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The regulatory mechanisms in cellular signaling systems have been studied intensively from the viewpoint that the malfunction of the regulation is thought to be one of the substantial causes of cancer formation. On the other hand, it is rather difficult to develop the theoretical framework for investigation of the regulatory mechanisms due to their complexity and nonlinearity. In this study, more general approach is proposed for elucidation of characteristics of the stability in cellular signaling systems by construction of mathematical models for a class of cellular signaling systems and stability analysis of the models over variation of the network architectures and the parameter values. The model system is formulated as regulatory network in which every node represents a phosphorylation-dephosphorylation cyclic reaction for respective constituent enzyme. The analysis is performed for all variations of the regulatory networks comprised of two nodes with multiple feedback regulation loops. It is revealed from the analysis that the regulatory networks become mono-stable, bi-stable, tri-stable, or oscillatory and that the negative mutual feedback or positive mutual feedback is favorable for multi-stability, which is augmented by a negatively regulated node with a positive auto-regulation. Furthermore, the multi-stability or the oscillation is more likely to emerge in the case of low value of the Michaelis constant than in the case of high value, implying that the condition of higher saturation levels induces stronger nonlinearity in the networks. The analysis for the parameter regions yielding the multi-stability and the oscillation clarified that the stronger regulation shifts the systems toward multi-stability.

Cellular signaling systems have been studied extensively from the recent viewpoint that their disorder is thought to be one of the causes for cancer formation since the systems are known to regulate biochemical reactions operating in cells for various functions such as cell differentiation, cell proliferation, and homeostasis.

seems to be primary reaction system for other cellular signaling systems such as Rac1, PAK, and RhoA signaling networks [

Many studies have employed simulation analysis with the given values for the parameters in the signaling systems because of the difficulty in developing the analytical method for the systems analysis due to their nonlinearity. In this study, more general approach is proposed for characterization of the stability in cellular signaling systems by construction of mathematical models for a class of cellular signaling systems and simulation analysis of the models over variation of the network architectures and the values of parameters. The model system is formulated as regulatory network in which every node represents an activation-inactivation cyclic reaction for respective constituent enzyme of the network and the regulatory interactions between the activated enzyme and the reaction are depicted by arcs between nodes. The Michaelis-Menten mechanism is assumed for the reaction paths in each cyclic reaction and the emergence of the stable point in steady states of the network is analyzed.

It is biologically significant to analyze the characteristics of the stability in cellular signaling systems since stable points are convergent states of the relaxation process in dynamic changes due to random noises, and seem to correspond to the distinct biochemical states such as normal states or malfunctional states. Similar approaches have been taken in several studies [

In this study the stability analysis is performed for all variations of the regulatory networks comprised of two nodes with multiple feedback regulation loops.

The regulatory network is formulated as follows. Every node represents an activation-inactivation cyclic reaction for respective constituent enzyme of the network. The activated enzyme in a node acts on another node for the positive regulation which increases the active enzyme or for the negative regulation which increases the inactive enzyme through the reverse path. The regulatory interactions between the activated enzyme and the reaction are depicted by arcs between nodes. The MAPK cascade is comprised of several cyclic reactions and has a feedback regulation from MAPK-PP as shown in

It should be noted that the Michaelis-Menten equation is adopted as the reaction mechanism, and therefore, the enzyme-substrate complex does not appear in the reaction rate equations.

We analyze the characteristics of stability for all variations of two-node regulatory networks with multiple feedback regulation loops under the restriction that each node has at most one positive regulation and one negative regulation as designated in

In

In

It should be noted that, as the networks G, H, and J are symmetric for each node, the graphs for these networks are symmetric to the diagonal line. It is revealed that bi-stable states appear in the area with low values of

networks H and J which contain mutual positive feedbacks. Since the parameter

The bi-stable area for lower value of the Michaelis constant include the area for higher value of the Michaelis constant for the network F, G, H, I, and J, that is consistent with the tendency that lower value of the Michaelis constant is favorable to emergence of multi-stable state shown in ^{0}.

It follows from

The white area in the graph at

area with small values of

It is found that all of the limit cycles are counterclockwise. The limit cycles arise in the case of

The characteristics of the emergence of stable equilibrium points are analyzed quantitatively by the emergence ratio and the sensitivity of equilibria which are newly proposed and defined in this study. These quantities seem to successfully detect the effects of the architectures and the values of parameters on the characteristics.

Comparison of the emergence ratios of bi-stable state for the regulatory networks as shown in

Mutual feedback | Auto-feedback | |
---|---|---|

Boost | ||

Decline |

The partial network structures in the upper row make the emergence ratio of multi-stable state higher, while those in the bottom row work reversely.

the other node could boost multi-stable systems, while a network containing node with both of negative auto regulation and positive regulation from the other node curtail the multi-stable systems.

As a common feature for the regulatory networks examined, smaller normalized Michaelis constant L yields higher emergence ratio of multi-stability or oscillation, implying that the condition of the higher saturation levels induces stronger nonlinearity. In the parameter spaces multi-stability arises in the region where

It is difficult to visualize and analyze the number of the stable equilibrium points and the values of the points, since high dimensional representation is required due to the existence of plural stable equilibrium points in two-dimen- sional space such as

Furthermore, the emergence of oscillations are analyzed to elucidate that the emergence of the oscillations are rare from the viewpoints of the regulatory structures and the values of parameters comparing to the high emergence of multi-stable states, such as 100% for the network J at the small values of L.

It is suggested that the method proposed in this study utilizing the emergence ratio and the sensitivity of equilibria are useful for this kind of analysis. On the other hand, it is unclear if these results are available to the higher order networks such as three-node or four-node regulatory networks due to the nonlinearity of the systems. Therefore, the similar analysis for the higher order networks are undergoing in our laboratory, facing the difficulty for the high dimension to visualize. It is also required to invent the new way to visualize and analyze the high dimensional dynamics.

The analysis is performed with variation of the parameter values in appropriate range to cover the assumed values for reactions of MAPK cascade described in other studies [^{2}).

Stability is assessed by the standard stability theory [

Furthermore, the sensitivity of equilibria (or equilibrium) is defined and utilized to quantify the effect of the change of parameter values on a set of stable equilibrium points. That is, the sensitivity denotes the change of numbers of the equilibrium points and how much the values of equilibria move when the values of parameters are changed. At first, we define the distance between two sets of points in two dimensional Euclid space as:

The defined distance

where

Sueyoshi, C. and Naka, T. (2017) Stability Analysis for the Cellular Signaling Systems Composed of Two Phosphoryltion-Dephosphorylation Cyclic Re-actions. Computational Molecular Bioscience, 7, 33-45. https://doi.org/10.4236/cmb.2017.73003