Anders Eriksson Complex Systems Group Dept. Energy and Environmental Research Chalmers EMBIO Cambridge July 2005 Complex Systems at Chalmers Information Theory and Multi-scale Simulations

Complex Systems — Chalmers 1 Outline People Information theory Based on presentation by Kristian Lindgren Hierarchical dynamics Based on presentation by Martin Nilsson Jacobi Discontinuous Molecular Dynamics

Complex Systems — Chalmers 2 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

Complex Systems — Chalmers 3 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)

Complex Systems — Chalmers 4 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

Complex Systems — Chalmers 5 Thermodynamic context 2 nd law of thermodynamics: in total, entropy is increasing Out-of-equilibrium, low-entropy state maintained by exporting more entropy than what is imported and produced Free energy (light, food, fuel, …) Low-value energy (waste, heat, …) Chemical self- organising system

Complex Systems — Chalmers 6 Gibb’s free energy and information The free energy E of a concentration pattern c i (x) can be related to the information-theoretic relative information K : where k B is Boltzmann’s constant and T 0 is the temperature. The free energy E is related to information content I (in bits) by

Complex Systems — Chalmers 7 Decomposition of information The information can be decomposed into two terms (quantifying deviation from equilibrium and spatial homogeneity, respectively): The spatial information K spatial can be further decomposed into contributions from different length scales (resolution) r, and further from positions x:

Complex Systems — Chalmers 8 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: High resolution ( r ≈ 0)

Complex Systems — Chalmers 9 Resolution and position x y r Resolution (length scale) K chem K spatial r k spatial (r)

Complex Systems — Chalmers 10 Gray-Scott self-replicating spots Reaction-diffusion dynamics: Gray & Scott, Chem Eng Sci (1984), Pearson, Science (1993), and Lee et al, (1993).

Complex Systems — Chalmers 11 Information density in the model Information density: k(r=0.01, x, t) k(r=0.05, x, t) Concentration of V: c V (x, t) The information density for two resolution levels r illustrate the presence of spatial structure at different length scales.

Complex Systems — Chalmers 12 Continuity equation for information x y r Resolution (length scale) K chem K spatial Inflow of chemical information (exergy) Destruction of information (entropy production) j(r, x, t) j r (r, x, t) k(r, x, t) J(r, x, t) Flow in scale Flow in space Sinks (open system)

Complex Systems — Chalmers 13 Outlook Generalised 2 nd 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

Complex Systems — Chalmers 14 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.

Complex Systems — Chalmers 15 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.

Complex Systems — Chalmers 16...or in a picture:

Complex Systems — Chalmers 17 Conceptual overview

Complex Systems — Chalmers 18 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

Complex Systems — Chalmers 19 Discontinuous Molecular Dynamics Discontinuous Molecular Dynamics (DMD) Estimating contact (free) energies Folding dynamics

Complex Systems — Chalmers 20 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

Complex Systems — Chalmers 21 Thermodynamic properties Discrete set of energy levels Only depends on which residue are in contact Can reproduce basic thermodynamic properties of clusters Zhou et al. (1997), J. Chem. Phys. 107(24), p

Complex Systems — Chalmers 22 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

Complex Systems — Chalmers 23 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

Complex Systems — Chalmers 24 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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Complex Systems — Chalmers 29 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.