Anders Eriksson

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Complex Systems at ChalmersInformation Theory
and Multi-scale Simulations

• Anders Eriksson
• Complex Systems Group
• Dept. Energy and Environmental Research
• Chalmers
• EMBIOCambridge July 2005

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Outline

• People
• Information theory
• Based on presentation by Kristian Lindgren
• Hierarchical dynamics
• Based on presentation by Martin Nilsson Jacobi
• Discontinuous Molecular Dynamics

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People

• 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

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Introduction 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)

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Information 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

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Thermodynamic 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

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Gibbs 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

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Decomposition 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

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Resolution 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

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Resolution and position
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Gray-Scott self-replicating spots
Gray Scott, Chem Eng Sci (1984), Pearson,
Science (1993), and Lee et al, (1993).
Reaction-diffusion dynamics
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Information 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)
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Continuity 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
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Outlook

• 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

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Hierarchical 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.

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Informal 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.

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...or in a picture
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Conceptual overview
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Equation-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

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Discontinuous Molecular Dynamics

• Discontinuous Molecular Dynamics (DMD)
• Estimating contact (free) energies
• Folding dynamics

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Discontinuous 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

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Thermodynamic 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
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Estimation 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

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The 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

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Summary

• 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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Future 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.