# Expanding the class of conditionally exactly solvable potentials

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VOLUME 50, NUMBER 5 PHYSICAL REVIEW A Expanding the class of conditionally NOVEMBER 1994 exactly solvable potentials A. de Souxa Dutra Uniuersidade Estadual Paulista Ca-mpus de Guaratingueta-DFQ, Auenida Dr. Ariberto Pereira da Cunha, 333, Codigo de Enderegamento Postal 12500-000, Guaratingueta, Sio Paulo, Brazil Frank Uwe Girlich Fachbereich Physik, Universitat Leipzig, Augustusplatz 10, 04109 Leipzig, Germany (Received 30 December 1993) Recently a class of quantum-mechanical potentials was presented that is characterized by the fact that they are exactly solvable only when some of their parameters are fixed to a convenient value, so they were christened as conditionally exactly solvable potentials. Here we intend to expand this class by introducing examples in two dimensions. As a byproduct of our search, we found also another exactly solvable potential. PACS number{s): 03.65.Ca, 03.65.Ge, 02.90.+ p Recently the existence of a whole class of exactly solvable potentials in quantum mechanics was discovered [1]. Furthermore it was observed that these potentials obey a supersymmetric algebra [2]. In fact the potentials discussed in Ref. [1] were also discussed independently in a previous work of Stillinger [3]. In that article the author did not perceive that such potentials were only the first representatives of an entire new class of potentials. What distinguishes these potentials from the usual exactly solvable potentials is the fact that at least one of their parameters must be fixed to a specific value, because if you do not do that the potentials no longer have exact solutions. On the other hand, they cannot be classified among the so called quasi exactly solvable potentials [4 —10], because these have only a finite number of exactly solvable energy levels, and this happens only when the potential parameters obey some constraint equations, so that only some of the parameters are free, and the others are fixed in terms of them. The continuous interest in obtaining and classipotentials is because of their fying quantum-mechanical importance as a basis for expanding more realistic cases, and also for use of their solutions to test numerical calculations. Another possibility is that of trying to use them as a kind of theoretical laboratory, searching for new interesting physical features. Some of the examples that we will study below have, in fact, the interesting characteristic of representing a kind of semi-infinite charged string, because, as can be seen in Fig. 1, the potential is infinitely attractive along the x semiaxis. Here in this work we intend to expand the class of conditionally exactly solvable (CES) potentials. This is going to be done through the presentation of two examples in two dimensions. As a byproduct of our search for CES potentials, we found another exactly solvable potential, whose solution will also be presented. The potentials that will be considered here are V, (x,y) = Vn(x, y) = 2/3 X A arctan— + 8 [arctan(y/x) ] go [ arctan(y /x) ] +C (lb) where the function arctan(y/x) is defined in the principal branch. It can be observed that the first case above has an exact solution for any value of Go if 8 =0. Otherwise 3irt /32@, the case 8%0 has an exact solution for Go= — and only for a restricted region of the space (x and y both bigger or both less than zero). The second potential has exact solutions only when go= —5trt /32@. In this case the solution is valid along all the x-y plane. It is not difBcult to verify that these potentials are singular along the positive semiaxis x, as can be seen from Fig. 1. So they represent a kind of "charged string. " 20 V p B 1 x +y and arctan(y /x + [arctan(y /x ) Go [arctan(y /x) ] +D(x +y 1050-2947/94/50(5)/4369(4)/$06. 00 }]' 2 +C FIG. 1. Plot of the potential CES II with ) 50 4369 1994 unitary parameters. The American Physical Society BRIEF REPORTS 4370 like multivaluedness of the wave function. However, one can see that this transformation is typical of a stereographic mapping, in this case of R onto 5 . This kind of compactification can be done provided that one has suitable asymptotic conditions [11].In fact, this implies that one will have a finite wave function throughout variation of the variables, and particularly that at infinity it should vanish. This imposes the condition that the parameter D be nonzero, because along the positive x semiaxis the wave function will be singular at infinity if D =0. Furthermore, there is no periodicity of the wave function in the variables x and y, as we will see