# Common Nearest Neighbor Clustering--A Benchmark
## 2 Molecular Dynamics Simulations  
Molecules are not rigid entities but fluctuate between different three-dimensional shapes, called conformations. To understand the properties and functions of complex molecules, such as, for example, proteins, one often needs to have a detailed understanding of their conformational dynamics.  
These conformational changes are difficult to characterize by experiments, but they can be simulated by classical MD simulations [30; 31; 32]. In these simulations, a conformation is represented by a position vector \(\mathbf{x}\in\Omega=\mathbb{R}^{3N}\), where \(N\) is the number of atoms in the system, and each atom is assigned a position in the three-dimensional space. The interactions between the atoms are represented by a predefined potential energy function \(V:\mathbb{R}^{3N}\rightarrow\mathbb{R}\) (Figure 1a). The dynamics are generated by assigning initial velocities to each atom and integrating [PERSON]'s equations of motion numerically on the potential energy surface (PES) \(V(\mathbf{x})\). This yields a trajectory, that is, a sequence of positional vectors \(S=\{\mathbf{x}_{1},\dots\mathbf{x}_{n}\}\), where \(n\) is the number of time steps in the simulations. The trajectory can be visualized as a trace in the conformational space (Figure 1b).  
If the system is coupled to a heat bath by a thermostat algorithm, the trajectory represents a sample of the Boltzmann probability density:  
\[p(\mathbf{x})=\frac{\exp\left(-\frac{1}{k_{B}T}V(\mathbf{x})\right)}{\int_{ \Omega}\exp\left(-\frac{1}{k_{B}T}V(\mathbf{x})\right)\,\mathrm{d}\mathbf{x}} \tag{1}\]  
where \(T\) denotes the absolute temperature, and \(k_{B}\) is the Boltzmann constant (Figure 1c). Equation (1) implies that regions with low energy are sampled much more frequently than regions with high potential energy. Of particular importance are the minima in the potential energy functions, which correspond to regions with a high data-point density that are separated from other frequently sampled regions (other minima) by regions of significantly low data-point density (transition states). In a simulation of the MD, a trajectory will stay for a long time within one minimum before it transitions to another minimum. Thus, the minima do not only correspond to frequently sampled conformations but concomitantly to long-lived conformations (metastable states).  
Figure 3 shows an example: the conformations of the C-type lectin receptor Langerin. Langerin is a mostly rigid protein (gray parts in Figure 3b), but it has a flexible short loop that assumes several distinct conformations (colored parts in Figure 3b). For details of the simulation protocol, see [33]. Figure 3a shows a projection of the Boltzmann distribution on a two-dimensional (2D) reduced space (obtained by principle component analysis of the trajectory). The conformations correspond to highly populated regions in this space. Figure 3a also shows that, in this reduced space, the conformations partially overlap. Knowing the core sets of these regions is thus much more useful for the characterization of the conformations than knowing the exact boundaries. In this case, the gap between the short loop and the long loop can open and close (Figure 3b), and the conformations are distinct states along this reaction coordinate (Figure 3c). Finally, changes in the environment or interactions with other molecules, for example, ligands, can induce shifts in the Boltzmann distribution of a molecule, which are critical to its functions [33; 34; 35]. For the analysis of simulations and the  
elucidation of MD, it is therefore of supreme importance that we can accurately identify the regions of high probability density (core sets).