Determining minimal output sets that ensure structural identifiability

Abstract

The process of inferring parameter values from experimental data can be a cumbersome task. In addition, the collection of experimental data can be time consuming and costly. This paper covers both these issues by addressing the following question: "Which experimental outputs should be measured to ensure that unique model parameters can be calculated?". Stated formally, we examine the topic of minimal output sets that guarantee a model's structural identifiability. To that end, we introduce an algorithm that guides a researcher as to which model outputs to measure. Our algorithm consists of an iterative structural identifiability analysis and can determine multiple minimal output sets of a model. This choice in different output sets offers researchers flexibility during experimental design. Our method can determine minimal output sets of large differential equation models within short computational times.

Fig 1. NF- κB model: Singular values of the output sensitivity matrix, S_norm, if we measure all states, {x₁, …, x₁₅}, as model output.

Singular values, arranged in descending order, reveal no gap. This suggests that the sensitivity matrix is of full rank and therefore the model is structurally identifiable.

Fig 2. NF- κB model: Singular values of the output sensitivity matrix, S_norm, if we measure all states apart from x₄.

Singular values, arranged in descending order, reveal a clear gap with σ₄₃ = 7.8 × 10⁻¹⁶. This indicates that the sensitivity matrix is rank deficient and so the model is structurally unidentifiable.

Fig 3. NF- κB model: Entries in the last right singular vector corresponding to the vanishing singular value, σ₄₃, in Fig 2.

The corresponding non-trivial null-space indicates that parameters θ₂, θ₃, θ₂₇ and initial condition x₄(0) are totally correlated.

Fig 4. Example 1: Structural identifiability results of a chemical reaction system: Singular values of the output sensitivity matrix, S_norm, when measuring the output {x₁ …, x₁₁} omitting sensors x₄ and x₅.

Singular values, arranged in descending order, reveal a clear gap. This gap, in conjunction with the smallest singular value, σ₁₇ = 2.4 × 10⁻¹⁷, indicate that the model is structurally unidentifiable when measuring this output.

Fig 5. Example 1: Structural identifiability results of a chemical reaction system: Non-zero entries in the last 4 columns of matrix V.

These indicate that initial conditions x₄(0) and x₅(0) and model parameters θ₂ and θ₃ are unidentifiable. Since x₄ and x₅ are defined in y_max, both of these sensors are essential.

Fig 6. Example 3: JAK-STAT model: Singular values of the output sensitivity matrix, S_norm, when measuring the model output {x₁, …, x₃₁}, omitting sensors x₁₀ and x₁₁.

Singular values reveal a clear gap and this, in conjunction with the smallest singular value of σ₈₂ = 7.3 × 10⁻¹⁶, indicates that S_norm is not of full rank and therefore the model is structurally unidentifiable.

Fig 7. Example 3: JAK-STAT model: Entries in the last right singular vector corresponding to the vanishing singular value, σ₈₂, in Fig 6.

The corresponding non-trivial null-space indicates that model parameters θ₁₄, θ₅₁ and initial conditions x₁₀(0) and x₁₁(0) are totally correlated and so the model is not identifiable when model states x₁₀ and x₁₁ are simultaneously omitted from the model’s output.

Fig 8. Example 4: Ligand binding model: Entries in the last right singular vector, corresponding to the smallest singular value of precisely zero, calculated for the measured output {x₁, x₂, x₃, x₄, x₆}.

The non-trivial null-space indicates that the initial condition of state x₅ is unidentifiable when this state is not measured. Accordingly, x₅ should be included into the model’s minimal output set.

Fig 9. Example 5: Simplified glycolytic reaction model: Entries in the right singular vectors corresponding to 2 vanishing singular values.

The non-zero values indicate that the initial condition x₁₀(0) and parameter θ₁₃ are unidentifiable when state x₁₀ is not measured.

Fig 10. Example 6: Goldbeter model: Entries in the last right singular vector corresponding to single vanishing singular value calculated.

The non-zero values indicate that parameters θ₁, θ₃, θ₄, θ₅ and initial condition x₁(0) are unidentifiable when state x₁ is not measured.

Fig 11. Example 7: JAK/STAT model with specific model output: Singular values of the output sensitivity matrix, S, when omitting sensor θ₁₇(x₄ + x₅) from y_max.

Singular values, arranged in descending order, reveal a clear gap. This gap in conjunction with the smallest singular value of 4 × 10⁻¹⁸, indicate that S is rank deficient.

Fig 12. Example 7: JAK/STAT model with specific model output: Entries in the last right singular vector corresponding to the vanishing singular value in Fig 11.

The non-trivial null-space indicates that model parameter θ₁₇ is not identifiable when sensor θ₁₇(x₄ + x₅) is not measured.