# What's the differences between one-dimensional spectrum and two-dimensional spectrum?

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mostly, we use the one-dimensional spectrum. But sometimes we use two-dimensional spectrum, what's the differences between them?

When you place a spectrograph slit on a source, the spectrum recorded can be thought of as lots of images of the slit at different wavelengths.

Ordinarily, you would sum up this spectrum along the direction of the slit images to give you a one dimensional spectrum. If however, you leave the image as recorded, then you have a two dimensional spectrum - intensity as a function of wavelength along one axis and as a function of position along the slit in the other.

Two dimensional spectra are used when we expect the spectrum to vary with position along the slit. Examples might include a spectrum recorded across a galaxy, or a spectrum of a binary star with the slit placed across both components.

An example is shown below. The inset image shows a broad-band image of V458 Vul - a classical nova that is surrounded by (not visible) shells of ionised material. The authors of this particular study lined up a spectrograph slit as shown in the inset and then obtained the two two-dimensional spectra shown in the main image. What you have to imagine is that each position on the slit produces a horizontal spectrum at a vertical position that corresponds to its position on the slit. Therefore we see a bright spectrum across the middle corresponding to the central source, but there are then "knots" of emission at particular wavelengths that are some distance away from the central star.

Slitless two dimensional spectroscopy is also possible using integral field spectrographs. Fibers record spectra over a two dimensional area. This can also be referred to as two dimensional spectroscopy.

## Analysis and Comparison of Spectral Extraction Algorithms in LAMOST ☆

In this paper, six spectral extraction algorithms are analyzed and compared in LAMOST (Large Sky Area Multi-Object Fiber Spectroscopic Telescope) two-dimensional (2D) spectral images. The compared algorithms include aperture method, profile-fitting method, direct deconvolution method, Tikhonov deconvolution method, deconvolution extraction based on adaptive Landweber iteration method, and deconvolution extraction based on Richardson-Lucy iteration method. The six algorithms are compared in the aspects of signal-to-noise ratio (SNR) and resolution, it can be found that Tikhonov deconvolution method, deconvolution extraction based on adaptive Landweber iteration method, and deconvolution extraction based on Richardson-Lucy iteration method are the most reliable algorithms among them. Finally, the future work is proposed.

## The Use of Fuzzy Graphs in Chemical Structure Research

### A. Independent Spin Coupling Networks

A structure can be deduced from a set of substructures. The latter is derived from multispectra. The structure elucidation involves extracting the set of substructures from multispectra and then assembling them. Let us consider two-dimensional Correlated SpectroscopY (COSY) as shown in Fig. 2 . A cross-peak indicates that two protons in atoms have a spin coupling, which implies that they are separated by two or three bonds

FIGURE 2 . The COSY spectrum has two frequency domains, F1 and F2. Both of them represent one-dimensional 1 H NMR spectra. The cross-peaks have coordinates in the form (f1,f2). The coordinate indicates that the two protons f1 and f2 are coupled. This coupling can be a two- or three-bond coupling. The diagonal peaks are not informative for the two frequencies if their coordinates are equivalent. The diagonal peaks are usually crowded and cannot be easily interpreted.

(geminal or vicinal). Accordingly, given a structure, these proton spin coupling relations can be predicted. Each spin coupling pair corresponds to a COSY cross-peak. These spin coupling relations can be connected to form spin coupling topological networks [or alternatively, independent spin coupling networks (ISNets)]. A molecular structure may have more than one ISNet because its spin coupling topology may be disconnected. For the peptide fragment ∼ Lys → Val ∼, we show in Fig. 3 the relevant ISNets.

FIGURE 3 . The Lys and Val residues are divided into two parts of a carbonyl group. The H α of the Lys can couple with the H 0 of the Val. Theoretically, all the edges in Lys ISNet or Val ISNet can be observed as COSY cross-peaks. These edges represent through-bond couplings.

An ISNet may be defined as

where CS is a set of chemical shifts and SC is a set of spin couplings (edges). Here csi indicates the ith chemical shift in CS.

Before two-dimensional NMR experiments were invented, a set of discrete data points (chemical shifts) was usually assigned to a molecular structure with little or no knowledge of the correlations among the data points. There was no direct way to observe these correlations. Now multiple-dimensional NMR experiments have opened up opportunities for the analysis of the correlations between the atoms in a structure. The relationship of subspectra-substructure is no longer the relationship of the discrete data points and the substructure. Instead, it is the relationship of the graph (data points plus their correlations) and the substructure. Thus, multiple-dimensional NMR (nD-NMR) experiments offer a lot of advantages in solving spectral data overlap problems.

The ISNet concept is different from the conventional description of a spin system. Just as an AX system implies that atom A couples with atom X, an ABX system implies that atom B couples withatom A and atom X. ISNets emphasize the global coupling network. However, a conventional spin system is usually just a fragment of an ISNet. More complicated ISNets have been discussed by Xu and Borer. 3 The cross-peaks in a three-dimensional spectrum (or three-dimensional hyperplane from a higher-dimensional spectrum) are represented by two edges sharing a common node, four-dimensional cross-peaks by three edges with two common nodes, etc. It should be noted that such a graph does not reflect the dimensionality of the nD-NMR signals. The formalism developed for two-dimensional NMR can be extended rather simply without needing to reinvent the two-dimensional formalism. The edges may also have a “color,” e.g., a range of coupling constant values. The certainty of mapping observed ISNets onto the family of allowed ideal ISNets is thereby increased. 3

Figure 4 summarizes some two-dimensional NMR experiments and their interpretations from which homonuclear ISNets and heteronuclear ISNets can be extracted. Duddeck and Dietrich have discussed the details of these spectra. 9

FIGURE 4 . Spin coupling correlations observed from some two-dimensional NMR experiments. There are three ways to classify them: (1) short distance correlations (such as, COSY, H, C-COSY, 2D INADEQUATE) or long distance correlations (such as relayed COSY, TOCSY, HMBC) (2) homonuclear correlations (such as H,H-COSY, relayed H,H-COSY, TOCSY) or heteronuclear correlations (such as HMQC, HMBC) and (3) through-bond correlations (such as all types of COSY, TOCST, HMBC, 2D INADEQUATE) or through-space correlations (such as, NOESY, ROESY). 2D INADEQUATE is an ideal experiment to determine the carbon backbone of an unknown compound. However, its signal is too weak to be useful, as are C-relayed or H-relayed 1 H, 13 C-COSY. Long distance correlation is ambiguous, but provides good additional evidence for adding a new spin to a spin coupling system.

In practice, no experiment can give a complete peak set for a given compound. However, combining these spectra will give sufficiently complete ISNets for them to be mapped onto the entire structure. This procedure needs a number of graph-theoretical algorithms. 5

As shown in Fig. 3 , different substructures may have different ISNet patterns. In the ideal case, the assignment can be done by graphic pattern recognition. The same substructure may render a relatively different ISNet pattern for differing chemical environmental changes. The ISNet is a fuzzy graph. Fuzzy graph pattern recognition involves mapping an ISNet graph onto its cluster center. 5

## Two Dimensional NMR Spectroscopy

Well, just have a look at this 1D NMR spectrum of a protein:

## Anatomy of a 2D experiment:

• Do something with the nulcei (preparation),
• let them precess freely (evolution),
• do something else (mixing),
• and detect the result (detection, of course).

After preparation the spins can precess freely for a given time t1. During this time the magnetization is labelled with the chemical shift of the first nucleus. During the mixing time magnetization is then transferred from the first nucleus to a second one. Mixing sequences utilize two mechanisms for magnetization transfer: scalar coupling or dipolar interaction (NOE). Data are acquired at the end of the experiment (detection, often called direct evolution time) during this time the magnetization is labelled with the chemical shift of the second nucleus.

Two dimensional FT yields the 2D spectrum with two frequency axes. If the spectrum is homonuclear (signals of the same isotope (usually 1 H) are detected during the two evolution periods) it has a characteristic topology:

The cross signals originate from nuclei that exchanged magnetization during the mixing time (frequencies of the first and second nucleus in each dimension, respectively). They indicate an interaction of these two nuclei. Therefore, the cross signals contain the really important information of 2D NMR spectra.

## Homonuclear 2D Experiments:

### 2D TOCSY:

Thus a characteristic pattern of signals results for each amino acid from which the amino acid can be identified. However, some amino acids have identical spin systems and therefore identical signal patterns. They are: cysteine, aspartic acid, phenylalanine, histidine, asparagine, tryptophane and tyrosine ('AMX systems') on the one hand and glutamic acid, glutamine and methionine ('AM(PT)X systems') on the other hand.

### 2D NOESY:

Here is a picture of a 2D NOESY spectrum (38 k)
Here is a scheme which shows the several spectral regions in the 2D NOESY (8 k).

## Heteronuclear NMR spectroscopy:

### The HSQC Experiment:

The spectrum contains the signals of the H N protons in the protein backbone. Since there is only one backbone H N per amino acid, each HSQC signal represents one single amino acid. The HSQC also contains signals from the NH2 groups of the side chains of Asn and Gln and of the aromatic H N protons of Trp and His. A HSQC has no diagonal like a homonuclear spectrum, because different nuclei are observed during t1 and t2. An analogous experiment ( 13 C-HSQC) can be performed for 13 C and 1 H.

