Single-variable reaction systems: deterministic and stochastic models.
Biochemical reaction networks are often described by deterministic models based on macroscopic rate equations. However, for small numbers of molecules, intrinsic noise can play a significant role and stochastic methods may thus be required. In this work, we analyze the differences and similarities between a class of macroscopic deterministic models and corresponding mesoscopic stochastic models. We derive expressions that provide a clear and intuitive view upon the behavior of the stochastic model. In particular, these expressions show the dependence of both the dynamics and the stationary distribution of the stochastic model on the number of molecules in the system. As expected, most properties of the stochastic model correspond well with those in the deterministic model if the number of molecules is large enough. However, for some properties, both models are inconsistent, even if the number of molecules in the stochastic model tends to infinity. Throughout this paper, we use a bistable autophosphorylation cycle as a running example. For such a bistable system, we give an explicit proof that the rate of convergence to the stationary distribution (or the second eigenvalue of the transition matrix) depends exponentially on the number of molecules.
Steijaert
M N
MN
Department of Biomedical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
Liekens
A M L
AM
Bosnacki
D
D
Hilbers
P A J
PA
ten Eikelder
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HM
D000465:Algorithms; D007700:Kinetics; D008390:Markov Chains; D053858:Metabolic Networks and Pathways / Q000502:physiology*; D008954:Models, Biological; D015233:Models, Statistical; D010744:Phosphoric Monoester Hydrolases / Q000378:metabolism; D010766:Phosphorylation; D010770:Phosphotransferases / Q000378:metabolism; D011499:Protein Processing, Post-Translational; D013269:Stochastic Processes
D010770:Phosphotransferases; D010744:Phosphoric Monoester Hydrolases