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EXPLORING PROTEIN FOLDING TRAJECTORIES USING

GEOMETRIC SPANNERS

D. RUSSEL and L. GUIBAS

Computer Science Department

353 Serra Mall

Stanford, CA 94305, USA

E-mail: {drussel,guibas}@cs.stanford.edu

We describe the 3-D structure of a protein using geometric spanners — geometric

graphs with a sparse set of edges where paths approximate the n2inter-atom

distances. The edges in the spanner pick out important proximities in the structure,

labeling a small number of atom pairs or backbone region pairs as being of primary

interest. Such compact multiresolution views of proximities in the protein can

be quite valuable, allowing, for example, easy visualization of the conformation

over the entire folding trajectory of a protein and segmentation of the trajectory.

These visualizations allow one to easily detect formation of secondary and tertiary

structures as the protein folds.

1Introduction

There has been extensive work on visualizing the 3-D structure of proteins

in ways that attempt to make the certain aspects of the structure more ap-

parent. For example, commonly used software packages such as RasMol [10],

ProteinExplorer

[9], or SPV [8], among others, permit visualizations via

hard-sphere models, stick models, and ribbon models that emphasize differ-

ent aspects of the protein surface or secondary structure. Even more abstract

visualizations have been used as a tool for understanding intra-molecular prox-

imities, including contact maps and distance matrix images [13]. None of these

approaches work very well, however, if the goal is to visualize proteins in mo-

tion and not just their static conformations.

Large corpora of molecular trajectories are becoming available through

efforts such as Folding@Home [12] where molecular simulations are carried

out on distributed networks of many thousands of computers. There is an

increasing need to compare, classify, summarize, and organize the space of such

protein trajectories with an eye toward advancing our understanding of protein

folding by studying their ensemble behaviors. Most currently used methods for

understanding such data revolve around computing a few summary statistics

for each conformation, such as radius of gyration or number of native contacts

and watching how these evolve during each trajectory. More similarly, the

chemical distance, a statistic of an adjacency graph of the amino acids, was

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use to differentiate folded and unfolded states [4]. In this paper we explore the

use of a more rich and abstract representation of the protein structure, based

on spanners, which makes the task of understanding and exploring the space

of protein motions easier.

Our basic idea is to take the continuous folding process and map it to

a more discrete combinatorial representation. This representation focuses on

higher-level geometric proximities that tend to form and be more stable over

time rather than atom coordinates or specific aspects of secondary/tertiary

structure. Specifically, we look at the formation of proximities between differ-

ent parts of the protein across a range of scales, and track the changes of such

proximities over time. Our more abstract description of the folding process is

in terms of ‘proximity events’ — when certain proximities are formed or de-

stroyed. Together, these characterize the folding process in a qualitative way

and capture the important aspects of the trajectory, the sequence of conforma-

tions adopted by a protein in a particular folding path. Just as an algebraic

topologist captures the essence of the connectivity of a continuous space in a

few discrete invariants (the homology groups), we aim to capture the signifi-

cant conformational changes during motion through a discrete representation

of proximities that form and break.

We use geometric spanners to accomplish this goal.

abstract graph with weights on its edges, a spanner is a sparse subgraph (in

the sense of having a number of edges roughly proportional to the number of

vertices), such that all edges in the full graph can be well approximated by

paths in the spanner (in the sense that the sum of the weights of edges of the

path in the spanner is very close to the weight of the original graph edge). In

the geometric setting the vertices in the original graph are points each pair

of which is connected by an edge with weight equal to the Euclidean distance

between the corresponding pair of points. The quality of the approximation

can be controlled by varying the number of edges in the spanner.

Note that spanners are at once generalizations of contact maps as well

as compressions of distance matrices. One can think of a spanner as a mul-

tiresolution contact map that allows an approximate reconstruction of the full

distance matrix (and therefore the full 3-D structure as well).

We propose to use these combinatorial structures as a tool for capturing

the important proximities of a protein conformation and, in this paper, for

comparing and visualizing sequences of protein conformations from molecular

trajectories. Key properties of the spanner that facilitate these goals include:

Starting from an

• Spanners are proximity based — this parallels proteins where local in-

teractions determine the behavior.

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• Spanners are discrete — they have a combinatorial structure whose de-

scription does not include any geometric coordinates.

• Spanners are controllable — we can produce descriptors that more loosely

or more tightly capture the shape of the protein, converging to distance

matrices as the approximation gets tighter.

• Spanners are uniform — there is only one type of combinatorial element,

namely an edge. This makes comparison, processing and display simpler.

• Spanners can be made smooth — small changes in the protein confor-

mation generally result in few large changes in the spanner, enabling

tracking of the spanner structure over time.

• Spanners are local — the combinatorial features, edges, are affected by a

small subset of the total point set. This means that changes in one part

of the protein do not generally affect the spanner edges in other parts.

As a result the edges can be assigned semantic meaning based on their

endpoints, rather than on larger regions of the protein.

We use our spanners to investigate the folding of the protein BBA5 [11]

using simulation data produce produced by the Folding@Home project. The

spanners enable us to produce diagrams which show the formation (and some-

times dissolution) of secondary and tertiary structure during a whole folding

trajectory and allow us to segment these trajectories into logical parts. We

expect that the spanner approach will provide a valuable toolkit for the un-

derstanding and visualization of protein trajectories.

