Microsoft Word - CCA_2009_082_A042.docx* Dedicated to Professor
Emeritus Drago Grdeni, Fellow of the Croatian Academy of Sciences
and Arts, on the occasion of his
90th birthday. ** Author to whom correspondence should be
addressed. (E-mail:
[email protected]) (a) Iterating a map
gives rise to a memory-less trajectory or Markov chain, because
only the last point determines the next. It can
be argued that, in this way, some useful information from the whole
past trajectory remains unexploited. This does not hold, for
instance, in protein crystallography when new phases are combined
with a previous set of phases.
CROATICA CHEMICA ACTA CCACAA, ISSN-0011-1643, ISSN-1334-417X
Croat. Chem. Acta 82 (2) (2009) 421–432 CCA-3331
Review
Iterative Methods for the Solution of the Phase Problem in Protein
Crystallography*
Anton Thumigera and Giussepe Zanottia,b,**
aDepartment of Biological Chemistry, University of Padua, Viale G.
Colombo 3, 35131 Padua, Italy bVenetian Institute of Molecular
Medicine (VIMM), Via Orus 2, 35129 Padua, Italy
RECEIVED APRIL 7, 2008; REVISED JUNE 17, 2008; ACCEPTED JULY 15,
2008
Abstract. The phase problem is a major challenge when using X-ray
crystallography for structure de- termination. This is especially
true when the objects studied are macromolecular crystals, which
contain many atoms and diffract quite poorly. For this reason,
conventional direct methods, which are very suc- cessful for small
and medium-sized molecule crystals, generally fail with protein
crystals that do not dif- fract to atomic resolution. In this
paper, we review some of the iterative phase retrieval methods used
in optics, and present our own results obtained while trying to
extend these methods to the field of macro- molecular
crystallography. A binary constraint on density has been
incorporated in a new iterative algo- rithm, as well as into an
existing Difference Map, in order to attempt crystallographic phase
retrieval. Another existing algorithm, Charge Flipping, has been
modified to test a connectivity-based phasing ap- proach. While the
results on binary densities could not be extended to realistic
cases, the connectivity cri- terion has shown to possess some phase
extension power.
Keywords: phase problem, iterative methods, flipping algorithm,
binary approximation, density modifi- cation
INTRODUCTION
Many iterative methods exist for non-periodic object
reconstruction. From a general point of view, all these methods
operate by creating some succession of points in phase (or density)
space, i.e., in the space where pos- sible solutions are defined.
Each point represents a set of phases { },h or, equivalently, the
corresponding density function ρ(x). Usually, a starting point is
chosen at random and the succession is constructed in such a way
that, almost for an appreciable percentage of start- ing points,
convergence to the solution occurs. This solution, satisfying all
the constraints simultaneously, must lie at the intersection
between two constraint sub- sets: one defined by the experimental
moduli and the other determined by a priori constraints (which are
often easier to express in real space). The generator of the
succession is a map
1: n nρ ρ (a) (1)
usually devised in such a way that the solution ρ is a fixed point
attractor for the iterations:
ˆ ˆ( )ρ ρ (2)
(in some cases, the attractor can be a limiting cycle ˆ ˆ( ) ).n ρ
ρ A fixed point is left unchanged by the ap-
plication of the map, so that once the iterations have converged to
it, no further evolution occurs. Neverthe- less, the existence of
fixed points does not suffice per se to ensure convergence, and it
is not possible to set an upper bound to the number of iterations
needed to reach the solution. In this sense, a completely
satisfactory phase retrieval algorithm has not been proposed
yet.
Given an N-point sampling, a generic density is represented by a
vector in .N If we call CR and CMOD the two subsets corresponding
to the densities consistent respectively with real-space
constraints and observed moduli, the solution must belong to their
intersection
422 A. Thumiger and G. Zanotti, Iterrative Methods in
Crystallography
Croat. Chem. Acta 82 (2009) 421–432
* .R MODC C C In absence of supplementary data, the starting point
is a randomly chosen element in CMOD, which is generated simply by
Fourier transforming the known moduli with random phases. A
repeated appli- cation of the map generates a trajectory in phase
space, which in favorable conditions is likely to end in the
intersection. When the origin is not fixed from the beginning (for
example, by specifying some region in which the object density has
known values) the intersec- tion is not represented by a point, but
rather by a con- tinuous or a discrete set of points according to
the space group symmetry (a three-dimensional submanifold of
),N since all the possible choices for origin and enan- tiomorph
are equally valid. The trajectory can be thought to evolve in real
space (object density) as well in phase space, since for a given
set of moduli there is a one-to-one correspondence between points
in the two spaces.
Usually, the map used in iterative phasing can be constructed by
composing elementary operations known as vectorial subset
projections. The projection of an element x U on a subset of ,U Y U
is written as
: { }Y x y and associates to x the set { }y of its near- est
elements in Y:
( ) { : inf }Y y Y x y Y y x y x
(3)
The set { }y always contains a single element when the subset Y is
convex. A set Y is said to be con- vex when, for every arbitrary
pair of points 1 2, ,x x A all the points xμ in the segment
2 1{ (1 ) ,0 1}μx μ x μx μ (4)
also belong to Y. For subsets of the euclidean plane 2 the meaning
is intuitive (see Figure 1).
