**A Beginner's Guide to radiation damage.**

James M. Holton^1,2^

^1^Department of Biochemistry and Biophysics, University of California,
San Francisco, CA 94158-2330, USA

^2^Advanced Light Source, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA

# Synopsis

Radiation damage considerations affecting data collection by more than a
factor of two are summarized and damage avoidance strategies are
suggested.

# Abstract 

Many advances in the understanding of radiation damage to protein
crystals, particularly at cryogenic temperatures have been made in
recent years, but with this comes an expanding literature, and to the
new breed of protein crystallographer who is not really interested in
X-ray physics or radiation chemistry, but just wants to solve a
biologically relevant structure, the technical nature and breadth of
this literature can be daunting. The purpose of this paper is to serve
as a rough guide to radiation damage issues, and to provide references
to the more exacting and detailed work. No attempt has been made to
report precise numbers (a factor of two is considered satisfactory), and
since there are aspects of radiation damage that are demonstrably
unpredictable, the "worst case scenario" as well as the "average
crystal" are discussed in terms of the practicalities of data
collection.

**1 Introduction.**

It is not uncommon for radiation damage to prevent the solution of a
structure. Diffraction spots can fade away before the data set is
complete and heavy atoms sites can become disordered before sufficient
anomalous signal is measured. The former is easy to detect by eye during
data collection, but the latter is more insidious (Holton, 2007; Oliéric
*et al.,* 2007). In either case, however, significant damage by the
middle of the data set generally means that the data set is already
ruined. What is more, the specific chemical changes induced by radiation
damage can change the structure from the biologically relevant form, and
this sometimes leads to wrong conclusions about function.

Radiation damage can be avoided by keeping the accumulated X-ray
exposure "short", but how short is short enough? How short is too short
to solve the structure? Does it matter how bright the beam is? Is there
anything that can be done during sample preparation or data collection
that can mitigate or at least predict radiation damage?

There are several good reviews and broad reports on the field of
radiation damage (Garman & McSweeney, 2007; Ravelli & Garman, 2006;
Carugo & Carugo, 2005; Nave & Garman, 2005; Garman, 2003; Garman & Nave,
2002; Ravelli & McSweeney, 2000; Burmeister, 2000), and the reader is
referred to these works for comprehensive coverage of the field. What
follows is a rough guide focusing on damage avoidance.

**2 The factor of two.**

The cut-off of a factor of two for a radiation damage effect to be
considered "significant" here was chosen because a factor of two in
scattering power corresponds to a relatively small change in crystal
size. Specifically, increasing all three linear dimensions of a crystal
by 26% will double the volume of scattering matter (1.26^3^ = 2). Such a
change in size appears to be a typical "error bar" when examining
crystals under a microscope, as most crystallographers will not
distinguish between an 88 μm crystal and a 110 μm crystal, but the
latter has twice the volume of the former and the number of photons a
crystal will diffract before it is "dead" is proportional to volume (see
Appendix A). Perhaps more attention should be paid to crystal size, but
a factor of two can also be the uncertainty in X-ray beam intensity,
especially if parameters such as the variability in storage ring current
over a refill cycle are not taken into account. Increasing the exposure
time by a factor of two will double the damage but increase the
signal/noise ratio of the data by only \~42% (1 vs. √2), an improvement
which can be difficult to detect by inspecting a diffraction image, and
in practice a factor of four in exposure time is generally needed to see
new spots. A factor of two is also roughly the difference between
collecting one wavelength or two (all other things being equal, such as
the per-image exposure time), collecting the inverse beam wedge or not,
and also the difference between collecting from one crystal or merging
data from two crystals. So, a factor of two roughly corresponds to the
decision thresholds that must be faced in data collection strategy.

**3 What is a MGy?**

The flux density (photons/μm^2^/s) of current synchrotron X-ray beams
varies by a factor of ten thousand (for a table of expected crystal
lifetime at different beamlines, see http://biosync.rcsb.org/), so
describing radiation damage in terms of "frames" is not useful when
trying to apply strategies learned using one beamline at another. A more
transferrable unit is needed.

The Gy (J/kg) is the SI unit of dose, which is the amount of energy
absorbed by something per unit of mass, and protein crystals are
typically given millions of Gy, so the MGy is a convenient unit.
Obviously, the extent of radiation damage will depend on the number of
photons absorbed, but it is not so much the photons themselves as the
energy they carry that drives the chemical reactions of damage (Newton,
1963; Myers, 1973; Box, 1977), so it is most relevant and useful to
describe damage in terms of dose. Note that fluence (incident
photons/μm^2^) is sometimes incorrectly referred to as a "dose", but the
SI definition of dose is absorbed energy, not incident energy, and the
correct meaning of the word "dose" will be used here. The ratio of
fluence to dose (k~dose~) depends on the X-ray wavelength and the atomic
composition of the sample (see §6), but typically only a tiny fraction
of the X-ray beam is absorbed by a protein crystal (usually \~2%) so
dose is generally independent of crystal size and directly proportional
to incident photons/μm^2^. (Note that if the crystal is bigger than the
beam, then the dose to the part exposed to the beam will be proportional
to incident photons/μm^2^; see §7). Typically, the "dose ratio"
(k~dose~) is \~2000 photons/μm^2^/Gy. That is, a dose of 1 MGy will be
deposited in a metal-free crystal after 20 s in a 100 μm x 100 μm beam
of 1 Å X-rays with a flux of 10^12^ ph/s.

Neglecting the crystal thickness does introduce a small error. The
actual dose will always be a little less than that given by k~dose~, but
the error introduced is less than a factor of two as long as the crystal
is thinner than the attenuation depth of the X-rays, which is 3600 μm in
the above case. The error is less than 5% if the crystal is smaller than
370 μm thick.

The assumption k~dose~ = 2000 photons/μm^2^/Gy is also accurate within a
factor of two between 1.1 and 0.9 Å X-ray wavelengths if the overall
heavy (heavier than sulphur) atom concentration in the crystal is \<
\~100 mM (see §5, §6 and Table 1). For wavelengths and crystal
formulations outside this range, or simply for better accuracy, use
RADDOSE (Murray *et al.*, 2004; Murray *et al.*, 2005; Paithankar *et
al.*, 2009) but even if k~dose~ is uncertain, the lifetime (in seconds)
of crystals with given elemental composition at a given wavelength will
always be inversely proportional to flux density (photons/μm^2^/s) when
moving from one beamline to another, or as a given beamline is
attenuated. Thus it is important to know the beamline flux (photons/s),
as well as the size of the beam at the crystal (μm^2^) (see §11).

