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Stability analysis and dynamic simulation are important for researchers to capture the performance and the properties of underling systems. S-systems have good potential for characterizing dynamic interactive behaviour of large scale metabolic and genetic systems. It is important to develop a platform to achieve timely dynamic behaviour of S-systems to various situations. In this study, we first set up the respective block diagrams of S-systems for module-based simulation. We then derive reasonable theorems to examine the stability of S-systems and find out what kinds of environmental situations will make systems stable. Three canonical systems are used to examine the results which are carried out in the Matlab/Simulink environments.

A model in state space representation described as nonlinearly coupled ordinary differential equations (ODEs) is able to extract biologically information of underlying systems. S-systems [

System identification is reduced to infer the structure and estimate the parameters in the case that a given family of ODEs is chosen. One-stage identification carries out these two things at the same time. The identification becomes a multiobjective optimization problem. Extremely good performances in globally and locally searching abilities for computational modelling challenge researchers. We previously proposed various methods to smarten up the existing intelligent technologies for S-system modelling. A self-interactive learning was proposed to integrate an error performance, a skeleton structure index and a smooth evolution index [

Some researchers divided S-system modelling into two stages (structure identification and parameter estimation). In this way a multiobjective optimization problem becomes two single-objective optimization problems (two-stage system identification). S-system modelling also reduces to parameter estimation when the interactive relation of constitutes is partially known, or the static biological pathway is known. Therefore, parameter estimation for S-systems is required in the case that (a) system structures are inferred, (b) the relationship between genes and/or proteins is known, or (c) the qualitative pathways of underlying systems are known. Two review articles for metaheuristic developments in systems biology were published recently [

Chowdhury and coworkers currently introduced time delay into S-systems, wherein kinetic constants become real numbers instead of integer values [

The steady state values and the sensitivity at equilibrium points of S-systems were theoretically derived through algebraic equations because of the special power-law structures [

S-systems derived from biochemical system theory (BST) are used to describe interactive behaviour of metabolites. The rate change of constitutes (metabolites, proteins or genes) _{i} and _{i} are the rate constants, and

There are one constant source

Steady state values of the branch pathway.

Case | |||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

initial values | | 1.6 | 1.6 | 2.7 | 2.7 | 1.6 | 0.4 | 2.2 | 2 |

| 1.6 | 1.6 | 1.6 | 0.4 | 1.6 | 2.7 | 2 | 1.2 | |

| 0.4 | 0.4 | 2 | 0.4 | 0.4 | 0.4 | 1.6 | 2.3 | |

| 1.6 | 1.6 | 0.4 | 1.6 | 1.6 | 2.7 | 2.3 | 0.8 | |

| |||||||||

Independent variables | | 0.6 | 0.3 | 0.3 | 0.9 | 0.9 | 0.9 | 0.6 | 0.6 |

| |||||||||

steady-state values | | 0.3995605 | 0.2127742 | 0.2127742 | 0.5776509 | 0.5776509 | 0.5776509 | 0.3995605 | 0.3995605 |

| 2.006073 | 1.317966 | 1.317966 | 2.564889 | 2.564889 | 2.564889 | 2.006073 | 2.006073 | |

| 2.228371 | 1.38912 | 1.38912 | 2.937986 | 2.937986 | 2.937986 | 2.228371 | 2.228371 | |

| 0.1427537 | 0.0962826 | 0.0962826 | 0.1797378 | 0.1797378 | 0.1797378 | 0.1427537 | 0.1427537 |

Cascade pathways [

The block diagrams for the cascade pathway.

(3 genes) simulation results of Case 8 in Table

The branch pathway in Figure

Steady state values of the cascade pathway.

Case | |||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

initial values | | 0.1 | 0.4 | 0.6 | 0.6 | 0.6 | 0.6 | 0.4 | 0.2 |

| 0.1 | 0.6 | 0.1 | 0.6 | 0.1 | 0.6 | 0.1 | 0.5 | |

| 0.1 | 0.6 | 0.4 | 0.4 | 0.6 | 0.1 | 0.6 | 0.1 | |

| |||||||||

Independent variables | | 0.75 | 0.9 | 0.6 | 0.6 | 0.9 | 0.75 | 0.75 | 0.75 |

| |||||||||

steady-state values | | 1.834666 | 2.428703 | 1.301562 | 1.301562 | 2.428703 | 1.834666 | 1.834666 | 1.834666 |

| 3.539094 | 4.684998 | 2.510729 | 2.510729 | 4.684998 | 3.539094 | 3.539094 | 3.539094 | |

| 0.6144261 | 0.8133678 | 0.4358904 | 0.4358904 | 0.8133678 | 0.6144261 | 0.6144261 | 0.6144261 |

Genetic branch pathways [

The block diagrams for the genetic branch pathway.

