## Project-Team : modbio

## Section: New Results

**Keywords: ***Integer programming*, *crystallography*, *phase problem*.

## Integer programming and the phase problem in crystallography

**Participants:**Alexander Bockmayr, Eric Domenjoud.

The electronic density in a crystal is a non-negative
periodic function which may be written as a Fourier series with
complex coefficients .
X-ray crystallography provides us with the magnitudes F_{n} but this information is only partial in
that the phases remain unknown. In order to rebuild the
function, we must determine these phases by other means.
This step constitutes the *phase problem* in crystallography.
We have to determine a non-negative function such that
the magnitudes of its Fourier coefficients match the measured values.
We address a discrete version of the problem where we are interested
in the value of only at the vertices of a grid on the unit
cell of the crystal. The function itself is taken of the
form , where is a real coefficient and
takes its values in .

In [10], we have shown how this problem can be modeled and
solved with binary integer programming (cf. Sect.
3.3).
During this year, we started to investigate another approach based
on the *Patterson function* [42]. This function links directly the
electron density to the magnitudes F_{n} without involving
the phases. We use then local search techniques to minimize an
objective function defined as the square of the norm of the
difference between the Patterson function computed from the measured
values and the one computed from a candidate solution.

The possible benefits are as follows. First, we get rid of the phases which never explicitely appear in the equations. Second most handled values are naturally integers, which allows for efficient computations and updates of the objective function. In addition, at each search step, an optimal value for the coefficient may be deduced directly. This method is still under investigation, but first results obtained on some real examples like the G-protein are very promising.