Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



What can the computer say about the Randić index?


Mustapha Aouchiche1, Gilles Caporossi2, Pierre Hansen2 and Marie Laffay3


1Ecole Polytechnique  de Montréal,  Canada       


2GERAD and HEC Montréal,  Canada


3CUST,  Clermont-Ferrand, France




In  addition  to  performing  numerous  computational  tasks  in chemical  graph  theory,  as  e.g computing  chemical  invariants  or  enumerating  families  of  graphs,  computers  are  being increasingly  used  for  advancing  the  theory  itself1,  i.e., finding  conjectures,  proofs  and refutations , in  an  assisted  and sometimes  fully  automated  way.

The  Randić  index2  is  one  of  the  most  studied  among  chemical  invariants,  both  without  and with  computers.  In  this  paper  we  survey  computer-based  work  on  the  Randić  index  and  its properties,  and  give  some  new  results.

We  focus  on:


(i) finding  and  characterizing  extremal  graphs  for  the  Randić  index3;

(ii) refuting,  repairing,  corroborating  and  strengthening  Graffiti4 conjectures;

(iii) proving  automatically  new  conjectures  and  providing  ideas  of  proof  for  more  difficult  ones with  AGX5,6;

(iv) making  a  systematic  comparison  of  the  Randić  index  and  other  indices  for  various  families of  graphs,  through  conjecture  finding  with  AGX.


To  complete  the  paper  we  prove,  by  computer  or  by  hand,  several  conjectures,  and present  a  list  of  open  ones.


1. P. Hansen (2002) Graph Theory Notes of New York  53: 20-34.

2. M. Randić (1975) J. Am. Chem. Soc. 97: 6609-6615.

3. G. Caporossi, I. Gutman, P. Hansen (1999) Comp. Chem. 23: 969-977.

4. S. Fajtlowicz (1988) Discrete Math. 72: 113-118.

5. G. Caporossi, P. Hansen (2000) Discrete Math. 212: 29-44.

6. G. Caporossi, P. Hansen (2004) Discrete Math. 276: 81-94.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Fractal kinetics and its application to

BIACORE binding data analysis


Željko Bajzer, Yves Nominé and Georges Mer


Department of Biochemistry and Molecular Biology, Mayo Clinic College of Medicine, Rochester, MN, USA




The reaction environments of in vivo conditions are characterized by spatial non-uniformity, macromolecular crowding and small volumes1. In these situations it is assumed that fractal geometry of the environment plays an essential role and as a consequence the law of mass action valid for uniform environments has to be modified. This leads to fractal–like kinetics1,2 which implies time-dependent “rate constants”. We present a simple derivation of law of mass action for reactions on fractals, based on expression for the mean number of sites on the fractal visited by a random walker in a given time period3. The model for bimolecular binding

A + B → AB

is developed and compared to another model of fractal kinetics4 based on fractional powers in classical low of mass action:

k [A]α [B]β

In the recent years measurement of binding of macromolecules have been performed by BIACORE. This is an instrument where binding of molecules (A) in circulating solution to molecules (B) attached to the surface of a sensor chip is recorded as a function of time. The increase of mass on the chip due to binding is detected by using surface plasmon resonance technology (www.biacore.com). It has been argued5,6 that molecules attached on the chip surface constitute a fractal and that fractal kinetics should be applied. Based on our equations for fractal bimolecular binding we derive a model for binding in BIACORE instrument which also includes the effects of flow. This model appears to be more realistic than the standard model in which neither fractal surfaces, nor flow are taken into account. We apply our model to reanalyze BIACORE data from literature7,8 and we further analyze binding of breast-cancer-associated protein  BRCA1 to phosphorylated  DNA repair helicase BACH19,10.


 1. S. Schnell, T.E. Turner (2004) Prog. Biophys. Molec. Biol. 85: 235-260.

 2. R. Kopelman (1988) Science 241: 1620-1626.

 3. H.Q. Li, S.H. Chen, H.M. Zhao (1990) Biophys. J. 58: 1313-1320.

 4. M.A. Savageau (2995) J. Theor. Biol. 176: 115-124.

 5. A. Sadana (2001) Analy. Biochem. 291: 34-47.

 6. Ž. Bajzer, J.D. Orth (20002) Biophys. J. 82: 481a.

 7. R. Karlsson, A. Michaelsson,  Mattsson (1991) J. Immunol. Methods 145: 229-240.

 8. H. Houshmand, G. Fröman, G. Magnusson (1999) Anal. Biochem. 268: 363-370.

 9. X. Yu, C.C. Chini, M. He, G. Mer, J. Chen (2003) Science 302: 639-642.

10. M.V. Botuyan, Y. Nominé, X, Yu, N. Juranić, S. Macura, J. Chen, G. Mer (2004) Structure (Camb.) 12: 1137-1146.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Heuristics for the Reversal Median Problem:

How to select good reversals?


Matthias Bernt, Daniel Merkle  and Martin Middendorf


Department of Computer Science, University of Leipzig, Germany




The Reversal Median Problem (RMP) is to find for three given signed permutations (genomes) a signed permutation such that the number of reversals needed to transform it into the given signed permutation is minimal. In Moret and Siepel1 an exact branch-and-bound algorithm was presented for RMP which can be applied in reasonable time only to small problem instances. Other algorithms like MGR2 and rEvoluzer I3 use heuristic approaches that iteratively apply promising reversals starting with the given signed permutations to find a solution.

A new algorithm called rEvoluzer II4 applies several promising reversals in parallel, which leads to fronts of permutations approaching each other. Here we investigate several strategies to select the candidate reversals for obtaining the permutations in the front. Moreover, we study the properties of the resulting solution sets for the RMP.


1. B. Moret, A. Siepel (2001) Finding an optimal inversion median: Experimental results. In: Proc. of the 1st International Workshop on Algorithms in Bioinformatics (WABI 2001), No. 2149 in LNCS, Springer, pp. 189-203. 

2. G. Bourque, P. Pevzner (2002) Genome-scale evolution: Reconstructing gene orders in the ancestral species. Genome Res. 12(1): 26-36.

3. M. Bernt, D. Merkle, M. Middendorf (2004) Genome rearrangement based on reversals that preserve conserved intervals. (submitted).

4. M. Bernt, D. Merkle, M. Middendorf (2005) A parallel algorithm for solving the reversal median problem. (submitted).




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Yb3+ as an origin of the strong anti‑Stokes luminescence in the NIR FT‑Raman spectra

of some lanthanide sesquioxides


Tomislav Biljan and  Zlatko Meić


Faculty of Science, Department of Chemistry, Strossmayerov trg 14, 10000 Zagreb, Croatia




Strong anti-Stokes bands present in FT-Raman spectra of Y2O3, La2O3, Gd2O3 and Lu2O3 are explained by the NIR luminescence of Yb3+ impurities present in sesquioxides1 after the excitation with the 1064 nm line of a Nd:YAG laser. Samples of Y2O3:Yb, La2O3:Yb, Ga2O3:Yb, CeO2:Yb, Gd2O3:Yb and Lu2O3:Yb were prepared by solution combustion synthesis procedure using urea. All materials were investigated by FT-Raman and FT‑NIR spectroscopy and characterized by X-ray powder diffraction. Strong anti-Stokes luminescence caused by Yb3+ ions in FT-Raman spectra has a potential application of qualitative and quantitative determination of ytterbium in solid materials.


1. T. Biljan, S. Rončević, Z. Meić, K. Kovač (2004) Chem. Phys. Lett. 395: 246–252.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



I-graphs and the Corresponding Configurations


Marko Boben1, Tomaž Pisanski2 and Arjana Žitnik3


1University of Ljubljana, IMFM, FRI, Tržaška 25, SI-1000 Ljubljana, Slovenija


2 University of Ljubljana, IMFM, FMF and University of Primorska, Jadranska 19,
SI-1000 Ljubljana, Slovenija


3 University of Ljubljana, IMFM, FMF, Jadranska 19, SI-1000 Ljubljana, Slovenija




Trivalent or cubic graphs form an extensively studied class of graphs. Since they are sparse, trivalent graphs can be readily drawn and visualized. Many graph theoretical problems can be reduced to the trivalent case. The purpose of this presentation is the study of I-graphs, a special class of trivalent graphs. I-graphs were introduced in ref. 1 and form a natural generalization of generalized Petersen graphs4. An I-graph is described by three integer parameters. We determine the necessary and sufficient conditions for testing whether two I‑graphs are isomorphic or not. We also classify I-graphs in terms of girth, bipartiteness, and automorphism group.

Bipartite cubic graphs with girth at least 6 can be considered as incidence graphs (or Levi graphs) of combinatorial configurations. Although configurations are mathematical objects known for more than 150 years, the connection between them and certain classes of graphs has not been widely investigated. But, for example, the connection between configurations and cages was established in ref. 3. Here we contribute some results concerning configurations arising from I-graphs.

From the combinatorial point of view, some results follow from the properties of their Levi graphs, for example about symmetry and about being triangle- or quadrangle-free, etc. From the geometric point of view, there is an interesting connection to (cyclic) astral configurations introduced in ref. 2. These configurations can be realized in the Euclidean plane with maximal possible cyclic symmetry.


1. I.Z. Bouwer, W.W. Chernoff, B. Monson, Z. Star (1988) The Foster Census, Charles Babbage Research Centre.

2. B. Grünbaum (1993) Astral (nk) configurations. Geombinatorics 3: 32–37.

3. T. Pisanski, M. Boben, D. Marušič, A. Orbanić, A. Graovac (2004) The 10-cages and derived configurations. Discrete Math. 275: 265–276.

4. M. Watkins (1969) A theorem on Tait colorings with an application to the generalized Petersen graphs. J. Combin. Theory 6: 152–164.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Mutability of short sequence repeats in

human genome


Branko Borštnik, Borut Oblak and Danilo Pumpernik


National Institute of Chemistry, Ljubljana, Slovenia




The availability of entire genomic sequences and many millions of single nucleotide polymorphisms represents a challenge to computational chemists and biologists.

We have combined both categories of information in order to unveil the susceptibility of specific regions of human genome towards the alterations which lead to genetic polymorphism. Since short sequence repeats are highly prone to elongation and shortening process1,2 the attention was focused to polyadenine repeats which we found several millions in human genomic sequences and 55.000 among the single polymorphism entries. The polyadenines were grouped in several categories and it was found that the short sequence repeats in regions which are poor in genes represent the most mutable part of human genome.


1. B. Borštnik, D. Pumpernik (2004) Europhys. Lett. 65: 290-296.

2. B. Borštnik, D. Pumpernik (2005) Phys. Rev. E  71: 031913(7 pages).




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



MixeR package for compositional data analysis


Matevž Bren1,2 and Vladimir Batagelj 2,3


1Faculty of Organizational Sciences, University of Maribor, Kidričeva 55a, Kranj, Slovenia


2Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Ljubljana, Slovenia


3Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia




R (http://www.r-project.org/) is `GNU S' - a language and environment for statistical computing and graphics. R is similar to the award-winning S system, which was developed at Bell Laboratories by John Chambers et al. It provides a wide variety of statistical and graphical techniques (linear and nonlinear modelling, statistical tests, time series analysis, classification, clustering...). Further extensions can be provided as packages.

We started to develop a library of functions in R to support the analysis of mixtures and our goal is a MixeR package for compositional data analysis that  provides support for

operations on compositions:  perturbation and power multiplication, subcomposition with or without residuals, centering of the data, computing Aitchison's, Euclidean, Bhattacharyya distances,  compositional Kullback-Leibler divergence etc.

graphical presentation of compositions in ternary diagrams and tetrahedrons with additional features:   barycentre the geometric mean of the data set,   the percentiles  and ratio lines, marking and coloring of subsets of the data set, notation of individual data in the set etc.

dealing with zeros and missing values in compositional data sets  with R procedures for simple and multiplicative replacement strategy.

We'll present the current status of MixeR development and illustrate its use on selected data sets.


J. Aitchison (1986) The Statistical Analysis of Compositional Data, Chapman & Hall, New York.

J.A. Martin-Fernandez, C. Barcelo-Vidal, V. Pawlowsky-Glahn (2003) Dealing with zeros and missing values in compositional data sets using nonparametric imputation. Math. Geology 35 (3): 253-278.

J.A. Martin-Fernandez, C. Barcelo-Vidal, M. Bren, V. Pawlowsky-Glahn (1999) A measure of difference for compositional data based on measures of divergence. In: Proceedings of the 5th Annual Conference of the International Association for Mathematical Geology (S.J. Lippard, A. Naess, R. Sinding-Larsen, eds.),  Trondheim, Norway, vol. 1, pp. 211-215.

H. Von Eynatten, C. Barcelo-Vidal, V. Pawlowsky-Glahn (2003) Modelling compositional change: The example of chemical weathering of granitoid rock. Math. Geology 35 (3): 231-251.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Decomposition of the Free Energy Using

the Free Energy Perturbation Method


Urban Bren1 and Jan Florian2


1Centre for Molecular Modelling, National Institute of Chemistry, Hajdrihova 19, SI-1000 Ljubljana, Slovenia


2Department of Chemistry, Loyola University Chicago, 6525 N. Sheridan Road, Chicago, IL 60660, USA




The partitioning of the free energy into additive contributions originating from groups of atoms or force field terms has a potential to provide free energy based relationships between structure and biological activity of molecules and was used as a background for many new methods of Bioinformatics like scoring functions or QSAR. Whether such decomposition is justified has been a subject of vigorous debate in the molecular modeling society. This question was never before addressed in terms of the free energy perturbation (FEP) method, which represents the most robust methodology for calculations of the free energy differences.

Using the FEP methodology we established, that every dissecting of the free energy into specific components possesses an inherent error (termed the nonadditivity error) arising from coupling between corresponding energy contributions. This effect is seen as the appearance of mixing terms in the equation for the free energy difference. Good news is that these mixing terms increasingly loose importance as the change in the energy becomes smaller and smaller. Therefore we are able to decrease the nonadditivity error to arbitrary value just by increasing the number of FEP windows. At the same time one has to be aware, that the free energy components are not state functions and should consequently be calculated on the most natural trajectory possible.

We confirmed our theoretical findings via molecular dynamics calculations of hydration free energies of nucleobases. Although our simulation was based on the decomposition of the free energy, it yielded results surprisingly similar to those obtained from the usual FEP procedure. Finally, we studied substituent effects on 2’-deoxynucleosides with different charged substituents attached at the 3’ position of deoxyribose. The obtained solvation free energies of the base moieties exerted by their surroundings were only negligibly affected by the structure and the charge of the substituent. The nonadditivity error presented less than 0.1 % of the overall result.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005






Patrick Bultinck1*, Ana Gallegos1,2, Sofie Van Damme1, Robert Ponec3 and Ramon Carbó‑Dorca1, 2


1Ghent University, Department of Inorganic and Physical Chemistry, Krijgslaan 281 (S3), 9000 Gent, Belgium


2Institute of Computational Chemistry, University of Girona, Campus de Montilivi, 17005 Girona, Catalonia, Spain


3Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Prague 6, Suchdol, 165 02 Czech Republic.




Aromaticity is a very wide spread concept, although without strict quantum chemical definition. This has led to many different measures of molecular aromaticity1,2, including structural, energetic and magnetic criteria. It is well-known that different classes of aromaticity descriptors can even yield divergent conclusions about aromaticity2.

A noteworthy case where different aromaticity measures yield different conclusions is the case of polycyclic aromatic hydrocarbons. In the present study we introduce a new measure for benzenoid ring aromaticity, based on the molecular quantum similarity theory3.For a benzenoid ring L’ in an organic molecule M and the ring L in benzene (M’), this requires solving integrals of the type:

where  is an overlap integral over the four involved basis functions.

It will be shown that this method offers new insights into aromaticity and relates to the old Hückel MO based aromaticity measure introduced in 1967 by Polansky and Derflinger4.


