atoms and molecules have been used as precision standards for measurements of fundamental physical quantities such as time , length , mass , and the gravitational constant .the measurement of electric and magnetic fields has also been improved by using atoms as sensors .radio frequency ( rf ) electric field ( e - field ) measurements are important because the rf spectrum is the foundation of modern communications , remote sensing and many other applications , including some in medical science .recently , atom - based sensors have been used for rf e - field measurements using the concept of electromagnetically induced transparency ( eit ) with rydberg atoms in room temperature vapor cells .rydberg atoms in vapor cells can also be used for quantum information science .rf e - fields can be used to coherently manipulate rydberg atoms for these purposes which significantly increases the scientific interest in studying the interaction between rf e - fields and atoms in vapor cells .rydberg atom - based rf e - field measurements are superior to conventional antenna - based techniques and can be used , in principle , over the range of ghz - thz .it has been shown that rydberg atom - based methods have surpassed the sensitivity limit , , and accuracies , depending on frequency , of current dipole antenna - based traceable rf e - field measurements .rydberg atom - based electrometry can also reach frequencies above for which there is no current traceable standard .we have demonstrated in our prior work that it is possible to achieve a sensitivity of 5 and accuracies of using rydberg atom - based sensors at ghz frequencies .imaging with sub - wavelength resolution and vector rf e - field measurements can be used to extend the utility of the technique . in this paper, we study frequency modulated ( fm ) spectroscopy as a low noise and , in principle , compact way to read - out the probe laser signal of the eit system that carries the information about the rf e - field .this study is important for pushing the sensitivity of the technique to the atomic projection noise limit by eliminating technical noise in the probe laser read - out and elucidating the role of probe laser shot noise .the projection noise limit of the rydberg atom - based e - field measurement can be on the order of  .nm while the coupling laser wavelength is nm . these lasers are nearly resonant with both transitions .( b ) shows the probe transmission signal vs probe laser detuning for different rf e - field strengths .these measurements show the at regime .the measurements were carried out with amplitude modulation of the coupling laser beam similar to .,width=3 ]  at the heart of rydberg atom - based rf electrometry is the large transition dipole moment , , between two adjacent rydberg states , e.g. 100 to 10000 times the alkali d transition , which allows one to detect absolute rf e - fields with high sensitivity . when the rf e - field is at the level of mv , autler - townes ( at ) splitting is generated in the eit spectrum by the rf e - field coupling between the rydberg states , fig .[ fig0 ] .the at splitting , , is proportional to the transition dipole moment between the rydberg states and the rf e - field amplitude , i.e. , where is planck s constant .the principle quantum number , , of the rydberg states can be changed to cover a broad frequency spectrum from ghz to thz , with varying sensitivity depending primarily on the transition dipole moments and collision rates .when the at splitting can not be observed due to the spectral resolution , the amplitude of the eit probe transmission can be used to determine the e - field .the latter measurement is more sensitive to noise because the probe transmission on resonance and/or the spectral lineshape must be accurately determined . to increase the sensitivity , two goals present themselvesfirst , it is important to increase the signal - to - noise ratio ( snr ) of the measurement .second , it is advantageous to increase the spectral resolution to expand the at regime .this paper addresses the first goal using a method , fm spectroscopy , that can be integrated into a portable , compact package .the sensitivity of the rf e - field measurement is limited by three main factors from reaching the projection noise limit of the atomic sensor .these factors are : the technical noise of the probe laser , the residual doppler effect due to the wavelength mismatch of the eit probe and coupling lasers , and the photon shot noise of the probe laser on the photodetector .recently , we have utilized a homodyne detection technique with a mach - zehnder interferometer ( mzi ) to suppress the technical noise of the probe laser .we achieved a new sensitivity limit for atom - based rf e - field sensing of which was determined to be photon shot noise limited .the mzi is experimentally more complicated than fm spectroscopy because it requires an additional frequency stabilized laser to control the phase of the interferometer .fm spectroscopy is an alternative , less complex method , that can reduce the technical noise of the probe laser so that photon shot noise limited performance can be realized .fm spectroscopy has been used for precision measurements such as gravitational wave detection interferometry , cavity - based laser frequency stabilization , and high - speed detection of weak absorption signals .this method has also been used to suppress light shifts in optical pumping systems . in order to achieve high sensitivity with fm spectroscopy residual amplitude modulation ( ram ) must be addressed .ram is caused by etalon effects , variations of the birefringence of the electro - optic modulator ( eom ) used for frequency modulation , fluctuations in the rf drive power , spatial variation of the field inside the eom crystal , etc that lead to unequal magnitude and/or unwanted phase shifts of the modulation sidebands .active control and cancellation of ram improves the readout snr and is required to optimize fm spectroscopy . recently, an alternative method has also been proposed to passively control ram .a wedged electro - optic crystal was used to reduce ram caused by the input polarization misalignment and the etalon effects .this method is useful for the reduction of baseline drift of the pound drever hall ( pdh ) error signal for stable squeezed light generation .a wedged electro - optic crystal is not required with active control of the ram .  in this work ,we show that we can achieve a sensitivity of for rydberg atom - based e - field measurement with fm spectroscopy .we actively stabilize the ram in our setup .our sensitivity using fm spectroscopy is also photon shot noise limited and virtually the same as that obtained with the mzi .these results support our work using the mzi and show that photon shot noise is a challenge for rydberg atom - based e - field measurements since it is difficult to increase the probe laser power without increasing collision rates and introducing unwanted amounts of power broadening .fig . [ fig0](a ) shows the four - level atomic energy level scheme we used for the experiments .the probe laser is tuned near the cesium ( cs ) d transition , , while the coupling laser excites atoms to a rydberg state , .the three - level ladder system with the couplings shown in fig . [ fig0 ] leads to the cancellation of absorption of a resonant probe laser in the presence of a resonant coupling laser .the enhanced transmission of the probe laser is known as eit .a rf e - field at a frequency of , resonant with two adjacent rydberg levels , , can cause the probe transmission window to split via the at effect with proper selection of laser polarization .the rf e - field is determined by how it modifies the probe laser transmission .[ fig0](b ) shows the typical probe transmission signal as a function of probe laser detuning in the at regime .the traces shown in fig .[ fig0 ] were obtained using amplitude modulation of the coupling laser as used for our previous works .  the experimental setup for the implementation of fm spectroscopy is shown in fig .a cm long , cross - sectional area rectangular vapor cell filled with cs atoms at room temperature is used to perform the rf e - field measurements .the lasers are offset locked to an ultra - stable fabry prot cavity using the pdh technique .the estimated linewidth of the lasers is based on the cavity locking error signal .the two laser beams interact with the cs atoms in a counter - propagating geometry .the intensity fluctuations of both lasers are stabilized using a feedback loop to acousto - optic modulators ( aom ) for the sensitivity measurements .rf absorbing material is placed around the setup to suppress reflections of the rf e - field .  to modulate the probe laser ,a linear polarized probe laser beam is incident on a fiber - coupled waveguide - based eom with small angle relative to the crystal axis . to produce high purity linear polarized light , two glan - thompson prisms ( gtp )are placed at the input and output of the eom to act as a polarizer and analyzer , respectively . the modulation frequency that we used for the experimentsis which is larger than the spectral width of the eit probe transmission window .half ( quarter ) -waveplate ; pd , photodetector ; bpf , band pass filter ; lpf , low pass filter ; , phase shifters ; dbm , double balanced mixer ; ios , optical isolator ; dm , dichroic mirror.,width=384 ]  ram causes the fm sidebands of the probe laser to vary in relative magnitude and phase .ram can be actively compensated by rotating the principal axes of the eom crystal by the application of an electric field in order to align the linear polarized probe laser with one of the principal axes of the crystal . by aligning the probe laser polarization with one of the principal axes of the crystal, the probe light propagates as if in a homogeneous material .a waveguide - based eom is advantageous because the applied voltages needed to produce the necessary optical phase shift are 2 orders of magnitude smaller and the input fiber acts as a spatial mode filter , which reduces the ram contributions arising from the spatial inhomogeneity of the optical field .we temperature stabilized the waveguide - based eom to reduce drift due to the temperature fluctuations .ram can be measured and actively reduced by determining the power falling on a fast photodiode at the modulation frequency and using that signal to correct the principal axes of the crystal using a feedback loop .the relative phase shift due to birefringence of the crystal is , where is the wave number of the probe light with wavelength , is the length of the crystal , and ( ) is the refractive index of the extraordinary ( ordinary ) wave .the phase shift can be externally controlled because and depend on the electric field present inside the crystal via the pockels effect .phase changes can then be due to a control electric field applied to the crystal , , or from unwanted variations as mentioned in the introduction , .the correction phase needed to compensate the ram can be determined by the condition that all odd harmonics of of the laser power falling on a photodiode placed after the eom are zero for perfect phase modulation ( alignment of the probe electric field vector along one of the principal axes of the crystal ) . for odd harmonics of ,the measured photocurrent will be  where is an odd integer for . is the e - field amplitude of the probe laser . and are the relative angles of the gtp polarizer and analyzer with respect to the crystal axes , respectively . is the -order bessel function with defined as the difference between the modulation indices of the ordinary and extra - ordinary waves . depends on and the rf e - field applied to the eom .pure phase modulation is realized by satisfying the condition = 0 . to correct for the ram and obtain ideal phase modulation ,a fast photodiode is used to measure the photocurrent given in eqn .