(

The genetic system in Figure

Steady state values of the small scale network.

Case | |||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

initial values | | 0.4 | 0.7 | 0.1 | 0.4 | 0.4 | 0.7 | 0.1 | 0.1 |

| 0.4 | 0.4 | 0.4 | 0.4 | 0.1 | 0.1 | 0.1 | 0.7 | |

| 0.7 | 0.1 | 0.1 | 0.4 | 0.7 | 0.1 | 0.4 | 0.4 | |

| 0.1 | 0.4 | 0.4 | 0.7 | 0.4 | 0.1 | 0.4 | 0.1 | |

| 0.7 | 0.1 | 0.4 | 0.4 | 0.7 | 0.4 | 0.7 | 0.7 | |

| |||||||||

Independent variables | | 1 | 1 | 1.25 | 1.25 | 1.25 | 0.75 | 0.75 | 0.75 |

| 0.75 | 1 | 1.25 | 1 | 1.25 | 1 | 0.75 | 1 | |

| 0.75 | 1.25 | 1.25 | 0.75 | 1 | 0.75 | 1 | 1.25 | |

| |||||||||

steady-state values | | 0.7517064 | 0.7476744 | 0.7760044 | 0.7718421 | 0.7617077 | 0.6509968 | 0.6995416 | 0.679307 |

| 0.6509968 | 0.7476744 | 0.8675993 | 0.7718421 | 0.8516151 | 0.6509968 | 0.6058208 | 0.679307 | |

| 0.8660254 | 1.118034 | 1.118034 | 0.8660254 | 1 | 0.8660254 | 1 | 1.118034 | |

| 0.8848579 | 1 | 1.037891 | 0.9085603 | 0.9634925 | 0.7663094 | 0.8848579 | 0.90856 | |

| 0.7663094 | 1 | 1.160397 | 0.9085603 | 1.077217 | 0.7663094 | 0.7663094 | 0.90856 |

Small scale genetic networks [

The block diagrams for the small scale genetic network.

(

The S-system in (

The change of the values of independent variables denotes that a cell faces persistent changes in survival environments. Systems will show different dynamic evolution in response to such a change, which may be induced by such a stress environment as heat shock. Lee and coworkers [

As we know a biological system always operates at a steady state and will be temporarily deviated from the state. For example, blood serum undergoes short term changes for the intake and absorption of water and food and for kidney and liver’s operations. In a living system there exist regulatory mechanisms to effectively regulate various concentrations back to their nominal steady state levels. So what we concern is whether a biological system is locally stable. What kinds of independent variables can generate an equilibrium state such that a system can asymptotically stabilized to the state.

If all of the eigenvalues of the Hadamard product

By the converse theorem [

If

We shall derive Theorem

In the case of the system parameter

If

We know

“

Through (

For

Now we shall use the derived theorems to discuss the stability of our systems. The kinetic-order parameter of the branch system is

_{d} is -0.2705 ± 0.5152

Stability analysis for the three biological systems. The independent variables in the _{6}, x_{7}, x_{8}) = (1, 0.75, 0.75) for Case

cases | ||||||
---|---|---|---|---|---|---|

a branch pathway (4-gene) | A cascade pathway (3-gene) | A small-scale network (5-gene) | ||||

| stable | | stable | | stable | |

| ||||||

1 | -1.0599 ±1.7013i | yes | -0.5543 | yes | -13.0199 | yes |

-7.9502 ± 1.6192i | -1.6468 | -15.0341 | ||||

-4.6200 | -26.6061 | |||||

-16.5117 ± 11.5849 | ||||||

| ||||||

2 | -1.2373±1.9465i | yes | -0.4818 | yes | -14.9535 | yes |

-9.7292 ± 1.0894i | -1.4313 | -14.9535 | ||||

-4.0154 | -29.9070 | |||||

-20 ± 14.1421 | ||||||

| ||||||

3 | -0.9651 ± 1.5704i | yes | -0.6581 | yes | -17.3520 | yes |

-7.1134 ± 1.5568i | -1.9552 | -15.5201 | ||||

-5.4851 | -25.7730 | |||||

-21.9829 ± 15.4717i |

Figure

Simulation results for a

The generalized mass action model (another popular BST-based model in biological systems) is composed of a set of coupled nonlinearly differential equations. However, each equation includes more than two power-law functions. For a generalized mass action model with

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This research was supported by the Ministry of Science and Technology of Taiwan, Grant no. MOST 107-2221-E-212-013.