1. F. De Proft, P. Geerlings (2001) Chem. Rev. 101: 1451-1464.

2. V.I. Minkin, M.N. Glukhovtsev, B.Ya  Simkin (1994) Aromaticity and Antiaromaticity, Wiley Intersci., New York.

3. P. Bultinck, X. Gironés, Ramon Carbó-Dorca (2005) Rev. Comput. Chem. 21: 127-207.

4. O. Polansky, G. Derflinger (1967) Int. J. Quant. Chem. 1: 379-401.






Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Studies of Homogeneous Electron Gas


Jerzy Cioslowski


Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland




Emphasizing a proper description of short-range interactions, the ladder theory (LT) is uncapable of reliably reproducing any property of the three-dimensional electron gas except for the correlation function at the electron coalescence limit (the on-top density) g(0) and the related large-k tail of the momentum distribution n(k). Because of the violation of the cusp condition, poor accuracy of the predicted g(r) is expected for any nonvanishing r.  Although LT yields components of the correlation energy that satisfy the virial theorem for homogeneous interaction potentials, in the case of the Coulomb potential these components turn out to be infinite. A straightforward analysis shows that any effort at alleviating this problem by introducing a long‑range screening is bound to violate the virial condition. A commonly employed approximate version of LT, which avoids Coulomb singularities, yields incorrect energy components and an unphysical momentum distribution despite producing reasonable values of g(0). Since lessening of the approximation worsens the accuracy of the high-density limit of g(0), this result appears to be due to a fortuitous cancellation of errors.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



On the use of ICS and CSA NMR data for predicting secondary structures of peptides and proteins


Attila G. Császár and Eszter Czinki


Department of Theoretical Chemistry, University of Eötvös Lorand, Pázmány sétány 1/A,

H-1117 Budapest, Hungary




Nuclear magnetic resonance (NMR) chemical shielding tensors, computed at different levels of Hartree-Fock, density functional (DFT), and even coupled-cluster [CCSD(T)] theory within the gauge-including atomic orbital (GIAO) formalism, have been investigated as a function of the backbone dihedral angles φ and ψ. The changes in  chemical shieldings due to different residues and to the increase in model size are investigated in detail. The computed surfaces are compared to results corresponding to an experimental database.

It appears that two computational routes to structure determinations from NMR chemical shielding information can be pursued. The perhaps simpler one concentrates on those limited regions of the Ramachandran surface that characterize the dominant peptide conformations. In this approach first all (or most of) the possible conformers are obtained, and the NMR computations are performed at these reference structures. This approach has been pursued, for example, for β-hairpin conformers and dipeptide models involving diverse amino acid residues. It seems that the most useful utilization of the corresponding results is through the construction of multidimensional chemical shift/chemical shift plots.

The second route does not discriminate on the basis of conformers but defines complete isotropic chemical shift (ICS) and chemical shielding anisotropy (CSA) surfaces for all relevant nuclei as a function of dihedral angles.

The periodic 3-dimensional ICS(φ,ψ) and CSA(φ,ψ) surfaces, in the latter case including both direction‑independent as well as directional definitions, were fitted employing a number of suitable mathematical functions. The best representation of the computed data is provided by a 10th-order cosine expansion. Although the CSA surfaces, when non‑directional CSA definitions were used to generate them are well structured at the levels of theory probed, the results suggest that these surfaces have only limited utility in distinguishing between the major conformations, α-helix and β-sheet, of peptides and most likely proteins. The orientation of NMR chemical shielding tensors seems to depend substantially on the backbone dihedral angles, most prominently in case of 13Cα, and thus promises to provide useful information for distinguishing between α-helix and β-sheet regions. For other nuclei, 1HN, 15NH, 13C, the orientation of the tensor is affected by other factors, like H-bonding, and, consequently, the prediction of backbone angles using tensor orientation information alone is limited.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Frequency analysis of photochemical

pollution data


Tomislav Cvitaš1, Nenad Kezele1, Leo Klasinc1,

Matevž Pompe2 and Marjan Veber2


1Ruđer Bošković Institute, Bijenička 54, Zagreb


2Faculty of Chemistry and Chemical Technology, University of Ljubljana, Slovenia




Frequency analysis of ozone data were recently reported for 12 European ozone monitoring sites1. The same procedure was applied to the ozone and other pollutants data from six slovenian and croatian urban stations. As expected, strong frequency signals are found for the 1 year and 1 day periods. The remaining peaks are selected statistically by checking whether the examined point exceed the value of  of 10 neighboring points, five on each side. The periods corresponding to this condition were chosen as significant.

The most significant frequency peaks (Fig.1.) are located at 365 and 1 day periods (natural cycles) and at 7 days period (anthropogenic cycle). The relative intensity of 7 days peak could serve as a indication of anthropogenic impact on measuring site. A quantification of the degree of anthropogenic influences on ozone values was also given recently by analysis of 7 days, 1 day and ½ day periodicities2.

Some peaks in the spectra could correspond to exact frequencies of the second, third and fourth harmonics of the base peak of 365.25 days. In most of the spectra, these peaks can be found on 6, 4 and 3 month periods. These harmonic cycles appear because the 1-year period is not purely sinusoidal. Other important peaks for each station are found to describe quasi periods ranging between 7 and 44 days and are listed in Table 1.
























































+10 10.5 13.5 15.5

11.5 15.5








8.2 10 10.7

13.5 15.5 16.7

11.7 13.5 15.7

8; 9.5





8.4; 8.7

9.3; 9.6

15; 15.8 17.4


 20 22.5 25









22.5 24.5


















































Table 1. Significant frequency signals given in days.


1. T. Cvitaš et al. (2004) J. Geophys. Res. 109: 2302-2311.

2. N. Audiffren, C. Duroure, G. Le Nir (2003) In: TOR-2 Final Report, EUROTRAC-2 ISS, GSF, Munich, pp 59-62.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Ab initio Studies on Low-Energy

Conformers of Oligopeptides


Jiří Czernek


Department of Bioanalogous and Special Polymers, Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, Heyrovsky Square 2, Prague 16206, The Czech Republic




An accurate description of the conformational space of peptides is of key interest in biomolecular science. Thanks to the progress in the computer hardware and in the efficiency of the relevant software, ab initio quantum chemical methods can now routinely be applied to quite sizeable peptide structures (hundreds of atoms). In particular, the resolution-of-identity (RI) approximation elaborated by Ahlrichs and coworkers and implemented in the TURBOMOLE program package enables one, when combined with some density functional theory (DFT)-based method, to perform computationally very efficient geometrical optimizations. Moreover, RI approximation coupled with the second-order Møller–Plesset perturbational theory (MP2) can account for a large portion of the correlation energy with reasonable computer time, memory, and disc space requirements. A number of methods, both RI-based and conventional, have been applied to describe the geometries and total energies of four low-energy conformers of an important model system, N‑acetyl‑N’‑methylalanineamide (“dialanine”), which has been studied at the Hartree–Fock level previously1. The basis set dependence of the results has been addressed by performing each calculation with a smaller (i.e., SVP) and an extended (i.e., TZVP) set of atomic orbitals. The accuracy of the relative energies provided by respective approaches has been assessed by a comparison with the benchmark RI-CC2 (the second order approximate coupled cluster method)/TZVP results. Seven conformers of the heptapeptide Ac‑VVVE(Hnb)(tBu)VVV‑OH (Ac, V, and E denote acetyl group, valine, and glutamic acid, respectively, Hnb is 2-hydroxy-6-nitrobenzyl group used to substitute the amide proton of E, and tBu is tert-butyl group protecting the carboxylic function of E), which consists of 148 atoms and has been prepared in our Department and studied experimentally and by molecular dynamics, have been subsequently optimized by RI‑DFT methods and their final energies have been calculated using the RI-MP2 technique. Various factors contributing to each conformer’s stability are compared. The importance of the inclusion of the correlation energy in the calculations is highlighted.


Acknowledgements. This research has been supported by the Grant Agency of the Academy of Sciences of the Czech Republic, Grant KJB4050311. Time allocation in the Czech Academic Supercomputer Centre and the Mississippi Center for Supercomputing Research is gratefully acknowledged.


1. H.-J. Böhm, S. Brode (1991) J. Am. Chem. Soc. 113: 7129–7135.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Testing of a priori vapor pressure and boiling point predictive ability of COSMOtherm


András Dallos and R. Kresz


University of Veszprém, H-8201 Veszprém, P.O. Box 158, Hungary




The COSMOtherm program is based on COSMO-RS theory of interacting molecular surface charges1. The chemical potential of the compounds in the system, which allows for the prediction of almost all thermodynamic properties of compounds or mixtures, is available from integration of the screening charge density over the surface of the compounds2. In addition to the prediction of thermodynamics of liquids COSMO-RS is also able to provide a reasonable estimate of a pure compound’s chemical potential in the gas phase and therefore it is possible to a priori predict vapor pressures of pure compounds.

For testing of vapor pressure and boiling point predictive ability of COSMOtherm we compared the experimental vapor pressure values at 298.15 K and normal boiling points of 139 organic compounds of 20 molecular classes, mostly homologues (normal-, iso-, cyclo-, 1-chloro-, 1-bromo-, 1-fluoro-, 1-cyano-, 1‑nitro-, 1-acetoxy-alkanes, alkenes, alkynes, aromatics, 1-alkanols, 2-alkanols, 1-thiols, 2-alkanones, aldehydes, esters, pyridins and others) with those estimated by COSMOtherm. It was concluded that the estimation power of COSMOtherm is close to the vapor pressure calculation methods with one adjustable parameter, like Thek-Stiel extrapolation method.

                                    (a)                                                                    (b)

Figure 1. Comparison of measured and estimated vapor pressure of benzyl acetate at different temperatures (a) and 139 organic compounds at 298.15 K (b).


1. A. Klamt (1995) J. Phys. Chem. 99: 2224-2235.

2. A. Klamt, F. Eckert (2000) Fluid Phase Equilibr. 172: 43-72.



Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Chemical Ontologies for Bioinformatics


Kirill Degtyarenko


European Bioinformatics Institute, Wellcome Trust Genome Campus, Hinxton, Cambs CB10 1SD, United Kingdom




Although biological macromolecules (nucleic acids and proteins) are biochemical entities, the biological sequence databases often lack meaningful (bio)chemical annotation. To process the ever-growing volume of biological sequence data, the standardization and organization on different levels is required, from controlled vocabularies to dictionaries and thesauri to taxonomies and formal ontologies. Ontology of some domain of knowledge is defined here as a controlled vocabulary of terms with defined logical relationships to each other. The unique types of relationships between terms have to be included in chemical ontologies. To illustrate the relevance of chemical ontologies to bioinformatics, the current resources at the European Bioinformatics Institute are presented, such as ChEBI (Chemical Entities of Biological Interest), COMe (the ontology of bioinorganic proteins) and the ontologies for physico-chemical methods and processes.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Special properties of density matrices derived

from an arbitrary full-CI wave function


Gergey Dezsö1,2 and Iván Gyémánt1


1Department of Theoretical Physics, University of Szeged, H-6720 Hungary


2Department of Technology and Production Engineering, College of Nyíregyháza, H‑4401 Hungary




The importance of density matrices in physics and chemistry is known since the dawn of quantum mechanics. The main advantage of describing a system involving at most two‑particle interactions by density matrices, particularly second-order reduced density matrices is, that it contains all relevant information in the most compact form. Unfortunately, second-order reduced density matrices can not be optimized directly because of the so-called N-representability problem.

Since the N-representability problem is realized, there were a lot of attempts to formulate necessary and sufficient conditions for density matrices ensuring their N-representability without the knowledge of the underlying wave function. Nowadays we have either necessary or sufficient conditions, excellent approximate methods, but the exact solution of the N-representability problem is not known.

Our approach works within the frame of a finite one-particle basis, but the dimension of this basis is arbitrary. Density matrices can be expanded in geminal basis, where geminals are antisymmetric two-particle functions constructed over the one-particle basis. N-representable density matrices must be derivable from a FCI-type wave function, which is a linear combination of Slater-determinants over the one-particle basis. In previous papers the connection between the wave function and the density matrix was exactly described by the so-called structure matrix1. This enables one to optimize the density matrice2-4, but configurational coefficients are needed in each iteration step, when building the density matrix and the gradients of the matrix elements.

Elements of the density matrix (in the geminal basis) are pointed out to be written exactly as scalar products of specially defined vectors. These vectors contain the configurational coefficients. Using basic rules of vector operations, conditions can be formulated including only elements of density matrix. Necessary conditions formulated have the form of inequalities.

Case studies for small atoms and molecules are also presented.


1. I. Bálint, G. Dezsö, I. Gyémánt  (2000) J. Mol. Struct.,THEOCHEM 501-502: 125-132.

2. I. Bálint, G. Dezsö, I. Gyémánt (2001) J. Chem. Inf. Comp. Sci. 41: 806-810.

3. G. Dezsö, I. Bálint, I. Gyémánt (2001) J. Mol. Struct. THEOCHEM 542: 21-23.

4. I. Bálint, G. Dezsö, I. Gyémánt (2001) Int. J.Quant. Chem. 84(1): 32-38.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Generalized Operations on Maps


Mircea V. Diudea1, Monica Ştefu1, Peter E. John2 and Ante Graovac3,4


1Faculty of Chemistry and Chemical Engineering, “Babeş-Bolyai” University, 400028 Cluj, Romania


2Technical University Ilmenau, Institute of Mathematics, PSF 100565, D-98684 Ilmenau, Germany


3 Ruđer Bošković Institute, HR-10002 Zagreb, POB 180, Croatia


4Faculty of  Natural Science, Mathematics and Education, University of Split, N. Tesle 12, HR-21000  Split, Croatia



A map M is a combinatorial representation of a closed surface. Convex polyhedra, starting from the Platonic solids and going to spherical fullerenes, can be operated to obtain new objects, with a larger number of vertices and various tiling.

Three composite map operations: leapfrog, chamfering and capra, play a central role in the fullerenes construction and their electronic properties.

Generalization of the above operations leads to series of transformations, characterized by distinct, successive pairs in the Goldberg multiplication formula m(a,b) see Figure 1.

Parents and products of most representative operations are illustrated. 








Figure 1. Generalized (a, b) operations



Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Transcription Factor Target Detection

in Comparative Genomics


Claudia Fried and  Peter F. Stadler


Bioinformatics Group, Department of Computer Science, University of Leipzig, Hätelstraβe 16-18, D-04107 Leipzig, Germany




Cellular signaling pathways induce gene expression by activating specific transcription factors. Errors in the activation of transcription factors are common events in cancer as several of this factors act on genes involved in ellular proliferation, survival and differentiation. The identification of the targets of  tumorigenic transcription factors that cause these changes is an important task in cancer research. One way to identify those target genes is detection of transcription factor binding sites. These binding sites can be predicted by the search of recurring motifs in the regulatory regions of co-expressed genes1. This has been shown to be feasible in yeast where the intergenic regions are very small. On the other hand, intergenic regions in the genomes of vertebrates can be very large. A simple search for exact string matches of experimentally verified binding sides on a genome wide level in vertebrates thus leads to a high number of false positives. To overcome this problem, only regions might be taken into account that are evolutionarily conserved. Conserved regions can be detected by phylogenetic footprinting with the program tracker2 that compares non-coding sequences surrounding a set of orthologous genes. The aim of our study, is to find target genes of the transcription factor STAT3  (Signal transducer and activator of transcription 3), a member of the STAT family of transcription factors that act as signal transducers of cytokines, hormones and growth factors. Stat3 is involved in the regulation of cell growth, survival and differentiation3. Analysis of STAT3 targets found by this study can provide new insight into mechanisms of cancer and may shed light on strategies for targeted therapy.


1. C. Dieterich, R. Herwig, M. Vingron (2003) Bioinformatics 19.

2. S. Prohaska, C. Fried, C. Flamm, G. P. Wagner, P.F. Stadler (2004) Mol. Evol. Phylog. 31: 581-604.

3. J. Turkson (2004) Expert. Opin. Ther. Targets. 8 (5): 409-422.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005





Josipa Friščić1, Sonja Nikolić1 and Marica Medić-Šarić2


1Ruđer Bošković Institute, HR-10002 Zagreb, POB 180, Croatia


2Faculty of Pharmacy and Biochemistry, University of Zagreb, HR-10000 Zagreb, Croatia



Flavonoids are a group of coumpounds widely distributed in plants, and due to their poliphenolic structure on flavan nucleus (Fig.1.) they have significant antioxidant and chelating properties.

