Title: Anders Eriksson
1Complex Systems at ChalmersInformation Theory
and Multi-scale Simulations
- Anders Eriksson
- Complex Systems Group
- Dept. Energy and Environmental Research
- Chalmers
- EMBIOCambridge July 2005
2Outline
- People
- Information theory
- Based on presentation by Kristian Lindgren
- Hierarchical dynamics
- Based on presentation by Martin Nilsson Jacobi
- Discontinuous Molecular Dynamics
3People
- Kristian Lindgren
- Information dynamics
- Martin Nilsson Jacobi
- Hierarchical dynamics
- Non-equilibrium statistical mechanics
- Kolbjørn Tunstrøm
- Multi-scale simulations
- Olof Görnerup
- Coarse-grained molecular dynamics
- Anders Eriksson
- Folding dynamics of simplified protein models
4Introduction to information dynamics
- Adapted from presentation by Kristian Lindgren
- Information and self-organisation
- Thermodynamic context
- Geometric information theory
- Continuity equation for information
- Example system Gray-Scott model
(self-reproducing spots system)
5Information in self-organisation
- Three types of information characteristics
- Information on dynamics (genetics), IG
- Information from fluctuations (symmetry
breaking), IF - Information in free energy (driving force), ITD
- Typically IG ltlt IF ltlt ITD
6Thermodynamic context
- 2nd law of thermodynamics in total, entropy is
increasing - Out-of-equilibrium, low-entropy state maintained
byexporting more entropy than what is imported
and produced
7Gibbs free energy and information
- The free energy E of a concentration pattern
ci(x) can be related to the information-theoretic
relative information K where kB is
Boltzmanns constant and T0 is the temperature. - The free energy E is related to information
content I (in bits) by
8Decomposition of information
- The information can be decomposed into two terms
(quantifying deviation from equilibrium and
spatial homogeneity, respectively) -
- The spatial information Kspatial can be further
decomposed into contributions from different
length scales (resolution) r, and further from
positions x
9Resolution length scale
- We define the pattern of a certain component i
at resolution r by the following convolution of
ci(x) with a Gaussian of width r - This has the properties For simplicity we
write
10Resolution and position
11Gray-Scott self-replicating spots
Gray Scott, Chem Eng Sci (1984), Pearson,
Science (1993), and Lee et al, (1993).
Reaction-diffusion dynamics
12Information density in the model
The information density for two resolution levels
r illustrate the presence of spatial structure at
different length scales.
Information density k(r0.01, x, t)
Concentration of VcV (x, t)
k(r0.05, x, t)
13Continuity equation for information
Inflow of chemical information (exergy)
r
Kchem
j(r, x, t)
k(r, x, t)
Resolution (length scale)
Kspatial
jr(r, x, t)
J(r, x, t)
y
Destruction of information (entropy production)
x
Flow in scale
Sinks (open system)
Flow in space
14Outlook
- Generalised 2nd law of information destruction
flow of information from larger to smaller
scales - Small characteristic length scale of free energy
inflow may imply limited possibilities to support
meso-scale concentration patterns - Illuminate stability of dissipative structures
15Hierarchical dynamics
- Adapted from presentation by Martin Nilsson
Jacobi - Main goals
- Develop a mathematical framework to describe
hierarchical structures in (smooth) dynamical
systems. - Tool for multi-scale simulations.
- Address the emergence of objects and natural
selection in dynamical systems. - Understand the transition from nonliving to
living matter from a dynamical systems
perspective.
16Informal definition
- Each level in the hierarchy should be
deterministic when described in isolation. - A higher level in the hierarchy should be derived
from a lower through a smooth projective map. - Arbitrary nonlinear projective maps should be
allowed, and thereby allow for highly
heterogeneous (or functional'') course graining.
17...or in a picture
18Conceptual overview
19Equation-free simulation
- Coarse-graining method that relies on the
separation between fast and slow manifolds - Basic idea
- Kevrekidis et al. (2002), Hummer and Kevrekidis
(2003) - Identify slow variables, which span important
parts of the slow manifold - Estimate the rate of change of these variables
from bursts of short simulations on the
fine-grained (MD) level. - Most difficult part how to find initial state on
the fine-grained level, consistent with the
coarse-grained description of the system
20Discontinuous Molecular Dynamics
- Discontinuous Molecular Dynamics (DMD)
- Estimating contact (free) energies
- Folding dynamics
21Discontinuous Molecular Dynamics
- Linear chain of spheres, connected by bonds
- Bonds are hard-sphere
- Contact potential
- Piecewise constant
- Hard-sphere core
- Potential well for residue-residue contact
energy gain - Finite range
- Heat bath
- Boltzmann distributed impulses
- Provides temperature
- Independent heat bath for each bead
22Thermodynamic properties
- Discrete set of energy levels
- Only depends on which residue are in contact
- Can reproduce basic thermodynamic propertiesof
clusters
Zhou et al. (1997), J. Chem. Phys. 107(24), p.
10697
23Estimation of contact energies
- Miyazawa and Jernigan (J. Mol. Biol., 1996, 256,
p. 623) - Based on the native state of proteins X-ray
data from the Protein Data Bank (NMR excluded) - Each protein is mapped onto a lattice
- Quasi-chemical approximation gives the free
energies from counts of contacts in this
gridwhere i and j are residues, 0 is a
solvent volume element - The total free energy of a protein is
24The path to equilibrium
- Use this simplified dynamics to study the road to
equilibrium - Do these systems exhibit a folding funnel?
- If so, is it consistent with the free energy
landscape of real proteins? - Questionable far from equilibrium needs
validation - May learn mechanisms
25Summary
- Information dynamics and qualitative models can
give insight into the mechanisms of folding - A theory for hierarchical dynamics allows proper
coarse-grained dynamics
The End
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30Future work
- Generalised 2nd law of information destruction
flow of information from larger to smaller
scales - Small characteristic length scale of free energy
inflow, may imply limited possibilities to
support meso-scale concentration patterns - Possible application the fan reactor
- The inflow in the fan reactor has a small
characteristic length scale, indicating that
there may be limitations on what meso-scale
(concentration) patterns that can be supported in
that system.