7 research outputs found
Mathematical diffraction of aperiodic structures
Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details
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Predicting multibody assembly of proteins
textThis thesis addresses the multi-body assembly (MBA) problem in the context of protein assemblies. [...] In this thesis, we chose the protein assembly domain because accurate and reliable computational modeling, simulation and prediction of such assemblies would clearly accelerate discoveries in understanding of the complexities of metabolic pathways, identifying the molecular basis for normal health and diseases, and in the designing of new drugs and other therapeutics. [...] [We developed] F²Dock (Fast Fourier Docking) which includes a multi-term function which includes both a statistical thermodynamic approximation of molecular free energy as well as several of knowledge-based terms. Parameters of the scoring model were learned based on a large set of positive/negative examples, and when tested on 176 protein complexes of various types, showed excellent accuracy in ranking correct configurations higher (F² Dock ranks the correcti solution as the top ranked one in 22/176 cases, which is better than other unsupervised prediction software on the same benchmark). Most of the protein-protein interaction scoring terms can be expressed as integrals over the occupied volume, boundary, or a set of discrete points (atom locations), of distance dependent decaying kernels. We developed a dynamic adaptive grid (DAG) data structure which computes smooth surface and volumetric representations of a protein complex in O(m log m) time, where m is the number of atoms assuming that the smallest feature size h is [theta](r[subscript max]) where r[subscript max] is the radius of the largest atom; updates in O(log m) time; and uses O(m)memory. We also developed the dynamic packing grids (DPG) data structure which supports quasi-constant time updates (O(log w)) and spherical neighborhood queries (O(log log w)), where w is the word-size in the RAM. DPG and DAG together results in O(k) time approximation of scoring terms where k << m is the size of the contact region between proteins. [...] [W]e consider the symmetric spherical shell assembly case, where multiple copies of identical proteins tile the surface of a sphere. Though this is a restricted subclass of MBA, it is an important one since it would accelerate development of drugs and antibodies to prevent viruses from forming capsids, which have such spherical symmetry in nature. We proved that it is possible to characterize the space of possible symmetric spherical layouts using a small number of representative local arrangements (called tiles), and their global configurations (tiling). We further show that the tilings, and the mapping of proteins to tilings on arbitrary sized shells is parameterized by 3 discrete parameters and 6 continuous degrees of freedom; and the 3 discrete DOF can be restricted to a constant number of cases if the size of the shell is known (in terms of the number of protein n). We also consider the case where a coarse model of the whole complex of proteins are available. We show that even when such coarse models do not show atomic positions, they can be sufficient to identify a general location for each protein and its neighbors, and thereby restricts the configurational space. We developed an iterative refinement search protocol that leverages such multi-resolution structural data to predict accurate high resolution model of protein complexes, and successfully applied the protocol to model gp120, a protein on the spike of HIV and currently the most feasible target for anti-HIV drug design.Computer Science
Logic and intuition in architectural modelling: philosophy of mathematics for computational design
This dissertation investigates the relationship between the shift in the focus of architectural modelling from object to system and philosophical shifts in the history of mathematics that are relevant to that change. Particularly in the wake of the adoption of digital computation, design model spaces are more complex, multidimensional, arguably more logical, less intuitive spaces to navigate, less accessible to perception and visual comprehension. Such spatial issues were encountered much earlier in mathematics than in architectural modelling, with the growth of analytical geometry, a transition from Classical axiomatic proofs in geometry as the basis of mathematics, to analysis as the underpinning of geometry. Can the computational design modeller learn from the changing modern history, philosophy and psychology of mathematics about the construction and navigation of computational geometrical architectural system model space? The research is conducted through a review of recent architectural project examples and reference to three more detailed architectural modelling case studies. The spatial questions these examples and case studies raise are examined in the context of selected historical writing in the history, philosophy and psychology of mathematics and space. This leads to conclusions about changes in the relationship of architecture and mathematics, and reflections on the opportunities and limitations for architectural system models using computation geometry in the light of this historical survey. This line of questioning was motivated as a response to the experience of constructing digital associative geometry models and encountering the apparent limits of their flexibility as the graph of dependencies grew and the messiness of the digital modelling space increased. The questions were inspired particularly by working on the Narthex model for the Sagrada Família church, which extends to many tens of thousands of relationships and constraints, and which was modelled and repeatedly partially remodelled over a very long period. This experience led to the realisation that the limitations of the model were not necessarily the consequence of poor logical schema definition, but could be inevitable limitations of the geometry as defined, regardless of the means of defining it, the ‘shape’ of the multidimensional space being created. This led to more fundamental questions about the nature of Space, its relationship to geometry and the extent to which the latter can be considered simply as an operational and notational system. This dissertation offers a purely inductive journey, offering evidence through very selective examples in architecture, architectural modelling and in the philosophy of mathematics. The journey starts with some questions about the tendency of the model space to break out and exhibit unpredictable and not always desirable behaviour and the opportunities for geometrical construction to solve these questions is not conclusively answered. Many very productive questions about computational architectural modelling are raised in the process of looking for answers