Hessian geometry of nonequilibrium chemical reaction networks and entropy production decompositions

We derive the Hessian geometric structure of nonequilibrium chemical reaction networks on the flux and force spaces induced by the Legendre duality of convex dissipation functions and characterize their dynamics as a generalized flow. With this geometric structure, we can extend theories of nonequilibrium systems with quadratic dissipation functions to more general cases with nonquadratic ones, which are pivotal for studying chemical reaction networks. By applying generalized notions of orthogonality in Hessian geometry to chemical reaction networks, two generalized decompositions of the entropy production rate are obtained, each of which captures gradient-flow and minimum-dissipation aspects in nonequilibrium dynamics.

Figure 1

Diagrammatic illustration of CRN (a) Diagrammatic representation of chemical reaction equations with reaction rate constants ${\left\{k_{\pm i}\right\}}_{i=1,2,3}$. (b) The graph-theoretic structure of the CRN in (a). ${\mathbb{v}}_1, {\mathbb{v}}_2$, and ${\mathbb{v}}_3$ correspond to complex $2{\mathbb{X}}_1, 2{\mathbb{X}}_2$, and ${\mathbb{X}}_1+{\mathbb{X}}_2$, respectively. Each directed edge represents a pair of forward and reverse reactions, and the direction of the edge indicates the direction of the forward reaction.

Figure 2

(a) Linear-algebraic dual spaces $(\mathcal{J},\mathcal{F})$ with the inner product structure defined by a metric via $M$ and $M^{*}=M^{-1}$. Linear subspaces in $\mathcal{J}$, e.g., blue-solid and red-dashed lines, are mapped to linear subspaces in $\mathcal{F}$. (b) Linear-algebraic dual spaces $(\mathcal{J},\mathcal{F})$ with the Hessian geometric structure defined by Legendre transformations $\partial \mathrm{\Psi }$ and ${\partial }^{*}\mathrm{\Psi }$. A linear subspace in $\mathcal{J}$, e.g., red-dashed line, is mapped to a curved subspace in $\mathcal{F}$ whereas a linear subspace in $\mathcal{F}$, e.g., blue-solid line, is mapped to a curved subspace in $\mathcal{J}$.

Figure 3

Schematic representation of the flux-force relationship for CRN with mass action kinetics and induced CRE [Eq. (18)]. Depending on the current state $\mathit{x}\left(t\right)$, (a), the force ${\mathit{f}}_0\left({\mathit{x}}_t\right)$ is induced (b). The force is mapped to the corresponding flux ${\mathit{j}}_0\left({\mathit{x}}_t\right)$ by the Legendre transformation $\partial {\mathrm{\Psi }}_{{\mathit{\omega }}_0\left({\mathit{x}}_t\right)}^{*}\left[{\mathit{f}}_0\left({\mathit{x}}_t\right)\right]$ (c). The flux induces the change in ${\mathit{x}}_t$ via the continuity equation (d).

Figure 4

Schematic illustration of the Hilbert orthogonality between ${\mathit{f}}_S\left({\mathit{x}}_t\right)$ (blue arrows) and ${\mathit{f}}_A$(magenta arrows). The green-solid curve represents the isodissipation hypersurface: $\left\{\mathit{f}\right|{\mathrm{\Psi }}_{{\mathit{x}}_t}^{*}\left(\mathit{f}\right)={\mathrm{\Psi }}_{{\mathit{x}}_t}^{*}({\mathit{f}}_0\left({\mathit{x}}_t\right))=\mathrm{const}.\}$. The black circle, red square, and green star in $\mathcal{F}$ are ${\mathit{f}}_0\left({\mathit{x}}_t\right), {\mathit{f}}_S\left({\mathit{x}}_t\right)$, and ${\mathit{f}}_{\mathrm{iso}}\left({\mathit{x}}_t\right)$, respectively. The black circle, red square, and green star in $\mathcal{J}$ are their Legendre transform, i.e., ${\mathit{j}}_0\left({\mathit{x}}_t\right), {\mathit{j}}_S\left({\mathit{x}}_t\right)$, and ${\mathit{j}}_{\mathrm{iso}}\left({\mathit{x}}_t\right)$, respectively. The light blue lines in $\mathcal{F}$ are the NEF subspaces, ${\mathcal{P}}^f\left({\mathit{f}}_A\right)$(solid line) and ${\mathcal{P}}^f(-{\mathit{f}}_A)$ (dashed line), and the EF subspace ${\mathcal{P}}_{\mathrm{eq}}^f$ (dotted line). In $\mathcal{J}$, the corresponding NEF manifolds, ${\mathcal{M}}_{{\mathit{x}}_t}^f\left({\mathit{f}}_A\right)$ (the light-solid blue curve) and ${\mathcal{M}}_{{\mathit{x}}_t}^f(-{\mathit{f}}_A)$ (the light-dashed-blue curve), and the EF manifold ${\mathcal{M}}_{{\mathit{x}}_t,eq}^f$ (the light-dotted-blue curve) are also depicted.

