# C03 - Multiscale modelling and simulation for spatiotemporal master equations

### Project Details

**Status:**In Progress**Head(s):**Prof. Schütte, Prof. Noé, Prof. Höfling**Project member(s):**Dr. Stefanie Winkelmann, Fabian Paul, Luigi Sbailò**Participating Institution(s):**FU Berlin**Area:**C: Bridging the micro-macro scale range

### Project Summary

Accurate modeling of reaction kinetics is important for understanding the functionality of biological cells and the design of chemical reactors. Depending on the particle concentrations and on the relation between particle mobility and reaction rate constants, different mathematical models are appropriate.

In the limit of slow diffusion and small concentrations, both discrete particle numbers and spatial inhomogeneities must be taken into account. The most detailed root model consists of particle-based reaction-diffusion dynamics (PBRD), where all individual particles are explicitly resolved in time and space, and particle positions are propagated by some equation of motion, and reaction events may occur only when reactive species are adjacent.

For rapid diffusion or large concentrations, the model may be coarse-grained in different ways. Rapid diffusion leads to mixing and implies that spatial resolution is not needed below a certain lengthscale. This permits the system to be modeled via a spatiotemporal chemical Master equation (STCME), i.e. a coupled set of chemical Master equations acting on spatial subvolumes. The STCME becomes a chemical Master equation (CME) when diffusion is so fast that the entire system is well-mixed. When particle concentrations are large, populations may be described by concentrations rather than by discrete numbers, leading to a PDE or ODE formulation. See Fig. C03-1 for an overview.

Figure C03-1. Illustration of reaction kinetics formulations. The blue models and the switching between them will be investigated in this project.

Many biological processes call for detailed models (PBRD, ST-CME or CME), but these models are extremely costly to solve. Ef.cient mathematical and computational methods are needed in order to approximate the solutions of these models with some guaranteed accuracy level. An approach to optimal or ef.cient switching between different models is, as yet, missing.

In this project, we will set out to develop a multiscale theory for reaction kinetics processes, starting from a consistent and well-de.ned formulation of PBRD models, and including spatial scaling (PBRD <-> ST-CME <-> CME) coupled to population scaling (CME <-> ODE). In particular, we aim at providing solutions for the problematic cases of having particles at diverse copy numbers (CME . ODE) and at least some slowly diffusing particles (PBRD <-> CME <-> STCME). The cascades of scales in these scenarios and efficient approximation strategies will be explored.

### Project Publications

Paul, F. and Wu, H. and Vossel, M. and de Groot, B.L. and Noé, F. (2019) *Identification of kinetic order parameters for non-equilibrium dynamics.* J. Chem. Phys., 150 (16). p. 164120. ISSN 0021-9606, ESSN: 1089-7690

Pinamonti, G. and Paul, F. and Noé, F. and Rodriguez, A. and Bussi, G. (2019) *The mechanism of RNA base fraying: Molecular dynamics simulations analyzed with core-set Markov state models.* J. Chem. Phys., 150 (15). p. 154123. ISSN 0021-9606, ESSN: 1089-7690

Wang, J. and Olsson, S. and Wehmeyer, C. and Perez, A. and Charron, N.E. and de Fabritiis, G. and Noé, F. and Clementi, C. (2019) *Machine Learning of coarse-grained Molecular Dynamics Force Fields.* ACS Cent. Sci., 5 (5). pp. 755-767. ISSN 2374-7943, ESSN: 2374-7951

Hoffmann, M. and Fröhner, Chr. and Noé, F. (2019) *ReaDDy 2: Fast and flexible software framework for interacting-particle reaction dynamics.* PLoS Computational Biology, 15 (2). e1006830. ISSN 1553-7358

Hoffmann, M. and Fröhner, Chr. and Noé, F. (2019) *Reactive SINDy: Discovering governing reactions from concentration data.* J. Chem. Phys., 150 (2). 025101. ISSN 0021-9606, ESSN: 1089-7690

Scherer, M. K. and Husic, B.E. and Hoffmann, M. and Paul, F. and Wu, H. and Noé, F. (2018) *Variational Selection of Features for Molecular Kinetics.* SFB 1114 Preprint in arXiv:1811.11714 . pp. 1-12. (Unpublished)

