Celebratio Mathematica

P. Francis-Lyon, S. Gu, J. Hass, N. Amenta, and P. Koehl: “Sampling the conformation of protein surface residues for flexible protein docking,” BMC Bioinform. 11 (2010), pp. 575–588. article

Nina Amenta	Related
Patricia Francis-Lyon	Related
Shengyin Gu	Related
Patrice Koehl	Related

S. Gu, P. Koehl, J. Hass, and N. Amenta: “Surface-histogram: A new shape descriptor for protein-protein docking,” Proteins Struct. Funct. Bioinf. 80 : 1 (January 2012), pp. 221–238. article

Determining the structure of protein-protein complexes remains a difficult and lengthy process, either by NMR or by X-ray crystallography. Several computational methods based on docking have been developed to support and even serve as possible alternatives to these experimental methods. In this article, we introduce a new protein-protein docking algorithm, shDock, based on shape complementarity. We characterize the local geometry on each protein surface with a new shape descriptor, the surface-histogram. We measure the complementarity between two surface-histograms, one on each protein, using a modified Manhattan distance. When a match is found between two local protein surfaces, a model is generated for the protein complex, which is then scored by checking for collision between the two proteins. We have tested our algorithm on Version 3 of the ZDOCK protein-protein docking benchmark. We found that for 110 out of the 124 test cases of bound docking in the benchmark, our algorithm was able to generate a model in the top 3600 candidates for the protein complex within an root-mean-square deviation of \( 2.5\,Å \) from its native structure. For unbound docking predictions, we found a model within \( 2.5\,Å \) in the top 3600 models in 54 out of 124 test cases. A comparison with other shape-based docking algorithms demonstrates that our approach gives significantly improved performance for both bound and unbound docking test cases.

Nina Amenta	Related
Shengyin Gu	Related
Patrice Koehl	Related

A. Tsui, D. Fenton, P. Vuong, J. Hass, P. Koehl, N. Amenta, D. Coeurjolly, C. DeCarli, and O. Carmichael: “Globally optimal cortical surface matching with exact landmark correspondence,” pp. 487–498 in Information processing in medical imaging: 23rd international conference, IPMI 2013 (Asilomar, CA, 28 June–3 July 2013). Edited by J. C. Gee, S. Joshi, K. M. Pohl, W. M. Wells, and L. Zöllei. Lecture Notes in Computer Science 7917. Springer (Berlin), 2013. incollection

We present a method for establishing correspondences between human cortical surfaces that exactly matches the positions of given point landmarks, while attaining the global minimum of an objective function that quantifies how far the mapping deviates from conformality. On each surface, a conformal transformation is applied to the Euclidean distance metric, resulting in a hyperbolic metric with isolated cone point singularities at the landmarks. Equivalently, each surface is mapped to a hyperbolic orbifold: a pillow-like surface with each point landmark corresponding to a pillow corner. An initial surface-to-surface mapping exactly aligns the landmarks, and gradient descent is used to find the single, global minimum of the Dirichlet energy of the remainder of the mapping. Using a population of real MRI-based cortical surfaces with manually labeled sulcus endpoints as landmarks, we evaluate the approach by how much it distorts surfaces and by its biological plausibility: how well it aligns previously-unseen anatomical landmarks and by how well it promotes expected associations between cortical thickness and age. We show that, compared to a painstakingly-tuned approach that balances a tradeoff between minimizing landmark mismatch and Dirichlet energy, our method has similar biological plausibility, superior surface distortion, a better theoretical foundation, and fewer arbitrary parameters to tune. We also compare to conformal mapper in the spherical domain to show that sacrificing exact conformality of the mapping does not cause noticeable reductions in biological plausibility.

