Anders ErikssonComplex Systems Group

Dept. Energy and Environmental Research

Chalmers

EMBIOCambridge July 2005

Complex Systems at Chalmers

Information Theory and Multi-scale Simulations

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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 << IF << 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

Free energy (light, food, fuel, …)

Low-value energy (waste, heat, …)

Chemical self-organising system

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Gibb’s 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 Boltzmann’s 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:

QuickTime och enTIFF (LZW)-dekomprimerarekrävs för att kunna se bilden.

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QuickTime och enTIFF (LZW)-dekomprimerare

krävs för att kunna se bilden.

High resolution (r ≈ 0)

€

˜ c i(r,x)

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Resolution and position

x

y

r

Res

olut

ion

(leng

th s

cale

)

Kchem

Kspatial

r

kspatial(r)

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Gray-Scott self-replicating spots

in-flow of U

out-flow of V

GV

VVU

→→+ 32

VFVkVVkUVDt

V

UFVVkUUDt

U

backv

backu

−−−+∇=

−+−−∇=

22

22

)(

)1()(

∂∂∂∂

Reaction-diffusion dynamics:

Gray & Scott, Chem Eng Sci (1984),Pearson, Science (1993), and Lee et al, (1993).

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Information density in the model

in-flow of U

out-flow of V

GV

VVU

→→+ 32

Information density: k(r=0.01, x, t)

QuickTime och en-dekomprimerare

krävs för att kunna se bilden.

QuickTime och en-dekomprimerare

krävs för att kunna se bilden.

QuickTime och en-dekomprimerare

krävs för att kunna se bilden.

k(r=0.05, x, t)Concentration of V:cV (x, t)

The information density for two resolution levels r illustrate the presence of spatial structure at different length scales.

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Continuity equation for information

x

y

r

Res

olut

ion

(leng

th s

cale

)

Kchem

Kspatial

Inflow of chemical information (exergy)

Destruction of information (entropy production)

j(r, x, t)

jr(r, x, t)

k(r, x, t)

J(r, x, t)

Flow in scale

Flow in space

Sinks (open system)

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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

Effective potential

Distance

• Contact potential Piecewise constant Hard-sphere core Potential well for

residue-residue contact energy gain

Finite range

Bond potential

Contact potential

• Linear chain of spheres, connected by bonds Bonds are hard-sphere

• 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 grid:

where 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.

fan flow circular flow