# Chemical Boltzmann Machines

## Abstract

How smart can a micron-sized bag of chemicals be? How can an artificial or real cell make inferences about its environment? From which kinds of probability distributions can chemical reaction networks sample? We begin tackling these questions by showing three ways in which a stochastic chemical reaction network can implement a Boltzmann machine, a stochastic neural network model that can generate a wide range of probability distributions and compute conditional probabilities. The resulting models, and the associated theorems, provide a road map for constructing chemical reaction networks that exploit their native stochasticity as a computational resource. Finally, to show the potential of our models, we simulate a chemical Boltzmann machine to classify and generate MNIST digits in-silico.

## Notes

### Acknowledgements

This work was supported in part by U.S. National Science Foundation (NSF) graduate fellowships to WP and to AOM, by NSF grant CCF-1317694 to EW, and by the Gordon and Betty Moore Foundation through Grant GBMF2809 to the Caltech Programmable Molecular Technology Initiative (PMTI), by a Royal Society University Research Fellowship to TEO, and by a Bharti Centre for Communication in IIT Bombay award to AB.

## References

- 1.Bray, D.: Protein molecules as computational elements in living cells. Nature
**376**(6538), 307 (1995)CrossRefGoogle Scholar - 2.Bray, D.: Wetware: A Computer in Every Living Cell. Yale University Press, New Haven (2009)Google Scholar
- 3.McAdams, H.H., Arkin, A.: Stochastic mechanisms in gene expression. Proc. Natl. Acad. Sci.
**94**(3), 814–819 (1997)CrossRefGoogle Scholar - 4.Elowitz, M.B., Levine, A.J., Siggia, E.D., Swain, P.S.: Stochastic gene expression in a single cell. Science
**297**(5584), 1183–1186 (2002)CrossRefGoogle Scholar - 5.Perkins, T.J., Swain, P.S.: Strategies for cellular decision making. Mol. Syst. Biol.
**5**(1), 326 (2009)Google Scholar - 6.Muroga, S.: Threshold Logic and Its Applications. Wiley Interscience, New York (1971)zbMATHGoogle Scholar
- 7.Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci.
**79**(8), 2554–2558 (1982)MathSciNetCrossRefGoogle Scholar - 8.Hinton, G.E., Sejnowski, T.J., Ackley, D.H.: Boltzmann Machines: Constraint Satisfaction Networks that Learn. Department of Computer Science, Carnegie-Mellon University, Pittsburgh (1984)Google Scholar
- 9.Bray, D.: Intracellular signalling as a parallel distributed process. J. Theor. Biol.
**143**(2), 215–231 (1990)CrossRefGoogle Scholar - 10.Hellingwerf, K.J., Postma, P.W., Tommassen, J., Westerhoff, H.V.: Signal transduction in bacteria: phospho-neural network(s) in Escherichia coli. FEMS Microbiol. Rev.
**16**(4), 309–321 (1995)CrossRefGoogle Scholar - 11.Mjolsness, E., Sharp, D.H., Reinitz, J.: A connectionist model of development. J. Theor. Biol.
**152**(4), 429–453 (1991)CrossRefGoogle Scholar - 12.Mestl, T., Lemay, C., Glass, L.: Chaos in high-dimensional neural and gene networks. Physica D: Nonlin. Phenom.
**98**(1), 33–52 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Buchler, N.E., Gerland, U., Hwa, T.: On schemes of combinatorial transcription logic. Proc. Natl. Acad. Sci.
**100**(9), 5136–5141 (2003)CrossRefGoogle Scholar - 14.Deutsch, J.M.: Collective regulation by non-coding RNA. arXiv preprint arXiv:1409.1899 (2014)
- 15.Deutsch, J.M.: Associative memory by collective regulation of non-coding RNA. arXiv preprint arXiv:1608.05494 (2016)
- 16.Hjelmfelt, A., Weinberger, E.D., Ross, J.: Chemical implementation of neural networks and turing machines. Proc. Natl. Acad. Sci.
**88**(24), 10983–10987 (1991)CrossRefzbMATHGoogle Scholar - 17.Hjelmfelt, A., Ross, J.: Chemical implementation and thermodynamics of collective neural networks. Proc. Natl. Acad. Sci.
**89**(1), 388–391 (1992)CrossRefGoogle Scholar - 18.Kim, J., Hopfield, J.J., Winfree, E.: Neural network computation by in vitro transcriptional circuits. In: Advances in Neural Information Processing Systems (NIPS), pp. 681–688 (2004)Google Scholar
- 19.Napp, N.E., Adams, R.P.: Message passing inference with chemical reaction networks. In: Advances in Neural Information Processing Systems (NIPS), pp. 2247–2255 (2013)Google Scholar
- 20.Gopalkrishnan, M.: A scheme for molecular computation of maximum likelihood estimators for log-linear models. In: Rondelez, Y., Woods, D. (eds.) DNA 2016. LNCS, vol. 9818, pp. 3–18. Springer, Cham (2016). doi: 10.1007/978-3-319-43994-5_1 CrossRefGoogle Scholar