It is very important to remember that, like dose, diffracted intensities
are proportional to fluence (photons/μm^2^), but relatively insensitive
to heavy atom content. That is, for a given exposure time, the amount of
information obtained will depend on how many photons were thrown at the
crystal, but the amount of damage inflicted will depend on how many were
absorbed. The value of k~dose~ is therefore a good indicator of how much
data a crystal will yield in its useful life relative to another crystal
of the same size and type but different heavy atom content (see §6).
Lower k~dose~ is better.

**4 There are two kinds of radiation damage: global and specific.**

Irradiated protein crystals suffer an overall loss of resolution as
high-angle spots fade away which is referred to as global damage. There
are also specific chemical changes that can be seen in the electron
density maps, such as side chains popping off. Specific damage can be up
to \~60x faster than global damage (see below), but the good news is
that at cryogenic temperatures the global damage "rate" appears to be
essentially the same for every protein crystal, once "lifetime" has been
normalized to dose (Garman & McSweeney, 2007; Ow*en et a*l., 2006;
Leir*os et a*l., 2006; Sl*iz et a*l., 2003). In fact, the term "damage
rate" is something of a misnomer since the words "rate" and "lifetime"
imply progression with time but the fundamental coordinate of cryogenic
damage is dose. For this reason, we introduce the term "lifedose" to
refer to the amount of dose a crystal can endure, and the word
"lifetime" will be used to indicate time.

Owen *et al.* (2006) recommended a general maximum tolerable dose
("lifedose") of 30 MGy but noted there was also some resolution
dependence to this as high-angle spots faded first. In fact, there is a
remarkably linear relationship between scaling B-factor and dose (Kmetko
*et al.,* 2006; Borek *et al.,* 2007), but Howells *et al.* (2005)
proposed a resolution-dependent dose limit criterion of 10 MGy per Å of
resolution. For example, if a resolution of 3 Å is desired, the Howells
criterion suggests a lifedose of 30 MGy. Since most of the dose limits
used to derive this criterion used spot fading to half intensity as the
indication of a dose limit, the fading of a spot at a given resolution
can be supposed to follow an exponential decay:

(1) $$I = I_{0}\text{ exp}\left( - \text{ln}(2)\frac{D}{H \cdot d} \right)$$

Where:

I radiation-damaged spot intensity

I~0~ is the spot intensity at zero dose

ln(2) natural log of two (\~0.7)

D dose in MGy

d d-spacing in Å

H Howells *et al.* (2005) criterion (10 MGy/Å)

Note that I = 0.5 \* I~0~ when D in MGy is 10x the d-spacing in Å. This
is not exactly the definition of "d" given in Howells *et al.* (2005),
but Equation (1) agrees remarkably well with recent damage studies. For
example, applying Equation (1) to the square structure factors of
apoferritin (PDB ID: 2clu) results in a fairly linear falloff of total
intensity with dose that reaches half intensity at 42 MGy (not shown)
which is consistent with 43 +/- 3 MGy observed at half total intensity
by Owen *et al*. (2006). In addition, scaling these same
exponentially-modified apoferritin data to unmodified intensities with
SCALEIT (Howell & Smith, 1992) results in a best-fit relative B-factor
that increases linearly with dose having a slope of 1.3 B/MGy, which is
identical to the slope reported by Kmetko *et al.* (2006) for their
apoferritin observations. Application of this same resolution-dependent
spot-fading rate to lysozyme data (PDB ID: 2blx) reproduces the 1.03
B/MGy Kmetko *et al.* (2006) reported for lysozyme, and thus explains
the apparent protein-to-protein variability they observed. It is
worthwhile noting that the sum of all diffracted intensities reduces by
half if one applies a B-factor of 15 Å^2^ to the apoferritin data, but
this corresponds to a dose of 11 MGy using the ratio 1.3 B/MGy; a result
that would be inconsistent with those of Owen *et al.* (2006) if damage
manifested as a simple B-factor. This is because the B-factor has a
resolution dependence of exp(-1/d^2^), not the exp(-1/d) found by
Howells *et al.* (2005). The Howells criterion of 10 MGy/Å therefore
appears remarkably consistent with the observations of recent damage
studies and is recommended as a good rule of thumb for predicting the
lifedose of spots at a given d-spacing.

The bad news is that the rates of specific damage reactions are variable
and depend on many factors including the folded structure of the protein
(Holton, 2007) so there will probably never be a way to predict them
before the structure is solved. This problem is exacerbated by the fact
that "interesting" parts of the molecule such as active sites, bound
ligands and heavy atom sites are particularly prone to specific damage.
One might presume that this trend has anthropogenic origins because
these are the parts of the protein where people spend the most time
looking, but many systematic studies have now been carried out, and the
trend does appear to be real (Burmeister, 2000; Ravelli & McSweeney,
2000; Weik *et al.,* 2000; Leiros *et al.*, 2001; Weik *et al.*, 2001;
O'Neill *et al.,* 2002; Nukaga *et al.*, 2003; Fuhrmann *et al.*, 2004;
Carugo & Carugo, 2005; Dubnovitsky *et al.*, 2005; Roberts *et al.*,
2005; Yano *et al.*, 2005; Leiros *et al.*, 2006; Fioravanti *et al.*,
2007; Schiltz & Bricogne, 2007). The mechanism of site-specific
radiosensitivity is still unclear. Several mechanisms for the high
sensitivity of particular sites have been proposed, such as solvent
accessibility (Burmeister, 2000; Weik *et al.,* 2002; Garman & Nave,
2002), the high X-ray cross sections of heavy atoms (refuted in
Southworth-Davies *et al.,* 2007), chemical bond strain (Weik *et al.,*
2000; Fuhrmann *et al.*, 2004; Dubnovitsky *et al.*, 2005; Fioravanti
*et al.,* 2007), and electrostatic field lines (Holton, 2007). For each
of these models, there is both evidence and counterexamples, and the
reader is referred to the above references for details. It will suffice
here to say that there are probably at least two different radiochemical
mechanisms at work, and the lifedose of any given site depends on its
location in the structure.

