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The funnel trap paradox
(2012)
In this article, an illustrative example is given for the coarse-graining of a
Markov process which leads to a shift in the statistical weights of a two-states-system.
The example is based on a 2D-funnel trap. The funnel trap is constructed in such a
way, that the area inside and outside of the trap is identical. However, observing the
flight of the insect as a Markov process, the probability for being “in the trap” is higher.
This example can be transferred to several kinds of processes (like receptor-ligandbinding
processes in chemistry) and describes the influence of “re-entering events”.
Supercomputers can simulate complex molecular systems. However, there is a very large gap between the fastest oscillations of covalent bonds of a molecule and the time-scale of the dominant processes. In order to extract the dominant time-scales and to identify the dominant processes, a clustering of information is needed. This thesis shows that only the subspace-based Robust Perron Cluster Analysis (PCCA+) can solve this problem correctly by the construction of a Markov State Model. PCCA+ allows for time-extrapolation in molecular kinetics. This thesis shows the difference between molecular dynamics and molecular kinetics. Only in the molecular kinetics framework a definition of transition rates is possible. In this context, the existence of an infinitesimal generator of the dynamical processes is discussed. If the existence is assumed, the Theorem of Gauß can be applied in order to compute transition rates efficiently. Molecular dynamics, however, is not able to provide a suitable statistical basis for the determination of the transition pattern.
The rebinding effect is a phenomenon which occurs when observing a ligand-receptor binding process.
On the macro scale this process comprises the Markov property.
This Makovian view is spoiled when switching to the atomistic scale of a binding process.
We therefore suggest a model which accurately describes the rebinding effect on the atomistic scale by allowing ''intermediate'' bound states.
This allows us to define an indicator for the magnitude of rebinding and to formulate an optimization problem.
The results form our examples show good agreement with data form laboratory.