Modelling the Dynamics of Complex Molecular Systems

Description and aim

Mathematics and physical chemistry have a long history of collaboration and interaction, which has been of great benefit to both communities. In recent years, the need for a more intense partnership has become evident from the development of sensible computer algorithms and theories which can be "scaled up" to address challenging problems, such as the simulation of proteins and nucleic acids, or prediction of structures in nano-engineered materials. It is no longer possible to make progress by simply using a larger computer; new principles and methodologies are needed. Moreover, as scale boundaries are crossed, mathematical approaches must provide a seamless transition from one formulation to another. This 4-week meeting brings together researchers interested in the design of new theories and algorithmic principles for a broad range of phenomena formulated at atomistic and coarsened scale regimes, and integrated multiscale approaches. In addition, the meeting aims to establish real collaborations with potential for long lasting impact on both fields.

This 4-week program is divided in two workshops and two focus groups, each lasting one week and alternating with each other. In the workshops we will address theoretical challenges that physical chemists face in the field of molecular dynamics. Workshop 1 will be devoted to the theory of molecular dynamics, reaching long time scale limits. Workshop 2 will focus on coarse graining and multiscale algorithms. As these challenges are actually intertwined, special attention will be given to establishing the links between the two workshops. In the focus groups, scientists will work on dedicated problems that have been identified in the preceding workshop.

Explanation Program

Introduction

The need for multiscale simulation approaches to challenges such as the dynamics of complex chemical reactions, biomolecular (proteins, DNA) conformational changes, (ligand) binding and self-assembly, has recently led to a surge in interest in novel mathematics and algorithms for molecular modelling and simulation, ranging from all-atom molecular dynamics to coarse-grained ‘mesoscale’ molecular simulation (and beyond, to continuum mechanics regimes).

The meeting consists of two workshops, each followed by a smaller focus group. For the workshops (1st and 3rd week), the program consists of a limited number of lectures, leaving ample time for plenary discussions as well as informal interactions. During each workshop, the state of the field will be discussed and questions in the field will be formulated. In the subsequent week, the focus group will work on the questions identified during the workshop. The program of the focus group may "go-with-the-flow", with each day starting with a lecture and ending with a discussion session. The first workshop and focus group focus to long term dynamics and the second workshop and focus group focus on coarse graining and concurrent multiscale modeling. These topics are very much intertwined and participants are therefore encouraged to apply for all four weeks.

Program Layout

Workshop and Focus Group 1: Long time dynamics (week 1 and 2)

Week 1, day 1: Mathematics of molecular and stochastic dynamics

Based on the numerical integration of the equations of motion, molecular dynamics simulation is the primary tool for creating a time evolution of a mechanical model for molecular based systems. Stochastic perturbations are frequently used to model missing degrees of freedom or to enhance the flow of energy, but many theoretical and numerical analysis issues remain open, such as the importance of smoothness in molecular trajectories. More questions arise in the area of ergodicity of various methods and in various ensembles, and rates of convergence to equilibrium. After we have laid out a general overview of the workshops, we will start discussing such issues.

Week 1, days 2-3: Accessing long time scales by rare event simulation

One of the main challenges in current atomic level simulations is to push them to time-scales on which rare but important reactive events arise. Many crucial processes such as chemical reactions, nucleation events in kinetic phase transitions, vacancy diffusion in crystals, crack propagation in solids, and conformational changes in proteins involve such reactive events, and they typically occur on time-scales which are much too long to be accessed by direct simulations.

Techniques that aim to address rare events can search for saddle points in the potential energy surface, or, in case of a rough free energy landscape, apply a biasing potential along a collective variable towards the dynamical bottleneck. However, this requires prior knowledge of the reaction coordinate, which is often incomplete or even lacking. Trajectory based techniques such as transition path sampling, the string method or Mile stoning mitigate this to some extent. In addition, complex reaction networks can be analysed with Markov State Models. While the recently developed transition path theory provides a theoretical framework for these approaches, much work needs to be done. We will debate on the need to push these long time scale approaches.

