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56
Using Motion Planning to Study Protein Folding Pathways
 Journal of Computational Biology
, 2001
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Straightening polygonal arcs and convexifying polygonal cycles
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2000
"... Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles ..."
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Cited by 78 (30 self)
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Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewisedifferentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the wellstudied carpenter’s rule conjecture.
Using motion planning to map protein folding landscapes and analyze folding kinetics of known native structures
 In Proc. ACM Int. Conf. on Computational Biology (RECOMB
, 2002
"... We present a novel approach for studying the kinetics of protein folding. The framework has evolved from robotics motion planning techniques called probabilistic roadmap methods (PRMS) that have been applied in many diverse fields with great success. In our previous work, we used a PRMbased techniqu ..."
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Cited by 61 (14 self)
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We present a novel approach for studying the kinetics of protein folding. The framework has evolved from robotics motion planning techniques called probabilistic roadmap methods (PRMS) that have been applied in many diverse fields with great success. In our previous work, we used a PRMbased technique to study protein folding pathways of several small proteins and obtained encouraging results. In this paper, we describe how our motion planning framework can be used to study protein folding kinetics. In particular, we present a refined version of our PRMbased framework and describe how it can be used to produce potential energy landscapes, free energy landscapes, and many folding pathways all from a single roadmap which is computed in a few hours on a desktop PC. Results are presented for 14 proteins. Our ability to produce large sets of unrelated folding pathways may potentially provide crucial insight into some aspects of folding kinetics, such as proteins that exhibit both twostate and threestate kinetics, that are not captured by other theoretical techniques. 1.
A motion planning approach to folding: From paper craft to protein folding
 In Proc. IEEE Int. Conf. Robot. Autom. (ICRA
, 2001
"... In this paper, we present a framework for studying folding problems from a motion planning perspective. In particular, all folding objects are modeled as treelike multilink articulated `robots', where fold positions correspond to joints and areas that cannot fold correspond to links. This for ..."
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Cited by 33 (9 self)
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In this paper, we present a framework for studying folding problems from a motion planning perspective. In particular, all folding objects are modeled as treelike multilink articulated `robots', where fold positions correspond to joints and areas that cannot fold correspond to links. This formulation allows us to apply recent techniques developed in the robotics motion planning community for articulated objects with many degrees of freedom (many links) to folding problems. An important bene t of this approach is that it not only allows us to study foldability questions, such as, can one object be folded (or unfolded) into another object, but also enables us to study the dynamic folding process itself. The framework proposed here has application in traditional motion planning areas such as automation, teaching through demonstration, animation, and most importantly, presents a di erent approach to the most profound problem in computational biology: protein struction prediction. Indeed, our preliminary experimental results with traditional paper crafts (e.g., box folding) and a relatively small protein are quite promising.
Ununfoldable polyhedra with convex faces
 COMPUT. GEOM. THEORY APPL
, 2002
"... Unfolding a convex polyhedron into a simple planar polygon is a wellstudied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex fa ..."
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Cited by 26 (11 self)
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Unfolding a convex polyhedron into a simple planar polygon is a wellstudied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut.
Recent Results in Computational Origami
 In Proceedings of the 3rd International Meeting of Origami Science, Math, and Education
, 2001
"... Computational origami is a recent branch of computer science studying efficient algorithms for solving paperfolding problems. This field essentially began with Robert Lang's work on algorithmic origami design [25], starting around 1993. Since then, the field of computational origami has grown ..."
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Cited by 18 (3 self)
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Computational origami is a recent branch of computer science studying efficient algorithms for solving paperfolding problems. This field essentially began with Robert Lang's work on algorithmic origami design [25], starting around 1993. Since then, the field of computational origami has grown significantly. The purpose of this paper is to survey the work in the field, with a focus on recent results, and to present several open problems that remain. The survey cannot hope to be complete, but we attempt to cover most areas of interest.
Geometric and Computational Aspects of Polymer Reconfiguration
, 2000
"... We examine a few computational geometric problems concerning the structures of polymers. We use a standard model of a polymer, a polygonal chain (path of line segments) in three dimensions. The chain can be reconfigured in any manner as long as the edge lengths and the angles between consecutive edg ..."
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Cited by 18 (3 self)
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We examine a few computational geometric problems concerning the structures of polymers. We use a standard model of a polymer, a polygonal chain (path of line segments) in three dimensions. The chain can be reconfigured in any manner as long as the edge lengths and the angles between consecutive edges remain fixed, and no two edges cross during the motion. We discuss preliminary results on the following problems. Given a chaib, select...
Ununfoldable Polyhedra
, 1999
"... A wellstudied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can inde ..."
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Cited by 16 (9 self)
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A wellstudied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that "open" polyhedra with convex faces may not be unfoldable no matter how they are cut.
Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings
, 2003
"... Abstract. Let S be the boundary of a convex polytope of dimension d + 1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into R d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R d by id ..."
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Cited by 14 (4 self)
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Abstract. Let S be the boundary of a convex polytope of dimension d + 1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into R d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v ∈ S, which is the exponential map to S from the tangent space at v. We characterize the cut locus (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3polytopes into R 2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic nonpolynomial complexity of nonconvex manifolds.
VertexUnfoldings of Simplicial Manifolds
"... We present an algorithm to unfold any triangulated 2manifold (in particular, any simplicial polyhedron) into a nonoverlapping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles ..."
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Cited by 14 (3 self)
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We present an algorithm to unfold any triangulated 2manifold (in particular, any simplicial polyhedron) into a nonoverlapping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles are connected at vertices, but not necessarily joined along edges. We extend our algorithm to establish a similar result for simplicial manifolds of arbitrary dimension.