Some problems in the area of computational biology seem suitable for practical applications that are not artificial - or at least not as artificial as the problems mentioned by Jukka Suomela.
For instance, people often mention the work by F. Abu-Khzam, R. Collins, M. Fellows, M. Langston, W. Suters C. Symons, Kernelization Algorithms for the Vertex Cover Problem: Theory and Experiments, Proceedings of the 6th Workshop on Algorithm Engineering and Experiments (ALENEX), ACM/SIAM, Proc. Applied Mathematics 115, 2004.
As the authors state, "One of the applications to which we have applied our methods involves finding phylogenetic trees based on protein domain information, ..." (section 8 of above paper).
A subset of the authors have similar papers on this topic, see, e.g., Faisal N. Abu-Khzam, Michael A. Langston, Pushkar Shanbhag and Christopher T. Symons, Scalable Parallel Algorithms for FPT Problems, Algorithmica, Volume 45, Number 3, 269-284.
I'm not sure whether the instances used in the experiments were real-world instances or artificial, but I hope the two references give you a good starting point.