Abstract This manuscript investigates the level of complexity and thermodynamic properties of the... more Abstract This manuscript investigates the level of complexity and thermodynamic properties of the real RNA structures and compares the properties with the random RNA sequences. A discussion on the similarities of thermodynamical properties of the real structures with the non linear random matrix model of RNA folding is presented. The structural information contained in the PDB file is exploited to get the base pairing information. The complexity of an RNA structure is defined by a topological quantity called genus which is calculated from the base pairing information. Thermodynamic analysis of the real structures is done numerically. The real structures have a minimum free energy which is very small compared to the randomly generated sequences of the same length. This analysis suggests that there are specific patterns in the structures which are preserved during the evolution of the sequences and certain sequences are discarded by the evolutionary process. Further analyzing the sequences of a fixed length reveal that the RNA structures exist in ensembles i.e. although all the sequences in the ensemble have different series of nucleotides (sequence) they fold into structures that have the same pairs of hydrogen bonding as well as the same minimum free energy. The specific heat of the RNA molecule is numerically estimated at different lengths. The specific heat curve with temperature shows a bump and for some RNA, a double peak behavior is observed. The same behavior is seen in the study of the random matrix model with non linear interaction of RNA folding. The bump in the non linear matrix model can be controlled by the change in the interaction strength.
The volatility of an Indian stock market is examined in terms of aspects like participation, sync... more The volatility of an Indian stock market is examined in terms of aspects like participation, synchronization of stocks and quantification of volatility using the random matrix approach. Volatility pattern of the market is found using the Bombay Stock Index for the three-year period 2000−2002. Random matrix analysis is carried out using daily returns of 70 stocks for several time windows of 85 days in 2001 to (i) do a brief comparative analysis with statistics of eigenvalues and eigenvectors of the matrix C of correlations between price fluctuations, in time regimes of different volatilities. While a bulk of eigenvalues falls within Random Matrix Theory bounds in all the time periods, we see that the largest (deviating) eigenvalue correlates well with the volatility of the index (ii) observe the corresponding eigenvector clearly shows a shift in the distribution of its components from volatile to less volatile periods and verifies the qualitative association between participation and volatility (iii) set up a variability index, V whose temporal evolution is found to be significantly correlated with the volatility of the overall market index.
We apply RMT, Network and MF-DFA methods to investigate correlation, network and multifractal pro... more We apply RMT, Network and MF-DFA methods to investigate correlation, network and multifractal properties of 20 global financial indices. We compare results before and during the financial crisis of 2008 respectively. We find that the network method gives more useful information about the formation of clusters as compared to results obtained from eigenvectors corresponding to second largest eigenvalue and these sectors are formed on the basis of geographical location of indices. At threshold 0.6, indices corresponding to Americas, Europe and Asia/Pacific disconnect and form different clusters before the crisis but during the crisis, indices corresponding to Americas and Europe are combined together to form a cluster while the Asia/Pacific indices forms another cluster. By further increasing the value of threshold to 0.9, European countries France, Germany and UK constitute the most tightly linked markets. We study multifractal properties of global financial indices and find that financial indices corresponding to Americas and Europe almost lie in the same range of degree of multifractality as compared to other indices. India, South Korea, Hong Kong are found to be near the degree of multifractality of indices corresponding to Americas and Europe. A large variation in the degree of multifractality in Egypt, Indonesia, Malaysia, Taiwan and Singapore may be a reason that when we increase the threshold in financial network these countries first start getting disconnected at low threshold from the correlation network of financial indices. We fit Binomial Multifractal Model (BMFM) to these financial markets.
