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The active loop extrusion hypothesis proposes that chromatin threads through the cohesin protein complex into progressively larger loops until reaching specific boundary elements. We build upon this hypothesis and develop an analytical theory for active loop extrusion which predicts that loop formation probability is a nonmonotonic function of loop length and describes chromatin contact probabilities. We validate our model with Monte Carlo and hybrid Molecular Dynamics–Monte Carlo simulations and demonstrate that our theory recapitulates experimental chromatin conformation capture data. Our results support active loop extrusion as a mechanism for chromatin organization and provide an analytical description of chromatin organization that may be used to specifically modify chromatin contact probabilities. 

Slide-ring gels are polymer networks with cross-links that can slide along the chains. In contrast to conventional unentangled networks with cross-links fixed along the chains, the slide-ring networks are strain-softening and distribute tension much more uniformly between their strands due to the so-called “pulley effect”. The sliding of cross-links also reduces the elastic modulus in comparison with the modulus of conventional networks with the same number density of cross-links and elastic strands. We develop a single-chain model to account for the redistribution of monomers between network strands of a primary chain. This model takes into account both the pulley effect and fluctuations in the number of monomers per network strand. The pulley effect leads to modulus reduction and uniform tension redistribution between network strands, while fluctuations in the number of strand monomers dominate the strain-softening, the magnitude of which decreases upon network swelling and increases upon deswelling. 

We present a modified Lake–Thomas theory that accounts for the molecular details of network connectivity upon crack propagation in polymer networks. This theory includes not only the energy stored in the breaking network strands (bridging strands) but also the energy stored in the tree-like structure of the strands connecting the bridging strands to the network continuum, which remains intact as the crack propagates. The energy stored in each of the generations of this tree depends nonmonotonically on the generation index due to the nonlinear elasticity of the stretched network strands. Further, the energy required to break a single bridging strand is not necessarily dominated by the energy stored in the bridging strand itself but in the higher generations of the tree. We describe the effect of mechanophores with stored length on the energy stored in the tree-like structure. In comparison with the “strong” mechanophores that can only be activated in the bridging strand, “weak” mechanophores that can be activated both in the bridging strand and in other generations could provide more energy dissipation due to their larger contribution to higher generations of the tree. 

Weak cross-linkers can improve the ultimate strength and tear resistance of polymer networks.