Tsallis entropy and a maximum entropy principle allows to reproduce experimental data of DNA double strand breaking by electron and neutron radiation. Analytic results for the probability of finding a DNA segment of length l are obtained reproducing quite well the fragment distribution function experimentally obtained

Antoranz, JC

Arruda Neto, JDT

Deepman, A

Guzman, F

Rodgers, GJ

Rodriguez, O

Sotolongo-Costa, O

English

arXiv

Brunel University Research Archive

arXiv:cond-mat/0201289 v1   16 Jan 2002October 24, 2006A non extensive approach for DNA breakingby ionizing radiation.O. Sotolongo-Costa1, F. Guzman2, J. C. Antoranz3,G. J. Rodgers4, O. Rodriguez2, J.D.T. Arruda Neto5,A. Deepman51.- Department of Theoretical Physics. U.H., Havana 10400,Cuba. <oscarso@ff.oc.uh.cu>2.- Institut of Nuclear Sciences and Technology, Havana 10600,Cuba.<guzman@info.isctn.edu.cu>3.- Departamento de Fisica Matematica y Fluidos, UNED, 28040Madrid. <antoranz@apphys.uned.es>4.-Department of Mathematical Sciences. Brunel University,Uxbridge, UK.<g.j.rodgers@brunel.ac.uk>5.- Instituto de F´isica, Universidade de Sao Paulo, Brasil, CEP:05315-970 <arruda@usp.br>AbstractTsallis entropy and a maximum entropy principle allows to reproduceexperimental data of DNA double strand breaking by electron and neutronradiation. Analytic results for the probability of finding a DNA segmentof length l are obtained reproducing quite well the fragment distributionfunction experimentally obtained.1Atomic force microscopy (AFM) has revealed itself as an extremely usefuldevice in the analysis of very small structures and specially in DNA fragmentanalysis as it has been shown in.1It is interesting to study the production of fragments in DNA as a resultof radiation, since the presence of radiation interacting with DNA moleculescan influence the properties of living cells up to a lethal extreme.On the other hand, DNA fragment analysis may help in the study ofthe structural properties of genome texts, and then to understand generalprinciples of genetic sequences.The process of DNA double strand breaking was performed by irradia-tion of DNA molecules with electrons and neutrons at different doses ( See1). Then, the length of the resulting fragments was measured. As a result, thecollection of fragments was found to obey a fragment size distribution func-tion (FSDF), which presents important characteristics from the viewpoint ofcomplexity.The main fact, which will be focused in this paper, is that the collectionof fragments is such that there is not a “characteristic” size of the fragments,i.e., the smaller the fragment, the more abundant is it. The FSDF in this casedoes not present a definite local maximum, resembling more to an inversepower law, i.e., a distribution function in the basin of attraction of a stable(Le´vy) distribution.2The main distinction of Le´vy distributions lies in the fact that their vari-ance is divergent. Maybe because of it, scientists have paid attention to themonly recently.This feature of the FSDF is not new. It has been reported in3 the oc-currence of transition to scaling in FSDF during glass rods breaking, in4 thepower law distribution of fragments was related to self-organized criticality(SOC). Matsushita5 proposed a fractal representation for a general processof fragmentation.Our group6,7 detected power law behavior in the process of liquid dropfragmentation and we proposed a Bethe lattice representation to interpretFSDF in these experiments.Some attempts to relate FSDF to first principles in physics like the maxi-mum entropy principle are present in8,9 with results that, at the best, do notcover the process in which scaling in FSDF is present. (i.e., when the energyof the fragmentation process is high).The universal nature and almost unlimited range of applicability of the2maximum entropy principle leads us to expect it to be useful in describingscaling in FSDF even at DNA scale.But the process of fractionating, by its own nature, is a paradigm ofphenomena in which interactions are long-range correlated among all partsof the object under fragmentation. Then, though the maximum entropyprinciple is expected to have an unlimited range of application, in the processof breaking the expression for the entropy in its Shannon form:S = −k∫p(x) log p(x)dx (1)-where p(x)dxis the probability of finding the system magnitude x in theinterval [x, x+ dx] , and k is Boltzmann’s constant- is not applicable.This is because this formula, based in Boltzmann-Gibbs statistics, isexpected to be valid when the effective microscopic interactions are short-ranged, and this gives to this entropy its extensive character (The entropy ofthe whole object equals the sum of the entropies of its constituent indepen-dent parts).Since, as we already pointed out, all parts of the fractionating objectduring the process of violent breakage are correlated, then the entropy ofthe object being fractionated is smaller tan the sum of the entropies of theparts in which the object divides, defining this way a “superextensivity” inthis system. This suggests that it may be necessary to use non-extensivestatistics, instead of the Boltzmann-Gibbs one.This kind of theory has already been proposed by Tsallis,10 who postu-lated a generalized form of entropy, given bySq = k1−∫pq(x)dxq − 1(2)where q is a real number.This entropy can also be expressed as:Sq = −k∫p(x)lqp(x) (3)where the generalized logarithm lqp(x) is defined as (See11):lq(p) =p1−q − 11− q(4)3It is straightforward to see that Sq −→ S when q −→ 1 , recoveringBoltzmann-Gibbs statistics.It is our goal to