# B05 - Origin of scaling cascades in protein dynamics

**Head(s):** Prof. Dr. Bettina Keller (FU Berlin), PD Dr. Marcus Weber (ZIB), Prof. Dr. Karsten Heyne (FU Berlin)**Project member(s):** Dr. Daniel Baum, Dr. Luca Donati, Irtaza Hassan, Dr. Shreetama Karmakar**Participating institution(s):** FU Berlin, ZIB

### Project Summary

In proteins, small chemical changes or the non-covalent binding of a ligand can cause major changes in their dynamics on all scales ranging from picoseconds to minutes. Mathematically, these modifications correspond to small and local variations in the potential energy function *V* (*x*). By modelling the time evolution of the system as a diffusion process in *V* (*x*), one can analyse the dynamics in terms of the eigenspace of the corresponding transfer operator. Specifically, the long-scale dynamics are captured by the dominant eigenspace *H*_{dom}. The aim of B05 is to develop a mathematical understanding of how a small and local variation in *V* (*x*) gives rise to a cascade of processes which ultimately affects *H*_{dom} – and to test this understanding in numerical and laboratory experiments.

During the first funding period, we developed a numerical experiment which is precise enough to measure changes in *H*_{dom} even for small variations of *V* (*x*), and with which one can efficiently test a whole series of variations. Precision is achieved by ansatz functions for the discretisation of the transfer operator which are customised for peptide dynamics (variational peptide dynamics). Efficiency is achieved by the Girsanov reweighting method. With this method, one can estimate *H*_{dom} (via Markov models of the dynamics) for a series of perturbed potential energy functions from molecular-dynamics (MD) simulations at a single reference potential energy function. In collaboration with A05 and C05, we devised a discretisation for the infinitesimal generator of the transfer operator, which establishes an algebraic link between *V* (*x*) and the discretised generator (square-root approximation, SQRT-A). To link numerical and laboratory experiments, we used hidden Markov model analysis to elucidate the dynamic response of a riboswitch upon ligand binding. We also developed a method to interpret Infrared (IR) spectra of conformational ensembles with a combination of classical and first-principle MD simulations and Markov state models (MSMs). Finally, we built a spectroscopic experiment in which vibrational modes are selectively excited such that the dynamics along a chosen reaction coordinate (RC) are sped up. Equipped with these tools, we will take on the following challenges in the second funding period:

- We will extend the numerical experiment to alchemical changes of *V* (*x*) and use it to study the sensitivity of *H*_{dom} with respect to small chemical changes. In particular, we will investigate which force field terms dominate the observed variation in *H*_{dom}.

- We will aim at deriving an algebraic relation between *V* (*x*) and *H*_{dom}. We will build on our SQRT-A of the infinitesimal generator, but will also consider alternative approaches such as homotopy or a scale analysis of the Fokker–Planck equation.

- Starting from the SQRT-A of the infinitesimal generator for reversible and time-independent processes, we will extend the mathematical framework to non-equilibrium processes.

- We will link our numerical experiments to spectroscopic experiments. Non-equilibrium processes will be measured in time-dependent IR and circular dichroism (CD) experiments and intepreted by a DMD-PCCA+ analysis and MD simulations. We will extract reaction coordinates from our MD simulations. The reaction coordinates will be tested by mapping them to vibrational modes and exciting them selectively in an IR experiment.