[Paper Review] Determination of Functional Network Structure from Local Parameter Dependence Data

Boris Ν. Kholodenko, Eduardo D. Sontag|ArXiv.org|May 2, 2002

Bioinformatics and Genomic NetworksBiochemistry, Genetics and Molecular Biology4 references21 citations

TL;DR

This paper proposes a method to reconstruct the functional interaction network of a dynamical system from local parameter sensitivity data at steady state. By leveraging known structural constraints (which parameters directly influence which equations) and orthogonality between sensitivity vectors and the system's Jacobian rows, it recovers the Jacobian matrix up to scalar multiples, enabling inference of interaction structure and relative influence strengths in biochemical and other complex networks.

ABSTRACT

In many applications, such as those arising from the field of cellular networks, it is often desired to determine the interaction (graph) structure of a set of differential equations, using as data measured sensitivities. This note proposes an approach to this problem.

Motivation & Objective

• To infer the interaction graph and relative influence strengths of a system of differential equations when the functional forms of the equations are unknown.
• To determine the Jacobian matrix of the system up to scalar multiples using only local sensitivity data at steady state.
• To leverage prior knowledge about which parameters directly influence which equations (structural constraints) to resolve the inherent ambiguity in sensitivity-based inference.
• To extend the method to non-steady state data using time-derivative sensitivities and orthogonality in higher-dimensional spaces.

Proposed method

• Uses experimental sensitivity data Σ, where σkj = ∂ξk/∂pj(ṗ), representing how steady-state concentrations change with parameter perturbations.
• Applies structural assumptions: for each equation i, a subset Si of parameters p_j are known to have no direct influence on fi, so ∂fi/∂pj(ẋ,ṗ) = 0 for j ∈ Si.
• Derives that the Jacobian row Ai must be orthogonal to the sensitivity vectors Γj = ∂ξ/∂pj(ṗ) for all j ∈ Si, leading to the condition Ai · Γj = 0.
• Reconstructs Ai up to scalar multiple by solving the orthogonality condition within the linear subspace Ki spanned by {Γj | j ∈ Si}, assuming dim(Ki) ≥ n−1.
• Extends the method to non-steady state by incorporating time-derivative sensitivities ηij = ∂²ξi/∂t∂pj, forming augmented vectors Σij = (ηij, σ1j, ..., σnj).
• Uses orthogonality of [−1, Ai] to the subspace Hi spanned by {Σij | j ∈ Si} to constrain Ai, with uniqueness under genericity conditions (e.g., dim(Hi) = n).

Experimental results

Research questions

• RQ1 Can the functional interaction structure of a system of differential equations be inferred from local sensitivity data at steady state?
• RQ2 How can prior knowledge about which parameters directly influence which equations be used to resolve the ambiguity in sensitivity-based Jacobian reconstruction?
• RQ3 What is the minimal set of structural assumptions needed to recover the Jacobian up to scalar multiples from sensitivity data?
• RQ4 Can the method be extended to non-steady state measurements using time-derivative sensitivities?
• RQ5 Under what conditions does the orthogonality condition uniquely determine the Jacobian rows?

Key findings

• The Jacobian row Ai can be reconstructed up to a scalar multiple using only sensitivity data Σ and structural constraints on parameter dependencies.
• The method relies on the orthogonality between the Jacobian row Ai and the sensitivity vectors Γj for parameters j that do not directly affect fi.
• When the subspace Ki spanned by relevant sensitivity vectors has dimension at least n−1, Ai is uniquely determined up to scalar multiple.
• For non-steady state data, the method uses augmented vectors Σij = (ηij, σ1j, ..., σnj) to enforce orthogonality of [−1, Ai] to the subspace Hi.
• Under genericity conditions such as dim(Hi) = n, the vector [−1, Ai] is uniquely determined, enabling full reconstruction of the interaction structure.
• The approach is robust to small errors and can be implemented using finite differences in place of exact derivatives.