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Comparative Analysis of Free Energy Calculation Methods

This document provides a detailed mathematical comparison of free energy calculation and interpolation methods between the custom Python script freeEnergyLandscape.py and the method of GROMACS tools gmx_sham and gmx_wham.

GMX SHAM Mechanism Description

The gmx_sham tool is part of the GROMACS suite, designed to analyze free energy landscapes from simulation data. The core functionality revolves around creating multidimensional histograms from the provided data and calculating the free energy landscape. Here's a detailed breakdown of the process, including the improved explanation of "accumulated probability of all data points":

Data Preparation

  1. Extremes Determination: For each eigenvector, the code calculates minimum and maximum values across all data points, adding a small buffer to ensure all data is encompassed (find_extremes function).

  2. Normalization and Volume Correction: Data undergo normalization and volume correction if necessary, based on user-specified dimensions. This step adjusts the probability calculations for spatial dimensions, correcting for volume effects that increase with distance in 2D and 3D spaces, as seen in the correct_for_volume_effects method.

Histogram Creation and Free Energy Calculation

  1. Binning: Data is sorted into multidimensional bins, each representing specific intervals of the eigenvectors (binning_data function). Points are allocated to bins based on their eigenvector values.

  2. Accumulated Probability Calculation: The accumulated probability ($P_{acum}(bin)$) for each bin is calculated as the number of data points within the bin ($(N_{bin})$) divided by the total number of data points ($(N_{total})$), reflecting the proportion of occurrences for each bin relative to the entire dataset.

$$P_{acum}(bin) = \frac{N_{bin}}{N_{total}}$$

  1. Free Energy Calculation: Free energy ($G$) for each bin is determined using the inverse Boltzmann relation, where the accumulated probability informs the energy level:

$$G(bin) = -kT \ln(P_{acum}(bin))$$

Here, ($k$) is Boltzmann's constant, and ($T$) is the system's temperature. This step is executed within the calculate_free_energy method.

  1. Probability Normalization and Energy Adjustment: Finally, the probability in each bin is normalized, and the free energy values are adjusted so the minimum free energy across the landscape is set to zero, facilitating easier interpretation and visualization.

GMX WHAM Mechanism Description

The gmx_wham tool within the GROMACS suite utilizes the Weighted Histogram Analysis Method (WHAM) to derive free energy landscapes from multiple simulations. This sophisticated statistical approach combines data from various histograms to estimate the system's free energy landscape accurately. Additionally, gmx_wham employs linear interpolation for handling tabulated potentials. Below is a detailed explanation of WHAM's process, incorporating the clarification regarding interpolation methods:

Data Preparation and Histogram Creation

  1. Histogram Generation: Each simulation dataset contributes a histogram based on a specific reaction coordinate, representing the frequency distribution of system states across the sampled parameter space.

  2. Bias Correction: To effectively explore specific regions of the parameter space, simulations often apply a bias. WHAM corrects these biases across all histograms, ensuring equitable contributions to the final analysis.

Statistical Combination and Free Energy Calculation

WHAM utilizes a statistical method to combine histograms and derive the free energy landscape:

  1. Combining Histograms: The method combines biased histograms using iteratively adjusted weights. Each histogram's weight reflects its contribution to the overall free energy calculation, considering the applied bias during simulation.

  2. Iterative Solution: WHAM solves a set of self-consistent equations to find the weights that maximize the likelihood of the combined histogram data:

    • Let $N_i(j)$ be the number of counts in the $j^{th}$ bin of the $i^{th}$ histogram, with $n_i$ total observations and a biasing energy $U_i(j)$.
    • The unbiased probability distribution $P(j)$ for the $j^{th}$ bin is estimated by:

$$P(j) = \frac{\sum_{i} N_i(j)}{\sum_{i} n_i \exp\left[-\beta (U_i(j) - F_i)\right]}$$

Here, $\beta = 1/(k_BT)$, $k_B$ is Boltzmann's constant, $T$ is the temperature, and $F_i$ is the iteratively adjusted free energy of the $i^{th}$ simulation.

  1. Free Energy Landscape: The free energy $G$ for each state is calculated from the probability distribution $P(j)$:

$$G(j) = -k_BT \ln(P(j))$$

This reveals the energetically favorable states and barriers between them.

Linear Interpolation for Tabulated Potentials

In addition to the statistical combination of histograms, gmx_wham uses linear interpolation within the tabulated_pot function for tabulated potentials. This method estimates potential energy values at intermediate distances, providing a continuous potential energy landscape.

However, this linear interpolation method is specifically applied to the scenario of dealing with tabulated potentials and does not directly influence the primary WHAM algorithm's statistical combination of histograms for free energy calculation. The WHAM methodology itself does not inherently use linear interpolation as part of its core algorithm for combining histograms or calculating the free energy landscape. Instead, WHAM relies on a statistical approach to optimally combine data from multiple biased simulations to reconstruct the unbiased free energy profile.

Free Energy Landscape Calculation with Python Script

This document outlines the use of Kernel Density Estimation (KDE) and Boltzmann Inversion by the freeEnergyLandscape.py script to estimate the free energy landscape of a system from its collective variables.

Kernel Density Estimation (KDE) Method

KDE smooths the distribution of collective variables to enhance the estimation of probability densities. It employs Scott's rule for bandwidth selection, ensuring an appropriately smooth density estimate for the data's scale.

$$ \text{Bandwidth (Scott's Rule)} = \sigma \cdot n^{-1/5} $$

  • $\sigma$ is the standard deviation of the sample.
  • $n$ is the sample size.

KDE for a set of points is given by:

$$ \hat{f}(x) = \frac{1}{n \cdot h} \sum_{i=1}^{n} K\left( \frac{x - x_i}{h} \right) $$

  • $n$ is the number of points
  • $K$ is the kernel function, typically Gaussian.
  • $h$ is the bandwidth.

Boltzmann Inversion

The free energy $G(x)$ of a state is calculated from the probability density $\hat{f}(x)$, obtained via KDE:

$$ G(x) = -k_BT \ln(\hat{f}(x)) $$

  • $k_B$ is the Boltzmann constant.
  • $T$ is the temperature.

Probability Normalization

To ensure accurate energy calculation, the probability densities are normalized over the entire range of possible values, ensuring they sum to one across the configurational space:

$$ \int_{-\infty}^{\infty} \hat{f}(x) , dx = 1 $$

Interpolation

The smoothly interpolated probability density function provided by KDE enables calculating the free energy landscape over a continuous range of collective variable values, highlighting energetically favorable states and barriers between them.

Conclusion

In conclusion, while gmx_sham and gmx_wham are powerful tools for free energy analysis within the GROMACS environment, freeEnergyLandscape.py offers a more user-friendly and customizable approach, particularly beneficial for those seeking immediate visualization and flexible data analysis options.