Figure 1. Flavan nucleus.


Their antibacterial, antiviral, antimycotic, antiinflammatory and antithrombotic effects have been proven, as well as their positive effects on tumors, cardiovascular, immunological and many other diseases. Structure diversity and various mechanisms of actions, as well as great number of methods for establishing their activity, makes flavonoids a challenge for making a complete QSPR/QSAR profile1.

In this work, several QSPR (Quantitative Structure-Property Relationship) models in study of some physical‑chemical properties (partition coefficient logP, Van der Waals volume Vw, molecular weight, and melting point) have been investigated for group of 30 flavonoids. For calculating the molecular descriptors we used 2 computer programs – TAM2 (University of Zagreb) and DRAGON 3.03 (University of Milan). Models were calculated using program CROMRsel4 (Ruđer Bošković Insitute in Zagreb).

All models were tested for their stability and predictability, and we compared models obtained with molecular descriptors from TAM and from DRAGON 3.0.

It has been shown that for simple models we can use molecular descriptors calculated with TAM (molecular weight), while molecular descriptors from DRAGON 3.0 give more reliable models. For predicting melting point no reliable models were obtained. For partition coefficient, log P, and Van der Waals volume, Vw, best models were those with 2 molecular descriptors calculated with DRAGON 3.0 (Fig. 2.).


 Figure 2. QSPR models for log P and Vw.


1. K.E. Heim, A.R. Tagliaferro, D.J. Bobilya (2002) J. Nutr. Biochem. 13: 572-584.

2. S. Marković, M. Vedrina, M. Medić-Šarić, N. Trinajstić (1997) Comput. Chem., 21: 355-361.

3. R. Todeschini, V. Consonni (2000) Handbook of Molecular Descriptors, Wiley-VCH, Weiheim.

4. B. Lučić, N. Trinajstić (1999) J. Chem. Inf. Comput. Sci. 39: 121-122.



Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Partial Orders, Molecular Phylogenies,

and Migration


Guido Fritzsch1,2, Sonja J. Prohaska 1,3 and Peter F. Stadler1,3,4,5


1Interdisciplinary Center of Bioinformatics, University of Leipzig, Germany


2Molecular Evolution and Animal Systematics, Departmen of Biology II, University of Leipzig


3Bioinformatics, Department of Computer Science, University of Leipzig, Germany


4Institute for Theoretical Chemisty, University of Vienna, Austria


5The Santa Fe Institute, Santa Fe, New Mexico


The expansion of a species in a heterogeneous environment can be correlated with relative rates of evolution in geographically separated subpopulations. The rate variation may be due to adaptation to different environmental conditions and due to changes in population size or structure1.

Given a phylogenetic tree, Tajima's relative rate test2 can be used to identify pairs of taxa that have evolved with significantly different rates since their divergence from a common ancestor. The ratio of the inferred number of substitutions since this last common ancestor can then be used as an estimate for the ratio of  their speed of evolution. We use the quantity exy which is defined as the average of the logarithms of the ratios over all possible choices of the outgroup using the chi-squared value of the Tajima tests as weights.

One can show  that the matrix (exy) is anti-symmetric and transitive, i.e., exy >0 and eyz >0 implies exz >0. Thus it defines a partial order on the set of taxa that faithfully reflects the relatives speed of evolution:




Figure 1. Example of a relative rate poset. Data are 5'UTRs of HIV-1. Thin lines in the l.h.s. panel indicate significant Tajima tests, the thick lines represent the associated Hasse diagram of  the partially ordered set.








We analyse the relative rate posets of a variety of cases in which recent migrations are well documented and for which a fairly complete taxon sampling is available, among them the European pond turtle Emys orbicularis3.


Acknowledgements. This work was supported in part by the DFG Bioinformatics Initiative.


1. C. Stringer, R. McKie (1996) African Exodus: The Origins of Modern Humanity. John MacRae Book/
Henry Holt & Co.,
New York.

2. F. Tajima (1993) Genetics 135: 599-607.

3. P. Lenk, U. Fritz, U. Joger, M. Wink (1999) Mol. Ecol. 8:1911-1922.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Sphericities of Cycles.

What Pólya’s Theorem is Deficient in


Shinsaku Fujita


Department of Chemistry and Materials Technology, Kyoto Institute of Technology, Matsugasaki, Sakyoku, Kyoto, 606-8585 Japan



1. Introduction

The concepts of sphericity, sphericity indices, and unit subduced cycleindices with chirality fittingness (USCI-CFs), which have been proposed by Fujita on the basis of coset rep­resentations and their subductions1, are versatile to discuss stereochemistry in molecule as well as to enumeratestereoisomers.This USCI approach is capable of enumerating isomers as 3D chemical structures (stereoisomers), where theyare itemized with respect to point-group symmetries. However, the USCI approach requires mark tables, USCI tables, and related group theoretical tools, which are not so easily obtained. On the other hand, Pólya’s Theorem, which has been widely used from 1930s2, is simple and convenient to calculate gross isomer numbers without taking account of symmetry-itemization. Because Pólya’s Theorem enumerates graphs, not 3D chemical structures (stereoisomers) from the viewpoint of the USCI approach, its application has been restricted within graph-theoretical problems. In order to treat such stereochemical problems as solved bythe USCI approach, what is Pólya’s Theorem deficient in?


2. The Proligand Method

The deficiency of Pólya’s Theorem is now concluded to be the concept of sphericities of cycles3. For example, let us consider an enumeration problem in which the three hydrogen atoms of acetic acid are replaced by Fand p (andp), where chiral proligands pandpare enantiomeric to each other. After the three hydrogens are numbered sequentially, the relationship between each symmetry operation and the resulting permutation (a product of cycles) is examined to give the following correspondence:


proper permutations

improper permutations


I (1)(2)(3)    b13


σd(1) (1)(2 3)   a1c2


C3 (1 2 3)     b3


σd(2) (1 3)(2)   a1c2


C3 (1 3 2)     b3


σd(2) (1 2)(3)   a1c2


In this table, a permutation corresponding to an improper rotation is called an improper pemutation and designated by an overbar. The cycles are divided into three categories and char­acterized by sphericities and sphericityindices. Thus, an odd-membered cycle contained in an improper permutation is called a homospheric cycle and characterized by a sphericity index ad (d: the length of the cycle), while an even-membered cycle contained in an improper permuta­tion is called an enantiospheric cycle and characterized by a sphericity index cd. On the other hand, a cycle contained in a proper permutation is called a hemispheric cycle and characterized by a sphericity index bd whether d is odd or even.Thereby, the permutations listed above are characterized by products of such sphericity indices, which are summed up to give a CI-CF

(cycle index with chirality fittingness):

The CI-CF provides us with a tool (the proligand method) for stereoisomer enumeration, in which the chirality fittingness controls the mode of accommodation ofchiral and achiral ligands. By placing ad = cd = bd = sd, we can obtain the cycle index of Pólya’s Theorem (i.e., CI = (1/6)(s13+2s3+3s1s2)). This means that the proligand method is more informative than Pólya’s theorem so as to be capable of solving stereochemical problems.


1. S. Fujita (1991) Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag.

2. G. Pólya, R. C. Read (1987) Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds, Springer-Verlag.

3. S. Fujita (2005) Theor.Chem .Acta 113: 73–79; ibid 113: 80–86.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Solid-State and Solution-State Conformational Differences Solution-State and 13C CP/MAS NMR Studies on Conformational Polymorphic Crystals Assisted bQuantum-Mechanical Calculations

of Shielding and Spin-Spin Coupling Constants


 Robert Glaser and Itzhak Ergaz


Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva  84105, Israel




The molecular environments in solution, in the crystalline state, and in the 'computer' [gas phase, zero deg. K] are obviously all different. Thus, in principle, all four 'important' conformations [crystalline state, solution-state, global minimum, and bioactive conformation] have no symmetry obligation to be one and the same. This is all the more apparent when one considers that a change in solvent may also greatly effect a conformational equilibrium. Moreover, conformational polymorphism in crystals testify that different packing motifs and space groups can also afford different solid-state conformations [conformational polymorphism in crystals].  In other words, one is obligated to undertake structural studies in each case.

Subtle structural changes can greatly effect the relationships between the most stable conformation in solution versus that in the crystal. Medium ring secondary, tertiary, and quaternary ammonium salts will be used to illustrate cases where x-ray crystallography together with solid-state cp/mass NMR can be used to confirm or disprove a solution-state structural hypothesis. This is particularly useful when the conformational choices involve carbon atoms having a clearcut γ-gauche effect on their relative chemical shifts.  

In addition, while quantum mechanical calculations (Gaussian) have long been able to calculate shielding constants, these predictions have improved in quality in version 03, and now have reached the stage where desktop computers can be readily used to calculate spin-spin coupling constants based on geometry optimized conformational models of fairly large molecules. When the solid-state cp/mass 13C chemical shifts are markedly different from those in solution, and the solution-state vicinal coupling constants clearly do not fit those expected from dihedral angles in the x-ray crystallographically determined conformation, molecular modeling can provide provisional solution-state geometrical hypotheses. Using these as input, comparisons of quantum mechanical calculated 13C chemical shifts (based on scaled shielding constants) and 3JHH coupling constants with those measured in solution can greatly strengthen or weaken the validity of our new proposed structure. This insight can be further bolstered by using the geometry optimized x-ray crystallographically conformation as input to now calculate 13C chemical shifts for comparison with those found in the cp/mass spectrum.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



The Evolution of Animal miRNAs


Jana Hertel1, Claudia Fried1, Manja Lindemeyer1, Kristin Missal1,

Sonja J. Prohaska1, Andrea Tanzer1,2, Christoph Flamm2, Ivo L. Hofacker2, Peter F. Stadler1,2,3, and the students of the bioinformatics computer labs 2004 and 2005


1Bioinformatics Group, Department of Computer Science, University of Leipzig, Germany


1      2 Institute for Theoretical Chemistry and Structural Biology, University of Vienna, Austria


3      3The Santa Fe Institute, Santa Fe, New Mexico, USA




MicroRNAs are short (~22nt) non-coding RNAs, that act as factors in the degeneration or translational repression of their mRNA targets1. We clustered the known miRNAs due to their sequence similarity and searched all available metazoan genomes for homologs. Based upon those hits we constructed gene phylogenies. All of the resulting trees are consistent with the genome duplication history at the origin of the vertebrates and the teleost lineage, respectively. Tandem duplications of microRNAs typically preceded these genome-wide events.

We furthermore investigated distant homologies between established microRNA families. To do that we aligned their consensus sequences and examine the resulting phylogeny for the known phylogenetic theories. The intention to this is to identify larger groups that could be distant homologs.

Recently, genome-wide surveys have identified thousands of putative non-coding RNAs2. Our exhaustive analyses provide a phylogenetic grouping of microRNAs that serves a basis for the development of computational methods for microRNA recoginition. This will lead us to the classification of microRNAs within a large collection of non-coding RNAs.


Acknowledgements.  This work was supported in part by the DFG Bioinformatics Initiative.


1. P. Nelson, M. Kiriakidou, A. Sharma, E. Maniataki, Z. Mourelatos (2003) The microRNA world: small is mighty.Trends Biochem Sci. 28(10): 534-540.

2. S. Washietl, I.L. Hofacker, P.F. Stadler (2005) Thousands of noncoding RNAs with conserved structure in mammalian genomes. (in review)




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



The Redundancy of Topological Indices


Boris Hollas


Theoretische Informatik, Universität Ulm, D-89069 Ulm




One approach to predict or compare molecular properties is to encode a molecule by a set of numbers and to use these molecular descriptors for regression analysis or clustering. Topological indices (TIs) are molecular descriptors based on the graph of a molecule. A large number of TIs has been proposed and applied in QSAR and QSPR studies, a method to relate the structur of a molecule to a property (e.g. boiling point or toxicity). Many TIs are mutually correlated1, which causes major problems in QSAR and QSPR studies. Most QSAR/QSPR approaches use molecular descriptors to establish a linear regression model. If descriptors are correlated, the outcome of this model becomes meaningless or modeling may fail completely.

In a series of papers2-6, we consider TIs of the general form



where  is an edge in the graph,  is a numerical property of vertex  (vertex property) and  is a vector of vertex properties. For example, gives the connectivity index. If  is a physicochemical property of atom , is the Moreau-Broto autocorrelation of distance 1.

It turns out that TIs  may be linearly correlated even if are vectors of independent vertex properties. In this case, the information provided by either TI is completely redundant. The observed correlations among TIs are not caused by similar vertex properties but a result of  the above encoding of vertex properties in . On the other hand, by a simple modification these correlations can be reduced or eliminated. This we demonstrate for both artificial TIs with independent vertex properties as well as for a number of TIs that depend on the degree of a vertex, such as the connectivity index, the Zagreb indices, or the Platt number.

Another issue is the variance of  . If descriptor data are clustered by neural nets, a uniform variance throughout the data set is important. While  increases linearly in the number of vertices, a simple modification assures a bounded variance. We demonstrate this for a number of TIs using chemical graphs as well as random graphs.


1.     I. Motoc, A.T. Balaban, O. Mekenyan, D. Bonchev (1982) MATCH Commun. Math. Comput. Chem. 13: 369-404.

2.     B. Hollas (2002) MATCH Commun. Math. Comput. Chem. 45: 27-33.

3.     B. Hollas (2003) J. Math. Chem. 33: 91-101.

4.     B. Hollas (2003) MATCH Commun. Math. Comput. Chem. 47: 79-86.

5.     B. Hollas (2005) MATCH Commun. Math. Comput. Chem. 54: 177-187.

6.     B. Hollas (2005) MATCH Commun. Math. Comput. Chem. 55 (to appear)




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



On the Number of Hamiltonian Groups


Boris Horvat, Gašper Jaklič and Tomaž Pisanski


IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia




Subgroups of abelian groups are abelian and hence self-conjugate or normal. A nonabelian group all of whose subgroups are normal is called hamiltonian1.

In topological graph theory2,5, hamiltonian groups have been studied in the past3,4. For several classes of hamiltonian groups the genus is known exactly. For abelian and hamiltonian groups, there are structural theorems available.

In this paper we determine the number h(n) of hamiltonian groups of order n and the number b(n) of all groups of order n with the property, that all their subgroups are normal. We also determine the number v(n) of all hamiltonian groups of order ≤ n and the number w(n) of  all groups of order ≤ n with the property, that all their subgroups are normal.


1. R.D. Carmichael (1956) Introduction to the Theory of Groups of Finite Order, Dover Publishing Co., New York.

2. J.L. Gross, T.W. Tucker (1987) Topological Graph Theory, Wiley Intersci., New York.

3. T. Pisanski, T.W. Tucker (1989) The genus of the product of a group with an abelian group. Eur. J. Combin.10: 469‑475.

4. T. Pisanski, T.W. Tucker (1989) The genus of low rank hamiltonian groups. Discrete Math. 78: 157-167.

5. A.T. White (2001) Graphs of Groups on Surfaces, North-Holland Publishing Co., Amsterdam.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



The SOS response signalling mechanism in bacteria Escherichia coli: Involvement of RecA loading activity


Ivana Ivančić-Baće1, Ignacija Vlašić2, Boris Mihaljević2, Mirna Imešek2,

Erika Salaj-Šmic2 and Krunoslav Brčić-Kostić2


Department of Molecular Biology, Faculty of Science, Rooseveltov trg 6, Zagreb, Croatia



Department of Molecular Biology, Ruđer Bošković Institute, Bijenička 54, Zagreb, Croatia





The SOS response involves many bacterial functions that are induced in response to damage of chromosomal DNA. The SOS response is regulated by the LexA repressor and the RecA nucleofilament which is formed at single stranded DNA. Such RecA nucleofilaments display a coprotease activity that stimulates self-cleavage of the LexA repressor. This cleavage enables transcription of more that 40 genes in the SOS pathway.

The exact mechanism whereby initial damage leads to induction of SOS response is still unknown. It has been proposed that the signal for SOS induction is single-stranded (ss) DNA formed as a result of DNA damage. This ssDNA can be produced either by DNA polymerase arresting at sites of DNA damage or by RecBCD enzyme unwinding at dsDNA breaks. RecA protein then probably forms presynaptic filaments on these regions of ssDNA, resulting in repressor cleavage.