Despite this inherent unpredictability, there are "world records" for
lowest lifedose of specific damage reactions: 2 MGy for selenomethionine
(Holton, 2007), 0.5 MGy for bromouracil (Oliéric *et al.,* 2007) and as
little as \~0.3 MGy for the metalloprotein putidaredoxin (Corbett *et
al.,* 2007), and these "worst case scenarios" can be used when planning
data collection (see §11).

**5 Crystals are killed by photons/μm^2^, not time.**

One of the most remarkable findings about cryogenically cooled protein
crystals is that global damage is proportional to dose, but not how fast
that dose was delivered: the "dose rate" (Garman & McSweeney, 2007; Sliz
*et al.,* 2003; Leiros *et al.,* 2006; Owen *et al.,* 2006). This is
certainly not the case at room temperature, where the extent of damage
inflicted by a given dose does depend on the dose rate (Blake &
Phillips, 1962; Southworth-Davies *et al.,* 2007), varies from protein
to protein and even continues damaging the crystal after the X-rays have
been turned off (Blundell & Johnson, 1976). However, unless stated
otherwise, the discussion in this paper is about damage at cryogenic
temperatures.

The exact timescale (and indeed the nature) of the cryogenic global
damage reaction is not clear, but it must be very fast to have no
dose-rate dependence and a lack of any demonstrable "dark progression".
Whatever reactive species are generated by the beam, they must be
consumed as fast as they are made, or their concentration would build up
at high dose rates, saturating the downstream reactions. This means that
as long as a partially "burnt" crystal is stored properly (such as
described in Owen *et al.,* 2004) it will stay at liquid nitrogen
temperatures and diffract to the same resolution at a later time, even
when restored to the beamline months later (not shown).

Specific damage is more complicated as a dose-rate dependence has been
demonstrated (Leiros *et al.,* 2006) and damaged species have also been
observed spectroscopically to disappear with a time scale of minutes
after the beam has been turned off (Weik *et al.,* 2002;
Southworth-Davies & Garman, 2007, McGeehen *et al.*, 2009). Nevertheless
no reported dose-rate dependence has been more than a factor of two.
That is, the ratio of accumulated dose to a given damage metric such as
site occupancy, scaling B-factor, or total diffracted intensity has not
been shown to change by more than a factor of two as a result of
changing the dose rate. Consequently, dose rate dependence will be
considered "insignificant" here, and the reader is referred to the above
references to learn more about it.

A very important consequence of the dose-rate independence of radiation
damage is that the quality of data that can be obtained from a crystal
before it is "dead" will not change no matter how rapidly the photons
are applied (flux or photons/s). This is because the data are scattered
photons and scattering is exactly proportional to fluence
(photons/μm^2^), which has no dimensions of time. This fact can be found
in any of the many good books describing the physics of X-ray
diffraction, such as Blundell & Johnson (1976) or Drenth (1999), with
rigorous proofs given by M. M. Woolfson (1970) or R. W. James (1962).
There are other parameters of beam quality such as collimation, spectral
purity, crossfire and flicker noise which can impact data quality in
various ways that will not be covered here, but flux (photons/s) by
itself does not affect data quality until it is converted into fluence
(photons/μm^2^). Therefore, since there appears to be no significant
dose-rate dependence to damage, the extent of damage is also
proportional to fluence (photons/μm^2^), and the data-to-damage ratio is
independent of the time taken to collect the data.

In the future, it may eventually become possible to collect data on
timescales faster than the chemical reaction rates involved in damage,
which will introduce a beneficial dose-rate effect. Exactly what this
timescale must be is not presently clear as the rate constants (and
indeed the mechanisms) of these reactions are not known.

**6 Beware of high atomic numbers.**

The lifetime of a crystal in a given X-ray beam depends on the elemental
composition of the crystal and the solvent inside it. Unlike visible
light, X-rays are absorbed by all of the electrons in an atom, and the
cross-section increases roughly as Z^2.7^ where Z is the atomic number.
This means that absorption goes up *very* steeply moving across and down
the periodic table! For example, 600 mM NaI instead of 600 mM NaCl in
solvent channels (assuming solvent is 50% of the total volume) will cut
the lifetime of a crystal roughly in half, but since absorption is
proportional to concentration, lower concentrations of NaI are less of a
problem. Note that we are discussing lifetime and not lifedose, as the
latter is remarkably unaltered by heavy atom content.

Precise X-ray absorption cross sections have been tabulated (McMaster
*et al.*, 1969; Hubbell, 1982; Hubbell, 2006), and (Seltzer, 1993)
generated effective cross sections accounting for energy-loss mechanisms
such as X-ray fluorescence (which is \~40% in the case of Se atoms) and
assumed no self-absorption of fluorescent X-rays. The Seltzer (1993)
tabulations were used to produce Table 1 because the error introduced by
neglecting self-absorption is usually quite small (much less than a
factor of two). This correction was discussed in detail by Paithankar
*et al.*, (2009). Table 1 was also made with the assumption that protein
crystals and water have the same absorption. Although the average X-ray
cross section of protein atoms is lower than that of water, the higher
density of protein (\~1.34 g/cm^3^) tends to make up the difference. As
long as the sample is free of heavy atoms, the dose (energy absorbed per
unit mass) deposited by a given beam of X-rays in a thin layer of
protein crystal is within 5% of that deposited in a thin layer of pure
water for all photon energies between 5 and 50 keV.

Using these assumptions, the concentration of a given element that will
cut a protein crystal lifetime in half (by doubling k~dose~) may be
calculated. This concentration is moles of the element per liter of
sample, not just the solution in the solvent channels. For example, if
we fill the solvent channels of a protein crystal with a solution
containing 680 mM of selenium, then the total concentration will be
roughly 340 mM (assuming 50% solvent content and that the selenium
compound does not bind to the protein).

