquisition length. 726 \n 727 \nMechanical power calculation using PIV velocity field 728 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nDefining 𝜂′(𝜔)= \n𝐺′′(𝜔)\n𝜔 , this is: 729 \n⟨𝑝⟩ = ∫ 𝜂′(2𝜋𝑓)\n∞\n0\n 𝑆𝛾̇(𝑓) 𝑑𝑓 \n \n \n(6) \n 730 \nFor a 2D/3D velocity field, using the strain-rate tensor 𝐷𝑖𝑗(𝑟⃑,𝑡)  and its PSDs: 731 \n⟨𝑝⟩ ≈ 2𝛽 ∫ 𝜂′(2𝜋𝑓)\n𝑓𝑚𝑎𝑥\n0\n [𝑆𝑥𝑥(𝑓)+ 𝑆𝑦𝑦(𝑓)+ 2𝑆𝑥𝑦(𝑓)] 𝑑𝑓 \n \n \n(7) \nFor more details, please refer to Supplemental Materials. 732 \n 733 \nVelocity spatial correlation analysis 734 \nEqual-time spatial velocity correlations were computed from particle image velocimetry (PIV) 735 \ndata using a custom MATLAB pipeline. The velocity correlation function was defined as 736 \n𝐶𝑣𝑣(𝑟)= ⟨𝐯(𝐫′,𝑡)⋅ 𝐯(𝐫′ + 𝐫,𝑡)⟩𝐫′\n⟨∣ 𝐯(𝐫′,𝑡)∣⟩𝐫′\n2 , \n \n(8) \nwhere 𝐯(𝐫′,𝑡)denotes the local velocity fluctuation at position 𝐫′and time 𝑡, and ⟨⋅⟩𝐫′indicates 737 \nspatial averaging over all interrogation windows. 738 \nVelocity fields in the 𝑥and 𝑦directions (𝑣𝑥,𝑣𝑦) were first preprocessed to handle missing values 739 \narising from PIV failures. NaN entries were replaced using MATLAB’s fillmissing function with 740 \nlinear interpolation to ensure spatial continuity of the velocity field. 741 \nTo compute the correlation efficiently and without directional bias, the calculation was 742 \nperformed in Fourier space. For each time point, the mean velocity was subtracted from the raw 743 \nvelocity field 𝐯0(𝐫′,𝑡)to obtain velocity fluctuations, 744 \n𝐯(𝐫′,𝑡)= 𝐯0(𝐫′,𝑡)− ⟨𝐯0(𝐫′,𝑡)⟩𝐫′. \n \n \n(9) \nThe resulting field was then normalized by its spatial root-mean-square magnitude, 745 \n𝐯norm(𝐫′,𝑡)= 𝐯(𝐫′,𝑡)/√∑ ∣ 𝐯(\n𝐫′\n𝐫′,𝑡)∣2. \n \n \n(10) \nA two-dimensional fast Fourier transform (FFT) was applied to the normalized velocity field, 746 \nand the autocorrelation was obtained via inverse FFT of the power spectrum: 747 \n𝐶𝑣𝑣(𝐫)= Re[fftshift(ifft2(𝐯̂norm(𝐤,𝑡) 𝐯̂norm\n∗ (𝐤,𝑡)))], \n \n \n(11) \nwhere 𝐯̂norm(𝐤,𝑡)denotes the Fourier-transformed velocity field and ∗ indicates complex 748 \nconjugation. The fftshift operation was used to center the zero-frequency component. 749 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nThe resulting two-dimensional correlation map was radially averaged using a custom radialavg 750 \nroutine to obtain 𝐶𝑣𝑣(𝑟). The decay of the correlation function was quantified by fitting to a 751 \nsingle-exponential form, 752 \n𝐶𝑣𝑣(𝑟)= 𝑏1exp (−𝑟/𝜆𝑣)+ 𝑏2, \n \n \n(12) \nwhere 𝜆𝑣 defines the characteristic velocity correlation length. 753 \n 754 \nEntropy production calculation 755 \nThe entropy production factor (EPF) and entropy production rate (EPR) are: 756 \n  757 \n𝐸𝑃𝐹 (𝒒,𝜔)= 1\n2[𝐶−1(𝒒,−𝜔)− 𝐶−1(𝒒,𝜔)]𝑖𝑗𝐶𝑗𝑖(𝒒,𝜔) \n \n(13) \n𝐸𝑃𝑅 = ∫𝑑𝜔\n2𝜋\n𝑑2𝒒\n(2𝜋)2 𝐸𝑃𝐹 (𝒒,𝜔) \n \n(14) \n, where 𝐶𝑖𝑗(𝒒,𝜔) is the dynamic structure factor (for more derivations, please refer to Seara, 758 \nMachta, and Murrell, 2021). The entropy calculation and analyses are carried out using customized 759 \nPython script utilizing frequent and freqentn package developed by Dr. Daniel S. Seara 760 \n(https://github.com/lab-of-living-matter/freqent/tree/epf_paper/freqent).  761 \n  762 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint \n\nData availability 763 \nSource data are provided with this paper. Raw data supporting the findings of this manuscript are 764 \navailable from the corresponding authors upon reasonable request. A reporting summary for this 765 \nArticle is available as a Supplementary Information file. 766 \n 767 \nCode availability 768 \nCode supporting the findings of this manuscript are available from the corresponding authors upon 769 \nreasonable request. A reporting summary for this Article is available as a Supplementary 770 \nInformation file. 771 \n 772 \n.CC-BY 4.0 International licenseavailable under a \n(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 \nThe copyright holder for this preprintthis version posted April 25, 2026. ; https://doi.org/10.64898/2026.04.22.720181doi: bioRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}