8.3: Conformational Changes with Temperature
- Page ID
- 294308
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Four bead polymer on a two‐dimensional lattice
Place polymer on lattice \(z = 4\) \(n = 4\) in 2D (with distinguishable end beads):
Configurational Partition Function
Number of thermally accessible microstates.
\[\begin{align*} Q & = (q_{conf})^N \\[4pt] q_{conf} & = \underbrace{\sum_{i\ states = 1}^9 e^{-E_i/kT}}_{\text{sum over microstates}} \\[4pt] &= \underbrace{\sum_{j\ levels = 1}^2 g_j e^{-E_j/kT}}_{\text{sum over energy levels}} \\[4pt] & = 2 + 7e^{-\varepsilon /kT} \end{align*} \]
Probability of Being "Folded"
Fraction of molecules in the folded state
\[P_{fold} = \dfrac{g_{fold} e^{-E_{fold}/kT}}{q_{conf}} = \dfrac{2}{2 + 7e^{-\varepsilon /kT}} \nonumber\]
Mean End-to-End Distance
\[\begin{array} {rcl} {\langle r_{ee} \rangle } & = & {\sum_{i = 1}^{9} \dfrac{r_i e^{-E_i/kT}}{q_{conf}}} \\ {} & = & {\dfrac{(1)(2) + (\sqrt{5}) 6e^{-\varepsilon /kT} + 3e^{-\varepsilon /kT}}{q_{conf}}} \\ {} & = & {\dfrac{2 + (6\sqrt{5} + 3) e^{-\varepsilon /kT}}{q_{conf}}} \end{array} \nonumber\]
Also, we can access other thermodynamic quantities:
\[F = -k_B T \ln Q \ \ \ \ \ U = \langle E \rangle = k_B T^2 \left (\dfrac{\partial \ln Q}{\partial T} \right )_{V, N} \nonumber\]
\[S = - \left (\dfrac{\partial F}{\partial T} \right )_{V, N} = k_B \ln Q + k_B T \left (\dfrac{\partial \ln Q}{\partial T} \right )_{V, N} \nonumber\]