High throughput measurement of duplex, triplex and quadruplex melting curves using molecular beacons and a LightCycler

We have used oligonucleotides containing molecular beacons to determine melting profiles for intramolecular DNA duplexes, triplexes and quadruplexes (tetraplexes). The synthetic oligonucleotides used in these studies contain a fluorophore (fluorescein) and quencher (methyl red) attached either to deoxyribose or to the 5 position of dU. In the folded DNA structures the fluorophore and quencher are in close proximity and the fluorescence is quenched. When the structures melt, the fluorophore and quencher are separated and there is a large increase in fluorescence. These experiments were performed in a Roche LightCycler; this requires small amounts of material (typically 4 pmol oligonucleotide) and can perform 32 melting profiles in parallel. We have used this technique to compare the stability of triplexes containing different base analogues and to confirm the selectivity of a triplex-binding ligand for triplex, rather than duplex, DNA. We have also compared the melting of inter- and intramolecular quadruplexes.     

Spreadsheet software for thermodynamic melting point prediction of oligonucleotide hybridization with and without mismatches

The use of thermodynamic parameters for the calculation of oligonucleotide duplex stability provides the best estimates of oligonucleotide melting temperatures (Tm). Such estimates can be used for evidence-based design of molecular biological experiments in which oligonucleotide melting behavior is a critical issue, such as temperature or denaturing gradient gel electrophoreses, Southern blotting or hybridization probe assays on the LightCycler. We have developed a user friendly program for Tm calculation of matched and mismatched probes using the spreadsheet software Microsoft Excel. The most recently published values for entropy and enthalpy of Watson-Crick paris are used, and salt and oligonucleotide concentrations are considered. The 5' and 3' end stability is calculated for the estimation of primer specificity. In addition, the influence of all possible mutations under a given probe can be calculated automatically. The experimental evaluation of predicted Tm with the LightCycler, based on 14 hybridization probes for different gene loci, showed an excellent fit between measured results and values predicted with the thermodynamic model in 14 matched, 25 single mismatched and 8 two-point mismatched assays (r = 0.98; Sy. x = 0.90; y = 1.01 x -0.38). This program is extremely useful for the design of oligonucleotide probes because the use of probes that do not discriminate with a reasonable Tm difference between wild-type and mutation can be avoided in advance.     

Electrostatics dominate quadruplex stability

Stabilization of nucleic acid structures results from a balance of multiple interactions, including electrostatics, base stacking, hydrophobic interactions, hydrogen bonding, van der Waals forces, etc. Nucleic acid quadruplexes are unusual structures in that their formation is driven by specific binding of metal ions. This unique mode of metal binding, which is tightly coupled to oligonucleotide folding, can engender correspondingly unique solution behavior. In particular, we show that addition of many cosolvents, such as primary aliphatic alcohols, increases the thermal stability of quadruplexes, as determined by melting temperature, Tm, in direct contrast to the response of duplexes to the same admixture of solvents. Thermal stability is observed to increase as the dielectric constant of the composite solvent decreases. This behavior suggests a dominant role for electrostatics in quadruplex formation and stability. Additional studies done with other cosolvents and solutes suggest that, in some cases, other forces may come into play, including the possibility of direct interaction with the quadruplex structure. Nonetheless, many cosolvents and small molecules, such as ethanol, dimethylformamide, and betaine, stabilize the quadruplex conformation in sharp distinction to their destabilization of DNA duplexes.     

Sequence effects in single-base loops for quadruplexes

Intramolecular G-quadruplexes formed by a single DNA strand have attracted much interest due to the possibility that they may form in telomeres, oncogene promoter sequences and other biologically relevant regions of the genome. Therefore, it is important to understand the rules that govern the formation of these intramolecular structures and to determine their stabilities. We compared here 27 different sequences containing four tracts of three guanines with a medium (3) or relatively long (6-9 bases) central loop and two loops composed of a single nucleotide H (A, T or C) corresponding to the GGGHGGGN3-9GGGHGGG motif. These sequences are similar to sequence motifs that can be found in repeated and promoter sequences. Several conclusions were reached: (i) all sequences are prone to form very stable quadruplexes in potassium (Tm between 55 degrees C and 83 degrees C); (ii) these quadruplexes also form in sodium but with a strongly decreased Tm, with a 21 to 36 degrees C difference in melting temperature between Na+ and K+; (iii) a long (9 bases) central loop had a minimal deleterious impact on the stability of the quadruplex; (iv) pyrimidines are preferred over adenine in both single-base loops; (v) the folding topology is relatively robust in potassium: the CD spectra of all oligonucleotides matched the one of all-parallel stranded reference quadruplexes; (vi) conversely, in sodium the folding is diverse and sequence-dependent, as judged from CD and electrophoresis results.     

