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December  2017, 10(4): 1055-1087. doi: 10.3934/krm.2017042

The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks

Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36,8010 Graz, Austria

* Corresponding author: Klemens Fellner

Received  April 2015 Revised  November 2016 Published  March 2017

In this paper, the applicability of the entropy method for the trend towards equilibrium for reaction-diffusion systems arising from first order chemical reaction networks is studied. In particular, we present a suitable entropy structure for weakly reversible reaction networks without detail balance condition.

We show by deriving an entropy-entropy dissipation estimate that for any weakly reversible network each solution trajectory converges exponentially fast to the unique positive equilibrium with computable rates. This convergence is shown to be true even in cases when the diffusion coefficients of all but one species are zero.

For non-weakly reversible networks consisting of source, transmission and target components, it is shown that species belonging to a source or transmission component decay to zero exponentially fast while species belonging to a target component converge to the corresponding positive equilibria, which are determined by the dynamics of the target component and the mass injected from other components. The results of this work, in some sense, complete the picture of trend to equilibrium for first order chemical reaction networks.

Citation: Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042
References:
[1]

F. AchleitnerA. Arnold and D. Stürzer, Large-time behavior in non-symmetric Fokker-Planck equations, Riv. Mat. Univ. Parma, 6 (2015), 1-68.   Google Scholar

[2]

D. F. Anderson, A proof of the Global Attractor Conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508.  doi: 10.1137/11082631X.  Google Scholar

[3]

A. ArnoldP. MarkowichG. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. PDE, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246.  Google Scholar

[4]

M. Bisi and L. Desvillettes, Global existence and large-time behavior of solutions to the threedimensional equations of compressible magnetohydrodynamic flows,, J. Stat. Phys., 125 (2006), 249-280.  doi: 10.1007/s10955-005-8075-x.  Google Scholar

[5]

J. Bang-Jensen and G. Z. Gregory, Digraphs: Theory, Algorithms and Applications Springer Science & Business Media, 2009. doi: 10.1007/978-1-84800-998-1.  Google Scholar

[6]

G. BokinskyD. RuedaV. K. MisraM. M. RhodesA. GordusH. P. BabcockN. G. Walter and X. Zhuang, Single-molecule transition-state analysis of RNA folding, Proc. Natl. Acad. Sci. USA, 100 (2003), 9302-9307.  doi: 10.1073/pnas.1133280100.  Google Scholar

[7]

J. CarrilloA. JüngelP. MarkowichG. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.  doi: 10.1007/s006050170032.  Google Scholar

[8]

G. CraciunA. DichkensteinA. Shiu and B. Sturmfels, Toric dynamical systems, J. Symb. Comput., 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006.  Google Scholar

[9]

L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction–diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.  doi: 10.1016/j.jmaa.2005.07.003.  Google Scholar

[10]

L. Desvillettes and K. Fellner, Entropy methods for reaction–diffusion systems, Discrete Contin. Dyn. Syst. Issue Special, (2007), 304-312.   Google Scholar

[11]

L. Desvillettes and K. Fellner, Entropy methods for reaction–diffusion equations: Slowly growing a-priori bounds, Rev. Mat. Iberoamericana, 24 (2008), 407-431.  doi: 10.4171/RMI/541.  Google Scholar

[12]

L. Desvillettes and K. Fellner, Exponential convergence to equilibrium for a nonlinear reaction-diffusion systems arising in reversible chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014), 96-104.  doi: 10.1007/978-3-662-45504-3_9.  Google Scholar

[13]

L. Desvillettes, K. Fellner and B. Q. Tang, Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks, arXiv: 1604. 04536, to appear in SIAM J. on Mathematical Analysis. Google Scholar

[14]

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.  doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[15]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Inventiones Mathematicae, 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[16]

M. Di FrancescoK. Fellner and P. Markowich, The entropy dissipation method for inhomogeneous reaction–diffusion systems, Proc. Royal Soc. A, 464 (2008), 3272-3300.  doi: 10.1098/rspa.2008.0214.  Google Scholar

[17]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[18]

M. Feinberg, Lectures on Chemical Reaction Networks, available online at http://www.crnt.osu.edu/LecturesOnReactionNetworks, 1979. Google Scholar

[19]

M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors. Ⅰ. The deficiency zero and deficiency one theorems, Chem. Eng. Sci., 42 (1987), 2229-2268.  doi: 10.1016/0009-2509(87)80099-4.  Google Scholar

[20]

M. Feinberg and F. J. M. Horn, Dynamics of open chemical systems and the algebraic structure of the underlying reaction network, Chem. Eng. Sci., 29 (1974), 775-787.  doi: 10.1016/0009-2509(74)80195-8.  Google Scholar

