Hybrid Modeling in Biochemical Systems Theory by Means of Functional Petri Nets

Many biological systems are genuinely hybrids consisting of interacting discrete and continuous components and processes that often operate at different time scales. It is therefore desirable to create modeling frameworks capable of combining differently structured processes and permitting their analysis over multiple time horizons. During the past 40 years, Biochemical Systems Theory (BST) has been a very successful approach to elucidating metabolic, gene regulatory, and signaling systems. However, its foundation in ordinary differential equations has precluded BST from directly addressing problems containing switches, delays, and stochastic effects. In this study, we extend BST to hybrid modeling within the framework of Hybrid Functional Petri Nets (HFPN). First, we show how the canonical GMA and S-system models in BST can be directly implemented in a standard Petri Net framework. In a second step we demonstrate how to account for different types of time delays as well as for discrete, stochastic, and switching effects. Using representative test cases, we validate the hybrid modeling approach through comparative analyses and simulations with other approaches and highlight the feasibility, quality, and efficiency of the hybrid method.

[1]  E O Voit.  Smooth bistable S-systems. , 2005, Systems biology.

[2]  D. Gillespie.  Approximate accelerated stochastic simulation of chemically reacting systems , 2001 .

[3]  D. Gillespie.  The Chemical Langevin and Fokker−Planck Equations for the Reversible Isomerization Reaction† , 2002 .

[4]  M A Savageau,et al.  A theory of alternative designs for biochemical control systems. , 1985, Biomedica biochimica acta.

[5]  D. Gillespie.  The chemical Langevin equation , 2000 .

[6]  J. Heijnen,et al.  The mathematics of metabolic control analysis revisited. , 2002, Metabolic engineering.

[7]  Robert M. May,et al.  Theoretical Ecology: Principles and Applications , 1977 .

[8]  Ramón Varón,et al.  Kinetics of a general model for enzyme activation through a limited proteolysis , 1987 .

[9]  Atsushi Doi,et al.  Biopathways representation and simulation on hybrid functional Petri net , 2003, Silico Biol..

[10]  Eberhard O. Voit,et al.  Computational Analysis of Biochemical Systems: A Practical Guide for Biochemists and Molecular Biologists , 2000 .

[11]  E O Voit,et al.  Approximation of delays in biochemical systems. , 2005, Mathematical biosciences.

[12]  Daniel T Gillespie,et al.  Stochastic simulation of chemical kinetics. , 2007, Annual review of physical chemistry.

[13]  K. Burrage,et al.  Stochastic models for regulatory networks of the genetic toggle switch. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[14]  Kwang-Hyun Cho,et al.  Modeling and simulation of intracellular dynamics: choosing an appropriate framework , 2004, IEEE Transactions on NanoBioscience.

[15]  E. Voit,et al.  Recasting nonlinear differential equations as S-systems: a canonical nonlinear form , 1987 .

[16]  A. J. Lotka.  Elements of Physical Biology. , 1925, Nature.

[17]  Masao Nagasaki,et al.  Genomic data assimilation for estimating hybrid functional Petri net from time-course gene expression data. , 2006, Genome informatics. International Conference on Genome Informatics.

[18]  D. Fell.  Understanding the Control of Metabolism , 1996 .

[19]  M. Savageau.  Biochemical Systems Analysis: A Study of Function and Design in Molecular Biology , 1976 .

[20]  Eberhard O. Voit,et al.  S-system modelling of complex systems with chaotic input , 1993 .

[21]  Wolfgang Reisig.  Petri Nets: An Introduction , 1985, EATCS Monographs on Theoretical Computer Science.

[22]  J E Bailey,et al.  MCA has more to say. , 1996, Journal of theoretical biology.

[23]  Michael A. Savageau,et al.  Design principles for elementary gene circuits: Elements, methods, and examples. , 2001, Chaos.

[24]  K. Burrage,et al.  Stochastic delay differential equations for genetic regulatory networks , 2007 .

[25]  A. J. Lotka,et al.  Elements of Physical Biology. , 1925, Nature.

[26]  Hiroshi Matsuno,et al.  Structural Modeling and Analysis of Signaling Pathways Based on Petri Nets , 2006, J. Bioinform. Comput. Biol..

[27]  M A Savageau,et al.  Effect of overall feedback inhibition in unbranched biosynthetic pathways. , 2000, Biophysical journal.

[28]  Marek Kimmel,et al.  Transcriptional stochasticity in gene expression. , 2006, Journal of theoretical biology.

[29]  M. Savageau.  Biochemical systems analysis. II. The steady-state solutions for an n-pool system using a power-law approximation. , 1969, Journal of theoretical biology.

[30]  W. S. Hlavacek,et al.  Rules for coupled expression of regulator and effector genes in inducible circuits. , 1996, Journal of molecular biology.

[31]  P J Goss,et al.  Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[32]  J. Hasty,et al.  Dynamics of single-cell gene expression , 2006, Molecular systems biology.

[33]  N. Monk.  Oscillatory Expression of Hes1, p53, and NF-κB Driven by Transcriptional Time Delays , 2003, Current Biology.

[34]  Michael L. Mavrovouniotis,et al.  Petri Net Representations in Metabolic Pathways , 1993, ISMB.

[35]  A. Thiel.  Lois générales de l'action des diastases, par VICTOR HENRI. VIII und 129 Seiten. (Paris, A. HERMANN, 1903.) , 1903 .

[36]  Pierre N. Robillard,et al.  Modeling and Simulation of Molecular Biology Systems Using Petri Nets: Modeling Goals of Various Approaches , 2004, J. Bioinform. Comput. Biol..

[37]  Savageau Ma,et al.  A theory of alternative designs for biochemical control systems. , 1985 .

[38]  M. Savageau.  Biochemical systems analysis. II. The steady-state solutions for an n-pool system using a power-law approximation. , 1969, Journal of theoretical biology.

[39]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[40]  M A Savageau,et al.  Network regulation of the immune response: alternative control points for suppressor modulation of effector lymphocytes. , 1985, Journal of immunology.

[41]  Pei Yee Ho,et al.  Multiple High-Throughput Analyses Monitor the Response of E. coli to Perturbations , 2007, Science.

[42]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[43]  E. O’Shea,et al.  Living with noisy genes: how cells function reliably with inherent variability in gene expression. , 2007, Annual review of biophysics and biomolecular structure.