derive, starting from first principles, a functional depen-dence to describe the DNA DSDF obtained in.1Starting from equation 2 we may follow the method of Lagrange multipli-ers to apply the maximum entropy principle to the fragmentation of DNA.To do this, we impose two constraints: The first is the trivial one of normal-ization of the probability:∫p(l)dl = 1 (5)i.e., the sum of the probabilities of finding a fragment of any length isequal to unity.As a second constraint we may choose to adopt a “q-mean value” as:∫pq(l)ldl = 1 (6)Which reduces to the classical mean value when q −→ 1 .In this formu-lation the length l of the fragments has been referred to a unit adequatelychosen as to choose the ”q-mean value” equal to one.It may seem strange to introduce a “q-mean value” , also known as “un-normalized mean value” in this formulation. Really, this choice is not uniquebut for our purposes and for simplicity reasons we will choose this formula-tion. In11 a detailed discussion of the possible choices for the second con-straint can be found. The one here chosen showed to be particularly usefulin describing anomalous diffusion and was also employed by us in12,13 dealingwith problems of fragmentation.Now we use the method of Lagrange multipliers by means of the con-struction of the functional:£(pi, α, β) = Sq − α∫p(l)dl + β∫pq(l)ldl (7)being α and β the Lagrange multipliers.The extremization of this functional leads to:p(li) =β(2− q)dl[1 + β(q − 1)l]1/(q−1); (8)4Alternatively, the same method when applied to the Boltzmann entropy(q = 1) givesp(l)dl = βe−βldl. (9)Equation 8 is the expression for the probability of finding a fragment of lengthli and depends on three coefficients to adjust. In this case we can apply thisexpression to fit it to the experimental data reported in.1, where methods ofatomic force microscopy were applied to measure FSDF of irradiated DNA.Figure 1 shows the experimental results for DNA breaking with electronsat doses of 5000 and 7000 Gy. Both are fitted with Eq. 8. Figure 2 representsFSDF for DNA breaking with neutrons at the same doses. In both cases thelength of the fragments was normalized to the length of the largest one, andthe number of fragments was normalized to the total number of fragments.As it can be seen, the agreement is very good.More experimental data for electrons from 50 to 200 Gy and neutrons atdoses of 900, 7500, 2000 and 10000 Gy were also fitted with good results. Inthis paper we are reporting the results of the coincident doses of electrons andneutrons of 5000 and 7000 Gy to illustrate the application of this viewpoint.Only in the cases of very low doses of electrons (50 and 100 Gy, wherefluctuations in FSDF are important) the results are not as good as the onesbefore.This fact reveals the non extensive nature of DNA breaking, as it wasshown for macroscopic objects in.12,13 So, This characteristic of breaking isnot exclusive of macroscopic bodies.Use of Boltzmann´s entropy to describe FSDF obtained in these experi-ments leads to incorrect results, (i.e.,) impossible to fit with the data, whichshows power law behavior.On the other hand, this non extensivity may also reflect an intrinsicnature of the very DNA chain. The presence of 1/f spectrum in sequencesand long-range correlations in the DNA sequences14 supports this assertion.This work has been partially supported by the “Alma Mater” contest,Havana University. One of us (O.S.) is indebted to the Department of Math-ematical Physics and Fluids, UNED, for kind hospitality.5References1D. Peng Atomic Force Microscopy investigation of electron and neutron in-duced DNA double strand breaks. Ph. D. Thesis. The George WashingtonUniversity, Washington D.C. 1997.2Feller, W. An Introduction to Probability Theory and Applications Vol. 2Wiley, New York, 1966.3T. Ishii and M. Matsushita, J. Phys. Soc. Jpn. 61, 3474 (1992).4L. Oddershede, P. Dimon and J. Bohr, Phys. Rev. Lett.71, 3107 (1993).5M. Matsushita J. Phys. Soc. Jpn. 54, 857 (1985).6O. Sotolongo-Costa, E, Lopez, F. Barreras, J. Marin Phys. Rev. E. 49,4027 (1994).7O. Sotolongo-Costa, Y. Moreno, J. Lloveras and J. C. Antoranz Phys. Rev.Lett. 76, 42 (1996).8R. Englman, N. Rivier and Z. Jaeger Phil. Mag.65, 751 (1987).9X. Li, R. S. Tankin Combust. Sci. and Tech. 56, 65 (1987).10C. Tsallis J. Stat. Phys. 52, 479 (1988).11C. Tsallis Brazilian J. Phys. 29, 1 (1999).12O. Sotolongo-Costa, Arezky H. Rodriguez and G. J. Rodgers Entropy. 2,172 (2000) (http://www.mdpi.org/entropy)13O. Sotolongo-Costa, Arezky H. Rodriguez and G. J. Rodgers Physica A286, 210-214 (2000).14R. F. Voos Phys. Rev. Lett. 68, 3805 (1992). See also M. de Sousa Vieiracond-matt/9905074.6A. Figure CaptionsFig.1: Normalized DSDF for electron irradiation of DNA at 5000 Gy and7000 Gy. The solid squares represent the experimental results at 7000 Gyand solid circles at 5000 Gy. Fitting was made with Equation 8. Solid curveis for 5000 Gy. Curve with open squares is for 7000 Gy.Fig. 2: Normalized DSDF for neutron irradiation of DNA at 5000 Gyand 7000 Gy. The solid squares represent the experimental results at 7000Gy and solid circles at 5000 Gy. Fitting was made with Equation 8. Solidcurve is for 5000 Gy. Curve with open squares is for 7000 Gy.7This figure "electr1.jpg" is available in "jpg" format from:http://uk.arXiv.org/ps/cond-mat/0201289This figure "neutrons.jpg" is available in "jpg" format from:http://uk.arXiv.org/ps/cond-mat/0201289

A non extensive approach for DNA breaking by ionizing radiation

A non extensive approach for DNA breaking by ionizing radiation

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