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## Modelling the Dynamics of Complex Molecular Systems |

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.
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.
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.
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.
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.
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?
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.
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.
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.
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.
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?
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?
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. [Back] |