Consider the photon energy to be 12680 eV. The mass energy absorption
cross section Seltzer (1993) provided for water at this energy is 2.3
cm^2^/g, so exposing a thin layer of water to a fluence of 10^6^
photon/cm^2^ will deposit 2.9x10^10^ eV/cm^3^ (10^6^ photon/cm^2^ x 2.3
cm^2^/g x 12680 eV/photon x 1.0 g/cm^3^) of energy. The mass energy
absorption cross section of selenium is at this energy is 87 cm^2^/g, so
a gas of selenium atoms at 27 mg/cm^3^ will absorb the same amount of
energy per unit volume as water (10^6^ photons/cm^2^ x 87 cm^2^/g x
12680 eV/photon x 0.027 g/cm^3^ = 3.0x10^10^ eV/cm^3^). Combining these,
we conclude that an aqueous solution of 27 mg/cm^3^ (340 mM) selenium in
water will absorb roughly twice as much energy as pure water or a native
protein crystal. If we consider lighter elements such as sodium, then a
14 mol/L gas will absorb as much energy as an equal volume of water, but
a 14 M aqueous solution of sodium will contain significantly less water
per unit volume than pure water, and therefore absorb fewer X-rays. If
we assume that each sodium atom displaces one water molecule, then a 19
M sodium is required to double the dose.

Table 1 was calculated as above using X-rays at the Se edge: 0.9793 Ǻ
and these dose-doubling concentrations will be different at other
wavelengths. As an extreme example, the dose-doubling concentration of
Br is \~1.2 M at 0.9793 Ǻ ("below" the Br K-edge) but this will drop to
320 mM at 0.9193 Ǻ ("above" the Br K-edge). To calculate exactly how a
particular crystal composition will behave at a particular wavelength,
use RADDOSE (Murray *et.al.* 2004; Murray *et.al.* 2005; Paithankar *et
al.*, 2009). However, to within a factor of two anything at less than
\~100 mM is "safe". Note however that this does not mean 99 mM is the
same as 0 mM; the effect is proportional to concentration.

**7 The "spreading" of radiation damage is \~3 μm**

Cryo-cooled protein crystals are solids, and this is evident if they are
mounted with the wrong size cryo tongs: the droplet containing the
crystal can be crushed or shattered like any other glass. Since the
viscosity of a glass is incredibly high, there can be no "diffusion" of
radicals in the traditional sense (Douzou,1977). That is, there is no
mass transport except on geological timescales (thousands to millions of
years). There is also not enough energy in available X-ray beams to heat
a crystal by more than a few degrees (Snell *et al.,* 2007). This is not
to say that massless reactive species cannot move as much as a few dozen
Å or more in a glass, they can (Box, 1977; Petrik & Kimmel, 2003,2004),
but this is solid-state chemistry, which is very different from
radiation chemistry in aqueous solution at room temperature (Zagórsky,
1999).

It is possible to shoot one part of the crystal so much that it expands
physically and the strain induced by this expansion distorts the lattice
of neighbouring regions of the crystal (not shown), but the bulk of data
collected at the Advanced Light Source beamline 8.3.1 (MacDowell *et
al.,* 2004) by "walking" down needle crystals shows no signs of damage
"spreading" any more than a few μm, and a systematic study by
Schulze-Briese *et al.* (2005) found that damage is indeed limited to
the irradiated area. Since the range of a \~10 keV photoelectron in
organic matter is \~3 μm (Cole, 1969), there is no physical reason to
think that damage "travels" any further than that. Conversely, half of
the absorbed energy should escape a crystal which is smaller than \~40%
of the photoelectron range (Nave & Hill, 2005), but a practical
demonstration of this effect has yet to appear in the literature.

**8 Scavengers and "radioprotectants"**

There are now seven molecules that have been shown to have a protective
impact on specific radiation damage including ascorbate (Murray &
Garman, 2002; O'Neil *et al.,* 2002; Betts *et al.* 2004;
Southworth-Davies & Garman, 2007; Holton, 2007), nicotinic acid and
DTNB[^1] (Kauffmann *et al.*, 2006), nitrate ion (Borek *et al.,* 2007;
Holton, 2007) and 1,4-benzoquinone, TEMP[^2] and DTT[^3]
(Southworth-Davies & Garman, 2007). These workers also found a much
longer list of substances that have no effect.

As for global damage, Kauffmann et al. (2006) and Murray & Garman (2002)
both reported a protective effect from their additives. Murray & Garman
(2002) did not claim more than a factor of two change in any of the
metrics used, but Kauffmann et al. (2006) reported better than a factor
of two impact on both global and specific damage. Unfortunately, the
results reported by Kauffmann et al. (2006) were not normalized for dose
nor for scattering power (see §10), so it is difficult to asses the
impact of these additives on a transferrable scale. Nevertheless, if a
crystal is found to tolerate the presence of any of these additives, it
is advisable to try them.

**9 Helium**

At any temperature, using helium gas instead of nitrogen will reduce
background scattering (Polentarutti *et al.*, 2003), and this effect is
particularly prominent inside the "water ring" (d-spacings between 3.8 Å
and the beam stop) where helium scatters 1/49^th^ as much as air. At the
"water ring" and higher angles, the scattering from the sample is
roughly equivalent to that of 1000 times its thickness of air, which
tends to overwhelm air scatter unless the air path is more than 1000x
longer than the sample is thick. For example, a 3 cm air path is
significant if the sample (crystal + cryosolvent) is 20 μm thick, but
not if the sample is 200 μm thick. Some of the historical confusion
about the benefits of helium arose from the need to de-convolute this
effect, but the more recent studies have taken it into account.
Nevertheless, from a data collection standpoint, the background
reduction alone can be a good enough reason to use helium, even at 100
K.

As with radioprotectants, there have been plenty of negative results
attempting to use temperatures lower than 100 K to reduce radiation
damage and a few have even been published (Meitzner *et al.,* 2005), but
there are now several reports of significant reduction of specific
damage at temperatures from 7 to 40 K (Yano *et al.*, 2005; Grabolle *et
al.*, 2006; Corbett *et al.,* 2007). All of these reports were
reductions in active site damage of metalloproteins ranging from factors
of two to a factor of 30 in Corbett *et al.*, but this does not mean
that helium temperatures will not slow down other specific damage
reactions.