[ photocurrent ] by demodulating it at and actively feeding back this error signal into the eom via the dc port of a bias tee as shown in fig.[fig1 ] . a pid loop based on a field programmable gate arrayis used to generate the signal that is applied to the bias tee . the laser power used forthe ram correction setup can be relatively large .typically we used .  a miniature rubidium atomic clock ( microsemi sa.35 m )is used to synchronize the three function generators used to control the modulation and demodulation in the setup .one function generator is used for the rf modulation of the eom .the other two function generators are used as local oscillators ( lo ) for the ram servo and the probe fm signal .the time drift of the clock in a 24 hour period is 7 .the demodulation of the error signal for the ram servo and the fm signal are performed using two double - balanced mixers . a band - pass filter ( bpf )is used to suppress signal at other frequencies , such as higher order sidebands .the phase of the lo for the ram servo is adjusted to obtain a maximum error signal from the mixer output .the phase of the lo used to demodulate the fm signal can produce in - phase or quadrature phase probe laser transmission signals .the stability of the ram and the probe transmission signal can be studied in a log - log plot of the allan deviation versus sampling time .the allan deviation is the square root of the allan variance defined as , where is the average fluctuation over the sampling time .[ fig2](a ) shows an example of the allan deviation of the ram signal plotted against sampling time with the ram lock on ( red curve ) and off ( black curve ) .the inset of fig .[ fig2](a ) shows the corresponding ram signal as a function of time with the ram lock on ( red ) and off ( black ) .the modulation frequency was with a modulation depth .the lo modulation depth was .the allan deviation is larger for short sampling times but decreases as the sampling time increases due to averaging . at larger sampling times , the allan deviation of the unlocked signal increases due to slow drifts , such as polarization changes caused by temperature changes of the optics .the allan deviation of the locked signal continues to decrease over the entire range of sampling times .the decrease in both traces shown in fig .[ fig2](a ) is characteristic of electronic noise , sometimes referred to as flicker phase noise , as well as noise due to the probe laser locking , white phase and frequency noise .[ fig2](a ) shows that the ram is largely eliminated and the fm sidebands are stable over long intervals of time due to the active stabilization of the fm sidebands .to further illustrate the point , fig .[ fig2](b ) shows an example of the allan deviation of the probe transmission signal acquired using fm spectroscopy as a function of sampling time at two - photon resonance , the eit resonance , with the ram lock on ( red curve ) and off ( black curve ) .the inset of fig .[ fig2](b ) shows the corresponding probe transmission signal in the time domain with ram lock on ( red curve ) and off ( black curve ) . for the eit signal, we used of probe laser power and of coupling laser power .the probe laser beam had a diameter mm while the coupling beam diameter was mm .the corresponding probe and coupling laser rabi frequencies are and , respectively . note that this is a fundamentally different measurement than shown in fig .[ fig2](a ) because now the signal depends on other parameters like the density of atoms in the vapor cell and the laser detunings .consequently , we observe similar but not identical behavior as that shown in fig .[ fig2](a ) . at first , ,both the locked and unlocked allan variances increase .after the initial increase , the allan deviation decreases with sampling time . for longer sampling times , the allan deviation starts increasing again .the behavior in this example is characteristic of white noise from electronics and then , most likely , slow environmental drifts of which there are more , since the experiment is more complicated than the measurement of the ram on a single photodiode . for ,both curves most likely show random walk frequency noise due to the laser locking .[ fig2](b ) demonstrates how sensitive these measurements are , particularly with respect to stabilizing the frequency and intensity of the lasers , and the utility of using the allan deviation to determine the noise sources .the probe laser is resonant while the coupling laser is detuned .( b ) probe laser transmission as a function of rf electric field strength for the coupling laser on resonance ( black curve ) and detuned by 1 mhz ( red curve ) .the data was taken with a detection bandwidth.,title="fig:",width=211 ] v / cm .the probe laser is resonant while the coupling laser is detuned .( b ) probe laser transmission as a function of rf electric field strength for the coupling laser on resonance ( black curve ) and detuned by 1 mhz ( red curve ) .the data was taken with a detection bandwidth.,title="fig:",width=211 ]  we studied the effect of the modulation depth on the probe laser transmission as a function of probe detuning in order to optimize the probe laser transmission signal .[ fig3 ] , shows the probe transmission signal obtained with fm spectroscopy for the three - level ladder system shown in fig .[ fig0 ] with no rf e - field present .the slope of the probe transmission signal close to the probe resonance depends on the modulation frequencies and modulation depth . for a small modulation depth, the amplitude of the fm sideband is very small compared to the carrier wave .as we increase the modulation depth , the amplitude of the sideband grows larger which increases the slope of the fm signal . for a large modulation depth, we also observe that the amplitude of the higher - order harmonics grows larger and the slope of the fm signal decreases . to optimize the slope of the fm probe laser transmission signal at a modulation frequency of , we used a modulation depth of for the eom and a lo modulation depth of to demodulate the fm signal .[ fig3](b ) shows the probe transmission as a function of the lo phase at the eit probe transmission resonance .for quadrature lo phases , we observe a antisymmetric fm spectral lineshape about the line center .the steepness of the probe transmission is maximum at these quadrature phases consistent with prior work .we use the quadrature phase of the fm derived probe laser transmission to perform the rf e - field measurements shown in fig .[ fig4 ] and fig .[ fig5 ] .we also optimized the detuning of the coupling laser so that the applied rf e - field would cause the maximum change in probe laser transmission .[ fig4](a ) shows probe laser transmission as a function of rf detuning for several different coupling laser detunings .the probe laser is tuned to resonance .we used of probe laser power and of coupling power which corresponds to rabi frequencies of and , respectively . in fig .[ fig4](a ) , the rf e - field strength is .the figure shows that the amplitude of the probe transmission is optimized for a detuned coupling field for our eit parameters and modulation frequency . at coupling fielddetuning , we observed maximum probe transmission . fig .[ fig4](b ) shows the probe laser transmission plotted against rf e - field strength for two different coupling detunings .fig [ fig4](b ) illustrates the effect of optimizing the coupling detuning for the measurements .one can observe from the graph that the probe transmission as a function of rf e - field amplitude is more sensitive for the coupling laser detuning .[ fig5](a ) shows weak rf e - field measurements performed by detecting the probe laser transmission as a function of rf e - field detuning at a coupling laser detuning .the probe laser is resonant .the rabi frequencies of the probe and coupling lasers are the same as in fig .the rf e - field amplitude was calibrated by measuring it in the at regime using the rydberg atom - based sensor and extrapolating the power reading on the rf generator used to drive the antenna . in fig .[ fig5](a ) , the black curve shows the measured probe laser transmission while the red curve shows a lorentzian fit to the lineshape .we used a lorentzian fit because the lineshape is expected to be dominated by power broadening since we increased the probe laser power to optimize the signal with regards to photon shot noise and lineshape .the full width at half maximum was for each fit within the fitting error .the lorentzian fit the data curves much better than a gaussian particularly in the wings of each peak . fig .[ fig5](b ) shows the same plots as fig .[ fig5](a ) after processing the data using a matched filter .matched filtering is useful for improving the snr when the form of the signal is known . it detects or extracts a known signal with high sensitivity that has been contaminated by noise . to implement the matched filter , we convolve a lorentzian distribution function , with the measured signal , where is the full width at half maximum , is the amplitude , and is the line center .the matched filter improves the visibility of the signal , but the signal to noise level is approximately the same at the smallest e - field amplitudes , presumably because photon shot noise is limiting the sensitivity . the results in fig .[ fig5]a show a sensitivity .each data point corresponds to a detection bandwidth . with matched filtering, one can detect a rf electric field amplitude of 1.8 .mhz and , respectively .( b ) the plot shows probe signal vs rf detuning using matched filter at different values of rf e - field strengths .all parameters are same as ( a ) .the data was acquired using a detection bandwidth .each lorentzian fit gave a full width at half maximum of .,title="fig:",width=240 ] mhz and , respectively .( b ) the plot shows probe signal vs rf detuning using matched filter at different values of rf e - field strengths .all parameters are same as ( a ) .the data was acquired using a detection bandwidth .each lorentzian fit gave a full width at half maximum of .,title="fig:",width=240 ]  the current sensitivity achieved with fm spectroscopy is about three orders of magnitude worse than what can be achieved for an optimized projection noise limit of the atomic sensor . for the parameters used in this effort ,the projection noise limit is about 20 times better than what we demonstrated .the minimum rf e - field that an atomic sensor can measure depends on the rf transition dipole moment , ; the effective number of atoms participating , ; and the dephasing time ,  .the atomic projection noise limited sensitivity of the rydberg atom - based e - field sensor for atoms can be calculated as where is the planck s constant .for the parameters used in our experiment ; , participating atoms and , eqn .