Week 1, day 4: Non-equilibrium dynamics

The correct description of non-equilibrium dynamics is problematic: the very formulation of a correct – and realistic – approach to study systems that are driven out of equilibrium is, at the very least, the subject of heated controversies. Yet for many applications (e.g. the modelling of living systems, glasses, aging of materials), it is crucial to take account of the fact that the system under study is out of equilibrium. We will discuss several approaches to these problems.

Week 1, day 5: Wrap up discussions and explicit link to the next workshop

The outcome of the discussions will be summarized and issues for the following focus group will be specified. We will link the first workshop to the second workshop, to be held in week 3. For example, how does Markov state modeling connect the rare event problem to coarse-graining?

Week 2, day 1-5: Focus group on long time dynamics

The focus group will work on specific issues that are identified during the workshop in the first week. Examples could include: developing of methods that allow studying non-equilibrium rare events, developing of methods that make large time steps possible, combination of Markov state modeling and rare event techniques.

Workshop and Focus Group 2: Coarse graining and concurrent multiscale modeling (week 3 and 4)

Week 3, day 1:  Accessing larger systems via coarse grained (mesoscale) models

While much effort is undertaken to develop coarse-grained models that reproduce certain properties of the underlying level (e.g. quantum calculations, or all atom semi-empirical force fields), or even experimental data, not much is known about the mathematical foundations of the procedures. Yet, in order to make progress, it is essential to know what the theoretical limits are on models constructed by coarse-graining, what criteria we should use to define “optimal” coarse-graining procedures, and how models for simulations should be constructed in a more Bayesian way: i.e. as the optimal representation of our knowledge of the system.

Week 3, day 2: Dynamical coarse graining

Because coarse-grained models smoothen out the interaction they usually do not preserve the true dynamics. Recently, several attempts have been made to correctly reproduce the dynamics in coarse-grained systems by including memory terms. A deeper mathematical understanding and foundation of this type of dynamical coarse graining is needed, as well as novel numerical schemes.

Week 3, day 3: Integrating different levels in multiscale simulations

The specific problem that occurs when modelling the large biomolecular systems (our main theme) is that dynamical simulation on a mesoscopic coarse-grained level is not sufficient. For some properties, atomistic details are crucial, for others not. To simulate such complex processes through molecular dynamics, we need a consistent integration of different levels of modelling. The prime example of such integration is the QMMM approach in which a chemical reaction in e.g. a biomolecule is treated by quantum mechanics, whereas the environment is treated by classical force fields.

Recently, concurrent approaches for all-atom and coarse-grained models have been developed. Particular attention is given to the open, adaptive nature of the all-atom part, in which atoms can move in and out. The integration of the different levels of description has been mostly “ad hoc” and there is a great need to put this discussion on a more formal basis.

Week 3, day 4: Reaction coordinates and coarse graining

When describing a dynamical system on a coarse level one needs to know which degrees of freedom to keep, and which can be integrated out. This question is much related to the problem of finding the reaction coordinate, which describes a dynamical (rare event) process. What are the relations between the two, and what can we learn from both?

Week 3, day 5: Wrap-up discussions and explicit link to the previous workshop

The outcome of the discussions will be summarized and issues for the following focus group will be specified. We will link this workshop to the first workshop, held in week 1. For instance, can coarse graining help to solve the rare event (long time scale) problem? How do we reach long time dynamics in a concurrent multiscale approach such as QMMM?

Week 4, days 1-5: Focus group on coarse graining and concurrent multiscale modeling

This focus group will work on specific issues that are identified during the workshop in the third week. Examples could include: developing of methods that allow to coarse grain while preserving dynamics, methods that allow multiscale scale modelling to reach long time scales, finding procedures to estimate which degrees of freedom can be missed in coarse graining.