The umbrella integration algorithm.

## Project description

Umbrella Integration[1] algorithm of calculating PMF using Python.

## Dependence

• Python3
• Numpy
• pandas for parsing metafile
• Scipy for trapz integration

## Usage:

See help:

```python3 ubint.py -h
```

### Input

#### Metafile

The <your-metafile> should be in fellowing form:

```/path/to/your/window/file window_center spring_constant [temperature]
```

There is a variable of T in ubint.py, if the temperature left blank in the metafile, the default temperature would be variable T in the ubint.py, or you can set specific temperature for some window.

#### Data file for each window

The data file of each window need to be a 2-column file with time reaction_coordinate, the coordinate should be 1-dimensional.

### Output

The output file is free_py.txt with 2-column reaction_coordinate free_energy

## Warning

### Unit

I use kJ/mol in this program.

### Spring constant K

In your simulation, the biased spring potential shoud be in form of 0.5 * K * (r - r0) ** 2, here K is the parameter set in your <your-metafile>, for some simulation program, there is no 0.5 in the biased spring potential.

## Screen shots

Raw data was generated by Gaussian distribution for each window with MEAN=window_center and STD=0.8, the centers are in range of 0.0 ~ 19.5 by step of 0.5, here is the result compare with WHAM[2]:

• Raw Data
• Compare with WHAM

The zero point in WHAM is the minimum value and the zero point in UI is 0.

## TO DO

The UI algorithm with higher oder terms[3] of A(xi) is ubint_ho_devel.py, the result is not ideal using previous data, still in development.

Problems occurred at standard normal distributions, maybe the quadruplicate term which even possesses a small value could cause a huge deviation. I should try some systems with non-quadratic potentials.

The function ``exp(-beta(a1*xi+a2*xi^2+a3*xi^3+a4*xi^4))`` and its integration (Normalization factor) give very large value (even inf), this is unable to solve yet.

## Ref

1. Kästner, Johannes, and Walter Thiel. “Bridging the Gap between Thermodynamic Integration and Umbrella Sampling Provides a Novel Analysis Method: ‘Umbrella Integration.’” The Journal of Chemical Physics 123, no. 14 (October 8, 2005): 144104. doi:10.1063/1.2052648.
2. http://membrane.urmc.rochester.edu/content/wham
3. Kästner, Johannes. “Umbrella Integration with Higher-Order Correction Terms.” The Journal of Chemical Physics 136, no. 23 (June 21, 2012): 234102. doi:10.1063/1.4729373.