[ eq3 ] gives a projection noise limited sensitivity of .for , we made a conservative estimate which takes into account the doppler averaging and relative rydberg excitation of the atomic sample based on a density matrix calculation of the system shown in fig .the result of the projection noise limited sensitivity estimate shows that the shot noise of the atoms in the vapor cell is not limiting the sensitivity , since it is a factor of 20 better than what was measured in the experiments . as in our prior work, photon shot noise on the photo - detector is limiting the sensitivity .the photon shot noise limited snr of a detector is , where is the quantum efficiency , is the detection bandwidth , is the power falling on the detector , and is the frequency of the light . for the experiments , a total probe laser power of fell on the detector , but less than carried the signal because of the size of the laser beam overlap , necessitated in our setup by the small dipole moment of the coupling laser transition , and modulation . for a detection bandwidth , as used in the prior analysis , the shot noise limited snr is using this reasoning .the smallest probe transmission signals shown in this paper are of the eit signal while the overall eit signal without the rf electric field is of the absorption signal .these estimates show that photon shot noise is the primary noise source for low probe laser transmission signals .surpassing the photon shot noise limit in cases where sub- sensitivities need to be achieved is nontrivial for rydberg atom - based rf e - field sensing because increasing the probe laser power leads to power broadening as well as the possible reduction of depending on how the other parameters of the sensor are constrained .we have shown that fm spectroscopy with active control of ram improves the readout snr of rydberg atom - based rf e - field sensing when compared to our prior approach .we have achieved a sensitivity of 3 and demonstrated the detection of weak rf e - fields , 1.8 , using matched filtering .this sensitivity limit is the same as what we achieved recently utilizing a mzi .the current experimental sensitivity of the rf e - field measurement is worse than the atomic projection noise limit for realistic cases where and are not too small because of photon shot noise .it is possible to change the parameters of the system to incrementally improve the sensitivity , however , photon shot noise is a barrier that must be overcome to achieve optimal projection noise limited sensitivity .the mzi read - out method was similarly bounded because the laser rabi frequencies are determined by a desire to maximize coherence times and avoid power broadening .the agreement between the fm spectroscopy and mzi results supports the idea that photon shot noise is an important source of noise to address if the sensitivity of rydberg atom - based rf e - field measurement is to be significantly improved . some approaches that we are investigating to surpassthe photon shot noise limit are to utilize non - resonant two - photon excitation or squeezed light for the probe transition .fm spectroscopy is very versatile and can be used in a portable , compact package that is less complicated than an interferometer but can reach similar performance levels .rf e - field measurements at sensitivities below the photon shot noise limit can possibly be used in a wide range of applications in medical science , such as detection of early stage breast cancer , and as a tool for radio astronomy to detect absolute levels of thermal background radiation and explore fundamental physics of the universe . at present ,rydberg atom - based sensing is well - poised to become the standard for electric field measurement , particularly useful for calibration of rf and terahertz devices , from ghz to thz .the authors would like to thank jiteng sheng for the fruitful discussions .this work was supported by the darpa quasar program by a grant through the aro ( 60181-ph - 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wavelength imaging and field mapping via electromagnetically induced transparency and autler - townes splitting in rydberg atoms , " , 244102 ( 2014 ) .g. d. vine , d. s. rabeling , b. j. j. slagmolen , t. t .- y .lam , s. chua , d. m. wuchenich , d. e. mcclelland , and d. a. shaddock , picometer level displacement metrology with digitally enhanced heterodyne interferometry , " opt . exp . *17 * , 828 ( 2009 ) .j. ye and t. w. lynn , applications of optical cavities in modern atomic , molecular , and optical physics , " in advances in atomic , molecular , and optical physics , vol .49 , edited by b. bederson and h. walther ( academic press , 2003 ) .w. zhang , m. j. martin , c. benko , j. l. hall , j. ye , c. hagemann , t. legero , u. sterr , f. riehle , g. d. cole , and m. aspelmeyer , reduction of residual amplitude modulation to 1 for frequency modulation and laser stabilization , " , 1980 ( 2014 ) .z. li , w. ma , w. yang , y. wang , and y. zheng , reduction of zero baseline drift of the pound - 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Abstract: rydberg atom - based electrometry enables traceable electric field measurements with high sensitivity over a large frequency range , from gigahertz to terahertz . such measurements are particularly useful for the calibration of radio frequency and terahertz devices , as well as other applications like near field imaging of electric fields . we utilize frequency modulated spectroscopy with active control of residual amplitude modulation to improve the signal to noise ratio of the optical readout of rydberg atom - based radio frequency electrometry . matched filtering of the signal is also implemented . although we have reached similarly , high sensitivity with other read - out methods , frequency modulated spectroscopy is advantageous because it is well - suited for building a compact , portable sensor . in the current experiment ,   hz sensitivity is achieved and is found to be photon shot noise limited .


there is a long - standing interest in knowing and understanding the correlation between bases in dna sequences . before the human genome project era, there were no long , continuous dna sequences .the study then concerned either the nearest - neighbor base - base correlations [ josse , et .al . , 1961 ] or the base density heterogeneity in digested dna segments [ sueoka , 1959 ] .a more complete characterization of correlation between base pairs at both short and long distances became possible only as long dna sequences became more commonly available .not all studies of a complete characterization of correlation structure of dna sequences were motivated by biology .rather , many such studies were motivated by the issues of mathematical modeling , dynamical systems , stochastic processes , and noise .perhaps due to this reason , this study has not yet become part of the toolbox in the mainstream " dna sequence analysis .this review is an attempt to summarize the current status of this study .there are at least two goals for this review .first , there have been disagreements on the result of correlation structure in dna sequences . due to this uncertainty of what the actual result is, some people still believe that dna sequences do not exhibit any feature which can not be explained by the basic stochastic processes such as random sequence or markov chain - with the first process having no correlation and the second one having only short - range correlations .resolving this disagreement can be straightforward once everybody agrees to use the same measure of correlation , use the same estimator , and apply this estimator of the correlation to the same sequence .the second goal is to promote more biologically - motivated study of correlation structure of dna sequences .although this paper does not accomplish this task , the intention is to at least raise the issue .most of the current studies of correlation in dna sequences are based on base - base statistical correlations .this base - base correlation may not be a powerful way to reveal the correlation on a global scale or between larger units .using an analogy of the natural language texts : statistical correlation between letters in an english text rarely reveals correlations at the syntax level in a sentence , or the correlation between sections , or the overall organization of the text , because the linguistically meaningful units are words , sentences , paragraphs , instead of letters .  besides the above note that most correlations studied are base - base correlation, another caution is that these are _ statistical _ correlations .a statistical correlation between two events exists if and only if the joint probability of the two events is not equal to the product of the two probabilities for each event .in other analysis of correlations , such as the one between a risk factor ( smoking ) and a disease ( cancer ) , the distinction is sometimes made between _ causal correlation _ and _ spurious correlation_. the causal correlation is the correlation between true cause and the effect . the spurious correlation is the non - causal part of the statistical correlation . without a definition of the causal base - base correlation , it is not clear whether the above distinction is applicable here .the claim of ( base - base , statistical ) correlation at long distances in dna sequences is still a few steps away from finding an organization principle of the genome .the quote from [ ohno,1993 ] for example , this would suggest the existence of a ` grand design ' in the construction of dna sequences " , represents such a misunderstanding . in order to discover a true organization principle " , one might start from the biologically meaningful units , arrange both the structurally functional regions and the genetically functional regions in place , and see how the organization helps the living of the cell .there is another confusion related to this study .the term long - range " in the long - range ( base - base , statistical ) correlation " was meant to be longer than ( 1 ) 3 - 6 bases [ li , 1992 ] ; or ( 2 ) 800 bases [ li and kaneko , 1992a ] ; or ( 3 ) 1 -10 kb ( 1 kb = 1000 bases ) [ peng , et .al . 1992 ] . a recent analysis of complete dna sequences of budding yeast chromosomes ( _ saccharomyces cerevisiae _ ) also suggested that the correlation exists up to more than 10 kb , but is absent at 100 kb range ( w.li , unpublished results ) . on the other hand , the long - range " in long - range physical mapping "typically means the range covered by the current physical mapping techniques is as large as 100 kb - 1 mb ( 1 mb = 1,000,000 bases ) .clearly , what is considered to be long is relative to what is considered to be short .for this reason , the term long - range " will be used with caution in this review .more often , the term correlation structure " is used instead to indicate the pattern of correlation at all distances .  this paper is organized as follows : section 1 describes the original motivation for a complete characterization of correlation structure in dna sequences ; section 2 reviews both the measures used in characterizing correlation structure ( in spatial as well as spectral domain ) and some published results on these correlation structures of dna sequences ; two special subsections are included : one on the estimation of correlations ( sec .2.2 ) , and another on characterizing heterogeneity of dna sequences ( sec . 