A method to compute the volume of a molecule

Pavani, Raffaella; Ranghino, Graziella

doi:10.1016/0097-8485(82)80006-5

Cited by 34 publications

(18 citation statements)

References 7 publications

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“…Outside the completely molten zone are concentric spheres of partial melting; the critical velocity for partial melting in dunite is 6.5 km s −1 (P H = 102 GPa). For large amounts of melting geometric corrections become important: one must find the overlapping volume of the melt sphere and the planetary sphere [25]. Results show that substantial melting is expected during the accretion of rocky planets.…”

Section: −2β Pmentioning

confidence: 99%

Melting in super-earths

Stixrude

2014

Phil. Trans. R. Soc. A.

104

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We examine the possible extent of melting in rockiron super-earths, focusing on those in the habitable zone. We consider the energetics of accretion and core formation, the timescale of cooling and its dependence on viscosity and partial melting, thermal regulation via the temperature dependence of viscosity, and the melting curves of rock and iron components at the ultra-high pressures characteristic of superearths. We find that the efficiency of kinetic energy deposition during accretion increases with planetary mass; considering the likely role of giant impacts and core formation, we find that super-earths probably complete their accretionary phase in an entirely molten state. Considerations of thermal regulation lead us to propose model temperature profiles of super-earths that are controlled by silicate melting. We estimate melting curves of iron and rock components up to the extreme pressures characteristic of superearth interiors based on existing experimental and ab initio results and scaling laws. We construct superearth thermal models by solving the equations of mass conservation and hydrostatic equilibrium, together with equations of state of rock and iron components. We set the potential temperature at the core-mantle boundary and at the surface to the local silicate melting temperature. We find that ancient (∼4 Gyr) super-earths may be partially molten at the top and bottom of their mantles, and that mantle convection is sufficiently vigorous to sustain dynamo action over the whole range of super-earth masses.

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Section: −2β Pmentioning

confidence: 99%

Melting in super-earths

Stixrude

2014

Phil. Trans. R. Soc. A.

104

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“…For example, Pavani and Ranghino [51] proposed a method for computing the volume of a molecule by inclusion-exclusion. In their implementation, only intersections of up to three balls were considered.…”

Section: Measuring Macromoleculesmentioning

confidence: 99%

Measuring the Shapes of Macromolecules – And Why It Matters

Mach

Koehl

2013

Computational and Structural Biotechnology Journal

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The molecular basis of life rests on the activity of biological macromolecules, mostly nucleic acids and proteins. A perhaps surprising finding that crystallized over the last handful of decades is that geometric reasoning plays a major role in our attempt to understand these activities. In this paper, we address this connection between geometry and biology, focusing on methods for measuring and characterizing the shapes of macromolecules. We briefly review existing numerical and analytical approaches that solve these problems. We cover in more details our own work in this field, focusing on the alpha shape theory as it provides a unifying mathematical framework that enable the analytical calculations of the surface area and volume of a macromolecule represented as a union of balls, the detection of pockets and cavities in the molecule, and the quantification of contacts between the atomic balls. We have shown that each of these quantities can be related to physical properties of the molecule under study and ultimately provides insight on its activity. We conclude with a brief description of new challenges for the alpha shape theory in modern structural biology.

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“…For example, Pavani and Ranghino [16] proposed a method for computing the volume of a molecule by inclusion-exclusion. In their implementation, only intersections of up to three balls were considered.…”

Section: Introductionmentioning

confidence: 99%

Geometric measures of large biomolecules: Surface, volume, and pockets

Mach

Koehl

2011

J Comput Chem

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Geometry plays a major role in our attempt to understand the activity of large molecules. For example, surface area and volume are used to quantify the interactions between these molecules and the water surrounding them in implicit solvent models. In addition, the detection of pockets serves as a starting point for predictive studies of biomolecule-ligand interactions. The alpha shape theory provides an exact and robust method for computing these geometric measures. Several implementations of this theory are currently available. We show however that these implementations fail on very large macromolecular systems. We show that these difficulties are not theoretical; rather, they are related to the architecture of current computers that rely on the use of cache memory to speed up calculation. By rewriting the algorithms that implement the different steps of the alpha shape theory such that we enforce locality, we show that we can remediate these cache problems; the corresponding code, UnionBall has an apparent (n) behavior over a large range of values of n (up to tens of millions), where n is the number of atoms. As an example, it takes 136 seconds with UnionBall to compute the contribution of each atom to the surface area and volume of a viral capsid with more than five million atoms on a commodity PC. UnionBall includes functions for computing the surface area and volume of the intersection of two, three and four spheres that are fully detailed in an appendix. UnionBall is available as an OpenSource software.

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A method to compute the volume of a molecule

Cited by 34 publications

References 7 publications

Melting in super-earths

Melting in super-earths

Measuring the Shapes of Macromolecules – And Why It Matters

Geometric measures of large biomolecules: Surface, volume, and pockets

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