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Abstract
This thesis addresses two important problems existing in the Laue crystallography - energy overlap and low-resolution hole. ... A new method for this deconvolution is presented which is based on maximising the entropy of the Patterson subject to the constraints imposed by the observed intensities of single and overlapping reflections. ... Some thoughts on the future development of the maximum entropy method are also discussed. ...
Chapter 1 Introduction to X-ray Crystallography And Laue Method 1
1. ... 2 Synchrotron Radiation, Time-Resolved Studies and Laue Crystallography 4
1. ... 5 Two of the Major Problems Existing in Current Laue Crystallography 9
1. ... 3 Bayesian method 13
Chapter 2 Maximum Entropy Method And Its Use In Crystallography 15
2. ... 3 Entropy Maximisation Algorithms 18
2.4 Maximum Entropy in Crystallography 20
2.5 The future Of Implementing Maximum Entropy Method To Crystallography 27
Chapter 3 Using Maximum Entropy to Solve Energy Overlap Problem in Laue Crystallography 28
3. ... Using Maximum Entropy Method To Evaluate Unobserved Reflection Intensities In X-ray Crystallography Data Collection 55
4. ... Some Thoughts About Future Development 72
Of Maximum Entropy Method 72
Chapter 1 Introduction to X-ray Crystallography And Laue Method
My thesis will concern on two important problems existing in current Laue crystallography—energy overlap and low-resolution hole. ... There are all together 230 distinctive space groups in the three-dimensional space (International Tables for Crystallography, Kluwer Academic Publishers). ... ) will have a maximum point in the spot where scattered lights contact the detector. ... This industry requirement has also helped in development and popularity of X-ray crystallography and it is this that has provided the tremendous encouragement for the development of this field. ... 2 Synchrotron Radiation, Time-Resolved Studies and Laue Crystallography
Since the first introduction of synchrotron radiation for structure studies by Rosenbaum and his co-workers (Rosenbaum, and etc. 1971), this new X-ray source has brought important advances in the field of macromolecular crystallography. ... 3 Reciprocal Lattice, Structure Factors, Fourier Transform and Phasing Problem
Before we go any further in this Laue topic, it is needed to introduce some basic crystallography concepts. ... To have the maximum constructive interference from other directions (b and c axis) also, s should satisfy (which is called Laue equation)
1. ... This is the long existing phase problem in crystallography. ... It is lucky that we could manage to avoid phasing--the most complex problem in crystallography. In Chapter 2 and 5, the use of Maximum Entropy in phasing problem will be discussed. ... 5 Two of the Major Problems Existing in Current Laue Crystallography
Although Laue method has the advantage of speed over monochromatic method, some features of this experimental method make it less popular. ... Chapter 3 will be devoted to a new method which uses Maximum Entropy to solve these two problems (at the same time). ... , 1987) in Laue crystallography. ... 3 Bayesian method
The Bayesian method is said to have some internally close relation with the maximum entropy method. The use of Bayesian theory in crystallography goes back to 1970 when Kheiker and Nekrasov (1970) first proposed it for analysing the monochromatic single crystal diffraction data. ...
Chapter 2 Maximum Entropy Method And Its Use In Crystallography
2.1Introduction
The theory of maximum entropy originated from information theory. The basic equation of an entropy function is defined as
2. ... Several scientists (Gull & Daniell, 1978; Collins, 1982; Wilkins, Varghese & Lehmann, 1983; Skilling & Bryan, 1984; Bricogne, 1984) pioneered the implementation of maximum entropy method in crystallography. One of the earliest results of these works is that maximum entropy method can refine electron density map to a higher resolution (Collins, 1982). In a paper by Bricogne(1984), he pointed out that maximum entropy theory, in basic principle, is deeply related to direct method. ... In the beginning of 1990s, several successful maximum entropy applications in small molecule structure phasing (Gilmore and et al. ... Since then maximum entropy has been used in small molecule structure refinement, powder diffraction intensity deconvolution, charge density distribution calculation, electron crystallography and Laue crystallography. Maximum entropy method is experiencing a period of converting from mathematical theory to a practical tool for chemical applications. ... 2The Basic Mathematical Principle
The solution of maximisation entropy under general linear constraints was first proposed by Jaynes (1957) .
Given probability density function q(x) and the prior density function, which is obtained from partial structure, heavy atom positions, protein mask…, m(x), the entropy function is defined as
2. ... When F(q) reaches its maximum
2.6
so the maximum entropy density qME(x) satisfy the relation
2. ... 12) give a full solution to the maximum entropy problem under linear constraints.