We wanted to test our hypothesis that RecA loading is essential step in creating the SOS inducing signal. In order to test this, we measured SOS response in a strain in which the lacZ gene (expressing b-galactosidase) is fused with the regulatory region of a sfiA SOS gene. We introduced additional mutations into this strain that inhibit RecA loading and nuclease activity but retain helicase activity (recB1080 and recB1067 mutations). We also wanted to test whether different types of DNA damage require different enzyme processing. Therefore we measured SOS induction after UV and gamma irradiation, and introduction of double strand breaks by endonuclease.



Bacteria were grown at 30°C in LB broth. When cultures reached O.D.600=0.2, cells were either UV‑irradiated, g-irradiated or arabinose was added at final conc. 0.2%. Addition of arabinose induces SceI endonuclease which introduces dsDNA breaks in DNA. The cultures were further incubated for 180 minutes and assayed for β-galactosidase activity.

Our results indicate that RecA loading activity is required for SOS inducing signal after UV, γ-irradiation and dsDNA breaks. When RecA loading activity is inactivated in RecB1080CD enzyme, this activity is provided by RecFOR.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Analytical treatment of hydrogen vibrations in molecular dynamics simulation


Dušanka Janežič, Franci Merzel and Matej Praprotnik


Laboratory for Molecular Modeling and NMR, National Institute of Chemistry, Ljubljana, Slovenia



We propose an analytical treatment of hydrogen bond-stretching vibrations in molecular dynamics simulations using a new form of the classical Liouville propagator. We construct a second-order integrating algorithm which is useful for all-atom molecular dynamics simulations of molecular systems described by flexible models. We apply this algorithm to MD simulation of the liquid water providing the evidence about its superior numerical characteristics over the standard approach.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



A Multi-Objective Particle Swarm Optimization
Algorithm Applied to Molecular Docking


Stefan Janson and Daniel Merkle


Department of Computer Science, University of Leipzig, Germany




In the molecular docking problem1 the three dimensional structure and the affinity of a binding of a target receptor and a ligand has to be determined. This problem is too complex to be solved optimally and is therefore commonly tackled by stochastic local search procedures. Molecular docking requires intra- and inter-molecular energies to be minimized. In our algorithm we consider different energies as different optimization objectives. We introduce a new hybrid Particle Swarm Optimization approach that uses clustering to divide all of the individuals into several subswarms within separate regions of the search space. Our results show that our new algorithm clearly outperforms a well-known Lamarckian Genetic Algorithm2 for the problem.


Figure 1. Convergence behavior of a Simulated Annealing algorithm (SA), a Lamarckian Genetic Algorithm (LGA) as used in2, and our proposed algorithm (ClustMPSO): shown are the worst, average, and best found docking energy (10 runs); instance 4cha.


1.     I. Halperin, B. Ma, H. Wolfson, R. Nussinov (2002). Principles of docking: An overview of search algorithms and a guide to scoring functions. Proteins 47 (4): 409-443.

2.     G.M. Morris, D.S. Goodsell, R.S. Halliday, R. Huey, W.E. Hart, R.K. Belew, A.J. Olson (1998) Automated docking using a lamarckian genetic algorithm and an empirical binding free energy function. J. Comput. Chem. 19 (14): 1639‑1662.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



A 2-parametric class of transformations for fullerenes and other polyhedra


Peter E. John  and Horst Sachs


Institute of Mathematics, Ilmenau Technical University, P.O. Box 100565, D-98384 Ilmenau, Germany




Let P denote any polyhedron all of whose vertices have valency 3, and let S denote the set of all such polyhedra. A 2-parametric class of transformations T, generalizing the well-known leapfrog procedure and related operations, is defined which maps S into S and, in a way, preserves the relative position of the non‑hexagonal faces of P.

Every such transformation T – except for two “small” cases – has the property that each of the polyhedra T(P) possesses some perfect matching that avoids any edge that belongs to the boundary of some non‑hexagon.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



De Novo Determination of Protein Substructures

from NMR Couplings of Peptide Groups


Nenad Juranić, Franklyn G. Prendergast and Slobodan Macura


Departments of Biochemistry and Molecular Biology , Mayo College of Medicine, Mayo Clinic and Foundation, Rochester, Minnesota, 55905 U.S.A





Multi-dimensional NMR spectroscopy has been an effective tool for determining three-dimensional structures of relatively small proteins (<25 kD). Its application to larger proteins faces several challenges the most serious of which is that the fast relaxation of nuclear spins in such samples makes  deuteration (proton-depletion) an absolute requirement1. While proton-depletion enables the use of heteronuclear correlation experiments to determine backbone assignments, it severely reduces the structural information obtained regarding interproton distances, traditionally the major source of structural constraints2. However, new structural constraints, that can be obtained from residual dipolar couplings3-5 and hydrogen-bond spin-spin couplings6-8, have been introduced, and these offer the possibility for atomic-resolution structure determination of  the protein backbone.  In principle, protein structures can be solved from residual dipolar couplings alone, but that requires multiple orientation media (needs five independent alignments)9 which is impractical and often impossible because of the adverse protein-media interactions. Consequently, methods that rely on trial substructures are currently used. A more objective approach requires the experimental determination of an initial substructure and a protocol for direct structure building. 

We have developed method to solve an initial substructure by utilizing sequential and hydrogen-bond connectivity of peptide groups (as detected by NMR spin-spin couplings). The initial substructure has such accuracy that it allows determination of protein alignment tensor using a single orientation media. Once the alignment tensor is known, we propose direct protein structure building utilizing NMR detected hydrogen bond chains8, which are essential elements of protein secondary structure.



Figure 1: Utilizing the ideal hydrogen bond chain motifs (HBCs), a model of the initial sub-structure was obtained from three neighboring HBCs containing high values of the H-bond couplings, while the connectivity between them required an anti-parallel type of HBC




1. R. A.Venters, R. Thompson, J. Cavanagh (2002) J. Mol. Struct.  602: 275.

2. S. Macura, R. R. Ernst (2002) (reprinted from Mol. Physics (1980) 41: 95-117), Mol.Physics 100: 135.

3. J. R. Tolman, J. M. Flanagan, M. A. Kennedy, J. H. Prestegard (1995) P. Natl. Acad. Sci.USA 92: 9279.

4. N. Tjandra, A. Bax (1997) Science 278: 1111.

5. A. Bax, G. Kontaxis, N. Tjandra (2001) Pt B. 339: 127.

6. E. Cordier, S. Grzesiek (1999) J. Am. Chem. Soc. 121: 1601.

7. G. Cornilescu, J. S. Hu, A. Bax (1999) J. Am. Chem. Soc. 121: 2949.

8. N. Juranic, M. C. Moncrieffe, V. A. Likic, F. G. Prendergast, S. Macura (2002) J. Am. Chem. Soc. 124: 14221.

9.  K. B. Briggman, J. R. Tolman (2003) J. Am. Chem. Soc. 125: 10164.



Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



The maximum entropy production principle

as the guideline for predicting evolution

of complex systems


Davor Juretić1, Paško Županović1 and Srećko Botrić2


1Faculty of Natural Sciences, Mathematics and Education, University of Split


2Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split





Maximum entropy production (MEP) principle has been derived recently in a ground‑breaking work of Dewar1, who used an information theoretical formulation of non‑equilibrium statistical mechanics. The MEP principle states that the MEP state is reproducibly selected because it is the most probable non‑equilibrium steady state compatible with given external constraints.

In the linear region, for the case of an electric network, MEP is equivalent to the Kirchhoff’s loop law when overall energy conservation is assumed2. In the case of heat conduction in an anisotropic crystal, MEP is equivalent to the Onsager-Rayleigh principle of the “least dissipation of energy”3. In the nonlinear region, MEP provides generalization of Onsager’s theory for situations and systems far from equilibrium. For instance, when modeling global climate MEP predicts that winds and currents driven by thermal gradients establish themselves as the most effective heat transport from the warmer tropics to the colder poles which maximizes the entropy production.

We shall discuss what would be the best mode of MEP application in biochemistry4,5 and how realistic are selected states with maximum information entropy and maximum dissipation. Natural selection for the coexistence of ordered and dissipative regions far from equilibrium can be understood as an expression of the same basic concept, namely, selection of the most probable state, which produces and exports more entropy to the environment than a purely dissipative 'soup'. To conclude, recent research at the University of Split points out toward maximum entropy production principle as a guideline to modeling complex system, including predicting its evolution, irrespective of system scale, nature, or its distance to equilibrium.


1. R.C. Dewar (2005) Maximum entropy production and the fluctuation theorem, preprint.

2. P. Županović, D. Juretić, S. Botrić (2004) Phys. Rev. E. 70: 0561108.

3. P. Županović, D. Juretić, S. Botrić  (2005) Fizika (in press).

4. P. Županović, D. Juretić (2004) Croat. Chem. Acta 77: 561-571.

5. P. Županović, D. Juretić (2003) Comput. Biol.  Chem. 27: 541-553.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Applicability of the SQM Force Field Method

to the Vibrational Spectra of charged systems:

Sodium Acetate


Gábor Keresztury1, Krisztina István1 and Tom Sundius2


1Chemical Research Center, Hungarian Academy of Sciences, P.O. Box 17, H-1525 Budapest, Hungary


2Department of Physical Sciences, University of Helsinki, P.O. Box 64, FIN-00014 Helsinki, Finland




The applicability of the scaled quantum mechanical force field (SQM FF) method1,2 to the prediction of the vibrational spectra of a charged molecule has been studied by the example of the acetate ion (CH3CO2) in sodium acetate for which an efficient empirical valence force field (SVFF) based on observed IR spectra of six isotopomers of sodium acetate is available in the literature. Standard SQM FF calculations done on a free acetate ion at the B3LYP/6-31G* level failed to give an acceptable estimation of even the most characteristic features of the observed spectra, which can be exemplified by the gross overestimation of the frequency separation of the naCO2 and nsCO2 vibrations.

In search for a better description, SQM calculations were done for three simple structural models of sodium acetate, testing different QM methods. The results indicate that in addition to taking into account the dielectric field effect of the surrounding medium, incorporation of a Na+ counterion is necessary to achieve a realistic simulation of the IR and Raman spectra. Satisfactory results were obtained with a bidentate Na-acetate complex by the SQM method coupled with a continuum model at the B3LYP/6-31+G** level, whereas the use of the Onsager-type spherical cavity model and the polarizable continuum model (PCM) were found preferable over SCI-PCM.


Acknowledgements. The authors are grateful to Dr. O. Berkesi (Szeged, Hungary) for initiating this study.


1. P. Pulay, G. Fogarasi, G. Pongor, J.E. Boggs, A. Vargha (1983) J. Am. Chem. Soc. 105: 7037.

2. G. Fogarasi, X. Zhou, P.W. Taylor, P. Pulay (1992) J. Am. Chem. Soc. 114: 8191.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Gas phase reactions of ambiental CO and SO2 with ozone 


Nenad Kezele1, Glenda Šorgo1, Leo Klasinc1,2 and William A. Pryor2


1Ruđer Bošković Institute, POB 180, HR-10002 Zagreb, Croatia


2Biodynamics Institute, Louisiana State University, Baton Rouge LA70803, USA




Scientific cooperation with Biodynamics Institute exists now for more than a decade and is documented by several joint publications1-6. Thus, peroxynitrite and peroxynitrous acid, nitrogen oxide anions and nitrogen oxyradicals, toxicity of ozone and synergistic effects of nitrogen oxides and other oxidants as well as biological and health effects of air pollution were addressed. The research part in Zagreb is more oriented to the gaseous phase and atmospheric measurements. Here we report the influence of elevated atmospheric ozone concentrations on readings of ambiental CO and SO2 pollution e.g. from combustion sources.


1. L. Klasinc, D. Srzić, Lj. Paša-Tolić, S. Martinović (1996) Gas Phase Properties of ONOO-anion and ONOO-radical. Croat. Chem. Acta 69: 1007-1011.

2. B. Juršić, L. Klasinc, S. Pečur, W.A. Pryor (1997) On the Mechanism of HOONO to HONO2 Conversion. Nitric Oxide Biol. - Ch. 1: 494-501.

3. M. Friedman, S. Kazazić, N. Kezele, L. Klasinc, S.P. McGlynn, S. Pečur, W. A. Pryor (2000) Role of Nitrogen Oxides in Ozone Toxicity. Croat. Chem. Acta 73: 1141-1151

4. S. Kazazić, S.P. Kazazić, L. Klasinc, S.P. McGlynn, W.A. Pryor (2002) Proton Affinities of N-O Anions and their Protonated Forms. J. Phys. Org. Chem. 15: 728-731.

5. S.P. Kazazić, L. Klasinc, S.P. McGlynn, W.A. Pryor (2004) Proton Affinities of Nitrogen Oxyradicals. Croat. Chem. Acta, 77: 465-468.

6. T. Cvitaš, L. Klasinc, N. Kezele, S.P. McGlynn, W.A. Pryor (2005) How Dangerous is Ozone? Atmos. Environ. (in press).




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



A New Approach to NMR Chemical Shift

Additivity Parameters Using Simultaneous

Linear Equation Method


Rabah A. Khalil and Yosif A. Shahab


Department of Chemistry, College of Science, University of Mosul, Mosul, Iraq




A new approach to nmr chemical shift additivity parameters using simultaneous linear equation method has been introduced. Three general 15N nmr chemical shift additivity parameters with physical significance for 193 compounds of aliphatic amines in methanol and cyclohexane and their hydrochlorides in methanol have been derived. A characteristic feature of these additivity parameters is the individual equation can be applied to both open‑chain and rigid systems. The factors that influence the 15N chemical shift of these substances have been determined. A new method for evaluating conformational equilibria at nitrogen in these compounds using the derived additivity parameters has been developed. Conformational analyses of these substances have been worked out. In general, the results indicate that there are four factors affecting the 15N chemical shift of aliphatic amines, paramagnetic term (p-character), lone pair-proton interactions, proton-proton interactions, and molecular association. 




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



The cataloguing and encoding of toroidal fully resonant azulenoids  by the use of 

toroidal fully resonant benzenoid templates


Edward C. Kirby


Resource Use Institute, Darachbeag, 14 Lower Oakfield, Pitlochry, Perthshire PH16 5DS, Scotland UK




The azulene graph is a pentagon‑heptagon pair that has one edge in common. A fully resonant azulenoid is one for which a set of disjoint azulenes may be drawn that accounts for every vertex, and a considerable number of toroidal network patterns meet this definition.

For a meaningful reference code, especially for the simpler members, it is helpful to relate the structures to appropriately sized fully resonant toroidal benzenoids. To illustrate this procedure pictorially, an azulene graph is scaled to fit each >full= hexagon of the benzenoid, four vertices of the azulene and of the benzene being congruent in each case. By this means, if all local inter‑azulene patterns are the same, then eleven integers suffice to encode connectivity information. Otherwise, the method, while usually still useful, becomes more complex.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Electronic Structure of

halogenated diphenylmethanones


Leo Klasinc1, Berta Košmrlj2, Branka Kovač1 and Boris Šket2


1Ruđer Bošković Institute, Bijenička 54, Zagreb, Croatia


2University of Ljubljana, Faculty of Chemistry and Chemical Technology, Aškerčeva 5, Ljubljana, Slovenia




HeI photoelectron spectra of a series of 2-halo-substituted 1,3-diphenyl-1,3-diketones have been measured. The assignment of the spectra was made by comparison with photoelectron spectra of related compounds and by density functional theory calculations with the B3LYP hybrid functional. The effect of halogen substitution is discussed, and the results are correlated with the data obtained from their uv spectra. Namely, dibenzoylmethanes are known to exhibit strong absorption in the uv region and almost no fluorescence, which makes them good photostabilization agents.

Text Box:










Figure 1. 2-F 1,3-diphenyl-1,3-diketone.


