Skip to main content

GPU-accelerated steady-state computation of large probabilistic Boolean networks

Abstract

Computation of steady-state probabilities is an important aspect of analysing biological systems modelled as probabilistic Boolean networks (PBNs). For small PBNs, efficient numerical methods to compute steady-state probabilities of PBNs exist, based on the Markov chain state-transition matrix. However, for large PBNs, numerical methods suffer from the state-space explosion problem since the state-space size is exponential in the number of nodes in a PBN. In fact, the use of statistical methods and Monte Carlo methods remain the only feasible approach to address the problem for large PBNs. Such methods usually rely on long simulations of a PBN. Since slow simulation can impede the analysis, the efficiency of the simulation procedure becomes critical. Intuitively, parallelising the simulation process is the ideal way to accelerate the computation. Recent developments of general purpose graphics processing units (GPUs) provide possibilities to massively parallelise the simulation process. In this work, we propose a trajectory-level parallelisation framework to accelerate the computation of steady-state probabilities in large PBNs with the use of GPUs. To maximise the computation efficiency on a GPU, we develop a dynamical data arrangement mechanism for handling different size PBNs with a GPU. Specially, we propose a reorder-and-split method to handle both large and dense PBNs. Besides, we develop a specific way of storing predictor functions of a PBN and the state of the PBN in the GPU memory. Moreover, we introduce a strongly connected component (SCC)-based network reduction technique to further accelerate the computation speed. Experimental results show that our GPU-based parallelisation gains approximately a 600-fold speedup for a real-life PBN compared to the state-of-the-art sequential method.

This is a preview of subscription content, access via your institution.

References

  1. Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7(4): 457–472

    Article  MATH  Google Scholar 

  2. Lähdesmäki H, Hautaniemi S, Shmulevich I, Yli-Harja O (2006) Relationships between probabilistic Boolean networks and dynamic Bayesian networks as models of gene regulatory networks. Signal Process 86(4): 814–834

    Article  MATH  Google Scholar 

  3. Kauffman SA (1969) Homeostasis and differentiation in random genetic control networks. Nature 224: 177–178

    Article  Google Scholar 

  4. Mizera A, Pang J, Yuan J (2015) ASSA-PBN: An approximate steady-state analyser of probabilistic Boolean networks. In: Proceedings of 13th international symposium on automated technology for verification and analysis, volume 9364 of LNCS. Springer, pp 214–220

  5. Mizera A, Pang J, Yuan Q (2015) Reviving the two-state markov chain approach (technical report). http://arxiv.org/abs/1501.01779

  6. Mizera A, Pang J, Yuan Q (2016) ASSA-PBN 2.0: A software tool for probabilistic Boolean networks. In: Proceedings of 14th international conference on computational methods in systems biology, volume 9859 of LNCS. Springer, pp 309–315

  7. Mizera A, Pang J, Yuan Q (2016) Fast simulation of probabilistic Boolean networks. In: Proceedings of 14th international conference on computational methods in systems biology, volume 9859 of LNCS. Springer, pp 216–231

  8. Mizera A, Pang J, Yuan Q (2016) GPU-accelerated steady-state computation of large probabilistic Boolean networks. In: Proceedings of 2nd international symposium on dependable software engineering: theories, tools, and applications, volume 9984 of LNCS. Springer, pp 50–66

  9. Mizera A, Pang J, Yuan Q (2016) Parallel approximate steady-state analysis of large probabilistic Boolean networks. In: Proceedings of 31st ACM symposium on applied computing, pp 1–8

  10. Mizera A, Pang J, Yuan Q (2017) Reviving the two-state Markov chain approach. In: IEEE/ACM transactions on computational biology and bioinformatics

  11. Raftery AE, Lewis S (1992) How many iterations in the Gibbs sampler ?. Bayesian Stat 4: 763–773

    Google Scholar 

  12. Shmulevich I, Dougherty ER (2010) Probabilistic Boolean networks: the modeling and control of gene regulatory networks. SIAM Press, Auckland

    Book  MATH  Google Scholar 

  13. Shmulevich I, Dougherty ER, Kim S, Zhang W (2002) Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics 18(2): 261–274

    Article  Google Scholar 

  14. Shmulevich I, Gluhovsky I, Hashimoto RF, Dougherty ER, Zhang W (2003) Steady-state analysis of genetic regulatory networks modelled by probabilistic Boolean networks. Comp Funct Genom 4(6): 601–608

    Article  Google Scholar 

  15. Schlatter R, Schmich K, Vizcarra IA, Scheurich P, Sauter T, Borner C, Ederer M, Merfort I, Sawodny O (2009) ON/OFF and beyond—a Boolean model of apoptosis. PLOS Comput Biol 5(12): e1000595

    Article  Google Scholar 

  16. Trairatphisan P, Mizera A, Pang J, Tantar A-A, Schneider J, Sauter T (2013) Recent development and biomedical applications of probabilistic Boolean networks. Cell Commun Signal 11: 46

    Article  Google Scholar 

  17. Trairatphisan P, Mizera A, Pang J, Tantar A-A, Sauter T (2014) optPBN: An optimisation toolbox for probabilistic boolean networks. PLoS ONE 9(7): e98001

    Article  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees who read carefully the previous versions of this paper and gave a lot of valuable comments. Those comments helped us to greatly improve the quality of our paper both in content and presentation. Qixia Yuan was supported by the National Research Fund, Luxembourg (Grant 7814267). Jun Pang was partially supported by the project SEC-PBN (funded by the University of Luxembourg) and the ANR-FNR Project AlgoReCell (INTER/ANR/15/11191283). Andrzej Mizera contributed to this work while holding a postdoctoral researcher position at the Computer Science and Communications Research Unit, University of Luxembourg.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jun Pang.

Additional information

Naijun Zhan, Heike Wehrheim, Martin Fränzle, and Deepak Kapur

A preliminary version of this work was presented at the 2nd International Symposium on Dependable Software Engineering: Theories, Tools, and Applications [MPY16c]

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mizera, A., Pang, J. & Yuan, Q. GPU-accelerated steady-state computation of large probabilistic Boolean networks. Form Asp Comp 31, 27–46 (2019). https://doi.org/10.1007/s00165-018-0470-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00165-018-0470-6

Keywords

  • Probabilistic Boolean networks
  • Biological networks
  • Computational modelling
  • Discrete-time Markov chains
  • Simulation
  • Statistical methods
  • Graphics processing unit (GPU)