As for global damage, there certainly are reports of a positive impact
from helium cryostats (Teng & Moffat, 2002; Hanson *et al.,* 2002;
Chinte *et al.,* 2007; Borek *et al.,* 2007; Meents *et al.,* 2007), but
no reports thus far have claimed better than a factor of two in sample
lifedose. Greater than a factor of two reductions in global damage has
been reported in cryo-electron microscopy, but the benefits of helium
are still controversial in this field (Massover, 2007; Glaeser, 2008).

**10 The minimum crystal size to solve a structure**

It was shown in §4 that diffraction spots from any protein crystal at a
given d-spacing will fade with essentially the same lifedose, so the
amount and quality of data that can be collected before a crystal "dies"
will depend on the initial scattering power of the crystal. For example,
given a particular data-quality goal, crystals with larger unit cells
will have to be bigger than those with smaller unit cells because spot
intensity is proportional to the number of unit cells in the beam. The
relationship between all the factors affecting scattering power and
radiation damage is detailed in Appendix A, where the number of crystals
of a given type and size that are needed to solve a structure is
derived:

\(2\)
$n_{\text{xtals}} = n_{0}\frac{\text{MW} \cdot V_{M^{2}}}{\ell_{x}\ell_{y}\ell_{z}(d^{3}\text{-1}\text{.}\text{53})\text{ exp}(\text{-0}\text{.}5 \cdot \text{B/d}^{2})}$

Where:

n~xtals~ - number of crystals required to solve the structure

n~0~ - number of crystals required in a unitary reference case (see
below and Table 2)

ℓ~xyz~ - crystal size in each direction (microns)

MW - molecular weight (kDa)

V~M~ - Matthews's coefficient (Å^3^/Da)

d - d-spacing of interest (Å)

B - Wilson B factor (Å^2^)

The crystal size, molecular weight, approximate V~M~ and resolution of
interest are all readily available quantities to the crystallographer
screening crystals, but it can be difficult to know the Wilson B factor
*a priori*. A survey (not shown) of the Protein Data Bank (PDB; Berman
*et al.*, 2000) revealed a simple relationship between the resolution
limit (d~min~) and the average atomic B factor of entries claiming that
resolution limit:

\(3\) B~avg~ ≈ 4 (d~min~)^2^ + 12

which holds in the range 1.5 Å \< d~min~ \< 4 Å. Whatever the reason
behind this trend, B~avg~ is empirically a rough estimate of the Wilson
B factor of a crystal that appears to diffract to d~min~ on initial
screening diffraction images and this can be substituted into Equation
2.

The value of the empirical "difficulty parameter" n~0~ will depend on
the type of experiment to be attempted, and it can be interpreted as the
number of crystals required when the fraction in Equation (2) is equal
to 1.0, such as 5x5x5 μm lysozyme crystals with Wilson B = 21 and d = 2
Å. Experiments with higher data quality requirements (see below) will
have larger values of n~0~, but improvements in methodology and
equipment are expected to lower n~0~ for a given experiment.

The usefulness of Equation (2) is demonstrated in Table 2. It can be
seen that n~0~ for the goal of obtaining a complete native data set has
been decreasing over time but has never been less than 3. Therefore,
using n~0~ = 3 in Equation 2 is a good way to gauge if trying to obtain
a complete data set is hopeless. That is, if plugging in the parameters
of a crystal of interest and n~0~ = 3 results in n~xtals~ greater than
one, then obtaining a complete data set from a single crystal would be a
record-setting feat. It is not recommended to try and set new records
with projects that might require more than just a complete data set.
That is, 90% complete data may be sufficient for molecular replacement
and refinement to succeed, but the integrity of "important" chemical
bonds can be questionable at n~0~ = 3.

A case in point is MAD/SAD structure solution, as it was noted in §4
that metal sites can break down up to 60x faster than the observed decay
of spot intensities. This implies that for "safe" structure
determination by MAD or SAD, n~0~ should be increased \~60-fold to n~0~
= 180. Covalent bonds in active sites and ligands can break rapidly as
well, but these decay rates are currently not well characterized, so if
breaking a particular bond would change the conclusions derived from the
structure, a recommended strategy is to use n~0~ \~ 180. For example,
the high value of n~0~ for the 3.5 Å structure of the ribosome
(Schuwirth *et al.*, 2005 in Table 2) is due to the fact that care was
taken to avoid any signs of radiation damage in those data (\~2
MGy/xtal) and that the dose ratio (k~dose~) of nucleic acids is about
twice that of protein (due to phosphorous content). In general, addition
of any of the dose-doubling concentrations listed in Table 1 requires
doubling the value of n~0~. This is because n~0~ is linear with dose.

Note that a value of n~xtals~ greater than one does not mean that the
crystal should be discarded. Walking a small beam down a needle crystal
to n~xtals~ different locations and merging the data will satisfy
Equation (2). Remember that the beam size will "cut off" the effective
crystal dimensions (see §7) and that should be reflected in the values
of ℓ~x~, ℓ~y~ and ℓ~z~ because the product of these three numbers is the
volume of scattering matter. For example, a 10 x 10 μm beam on a crystal
100 μm in all dimensions is equivalent to a crystal and beam which are
22 μm in all dimensions, because 10 x 10 x 100 =10^4^, and 22 x 22 x 22
≈ 10^4^.

**11 Summary and general strategy recommendations**

Radiation damage begins with the first X-ray photon absorbed in the
crystal, so data collection strategy must always be a balance between
the data quality required and the amount of damage than can be
tolerated. It is often the case that this balance cannot be struck with
just one crystal and perhaps not even with many if the crystals are very
small and very weakly-diffracting. So, given a new and very precious
protein crystal, how shall one proceed with data collection?

The first thing that must be known about a beamline or other X-ray
source is the flux (φ) in photons/s, and this can be obtained using a
calibrated photodiode (Owen *et al.,* 2009). The next parameters to
obtain are the dimensions of the beam at the crystal (ℓ~beamx~ and
ℓ~beamy~), as these are needed to compute the flux density
(photons/μm^2^/s). If these are not readily available, they can be
obtained by exposing a small piece of silica glass (such as a cover
slip) which will turn brown in the X-ray beam. If necessary, this
darkened glass can be transferred to a microscope with a calibrated
reticule for measurement of the beam size.

