Synergistic system

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]

One variable S-system

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.

Two variables S-system

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:

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

Applications

Mass-action Law^[2]

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 (Bone Remodeling)^[3][4]

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 ${\displaystyle y_{i}:={\frac {1}{2}}\left[\left(x_{i}-{\bar {x_{i}}})+(x_{i}-{\bar {x_{i}}})\right)\right]}$ 

Modified Komarova Model (Bone Remodeling with Tumor affecting, Bone metastasis)^[5]

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

References

1. Savageau, Michael A. (1988-01-01). "Introduction to S-systems and the underlying power-law formalism". Mathematical and Computer Modelling. 11: 546–551. doi:10.1016/0895-7177(88)90553-5. ISSN 0895-7177.
2. Tournier, Laurent (2005-07-24). "Approximation of dynamical systems using s-systems theory: application to biological systems". Proceedings of the 2005 international symposium on Symbolic and algebraic computation. ISSAC '05. New York, NY, USA: Association for Computing Machinery: 317–324. doi:10.1145/1073884.1073928. ISBN 978-1-59593-095-8.
3. Komarova, Svetlana V.; Smith, Robert J.; Dixon, S. Jeffrey; Sims, Stephen M.; Wahl, Lindi M. (August 2003). "Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling". Bone. 33 (2): 206–215. doi:10.1016/s8756-3282(03)00157-1. ISSN 8756-3282. PMID 14499354.
4. Ramtani, Salah; Sánchez, Juan Felipe; Boucetta, Abdelkader; Kraft, Reuben; Vaca-González, Juan Jairo; Garzón-Alvarado, Diego A. (June 2023). "A coupled mathematical model between bone remodeling and tumors: a study of different scenarios using Komarova's model". Biomechanics and Modeling in Mechanobiology. 22 (3): 925–945. doi:10.1007/s10237-023-01689-3. ISSN 1617-7940. PMC 10167202. PMID 36922421.
5. Ayati, Bruce P.; Edwards, Claire M.; Webb, Glenn F.; Wikswo, John P. (2010-04-20). "A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease". Biology Direct. 5: 28. doi:10.1186/1745-6150-5-28. ISSN 1745-6150. PMC 2867965. PMID 20406449.