The organisation of bio-molecules, in particular proteins, in the sequence and structure space has recently been attracting increased attention. Particularly questions concerning finding the native structure or investigating the kinetics and evolution of proteins have been widely studied. These problems are often tackled using simplified models such as the Hydrophobic-Polar (HP) model (e.g. Jacob et al. 2). Though abstract, these models are computationally feasible and do allow for deeper insights into fundamental and general principles 234.

Several recurring tasks can be identified in such studies using simplified models. Namely, predicting the native structure, classifying whether a sequence is protein-like, calculating its degeneracy and stability, or the design of sequences that optimally fold to a given structure. The problems associated with these tasks are computationally very hard (NP-complete) 567. Nevertheless, these tasks demand for exact and complete (i.e. non-heuristic) methods. It is important to note that stochastic methods cannot be used for proving optimality and in particular proving that a sequence has a unique lowest energy (protein-like) fold 8.

Consequently, with the exception of Yue and Dill 9, all studies requiring complete and exact answers to optimal structure prediction were based on exhaustive enumeration. These studies were, hence, confined to small sequence lengths. In other approaches, structures are artificially restricted to be maximally compact (e.g. filling a 3 × 3 × 3 cube) 10. This allows for complete enumeration but artificially biases the energy function towards overall hydrophobicity.

Furthermore, many studies are confined to extremely simplified models on the 2D-square or 3D-diamond-lattice 211. The coordination number, a measurement of lattice complexity, is four in both cases. The use of lattices with such a low complexity may lead to oversimplified models that are not able to reproduce real world properties. Park and Levitt 12 have shown that lattices with higher coordination number provide a much better fit to real protein structures. A further hint toward the simplicity of the 2D-lattice is the low computational complexity of inverse folding when compared to the 3D-cubic lattice 7. The Constraint-based Protein Structure Prediction (CPSP) approach by Backofen and Will 13 provides a way to overcome the aforementioned obstacles. The method is tailored to the HP model introduced by Lau and Dill 14. This model is widely used in the literature 1516. CPSP supports complex 3D lattices (currently cubic and face centered cubic) without artificial restrictions (e.g. to be maximally compact). The approach predicts all globally optimal structures together with a proof of optimality. No naive, exhaustive enumeration of all structures is performed and it is as fast as stochastic methods that cannot prove optimality. Backofen and Will 13 showed that the CPSP-approach could fold even sequences of length 200 to optimality within seconds. In contrast, exhaustive structure enumeration as e.g. done by Blackburne and Hirst 17 is restricted to short sequence lengths. For instance, on a 3D-cubic lattice it is only viable to enumerate up to about length 20. In fact, the exact number of structures is only known up to length 23 where there are already more than 5 × 10¹⁵ 18. CPSP uses constraint programming that is commonly applied to hard (NP-complete) problems and, thus, avoids the complete expansion of the whole search space. Hence, constraint-programming techniques are a powerful tool to handle the high complexity that typifies problems related to protein structure. Constraint-programming techniques have successfully been applied to structure prediction with given secondary structure information 19, analysis of NMR data 20, and modeling of protein complexes 21.

Currently, we are not aware of any other complete approach that ensures optimality of the predicted structures in different lattices. There is an alternative to CPSP for the 3D-cubic lattice, the constraint-based hydrophobic core construction method by Yue and Dill 9. This allows the prediction of optimal structures and proves their optimality. However, using the CPSP-approach, Backofen and Will showed that the method developed by Yue and Dill is not always complete in enumerating all optimal structures 13.

CPSP-tools provides a set of programs that enable typical, modern research tasks to be calculated efficiently and accurately. Here we list the programs each with a typical example application. HPSTRUCT predicts (all) optimal and suboptimal structures as required for investigating properties of low energy conformations, as e.g. studied by Jacob and Unger 16. The statistical analysis of protein-like sequences, see Blackburne and Hirst 11, requires a degeneracy-based classification of sequences. This is possible with HPDEG. For the exploration of protein evolution, similar to Wroe and Chan 24, one needs to investigate the sequence-structure space. We provide HPDESIGN for sequence design and HPNNET for neutral network computation.

All methods can be applied to HP-sequences in the cubic and the more complex face centered cubic lattice model. Before giving a detailed description of the tools, we first introduce the idea of H-cores, central to these methods.

