Bulletin of Mathematical Biology

, Volume 77, Issue 6, pp 1013–1045 | Cite as

Laplacian Dynamics with Synthesis and Degradation

  • Inom Mirzaev
  • David M. Bortz
Original Article


Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph \(G\) and the associated system of linear non-homogeneous differential equations with first-order degradation and zeroth-order synthesis. We also present a theorem which provides necessary and sufficient conditions for the dynamics to engender a unique stable steady state. Although the dynamics are linear, one can apply this framework to nonlinear systems by encoding nonlinearity into the edge labels. We answer an open question from our previous work concerning the non-positiveness of the elements in the inverse of a perturbed Laplacian matrix. Moreover, we provide a graph theoretical framework for the computation of the inverse of such a matrix. This also completes our previous framework and makes it purely graph theoretical. Lastly, we demonstrate the utility of this framework by applying it to a mathematical model of insulin secretion through ion channels in pancreatic \(\beta \)-cells.


Laplacian dynamics Biochemical networks Synthesis and degradation Matrix-Tree Theorem Insulin secretion 

List of abbreviations

\(\bar{\varvec{\rho }}^{C_{i}}\)

A column vector, which is the extension of \({\rho }^{C_{i}}\), see Sect. 2

\(\varvec{\rho }^{G}\)

Kernel element of strongly connected graph \(G\) calculated by MTT

\(\varDelta \)

A diagonal matrix with nonnegative entries

\(\frac{\mathrm{d}{\mathbf {x}}}{\mathrm{d}t}={\mathcal {L}}(G){\mathbf {x}}\)

Laplacian dynamics defined on the graph \(G\)

\(\frac{\mathrm{d}{\mathbf {x}}}{\mathrm{d}t}={\mathcal {L}}(G){\mathbf {x}} -D{\mathbf {x}}+{\mathbf {s}}\)

Synthesis and degradation dynamics

\({\mathbb {R}}^{m\times n}_{>0}\)

Set of all \(m\times n\) matrices with strictly positive entries

\({\mathbf {s}}\)

Synthesis vector: a column vector with synthesis edges as entries

\({\mathbf {x}}_{\mathrm{s}}\)

Steady state solution

\({\mathcal {L}}(G)\)

Laplacian matrix of the graph \(G\)

\({\mathcal {L}}_{i}\)

Perturbed matrix corresponding to SCC \(C_i, {\mathcal {L}}_{i}={\mathcal {L}}(C_{i})-\varDelta _{i}\), where \(\varDelta _i\) diagonal matrix corresponding to outgoing edges of SCC \(C_i\); if \(C_i\) is tSCC, then \(\varDelta _i\equiv 0\)

\(\mathcal {V}(X)\)

The set of vertices of graph \(X\)


Directed spanning tree

\(\varTheta _i(G)\)

Set of DSTs of graph \(G\) rooted at vertex \(i\)


Number of vertices in SCC \(C_i\)


\(ij\)th minor of Laplacian matrix \(A\) and is the determinant of \((n-1)\times (n-1)\) matrix that results from deleting row \(i\) and column \(j\)


Algebraic multiplicity of zero eigenvalue of matrix \(A\)


SCC containing \(i\)

\(C[i]\preceq C[j]\)

SCC containing vertex \(j\) can be reached from SCC containing \(i\)

\(C_{i}\preceq C_{j}\)

SCC \(C_{j}\) can be reached from SCC \(C_{i}\).


Degradation matrix, which is a diagonal matrix with degradation edges as diagonal entries


Label of degradation edge at vertex \(i\)


an edge from vertex \(j\) to vertex \(i\)


Labeled directed graph

\(G^{\star }\)

Complementary graph of \(G\), which is formed by directing all synthesis and degradation edges to new vertex \(*\)


Geometric multiplicity of zero eigenvalue of matrix \(A\)

\(i\Longleftrightarrow j\)

There exists a directed path from vertex \(i\) to vertex \(j\) and a directed path from vertex \(j\) to vertex \(i\)

\(i\Longrightarrow j\)

There exists a directed path from vertex \(i\) to vertex \(j\)


\(i\)th partial sum of \(c_i\)’s, \(m_{i}=\sum _{k=1}^{i}c_{k}\)


Lower-block diagonal submatrix of \({\mathcal {L}}(G)\) corresponding to non-terminal SCCs