ABSTRACT Certain models of structural glasses ref. [1, 2] map onto random matrix models. These ra... more ABSTRACT Certain models of structural glasses ref. [1, 2] map onto random matrix models. These random matrix models have gaps in their eigenvalue distribution. It turns out that matrix models with gaps in their eigenvalue distributions have the unusual property of multiple solutions or minimas of the free energy at the same point in phase space. I present evidence for the presence of multiple solutions in these models both analytically and numerically. The multiple solutions have different free energies and observable correlation functions, the differences arising at higher order in 1/N. The system can get trapped into different minimas depending upon the path traversed in phase space to reach a particular point. The thermodynamic limit also depends upon the sequence by which N is taken to infinity (e.g. odd or even N), reminicent of structure discussed in another model for glasses ref. [3]. Hence it would be of interest to study the landscape of these multiple solutions and determine whether it corresponds to a supercooled liquid or glass.
We study the statistical properties of eigenvalues of the Hessian matrix H (matrix of second deri... more We study the statistical properties of eigenvalues of the Hessian matrix H (matrix of second derivatives of the potential energy) for a classical atomic liquid, and compare these properties with predictions for random matrix models (RMM). The eigenvalue spectra (the Instantaneous Normal Mode or INM spectra) are evaluated numerically for configurations generated by molecular dynamics simulations. We find that distribution of spacings between nearest neighbor eigenvalues, s, obeys quite well the Wigner prediction s exp(−s 2), with the agreement being better for higher densities at fixed temperature. The deviations display a correlation with the number of localized eigenstates (normal modes) in the liquid; there are fewer localized states at higher densities which we quantify by calculating the participation ratios of the normal modes. We confirm this observation by calculating the spacing distribution for parts of the INM spectra with high participation ratios, obtaining greater conformity with the Wigner form. We also calculate the spectral rigidity and find a substantial dependence on the density of the liquid.
The current characteristics of DNA decorated carbon nanotubes for different gas odors are studied... more The current characteristics of DNA decorated carbon nanotubes for different gas odors are studied. A simple model of charge transfer between the Gas-DNA-base complex and single wall carbon nanotube (SWCN) is proposed to explain the current response for different odors. The autocorrelation and two-point correlation functions are calculated for the current sensitivity curves. These correlation functions together with the current characteristics form fingerprints for detection of the odor and DNA sequence.
We calculate the two-time current correlation function using the experimental data of the current... more We calculate the two-time current correlation function using the experimental data of the current-time characteristics of the Gas-DNA-decorated carbon nanotube field effect transistor. The pattern of the correlation function is a measure of the sensitivity and selectivity of the sensors and suggest that these gas flow sensors may also be used as DNA sequence detectors. The system is modelled by a one-dimensional tight-binding Hamiltonian and we present analytical calculations of quantum electronic transport for the system using the time-dependent nonequilibrium Green's function formalism and the adiabatic expansion. The zeroth and first order contributions to the current I (0) (t) and I (1) (t) are calculated, where I (0) (t) is the Landauer formula. The formula for the time-dependent current is then used to compare the theoretical results with the experiment.
We study the statistical and spectral properties of the foreign exchange of 21 different currenci... more We study the statistical and spectral properties of the foreign exchange of 21 different currencies from January 4, 1999 to March 30, 2018. The correlation matrix is calculated for different periods with a rolling window method and the properties are studied for each window. The basic statistics on the correlation matrix shows that the currencies are more and more correlated with times. The distribution of the correlation matrix was very asymmetric with non zero skewness which shows a fat tail behavior for the initial years but approach Gaussian distribution for the later time. The spectral properties of the correlation matrices for each window when compare with the properties of the correlation matrix formed for the complete period and with analytical results for Wishart matrices shows that the distribution is different for the windows comprising the calm and the crisis period. The study of the number of eigenvalues which are outside the random matrix bounds for each window on both sides of spectrum reveals that for the crisis period, the number of eigenvalues outside the lower bound increases as compared to the calm period. This increase in the number of eigenvalues on the lower side of the spectrum for a window also implies a crisis in the near future. The lower end of the spectra contains more information than the higher side as revealed by the entropic measures on the eigenvalues. This entropic measure shows that the eigenvectors on the lower side are more informative and localized. In this work, the analysis of individual eigenvector captures the evolution of interaction among different currencies with time. The analysis shows that the set of most interacting currencies that are active during the calm period and the crisis period are different. The currencies which was dominating in the calm period suddenly lose all weight and new set of currencies become active at the onset and during the crisis. The largest eigenvector of the correlation matrix can separate currencies based on their geographical location.
