Alane Cabral

Emanuella De Oliveira

Emanuella Gomes

Da Silva

Dr Danilo Damasceno Rocha

Elisabeth A Hillard
email: hillard@crpp-bordeaux.cnrs.fr

Dr Pascal Pigeon

Gérard Jaouen

Felipe A R Rodrigues

Dr Fabiane C De Abreu

Fabricia Da

Rocha Ferreira

Marilia O F Goulart

Alane Cabral De Oliveira

Emanuella Gomes Da Silva

Elizabeth A Hillard

Fabrícia Da

Letícia V Costa-Lotufo
email: costalotufo@gmail.com

Dr E A Hillard

Prof L V Costa-Lotufo

Dr A C De Oliveira

Prof. F E G C Da Silva

Rocha Da

Prof. M O F Ferreira

Goulart

Prof G Jaouen

Molecular Mechanism of Action of 2-Ferrocenyl-1,1-diphenylbut-1-ene on HL-60 Leukemia Cells

Keywords: antitumor agents, apoptosis, bioelectrochemistry, ferrocenes, HL-60 cells

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Ferrocene (Fc) is receiving considerable attention as a bioisostere in medicinal chemistry, as it can potentiate the biological activity of small molecules used in treatment against cancer [START_REF]Ferrocene-functionalized endocrine modulators as anticancer agents[END_REF] and contagious and parasitic diseases, [2] among others. [3] The ferricenium cation is known to be moderately cytotoxic, [4] and the combination of Fc with small molecules that target receptors or enzymes has been shown to be an advantageous strategy in experimental chemotherapy. [5] A series of ferrocenyl compounds, based on the tamoxifen skeleton, show antiproliferative effects against a variety of cancer cell lines, including hormone-dependent and -independent breast carcinoma. [6] Derivatives in which one or both of the geminal phenyl rings are functionalized with an oxidizable group have shown the best (nanomolar) activity, and reactive quinone or aminoquinone methide intermediates have been implicated in antiproliferative effects. [6f] Lacking such a functional group, 2-ferrocenyl-1,1-diphenylbut-1ene (1), is only moderately active against breast cancer cells [7] and thus has not been studied further in this context. However, we recently showed that 1 demonstrates IC 50 values below 2 µM for SF-295 human glioblastoma, HCT-8 human colon cancer, and HL-60 acute promyelocytic leukemia cell lines after 72 h. [8] This cytotoxicity, coupled with low hemolytic activity, prompted this further study of 1.

The present work is devoted to the investigation of the mechanism of biological action of 1 as a cytotoxic agent against HL-60 cells. The influence of 1 on cell morphology, cell membrane integrity, DNA fragmentation, cell cycle, phosphatidylserine (PS) externalization, and activation of caspases 3 and 7 was studied. The ability of 1 to interact with DNA was assessed through electrochemical and spectrophotometric experiments using double-stranded DNA (dsDNA)-modified electrodes and dsDNA and single-stranded DNA (ssDNA) in solution. To elucidate the role of the ferrocenyl group, the antiproliferative properties of the phenyl analogue, 1,1,2-triphenylbut-1-ene (2), were also evaluated.

Results

Antiproliferative effects

In the present study, we compared the antiproliferative properties of 1 with its organic analogue, 2, against cancerous and noncancerous cell lines (Table 1). Compound 1 was not cytotoxic against the glomerular basement membrane (GBM) noncancerous cell line, with an IC 50 value higher than the maximum concentration tested (343 µM). With IC 50 values below 4 µM for HL-60 acute promyelocytic leukemia, HCT-8 human colon cancer, SF-295 human glioblastoma, [8] and OVCAR-8 ovarian carcinoma, 1 can be considered selective for tumorigenic cells. Compound 2, lacking the ferrocenyl moiety, was likewise inactive against the GBM cell line. However, 2 was also inactive against HL-60, SF-295, and OVCAR-8 cell lines, with IC 50 values above the maximum concentration tested (25 µM) for all cell lines, showing that the Fc group is essential for antiproliferative effects.

Mechanistic studies in HL-60 cells

To characterize the cytotoxic effect of 1, a phenotypic evaluation of treated HL-60 cells after 24 h incubation was performed. The tested concentrations were chosen as 1, 2, and 4 µM, based on 24 h IC 50 values. Negative control cells exhibited a typical non-adherent round morphology after 24 h in culture, while several 1-induced morphological changes typical of apoptosis, including a decrease in cell size and the formation of apoptotic bodies and intracellular vacuoles, were observed (Figure 1 A). The severity of the effects increased with concentration, and at 4 µM, evidence of necrotic cell death was noted. The cytotoxic effects of compound 1 were confirmed by flow cytometry, demonstrating a concentration-dependent decrease in cell density (Figure 1 B). The mean number of cells decreased from 4.9 x 10 5 cells mL -1 in the negative control to 1.5 x 10 5 cells mL -1 under exposure to 1 at 4 µM; at this concentration, the effect was similar to that observed in the presence of the positive control doxorubicin at 0.5 µM (1.4 x 10 5 cells mL -1 , Figure 1 C). The effect on cell density was accompanied by a decline in cell membrane integrity (Figure 1 C), although the effects on proliferation seemed to be more marked and likely related to more specific pathways than loss of membrane integrity. At the highest tested concentration, 1 decreased the cell number by 70%, whereas cell membrane integrity was affected in only 25% of treated cells.

Compd

IC 50 [µM] [a] HL-60 HCT-8 SF-295 MDA-MB-435 OVCAR-8 GBM 1 1.040±04 [8] 0.90±1 [8] 1.00±2 [8] 16±4 [8] 3.4±0.6 2 >25 ND >25 ND >25 >343.0 [a] Data are the 95% confidence interval obtained by nonlinear regression for all cell lines from two independent experiments, performed in duplicate, after 72 h incubation; ND: not determined. The cell-cycle distribution of HL-60 cells treated with 1 was analyzed by flow cytometry.

Results demonstrated that cells were arrested at G 0 /G 1 , followed by a progressive increase in the number of cells with DNA fragmentation with increasing concentrations of compound (Figure 2). Cell-cycle arrest was most evident at 1 µM, with 74% of cells in G 0 /G 1 compared with 52% in the negative control, while DNA fragmentation was maximum at the highest tested concentration (4 µM), corresponding to 34%, relative to 4% in the negative control.

DNA fragmentation is evidence for an increased number of dead cells, and is a signature of both necrotic and apoptotic processes. The externalization of PS, an important biomarker characteristic of the early stages of cell death by apoptosis, was observed in treated cells in a concentration-dependent manner. After 24 h of incubation, only 3% of the negative control cells presented PS externalization, while in cells treated with 1 at 4 µM, this number increased to 38% (Figure 3 A), suggesting that the effects of 1 are related to apoptosis induction.

Apoptosis was also reinforced by the analysis of caspase 3 and 7 activation (Figure 3 B).

After 24 h of exposure to 4 µM of 1, HL-60 cells presented an increase from 1.2 to 14% in caspase 3 and 7 activation, relative to the negative control.

DNA interaction

Evaluation of the interaction of 1 with dsDNA-modified electrodes and with ssDNA in solution was carried out using electrochemical techniques. For dsDNA (Figure 4 A), the blank (grey line) showed no redox processes, as to be expected when the nucleoside bases are primarily paired. In the presence of 1 (black line), the appearance of diagnostic peaks Compound 2 did not interact with ssDNA, behavior that was confirmed by the lack of modification of the oxidation currents of guanosine and adenosine in ssDNA, similar to the behavior of Fc (Figure 4D). The result was ambiguous toward dsDNA, as the oxidation of compound 2 occurred at a similar potential to that of adenosine in dsDNA; however, the small difference in currents in both voltammograms suggests a slight interaction at most (Figure S2, Supporting Information). Spectrophotometric studies of the effects of compound 1 in calf thymus dsDNA were carried out to compare with the electrochemical investigation and to clarify the role of the preoxidation of Fc (Figure 5). In this experiment, a fixed concentration of DNA (0.130 mM) in 30% ethanol/Tris solution was treated with different concentrations of 1 (3.2-8 mM). Figure 5A shows the spectra of the free dsDNA and of the complex formed by compound 1 and dsDNA. In order to display only the dsDNA-1 interaction, the spectrum of 1 was subtracted from the spectrum of the complex at the same concentration. An increase in the concentration of 1 provoked a hyperchromic effect on the dsDNA peak at 260 nm (Figure 5A).

