Robustness in Power-Law Kinetic Systems with Reactant-Determined Interactions

Abstract

Robustness against the presence of environmental disruptions can be observed in many systems of chemical reaction network. However, identifying the underlying components of a system that give rise to robustness is often elusive. The influential work of Shinar and Feinberg established simple yet subtle network-based conditions for absolute concentration robustness (ACR), a phenomenon in which a species in a mass-action system has the same concentration for any positive steady state the network may admit. In this contribution, we extend this result to embrace kinetic systems more general than mass-action systems, namely, power-law kinetic systems with reactant-determined interactions (denoted by “PL-RDK”). In PL-RDK, the kinetic order vectors of reactions with the same reactant complex are identical. As illustration, we considered a scenario in the pre-industrial state of global carbon cycle. A power-law approximation of the dynamical system of this scenario is found to be dynamically equivalent to an ACR-possessing PL-RDK system.

Appendices

Appendix

A Pre-industrial Carbon Cycle Model of Anderies et al.

The complete set of ODEs for the pre-industrial state is given by

$$\begin{aligned} \left. \begin{array}{cl} \dot{A}_1 &{}= r_{tc} [P(t)-R(t)] A_1 \left[ 1- \frac{A_1}{k} \right] - \alpha A_1 \\ \dot{A}_2 &{}= r_{tc} [R(t)-P(t)] A_1 \left[ 1- \frac{A_1}{k} \right] + \alpha A_1 - a_m A_2 + a_m \beta A_3 \\ \dot{A}_3 &{}= a_m A_2 - a_m \beta A_3 . \end{array} \right\} \end{aligned}$$

(22)

where

$$\begin{aligned} P(t)&= a_f A_2(t)^{b_f} \cdot \left[ a_p \cdot (a_T A_2(t) + b_T)^{b_p} \cdot e^{-c_p \cdot (a_T A_2(t) + b_T)} \right] \\ R(t)&=\left[ a_r \cdot (a_T A_2(t) + b_T)^{b_r} \cdot e^{-c_r \cdot (a_T A_2(t) + b_T)} \right] . \end{aligned}$$

For the description of the parameters, the reader is referred to [1] and the Appendix of [13]. The parameter values are identical to the values used in [13] but with $\alpha =0$. This particular parameter is assigned to the human terrestrial carbon off-take rate. It is associated with human activities such as clearing, burning or farming, which reduce the capacity of land to capture carbon.

A power-law approximation of the ODE system at an operating point is obtained to generate a Generalized Mass Action (GMA) System [28, 29]. Mathematically, GMA system approximation is equivalent to Taylor approximation up to the linear term in logarithmic space. The function $V(X_1,X_2,\dots ,X_m)$ can be approximated by $\displaystyle {V = \alpha X_1^{p_1} X_2^{p_2} \cdots X_m^{p_m}}$ at an operating point where

$$\begin{aligned} p_i =\dfrac{\partial V}{\partial X_i} \cdot \dfrac{X_i}{V} \text { and } \alpha = V(X_1,X_2,\dots ,X_m)X_1^{-p_1}X_2^{-p_2}\cdots X_m^{-p_m}. \end{aligned}$$

(23)

Table 1. Power-law approximation of the process rates.

Table 1 presents the four carbon fluxes present in the pre-industrial state of the Anderies et al. model, and their corresponding rate functions. Furthermore, the last column lists their respective target power-law approximation. The last two functions, $a_m A_2$ and $a_m \beta A_3$, are already in the desired format and are thus, kept as is. To compute for the kinetic orders (and rate constants), we apply (23). By taking the parameter values used in [13] but with $\alpha =0$, and assuming the initial values to be $A_1 =2850/4500$, $A_2=750/4500$ and $A_3=900/4500$ (as in [1]), the ODE system in (22) reaches the following steady state: $A_1 = 0.7, \quad A_2 = 0.15\text { and } A_3 = 0.15$.

The algebraic calculations are implemented in Mathematica as shown in Fig. 2. When $\alpha =0$ (i.e., the human off-take term vanishes),

$$ p_1=p_2=\dfrac{2A_1-k}{A_1-k}. $$

For the power-law approximation, we choose values close to the equilibrium point as operating point: $A_1=0.69$, $A_2=0.155$ and $A_3=0.155$. Consequently, we obtain

$$\begin{aligned} \left. \begin{array}{clcl} p_1 &{}=-68, &{}\quad p_2 &{}=-68, \\ q_1 &{}=0.580148, &{}\quad q_2 &{}=0.910864. \end{array} \right. \end{aligned}$$

(24)

Fig. 2.

Mathematica codes.