Statistical Mechanics of Binding

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Let us now calculate the binding probability of a TF to a binding site on the DNA as a function of the TF concentrations. We assume that the system is in thermodynamic equilibrium and use the framework of statistical mechanics. To this end, consider the more detailed view on the events that determine the kinetics we discussed above. Namely we are going to have a look on TF binding to TF binding site in the light of equilibrium thermodynamics, that will then lead us to the description of RNA polymerase binding to DNA. For this purpose, suppose a box filled with multiple molecules and a receptor.

Illustration of the two different macro states in the "lattice model" (i) the binding site ether does bind a TF or (ii) does not bind a TF

Assume, that there are $L$ different TFs in the box. Partition the box in $\Omega$ rectangular segments, so that every TF can be assigned to one rectangle. Each way of assembling the $L$ TFs in the $\Omega$ rectangles is a microstate. There are \begin{equation} \frac{\Omega !}{L!(\Omega -L)!} \end{equation}
different microstates. The probability that the system is in one of the two states (i) or (ii) depends on the associated energies of the state. Let $\varepsilon_{b}$ be the energy of the bound state (ii) and $\varepsilon_{u}$ the energy of the unbound state (i). We can now formulate the partition function for the system.
\begin{equation} Z(L,\Omega)=\left( \sum_{unbound} e^{-\beta L \varepsilon_u}\right) + \left( e^{-\beta \varepsilon_b} \sum_{bound} e^{-\beta (L-1) \varepsilon_u}\right) \end{equation}
There are \begin{equation} \frac{\Omega !}{L!(\Omega -L)!}\simeq \frac{\Omega^L}{L!} \end{equation}
different microstates with the same energy $\varepsilon_{u}$ in state (ii) and
\begin{equation} \frac{\Omega !}{(L-1)!(\Omega -(L+1))}\simeq \frac{\Omega^{(L-1)}}{(L-1)!} \end{equation}
different microstates with energy $\varepsilon_{u}$ and one microstate with energy $\varepsilon_{b}$ in macrostate (i). Summing over all microstates we can write the partition function
\begin{equation} Z(L,\Omega)=\left( \frac{\Omega^L}{L!} e^{-\beta L \varepsilon_u}\right) + \left( \frac{\Omega^{(L-1)}}{(L-1)!} e^{-\beta (L-1) \varepsilon_u} e^{-\beta \varepsilon_b} \right) \end{equation} The probability, that the TF binds the binding site can be found by dividing the sum of microstates belonging to the macrostate of interest (in our case state (i) - TF bound) by the the partition function \begin{equation} p_b=\frac{ \frac{\Omega^{(L-1)}}{(L-1)!} e^{-\beta [(L-1) \varepsilon_u + \varepsilon_b]}} { \frac{\Omega^L}{L!} e^{-\beta L \varepsilon_u} + \frac{\Omega^{(L-1)}}{(L-1)!} e^{-\beta [(L-1) \varepsilon_u + \varepsilon_b]}} = \frac{(L/\Omega)e^{- \beta \Delta \varepsilon}}{1+(L/\Omega)e^{- \beta \Delta \varepsilon}} \end{equation} You get the last term after multiplying by $(L!/\Omega ^L)e^{\beta L \varepsilon_u}$ and replacing $\varepsilon_b - \varepsilon_u$ by $\Delta \varepsilon$. We now introduce the TF concentration $c=L/V_{box}$ and the reference concentration $c_0=\Omega/V_{box}$ that corresponds to the case, were all sites in the lattice are occupied. Rewriting yields

\begin{equation} p_b=\frac{(c/c_0)e^{-\beta \Delta \varepsilon}}{1+(c/c_0)e^{-\beta \Delta \varepsilon}} \tag{1} \end{equation}

this is the celebrated Hill function

The graphic shows the binding probability of the ligand (concentration in $\mu M$) to the promoter for three energy differences $\Delta \varepsilon$. The bigger the energy difference $\Delta \varepsilon$, the bigger is the slope (in the limit $\underset{c \rightarrow 0}[[:Template:Lim]]\hspace{0.1cm} p_b'(c)$) of the graph. The reference concentration was set to $c_0=0.6M$ that corresponds to a lattice with elementary boxes of $\sim 1nm^3$. [Garcia, Phillips - Thermodynamics of Biological Processes]