[21]

K. Fellner, E. Latos and B. Q. Tang, Well-posedness and exponential equilibration of a volumesurface reaction-diffusion system with nonlinear boundary coupling, arXiv: 1404.2809, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire. Google Scholar

[22]

J. Fontbona and B. Jourdain, A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations, Annals of Probability, 44 (2016), 131-170.  doi: 10.1214/14-AOP969.  Google Scholar

[23]

C. GadgilC. H. Lee and H. G. Othmer, A stochastic analysis of first-oder reaction networks, Bull. Math. Biology, 67 (2005), 901-946.  doi: 10.1016/j.bulm.2004.09.009.  Google Scholar

[24]

A. GlitzkyK. Gröger and R. Hünlich, Free energy and dissipation rate for reaction-diffusion processes of electrically charged species, Appl. Anal., 60 (1996), 201-217.  doi: 10.1080/00036819608840428.  Google Scholar

[25]

A. Glitzky and R. Hünlich, Energetic estimates and asymptotics for electro-reaction-diffusion systems,, Z. Angew. Math. Mech., 77 (1997), 823-832.  doi: 10.1002/zamm.19970771105.  Google Scholar

[26]

M. Gopalkrishnan, On the Lyapunov function for complex-balanced mass-action systems, arXiv: 1312.3043. Google Scholar

[27]

K. Gröger, Free energy estimates and asymptotic behaviour of reaction-diffusion processes, Preprint 20, Institut f¨ur Angewandte Analysis und Stochastik, Berlin, 1992. Google Scholar

[28]

F. J. M. Horn and R. Jackson, General mass action kinetics, Arch. Rational Mech. Anal., 47 (1972), 81-116.  doi: 10.1007/BF00251225.  Google Scholar

[29]

F. J. M. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Rational Mech. Anal., 49 (1972), 172-186.  doi: 10.1007/BF00255664.  Google Scholar

[30]

F. J. M. Horn, The dynamics of open reaction systems, SIAM-AMS Proceedings, SIAM, Philadelphia, 3 (1974), 125–137.  Google Scholar

[31]

U. MayorN. R. GuydoshC. M. JohnsonJ. G. GrossmannS. SatoG. S. JasS. M. FreundD. O. AlonsoV. Daggett and A. R. Fersht, The complete folding pathway of a protein from nanoseconds to microseconds, Nature, 421 (2003), 863-867.   Google Scholar

[32]

A. MielkeJ. Haskovec and P. Markowich, On uniform decay of the entropy for reaction-diffusion systems, J. Dynam. Differential Equations, 27 (2015), 897-928.  doi: 10.1007/s10884-014-9394-x.  Google Scholar

[33]

H. Minc, Nonnegative Matrices Wiley Interscience in Discrete Mathematics and Optimization, John Wiley 1988.  Google Scholar

[34]

M. Mincheva and D. Siegel, Stability of mass action reaction–diffusion systems, Nonlinear Analysis TMA, 56 (2004), 1105-1131.  doi: 10.1016/j.na.2003.10.025.  Google Scholar

[35]

B. Perthame, Transport Equations in Biology Birkhäuser, Basel, 2007.  Google Scholar

[36]

F. Rothe, Global Stability of Reaction-Diffusion Systems Springer, Berlin, 1984.  Google Scholar

[37]

E. Seneta, Non-negative Matrices and Markov Chains Springer Series in Statistic, 2nd Edition, Springer, 1981. doi: 10.1007/0-387-32792-4.  Google Scholar

[38]

D. Siegel and D. MacLean, Global stability of complex balanced mechanisms, Journal of Mathematical Chemistry, 27 (2000), 89-110.  doi: 10.1023/A:1019183206064.  Google Scholar

[39]

J. Smoller, Shock Waves and Reaction-Diffusion Equations Springer, New York, 1994.  Google Scholar

[40]

M. Thattai and A. van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci. USA, 98 (2000), 8614-8619.  doi: 10.1073/pnas.151588598.  Google Scholar

[41]

G. Toscani, Kinetic approach to the asymptotic behaviour of the solution to diffusion equations, Rend. Mat., 16 (1996), 329-346.   Google Scholar

[42]

G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation, Comm. Math. Phys., 203 (1999), 667-706.  doi: 10.1007/s002200050631.  Google Scholar

[43]

C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.  doi: 10.1007/s00220-002-0777-1.  Google Scholar

[44]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp.  doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[45]

