Synergistic system

Template:Short description Template:Multiple issues A Synergistic system (or S-system) is a collection of ordinary nonlinear differential equations

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

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

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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

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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

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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)

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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)

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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