Hyperchromicity is typical of DNA interactions [9,10] and has been attributed, for instance, to intercalation interactions with small molecules, which results in exposure of the base pairs to UV/Vis radiation and increasing absorbance. [9c, 11, 12] The binding constant (K) of the 1-DNA complex was obtained by applying the Benesi-Hildebrand equation [Eq. ( 1)] to the band at 260 nm: [9c] in which l is the path length (1 cm), D t is the total DNA concentration,  DS is the molar absorptivity of the interaction complex, [S] is the analyte concentration, and K DS is the stability constant for complex formation (DS). Thus, the ratio of the intercept to the slope in the double reciprocal plot in Figure 5 B gave the binding constant (K DS ). [9c] The binding constant for this interaction was found to be 2 x 10 4 M -1 , consistent with literature data for interaction of dsDNA with other intercalators, [9,13] including ferrocenyl compounds. [9a, 14] (1)

Discussion

The antitumor properties of the ferrocifens and derivatives have been well studied, particularly against ER+ (MCF-7) and ER-(MDA-MB-231) breast cancer cells. [6] The antiproliferative activity of some of the most potent compounds has been attributed to formation of electrophilic intermediates and requires the presence of an oxidizable group conjugated to the ferrocenyl group. [6f] In this case, 1 possesses no such functional groups and was indeed less active on MDA-MB-231 cells (IC 50 = 7.54 µM) [15] than those compounds possessing phenol functionalities (e.g., IC 50 =0.6 µM for the 4,4'-dihydroxy analogue). [7] Nonetheless, we felt that 1 merited further study, based on a recent discovery of its antiproliferative activity against SF295, HCT-8, and HL-60 tumor cells in the low micromolar range, the lack of unspecific toxicity on erythrocytes, and its lack of oxidative stress induction properties. [8] Compound 1 was selective toward tumor cells when compared with noncancerous cells;

the mean IC 50 value (4.5 µM) against HL-60, HCT-8, SF-295, MDA-MB-435, [8] and OVCAR-8 tumor cells compared favorably with that obtained using the noncancerous GBM monkey cell line (IC 50 > 343.00 µM). Such selectivity toward tumor cells is a key factor in the development of new anticancer agents, with the goal of a decrease in side effects and an improvement in the quality of life of patients under chemotherapy. Often, such selectivity is suggestive of specific interactions with receptors present in cancer cells, such as the estrogen receptor (ER), which is targeted by hydroxytamoxifen. Although there is evidence of the presence of functional membrane ERα and ERβ in HL-60 leukemia cells, [16] as well as functional ERβ for colon cancer cells, including HCT-8, [17] analysis of the molecular data from the NCI [18] showed that all cell lines used in the present work indeed express low levels of ERs relative to some breast (MCF-7 and T47D) and ovarian (SK-OV-3) overexpressing cells. Based on the present findings, the activity of 1 seems to be independent of an ER expression pattern.

Studies on the mechanism of action of 1 suggest that it induces cell-cycle arrest at the G 0 /G 1 phase, followed by an increasing number of cells with DNA fragmentation. Usually, in the presence of DNA damage, cells can trigger a complex sequence of events, including checkpoint activation, leading to cell-cycle arrest, allowing DNA repair. Depending on the severity of the damage, cell death pathways can be activated. [19] Indeed, cells treated with 1 underwent characteristic apoptotic cell death, as demonstrated by the morphology of the treated cells, the increasing number of cells with DNA fragmentation, and by PS externalization and caspase 3 activation, all phenotypic and biochemical features related to apoptosis. [20] It is useful to compare these results with what is already known about the biochemical mechanism of the ferrocifens. The effect on cell-cycle progression of the ferrocenyl analogue of hydroxytamoxifen (Fc-OH-TAM) depends on the ER expression pattern of treated cells; [21, 6e] in ER+ cells, Fc-OH-TAM provokes arrest in the G 0 /G 1 phase, while ER-cells are arrested in the S phase. The authors furthermore linked the antiproliferative effects of Fc-OH-TAM with ROS production and senescence, which apparently is not the case for 1, as previous studies demonstrated that it was unable to elicit ROS production in HL-60 cells. [8] This observation, as well as the observation that 1 provoked G 0 /G 1 arrest, in spite of the low ER expression in HL-60 cells, suggest that the mechanisms of action for 1 and Fc-OH-TAM differ significantly.

Interestingly, Osella et al. [22] showed that Fc-OH-TAM cytotoxicity cannot be directly attributed to oxidative stress, although it is known to provoke DNA damage. Indeed, one of the most important aspects in the discovery of antineoplastic agents is the interaction of these substances with DNA. Electrochemical experiments on 1 using a dsDNA biosensor or ssDNA in solution evidenced such an interaction. The obtained binding constant of 2 x 10 4 M -1 , as well as the hyperchromic effect upon addition of 1 to dsDNA, is consistent with weak intercalation between base pairs. [9a, 11, 12, 23] Studies to evaluate the interaction of Fc or the organic moiety 2 with dsDNA and ssDNA gave negative results, reinforcing the importance of the hybrid skeleton on the activity of 1. [START_REF] Kerns | Drug-like Properties: Concepts, Structure, Design and Methods[END_REF] Conclusions 2-Ferrocenyl-1,1-diphenylbut-1-ene ( 1) was not cytotoxic toward GBM monkey cells but was able to decrease, in a concentration-dependent manner, the number of viable cells in the HL-60 line. Cell death was caused by apoptosis, as evidenced by morphological analyses, activation of effector caspases 3 and 7, increased DNA fragmentation, externalization of PS, and a necrosis process at the highest concentration. Furthermore, 1 interfered with the cell cycle, leading to accumulation of cells in the G 0 /G 1 phase. Interaction with dsDNA and ssDNA suggests that the mechanism of action involves molecular damage to DNA intercalation.

Experimental Section

Preparation of 2-ferrocenyl-1,1-diphenylbut-1-ene (1) and 1,1,2-triphenylbut-1-ene (2)

The synthesis of 1 has been previously reported. [7] Briefly, 1 was obtained by a Friedel-Crafts acylation of Fc, followed by a McMurry cross-coupling of propionyl Fc with benzophenone.

The preparation and characterization of 2 has been reported. [START_REF] Schneider | [END_REF] However, we here used a onestep McMurry method, starting from commercially available propiophenone and benzophenone. Titanium chloride (7.6 g, 4.4 mL, 40 mmol) was added dropwise to a suspension of zinc powder (3.9 g, 60 mmol) in dry THF at 10°C. The mixture was heated at reflux for 2 h. A second solution was prepared by dissolving propiophenone (1.34 g, 1.33 mL, 10 mmol) and benzophenone (1.82 g, 10 mmol) in dry THF. This latter solution was added dropwise to the first solution, and the solution was stirred at reflux for an additional 2 h. After cooling to room temperature, the mixture was stirred with water and dichloromethane, acidified with dilute hydrochloric acid until the dark color disappeared, and decanted. The aqueous layer was extracted with dichloromethane, and the organic layers were combined and dried over magnesium sulfate. After concentration under reduced pressure, the crude product was separated by preparative HPLC with acetonitrile as the eluent to yield pure 2 in 55% yield. Physicochemical data were identical to those in the literature. [START_REF] Schneider | [END_REF] Compounds 1 and 2 were purified by semi-preparative HPLC before use. Elemental analysis of 1 and 13 C NMR of 2 confirmed >95% purity.

Biochemistry

Cell lines: HL-60 (acute promyelocytic leukemia), OVCAR-8 (ovarian carcinoma), and SF-295 (glioblastoma) cell lines were provided by the U.S. National Cancer Institute (Bethesda, MD). Glomerular basement membrane (GBM) monkey cells were used as a noncancerous cell line for comparison. Cells were maintained in RPMI 1640 medium supplemented with 10% fetal bovine serum and 1% antibiotics at 37°C in a humidified atmosphere of 5% CO 2 and 95% air.

Cytotoxicity: Cytotoxicity was determined using the MTT (3-(4,5-dimethylthiazol-2-yl)-2,5diphenyltetrazolium bromide) assay. [26] Accordingly, GBM, HL-60, OVCAR-8, and SF-295 cell lines were seeded in 96-well plates and incubated with 1 (0.02-343 µM) or vehicle control (DMSO, 0.5%) for 72 h. After the incubation period, a solution of MTT (1 µM) was added to each well and incubated for 3 h. Plates were spectrophotometrically evaluated at 570 nm, and IC 50 values were determined using the GraphPad Prism 5.0 software.