[1D NMR Spectroscopy] [Index] [PPS2 Projects] [3D NMR Spectroscopy] Horst Joachim Schirra's PPS2 project
Determination of Protein Structure with NMR Spectroscopy
last updated 281196

## Contents

Dimensional modals are intended to reflect what constitutes personality disorder symptomology according to a spectrum, rather than in a dichotomous way. As a result of this they have been used in three key ways firstly to try to generate more accurate clinical diagnoses, secondly to develop more efficacious treatments and thirdly to determine the underlying etiology of disorders. [4]

### Clinical diagnosis Edit

The "checklist" of symptoms that is currently used is often criticized for a lack of empirical support [5] and its inability to recognize personality-related issues that do not fit within the current personality disorder constructs or DSM criteria. [6] It has also been criticized for leading to diagnoses that are not stable over time, have poor cross-rater agreement and high comorbidity [7] suggesting that they do not reflect distinct disorders. [8] In contrast the dimensional approach has been shown to predict and reflect current diagnostic criteria, but also add to them. [9] It has been argued to be especially useful in explaining comorbidity which is often high for patients diagnosed with a personality disorders. [7] Following from these claims, the fifth edition of the Diagnostic and Statistical Manual of Mental Disorders (DSM-5) incorporates a combined categorical-dimensional approach to diagnosing personality disorders [5] based on the degree to which a person shows elevated levels of particular personality characteristics. However one of the issues in using a dimensional approach to diagnosis has been determining appropriate cut off points so as to know who belongs to the category of people requiring treatment, this is partly why both categorical and dimensional diagnoses are included. [10]

Since the categorical model is widely used in clinical practice and has a significant body of research supporting it, its common usage is compelling to laypeople when they are judging the credibility of professional opinion. Therefore, the dimensional approach is often further criticized for being difficult to interpret and less accessible. It is however widely used in some professional settings as the established approach, for example by forensic psychologists. [11]

### Treatment efficacy Edit

Another suggested usage of the dimensional approach is that it can aid clinicians in developing treatment plans and assessing other mechanisms contributing to patient’s difficulty in functioning within the social, personal, or occupational domains. The approach can improve treatment in two ways. Firstly it can enable development of more personalized care plans for individuals based on their adaptive and maladaptive characteristics. Secondly, it means that relevant symptomology which is not considered maladaptive can be considered when developing and evaluating general therapeutic and medical treatment. [4]

### Determining cause Edit

Attempts at presenting an etiological description of personality disorders have been avoided due to the influence of the DSM and its principles in psychiatric research (See history section). However some techniques are looking at potential interrelated causalities between symptoms of personality disorders and broader influences including aspects of normal personality (See integrated approaches section).

### Initial development of a categorical model Edit

The adoption of a categorical approach to personality disorders can be understood in part due to ethical principles within psychiatry. The ‘do no harm principle’ led to Kraepelinian assumptions about mental illness and an emphasis on empirically grounded taxonomic systems that were not biased by unsubstantiated theories about etiology. [12] A taxonomic checklist based on empirical observations rather than bias prone theoretical assumptions developed. It was both categorical and hierarchical, with the diagnosis of a disorder being dependent of the presence of a threshold number categories (usually five) out of a total number (seven to nine) [12] Disorders were organized into three clusters, existing purely to make the disorders easier to remember by associating them with others that have similar symptoms, not based on any theory about their relatedness. [10]

### Emerging problems with the categorical model Edit

The dimensional model was developed in response to the limitations of this standard categorical model. [9] The expectations from a Kraepelinian approach were that as systematic research into psychiatric health increased diagnostic categories would be refined and targeted reliable treatments would be developed. [13] However this reductionist approach to diagnostic categorization has led to disorders with high comorbidity, life course instability, poor treatment efficacy and poor diagnostic agreement. [1] In addition the findings from psychopathological research have led to an increasing body of evidence suggesting overlaps between normal and maladaptive personality and interrelatedness across disorders. [7] These findings have been further supported by genetic [14] and developmental studies [15] which have constantly pointed towards greater interrelatedness then the diagnostic categories can offer. These consistently disconformity findings, alongside the successful shift to a continuous rather than categorical approach in other areas of research, such as regarding ASD, led to consideration of alternative approaches. [16]

### Development of methodological techniques Edit

#### Factor analysis Edit

The development of factor analysis as a popular statistical technique in differential psychology has led to an increase in attempts at finding underlying traits. More recently this has been used in the context of personality disorders both as a means of looking at which personality traits current categorical diagnoses are related to and also as a method of looking for new psychopathological latent variables. Factor analysis has helped illustrate that the full range of relevant personality pathology is not included in the DSM psychiatry nosology. However the technique does not show information about a continuum from normal to clinically relevant personality. [9]

#### Dimensional analysis Edit

Dimensional classification techniques show individual multidimensional profiles and therefore they can show information about a personality continuum (from normal to atypical), one such technique is Hybrid modeling. [17] Cut off points can be introduced into these modals to show where a diagnosis may lie. However the number of different rating scales that need to be looked at and the lack of interdisciplinary research between statisticians and psychologists has meant that attempts at finding a ‘worldwide’ criteria for dimensional diagnosis using this method has been of limited success. [17]

#### Comparative analysis Edit

Analyses have been conducted to test the relative fit of categorical and dimensional modals to evaluate whether single diagnostic categories are suited to either status. These types of analysis can include a range of data, including endophenotypes or other genetic or biological markers which increases their utility. Multivariate genetic analysis helps establish how well the current phenotypically developed structure of personality disorder diagnosis fits with the genetic structure underlying personality disorders. Results from these types of analysis support dimensional over categorical approaches. [10]

#### Network analysis Edit

Network Analysis has been used as a means of integrating information about personality with personality disorders and as well as information about other genetic, biological and environmental influences into a single system and looking at interrelated causalities between them (See integrated modals).

There are different ways to ‘dimensionalize’ personality disorders, these can be summarised into two categories.

1. The first involves quantifying DSM-5 pathology. This can be done either based on the degree to which symptoms are present or on how close to a prototypic presentation a patient's presentation may be. The prototype approach includes features not present in the DSM. [18]
2. The second approach involves identification of DSM disorder traits by means of factor analysis to show underlying dimensions of the personality disorder criteria, this method may also include relevant psychopathology. [17]

### Normal personality models Edit

#### Five-factor model Edit

The Five-Factor model of personality, which is the most dominant dimensional model, [19] has been used to conceptualize personality disorders and has received various empirical support. Under this approach, extreme levels of the basic personality traits identified by the FFM are what contributes to the maladaptive nature of personality disorders. [20] Over 50 published studies supporting this model have been identified, providing much empirical support for this approach. Most of these studies examine the relationship between scores on separate measures of Big Five trait and personality disorder symptoms. [20]

The Five-Factor model was first extended to personality disorders in the early 1990s, when it was established that a satisfactory profile of each personality disorder in the DSM-III-R could be created through various levels of Big Five traits. [5] Thomas Widiger and his colleagues have demonstrated that many of the central elements of personality disorders can be explained in terms of Big Five traits – for example, borderline personality disorder is characterized by high levels of hostility, trait anxiety and depression, and vulnerability, all of which are facets of neuroticism. [5] This approach also helps to differentiate characteristics of disorders that overlap under the current categorical model, such as avoidant and schizoid personality disorders. The Five-Factor-based approach explains much of that overlap as well as the ways in which they are different. [5] For example, both are characterized primarily by maladaptive excessive introversion, but antisocial personality disorder also includes high levels of facets of neuroticism (such as self-consciousness, anxiety, and vulnerability), while schizotypal personality disorder includes the addition of low assertiveness. The Five-Factor approach also resolves previous anomalies in factor analyses of personality disorders, which makes it a more explanatory model than the current categorical approach, which only includes three factors (odd-eccentric, dramatic-emotional, and anxious-fearful). [5]

A prototype diagnostic technique has been developed in which Five-Factor-based prototypes for each disorder were created, based on the aggregated ratings of personality disorder experts. These prototypes agree well with DSM diagnostic criteria. [20] The Five-Factor prototypes also reflected the high comorbidity rates of personality disorders. This is explained by the idea that various other disorders tap into dimensions that overlap with those of the primary diagnosis. [20]

Another Five-Factor based technique involves diagnosing personality disorders based on clinician ratings of various facets of the five factors (e.g. self-consciousness, which falls under the neuroticism factor excitement seeking, which falls under the extraversion factor). This technique is partially based on the prototype model, as each facet's "score" is based on its rating of how prototypical it is of each personality disorder, with prototypically low facets (with a score less than 2) reverse-scored. Using this technique, diagnosis is based on an individual's summed score across relevant facets. This summed-score technique has been shown to be as sensitive as the prototype technique, and the easier computation method makes it a useful suggested screening technique. [6]

The Five-Factor assessment of personality disorders has also been correlated with the Psychopathy Resemblance Index of the NEO Personality Inventory, as well as with the individual personality dimensions of the NEO-PI-R. [21] It also resolves several issues regarding the PCL-R psychopathy assessment, as a Five-Factor-based re-interpretation of the PCL-R factor structure shows that the “Aggressive Narcissism” factor taps into facets of low agreeableness (with some contribution of facets of neuroticism and extraversion), and the “Socially deviant lifestyle” factor represents facets of low conscientiousness and low agreeableness. It has also been shown that the sex differences in personality disorders can be reasonably predicted by sex differences in Big 5 traits. [22]