In this article we study the thermodynamic properties of a substrate-induced graphene superlattic... more In this article we study the thermodynamic properties of a substrate-induced graphene superlattice in the presence of uniform magnetic field B. We derive an expression for the partition function of the system with uniform magnetic field B using the zeta function approach and find all related thermodynamic functions such as Helmholtz free energy, total energy, specific heat capacity and entropy for three cases related to the sublattice potential. We also show that the Dulong-Petit law is verified for all cases in the high-temperature limit. These results may shed light on the study of graphene superlattices in the developement of thermo-electric devices.
This thesis studies the structure of local and global anomalies in certain systems and examines t... more This thesis studies the structure of local and global anomalies in certain systems and examines the conditions for their cancellation. Gauge anomalies-abelian and non-albelian-antisymmetric tensor, and gravitational anomalies in simple spinor theories with background fields have been analyzed by perturbative methods and local counterterms have been constructed to cancel the anomalies wherever possible. Anomalies occurring in supersymmetric theories in (2 + 1)-dimensions have also been calculated using both perturbative and heat kernel techniques, here again counterterms have been constructed to cancel these parity violating anomalies for certain gauge field configurations. (i) For gauge theories in four dimensions which contain couplings of fermions to a non-abelian antisymmetric tensor field, the contribution of the later to anomalies in the non-abelian chiral Ward identity is computed. It is shown by explicit construction of suitable counterterms that these anomalies can all be ca...
The volatility of an Indian stock market is examined in terms of aspects like participation, sync... more The volatility of an Indian stock market is examined in terms of aspects like participation, synchronization of stocks and quantification of volatility using the random matrix approach. Volatility pattern of the market is found using the Bombay Stock Index for the three-year period 2000−2002. Random matrix analysis is carried out using daily returns of 70 stocks for several time windows of 85 days in 2001 to (i) do a brief comparative analysis with statistics of eigenvalues and eigenvectors of the matrix C of correlations between price fluctuations, in time regimes of different volatilities. While a bulk of eigenvalues falls within Random Matrix Theory bounds in all the time periods, we see that the largest (deviating) eigenvalue correlates well with the volatility of the index (ii) observe the corresponding eigenvector clearly shows a shift in the distribution of its components from volatile to less volatile periods and verifies the qualitative association between participation and volatility (iii) set up a variability index, V whose temporal evolution is found to be significantly correlated with the volatility of the overall market index.
We propose a physiochemical property based analysis, that represent a protein sequence as a multi... more We propose a physiochemical property based analysis, that represent a protein sequence as a multidimensional time-series from which residue positions controlling protein function can be extracted. We observe that the favorable substitutions at a position are the ones which preserve the property crucial for functioning of that site which is quantified by the entropy and Kullback-Leibler divergence. The entropic measures shows that during the evolutionary history of the protein family, it is the certain physiochemical properties that are conserved rather than the type of amino acids. For each physiochemical property, the correlation matrix between positions is calculated, and using an ensemble of Wishart matrices from the random matrix theory for the noise estimation and information filtering. The spectral properties of correlation matrices are calculated and compared with the analytical results for the Wishart matrices.
The structural organization of a protein family is investigated by devising a method based on the... more The structural organization of a protein family is investigated by devising a method based on the random matrix theory (RMT), which uses the physiochemical properties of the amino acid with multiple sequence alignment. A graphical method to represent protein sequences using physiochemical properties is devised that gives a fast, easy, and informative way of comparing the evolutionary distances between protein sequences. A correlation matrix associated with each property is calculated, where the noise reduction and information filtering is done using RMT involving an ensemble of Wishart matrices. The analysis of the eigenvalue statistics of the correlation matrix for the β-lactamase family shows the universal features as observed in the Gaussian orthogonal ensemble (GOE). The property-based approach captures the short-as well as the long-range correlation (approximately following GOE) between the eigenvalues, whereas the previous approach (treating amino acids as characters) gives the usual short-range correlations, while the long-range correlations are the same as that of an uncorrelated series. The distribution of the eigenvector components for the eigenvalues outside the bulk (RMT bound) deviates significantly from RMT observations and contains important information about the system. The information content of each eigenvector of the correlation matrix is quantified by introducing an entropic estimate, which shows that for the β-lactamase family the smallest eigenvectors (low eigenmodes) are highly localized as well as informative. These small eigenvectors when processed gives clusters involving positions that have well-defined biological and structural importance matching with experiments. The approach is crucial for the recognition of structural motifs as shown in β-lactamase (and other families) and selectively identifies the important positions for targets to deactivate (activate) the enzymatic actions.