Padmini Joshi	Related
Alex Tsui	Related
Phong Vuong	Related
Nina Amenta	Related
Owen Carmichael	Related
David Coeurjolly	Related
Charles DeCarli	Related
David Fenton	Related
Patrice Koehl	Related

@incollection {key76348359,
	AUTHOR = {Tsui, Alex and Fenton, Devin and Vuong,
		 Phong and Hass, Joel and Koehl, Patrice
		 and Amenta, Nina and Coeurjolly, David
		 and DeCarli, Charles and Carmichael,
		 Owen},
	TITLE = {Globally optimal cortical surface matching
		 with exact landmark correspondence},
	BOOKTITLE = {Information processing in medical imaging:
		 23rd international conference, {IPMI}
		 2013},
	EDITOR = {Gee, James C. and Joshi, Sarang and
		 Pohl, Kilian M. and Wells, William M.
		 and Z\"ollei, Lilla},
	SERIES = {Lecture Notes in Computer Science},
	NUMBER = {7917},
	PUBLISHER = {Springer},
	ADDRESS = {Berlin},
	YEAR = {2013},
	PAGES = {487--498},
	DOI = {10.1007/978-3-642-38868-2_41},
	NOTE = {(Asilomar, CA, 28 June--3 July 2013).},
	ISSN = {0302-9743},
	ISBN = {9783642388675},
}

P. Koehl and J. Hass: “Automatic alignment of genus-zero surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 36 : 3 (March 2014), pp. 466–478. article

A new algorithm is presented that provides a constructive way to conformally warp a triangular mesh of genus zero to a destination surface with minimal metric deformation, as well as a means to compute automatically a measure of the geometric difference between two surfaces of genus zero. The algorithm takes as input a pair of surfaces that are topological 2-spheres, each surface given by a distinct triangulation. The algorithm then constructs a map \( f \) between the two surfaces. First, each of the two triangular meshes is mapped to the unit sphere using a discrete conformal mapping algorithm. The two mappings are then composed with a Mobius transformation to generate the function \( f \). The Mobius transformation is chosen by minimizing an energy that measures the distance of \( f \) from an isometry. We illustrate our approach using several “real life” data sets. We show first that the algorithm allows for accurate, automatic, and landmark-free nonrigid registration of brain surfaces. We then validate our approach by comparing shapes of proteins. We provide numerical experiments to demonstrate that the distances computed with our algorithm between low-resolution, surface-based representations of proteins are highly correlated with the corresponding distances computed between high-resolution, atomistic models for the same proteins.

J. Hass and P. Koehl: “How round is a protein? Exploring protein structures for globularity using conformal mapping,” Front. Mol. Biosci. 1 (December 2014), pp. 1–26. article

We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map \( f \) between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy \( E_S(f) \) that measures the distance of \( f \) from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that \( E_S(f) \) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy \( E_S(f) \) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that \( E_S(f) \) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry.

J. Hass and P. Koehl: A metric for genus-zero surfaces. Preprint, July 2015. ArXiv 1507.00798 techreport

P. Koehl and J. Hass: “Landmark-free geometric methods in biological shape analysis,” J. R. Soc. Interface 12 : 113 (2015), pp. 1–11. article

In this paper, we propose a new approach for computing a distance between two shapes embedded in three-dimensional space. We take as input a pair of triangulated genus zero surfaces that are topologically equivalent to spheres with no holes or handles, and construct a discrete conformal map \( f \) between the surfaces. The conformal map is chosen to minimize a symmetric deformation energy \( E_{sd}(f) \) which we introduce. This measures the distance of \( f \) from an isometry, i.e. a non-distorting correspondence. We show that the energy of the minimizing map gives a well-behaved metric on the space of genus zero surfaces. In contrast to most methods in this field, our approach does not rely on any assignment of landmarks on the two surfaces. We illustrate applications of our approach to geometric morphometrics using three datasets representing the bones and teeth of primates. Experiments on these datasets show that our approach performs remarkably well both in shape recognition and in identifying evolutionary patterns, with success rates similar to, and in some cases better than, those obtained by expert observers.

J. Hass and P. Koehl: “Comparing shapes of genus-zero surfaces,” J. Appl. Comput. Topol. 1 : 1 (2017), pp. 57–87. MR 3975549 article

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