1. S. Tobita, J. Ohba,  K. Nakagawa, H. Shizuka (1995) J. Photochem. Photobio. A  92: 61-67.

2. S.S. Kim, J.S. Lim, J.M. Lee, S.C. Shim (1999) Bull. Korean Chem. Soc. 20: 531–534.

3. B. Košmrlj, B. Šket (2000) J. Org. Chem. 65: 6890-6896.

4. T. Pasinszki, T. Veszpremi, M. Feher, B. Kovač, L. Klasinc, S.P. McGlynn (1992) Int. J. Quantum. Chem. Symp. 26: 443-453.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Metal Complexation of Thiacrown Ether Macrocycles by Mass Spectrometry

in Liquid and Gas Phase


Leo Klasinc1, Bogdan Kralj2, Kata Mlinarić-Majerski1, Marko Rožman1, Dunja Srzić1, Ines Vujasinović1 and Dušan Žigon2


1Ruđer Bošković Institute, HR-10002 Zagreb, Croatia


2 Jožef Stefan Institute, SI-1000 Ljubljana, Slovenia




Crown and thiacrown ethers are an important and widely used class of organic molecules particularly for chemical separations because of their selective binding of metal ions. Here we investigate the complexation of Cu, Cd and Hg cations with following structurally related thiacrown ether macrocycles:

Text Box: Text Box: Text Box:










in the liquid phase (fast atom bombardment, FAB, and electrospray ionization1, ESI) and gas phase (laser desorption/ionization, LDI) mass spectrometry. Special attention was paid to selectivity of formation and stability of the monomeric products, as well as to the addition of a second crown ether molecule.


1. S.M. Williams, J.S. Brodbelt, A.P. Marchand, D. Cal, K. Mlinarić-Majerski (2002) Anal. Chem. 74: 4423-4433.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Data mining and identification of

drug side-effects during clinical trials


Paško Konjevoda

Ruđer Bošković Institute, POB 180, Bijenička 54, HR-10002 Zagreb, Croatia




Clinical trials are important part of new drug testing. However, side-effects are common finding during clinical trials. The type and level of observed side-effects must be precisely defined and explained. Data mining techniques are efficient and robust systems capable of proactively gathering informations about the safety of new drugs. In our presentation we explain the concrete example of identification and explanation of side-effects during bioequivalence testing, and capability of data mining techniques to analyse this data and use it more effectively in comparison to classical statistics.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Progressive Multiple Sequence

Alignments from Triplets


Matthias Kruspe and Peter F. Stadler


Bioinformatics Group, Department of Computer Sciences, University of Leipzig, Härtelstraβe 16/18, D-04107 Leipzig, Germany




The quality of progressive sequence alignments strongly depends on the performance of the pairwise alignment steps which are necessary to obtain the final alignment. The correct identification of insertions or deletions that occurred during the evolutionary history of the taxa considered is crucial. This becomes fundamental particularly for rather divergent nucleotide sequences where the information content within the sequences is low. The pairwise examination of the sequences thus is not satisfactory in many cases.
We present here a novel multiple alignment tool which simultaneously aligns three sequences in every step and so increases the information transfer to the alignment. This additional information is used to reliably derive the underlying phylogeny of the sequences under consideration.

We use a natural gap cost model which deals with all different variations of gap openings or extensions for three sequences. Gap penalties are adjusted to global and local sequence properties. The phylogenetic history of the sequences is represented by a network which also determines the order of the three-way alignments.  To speed up computation time and limit memory equirements a divide-and-conquer approach is used. We find the obtained alignments more reliable compared to alignments built with other alignment tools such as ClustalW.  Because of the lesser information content compared to amino acid sequences, the benefit of our tool applies especially to RNA sequences. Furthermore we figure out that the natural gap costs model has a benefit compared to the quasi-natural gaps costs used in other algorithms.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Geometric and electronic structure of

carbon nanotube junctions


István László


Department of Theoretical Physics, Institute of Physics and Center for Applied Mathematics and Computational Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary




The electronic properties of carbon nanotubes are usually obtained with the help of the zone folding method, which   is based on the graphene electronic structure. The rolling up of the graphene changes, however, the angles and distances between the carbon atoms in the hexagonal network and in a more sophisticated calculation curvature effects and the corresponding hybridizations  must be taken into count. The electronic structure is modified also by possible non hexagonal polygons in the hexagonal network.  This is the case if we study the electronic properties of a nanotube junction. Nanotube junctions are possible candidates for building blocks in nanoscale electronic devices and in an ideal case each nanotube junction must contain at least six heptagons if other polygons are not allowed. If the chirality of the tubes of a junction is given there are also various possibilities for the positions of the heptagons and each of them must have different electronic properties. In this work the connection between the geometric and electronic structure of a junction will be studied using various tight-binding methods.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



An Introduction to Nonuniform

Random Variate Generation


Josef Leydold


Department for Statistics and Mathematics, University of Economics and Business Administration, Augasse 2-6, A-1090 Wien, Austria




This short course gives an comprehensive survey on nonuniform random variate generation. Besides basic principles for the univariate case we also describe methods for sampling from multivariate distributions. The emphasis of the course will be on so called automatic methods (also known as black box algorithms).




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



snRNA search without a magnifying glass


Manuela Lindemeyer1,2 and Peter F. Stadler1,2,3.4


1Interdisciplinary Center of Bioinformatics, University of Leipzig, Germany


2Bioinformatics, Department of Computer Science, University of Leipzig, Germany


3Institute for Theoretical Chemistry, University of Vienna, Austria


4The Santa Fe Institute, Santa Fe, New Mexico, USA




Everyone knows them: Introns. Eucaryots are very proud about them, because Procaryots do not have them. Higher Eucaryots have most reason for boasting, because their introns are longer and occur more often than in other Eucaryots. In evolution the step of processing transcribed DNA took a long way. Nowadays we know that producing mature mRNA requires at least three processes: capping 5'-end, intron splicing, and polyadenylating procedure. The spliceosom consists of proteins and RNA components. There are three mechanisms of splicing: The most common one consists out of snRNAs U1, U2, U5, U4/U6 and some different proteins. The minor spliceosome contains the snRNAs U11, U12, U5, U4atac/U6atac. The third type of splicing, trans-splicing, can be found in trypanosomes, nematodes and some urochordates, which uses SL RNA in addition to some of the normal spliceosomal RNA. All of them have a conserved Sm-binding site.

The sequence of snRNAs evolves much faster than their secondary structure1. The structure of snRNAs is consequently much more conserved than their sequences. Therefore it is useful to detect snRNAs in silico with pattern search algorithms. Programs using structure and  sequence informations will be presented and compared2,3. We will discuss which programs can be used efficently for snRNA search.

To discover more features of snRNAs we will answer different questions: How many copies of the various snRNAs can be found in different genomes? How many of them are pseudogenes? Can we find clusters?

We will present a first overview of the distribution of paralogous snRNAs genes in metazoa.


Acknowledgements. This work was supported in part by the DFG Bioinformatics Initiative.


1. L.J. Collins (2004) Lost in the RNA world. Ph.D. Thesis, Allan Wilson Center, Massey University, Palmerston North, New Zealand.

2. L.J. Collins, T.J. Macke, D. Penny (2004) Searching for ncRNAs in eukaryotic genomes: Maximizing biological input with RNAmotif. J. Integrated Bioinform. 6: 15.

3. A.F. Bompfünewerer, Ch. Flamm, C. Fried, G. Fritzsch, I.L. Hofacker, J. Lehmann, K. Missal, A. Mosig, B. Müller, S.J. Prohaska, B.M.R. Stadler, P.F. Stadler, A. Tanzer, S. Washietl, Ch. Witwer (2005) Evolutionary patterns of non‑coding RNA. Theor. Biosci. 123: 301-369.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005





István Lukovits


Surface Modification and Nanostructures, Chemical Research Center, Pusztaszeri ut 59-67, H-1025 Budapest, Hungary




Graphite seems to be the most "aromatic" structure, because it is an infinitely large, peri-condensed, benzenoid. Obviously the enumeration of the Kekulé structures in graphite cannot be accomplished by applying constructive methods. In this work various algorithms were proposed in order to determine the number of Kekulé structures in different graphite sheets. Based on the techniques used to enumerate Kekulé structures, the number of conjugated circuits was also determined. The results indicate that the numerical value of the resonance energy per electron (REPE) strongly depends on the shape of the actual graphite sheet. The values of the REPE as obtained for graphite sheets were compared with values obtained for nanotubes and nanobelts. The latter structures seem to be somewhat less "aromatic" than infinite graphite.


1. I. Lukovits (2004) Resonance energy in graphite. J. Chem. Inf. Comput. Sci. 44: 1565-1570.

2. I. Lukovits, A. Graovac, E. Kálmán, Gy. Kaptay, P. Nagy, S. Nikolić, J. Sytchev, N. Trinajstić (2003) Nanotubes: Number of Kekulé structures and aromaticity. J. Chem. Inf. Comput. Sci. 43: 609-614.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Stabilizing Residues in Proteins


Csaba Magyar1, Michael M. Gromiha2 and István Simon1


1Institute of Enzymology, Karolina 29, Budapest, H-1113 Hungary


2CBRC AIST, 2-42 Aomi, Koto-ku, Tokyo 135-0064 Japan



Stabilizing Residues (SRs) are defined by combining several methods based mainly on the interactions of a given residue with its spatial, rather than its sequential neighborhood and by considering the evolutionary conservation of the residues. SRs are expected to play key roles in the stabilization of proteins. A residue is defined as a Stabilizing Residue if it is highly conserved, it has high Surrounding Hydrophobicity, high Long Range Order values and it belongs to a Stabilization Center. As thermodynamic and kinetic experiments showed, stabilizing residues identified by our algorithm have a significant role in the stabilization of protein structures. We present a public web server which identifies SRs in proteins with 3D structures and some applications of the SR concept.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Application of Random Matrix Theory

to BioSignal Analysis


Mladen Martinis and Vesna Mikuta-Martinis


Theoretical Physics Division, Ruđer Bošković Institute, Zagreb, Croatia




Some characteristic properties of fluctuating biosignals are compared with the predictions of the Random Matrix Theory (RMT). The aim is to connect the structures of these fluctuations with the universality classes of RMT, in order to distinguish healthy and unhealthy states of the human body. In particular, the statistical properties of random matrices with critical spectra will be considered, as well as the properties of random determinants and random distances between eigenvalues. A possible application of the RMT to ECG, EEG signals and to DNA microarray data will be indicated.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Non-vibrational features in  NIR FT-Raman

spectra of lanthanide sesquioxides


Zlatko Meić and Tomislav Biljan


Faculty of Science, Department of Chemistry, Strossmayerov trg 14, 10000 Zagreb, Croatia




The majority of lanthanide sesquioxides and yttrium sesquioxide show additional bands in FT-Raman spectra1 (after the excitation with the 1064 nm line of a Nd:YAG laser) that cannot be explained by vibrational origin. Additional bands in the FT-Raman spectra of heavy lanthanide sesquioxides appear in the Stokes region of the spectrum, but there are also some very strong unexpected bands in the anti-Stokes region of some light lanthanide sesquioxides and in yttrium sesquioxide, notably around 800 and 1100 cm-1. The non-vibrational bands observed in FT-Raman spectra of lanthanide sesquioxides and yttrium sesquioxide are not seen in Raman spectra excited in the visible region. A possible origin of these additional bands is in luminescence of lanthanide ions. Strong anti-Stokes bands observed in FT-Raman spectra of Y2O3, La2O3, Gd2O3 and Lu2O3 are explained by NIR luminescence of Yb3+ impurities present in sesquioxides after the excitation with the 1064 nm line of a Nd:YAG laser.

Figure 1. FT-Raman spectra of some lanthanide  sesquioxides.



1. T. Biljan, S. Rončević, Z. Meić, K. Kovač (2004) Chem. Phys. Lett. 395: 246–252.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



CASE via MS: Generation of Molecular Structures and Structure Ranking by Mass Spectra


Markus Meringer


Department of Medicinal Chemistry, Kiadis B.V., Zernikepark 6-8, 9747 AN Groningen, The Netherlands





Computer-aided structure elucidation (CASE) is of immense importance for present-day drug discovery programs. Thanks to modern screening methods a large number of biologically active compounds can be found in a short time. Structure elucidation then becomes a serious bottleneck in the drug discovery workflow. Due to its high sensitivity mass spectrometry is still one of the analytical key methods for elucidation of unknown structures.

The following computational experiment is based on low-resolution electron impact mass spectra of small organic compounds, taken from the NIST MS library1. The aim was to evaluate the accuracy of a structure-ranking algorithm included in MOLGEN-MS2. The method is based on virtual fragmentation of a candidate structure and comparison of the fragments’ isotopic patterns against the spectrum of the unknown compound. This way a structure-spectrum compatibility matchvalue is computed, ranging from 0 (no match) to 1 (perfect match). Of special interest was the matchvalue’s ability to distinguish between the correct and false constitutional isomers.

Therefore a quality score was computed in the following way: For a (randomly selected) spectrum-structure pair from the MS library all constitutional isomers were generated using the structure generator MOLGEN3. For each isomer the matchvalue with respect to the library spectrum is calculated4, and isomers are ranked according to their matchvalues. The quality of the ranking can be quantified in terms of the relative ranking position:


This procedure was repeated for 100 randomly selected spectrum-structure pairs that fulfilled certain conditions (molecular mass ≤ 200 amu, 1 < number of isomers ≤ 10000). In this first approach an average RRP of 0.30 was computed. More sophisticated algorithms for virtual fragmentation5,6 raise hope for better ranking results. In combination with more accurate high‑resolution MS/MS techniques this could pave the way towards automated structure elucidation via mass spectrometry.


9.     NIST/EPA/NIH Mass Spectral Library, NIST '98 version. U.S. Department of Commerce, National Institute of Standards and Technology.

10.  A. Kerber, R. Laue, M. Meringer, K. Varmuza (2001) MOLGEN-MS: Evaluation of Low Resolution Electron Impact Mass Spectra with MS Classification and Exhaustive Structure Generation. Advances in Mass Spectrometry, Vol. 15, Wiley, pp. 939-940.

11.  T. Grüner, A. Kerber, R. Laue, M. Meringer (1998) MOLGEN 4.0. MATCH Commun. Math. Comput. Chem. 37: 205‑208.

12.  M. Meringer (2004) Mathematical Models for Combinatorial Chemistry and Molecular Structure Elucidation. Logos‑Verlag Berlin (in German).

13.  MassFrontier 4.0. HighChem, Ltd., Bratislava, Slovakia.

14.  ACD/MS Fragmenter. Advanced Chemistry Development, Inc., Toronto, Canada.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005







Institute for Medical Research and Occupational Health, HR-10001 Zagreb, P. O. B. 291, Croatia




The method of overlapping spheres (OS) for the estimation of the stability constants of coordination compounds is based on the calculation of the overlapping volume, V*, of the central sphere (situated at the central or a bite atom) and the van der Waals spheres of surrounding atoms. The most critical parameter in the model is the radius of the central sphere, Rv, which is usually taken to be 3 or 4 Å.

Here we propose the model with variable value of the Rv parameter, making it to correspond to its value at the maximum of the volume density, V*(Rv)/ (4/3 Rv3 π), for each molecule. For the training set (N = 14) the range of maximum volume densities and Rv are 0.556 – 0.722 and 2.7 – 3.0 Å, respectively.