Abstract This manuscript investigates the level of complexity and thermodynamic properties of the... more Abstract This manuscript investigates the level of complexity and thermodynamic properties of the real RNA structures and compares the properties with the random RNA sequences. A discussion on the similarities of thermodynamical properties of the real structures with the non linear random matrix model of RNA folding is presented. The structural information contained in the PDB file is exploited to get the base pairing information. The complexity of an RNA structure is defined by a topological quantity called genus which is calculated from the base pairing information. Thermodynamic analysis of the real structures is done numerically. The real structures have a minimum free energy which is very small compared to the randomly generated sequences of the same length. This analysis suggests that there are specific patterns in the structures which are preserved during the evolution of the sequences and certain sequences are discarded by the evolutionary process. Further analyzing the sequences of a fixed length reveal that the RNA structures exist in ensembles i.e. although all the sequences in the ensemble have different series of nucleotides (sequence) they fold into structures that have the same pairs of hydrogen bonding as well as the same minimum free energy. The specific heat of the RNA molecule is numerically estimated at different lengths. The specific heat curve with temperature shows a bump and for some RNA, a double peak behavior is observed. The same behavior is seen in the study of the random matrix model with non linear interaction of RNA folding. The bump in the non linear matrix model can be controlled by the change in the interaction strength.
Abstract This manuscript investigates the level of complexity and thermodynamic properties of the... more Abstract This manuscript investigates the level of complexity and thermodynamic properties of the real RNA structures and compares the properties with the random RNA sequences. A discussion on the similarities of thermodynamical properties of the real structures with the non linear random matrix model of RNA folding is presented. The structural information contained in the PDB file is exploited to get the base pairing information. The complexity of an RNA structure is defined by a topological quantity called genus which is calculated from the base pairing information. Thermodynamic analysis of the real structures is done numerically. The real structures have a minimum free energy which is very small compared to the randomly generated sequences of the same length. This analysis suggests that there are specific patterns in the structures which are preserved during the evolution of the sequences and certain sequences are discarded by the evolutionary process. Further analyzing the sequences of a fixed length reveal that the RNA structures exist in ensembles i.e. although all the sequences in the ensemble have different series of nucleotides (sequence) they fold into structures that have the same pairs of hydrogen bonding as well as the same minimum free energy. The specific heat of the RNA molecule is numerically estimated at different lengths. The specific heat curve with temperature shows a bump and for some RNA, a double peak behavior is observed. The same behavior is seen in the study of the random matrix model with non linear interaction of RNA folding. The bump in the non linear matrix model can be controlled by the change in the interaction strength.
The volatility of an Indian stock market is examined in terms of aspects like participation, sync... more The volatility of an Indian stock market is examined in terms of aspects like participation, synchronization of stocks and quantification of volatility using the random matrix approach. Volatility pattern of the market is found using the Bombay Stock Index for the three-year period 2000−2002. Random matrix analysis is carried out using daily returns of 70 stocks for several time windows of 85 days in 2001 to (i) do a brief comparative analysis with statistics of eigenvalues and eigenvectors of the matrix C of correlations between price fluctuations, in time regimes of different volatilities. While a bulk of eigenvalues falls within Random Matrix Theory bounds in all the time periods, we see that the largest (deviating) eigenvalue correlates well with the volatility of the index (ii) observe the corresponding eigenvector clearly shows a shift in the distribution of its components from volatile to less volatile periods and verifies the qualitative association between participation and volatility (iii) set up a variability index, V whose temporal evolution is found to be significantly correlated with the volatility of the overall market index.
