Analysis of the Dynamics of Protein Conformations: Structural and Mechanistic Aspects of Protein Folding

(funded by Zukunftskolleg)

The purpose of the proposed project is to address the folding problem for intrinsically disordered proteins and peptides from a mathematical point of view. According to the classical protein folding concept, a protein can perform its biological function(s) only after folding into a unique three-dimensional structure. With the discovery of intrinsically disordered proteins and peptides (IDPs), researchers realized that proteins do not necessarily adopt a well-defined secondary and tertiary structure in order to perform biological functions.

The protein folding problem for proteins with a well-defined secondary and tertiary structure is already one of the most challenging problems in current biochemistry. Therefore, the analysis of IDPs is a very rich source of interesting problems in mathematical modeling and numerical analysis, requiring an interplay of techniques in eigenvalue calculations, stiff differential equations, stochastic differential equations, local and global optimization, nonlinear least squares, multidimensional approximation of functions, dimensionality reduction, design of experiment, and statistical classification of data obtained from molecular dynamics (MD) simulations. The energetically accessible regions of the phase space of large biomolecules have a highly complex, non-linear and possibly fractal structure, so the methods of its analysis are gaining even more importance as a challenging topic both for mathematical as well as natural scientists.

The overall goals of the project are by combining MD simulations and mathematical methods to 1) develop methods for analysis of simulation data that will allow determining structures and transitional states for IDPs; 2) develop kinetic models of the transitions for selected IDPs. Here, in the further course of the project, special attention will be paid to the question how IDPs undergo a structuring transition when their environment changes (for example when they encounter aggregate seeds, surfaces or biomembranes).

Publication, related to this project.

Reservoir Computing with Application to Molecular Dynamic Simulation Data

(Done in collaboration with Prof. Dr. Lyudmila Grigoryeva, funded via Interdisciplinary collaborative project Programme of the Zukunftskolleg )

The goal of this project is to implement novel techniques of statistical/machine learning (ML) in order to address some of the problems that arise in the field of computational chemistry. We will focus on a particularly promising ML approach called reservoir computing (RC) (also known in the literature as echo state networks or liquid state machines). RC combines the computational power of dynamical systems with an easy-to-implement supervised learning scheme. The RC model consists, roughly speaking, of a fixed dynamical system (deterministic or random and usually high- or infinite-dimensional) called reservoir, into which a lower-dimensional input is injected, and of the output layer (readout) which is trained (estimated) via a Tikhonov regularized regression. This novel ML approach is free from many of the complications that arise when using standard artificial neural networks (ANN) and that have become the main reason for their limited use in certain realms of application. RC has already shown to outperform conventional methods in many computational tasks: chaotic path continuation, stochastic time series forecasting, reconstruction of nonlinear functions, or memory tasks. 

We intend to tackle with this ML methodology several crucial problems that theoretical/computational chemistry is facing in the context of protein studies. More specifically, the idea of this project is to use the information processing ability of certain dynamical systems in connection with the reservoir computing approach, to reconstruct, explain, and predict some salient static and dynamic features of molecular structures of interest.

Quantitative Analysis of Linker Histones Ubiquitylation based on Molecular Simulation Data

(funded by Carl Zeiss Stiftung)

This project focuses on the development and application of computational methods and mathematical tools to study functional conformational changes of complex biological systems, in particular the influence of mono-ubiquitylation in linker histones on DNA interaction and their ability to form higher order chromatin structures. Investigation of biological systems of this size with classical simulations is limited due to the timescale problem which becomes substantially large and cannot be addressed straightforwardly. Therefore the use of different resolution levels (multiscale models) in combination with novel machine learning techniques can help to simulate efficiently functional conformational changes of the ubiquitinated histone and to characterize its phase space. Furthermore, obtained kinetic and thermodynamic information will be used to model the behavior of ubiquitinated histone. Comprehensive information of this kind will be of fundamental importance in the interpretation of experimental data and the planning of future experiments. In addition, developed and modified theoretical methods from this project will be used for the investigation of other multi-body systems.