Label of synthesis edge at vertex \(i\)


Block diagonal submatrix of \({\mathcal {L}}(G)\) corresponding to tSCCs


Directed spanning tree


Matrix-Tree Theorem


Strongly connected component

SD dynamics

Synthesis and degradation dynamics


Terminal strongly connected component



Funding for this research was supported in part by Grants NIH-NIGMS 2R01GM069438-06A2 and NSF-DMS 1225878. The authors would also like to thank Dr. Clayton Thompson (Systems Biology Group, Pfizer, Inc.) for his suggestion of the insulin synthesis example used in Sect. 5 and the two anonymous reviewers for their valuable comments on our manuscript.


  1. Ackers GK, Johnson AD, Shea MA (1982) Quantitative model for gene regulation by lambda phage repressor. Proc Natl Acad Sci USA 79(4):1129–1133CrossRefGoogle Scholar
  2. Agaev R, Chebotarev P (2006) The matrix of maximum out forests of a digraph and its applications. Autom Remote Control 61(9):27MathSciNetGoogle Scholar
  3. Ahsendorf T, Wong F, Eils R, Gunawardena J (2014) A framework for modelling gene regulation which accommodates non-equilibrium mechanisms. BMC Biol 12(1):102CrossRefGoogle Scholar
  4. Barg S, Olofsson CS, Schriever-Abeln J, Wendt A, Gebre-Medhin S, Renström E, Rorsman P (2002) Delay between fusion pore opening and peptide release from large dense-core vesicles in neuroendocrine cells. Neuron 33(2):287–299CrossRefGoogle Scholar
  5. Bérenguier D, Chaouiya C, Monteiro PT, Naldi A, Remy E, Thieffry D, Tichit L (2013) Dynamical modeling and analysis of large cellular regulatory networks. Chaos 23(2):025114MathSciNetCrossRefGoogle Scholar
  6. Bronski JC, DeVille L (2014) Spectral theory for dynamics on graphs containing attractive and repulsive interactions. SIAM J Appl Math 74(1):83–105MathSciNetCrossRefzbMATHGoogle Scholar
  7. Chebotarev P, Agaev R (2002) Forest matrices around the Laplacian matrix. Linear Algebra Appl 1:1–19MathSciNetCrossRefGoogle Scholar
  8. Chen Y-D, Wang S, Sherman A (2008) Identifying the targets of the amplifying pathway for insulin secretion in pancreatic beta-cells by kinetic modeling of granule exocytosis. Biophys J 95(5):2226–2241CrossRefGoogle Scholar
  9. Chou K-C, Min LW (1981) Graphical rules for non-steady state enzyme kinetics. J Theor Biol 91(4):637–654CrossRefGoogle Scholar
  10. Chou K-C (1981) Two new schematic rules for rate laws of enzyme-catalysed reactions. J Theor Biol 89(4):581–592CrossRefGoogle Scholar
  11. Chou K-C (1983) Advances in graphic methods of enzyme kinetics. Biophys Chem 17(1):51–55CrossRefGoogle Scholar
  12. Craciun G, Feinberg M (2005) Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J Appl Math 65(5):1526–1546MathSciNetCrossRefzbMATHGoogle Scholar
  13. Craciun G, Feinberg M (2006) Multiple equilibria in complex chemical reaction networks: II. The species-reaction graph. SIAM J Appl Math 66(4):1321–1338MathSciNetCrossRefzbMATHGoogle Scholar
  14. Craciun G, Tang Y, Feinberg M (2006) Understanding bistability in complex enzyme-driven reaction networks. Proc Natl Acad Sci USA 103(23):8697–8702CrossRefzbMATHGoogle Scholar
  15. Craciun G, Feinberg M (2010) Multiple equilibria in complex chemical reaction networks: semiopen mass action systems. SIAM J Appl Math 70(6):1859–1877MathSciNetCrossRefzbMATHGoogle Scholar
  16. Dasgupta T, Croll DH, Owen JA, Vander Heiden MG, Locasale JW, Alon U, Cantley LC, Gunawardena J (2014) A fundamental trade-off in covalent switching and its circumvention by enzyme bifunctionality in glucose homeostasis. J Biol Chem 289(19):13010–13025CrossRefGoogle Scholar
  17. Domijan M, Kirkilionis M (2008) Graph theory and qualitative analysis of reaction networks. Netw Heterog Media 3(2):295–322MathSciNetCrossRefzbMATHGoogle Scholar