later. So we see that the original potentials were, in some cases, reduced to one-dimensional CES potentials, whose solutions were obtained previously. Now we proceed to the solution of the above mentioned possibilities. The fi. rst example is Case I for 8 =0 and Gu arbitrary that of an exactly solvable potential. In this case the potential V, (x, y) is mapped into a harmonic oscillator with centrifugal barrier in the variable u, and a Coulomb plus centrifugal barrier in the variable U. So we only need to identify the corresponding parameters, substitute them in the equation for the "energy, which in the transformed equation is the parameter K, and solve it for the original The solution for these potentials can be obtained through a suitable change of coordinates: x =u cos(u), y =u sin(u) . After this change of variables, and their separation, the Schrodinger equation can be split into two others. In the variable u we have that + V, (u) —6' '4(u)=0 d 2p du t where (C+K —iri /8p, ) with K being the separation constant and p the mass of the particle. The subscript e in the potential stands for its exact solvability. In the variable v we have d 2p dU + VCEs(u) Eg(v— ) =0, & " with VCEs(u) having two possibilities: VcEs(u) + 8 + = Go eAergp (4a) U '„„(a=0)=4CD- (, , and II VCEs(u) = Av 2/3 + + 8 A D [2m+ I+a, ]' A' (4b) (2n The wave function can be obtained by returning %(u, u) =u ' 4(u)y(u) to the original variables. At this point, we shall observe that the transformation used leads us to a mapping into an angle variable u = arctan(y /x ), which can, in principle, create problems 1/2 +1), —(I+8pGO/fi )'~ . It can be seen that the where a& = above spectrum is a kind of mixing of the Coulomb and harmonic oscillator spectra. The normalized wave functions are I qI, B =0( 32pD f2 ) 1/8 (a&+m+1) 1 4C D ( "' arctan— arctan-y X . —exp pD arctan y X 1/2 ( AD j ( fi 2D p p 1/2 2pA + a, + I )iri z) (X2+y )— —2pE 1/2 $2 arctan— X real. So, as the left hand of the above equation has its minimum value when m =0 (n arbitrary), we must have C ~ 2@A /(a +1)A' . Apparently if C is less than this we should impose the side & 2 (2n z+ J 1/2 2%2 +1) —C, (7b) and L„'(x, y) is the generalized Laguerre polynomial. It is remarkable that, as can be seen from Eq. (5), the domain of validity of the parameter C is defined through t equation (2m 01 /2 +y2) (X (2~) + 1)/4 —(2n +1), p x=- 1/2 in order to keep the energies 1/2 2A $2 1/2 ' AD $2 —SpK 1/2 +1) —8@K where ' 1/2 +1) 1(a, +n+1) I (m I 21'(n condition that where Int( ) stands for taking the first integer greater than or equal to the argument. This is a strange feature because, in this case, the ground state would have nodes looking like an excited state. This characteristic should BRIEF REPORTS 50 be verified through numerical calculation. Perhaps we would need some kind of analytical continuation for the energy in such cases. Now, the next case. 3A' /32p. In this case we Case for 8%0 and Go= — have our first example of a two-dimensional CES potential. After repeating the previous procedure we get the following equation for the energy eigenstates: I i' (m+ —,')K +2pA K +pAB K with ' K 2A 1 1/2 D p + pB4 =0, (10) 4371 once more we have a limitation on the parameter C. The solution of the equation for the parameter E is obtained following the procedure appearing in Ref. [1]. This gives us : D—+Q +R, Q:—(3az —a —27a —2a, ) /54, and with & (2n+1) —C 8'„, . 2A D (2n ' tg (x, y) = 1/8 2 XL: Xexp (x +y f2 — ' z ) (x +y H ) — pAB 2@A ') (m + — —, The wave function is given by ' 1/2 1/2 $2 ('+') ' a/2 arctan- x B &arctan(y/x)— 2 B 2E (14) ' 1/2 2A2D +4v'A —3 (m +1/2)iri 2R and ' 1/4 $2 D ' n, m 1/2 (15a) 1/2 1/2 1/2 + +4CD + 98 4 2p This should be expected due to the singularity from the wave function (14) that, as observed previously when we presented the potential, one must restrict the region of space such that x and y are both positive or both negative. Furthermore, the inverse tangent function is defined in the principal branch, as said previously. Finally we treat the second case. Case II (go= — 5R /72p). Now we present the last exCES potential that will be ample of a two-dimensional discussed in this work. If we repeat once more the same procedure we get the following equation for the energy in this last case: + 1) —C, (2n 9iri 2 A H (x, y) is the Hermite polynomial. It is easy to verify that the wave function, being proportional to arctan(y/x), vanishes along the positive x of the potential in this region. Moreover, it is easy to see 4 1/2 with the solution and semiaxis. 