It is easy to show that the subset CMOD is not con- vex. In fact,
given two densities ρ1,ρ2 corresponding to the observed moduli
{F(h)} with the phase sets
1 2{ },{ }, the densities on the segment 2(1 )μρ μ ρ 1μρ in general
will not belong to CMOD, since they will
not correspond to the moduli {F(h)} unless a very spe- cial choice
for 1 2{ },{ }, is made. As a consequence, the projection on CMOD
is not uniquely defined, since a zero-valued F(h) is projected onto
the set of points
lying on the circle of radius Fobs(h) (Figure 2). In Fouri- er
space the projection of a generic element of {F(h)} on CMOD can be
written:
1
(5)
where the function Ψh is an arbitrary one. It is common to select
among the many possibilities the projection with Ψh = 0, which will
be called ΠMOD in the following.
Another drawback due to non-convexity of the CMOD subset is the
presence of traps in a sequence of iterated projections.1 Traps are
fixed points which do not correspond to an intersection between the
subsets. When the map is a simple alternation of projections, =
Π1Π2, and the constraints are non-convex, traps can represent a
serious problem. If the trajectory of the re- presentative point
gets to a trap, in each successive iteration the density will
oscillate between 1 1ρ C and
2 2 ,ρ C each being the projection of the other, i.e. Π1(ρ2) = ρ1
and Π2(ρ1) = ρ2 (Figure 3). This can be
Figure 1. Convex and concave sets in the euclidean plane. Every
point lying on the segment drawn between any two points of a convex
set belongs to the set itself.
Figure 2. The Fourier modulus projection represented on the Argand
plane. The correct modulus subset is a circle of radius | |;obsFh a
generic Fh is projected on it by leaving the phase angle unchanged
and substituting the modulus with the correct one. A null vector Fh
= 0 would lie at the same distance from any point of the circle,
and the arbitrary choice made in defin- ing ΠMOD is to project it
with zero phase.
Figure 3. Two trajectories constructed by alternated projec- tions
on non-convex subsets. The succession of points starting from a
converges to the intersection, while the one beginning in b ends in
a trap. This means that the representative point is projected back
and forth between two points lying at a local minimum of distance
between the sets.
A. Thumiger and G. Zanotti, Iterrative Methods in Crystallography
423
Croat. Chem. Acta 82 (2009) 421–432
viewed as a consequence of the two subsets attaining a local
minimum of distance; if their boundaries are con- tinuous, the
surface of the subset C1 in ρ1 and that of subset C2 in ρ2 will be
parallel. In cases of nearly paral- lel surfaces the evolution is
not completely blocked but becomes very slow; in that case we say
the algorithm has entered a tunnel. These undesirable phenomena are
known as stagnation.
AN OVERVIEW OF SOME EXISTING PHASING ALGORITHMS
Traps and tunnels potentially occur in phasing when using the
Gerchberg-Saxton (GS) algorithm,2 which constitutes the first
non-crystallographic phase retrieval algorithm ever proposed. The
real space constraint al- lowing image reconstruction is
represented by the knowledge of the object support S, defined as
the region in which the density is expected to be non zero. The GS
map is simply the repeated projection on support and moduli
subsets:
GS S MOD (6)
The projection onto the correct support subset is obtained by
simply setting to zero the density values outside the region
S:
if
(7)
The success of the reconstruction obviously relies on some
knowledge about the object size and shape. An upper bound for the
support can be inferred from its autocorrelation function, directly
computable from Fourier moduli. In general, for an N-point
sampling, a necessary (but not sufficient) condition for solution
uniqueness is that the sum of the dimensions of the two subspaces
must not exceed the dimension of the search space, that is, dim dim
,S MODC C N otherwise the intersection cannot be empty. For this to
be true, since the subset defined by known moduli has a dimension
of N, we must have dim / 2;SC N in other words, the problem is well
posed only when the object is smaller than half of the image.
The progress of the iterations can be followed by monitoring the
summed distance error J, which corres- ponds to the sum of the
distances between the current density and its projections on the
two subsets:
( ) ( ) ( )MOD SJ ρ ρ ρ ρ ρ (8)
Since this quantity can vanish only at the intersection of the
subsets, a trap is characterized by the fact that J stabilizes on a
non-zero value. A powerful alternative to
the GS map was introduced by Fienup algorithms,3 the most effective
being the so-called Hybrid Input-Output (HiO):
( ) if
: ( ) if
(9)
Density within the support is modified by imposing the observed
moduli, like in the GS algorithm; the dif- ference lies in the
outside region, where the density is no more set to zero but rather
to its previous value di- minished by the feedback term ( ),MOD xβ
ρ which increases with the difference between the projected density
outside the support and its expected value of zero. When the
intersection has been found, the result- ing density ρ is
consistent with the observed moduli and is also zero outside the
support, so that no further evolution is observed:
ˆ ˆ( ) 0 MOD xρ ρ x S (10)
Compared to GS, the HiO algorithm does not suf- fer from traps, and
the convergence is faster. In terms of projections, the HiO map can
be written as
(1 ) (1 )MOD S MOD S MODβ (11)
Recently a general form of map has been proposed,4 the difference
map (DM), which avoids stagnation and can be applied to any kind of
non-convex constraints. The HiO algorithm turns out to be a
particular case of DM in which the support constraint is used and a
given choice of the parameters is made. The DM operator is defined
by
1DM β (12)
1 2 2 1f f (13)
The operator DM adds to the density a quantity Δ pro- portional to
the difference of two composed maps. Each of these two maps results
from the successive appli- cation of a map fi and a projection Πj
on one of the two constraint subsets.