Effects of monofunctional platinum binding on the thermal stability and conformation of a self-complementary 22-mer

We investigated the effect of various monofunctional platinum complexes on the thermal stability and conformation of a self-complementary 22-mer duplex oligonucleotide by means of CD and UV melting profiles. We studied several families of triamine complexes of the general formula PtA2AmCl where A2=(NH3)2 and ethylenediamine and where Am=N1-4-methyl-pyridine, N7-guanosine, and 9-ethyl-guanine. Platination by the N1-4-methyl-pyridine and 9-ethyl-guanine complexes led to a decrease in the Tm of the oligonucleotide by 2-11.5 degrees C while platination with the N7-guanosine complexes led to a rise in the melting temperature of the oligonucleotides by 4.5 degrees C. A similar inverse correlation between the two groups of platinum compounds was found in the CD spectra. In all cases, the cis isomer had a more pronounced effect on both the melting curve and the CD spectrum. The cis isomer was found to have a more destabilizing effect than its trans counterpart. This indicates that the cis geometry in fact forces a greater structural constraint on the backbone of the double helix. We have also found that the sugar of the guanosine has a significant influence on both the Tm and CD spectra; the sugar moiety contributes to the stability of the double helix, probably through the formation of hydrogen bonds.     

Sequence dependent effects in methylphosphonate deoxyribonucleotide double and triple helical complexes.

Deoxyribooligonucleotides containing 19 repeating bases of A, T or U were prepared with normal phosphodiester (dA19, dT19, dU19) or methylphosphonate (dA*19, dT*19, dU*19) linkages. Complexes of these strands have been investigated at 1:1 and 1:2 molar ratios (purine:pyrimidine) by thermal melting and gel electrophoresis. There are dramatic sequence dependent differences in stabilities of complexes containing methylphosphonate strands. Duplexes of dA*19 with dT19 or dU19 have sharp melting curves, increased Tm values, and slopes of Tm versus log (sodium ion activity) plots reduced by about one half relative to their unmodified 'parent' duplexes. Duplexes of dA19 with either dT*19 or dU*19, however, have broader melting curves, reduced Tm values at most salt concentrations and slopes of less than one tenth the values for the unmodified duplexes. Duplex stabilization due to reduced phosphate charge repulsion is offset in the pyrimidine methylphosphonate complexes by steric and other substituent effects. Triple helical complexes with dA19 + 2dT19 and dA19 + 2dU19, which can be detected by biphasic melting curves and gel electrophoresis, are stable at increased Na+ or Mg+2 concentrations. Surprisingly, however, no triple helix forms, even at very high salt concentrations, when any normal strand(s) is replaced by a methylphosphonate strand. Since triple helical complexes with methylphosphonates have been reported for shorter oligomers, inhibition with larger oligomers may vary due to their length and extent of substitution.     

5-(1-propargylamino)-2'-deoxyuridine (UP): a novel thymidine analogue for generating DNA triplexes with increased stability

We have used quantitative DNase I footprinting and UV-melting studies to examine the formation of DNA triplexes in which the third strand thymines have been replaced by 5-propargylamino-dU (UP). The intra-molecular triplex A6-L-T6-L-(UP)5T (L = two octanediol residues) shows a single UV-melting transition which is >20 degrees higher than that of the parent triplex A6-L-T6-L-T6at pH 5.5. Although a single transition is observed at all pHs, the melting temperature (Tm) of the modified oligonucleotide decreases at higher pHs, consistent with the requirement for protonation of the amino group. A similar intramolecular triplex with a longer overhanging duplex shows two melting transitions, the lower of which is stabilised by substitution of T by UP, in a pH dependent fashion. Triplex stability increases by approximately 12 K for each T to UP substitution. Quantitative footprinting studies have examined the interaction of three UP-containing 9mer oligonucleotides with the different portions of the 17mer sequence 5'-AGGAAGAGAAAAAAGAA. At pH 5.0, the UP-containing oligo-nucleotides footprint to much lower concentrations than their T-containing counterparts. In particular (UP)6CUPT binds approximately 1000-fold more tightly than the unmodified oligonucleotide T6CTT. Oligonucleotides containing fewer UP residues are stabilised to a lesser extent. The affinity of these modified third strands decreases at higher pHs. These results demonstrate that the stability of DNA triplexes can be dramatically increased by using positively charged analogues of thymine.     