The Hill equation can also be derived by use of calculus of equilibrium. Having in mind the following reaction (R: binding site; L: transcription factor) \begin{equation} n \cdot L+R \overset{ k_f}{\underset{ k_b }{ \rightleftharpoons}} LR \end{equation} with the rate constant $k_f$ (forward reaction) and $k_b$ (backward reaction) the dynamics are described by the ODE \begin{equation} \frac{d[LR]}{dt} = k_f \cdot [L]^n[R] - k_b [L_nR] \end{equation} in equilibrium ($\frac{d[LR]}{dt} =0$) we get \begin{equation} \frac{k_b}{k_f}=(K_d)^n= \frac{[L]^n[R]}{[L_nR]} \tag{2} \end{equation} The probability $p_b$ for the receptor to be bound, is the ratio between the concentration of ligand-bound receptors and the total receptor concentration. \begin{equation} p_b=\frac{[L_nR]}{[R]+[L_nR]} \end{equation} replacing $[L_nR]$ with $\frac{[L]^n[R]}{K_d^n}$ (from equ. (2)) and canceling we get the familiar result for $n=1$ \begin{equation} p_b=\frac{([L]/K_d)^n}{1+([L]/K_d)^n} \end{equation} The power $n$ is known as the Hill coefficient and describes cooperativity effects between TFs. Comparing the last equ. with equ. (1), we find the following relation between the dissociation constant $K_d$ and the microscopic parameters \begin{equation} K_d=\frac{e^{\beta \Delta \varepsilon}}{V_{\Omega}} \end{equation} $V_{\Omega}$ represents the volume of a single lattice box.

The concept of RNA polymerase binding to DNA is similar to that of receptor-ligand binding. The promoter plays the role of the receptor and RNA polymerase takes the place of the ligand. The RNA polymerase can also bind to other non specific DNA sites than the promoter. Despite free ligands in solution, we now consider the RNA polymerases bound to the DNA. This is a simplification justified by experiments on DNA-free mini-cells, that contain no copy of the genome and rarely contain any polymerases. Imagine $N$ non specific binding sites and $P$ RNA polymerase molecules, then there are \begin{equation} \frac{N !}{P!(N -P)!} \sim \frac{N^P}{P!} \tag{3} \end{equation} ways of arranging the molecules on the non specific binding sites. Or likewise \begin{equation} \frac{N !}{(P-1)!(N -(P-1))!}= \frac{N^{P-1}}{(P-1)!} \tag{4} \end{equation} arrangements if one polymerase is bound to the promoter. Knowledge of the associated energies $\varepsilon_S$ (Energy for specific binding on the promoter) and $\varepsilon_N$ (Energy for non specific binding to DNA), enables us to write down the partition function. \begin{equation} Z(P,N)= \frac{N !}{P!(N -P)!} e^{- \beta P \varepsilon _N} + \frac{N !}{(P-1)!(N -(P-1))!} e^{- \beta (P-1) \varepsilon _N} e^{- \beta \varepsilon _S} \end{equation}

Illustration: Binding of RNA polymerase (i) to non specific binding site (ii) to promoter. Weights: Multiplicity x Boltzmann Weight. We consider only one promoter, $N$ non specific binding sites and $P$ RNA polymerases. Energy for specific binding on the promoter: $\varepsilon_S$; Energy for non specific binding to DNA: $\varepsilon_N$

The general formula for calculating the probability of being in the $i$th marcrostate is \begin{equation} p_i=\frac{1}{Z} \cdot \Omega _i(N,P) \cdot e^{- \beta E_i} \end{equation} It is convenient to name the states belonging to the same energy level $E_i$, thus we denote the degeneracy (or multiplicity) of the energy level $E_i$ by $\Omega _i(N,P)$. Using this formula applying the approximation from equ. (3) and equ. (4) for $P \ll N$ and multiplying by $(P! / N^P)e^{\beta P \varepsilon_N}$ we find for the binding probability \begin{equation} p_b= \frac{\frac{P}{N}e^{- \beta \Delta \varepsilon}}{1+\frac{P}{N}e^{- \beta \Delta \varepsilon}}=\frac{1}{1+ \frac{N}{P} e^{ \beta \Delta \varepsilon}} \end{equation}
were $\varepsilon_S - \varepsilon_N$ was replaced by $\Delta \varepsilon$.

Further Reading:

  • Uri Alon - An Introduction to Systems Biology
  • Rob Phillips et. al. - Physical Biology of the Cell
  • Ken A. Dill und Sarina Bromberg - Molecular Driving Forces

  • Bintu, L. et al. - Transcriptional regulation by the numbers
  • Garcia, H. G., Kondev, J., Orme, N., Theriot, J. A. $\&$ Phillips, R. - Thermodynamics of biological processes