- 21.Hjelmfelt, A., Schneider, F.W., Ross, J.: Pattern recognition in coupled chemical kinetic systems. Science
**260**, 335–335 (1993)CrossRefGoogle Scholar - 22.Kim, J., White, K.S., Winfree, E.: Construction of an in vitro bistable circuit from synthetic transcriptional switches. Mol. Syst. Biol.
**2**, 68 (2006)CrossRefGoogle Scholar - 23.Kim, J., Winfree, E.: Synthetic in vitro transcriptional oscillators. Mol. Syst. Biol.
**7**, 465 (2011)CrossRefGoogle Scholar - 24.Qian, L., Winfree, E., Bruck, J.: Neural network computation with DNA strand displacement cascades. Nature
**475**(7356), 368–372 (2011)CrossRefGoogle Scholar - 25.Lestas, I., Paulsson, J., Ross, N.E., Vinnicombe, G.: Noise in gene regulatory networks. IEEE Trans. Autom. Control
**53**, 189–200 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 26.Lestas, I., Vinnicombe, G., Paulsson, J.: Fundamental limits on the suppression of molecular fluctuations. Nature
**467**(7312), 174–178 (2010)CrossRefGoogle Scholar - 27.Veening, J.W., Smits, W.K., Kuipers, O.P.: Bistability, epigenetics, and bet-hedging in bacteria. Annu. Rev. Microbiol.
**62**, 193–210 (2008)CrossRefGoogle Scholar - 28.Balázsi, G., van Oudenaarden, A., Collins, J.J.: Cellular decision making and biological noise: from microbes to mammals. Cell
**144**(6), 910–925 (2011)CrossRefGoogle Scholar - 29.Tsimring, L.S.: Noise in biology. Rep. Prog. Phys.
**77**(2), 26601 (2014)CrossRefGoogle Scholar - 30.Eldar, A., Elowitz, M.B.: Functional roles for noise in genetic circuits. Nature
**467**(7312), 167–173 (2010)CrossRefGoogle Scholar - 31.Mansinghka, V.K.: Natively probabilistic computation. Ph.D. thesis, Massachusetts Institute of Technology (2009)Google Scholar
- 32.Wang, S., Zhang, X., Li, Y., Bashizade, R., Yang, S., Dwyer, C., Lebeck, A.R.: Accelerating Markov random field inference using molecular optical Gibbs sampling units. In: Proceedings of the 43rd International Symposium on Computer Architecture, pp. 558–569. IEEE Press (2016)Google Scholar
- 33.Fiser, J., Berkes, P., Orbán, G., Lengyel, M.: Statistically optimal perception and learning: from behavior to neural representations. Trends Cogn. Sci.
**14**(3), 119–130 (2010)CrossRefGoogle Scholar - 34.Pouget, A., Beck, J.M., Ma, W.J., Latham, P.E.: Probabilistic brains: knowns and unknowns. Nat. Neurosci.
**16**(9), 1170–1178 (2013)CrossRefGoogle Scholar - 35.Ackley, D.H., Hinton, G.E., Sejnowski, T.J.: A learning algorithm for Boltzmann machines. Cogn. Sci.
**9**(1), 147–169 (1985)CrossRefGoogle Scholar - 36.Tanaka, T.: Mean-field theory of Boltzmann machine learning. Phys. Rev. E
**58**(2), 2302 (1998)CrossRefGoogle Scholar - 37.Tang, Y., Sutskever, I.: Data normalization in the learning of restricted Boltzmann machines. Department of Computer Science, University of Toronto, Technical report UTML-TR-11-2 (2011)Google Scholar
- 38.Taylor, G.W., Hinton, G.E.: Factored conditional restricted Boltzmann machines for modeling motion style. In: Proceedings of the 26th Annual International Conference on Machine Learning (ICML), pp. 1025–1032. ACM (2009)Google Scholar
- 39.Casella, G., George, E.I.: Explaining the Gibbs sampler. Am. Stat.
**46**(3), 167–174 (1992)MathSciNetGoogle Scholar - 40.Gillespie, D.T.: Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem.
**58**, 35–55 (2007)CrossRefGoogle Scholar - 41.Qian, H.: Phosphorylation energy hypothesis: open chemical systems and their biological functions. Annu. Rev. Phys. Chem.
**58**, 113–142 (2007)CrossRefGoogle Scholar - 42.Beard, D.A., Qian, H.: Chemical Biophysics: Quantitative Analysis of Cellular Systems. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
- 43.Ouldridge, T.E.: The importance of thermodynamics for molecular systems, the importance of molecular systems for thermodynamics. arXiv preprint arXiv:1702.00360 (2017)
- 44.Joshi, B.: A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced. arXiv preprint arXiv:1312.4196 (2013)
- 45.Erez, A., Byrd, T.A., Vogel, R.M., Altan-Bonnet, G., Mugler, A.: Criticality of biochemical feedback. arXiv preprint arXiv:1703.04194 (2017)
- 46.Anderson, D.F., Craciun, G., Kurtz, T.G.: Product-form stationary distributions for deficiency zero chemical reaction networks. Bull. Math. Biol.
**72**(8), 1947–1970 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 47.LeCun, Y., Cortes, C., Burges, C.J.C.: The MNIST database of handwritten digits (1998)Google Scholar