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Laplacian Dynamics with Synthesis and Degradation

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Abstract

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.

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Notes

  1. If a negative edge weight is encountered in applications, one can reverse orientation of that edge, hence preserving positivity of edge labels.

  2. For the convenience of the reader and to promote clarity, we include at the end of this document a list of nomenclature used throughout this work.

  3. Available at http://vcp.med.harvard.edu/software.html.

  4. For more generalized versions of MTT such as all-minors Matrix-Tree Theorem and Matrix-Forest Theorem, we refer reader to Chebotarev and Agaev (2002) and Agaev and Chebotarev (2006).

  5. Later, we will define \(\rho _{i}^{G}\) as an entry of the kernel element of Laplacian.

    Fig. 1
    figure 1

    (Color Figure Online) Illustration of Matrix-Tree Theorem. a A strongly connected graph \(G\) with its set of spanning trees rooted at each of its vertices. The root node is bolded in each spanning tree. b Associated Laplacian matrix of \(G\). Two minors of the Laplacian matrix, \({\mathcal {L}}(G)_{(23)}\) and \({\mathcal {L}}(G)_{(32)}\), are calculated using the Matrix-Tree Theorem

    Full size image

  6. We have devoted all of Sect. 4 for the proof of this fact as well as presenting a graph theoretical algorithm for computation of \(N^{-1}\).

  7. Recall that dimensions of row and column null spaces of a matrix are same. In fact, from Corollary 1, we have this dimension equal to number of tSCC of graph \(G^{\star }\).

  8. Note that \(G^{\star }({\mathbf {e}}^{(i)})\) is not always strongly connected.

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 \(X\)

\({\fancyscript{T}}\) :

Directed spanning tree

\(\varTheta _i(G)\) :

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

\(a_i\) :

Number of vertices in SCC \(C_i\)

\(A_{(ij)}\) :

\(ij\)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 \(j\)

\(\mathrm{alg}_A(0)\) :

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

\(C[i]\) :

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 \(C_{i}\).

\(D\) :

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

\(d_i\) :

Label of degradation edge at vertex \(i\)

\(e_{ij}\) :

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

\(G\) :

Labeled directed graph

\(G^{\star }\) :

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

\(\mathrm{geo}_A(0)\) :

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 \(j\)

\(m_i\) :

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

\(N\) :

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

\(s_i\) :

Label of synthesis edge at vertex \(i\)

\(T\) :

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

DST:

Directed spanning tree

MTT:

Matrix-Tree Theorem

SCC:

Strongly connected component

SD dynamics:

Synthesis and degradation dynamics

tSCC:

Terminal strongly connected component

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Acknowledgments

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.

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  1. Applied Mathematics, University of Colorado, Boulder, CO, 80309-0526, USA

    Inom Mirzaev & David M. Bortz

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Mirzaev, I., Bortz, D.M. Laplacian Dynamics with Synthesis and Degradation. Bull Math Biol 77, 1013–1045 (2015). https://doi.org/10.1007/s11538-015-0075-7

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