Inria / Raweb 2004
Project-Team: MODBIO

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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 $ \rho$ in a crystal is a non-negative periodic function which may be written as a Fourier series with complex coefficients Im2 ${\#8497 _n=F_ne^{i\#981 _n}}$. X-ray crystallography provides us with the magnitudes Fn but this information is only partial in that the phases Im3 $\#981 _n$ remain unknown. In order to rebuild the $ \rho$ 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 $ \rho$ 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 $ \rho$ only at the vertices of a grid on the unit cell of the crystal. The function $ \rho$ itself is taken of the form $ \alpha$$ \chi$, where $ \alpha$ is a real coefficient and $ \chi$ takes its values in Im4 ${{0,\#8943 ,K}}$.

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 $ \rho$ to the magnitudes Fn 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 $ \alpha$ 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.