Once the flux density (photons/μm^2^/s) is known, all that remains is
the dose ratio (k~dose~) which is usually 2000 photons/μm^2^/Gy, but can
also be calculated more precisely for a given wavelength and chemical
composition of the crystal using RADDOSE (Murray *et al.*, 2004; Murray
*et al.*, 2005; Paithankar *et al.*, 2009). Given all these, the maximum
recommended shutter-open time a crystal will endure can be calculated:

\(4\)
$t_{\text{xtal}} = D \cdot k_{\text{dose}}\frac{\ell_{\text{beamx}}\ell_{\text{beamy}}}{\phi}$

t~xtal~ maximum shutter-open time for a data set (seconds)

D expected lifedose of the crystal (Gy)

k~dose~ dose ratio (\~2000 photons/μm^2^/Gy)

φ beam flux (photons/s)

ℓ~beamx~ one dimension of the X-ray beam spot (μm)

ℓ~beamy~ other dimension of the X-ray beam spot (μm)

Some data collection strategy programs such as BEST (Bourenkov & Popov,
2006) can take t~xtal~ as input, but it is instructive to examine its
meaning here. For example, if it is desired to collect a complete data
set to 2 Å resolution, then D = 20x10^6^ Gy can be obtained using the
Howells criterion (see §4). Assuming the X-ray wavelength is \~1 Å (see
§3) and no heavy-atom concentration is above 100 mM (see §6), the
k~dose~ given above will roughly apply, and using a beam 100 μm high and
200 μm wide with flux 1x10^12^ photons/s, we obtain t~xtal~ = 800
seconds. This means that a data set of 100 images should use a per-image
exposure time of 8 seconds or less. A 3 Å data set, however, could use
12 second exposures.

Note that t~xtal~ is proportional to d via the Howells criterion, but
this does not imply that shorter t~xtal~ will yield better resolution;
rather measuring high-angle spots must be done "quickly" because they
will endure less dose than low-angle spots. Increasing flux (photons/s)
will also reduce t~xtal~, which means that the data can be collected
faster, but the number of photons scattered into the detector before the
spots fade away will not change. This is a consequence of the lack of a
significant dose-rate dependence to radiation damage (see §5).

It is also important to note that Equation (4) takes the size of the
X-ray beam and care must be taken when considering a mismatch in size
between the beam and the crystal. A crystal much larger than the beam
and rotating will complicate the dose calculation as "fresh" and
"partially burnt" crystal matter moves in and out of the beam. However,
this effect can be roughly accounted for by assuming that the average
dose (total energy absorbed divided by the total mass that absorbed it)
applies. That is, estimate the total volume of crystal that will ever
experience the beam (this will require some knowledge of geometry) and
then take the cube root of this volume as the value for ℓ~beamx~ and
ℓ~beamy~. However, the flux φ (photons/s) used should be kept the same.
For example, if the crystal is a rod 50 x 80 μm thick and 1 mm long with
the long dimension oriented along the rotation axis and the data
collection rotates it 180° in a 10 x 10 μm beam, then a 50 x 80 x 10 μm
volume is illuminated and effectively ℓ~beamx~ = ℓ~beamy~ = 34 μm
because 50 x 80 x 10 = 4x10^5^ and 34 x 34 x 34 = 3.9x10^5^.

If the beam profile is a "top hat" with even illumination (constant flux
density or photons/s/μm^2^) at every point, then crystals smaller than
the beam will still obey Equation (4). However, if a small crystal is in
a "hot" central region of a larger beam, it will absorb a larger dose
than that given by Equation (4), since it is actually experiencing the
flux density (photons/s/μm^2^) passing through its own cross-sectional
area. For example, a Gaussian beam shape will have a peak flux density
that is 88% of the "average" flux density assumed by entering the
whole-beam flux (photons/s) as φ and the two full-width at half-maximum
(FWHM) dimensions of the beam profile as ℓ~beamx~ and ℓ~beamy~ in
Equation (4). Similarly, a "round" crystal that exactly matches the FWHM
dimensions of a Gaussian beam will experience 50% of the whole-beam flux
φ (photons/s). Crystals in such beams can therefore be expected to last
roughly 13% longer or twice as long as predicted by Equaiton (4),
respectively. This is not a radiation-protective effect of Gaussian
beams, but merely an artefact of computing flux density. On general
terms it is recommended to try to match the size of the beam to the size
of the crystal as this optimizes the ratio between diffracted
intensities and background scattering.

The above example uses Howells's criterion for the lifedose of general
damage, but if specific damage is a concern then the lifedose of the
relevant reaction should be used. For example, selenomethionine sites
have never been observed to be more than half-decayed after a dose of 2
MGy (Holton, 2007), and using this value for D in the example above, we
obtain a maximum per-image exposure time of 0.8 seconds. This is an
example of why it is generally necessary to grow larger and
higher-quality crystals for MAD/SAD structure solution than it is to
obtain a complete data set, as such crystals will diffract more photons
into the detector for a given exposure time.

It will often be the case that a diffraction image taken with the
exposure time recommended by Equation (4) will have an unsatisfactory
appearance, such as the absence of high-angle spots that were clearly
evident on a longer exposure. This generally means that the crystal at
hand will not yield enough data to meet the specified goals. The next
step is therefore either to change the crystal or change the goals. For
example, if it is decided that only 10 second exposures are
satisfactory, but selenomethionine decay is a concern, then t~xtal~ = 80
seconds implies that only 8 images can be collected before there is a
danger that some of the sites have been disordered. However, some
selenomethionine side chains are particularly hardy, so it is prudent to
continue collecting data until the spots fade away with the caveat that
only the first 8 images may contain anomalous signal. There are two ways
to address this caveat:

If more than one crystal is available, then an advisable strategy is to
combine the first \~2 MGy worth of data from several crystals into a
composite data set. In this case care must be taken to collect the
"early" data from each crystal in different regions of reciprocal space.
This technique was developed by Kendrew *et al.* (1960) and employed
more recently by Facciotti *et al.* (2003).

