Topographies, Dynamics and Kinetics on the Landscape of Multidimensional Potential

SurfacesR. Stephen Berry

The University of ChicagoGlobal Optimiization Theory Institute

Argonne National Laboratory

8-10 September 2003

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?

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?

What are some of the steps we’ve 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

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

First example: Ar19

Samples of its monotonic sequences

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

Ah, but then there’s (KCl)32!A very different beast

(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

What about proteins? Shouldn’t 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 -barrel efficiently

• The long-range character of its forces comes from the constraint of retaining the integrity of the polymer chain

So what’s its topography?

Not a bad staircase at all, but...

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?

Push that question further:Could there be more than one

“there”?• Do we know whether native structures are

really unique? NO! Active sites may well have unique structures, but we don’t know whether variability may occur in the outer scaffolding. There is some evidence that it may, but nothing definite.

• Experimental tests might be possible.

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 doesn’t give clean, bright, interpretable spots?

Return to what is established: we can sometimes infer topographies

from 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.

Example: Bovine Pancreatic Trypsin Inhibitor (BPTI)

(Fernández, Kostov, RSB)

The effective potential, found by a kind of Monte Carlo search procedure with folding and

unfolding, is indeed staircase-like

• So let’s generalize:

• Structure-seekers, vs.

• Glass-formers

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)

Here are disconnection diagrams for LJ13 and LJ19, two examples of

palm trees (Wales)

And a pathological case, LJ38

Why pathological? One close-packed structure, the deepest,

in a sea of icosahedra

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?

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?

And then,

• What should our priorities be now, and

• How should we set them? How should we balance what’s important, with what’s possible?

Connect topography with dynamics: Ar55 simulated @15, 20, 25 K

Likewise, (KCl)32 @ 350, 550 and 600 K: High T => fast, deep