Information geometric bound on general chemical reaction networks

We investigate the convergence of chemical reaction networks (CRNs), aiming to establish an upper bound on their reaction rates. The nonlinear characteristics and discrete composition of CRNs pose significant challenges in this endeavor. To circumvent these complexities, we adopt an information geometric perspective, utilizing the natural gradient to formulate a nonlinear system. This system effectively determines an upper bound for the dynamics of CRNs. We corroborate our methodology through numerical simulations, which reveal that our constructed system converges more rapidly than CRNs within a particular class of reactions. This class is defined by the count of chemicals, the highest stoichiometric coefficients in the reactions, and the total number of reactions involved. Further, we juxtapose our approach with traditional methods, illustrating that the latter falls short in providing an upper bound for CRN reaction rates. Although our investigation centers on CRNs, the widespread presence of hypergraphs across various disciplines, ranging from natural sciences to engineering, indicates potential wider applications of our method, including in the realm of information science.

Figure 1

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ on $t$ for the CRN in Eq. (3.4) and their upper bound in the case of ${\mathit{x}}_{\mathrm{eq}}=\widehat{\mathit{x}}$.

Figure 2

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ on time $t$ for the CRN in Eq. (6.1) and its upper bound in the case of ${\mathit{x}}_{\mathrm{eq}}=\widehat{\mathit{x}}$.

Figure 3

Relationship between ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ and $-\mathrm{\Delta }{\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ for the CRN in Eq. (6.1) and its upper bound in the case of ${\mathit{x}}_{\mathrm{eq}}=\widehat{\mathit{x}}$. We have used ${\mathit{x}}_t$ on the solution of Eq. (6.1).

Figure 4

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ on time $t$ for several CRNs in Eq. (6.2) and their upper bound in the case of ${\mathit{x}}_{\mathrm{eq}}\ne \widehat{\mathit{x}}$.

Figure 5

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ on time $t$ for several CRNs in Eqs. (6.3) and their upper bound in the case of ${\mathit{x}}_{\mathrm{eq}}\ne \widehat{\mathit{x}}$.

Figure 6

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ on time $t$ for the CRNs in Eq. (6.4) and its upper bound in the case of ${\mathit{x}}_{\mathrm{eq}}\ne \widehat{\mathit{x}}$.

Figure 7

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ on time $t$ for the CRN in Eq. (6.4c) and its upper bound in the case of ${\mathit{x}}_{\mathrm{eq}}=\widehat{\mathit{x}}$.

Figure 8

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ on time $t$ for the CRN in Eq. (6.4c) and its upper bound in the case of ${\mathit{x}}_{\mathrm{eq}}=\widehat{\mathit{x}}$.

Figure 9

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ on time $t$ for the CRN in Eq. (6.5) and its upper bound.

Figure 10

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ on time $t$ for the CRN in Eq. (6.6) and its upper bound.

Figure 11

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ for $t$ for $m=1,2,3,4$. We set $N_{\mathbb{X}}=4$ and $N_{\mathbb{e}}=1$.

Figure 12

Relationship between ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ and $-\mathrm{\Delta }{\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ for $N_{\mathbb{X}}=4$ and $N_{\mathbb{e}}=1$.

Figure 13

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ for $t$ for $m=1,2,3,4$. We set $N_{\mathbb{X}}=1$ and $m=1$.

Figure 14

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ for $t$ for $m=1,2,3,4$. We set $N_{\mathbb{X}}=1$ and $N_{\mathrm{e}}=1$.

Figure 15

Dependence of ${\mathcal{D}}_{\mathrm{KL}}({\mathit{x}}_t\parallel \widehat{\mathit{x}})$ for $t$ for $m=1,2,3,4$. We set $N_{\mathbb{e}}=1$ and $m=1$.