Mechanistic studies: For the following experiments, HL-60 cells were seeded at a density of 3

x 10 5 cells mL -1 and incubated with 1, 2, or 4 µM of 1 for 24 h. The vehicle (DMSO, 0.5%) and doxorubicin (0.5 µM) were used as negative and positive controls, respectively. Statistical analyses of results were performed using the GraphPad Prism 5.0 software.

Analysis of morphological changes-(H/E stain):

Untreated or 1-treated cells were examined for morphological changes by light microscopy (Olympus, Tokyo, Japan). To evaluate nuclear morphology, cells were harvested, transferred to cytospin slides, fixed with methanol for 1 min, and stained with hematoxylin and eosin.

Flow cytometry analysis: All experimental procedures adopted in flow cytometry analyses essentially followed the methodology described by Montenegro et al. [27] The evaluated parameters were cell membrane integrity, DNA fragmentation, cell cycle, PS externalization and caspase 3/7 activation. For all experiments, 5 000 events were evaluated using a Guava

EasyCyte Mine flow cytometer and Guava Express Plus software. DNA fragmentation and the cell cycle were analyzed by ModFit LT for Win32 version 3.1.

DNA interaction studies

Electrochemistry with dsDNA biosensors: The ability of 1 and 2 to interact dsDNA was investigated using an electrochemical approach employing DNA-modified glassy carbon electrode (GCE) biosensors. [28,29] The electrochemical experiments were performed with a conventional undivided three electrode cell using an Autolab PGSTAT-30 potentiostat (Echo Chemie, Utrecht, the Netherlands) coupled to a microcomputer, interfaced by GPES 4.9 software. A working GCE (diameter=3 mm), a Pt wire counter electrode, and a Ag|AgCl, Cl - (saturated) reference electrode were used. All experiments were conducted at room temperature (25±2 °C) after purging with argon. The DNA used was type I calf thymus (Sigma Aldrich, Saint Louis, USA), highly polymerized, containing 6.2% Na and 13% H 2 O, dried, and stored at 8 °C. The electrochemical procedure for the investigation of dsDNA interaction with compounds 1 and 2 involved three steps: preparation of the GCE surface, immobilization of dsDNA gel, and voltammetric transduction. Initially, the GCE was polished with alumina (BAS polishing kit). The electrode was then electrochemically pre-treated with a sequence of 5 cyclic potential scans from 0 to +1.4 V versus Ag|AgCl, KCl (0.1 M) in acetate buffer, washed thoroughly with distilled/deionized water, and dried. In order to immobilize the dsDNA, the surface of the electrode was coated with 10 mL of calf thymus DNA solution (containing 12.0 mg of dsDNA in 1.0 mL of acetate buffer, pH 4.5, stored at 20°C for 48 h), the gel was allowed to dry at room temperature under a stream of nitrogen, and the biosensor was subsequently immersed in 5 mL of aqueous-ethanolic (30%) acetate buffer, pH 4.5. For each series of experiments, an identical dsDNA-GCE was prepared as a reference blank.

Before analysis, the modified electrode was left for 10 min in contact with the buffer solution containing either 20 mL of ethanol (negative control), Fc (10 µM), 1 (20 µM), or 2 (10 µM).

DPV scans (=10 mV s -1 , pulse amplitude=50 mV and pulse width=70 ms) and baseline corrections were performed using GPES 4.9. [28,29] Electrochemistry of ssDNA in solution: To produce ssDNA by acid-base treatment, dsDNA (1 mg) was dissolved in 1 M HCl (100 mL) by heating at 100°C in a sealed glass tube immersed in a boiling water bath for 1 h, followed by neutralization with 1 M NaOH. [29,30] A freshly prepared solution, consisting of 4 mL ssDNA and 950 mL of ethanol, [28][29][30] was added to the electrochemical cell. Single-scan DPV experiments were conducted between 0 and +1.4 V versus Ag|AgCl, KCl (0.1 M), (=10 mV s -1 , pulse amplitude=50 mV and pulse width=70 ms). Peaks corresponding to the oxidation of guanine and adenine appeared at +0.815 V and +1.164 V, respectively. After rinsing the surface, the GCE was inserted into a solution containing 1 or 2 (50 mL, corresponding to a concentration of 10 µM) or Fc (10 µM), and the DPV experiment was repeated. A clean GCE was also employed in DPV experiments involving a 0.1 mM buffered solution of 1 alone (pH 4.5), in order to observe possible interference of oxidation waves from compounds 1 or 2. Baseline corrections were made using the software GPES 4.9.

Spectrophotometric evaluation of interaction with dsDNA: Spectrophotometric studies were carried out with a Shimadzu Multispec 1501 or diode array Hewlett Packard 8453 spectrophotometer. The interaction of compound 1 with dsDNA was monitored by the absorption titration method, [9,10] in which a fixed concentration of DNA (130 µM) was treated with different concentrations of compound 1 (3.2-8 µM) in Tris-HCl buffer (pH 7.4, 0.1 M, 30% ethanol). A stock solution of DNA was prepared by dissolving 12 mg of calf thymus DNA per mL of acetate buffer. An aliquot (10 mL) was then dissolved in Tris-HCl buffer and kept at 8 °C for 24 h, stirring at frequent intervals to ensure the homogeneity of the solution.

The final concentration of the dsDNA solution was measured spectrophotometrically at 260 nm, using the molar extinction coefficient of  260 =6600 cm -1 M -1 per nucleotide for calf thymus DNA. [9a,d] The concentration of the DNA solution was adjusted to 130 µM. The ratio between the absorbance values at 260 and 280 nm in the DNA spectra (A260/A280) was measured as 1.81, indicative of DNA free of protein contamination. [9a, 31] The same procedure was performed for compound 2.

Figure 1 .

 1 Figure 1. Effect of 1 on A) HL-60 cell morphology, B) membrane integrity, and C) cell number, as determined by application of hematoxylin/eosin (H/E) staining and flow cytometry. Data are presented as mean values ± SEM of three independent experiments performed in triplicate; *p < 0.05 relative to negative control by ANOVA, followed by Dunnett's test. HL-60 cells were treated with controls or 1 for 24 h at 1, 2, and 4 µM. DMSO (0.2%) and 0.5 µM doxorubicin were used as the negative and positive controls, respectively. For H/E staining in panel A), solid-head arrows denote necrotic cells, dashed arrows denote apoptotic bodies, and openhead arrows denote intracellular vacuoles. Scale bars: 20 µM.

Figure 2 .

 2 Figure 2. Effect of 1 on the cell-cycle distribution of HL-60 cells treated with controls or 1 for 24 h at 1, 2, and 4 µM. Cell cycle was determined by flow cytometry, and pie graphs depict the percentage of cells in G 0 /G 1 , S, or G 2 /M phases; sub-G 1 denotes fragmentation of DNA. DMSO (0.2%) and 0.5 µM doxorubicin were used as the respective negative and positive controls.

Figure 3 .

 3 Figure 3. Effect of 1 on: A) phosphatidylserine externalization and B) activation of caspases 3 and 7 of HL-60 cells treated with controls or 1 for 24 h at 1, 2, and 4 µM. DMSO (0.2%) and 0.5 µM doxorubicin were used as the respective negative and positive controls. Cells were analyzed by flow cytometry, and data are presented as mean values SEM of two independent experiments performed in triplicate; *p < 0.05 relative to negative control by ANOVA, followed by Dunnett's test.

  corresponding to guanosine and adenosine oxidation represented a conformational change in the dsDNA, resulting from an interaction between 1 and dsDNA. Compound 1 could also interact with ssDNA (Figure4B); here, the intensities of the adenosine and guanosine current peaks were diminished upon addition of 1 to the solution, without displacement in terms of peak potential. It is worth mentioning that the Fc moiety in 1 underwent oxidation in these experimental conditions, generating a positive charge on the molecule. This might cause an electrostatic attraction toward the phosphate backbone of DNA and could be a drawback in the use of electrochemical sensors for this system. Therefore, we investigated the possible interaction of Fc with dsDNA and ssDNA. Figure4C shows the differential pulse voltammetry (DPV) scans of a solution containing ssDNA, with and without Fc. In this case, there was only a slight alteration in the oxidation currents of guanosine, suggesting negligible interaction with the ferricenium cation. Furthermore, no interaction of the ferricenium cation with dsDNA (FigureS1, Supporting Information) was observed, suggesting that the oxidation of the Fc moiety, per se, did not cause a false positive result.

Figure 4 .