Replace Cj(x) and cj with exp(-2pih×r) and structure factors F(h) respectively, we have the maximum entropy solution in a crystallographic environment. ... 3Entropy Maximisation Algorithms
Since we have the mathematical solution of maximum entropy, the next step is to solve it by a computer. ... 4Maximum Entropy in Crystallography
Maximum Entropy has been used by Bricogne, Gilmore, Prince, Navaza and other crystallographers for about 20 years. The major work in this field is to solve the phase problem in X-ray crystallography. ...
Collins pioneered the use of Maximum Entropy in crystallography. In his paper published in 1982 (Collins, 1982), he used a special maximum entropy formula to refine the electron density map. ...
From today’s point based on current maximum entropy research in this field, his algorithm was too simple and couldn’t reflect the effect of maximising the entropy on the improvement of electron density map. However his deduction sets up an example of how to use maximum entropy theory in a crystallographic problem.
After Collins’ work, several papers on research of the basic Maximum Entropy theory appeared in crystallographic journals in the middle of 80s, including Wilkins (1983a, b) and his colleagues(1983), Bricogne(1984), Navaza(1985) and Livesey and Skilling’s (1985) work. ... This paper connected Maximum Entropy with Direct Method¾the long-term example of crystallographers’ success in solving small molecule phase problem. ... Maximum entropy, by using a different approximation, solves this problem naturally.
So maximum entropy method will exceed and replace direct method one day, according to Bricogne’s analysis. ... Until now, 18 years after his statement, maximum entropy is still struggling among all phasing methods in crystallography. ... , 1990) on using maximum entropy to solve small molecule phase problem appeared in Acta Crystallography A in 1990. The work was based on likelihood optimisation and entropy maximisation. ...
There was some post-processing after a maximum entropy map was acquired and initial phases were obtained. ... In this paper, they proposed a method, which combined solvent flattening and entropy maximisation. ...
Another major branch of crystallographer’s job of developing maximum entropy method was done by Prince and his co-workers(Prince et al. ...
In recent years another maximum entropy program—MEED appeared in crystallographer’s scope. ... This program uses single-pixel approximation to do the entropy maximisation, which prevents its further use in phasing field (Sakata and Sato, 1990). ... His main idea was that all information involved in phases and structures can be used as entropy maximisation constraints. This was not a new idea because as early as in Bricogne’s 1984 paper, he stated that one of the natural advantages of maximum entropy is that all kinds of information, including mathematical, physical and chemical information (like anomalous scattering phases), can be included in the entropy constraints.
Some other implementations of maximum entropy to crystallography were done by David(David, 1987). ... 5The future Of Implementing Maximum Entropy Method To Crystallography
Maximum Entropy method is still a hot point in crystallography. ... However until now maximum entropy hasn’t made any real big impact on modern crystallography and so is not so popular in crystallographers. One of the reasons exists within the maximum entropy algorithm itself. ... However, compared to conventional phasing method, maximum entropy has its own advantages. ... The future of maximum entropy looks promising, but is still unknown.
Chapter 3 Using Maximum Entropy to Solve Energy Overlap Problem in Laue Crystallography
3.1Continuation of the Discussion in Chapter 1 and Chapter 2
As discussed in Chapter 1, energy overlaps in the Laue technique, especially those in low-resolution, seriously limit its use in modern crystallography. ... Maximum entropy was selected as a solution to this problem.
The concept of applying the maximum entropy method to the Laue diffraction is both novel and challenging:
The maximum entropy is a fundamental information theoretical principle and its application should yield an unbiased solution if you have the correct probability model.
The overlapped Laue intensities can be formulated as constraints to maximise the entropy. The equations describing the overlaps are linear, which is the most suitable condition to be used with maximum entropy. ... All maximum entropy formulas are different for different cases. ...
The initialisation problem that is usual a bottleneck in the maximum entropy is uniquely addressed by utilising the available intensities from the single (i. ... 2 CCP4 Program Suite and O Program
Before discussing the application of maximum entropy to Laue X-ray crystallography, I need to describe the software package that I used to do all crystallographic calculations, and the function and subroutine libraries which are extensively used in the programming of ME. ... html) as:
The CCP4 program suite is a collection of disparate programs covering most of the computations required for macromolecular crystallography. They have been collected and developed under the auspices of the Collaborative Computing Project Number 4, in Protein Crystallography, supported by the UK Science and Engineering Research Council (SERC) since 1979 and currently the Biotechnology and Biological Sciences Research Council (BBSRC), and co-ordinated at Daresbury Laboratory. ... 3Mathematical Analysis
Following the discussion in Chapter 2, more specific details of implementation of maximum entropy to overlap deconvolution are described here. ... For this reason, I chose Patterson function as the probability distribution in entropy function.
