time¶

hgana.affinity.time(data_link, T, V, dt, len_frame=1, is_std=False)¶

This function calculates the binding affinity \(\Delta G_T\). This is done by determining the association rate constant \(k_\text{On}\) and dissociation rate constant \(k_\text{Off}\)

\[\begin{array}{cc} k_\text{On}=\dfrac{1}{\langle t_u\rangle\cdot C_g},& k_\text{Off}=\dfrac{1}{\langle t_b\rangle} \end{array}\]

with solute concentration \(C_g\) of the free state in the complex

\[C_g=\frac{N}{V},\]

number of solute molecules \(N\), box volume \(V\), average bound time \(\langle t_b\rangle\) and average unbound time \(\langle t_u\rangle\). The two averages are calculated by determining the time of the host molecule being in a bound state before changing to an unbound one and vice versa. The resulting bound \(t_b\) and unbound \(t_u\) time instances, with instance quantities \(M_b\) and \(M_u\), are then filtered by a minimal dwelling time \(c_{min}\) resulting in the time average by normalizing the sum of all instances by their count

\[\begin{array}{cc} \langle t_b\rangle=\dfrac1{M_b}\sum_{i=1}^{M_b}t_{b,i},& \langle t_u\rangle=\dfrac1{M_u}\sum_{i=1}^{M_u}t_{u,i}. \end{array}\]

Finally the binding affinity results from

\[\begin{split}\Delta G_T&=-RT\ln\frac{k_\text{On}C_0}{k_\text{Off}} =-RT\ln\frac{\langle t_b\rangle}{\langle t_u\rangle}-RT\ln\frac{C_0}{C_g}\\ &=-RT\ln\frac{\langle t_b\rangle}{\langle t_u\rangle}-RT\ln\frac{V}{NV_0}\end{split}\]

with standard gas constant \(R\), temperature \(T\), standard state concentration \(C_0=1\frac{\text{mol}}{\text{l}}=V_0^{-1}\) and standard state volume \(V_0=1.661nm\).

The standard deviation can be determined by creating permutations containing a percentage of elements from the time arrays \(t_{b,i}\) and \(t_{u,j}\)

\[\begin{array}{cc} \boldsymbol{P}_u=\sum_k^{N_p}p_{u,k},& p_{u,k}=[t_{u,0},\dots,t_{u,N_u-x}] \end{array}\]

with permutation matrix \(\boldsymbol{P}\), permutation \(p\), number of permutations \(N_p\) and element percentage \(1-x\). This permutation matrix is then used to calculate multiple association and dissociation rates creating a pool of values, of which the standard deviation can be determined using

\[\text{std}(k_\text{On})=\sqrt{\frac{\sum_{k=1}^{N_p}\left(k_{\text{On},k}-\bar k_\text{On}\right)^2}{N_p-1}}\]

with mean value \(\bar k_\text{On}\) of the association rates calculated from the permutation matrix. Similarly, the standard deviation of the binding affinity can be determined by

\[\text{std}(\Delta G_T)=\sqrt{\frac{\sum_{k=1}^{N_p}\left(\Delta G_T^k-\Delta\bar G_T\right)^2}{N_p-1}}\]

with mean value \(\Delta\bar G_T\).

Parameters:	data_link : string Sampled affinity object T : float Simulated temperature in \(\text{K}\) V : float Simulation box volume in \(\text{m}^3\) dt : integer Time calculation cut-off in ps len_frame : float, optional Length of a frame in ps is_std : bool, optional True to calculate standard deviation
Returns:	table : DataFrame Pandas DataFrame of binding affinity in \(\frac{\text{kJ}}{\text{mol}}\) and \(\frac{\text{kcal}}{\text{mol}}\), \(k_\text{On}\) in \(\frac{1}{\text{s}}\) and \(k_\text{Off}\) in \(\frac{\text{dm}^3}{\text{mol}\cdot\text{s}}\), and optionally the standard deviations of these values