 4 Figure 4. A) Differential pulse voltammetry at a dsDNA-modified GCE in acetate buffer+EtOH (4:1, pH 4.5), in the presence and absence of 1 (20 µM). B) Differential pulse voltammetry at a GCE in acetate buffer+EtOH (7:3, pH 4.5), and ssDNA in the presence and absence of 1 (10 µM). C) Differential pulse voltammetry at a GCE in acetate buffer+EtOH (7:3, pH 4.5), and ssDNA in the presence and absence of ferrocene (10 µM). D) Differential pulse voltammetry at a GCE in acetate buffer+EtOH (7:3, pH 4.5), and ssDNA in the presence and absence of 2 (5 and 10 µM).

Figure 5 .

 5 Figure 5. A) UV/Vis spectra for dsDNA (130 µM), in the absence and presence of 1. Inset: The effect of the concentration of 1 (3.2, 4.0, 4.8, 5.6, 6.4, 7.2, and 8 µM) on the spectral profile of dsDNA. B) Double reciprocal plot of 1 binding to DNA, where A0 is the initial absorption of free DNA and A is the absorption at various concentrations (c) of 1 at 260 nm.

Table 1 .

 1 Antiproliferative properties of compounds 1 and 2 after 72 h.

Acknowledgements

This research was supported by grants from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), CAPES, CAPES/PNPD, RENORBIO, UFAL, and the National Cancer Institute of Brazil. The authors are also indebted to PRONEX-CNPq-FAPEAL and the CNRS.

Supporting Information

Here, the blank showed small peaks for guanosine and adenosine, suggesting that the bases were not completely paired. Thus, the blank was subtracted from both DPVs for visual clarity.


B R Barrett

B G Giraud

B K Jennings

N P Toberg

Concavity for nuclear bindings, thermodynamical functions and density functionals

Keywords: 21.10.Dr, 21.10.-k, 21.60.-n, 24.10.Pa

Sequences of experimental ground-state energies for both odd and even A are mapped onto concave patterns cured from convexities due to pairing and/or shell effects. The same patterns, completed by a list of excitation energies, give numerical estimates of the grand potential Ω(β, µ) for a mixture of nuclei at low or moderate temperatures T = β -1 and at many chemical potentials µ. The average nucleon number A (β, µ) then becomes a continuous variable, allowing extrapolations towards nuclear masses closer to drip lines. We study the possible concavity of several thermodynamical functions, such as the free energy and the average energy, as functions of A . Concavity, which always occurs for the free energy and is usually present for the average energy, allows easy interpolations and extrapolations providing upper and lower bounds, respectively, to binding energies. Such bounds define an error bar for the prediction of binding energies. Finally we show how concavity and universality are related in the theory of the nuclear density functional.

Introduction

The observation of a valley of stability and the search for mass formulae belong to the oldest subjects studied in nuclear physics [1] and continue to be of great interest today [2]. Given the neutron and proton numbers N and Z as independent variables and the corresponding atomic number, A ≡ N + Z, terms such as volume energy ∝ A, surface tension ∝ A 2 3 , Coulomb energy ∝ Z(Z -1)/A 1/3 , symmetry energy ∝ (N -Z) 2 /A, etc., flourish in the literature, and a great deal of attention has been dedicated to the consideration of finer corrections, such as, for instance, terms s(N, Z) and p(N, Z) that account for shell and pairing effects, respectively, and further correlations. This work is motivated by the observation that the dominant terms, such as ∝ A, ∝ Z(Z -1) and ∝ (N -Z) 2 , define a notoriously concave energy surface.

Upper and lower bounds to nuclear binding energies can be deduced from such a concavity, provided that deviations from concavity, possibly induced by subdominant terms like ∝ A 2 3 , s(N, Z), p(N, Z), etc., can be corrected. For the sake of simplicity this paper first considers only sequences of isotopes and, thus, takes advantage of concavity with respect to N only; Z is frozen. In Section 2 we begin with a theory at zero temperature and show how elementary, invertible transformations of data can generate truly concave patterns. This is obtained by an analysis of the table of second differences between binding energies, then by a removal of the pairing energy, and finally by an ad hoc, but minimal, parabolic term added to the nuclear energies, if necessary.

Since concavity is also a property of several thermodynamical functions, an extension of the zero temperature analysis to finite temperatures is in order. In Section 3 we discuss properties of that grand potential, Ω(β, µ), which can be deduced from the experimental data after their tuning. Other thermodynamical functions are also considered, and their concavity is tested. Bounds are found, and an error bar for predictions is estimated.

A generalization to concavity with respect to both N and Z is briefly studied in Section 4. An additional motivation for our investigation of concavity properties is the need, in density functional theories, of concavity, if the universality [3] of a density functional must be obtained. A solution will be shown.

Finally, a discussion and conclusion are given in Section 5.

Concavity for experimental ground-state energies

Our argument is best illustrated numerically, by using a sequence of isotopic ground-state binding energies, -E A . We choose to work with the tin isotopes (Sn), because they provide a large number of known isotopes for testing our method. In addition, in Section 3, we will extend our approach to finite temperatures using a grand potential, which requires that a sufficiently large number of excited states are also known. For instance, a Despite a standard linear trend because of a "not too much fluctuating average energy per nucleon", this list is not making a smooth pattern, even less of a concave one. The full line in Fig. 1, where, for graphical convenience, we have added to each E A a constant term, 115 MeV and a linear term, 7.5 × A MeV, shows the amount of irregularity in the pattern. The main source of irregularity is, obviously, the pairing effect. If it can be removed, concavity emerges. Concavity for a sequence of isotones is not a surprise, because of the Z(Z -1) nature of the Coulomb term. For a sequence of isotopes, however, there is no such obviously quadratic term available, notwithstanding the empirical modelization of a symmetry energy ∝ (N -Z) 2 . Our choice of Sn, because of its long isotope sequence, illustrates this concavity more dramatically. Most energies stated in this paper are in units of keV. Our data come from Ref. [4]. See also [5,6,[START_REF] Ekström | Nuclear Structure and Decay Data[END_REF]. These sources usually quote the binding energy per nucleon instead of the total binding energy itself and such values per nucleon are given to varying numbers of significant figures, from six to seven for the Sn isotopes [4] that we consider. Consequently, we quote and use all our binding energies to six or seven significant figures. Our values are generally accurate to the order of two to three keV near the center of the sequence, and tens of keV at both ends for the Sn isotopes. See, in particular, [4] for error bars. The staggering effect is clear from the 24 first differences, E A+1 1450}, is insensitive to the constant and linear terms we used for graphical convenience. It gives estimates of the "curvatures" of the pattern. It turns out to be far from containing only positive numbers. The nonconnected points shown in Fig. 3 represent this pattern of SDs. A systematic oscillation, reflecting the staggering effect, is found. Alternating signs are obviously due to the gains of binding for even Sn nuclei because of pairing. The oscillation between SDs centered at odd and even nuclei has, roughly speaking, a constant amplitude. Notice, however, the maximum in the list, 4842, due to the shell closure at 132 Sn. Add now to each even nucleus energy a fixed number, for example p(N, Z) = 1250 keV, to tentatively suppress the increase of binding due to pairing. The resulting list of SDs is attenuated by an amount equal to ±2p, as illustrated by the full line in Fig. 3. The attenuated list reads, {-114, 544, -57, 254, 482, 121, 116, 343, -124, 438, -143, 367, -42, 255, 40, 143, 144, 75, 214, -130, 436, 2342, 1050}. All numbers are now significantly smaller than their partners in the previous list of SDs, except for the smaller, but still large maximum at 132 Sn. This maximum is positive, however, and causes no difficulty for concavity. The interesting point is rather the most negative number in the list, namely -143 keV. All negative curvatures can be converted into positive ones if we add to every energy an artificial, parabolic correction, P × (A -122) 2 , with P = 75 keV. Incidentally, the lowest point, A 0 = 122, of the "added parabola" is arbitrary, because SDs will increase by just a constant, namely twice the coefficient P of the A 2 term. After this 2P = 150 keV shift, the whole sequence of SDs becomes positive, {36, 694, 93,404,632,271,266,493,26,588,[START_REF] Ekström | Nuclear Structure and Decay Data[END_REF]517,108,405,190,293,294,225,364,20,586,2492, 1200}, see the dashed line shown in Fig. 3.