The entropy function will look like
3. ...
We hope to maximise the entropy function under these (multiple) constraints. ... This can be explained like this: imagine the entropy maximisation of a Patterson map is similar to a density-modification procedure, during this procedure, the roughly shape of Patterson map is decided by single reflection intensities because single reflections stands for a great percentage of all reflections. ... It is also noticed that calculation from zero information will tend to converge to a local maximum more than calculation from an already approximate map. ... According to Lagrange theory, the maximum point of the function
3. ... 5
and x is Lagrange multipliers, also gives the maxim point of entropy function under the constraints.
Following the analysis in Chapter 2, when Lagrange function reaches its maximum, the first partial derivatives of it with respect to p(r) will all be zero, which are (some constants are ignored in this equation) (see Eq. ... Obviously, this problem can be solved by comparing the entropy of different solutions. The one with the biggest entropy will be the solution of our problem.
Searching the maxim of Lagrange function doesn’t equal to searching the maxim of entropy function under constraints. ... 8) cannot give all the maxims of entropy function. This could result in being trapped in a local maximum entropy solution instead of a global one. ... Since the real map is approximated by the map with a global maxim of entropy function, it is reasonable to assume that the initial (prior) Patterson map is close to the maximum entropy solution. ... In crystallography the accuracy of data is not in absolute scale (like, all Patterson functions are affected by I(000)). ... Prince(1993) uses entropy, while Gilmore(1990) uses LLG(log-likelihood gain). ...
All intensities of the multiple spots are then processed by the maximum entropy program ME. ... There are two ways to prove the effectiveness of maximum entropy. ... 1 Analysis of the deconvoluted multiple reflections (lysozyme) obtained by the maximum entropy method as a function of resolution and intensities. ... 2) against the monochromatic data was much higher than the maximum entropy result. ... 3) appears slightly better than that from the maximum entropy method but for many fewer reflections. ... 4 Analysis of the basic deconvoluted multiple reflections (lysozyme) obtained by the maximum entropy method as a function of resolution and intensities. ... 5 Analysis of the higher order deconvoluted multiple reflections (lysozyme) obtained by the maximum entropy method as a function of resolution and intensities. ... Using Maximum Entropy Method To Evaluate Unobserved Reflection Intensities In X-ray Crystallography Data Collection
4.1Some Background and Other People’s Work in This Field
In data collection of X-ray crystallography experiment, very often part of the reflection data is missing. ... In protein crystallography, low resolution reflections is important in defining molecular mask and polypeptide backbone. ... Patterson map built from a maximum entropy method can have a higher resolution than the original map calculated from only observed reflection data. ... 2Mathematical Analysis
Continuing the mathematical analysis developed in the last two chapters, here I examine the mathematical principle of maximum entropy and use it to the problem of missing reflection data. ...
Because the Patterson map is built from a uniform distribution, the prior Patterson function is ignored in the entropy function. The problem is to maximise entropy
4. ... 4
reaches its maximum, Patterson function should satisfy
4. ... 1 Analysis of the evaluated resolution intensities by the maximum entropy method at different resolution; the R factor as defined in equation (3. ...
This may prove that maximum entropy is a good algorithm to evaluate the unobserved intensities. ... However the Patterson function in a maximum entropy solution is an exponential function, which means only multiplying a constant won’t change its shape. ... So the smaller the I(000), the less effect on Patterson map, and the better result of maximum entropy solution. ...
Another interesting topic is how maximum entropy can help in evaluating reflection intensities outside the observed range. ...
All the above discussion is based on the mathematical nature of maximum entropy method and Newton-Raphson algorithm. ... 5Concluding remarks
It is amazing that with only reflection intensity information and a simple assumption that the entropy must be maximised, the data beyond the experimental resolution limit can be calculated with a reasonably good R-factor and the quality of the electron density map can be improved. This may reflect such a fact that the real map has got the most information comparing to other maps calculated from reflection data, because entropy is a function to describe the information amount contained in a probability distribution. ... Entropy of an electron density map, is the information contained in this probability distribution. The map will be a real structure when the entropy reaches its maximum. ... Some Thoughts About Future Development
Of Maximum Entropy Method
As we see in the previous two chapters, maximum entropy method shows important success in both the Laue and monochromatic Bragg X-ray crystallography. ... what makes maximum entropy method succeed in these applications? ... The most difficult problem in crystallography has been avoided, the success is based on a simplification of real problem. ... Because the maximum entropy solution of Patterson function is an exponential function, addition of a small Dx in the Lagrange multipliers means multiplying a real value through all Patterson function.
Approximate Word count = 12438
Approximate Pages = 49.8
(250 words per page double spaced)
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