1/2 + 2p —(2n +1), 1/2 p 8pK 1/4 P &arctan(y/x)— 2E where we defined ' ') +— fi (m 8' (m+ ') 1/2 AD R:—(9a&a2 B4 +1); En+1) + m!I (a+n+1) )/9, & (13b) 1/2 (12) 32p, D vari E, =+2&D (K +C)' + (13a) (11) As we are looking for we see that it is necessary to solve a third order polynomial equation for and then use this solution in the expression for the energy ' a1 3 )' D— Km =(R +D)' +(R p (2n + 1) . (15b) This time we need to do a careful analysis of the range In the case of of validity for the potential parameters. the parameter B, we have the restriction 1/2 B~— (16) in order to include m =0 in the solution. rameter C is also restricted by C) 4&A 3 ( +, ) 9A' A 2p 1/2 Besides, the pa1/2 (17) BRIEF REPORTS So we see that the negative solution (positive in the above equation) introduces a limit on the number of bound states. The positive case (negative above) is such that we only need impose that C is bigger than the left hand side tl, 32 D = m (x, y) ] /8 3r( +1) 'm!I (a +n +1) 2 1/2 XL„' (x'+y ][ X exp pD ) 8 /2 2A (x+y ) of Eq. (17) with m =0. After this brief analysis of the region of validity of the potential parameters, we present the corresponding wave functions: ~ /2 2pa (x +y $2 ) arctan — 2/3 P ] 1/4 z~ 2/3 ~P arctan— —— arctan— 2 3' E X 2A ' I where —(2n +1) p 2A D (19) and ' 9@A ]/4 2A This finishes the presentation of examples of CES potentials, now in two-dimensional space. The principal feature presented by these potentials, besides belonging to the CES class of potentials, is that a stringlike singularity of the potential appears. This is corroborated by the fact that the probability of finding the particle over this singularity is zero. This accounts for the interest in doing further investigations of the characteristics of such poten- [1] A. de Souza Dutra, Phys. Rev. A 47, R2435 (1993). [2] E. Papp, Phys. Lett. A 17$, 231 (1993). [3] F. H. Stillinger, J. Math. Phys. 20, 1891 (1979). [4] G. P. Flessas, Phys. Lett. 72A, 289 (1979); 78A, 19 (1980); 81A, 17 (1981);J. Phys. A 14, L209 (1981). [5] P. G. L. Leach, J. Math. Phys. 25, 974 (1984); Physica D 17, 331 (1985). [6] A. de Souza Dutra, Phys. Lett. A 142, 319 (1988). [7] R. S. Kaushal, Phys. Lett. A 142, 57 (1989). [8] M. A. Shiffman, Iut. J. Mod. Phys. A 4, 2897 (1989). tials. It would be interesting to study the behavior of their corresponding coherent states, and also see what happens with wave packets around the linear singularity of the potential. Furthermore, in analogy with what has been done with the exact potentials [12,13], and also with the quasiexact ones [8,9], it is our intention to look for the dynamical algebra behind the solvability of the CES potentials. This study is presently under development and we hope to report on it in the near future. One of the authors (A. S.D. ) thanks CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico) of Brazil for partial financial support and FAP ESP (Fundal'Ko de Amparo a Pesquisa no Estado de Sao Paulo) for providing computer facilities. [9] A. de Souza Dutra and H. Boschi Filho, Phys. Rev. A 44, 4721 (1991). [10] L. D. Salem and R. Montemayor, Phys. Rev. A 43, 1169 (1991). [11]T. Eguchi, P. B. Gilkey, and A. J. Hanson, Phys. Rep. 66, 213 (1980). [12] A. O. Barut, A. Inomata, and R. Wilson, J. Phys. A 20, 4075 (1987); 20, 4083 (1987}. [13] H. Boschi-Filho and A. N. Vaidya, Ann. Phys. (NY) 212, 1 (1991). equilibrium value of MeCpG steps (,+14 deg.) [31,44]. In comparison, methylation has a significantly lower stability cost when happening at major groove positions, such as 211 and 21 base pair from dyad (mutations 9 and 12), where the roll of the nucleosome bound conformation (+10 deg.) is more compatible with the equilibrium geometry of MeCpG steps. The nucleosome destabilizing effect of cytosine methylation increases with the number of methylated cytosines, following the same position dependence as the single methylations. The multiple-methylation case reveals that each major groove meth- PLOS Computational Biology | www.ploscompbiol.org 3 November 2013 | Volume 9 | Issue 11 | e1003354 DNA Methylation and Nucleosome Positioning ylation destabilizes the nucleosome by around 1 kJ/mol (close to the average estimate of 2 kJ/mol obtained for from individual methylation studies), while each minor groove methylation destabilizes it by up to 5 kJ/mol (average free energy as single mutation is around 6 kJ/mol). This energetic position-dependence is the reverse of what was observed in a recent FRET/SAXS study [30]. The differences can be attributed to the use of different ionic conditions and different sequences: a modified Widom-601 sequence of 157 bp, which already contains multiple CpG steps in mixed orientations, and which could assume different positioning due to