Topographies, Dynamics and Kinetics on the Landscape of Multidimensional Potential Surfaces

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Topographies, Dynamics and Kinetics on the
Landscape of Multidimensional Potential Surfaces

• R. Stephen Berry
• The University of Chicago
• Global Optimiization Theory Institute
• Argonne National Laboratory
• 8-10 September 2003

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An Overview

• First, identify the issues and the problems What
  are the important, challenging problems from the
  perspective of the physicist or chemist? What
  steps have we made toward elucidating them? What
  tools have we used?
• Then, what lies ahead What kinds of known
  problems have resisted explication? What new
  directions might we explore?

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What are obvious, big problems?

• Dealing with incredibly complex landscapes with
  all sorts of topographies
• Deciding what information is useful (Wayne Booth
  What information is worth having?)
• Connecting topographies with kinetics and
  dynamics how can we infer about these from
  knowledge of topography?

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What are some of the steps weve made toward
elucidating these?

• Inventing efficient algorithms for finding
  stationary points, even in many dimensions
• Inventing ways to identify sequences of
  geometrically-linked stationary points
• Inventing patterns of topographies by using
  disconnection diagrams
• Learning how to construct reliable master
  equations

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Some more steps accomplished

• Devising ways to simplify multidimensional
  surfaces, such as smoothing bumps and
  characterizing gross structure (Scheraga)
• Finding ways to extract key variables, e.g.
  principal components principal coordinates
• Linking dynamics with character of
  topography--but just qualitatively, so far

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First example Ar19Samples of its monotonic
sequences
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Ar19 has a sawtooth topography!

• This makes it a glass-former quenched from
  liquid, it becomes amorphous
• The topography is a consequence of short-range
  interparticle forces
• Hence few particles move when the cluster passes
  from one local minimum to the next

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Ah, but then theres (KCl)32!A very different
beast
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(KCl)32 is a structure-seeker with a staircase
topography!

• (KCl)32 finds a rocksalt structure when quenched
  from liquid in more than ca. 5 vibrations,
  against naïve odds of 1/1011
• Characterized by long-range or effective
  long-range interparticle forces
• Many particles move in most well-to-well passages

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What about proteins? Shouldnt they be
structure-seekers?

• Look first at the topography of a protein model,
  a 46-bead object developed by Skolnick and then
  Thirumalai, a system that forms a b-barrel
  efficiently
• The long-range character of its forces comes from
  the constraint of retaining the integrity of the
  polymer chain

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So whats its topography?
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Not a bad staircase at all, but...
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This model system, like the alkali halide
cluster, has lots of deep basins, very much alike

• The pathways down into one look about the same as
  those in all of the others
• Puzzle In a real protein, what makes the native
  structure so special? How does the topography
  lead the system there?

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Push that question furtherCould there be more
than onethere?

• Do we know whether native structures are really
  unique? NO! Active sites may well have unique
  structures, but we dont know whether variability
  may occur in the outer scaffolding. There is
  some evidence that it may, but nothing definite.
• Experimental tests might be possible.

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What is the evidence for uniqueness?

• First and foremost, crystal structures.
• But crystals are selective, and may only admit
  molecules with the same structure as those
  already there.
• Moreover crystallographers are also selective.
  Who wants to take an X-ray picture of a crystal
  that doesnt give clean, bright, interpretable
  spots?

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Return to what is established we can sometimes
infer topographiesfrom kinetics

• Forward and backward rates, and microscopic
  reversibility, allow us to infer barrier heights,
  for effective potential landscapes as well as for
  real and explicitly simulated ones.

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Example Bovine Pancreatic Trypsin Inhibitor
(BPTI)(Fernández, Kostov, RSB)
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The effective potential, found by a kind of Monte
Carlo search procedure with folding and
unfolding, is indeed staircase-like

• So lets generalize
• Structure-seekers, vs.
• Glass-formers

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Now what are some problems that have resisted
explication?

• Simply classifying and quantifying the kinds of
  complexity of surfaces (but the classification of
  disconnection diagrams is a significant step in
  this direction)
• From this, determining the gross basin structure
  (again, the kind of disconnection diagrams tells
  much)

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Here are disconnection diagrams for LJ13 and
LJ19, two examples of palm trees (Wales)
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And a pathological case, LJ38
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Why pathological? One close-packed structure,
the deepest,in a sea of icosahedra
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More open problems

• How can we construct efficient, reliable
  simplified representations of kinetics, e.g. from
  simplified master equations?
• How can we determine the reliability of a method
  of simplification, e.g. a statistically-based
  master equation, or a principal component
  representation or some combination of these?

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Still more and more...

• How can we coarse-grain mechanical
  representations in ways that give reliable
  results for long-time processes, such as those
  taking milliseconds?
• How can we integrate coarse-grained and
  finer-grained approaches?
• How can we characterize the variety and
  multiplicity of folding or relaxation paths?

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And then,

• What should our priorities be now, and
• How should we set them? How should we balance
  whats important, with whats possible?

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Connect topography with dynamics Ar55 simulated
_at_15, 20, 25 K
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Likewise, (KCl)32 _at_ 350, 550 and 600 K High T gt
fast, deep