##### Criticism Edit

The dimension of openness to experience of the Five-Factor model has been criticized for not directly relating to any of the major characteristics of personality disorders in the same way as do the other four dimensions [ citation needed ] . It has been suggested that schizotypal and histrionic personality disorders could be partially characterized by high levels of openness to experience (in the forms of openness to ideas and feelings, respectively) [ citation needed ] , while obsessive-compulsive, paranoid, schizoid, and avoidant personality disorders can all be conceptualized by extremely low levels of openness [ citation needed ] . However, there is little to no empirical support for this hypothesis, particularly with schizotypal personality disorder. Additionally, the Openness scale of the NEO-PI-R, which is one of the most widely used measures of Big Five traits, was based on research and theory which viewed openness (such as self-actualization and personal growth) as beneficial, so measurement of extreme openness using the NEO-PI-R, is actually a marker of good mental health. [5]

#### Seven factor model Edit

The Five-Factor approach has been criticized for being limited in some respects in its conceptualization of personality disorders. This limitation is due to the fact that it does not include evaluative trait terms such as “bad”, “awful”, or “vicious”. Some research has suggested that two evaluative dimensions should be added to the Five-Factor model of personality disorders. Empirical support for this approach comes from factor analyses that include the Big Five factors and evaluative terms. These analyses show that the evaluative terms contribute to two additional factors, one each for positive and negative valence. The addition of these two factors resolves much of the ambiguity of the openness dimension in the Five-Factor approach, as the openness factor changes to a conventionality factor, and adjectives such as “odd”, “strange”, and “weird” (which all characterize schizotypal personality disorder) fall onto the negative valence factor. These results indicate that the inclusion of evaluative terms and valence dimensions can be valuable for better describing the extreme and maladaptive levels of personality traits that comprise personality disorder profiles. [5]

#### Internalizing/Externalizing model Edit

A two-factor model of psychopathology in general has also been suggested, in which most disorders fall along internalizing and externalizing dimensions, [23] [24] which encompass mood and anxiety disorders, and antisocial personality and substance use disorders, respectively. [24] Although this approach was originally developed to understand psychopathology in general, it has often been focused to apply to personality disorders, such as borderline personality disorder to help better understand patterns of comorbidity. [25]

#### Szondi drive theory Edit

Hungarian psychiatryst Léopold Szondi formulated in 1935 a dimensional model of personality comprising four dimensions and eight drives ("facets" in DSM V terminology). It was based on a drive theory, in which the four dimensions correspond to the independent hereditary circular mental diseases established by the psychiatric genetics of the time: [26] the schizoform (containing the paranoid and the catatonic drives), the manic-depressive (for the "contact" dimension), the paroxysmal (including the epileptic and hysteric drives), and the sexual drive disorder (including the hermaphrodite and the sadomasochist drives). [27] The Sex (S) and Contact (C) dimensions can be further grouped as representing pulsions at the border with the outer world, while the Paroximal (P) and Schizoform (Sch) dimensions at the inner part of the psyche.

### Integrated models Edit

#### Network analysis Edit

Network analysis diverts most strongly from the categorical approach because it assumes that the symptoms of a disorder have a causal relationship to each other. This theoretical assumption is made because no mental disorder can currently be understood as existing independently from its symptoms, as other medical diseases can be. According to the network approach symptoms are not looked at as the product of a set of latent disorders, instead they are looked at as mutually interacting and reciprocally reinforcing elements within a wider network. [28] Therefore, a diagnosis is not needed to understand why the symptoms hang together. Clusters of densely connected symptoms can be defined as disorders, but they are inevitably intertwined with related symptoms and cannot be entirely separated. This helps explain the growing body of research showing comorbidity, co-occurring genetic markers and co-occurring symptoms across personality disorders. [29]

##### Therapeutic consequences Edit

The therapeutic consequence of this is that treatment is targeted at the symptoms themselves and the causal relations between them, not the overarching diagnosis. This is because targeting the diagnosis is trying to treat an unspecified summary of a complex collection of causes. Adopting this attitude sits well with the therapeutic treatments in use at the moment that have the strongest evidence base. [29]

##### Network construction Edit

Network analysis has its roots in mathematics and physics but is increasingly being used in other areas. Essentially it is a method of analyzing mutually interacting entities by represented them as nodes which are connected to through relations called edges. Edges represent any sort of relation such as a partial correlation. Complex network analyses of other subjects have looked at tipping points, where one system suddenly transitions into another, such as when a tropical forests goes into a savannah. If these could be identified in individual’s psychopathological dynamic networks then they could be used to determine when a person’s network is on the brink of collapse and what can be done to alter it. [29]

##### Criticisms Edit

There are concerns that the network modal does not have enough parsimony and is too difficult to interpret. [30]

The Personality and Personality Disorder Work Group proposed a combination categorical-dimensional model of personality disorder assessment that will be adopted in the DSM-5. The Work Group's model includes 5 higher-order domains (negative affectivity, detachment, antagonism, disinhibition, and psychoticism) and 25 lower-order facets, or constellations of trait behaviors that constitute the broader domains. The personality domains can also be extended to describe the personality of non-personality disorder patients. Diagnosis of personality disorders will be based on levels of personality dysfunction and assessment of pathological levels of one or more of the personality domains, [31] resulting in classification into one of six personality disorder "types" or Personality Disorder Trait Specified (depending on the levels of traits present), in contrast to the current traditional categorical diagnoses of one of 10 personality disorders (or personality disorder not otherwise specified) based on the presence or absence of symptoms. [32]

### Criticism Edit

There are concerns that the addition of dimensional models to DSM-5 may raise confusion. Carole Lieberman has stated that "As it is now, people don't really make use of the subcategories that there are to describe severity of symptoms. Instead, I see this as a tool that insurance companies could well co-opt to try to deny benefits." [33]

## What's the differences between one-dimensional spectrum and two-dimensional spectrum? - Astronomy

Fourier transform can be generalized to higher dimensions. For example, many signals are functions of 2D space defined over an x-y plane. Two-dimensional Fourier transform also has four different forms depending on whether the 2D signal is periodic and discrete.

Aperiodic, continuous signal, continuous, aperiodic spectrum

where and are spatial frequencies in and directions, respectively, and is the 2D spectrum of .

where and are the spatial intervals between consecutive signal samples in the and directions, respectively, and and are sampling rates in the two directions, and they are also the periods of the spectrum .

where and are periods of the signal in and directions, respectively, and and are the intervals between consecutive samples in the spectrum .

where and are the numbers of samples in and directions in both spatial and spatial frequency domains, respectively, and is the 2D discrete spectrum of . Both and can be considered as elements of two by matrices and , respectively.

Consider the Fourier transform of continuous, aperiodic signal (the result is easily generalized to other cases):

The inverse transform represents the spatial function as a linear combination of complex exponentials with complex weights .

The Complex weight can be represented in polar form as

in terms of its amplitude and phase :

• is the unit vector along direction ,
• is a vector along the direction in the 2D spatial domain.

In the function on top, (2 cycles per unit distance in x) and and (3 cycles per unit distance in y), while in the function at bottom, (3 cycles per unit distance in x) and (2 cycles per unit distance in y). But along their individual directions ( and respectively), their spatial frequencies are the same .

Now the 2DFT of a signal can be written as:

The 2D function shown below contains three frequency components (2D sinusoidal waves) of different frequencies and directions:

and rewrite the 2D transform as

First consider the expression for . As the summation is with respect to the row index of , the column index can be treated as a parameter, and the expression is the 1D Fourier transform of the nth column vector of , which can be written in column vector (vertical) form for the nth column:

i.e., the nth column of is the 1D FT of the nth column of . Putting all columns together, we have

where is a by Fourier transform matrix.

Now we reconsider the 2DFT expression above

As the summation is with respective to the column index n of , the row index can be treated as a parameter, and the expression is the 1D Fourier transform of the kth row vector of , which can be written in row vector (horizontal) form for the kth row:

i.e., the kth row of is the 1D FT of the kth row of . Putting all rows together, we have

Similarly, the inverse 2D DFT can be written as

It is obvious that the complexity of 2D DFT is (assuming ), which can be reduced to if FFT is used.

Consider a real 2D signal:

The imaginary part . The 2D Fourier spectrum of this signal can be found by 2D DFT. The real part of the spectrum is:

## Relation between one- and two-dimensional noise power spectra of magnetic resonance images

Our purpose in this study was to elucidate the relation between the one-dimensional (1D) and two-dimensional (2D) noise power spectra (NPSs) in magnetic resonance imaging (MRI). We measured the 1D NPSs using the slit method and the radial frequency method. In the slit method, numerical slits 1 pixel wide and L pixels long were placed on a noise image (128 × 128 pixels) and scanned in the MR image domain. We obtained the 1D NPS using the slit method (1D NPS_Slit) and the 2D NPS of the noise region scanned by the slit (2D NPS_Slit). We also obtained 1D NPS using the radial frequency method (1D NPS_Radial) by averaging the NPS values on the circumference of a circle centered at the origin of the original 2D NPS. The properties of the 1D NPS_Slits varied with L and the scanning direction in PROPELLER MRI. The 2D NPS_Slit shapes matched that of the original 2D NPS, but were compressed by L/128. The central line profiles of the 2D NPS_Slits and the 1D NPS_Slits matched exactly. Therefore, the 1D NPS_Slits reflected not only the NPS values on the central axis of the original 2D NPS, but also the NPS values around the central axis. Moreover, the measurement precisions of the 1D NPS_Slits were lower than those of the 1D NPS_Radial. Consequently, it is necessary to select the approach applied for 1D NPS measurements according to the data acquisition method and the purpose of the noise evaluation.