Linear regressions of stability constants (log K1) for mono-complexes of diamines with copper(II) and nickel(II) measured at 0 and 25 oC on overlapping spheres volume, V*, yielded slightly, but significantly better results than calculations done at Rv = 3 Å. The four regression yielded <r> = 0.786 (r = 0.677 – 0.858) and <r> = 0.796 (r = 0.681 – 0.878) for the constant and variable Rv value, respectively. Also, test on five copper(II) mono-complexes (25 oC), not included in the training set, gave residuals in the range 0.01 – 0.71 (r.m.s.  = 0.33) log K units, which is slightly better than the results for Rv = 3 Å (range = 0.02 – 0.75, r.m.s. = 0.34 log K units), but substantially better than results obtained with Rv = 4 Å (range 0.15 – 0.91, r.m.s. = 0.48 log K units). In spite of the slight differences, which should be mostly attributed to Rvvar ≈ 3Å, the proposed approach seems to be sound for the further improvement of the model.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Computational identification of ncRNAs in Invertebrates by comparative genomics


Kristin Missal1, Dominic Rose1 and Peter F. Stadler1,2


1Bioinformatics Group, Department of Computer Science, University of Leipzig, Germany


2Institute for Theoretical Chemistry, University of Vienna, Austria




The analysis of animal genomes showed that only a minute part of their DNA codes for proteins.  Recent experimental results agree, however, that a large fraction of these genomes is transcribed and hence is probable functional at the RNA level1. A computational survey of vertebrate genomes had predicted thousands of previously unknown ncRNAs with evolutionary conserved secondary structures4. To extend this comparative studies to invertebrates is difficult, since most ncRNAs evolve relatively fast at the sequence level while conserving their characteristic secondary structures. Hence, independent screens in invertebrates are necessary. Urochordates, as the sister group of vertebrates, that do not share the genome duplications, are of particular interest in this context. The genomes of two ascidians, Ciona intestinalis and Ciona savignyi have been sequenced, and a third project for the larvacean Oikopleura dioica is on the way. This gives us sufficient data and annotation to screen for evolutionary conserved ncRNAs in urochordates independent of vertebrates. Further the genomes of two nematodes, Caenorhabditis elegans and Caenorhabditis briggsae, are also available providing us with an additional separate set of prostostomia which gives us an extended spot of ncRNA evolution. 

Here, we report computational screens for evolutionary conserved ncRNAs in two independent sets of invertebrates: C. intestinalis, C. savignyi, O. dioica and C. elegans, C. briggsae. First, we identify conserved ncDNA regions (excluding regions annotated to be repetitive) by BLAST alignments. Conserved ncDNA regions with short distance between are combined considering consistence checks. Global alignments of those regions using CLUSTALW are computed. These alignments are screened with RNAz3 to detect regions that are also conserved at the secondary structure level.


1. J.S. Mattick (2004) RNA regulation: a new genetics?. Nat. Rev. Genet. 5(4): 316-323.

2. T.M. Lowe, S.R. Eddy (1997) tRNAscan-SE: A program for improved detection of transfer RNA genes in genomic sequence. Nucl. Acids Res. 25: 955-964.

3. S. Washietl, I.L. Hofacker, P.F. Stadler (2005a) Fast and reliable prediction of noncoding RNAs. Proc. Natl. Acad. Sci. USA 102: 2454-2459.

4. S. Washietl, I.L. Hofacker, P.F. Stadler (2005b) Thousands of noncoding RNAs with conserved structure in mammalian genomes. (in review).




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Symmetry Properties of Tetraammine

Platinum(II)  with C2v and C4v point groups


Ghorban Ali Moghani and Ali Reza Ashrafi


Department of Mathematics, Faculty of Science, Payame Noor, University, Tehran, Iran




Let G be a weighted graph with the adjacency matrix A = [aij]. An Euclidean graph associated to a molecule is defined by a weighted graph with the adjacency matrix D = [dij], where for ij, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent.

Otherwise, one may introduce different weights for different nuclei. Balasubramanian computed the Euclidean graphs and its automorphism groups for benzene, eclipsed and staggered forms of ethane and eclipsed and staggered forms of ferrocene (see Chem. Phys. Letters (1995) 232: 415.). In this work a simple method is described, by means of which it is possible to calculate the automorphism group of weighted graphs. We apply this method to compute the symmetry of tetraammine platinum(II) with C2v and C4v point groups.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Optimizing Barbeques and Tanimoto Scores with Applications in Regulatory Motif Discovery


Axel Mosig1,2, Sonja J. Prohaska1,2 and Peter F. Stadler1,2,3,4


1Interdisciplinary Center of Bioinformatics, University of Leipzig, Germany


2Bioinformatics, Department of Computer Science, University of Leipzig, Germany


3Institute for Theoretical Chemistry, University of Vienna, Austria


4The Santa Fe Institute, Santa Fe, New Mexico, USA




Understanding the mechanisms of gene expression is a major challenge of current genomics. On the level of transcription, gene expression in Eukaryotes is known to be regulated by several transcription factors binding to clusters of respective binding sites. These binding site clusters are also referred to as cis-regulatory modules.

We propose multiple-alignment like pattern discovery methods that allow to discover putative regulatory modules. Our algorithms are based on detecting clustered occurrences of binding sites that occur simultaneously in several genomic sequences. Computationally, this is achieved through solving the so-called best barbeque problem1 in combination with Tanimoto scores, which were originally introduced as a measure for feature-based molecular similarity2 and have recently been proposed as a similarity measure between regulatory modules3.


Acknowledgements. This work was supported by the DFG Bioinformatics Initiative.


1. A. Mosig, T. Biyikoglu, S. J. Prohaska, P. F. Stadler (2004) Discovering cis-regulatory modules by Optimizing Barbeques. (submitted)

2. P. Perco, A. G. Mayer, A. Lukas, R. Oberbauer, B. Mayer, (2005) A genetic algorithm to derive joint promoter modules in coexpressed genes. (submitted)

3. J.B.O. Mitchell (2001) The relationship between the sequence identities of α-helical proteins in the PDB and the molecular similarities of their ligands. J. Chem. Inf. Comp. Sci. 41: 1617-1622.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Relinking Marriages in Genealogy of Ragusan Noble Families between 12th and 16th Century


Andrej Mrvar and Vladimir Batagelj


University of Ljubljana, Slovenia




Genealogies can be represented as graphs in different ways: as Ore graphs, as p-graphs, or as bipartite p‑graphs.

Advantages and disadvantages of all three presentations will be discussed. The emphasis will be given to all possible relinking marriages (blood and non-blood marriages) in p‑graphs containing 2 up to 6 vertices.

As an example we will take the genealogy of Ragusan (Dubrovnik) noble families between 12th and 16th century which contains records about around 6000 individuals.

We will compare the frequency distribution of  different types of relinkings to some other genealogies (genealogy of island Silba, genealogy of European royal families).

Analyses were done using program for analysis and visualization of large networks Pajek, which can be downloaded for free: http://vlado.fmf.uni‑lj.si/pub/networks/pajek/default.htm




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Toxicity of Aliphatic Ethers Revisited


Sonja Nikolić,1 Ante Miličević2 and Nenad Trinajstić1


1Ruđer Bošković Institute, P.O.B. 180, HR-10002 Zagreb, Croatia


2Institute of Medical Research and Occupational Health, P.O.B. 291, HR-10001 Zagreb, Croatia




There are in the literature several reports on QSAR study of toxicity of aliphatic ethers.  Authors of these reports used a variety of molecular descriptors: vertex-connectivity index c, the first-order and the second-order valence vertex-connectivity indices 1cv and 2cv, weighted identification number WID, Balaban indices J and Jhet and edge-connectivity index e.   We collected all these indices for 21 aliphatic ether and applied our CROMRsel modeling procedure.1,2  This is a multivariate procedure that has been designed to select the best possible model among the set of models obtained for a given number of descriptors, the criterion being the standard error of estimate.  The quality of models is expressed by fitted (descriptive) statistical parameters: the correlation coefficient (Rfit), the standard error of estimate (Sfit) and the Fisher's test (F).  The models are also cross(internally)-validated by a leave-one-out procedure. Statistical parameters for the cross-validated models are symbolized by Rcv and Scv, where subscript cv denotes the cross-validation.

                The best single-descriptors model is based on WID (Rfit=0.942, Rcv=0.920, Sfit=0.14, Scv=0.16), the best two-descriptor model is based on c and J (Rfit=0.966, Rcv=0.951, Sfit=0.10, Scv=0.13), the best three-descriptor model is based on 1cv, Jhet and e (Rfit=0.981, Rcv=0.968, Sfit=0.08, Scv=0.10) and the best four-descriptor model is based on c,1cv, J and Jhet (Rfit=0.983, Rcv=0.969, Sfit=0.08, Scv=0.10).  These last two models are comparable among themselves – our choice is the model with less descriptors.  The best models from the literature3,4 are based on the variable connectivity index 1 cf (Rfit=0.975, Sfit=0.10), and on Xu index5 and atom type descriptors (Rfit=0.987, Rcv=0.965, Sfit=0.07, Scv=0.12).


1. B. Lučić, N. Trinajstić (1999) J. Chem. Inf. Comput. Sci. 39: 121-132.

2. A. Miličević, S. Nikolić (2004) Croat. Chem. Acta 77: 97-101.

3. M. Randić, S. C. Basak (2001) J. Chem. Inf. Comput. Sci. 41: 614-618.

4. L. Xu (1996) Chemometrical Method, Scientific Press of China, Beijing.

5. B. Ren (unpublished).




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Tandem repeats in three dimensional

structures of proteins


Borut T. Oblak and Branko Borštnik


National Institute of Chemistry, Ljubljana, Slovenia




With growing amount of knowledge about the primary structure and three dimensional structure of proteins it is becoming more and more evident that an appreciable fraction of proteins does not possess well defined three dimensional structure. Such proteins usually contain tandem repeats which are usually associated with other sequential and functional features which are the consequence of rapid evolution on molecular scale. We searched the amino acid sequence file of the proteins whose structures are deposited in the Protein Data Bank for the tandem repeat patterns. Several filamentous proteins and other proteins were found which do not possess necessarily well defined three dimensional structure for the entire amino acid sequence. Tandem repeats are located by rule in the regions where increased mutability is exhibited. The repeats are the results of the repeat amplification mechanisms which are the fundamental driving force of molecular evolution. Slippage mutations, transpositions and related events generate repetitious sequences in the coding regions of genes.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



A Calculation and a Representation of Edge‑Transitive Maps of Small Order


Alen Orbanić and Tomaž Pisanski


IMFM, Jadranska 19, SI- 000 Ljubljana, Slovenia




We present a calculation of non-degenerate edge-transitive maps2 up to 100 edges which is at the present time a computationally hard problem. A computation was performed by use of program LOWX made by Conder and Dobcsanyi1. We have built in an optimization into the procedure for searching  factor groups up to some order, which allows exclusion of factor groups having some forbidden relations and thus optimizing a running time of the algorithm for up to 50%. 

Since these maps have a high symmetry, we consider several automated procedures to draw their graphical representations with highly symmetric drawings3. Good drawings are a very useful tool in the research of the maps.

Some chemical structures, like fullerenes, can be described as maps of high symmetry. A calculation and determination of such maps can yield to better understanding of topology of some chemical structures. 


1. M. Conder, P. Dobcsanyi (2001) Determination of all regular maps of small genus. J. Comb. Theory B 81: 224-242.

2. J.E. Graver, M.E. Watkins (1997) Locally finite, planar, edge-transitive graphs. Mem. Am. Math. Soc. 126: 601.

3. T. Pisanski et al. Project Vega, http://vega.ijp.si




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Constructing convex polyhedra


Igor Pak


Department of Mathematics, MIT, Cambridge, MA 02139, USA




We investigate the problem of constructing polyhedra from their edge length.  In a joint work with Fedorchuk we obtained sharp upper bounds for the degrees of Sabitov polynomials having the diagonal lengths as its roots. Knowing all the diagonal lengths one can easily construct the polyhedron. We then ask about flexible polyhedra and their structure, discuss several constructions and how they can be generalized. 


Acknowledgements. We would like to thank Bob Connelly and Maksym Fedorchuk for helpful discussions. The author received funding from the NSF.  


V. Alexandrov (1995) New example of a flexible polyhedron, Sib. Math. J. 36: 1215-1224.

B. Connelly (1979) The rigidity of polyhedral surfaces, Math. Mag. 52: 275-283.

M. Fedorchuk, I. Pak (2005) Rigidity and polynomial invariants of convex polytopes Duke Math. J. (to appear).

I. Pak (2005) The area of cyclic polygons: recent progress on Robbins conjectures Adv. Applied Math.(to appear).

I. Kh. Sabitov (1998) The volume as metric invariant of a polyhedra Disc. Comp. Geom. 20: 405-425.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



An effective potential for the natural spinorbitals


Katarzyna Pernal1,2


1Section Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands


2Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland




It is well known that in the Kohn-Sham (KS) formulation of the density functional theory the variational principle leads to one-electron equations for the KS orbitals with the effective local potential. Until now it has not been clear if an analogous effective potential exists for the natural spinorbitals and how to construct it explicitly. This problem is of great importance for the density matrix functional theory (DMFT), where the total energy is expressed in terms of the one-electron reduced density matrix (1-matrix).

It will be presented how to construct an effective nonlocal potential that, for a given set of occupation numbers, produces an optimal set of natural spinorbitals. A problem of its nonuniqueness in the case of degenerate 1-matrices will be discussed. If the Hamiltonian is perturbed with a static perturbation, one‑electron equations for the natural spinorbitals together with the Lagrange equations for the natural occupancies serve as a basis to derive coupled-perturbed equations that yield a response of the 1-matrix to a perturbation. This leads in a straightforward manner to calculations of the second-order response properties of a system described by a given density matrix functional. Some results of the dipole-dipole polarizabilities will be shown for the recently proposed density matrix functionals.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Einstein: from Special Theory of Relativity (1905), through General Theory of Relativity (1916),

until Cosmological Term (1917)


Krunoslav Pisk


Ruđer Bošković Institute, POB 180, HR-10002 Zagreb, Croatia



1905 – Special  Theory of Relativity (STR)

About uniform motion in empty space and relative motions with uniform translations among frames of reference. The speed of light is constant in inertial frames. A new concept of space and time: space-time continuum as an integral entity. Lorentz transformations as coordinate links among different frames. Relations among observables. Relativity of lengths and time intervals. Equations which express physical laws are form-invariant (covariant) against Lorentz transformations. A new relation between mass and energy, E = mc2, emerges naturally.


1916 – General Theory of Relativity (GTR)

Extension of STR by imposing that equations of law of physics are form-invariant against general space-time transformations (general covariance), which also include coordinate relations among accelerated frames. Accelerated frames and gravitational fields (principle of equivalence). Gravitation and curvature of space. Equations of GTR:

a) Einstein’s field equations - energy (mass) – distribution and corresponding pressures, as a source of curved space;

b) Einstein’s equations of motion – motion of material particle in gravitational field. GTR and Newton’s laws of motion and gravitation. 


1917- Cosmological Term

When applied to the Universe, solution of field equations of GTR are not static. Einstein modifies original equations by introducing a cosmological term (cosmological constant today) - λ, which is consistent with relativity  postulates. Now, a quasi-static distribution of matter in Universe is possible. The Universe appears a 3-dimensional curved space (with constant radius of curvature) and finite volume of 2π2R3, with λ = 1/R2.



Thirteen years ago (1930), after Hubble’s discovery of Universe expansion Einstein said that introduction of cosmological term was “the biggest blunder of my life”.

Recent measurements indicate that Universe accelerates and abandoned cosmological term could play important role – “dark energy”4.


1. A. Einstein (1905) On the Electrodynamics of Moving Bodies, Annalen der Physik 17: 891.

2. A. Einstein (1916) The Foundation of the GTR, Annalen der Physik 49: 769.

3. A.Einstein (1917) Cosmological Considerations of GTR, Sitzungsberichte der Preussischen Akad. d. Wissenschaften.

4. P.J.E. Peebles, B. Ratra (2003) Rev. Mod. Phys. 75: 607.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Variable topological indices in drug design


Matevž Pompe1 and Milan Randić2


1University of Ljubljana, Faculty of Chemistry and Chemical Technology, Aškerčeva 5, 10000 Ljubljana, Slovenia


2National Institute of Chemistry, Hajdrihova 19, 1000 Ljubljana, Slovenia




Quantitative structure-activity relationship (QSAR) studies have been successfully applied for the modeling of different important biopharmaceutical properties. During the first stage of drug design large data bases are usually screened for suitable candidates. In order to complete screening in the reasonable time a simple structural representation should be used which at the same time enables creation of reasonably accurate prediction models. Variable topological indices can be considered as good candidates because they are easily calculated, give good prediction models and enables structural interpretation of the obtained models. Several modifications of variable topological indices will be presented to account for modeling of large molecules as well as compounds where certain part of the molecule suppresses the modeled activity.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005







Robert Ponec


Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Prague 6, Suchdol 2, 165 02, Czech Republic




A simple model was proposed1 allowing to estimate the Brönsted exponents in acidobasic catalysis on the basis of the pK values of the species participating in the proton transfer process. The approach was tested using the experimental data on the basically catalysed halogenation of carbonyl compounds and on the proton removal from nitroalkanes. It has been shown that the model is able to reproduce the Brönsted exponents not only in the case of "ordinary" Brönsted plots with the slope within the expected range 0 – 1 but also for unusual plots with negative slope. In addition to this the proposed model opens the possibility of the calculation of the activation energies of a given proton transfer reaction and also provides straightforward theoretical justification for the validity of Hammond postulate in these reaction.


