pyemma.thermo.wham¶
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pyemma.thermo.wham(ttrajs, dtrajs, bias, maxiter=100000, maxerr=1e-15, save_convergence_info=0, dt_traj='1 step')¶ Weighted histogram analysis method
Parameters: - ttrajs (numpy.ndarray(T) of int, or list of numpy.ndarray(T_i) of int) – A single discrete trajectory or a list of discrete trajectories. The integers are indexes in 0,...,num_therm_states-1 enumerating the thermodynamic states the trajectory is in at any time.
- dtrajs (numpy.ndarray(T) of int, or list of numpy.ndarray(T_i) of int) – A single discrete trajectory or a list of discrete trajectories. The integers are indexes in 0,...,num_conf_states-1 enumerating the num_conf_states Markov states or the bins the trajectory is in at any time.
- bias (numpy.ndarray(shape=(num_therm_states, num_conf_states)) object) – bias_energies_full[j, i] is the bias energy in units of kT for each discrete state i at thermodynamic state j.
- maxiter (int, optional, default=10000) – The maximum number of dTRAM iterations before the estimator exits unsuccessfully.
- maxerr (float, optional, default=1e-15) – Convergence criterion based on the maximal free energy change in a self-consistent iteration step.
- save_convergence_info (int, optional, default=0) – Every save_convergence_info iteration steps, store the actual increment and the actual loglikelihood; 0 means no storage.
- dt_traj (str, optional, default='1 step') –
Description of the physical time corresponding to the lag. May be used by analysis algorithms such as plotting tools to pretty-print the axes. By default ‘1 step’, i.e. there is no physical time unit. Specify by a number, whitespace and unit. Permitted units are (* is an arbitrary string):
‘fs’, ‘femtosecond*’‘ps’, ‘picosecond*’‘ns’, ‘nanosecond*’‘us’, ‘microsecond*’‘ms’, ‘millisecond*’‘s’, ‘second*’
Returns: wham_estimator – A stationary model which consists of thermodynamic quantities at all temperatures/thermodynamic states.
Return type: Example
Umbrella sampling: Suppose we simulate in K umbrellas, centered at positions \(y_0,...,y_{K-1}\) with bias energies
\[b_k(x) = \frac{c_k}{2 \textrm{kT}} \cdot (x - y_k)^2\]Suppose we have one simulation of length T in each umbrella, and they are ordered from 0 to K-1. We have discretized the x-coordinate into 100 bins. Then dtrajs and ttrajs should each be a list of \(K\) arrays. dtrajs would look for example like this:
[ (0, 0, 0, 0, 1, 1, 1, 0, 0, 0, ...), (0, 1, 0, 1, 0, 1, 1, 0, 0, 1, ...), ... ]
where each array has length T, and is the sequence of bins (in the range 0 to 99) visited along the trajectory. ttrajs would look like this:
[ (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...), (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...), ... ]
Because trajectory 1 stays in umbrella 1 (index 0), trajectory 2 stays in umbrella 2 (index 1), and so forth. bias is a \(K \times n\) matrix with all reduced bias energies evaluated at all centers:
\[\begin{split}\left(\begin{array}{cccc} b_0(y_0) & b_0(y_1) & ... & b_0(y_{n-1}) \\ b_1(y_0) & b_1(y_1) & ... & b_1(y_{n-1}) \\ ... \\ b_{K-1}(y_0) & b_{K-1}(y_1) & ... & b_{K-1}(y_{n-1}) \end{array}\right)\end{split}\]Let us try the above example:
>>> from pyemma.thermo import wham >>> import numpy as np >>> ttrajs = [np.array([0,0,0,0,0,0,0,0,0,0]), np.array([1,1,1,1,1,1,1,1,1,1])] >>> dtrajs = [np.array([0,0,0,0,1,1,1,0,0,0]), np.array([0,1,0,1,0,1,1,0,0,1])] >>> bias = np.array([[0.0, 0.0], [0.5, 1.0]]) >>> wham_obj = wham(ttrajs, dtrajs, bias) >>> wham_obj.log_likelihood() -6.6... >>> wham_obj.state_counts array([[7, 3], [5, 5]]) >>> wham_obj.stationary_distribution array([ 0.5..., 0.4...])
References
[1] Ferrenberg, A.M. and Swensen, R.H. 1988. New Monte Carlo Technique for Studying Phase Transitions. Phys. Rev. Lett. 23, 2635–2638 [2] Kumar, S. et al 1992. The Weighted Histogram Analysis Method for Free-Energy Calculations on Biomolecules. I. The Method. J. Comp. Chem. 13, 1011–1021