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## Convolution Neural Network Based on Two-Dimensional Spectrum for Hyperspectral Image Classification

Inherent spectral characteristics of hyperspectral image (HSI) data are determined and need to be deeply mined. A convolution neural network (CNN) model of two-dimensional spectrum (2D spectrum) is proposed based on the advantages of deep learning to extract feature and classify HSI. First of all, the traditional data processing methods which use small area pixel block or one-dimensional spectral vector as input unit bring many heterogeneous noises. The 2D-spectrum image method is proposed to solve the problem and make full use of spectral value and spatial information. Furthermore, a batch normalization algorithm (BN) is introduced to address internal covariate shifts caused by changes in the distribution of input data and expedite the training of the network. Finally, Softmax loss models are proposed to induce competition among the outputs and improve the performance of the CNN model. The HSI datasets of experiments include Indian Pines, Salinas, Kennedy Space Center (KSC), and Botswana. Experimental results show that the overall accuracies of the 2D-spectrum CNN model can reach 98.26%, 97.28%, 96.22%, and 93.64%. These results are higher than the accuracies of other traditional methods described in this paper. The proposed model can achieve high target classification accuracy and efficiency.

#### 1. Introduction

Hyperspectral images (HSIs) are typically composed of hundreds of spectral data channels in the same scene. HSIs can provide continuous data in space and spectrum through combined imaging and spectrum technology. Hyperspectral data are important in monitoring information of the Earth’s surface because the spectral information provided by the hyperspectral sensor increases the accuracy of the resolution of target materials and thus improves classification accuracy [1].

At first, scholars mainly use artificial extraction of image features for object identification classification of remote sensing images using a local binary pattern, histogram of oriented gradient [2], and Gabor filter [3]. However, this method is ineffective in processing hyperspectral data with the increase in dimension. Thus, feature extraction and classifier are combined, thereby yielding a satisfactory classification effect. Methods for feature extraction include principal component analysis (PCA) [4], independent component analysis (ICA) [5], and linear discriminant analysis (LDA) [6] and robust PCA. The classifier went through a process from fuzzy K-nearest neighbor algorithm [7], naive Bayes with deep feature weighting [8], and logistic regression [9] to support vector machine (SVM) [10]. SVM improves classification performance by extending classification kernel [11]. However, these combinatorial methods demonstrate the following significant limitations. (1) Feature extraction uses linear transformations to extract potentially useful features from the input data. Hyperspectral data are essentially nonlinear considering a complex light-scattering mechanism [12]. (2) Most traditional classification methods only consider single-layer processing, which reduces the capability of feature learning, and are unsuitable for high-dimensional data.

Neural networks (NNs) with multiple layers and hidden nodes are more suitable than shallow classifiers, such as SVM, in building an HSI data model [13]. The NNs, including multilayer perceptron [14] and radial basis function [15], have been studied for classifying remote sensing data. Researchers have proposed a semisupervised NN framework for large-scale HSI classification [16]. Various deep NNs (DNN) have been developed according to system architecture and activation functions these networks include deep belief network (DBN) [17], deep Boltzmann machine [18], and AutoEncoder (AE) [19]. In 2014, a stacked AE (SAE) was used for HSI classification [20]. An improved AE based on sparse constraints was then proposed [21]. The DBN is another DNN model that was proposed in 2015 [22]. The depth model can extract robust features and is superior to other methods in terms of classification accuracy.

Convolution NN (CNN) [23] uses local receptive fields in efficiently extracting spatial information and sharing weights to significantly reduce the number of parameters. CNNs are used to extract the spatial spectral features of hyperspectral images for classification [24], and their performance was better than that of traditional classifiers such as SVM. In addition, a method using a virtual sample enhanced to limited labeled samples was proposed in [25]. A previous study proposed the use of a greedy layer unsupervised pretraining to form a CNN model [26]. However, the application technology of CNN in hyperspectral classification remains imperfect, and several shortcomings, such as easy saturation of the training gradient, low classification accuracy, and poor model generalization, should be addressed.

The spectral values of the HSIs in the third dimension are approximately continuous, and the curves of each feature possess a unique spectral plot that is different from those of other classes. In the traditional classification methods, one-dimensional spectral vectors are used as the final form of input data [27, 28] or neighboring pixels are used to form small regional pixel blocks as input data [29, 30]. Although the former simplifies the complexity of deep learning network training, it omits spatial dimension information of spectral values at the same time. The latter combines multiple pixels into one sample, which introduces heterogeneous noises and aggravates the problem of missing hyperspectral data.

Compared with the traditional CNN methods, this study designs a 2D-spectrum CNN model as follows: (i) Hyperspectral pixels have rich spectral information. The traditional data processing methods which use small area pixel block or one-dimensional spectral vector as input unit bring many heterogeneous noises. In this paper, we convert the spectral value vector to 2D-spectrum image, so that the optimization of all CNN model parameters (including the BN parameters) is based on the spectral values of the pixel points and spectral space information. The target of fully extracting spectral spatial information can be achieved while heterogeneous noises are also avoided. In addition, a multilevel BN algorithm is achieved for the first time, and the effect of network acceleration is obvious. (ii) A BN algorithm is introduced to reduce the vanishing gradient problem and dynamically accelerate the training speed of the DNN by reducing the scaling and initialization of the dependent parameters. A small area pixel block was selected as the input unit. Liu et al. [30] used the BN algorithm to the CNN for the HIS. However, the introducing heterogeneous noises and wasting scarce samples will weaken the BN algorithm’s role in network regularization and accelerated training. (iii) Softmax loss models are used instead of combining Softmax regression and multinomial logistic loss models thus, the output of the last layer competes with one another to improve the classification accuracy. The experimental results show that the proposed CNN-based HSI classification model exhibits high accuracy and efficiency in the HSI dataset.

#### 2. CNN-Based Classification Model

The researchers found that the human visual system can effectively solve the problem of classification, detection, and identification, with the rapid development of modern nervous systems. This development motivates researchers on biological visual systems to establish advanced data processing methods [31]. Cells in the cortex of a human visual system are only susceptible to small areas, and accepting cells in the field can exploit the local spatial correlation in the image.

The CNN architecture uses two special methods, namely, local receptive field and shared weights. The activation value of each convolution neuron is calculated by multiplying the local input with weight

, which is shared in the entire input space (Figure 1). Neurons that belong to the same layer share the same weight. The use of specific architectures, such as local receptive field and shared weights, reduces the total number of training parameters and facilitates the development of an efficient training model.

## Fit the background¶

To subtract the background signal from the source region we want to fit a quadratic to the background pixels, then subtract that quadratic from the entire image — including the source region.

Let’s tackle a simpler problem first and fit the background for a single column. From visual inspection of the 2D spectrum, we have decided to isolate rows 10–199 and 300–479 as ones containing pure background signal:

Now let’s do this for every column and store the results in a background image:

Finally let’s subtract this background and see how the results look:

Detour: vector operations versus looping

If you are used to C or Fortran you might be wondering why jump through these hoops with slicing and making sure everything is vectorized. The answer is that pure Python is an interpreted dynamic language and hence doing loops is slow. Try the following:

Now compare to the vectorized NumPy solution:

Sometimes doing things in a vectorized way is not possible or just too confusing. There is an art here and the basic answer is that if it runs fast enough then you are good to go. Otherwise things need to be vectorized or maybe coded in C or Fortran.

## What's the differences between one-dimensional spectrum and two-dimensional spectrum? - Astronomy

rr, and hisactophilin. an actin binding protein of 118 amino acids) have been determined based on t’he NOE data derived solely from the homonuclear 3D NOE-NOE magnetic resonance spectroscopy. Two different approaches for extract’ion of the structural information from the 31) NOE-NOE experiment, were tested. One approach was based on the transformation of the 31) intensities into distance constraints. Tn the second, and more robust approach. the 31) NOE intensities were used directly in structure calculations, without the need to transform them into distance constraints. A new 2D potential function representing t,he 311 NO&NOE intensity was developed and used in the simulated annealing protocol. For CMTT-I, a comparison between structures det’ermined with the 3D NOE-NOE method and of’ various 21) NOE approaches was carried out. The 3D data set allowed better definition t,he structures than was previously possible with the 2D XOE procedures that used thf isolated two-spin approximation to derive distance information.

1. Introduction Multidimensional r1.m.r.t spectroscopy has heen shown to be of great) value in extending the methodology of protein structure determination by t1.rn.r. The main advantage of multidimensional over 21) n.m.r. experiments lies in their potential t,o alleviate resonance overlap. A number of homonuclear (Griesinger rt ccl.. 1989 Vuister fd nl., 1989: Oschkinat et al.. 1990) and het,eronuc4ear (Fesik B Zuiderweg, 1988 Marion et al., 1989: Zuiderwrg 8 Fesik. 1989 Kay et

2.. 1990) 3D and 11) n.m.r. experiments have been described recently. For the purposes of a,ssignment of proteins larger t,han approximately 130 amino aclids. t hr heteronurlrar i- Akthrrviationa used n.m.r., nu

opy: %I). trio-dimensional: 41). thrfvtlitnf

nsional 1T). fi)ur-dimensional: (‘JITT-T. (‘cnc

mrr.ri,mrr t,rypsin inhibit,or I X‘OE. Xurlrar Overhausrt effect: IL’OESY. two-dimensional NOE: spfx

troscwpy: FYI). f’rrr induc%ion decay: tJR. jurnprpt,urn: TO(‘87. total cwrrelation spectroscopy HMQC. heteronuclrat multiplr quantum wrrrlation 31) NOE- NOE, thrrr dimrnsional h’OF:- SOE: L)IS(:E:O, tlistjancr gromet,rj prcjgratn E.c’OSY. exclusivf

cwrrrlation spectroscopy: SA. simulatfd annealing: r.m.s.. root mran square: r.tn.s.tl.. r.m.s. difference.

