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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[30]

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

[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.

[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.

[33]

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

[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.

[35]

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

[36]

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

[37]

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

[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.

[39]

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

[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.

[41]

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

[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.

[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.

[44]

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

[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).

[46]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[30]

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

[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.

[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.

[33]

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

[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.

[35]

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

[36]

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

[37]

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

[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.

[39]

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

[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.

[41]

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

[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.

[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.

[44]

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

[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).

[46]

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

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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