2.7 ) ; section 3 discusses different approaches to the modeling of the correlation structure of dna sequences ; and finally , section 4 discusses some biological issues related to the correlation study .a review of this sort is inevitably biased , in the sense that the material chosen reflects the author s interests and experience , and the opinions expressed are mainly the author s .nevertheless , i intend to make the review as balanced and as fair as possible .an early review covers the study on this topic before 1993 can be found in [ li , et .al . , 1994 ] .as early as the sixties , there were attempts to characterize the statistical properties of dna sequences : for example , the correlations between the nearest - neighbor bases [ josse , et .al . , 1961 ;gatlin,1966 ] and the heterogeneity of base density in fragmented dnas [ sueoka , 1959 ] .statistical regularities were used to detect coding regions [ shulman , et .al . , 1981 ;shepherd , 1981a,1981b ; staden and mclachlan , 1982 ; fickett , 1982 ] , and to study the nucleosome formation [ trifonov and sussman , 1980 ] .all these studies focus on a particular aspect of the correlation structure of dna sequences in relation to a particular biological question .  for mathematically - oriented researchers ,a dna sequence might be considered as a string of symbols whose correlation structure can be characterized completely by all possible base - base correlation functions or their corresponding power spectra . for people interested in dynamical systems ,there is also a dynamical " aspect to this otherwise static problem : the change of dna sequences ( even without a reference to the natural selection ) can be considered as the updating of the symbolic sequences .this updating , once described by a mathematical model , can be studied easily on a computer .one question is which model will generate what types of correlation structure in a symbolic sequence .a few years ago , i was studying such a problem .the dynamical system being studied which updates a symbolic sequence is the so - called cellular automata " [ toffoli and margolus , 1987 ; wolfram,1986 ] .these systems update sequence locally .the question asked was whether these locally - operated systems are able to generate global correlations .the answer turned out to be negative for simple cellular automata [ li , 1987 ] .two events turned my attention to dna sequences .one was a publication trying to model the dna evolution by cellular automata [ burks and farmer , 1984 ] .another was the discovery that long - range correlation in a sequence could be easily generated if the sequence is allowed to increase [ li , 1989 ] . in what i called the expansion - modification system " [ li , 1989 ; 1991 ] , there are only two processes : expansion " and modification " .when the expansion rate far exceeds the modification rate , the sequences generated by the system exhibit a long - range correlation called 1/f spectrum " ( to be discussed more in the sec .it was pointed out to me ( k. kaneko , private communication , 1988 - 89 ) that the expansion - modification model i was studying was reminiscent of the biological processes : expansion being the base or oligonucleotide duplication process , and modification being the point mutation .this led us to a study of the correlation structure in dna sequences [ li , 1992 ; li and kaneko , 1992a ] .other groups interested in studying correlation structure in dna sequences were probably motivated by different reasons .for example , in [ peng , et . al . 1992 ] , the interest was perhaps to compare power - law functions observed in a representation of dna sequences to other self - similar phenomena in nature .the interest in [ voss , 1992 ] was to compare the 1/f spectra observed in dna sequences to other 1/f noise in nature such as musical signals [ voss and clarke , 1975 ] . [ voss , 1992 ] also suggested that the information storage in dna sequences is between efficient ( as in random sequences ) and redundant ( as in repetitive sequences ) .the basic measure of correlation among bases in a dna sequence are the 16 correlation functions between all 16 possible base - pairs : each defined as the correlation between nucleotide and another nucleotide separated by a distance : where is the joint probability of observing and separated by a distance , and are the density for nucleotide and , respectively . because ofthe relations , the number of independent correlation functions ( not yet considering any other symmetries ) is actually 9 [ herzel and groe , 1995 ] .  following are comments concerning these correlation functions :  1 .* statistics are taken along a sequence * : suppose is 1 when the nucleotide at position is , and 0 otherwise . is the count of pair separated by a distance ( the sampling is stopped when the second base reaches the end of the sequence ) . if one uses the periodic boundary condition , ( where indicating circular " ) .+ is estimated _ along _ the sequence , either by or by .( note : since the number of counts can either be or , depending on whether the periodic boundary condition is used or not , the symbol used in eqs.([eq4],[eq5],[eq7],[eq8 ] , [ eq9 ] , [ eq13 ] , [ eq15 ] ) can be either or . ) because statistics are taken along the sequence , all heterogeneity in along the sequence will be averaged out .for example , counts may mainly come from one region of the sequence , or from throughout the sequence .but from the number , we do not know which is the case .consequently , at a fixed distance does not contain information on heterogeneity . nevertheless , _ as a function of does reveal heterogeneity in the sequence_. 2 .  * strand complementarity *: dna sequences are double - stranded with nucleotides on one strand complementary to those on the other . as a result , on one strand ( in 5 3 direction )is exactly the same with the on the opposite strand in the opposite direction ( but also in 5 3 direction viewed from that strand ) .for example , in fig.1 , 3 .  * strand symmetry * : it was observed that on one strand is _ approximately _ equal to on the opposite strand in the opposite direction .this idea of strand symmetry " was suggested in [ fickett , et .1992 ] , though they actually suggested the symmetry for base density and density - density correlation , not base - base correlation as explicitly written here .combining strand symmetry with strand complementarity , we have on one strand .for example , in fig.1 , .this approximate symmetry reduces the number of independent correlation functions from 9 to 6 . + there were suggestions of other symmetries .for example , in [ teitelman and eeckman , 1996 ] , it was suggested that correlation is almost the same under simultaneous and transformation ( e.g. ) .this symmetry reduces the number of independent correlation functions from 9 to 5 .if the correlation matrix is invariant under simultaneous , , , transformation , the number of independent correlations is reduced from 9 to 3 .2 shows the 16 correlation functions for from 1 to 1000 , determined from the budding yeast chromosome 1 ( using the bayesian estimator as will be discussed in the next section ) .it is clearly seen that and , with all other cross - correlations roughly similar to each other .* relative contribution to the correlation from different pairs * : among the 9 independent correlation functions , some contribute more to the overall correlation than others . in [ teitelman and eeckman , 1996 ] , the data showed that is the largest among all correlation functions . is the second largest .the calculation from budding yeast chromosome 1 ( fig.2 ) confirms this .+ also note that the correlations between the same nucleotide ( e.g. , , ) are always positive , and those between different nucleotides ( e.g. , ) are usually negative ( m. zhang , unpublished draft , 1992 ; also see fig.2 ) .this can actually be explained in the case of simple domains structure ( appendix a of [ li , et .al . , 1994 ] ) . 5 .  *average over all correlation functions * : rather than calculate all independent correlation functions , we can define one measure which takes into account all individual correlation functions .for example , the mutual information function is defined as ( [ shannon , 1948 ; li , 1990 ] ) ( the base of the log term can also be 4 , 10 , or , rather than 2 ) : or ( see , e.g. , sec.13.4 of [ rice , 1995 ] ) :  should obey the distribution with 9 degrees of freedom under the zero - correlation hypothesis [ kullback , 1959 ] . + the sufficient and necessary condition for no correlation between two bases is that all correlation functions are equal to 0 .it is equivalent to or .in fact , in a first approximation , [ li , 1990 ; herzel and groe , 1995 ] .since the correlation at longer distances is typically small , it is important to use the best possible estimator to measure the correlation .otherwise , the error due to a finite sample size can be as large as the correlation value itself .we discuss three estimators here :  1 .  * frequency - count estimator * : it is usually assumed that the best way to estimate the probability of an event ( ) is to divide the number of count for ( ) by the total number of count ( ) , i.e. , the frequency - count estimator : this estimator is obtained by maximizing ( here data " is and parameter " is ) to get the parameter . in our case ,when is maximized with respect to , we obtain eq.([eq5 ] ) .+ although is an unbiased estimator of , inserting these estimators to a derived quantity of may not be an unbiased estimator of that quantity .for example , if entropy is estimated by , there is an underestimation [ basharin , 1959 ; herzel , 1988 ; herzel and groe , 1995 ; groe , 1995 ] .similarly , usually overestimates [ li , 1990 ; herzel and groe , 1995 ; groe , 1995 ] .( note : the eq.(5.10 ) in [ li , 1990 ] should be instead of . )the bias in the frequency - count estimator can be corrected when it is approximately calculated .nevertheless , the variance of the estimator around the true value is not reduced by this correction .+ the frequency - count estimator for correlation function would be : 2 .  * indirect bayesian estimator * : a completely different type of estimator is the bayesian s estimator ( see , e.g. sec.15.2.4 [ rice,1995 ] ) .the bayesian estimator of an event out of possible states is : this estimator is obtained by first calculating the posterior probability for the parameter : then using this probability as the weight to calculate the average value of the parameter . in the case of two possible outcomes , and assuming ( i.e. , uniform prior distribution ), we have which leads to eq.([eq9 ] ) ( set ) . +so the second estimator of a derived quantity of the probabilities is obtained by inserting the bayesian estimator of the probabilities to the function [ p. grassberger , unpublished result , 1994 ] .indirect " bayesian estimator reduces the variance , but there is still a substantial bias [ groe , 1996 ] .the indirect bayesian estimator for correlation function is ( for ) : 3 .  *direct bayesian estimator * : a direct bayesian estimator for a quantity does not rely on the bayesian estimator of the probabilities , but a direct average of the quantity using the posterior probability .for example , the direct bayesian estimator for correlation function is : for , it can be calculated to be [ wolpert and wolf , 1995 ; groe , 1995 ] ( using the multinomial distribution , and notice and ): although the direct bayesian estimator is still biased , the bias is much smaller than the frequency - count estimator and the indirect bayesian estimator , and the variance is also reduced as compared with the frequency - count estimator [ groe , 1996 ] .it is straightforward to apply the direct measure of correlation to dna sequences , as was done , for example , in [ shepherd , 1981a , 1981b ; fickett,1982 ; konopka and smythers , 1987 ; konopka , et .al . 