We apply RMT, Network and MF-DFA methods to investigate correlation, network and multifractal pro... more We apply RMT, Network and MF-DFA methods to investigate correlation, network and multifractal properties of 20 global financial indices. We compare results before and during the financial crisis of 2008 respectively. We find that the network method gives more useful information about the formation of clusters as compared to results obtained from eigenvectors corresponding to second largest eigenvalue and these sectors are formed on the basis of geographical location of indices. At threshold 0.6, indices corresponding to Americas, Europe and Asia/Pacific disconnect and form different clusters before the crisis but during the crisis, indices corresponding to Americas and Europe are combined together to form a cluster while the Asia/Pacific indices forms another cluster. By further increasing the value of threshold to 0.9, European countries France, Germany and UK constitute the most tightly linked markets. We study multifractal properties of global financial indices and find that financial indices corresponding to Americas and Europe almost lie in the same range of degree of multifractality as compared to other indices. India, South Korea, Hong Kong are found to be near the degree of multifractality of indices corresponding to Americas and Europe. A large variation in the degree of multifractality in Egypt, Indonesia, Malaysia, Taiwan and Singapore may be a reason that when we increase the threshold in financial network these countries first start getting disconnected at low threshold from the correlation network of financial indices. We fit Binomial Multifractal Model (BMFM) to these financial markets.
ABSTRACT Certain models of structural glasses ref. [1, 2] map onto random matrix models. These ra... more ABSTRACT Certain models of structural glasses ref. [1, 2] map onto random matrix models. These random matrix models have gaps in their eigenvalue distribution. It turns out that matrix models with gaps in their eigenvalue distributions have the unusual property of multiple solutions or minimas of the free energy at the same point in phase space. I present evidence for the presence of multiple solutions in these models both analytically and numerically. The multiple solutions have different free energies and observable correlation functions, the differences arising at higher order in 1/N. The system can get trapped into different minimas depending upon the path traversed in phase space to reach a particular point. The thermodynamic limit also depends upon the sequence by which N is taken to infinity (e.g. odd or even N), reminicent of structure discussed in another model for glasses ref. [3]. Hence it would be of interest to study the landscape of these multiple solutions and determine whether it corresponds to a supercooled liquid or glass.
We study the statistical properties of eigenvalues of the Hessian matrix H (matrix of second deri... more We study the statistical properties of eigenvalues of the Hessian matrix H (matrix of second derivatives of the potential energy) for a classical atomic liquid, and compare these properties with predictions for random matrix models (RMM). The eigenvalue spectra (the Instantaneous Normal Mode or INM spectra) are evaluated numerically for configurations generated by molecular dynamics simulations. We find that distribution of spacings between nearest neighbor eigenvalues, s, obeys quite well the Wigner prediction s exp(−s 2), with the agreement being better for higher densities at fixed temperature. The deviations display a correlation with the number of localized eigenstates (normal modes) in the liquid; there are fewer localized states at higher densities which we quantify by calculating the participation ratios of the normal modes. We confirm this observation by calculating the spacing distribution for parts of the INM spectra with high participation ratios, obtaining greater conformity with the Wigner form. We also calculate the spectral rigidity and find a substantial dependence on the density of the liquid.
The current characteristics of DNA decorated carbon nanotubes for different gas odors are studied... more The current characteristics of DNA decorated carbon nanotubes for different gas odors are studied. A simple model of charge transfer between the Gas-DNA-base complex and single wall carbon nanotube (SWCN) is proposed to explain the current response for different odors. The autocorrelation and two-point correlation functions are calculated for the current sensitivity curves. These correlation functions together with the current characteristics form fingerprints for detection of the odor and DNA sequence.
