# B02 - Polymer dynamic theories, Markov state models and protein folding free energy landscapes

### Project Details

**Status:**In Progress**Head(s):**Prof. Netz, Prof. Noe**Project member(s):**Julian Kappler, Dr. Won Kyu Kim, Sina Zendehroud**Participating Institution(s):**FU Berlin**Area:**B: Uniform meso scale behavior in scaling cascades**Positions available?:**

### Project Summary

In this CRC Markov State Models (MSMs) play a central role as a tool for the efficient study of the kinetics of protein folding and other molecular processes. These models and techniques more and more replace the traditional view of a protein diffusing in a one-dimensional folding free energy landscape[9, 8]. In fact, within the context of a sim

plified polymer model, there is a formal one-to-one correspondence between MSMs and the so-called Rouse or Zimm theories, the classical theories for the dynamics of polymers. The Rouse theory considers the dynamics of an ideal (i.e., non-interacting) Gaussian polymer (consisting of monomers connected by harmonic springs) in the absence of hydrodynamics, the Zimm theory adds hydrodynamic effects in an approximate fashion [DE86]. In both theories the Langevin equation describing the over-damped polymer dynamics can be diagonalized, they thus represent an essentially exact solution of the multiscale problem constituted by the polymer dynamics [6]. The main difference to MSMs is that in the Rouse/Zimm description, the state variables are the positions of all monomers and thus completely specify the polymer conformation; due to the simplicity of the underlying model and the absence of interactions between monomers, the resulting calculations can be done analytically [6]. In the MSM approach, on the other hand, one is more flexible in choosing the state variables and typically no complete description is attempted but rather some coarse-graining is performed prior to the dynamic analysis. But the relation between Rouse/Zimm theories on the one hand and MSMs on the other hand has not been explored in depth, neither has the connection to the chemist’ view of protein folding and recon.guration as a diffusion in a free energy landscape been fully unravelled [CUPM09, PB09]. The central questions of this proposal are:

(i) How has an MSM (and in particular the discretized state variables) to be designed that – when applied to a simple polymer chain as treated within the Rouse/Zimm theories – recovers the known dynamic spectrum of the corresponding Langevin equation and in particular the full dynamic polymer behavior? polymer dynamics will be exempli.ed by, e.g., the end-to-end-distance, which experimentally and theoretically is a central quantity and exhibits complex dynamic behavior and in particular sub-diffusion at intermediate time scales up to the longest internal polymer relaxation time. This comparison will in turn be helpful in developing stringent tests on the accuracy of MSMs in the form of a convenient benchmark [11, 10].

(ii) How does the eigenvalue spectrum of the Rouse/Zimm and MSM modes change when the underlying physical model is made more complex, i.e. when one goes from idealized polymer models to realistic models that can describe protein folding (in which case simulations have to be used)? In specific, what is the most economic way of representing the conformational space within MSMs? In this part we will in a bidirectional approach explain dynamic features displayed by MSMs in terms of the underlying Hamiltonian that governs the Langevin equation (research direction I of the CRC) but also optimize the MSM formulation and thereby improve the ef.ciency of a numerical approximative scheme (research direction III of the CRC).

(iii) As is known from classical polymer theory, the sub-diffusive behavior of the polymer end-to-end-distance, which conceptually is very much related to the sub-diffusive behavior induced by turbulent fluctuations in fluid flow, results from the coupling of the entire dynamic mode spectrum of the polymer chain. It is not straightforward to obtain sub-diffusive behavior in the framework of a polymer diffusing in a one-dimensional free-energy landscape, which is the traditional view in the field of protein folding. An important conceptual question is therefore how and whether the intuitively appealing picture of a protein or polymer diffusing in a low-dimensional space of suitably de.ned reaction coordinates can be brought into accordance with the sub-diffusive behavior known from traditional polymer or MSM theories.

### Project Publications

Kappler, Julian and Noe, Frank and Netz, Roland R. (2019) *Cyclization and Relaxation Dynamics of Finite-Length Collapsed Self-Avoiding Polymers.* Phys. Rev. Lett., 122 (6). 067801.

Kappler, J. and Noé, F. and Netz, R.R. (2018) *Cyclization dynamics of finite-length collapsed self-avoiding polymers.* SFB 1114 Preprint 02/2018 . (Unpublished)

Gerber, S. and Olsson, S. and Noé, F. and Horenko, I. (2018) *A scalable approach to the computation of invariant measures for high-dimensional Markovian systems.* Sci. Rep., 8 (1796). ISSN 2045-2322

Kappler, J. and Daldrop, J.O. and Brünig, F.N. and Boehle, M.D. and Netz, R.R. (2018) *Memory-induced acceleration and slowdown of barrier crossing.* J. Chem. Phys., 148 (1). 014903.

Daldrop, J.O. and Kowalik, B.G. and Netz, R.R. (2017) *External Potential Modifies Friction of Molecular Solutes in Water.* Phys. Rev. X, 7 (4). 041065.

Netz, R.R. (2017) *Fluctuation-dissipation relation and stationary distribution for an exactly solvable many-particle model far from equilibrium.* SFB 1114 Preprint 12/2017 . (Unpublished)

Gerber, S. and Horenko, I. (2017) *Toward a direct and scalable identification of reduced models for categorical processes.* Proceedings of the National Academy of Sciences, 114 (19). pp. 4863-4868.

Kim, W.K. and Netz, R.R. (2015) *The mean shape of transition and first-passage paths.* J. Chem. Phys., 143 (224108).

Rinne, K.F. and Schulz, J.C.F. and Netz, R.R. (2015) *Impact of secondary structure and hydration water on the dielectric spectrum of poly-alanine and possible relation to the debate on slaved versus slaving water.* J. Chem. Phys., 142 (215104).

Schulz, J.C.F. and Miettinen, M.S. and Netz, R.R. (2015) *Unfolding and Folding Internal Friction of β‑Hairpins Is Smaller than That of α‑Helices.* Journal of Physical Chemistry B, 119 (13). pp. 4565-4574.