A. I. Volpert, Differential equations on graphs, Mat. Sb. , 88 (1972), 578–588 (in Russian); Math. USSR-Sb. , 17 (1972), 571–582 (in English).  Google Scholar

[46]

A. I. Volpert, V. A. Volpert and V. L. A. Volpert, Traveling Wave Solutions of Parabolic Systems American Mathematical Society, Providence, RI, 1994.  Google Scholar

show all references

References:
[1]

F. AchleitnerA. Arnold and D. Stürzer, Large-time behavior in non-symmetric Fokker-Planck equations, Riv. Mat. Univ. Parma, 6 (2015), 1-68.   Google Scholar

[2]

D. F. Anderson, A proof of the Global Attractor Conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508.  doi: 10.1137/11082631X.  Google Scholar

[3]

A. ArnoldP. MarkowichG. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. PDE, 26 (2001), 43-100.  doi: 10.1081/PDE-100002246.  Google Scholar

[4]

M. Bisi and L. Desvillettes, Global existence and large-time behavior of solutions to the threedimensional equations of compressible magnetohydrodynamic flows,, J. Stat. Phys., 125 (2006), 249-280.  doi: 10.1007/s10955-005-8075-x.  Google Scholar

[5]

J. Bang-Jensen and G. Z. Gregory, Digraphs: Theory, Algorithms and Applications Springer Science & Business Media, 2009. doi: 10.1007/978-1-84800-998-1.  Google Scholar

[6]

G. BokinskyD. RuedaV. K. MisraM. M. RhodesA. GordusH. P. BabcockN. G. Walter and X. Zhuang, Single-molecule transition-state analysis of RNA folding, Proc. Natl. Acad. Sci. USA, 100 (2003), 9302-9307.  doi: 10.1073/pnas.1133280100.  Google Scholar

[7]

J. CarrilloA. JüngelP. MarkowichG. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.  doi: 10.1007/s006050170032.  Google Scholar

[8]

G. CraciunA. DichkensteinA. Shiu and B. Sturmfels, Toric dynamical systems, J. Symb. Comput., 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006.  Google Scholar

[9]

L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction–diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.  doi: 10.1016/j.jmaa.2005.07.003.  Google Scholar

[10]

L. Desvillettes and K. Fellner, Entropy methods for reaction–diffusion systems, Discrete Contin. Dyn. Syst. Issue Special, (2007), 304-312.   Google Scholar

[11]

L. Desvillettes and K. Fellner, Entropy methods for reaction–diffusion equations: Slowly growing a-priori bounds, Rev. Mat. Iberoamericana, 24 (2008), 407-431.  doi: 10.4171/RMI/541.  Google Scholar

[12]

L. Desvillettes and K. Fellner, Exponential convergence to equilibrium for a nonlinear reaction-diffusion systems arising in reversible chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014), 96-104.  doi: 10.1007/978-3-662-45504-3_9.  Google Scholar

[13]

L. Desvillettes, K. Fellner and B. Q. Tang, Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks, arXiv: 1604. 04536, to appear in SIAM J. on Mathematical Analysis. Google Scholar

[14]

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.  doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[15]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Inventiones Mathematicae, 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[16]

M. Di FrancescoK. Fellner and P. Markowich, The entropy dissipation method for inhomogeneous reaction–diffusion systems, Proc. Royal Soc. A, 464 (2008), 3272-3300.  doi: 10.1098/rspa.2008.0214.  Google Scholar

[17]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[18]

M. Feinberg, Lectures on Chemical Reaction Networks, available online at http://www.crnt.osu.edu/LecturesOnReactionNetworks, 1979. Google Scholar

[19]

M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors. Ⅰ. The deficiency zero and deficiency one theorems, Chem. Eng. Sci., 42 (1987), 2229-2268.  doi: 10.1016/0009-2509(87)80099-4.  Google Scholar

[20]

M. Feinberg and F. J. M. Horn, Dynamics of open chemical systems and the algebraic structure of the underlying reaction network, Chem. Eng. Sci., 29 (1974), 775-787.  doi: 10.1016/0009-2509(74)80195-8.  Google Scholar

[21]

K. Fellner, E. Latos and B. Q. Tang, Well-posedness and exponential equilibration of a volumesurface reaction-diffusion system with nonlinear boundary coupling, arXiv: 1404.2809, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire. Google Scholar

[22]

J. Fontbona and B. Jourdain, A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations, Annals of Probability, 44 (2016), 131-170.  doi: 10.1214/14-AOP969.  Google Scholar

[23]

