Problem \#3 (The J-factor connection): Derive the relation between the J-factor and the free energy of cyclization described in class. In particular, construct a lattice model of the cyclization process by imagining a box of \( \Omega \) lattice sites each with volume \( \mathrm{v}(\Omega>>1) \). We consider three species of DNA: monomers with sticky ends that are complementary to each other; dimers, which reflect two monomers sticking together; and DNA circles in which the two ends on the same molecule have stuck together. The number of molecules of each species is \( \mathrm{N}_{1}, \mathrm{~N}_{2} \), and \( \mathrm{N}_{\mathrm{c}} \), respectively.
(a) Use the lattice model to write down the free energy of this assembly and attribute an energy \( \varepsilon_{b} \) to the binding of complementary ends, \( \varepsilon_{\text {loop }} \) to the looped configurations, and \( \varepsilon_{\text {sol }} \) as the energy associated with DNA-solvent interactions when the length of the DNA corresponds to one monomer. Further, assume that the solvent energy for a dimer is \( 2 \varepsilon_{\text {sol }} \) and for a looped configuration is identical to that of a monomer. Use a Lagrange multiplier \( \mu \) (chemical potential for each molecular species) to impose the constraint that \( \mathrm{N}_{\text {tot }}=\mathrm{N}_{1}+2 \mathrm{~N}_{2}+\mathrm{N}_{\mathrm{c}} \).
(b) Minimize the free energy with respect to \( \mathrm{N}_{\mathrm{l}}, \mathrm{N}_{2} \), and \( \mathrm{N}_{\mathrm{c}} \) to find expressions for the concentrations of the three species. Then use the fact that J is the concentration of monomers at which \( \mathrm{N}_{2}=\mathrm{N}_{\mathrm{c}} \) to solve for the unknown chemical potential \( \mu \) and to obtain an expression for the J -factor in terms of the looping free energy.
[Hint: Use \( \mathrm{G}_{\mathrm{tof}}=\Sigma \mathrm{N}_{\mathrm{i}} \varepsilon_{\mathrm{i}} \) (internal energy/binding)- \( \mathrm{T}^{-2} \mathrm{~S}_{\mathrm{i}}-\mu \mathrm{N}_{\text {tot }} \) and to find \( \mathrm{S}_{\mathrm{i}} \) for each molecular species, refer to the class slides on Entropy].