3D experiment seems t’o be more useful than thr homonuclear experiment. The heteronuclear experimerit’ has higher sensitivity as it involves large heteronuclear coupling constants compared to the proton linewidth. It also has higher spectral resolut)ion due t,o t,he larger chemical shift dispersions of’ “N and 13(” nuclei compared to t)hc ‘H shifts. The number of cross-peaks in a heteronuclear 31) spe’c’

trum is equal to t)hat of the corresponding homonuclear PI) spectrum, whilst additional csross-peaks complicate a homonuclear 31) sprc%rum. However. on(ae the assignments have heen made, the homonuclear 31) NOE spect’ra contain more information relating to distance criteria than the corresponding 31) hetrronuclear spectra (Fairbrother et trl.. 1!)92). This informat,ion is crit’ical for an accurate st ruc+urr determination. Rrc

ntly t hr pot’ential of homonuclear 31) SOLNOE experiment has I)een dt

tnonstrated for the differentiat

iori of spin diffusion pat’hways (Boelens rt crl.. 1989: Krrg rf r/l.. 1990: Kessler rt (

1.. 1991). Also. 31) NOE--NOE spectra (San be used to assess t’he amount of’spin diffusion in 2D POESY spectra (Habnzettl et nl., I!Nl). Tn this paper, we describe two different approachcas for the structure dctermination of proteins based on t,he information derived from the homonuclear 3D SOE-NOE experiment. One

approach is based on the transformation of the 3D intensities into distance constraints using the approximation lijk cc rij6r,i6, where Zijk is the intensity of the 3D NOE-NOE peak, and the rs are distance constraints extracted from the 3D peak. In a preliminary report on this approach, we already showed that a large number of distance constraints can be extract,ed from the homonuclear 3D spectrum to provide sufficient input data for “the highresolut’ion structures” (Holak et al., 1991). A detailed description of t’his approach is given in the present paper, Tn the second approach, the 3D NOE intensities are used directly in structure calculations. without being transformed into distance constraints. A new 2D potential function representing t’hr 31) NO%NOE intensity is developed and used in the simulated annealing protocol. The two methods are test’ed on two proteins: CMTI-I, a trypsin inhibitor from Cucurbita maxima. and hisactophilin, an a&in binding prot’ein of 118 amino acids from Dictpsfeliwn discoideurn.

and Methods of spectra and inteption

The humonuclear 31) NOE-NOE spectra were acquired from a 15 mM sample of CMTI-I at pH 4.3 in 2 mM-sodium acetate. 9O’jb H,O/lOqb ‘H,O. CMTI-I was isolated from the se& of’ C’ucurbita maxima and purified as described previously (Holak it al.. 1989a). The spectra were recorded at 25”(: on a Bruker AMX-600 spectrometer. The 31) experiments were carried out using a pulse sequence described by Boelens ef al. (1989). Two 3D NOE-XOE spert ra were acquired for CMTI-I. In the first 31) NOEL-NOE (Axperiment. 2 identical mixing times of 140 ms each were used. In the second. both mixing times were 50 ms. Each FIU consisted of 8 scans. The data set, consisted of t, x t, x t, = 256 x 256 x 512 points over a spectral width of 7400 Hz in all 3 dimensions, resulting in a t,ot,al measurement t,ime of 6 days. Only subvolumes containing t)he ?jH resonances in F3 were processed. Appropriate Lorent,z-to-Gaussian transformations and a I -time zero-filling in all 3 dimensions were applied. together with a baseline correction by a 3rd order polynomial in the F3 dimension. Hisactophilin was rxpressed in E. coli and purified essentially as described by Scheel et al. (1989). Compared to hisactophilin of I)irtyosfrlium discoideum, the E. coli pol,vpeptide cont,ains an insertion of 4 additional amino acids Glp-Glu-Pha-(:ly after the initial methionine, i.e. the total numbclr of residues is 122 (Scheel et al.. 1989). As there were no trivial NOE connectivities assignable to the insertion amino acids. we numbered the residues acrordiny to t)hr amino sequence of hisartophilin from nictyoatrii,,

m discoidcxm. The 3D NOE--NOE spectra for hisactophilin were acquired by a JR pulse sequence as descbribrd b,v FLOSSuf (II.. 1991). The mixing t,imes were 100 ms anti the dataset consisted of I, x t, x t, = 240 x 256 x I K (K, kilobyte 1027 points) over a spectral width of 7400 Hz. Eac*h FTD consisted of 8 scans resulting in a total measurement time of 4 days. Other spectral and processing parameters were similar t,o t,hosr used for t,hr (‘MTI-I spect,runl. The spectra were acquired from a I.7 mM sample of hisactophilin at pH A5 in 50 rn3r-KH,PO,. Tt, c.an be appreciated that,, with 31 histidines and 15

addit,ional glycines. the assignment of the spectra was not. an easy task. The n.m.r. spectra of hisactophilin were assigned with the 2D and 3D n.m.r. methods based on the following spectra: 2D NOESY, 2D TOCSY. 3D NOESYHMQC, 3D TOCSY-HMQC. 3D NOE-NOE JR. Prot)on and nitrogen resonances of most backbone and side-chain atoms were assigned. except those at the amino terminus of the protein (3 first-insertion amino acids and Metl) and in the loop between residues 26 to 30, whose assignments could not be confirmed because of the paucity of NOE connectivities to these residues. The heteronuclear 3D techniques proved to be very useful for the assignment of glycines because of their unique 15N chemical shifts. Also, the overlap of the proton NHs in the most crowded region. between 8.3 and 8.9 p.p.m., was most,ly removed by spreading resonances into the “N dimension. The 2D pot)ential function for the 3D NOE

NOE intensit,ies has been incorporated into the X-PLOR program (Briinger, 1988). Data processing, peak-picking and simulation of the spectra was performed with our own software. which is available upon request. The 31) cross-peak int,ensities in the sprcatra were semi-quantified by measuring the int,ensit). of the highest point in a volume around the 3D crosspeak. This intensity is directly proportional to the volume of the 31) cross-peak provided that t,he linewidt,h of every peak in all 3 dimensions is similar and larger than the multiplicit,ies of the signals. This is the, (‘asp in the RD SOE

(b) Th tory The intensity of a 31) NO&SOE (aross-peak, ), bet,ween spins i. j and k. is proportional to fijk(Trnl. Tm2 thr product of the individual NOE transfer eficirncies of each mixing time (Boelens rt 01.. 1989: Grirsingtr et nl., 1989): Iijt(Tmt’ Tm2) = clrxP(-Rz,z)lijlexP(-RT,, )Ijk-Akk((l). (1) where T,, is the first and T.

the second mixing time. R represents the cross-relaxation matrix. d,,(O) the equilibrium magnetization of spin k. and c is a constant. Tf %I = 7m2. and as R is symmetric. the intensity of a bac.k-transfer peak is given by: I. (L) = c[‘=p( -&,,)]

&.(W (2) which is proportional to t,he squarcl of the intensity of the corrrsponding XD peak Ii, with mixing titnr T,. Expanding the exponentials in eqn (1) and neglecting all trrms higher than 2nd order in T.

. a cnross-p(Lak (i #,j # k) is thrn given by: ‘tj/c(Tml, Tm2)=

As the cross-relaxation rate. Rij. is proportional to the inverse of the 6th power of the distance separation, rij, of the 2 prot,ons i and j, the distance constraints can be derived from 31) NOE-NOE intensities by appropriate scaling with knnu-n distances using the al)l)rosimatioll: lijk = Kr,g%,“, (4) where rij and rjk are the distances between protons i. j and j, k respectively. and h’ is the scaling c>onst,ant For (i #,j = k). the expansion results in:

whic.h, after neglect,ing 2nd o&r proportional to Rij.

