FÜR INFORMATIONSTECHNIK
BERLIN

# MATHEON-A19

## Towards a mathematics of biomolecular flexibility

Description:

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 Membership Functions

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. In contrast, the  soft conformation aim at a degree of membership, which is assigned to each state. As a consequence, each state can belong to more than one conformation and allows for a faithful representation of 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 but they are also present in other phenomena, like the rebinding effect.

Bivalent Docking and Rebinding

Left: unbounded bivalent ligand and a receptor. Middle: bounded ligand and receptor. Right: Rebinding

In a bivalent binding process the ligand and the receptor have more than one binding site. As a consequence between the bounded and the unbounded state an intermediate state can be found wich is neither bound nor unbound. Moreover, having bounded on one site, the other site of the ligand switches between the bound and unbound state (rebinding). One possible approach might be to define the single bounded state as another metastable state, however in many applications this state is not metastable and is difficult to measure in laboratory.

Mathematical Modelling of the Rebinding Effect

The soft membership functions allow for such an intermediate state which is not metastable, by assigning a "degree of boundness" to each state. As a consequence the memebershipf functions have not a characteristic structure but an overlap. These overlap regions represent the intermediate states which can be considered as bound and unbound.

Soft Conformations allowing for intermediate states.

Mathematically the soft conformations are a linear combination of the eigenvectors of the infinitesimal generator Q. The coefficients in this linear combination are  computed by PCCA+. We also developed a linear program which enables us, to compute the minimal rebinding effect in a given kinetics. We applied this method to a bivalent docking process and to the flame retardant HBCD.

Publications:

M. Weber, K. Fackeldey: Computing the Minimal Rebinding Effect Included in a Given Kinetics. Submitted for publication.

H. C. Lie, K. Fackeldey, M. Weber: A square root approximation for transition rates in Markov state models. Accepted for publication.

M. Weber, A. Bujotzek, R. Haag: Quantifying the rebinding effect in multivalent chemical ligand-receptor systems. Journal of Chemical Physics, 137(5):054111, 2012.