A fixed point ρ of the difference map is characte- rized by Δ = 0,
so that
1 2 2 1 1 2ˆ ˆ( ) ( )f ρ f ρ ρ (14)
where the element 1 2 ,ρ lying at the intersection be- tween the
subsets C1 and C2, represents the solution to the phase problem. It
should be pointed out that here the solution does not coincide with
the fixed point ˆ.ρ Since in a fixed point Δ must vanish, its
norm
( )i iε ρ (15)
can be used to follow the progress of the iterations.
424 A. Thumiger and G. Zanotti, Iterrative Methods in
Crystallography
Croat. Chem. Acta 82 (2009) 421–432
While the global behavior of the algorithm does not depend on the
nature of the fi, a careful choice of them is necessary to allow
convergence. Setting for instance f1 = f2 = 1 (the identity map)
does not give attractive fixed points. A possible choice is to
construct fi in a way that its operation on ρ produces a point on
the line joining ρ to Πi(ρ):
( ) (1 ) ( )i i i if ρ γ ρ γ ρ (16)
The optimal parameter values are γ1 = –β–1, γ2 = β–1, as found by
taking into account the local behavior in the proximity of a fixed
point. It can be shown that the difference map can escape traps;
these cannot behave like fixed points because they do not allow the
quantity Δ to vanish.
THE BINARY APPROXIMATION
A possibility for restraining the number of solutions is to
approximate the electron density in the unit cell to a binary
function. This approximation is motivated by the physical reality
of separated solvent and protein regions. The densities of the two
zones differ in average value and in variance, both quantities
being greater in the protein region. The solvent density can be
assumed to be flat to a good approximation, while in the protein
region the density can deviate much from its average value.5
Numerical tests show that approximating an image with a binary one
leads, in Fourier space, to es- sentially correct phases, while the
moduli are more seriously affected. In terms of constraint subsets,
the binary densities subset is not expected to intersect the moduli
subset, so that an approximate solution would lie between the
closest points of the two sets. Moreover, the (euclidean) distance
between these two elements of the two sets should be appreciable.
However, the two- value approximation can be justified to some
extent if the resolution is low (> 4 Å). A search for a binary
mask has been successful in reconstructing the density at a
resolution of about 12 Å.6 In that case, a Binary Integer
Programming (BIP) approach was used, where the main drawback is
that the computing time grows exponen- tially with the complexity
of the problem (i.e. with the number of grid points chosen to
sample the electron density). In this perspective a more efficient
search method, as an iterative one, could perhaps help in ex-
tending the resolution limit (at least in the range where the
binary approximation is justified). A two-valued function can be
scaled to a binary one (having only 0 and 1 as possible values), by
shifting and scaling its values. To operate this scaling in Fourier
space one needs to know the expected fraction of ones in the unit
cell, that is, the volume defined by the molecular envelope that is
to be searched for.
TWO BINARY ALGORITHMS
The subset of binary densities 01 { ( ) {0,1} }C ρ x x is formed by
disjoint points (the corners of an hypercube) and so it is not
convex. The projection of ρ on C01 is the element 01 01ρ C which
minimizes the distance
2 01 01[ ( ) ( )]k k
k
ρ ρ ρ ρ x x (17)
and this means that the quantities 01| ( ) ( ) |k kρ ρ xx must be
minimum for every pixel k. This leads to the simple expression for
the binary projector:
01
x
(18)
This projector is not single-valued and some arbi- trary choice has
to be made about the treatment of den- sities with value ½ , since
they can be indifferently set to 0 or 1.
Here both subsets are non-convex, so that alternate projections
will fail. In fact, iteration of a map Π01ΠMOD rapidly gets to a
trap, because many different ρ(x) pos- sess the same projection.
Once ΠMOD(ρn+1) becomes too close to ΠMOD(ρn) the evolution stops,
since Π01 projects both of them on the same point of C01.
To find a solution to the binary phase problem it is thus necessary
to avoid that any iteration ρn exactly belongs to the subset C01.
For this reason in the present work a heuristic algorithm inspired
to the HiO map, and in particular to the feedback concept, was
conceived. It is based on a map B, consisting in the alternate
appli- cation of the two operations ( )γ
MOD and ( , ) 01 ,β δ each one
flipping the density or the moduli about their 'expected
values':
( ) ( , ) 01 , , , 0γ β δ
01
β δ ρ δ
h
(21)
(The symbols T and T–1 stand for direct and inverse Fourier
transform, respectively). The ( , )
01 β δ operator
leaves unchanged the density values falling into the interval [δ,1
– δ], while the remaining are flipped about the nearest expected
value (0 or 1) (Figure 4); the extent by which each pixel value is
flipped is proportional to the parameter β. A similar operation is
carried out in reciprocal space on the values of the moduli by
the
A. Thumiger and G. Zanotti, Iterrative Methods in Crystallography
425
Croat. Chem. Acta 82 (2009) 421–432
operator ( ) ;γ MOD in that case, the expected value of each
Fourier modulus |Fh| is simply the known quantity 0| |,Fh and every
modulus is flipped by a quantity pro-
portional to γ. It must be noted that both flipping opera- tions,
in real and reciprocal space, are needed for the iterations to
converge. Moreover, the previous know- ledge of the zero-frequency
term F0 (which usually is experimentally unmeasurable) is also
necessary, and a separated flipping parameter γ0 was introduced for
it. It must be observed that, in terms of elementary projec- tions,
the map B results to be a rather complex one. The real space
operation can be written as
01 01
(22)
where the domain Zδ is defined as the set of points with a density
falling outside the range [ ,1 ].δ δ The flip- ping in Fourier
space, in turn, can be expressed as:
1
(1 )
(23)
The action of the operators 01 and MOD is to move the density on a
point which lies on the segment joining the starting density with
the projected one (in the case of
01 this is only an approximate picture).(a)
The progress of the iterations can be followed by means of a type
of a summed distance error (SDE):
(a)These expressions show an interesting similarity with the
dif-
ference map algorithm discussed below.
