Towards a mathematics of biomolecular flexibility
In this project we employ function space oriented methods for thermodynamical simulation and analysis of drug-like molecules. Especially in non equilibrium methods, we aim to give phenomenolgical approaches a strong mathematical background. This is then the starting point to develop reliable and robust algorithms for molecule design. One particular aspect of interest is the influence of explicit solvent on the dynamics of the molecule.
Markov State Models and Explicit Solvent
In the classical theory of Markov State Models, the clusters, corresponding to the molecular conformations of the system, are represented by a collection of characteristic basis functions that yield 1 if the state belongs to the one conformation and zero otherwise. Here, we relax the
condition, by softening the hard clustering. More precisely, we allow a state to belong to more than one conformation and assign a degree of membership to each state. As a consequence, this soft clustering allows for a faithful representation of intermediate states, i.e. states
which lie in transition regions between metastable conformations. These intermediate states are prevalent in simulation of biomolecules in water, where hydrogen bonds of the surrounding liquid can influence the stability of the molecule. The existence of intermediate states allows for a correct representation of the long-term behavior of molecular systems including explicit solvent.
Solid line is a double welled potential, dashed lines represent the hard (left) and the soft (right) metastbilities.
This is exemplified with Trialanine in explicit water (Tip 4 Ew): For the explicit solvent system, we used the TIP4P-Ew water model. Trialanine was placed in a rhombic dodecahedron solvent box of about 4.2 nm side length. In order to neutralize the overall charge, we placed a single negative counter ion in the box. The energy of the system was minimized by using the steepest descent algorithm. Afterwards, a 200 ps simulation was performed during which the position of all heavy atoms of Trialanine was restrained in order to settle the solvent molecules.
The output of this run was used as starting configuration for a molecular dynamics run of 100 ns. In order to maintain a constant temperature of 300 K and a pressure of 1 bar, velocity rescaling and Berendsen weak coupling were applied.
A twin range cut-off of 1.0/1.4 nm for van-der-Waals interactions was applied and the smooth particle mesh Ewald algorithm was used for Coulomb interactions, with a switching distance of 1.0 nm. Bond length oscillations of bonds involving hydrogen atoms were contrained using the
LINCS algorithm, allowing for an integration step of 1 fs.
Histogramm of the dihedrals of trialanine. LEFT: Computed by trajectory based methods. RIGHT: computed on the basis of Markov State Models.
K. Fackeldey, M. Klimm and M. Weber: A Coarse Graining Method for the Dimension Reduction of the State Space of Biomolecules, Journal of Mathematical Chemistry, Vol 5 (9), p. 2623 - 2635 (2012)
K. Fackeldey, A. Bujotzek and M. Weber: A meshless discretization method for Markov State Models applied to explicit water peptide folding simulations, accepted for Proceedings of the Sixth International Workshop - Meshfree Methods for Partial Differential Equations, Editors: M. Griebel M. A. Schweitzer (2012)
K. Fackeldey: Multiscale Methods in Time and Space, in Progress in Industrial Mathematics at ECMI 2010, pages: 619-626, Editor: Günther, M.; Bartel, A.; Brunk, M.;Schöps, S.; Striebel, M., Springer (2012)