Synergistic system

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A Synergistic system (or S-system)^[1] is a collection of ordinary nonlinear differential equations

${\displaystyle {\frac {dx_{i}}{dt}}=\alpha _{i}\prod _{j=1}^{n+m}x_{i}^{g_{ij}}-\beta _{j}\prod _{j=1}^{n+m}x_{i}^{h_{ij}}~~~~~(i=1,...,n)}$

where the ${\displaystyle x_{i}}$ are positive real, ${\displaystyle \alpha _{i}}$ and ${\displaystyle \beta _{i}}$ are non-negative real, called the rate constant(or, kinetic rates) and ${\displaystyle g_{ij}}$ and ${\displaystyle h_{ij}}$ are real exponential, called kinetic orders. These terms are based on the chemical equilibrium^[2]

In the case of ${\displaystyle n=1(i=1)}$ and ${\displaystyle m=0}$, the given S-system equation can be written as

${\displaystyle \ {\frac {dx}{dt}}=\alpha x^{g}-\beta x^{h}\ }$

Under the non-zero steady condition, ${\displaystyle {\frac {dx_{0}}{dt}}=0}$, 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 ${\displaystyle x=e^{\operatorname {log} x}=e^{y}}$(with ${\displaystyle x_{0}=e^{y_{0}}}$) Then, given a one-variable S-system is

${\displaystyle {\frac {dy}{dt}}=\alpha e^{(g-1)y}-\beta e^{(h-1)y}}$

Apply a non-zero steady condition to the given equation

${\displaystyle 0={\frac {dy_{0}}{dt}}=\alpha e^{(g-1)y_{0}}-\beta e^{(h-1)y_{0}}}$, or equivalently ${\displaystyle \alpha e^{(g-1)y_{0}}=\beta e^{(h-1)y_{0}}}$

Thus, ${\displaystyle y_{0}={\frac {\operatorname {log} \beta -\operatorname {log} \alpha }{g-h}}}$(or, ${\displaystyle x_{0}=\left({\frac {\beta }{\alpha }}\right)^{\frac {1}{g-h}}}$)

If ${\displaystyle {\frac {dy}{dt}}}$ can be approximated around ${\displaystyle y=y_{0}}$, remaining the first two terms,

${\displaystyle {\frac {dy}{dt}}\simeq \alpha e^{(g-1)y_{0}}+\alpha e^{(g-1)y_{0}}(g-1)(y-y_{0})-\beta e^{(g-1)y_{0}}-\beta e^{(h-1)y_{0}}(h-1)(y-y_{0})}$

By non-zero steady condition, ${\displaystyle \alpha e^{(g-1)y_{0}}=\beta e^{(h-1)y_{0}}}$, a nonlinear one-variable S-system can be transformed into a first-order ODE:

${\displaystyle {\frac {du}{dt}}\simeq \left(\alpha e^{(g-1)y_{0}}(g-h)\right)u=\left(Fa\right)u}$

where ${\displaystyle F=\alpha e^{(g-1)y_{0}}}$, ${\displaystyle a=g-h}$, and ${\displaystyle u=y-y_{0}\simeq {\frac {x-x_{0}}{x_{0}}}}$, called a percentage variation.

In the case of ${\displaystyle n=2(i=1,2)}$ and ${\displaystyle m=0}$ , the S-system equation can be written as system of (non-linear) differential equations.

${\displaystyle \left\{{\begin{matrix}{\frac {dx_{1}}{dt}}=\alpha _{1}x_{1}^{g_{11}}x_{2}^{g_{21}}-\beta _{1}x_{1}^{h_{11}}x_{2}^{h_{21}}\ \\{\frac {dx_{2}}{dt}}=\alpha _{2}x_{2}^{g_{21}}x_{2}^{g_{22}}-\beta _{2}x_{1}^{h_{21}}x_{2}^{h_{22}}\end{matrix}}\right.}$

Assume non-zero steady condition, ${\displaystyle {\frac {dx_{i0}}{dt}}=0}$.

Transformation two variables S-system into a second-order ODE

By putting ${\displaystyle x_{i}=e^{\operatorname {log} x_{i}}=e^{y_{i}}}$ . The given system of equations can be written as

${\displaystyle \left\{{\begin{matrix}{\frac {du_{1}}{dt}}=c_{11}u_{1}+c_{12}u_{2}\ \\{\frac {du_{2}}{dt}}=c_{21}u_{1}+c_{22}u_{2}\end{matrix}}\right.}$

(where ${\displaystyle u_{i}=y_{i}-y_{i0}}$, ${\displaystyle u_{i}=y_{i}-y_{i0}}$ and ${\displaystyle c_{ij}(i,j=1,2)}$ are constant.