If only one crystal is available, or non-isomorphism between crystals is
problematic, then resolution must be sacrificed in order to obtain
complete data. In this case, it is recommended to collect a complete
data set with short exposures first (i.e. the 0.8 second exposures in
the above example) and then keep collecting additional complete data
sets, multiplying the exposure time by a factor of three each time. This
protocol not only extends the effective dynamic range of the detector,
but roughly doubles the overall signal/noise ratio with each new data
set if everything is merged together in the end. This is recommended if
there are no signs of damage. Signs of global damage are an increasing
scaling B-factor (Kmetko *et al.* 2006; Borek *et al.* 2007) and a
reduced anomalous signal (R~anom~ or CC~anom~ ; Evans, 2006) is an
excellent indicator of specific damage to metal sites. If rapid specific
damage is evident, then merge the early and late data separately and
treat them as RIP data (Ravelli *et al.*, 2003; Ravelli *et al.*, 2005;
Nanao *et al.*, 2005; Banumathi *et al.* 2004; Zwart *et al.* 2004). In
this way, the same collection strategy can be used for two different
structure-solving pathways: RIP will be appropriate if the damage was
fast, and MAD/SAD if it was slow.

**Table captions:**

**Table 1: dose-doubling concentration**

A protein crystal containing the indicated element at the concentration
shown will absorb roughly twice as much energy (dose) as a metal-free
protein crystal when exposed to an X-ray beam with photon energy 12680
eV. Bear in mind that the concentrations shown are in terms of moles of
the indicated atom per unit volume of sample, so soaking a protein
crystal with 50% solvent with a 1 M solution with result in a total
concentration of roughly 0.5 M. This calculation assumed that protein
has roughly the same energy absorption as water and that one water
molecule was replaced by each atom of the indicated element, which
becomes important for high concentrations. Details of the calculation
are explained in §6. The (\*) on the Br entry is a reminder that the
dose-doubling concentration of Br is high for 12680 eV, but drops to 320
mM at 13486 eV.

**Table 2: Experimental determinations of minimum crystal size**

This table lists values for n~0~ determined from the scattering power of
crystals for which the minimum size required for a complete data set has
been reported. The parameters listed in the first two rows are examples
that both use the same value of n~0~ for Equation 2 and demonstrate that
the size requirement of different crystal types can still be governed by
a single n~0~ parameter. Note that n~0~ appears to be restricted to a
relatively small range when compared to the variety of molecular weights
and crystal sizes shown, and that n~0~ has been decreasing over time,
perhaps as instrumentation and algorithms have improved. Footnotes
indicate derived parameters and a (\*) indicates that equation 3 was
used to estimate the Wilson B factor. A "?" indicates that the parameter
was not provided in the given reference, but a reasonable average value
for protein crystals was substituted.

**Table 1: Dose-doubling concentration**

At 12680 eV/0.9793 Å (the Se edge)

  ---------------- ---------------- ---- ----------------- -----------------
  **Na**           **19 M**              **As**            **350 mM**

  **Mg**           **12 M**              **Se**            **340 mM**

  **P**            **4 M**               **Br\***          **1.2 M**

  **S**            **3 M**               **I**             **230 mM**

  **Cl**           **2.5 M**             **Gd**            **110 mM**

  **K**            **1.6 M**             **Ta**            **75 mM**

  **Ca**           **1.3 M**             **Pt**            **100 mM**

  **Fe**           **560 mM**            **Au**            **100 mM**

  **Cu**           **430 mM**            **Hg**            **88 mM**

  **Zn**           **400 mM**            **U**             **100 mM**
  ---------------- ---------------- ---- ----------------- -----------------

**Table 2** Experimentally determined minimum crystal sizes

+-----+---------+-------+------+---------+----+----+---------------+
| MW  | Res     | V~M~  | Wi   | Crystal | N  | n~ | reference     |
| (k  | olution | (Å^3  | lson | size    | o. | 0~ |               |
| Da) | (Å)     | ^/Da) | B    | (μm)    | of |    |               |
|     |         |       | (Å   |         | x  |    |               |
|     |         |       | ^2^) |         | ta |    |               |
|     |         |       |      |         | ls |    |               |
+-----+---------+-------+------+---------+----+----+---------------+
| 14  | 1.5     | 2.0   | 20   | 20      | 1  | 3  | example       |
|     |         |       |      |         |    | .1 |               |
+-----+---------+-------+------+---------+----+----+---------------+
| 100 | 2.5     | 2.4   | 40   | 15      | 1  | 3  | example       |
|     |         |       |      |         |    | .1 |               |
+-----+---------+-------+------+---------+----+----+---------------+
| 62  | 1.9     | 2.4?  | 20\* | 30      | 13 | 1  | Gonzalez &    |
|  ?[ |         |       |      |         |    | 30 | Nave 1994     |
| ^4] |         |       |      |         |    |    |               |
+-----+---------+-------+------+---------+----+----+---------------+
| 14  | 1.6     | 2.0   | 22\* | 35      | 1  | 25 | Teng & Moffat |
|     |         |       |      |         |    |    | 2000          |
+-----+---------+-------+------+---------+----+----+---------------+
| 28  | 2.1     | 2.5   | 30   | 20      | 1  | 12 | Glaeser *et   |
|     |         |       |      |         |    |    | al.* 2000     |
+-----+---------+-------+------+---------+----+----+---------------+
| 24  | 2.0     | 2.5   | 22   | 5x30x30 | 5  | 9  | Facciotti *et |
|     |         |       |      |         |    | .8 | al.* 2003     |
+-----+---------+-------+------+---------+----+----+---------------+
| 400 | 3.5     | 2.5   | 65\* | 20      | 1  | 9  | Sliz *et al.* |
|     |         |       |      |         |    | .3 | 2003          |
+-----+---------+-------+------+---------+----+----+---------------+
| 2   | 1.98    | 1.58  | 11   | 5       | 2  | 5  | Coulibaly     |
| 8.6 |         |       |      |         |    | .2 | *et. al.*     |
|     |         |       |      |         |    |    | 2007          |
+-----+---------+-------+------+---------+----+----+---------------+
| 0.8 | 1.3     | 1.5   | 10   | 1.      | 3  | 3  | Nelson *et    |
|     |         |       |      | 5x1.5x5 |    | .7 | al.* 2005     |
|     |         |       |      |         |    |    |               |
|     |         |       |      |         |    |    | Sawaya *et    |
|     |         |       |      |         |    |    | al.* 2007     |
+-----+---------+-------+------+---------+----+----+---------------+
| 78  | 2.65    | 3.06  | 56   | 16[     | 4  | 3  | Li et al.     |
|     |         |       |      | ^5]x5x5 |    | .6 | 2004          |
+-----+---------+-------+------+---------+----+----+---------------+
| 73  | 3.4     | 3.67  | 69   | 5       | 13 | 3  | Standfuss *et |
|     |         |       |      |         |    | .2 | al.* 2007     |
+-----+---------+-------+------+---------+----+----+---------------+
| 21  | 1.5     | 2.4   | 11.4 | 1x1x20  | 90 | 3  | Mo            |
|     |         |       |      |         |    | .1 | ukhametzianov |
|     |         |       |      |         |    |    | *et al.* 2008 |
+-----+---------+-------+------+---------+----+----+---------------+
| 6   | 3.46    | 3.4   | 70   | 70      | 17 | 1  | Schuwirth *et |
| 000 |         |       |      |         |    | 80 | al.* 2005     |
+-----+---------+-------+------+---------+----+----+---------------+