R. Ponec (2004) Collect. Czech. Chem. Commun. 69: 2121.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Rearrangement of Regulatory Elements


Sonja J. Prohaska1,2, Axel Mosig1,2 and Peter F. Stadler1,2,3.4


1Interdisciplinary Center of Bioinformatics, University of Leipzig, Germany


2Bioinformatics, Department of Computer Science, University of Leipzig, Germany


3Institute for Theoretical Chemistry, University of Vienna, Austria


4The Santa Fe Institute, Santa Fe, New Mexico, USA



"Phylogenetic footprinting"1 is the latest method to find regulatory elements, which are conserved in non-coding sequences around orthologous and functionally equivalent genes. Therefore, these elements can be detected by a comparative genomic approach based on sequence alignment. The result is a set of conserved 'footprints' that are in the same order and orientation in at least two of the input sequences. Since regulatory elements are rather insensitive to shuffling of single modules, putative protein binding sites might be missed by a simple comparative approach. We expanded the footprinting approach and look for rearranged footprint clusters, disregarding the order and orientation of the individual motifs. The main purpose of this contribution will be to interpret the biological relevance of this approach when applied to the duplicated Hox cluster sequences, that show a high rate of footprint remodeling.

We detect rearranged footprint clusters by solving the best barbeque problem2 for a given set of motifs and a set of orthologous sequences, that are supposed to share a common set of motifs. This means, we use bbq to look for the maximal set of footprints within a cluster of defined length L that occurs in all N sequences. The required set of motifs is either taken from transfac matches to the sequences or decomposed footprint clusters.

An application to the duplicated Hox cluster sequences – up to eight in teleost fishes and four in all other vertebrates – will shed light on the importance of methods like bbq for the detection of reorganized regulatory elements and rise ideas on the dynamics of structural organization of these elements.


Acknowledgements. This work was supported in part by the DFG Bioinformatics Initiative.


1. S.J. Prohaska, C. Fried, C. Flamm, G.P. Wagner, P.F. Stadler (2004) Mol. Evol. Phylog. 31: 581­-604 .

2. A. Mosig (2005)  A tool for discovering regulatory modules using weighted barbeques. (submitted).




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Graphical Representation of Proteins as Four-Color Maps and Their Numerical Representation


Milan Randić1,, Alexandru T. Balaban2 and Dejan Plavšić3


1National Institute of Chemistry, P.O. Box 3430, 1001 Ljubljana, Slovenia


2Texas A & M University at Galveston, Galveston, TX 77553, USA


3The Ruđer Bošković Institute, P.O.Box 180, HR-10002 Zagreb, Croatia




We put forward a novel 2-D graphical representation of primary structure of proteins that not only allows visual inspection of similarities and dissimilarities between proteins, but also leads to their numerical characterization. The novel representation is based on the four‑color map obtained from consideration of the virtual genetic code. The approach is illustrated with the A-chain of human insulin.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005





Guillermo RESTREPO


Laboratorio de Química Teórica, Universidad de Pamplona, Pamplona, Colombia




A general problem of the studies of Cluster Analysis (CA) is the one related to the selection of the clusters in the dendrogram (tree). There are several methods for selecting clusters in a tree, some of them are the “Phenon line”1, the amalgamation coefficient2 and some others.  However, the most common method in order to look for clusters is the Phenon line but it is a subjective procedure since it includes the previous knowledge of the researcher regarding the set Q under study. On the other hand, taking into consideration the uses of CA as a tool for determining similarity relationships among the elements of Q, we consider that a criterion for selecting the clusters of a dendrogram should include the similarities among the elements3. In order to build up a method for selecting clusters in a tree we developed two mathematical procedures free of subjectivities. Both of them are based on the chemotopological concept of maximal n-subtree4. This concept allows selecting a particular sort of branches (clusters) of the dendrogram for a given value of  (|Q| means the cardinality of the set Q).  Thus, each selection of an n generates a partition5 on the original set of elements.  The first method3 considers the number of clusters for a given n (TSPn) and their geometrical population (GPn). In this way we build up the selection number3 , whose maximum value allows determining the partition of the set Q that offers the clusters of Q. The second procedure is based on Mathematical Information Theory6. This approach allows selecting the clusters of a dendrogram taking into account the information content of each possible cluster of the dendrogram. Finally, we compare both methods for selecting clusters applying each of them to the CA results of 72 chemical elements (Z=1-86, omitting Z=58-71). Thus, we show that the Mendeleevian groups of the periodic table correspond to the clusters that show the maximum value of similarity information content.


Acknowledgements. We thank Dr. Mesa from the Universidad del Valle (Colombia) for his mathematical support and Dr. Llanos from the Observatorio Colombiano de Ciencia y Tecnología (Colombia) for programming some of these ideas.  Finally, we thank the Universidad de Pamplona in Colombia for its financial support and especially, Dr. A. González, Head Dean of this University, for his interest in the development of the mathematical chemistry.


1. P.H.A. Sneath,  R.R. Sokal (1973) Numerical Taxonomy: The Principles and Practice of Numerical Classification., W.H. Freeman Comp., San Francisco, p. 573.

2. Q. Liao, B. Manteuffel, C. Paulic, D.J. Sondheimer (2001) Emot. Behav. Disor. Spring.

3. G. Restrepo, E.J. Llanos, J.L. Villaveces (2005) In: Fourth Indo-US Workshop on Mathematical Chemistry (S. Basak, D.K. Sinha, eds.), University of Pune, Pune, India, pp. 39-40.

4. G. Restrepo, H. Mesa, E.J. Llanos, J.L. Villaveces (2004) J. Chem. Inf. Comput. Sci. 44: 68-75.

5. S. Lipschutz (1965) General Topology, McGraw-Hill, New York, p. 226.

6. C.E. Shannon (1948) Bell Syst. Tech. J. 27: 379-423, 623-656.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



The Preparation of  Some Novel Indazole

Derivatives by Using Chalcones


Javad Safaei-Ghomi and Zohreh Alishahi


Department of Chemistry, Faculty of Science, University of Kashan, 87317-51167  Kashan, I. R. Iran



Indazole and its derivatives have little biological significance and have not been found in natural products due to the difficulty for living organisms to construct an N-N bond. Indazole derivatives exhibit variety of pharmacological properties such as anti-inflammatory, antidepressant, antitumor, antiarthritic and analgesic activities1. Different synthetic pathways generate these compounds. For instance, ring closure of pyrazole moiety, addition of hydrazine derivatives to carbonyl compounds2 and cycloaddition reaction3. Herein, we report the synthesis of some new indazole derivatives by using chalcones as starting materials.

In this procedure chalcones were reacted with ethyl acetoacetate in a convenient method for the Michael addition followed by intramolecular aldolization catalyzed by K2CO3 under ultrasound irradiation. This procedure brings advantages such as mild reaction conditions, shorter times and reducing of starting materials. Reaction of prepared 3,5-diaryl-6-ethoxycarbonyl-2-cyclohex-1-one derivatives 1 with hydrazine hydrate or phenyl hydrazine gave corresponding indazole derivatives 2. The structures of the synthesized compounds were assigned on the basis of spectral data.



1. P.G. Balardi et al., Synthesis 1997: 1140.

2. E.B. Usova, L.I. Lysenko, G.D. Krapivin (2000) Molecule 5: M 128.

3. S. Matsugo, A. Takamisawa, Synthesis 1983: 852.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



De Revolution VII Six Dimensions


James Sawyer


Six Dimension Design 241 1/2 Lexington Ave., Buffalo, 14222 New York, USA



To His Holiness, Pope John Paul II in Heaven & Pope Benedict XVI on Earth & Pontifical Academy of Sciences




De Revolution I – XVII Series of Books is inspired by the teachings of Nicholas Copernicus and his original De Revolution I thru VI series of Books. His high level of scholarly development of the revolution of the Heliocentric Universe Theory used a foundation of triangular spherical geometry.  This geometry aided in the discovery of the present day space theory. This geometry can be advanced to the present day conception of atomic space within the atom and applied to mapping space travel for the next 500 years. This book opens the mind to triangular geometry and space in nature thru my eyes.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Software implementation for covariance

 NMR spectroscopic investigations


Wolfgang Schöfberger1, Vilko Smrečki2 and Norbert Müller1


1Johannes Kepler University Linz, Institute of Organic Chemistry, Altenbergerstraße 69 4040 Linz,  Austria


2Ruđer Bošković Institute, NMR-Center, Bijenička c. 54, HR-10002 Zagreb, Croatia




Covariance nuclear magnetic resonance (NMR) spectroscopy1,2 provides an effective way for establishing nuclear spin connectivity in molecular systems. The method, which identifies correlated spin networks in terms of covariance between 1D traces, facilitates delineation of spin networks and allows a probable interpretation of correlation reliability. We have implemented software that integrates into the routine workplan. The application of covariance NMR spectroscopy applied to HMBC experiments of novel carbohydrate derivatives is presented.


Acknowledgements. This work was supported by the Austrian Science Fund, project P15380, the Austrian-Croatian Joint Research Project 91102/03-06 (ÖAD-Project 13/04), and the Ministry of Science, Education and Sport of the Republic of Croatia Project 0098059.


1. R. Brüschweiler, F. Zhang (2004) Covariance nuclear magnetic resonance spectroscopy,  J. Chem. Phys. 120 (11): 5253-5260.

2. R. Brüschweiler (2004) Theory of covariance nuclear magnetic resonance spectroscopy, J. Chem. Phys. 121 (1): 409‑414.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Algebraic Kekulé Structures

and Bond Orders in Benzenoids


Jelena Sedlar1, Ivana Anđelić2, Damir Vukičević3,

Ante Graovac2, 3 and Ivan Gutman4


1Faculty of Civil Engineering, Matice Hrvatske 15, HR-21000 Split, Croatia


2Ruđer Bošković Institute,POB 180,  Bijenička c. 54, HR-10000 Zagreb, Croatia


3Faculty of Natural Sciences, Mathematics and Education, Nikole Tesle 12, HR-21000 Split, Croatia


4Faculty of Science, POB 60, 34000 Kragujevac, Serbia and Montenegro




Recently, a new interpretation of the Kekulé structures (KS) in benzenoid systems was proposed. A new quantity, Algebraic Kekulé Structure (AKS), that counts how many p electrons for a given KS belong to each hexagon has been defined1. In this way a possibility to characterize hexagons numerically was offered for the first time.

By knowing the AKS for all Kekulé structures gives us a chance to define a new, AKS bond order2. This new order is compared with the classical Pauling bond order and experimentally measured bond lengths in a series of benzenoid systems. Preliminary results show that both bond orders are approximately of the same quality.


1. I.Gutman, D. Vukičević, A. Graovac, M. Randić (2004) J. Chem. Inf. Comp. Sci. 44:  296-299.

2. J. Sedlar, I. Anđelić, D. Vukičević, A. Graovac, I. Gutman (in preparation).




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Intrinsically Unstructured Proteins


István Simon


Institute of Enzymology, Biological Research Center, Hungarian Academy of Sciences,

H-1518 Budapest, PO Box 7, Hungary




There is fast growing evidence that the unstructured state, common to all living organisms, is essential for basic cellular functions; thus it deserves to be recognized as a separate functional and structural category within the protein kingdom. In this lecture recent findings in this area are surveyed including some our recent work about preformed structural elements involved in partner recognition by intrinsically unstructured proteins, the way pairwise energy content estimated from amino acid composition discriminates between folded and intrinsically unstructured proteins and about our web server, IUPred, for the prediction of intrinsically unstructured regions of proteins based on estimated energy content.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



RNA Consensus Structures in

Molecular Morphology


Roman R. Stocsits1 and  Peter F. Stadler1,2


1Interdisciplinary Centre for Bioinformatics, Härtelstrasse 16-18, D-04107 Leipzig, Germany


2Bioinformatics Group, Department of Informatics, University of Leipzig, Germany



Functional non-coding RNA molecules perform vital functions in the cell and are a major source of data in molecular phylogenetics. But the peculiarities of RNA evolution, and in particular the long term conservation of their secondary structures, have been taken into account only recently. Because the functions of these molecules are mediated by their structures, selective pressure acts on these structures. But structural conservation is not necessarily mediated by sequence conservation: Many different sequences can fold into the same (functional) structure. On the other hand, only little changes in the nucleic acid sequences may cause large changes in secondary structures. Assuming independent evolution of each sequence position is not a good approximation because the highly conserved secondary structure of these molecules introduces strong correlations between the two strands of a helix.

The problem arises that variable sequences may obscure phylogenetic signals and lead to overestimated reliability of sequence based trees and artefacts in phylogeny reconstruction. Therefore, phylogeny reconstruction methods can be extended to RNA secondary structure, incorporating the slower evolution of the structural features, provided a good model for the secondary structure elements of the RNAs.

For obtaining the best estimate of the secondary structure of each individual sequence we use a consensus structure of the complete set of related sequences as a constraint for the RNA folding algorithm: basic structural features that are represented in all sequences guide the individual sequence folding process to a stabilized, more realistic prediction.

Figure 1. Individual sequences are folded with support by a consensus constraint, a set of basic structural features that are present in all sequences (mitochondrial trp-tRNA in this example).

Left: wrong prediction without constraint. Right: correct prediction after constraint directed folding.


In this talk I present first results of our pilot-studies with the aim to utilize the evolution of secondary structures when reconstructing phylogenies. Our goal is a systematic investigation: Starting from mitochondrial tRNA and rRNA with high phylogenetic information content, we plan to extend our methods to other types of non-coding RNAs that need to be re-evaluated by explicitly taking into account conserved functional secondary structure elements.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Intrinsic proofs for area and

circumradius of  cyclic hexagons.

Solving equations for arbitrary cyclic polygons


Dragitin Svrtan


Department of Mathematics, University of Zagreb, Bijenička c. 30, HR-10002 Zagreb, Croatia





Finding formulas for the area or circumradius of polygons inscribed in a circle in terms of side lengths is a classical subject. For the area of a triangle we have famous Heron formula and for cyclic quadrilaterals we have Brahmagupta’s formula. A decade ago D. P.Robbins3 found a minimal equations  satisfied by the area of cyclic pentagons and hexagons by a method of undetermined coefficients and he wrote the result in a nice compact form. The method he used could hardly be used for heptagons due to computational complexity of the approach. In another approach with two collaborators2 a concise heptagon/octagon area formula was obtained recently (not long after D.P.Robbins premature death). This approach uses covariants of binary quintics. It is not clear if this approach could be effectively used for cyclic polygons with nine or more sides. A nice survey on this and other Robbins conjectures is written by I. Pak4.

In this talk we shall present  an intrinsic proof of  the Robbins formula for the area (circumradius and area times circumradius) of cyclic hexagon based on an intricate direct elimination of diagonals (the case of  pentagon was treated in Ref. 5) and using a new algorithm from Ref. 6. In the early stage we used computations with MAPLE (which sometimes lasted several days!). Next we shall explain a simple quadratic system, which seems to be new, for the circumradius and area of arbitrary cyclic polygons based on a Wiener‑Hopf factorization of a new Laurent polynomial invariant of cyclic polygons. Explicit formulas for the circumradius (and less explicit for the area) of cyclic heptagons and cyclic octagons are obtained. We hope to apply recent new resultant formulas of  Eisenbud  et al. in our approach to cyclic polygons.


Acknowledgements.  We would like to thank Darko Veljan and Vladimir Volenec for helpful discussions.


1. A.F. Möbius (1828) Über die Gleichungen, mittelst welcher aus der Seiten eines in einen Kreis zu  beschriebenden Vielecks der Halbmesser des Kreises un die Flahe des Vielecks gefunden werden, Crelle's J. 3: 5-34.