11
x x (24)
The algorithm was implemented in Fortran 90 for the two-dimensional
case, using the static libraries GFT7 for FFT computation. Its
behaviour has been studied for different values of β, γ, γ0, δ, in
order to identify the set of parameters giving the quickest
convergence. Some test results are reported with a 2D trial density
(20×20 pixels). In Figure 5a the SDE plots are shown for 20
independent runs of the algorithm (each relates to a
Figure 4. The flipping operation in real space. The values for a
one-dimensional density are reported as a function of a spatial
coordinate. Values greatest than 1 or smaller than 0 are inverted
with respect to their nearest binary value.
(a)
(b)
Figure 5. (a) SDE versus iteration number for a non-optimal
parameter setting (β = 0.5, δ = 0.2, γ0 = 1.2, γ = 1.3). 20 plots,
corresponding to different runs, are displayed. Three behaviors
a,b,c can be observed, as discussed in the main text; (b) SDE for
optimized parameters (β = 0.5, δ = 0.2, γ0 = 1.2, γ = 1.6). The
plots for 100 different runs are displayed. Of the three behaviors
shown in Figure 8, a (quick convergence to the true solution) has
become the preferred one.
426 A. Thumiger and G. Zanotti, Iterrative Methods in
Crystallography
Croat. Chem. Acta 82 (2009) 421–432
different starting set of random phases). In each run, 1000
iterations were performed. Three cases can be identified:
(a) convergence to the true solution. It occurs sud- denly, once
the algorithm enters the basin of attraction of the solution after
a chaotic trajectory. Very low val- ues of SDE are attained (≈
0.01). The density evolution during a converging run is shown in
Figure 6.
(b) stagnation. At some moment the figure of merit begin to
decrease, but slowly sets to a non-zero value (≈ 0.1) because some
kind of trap has been en- tered.
(c) the trajectory extends over the performed 1000 iterations
without entering any basin of attraction.
The dependence of the behaviour on the different parameters can be
rationalized as:
• δ affects mostly the speed of convergence, which increases with δ
until it rapidly goes to zero above δ ≈ 4, probably because the
basins of attraction of the fixed points become very small.
• β and γ, since they determine the flipping magnitude, influence
the ability of the algorithm to 'jump over' local minima (traps).
Setting these para- meters to small values leads to stagnation,
while, at the other extreme, too high values prevent convergence.
Since these two quantities play a similar role, they can- not be
optimized independently; in fact, for each β value there exists a
given range of γ in which convergence is possible (Figure 7).
The situation after choosing the optimal para- meters can be seen
in Figure 5b. Traps are avoided, and at the same time the basin of
attraction of the true solu- tion has been enlarged, so that the
two unwanted situ- ations (b) and (c) of Figure 5a are both much
less proba- ble. The number of iterations before convergence (IC)
probably depends on the ratio between the volume of attraction
basins and the total volume of the search space; the IC
distribution (shown in Figure 8) is an expo- nential one, as
expected for a memory-less process.
The algorithm does not need any knowledge about the support, but
only about the fraction κ1 of non-zero pixels in the solution
(which relates to the zero frequen- cy term through κ1 = F0/N,
where N is the number of pixels); the object can appear anywhere in
the cell and obviously the two possible enantiomorph choices are
equally probable. Since the origin cannot be fixed a priori, such a
kind of algorithm will always work with a P1 cell, independently
from crystallographic symmetry, which cannot be taken into account.
Symmetry can only emerge by itself and for this reason it could be
used to test the correctness of the solution. For other phase
retrieval algorithms without support it has been shown
Figure 6. Snapshots of the density during its evolution, taken
every 15 iterations. The abrupt change (Figure 6, case (a)) in the
figure of merit (SDE) occurs near cycle 320, when the density
suddenly begins to converge to the correct (binary) one. According
to the color scale used here, negative values are represented in
blue, and positive ones in red. Zero valued pixels are green.
Figure 7. Optimization plot for the parameter γ0 for fixed values
of the other three parameters. A similar trend is ob- served for
the general parameter γ.
A. Thumiger and G. Zanotti, Iterrative Methods in Crystallography
427
Croat. Chem. Acta 82 (2009) 421–432
that any attempt to fix the origin results in a reduced convergence
speed, probably because the solution space collapses to a single
point.
