LeastSquares Formulation

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Least-Squares Formulation
http//www.math.iastate.edu/wu/math597.html
http//www.math.iastate.edu/wu/Math597HW0000/index
.htm

• Math/BCB/ComS597
• Zhijun Wu
• Department of Mathematics

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Given a set of distance constraints obtained from
NMR, find the coordinates of the atoms (and hence
the structure of the protein) satisfying the
distance constraints.
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Ensemble of Structures
Representative Structure
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Problems with Exact Distances
When all distances are available
can be solved in P
When only a subset of distances is available
NP-complete
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Least-Squares Formulation
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Linear Least-Squares
Curve Fitting
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Nonlinear Least-Squares
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Distance Geometry Optimization
1 2 1.4626445227 1 3 2.4141870267 1
4 3.0218931152 2 3 1.5333694923 2
4 2.4389840097 2 5 3.7816030992 3
4 1.3243628656 3 5 2.3818337473 4 5
1.4461262047
x1
x4
x2
x5
x3
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Simple Matlab
x1 x2 x3 x4 x5
x11 x12 x13 x21 x22 x23 x31 x32 x33 x41 x42 x43 x5
1 x52 x53
1 2 1.4626445227 1 3 2.4141870267 1
4 3.0218931152 2 3 1.5333694923 2
4 2.4389840097 2 5 3.7816030992 3
4 1.3243628656 3 5 2.3818337473 4 5
1.4461262047
x

data.dat
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The Least-Squares Function
function f fd (x) global data n, m size
(data) f 0 for k 1 n i data (k, 1)
j data (k, 2) d data (k, 3) i1 3(i-1)
1 i2 i1 1 i3 i2 1 j1 3(j-1)
1 j2 j1 1 j3 j2 1 h
(x(i1)-x(j1))2 (x(i2)-x(j2))2
(x(i3)-x(j3))2 h h - d d f f h
h End
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global data load data.dat load x0.dat rand
('state', 0) x0 rand (size (x0)) . x0 f0
fd (x0) x fmins ('fd', x0) f fd (x)
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32.5458 5.2693 61.7398 32.7025
4.0762 62.5715 32.1670 2.8851 61.7670
32.9345 2.4270 60.7896 32.4307
1.3513 59.9636
34.8422 17.2734 136.0620 17.7175
67.4883 171.6783 15.9891 1.4110
184.9119 15.1395 46.4418 178.5945
30.1267 56.0469 39.7237
x0
x
f (x0) 1.3453e009
f (x) 8.0077e-006