In the first method of the structure determination from 31)

?U‘OE data. only distant constraints tlrrivcvl from rqns (2). (4) arid (5) verp used in the c

-llout 10 t,o 50”,, of the distance wnst,raints vew c

NOE spwtrum ivith both mixing times of iO 111s.thr rest of the distance constraints from a spectrum wit,h both mixing times of 110 ms. Thv litttw slw’trum rxhihitrd a bflttclr signal-to-noiw ratio. Three srpratr c4ibration cwnstants (KS) wcxre tlrtrrv minrd using the cvnnwtivitiw that involved known inter f)roton distance: I caonstant for the cross-peaks with 2 of the 3 F,. F2 and b frequencies equal (the 2Y, ant1 AT1 cw

ss-p&s). I for the back transf’rr peaks ant1 I for th(b “rtlal” 31) 9OE SOE cwss-peaks with 3 different frrqurncirs. The S, atld ‘T2 vross-peaks are tlw to t hr ditwt XL’obZs during the tirst ant1 srconcl fnisitlg titrlw. wsprctivcJly (I

1.. 1990). ‘I’ht, .V, and ‘V2 peaks art’. in thr lincsar af)l,l,oxirnation in t,, and z,z. proportional to the tlistanvv. rL“. of rq

l (-5). Thc,rttforr thr valihration constant for the S, and .V2 praks (‘an he usfvi dirrc+ly to cA(

uprrimetital intrnsitirs on thr .VI anti s, lines. Fol, c

. thts .Y2 c.onnwtivit>, 22/jz “-” l-4 222 in Fig. I gives the distanctn cx)nstraint twtvwn (‘K

*H and (‘vs22HS (the Ilotat,ion L’g, z/!l, ““N c.,r

ls to thv mss-peak (14’, = ( ‘vsP2(‘P2H) -( Fz = (‘vs

‘H) - (I = C’ysL’dHN) on thtk A?1 linrs at the NH lm’idr plane of rrsitlur C1ys22). As thr hsc

peaks arv proportional to r I2 In the 2nd ordrr approximat ion in T, (ec4n (2)). the ttistanc,r wnstrxint c’ibn tlso tw c

ulatrtl directly from the 31) intensity. For the 91) wowpeaks with 3 different frequencies. the tlistanw cxjnstraint vorrrspontling to I of the 2 NOlC transfers hts to Iw known in order to c*alculatr the tlistancar cwnstl,aint irl-olvrd in thr ot1lc.r transfer (rqns (3) anti (4)). Thew rr+rrncv tlist,atic.r c.onstraints vere acyuiwtl t’rottl tlrt* tiistanw c.onstraints ohtainetl from thtl AV, and S, (‘row lwaks and from thtl hac*ktransfer ptsaks M ith I c,alihration c*onstarlt. The interproton distancrs tirrived from known amino avid gromrtries wvre also used. For example. the tlistanw (‘“H(i) NH(i), vhic*h is usually equal to . I24 (I .% = I()-’ nnr) for the amino witis not in the r-hrlic4 c

ioti (thr I)rotvins studirti hew) (1j’iithric.h use of such caalihrations is discussed in morr dthtail in tht I)iscussion (Holak rf

). Altogether. 541 tlistanw rvmstraints wrre obt,ained from the 37) XOE SOE HJMYtrum. A c

atalogur of‘ the number anti tvI)t’s of the c*onstraints is given in Table 2. Of the tot,al h4l clistlnw cwnstraints. 312 vt’rti derivtvi from 3-f’requtwy peaks

Irrtrarrsidur Long (Ii-,jl L 5) Medium (Ii-,jl I$- where k, is the force constant, and A, and A, are the lower and upper error bounds. respectively. The intensity Z$ is

the measured 31) X0& NOE intensity divided by thr calibration constant K determined after eqn (4) and as described in section (b), above. The difference between t.h(s calculated and observed 3D h’OE-NOE intensitv is now a driving force on the coordinates of t)he protein in the dynamical or energy-minimizing step of the structure calculation. Tf the observed intensity Z$and the 31) NOI? intensity I$ calculated from the structure model are the same within the error bounds (A.. A,), the energ term ,q!ijk’ ,n,enSi,yis then zero and the force of this ronatraint on the co-ordinates of the model structure is also zero. Tf the difference between the observed and c*al(*ulated

Figure 2. Logarithmic plot of the 3D NOE-NOE intensity potential for a peak li+,+k. rij and rjr are the distances between protons i. j and j, k. respectively. The energy in the “white” area is zero and increases with increasing distance from this area. The lines in the plot. give the lower distance boundaries of the repel function (1.6 w).

intensities is not zero wit)hin the given error bounds. a force act.s on the co-ordinates in the direction to make the structurcb model fulfill the constraint. The negative 12th root of the intensities was taken to set this energy in comparable size to the other NOE distance constraints. The full potential for 1 intensity constraint is shown in Fig. 2. It is a 2D function with the 2 variables. rij and rjk, at the .c-axis and y-axis. respectively. The contours shown in Fig. 2 correspond to the energy heights. In the central ‘.empty” area in the contour plot. the energy is zero and increases as a square function with increasing distance from this area. The energy function in Fig. 2 corresponds t,o an observed intensit)- 1: of 8.72 x IV6 units. issuming that the 2 distances rij and rjk are equal, this would correspond to a distance of 2.64 a for both proton pairs (2.64- ” = X.72 x 1W6). The upper and lower errors in intensity, AU and A,. respectively. are chosen such that an upper and lower error in the distance of 0.3 A for both proton pairs at the same time is allowed. In the example above. it corresponds to AU = 2.84 x 11)-’ and A, = 6.32 x lW6 units. The error bounds to the distances were restrained t,o W3 A. as only more intense 31) cross-peaks were used in the calculations. The intensities used for the calculations corresponded to the maximum distancae constraint of 3.3 4. assuming that the distances involved in the 31) peak arc equal. In Fig. 3, the upper and lower errors in the experimental intensities are plotted against the obst>rred int,ensity. The plot shows that an upper bound of 0.3 .q of the distance corresponds to the lower bound error in the intensity up to 75%. The (k3 .& differencar in the lower bound of t,he distance corresponds to the error in t,hr upper bound of the intensity of 450°h. These relationships reflect t,hr dependence of the 31) XOE-KOE intensity on the distances through the equation Zijt cc rz“r,i6. In the c-asr of pseudoatoms, which represent methyl groups or st,rreospec

ifically unassigned mrthylenr

Figure 3. Upper (horizontal lines) error bounds A, and lower (vertical lines) error bounds A, in the rxperiment,al intensity plotted against the intensity I$. protons, the lower error bound of t,hr intensity Ar is increased so that the intensity error hounds rorrrspond t,o an upper distance error of 1.3 h. The example in Fig. 2 shows that. if the distance dij were 1 .%. the distance dj, could be verv large (for example 10 A) without violating the allowed intensity range. But with the repel funct,ion present. the lower distance is limited to 1.6 A. This allows only distances above and on the right side of the lines in Fig. 2. annealing protoml mnstrain,ts fmly The structure of CMTT-I. based on the distance constraints derived from the 31) NO%SOE spectra. was determined using the hybrid met,hod of DTSGEO and dynamic simulated annealing (Holak et al.. 1989a). The basic protocol used for the calculations has heen presented previously (Holak et al., 1989a.c). The prot,ocol cottsists of 4 st,ages. In stage 1. the c-o-ordinat,es of the subst,ruvtures are obtained from the distance geometry program DISGEO (Havel. 1986: Have1 6 li’iithrirh. 19X.5). In the 2nd st,agr. all the atoms missing in the substructures are added. Step 3 consists of dvnamical simulated annealing (Bilges et al.. 198X). i.e. riising the temperature of the system, followed by slowly cooling the system t,o overcome local minima and locate the region of the global minimum of the target function. The 4th stage involves 200 c ycles of constrained Powell nlitrimization. These last 3 stages were carried out with the program X-I’LOR (Briinger. 1988). ,411 protons were rxplicit,l> defined in the dynamic simulated annealing calculations. The mrthylene protons and methyl groups were assigned arbitrarily to H, and H,. or Me, and$lr,. for protons resonating at lower and higher t1.m.r. fields. respectively. In the protocol that uses substruc+urrs. the force constants for the bond lengths. bond angles and planarity art> the same as t)hose used in the refinement of the structures using NOE-derived distance c-onstraints (Holak et al.. 1989a). The pseudoenergy target fun&on used in the calculations was a square-well potential. The 21) NOESY distance constraints have not been used in the

onstants bt’I’v iO IiC’ill lllol

1 .p-* for ihe tlistant.ts t,(mstraints and 30 kt,al mol ’ .r * for thr inttJnsit>t

tol)hilin with all tht, availahlt

were also t*arrird out with the ,Y-I’LOR

In-ograrn and were based on a protocol similar to that used for the distance const’raints only. The psrutlor

nergy target funcation was the 31) int,eniity funcAtion in rqn (9). As t’hrre are no distance ctmstraint,s. it was not J)ossihlr to use I)TS(:EO to create starting structures. Thr starting strut*tures were therefort, built, with X-I’T,C)R I)? randomly grilrratinp 4 and li/ angles. 1n the c

ulat,itms startirlg from random structures. NY used higher forty vonstant,s for thr bond lengths. bond angles. improl)ers atrtl [Aanarity. t)ecause high forces originating from at+ t

)orarily on thta I)rotriri. esprvially at ‘experimental thtl beginning of the folding. The t

ulations start)4 with an initial minimization of 60 stq)s and with a loa- van tier 2'dS ft,rtY? tY)nstant of lllt

l-l 4-* 0.1 kcal (I t.aI = 4.2 *I). ill1 Ei”,e”si,y k, forcat, constant of 1 kt,al lllOl ' .A-