An alternative algorithm can be derived as a spe- cial case of the
difference map D = 1 + βΔ with
1 1 01
1 1 01
(25)
where the binary projector Π01 has been defined accord- ing to one
of the two possible choices in Eq. (18). An advantage over the
binary flipping algorithm is that the zero-frequency term F0 can be
unknown, as it will be found automatically by the algorithm itself;
moreover, there is one single parameter to be optimized.
Various experiments have been conducted with different trial
densities to determine the influence of β on the speed of
convergence and to compare the be- havior of the two algorithms.
Two different optimal ranges of β have been found, one centered
about –1 and the other about 0.8 (Figure 9). This is in agreement
with the literature,4 where the optimum values for the β para-
meter are found to be close to ±1. The comparison be- tween binary
flipping and difference map shows that their effectiveness varies
greatly with the nature of the object to be reconstructed, but the
dependence differs from one algorithm to the other. The two methods
are, to some degree, complementary; putting aside very simple
cases, often one of the two appears to perform well in those
situations where the other exhibits a very slow convergence.
BINARY APPROXIMATIONS AND REAL CASES
Once established that a method existed to solve the binary problem,
a more realistic case was considered, consisting in
pseudo-molecular data in two dimensions. The moduli were obtained
by Fourier transforming the density of benzene molecules projected
onto the mole- cular plane. The cell was a square of 10 Å edge in
which one, two or four benzene molecules had been placed. Data were
used up to a resolution of 2 Å.
A binary approximation to the real density can be constructed by
scaling the density and then setting a threshold z. The points with
values higher than z are given the new value of 1 and the others of
0.
01 01 01
ρ z
x (26)
The structure factors corresponding to the binary density, F01(h),
can be assumed proportional to the true ones, as in Ref. 6:
01( ) ( )F kFh h (27)
where the constant k can be calculated from the know- ledge of the
fraction κ1 of non-zero pixels in ρ01:
012
h (28)
The data from molecular structures were scaled in this way and then
given as input to the binary flipping algorithm. No convergence was
observed, for none of
Figure 8. A histogram showing the distribution of number of
iterations needed for convergence (a sort of trajectory length) for
the binary flipping algorithm. The distribution has an
approximately exponential decay, suggesting a memory-less
process.
Figure 9. Optimization plot for the binary difference map. The
average number of iterations needed for convergence is shown as
function of the single parameter β. Two optimal ranges are found,
the first (centered on β = –1, the global minimum) being larger and
deeper.
428 A. Thumiger and G. Zanotti, Iterrative Methods in
Crystallography
Croat. Chem. Acta 82 (2009) 421–432
the β, γ, δ parameter sets that had worked better for the ideal
binary cases. This can be explained assuming that there is no
intersection between the two constraint sub- sets, that is, no
binary density exists that could repro- duce the non-binary moduli.
In fact, binarization of a density not only will affect the moduli
in the chosen resolution sphere (in 2D, a circle), but it will
create non- zero frequency components outside the sphere (where the
original moduli had been set to be zero). To allow the two subsets
to intersect in some point, out-of-sphere moduli should be allowed
to deviate to some extent from their expected value of zero; it is
not clear, how- ever, if any physically meaningful solution could
be found in this way.
A SIMPLIFIED SAYRE EQUATION FOR BINARY IMAGES
Another possibility for phasing diffraction data from a binary
object can be derived outside the iterative me- thods context,
taking inspiration from the Sayre equa- tion.8 While this
relationship has been derived to exploit the atomicity property, it
can be shown that it holds, in a simplified form, for binary
densities too. In fact, the Sayre equation presupposes that density
and squared density are related by convolution with a spread func-
tion g:
2ρ g ρ (29)
This is true for a density made of identical, well re- solved,
spherical peaks (equal atom structure); neverthe- less, it is also
consistent with a binary function, in which case g reduces to a
constant. Assuming the density can take only the values 0 or a, we
have
2a ρ ρ (30)
1( )hF aV k h k
k
F F (31)
where V is the unit cell volume (in the 3D case). This convolution
relationship would allow the solution search to be carried out
entirely in reciprocal space, borrowing a variety of existing
algorithms from the field of direct methods. Moreover, a binary
approximation to a non-binary object can be found by minimizing the
deviation between the two sides of the equation, while iterative
algorithms fail in this task. In fact, from the lack of
intersection between the constraint subsets fol- lows that only a
global minimum of the distance be- tween the subsets can be
searched. But this minimum is not qualitatively different from
those non-meaningful local minima (traps) that a good algorithm is
expected to avoid.
MODIFICATIONS OF THE CHARGE FLIPPING ALGORITHM
A possible criticism to the application of the binary flipping
approach to non-binary density is that, while the lowest density
region (corresponding to solvent in protein structures and to
vacuum in small molecule structures) can be effectively assumed to
be sharply distributed around zero, the object (molecular) density
has a broader distribution. The behavior of the algo- rithm becomes
more interesting after suppression of the flipping about the upper
value of 1, letting β tend to 1 and γ to 0, and giving F0 the
freedom to vary during the iterations: the density of a single
benzene ring in the cell could be slowly reconstructed. With these
modifi- cations, the algorithm reduces to the known method of
charge flipping,9 which alternates moduli projection to a change in
sign of low-valued density:
0 δ
ρ ρ δ
In term of projections, the flip operator can be written
0 ( )2 1δ S δ (34)
where ΠS(δ) stands for support projection. The important thing is
that the support S(δ) is a dynamic one, being updated at each
iteration by selecting the points with ρ ≥ δ. The CF algorithm has
been proposed in crystallogra- phy for reconstructing atomic (<
1.2 Å) resolution struc- tures, but it has been shown to be also
applicable to the phase retrieval of non-periodic objects that lack
atomici- ty. In both cases, however, the uniqueness of solution is
guaranteed by the presence of extended regions of den- sity with
near-zero values and by (not strict) positivity. For non-atomic
objects the algorithm tends more to stagnation, so that it has been
used in conjunction with the HiO map: CF provides support
evolution, while HiO drives to convergence because it is
insensitive to traps.