We calculate the two-time current correlation function using the experimental data of the current... more We calculate the two-time current correlation function using the experimental data of the current-time characteristics of the Gas-DNA-decorated carbon nanotube field effect transistor. The pattern of the correlation function is a measure of the sensitivity and selectivity of the sensors and suggest that these gas flow sensors may also be used as DNA sequence detectors. The system is modelled by a one-dimensional tight-binding Hamiltonian and we present analytical calculations of quantum electronic transport for the system using the time-dependent nonequilibrium Green's function formalism and the adiabatic expansion. The zeroth and first order contributions to the current I (0) (t) and I (1) (t) are calculated, where I (0) (t) is the Landauer formula. The formula for the time-dependent current is then used to compare the theoretical results with the experiment.
We study the statistical and spectral properties of the foreign exchange of 21 different currenci... more We study the statistical and spectral properties of the foreign exchange of 21 different currencies from January 4, 1999 to March 30, 2018. The correlation matrix is calculated for different periods with a rolling window method and the properties are studied for each window. The basic statistics on the correlation matrix shows that the currencies are more and more correlated with times. The distribution of the correlation matrix was very asymmetric with non zero skewness which shows a fat tail behavior for the initial years but approach Gaussian distribution for the later time. The spectral properties of the correlation matrices for each window when compare with the properties of the correlation matrix formed for the complete period and with analytical results for Wishart matrices shows that the distribution is different for the windows comprising the calm and the crisis period. The study of the number of eigenvalues which are outside the random matrix bounds for each window on both sides of spectrum reveals that for the crisis period, the number of eigenvalues outside the lower bound increases as compared to the calm period. This increase in the number of eigenvalues on the lower side of the spectrum for a window also implies a crisis in the near future. The lower end of the spectra contains more information than the higher side as revealed by the entropic measures on the eigenvalues. This entropic measure shows that the eigenvectors on the lower side are more informative and localized. In this work, the analysis of individual eigenvector captures the evolution of interaction among different currencies with time. The analysis shows that the set of most interacting currencies that are active during the calm period and the crisis period are different. The currencies which was dominating in the calm period suddenly lose all weight and new set of currencies become active at the onset and during the crisis. The largest eigenvector of the correlation matrix can separate currencies based on their geographical location.
In this article we study the thermodynamic properties of a substrate-induced graphene superlattic... more In this article we study the thermodynamic properties of a substrate-induced graphene superlattice in the presence of uniform magnetic field B. We derive an expression for the partition function of the system with uniform magnetic field B using the zeta function approach and find all related thermodynamic functions such as Helmholtz free energy, total energy, specific heat capacity and entropy for three cases related to the sublattice potential. We also show that the Dulong-Petit law is verified for all cases in the high-temperature limit. These results may shed light on the study of graphene superlattices in the developement of thermo-electric devices.
This thesis studies the structure of local and global anomalies in certain systems and examines t... more This thesis studies the structure of local and global anomalies in certain systems and examines the conditions for their cancellation. Gauge anomalies-abelian and non-albelian-antisymmetric tensor, and gravitational anomalies in simple spinor theories with background fields have been analyzed by perturbative methods and local counterterms have been constructed to cancel the anomalies wherever possible. Anomalies occurring in supersymmetric theories in (2 + 1)-dimensions have also been calculated using both perturbative and heat kernel techniques, here again counterterms have been constructed to cancel these parity violating anomalies for certain gauge field configurations. (i) For gauge theories in four dimensions which contain couplings of fermions to a non-abelian antisymmetric tensor field, the contribution of the later to anomalies in the non-abelian chiral Ward identity is computed. It is shown by explicit construction of suitable counterterms that these anomalies can all be ca...