1987 ; arqus and michel , 1987 ; li , 1992 ; li and kaneko , 1992a ; mani , 1992 ] .besides the well - known period-3 oscillation in coding regions , a calculation of the complete correlation function gives us more information , such as whether the correlation function decays as a power - law function or as an exponential function or in - between . hereare some comments :  1 .* different sequences may exhibit different correlation functions * : when a correlation function is calculated for an individual dna sequence , its form may be different from one sequence to another . in [ li , 1992 ] , for example , it is shown clearly that the s for the 5 human exon sequences are different from the 5 human intron sequences . in another example , the bacteriophage lambda sequence used in [ karlin and brendel , 1993 ] has spectrum [ li , et.al . , 1994 ] ( to be discussed in sec .2.4 - 2.5 ) whereas the budding yeast chromosome 3 sequence exhibits 1/f - like spectrum [ li , et .al . , 1994 ] .as a result , any conclusion from the analysis of one sequence should be taken with care when generalized to another sequence .* correlation function obtained from the whole sequence may be different from that obtained from a sub - sequence * : closely related to the comment # 1 in sec .2.1 , since the statistic is sampled along the sequence , a correlation function based on the statistic sampled from the whole sequence may differ from that based on a statistic sampled from a sub - sequence .there have been statements such as : no long - range correlations are found in any of the studied dna sequences " [ azbel , 1995 ] , while the study was only carried out at the subsequence level . + it is also possible that a correlation is present at the subsequence but will not be extended to longer distances when the whole sequence is analyzed .in fact , no one currently has the sequence of a complete human chromosome thus the correlation structure at the length scale of the whole human chromosome is unknown .* correlation function from one sequence may be different from that averaged over many sequences * : + this is yet another seemingly trivial statement but can be overlooked in a debate on the nature of long - range correlation in dna sequences .when the correlation functions from many sequences are averaged , the one with the slowest decay rate dominates at the long distances .if the correlation function in each sequence decays exponentially but with different rates ( i.e. , different correlation lengths ) , the averaged correlation function may decay as an non - exponential function , such as a power - law function ( it is closely related to the comment # 6 in the next section ) .power spectra via fourier transform ( see , e.g. , [ percival and walden , 1993 ] ) is widely used in time series analysis .the estimator of the power spectrum for pair is defined as where is 1 if the symbol at position is and 0 otherwise ; the is the complex conjugate .the frequency is defined as .although can range from 0 to , due to the mirror symmetry around , typically only the spectrum from to is displayed .there is a one - to - one correspondence between the power spectrum and the circular correlation function [ chechetkin and turygin , 1994 ] ( note : strictly speaking , no such relation holds between and the non - circular " ) : the second expression is due to the mirror symmetry around .since power spectrum and correlation function are two representations of the same correlation structure , power spectrum does not provide any new information which is not described by the correlation function . nevertheless , the visual representation of a power spectrum sometimes can more easily reveal patterns which are harder to discern in the correlation function .the following comments relate to power spectra :  1 .* averaged power spectrum * : similar to the case of correlation functions , there are many ways to average or sum power spectra .for example , or ( e.g. , [ voss , 1992 ; li , et .al . , 1994 ] ) or to assign four nucleotides to the four vertices of a tetrahedron , use the 3 coordinates of a vertex to represent a nucleotide , then calculate the sum of power spectra of sequences from each coordinate [ silverman and linsker , 1986 ; li and kaneko , 1992a ] .it seems that different projections of the power spectrum do not alter the general shape of the power spectrum .* exponential - decaying correlation functions correspond to lorentzian spectrum with a tail * : in eq.([eq17 ] ) , if the ( circular ) correlation function decays exponentially : , where , it can be shown that in the limit of [ borstnik , et .al . , 1994 ] ( if is finite , there are many more terms in the expression , but the main conclusion remains the same ) : using the taylor expansion of , the above expression can be approximated as where is the frequency . this spectral form is called a lorentzian spectrum .+ if the correlation function does not decay monotonically , but is oscillational , the lorentzian spectrum will be centered around a non - zero frequency due to the periodicity .* step functions exhibit power spectrum * : it is very easy to show that the correlation function of a step function ( e.g. , if and if ) decays linearly ( see , e.g. , appendix a of [ li , 1991 ] and appendix a of [ li , et .al . , 1994 ] ) . by using eq.([eq17 ] ) , the corresponding power spectrum contains a term .the implication of this almost trivial result is that for dna sequences which are c+g - rich on one half but c+g - poor on the other half , the power spectrum is expected to be of the form .* when both the correlation function and the power spectrum are power - law functions * : since the fourier transform of a power - law function is still a power - law function , we have ( using eq.([eq17 ] ) in the limit and approximate the sum by an integral ) suppose we change to  in other words , .this scaling argument " or dimension analysis " has been frequently used in physics .the step function discussed in comment # 3 confirms this relationship since and .+ caution should be taken for many real situations .for example , there can be cutoffs of the power - law function at both high- and low- frequencies [ theiler , 1991 ] .it is also possible that the power spectrum is only a stepwise power - law function .* 1/f spectra ( 1/f noise , 1/f fluctuation , flicker noise ) * : a particular interesting situation is when which implies , or the correlation function decay to zero very slowly . what makes spectra " or noise " interesting is that this type of fluctuation is very common in nature [ press , 1978 ] .1/f noise was perhaps first observed and studied in the noise of electronic systems [ johnson , 1925 ; schottky , 1926 ] , but it appears in numerous other phenomena , ranging from star luminosity to traffic flow ( an online bibiliography on 1/f noise can be found at url : _http://linkage.rockefeller.edu/wli/1fnoise_ ) .* 1/f spectra as a superposition of lorentzian spectra * : a natural and popular explanation of 1/f noise is that these are superpositions of many series with exponentially - decaying correlation function , each is sampled from a broad distribution of the correlation length ( in ) . one specific probability density function of the correlation length that leads to 1/f spectra is [ van der ziel , 1950 ] .the log - normal distribution can approximately lead to the distribution [ montroll and shlesinger , 1982 ] .+ since many so - called 1/f noise are only 1/f - like " , meaning these spectra are not perfect power - law functions , it is very likely that the requirement in eq.([eq24 ] ) is too strong .a reasonably broad distribution of correlation lengths might explain the data equally well .+ this point made here is very important to the understanding of the long - range correlation in dna sequences .there is nothing magic about the long - range correlation or 1/f spectra which could in principle be explained by the co - existence of many different length scales .what is essentially needed is a broad distribution of these different length scales ( to be discussed more in comment # 3 of sec.4 ) .although this fact is well known in the 1/f - noise community , its relevance to the correlation structure in dna sequence should be emphasized here .similar to the direct calculation of correlation function of dna sequences , the application of spectral analysis to dna is straightforward . as mentioned in the previous section ,the advantage of spectral analysis is to reveal patterns hidden in a directly correlation function .but the early applications of spectral analysis were mainly focused on revealing periodic signals [ mclachlan and karn , 1983 ; tavar and giddings , 1989 , and the references therein ] . only recently , attention turned to the functional shape of the power spectrum at all frequency ranges [ li , 1992 ; li and kaneko , 1992a ; voss,1992 ; buldyrev , et .al . 1995 ] . following two commentsare related to this topic :  1 .  *1/f - like power spectra were observed in dna sequences * : 1/f - like power spectra were indeed observed in dna sequences [ li and kaneko , 1992a ; voss , 1992 ; li , et .al . , 1994 ]. these are not white noise , indicating the existence of correlation .these are not lorentzian spectra either , indicating that there is a broad distribution for the correlation lengths in these sequences .it was questioned in [ borstnik , et .al . , 1993 ] whether the apparent " spectra are actually lorentzian spectra . by the comment # 6 in the sec .2.4 , we see that superposition of lorentzian spectra can lead to a 1/f - like spectrum .the question is whether there is a single length scale ( as in the case of lorentzian spectrum ) or multi - length scales ( as in the case of 1/f spectra ) .the quality of the 1/f spectra differs greatly among sequences * : by the quality " of a 1/f spectrum , i mean a measure on how close the observed spectrum is to a perfect ( ) spectrum . as mentioned in comment # 1 in sec 2.3 , different dna sequencesdo not necessarily exhibit the same power spectrum .the comment # 3 in sec 2.3 says that the spectrum obtained by averaging many sequences ( e.g. , such as [ voss , 1992 ; buldyrev , et .al . 1995 ] ) may have a different spectral form from that obtained in an individual sequence .+ the exponent in in [ voss , 1992 ] is very close to 1 ( note he subtracted the flat spectrum from the overall spectral shape ) , whereas the in [ buldyrev , et .al . 1995 ] is much smaller : another indication of the wide variation of the quality of 1/f spectra .the power spectra of all sixteen budding yeast chromosomes are strikingly similar , all 1/f - like ( w.li , unpublished results and paper in preparation ) .the question still remains that to what extent the spectra of different dna sequences are similar or different to each other .* * dna walk * : in a controversial paper [ peng , et .al . 1992 ] ( controversial because some results could not be reproduced on the same data set by two other groups [ prabhu and claverie , 1992 ; chatzidimitriou - dreismann and larhammar , 1992 ] , a dna sequence is first converted to a binary sequence ( for example , g or c is converted to 1 , a and t converted to 0 ) , then the binary sequence is converted to a walk ( 1 for moving up , 0 for moving down ) .a random binary sequence leads to a random walk .the variance of this walk at certain distance is related to the correlation function of the original binary sequence [ peng , et .al . 