In short, a "concavity ensuring" manipulation for the isotope energies consists in replacing each energy E A by E ′ A = E A + p × M od[A + 1, 2] + P × (A -A 0 ) 2 + 115000 + 7500 A. (We recall that the terms which we have added are here just for graphical, and later, numerical convenience; they do not influence the theory.) With p = 1250, P = 75 and A 0 = 122, the list of such tuned energies E ′ A (shown by the nonconnected dots in Fig. 1) reads, {17479, 13831, 10219, 7301, 4476, 2055, 266, -1252, -2504, -3263, -3996, -4141, -4279, -3900, -3413, -2521, -1439, -64, 1605, 3499, 5757, 8035, 10899, 16255, 22811}. The choice of the two parameters, p=1250 keV and P=75 keV, is empirical: one looks for a pairing correction leading to a modest, if not minimal, parabolic correction inducing concavity. Other choices for {p, P } are possible, but, obviously, within a small range around 1250 and 75. Furthermore, such parameters must be readjusted for different regions of the table of nuclei, but it is again obvious that readjustments will be moderate; for instance, the order of magnitude for pairing will always be around ∼ 1.0 to ∼ 1.5 MeV. Analyzing short sequences obviously leaves fewer negative SDs to be compensated by the artificial, parabolic term, and, hence should induce smaller values of P. For this reason, we would expect short sequences to give often better extrapolations.

Once concavity is obtained, it is straightforward that extrapolations from two points on the concave pattern allow predictions of lower bounds to nuclear energies and interpolations give upper bounds. Then, from such bounds for energies E ′ , one recovers bounds, of strictly the same quality, for the physical energies E. This obtains by subtracting from each E ′ bound its "tuning term".

Assume now that 110 Sn were unknown and one had done a brute force extrapolation, 2E 111 -E 112 = -931957. Compared with E 110 = -934571, this prediction underbinds 110 Sn by 2614 keV. Consider rather 2E ′ 111 -E ′ 112 = 17443, to be compared with E ′ 110 = 17479. A slight overbinding, by 36 keV, is found. Although this small error is likely accidental, it is clear that systematic lower bounds will be found. Naturally, once a value for E ′

A is predicted, one recovers as good an estimate for E A after removing the "concavity manipulation terms", which are known explicitly. In the present case, the tuning of the data added 952050 to E 110 . The same 952050 must be subtracted from that value, 2E ′

111 -E ′ 112 = 17443, extrapolated from the concave pattern, yielding a final result of -934607, to be compared with E 110 = -934571, which obviously exhibits the same slight overbinding, 36 keV. Based on the data in [4], the experimentally measured energy for 110 Sn has an error bar ±14 keV; hence, in this case, the experimental error bar and the uncertainty in our theoretical value have the same order of magnitude.

If and(E ′ 116 + E ′ 118 )/2 -994375 = -995494, with now overbinding of 1192 and underbinding of 133, respectively. The poor results from raw data stress the importance of pairing corrections. The much better results from the concave pattern do take this pairing into account, but, furthermore, they again make it trivial that extrapolations give overbindings, while interpolations give underbindings.

For the sake of completeness, we now consider a case with a two-step extrapolation. If the tuned pattern followed a perfect parabolic form,

E ′ A = a A 2 + b A + c, the formula, E ′ A+2 -3 E ′ A + 2 E ′ A-1 = 6
a, predicts, for the lower bound, an error which is a little larger than desirable. This is because of the coefficient, 6, and the fact that the "curvature parameter", a, has here an order of magnitude in the few hundred keV range, see those residual second differences used for the dashed curve in Fig. 3. This indicates that further stages of the theory, with polynoms at least, are worthy of consideration. However, to remain within the "linear" scope of the present paper, we give the results obtained for 135 Sn, -1113208, and 136 Sn, -1115027, when we use the lower bound estimates from the formula, 3

E ′ A -2 E ′ A-1 . From Ref. [4]
, the predictions, which are based on neighbor nuclei with the same Z and N parities, are -1111320 and -1115064, respectively, with error bars of ∼ ±.5 MeV. We do find a lower bound for 135 Sn, with apparently a severe, but mostly expected, error of a couple of MeVs. For 136 Sn, we apparently fail to generate a lower bound, but by so small a difference that our results stay well inside the error bars.

To summarize Section 2, when concavity occurs, one concludes that extrapolations from two points on the concave pattern allow predictions of lower bounds to nuclear energies. In the same way, interpolations provide upper bounds. The quality of such "concavity bounds" from zero temperature data is found to be good; interpolations and extrapolations from the raw, nonconcave pattern, are clearly less satisfactory. But there is a more profound reason why a concave pattern is necessary. Indeed, several thermodynamical functions, governed by theorems proving their concavity, have a notoriously singular limit at zero temperature: they become nonanalytical and are just piecewise continuous. Their limit plots are made of segments; derivatives are discontinuous at turning points. Because of the staggering effect, the concave envelope of the raw pattern of E A 's would contain only the even isotopes. Concavity is, thus, necessary for a theory that must accommodate both odd and even nuclei. Therefore, the thermodynamical functions studied in Section 3 preferably use concave energies E ′ A and the corresponding excited state energies E ′ nA .

Concavity with thermodynamical functions

Consider the particle number operator A and a familiar nuclear Hamiltonian H = A i=1 t i + A i>j=1 v ij , where A, t and v are the mass number, one-body kinetic energy and two-body interaction, respectively. Nuclear data tables [START_REF]Table of Isotopes[END_REF]9] give precise values for a large number of lowest-lying eigenvalues E nA of H, for many nuclei. One may, thus, reasonably estimate the grand partition function,

Z(µ, β) = Tr exp [β (µA -H)] = nA (2j nA + 1) exp [β (µA -E nA )] , (1)
provided that i) the temperature, T = β -1 , is low enough to allow a truncation of the spectrum to include only those states provided by the tables and ii) the chemical potential, µ, selects mainly those nuclei in which we are interested. Let denote, as usual, a statistical average. The (equilibrium!) density operator in Fock space, ρ = Z -1 exp [β (µA -H)] , ensures that the following grand potential, Ω(µ, β) = (H -µA) -T S, is minimal in the space of many-body density matrices with unit trace, since, by definition, A = Tr ρ A, H = Tr ρ H, with the entropy, S = -Tr (ρ log ρ).

We shall rather use (-Ω) in the following, to make upcoming proofs of concavity slightly easier. This grand potential also reads,

-Ω(µ, β) = β -1 ln Z = β -1 ln nA (2j nA + 1) exp [β (µA -E nA )] , (2)
Simple manipulations then give the relevant statistical averages of particle numbers and energies, together with their derivatives and fluctuations,

∂(-Ω)/∂µ = A = A A p A , p A = Z -1 n (2j nA + 1) exp [β (µA -E nA )] , (3) and ∂ 2 (-Ω)/∂µ 2 = ∂ A /∂µ = β A 2 -A 2 , ( 4
) then ∂(-Ω)/∂T = S = ln Z -β (µA -H) , (5)
or as well,

H = Z -1 nA (2j nA + 1)E nA exp [β (µA -E nA )] . (6)
Furthermore,

∂ 2 (-Ω)/∂T 2 = β 3 (µA -H) 2 -(µA -H) 2 , (7)
and

∂ 2 (-Ω)/(∂µ ∂T ) = -β 2 [ A (µA -H) -A (µA -H) ] . (8)
For our investigations, we will freeze β as real and consider functions of a real µ. It is then well known that (-Ω) is a concave function of µ and T. In turn, the double Legendre transform, with respect to both µ and T,

µ ∂(-Ω)/∂µ + T ∂(-Ω)/∂T + Ω = H , (9)
shows that H is a concave function of both A and S, the conjugate variables of µ and T, respectively.

In the following, we do not perform the full, double Legendre transform. We rather retain an intermediate representation, with A and either T or β. We stay with real variables and functions. We stress that, while A has a discrete spectrum, A is continuous, a monotonically increasing function of µ, smooth provided β is finite. The monotonicity results from Eq. (4). Actually, at low temperatures, strong derivatives signal the onset of discrete jumps due to the integer spectrum of A, but we may stay away from this "jumpy" regime in the following, at least temporarily. Anyhow, at any fixed, finite β, the smoothness and monotonicity of A with respect to µ allows a reasonably easy numerical calculation of the inverse function µ ( A ) . Thus, a main argument of this Section is that at fixed temperatures we will use A as a continuous variable and attempt extrapolations towards unknown nuclei.

For this, given a value of T, we keep track of A and H as functions of µ. Since the functional inversion from A (µ) to µ( A ) is reasonably easy, we can plot H in terms of A and attempt an extrapolation for further values of A . This extrapolation can be considered as a "candidate" for a mass formula, at that finite temperature T.

