Theoretical approaches for the dynamics of complex biological systems from information of networks

Abstract

Modern biology has provided many examples of large networks describing the interactions between multiple species of bio-molecules. It is believed that the dynamics of molecular activities based on such networks are the origin of biological functions. On the other hand, we have a limited understanding for dynamics of molecular activity based on networks. To overcome this problem, we have developed two structural theories, by which the important aspects of the dynamical properties of the system are determined only from information on the network structure, without assuming other quantitative details. The first theory, named Linkage Logic, determines a subset of molecules in regulatory networks, by which any long-term dynamical behavior of the whole system can be identified/controlled. The second theory, named Structural Sensitivity Analysis, determines the sensitivity responses of the steady state of chemical reaction networks to perturbations of the reaction rate. The first and second theories investigate the dynamical properties of regulatory and reaction networks, respectively. The first theory targets the attractors of the regulatory network systems, whereas the second theory applies only to the steady states of the reaction network systems, but predicts their detailed behavior. To demonstrate the utility of our methods several biological network systems, and show they are practically useful to analyze behaviors of biological systems.

Figure 1.

Examples of small regulatory networks. The directed edges show the regulatory interactions between the nodes. The gray vertices highlight a selected minimal feedback vertex set in each case (a)–(e). Modified from Mochizuki et al. (2013).¹⁰⁾

Figure 2.

Intuitive interpretation of the theory. If the dynamics are known at the gray vertices, the dynamics of the remaining vertices are uniquely determined. The set of vertices on which the dynamics are given can be reduced to a minimal feedback vertex set (in this case, the single vertex marked by a red circle). Details are given in the text. Modified from Mochizuki et al. (2013).¹⁰⁾

Figure 3.

Gene regulatory network of Ascidiacea development. (a) Network based on Imai et al. (2006).³⁾ The original network includes 16 genes with self-repression. We removed these repressive self-loops, because self-repression is subsumed under degradation in our formulation [2]. (b) Reduced network after successive removal of nodes without input or output. The reduced network contains a minimal feedback vertex set with a single vertex, FoxD-a/b. Modified from Mochizuki et al. (2013).¹⁰⁾

Figure 4.

Sensitivity analysis of a single reaction pathway of chemical reactions with a feed 1 and an exit reaction 4. (a–d) Changes in concentrations and fluxes induced by perturbations of reaction rates k₁ to k₄ from top (a) to bottom (d). Red triangles indicate the perturbed reactions j = 1, ⋯, 4. The plus or minus signs adjacent to the circles indicate an increase or decrease in the concentrations of chemicals A, B, C, respectively. Red bold circles (and arrows) indicate increases in concentrations (and fluxes). Red dashed circles (and arrows) indicate decrease in concentrations (and fluxes). Modified from Mochizuki and Fiedler (2015).¹¹⁾

Figure 5.

Sensitivity analysis of a simple network with one feedback loop. (a–f) Perturbed reactions (indicated by red triangles) are j = 1, ⋯, 6. Changes in concentrations and fluxes induced by perturbations of reaction rates k₁, …, k₆ are indicated. See also the legend of Fig. 1. Modified from Mochizuki and Fiedler (2015).¹¹⁾

Figure 6.

Reaction network of the TCA cycle in the carbon metabolic network of E. coli. Colors summarize the influence patterns of the flux responses shown in Fig. 7, with some details omitted. The flux influence patterns are summarized as: i) yellow→yellow, ii) red→{red, yellow}, iii) brown→{red, yellow}, iv) light blue→light blue, v) blue→{blue (including dashed blue), light blue}, vi) green→{green (including dashed), blue, light blue, brown, red, yellow}, vii) black→{black (including dashed), green, blue, light blue, brown, red, yellow}. Modified from Mochizuki and Fiedler (2015).¹¹⁾

Figure 7.

Hierarchy of nonzero response patterns of the carbon metabolism network shown in Fig. 6. Modified from Mochizuki and Fiedler (2015).¹¹⁾