* and a dihedral angles (kdihedral) forvr tvnstant of 5 kt.al molrad ‘. The fortBe constants to maintain

tl Irngths (kbonda). bond angles (kanglea) and improl)ers (kimpr) were set to 1000 kcal mol ml &*. 500 ktaal mol I ratI- 2 ant1 500 kt-al mu-’ ratI-* rrsyW.ively. Tn tht, followinp 337.50 t,imrstel)s (2 fs) of high t,emperaturr dvnallliw. the forve c*onstants kbonds. lZangles and iimpr werf’ set to 500 kt,al rtrol

l. 400 and ZOO kval mol-’ rad-*. rvsl)ct

tively. In the simulat,rd-alinraline stag” of high tr

nryraturr dynamic3 (X)00 timestrl)s of 2 fS). thr forcv cY,

rcasrd to -C kcal molt’ -4-l. k, to 100 kt.aI mol-’ A-*. and kdihedra, to 200 kcal moi-’ rad ‘. . itbr caooling thr system. the final stage involved 200 t

s of c*onstrainrd Powell minimization with kbonds = IWO kval mom’ .A

2. kangles = ,500 kval mol ’ rad - 2. and The t*alculatitm of a strutakimpr = 500 kcal rnol- ’ ratI-‘. turta of (‘MTT-1 with Xti intensity t*onstraints took alyroximately I h caentral In-ocessor unit time on it (‘ONVEX (‘220 t*oml)utjer. The intensity constraints were oi)tainetl either from the 31) NOE

NOE q)ectrum with both mixing t,imes of .5Oms (40 to .50°, in number) 01’ front a sy)rt*trum vith both mixing times of 110 ms. The numl)c

r of intensity tvnstraints used in this arction was smallrr than that rxtract,ed from t,hti 110 ms syvtrum only (SW srtation (P)) because weaker intensities were not

Structures of (‘MTT-I and hisact,ophilin were also determint

tl from both dist,ancr and intensit,y constraints ]-)resrnt in the calculations. For (“MTT-1. the 3D NOES on t,hr LV, and 3, l’lanrs were introduced as distance taonstraints. The same protocol as in the calculations with 31) intensity tsonstraints was used. with the exception that the fort,r constant of the distance constraints was IO0 kval mol-’ &*. For hisactophilin. there were 1287 distance constraints derived from the 31) SO&I%OE eqeriment. during our earlier n.m.r. study on hisactophilin. Out, of this number. 1150 connr&ivit,irs vould also be identified in the 21)

taontains an ilrsrrtioli of’4 addititmal arniiio xt,itls afttsr t Iit> irlitial nit&i hionilrc

l out fiH thv amino ataitl st’tji

,/ium di.scGdetrm as thrrta vt-rt’ no trivitl SOE twnnrctivitif3 rssignal)lr to theatlditionLI amil10 irc,itl fragment

Three sets of structures were

alt*ulatc:cl for (MTI-I. In the first, set, the stereospecific* assignments of the prochiral tsentres obtain4 in the previous study (Holak rt al.. 1989

) and from the E.COSY spectrum were retained in the c

alculations. Fourteen x1 angle constraint,s with t,hr bounds + 30” were used in these calrulations. The second stlt was obtained using the floating chirality method in the simulated annealing stag distance (*onstrain&, was 0+49 for t)hese 229 distance caonst raint,s. Again. out of 336

Structure (‘ombinedt :%I)-Intensitv$Hisactophilin t (‘alculatrd with a wmbination of intensity rwnstraints and distanw constraints. 2.L9 distance constraints were derived from crosspeaks with the 2 E’ frrquencies rqual.$ (‘alculated with intensity constraints derived from 31) NOI%NOI? SJWtrUtn only.

Table 4 =ftowLic r.m.s. differences

between structures of CMTI-I and 3D n.m.r. data Hravy

3 I) distancet NY-sus 3D distance 31) intensityf

crsus 31) intensity (‘ombined§ versus combined 3D distance WGSUS2Dll (‘ombined UCTSUS 2D 21) twst

s rrfinedf 3D distance WTSUSrefined 31) intensity ZWSU,Srefined (‘om bined

rsus refined 21) I’cIs1L8X-ray 31) distance I’prs?LsS-ray 31) intensity IW

.E X-rag (‘ombined versus Y-ray Refined WTSUSX-ray 31) int,ensit,y W

SUScombined 3 1) distance PPISW c*ombined Ail r.m.s.d.s determined for residues 3 to 29. t Strncturrs calculated with distance constraints derived from 31) NOE-NOE sprcstrum. : (‘alc*ulated with intensity constraints derived from 31) NOE-

