Five-membered heterocyclic ring systems constitute an important part of many biologically relevant molecules. They occur in carbohydrates (furanoses), nucleosides and nucleotides, the amino acid proline and their many derivatives. In addition, they often occur as a moiety in complex natural products. Chemical modifications of nucleic acids, often driven by the needs of antisense research, target in part the five-membered cycle or its analogues in order to tailor their conformation towards the desired needs [1, 2].

The advent of combinatorial chemistry has also revived the interest in heterocyclic rings and their conformation [3]. As a result, scaffolds containing five-membered heterocycles have received much attention for the rapid generation of potential lead compounds in pharmaceutical research [4–11].

Typically, the chemical and conformational space is explored by introducing a diversity of substituents at varying positions around the cycle. Depending on the position and nature of these substituents, the cycle either adopts a single conformation or may be in equilibrium between two conformations. These conformations will in turn impact on the conformational space that will be covered by the substituents, making the determination of the cycle's conformation an issue of considerable interest.

Over the years, NMR has become a well-established technique for this purpose. In particular, ^{3}*J*_{
HH
}scalar coupling constants are well-suited as they are mainly determined by the torsion angle over which they are measured. In the case of ring systems, the vicinal ^{3}*J*_{
HH
}scalar coupling constants are directly correlated to their corresponding exocyclic torsion angles (*θ*_{
exo
}). These are related to the corresponding endocyclic torsion angles (*θ*_{
endo
}) by a simple equation (1) where *A* and *B* are constants determined by the geometry of the atoms linked to the common central bond.

*θ*_{
exo
}= *A θ*_{
endo
}+ *B*

As the set of all five endocyclic torsion angles in a five-membered ring fully determines its conformation, ^{3}*J*_{
HH
}scalar coupling constants provide a direct measure of the ring's conformation. The Haasnoot-Altona equation (3) [12] and the Diez-Donders equation (4) [13, 14], both based on the well known Karplus equation (2), describe the relation between a ^{3}*J*_{
HH
}coupling and the corresponding exocyclic torsion angle (*θ*_{
exo
}) to a high level of accuracy. In both equations this is mainly achieved by including a set of four parameters *λ*_{
i
}(*i* = 1, ..., 4) that account for the influence of electronic effects contributed by the substituents [15, 16]. In some studies, the set of experimental ^{3}*J*_{
HH
}scalar coupling constants is further extended by ^{3}*J*_{
HF
}scalar coupling constants [17–19] or interproton distances obtained by nOe NMR experiments. Here however, we assume that only ^{3}*J*_{
HH
}scalar couplings are available for conformational analysis.

^{3}*J*_{
HH
}= *P*_{1} *cos*^{2}(*θ*) + *P*_{2} *cos*(*θ*) + *P*_{3}

\begin{array}{c}{\phantom{\rule{0.5em}{0ex}}}^{3}{J}_{HH}={P}_{1}\phantom{\rule{0.5em}{0ex}}co{s}^{2}(\theta )+{P}_{2}\phantom{\rule{0.5em}{0ex}}cos(\theta )+{P}_{3}+\\ {\displaystyle \sum _{i=1}^{4}{\lambda}_{i}}\left\{{P}_{4}+{P}_{5}\phantom{\rule{0.5em}{0ex}}co{s}^{2}({(-1)}^{i}{\lambda}_{i}\theta +{P}_{6}\left|{\lambda}_{i}\right|)\right\}\end{array}

(3)

{\phantom{\rule{0.5em}{0ex}}}^{3}{J}_{HH}={\displaystyle \sum _{i=0}^{3}{C}_{i}}({\lambda}_{1\dots 4})\phantom{\rule{0.5em}{0ex}}cos(i\theta )+{\displaystyle \sum _{i=1}^{3}{S}_{i}}({\lambda}_{1\dots 4})\phantom{\rule{0.5em}{0ex}}sin(i\theta )

(4)

Altona and Sundaralingam showed that the description of a five-membered ring conformation can be reduced to a two-parameter pseudorotation model [20, 21] that fully describes its conformation. The first parameter, the pucker phase *P*, represents the phase of the conformation and indicates which ring atoms are positioned out of the ring plane. The second parameter, the pucker amplitude *ν*_{
max
}, corresponds to the amplitude of the conformation and describes the extent to which the atoms determined by *P* are out of the plane. The relationship with the endocyclic torsion angles *θ*_{endo, i}is shown in (5).

{\theta}_{endo,i}={\nu}_{max}\phantom{\rule{0.5em}{0ex}}cos(P+\frac{4\pi i}{5})

(5)

This well-known pseudorotation description, originally described for the furanose ring in nucleosides and nucleotides [20, 21], was further generalized to any five-membered heterocycle by Diez et al. [22–24] who introduced two additional parameters *α*_{
i
}and *ε*_{
i
}for each endocyclic bond to cope with differences in bond lengths in various types of five-membered rings (Equation 6). As the phase of the conformation *P* is a periodic variable, polar plots called pseudorotation wheels are mostly used to depict ring conformations.

{\theta}_{endo,i}={\alpha}_{i}{\nu}_{max}\phantom{\rule{0.5em}{0ex}}cos(P+{\u03f5}_{i}+\frac{4\pi i}{5})

(6)

Using the above equations, ^{3}*J*_{
HH
}couplings can be used to derive the pseudorotation parameters of the five-membered cycle. As mentioned previously, the cycle may be in equilibrium between two conformations. Thus, most generally two sets of pseudorotation parameters (*P* and *ν*_{
max
}) and the relative population (%_{1}, i.e. the percentage of the first conformation present with %_{2} = 1 - %_{1}) need to be fitted to the experimental NMR data. In order to avoid an under-determined model, experimental data measured at different temperatures is generally used. In such cases, the model assumes that only the relative population of the two conformations varies when changing the temperature. Thus *n* + 4 (*n* being the number of temperatures used) variables will be optimized to fit the experimental data in such cases. To the best of our knowledge, the program PSEUROT [25], originally developed by Altona et al., is still the only generally available program to perform this type of analysis. Written in FORTRAN, its interface as well as its output is purely text-based. In order to facilitate the analysis of the PSEUROT results, a post-processing feature has been included in the independently developed MULDER package [26] to generate a graphical output of the PSEUROT results. In this communication, we propose an integrated, user-friendly Matlab program, including a self-explanatory graphical user interface (GUI), to facilitate the set-up, execution and subsequent analysis of pseudorotation calculations for five-membered ring systems decorated with a variety of substituents. The use of Matlab as high-level programming language enables to create, within a limited time frame, high-quality plots that provide a graphical impression of the conformational space accessible. Furthermore, due to the open-source GNU GPL license, users have the opportunity to adapt the program to their specific needs.