Since ${\displaystyle {\frac {d^{2}u_{1}}{dt^{2}}}=c_{11}{\frac {du_{1}}{dt}}+c_{12}{\frac {du_{2}}{dt}}}$, the given system of equation can be approximated as a second-order ODE:

${\displaystyle {\frac {d^{2}u_{1}}{dt^{2}}}-\left(c_{11}+c_{22}\right){\frac {du_{1}}{dt}}+\left(c_{11}c_{22}-c_{12}c_{21}\right)u_{1}=0}$,

Consider the following chemical pathway:

${\displaystyle {\ce {A + 2B ->[k_1] C ->[k_2] 3D + E}}}$

where ${\displaystyle k_{1}}$ and ${\displaystyle k_{2}}$ are rate constants.

Then the mass-action law applied to species ${\displaystyle {\ce {C}}}$ gives the equation

${\displaystyle {\frac {d[C]}{dt}}=k_{1}[A][B]^{2}-k_{2}[D]^{3}[E]}$

(where ${\displaystyle [A]}$ 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.

${\displaystyle \left\{{\begin{matrix}\quad {\frac {dx_{1}}{dt}}=\alpha _{1}x_{1}^{g_{11}}x_{2}^{g_{21}}-\beta _{1}x_{1}\\\quad {\frac {dx_{2}}{dt}}=\alpha _{2}x_{1}^{g_{12}}x_{2}^{g_{22}}-\beta _{2}x_{2}\\{\frac {dz}{dt}}=-k_{1}y_{1}+k_{2}y_{2}\end{matrix}}\right.}$

Where

• ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$: The number of osteoclast/osteoblasts
• ${\displaystyle \alpha _{1}}$, ${\displaystyle \alpha _{2}}$: Osteoclast/Osteoblast production rate
• ${\displaystyle \beta _{1}}$, ${\displaystyle \beta _{2}}$: Osteoclast/Osteoblast removal rate
• ${\displaystyle g_{ij}}$: Paracrine factor on the ${\displaystyle j}$-cell due to the presence of ${\displaystyle i}$-cell
• ${\displaystyle z}$: The bone mass percentage
• ${\displaystyle y_{i}}$: Let ${\displaystyle {\bar {x_{i}}}}$ be the difference between the number of osteoclasts/osteoblasts and its steady state. Then ${\displaystyle y_{i}:={\frac {1}{2}}\left[\left(x_{i}-{\bar {x_{i}}})+(x_{i}-{\bar {x_{i}}})\right)\right]}$

The modified Komarova Model describes the tumor effect on the osteoclasts and osteoblasts rate. The following equation can be described as

${\displaystyle \left\{{\begin{matrix}{\frac {dx_{1}}{dt}}={\widehat {\alpha _{1}(\omega )}}x_{1}^{g_{11}}x_{2}^{g_{21}}-{\widehat {\beta _{1}(\omega )}}x_{1}\\{\frac {dx_{2}}{dt}}={\widehat {\alpha _{2}(\omega )}}x_{2}^{g_{22}}-{\widehat {\beta _{2}(\omega )}}x_{2}\\{\frac {d\omega }{dt}}=\mu \omega \operatorname {log} \left(\sigma {\frac {L_{\omega }}{\omega }}\right)\end{matrix}}\right.}$

(with initial condition ${\displaystyle x_{1}(0)=x_{10}}$, ${\displaystyle x_{2}(0)=x_{20}}$, and ${\displaystyle \omega (0)=\omega _{0}}$)

Where

• ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$: The number of osteoclast/osteoblasts.
• ${\displaystyle \omega =\omega (t)}$ : The tumor representation depending on time ${\displaystyle t}$
• ${\displaystyle {\widehat {\alpha _{1}(\omega )}}}$,${\displaystyle {\widehat {\alpha _{2}(\omega )}}}$: The representation of the activity of cell production
• ${\displaystyle {\widehat {\beta _{1}(\omega )}}}$,${\displaystyle {\widehat {\beta _{2}(\omega )}}}$: The representation of the activity of cell removal
• ${\displaystyle g_{ij}}$: The net effectiveness of osteoclast/osteoblast derived autocrine and paracrine factors
• ${\displaystyle \mu }$ : The tumor cell proliferation rate
• ${\displaystyle L_{\omega }}$: The upper limit value for tumor cells
• ${\displaystyle \sigma }$ : Scaling constant of tumor growth