1992 ; karlin and brendel , 1993 ] : for random sequences , when , so only the first term is non - zero .it leads to .any deviation from the linear relation indicates a deviation from the random sequence .+ one might be curious about whether the above relation can be generalized to cross correlation functions ( ) .indeed , using the identity , we have : \ ] ] where should be the covariance of two such converted walks , one for symbol and another for symbol .when , is negative . instead of linear increase , we have the case of linear decrease .+ converting a binary sequence to a walk is equivalent to carrying out an integral or summation .consequently , the statistics obtained from the walk ( such as the variance ) are related to the integral ( summation ) of the statistics from the original binary sequence ( such as the correlation function ) .the integral ( summation ) makes the statistics from the walk smoother . whether it is an advantage or disadvantage depends on the purpose of the studyactually , by using a better estimator or by averaging many sequences , the direct calculation of correlation function can also be very smooth .* * graphical representation of dna sequences * : similar to dna walk " ,there have been many other proposals to map a dna sequence to a graph , making visualization of the base density or correlation easier . rather than reviewing these graphical representations ,let me give a pointer to the original references : [ hamori and ruskin , 1983 ; hamori , 1985 ; gate , 1986 ; hamori , 1989 ; hamori , et .al . 1989 ; jeffrey , 1990 ; berthelsen , et .1992 ; pickover , 1992 ; wu , et . al . 1993 ; zhang and zhang , 1994 ] . +another type of graphical representation concerns the sequence - dependent bending / curving [ calladine and drew , 1990 ] .these graphical representations of a dna sequence require some biochemical modeling of the spatial bending of the double helix of the dna molecule ( e.g. [ tung and harvey , 1986 ; shpigelman et .al . 1993 ; tung and carter , 1994 ] ) .a review of this topic is outside the scope of this paper . ** wavelet analysis * : the wavelet transformation [ daubechies , 1988;1992 ] is a new type of transformation where localized wavelets " replace the sine / cosine functions in the fourier transform as the basis .this new method is ideal for studying heterogeneity in dna sequences , a topic to be discussed in the next subsection . for references on wavelet transformationapplied to dna sequences , see [ zhang , 1995 ; arneodo et .al . , 1995 ; 1996 ] .as pointed out in comment # 1 in sec 2.1 , statistics are sampled along dna sequences .if there is heterogeneity along the sequence , it is averaged out during the sampling process .this heterogeneity is intrinsically related to the slow - decay of correlation function : in order to have a significant correlation at longer distances , the correlated units should be larger than a few bases . these larger units , relatively homogeneous sub - sequences , can be called domains " .  heterogeneity of base density in dna sequences was observed in digested dna fragments ( e.g. , [ sueoka , 1959 ; filipski , et .al . 1973 ;thiery , et .al . 1976 ; macaya , et .al . 1976 ; cuny , et .al . 1981 ] ) .there were limitations in this experimental determination of the heterogeneity .first , only the c+g density of one dna fragment is determined .the spatial variation of c+g density within the fragment can not be detected .second , though a c+g - rich fragment is usually followed by a c+g - poor fragment , it is not known exactly how these fragments are assembled to the original long dna sequences , so the correlation among these fragments is not well known .these experimental studies inspired a few theoretical studies [ elton , 1974 ; churchill , 1989 ; kozhukhin and pevzner , 1991 ; fickett , et .al . , 1992 ] .the modeling of sequence heterogeneity will be discussed in the next section . here, let me address the issue on how to _ characterize _ the heterogeneity in a dna sequence .  consider the simplest situation of heterogeneity in a dna sequence : the left half of the sequence is highly c+g rich and the right half is c+g poor . a characterization of the heterogeneity includes the information on the boundary between the two homogeneous halves , the difference of c+g density ( or in general base densities ) between them , the size of each domain , etc .some potential problems can be seen immediately : what if the c+g density on the left half is only slightly different from that in the right half do we still consider the sequence to consist of two domains ?what if the left half can be decomposed to sub - domains itself ?the problem is that we may not always have a clear - cut domain structure .methods that identify and partition the whole sequence into homogeneous domains are _ segmentation algorithms _ , a term used in image processing . the basic idea in a segmentation algorithm is to use a measure of the degree of fluctuation ( such as variance ) , find a partition point which minimizes the fluctuation in either partition as compared with the degree of fluctuation in the original unpartitioned sequence .one conceptually simple segmentation algorithm was proposed and applied to dna sequences [ bernaola - galvn , et .al . , 1996 ] . in this segmentation algorithm , the single - base entropy , ,is used to measure the fluctuation ( measures the randomness at the base level ) .the weighted sum of the two entropies from both partitions is compared with the overall entropy from the original sequence .a partition point is determined when the difference between the two ( called jensen - shannon divergence " in [ lin , 1991 ] ) is maximized .an extension to the above approach is to use high - order entropies ( e.g. for 2-base entropy , for -base entropy , and the source entropy [ shannon , 1951 ] ) . typically , high - order entropies are difficult to calculate due to lack of sample size .one approach to extrapolate to the infinite - size limit is to use the regression analysis [ konopka , 1994 ] .another alternative is to use some measure of compressibility [ ziv and lempel , 1977 ] which is intrinsically related to the source entropy .both approaches are subject to finite - sample - size effect and may not give a reliable estimation of the source entropy . in the case of dna sequencesit has indeed been suggested [ konopka , 1994 ] that extrapolated source entropy carries a high sampling error and thereby is not a useful criterion to discriminate between introns and exons .there is an important parameter to be decided in a segmentation algorithm , which is the threshold of the divergence .in essence , the question is how different the two domains should be before we distinguish the two .a low threshold value makes it easier to partition , even for random sequences .a high threshold value leads to no partition at all . changing the threshold value ,we may change the size distribution of the partitioned domains . in [ bernaola - galvn , et .al . , 1996 ] , the size distributions of different dna sequences are compared and quantitative differences were observed . for future studies , such comparison might be carried out at each of the possible threshold values . a family of the size distributions is then compared with another family of distributions.  a crucial question concerning the heterogeneity of a dna sequence can be answered by these segmentation algorithms : whether there are domains within a domain . for simple heterogeneity , there are no domains within a homogeneous domain .for complex heterogeneity , the concept of homogeneity is only relative : a domain which is homogeneous under one threshold value can be heterogeneous under another slightly lower threshold value .this phenomenon was indeed observed in [ bernaola - galvn , et .al . , 1996 ] for one sequence , and not in another sequence .the claim that the mosaic character of dna consisting of domains of different composition can fully account for apparent long - range correlations in dna " [ karlin and brendel , 1993 ] underestimates the true complexity of the heterogeneity problem in dna sequences .characterizing the correlation structure of dna sequences does not involve any modeling of the sequence .the observation of the slower - than - exponential decay of correlation function and the 1/f - like power spectra in dna sequences does not require any assumption about the sequence .we notice a comment in [ karlin and brendel , 1993 ] : recent papers proffer the asymptotically weakly independent stationary process as a model to describe apparent long - range dependencies inherent to many dna sequences. however , the assumption of stochastic stationarity is problematic " .this comment confuses the two : by calculating the correlation function or the power spectrum , we only extract information from the dna sequence and summarize this information in a compact form ; we do not automatically assume the dna sequences to be homogeneous ( stationary ) .four different modeling strategies of dna sequences are reviewed below .one of the earliest attempts to model dna sequences was to use the 1-step markov chain [ gatlin , 1966 , 1972 ; elton , 1974 ] ) .it was then realized that 1-step markov chains are not good model for dna sequences .one reason is that in coding regions , the position within a codon is important , thus the markov transition probabilities depend on this position [ borodovskii , et .1986a , 1986b ; tavar and song , 1989 ] .this problem is relatively easier to fix : one might use markov models with cyclic , position - dependent , transition probabilities .another reason is due to heterogeneity : on the global scale the markov transition probabilities depend on which homogeneous domain the position falls in [ borodovskii , et .1986a , 1986b ] .this problem can not be solved within the framework of a markov model .the correlation structure of dna sequences gives another confirmation that 1-step markov chains are not good models for dna sequences .the correlation function for a 1-step markov chain decays exponentially .this can be easily proved : the joint probability can be obtained by calculating the power of the markov transition matrix , which is dominated by the power of the largest eigenvalue of the matrix . andthis leads to an exponential decay for ( see , e.g. , [ feller , 1968 ] ) .since we know from sec 2.3 that the correlation function in dna sequences does not usually decay exponentially , then a 1-step markov chain does not characterize the observed correlation functions .high - order markov chains certainly have more degrees of freedom to characterize a wider variety of correlation structures than the 1-step markov chains .in particular , with multiple eigenvalues for the transition matrix ( each related to a different decay rate of the exponential function ) , it is not impossible ( though practically unlikely ) to have such a distribution for these eigenvalues that the resulting mixture of exponential functions lead to a 1/f - like power spectrum [ li , 1987 ] .  