the introduction of new CpG steps and by effect of the methylation. The analysis of our trajectories reveals a larger root mean square deviation (RMSD) and fluctuation (RMSF; see Figures S2– S3 in Text S1) for the methylated nucleosomes, but failed to detect any systematic change in DNA geometry or in intermolecular DNA-histone energy related to methylation (Fig. S1B, S1C, S4–S6 in Text S1). The hydrophobic effect should favor orientation of the methyl group out from the solvent but this effect alone is not likely to justify the positional dependent stability changes in Figure 2, as the differential solvation of the methyl groups in the bound and unbound states is only in the order of a fraction of a water molecule (Figure S5 in Text S1). We find however, a reasonable correlation between methylation-induced changes in hydrogen bond and stacking interactions of the bases and the change in nucleosome stability (see Figure S6 in Text S1). This finding suggests that methylation-induced nucleosome destabilization is related to the poorer ability of methylated DNA to fit into the required conformation for DNA in a nucleosome. Changes in the elastic deformation energy between methylated and un-methylated DNA correlate with nucleosomal differential binding free energies To further analyze the idea that methylation-induced nucleosome destabilization is connected to a worse fit of methylated DNA into the required nucleosome-bound conformation, we computed the elastic energy of the nucleosomal DNA using a harmonic deformation method [36,37,44]. This method provides a rough estimate of the energy required to deform a DNA fiber to adopt the super helical conformation in the nucleosome (full details in Suppl. Information Text S1). As shown in Figure 2, there is an evident correlation between the increase that methylation produces in the elastic deformation energy (DDE def.) and the free energy variation (DDG bind.) computed from MD/TI calculations. Clearly, methylation increases the stiffness of the CpG step [31], raising the energy cost required to wrap DNA around the histone octamers. This extra energy cost will be smaller in regions of high positive roll (naked DNA MeCpG steps have a higher roll than CpG steps [31]) than in regions of high negative roll. Thus, simple elastic considerations explain why methylation is better tolerated when the DNA faces the histones through the major groove (where positive roll is required) that when it faces histones through the minor groove (where negative roll is required). Nucleosome methylation can give rise to nucleosome repositioning We have established that methylation affects the wrapping of DNA in nucleosomes, but how does this translate into chromatin structure? As noted above, accumulation of minor groove methylations strongly destabilizes the nucleosome, and could trigger nucleosome unfolding, or notable changes in positioning or phasing of DNA around the histone core. While accumulation of methylations might be well tolerated if placed in favorable positions, accumulation in unfavorable positions would destabilize the nucleosome, which might trigger changes in chromatin structure. Chromatin could in fact react in two different ways in response to significant levels of methylation in unfavorable positions: i) the DNA could either detach from the histone core, leading to nucleosome eviction or nucleosome repositioning, or ii) the DNA could rotate around the histone core, changing its phase to place MeCpG steps in favorable positions. Both effects are anticipated to alter DNA accessibility and impact gene expression regulation. The sub-microsecond time scale of our MD trajectories of methylated DNAs bound to nucleosomes is not large enough to capture these effects, but clear trends are visible in cases of multiple mutations occurring in unfavorable positions, where unmethylated and methylated DNA sequences are out of phase by around 28 degrees (Figure S7 in Text S1). Due to this repositioning, large or small, DNA could move and the nucleosome structure could assume a more compact and distorted conformation, as detected by Lee and Lee [29], or a slightly open conformation as found in Jimenez-Useche et al. [30]. Using the harmonic deformation method, we additionally predicted the change in stability induced by cytosine methylation for millions of different nucleosomal DNA sequences. Consistently with our calculations, we used two extreme scenarios to prepare our DNA sequences (see Fig. 3): i) all positions where the minor grooves contact the histone core are occupied by CpG steps, and ii) all positions where the major grooves contact the histone core are