In the next sections we describe how we construct and smooth our spanner-

based representation and how we use it to visualize trajectories. Then we

discuss our how we have used spanners to try to understand the folding of

BBA5. Finally we mention other promising applications of our spanner based

representations.

2Representing Proteins Using Spanners

We first provide a more rigorous definition of a geometric spanner. Let P be a

set of points in R3, Euclidean three-space, and G be a Euclidean graph on P

(graph whose vertices are points from P and whose edge weights are Euclidean

distances between the endpoints of the edge). For a parameter s > 1, known

as the stretch factor, G is a spanner for P, if for all pairs of points i and j

in P with Euclidean coordinates piand pj, πG(i,j) ≤ s||pipj|| where πG(i,j)

denotes the shortest path distance between i and j in the graph G. Thus,

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the spanner represents the quadratic number of interpoint distances in P by

the much sparser set of edges in G. There is a vast literature on spanners

that we will not attempt to review in here any detail; many different spanner

constructions are possible. It has been shown that for s arbitrarily close to 1,

there exist spanners whose number of edges is proportional to the size of P.

The reader is referred to a number of survey papers for background material

and additional references [1, 6].

For simplicity, we only use the backbone atoms of the protein. This allows

us to meaningfully identify each atom by its index along the backbone, so an

edge i,j connects the ith and jth atoms in the backbone. We can identify the

edge i,j with the point (i,j) where i < j. We define the distance, between two

edges i0,j0and i1,j1as the L1distance between the points (i0,j0) and (i1,j1),

namely |i0−i1|+|j0−j1|. We will write it d(i0,j0,i1,j1). Two edges are close

if they have a small L1distance between them the corresponding points. The

length as opposed to weight of an edge l(i,j) is defined as j − i. Throughout

the section s will designate the stretch factor. A s-spanner is a spanner with

stretch factor s.

2.1Computation

We use what is known in the literature as the ‘greedy’ spanner. Its computation

is conceptually very simple: starting with graph G initially containing only the

points P, test each of the?|P|

πG(i,j). If so, add the edge i,j to the G. We call this test the inclusion test.

The algorithm runs in O(n3) time due to the quadratic number of edges and

the worst case linear time required to evaluate the inclusion test.

This greedy spanner construction has been shown to have asymptotically

optimal complexity (number of edges) and weight (the sum of the lengths of the

edges) as well as good practical complexity and weight [2]. Having low weight is

important in our context since we want the spanner to consist of as many short

edges as possible in order to capture local interactions. Euclidean spanners can

be also be produced in O(nlog2n) time with the same asymptotic edge count

and weight bounds [3] although we have not implemented such methods.

If implemented naively, performing the inclusion test for long edges domi-

nates the running time as it requires a nearly linear time graph search for each

of these edges. However, such long edges are extremely unlikely to be in a

spanner of a packed protein. If we maintain an upper bound on the graph dis-

tance, dG(i,j) ≥ πG(i,j) between each pair of points, i,j, then we can quickly

eliminate any candidate edge for which s?ij? > dG(i,j). We can similarly

2

?interpoint candidate edges for inclusion, ordered

from shortest to longest. For each candidate edge i,j, check if s?pipj? <

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prune many of the paths while searching.

These upper bounds can be maintained lazily as graph searches are per-

formed. To tighten the upper bounds and further accelerate the process, it is

advantageous to periodically compute the all atoms shortest path distances in

the current spanner (an O(n2) process). In addition, we guide the search using

the Euclidean distance as a lower bound on the graph distance to bias our

search direction, as in the graph search algorithm A∗. Using these heuristics,

the 2-spanner of an 800 atom backbone of a protein can be computed in about

a second.

The kinetic spanner proposed in [7] is a possible alternative. It can be

cheaply maintained as the underlying points move around. However, it is non-

canonical, making comparison between trajectories tricky and it has more long

edges, which are hard to assign biological meaning.

2.2Spanners of Proteins

Figure 1 shows spanners computed using different expansion factors for single

protein and gives an estimate of the number of spanner edges per point for

typical proteins in their native state.

Expansion

|G|/|P|

1.25

4.5

1.5

1.5

2.0

.52

2.5

.31

3.0

.21

Example

Figure 1: Example spanners and average edge per point for various expansion factors. We

mostly use 2-3 spanners for out computations and visualizations as spanners below 2 get

very dense.

Secondary structure creates very well defined patterns in the spanner. If

each edge is visualized as a point (i,j), then α helices appear as a sequence of

points (i + kq,j + kq) where k is a counter variable and q is a stepsize which

depends on the expansion factor. For a expansion factors between 2 and 3 the

step size is 3, the edges are just longer when the expansion factor is larger.

For a expansion factor of 1.5, the stepsize is still 3 but there are several edges

leaving from each of the points. β hairpins appear as series of points heading in

an orthogonal direction to helices, namely, (i+kq,j−kq). k is 2 for 2 spanners

and rises to 4 or 6 for 3 spanners (depending on how the hairpin twists). Both

patterns are shown in Figure 2.