Stability of Control Networks in Autonomous Homeostatic Regulation of Stem Cell Lineages

Abstract

Design principles of biological networks have been studied extensively in the context of protein–protein interaction networks, metabolic networks, and regulatory (transcriptional) networks. Here we consider regulation networks that occur on larger scales, namely the cell-to-cell signaling networks that connect groups of cells in multicellular organisms. These are the feedback loops that orchestrate the complex dynamics of cell fate decisions and are necessary for the maintenance of homeostasis in stem cell lineages. We focus on “minimal” networks that are those that have the smallest possible numbers of controls. For such minimal networks, the number of controls must be equal to the number of compartments, and the reducibility/irreducibility of the network (whether or not it can be split into smaller independent sub-networks) is defined by a matrix comprised of the cell number increments induced by each of the controlled processes in each of the compartments. Using the formalism of digraphs, we show that in two-compartment lineages, reducible systems must contain two 1-cycles, and irreducible systems one 1-cycle and one 2-cycle; stability follows from the signs of the controls and does not require magnitude restrictions. In three-compartment systems, irreducible digraphs have a tree structure or have one 3-cycle and at least two more shorter cycles, at least one of which is a 1-cycle. With further work and proper biological validation, our results may serve as a first step toward an understanding of ways in which these networks become dysregulated in cancer.

Appendices

Appendix 1: Case 1 in Sect. 3.1

Case 1 in Sect. 3.2 allows for two different possibilities, depending on the matrix $\tilde{D}$, which contains 4 non-zero entries. The characteristic polynomial of $\tilde{J}=\tilde{D}\tilde{B}$ is given by

$$\begin{aligned} \lambda ^2-\lambda (\Delta _i^1Q_{ix} +\Delta _j^2Q_{jy})+Det(\tilde{D})Q_{ix}Q_{jy}. \end{aligned}$$

For both eigenvalues to have negative real parts, necessary and sufficient conditions are (as dictated by the Routh–Hurwitz conditions, see, e.g., Hershkowitz (2007)):

$$\begin{aligned}&\Delta _i^1Q_{ix} +\Delta _j^2Q_{jy}<0, \end{aligned}$$

(14)

$$\begin{aligned}&Det(\tilde{D})Q_{ix}Q_{jy}>0. \end{aligned}$$

(15)

Stability conditions resulting from the quadratic characteristic equation lead to the following two cases:

• Case 1(a). If $\Delta _i^1\Delta _j^2>0$ and $Det(\tilde{D})>0$, then the system is stable if the signs of the controls are assigned correctly. Similarly, with $\Delta _i^1\Delta _j^2<0$ and $Det(\tilde{D})<0$.

• Case 1(b). If $\Delta _i^1\Delta _j^2>0$ and $Det( \tilde{D})<0$, then the system allows two distinct sets of conditions that guarantee stability. The two sets of conditions imply different signs of the controls (and also contain restrictions on their magnitude), such that there are two alternative signings (or wirings) of the network compatible with stability. Similarly, if $\Delta _i^1\Delta _j^2<0$ and $Det(\tilde{D})>0$, then two alternative stable wirings are possible.

An example of case 1(a) above is given by the pair $(Q_{5x},Q_{1y})$. Case 1(b) is represented by the pair $(Q_{1x},Q_{5y})$. In both cases, $\Delta _i^1\Delta _j^2>0$ but the sign of $Det(\tilde{D})$ is, respectively, positive and negative in the two cases.

Appendix 2: Techniques

We wrote a program in Mathematica that for a given system, lists all stable minimal controls and classifies them in terms of reducibility. This Mathematica code is given in Supplementary Material available online.

The input includes the number of compartments ($n=3$ in the case considered) and the list of possible processes with the corresponding increments. The program includes a loop that goes over all possible n-tuples of controls. These are analyzed for stability and only those that can be stable are listed in the output.

To perform the analysis for the $n=3$ case, the following rules were used. These come from the Routh–Hurwitz conditions and results on potential stability (Grundy et al. 2012). Suppose the characteristic polynomial of J is denoted as

$$\begin{aligned} P(\lambda )= \lambda ^3+a_2 \lambda ^2+a_1 \lambda +a_0. \end{aligned}$$

A combination of three non-zero controls was discarded if:

• 1. Any of $a_i$ is zero.

• 2. The matrix $\tilde{J}$ consists of one simply connected component and contains fewer than 5 non-zero entries.

• 3. The matrix $\tilde{J}$ consists of one simply connected component, and $a_0-a_1a_2=0$.

For systems that are not rejected by these criteria, the characteristic polynomial is factored to determine the number of simply connected components, and also the stability conditions are determined by solving a set of inequalities.