nevertheless , there is a major drawback in using high - order markov chains : the number of free parameters increases with the square of the order of the markov chain . in the bayesian framework of model selection ,models with more free parameters are penalized ( see , e.g. [ sivia , 1996 ] ) , in the spirit of the ockham principle " . if one has to use a model with many parameters , it must fit the data comparatively much better to compensate for this penalty . in [ raffery and tavar , 1994 ] , a certain assumption is made concerning relationship among the parameters in the model , thus reducing the number of independent parameters . but this assumption might not be justified .the fundamental reason which casts doubt on 1-step markov model also negates the high - order markov chains : as long as the order of the markov chain is finite , it will not characterize the heterogeneity at the length scale larger than its order .the obvious heterogeneity in dna sequences motivated the introduction of other mechanisms that possibly describe the phenomenon .one powerful model is the hidden markov chain or hidden markov model [ baum and petrie , 1966 ; rabiner , 1989 ] .there are two layers of variables in a hidden markov model . when it is applied to dna sequences[ churchill , 1989 ; 1992 ] , these two layers of variables are : ( 1 ) the observed variable , which is the four nucleotides along a sequence ( ) ; and ( 2 ) the state variable ( unobserved ) , which is related to a description of the domain .the state variable sequence can be produced by a markov chain .thus , a hidden markov model is characterized by two sets of transition probabilities : one is the state transition probability ( suppose the markov chain is 1-step ) ; another is the mapping probability from the state variable to the observable .the state variable can be discrete or continuous .for an example of the discrete state variable , consider three possible values for : , for high , moderate , and low c+g density .the mapping probabilities are different for different state variables .for example , when , then and are higher than and .for an example of the continuous state variable , consider being the c+g density at the site , which can take any real number between 0 and 1 ( in practice , the lower and upper limits can be 1/3 and 2/3 [ fickett , et .al . , 1992 ] ) . with a given value of , say , , the mapping probability can be chosen as , .the continuous state variable version of the hidden markov chain was called walking markov chain " in [ fickett , et .al . , 1992 ] . comparing the continuous state variable version with the discrete version ,the former does not necessarily lead to a sharp boundary between domains .similar to the case of high - order markov chain , there is also an issue of estimating many parameters in the hidden markov chains , thus the fitting of the data must be very good in order to compensate the penalty for using too many parameters .but overall , hidden markov chains are a much better choice for characterizing heterogeneity in dna sequences than the high - order markov chains .hidden markov chain represents a major improvement over the regular markov chain in characterizing dna sequences because it is able to describe the heterogeneity well .but as discussed in sec 2.7 , when the heterogeneity becomes complex , a seemingly homogeneous region can be heterogeneous when the criterion for being heterogeneous is relaxed .the same phenomenon may reappear at a subdomain level .in other words , there can be subdomains within a domain , sub - sub - domains within a sub - domain , etc .rewriting systems are able to generate such hierarchical organization of domains and self - similarity .being rediscovered and renamed a few times , such as the context - free language [ chomsky , 1956 ] ( if the distinction between non - terminal symbols and terminal symbols is removed ) , l - systems [ lindenmayer , 1968 ; rozenberg and salomaa , 1980 ; prusinkiewicz and lindenmayer , 1990 ] , development systems [ wgrzyn , et .al . 1990 ] , substitutional sequences [ cheng and savit , 1990 ] , a rewriting system updates a sequence by replacing a symbol ( or a string ) by another string of symbols .for example , the famous fibonacci sequence is generated by repeated use of the following rewriting rule " ( replacing a symbol at with a string of symbols at ) : the expansion - modification system [ li , 1989 ; 1991 ] is a probablistic rewriting system : with the probability attached to each rule .one might easily generalize this model to a model of duplication and point mutation in dna sequences , such as if there is conflict among different rewriting rules , we might choose each one with certain probability .  rewriting systems operate quite differently from markov chain and its variants .markov chain moves along a sequence and generate new symbols sequentially .rewriting systems not only update symbols _ parallelly _ , but also _ repeatedly_. these two features give rewriting systems certain advantages over markov chains in modeling long dna sequences which might result from a long evolutionary process involving repeated duplications .there are also several parameters to be determined in a rewriting system .for example the initial sequences , the number of times the rule is applied , and the probability of applying certain rules when there is a conflict . it would be interesting to use a more rigorous model selection procedure to choose the type of rewriting system that might characterize dna sequences well , and to compare the rewriting system models with the hidden markov models .is the study of correlation structure of dna sequences useful for working biologists ?since most of the current dna sequence analysis is based on the knowledge of local signals ( e.g. consensus pattern of regulatory regions ) , the global view of the dna sequences takes a back seat .let me use an analogy of the natural language texts .understanding local signals is similar to the construction of a dictionary of words .the interaction among different regulatory regions is perhaps analogous to syntax of sentences .the evolutionary history of the dna sequences is analogous to the writing of a text ( with repeated redrafting ) . andthe resulting genome organization is analogous to a style of the text .if a dictionary has already been constructed , the next goal should be an understanding of the whole text .here are a list of biological issues related to the study of correlation structure of dna sequences :  1 .* concentration of genes and the c+g density * : + it was shown that the concentration of genes is correlated with the c+g density [ bernardi , et .al . 1985 ; bernardi , 1989 ; 1995 ; zoubak , et .al . , 1996 ;clay , et .al . 1996 ] .the concentration of genes in the c+g - richest region of human genome can be 5 - 10 [ bernardi , 1989 ] or 15 times [ zoubak , et .al . , 1996 ] higher than those of the c+g - poorest regions .if the connection between c+g density and the concentration of genes is proven general , spatial distribution of c+g density can be used to give an indication of the locations of genes .+ the base - base correlation structure reviewed here can be extended easily to the density - density correlation .one can partition a sequence into _ non - overlapping _ windows , and calculate the c+g density in each window .similarly , the spatial distribution of genes can be characterized by the binary sequence which assigns a 1 to a base in the gene and 0 to a base in the intergenic region .the cross - correlation between the density sequence and the binary sequence can answer quantitatively the question of whether the spatial distribution of genes is related to the spatial variation of c+g density .besides the period-3 structure in coding region , are the correlation structures in coding and non - coding regions different ? * : + due to the codon structure , the correlation function for coding sequences exhibits period-3 structure with a high - low - low pattern [ fickett , 1982 ] .eukaryotic intron sequences do not exhibit this period-3 pattern , but occasionally , they exhibit a period-2 pattern [ konopka and smythers , 1987 ; konopka , et .al . 1987 ; arqus and michel , 1987 ; li , 1992 ] ( unlike nuclear dna sequences , the non - coding regions in viral or mitochondrial sequences may still have period-3 patterns [ arqus and michel , 1987 ; konopka , 1994 ] ) .+ besides this well - known pattern , how does correlation function decay with distances in either coding and non - coding sequences , and how do the correlation structures differ in the two types of sequences ? despite several studies , this remains an open question . intuitively , because of the specific constraints imposed by the function of coding region , i.e. , the 3-nucleotide to 1-amino - acid translation as well as the ability for the translated amino acid sequence to fold successfully , evolutionary tinkering is mostly prohibited in the coding region . on the other hand ,non - coding regions ( both intron and the intergenic regions ) are open to changes , because these changes typically do not lead to fatal damage to the cell and the organism .+ one of the most important changes in dna sequences is duplication [ bridges , 1919 ; chapter 17 of morgan , et . al . , 1925 ] , either the oligonucleotide duplication or gene duplication . the gene duplication in particular ,is considered to be crucial for the generation of complexity and the acquisition of new functions in high organisms [ ohno , 1970 ; market , et .al . , 1975 ; macintype , 1976 ; doolittle , 1981 ] .to quote from [ doolittle , 1981 ] :  it is simpler to duplicate and modify proteins genetically than it is to assemble appropriate amino acid combinations de novo from random beginning " .+ if the changes in non - coding sequences are mainly duplications , the non - coding regions should be less - random , or more - redundant .this would predict a longer - ranged correlation in non - coding sequences . indeed , [ li , 1992 ] showed that this is true for a few human dna sequences .but [ voss , 1993 ; 1994 ] did not observe significant difference in power spectra between coding and non - coding sequences .more recently , another study shows that after the period-3 structure is subtracted , the mutual information functions for coding and non - coding regions are very similar to each other , both decay as an approximate power - law function in 1 - 1000 bases range ( i. groe , private communication , 1996 ) .this study paid particular attention to correcting the bias in the estimation due to finite sequence length to make sure the small correlation at long distances is estimated accurately .+ so which study is correct ? in light of the fact that the currently available dna sequences are biased towards coding sequences and the flanking sequences , i would like to caution that once the intergenic sequences are represented in the sample with a larger proportion , the conclusion might be modified accordingly .intergenic sequences may expose to a lesser degree of constraints as compared to the intron sequences ( with perhaps the exception of structural constraints ) , thus exhibiting different statistical features . indeed, intergenic sequences are analyzed separately from the intron sequence in , for example , [ guig and fickett , 1995 ] ) .+ human intergenic sequences , in particular , have more room " to accumulate changes due to their sheer size . with the combination of the two ( less constraint and larger size ), one can imagine that duplication , insertion and point mutation can play a major role in shaping the intergenic sequences ( this speculation is already supported by a study of intergenic sequences in maize genome [ sanmiguel , et .al . 1996 ] ) .thus , human intergenic sequences may more easily exhibit long - range correlation than other types of sequences .