Studies on interaction between poly(L-lysine58, L-phenylalanine42) and deoxyribonucleic acids.

A random copolymer of 58% L-lysine and 42% L-phenylalanine, poly(Lys58Phe42), was used as a model protein for studying the role of phenylalanine residues in protein-DNA interaction. Complexes between this copolypeptide and DNA, made by direct mixing, were studied by absorbance, circular dichroism (CD), fluorescence, and thermal denaturation. Complex formation results in an increase in absorbance, and an enhancement, red-shift, and broadening of phenylalanine fluorescence. The fluorescence enhancement is opposite to the quenching observed when a tyrosine copolypeptide is bound to DNA (R. M. Santella and H.J. Li (1974), Biopolymers 13, 1909). The positive CD band of DNA near 275 nm is reduced and red-shifted by the binding of the phenylalanine copolypeptide to a greater extent than by the tyrosine copolypeptide. Thermal denaturation of the complexes in 2.5 times 10(-4) M EDTA (pH 8.0) shows three characteristic melting bands. For complexes with calf thymus DNA, free base pairs melt at Tm,I (47-49 degrees) and copolypeptide-bound base pairs show two melting bands (Tm,II at 73-75 degrees, and Tm,III at 88 -90 degrees). Similar thermal denaturation results have been observed for complexes with Micrococcus luteus DNA. The fluorecence intensity of the complexes is greatly increased when the temperature is raised to the Tm,II region. In addition to fluorescence measurements, the effects of increasing temperature on absorption and CD spectra of the complexes were also studied. Stacking interaction between the phenylalanine chromophore and DNA bases, either partial or full intercalation, is implicated by the experimental results. Several mechanisms are proposed to describe the reaction between the copolypeptide and DNA, and thermal denaturation of the complex.     

Stable DNA triple helix formation using oligonucleotides containing 2'-aminoethoxy,5-propargylamino-U

We have prepared oligonucleotides containing the novel base analogue 2'-aminoethoxy,5-propargylamino-U in place of thymidine and examined their ability to form intermolecular and intramolecular triple helices by DNase I footprinting and thermal melting studies. The results were compared with those for oligonucleotides containing 5-propargylamino-dU and 2'-aminoethoxy-T. We find that the bis-substituted derivative produces a large increase in triplex stability, much greater than that produced by either of the monosubstituted analogues, which are roughly equipotent with each other. Intermolecular triplexes with 9-mer oligonucleotides containing three or four base modifications generate footprints at submicromolar concentrations even at pH 7.5, in contrast to the unmodified oligonucleotide, which failed to produce a footprint at pH 5.0, even at 30 microM. UV- and fluorescence melting studies with intramolecular triplexes confirmed that the bis-modified base produces a much greater increase in T(m) than either modification alone.     

Quadruplex melting

Melting curves are commonly used to determine the stability of folded nucleic acid structures and their interaction with ligands. This paper describes how the technique can be applied to study the properties of four-stranded nucleic acid structures that are formed by G-rich oligonucleotides. Changes in the absorbance (at 295nm), circular dichroism (at 260 or 295nm) or fluorescence of appropriately labelled oligonucleotides, can be used to measure the stability and kinetics of folding. This paper focuses on a fluorescence melting technique, and explains how this can be used to determine the T(m) (T((1/2))) of intramolecular quadruplexes and the effects of quadruplex-binding ligands. Quantitative analysis of these melting curves can be used to determine the thermodynamic (DeltaH, DeltaG, and DeltaS) and kinetic (k(1), k(-1)) parameters. The method can also be adapted to investigate the equilibrium between quadruplex and duplex DNA and to explore the selectivity of ligands for one or other structure.     

Effects of 1,3-chelation induced by cis-diamminedichloroplatinum(II) on the stability of DNA duplexes.

