# Talk:Chromosome structure via Euclidean Distance Matrices

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- | <pre> | ||

- | %%% Ronan Fleming, E.coli molecule data | ||

- | %%% -Jon Dattorro, August 9 2008 | ||

- | clear all | ||

- | load ecoli | ||

- | frame = 4; % 1 through 12 | ||

- | G = her49imfs12movfull(frame).cdata; % uint8 | ||

- | G = (double(G)-128)/128; % Gram matrix | ||

- | N = size(G,1); | ||

- | |||

- | Vn = [-ones(1,N-1); speye(N-1)]; | ||

- | [evec evals flag] = eigs(Vn'*G*Vn, [], 20, 'LA'); | ||

- | if flag, disp('convergence problem'), return, end; | ||

- | |||

- | close all | ||

- | Xs = [zeros(3,1) sqrt(real(evals(1:3,1:3)))*real(evec(:,1:3))']; % Projection of -VDV on PSD cone rank 3 | ||

- | plot3(Xs(1,:), Xs(2,:), Xs(3,:), '.') | ||

- | </pre> | ||

- | |||

- | == E.coli realization == | ||

- | [[Image:E.coli-4.jpg|thumb|right|560px|Test image E.coli]] | ||

- | I regard the [http://www.convexoptimization.com/TOOLS/ecoli.mat autocorrelation data] you provided as a Gram matrix. | ||

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- | Then conversion to a Euclidean distance matrix (EDM) is straightforward - <br>Chapter 5.4.2 of [http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Convex Optimization & Euclidean Distance Geometry]. | ||

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- | The program calculates only the first 20 eigenvalues of an oblique projection of the EDM on a positive semidefinite (PSD) cone - <br> | ||

- | Chapter 7.0.4 - 7.1 [http://meboo.convexoptimization.com/BOOK/ProximityProblems.pdf ''ibidem'']. | ||

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- | You can see at runtime that there are many significant eigenvalues; which means, the Euclidean body (the molecule) lives in a space higher than dimension 3, assuming I have interpreted the E.coli data correctly. | ||

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- | To get a picture corresponding to physical reality, we obliquely project the EDM on the closest rank-3 subset of the boundary of that PSD cone; this means, precisely, we truncate eigenvalues. | ||

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- | It is unlikely that this picture is an accurate representation unless the number of eigenvalues of that projection approaches 3 prior to truncation. | ||

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- | Matlab Figures allow 3D rotation in real time, so you can get a good idea of the body's shape. | ||

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- | I include a low-resolution figure here (frame 4) for reference. |