AdaptiveBandit, Methods, MD Simulations

MD Simulations

• The Configurational space of a Molecular System for MD Simulations is given by $\chi =\{x=(r_1,...r_n)\in {\mathbb{R}}^{3N}\}$, where $N$ is the number of atoms of the system
• Experimental observables $O$ are measured as equilibrium expectations $<O>=\int O(x)\mu (x)dx$, where $\mu (x)$ is the equilibrium distribution
• the form of this distribution is known
  • for instancem the Boltzmann distribution in the canonical ensemble at temperature $T$ is

$$\mu (x)=\frac{1}{Z}\exp -\frac{U(x)}{k_{B}T}Eq.1$$

$Eq.1$ Parameters

where

• $U(x)$ is the molecular potential energy
• $k_{B}T$ is the Boltzmann constant multiplied by the Temperature,
• and $Z$ is the normalization factor

• MD Numerically solves Newton’s eq. over the potential $U(x)$ for the variable $x$, plus a Langevin stochastic term accounting for thermal fluctuations
• Now consider the state $x(t)\in \chi$ as a specific conformation inside the configurational space $\chi$ at time $t$
• the probability of finding the molecule in configuration $x_{t+\tau }$ at a later time can be expressed by the conditional transition density function $p_{\tau }$, $x_{t+\tau }\sim p_{\tau }(x_{t+\tau }|x_t)$, which describes the probability of finding state $x_{t+\tau }$ given state $x_t$ at time $t$ after a time increment $\tau$

When performing an MD simulation,

• the dynamics of the molecular system propagates the state $x_t$ across time
• Therefore, MD samples from the transition density $p_{\tau }$ given discrete time steps $\tau$ to obtain the next state $x_{t+\tau }$
• this process is repeated for many steps, generating a trajectory of conformations

The main goal of performing MD simulations

• is to obtain a good representation of the system’s equilibrium distribution $\mu (x)$, i.e., the probability to find conformation $x$ under equilibtrium conditions, to measure the average of observable $<O>$
• if an MD trajectory $\tau$ is long enough, sampling from $p_{\tau }$ is equivalent to sampling from $\mu (x)$

$$\underset{\tau \to \infty }{\lim }{p}_{\tau }(x_{t+\tau }|x_t)=\mu (x)Eq.2$$

• generating long enough trajectories is computationally expensive, and often practically impossible when trying to sample slow events
• However, long trajectories can be substituted by short parallel trajectories
• while in principle one could model directly the conditional probability in $E1.2$, in practice this is not possible due to the very high dimensional space
• fortunately, it can be shown that the dynamics can be separated into a slow and fast set of variables, and b/c the contributions of fast variables decay exponentially in $\tau$, a reliable MSM can be constructed in terms of the slow variables to compute the thermodynamic averages
• usually, time-independent component analysis (tICA) and clustering methods are used to study this set of variable during sampling, which is necessary to build the MSM
• one we obtain the MSM, computed by estimating the transition probabilities from discrete conformational states, one can derive the thermodynamics and kinetic properties, just assuming local, not global, equilibrium (i.e., $\tau$ is much shorter than what is necessary to satisfy $Eq.2$)