Upon ligand binding or during chemical reactions the state of a molecular system changes in time. Usually we consider a finite set of (macro-) states of the system (e.g., 'bound' vs. 'unbound'), although the process itself takes place in a continuous space. In this context, the formula χ = XA connects the micro-dynamics of the molecular system to its macro-dynamics. χ can be understood as a clustering of micro-states of a molecular system into a few macro-states. X is a basis of an invariant subspace of a transfer operator describing the micro-dynamics of the system. The formula claims that there is an unknown linear relation A between these two objects. With the aid of this formula we can understand rebinding effects, the electron flux in pericyclic reactions, and systematic changes of binding rates in kinetic ITC experiments. We can also analyze sequential spectroscopy experiments and rare event systems more easily. This article provides an explanation of the formula and an overview of some of its consequences.Keywords: Robust Perron Cluster Analysis; membership function; invariant subspace.
MSC: 60J22
The Foundation of PCCA+Molecular simulation is used as a tool to estimate thermodynamical properties. Moreover, it could be used for investigation of molecular processes and their time scales. In computational drug design, e.g., binding affinities of ligand-receptor systems are estimated. In this case, trajectories in a 3N-dimensional space Ω of molecular conformations (N is the number of atoms) are generated. They will be denoted as micro-states x ∈ Ω in the following. After simulation the micro-states are classified as "bound" or "unbound". This classification can be achieved by projecting the state space Ω onto a low number n of macro-states of the system. Spectral clustering algorithms are one possible method in this context. Robust Perron Cluster Analysis (PCCA+) turned out to be a commonly used practical tool for this algorithmic step [1].Conformation Dynamics has been established in the late 1990s. From molecular simulations [2] transition matrices P ∈ R m×m were derived. The entries of P represent conditional transition probabilities between pre-defined m subsets of the state space Ω of a molecular process. Today P would be denoted as Markov State Model. According to the transfer operator theory and PCCA [3] the eigenvectors of these matrices were studied in the early 2000s leading to an empirical observation [4] and Figure 1: Given a matrix X ∈ R m×n of the n m leading eigenvectors of the m × m-transition matrix P, the m rows of X -plotted as n-dimensional points -seem to always form an (n − 1)-simplex. Some of those m points are located at the vertices of the simplices, other points are located on the edges. The points which are located at the n vertices of these simplices represent the thermodynamically stable conformations of the molecular systems. Whereas, the points which are located on the edges of the simplices represent transition regions. In order to transform the (n − 1)-simplex σ 1 ...