# 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.

### Project publications

- Hartmann, C. and Neureither, L. and Sharma, U. (2019)
*Coarse-graining of non-reversible stochastic differential equations: quantitative results and connections to averaging.*SIAM J. Math. Anal. . pp. 1-39. (Submitted) - Neureither, L. and Hartmann, C. (2019)
*Time scales and exponential trends to equilibrium: Gaussian model problems.*In: Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017. Springer Proceedings in Mathematics & Statistics, 282 . Springer, pp. 391-410. ISBN 978-3-030-15095-2 - Hartmann, C. and Schütte, Ch. and Zhang, W. (2019)
*Jarzysnki equality, fluctuation theorems and variance reduction: Mathematical analysis and numerical algorithms.*J. Stat. Phys., 175 (6). pp. 1214-1261. ISSN 0022-4715; ESSN: 1572-9613 - Keller, B.G. and Aleksic, S. and Donati, L. (2019)
*Markov State Models in drug design.*In: Biomolecular Simulations in Structure-based Drug Discovery. Methods and Principles in Medicinal Chemistry, 75 . Wiley-Interscience, Weinheim, pp. 67-86. ISBN 978-3-527-34265-5 - Fackeldey, K. and Koltai, P. and Névir, P. and Rust, H.W. and Schild, A and Weber, M. (2019)
*From Metastable to Coherent Sets – time-discretization schemes.*Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (1). 012101. ISSN 1054-1500 (print); 1089-7682 (online) - Witek, J. and Wang, S. and Schroeder, B. and Lingwood, R. and Dounas, A. and Roth, H.-J. and Fouché, M. and Blatter, M. and Lemke, O. and Keller, B. and Riniker, S. (2019)
*Rationalization of the Membrane Permeability Differences in a Series of Analogue Cyclic Decapeptides.*J. Chem. Inf. Model., 59 (1). pp. 294-308. ISSN 1549-9596, ESSN: 1549-960X - Kieninge, Stefanie and Donati, Luca and Keller, Bettina (2019)
*Dynamical reweighting methods for Markov models.*Current Opinion in Structural Biology, 61 . pp. 124-131. ISSN 0959-440X - Donati, L. and Heida, M. and Weber, M. and Keller, B. (2018)
*Estimation of the infinitesimal generator by square-root approximation.*Journal of Physics: Condensed Matter, 30 (42). p. 425201. ISSN 0953-8984, ESSN: 1361-648X - Donati, L. and Keller, B. (2018)
*Girsanov reweighting for metadynamics simulations.*Journal of Chemical Physics, 149 (7). 072335. ISSN 0021-9606 - Hassan, I. and Donati, L. and Stensitzki, T. and Keller, B. and Heyne, K. and Imhof, P. (2018)
*The Vibrational Spectrum of the hydrated Alanine-Leucine Peptide in the Amide region from IR experiments and First Principles Calculation.*Chem. Phys. Lett. . pp. 1-26. ISSN 0009-2614 - Stensitzki, T. and Yang, Y. and Kozich, V. and Ahmed, A.A. and Kössl, F. and Kühn, O. and Heyne, K. (2018)
*Acceleration of a ground-state reaction by selective femtosecond-infrared-laser-pulse excitation.*Nature Chemistry, 10 . pp. 126-131. - Manz, C. and Kobitski, A. and Samanta, A. and Keller, B.G. and Jäschke, A. and Nienhaus, G.U. (2017)
*Single-molecule FRET reveals the energy landscape of the full-length SAM-I riboswitch.*Nat. Chem. Biol., 13 . pp. 1172-1178. - Donati, L. and Hartmann, C. and Keller, B.G. (2017)
*Girsanov reweighting for path ensembles and Markov state models.*Journal of Chemical Physics, 146 (24). p. 244112. ISSN 0021-9606 - Quer, J. and Donati, L. and Keller, B.G. and Weber, M. (2017)
*An automatic adaptive importance sampling algorithm for molecular dynamics in reaction coordinates.*SIAM J. Sci. Comput. . pp. 1-19. ISSN 1064-8275 (print) (In Press) - Zhang, W. and Hartmann, C. and Schütte, Ch. (2016)
*Effective dynamics along given reaction coordinates, and reaction rate theory.*Faraday discussions, 195 . pp. 365-394. ISSN 1359-6640 - Lemke, O. and Keller, B.G. (2016)
*Density-based cluster algorithms for the identification of core sets.*Journal of Chemical Physics, 145 (164104). - Vitalini, F. and Noé, F. and Keller, B. (2016)
*Molecular dynamics simulations data of the twenty encoded amino acids in different force fields.*Data in Brief, 7 . pp. 582-590. - Bittracher, Andreas and Hartmann, C. and Junge, O. and Koltai, Péter (2015)
*Pseudo generators for under-resolved molecular dynamics.*The European Physical Journal Special Topics, 224 (12). pp. 2463-2490. ISSN 1951-6355 - Hartmann, C. and Delle Site, L. (2015)
*Scale Bridging in Molecular Simulation.*The European Physical Journal Special Topics, 224 (12). pp. 2173-2176. ISSN 1951-6355