The 2D benzene ring at 2 Å resolution does not display atomicity,
but the presence of a vast majority of pixels with small absolute
values of density still causes the solution to be unique. Because
of the lack of atomici- ty sudden convergence is never observed;
what happens is instead a slow, gradual approach to the solution.
This good behaviour is compromised in going from one mo- lecule to
two and four molecules per cell, because the ratio of null pixels
to the total number of pixels decrea- ses. With two molecules,
although the null pixels still occupy more than half of the cell,
the algorithm fails in reconstructing the rings, whose density is
rather flat, and shows a preference for 'peaky' solutions with
higher
A. Thumiger and G. Zanotti, Iterrative Methods in Crystallography
429
Croat. Chem. Acta 82 (2009) 421–432
density variance (Figure 10). The only way to find a solution with
the required characteristics is to introduce new restraints; for
example, an upper limit to density values can be used to force
density flatness.
A choice that has been proven to be effective is to set a
proportionality constant α between average density (calculated with
the values above the flipping threshold δ) and the maximum allowed
density s; at each CF cycle, the density values are modified by
inversion about the expected maximum (plateau) value.
( ), ρ δ
(35)
The value of s is calculated at each cycle. With this additional
restraint, correct solutions could be found for the cases of 2 and
4 molecules per cell (Figure 11). The best values for the
parameters were α ≈ 1.3, η ≈ 2; the first one depends on the
expected maximum value for the density, and can be varied only in a
very narrow range if wrong solutions are to be avoided. A 3D case
was then considered, to test if the modified CF algo- rithm with
upper bound restraint could phase bigger structures. Synthetic
trial data were calculated with the software SHELX10 from the PDB
coordinates of one molecule of Fatty Acid Binding Protein (FABP,
PDB
code 2HMB11). This protein comprises 131 aminoacids, organized in a
β structure which defines an internal cavity. The reflections were
computed from a single molecule positioned in a P1 cell (for
simplicity, a = b = c, α = β = γ = 90° were chosen). Since zero
density zones (which can be identified here with the solvent
regions) define the degree of determinacy of the prob- lem,
different tests have been carried out varying the length of the
cell edge, i.e., the unit cell volume. The effect of data
resolution was also investigated, across the range 20–2.5 Å.
It has been found that setting an upper bound for the density has
no or little effect on converging to the correct solution, which
could be retrieved in a small percentage of runs only when the
solvent content is very high (at least 85 % of the unit cell
volume, far too high to be found in any real crystal). This
probably means that, below a given fraction of null pixels, the
correct solution ceases to be a strong attractor for the CF algo-
rithm, and this happens well before the problem be- comes
underdetermined. In fact it was noted that, even starting from the
correct phases, there is a tendency to escape from the correct
solution; the rate of this process increases with the flipping
threshold δ. Another modifi- cation of the CF algorithm was tested
in the perspective of phase extension applied to protein
diffraction data. It consists in imposing on the electron density a
topologi- cal restraint motivated by very general features of pro-
tein structures. A key process consists in dividing the image in
the connected components, i.e., separated features appearing in
density when the isosurface for a given cutoff value is
constructed. For a given threshold κ a mask is defined as
{ : ( ) }k ρ κ x x (36)
the set of points k can be decomposed in a number M of connected
components ( ) ,i
kω each with a given vol- ume ( ).i
kν A subset of points ω is said to be a con- nected component when
every pair of points
1 2{ , } ωx x can be joined by a curve entirely contained in
ω.
While connected component analysis identifies volume segments,
without saying nothing about their shape, we can define a useful
quantity for estimating the linear length of density pieces. This
topological pro- perty, named connectivity, is computed by tracing
the skeleton, that is, the set of lines joining all neighboring
points above a given threshold.12 With density defined on a grid,
the procedure is to select grid points having density greater than
1.4 standard deviations above the mean, connecting by edges the
points which are nearest neighbors. Two grid points belong to the
same graph if they are connected by a continuous set of
edges.
Figure 10. Test of the CF algorithm (δ = 0.2) on the 2D pro-
jection of two benzene molecules (2 Å resolution). Upper plots:
density and its histogram for the true map. Lower plots: same for
the reconstructed map.
Figure 11. Densities reconstructed in four different runs of the
upper-bounded CF algorithm. Four molecules per cell are present.
Four unit cells are shown for clarity. Note the dif- ferent origin
positions, which depend on the (random) starting point.
