Hi Pavel,
I think your email highlights one of the differences between us and one
of the reasons for this discussion:
I am a scientist, not a mathematician - I want to improve
crystallographic methods because people who solve crystal structures
want an answer to a biological or chemical or physical question rather
than because they enjoy watching the realisation of a mathematical
definition. I like Ken Follett's definition of a physicist, for whom
reality is a poor approximation to theory, but the motivation for my
research runs the other way round.
Cheers,
Tim
On 11/29/2014 05:12 AM, Pavel Afonine wrote:
> Hi Tim,
>
> your examples are valid and valuable, and clearly exemplify existing
> problems, limitations as well as common misconceptions.
>
> However, if you follow mathematics and strict definitions thereof, then
> crystallographic structure refinement is nothing but an optimization
> problem that, fundamentally, to be defined requires: a) definition of model
> parameterization, b) definition of a function that relates experimental
> data and model parameters, and c) definition of a method that changes model
> parameters in a such a way that optimizes (most of the time minimizes) the
> chosen (at step "b") function.
>
> Please don't think that I've just made up or invented these "a)-b)-c)"
> steps above.. In fact, this has been published, for example, in
> *Acta Cryst.* (1985). A*41*, 327-333,
> and then reiterated using modern jargon, for example, in
> *Acta Cryst.* (2012). D*68*, 352-367.
>
> (I say "for example" above just to stick to the context and also point out
> that you can find more examples in crystallographic literature as well as
> in totally different disciplines such as economics, aerospace science etc.)
>
> Anyways, once all the above (a-b-c) are set and defined, then your only
> goal is as "simple" as finding the global minimum of the function that you
> have chosen to optimize.
>
> Anything else beyond that are either technical details or various
> inefficiencies related to improper model parameterization, improper target
> choice or using limited optimization tool.
>
> All the best,
> Pavel
>
>
> On Fri, Nov 28, 2014 at 11:40 AM, Tim Gruene <[log in to unmask]>
> wrote:
>
>> Dear Pavel,
>>
>> there is a beautiful paper called 'Where freedom is given, liberties are
>> taken' by Kleywegt and Jones, but also a wide variety of articles that
>> (fortunately) fought hard for the introduction of Rfree to the
>> (macro-)crystallographic community.
>>
>> In there is mentioned the threading of an amino acid chain backwards
>> into the density achieving (by refinement) a lower R-value than the
>> original one. Since this was achieved with refinement, the former
>> structure was closer to the global minimum than the latter one.
>> Apparently none of these authors had an idea how to modify the target
>> function so that this would not happen - whyfore they suggested to use
>> cross validation to avoid it.
>>
>> If you don't like this line of thought, I can offer a different one:
>>
>> there is a vast number of sets of parameters that ideally fit your data:
>> fill your asymmetric unit randomly with atoms so that your data to
>> parameter ratio is 1 or lower. Refine unrestrained and your are going to
>> end up with an R-value of 0. For unrestrained refinement, the formula
>> for the R-value corresponds (maybe not for maximum likelhood based
>> target functions, you may have to do some translation here) to the
>> target function, which usually has a lower bound of zero, hence this
>> vast number of "structures" all reached the global minimum. Note that
>> the deposited structure has an R value much greater than 0, i.e. it is
>> far away from the global minimum.
>>
>> In order to improve the situation, one modifies the target function by
>> adding restraints. They increase the target value of all "structures",
>> but in general those for the arbitrary solutions increase so much more
>> than that for an acceptable solution that most of those are lifted above
>> that of an acceptable solution.
>> As an example, one of the structures for the yeast polymerase I contains
>> about 34,500 atoms, i.e. the target function is minimised in a 138,000
>> dimensional space. I don't think there is a proof that any set of
>> restraints is ever so ideal that all false solutions are lifted above
>> the target value of the accepted solution. In fact, without being able
>> to proove it, I doubt that this the case, which lead me to the below
>> claim that we don,t necessarily want to reach the global minimum of the
>> target function.
>>
>> Of course an acceptable structure actually may have a target value
>> representing a global minimum, but I don't think this is always true.
>>
>> Best,
>> Tim
>>
>> On 11/28/2014 05:42 PM, Pavel Afonine wrote:
>>> Hi Tim,
>>>
>>> you don't necessarily want to find the global minimum (...)
>>>
>>>
>>> this contradicts the definition of crystallographic structure refinement.
>>> If finding the global minimum is not what you ultimately want then either
>>> the refinement target or model parameterization are poor.
>>>
>>> Clearly, given complexity of refinement target function profile (in case
>> of
>>> macromolecules) we unlikely to reach the global minimum; however,
>> reaching
>>> it is what we aim for (by definition and construction of refinement
>>> program) .
>>>
>>> Pavel
>>>
>>
>> --
>> Dr Tim Gruene
>> Institut fuer anorganische Chemie
>> Tammannstr. 4
>> D-37077 Goettingen
>>
>> GPG Key ID = A46BEE1A
>>
>>
>
--
Dr Tim Gruene
Institut fuer anorganische Chemie
Tammannstr. 4
D-37077 Goettingen
GPG Key ID = A46BEE1A
|