  18. Gunawardena J (2012) A linear framework for time-scale separation in nonlinear biochemical systems. PloS One 7(5):1–26CrossRefGoogle Scholar
  19. Gunawardena J (2014) Time-scale separation—Michaelis and Menten’s old idea, still bearing fruit. FEBS J 281(2):473–488CrossRefGoogle Scholar
  20. Kirchhoff G (1847) Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Annalen der Physik und Chemie 148(12):497–508CrossRefGoogle Scholar
  21. Lean AD, Stadel J, Lefkowitz R (1980) A ternary complex model explains the agonist-specific binding properties of the adenylate cyclase-coupled beta-adrenergic receptor. J Biol Chem 255(15):7108–7117Google Scholar
  22. Lin S-X, Lapointe J (2013) Theoretical and experimental biology in one. J Biomed Sci Eng 06(04):435–442CrossRefGoogle Scholar
  23. Marashi S-A, Tefagh M (2014) A mathematical approach to emergent properties of metabolic networks: partial coupling relations, hyperarcs and flux ratios. J Theor Biol 355:185–193MathSciNetCrossRefGoogle Scholar
  24. Mincheva M, Roussel MR (2007a) Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models. J Math Biol 55(1):61–86MathSciNetCrossRefzbMATHGoogle Scholar
  25. Mincheva M, Roussel MR (2007b) Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays. J Math Biol 55(1):87–104MathSciNetCrossRefzbMATHGoogle Scholar
  26. Mincheva M (2011) Oscillations in biochemical reaction networks arising from pairs of subnetworks. Bull Math Biol 73(10):2277–2304MathSciNetCrossRefGoogle Scholar
  27. Mirzaev I, Gunawardena J (2013) Laplacian dynamics on general graphs. Bull Math Biol 75(11):2118–2149MathSciNetCrossRefzbMATHGoogle Scholar
  28. Monod J, Wyman J, Changeux J (1965) On the nature of allosteric transitions: a plausible model. J Mol Biol 12:88–118CrossRefGoogle Scholar
  29. Olofsson CS, Göpel SO, Barg S, Galvanovskis J, Ma X, Salehi A, Rorsman P, Eliasson L (2002) Fast insulin secretion reflects exocytosis of docked granules in mouse pancreatic B-cells. Pflügers Archiv: Eur J Physiol 444(1–2):43–51CrossRefGoogle Scholar
  30. Rorsman P, Renström E (2003) Insulin granule dynamics in pancreatic beta cells. Diabetologia 46(8):1029–1045CrossRefGoogle Scholar
  31. Thomson M, Gunawardena J (2009a) The rational parameterization theorem for multisite post-translational modification systems. J Theor Biol 261(4):626–636MathSciNetCrossRefGoogle Scholar
  32. Thomson M, Gunawardena J (2009b) Unlimited multistability in multisite phosphorylation systems. Nature 460(7252):274–277CrossRefGoogle Scholar
  33. Tutte WT (2008) The dissection of equilateral triangles into equilateral triangles. Math Proc Camb Philos Soc 44(04):463MathSciNetCrossRefGoogle Scholar
  34. Uno T (1996) Algorithms and Computation, volume 1178 of Lecture Notes in Computer Science. Springer, Berlin, HeidelbergGoogle Scholar
  35. Voets T, Neher E, Moser T (1999) Mechanisms underlying phasic and sustained secretion in chromaffin cells from mouse adrenal slices. Neuron 23(3):607–615CrossRefGoogle Scholar
  36. Wollheim CB, Sharp GW (1981) Regulation of insulin release by calcium. Physiol Rev 61(4):914–973Google Scholar
  37. Xu Y, Gunawardena J (2012) Realistic enzymology for post-translational modification: zero-order ultrasensitivity revisited. J Theor Biol 311:139–152MathSciNetCrossRefGoogle Scholar
  38. Zhou G, Deng M (1984) An extension of Chou’s graphic rules for deriving enzyme kinetic equations to systems involving parallel reaction pathways. Biochem J 39(1):95–99Google Scholar

Copyright information

© Society for Mathematical Biology 2015

Authors and Affiliations

  1. 1.Applied MathematicsUniversity of ColoradoBoulderUSA

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