According to Eq. ( 7), the average, constrained energy, (H -µA) , is a monotonically decreasing function of β. Furthermore, at least for negative chemical potentials µ, and, more generally, if A has an upper bound, the operator, H-µA, is bounded from below. Therefore, there is a convergence of the process consisting in i) extrapolating with respect to µ both (H -µA) and A for fixed values of β, then ii) eliminating µ to generate the β-parametrized "mass formula" H ( A , β) , and finally iii) considering the limit of this mass formula when β → +∞. Alternately, it is equivalent, and maybe more efficient, to first eliminate µ and then extrapolate the "mass formula" H ( A , β) , first with respect to A , then with respect to β.

Is there concavity in this intermediate representation? Clearly, a simple Legendre transform of (-Ω), with respect to µ only, returns a free energy, F = H -T S, as a concave function of A and T. If T is low enough to allow the product T S to be neglected, then, at fixed T, one may accept that H is an "almost" concave function of A . This assumption will be tested by the numerical results which follow. Incidentally, a straightforward calculation of

A ′′ ( A , β) ≡ ∂ 2 H / (∂ A ) 2 yields, A ′′ ∝ A 2 -A 2 A 2 H + A A 2 -A 3 AH + A A 3 -A 2 2 H , with a positive factor, A 2 -A 2 -3 . More simply, ∂ 2 H ∂ A 2 ( A , β) ∝ (∆A) 2 (∆A) 2 ∆H -(∆A) 3 ∆A ∆H , (10)
if one uses the centered operators, ∆A = A -A and ∆H = H -H . From Eq. ( 10), concavity for H is unclear; we shall have to test it numerically. The results, which follow, use [START_REF]Table of Isotopes[END_REF]9] for the excited-state energies of the Sn isotopes. Using the first 10 levels of 110 Sn to 132 Sn, hence a maximum excitation of, typically, ∼ 1.2 MeV for odd isotopes and twice as much for even ones, we calculate Z, see Eq. (1). For those rare cases, where the spin j nA is ambiguous, we choose the lowest of the suggested spins. If the spin is completely unknown, we set it to be either 0 or 1/2, according to A. These tactics minimize the statistical influence of such rare cases. The highest excited levels that we use have an excitation energy of order a few MeV; hence, we restrict our thermodynamical analysis to temperatures of order ∼ 500 keV at most. It should be noted that we use the concavity tuned energies, E ′ nA = E nA + 115000 + 7500 A + 1250 M od[A + 1, 2] + 75 (A -122) 2 . Consequently, all our calculated quantities, A ′ , S ′ , H ′ , etc., derived from Eqs. (3), ( 5), and ( 6), for instance, carry a prime superscript. We want to stress that the spurious exponential factor created by the constant, 115000, appears in the same way in numerators and denominators and, hence, cancels out; it only helps the graphics and numerics. Similarly, the linear term, 7500 A, means but a change of reference for the chemical potential. Only the pairing and parabolic tuning terms make the primed quantities truly distinct from those "raw", unprimed ones, obtained with the untuned energies E nA .

As an example of our results we choose for Fig. 5 the sequence of six isotopes from 119 Sn to 124 Sn, because of its still severely contrasted list of second differences, {588, 7, 517, 108}, after tuning. This produces a serious geometrical constraint on the plots of H ′ and F ′ , as they reach their zero temperature limit, and better validates our approach. The upper curves in Fig. 5 are the plots of the function H ′ ( A ′ ) when T = 60 keV (lower full curve) and T = 150 keV (upper full curve), respectively. The increase of H ′ , when T increases is transparent. The striking result is the apparent concavity of both curves. It is also found that the lower temperature, 60 keV, is low enough to allow H ′ to run, in practice, almost through the ground-state energies of both even and odd nuclei. As a check of our results, we repeated our calculations using the untuned energies, i.e., those which lack concavity. We then found that the low temperature limit of the H curve went through the even nuclei only, namely, the limit curve follows the concave envelope of the experimental pattern.

For graphical and pedagogical convenience, Fig. 5 shows the plots for six nuclei only, but the same observations hold for full plots, with 110 ≤ A ′ ≤ 134. As a test, we also calculated H ′ and its low temperature limit when the levels of 132 Sn are omitted from the trace sum, Eq. ( 1). In that case, the concave envelope goes through the dots representing the two odd nuclei, 131 Sn and 133 Sn, and ignores the dot representing 132 Sn. Similar verifications of other concave envelopes were obtained by removing other nuclei.

To verify whether concave envelopes and concavity for H ′ result from negligible values of the entropy term in the free energy, or, more precisely, negligible values of its second derivative, a calculation of T S ′ is in order. Figure 6 shows the difference, T S ′ , between the energy and the free energy, as a function of A ′ , for T = 60 and 150 keV, respectively. At the higher temperature, 150 keV, wiggling effects seem to be small enough to allow for the concavity of H ′ . At the lower temperature, 60 keV, the wiggling is stronger. As a consequence of such a strong wiggling, possibly translating into strong second derivatives, it is not excluded that SDs coming from T S ′ might prevent H ′ from having the proven concavity property of the free energy. We must, therefore, numerically calculate second derivatives, see Eq. [START_REF] Eisenberg | Nuclear Models[END_REF]. In Fig. 7 we show, for tuned data, plots of the rhs of Eq. ( 10), divided by A 2 ′ -A ′2 3 . This quantity must remain positive if H ′ shows concavity. The full line represents the situation when T = 60 keV, the dashed line corresponds to T = 150 keV. We see that the second derivative, ∂ 2 H ′ /∂ A ′2 , remains positive almost always. For low temperatures, however, negative values may appear. For instance, the full curve, corresponding to T = 60 keV, indicates small, but definitely negative values around 119 Sn. Numerical tests, which are not easy because of a difference effect between the two terms present in Eq. ( 10), show that the occurrence of such negative, actually moderate, values for lower temperatures might be somewhat frequent, while not systematic. Furthermore, such "negativity accidents" turn out to be worse when we use untuned data, maybe because the untuned data lack concavity in the first place. A likely reason for the negativity accidents with tuned data might be that the fluctuation of A is not large enough to justify our use of A ′ as a continuous variable. Since it interpolates between integers, a fluctuation of order ∼ 1, or at least ∼ .5, might be necessary. As shown by the plots in Fig. 8, corresponding to T = 60 keV and 150 keV for the lower and upper curves, respectively, a minimum temperature of order ∼ 150 keV is needed to avoid too small a fluctuation of the particle number. Indeed, for, e.g. ∼ 100 keV, we found that the fluctuation is smaller than .5 for almost half of the calculated values of A ′ .

At this stage, the situation can be summarized as follows. On the one hand, the tuned pattern of experimental energies shows concavity, but the concavity of H ′ as a function of A ′ is not sure, although it seems to occur most of the time. On the other hand, we have a theorem proving concavity for the free energy, either

F ≡ H -T S = µ A +Ω or F ′ ≡ H ′ -T S ′ = µ A ′ +Ω ′ ,
as functions of T and A or A ′ , respectively. For instance, elementary derivations show that, in that representation where A (or A ′ ) and β are the primary variables,

∂F ′ ∂β = β -2 S ′ , ∂F ′ ∂ A ′ = µ and ∂ 2 F ′ ∂ A ′2 = T A 2 ′ -A ′2 . ( 11
)
To summarize this discussion, we see that the removal of the entropy term, leading from the free energy to just the energy, can sometimes destroy the concavity depending upon the temperature, but only weakly, e.g., see Fig. 7. A compromise may be found so that T is low enough to make T S ′ small with respect to F ′ and high enough to ensure both sufficient values of ∆A a tolerable oscillating of T S ′ . Our rationale will be that H ′ , even though it might deviate from concavity, will stay close enough to the concave F ′ . Their difference, T S ′ , a positive quantity, will define an error bar between a lower bound F ′ and an upper bound H ′ for ground-state tuned energies. Consider again Fig. 5. There is no need to stress how, given T, the energy and free energy curves make a band defining upper and lower bounds for the experimental energies. Moreover, a similar, but narrower band is obtained if T decreases. Figs. 5,6 and 8 suggests T ∼ 150 keV for an error bar ∼ 500 keV between energy and free energy, a smooth enough entropy energy and a sufficient fluctuation of the particle number. This order of magnitude for T is compatible with the average level splitting, of order ∼ 250 keV, that is observed from the first 10 levels of 110 Sn, ... , 132 Sn used in our calculations.

The following properties, i) the average energy and the free energy are increasing and decreasing, respectively, functions of increasing T, ii) the energy is larger than the free energy and iii) the entropy term by which they differ vanishes when T vanishes, are not big surprises. It can be concluded that, in so far as thermodynamical functions can be calculated at low enough temperatures, precise "accuracy bands" may be available. Their, hopefully analytic, continuation for higher and/or lower values of A than those available in nuclear tables provides a prediction scheme for exotic nuclei.