Several 1,3-intra-strand cross-linked decadeoxynucleotide duplexes, modified with cis-diamminedichloroplatinum(II) (cis-DDP), and their base substitution analogues at the complementary site to the intervening base of the coordination sites, were synthesized and measured for UV-melting profiles to determine melting temperature (Tm) values. The results indicated the thermal stability of the oligonucleotide duplexes containing Pt-induced 1,3-intra-strand cross-linking did not depend on the kind of intervening base of the coordination site but rather on its complementary base. These results may explain the mutagenicity of cis-DDP from a chemical aspect.     

Influence of loop size on the stability of intramolecular DNA quadruplexes

We have determined the stability of intramolecular DNA quadruplexes in which the four G₃-tracts are connected by non-nucleosidic linkers containing propanediol, octanediol or hexaethylene glycol, replacing the TTA loops in the human telomeric repeat sequence. We find that these sequences all fold to form intramolecular complexes, which are stabilized by lithium < sodium < potassium. Quadruplex stability increases in the order propanediol < hexaethylene glycol < octanediol. The shallower shape of the melting profile with propanediol linkers and its lower dependency on potassium concentration suggests that this complex contains fewer stacks of G-quartets. The sequence with octanediol linkers displays a biphasic melting profile, suggesting that it can adopt more than one stable structure. All these complexes display melting temperatures above 310 K in the presence of 10 mM lithium, without added potassium, in contrast to the telomeric repeat sequence. These complexes also fold much faster than the telomeric repeat and there is little or no hysteresis between their melting and annealing profiles. In contrast, the human telomeric repeat sequence and a complex containing two hexaethylene glycol groups in each loop, are less stable and fold more slowly. The melting and annealing profiles for the latter sequence show significant differences, even when heated at 0.2°C min^–1. CD spectra for the oligonucleotides containing non-nucleosidic linkers show positive maxima at 264 nm, with negative minima ~244 nm, which are characteristic of parallel quadruplex structures. These results show that the structure and stability of intramolecular quadruplexes is profoundly influenced by the length and composition of the loops.     

Thermal stability landscape for Klenow DNA polymerase as a function of pH and salt concentration

The thermal denaturation of Klenow DNA polymerase has been characterized over a wide variety of solution conditions to obtain a relative stability landscape for the protein. Measurements were conducted utilizing a miniaturized fluorescence assay that measures Tm based on the increase in the fluorescence of 1,8-anilinonaphthalene sulfonate (ANS) when the protein denatures. The melting temperature (Tm) for Klenow increases as the salt concentration is increased and as the pH is decreased. Klenow's Tm spans a range of over 20 degrees C, from 40 to 62 degrees C, depending upon the solution conditions. The landscape reconciles and extends previously measured Tm values for Klenow. Salt effects on the stability of Klenow show strong cation dependence overlaid onto a more typical Hofmeister anion type dependence. Cationic stabilization of proteins has been far less frequently documented than anionic stabilization. The monovalent cations tested stabilize Klenow with the following hierarchy: NH4+>Na+>Li+>K+. Of the divalent cations tested: Mg+2 and Mn+2 significantly stabilize the protein, while Ni+2 dramatically destabilizes the protein. Stability measurements performed in combined Mg+2 plus Na+ salts suggest that the stabilizing effects of these monovalent and divalent cations are synergistic. The cationic stabilization of Klenow can be well explained by a model postulating dampening of repulsion within surface anionic patches on the protein.     

Intramolecular recombination R-triplex in solution: stabilization by bis-intercalator YOYO

Recognition of double-stranded DNA with a mixed nucleotide sequence by oligonucleotide is a long-term challenge. This aim can be achieved via formation of the recombination R-triplex, accommodating two identical DNA strands in parallel orientation, and antiparallel complementary strand. In the absence of proteins the R-triplex stability is low, however, so that intermolecular R-triplex is not formed by three DNA strands in a ligand-free system. Recently, recognition of DNA with mixed base sequence by single-stranded oligonucleotide in the presence of bis-intercalator YOYO was reported. Here, we describe thermodynamic characteristics of YOYO complexes with the model oligonucleotides 5'-GT-2AP-GACTGAG TTTT CTCAGTCTACGC GAA GCGTAGACTGAG-3' (R(2AP)CW) bearing a single reporting 2-aminopurine (2AP) in place of adenine and 5'-CTCAGTCTACGC GAA GCGTAGACTGAG-3' (CW). We found that each oligonucleotide is able to bind two YOYO molecules via intercalation mode in 0.5 M LiCl. Fluorescence intensity of YOYO intercalated in triplex R(2AP)CW and in CW hairpin increased 40-fold compared to the free YOYO. Remarkably, the melting temperature of the triplex (determined using temperature dependence of the 2AP fluorescence) increased from 19 degrees C to 33 degrees C upon binding two YOYO molecules. Further increase in the YOYO concentration resulted in binding of up to five YOYO molecules to R(2AP)CW triplex and up to six YOYO molecules to CW hairpin.     