Figure 5

Schematic illustration of the information geometric orthogonalities. (Left) The orthogonality between ${\mathit{j}}_{\mathrm{st}}\left({\mathit{x}}_t\right)$ and ${\mathit{j}}_0\left({\mathit{x}}_t\right)-{\mathit{j}}_{\mathrm{st}}\left({\mathit{x}}_t\right)$ (blue arrows) and the dual orthogonality between ${\mathit{j}}_{\mathrm{eq}}\left({\mathit{x}}_t\right)$ and ${\mathit{j}}_0\left({\mathit{x}}_t\right)-{\mathit{j}}_{\mathrm{eq}}\left({\mathit{x}}_t\right)$ (red arrows) in $\mathcal{J}$. The blue planes are the isovelocity subspace ${\mathcal{P}}^v\left({\mathit{j}}_0\left({\mathit{x}}_t\right)\right)$ (the upper plane) and the zero-velocity subspace ${\mathcal{P}}_{\mathrm{st}}^v$ (the lower plane). The grey curves are the NEF manifold ${\mathcal{M}}_{\mathit{x}}^f\left({\mathit{f}}_0\left({\mathit{x}}_t\right)\right)$ and the EF manifold ${\mathcal{M}}_{\mathit{x},eq}^f$. (Right) The same orthogonalities shown in $\mathcal{F}$ space. The orthogonality between ${\mathit{f}}_{\mathrm{st}}\left({\mathit{x}}_t\right)$ and ${\mathit{f}}_0\left({\mathit{x}}_t\right)-{\mathit{f}}_{\mathrm{st}}\left({\mathit{x}}_t\right)$ (blue arrows) and the dual orthogonality between ${\mathit{f}}_{\mathrm{eq}}\left({\mathit{x}}_t\right)$ and ${\mathit{f}}_0\left({\mathit{x}}_t\right)-{\mathit{f}}_{\mathrm{eq}}\left({\mathit{x}}_t\right)$ (red arrows) in $\mathcal{F}$. The blue surfaces are the isovelocity manifold ${\mathcal{M}}_{{\mathit{x}}_t}^v({\mathit{j}}_0\left({\mathit{x}}_t\right))$ (the upper surface) and the zero-velocity manifold ${\mathcal{M}}_{{\mathit{x}}_t,st}^v$ (the lower surface). The grey lines are the NEF subspaces ${\mathcal{P}}^f({\mathit{f}}_0\left({\mathit{x}}_t\right))$ and the EF subspace ${\mathcal{P}}_{\mathit{x},eq}^f$.

Figure 6

Heatmap plots of EPR $\dot{\mathrm{\Sigma }}$ (left panels) and the two dissipation functions, ${\mathrm{\Psi }}_{\mathit{x}}({\mathit{j}}_0\left(\mathit{x}\right))$ (center panels) and ${\mathrm{\Psi }}_{\mathit{x}}^{*}({\mathit{f}}_0\left(\mathit{x}\right))$ (right panels) as functions of $\mathit{x}$ for the equilibrium parameter set (a), the complex balanced parameter set (b), and the noncomplex balanced parameter set (c) shown in Table 1. In each panel, gray and black points are the initial and the steady states, respectively. The white arrow is the trajectory of ${\mathit{x}}_t$. The black line in each panel is the set of equilibrium states (a), complex-balanced states (b), and noncomplex balanced steady states (c), respectively.

