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
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.
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:
,
Consider the following chemical pathway:
<chem>A + 2B ->[k_1] C ->[k_2] 3D + E </chem>
where and are rate constants.
Then the mass-action law applied to species <chem>C</chem> gives the equation
(where is a concentration of A etc.)
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 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