Talk:Chromosome structure via Euclidean Distance Matrices

<pre>

%%% Ronan Fleming, E.coli molecule data

%%% -Jon Dattorro, August 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);

D = diag(G)*ones(N,1)' + ones(N,1)*diag(G)' - 2*G; % EDM D

clear her49imfs12movfull G;

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

VDV = (-Vn'*D*Vn)/2;

clear D Vn

[evec evals flag] = eigs(VDV, [], 20, 'LR');

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

close all

Xs = 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 autocorrelation data you provided as a Gram matrix.

Then conversion to an EDM is straightforward - Chapter 5.4.2 of [http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Convex Optimization & Distance Geometry].

The program calculates only the first 20 eigenvalues of the projection of the EDM on a positive semidefinite (PSD) cone.

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.

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

It is unlikely that this picture is an accurate representation unless the number of eigenvalues of the EDM projection approaches 3 to begin with.

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

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