occupied by CpG steps. We then computed the elastic energy required to wrap the DNA around the histone proteins in unmethylated and methylated states, and, as expected, observed that methylation disfavors DNA wrapping (Figure 3A). We have rescaled the elastic energy differences with a factor of 0.23 to match the DDG prediction in figure 2B. In agreement with the rest of our results, our analysis confirms that the effect of methylation is position-dependent. In fact, the overall difference between the two extreme methylation scenarios (all-in-minor vs all-in-major) is larger than 60 kJ/mol, the average difference being around 15 kJ/ mol. We have also computed the elastic energy differences for a million sequences with CpG/MeCpG steps positioned at all possible intermediate locations with respect to the position (figure 3B). The large differences between the extreme cases can induce rotations of DNA around the histone core, shifting its phase to allow the placement of the methylated CpG steps facing the histones through the major groove. It is illustrative to compare the magnitude of CpG methylation penalty with sequence dependent differences. Since there are roughly 1.5e88 possible 147 base pairs long sequence combinations (i.e., (4n+4(n/2))/2, n = 147), it is unfeasible to calculate all the possible sequence effects. However, using our elastic model we can provide a range of values based on a reasonably large number of samples. If we consider all possible nucleosomal sequences in the yeast genome (around 12 Mbp), the energy difference between the best and the worst sequence that could form a nucleosome is 0.7 kj/mol per base (a minimum of 1 kJ/mol and maximum of around 1.7 kJ/mol per base, the first best and the last worst sequences are displayed in Table S3 in Text S1). We repeated the same calculation for one million random sequences and we obtained equivalent results. Placing one CpG step every helical turn gives an average energetic difference between minor groove and major groove methylation of 15 kJ/ mol, which translates into ,0.5 kJ/mol per methyl group, 2 kJ/ mol per base for the largest effects. Considering that not all nucleosome base pair steps are likely to be CpG steps, we can conclude that the balance between the destabilization due to CpG methylation and sequence repositioning will depend on the PLOS Computational Biology | www.ploscompbiol.org 4 November 2013 | Volume 9 | Issue 11 | e1003354 DNA Methylation and Nucleosome Positioning Figure 3. Methylated and non-methylated DNA elastic deformation energies. (A) Distribution of deformation energies for 147 bplong random DNA sequences with CpG steps positioned every 10 base steps (one helical turn) in minor (red and dark red) and major (light and dark blue) grooves respectively. The energy values were rescaled by the slope of a best-fit straight line of figure 2, which is 0.23, to source of circulating FGF-21. The lack of association between circulating and muscle-expressed FGF-21 also suggests that muscle FGF-21 primarily works in a local manner regulating glucose metabolism in the muscle and/or signals to the adipose tissue in close contact to the muscle. Our study has some limitations. The number of subjects is small and some correlations could have been significant with greater statistical power. Another aspect is that protein levels of FGF-21 were not determined in the muscles extracts, consequently we cannot be sure the increase in FGF-21 mRNA is followed by increased protein expression. In conclusion, we show that FGF-21 mRNA is increased in skeletal muscle in HIV patients and that FGF-21 mRNA in muscle correlates to whole-body (primarily reflecting muscle) insulin resistance. These findings add to the evidence that FGF-21 is a myokine and that muscle FGF-21 might primarily work in an autocrine manner. Acknowledgments We thank the subjects for their participation in this study. 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Gallego-Escuredo JM, Domingo P, Gutierrez MD, Mateo MG, Cabeza MC, et al. (2012) Reduced Levels of Serum FGF19 and Impaired Expression of Receptors for Endocrine FGFs in Adipose Tissue From HIV-Infected Patients. J Acquir Immune Defic Syndr 61: 527–534. 34. Domingo P, Gallego-Escuredo JM, Domingo JC, Gutierrez MM, Mateo MG, et al. (2010) Serum FGF21 levels are elevated in association with lipodystrophy, insulin resistance and biomarkers of liver injury in HIV-1-infected patients. AIDS 24: 2629–2637. PLOS ONE | www.plosone.org 7 March 2013 | Volume 8 | Issue 3 | e55632

Expanding the class of conditionally exactly solvable potentials Expanding The Class Of Conditionally Exactly Solvable Potentials