An estimate of the entropy term is now useful. Given µ and a large β, let A 0 and E ′ 0 correspond to that nucleus whose ground-state energy maximizes the exponential, exp[β (µA-E ′ 0A )]. Consider now the first subdominant exponential. It might be generated by the first excited state of the same nucleus, or by the ground-state of one of its neighbors. Let A 1 and E ′ 1 be its parameters and define

∆ = µ (A 1 -A 0 ) -(E ′ 1 -E ′ 0 )
. Concavity guarantees that ∆ < 0. Whenever β is large enough, it is trivial to reduce the grand canonical ensemble to a two state ensemble, and the entropy then boils down to s = -e β ∆ β ∆. Hence, the product, T s = -e β ∆ ∆, vanishes exponentially fast when β → ∞. The rate of decrease is governed by that scale defined by ∆, to be extracted from the tuned data. Then one can estimate an order of magnitude for the difference between the free energy and the energy. This estimate can be viewed as an error bar for the prediction of exotic nuclei via the present "concavity method".

We have applied the approach to several other isotopic regions, such as Sm, Hg, Pb. In all cases, making the switch to the concave shape improves our ability to make extrapolations and interpolations for unknown bindings. However, estimating the free energy and the average value of H, to set narrow upper and lower bounds on these binding energies, remains more difficult in drip line regions, because of the lack of experimental data on excited states in such regions. Very simple linear extrapolations still provide, nonetheless, error bars with an order of magnitude ∆E ∼ ±400 keV, similar to the best "band" seen in Fig. 5. A few examples of the kind of results that we can obtain from this thermodynamical error-bar analysis, when using the method described in Sec. 2, are these conservative estimates of the bindings of 171 Hg, 208 Hg and 209 Hg, namely 1314.33 MeV ±400 keV, 1629.13 MeV ±400 keV and 1632.16 MeV ±400 keV, respectively.

Two-dimensional analysis; density functional

We now want to consider binding-energy systematics in two dimensions, i.e., N and Z. The second differences studied in this Section are,

SD n = E N +1,Z + E N -1,Z -2 E N Z , SD p = E N,Z+1 + E N,Z-1 -2 E N Z , SD b = (E N +1,Z+1 + E N -1,Z-1 )/2 -E N Z and SD s = (E N +1,Z-1 + E N -1,Z+1 )/2 -E N Z .
If N and Z were continuous, and, furthermore, if E N Z were a smooth function of N and Z, the numbers, SD n , SD p , SD b and SD s , might be interpreted as estimates of second derivatives of the function E(N, Z) in the neutron, proton, constant N -Z and constant A directions, respectively. Their connections with neutron, proton and p-n pairings and with a symmetry energy are also transparent.

Setting aside what we consider to be "light" nuclei, namely those of the s-, p-, sd-and f 7/2 shells, we calculated these four SDs for the rest of the known nuclear table. Their maxima and minima are found to be of order ∼ {4840, 6180, 4530, 5900, -3670, -3080, -3160, -2150} keV, respectively. The lhs parts of Figs. 9-12 show, in scattered plots with respect to A, the patterns of those SD values. The negative SDs, contradicting concavity, are due mainly to pairing effects. An extension of the correction done in Sec. 2 is in order. Let p be a mass dependent [START_REF] Eisenberg | Nuclear Models[END_REF] estimate of pairing. Diminish the binding of doubly even nuclei by p, increase the binding of doubly odd ones by p, and leave odd nuclei untouched. We set, empirically, p = 1160 -A . There is no doubt that better parametrizations of p are possible [START_REF] Eisenberg | Nuclear Models[END_REF], but the present linear decrease suffices for our demonstration. With this correction, the worst SDs become ∼ {2790, 3970, 2470, 3680, -1500, -930, -980, -120} keV, respectively. As expected, this is a large reduction of both positive and negative amplitudes. This is shown by the rhs parts of Figs. 9-12, respectively. While the pairing correction has not completely suppressed negative SDs, note the worst minimum is still of order ∼ -1500 keV, concavity has been improved significantly and extrapolations are likely made more secure. Similar to our one-dimensional analysis in Sec. 2, one can now add to the energies a paraboloid term, of the form, u (N -N 0 ) 2 + 2v (N -N 0 ) (Z -Z 0 ) + w (Z -Z 0 ) 2 , in order to get rid of the residual, negative SDs. (We recall that SDs do not depend on N 0 and Z 0 , which are only parameters for graphical and/or numerical convenience.) Notice from Fig. 12, and the corresponding worse minimum, of order ∼ -120 keV, that the direction where A is constant does not deviate much from concavity. The dominant term in the paraboloid must, therefore, indicate a direction approximately orthogonal, and also take into account the bend of the global stability valley towards neutron excess. There is, furthermore, no doubt that the two-dimensional free energy, now a function of both N and Z made continuous by a grand canonical ensemble, remains a concave function, available for lower bounds. This is a theorem, which can be proven in a straightforward manner following the previous derivation for one dimension. Simultaneously, it is likely that the mean energy will be again "almost concave", to yield upper bounds, in which case the same error bar that was estimated in Sec. 3 remains valid.

A systematic use of such properties throughout the nuclear table is way beyond the scope of the present paper. Rather, we want to discuss the relation of concavity with the derivation of a nuclear density functional. Recall that, given an arbitrary many-body density matrix M in Fock space, the density functional derives from the definition [11],

F [ρ p , ρ n ] = Inf M→ρp,ρn Tr H M. ( 12
)
Here the symbol, M → ρ n , ρ p , means that a minimization of the energy is performed upon density matrices having the same neutron and proton profiles ρ n , ρ p . A nucleus is identified by the integrals, N = d r ρ n ( r) and Z = d r ρ p ( r), and a later minimization is performed with respect to the profiles ρ n , ρ p under such identification constraints,

E ZN = Inf ρn→N, ρp→Z F [ρ n , ρ p ]. (13)
The functional F should be universal, in the nuclear physics sense. Namely, it should not depend on N and Z. (In atomic and molecular physics, universality of the DF has a different meaning; it refers to arbitrary external potentials.) While BCS and Hartree-Bogoliubov calculations allow a distinction between even and odd particle numbers, universality in density profile space does not allow such a separate treatment. Observe, furthermore, that, because of the fact that there are many convexities in the pattern of raw energies, there exist many cases where, for instance, three nuclei,

A 1 , A 2 , A 3 , are such that N 2 = (N 1 +N 3 )/2 and Z 2 = (Z 1 + Z 3 )/2 and (E 1 + E 3 )/2 < E 2 .
Then, in a search for E 2 , the mixture density matrix,

M mix = ( |A 1 A 1 | + |A 3 A 3 |) /2
, provides the correct average particle numbers N 2 , Z 2 , but an absurd energy, lower than E 2 . Concavity is a mandatory condition for the universality of F . One way to simultaneously implement pairing corrections and use a minimal paraboloid term consists in adding to the physical H terms such as, p × [cos 2 (π N/2) + cos 2 (π Z/2) -1] + u N 2 + 2v N Z + w Z 2 . (Notice how the cosines take advantage of the continuous nature of N and Z at finite temperatures.) Concavity is achieved if u, v, w are tuned to compensate for the worst tuned SD minima listed above, namely {-1500, -930, -980, -120}. A rough solution consists in taking p ∼ 1100 and a "circular" ansatz, P × N 2 + Z 2 , with P ∼ 800, but less brutal solutions are obviously worthy of consideration, in order to minimize this tampering with the Hamiltonian. In any case, we stress that the operators, N and Z, commute with H, and, therefore, that the present counterterms do not perturb nuclear dynamics. Once again we emphasize that this approach unifies the treatment of odd and even nuclei. The fact that the nuclear density functional needs to originate from a Hamiltonian, completed by such counterterms, illustrates how important concavity is for making predictions for nuclei far from stability, i.e., for the so-called exotic nuclei.

Summary, discussion and conclusion

We have demonstrated how a list of ground-state energies for a sequence of isotopes can be turned into a concave pattern. This involves simple manipulations; for instance, an explicit term, accounting for pairing in even nuclei, can be subtracted from the bindings. This unifies the treatment of odd and even nuclei, a notoriously difficult problem. If needed, a small quadratic correction can also be added to guarantee concavity at all points along the sequence.