The volatility of an Indian stock market is examined in terms of aspects like participation, sync... more The volatility of an Indian stock market is examined in terms of aspects like participation, synchronization of stocks and quantification of volatility using the random matrix approach. Volatility pattern of the market is found using the Bombay Stock Index for the three-year period 2000−2002. Random matrix analysis is carried out using daily returns of 70 stocks for several time windows of 85 days in 2001 to (i) do a brief comparative analysis with statistics of eigenvalues and eigenvectors of the matrix C of correlations between price fluctuations, in time regimes of different volatilities. While a bulk of eigenvalues falls within Random Matrix Theory bounds in all the time periods, we see that the largest (deviating) eigenvalue correlates well with the volatility of the index (ii) observe the corresponding eigenvector clearly shows a shift in the distribution of its components from volatile to less volatile periods and verifies the qualitative association between participation and volatility (iii) set up a variability index, V whose temporal evolution is found to be significantly correlated with the volatility of the overall market index.
We propose a physiochemical property based analysis, that represent a protein sequence as a multi... more We propose a physiochemical property based analysis, that represent a protein sequence as a multidimensional time-series from which residue positions controlling protein function can be extracted. We observe that the favorable substitutions at a position are the ones which preserve the property crucial for functioning of that site which is quantified by the entropy and Kullback-Leibler divergence. The entropic measures shows that during the evolutionary history of the protein family, it is the certain physiochemical properties that are conserved rather than the type of amino acids. For each physiochemical property, the correlation matrix between positions is calculated, and using an ensemble of Wishart matrices from the random matrix theory for the noise estimation and information filtering. The spectral properties of correlation matrices are calculated and compared with the analytical results for the Wishart matrices.
The structural organization of a protein family is investigated by devising a method based on the... more The structural organization of a protein family is investigated by devising a method based on the random matrix theory (RMT), which uses the physiochemical properties of the amino acid with multiple sequence alignment. A graphical method to represent protein sequences using physiochemical properties is devised that gives a fast, easy, and informative way of comparing the evolutionary distances between protein sequences. A correlation matrix associated with each property is calculated, where the noise reduction and information filtering is done using RMT involving an ensemble of Wishart matrices. The analysis of the eigenvalue statistics of the correlation matrix for the β-lactamase family shows the universal features as observed in the Gaussian orthogonal ensemble (GOE). The property-based approach captures the short-as well as the long-range correlation (approximately following GOE) between the eigenvalues, whereas the previous approach (treating amino acids as characters) gives the usual short-range correlations, while the long-range correlations are the same as that of an uncorrelated series. The distribution of the eigenvector components for the eigenvalues outside the bulk (RMT bound) deviates significantly from RMT observations and contains important information about the system. The information content of each eigenvector of the correlation matrix is quantified by introducing an entropic estimate, which shows that for the β-lactamase family the smallest eigenvectors (low eigenmodes) are highly localized as well as informative. These small eigenvectors when processed gives clusters involving positions that have well-defined biological and structural importance matching with experiments. The approach is crucial for the recognition of structural motifs as shown in β-lactamase (and other families) and selectively identifies the important positions for targets to deactivate (activate) the enzymatic actions.
Abstract This manuscript investigates the level of complexity and thermodynamic properties of the... more Abstract This manuscript investigates the level of complexity and thermodynamic properties of the real RNA structures and compares the properties with the random RNA sequences. A discussion on the similarities of thermodynamical properties of the real structures with the non linear random matrix model of RNA folding is presented. The structural information contained in the PDB file is exploited to get the base pairing information. The complexity of an RNA structure is defined by a topological quantity called genus which is calculated from the base pairing information. Thermodynamic analysis of the real structures is done numerically. The real structures have a minimum free energy which is very small compared to the randomly generated sequences of the same length. This analysis suggests that there are specific patterns in the structures which are preserved during the evolution of the sequences and certain sequences are discarded by the evolutionary process. Further analyzing the sequences of a fixed length reveal that the RNA structures exist in ensembles i.e. although all the sequences in the ensemble have different series of nucleotides (sequence) they fold into structures that have the same pairs of hydrogen bonding as well as the same minimum free energy. The specific heat of the RNA molecule is numerically estimated at different lengths. The specific heat curve with temperature shows a bump and for some RNA, a double peak behavior is observed. The same behavior is seen in the study of the random matrix model with non linear interaction of RNA folding. The bump in the non linear matrix model can be controlled by the change in the interaction strength.
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