Figure 7

The EPR and its decompositions along the trajectory ${\mathit{x}}_t$ for complex balanced [(a),(b)] and noncomplex balanced [(c),(d)] parameter sets. The decomposition of EPR into two dissipation functions [(a),(c)]. The decompositions of EPR by Hilbert (${\dot{\mathrm{\Sigma }}}_{\mathrm{ex}}^{\mathrm{GF}}\left(\mathit{x}\right), {\dot{\mathrm{\Sigma }}}_{\mathrm{hk}}^{\mathrm{GF}}\left(\mathit{x}\right)$) and information geometric (${\dot{\mathrm{\Sigma }}}_{\mathrm{ex}}^{\mathrm{MD}}\left(\mathit{x}\right), {\dot{\mathrm{\Sigma }}}_{\mathrm{hk}}^{\mathrm{MD}}\left(\mathit{x}\right)$) orthogonalities [(b),(d)].

Figure 8

Computational visualization of the Hilbert orthogonality between ${\mathit{f}}_S\left({\mathit{x}}_t\right)$ and ${\mathit{f}}_A$ for the CRN in Fig 1. The green surface in $\mathcal{F}$ represents the isodissipation hypersurface: ${\mathrm{\Psi }}_{{\mathit{x}}_t}^{*}\left(\mathit{f}\right)={\mathrm{\Psi }}_{{\mathit{x}}_t}^{*}\left({\mathit{f}}_0\left({\mathit{x}}_t\right)\right)=\mathrm{const}.$. The black circle with yellow border, red square with black border, green star, and magenta circle with yellow border in $\mathcal{F}$ are respectively ${\mathit{f}}_0\left({\mathit{x}}_t\right), {\mathit{f}}_S\left({\mathit{x}}_t\right), {\mathit{f}}_{\mathrm{iso}}\left({\mathit{x}}_t\right)$, and ${\mathit{f}}_A$ evaluated at ${\mathit{x}}_t={(5/2,1/2)}^T$. The black circle, red square, green star, and magenta circle with border in $\mathcal{J}$ are their Legendre transform, i.e., ${\mathit{j}}_0\left({\mathit{x}}_t\right), {\mathit{j}}_S\left({\mathit{x}}_t\right), {\mathit{j}}_{\mathrm{iso}}\left({\mathit{x}}_t\right)$, and ${\mathit{j}}_A\left({\mathit{x}}_t\right)$, respectively. The black, red, and green circles without border in $\mathcal{F}$ are the trajectories of $\left\{{\mathit{f}}_0\left({\mathit{x}}_{t^{\prime }}\right)\right\}, \left\{{\mathit{f}}_S\left({\mathit{x}}_{t^{\prime }}\right)\right\}$, and $\left\{{\mathit{f}}_{\mathrm{iso}}\left({\mathit{x}}_{t^{\prime }}\right)\right\}$. The black, red, green, and magenta circles without border in $\mathcal{J}$ are the trajectories of $\left\{{\mathit{j}}_0\left({\mathit{x}}_{t^{\prime }}\right)\right\}, \left\{{\mathit{j}}_S\left({\mathit{x}}_{t^{\prime }}\right)\right\}, \left\{{\mathit{j}}_{\mathrm{iso}}\left({\mathit{x}}_{t^{\prime }}\right)\right\}$, and $\left\{{\mathit{j}}_A\left({\mathit{x}}_{t^{\prime }}\right)\right\}$, respectively. The light blue lines in $\mathcal{F}$ are the NEF subspaces, ${\mathcal{P}}^f\left({\mathit{f}}_A\right)$(solid line) and ${\mathcal{P}}^f(-{\mathit{f}}_A)$ (dashed line), and the EF subspace ${\mathcal{P}}_{\mathrm{eq}}^f$ (dotted line). In $\mathcal{J}$, the corresponding NEF manifolds, ${\mathcal{M}}_{\mathit{x}}^f\left({\mathit{f}}_A\right)$ (the light blue curve with dots), is also depicted. The blue plane in $\mathcal{J}$ is the zero-velocity subspace ${\mathcal{P}}_{\mathrm{st}}^v$, and the blue surface in $\mathcal{F}$ is its Legendre transformation: ${\mathcal{M}}_{{\mathit{x}}_t,st}^v$.