Similar arguments leading to concavity clearly hold for isotones as well, and, furthermore, for any other sequence of neighboring nuclei in any direction across the nuclear table. Once this empirical tuning has been implemented, linear (or more general) extra-and interpolations of the concave pattern can provide surprisingly accurate and robust estimates of, or bounds for, binding energies. These tuning terms, which are added to induce concavity, are, of course, subtracted in fine.

This work then defined a more ambitious extra-and interpolation scheme, involving thermodynamical functions from a grand canonical ensemble, because such functions may have rigorous concavity properties. Theorems are, indeed, available to prove such properties. For instance, the free energy is a concave function of the average particle number and is also a decreasing function of the temperature. We also found strong numerical evidences concerning the average energy. This average energy at nonzero temperature turns out, in general, to be a concave function of the average particle number, except for "minor accidents," which depend upon the temperature.

For every given, finite temperature, we found that the average energy and the free energy, as functions of the average particle number, give upper and lower bounds, respectively, for the concave envelope of the ground-state energies. When the temperature vanishes, both bounds converge to the exact results. At this vanishing temperature, however, the analyticity of such thermodynamical functions is lost, because their limit is only piecewise continuous. It is, therefore, necessary to retain a minimum temperature if one wants to obtain practical extrapolations for the prediction of exotic nuclei. A minimum amount of particle number fluctuation is necessary to validate the conversion of particle number, an initially discrete quantity, into a continuous variable.

We, therefore, implemented numerical estimates of several thermodynamical functions at moderate temperatures, a few hundred keV at most. This yields a first result, namely a "band", enclosing ground-state energies between the average energy and the free energy. The width of the band defines an error bar which can be trusted when extrapolations are made.

A difficulty arises, however, because of an insufficient number of excited states for nuclei at both ends of any sequence of isotopes. These excited-state energies are essential for calculating a meaningful value of the average energy for any value of A. More often, only the ground-state energy is known for such neutron very rich or neutron very poor nuclei. The calculation of the average and free energies is, thus, possible only in an interval smaller than the interval of masses where ground states are known. Two tactics are then available, namely i) an extrapolation only of the sequence of ground-state energies and ii) an extrapolation of the thermodynamical functions, starting from a smaller interval. We found that the first tactic, namely at zero temperature, is reliable, especially when completed by the error bar derived from the finite temperature considerations. This work also tested the second tactic, with some success but with the limitation due to the lack of known excited states. The lack leads to edge effects in the calculation of thermodynamical functions. Therefore, if linear, or polynomial, or more general analytical fits of such thermodynamical functions are attempted, it is better to restrict them from a slightly smaller mass interval. Then, extrapolations should be reliable within one and maybe two mass units towards drip lines.

We can make the strong conclusion that the combination of concavity and extrapolations of thermodynamical functions gives a systematic set of upper and lower bounds for the prediction of ground-state energies.

Our last, and perhaps most important result, is the connection between concavity and the universality of the density functional. In a two-dimensional analysis, we showed that a paraboloid term with respect to the proton and neutron number operators must be added to the Hamiltonian, so as to guarantee obtaining consistent energy minima everywhere in density space. This term can be made minimal if counterterms for pairing are also added.

Figure 1 :Figure 2 :

 12 Figure 1: Sn isotope energies: irregular line joining raw experimental groundstate energies E A + 7500 A + 115000; pairing and parabolic corrections give the nonconnected dots.

Figure 3 :

 3 Figure 3: Second differences SD = E A+1 -2E A + E A-1 . Dots, raw data; solid line, result of the pairing correction; dashed line, final result after the parabolic correction.

Figure 5 :

 5 Figure5: Energy H ′ (full curves) and free energy F ′ (dashed ones) as functions of A ′ , calculated from concavity tuned energies E ′ nA . Dots represent tuned, ground-state energies, E ′ 0A ≡ E ′ A , already used in Fig.1and the rhs part of Fig.4. Upper full and lower dashed curves, T = 150 keV. Lower full and upper dashed curves, T = 60 keV. Notice how the full curves turn out to be concave.

Figure 6 :

 6 Figure 6: Entropy energy T S ′ ( A ′ ). Upper (lower) curve, T = 150 (60) keV.

Figure 7 :

 7 Figure 7: (a) Behavior of H ′′ = ∂ 2 H ′ /∂ A ′2 ; full (dashed) curve, T = 60 (150)keV. Notice tiny excursions in negative value territory by the low T curve.

Figure 8 :

 8 Figure 8: Particle number fluctuation. Upper (lower) curve, T = 150 (60) keV.

Figure 9 :Figure 10 :

 910 Figure9: Scatter plot of convex second differences SD n in the neutron direction throughout the table of known nuclei for 27 < A < 251. Left: with bare energies. Right: after a pairing correction.

Figure 11 :Figure 12 :

 1112 Figure 11: Same as Fig. 9, but for SD b in the constant N -Z direction.

  table of 25 ground-state bindings for the tin isotopes from 110 Sn to 134 Sn reads, in keV,

	{ 934571, 942744, 953531, 961274, 971574, 979120, 988684, 995627, 1004954,
	1011438, 1020546, 1026716, 1035529, 1041475, 1049963, 1055696, 1063889,
	1069439, 1077345, 1082676, 1090293, 1095540, 1102851, 1105320, 1109239}.

  -E A + 7500,

	{-673, -3287, -243, -2800, -46, -2064, 557, -1827, 1016, -1608, 1330, -1313, 1554,
	-988, 1767, -693, 1950, -406, 2169, -117, 2253, 189, 5031, 3581}, see Fig. 2.
	The list of 23 second differences (SDs), SD = E A+1 -2E A + E A-1 ,
	{-2614, 3044, -2557, 2754, -2018, 2621, -2384, 2843, -2624, 2938, -2643, 2867,
	-2542, 2755, -2460, 2643, -2356, 2575, -2286, 2370, -2064, 4842, -

  132 Sn were unknown, the brute force extrapolation gives, 2E 131 -E 130 = -1100787, to be compared with E 132 = -1102851, showing an underbinding equal to 2064 keV, clearly failing to reproduce the shell closure effect. Now, from the concave pattern, we obtain 2E ′ 131 -E ′ 130 = 10313, to be compared with E ′ 132 = 10899, producing an overbinding by 586 keV. The same overbinding is obviously found if one subtracts from 2E ′ 131 -E ′ 130 = 10313, the tuning difference, E ′ 132 -E 132 = 1113750, with the result, -1103437, to be compared with E 132 = -1102851. See Fig.4, which also illustrates an extrapolation using E 133 and E 134 and an interpolation using E 131 and E 133 .

		keV
	-1.09 10	6					22500
		6
	-1.0925 10					20000
		6
	-1.095 10
							17500
		130	131	132	133	134	A
							15000
	6
	-1.1 10
		6					12500
	-1.1025 10
	-1.105 10	6					10000
		6
	-1.1075 10					130	131	132	133	134	A

keV Figure 4: Left, raw energies: full line, interpolation from 131 Sn and 133 Sn; long dashes, extrapolation from 130 Sn and 131 Sn; short dashes, extrapolation from 134 Sn and 133 Sn. Dots represent the experimental energies E A . Right: same inter-and extrapolations, concave data.

Let us now also compare for an odd rather than even isotope, e.g.,

117

Sn, two extrapolations and one interpolation from raw data and concave data, respectively. With our choice of parameters, the tuning difference, E ′ 117 -E 117 , is 994375, with E 117 = -995627. Extrapolations from lower masses give, 2E 116 -E 115 = -998248, and 2E ′ 116 -E ′ 115 -994375 = -995898, hence, overbindings of 2671 and 271, respectively. Extrapolations from higher masses give, 2E 118 -E 119 = -998470, and 2E ′ 118 -E ′ 119 -994375 = -996120, leading to overbindings of 2843 and 493, respectively. Interpolations from the nearest nuclei give, (E 116 + E 118 )/2 = -996819,

Acknowledgements: We thank I. Allison for helpful discussions and assistance with the management of the data sets. It is a pleasure for B. R. B. and B. G. G. to thank TRIUMF, Vancouver, B. C., Canada, where part of this work was done, for its hospitality. The Natural Science and Engineering Research Council of Canada is thanked for financial support. TRIUMF receives federal funding via a contribution agreement through the National Research Council of Canada. B. R. B. also thanks Institut de Physique Théorique, Saclay, France, for its hospitality, where another part of this work was carried out, and the Gesellschaft für Schwerionenforschung (GSI), Darmstadt, Germany, for its hospitality during the preparation of this manuscript, and acknowledges partial support from NSF grant PHY0555396 and from the Alexander von Humboldt Stiftung.