C. GadgilC. H. Lee and H. G. Othmer, A stochastic analysis of first-oder reaction networks, Bull. Math. Biology, 67 (2005), 901-946.  doi: 10.1016/j.bulm.2004.09.009.  Google Scholar

[24]

A. GlitzkyK. Gröger and R. Hünlich, Free energy and dissipation rate for reaction-diffusion processes of electrically charged species, Appl. Anal., 60 (1996), 201-217.  doi: 10.1080/00036819608840428.  Google Scholar

[25]

A. Glitzky and R. Hünlich, Energetic estimates and asymptotics for electro-reaction-diffusion systems,, Z. Angew. Math. Mech., 77 (1997), 823-832.  doi: 10.1002/zamm.19970771105.  Google Scholar

[26]

M. Gopalkrishnan, On the Lyapunov function for complex-balanced mass-action systems, arXiv: 1312.3043. Google Scholar

[27]

K. Gröger, Free energy estimates and asymptotic behaviour of reaction-diffusion processes, Preprint 20, Institut f¨ur Angewandte Analysis und Stochastik, Berlin, 1992. Google Scholar

[28]

F. J. M. Horn and R. Jackson, General mass action kinetics, Arch. Rational Mech. Anal., 47 (1972), 81-116.  doi: 10.1007/BF00251225.  Google Scholar

[29]

F. J. M. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Rational Mech. Anal., 49 (1972), 172-186.  doi: 10.1007/BF00255664.  Google Scholar

[30]

F. J. M. Horn, The dynamics of open reaction systems, SIAM-AMS Proceedings, SIAM, Philadelphia, 3 (1974), 125–137.  Google Scholar

[31]

U. MayorN. R. GuydoshC. M. JohnsonJ. G. GrossmannS. SatoG. S. JasS. M. FreundD. O. AlonsoV. Daggett and A. R. Fersht, The complete folding pathway of a protein from nanoseconds to microseconds, Nature, 421 (2003), 863-867.   Google Scholar

[32]

A. MielkeJ. Haskovec and P. Markowich, On uniform decay of the entropy for reaction-diffusion systems, J. Dynam. Differential Equations, 27 (2015), 897-928.  doi: 10.1007/s10884-014-9394-x.  Google Scholar

[33]

H. Minc, Nonnegative Matrices Wiley Interscience in Discrete Mathematics and Optimization, John Wiley 1988.  Google Scholar

[34]

M. Mincheva and D. Siegel, Stability of mass action reaction–diffusion systems, Nonlinear Analysis TMA, 56 (2004), 1105-1131.  doi: 10.1016/j.na.2003.10.025.  Google Scholar

[35]

B. Perthame, Transport Equations in Biology Birkhäuser, Basel, 2007.  Google Scholar

[36]

F. Rothe, Global Stability of Reaction-Diffusion Systems Springer, Berlin, 1984.  Google Scholar

[37]

E. Seneta, Non-negative Matrices and Markov Chains Springer Series in Statistic, 2nd Edition, Springer, 1981. doi: 10.1007/0-387-32792-4.  Google Scholar

[38]

D. Siegel and D. MacLean, Global stability of complex balanced mechanisms, Journal of Mathematical Chemistry, 27 (2000), 89-110.  doi: 10.1023/A:1019183206064.  Google Scholar

[39]

J. Smoller, Shock Waves and Reaction-Diffusion Equations Springer, New York, 1994.  Google Scholar

[40]

M. Thattai and A. van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci. USA, 98 (2000), 8614-8619.  doi: 10.1073/pnas.151588598.  Google Scholar

[41]

G. Toscani, Kinetic approach to the asymptotic behaviour of the solution to diffusion equations, Rend. Mat., 16 (1996), 329-346.   Google Scholar

[42]

G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation, Comm. Math. Phys., 203 (1999), 667-706.  doi: 10.1007/s002200050631.  Google Scholar

[43]

C. Villani, Cercignani's conjecture is sometimes true and always almost true, Comm. Math. Phys., 234 (2003), 455-490.  doi: 10.1007/s00220-002-0777-1.  Google Scholar

[44]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp.  doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[45]

A. I. Volpert, Differential equations on graphs, Mat. Sb. , 88 (1972), 578–588 (in Russian); Math. USSR-Sb. , 17 (1972), 571–582 (in English).  Google Scholar

[46]

A. I. Volpert, V. A. Volpert and V. L. A. Volpert, Traveling Wave Solutions of Parabolic Systems American Mathematical Society, Providence, RI, 1994.  Google Scholar

Figure 1.  A first-order chemical reaction network
Figure 2.  A reversible network
Figure 3.  A non-weakly reversible reaction network consisting of four strongly connected components
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