**Acknowledgements**

# I would like to thank Elspeth Garman, Frank Von Delft, Ana Gonzalez and Julie Lougheed for extremely helpful discussions of this manuscript. This work was supported by grants from the National Institutes of Health: GM074929 and GM082250, and the US Department of Energy under contract No. DE-AC03-76SF00098 at Lawrence Berkeley National Laboratory.

**Appendix A: Derivation of equation 2:**

The ratio of scattered to incident photons has been referred to as the
"scattering power" of a crystal (Sliz et al., 2003; Teng & Moffat,
2000), which is embodied in the classic formulae due to Darwin (1914),
one of which is simplified here:

A1)
$I_{\text{spot}} \propto t \cdot \frac{V_{\text{xtal}}}{V_{\text{cell}^{2}}} \cdot L \cdot P \cdot |F|^{2}$

I~spot~ integrated spot intensity

V~xtal~ volume of the crystal

V~cell~ volume of the crystal unit cell

t exposure time

L Lorentz factor

P polarization factor

F structure factor (equivalent electrons per unit cell)

Although Darwin (1914) showed that diffraction strength is inversely
proportional to the square of the unit cell volume, the unit cell volume
is, in turn:

A2) V~cell~ = MW∙n~symops~∙V~M~

MW molecular mass of the asymmetric unit (Da)

n~symops~ number of symmetry operators in the space group

V~M~ Matthews number (\~2.4 Å^3^/Da ; Matthews 1968)

In addition, Wilson (1949) showed that the average squared structure
factor is proportional to the number of atoms in the unit cell:

A3)
$\left\langle |F|^{2} \right\rangle \propto \text{MW} \cdot n_{\text{symops}} \cdot |f_{a}|^{2}\text{exp}(\text{-0}\text{.}5 \cdot \text{B/d}^{2})$

f~a~ average atomic structure factor of a protein atom

B average atomic B-factor

d d-spacing of the spot of interest

We will also assume that the full lifedose of the crystal is utilized,
which means that the effective total accumulated exposure time (t) will
be proportional to the d-spacing (d) of the spot because low-angle spots
last longer (Howells *et al.*, 2005). In addition, higher symmetry
crystals require less rotation range, so for any strategy that yields
complete data the total number of photons delivered to a given unique
hkl index will be proportional to the number of symmetry operators
(n~symops~):

A4) $t \propto d \cdot n_{\text{symops}}$

This effective exposure time is not the only parameter that varies with
resolution. The atomic structure factor (f~a~), the Lorentz (Darwin,
1914; Blundell & Johnson, 1976; Lipson & Langford, 2006) and
polarization (Azaroff, 1955; Kahn *et al.,* 1982; Drenth, 1999) factors
all depend on the d-spacing of the spot. The average product of all
these terms can be described by the empirical expression:

A5)
$\left\langle L \cdot P \right\rangle \cdot |f_{a}|^{2} \cdot d \propto d^{3} - 1\text{.}\text{53}$

Excluding a scale factor, this expression is accurate to within 10%
error for d-spacing between 1.5 and 4.5 Å.

Combining all the above expressions we arrive at an expression for the
average attainable spot intensity at a given resolution:

A4)
$\left\langle I(d) \right\rangle \propto \frac{V_{\text{xtal}}(d^{3}\text{-1}\text{.}\text{53})\text{ exp}(\text{-0}\text{.}5 \cdot \text{B/d}^{2})}{\text{MW} \cdot V_{M^{2}}}$

We now decompose the total volume of scattering matter (V~xtal~) into a
number of crystals (n~xtals~) with explicit dimensions (ℓ~x~, ℓ~y~, and
ℓ~z~), and solve for the required number of crystals.

A5)
$n_{\text{xtals}} \propto \frac{1}{I(d) >}\frac{\text{MW} \cdot V_{M^{2}}}{\ell_{x}\ell_{y}\ell_{z}(d^{3}\text{-1}\text{.}\text{53})\text{ exp}(\text{-0}\text{.}5 \cdot \text{B/d}^{2})}$

Assuming that some critical average spot intensity is required, we can
replace the intensity term with an arbitrary proportionality constant
(n~0~) to represent this requirement.

\(2\)
$n_{\text{xtals}} = n_{0}\frac{\text{MW} \cdot V_{M^{2}}}{\ell_{x}\ell_{y}\ell_{z}(d^{3}\text{-1}\text{.}\text{53})\text{ exp}(\text{-0}\text{.}5 \cdot \text{B/d}^{2})}$

The n~0~ parameter may now be obtained empirically by plugging in the
parameters from a previous experiment where the data were found to be
sufficient to reach a given goal, such as a complete data set or solving
a MAD structure. An accounting of the former is given in Table 2.

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[^1]: 5,5'-dithiobis-2-nitrobenzoic acid

[^2]: 2,2,6,6-tetramethyl-4-piperidone

[^3]: dithiothreitol

[^4]: Estimated for 100 Å unit cell in P4~3~2~1~2 with V~M~ = 2.4

[^5]: Taken from 400 um^3^ illuminated volume quoted by Moukhametzianov
    *et al*. (2008) and 5 um beam