Synergistic system

Short description: System of nonlinear differential equations

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

$\frac{d{x}_i}{dt}={\alpha }_i\prod _{j=1}^{n+m}{x}_i^{g_{ij}}-{\beta }_j\prod _{j=1}^{n+m}{x}_i^{h_{ij}}(i=1,...,n)$

where the $x_i$ are positive real, ${\alpha }_i$ and ${\beta }_i$ are non-negative real, called the rate constant(or, kinetic rates) and $g_{ij}$ and $h_{ij}$ are real exponential, called kinetic orders. These terms are based on the chemical equilibrium

One variable S-system

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

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

Under the non-zero steady condition, $\frac{d{x}_0}{dt}=0$, 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 $x=e^{\log x}=e^y$(with $x_0=e^{y_0}$) Then, given a one-variable S-system is

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

Apply a non-zero steady condition to the given equation

$0=\frac{d{y}_0}{dt}=\alpha e^{(g-1)y_0}-\beta e^{(h-1)y_0}$, or equivalently $\alpha e^{(g-1)y_0}=\beta e^{(h-1)y_0}$

Thus, $y_0=\frac{\log \beta -\log \alpha }{g-h}$(or, $x_0={\left(\frac{\beta }{\alpha }\right)}^{\frac{1}{g-h}}$)

If $\frac{dy}{dt}$ can be approximated around $y=y_0$, remaining the first two terms,

$\frac{dy}{dt}\simeq \alpha e^{(g-1)y_0}+\alpha e^{(g-1)y_0}(g-1)(y-y_0)-\beta e^{(g-1)y_0}-\beta e^{(h-1)y_0}(h-1)(y-y_0)$

By non-zero steady condition, $\alpha e^{(g-1)y_0}=\beta e^{(h-1)y_0}$, a nonlinear one-variable S-system can be transformed into a first-order ODE:

$\frac{du}{dt}\simeq (\alpha e^{(g-1)y_0}(g-h))u=\left(Fa\right)u$

where $F=\alpha e^{(g-1)y_0}$, $a=g-h$, and $u=y-y_0\simeq \frac{x-x_0}{x_0}$, called a percentage variation.

Two variables S-system

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

$\{\begin{array}{c}\frac{d{x}_1}{dt}={\alpha }_1{x}_1^{g_{11}}{x}_2^{g_{21}}-{\beta }_1{x}_1^{h_{11}}{x}_2^{h_{21}}\\ \frac{d{x}_2}{dt}={\alpha }_2{x}_2^{g_{21}}{x}_2^{g_{22}}-{\beta }_2{x}_1^{h_{21}}{x}_2^{h_{22}}\end{array}$

Assume non-zero steady condition, $\frac{d{x}_{i0}}{dt}=0$.

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

By putting $x_i=e^{\log x_i}=e^{y_i}$ . The given system of equations can be written as

$\{\begin{array}{c}\frac{d{u}_1}{dt}=c_{11}{u}_1+c_{12}{u}_2\\ \frac{d{u}_2}{dt}=c_{21}{u}_1+c_{22}{u}_2\end{array}$

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

Since $\frac{d^2{u}_1}{d{t}^2}=c_{11}\frac{d{u}_1}{dt}+c_{12}\frac{d{u}_2}{dt}$, the given system of equation can be approximated as a second-order ODE:

$\frac{d^2{u}_1}{d{t}^2}-(c_{11}+c_{22})\frac{d{u}_1}{dt}+(c_{11}{c}_{22}-c_{12}{c}_{21})u_1=0$,

Applications

Mass-action Law

Consider the following chemical pathway:

$\mathrm{A}+2\mathrm{B}\stackrel{\mathrm{k}{\phantom{A}}_1}{\to }\mathrm{C}\stackrel{\mathrm{k}{\phantom{A}}_2}{\to }3\mathrm{D}+\mathrm{E}$

where $k_1$ and $k_2$ are rate constants.

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

$\frac{d[C]}{dt}=k_1[A][B{]}^2-k_2[D{]}^3[E]$

(where $[A]$ is a concentration of A etc.)

Komarova Model (Bone Remodeling)

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.

$\{\begin{array}{c}\frac{d{x}_1}{dt}={\alpha }_1{x}_1^{g_{11}}{x}_2^{g_{21}}-{\beta }_1{x}_1\\ \frac{d{x}_2}{dt}={\alpha }_2{x}_1^{g_{12}}{x}_2^{g_{22}}-{\beta }_2{x}_2\\ \frac{dz}{dt}=-k_1{y}_1+k_2{y}_2\end{array}$

Where

• $x_1$, $x_2$: The number of osteoclast/osteoblasts
• ${\alpha }_1$, ${\alpha }_2$: Osteoclast/Osteoblast production rate
• ${\beta }_1$, ${\beta }_2$: Osteoclast/Osteoblast removal rate
• $g_{ij}$: Paracrine factor on the $j$-cell due to the presence of $i$-cell
• $z$: The bone mass percentage
• $y_i$: Let $\overline{x_i}$ be the difference between the number of osteoclasts/osteoblasts and its steady state. Then $y_i:=\frac{1}{2}\left[(x_i-\overline{x_i})+(x_i-\overline{x_i}))\right]$

Modified Komarova Model (Bone Remodeling with Tumor affecting, Bone metastasis)

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

$\{\begin{array}{c}\frac{d{x}_1}{dt}=\widehat{{\alpha }_1(\omega )}{x}_1^{g_{11}}{x}_2^{g_{21}}-\widehat{{\beta }_1(\omega )}{x}_1\\ \frac{d{x}_2}{dt}=\widehat{{\alpha }_2(\omega )}{x}_2^{g_{22}}-\widehat{{\beta }_2(\omega )}{x}_2\\ \frac{d\omega }{dt}=\mu \omega \log \left(\sigma \frac{L_{\omega }}{\omega }\right)\end{array}$

(with initial condition $x_1(0)=x_{10}$, $x_2(0)=x_{20}$, and $\omega (0)={\omega }_0$)

Where

• $x_1$, $x_2$: The number of osteoclast/osteoblasts.
• $\omega =\omega (t)$ : The tumor representation depending on time $t$
• $\widehat{{\alpha }_1(\omega )}$,$\widehat{{\alpha }_2(\omega )}$: The representation of the activity of cell production
• $\widehat{{\beta }_1(\omega )}$,$\widehat{{\beta }_2(\omega )}$: The representation of the activity of cell removal
• $g_{ij}$: The net effectiveness of osteoclast/osteoblast derived autocrine and paracrine factors
• $\mu$ : The tumor cell proliferation rate
• $L_{\omega }$: The upper limit value for tumor cells
• $\sigma$ : Scaling constant of tumor growth