* broad distribution of domain sizes as the origin of power - law decay of correlation function * : + there have been many papers discussing the biological origin " of long - range correlations .but what was mostly discussed was the known fact that the base density may be different at different regions , i.e. , the heterogeneity .what is non - trivial about the correlation structure in many dna sequences should be discussed along the line of complex heterogeneity versus simple heterogeneity " , spectra versus spectra " , broad versus narrow distribution of domain sizes " .+ for example , for the dna sequence of a complete chromosome , the sequence can be partitioned into coding and non - coding domains . assuming coding and non - coding domains are distinct in base density, the size distribution of coding as well as non - coding domains determines the correlation structure of the whole sequence .a study along this line was presented in [ herzel and groe , 1997 ] for budding yeast chromosomes , and a broad distribution for coding domains ( actually it is the distribution of the open reading frames " , i.e. , the potential or putative coding regions ) .note that this discussion does not apply to the correlation structure in intergenic sequences , and the assumption that a clear distinction between the base densities in coding and non - coding domains may not always hold .+ one can attribute the broad distribution to either internal or external factors .for external factors , it was shown that the size distribution of insertions and deletions of pseudogenes is approximately power - law functions [ gu and li , 1995 ] .if these insertions are distinct base - density - wise from the hosting dna sequence , the power - law distribution of the sizes is enough to lead to a complex heterogeneity .the power - law distribution of insertion sizes was used in [ buldyrev , et .al . 1993 ] by the assumption that the insertion is accomplished by a loop formation , and the size distribution of the loop length in a long polymer should be a power - law function [ des cloizeaux , 1980 ] .but it has not been examined whether this argument is correct . + duplication is a best example of the internal factors for broad distributions .the intuition can be again derived from the expansion - modification system [ li , 1989 ; 1991 ] .since a mutation followed by the repeated duplication of that mutated base generates a distinct domain whose size is proportional to the number of duplications , in order to have a broad distribution of sizes , one simply needs different duplication events to start at different times .then the broad distribution is just a historical profile of these duplication events . for a more realistic modeling of the ( gene ) duplication events ,see [ ohta , 1987a ; 1987b ; 1988a ; 1988b ; 1989 ; 1990 ] + using oligonucleotide repeats as an explanation of the 1/f - like spectra in dna sequence [ li and kaneko , 1992b ] does not mean that a single simple repeat can explain the correlation structure in dna sequences : it will not , because such a repeat manifests itself as a peak in the power spectrum instead of a broad band 1/f spectrum . rather , what we suggested in [ li and kaneko , 1992b ] was a series of duplications each occurring at a different historical period , and repeating a variety of times . + finally , it is also possible that both external and internal factors contribute to the broad distribution of domain sizes .* correlation between other units * : + there can be many variations to the average correlation between two bases calculated along a sequence .one example is the statistical correlation between two specific sites .the statistic in this case is no longer sampled along a sequence , but from a set of aligned sequences at these two specific sites .for example , the mutual information between two specific sites in hiv proteins was calculated to detect the co - varying mutation spots [ korber , et .1993 ] . in this example , a set of similar sequences is made possible by the high variability of the hiv .+ if there are only a few copies of a biologically meaningful unit on a dna sequence , it is difficult to obtain a statistic .take the replication origins on budding yeast chromosomes for example [ newlon , et .al . 1993 ] .the replication origins could be determined through a sequence analysis by searching the 10- or 11-base concensus pattern [ palzkill , et .al . , 1986 ] ( this might be called putative replication origin because of the possibility of false positives and the possibility of inactivation of the origin ) .one yeast chromosome may have only a limited number of replication origins ( around the order of 10 ) .thus , a proposition concerning these origins may be simply studied by an exhaustive listing without using any statistics .for example , in order to test the proposition that replication origins are always located inside a non - coding region [ murakami , et .1995 ] , we might just count how many replication origins are indeed located in the non - coding region .determining correlation structure of dna sequences on a more global scale reveals that the picture of simple heterogeneity is not enough to explain features of many sequences . a key tothe understanding of the complexity of correlation structure in dna sequences is the broad distribution of domain sizes .more quantitative measurement is necessary to characterize this feature more accurately . with even longer stretches of continuous dna sequences to be available in the future , it is conceivable that correlation structure at even larger scales can be studied .blurring the details at base level might be necessary in order to detect any significant correlation at these larger scales .our grand goal is to eventually learn the genome organization " principles , and explain this organization using our knowledge about evolution . on the latter aspect, this study may look similar to the study of molecular evolution and population biology ( see , e.g. , [ kimura , 1983 ; gillespie , 1991 ] ) , but with more emphasis on spatial correlation in dna sequences .an online bibliography on the topic of of correlation structure of dna sequences can be found at : _http://linkage.rockefeller.edu / wli / dna_corr_.this review would be impossible without extensive discussions with ivo groe , andrzej konopka , jse oliver , oliver clay , and michael teitelman , and i would like to thank them all .i also thank tara matise to proofreading the draft . a partial support from the aspen center for physics for attending the identifying features in biological sequences ( 1996 ) "is acknowledged .the author is supported by the grant hg00008 ( to j. ott ) from the national human genome research institute of nih .m. y. borodovskii , y.a .sprizhitskii , e.i .golovanov , a.a .aleksandrov ( 1986a ) , statistical patterns in the primary structures of functional regions of the genome in escherichia coli . i. frequency characterization " , molekulyarnaya biologiya , 20 , 1014 - 1023 ( russian ) ; molecular biology , 20 , 826 - 833 ( 1987 ) ( english translation ) .m. y. borodovskii , y.a .sprizhitskii , e.i .golovanov , a.a .aleksandrov ( 1986b ) , statistical patterns in the primary structures of functional regions of the genome in escherichia coli .ii . nonuniform markov models " , molekulyarnaya biologiya , 20 , 1024 - 1033 ( russian ) ; molecular biology , 20 , 833 - 840 ( 1987 ) ( english translation ) .    s.v .buldyrev , a.l .goldberger , s. havlin , c.k .peng , h.e .stanley , m.h .r. stanley , m. simmons ( 1993 ) , fractal landscapes and molecular evolution : modeling the myosin heavy chain gene family " , biophysical journal , 65 , 2673 - 2679 .buldyrev , a.l .goldberger , s. havlin , r.n .mantegna , m.e .masta , c.k .peng , m. simmons , h.e .stanley ( 1995 ) , long - range correlation properties of coding and non - coding dna sequences - genbank analysis " , physical review e , 51(5 ) , 5084 - 5091 .     v.r .chechetkin , a.y .turygin ( 1994 ) , on the spectral criteria of disorder in nonperiodic sequences - application to inflation models , symbolic dynamics and dna sequences " , journal of physics a , 27(14 ) , 4875 - 4898 .g. cuny , p. soriano , g. macaya , g. bernardi ( 1981 ) , the major components of the mouse and human genomes .i. preparation , basic properties , and compositional heterogeneity , " european journal of biochemistry , 115 , 227 - 233 .x. gu , w.h .li ( 1995 ) , the size distribution of insertions and deletions in human and rodent pseudogenes suggests the logarithmic gap penalty for sequence alignment " , journal of molecular evolution , 40 , 464 - 473 .e. hamori , g. varga , j.j .laguardia ( 1989 ) , hylas : program for generating h curves ( abstract three - dimensional representations of long dna sequences ) " , computer applications in the biosciences , 5(4 ) , 263 - 269 .j. josse , a.d .kaiser , a. kornberg ( 1961 ) , enzymatic synthesis of deoxyribonucleic acid : viii .frequencies of nearest neighbor base sequences in deoxyribonucleic acid " , journal of biological chemistry , 236 , 864 - 875 .korber , r.m .farber , d.h .wolpert , a.s .lapedes ( 1993 ) , covariation of mutation in the v3 loop of hiv-1 : an information theoretic analysis " , proceedings of national academy of sciences , 90 , 7176 - 7180 .w. li ( 1992 ) , generating non - trivial long - range correlations and 1/f spectra by replication and mutation " , international journal of bifurcation and chaos , 2(1 ) , 137 - 154 . [ in a more standard vocabulary in biology , the term replication " in the title should be duplication " . ]newlon , i. collins , a. dershowitz , a.m. deshpande , s.a .greenfeder , l.y .ong , j.f .theis ( 1993 ) , analysis of replication origin function on chromosome iii of saccharomyces cerevisiae " , in _ cold spring harbor symposium on quantitative biology _ , vol .58 , 415 - 423 .palzkill , s.g .oliver , c.s .newlon ( 1986 ) , dna sequence analysis of ars elements from chromosome iii of saccharomyces cerevisiae : identification of a new conserved sequence " , nucleic acids research , 14 , 6247 - 6264 .shepherd ( 1981b ) , method to determine the reading frame of a protein from the purine / pyrimidine genome sequence and its possible evolutionary justification " , proceedings of national academy of sciences , 78 , 1596 - 1600 .tung , s.c .harvey ( 1986 ) , computer graphics program to reveal the dependence of the gross three - dimensional structure of the b - dna double helix on primary structure " , nucleic acids research , 14 , 381 - 387 .m. zhang ( 1995 ) , exploratory analysis of long genomic dna sequences using the wavelet transform : examples using polyomavirus genomes " , in _ genome sequencing and analysis conference vi _ , page 72 , ( mary ann liebert ) .

Abstract: the study of correlation structure in the primary sequences of dna is reviewed . the issues reviewed include : symmetries among 16 base - base correlation functions , accurate estimation of correlation measures , the relationship between and lorentzian spectra , heterogeneity in dna sequences , different modeling strategies of the correlation structure of dna sequences , the difference of correlation structure between coding and non - coding regions ( besides the period-3 pattern ) , and source of broad distribution of domain sizes . although some of the results remain controversial , a body of work on this topic constitutes a good starting point for future studies .