NOE spectrum only. $(‘alrulated with intensity constraints and distance constraints of peaks wit,h 2 freyurn d from 31) NOE-SOE spectrum. Ij (‘alculat.ed with distance constraints of a 2D NOISY spectrum (H&k rt nl.. IWOn) c Rrtined with t,he frill relaxation matrix approach (Silgps of ctl.. lO!#l). intensity constraints. the ten st’ructures had no intensity violations that corresponded to a distance violation larger than 0.5 A for the t,wo distances defined by the 31) NOE-NOE intensity. The reproduction of the disulphide bridges was similar to the calculations with intensity constraint,s only. The deviations from idealized covalent bond geometry was also almost the sa,me. The average r.m.s. difference among the structures itself was 0.52 A f @I 4 a for the backbone atoms and 1.22 A&-019 A for all heavy at)oms. (d) Strwturr hisactophilin from a combination and i,ntensity constraints Ten structures were calculated for hisactophilin. as described in se&ion (g) above. The floating chira- lity method was used to obtain stereospecific assignments at prochiral centres in an identical manner as for CMTI-I. X11 ten structures satisfied the experiwere no distance constraints. There mental constraint violations greater than 05 A. The r.m.s. differences from the experimental constraints, which were calculated with respect to the upper and lower limits of the distance constraints, was 0.12 8. The structures, shown in Figures 7 and 8, also exhibited very small deviations from idealized covalent geometry (bonds, angles and impropers, standard deviations), and had very good non-bonded caontacts having negative Lennard-Jones-van der Waals’ energies. The average of the r.m.s. differences among the 3D NOE-XOE structures was 2.48 A iO36 A for t,he backbone atoms and 3.57 [email protected] A for all heavy atoms. The corre- Figure 7. Stereoview of the backbone atoms (N. CCL C) of the 10 hisactophilin structures best-fitted to all amino acids with the rxcrption of residues 25 to 33, 47 to 49, 55 to 60, 65 to 74 and I1 7 to 118. The amino acid polypept,ide of hisartophilin of Dictyostelium diacoideum was used in the calculations (see Material and Methods. srction (a)). Figure 8. Stereoview of thr (*Il-d&iwd parts of thr structure. t,he nholr hackbonr is shown. and for t,lre othrr t )tion of residues 25 to 33. 17 to 441.55 to 60. 65 to 74 least-squares fit of the parts of thra I)wkbonr atoms shown r.m.s.d.s werr sponding 2.52 -4 f 0.35 A, respectively, segments of a high variability iMtd 8: see also discussion). 143 A * 024 A and when residues in the were excaluded (Figs 7 4. Discussion (a) Structures SD LVOE-NOE constmints only The information in the 31) NO%NC)E experiment is overdetermined: many connectivities are observed at different NH planes, as illustrated for the connectivity 22CaHP28C”H. The assignments are therefore more reliable, in particular for the long-range connectivities between the side-chains of different residues. Around 200 such long-range NOI& could be obt,ained from the 3D SOE-NON spectrum. Such connectivities can also be obtained from 2D spectra of a sample of protein dissolved in *H,O they are. however, more dificult t’o assign unambigously for cases in which only a single COW nrctivity between protons distant in the primar) sequence is observed. Such was the case for the caonta& between Tyr27 and Leu23. The conneetivit? was observed in t,he Xl) NOES’ specs27&236. trum. but not used previously because of assignments ambiguity arising from partial overlap with the methyls of Leu7. This connectivity was resolved in t,he 31) spectrum further 27-23 contacts were observed i.e. 23y-27 gave three cross-peaks through different connectivities in the plane of 27 . The new distance constraints between 27 and 23 rrsult,ed in their side-chains coming closer together in the 31) NOti-NOT? structure than those based on the 21) NOE8Y spectra (Fig. 4). Apart from this difference. the structures are very similar (Fig. 4). More important new cross-peaks, not observed in the 2J> NOESY due to overlap, involved weak &-6X and 4/. - BN. The contacts are critical in resolving ambiguity in the conformation of the protease binding loop seen previously (Holak et al.. 19891’,). In conclusion. the 3D data set allowed Lwkbonr atoms of the IO hiwc+ophilin structrtrcss. Icor I structures. the I)ac*kl)onr atonrs of all atnino ac,itls with thr and I I5 to I IX arcs shot5 n. l’ht )osition VNSdone hy 8 for ail strucat urw. better detinition of’ the st ructurw than vas previously possible in the 21) spectra. The 21) n.rt1.r. struct,ures were based already on a vt’rv Iargc number of distance const*raints. totalling 3:L4. The 31) NOEm-XOE experiment provided ,541 tiistancsr csonstraints. In the third set of strnc tnrw. thrl disulphidc bridges were not defined either as bonds or as distance (aonstraints. These structures wcw almost identical to those calculated with the disulphide bonds specified. Qualit ativr analysis of the NOESY spectra can usually provide the information needed to establish the presence of disulphide bridges (LVilliamson et al.. 1985 Kline Pt (cl.. 198X). The C?H-(‘OH NO& bet.wren two eysteines usually indicates that, the eysteines pa’ticipate in the disulphide bridge. In (MTT-1. the three disulphidr bridges, whose positions are known from the X-ra) st,ructure. are very close to one another. and it is possible to intjerc hanpr them without any drastic changes in the secondary and tertiary structure. There are also several (‘“ll--(!aH NO& prwc’nt in the NOR spectra between cysteines, which arc trot involved in the disulphide bridge. For exampIt>, tht connec+vity between 1S/I and 20/I would suggest an incorrect pairing of the c -S pairing csould be determined unambiguously from structures. The S-S distanc*t is the n.rr1.r. 2.0 &2.2 ‘4 in t,he disulphide bridge, which vils the distance observed for the correct S -8 pairing of cystpines. In all other arrangement,s. the tlistance bet’ween sulphur atoms was greater than 5.5 :I. wit)h the exception of the pairing 10-28. for vhic*h the distance is 3% A. In the 211 NOESI stru(+urt’s. tht, conformat,ion of the disulphide bridge IO- %2 vas uniquely defined the two other disulphide bridges showed the presence of mirror images at the q-sulphur atoms (Holak et al.. 1989n). The present stru(.tures exhibit unique c,onformat,iotis for all t hrb disulphide bridges. The ability to obtain structures with the uniquely determined c,onformat,ions of the disulphide bridges could be traced to the presence of new NOES in the 3D spectrum involving cyst,eines. The quality of the present structures is such that a detailed comparison with the X-ray structure of the inhibitor in the trypsin complex is possible (Bode et al., 1989 Holak et aE.. 1989b). The n.m.r. and X-ray st,ructures are almost identical in terms of the global fold and secondary structure (Fig. 4). There are. however, a few small but distinct differences between the n.m.r. structures and the X-ray st,ructure at residues 17 and 24 to 27. The n.m.r. structures at these regions are more expanded than the X-ray structure this expansion is not an artifact of the n.m.r. data. For example, the interproton distances 17fl,-46, and 4fl,-17p,,2 are 2.2 a and 3.0 .A in the X-ray structure. respectively. No corresponding NO Es were seen in the 3D SOE-NOE spectrum. As (‘MTT-I has three disulphide bridges, which fix the whole structure, the protein is very rigid. A large reduction of ?OE intensities because of int’ernal movements is not probable for the p a,toms of the residue close to one of t,he disulphide bridges. There is also no evidence for any multiplicity in the conformation of R-04. Therefore in t’he n.rn.r. stru&ures these distances have to be longer than 3.5 I, a rather conservative lower estimate that would take into account increased flexibility of the segment (l’epermans et al., 1988). Some ot,her minor differences could be positively ident’ified between t,he two structures. A very close contact between 127, and 12N of 1.8 &A, as seen in the X-ray structure. should give rise to a strong SOE: in the 31) 90%NOE spectrum. However: this contact is weak in t)he spectrum. and consequently a longrr distanc*e is seen in t,he n.m.r. structures (Fig. 1). The difference between the n.m.r. struttures and the S-ray structure at residues 24 to 27 was noticed previously and was ascribed to the effect of crystal packing and interaction with the protease in the crystal structure (Holak et al.. 1989h). These examples show that. with the present structures. it is possible to make even small differentiations among the structures that are fully suppor?eti by the experiment,al data. (b) Strwtuws from thu 3D XOE-NOE inhsity umstraints or a combir ation of the intensity and distancf, constraints The csonclusions drawn in the previous section also hold for the structures calculated from the intensity constraints. The quality. in terms of geometry and agreement wit’h the KOE data, of the structures of (IMTI-I calculated with t)he int’ensity constraints was slightly. but noticeably. bett)er t’han t,ha.t of the structures calculated in the presence of distance constraints (Tables 3 and 4 Figs 4, 5 and 6). We have chosen t)hc structure of CMTI-T. which was refined with the relaxation matrix approach for a reference model that is the closest to the global minimum structure. For dist,ance constraints, the procedure of extracting the distances from the 31) NO&NOE cross-peaks is relatively time-consuming. as it requires a knowledge of one distance in each 3D cross-peak in order to extract the second distance. This stage is absent when working with the intensity const,raints. After an initial calibration of a few w’rsus the known distance constraints intensities (eqn (4)), the determination of the two distances in the 371 XOE--NOE peak is left t’o the caalculation itself. Within the int,ensity method. the error bounds of the distance constraints involved in the intensity constraints plays a smaller role in the final r.rn.s.d.s of the st,ructures t’han is the case when using dist,ance constraint’s directly during t.he calculat,ions. This is desirable, as t’he error bounds to the distance constraints are currently chosrn on the ba,sis of a rather subjective criteria. Another important advantage of the inbensity constraints is that the procedure is easily amenable to cbomput’er aut’omation, such as peak picking and sirnulation of the spectra. Tn fact. the cbalculation yields not only the structures, but also a simulated spectrum that) ca,n be directly compared with the experimental one (Fig. l(b)). Structures cualculated from a combination of the intensity constraints and the distance constraints (derived from peaks with 2 equal frequencies) exhibited energy and geometrical parameters t,hat were even closer t,o those of the reference CMTT-I structure. This is not surprising, as the quality of t.hr st#ru(stures increases with the increase of the number in non-trivial ronst8raints (Wiithrich. 1986). The calculation of the struct’ure of hisa.ctophilin off’rrs a practical example of an applicaation of the 311 SOE-NOE data t’o the det,erminat,ion of structures of larger proteins. Hisactophilin. an unique actin-binding prot,ein from I)ict!lostr/iurn discoiwhich dwm. is a submembraneous pH sensor. induces actin to polymerize at pH values below 7. and thus is a putative component of the signal transduction chain (Scheel et 01.. 1989). The protein has a molecaular weight of 13.5 kI)a. and its most characteristic feature-is the presence of 31 histidine residues out of 118 amino acids. Taking into ac*caount that histidine is t,he only amino acid with a pK, value close to physiological pH. the data indicate that the molecule senses the H+ c*oncentratJion 1.i the histidine residues and is active in its cationic form. A structural model was not previously available for the prot)ein. A superposition of the ten structures of hisactophilin is shown in Figures 7 and 8. A characteristic feature of the structures is the presence of 12 j-strands and no a-helix. The overall struct*ure can be best described as forming two four-St randed antiparallel fl-sheets and one six-stranded antiparallel ,&sheet. Figure 9 shows plots of the at)omic r.m.s. distributions for the structures. The backbone of hisactophilin within the 12 b&rands is very well determined. The t)urns and loops show greater variability. with the loop between residues 2.5 and 33 having the highest variability. Most, residues in this segment showed no KOEs in the ?‘OE spectra. It is interesting to note that all glycines and histidines, with the exception of His3.5, His75 and His78. are lo&ed in the loops or turns whose atomic r.m.s. Figure 9. Average pairwisc atomir r.m.s. differences among the hisactophilin structures. l represents t,hr residues that are involved in /?-sheet’s 0 represents residues of turns and loops 1 shows the standard deviations. The superposition of t,he strurturrs wsasas in Pip. 8. dif&rences for the backbone atoms are larger than 2.0 A (Fig. 9). Further details of the structure of hisactophilin, as well as t’he description of t,he assignments and secondary structure elements. will be presented elsewhere. Tt is clear from the presented data. however, that well-determined structures of large proteins can be obtained from the 31) NOE-NOE data even without the availability of the dihedral angle constraints. Since submission of our manuscrpt, a communcat,ion has appeared that describes a very similar approach to that of ours of a direct use of t,he 31) SO&NOE int’ensities for the structure calculations (Ronvin rt al.. 1991). The procedure described in this communication was tested on simulated n.m.r. data for an a-helix of eight residues. (c) Conclusions X’OE spectrum in H,O. t.hc SOEs between thta aliphatic J)rotons can be observed at, unique IrrlitI(b J>rot,on frequ( s. The oJ)timurn approach for reJjroduc:ing t,he strrrc* tural cbontent of the 31) NO&SOE cross-J)eak is t hta usage of the PI) pseudoenergy potential of rqn (9). which represents the intensity of the cross-peak. The difference bet,ween the caJculat,ed and obsrrvetl 31) NOE-SOE: intensitv is t,hrn used directly as a driving fi)rc:r on co-ordinates of it Jjrotcill in thtb structure c alcutation. The direct use of thch NOB intensities avoids the t ransformat.ion of SOE intensities into dist)ances. whic*h is i ime-c and introduces targer errors in thca distancatl caom&aints. because of t.he inclusion of an additional step that involves scaling thr data. As 1he drtermination of t,he two distances involved in thch 31) NOE-SOF: ?ross-J)eak is left to the caalculat ion itself, the error bounds to the tlist,ancet caonsi)raints involved in the intensity const,raint,s J)lag a smntlcr role in the final r.m.s.d.s of‘ the structures than is I hfa case when using distance (Lonstraints direcat Jy during the calculations. Vr thank Redmond Bernstein for his contribution to thr softwarcl drvrlopmrnt,. This work was supported by A researrh grant from the J lrll fiir Forschung und Tecxhnologir ((irant, no. OSIH!)O9A) and from t,hr I)eutsch Forsc haft ( f’rojrcts Ho-l Z69jl 1. Shl-dO4/%4 and SFR dO7). .J.O. has trcscstla recipient of a fellowship from the, Humboldt Foundation References Rode. I‘.. Creyling, H. ,J.. Hubrr. R.. Ottrwski. .J. X: Wilusz. T. (1989). The refined 2.0 AA x-ray c*r,vstal strucature of the clomplrx formed between bovincL S-trypsin and cmti-1. a trypsin inhibitor from squash seeds (C’ucurDitn ntn.rimn). 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