8.3: Conformational Changes with Temperature
- Page ID
- 294308
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\[F = -k_B T \ln Q \ \ \ \ \ U = \langle E \rangle = k_B T^2 \left (\dfrac{\partial \ln Q}{\partial T} \right )_{V, N} \nonumber\]
\[S = - \left (\dfrac{\partial F}{\partial T} \right )_{V, N} = k_B \ln Q + k_B T \left (\dfrac{\partial \ln Q}{\partial T} \right )_{V, N} \nonumber\]