430 A. Thumiger and G. Zanotti, Iterrative Methods in
Crystallography
Croat. Chem. Acta 82 (2009) 421–432
By means of the skeleton we define the connec- tivity as:
number of points in the longest graph Connectivity =
total number of points in graphs
(b) (37)
This quantity is obviously a function of the thresh- old and the
phase set. If the threshold is appropriately chosen, the global
maximum for connectivity should coincide with the true phases, for
which the electron density shows a single continuous polypeptide
chain. Connectivity values relative to random phase sets are
smaller than 0.1 while for correct phases a value above 0.9 is
expected. It has been shown that the addition of an increasing
phase error to the correct phase set always decreases the
connectivity in a gradual way.
The connectivity restraint could be exploited into an iterative
algorithm by selectively eliminating the densities that belong to
the shortest graphs. Although it is impossible to know if those
small segments would result to be correctly placed in the final
density, one surely knows that correct density should not show
small, isolated blobs. The idea is to force the density to evolve
by growth of the longest fragments rather than by fusion of many
small segments. An encouraging observation is that connectivity
only depends on strong reflections and it is preserved even if a
consistent frac- tion of moduli are given completely random phases
(up to 80 % of the weakest ones – test carried out at 4 Å
resolution).
An implementation was tried in this work using a weaker topological
constraint, based on the segment volume rather than on graph
length. The volume con- straint is expected to be weaker than
connectivity (as defined by Baker et al.) because it involves no
restric- tion on the shape of the density; there is no reason to
think that a general relationship between the volume of a connected
component and its skeleton length should exist. However, for a
densities in a neighborhood of the solution (so that the phase
error is acceptable and con- nectivity is not too low) some kind of
local relationship should arise, since the longest elements will
also be the largest ones. For that reason, one expects that the re-
quirement for the density to display a minimum number of volume
elements ( )i
kω could be used to improve or extend a set of known phases. Thus,
a modified CF algorithm was devised, introducing supplementary real
space operations:
• a binary mask is created to distinguish between points above and
below a fixed threshold.
• a segmentation algorithm is used to identify the
(b)An alternative definition for connectivity is the total
number
of graphs.
connected components into the density;
• a sorted list of segments is created on the basis of their volume
(number of voxels);
• segments with volume below a certain mini- mum value νmin are set
to zero in the density map.
The segmentation method used here was essential- ly the 'burning
grass' algorithm described by Lunina et al.,13 which consists in
the following steps (Figure 12):
• Initialization: the points above the threshold are given a value
1, the others 0. No found components are present.
• Search for a new component: the nodes of the grid are scanned
until a node with value '1' is found. The number of found
components is increased by one. A 'current front' is defined as a
set consisting of this node only. The new found component is marked
with a con- secutive number m. If no more '1' nodes are present the
algorithm stops.
• Isolation of a connected component: the 'future front' is defined
as the set of the nodes with value 1 that are neighbouring to one
of the nodes of the 'current front'.
• Propagation of the front: the nodes of the cur- rent front are
marked as belonging to the m-th compo- nent. The 'future front'
becomes the 'current front' and
Figure 12. Flow chart for the 'burning grass' segmentation
algorithm. Each m-th time a new initial point is found, the
propagation loop is entered. The loop defines a 'future front' as
the list of those points which are nearest-neighbors to points of
the 'current front'; these latter are then marked as belonging to
the m-th segment and the procedure is repeated until no more
nearest neighbors are found and all the m-th connected com- ponents
have been isolated.
A. Thumiger and G. Zanotti, Iterrative Methods in Crystallography
431
Croat. Chem. Acta 82 (2009) 421–432
the algorithm goes back to the preceding point. This loop is
repeated until the 'future front' is empty, then the search for a
new component is performed.
This last variant of the CF algorithm has shown some phase
extension power in a series of error-free tests conducted with a
starting set of exact phases (Figure 13), that were extended to
cover a larger sphere of ref-
lections. Phases not belonging to the starting set were initially
given random values, while known phases were kept constant in each
run. It must be noted, however, that the algorithm is not able to
improve a set of error- affected phases if these are given the
freedom to vary from one cycle to the other. In fact, a divergent
behavior was always observed in that case, probably because of
underdeterminacy. For this reason, a phase combination step should
be introduced; the best way to carry out the phase extension would
probably follow the density modification scheme.
Acknowledgement. We are grateful to Anke Seydel for care- fully
reading the manuscript.
REFERENCES
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Acta Crystal-
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Fourier Transform, copyright CNRS/
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V. Y. Lunin, Acta Crystallogr., Sect. A 41 (1985) 551–556. 9. G.
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phy, University of Göttingen, 1996. 11. G. Zanotti, G. Scapin, P.
Spadon, J. H. Veerkamp, and J. C.
Sacchettini, J. Biol. Chem. 267 (1992) 18541–18550. 12. D. Baker,
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SAETAK
Anton Thumigera i Giuseppe Zanottia,b
aDepartment of Biological Chemistry, University of Padua, Viale G.
Colombo 3, 35131 Padua, Italy bVenetian Institute of Molecular
Medicine (VIMM), Via Orus 2, 35129 Padua, Italy
Problem faze predstavlja najvei izazov prilikom odreivanja
strukture rentgenskom difrakcijom. To naroito do- lazi do izraaja
kad su objekti prouavanja makromolekulski kristali koji slabo
difraktiraju i sadre mnogo atoma. Zbog toga uobiajene direktne
metode, koje su pogodne za kristale malih i srednje velikih
molekula, uglavnom ne mogu rješiti strukturu proteina iji kristali
ne difraktiraju do atomske rezolucije. U ovom je radu dan pregled
nekih iterativnih metoda odreivanja faza koje se koriste u optici.
