A Synergistic system (or S-system) is a collection of ordinary nonlinear differential equations
where the are positive real, and are non-negative real, called the rate constant(or, kinetic rates) and and are real exponential, called kinetic orders. These terms are based on the chemical equilibrium
One variable S-system Source:
In the case of and , the given S-system equation can be written as
Under the non-zero steady condition, , the following non-linear equation can be transformed into an ordinary differential equation(ODE).
Transformation one variable S-system into a first-order ODE
Let (with ) Then, given a one-variable S-system is
Apply a non-zero steady condition to the given equation
, or equivalently
Thus, (or, )
If can be approximated around , remaining the first two terms,
By non-zero steady condition, , a nonlinear one-variable S-system can be transformed into a first-order ODE:
where , , and , called a percentage variation.
Two variables S-system Source:
In the case of and , the S-system equation can be written as system of (non-linear) differential equations.
Assume non-zero steady condition, .
Transformation two variables S-system into a second-order ODE
By putting . The given system of equations can be written as
(where , and are constant.
Since , the given system of equation can be approximated as a second-order ODE:
,
Applications
Mass-action Law Source:
Consider the following chemical pathway:
A + 2B ->[k_1] C ->[k_2] 3D + E
where and are rate constants.
Then the mass-action law applied to species C gives the equation
(where is a concentration of A etc.)
Komarova Model (Bone Remodeling) Sources:
Komarova Model is an example of a two-variable system of non-linear differential equations that describes bone remodeling. This equation is regulated by biochemical factors called paracrine and autocrine, which quantify the bone mass in each step.
Where
, : The number of osteoclast/osteoblasts , : Osteoclast/Osteoblast production rate , : Osteoclast/Osteoblast removal rate : Paracrine factor on the -cell due to the presence of -cell : The bone mass percentage : Let be the difference between the number of osteoclasts/osteoblasts and its steady state. Then
Modified Komarova Model (Bone Remodeling with Tumor affecting, Bone metastasis) Source:
The modified Komarova Model describes the tumor effect on the osteoclasts and osteoblasts rate. The following equation can be described as
(with initial condition , , and )
Where
, : The number of osteoclast/osteoblasts. : The tumor representation depending on time ,: The representation of the activity of cell production ,: The representation of the activity of cell removal : The net effectiveness of osteoclast/osteoblast derived autocrine and paracrine factors : The tumor cell proliferation rate : The upper limit value for tumor cells : Scaling constant of tumor growth
References