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Subsections


Gaussian Accelerated Molecular Dynamics

Gaussian accelerated molecular dynamics (GaMD) [76] is a type of accelerated molecular dynamics (aMD) calculation. It is an enhanced sampling method that works by adding a harmonic boost potential to smoothen the system's potential energy surface. By constructing a boost potential that follows Gaussian distribution, accurate reweighting of the GaMD simulations is achieved using cumulant expansion to the second order.

Please include the following two references in your work using the NAMD implementation of GaMD:

Theoretical background

GaMD enhances conformational sampling of biomolecules by adding a harmonic boost potential to smoothen the system's potential energy surface [76], as illustrated below:

Figure: Schematic illustration of GaMD. When the threshold energy $ E$ is set to the maximum potential ($ iE=1$ mode), the system's potential energy surface is smoothened by adding a harmonic boost potential that follows a Gaussian distribution. The coefficient $ k_0$ , which falls in the range of $ 0 - 1.0$ , determines the magnitude of the applied boost potential.
Image GaMD-scheme

Consider a system with $ N$ atoms at positions $ {\bf r} = \big\{{\bf r}_1,\cdots,{\bf r}_N \}$ . When the system's potential energy $ V({\bf r})$ is lower than a threshold energy $ E$ , the following boost potential is added:

$\displaystyle V^*({\bf r})= V({\bf r}) + \Delta V({\bf r}),$ (85)

where $ \Delta V({\bf r})$ is the boost potential,

$\displaystyle \Delta V({\bf r})= \left \{ \begin{array}{l l} \frac{1}{2} k \lef... ..., & \qquad V({\bf r})<E \\ 0, & \qquad V({\bf r})\geq E. \\ \end{array} \right.$ (86)

where $ k$ is the harmonic force constant.

As explained in reference [76], the two adjustable parameters $ E$ and $ k$ are automatically determined by the following three criteria. First, $ \Delta V$ should not change the relative order of the biased potential values, i.e., for any two arbitrary potential values $ V_1 (\bf r)$ and $ V_2 (\bf r)$ found on the original energy surface, if $ V_1 ({\bf r}) < V_2 ({\bf r})$ , then one should have $ V_1^* ({\bf r}) < V_2^* ({\bf r})$ . Second, the difference between potential energy values on the smoothened energy surface should be smaller than that of the original, i.e., if $ V_1 ({\bf r}) < V_2 ({\bf r})$ , then one should have $ V_2^* ({\bf r}) - V_1^* ({\bf r}) < V_2 ({\bf r}) - V_1 ({\bf r})$ . By combining the above two criteria and plugging in the formula of $ V^*({\bf r})$ and $ \Delta V$ , one obtains

$\displaystyle V_$max$\displaystyle \leq E \leq V_$min$\displaystyle + \frac{1}{k}$ (87)

where $ V_$min and $ V_$max are the system's minimum and maximum potential energies. To ensure that Eqn.(88) is valid, $ k$ needs to satisfy: $ k \leq \frac{1}{ V_\text{max} - V_\text{min} }$ . Define $ k \equiv k_0 \cdot \frac{1}{V_\text{max} - V_\text{min}}$ , then $ 0 < k_0 \leq 1$ . Third, the standard deviation of $ \Delta V$ needs to be small enough (i.e., narrow distribution) to ensure accurate reweighting using cumulant expansion to the second order: $ \sigma_{\Delta V} = k \left( E - V_\text{avg} \right) \sigma_V \leq \sigma_0$ , where $ V_$avg and $ \sigma_V$ are the average and standard deviation of the system's potential energies, $ \sigma_{\Delta V}$ is the standard deviation of $ \Delta V$ , while $ \sigma_0$ is a user-specified upper limit (e.g., $ 10 k_B T$ ) in order to achieve accurate reweighting.

iE = 1 mode: When $ E$ is set to $ E = V_$max according to Eqn.(88), $ k_0$ is calculated as:

$\displaystyle k_0 = \min(1.0, k'_0) = \min \left( 1.0, \frac{\sigma_0}{\sigma_V} \cdot \frac{V_\text{max} - V_\text{min}}{V_\text{max} - V_\text{avg}} \right)$ (88)

iE = 2 mode: Alternatively, when $ E$ is set to $ E = V_$min$ + \frac{1}{k}$ , $ k_0$ is calculated as:

$\displaystyle k_0 = k''_0 \equiv \left( 1 - \frac{\sigma_0}{\sigma_V} \cdot \frac{V_\text{max} - V_\text{min}}{V_\text{avg} - V_\text{min}} \right)$ (89)

If $ k''_0$ obtained from the above equation is smaller than 0 or greater than 1, then $ k_0$ will be calculated using Eqn.(89).

For more details on GaMD and the corresponding reweighting using cumulant expansion, see reference [76][85].

NAMD parameters

Same as aMD, three modes are available for applying boost potential in GaMD: (1) boosting the dihedral energy only, (2) boosting the total potential energy, and (3) boosting both the dihedral and total potential energy (i.e., ``dual-boost").

Some parameters from aMD, including: accelMD, accelMDdihe, accelMDdual, accelMDFirstStep, accelMDLastStep and accelMDOutFreq are shared by GaMD (see Section 11.1 for details). The following is a list of input parameters unique to a GaMD run:

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