Wehmeyer, C. and Scherer, M. K. and Hempel, T. and Husic, B.E. and Olsson, S. and Noé, F. (2018) *Introduction to Markov state modeling with the PyEMMA software — v1.0.* LiveCoMS, 1 (1). pp. 1-12. ISSN E-ISSN: 2575-6524

Fröhner, Chr. and Noé, F. (2018) *Reversible interacting-particle reaction dynamics.* J. Phys. Chem. B, 122 (49). pp. 11240-11250.

del Razo, M.J. and Qian, H. and Noé, F. (2018) *Grand canonical diffusion-influenced reactions: a stochastic theory with applications to multiscale reaction-diffusion simulations.* J. Chem. Phys., 149 (4). 044102. ISSN 0021-9606, ESSN: 1089-7690

Dibak, M. and del Razo, M.J. and De Sancho, D. and Schütte, Ch. and Noé, F. (2018) *MSM/RD: Coupling Markov state models of molecular kinetics with reaction-diffusion simulations.* Journal of Chemical Physics, 148 (214107). ISSN 0021-9606

Sadeghi, M. and Weikl, T. and Noé, F. (2018) *Particle-based membrane model for mesoscopic simulation of cellular dynamics.* J. Chem. Phys., 148 (4). 044901.

Sbailò, L. and Noé, F. (2017) *An efficient multi-scale Green's Functions Reaction Dynamics scheme.* J. Chem. Phys., 147 . p. 184106. ISSN 0021-9606, ESSN: 1089-7690

Paul, F. and Wehmeyer, C. and Abualrous, E. T. and Wu, H. and Crabtree, M. D. and Schöneberg, J. and Clarke, J. and Freund, C. and Weikl, T. and Noé, F. (2017) *Protein-peptide association kinetics beyond the seconds timescale from atomistic simulations.* Nat. Comm., 8 (1095).

Winkelmann, S. and Schütte, Ch. (2017) *Hybrid models for chemical reaction networks: Multiscale theory and application to gene regulatory systems.* Journal of Chemical Physics, 147 (11). pp. 1-18.

Olsson, Simon and Wu, H. and Paul, F. and Clementi, C. and Noé, F. (2017) *Combining experimental and simulation data of molecular processes via augmented Markov models.* Proc. Natl. Acad. Sci. USA, 114 . pp. 8265-8270.

Pinamonti, G. and Zhao, J. and Condon, D. and Paul, F. and Noé, F. and Turner, D. and Bussi, G. (2017) *Predicting the kinetics of RNA oligonucleotides using Markov state models.* J. Chem. Theory Comput., 13 (2). pp. 926-934.

Schöneberg, J. and Lehmann, M. and Ullrich, A. and Posor, Y. and Lo, W.-T. and Lichtner, G. and Schmoranzer, J. and Haucke, V. and Noé, F. (2017) *Lipid-mediated PX-BAR domain recruitment couples local membrane constriction to endocytic vesicle fission.* Nat. Comm., 8 . p. 15873.

Winkelmann, S. and Schütte, Ch. (2016) *The spatiotemporal master equation: approximation of reaction-diffusion dynamics via Markov state modeling.* Journal of Chemical Physics, 145 (21). p. 214107.

Albrecht, D. and Winterflood, C. M. and Sadeghi, M. and Tschager, T. and Noé, F. and Ewers, H. (2016) *Nanoscopic compartmentalization of membrane protein motion at the axon initial segment.* J. Cell Biol., 215 (1). pp. 37-46.

Wu, H. and Paul, F. and Wehmeyer, C. and Noé, F. (2016) *Multiensemble Markov models of molecular thermodynamics and kinetics.* Proceedings of the National Academy of Sciences, 113 (23). E3221-E3230 . ISSN 0027-8424

Trendelkamp-Schroer, B. and Wu, H. and Paul, F. and Noé, F. (2015) *Estimation and uncertainty of reversible Markov models.* J. Chem. Phys., 143 (17). p. 174101.

Scherer, M. K. and Trendelkamp-Schroer, B. and Paul, F. and Pérez-Hernández, G. and Hoffmann, M. and Plattner, N. and Wehmeyer, C. and Prinz, J.-H. and Noé, F. (2015) *PyEMMA 2: A Software Package for Estimation, Validation, and Analysis of Markov Models.* J. Chem. Theory Comput., 11 (11). pp. 5525-5542.