Prikazani su naši vlastiti rezultati pokušaja korištenja tih metoda
i u makromolekulskoj kristalografiji. Binarno ogranienje
elektronske gustoe ugraeno je u novi itera-
Figure 13. Starting and final correlation coefficient for some runs
of the connectivity-restrained CF performed on ideal data from the
protein FABP with an exact starting set. The corre- lation
coefficient was calculated according to the relationship:
2 1 2
The map correlation coefficient has been reported as a func- tion
of starting and final resolution. Several hundreds of cycles were
carried out, but in many cases the density ceased to evolve after
only 50–100 iterations.
432 A. Thumiger and G. Zanotti, Iterrative Methods in
Crystallography
Croat. Chem. Acta 82 (2009) 421–432
tivni algoritam kao i u diferentnu mapu elektronske gustoe s
namjerom odreivanja kristalografskih faza. Drugi postojei
algoritam, “charge flipping”, modificiran je da bi se ispitalo
odreivanje faza temeljeno na meusobnoj povezanosti atoma. Metoda
binarnih gustoa nije poluila rezultate u realnim sluajevima no
pokazalo se da krite- rij konektivnosti ima odreenog potencijala
kao metoda za proširenje faze.
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/GRE
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/HEB
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
/HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za
visokokvalitetni ispis prije tiskanja koristite ove postavke.
Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0
i kasnijim verzijama.) /HUN
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/ITA
<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>
/JPN
<FEFF9ad854c18cea306a30d730ea30d730ec30b951fa529b7528002000410064006f0062006500200050004400460020658766f8306e4f5c6210306b4f7f75283057307e305930023053306e8a2d5b9a30674f5c62103055308c305f0020005000440046002030d530a130a430eb306f3001004100630072006f0062006100740020304a30883073002000410064006f00620065002000520065006100640065007200200035002e003000204ee5964d3067958b304f30533068304c3067304d307e305930023053306e8a2d5b9a306b306f30d530a930f330c8306e57cb30818fbc307f304c5fc59808306730593002>
/KOR
<FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020ace0d488c9c80020c2dcd5d80020c778c1c4c5d00020ac00c7a50020c801d569d55c002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e>
/LTH
<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>
/LVI
<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>
/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken
die zijn geoptimaliseerd voor prepress-afdrukken van hoge
kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met
Acrobat en Adobe Reader 5.0 en hoger.) /NOR
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/POL
<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>
/PTB
<FEFF005500740069006c0069007a006500200065007300730061007300200063006f006e00660069006700750072006100e700f50065007300200064006500200066006f0072006d00610020006100200063007200690061007200200064006f00630075006d0065006e0074006f0073002000410064006f0062006500200050004400460020006d00610069007300200061006400650071007500610064006f00730020007000610072006100200070007200e9002d0069006d0070007200650073007300f50065007300200064006500200061006c007400610020007100750061006c00690064006100640065002e0020004f007300200064006f00630075006d0065006e0074006f00730020005000440046002000630072006900610064006f007300200070006f00640065006d0020007300650072002000610062006500720074006f007300200063006f006d0020006f0020004100630072006f006200610074002000650020006f002000410064006f00620065002000520065006100640065007200200035002e0030002000650020007600650072007300f50065007300200070006f00730074006500720069006f007200650073002e>
/RUM
<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>
/RUS
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/SKY
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/SLV
<FEFF005400650020006e006100730074006100760069007400760065002000750070006f0072006100620069007400650020007a00610020007500730074007600610072006a0061006e006a006500200064006f006b0075006d0065006e0074006f0076002000410064006f006200650020005000440046002c0020006b006900200073006f0020006e0061006a007000720069006d00650072006e0065006a016100690020007a00610020006b0061006b006f0076006f00730074006e006f0020007400690073006b0061006e006a00650020007300200070007200690070007200610076006f0020006e00610020007400690073006b002e00200020005500730074007600610072006a0065006e006500200064006f006b0075006d0065006e0074006500200050004400460020006a00650020006d006f0067006f010d00650020006f0064007000720065007400690020007a0020004100630072006f00620061007400200069006e002000410064006f00620065002000520065006100640065007200200035002e003000200069006e0020006e006f00760065006a01610069006d002e>
/SUO
<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>
/SVE
<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>
/TUR
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/UKR
<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>
/ENU (Use these settings to create Adobe PDF documents best suited
for high-quality prepress printing. Created PDF documents can be
opened with Acrobat and Adobe Reader 5.0 and later.) >>
/Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ <<
/AsReaderSpreads false /CropImagesToFrames true /ErrorControl
/WarnAndContinue /FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug
false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps
false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint
/Legacy >> << /AddBleedMarks false /AddColorBars false
/AddCropMarks false /AddPageInfo false /AddRegMarks false
/ConvertColors /ConvertToCMYK /DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK /Downsample16BitImages
true /FlattenerPreset << /PresetSelector /MediumResolution
>> /FormElements false /GenerateStructure false
/IncludeBookmarks false /IncludeHyperlinks false
/IncludeInteractive false /IncludeLayers false /IncludeProfiles
false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe)
(CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector
/DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling
/LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false >> ] >> setdistillerparams
<< /HWResolution [2400 2400] /PageSize [720.000 540.000]
>> setpagedevice