Cramer-Rao bound and absolute sensitivity in chemical reaction networks

TL;DR

The paper introduces basis-independent absolute sensitivities for chemical reaction networks (CRNs) by embedding CRN steady states in an information-geometric framework and focusing on how perturbations along conserved-quantity directions propagate across the network. For quasi-thermostatic CRNs, it derives a multivariate Cramer-Rao bound Cov(I_n) \geq diag(\frac{1}{x}) A and provides a linear-algebraic, exact characterization A = \mathrm{diag}(x) Cov(I_n) with Cov(I_n)_{ij} = ⟨π(e_i), π(e_j)⟩_{1/x}, where π is the projection onto the tangent space T_x\mathcal{V}^{ss} = diag(x) Ker[S^T]. The absolute-sensitivity entries satisfy α_{i\rightarrow j} = x_j ⟨π(e_j), π(e_i)⟩_{1/x}, and α_i = α_{i\rightarrow i} lie in [0,1], enabling a robust, parameter-free view of concentration robustness. The core example on the IDHKP-IDH glyoxylate bypass demonstrates both analytic and numerical confirmation of robustness and reveals a symmetry in the distribution of sensitivities not evident at the level of individual concentration vectors. The work also discusses how non-ideal thermodynamics can yield α_i > 1 or α_{i\rightarrow j} < 0, suggesting directions for future exploration of computation and control in CRNs via absolute-sensitivity analysis.

Abstract

Chemical reaction networks (CRN) comprise an important class of models to understand biological functions such as cellular information processing, the robustness and control of metabolic pathways, circadian rhythms, and many more. However, any CRN describing a certain function does not act in isolation but is a part of a much larger network and as such is constantly subject to external changes. In [Shinar, Alon, and Feinberg. "Sensitivity and robustness in chemical reaction networks." SIAM J App Math (2009): 977-998.], the responses of CRN to changes in the linear conserved quantities, called sensitivities, were studied in and the question of how to construct absolute, i.e., basis-independent, sensitivities was raised. In this article, by applying information geometric methods, such a construction is provided. The idea is to track how concentration changes in a particular chemical propagate to changes of all concentrations within a steady state. This is encoded in the matrix of absolute sensitivities. As the main technical tool, a multivariate Cramer-Rao bound for CRN is proven, which is based on the the analogy between quasi-thermostatic steady states and the exponential family of probability distributions. This leads to a linear algebraic characterization of the matrix of absolute sensitivities for quasi-thermostatic CRN. As an example, the core module of the IDHKP-IDH glyoxylate bypass regulation system is analyzed analytically and numerically by extensive random sampling of the concentration space. The experimentally known findings for the robustness of the IDH enzyme are confirmed and a hidden symmetry at the level of distributions is revealed, providing a blueprint for the analysis of the robustness properties of CRNs.

Figures (4)

• Figure 1: Illustration of the geometrical background for the case of quasi-thermostatic CRN. The concentration space $X$ is shown in two dimensions and the steady state manifold $\mathcal{V}^{ss}$ is represented by a one-dimensional curve. The linear algebra takes place on the tangent spaces $T_x X$ and $T_x \mathcal{V}^{ss} \subset T_x X$ at a given point $x \in X$. The vectors $e_i$ are the canonical basis vectors $\frac{\partial}{\partial x_i}$ of $T_xX$, and $\pi(e_i)$ is the $\langle ., . \rangle_{\frac{1}{x}}$-orthogonal projection of $e_i$ to $T_x \mathcal{V}^{ss}$. The absolute sensitivities $\alpha_{i \rightarrow j}$ are the respective components of the vector $\pi(e_i)$.
• Figure 2: Setup for the definition of absolute sensitivities. The map $U^T: X \rightarrow H$ gives a fibration of $X$ by stoichiometric polytopes (indicated by dotted lines) with base $H$. The differentiable parametrization of $\mathcal{V}^{ss}$ by the vectors of conserved quantities requires that the map $U^T: \mathcal{V}^{ss} \rightarrow H$ is locally invertible with a differentiable inverse. The latter condition is not satisfied at, e.g., bifurcation points ($\eta'$ in the figure) and such points have to be excluded by restricting the parameter space $\tilde{H} := H \setminus \{ \eta' \}$ accordingly. If $\mathcal{V}^{ss}$ has multiple intersection points with a stoichiometric polytope $P(\eta")$, then there are different possible sections $\beta$ which amount to choosing one of the respective branches of $\mathcal{V}^{ss}$.
• Figure 3: The two parametrization of $\mathcal{V}^{ss}$ for quasi-thermostatic CRN. The map $\gamma$ provides a direct parametrization of $\mathcal{V}^{ss}$ which is known as the exponential family in statistics and as a toric variety in algebraic geometry. The map $\beta$ characterizes points on $\mathcal{V}^{ss}$ through their vector of conserved quantities $\eta$, i.e., as the unique intersection point between $\mathcal{V}^{ss}$ and the stoichiometric polytope $P(\eta)$.
• Figure 4: Distributions of the absolute sensitivities $\alpha_i$ for randomly sampled concentration vectors in $\log x^i \in [-5,0]$. Each histogram consists of 1.000.000 data points, distributed in 100 bins.

Theorems & Definitions (18)

• Theorem
• Lemma
• Theorem
• Definition 1
• Remark 1
• Remark 2
• Theorem 1
• proof
• Remark 3
• Theorem 2