Question

Derive the relation between the \(J\) -factor and the free energy of cyclization described in Section \(10.3 .3 .\) In par. ticular, construct a lattice model of the cyclization pro. cess by Imagining a box of \(\Omega\) lattice sites each with volume \(v,\) 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 \(N_{1}, N_{2}\) and \(N_{c},\) respectively. (a) Use the lattice model to write down the free energy of this assembly and attribute an energy \(\&_{b}\) to the binding of complementary ends, reap to the looped configurations, and \(r_{\text {sol }}\) as the energy associated with DNA-solvent inter actions when the length of the DNA corresponds to one monomer. Further, assume that the solvent energy for a dimer is \(2 e_{\text {sal}}\) and for a looped configuration is identical to that of a monomer. Use a Lagrange multiplier \(\mu\) to impose the constraint that \(N_{\text {tot }}=N_{1}+2 N_{2}+N_{c}\) (b) Minimize the free energy with respect to \(N_{1}, \mathrm{N}_{2},\) and \(N_{c}\) to find expressions for the concentrations of the three species. Then use the fact that 1 is the concentration of monomers at which \(N_{2}=N_{c}\) to solve for the unknown Lagrange multiplier \(\mu\) and to obtain an expression for the J-factor in terms of the looping free energy.

Short answer

The free energy of the assembly is given by: \(F = \mu N_{tot} - r_{sol} (N_{1} + 2N_{2} + N_{c}) - \&_{b} (N_{2} + N_{c}) + kT [N_{1} \log(\Omega /N_{1}) + N_{2}\log(\Omega /N_{2}) + N_{c}\log(\Omega /N_{c})]\). Minimization of free energy gives expressions for concentrations of the three species. The unknown Lagrange multiplier \(\mu\) is then obtained, and the concentrations are used to derive the J-factor in terms of the looping free energy.

Step-by-step solution

Establish the free energy of the assembly

In the lattice model, the volume is filled with a given amount of DNA in the different species. The free energy of the assembly can therefore be written as: \[F = \mu N_{tot} - r_{sol} (N_{1} + 2N_{2} + N_{c}) - \&_{b} (N_{2} + N_{c}) + kT [N_{1} \log(\Omega /N_{1}) + N_{2}\log(\Omega /N_{2}) + N_{c}\log(\Omega /N_{c})]\] where \(kT\) is the term of entropy and \(r_{sol}\) refers to the energy associated with DNA-solvent interactions for one monomer molecule.

Minimize the free energy with respect to the number of molecules of each species

To find the concentrations of the three species, we need to minimize the free energy with respect to \(N_{1}, N_{2},\) and \(N_{c}\). This is achieved by differentiating our free energy equation with respect to all species and set equal to zero, and then solving for \(N_{1}, N_{2},\) and \(N_{c}\), respectively.

Solve for the unknown Lagrange multiplier and derive the J-factor expression

Given that 1 is the concentration of monomers at which \(N_{2} = N_{c}\), we can solve our equations for \(\mu\) and substitute it into our expressions for \(N_{2}\) and \(N_{c}\) to obtain the expressions for these species in terms of \(\mu\). Next, substitute this value into the J-factor equation: \(J = [c]N_{c}/N_{1}\) to derive the J-factor in terms of the looping free energy.

Key concepts

Lattice Model of Cyclization

The lattice model of cyclization is an essential framework used to understand the behavior of polymers like DNA when they form loops. This model assumes that the polymer chain occupies discrete sites on a lattice, which represents the possible locations the polymer can take in space. By considering the polymer as moving through a grid of points or 'lattice sites', we can calculate its behavior using statistical mechanics principles.

In the context of DNA cyclization, the lattice model helps us visualize and quantify how the chain can bend and its ends meet to form a loop. Imagine a box filled with a number of these lattice sites, each representing a volume a segment of DNA might occupy. We then introduce different species of DNA molecules: single strands ('monomers') with sticky ends, which can potentially bind together; pairs of these strands bound at their sticky ends ('dimers'); and DNA molecules whose ends have closed to form loops('DNA circles').

We attribute an energy \(e_b\) to the binding of complementary sticky ends and an energy \(r_{sol}\) to the interaction of the DNA with the surrounding solvent. These energies are critical for understanding the statistical mechanics of the assembly, as they will influence the likelihood of various configurations of DNA molecules within the lattice. The model allows researchers to predict the concentrations of each DNA species by relating them to the free energy of the entire system, a principle vital to molecular biology and the understanding of cellular processes.

Free Energy in Biological Systems

Free energy is a term used to describe the usable energy in a system that can be converted into work at a constant temperature and pressure. In biological systems, like within a cell, it’s fundamental to understanding how various molecular interactions occur, including DNA cyclization.

The free energy associated with the different species of DNA in our lattice model includes contributions from enthalpy, which accounts for interactions like the binding of DNA ends, and entropy, which considers the disorder or randomness of the system. The balance of these factors determines the stability and reactivity of molecular structures such as DNA loops. For example, a high free energy might suggest that the looped DNA structure is less stable or less likely to form under given conditions.

To quantify this, we consider the free energy equation given in the step-by-step solution. The equation combines both entropic terms (\(kT\)) - which incorporates the molecular distribution among the lattice sites, and the enthalpic contributions - like the energies from binding and solvent interactions. Minimizing the free energy of the system leads to the most probable distribution of DNA species, such as monomers, dimers, and looped configurations, which are crucial for the biological function of DNA.

J-Factor Calculation

The J-factor is a quantitative measure used to assess the probability of DNA loop formation. In the context of the lattice model for cyclization, it is a critical parameter representing the likelihood of a DNA molecule to spontaneously close into a circle. To calculate the J-factor, we utilize the free energy considerations for DNA loops and the concentration of DNA species within the lattice.

The relationship between the J-factor and the concentration of monomers, dimers, and looped DNA can be understood by setting the concentration of monomers at which the number of dimers equals the number of loops. This relationship allows us to solve for the unknown Lagrange multiplier, which maintains the constraint of constant total number of DNA segments within the lattice model. With the Lagrange multiplier found, an expression for the J-factor can be derived in terms of the looping free energy, as illustrated in the step-by-step solution.

In practical terms, the J-factor can help biologists and chemists understand how factors like the length of the DNA, the stiffness of the chain, and environmental conditions, such as the presence of certain ions or molecules, affect the looping dynamics of DNA. Ultimately, calculating the J-factor from the free energy provides insights into the mechanisms of gene regulation, as DNA looping plays a crucial role in how genes are expressed within cells.