Figure 9

Computational visualization of the information geometric orthogonality for the CRN in Fig 1. (Left) The orthogonality between ${\mathit{j}}_{\mathrm{st}}\left({\mathit{x}}_t\right)$ (cyan-dashed arrow) and ${\mathit{j}}_0\left({\mathit{x}}_t\right)-{\mathit{j}}_{\mathrm{st}}\left({\mathit{x}}_t\right)$ (cyan-solid arrow) and the dual orthogonality between ${\mathit{j}}_{\mathrm{eq}}\left({\mathit{x}}_t\right)$ and ${\mathit{j}}_0\left({\mathit{x}}_t\right)-{\mathit{j}}_{\mathrm{eq}}\left({\mathit{x}}_t\right)$ (purple-solid arrow) in $\mathcal{J}$ evaluated at ${\mathit{x}}_t={(5/2,1/2)}^T$. The blue planes are the isovelocity subspace ${\mathcal{P}}^v\left({\mathit{j}}_0\left({\mathit{x}}_t\right)\right)$ (the upper plane) and the zero-velocity subspace ${\mathcal{P}}_{\mathrm{st}}^v$ (the lower plane). The grey curve with dots is the NEF manifold ${\mathcal{M}}_{\mathit{x}}^f\left({\mathit{f}}_0\left({\mathit{x}}_t\right)\right)$. Black, red, and blue circles are the trajectories of $\left\{{\mathit{j}}_0\left({\mathit{x}}_{t^{\prime }}\right)\right\}, \left\{{\mathit{j}}_{\mathrm{eq}}\left({\mathit{x}}_{t^{\prime }}\right)\right\}$, and $\left\{{\mathit{j}}_{\mathrm{st}}\left({\mathit{x}}_{t^{\prime }}\right)\right\}$, respectively. (Right) The same orthogonalities shown in $\mathcal{F}$ space. The orthogonality between ${\mathit{f}}_{\mathrm{st}}\left({\mathit{x}}_t\right)$ (cyan-curved arrow) and ${\mathit{f}}_0\left({\mathit{x}}_t\right)-{\mathit{f}}_{\mathrm{st}}\left({\mathit{x}}_t\right)$ (cyan-solid arrow) and the dual orthogonality between ${\mathit{f}}_{\mathrm{eq}}\left({\mathit{x}}_t\right)$ (purple-dashed arrow) and ${\mathit{f}}_0\left({\mathit{x}}_t\right)-{\mathit{f}}_{\mathrm{eq}}\left({\mathit{x}}_t\right)$ (purple-solid arrow) in $\mathcal{F}$. Black, red circles are the trajectories of $\left\{{\mathit{f}}_0\left({\mathit{x}}_{t^{\prime }}\right)\right\}, \left\{{\mathit{f}}_{\mathrm{eq}}\left({\mathit{x}}_{t^{\prime }}\right)\right\}$, respectively. The blue surfaces are the isovelocity manifold ${\mathcal{M}}_{{\mathit{x}}_t}^v({\mathit{j}}_0\left({\mathit{x}}_t\right))$ (the upper surface) and the zero-velocity manifold ${\mathcal{M}}_{{\mathit{x}}_t,st}^v$ (the lower surface). The grey lines are the NEF subspaces ${\mathcal{P}}^f({\mathit{f}}_0\left({\mathit{x}}_t\right))$ and the EF subspace ${\mathcal{P}}_{\mathit{x},eq}^f$.