Stability of intramolecular DNA quadruplexes: comparison with DNA duplexes

We have determined the stability of intramolecular quadruplexes that are formed by a variety of G-rich sequences, using oligonucleotides containing appropriately placed fluorophores and quenchers. The stability of these quadruplexes is compared with that of the DNA duplexes that are formed on addition of complementary C-rich oligonucleotides. We find that the linkers joining the G-tracts are not essential for folding and can be replaced with nonnucleosidic moieties, though their sequence composition profoundly affects quadruplex stability. Although the human telomere repeat sequence d[G(3)(TTAG(3))(3)] folds into a quadruplex structure, this forms a duplex in the presence of the complementary C-rich strand at physiological conditions. The Tetrahymena sequence d[G(4)(T(2)G(4))(3)], the sequence d[G(3)(T(2)G(3))(3)], and sequences related to regions of the c-myc promoter d(G(4)AG(4)T)(2) and d(G(4)AG(3)T)(2) preferentially adopt the quadruplex form in potassium-containing buffers, even in the presence of a 50-fold excess of their complementary C-rich strands, though the duplex predominates in the presence of sodium. The HIV integrase inhibitor d[G(3)(TG(3))(3)] forms an extremely stable quadruplex which is not affected by addition of a 50-fold excess of the complementary C-rich strand in both potassium- and sodium-containing buffers. Replacing the TTA loops of the human telomeric repeat with AAA causes a large decrease in quadruplex stability, though a sequence with AAA in the first loop and TTT in the second and third loops is slightly more stable.     

Comparison of different melting temperature calculation methods for short DNA sequences

MOTIVATION: The overall performance of several molecular biology techniques involving DNA/DNA hybridization depends on the accurate prediction of the experimental value of a critical parameter: the melting temperature Tm. Till date, many computer software programs based on different methods and/or parameterizations are available for the theoretical estimation of the experimental Tm value of any given short oligonucleotide sequence. However, in most cases, large and significant differences in the estimations of Tm were obtained while using different methods. Thus, it is difficult to decide which Tm value is the accurate one. In addition, it seems that most people who use these methods are unaware about the limitations, which are well described in the literature but not stated properly or restricted the inputs of most of the web servers and standalone software programs that implement them. RESULTS: A quantitative comparison on the similarities and differences among some of the published DNA/DNA Tm calculation methods is reported. The comparison was carried out for a large set of short oligonucleotide sequences ranging from 16 to 30 nt long, which span the whole range of CG-content. The results showed that significant differences were observed in all the methods, which in some cases depend on the oligonucleotide length and CG-content in a non-trivial manner. Based on these results, the regions of consensus and disagreement for the methods in the oligonucleotide feature space were reported. Owing to the lack of sufficient experimental data, a fair and complete assessment of accuracy for the different methods is not yet possible. Inspite of this limitation, a consensus Tm with minimal error probability was calculated by averaging the values obtained from two or more methods that exhibit similar behavior to each particular combination of oligonucleotide length and CG-content class. Using a total of 348 DNA sequences in the size range between 16mer and 30mer, for which the experimental Tm data are available, we demonstrated that the consensus Tm is a robust and accurate measure. It is expected that the results of this work would be constituted as a useful set of guidelines to be followed for the successful experimental implementation of various molecular biology techniques, such as quantitative PCR, multiplex PCR and the design of optimal DNA microarrays.     

Thermal transitions of myosin and its helical fragments. I. Shifts in proton equilibria accompanying unfolding.