For illustration, we provide some scenarios that exemplify the use of CPSP-tools in extending known or enabling new studies. All examples are performed in the unrestricted 3D-cubic lattice with HP-sequences of length 27. Note that for this length there are already more than 10¹⁹ possible structures, which makes an exhaustive enumeration inapplicable. Table 1 outlines the performance of programs from CPSP-tools. Table 2 shows the sequences used for Table 1, their optimal energy (E), and degeneracy (deg). All tasks were performed on an Intel P4 3 GHz (using CPSP-2.0.0).

(1) Studies of sequence or structure features of proteins as done by Huard et al. 4 require a classification of sequences as protein-like. One way is to classify them by the number of optimal structures, i.e. their degeneracy. The fast calculation of this sequence property by HPDEG allows production of sufficiently large benchmark sets for detailed studies. To illustrate this, we run HPDEG for a random HP-sequence S₀ revealing an enormous degeneracy, which is a frequent finding in the HP-model. As a starting point for the following scenarios, we evaluate the degeneracy of S₁, a sequence with a single optimal structure. The very short runtimes for both checks are given in Table 1.

(2) Calculating the globally optimal structure for a given sequence is the main task in many studies, e.g. see Jacob and Unger 16. Furthermore, in stochastic folding simulation approaches knowing the minimal possible energy is favorable. Both can be calculated extremely rapidly using HPSTRUCT. Again, We demonstrate this with sequences S₀ and S₁. This results in an energy of -13 and -22 and the optimal structures X₀ and X₁, respectively. Both structures are visualized in Figure 2.

(3) To study protein evolution on the sequence level, neutral networks are widely utilized 17. Using HPNNET we can span the connected component of the neutral network for a given sequence with a unique optimal structure. Applied to S₁ with X₁ we find four sequences S₂ .. S₄ sharing X₁ as their unique optimal structure. Note, this can be done without exhaustive sequence enumeration for a given structure.

(4) The detailed study of neutral networks by Blackburne and Hirst 17 has shown that neutral networks may decompose into connected components. Their results are based on full enumeration of sequences and structures in the diamond lattice. This approach does not extend to complex lattice models due to the enormous size of the structure space as discussed above.

HPDESIGN can overcome that problem by directly designing sequences of the neutral network. Recall that the neutral network contains only sequences with the same unique optimal structure. The described design approach allows one to generate sequences of independent components in the neutral network without exhaustive enumeration. Afterwards, the full components can be expanded via HPNNET.

We apply this approach to the neutral network of the structure X₁. HPDESIGN calculates 12 members of the network (S₁ .. S₁₂), including the four sequences S₁ .. S₄ known from scenario (3). Expanding the network N from these sequences via HPNNET reveals two further sequences S₁₃, S₁₄ and two independent connected components as shown in Figure 3.

Preliminary studies performed with CPSP-tools indicate that neutral networks as large as N with several large independent components are rare in the unrestricted 3D-cubic model.

For complex 3D models, mainly heuristic and/or stochastic approaches to search for optimal structures of a given sequence are available 833. However, these methods are (a) incomplete and (b) cannot ensure the global optimality of the predicted structures. In consequence, the investigation of problems requiring this information was only possible using exhaustive enumeration, which is not possible for longer sequence lengths.

The CPSP approach is as fast as common stochastic methods but ensures that all predicted structures are globally optimal, and that none are missing. This is done without exhaustive structure space exploration applying constraint-programming techniques. Therefore, it is well suited to many studies in complex 3D models; especially for finding protein-like sequences, the investigation of neutral networks or sequence design. Further applications range from the generation of candidate sets to the validation of results of folding simulations and stochastic optimization methods.

The CPSP-tools package combines several applications in the field of bioinformatics concerning 3D lattice proteins. It allows advanced investigation of problems related to protein structure prediction, sequence evolution, inverse folding, and energy landscapes.

Project name: CPSP-tools

Project home page: http://www.bioinf.uni-freiburg.de/sw/cpsp/

Operating system(s): all Linux based systems (including Cygwin for MS Windows™)

Programming language: C++

Other requirements: Gecode and BIU library (a source bundle is provided)

License: BSD-style license

Any restrictions to use by non-academics: none

Implementation and software design was done by MM and SW. The CPSP method was developed by RB and SW and extended by SW and MM. The CPSP derived algorithms are designed by all authors. All authors have approved and contributed to the final manuscript.