Information geometric inequalities of chemical thermodynamics

We study a connection between chemical thermodynamics and information geometry. We clarify a relation between the Gibbs free energy of an ideal dilute solution and an information geometric quantity called an $f$ divergence. From this relation, we derive information geometric inequalities that give a speed limit for the changing rate of the Gibbs free energy and a general bound of chemical fluctuations. These information geometric inequalities can be regarded as generalizations of the Cramér–Rao inequality for chemical reaction networks described by rate equations, where un-normalized concentration distributions are of importance rather than probability distributions. They hold true for damped oscillatory reaction networks and systems where the total concentration is not conserved, so that the distribution cannot be normalized. We also formulate a trade-off relation between speed and time on a manifold of concentration distribution by using the geometrical structure induced by the $f$ divergence. Our results apply to both closed and open chemical reaction networks; thus they are widely useful for thermodynamic analysis of chemical systems from the viewpoint of information geometry.

Figure 1

Schematic of a stoichiometric compatibility class and the $f$ divergence for $N=3$. This $f$ divergence gives the difference of the Gibbs free energy $(G-G^{\mathrm{eq}})/RT$.

Figure 2

Geometric picture of $\mathit{q}$ and other quantities appearing in Sec. 5c for $N=3$. ${\overline{\mathit{q}}}^{min}$, which makes the right-hand sides of the inequalities in Eqs. (95) and (96) smallest, is obtained as a projection of $\mathit{q}$ onto $ker{\mathsf{S}}^{\mathsf{T}}$, which is perpendicular to the stoichiometric compatibility class. It is not the orthogonal projection in the Euclidean space. Instead, ${\mathsf{X}}^{1/2}\mathit{q}$ is orthogonally projected to ${\mathsf{X}}^{1/2}{\overline{\mathit{q}}}^{\min }$.

Figure 3

The time evolutions of $\mathrm{X}$'s and $\mathrm{Y}$'s concentrations obtained by integrating Eq. (124) with the parameters $k_1^{+}=1\times {10}^{-3}$, $k_1^{-}=k_2^{+}=k_2^{-}=1$, $k_3^{+}=1\times {10}^{-2}$, $k_3^{-}=1\times {10}^{-4}$, ${\left[\mathrm{X}\right]}_0=1$, ${\left[\mathrm{Y}\right]}_0=6$, and ${\left[\mathrm{A}\right]}_0={\left[\mathrm{B}\right]}_0=1\times {10}^3$. They oscillate and then relax to equilibrium.

Figure 4

The speed limit [Eq. (79)] with respect to the reaction system (123). The speed of the Gibbs free energy change cannot exceed $v_{\mathit{\mu }}(t,{\mathit{\mu }}^{\mathrm{eq}})$.

Figure 5

The speed limit [Eq. (88)] for a subset $S=\{\mathrm{X},\mathrm{Y}\}$. The speed of the partial Gibbs free energy change also does not exceed the corresponding function $v_{\mathit{\mu },S}(t,{\mathit{\mu }}^{\mathrm{eq}})$.

Figure 6

The time evolution of the concentrations in the CRN in Eq. (130) calculated with the parameters $k_1^{+}=k_1^{-}=k_2^{+}=1$, $k_2^{-}=1\times {10}^{-5}$, ${\left[\mathrm{A}\right]}_0=1$, and ${\left[\mathrm{B}\right]}_0={\left[\mathrm{C}\right]}_0=1\times {10}^{-5}$. The total concentration $\left[\mathrm{A}\right]+\left[\mathrm{B}\right]+\left[\mathrm{C}\right]$ (dashed line) is not constant.

Figure 7

Bounds on the changing rate of the total concentration $〈〈\mathit{q}〉〉$. The bound given by ${\overline{\mathit{q}}}^{\min }$ is much tighter than that given by $\overline{\mathit{q}}=\mathit{0}$.

Figure 8

The time evolution of the concentrations in the association reaction. The parameters are set to $k^{+}=k^{-}=1$, ${\left[\mathrm{A}\right]}_0=2.9$, and ${\left[\mathrm{B}\right]}_0=0.05$.

Figure 9

The trade-off relations in Eqs. (120) and (122) are verified for the CRN in Eq. (135). For fixed $\tau =0.1$, ${\mathcal{L}}_{\tau }^2/2{\mathcal{C}}_{\tau }$ and ${\mathcal{D}}_{\tau }^2/2{\mathcal{C}}_{\tau }$ give a lower bound. Because the concentration change is monotonic, ${\mathcal{L}}_{\tau }$ and ${\mathcal{D}}_{\tau }$ give the same bound.