The thermal transitions of myosin and its helical fragments have been studied with pH as the observable. Heating unbuffered solutions of these proteins near their pI values causes an abrupt rise in pH at a characteristic temperature (the "melting temperature," Tm) which is due to structural changes within the protein. Since the pH shift turns out to be insensitive to the degree of protein aggregation, we have obtained acceptable melting curves even under conditions where the protein coagulates during melting. The melting profiles and Tm vlaues of myosin, myosin rod, and light meromyosin have been found to be remarkably similar (Tm equal to 40 plus or minus 1 degree, 0.5 M KCl, pH 5.9). Proton binding which occurs during melting coincides with the unfolding of a section of myosin rod. Taken in the context of other studies, the proton binding is thought to occur near the "hinge region."     

van't Hoff enthalpies without baselines

Analysis of thermal melting curves represents one important approach for evaluating protein stability and the consequences of amino acid substitution on protein structure. By use of the van't Hoff relationship, the differential melting curve can be robustly fit to only three parameters, two of which are the underlying physical constants of melting temperature (Tm) and van't Hoff enthalpy (deltaHvH). Calculated Tm and deltaHvH values are insensitive to the choice of pre- and post-transition baselines. Consequently, the method accurately computes Tm and deltaHvH for extremely truncated data sets, in the complete absence of baseline information, and for proteins with low melting temperatures, where the traditional direct approach routinely fails. Moreover, agreement between deltaHvH values obtained using points derived from pre- vs. post-transition data provide an independent method for detecting some classes of non-two-state transitions. Finally, fitting of the differential denaturation curve should prove useful for analysis of abbreviated data sets obtained from high throughput array analysis of protein stability.     

Interactions of sodium and potassium ions with oligonucleotides carrying human telomeric sequence and pyrene moieties at both termini

The organization of human telomeric DNA is of intense interest because of its role in aging, cancer research and bioanalytical applications. The Htelom sequence 5'-G(3)(T(2)AG(3))(3)-3' has been use to prepare two pyrene-modified fluorescence probes with three- and six-carbon linkers: Py-Htelom-Py(C3) and Py-Htelom-Py(C6), respectively. Results of the circular dichroism (CD), native PAGE, steady-state fluorescence, and anisotropy measurements of sodium and potassium quadruplex formation by these pyrene-modified conjugates are presented and discussed in order to clarify which conformation facilitates or renders the pyrene/pyrene or G-tetrad/pyrene stacking interaction. The CD spectra and native PAGE images suggested that conjugation of pyrene moieties has negligible effect on the folding properties of Htelom oligonucleotide. CD melting profiles and thermodynamic parameters revealed that both sodium and potassium quadruplexes are stabilized by the anchoring of pyrene tags with potassium ion being more effective than its sodium counterpart. Monomer emission of pyrene dominated in all investigated systems with fluorescence intensity being sensitive to the nature and concentration of cation and this phenomenon was attributed to the quenching processes and to the particular topologies of sodium and potassium quadruplexes. Strong quenching observed in the presence of KCl was attributed to the peculiarity of the potassium hybrid-type quadruplex, which enables effective stacking of pyrene moieties on the exposed guanine tetrads, thus facilitating static or electron transfer quenching. Plausibility of stacking interactions between pyrene and G-tetrad in a hybrid-type potassium quadruplex was further supported by the anisotropy measurements and molecular modeling results.     

Prediction of collagen stability from amino acid sequence

An algorithm was derived to relate the amino acid sequence of a collagen triple helix to its thermal stability. This calculation is based on the triple helical stabilization propensities of individual residues and their intermolecular and intramolecular interactions, as quantitated by melting temperature values of host-guest peptides. Experimental melting temperature values of a number of triple helical peptides of varying length and sequence were successfully predicted by this algorithm. However, predicted T(m) values are significantly higher than experimental values when there are strings of oppositely charged residues or concentrations of like charges near the terminus. Application of the algorithm to collagen sequences highlights regions of unusually high or low stability, and these regions often correlate with biologically significant features. The prediction of stability from sequence indicates an understanding of the major forces maintaining this protein motif. The use of highly favorable KGE and KGD sequences is seen to complement the stabilizing effects of imino acids in modulating stability and may become dominant in the collagenous domains of bacterial proteins that lack hydroxyproline. The effect of single amino acid mutations in the X and Y positions can be evaluated with this algorithm. An interactive collagen stability calculator based on this algorithm is available online.