2. F. Miller Maley, D.P. Robbins, J.  Roskies, On the areas of cyclic and semicyclic polygons,  math. MG/0407300v1.

3. D.P. Robbins (1994) Areas of polygons inscribed in a circle, Discrete Comput. Geom. 12: 223-236, 1994.

4. I. Pak, The area of cyclic polygons: Recent progress on Robbins conjectures, Adv. Applied Math. (to appear).

5. D. Svrtan, D.Veljan and V. Volenec, Geometry of pentagons: From Gauss to Robbins, math. MG/0403503.

6. D.Svrtan (2005) A new approach to rationalization of surds. (submitted).

7. D.Svrtan (2005) Intrinsic proof of Robbins formula for the area of cyclic hexagons. (submitted).

8. D.Svrtan (2005) Equations for the circumradius and area of cyclic polygons via Wiener-Hopf factorization. Computational aspects and some new formulas. (in preparation).




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Molecular recognition of

the complementary peptide structures


Nikola Štambuk1, Paško Konjevoda1, Nikola Gotovac2 and Biserka Pokrić1


1 Ruđer Bošković Institute, Bijenička cesta 54, POB 180, HR-10002 Zagreb, Croatia


2Department of  Radiology, General Hospital Požega, Osječka bb, HR-34000 Požega, Croatia




Ligand-receptor peptide interaction is investigated considering Molecular Recognition Theory and the secondary protein structure. The theoretical concept of molecular recognition based on the side chain polarity modelling is discussed in the context of the protein structure and possible function.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005





Serge S. Tratch1, Marina S. Molchanova2 and Nikolai S. Zefirov1


1 Department of Chemistry, Moscow State University, Moscow 119992, Russia


2 Zelinskii Institute of Organic Chemistry, Russian Acad. Sci., Moscow 119991, Russia




Degenerate transformations, i.e., chemical interconversions with identical initial and final structures, are classified on the basis of some permutation groups associated with the corresponding bond redistributions. These groups are the automorphism groups of certain labeled graphs that can be constructed from the more simple "topology identifying" graphs; edges of the latter graphs represent all bonds changing their multiplicity. If the chemical nature of reaction centers is not explicitly taken into account, then three classes of degeneracy are distinguished. For regularly degenerate (R) transformations, at least one permutation converting the initial structure(s) into the final one(s) belongs to the group of the labeled graph (and consequently to the group of the topology graph); actually, all known degenerate rearrangements belong just to this class. In the case of semiregular (S ) or irregular (I ) degeneracy, no "producing" permutations belong to group of the labeled or both graphs, respectively. Explicit consideration of the chemical nature of participating atoms is shown to result in conservation (RR, SS, II) reduction (RS, RI, SI) or complete disappearance of degeneracy.

Although several examples of S- and I-degeneracy were found without the aid of computer, systematic investigation was performed by means of our reaction-generating program ARGENT-1. (This program, also applicable to search for new types of nondegenerate processes, is demonstrated at the conference.) Some of the results thus obtained are of interest for chemists; the corresponding processes are considered in detail. Among theoretical problems, the interrelationship between regularity and self-inverseness and also problems of the actual existence for some R-, S-, and I-degenerate transformations are briefly discussed. For example, for cyclic topology graphs with 4-12 vertices or cubic topology graphs with 4-10 vertices, no examples of S- and R-degeneracy, respectively, are detected. The search for a counterexample to another problem (associated with degeneracy in charged systems) required as many as 201,748,864 edge-labeled cubic graphs to be constructed by ARGENT-1.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005






Serge S. Tratch and Nikolai S. Zefirov


Department of Chemistry, Moscow State University, Moscow 119992, Russia


An interest to quantification of many molecular characteristics (such as connec­tivity, symmetry, chirality, etc) has been significantly increased during last decades. The distance-based method, making it possible to calculate symmetry measures for any fi­nite point systems, was suggested by Zabrodsky et al.1 and then somewhat improved by Alikhanidi and Kuz'min2. In the both approaches, symmetry measure is associated with some symmetry element and the result obtained always depends on the actual disposition of this element in 3D space. In contrast, the computationally very simple distance-based technique originated in this report can be applied to any individual symmetry operation or to an arbitrary set of such operations; the operations in question are represented by permutations of p points with known co-ordinates xi, yi, zi, i =1 – p. More specifically, for some permutation π (converting points i and j into π(i) and π(j)), its inverse permu­tation π-1 converts the pair (π-1 (i), π-1 (j)) into original pair (i, j) and this fact allows to compare the known distance d(i,j)

with the average value


calculated for any given set of permulations π12,...,πn. The Geometry Deviation Index Δ (DGI) of the p-point system with respect to these n permutations can be then easily computed; the expressions for average Δ' and normalized Δ'' indices are also shown below.

      Δ = ,   

In practice, the point systems are formed from (all or only skeleton) atoms of organic structures and from vertices of polygonal or polyhedral figures. The sets of per­mutations needed for calculation of GDIs are typically represented by symmetric groups Sp, by automorphism groups of (unlabeled or labeled) graphs or configurations, and by preselected subgroups or cosets of a given group. Some numerical results obtained for spatial models of mathematical and molecular graphs are finally discussed in the lecture.


1. H. Zabrodsky, S. Peleg, D. Avnir, J. Am. Chem. Soc. 114 (1992) 7843; ibid 115 (1993) 8278.

2. S. Alikhanidi, V.Kuz'min, Zh. Strukt. Khimii 39 (1998) 548; J. Mol. Model. 5 (1999) 116.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005





Oleg Ursu and Mircea V. Diudea


“Babes-Bolyai” University, Faculty of Chemistry and Chemical Engineering, Arany Janos 11, 400028  Cluj, Romania




Molecular similarity is one of the most subjective concepts in Chemistry and can be defined in a multitude of ways. Here we present a method for calculating the similarity between pairs of chemical structures represented by 3D molecular graphs. The method is based on a graph matching procedure using a maximum common subgraph (MCS) detection algorithm to compute the exact degree and composition of similarity. The proposed MCS algorithm involves the maximum clique detection problem and it is adapted for 3D molecular graphs. Graph matching procedure, used in our approach, accommodates the conformational flexibility by using distance constrains matrix, encoding distance ranges between atoms, rather than fixing the atom pair distances. The distance ranges are generated from the initial guess, followed by successive optimizations, using triangle and tetrangle bound smoothing algorithms from distance geometry. A set of dopamine receptor antagonists is taken into consideration for testing the proposed method; these compounds play a critical role in diseases such as Parkinson’s and schizophrenia.

Using the similarity scores from 3D molecular similarity procedure, we proposed an ordering for the studied compounds and compared them with the experimental biological activities. The procedure can be used as a pruning in the preliminary stage of rational drug design, in diseases involving dopamine neurotransmitters and other classes of biologically active compounds.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Multidimensional approach to

solid state reaction mechanisms


Hrvoj Vančik


Department of Chemistry, Faculty of Science, University of Zagreb, Horvatovac 102A, 10000 Zagreb, Croatia




Generally accepted method for studying solid state reaction mechanisms was until now based on kinetic methods and Avrami-Erofeev equations.1 In principle, these equations describe progress of the solid state reaction in terms of one-, two-, or three-dimensional growth of the reactant phase. Experiments from which such an approach is applicable were based mostly on the thermal analysis data.

In this work we represent reaction rate measurements on the basis of the time resolved powder X-ray diffraction, and vibrational spectroscopy.2 Combination of such kinetic data with the knowledge of the crystal structures of reactants, metastable intermediates and products, led to the new insight in the solid state reaction mechanism, in which one can follow not only the appearance of product, but also growth of the specific crystal planes. Such crystal planes, we call them critical planes, define the reaction mechanism and, in principle, they in theory have similar role which has reaction coordinate in the gas or liquid phase reactions.


1. A.K. Galwey (2004) Thermochim. Acta  413: 139-183.

2. H. Vančik, V. Šimunić-Mežnarić, E. Meštrović,, I.J. Halasz (2004) Org. Chem. 69: 4829‑4834.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Areas of some polygons and volumes of fullerenes


Darko Veljan


Department of Mathematics, University of Zagreb, Bijenička c. 30, HR-10002 Zagreb, Croatia




Problems of computing areas and volumes are among the oldest, hardest and most important in mathematics and its applications. In principle, the answer to these questions gave Newton in the 17th century by inventing integrals. However, computing integrals is very hard. Computations of areas of polygons could, again in principle, be reduced to triangles. But, it heavily depends on the data available. Here is an example. Suppose you want to compute the area of a (convex) pentagon, but you only know the areas of triangles with three consecutive vertices, and they are, say, 10, 20, 30, 40 and 40, respectively. What is the area of the pentagon? Answer: 100. Next problem. Let a convex polyhedron P be inscribed in a sphere of radius R, suppose further that the faces of P are only n-gons, 3 ≤ n ≤ 6, and suppose you only know (by measuring) the edge lengths of these faces (besides knowing how they are pasted together, i.e., the combinatorial structure). Then how to compute the volume, vol(P), or the surface area of P? And how to handle these problems when P is not inscribed, i.e., when P is a "potato-like", as genuine fullerenes are? In this context we discuss Gauss' pentagonal formula, Robbins' area formula, generalizations, consequences etc.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Deuterium Isotope Effects in NMR Spectra


Dražen Vikić-Topić


Ruđer Bošković Institute, NMR Center, P. O. Box 180, Zagreb, Croatia




Deuterium is often studied isotope due to the large fractional mass change on deuteration as well as because of easy of deuterium incorporation into the molecules. Deuterium substitution generates some peculiar effects into the molecules: optical activity in CXYHD: small dipole moment (10-2-10-4 D) in monodeuterated methane, acetylene and benzene, which enables the measuring of pure rotational spectra of these compounds. The most unusual feature of deuterium labeling of organic compounds is the existence of long range deuterium isotope effects on 13C NMR chemical shifts (LRDIE). The high magnetic field NMR spectrometers enabled detection of LRDIE even through 12 bonds, as small as 0.1 Hz, in p-electron systems.

The calculations of nuclear shileding changes1 due to LRDIE are still challenging because of their low magnitude (3-300 ppb) and still not enough developed theoretical approach for isotope interactions through many bonds.

Generally, isotope effects on chemical shifts are interpreted in terms of vibrational and rotational averaging of nuclear shielding. Changes in nuclear shielding with bond extension and/or bond angle deformation are introduced by deuteration. In the theoretical model of LRDIE two contributions have to be taken into account: (1) the secondary change in shielding at remote C-atom due to shorter C-D than C-H mean bond length at the site of deuteration and (2) the primary change in remote C-atom shielding, due to change in mean bond length at this remote site. We assumed that LRDIE predominantly arise from the change of bond length at the site of deuteration (1), since vibrational differences at remote C-atoms of isotopomeric molecules are practiacally undetectable. The C-D bond was modeled by reduction of the corresponding C-H bond in the range 0.003-0.018 Å. Nuclear shieldings (GIAO) and atomic charges (Löwdin, Mulliken) were calculated by different ab initio basis sets. For C-atoms more than four bonds away from the deuteration site the differences of shielding and charges between protonated and deuterated molecules correlate fairly good with experimental LRDIE. By this approach experimental LRDIE can be successfully calculated, even those over 10 and 12 C-C bonds.


1. D. Vikić-Topić, Lj. Pejov (2001) J. Chem. Inf. Comput. Sci. 41: 1478-1487.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Kekulé Structures of C70


Damir Vukičević1 and Milan Randić2


1Department of Mathematics, University of Split, Nikole Tesle 12, HR-21000 Split, Croatia


2National Institute of Chemistry, Hajdrihova 19, SI-1000 Ljubljana, Slovenia




The fullerene C70 is one of the two most important fullerenes. We present the following properties of this fullerene:

  1. Number of Kekulé structures
  2. Number of nonisomorphical Kekulé structures
  3. The number of conjugated hexagons in  Kekulé structures
  4. The maximal number of independent conjugated hexagons in  Kekulé structures
  5. Degree of freedom in Kekulé structures
  6. The description of Clar’s structures
  7. Resonance graph of Kekulé structures
  8. Partition of pi-electrons to 5 and 6 member rings




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



On generalization of the Hosoya-Wiener polynomial


Blaž Zmazek1,2 and Janez Žerovnik1,2


1FME, University of  Maribor, Smetanova 17, SI-2000 Maribor, Slovenia


2IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia




The Hosoya-Wiener polynomial1,2 of a graph G is defined as


Obviously,  .

It is well known that the (unweighted) Wiener number3 is the value of the first derivative of the Hosoya-Wiener polynomial at x=1, W(G) = W’(G;1).

We can generalize the  Hosoya-Wiener polynomial to weighted graphs as follows. Let G be a edge and vertex weighted graph. WW(x) is defined as



where d(u,v)  is the graph distance between u and v.

Clearly, the new definition is equivalent to original definition if all vertex weights are equal to 1. Note that WW(x) is in general not a polynomial if arbitrary edge lengths are allowed.

We will discuss algorithms for efficient computation of WW(x) on trees and on cacti. Our results include:

Theorem: The Wiener polynomial on a vertex weighted tree T can be computed in O(D Δ2 n) time, where D is the diameter of T and Δ  is the maximal degree of a vertex in T.


15.  H. Hosoya (1988) On some counting polynomials in chemistry. Discrete Appl. Math. 19: 239-257.

16.  B.E.Sagan, Y.-N.Neh, P.Zhang (1996) The Wiener polynomial of a graph. Int. J. Quantum Chem. 60: 959-969.

17.  H.Wiener (1947) Structural determination of paraffin boiling points. J. Amer. Chem. Soc. 69: 17-20.




Abstract: MATH/CHEM/COMP 2005, Dubrovnik, June 20-25, 2005



Exact quantum treatment of finite-dimensional system in the interaction with the known

infinite‑dimensional system


Tomislav P. Živković


Ruđer Bošković Institute, P.O.B. 180, HR-10002 Zagreb, Croatia




An important problem in science is to describe properties of relatively small systems (atoms, molecules or molecular fragments) that interact with surrounding media (other atoms, molecules, molecular fragments, electromagnetic fields, crystals, solute, etc.). Classical description of such systems is not sufficient, and one has to use quantum theory. One can obtain relatively reliable quantum description in two extreme cases. One extreme are small isolated quantum systems Sa (such as atoms and small molecules) which can be relatively well described (using modern fast computers) with the standard Schrödinger equation. Other extreme are some infinite quantum systems Sb (such as crystals – solid state, electromagnetic field etc.) which have some special regularity such as translational invariance. Utilizing this regularity one can obtain a relatively correct quantum description of those infinite systems. Exact quantum description of the combined system S=SaSb where a small quantum system Sa interacts with an infinite quantum system Sb is not so successful. A typical example of such a system is an isolated molecule (system Sa) in the interaction with the electromagnetic field (system Sb). Quantum description of this system is the main subject of spectroscopy. Another example is the interaction of an isolated molecule situated on the surface of some solid with this solid. This is the subject of the surface state physics. Yet another example is the interaction of an isolated molecule in a solution with the solute. There are generally two main methods how such combined systems are usually treated. If the interaction of a small system Sa with the infinite system Sb is relatively weak, the combined system can be treated within the formalism of the perturbation expansion. However, if this interaction is sufficiently strong, perturbation expansion may diverge. Also, if highly reliable results are needed, perturbation expansion may converge unacceptably slow in order to obtain required accuracy. Another approach is to use some semiclasical model. However, such an approach is only approximate and it can never completely replace exact quantum treatment. A new mathematical approach for the exact quantum treatment of such combined systems will be presented. In this approach, combined quantum system S can be exactly described by a (nonlinear) equation that has dimension of a (small) system Sa (however large the system Sb may be). Therefore, mathematical complexity of the problem (finding all properties of the system Sa subject to the interaction with the infinite system Sb) is essentially the same as mathematical complexity required to solve isolated system Sa. This applies to the time-independent as well as to the time-dependent case. The suggested method produces exact quantum description of the combined system, however strong the interaction between the two subsystems Sa and Sb. Preliminary results of this new mathematical approach will be presented.