The natural determinant reference (NDR) or principal natural determinant is the Slater determinant comprised of the N most strongly occupied natural orbitals of an N-electron state of interest. Unlike the Kohn–Sham (KS) determinant, which yields the exact ground-state density, the NDR only yields the best idempotent approximation to the interacting one-particle reduced density matrix, but it is well-defined in common atom-centered basis sets and is representation-invariant. We show that the under-determination problem of prior attempts to define a ground-state energy functional of the NDR is overcome in a grand-canonical ensemble framework at the zero-temperature limit. The resulting grand potential functional of the NDR ensemble affords the variational determination of the ground state energy, its NDR (ensemble), and select ionization potentials and electron affinities. The NDR functional theory can be viewed as an “exactification” of orbital optimization and empirical generalized KS methods. NDR functionals depending on the noninteracting Hamiltonian do not require troublesome KS-inversion or optimized effective potentials.

Background

Several dynamic models of a gene regulatory network of the light-induced floral transition process in Arabidopsis have been developed to capture the behavior of gene transcription and infer predictions based on experimental observations. It has been proven that the models can make accurate and novel predictions, which generate testable hypotheses.Two major issues were addressed in this study. First, construction of dynamic models for gene regulatory networks requires the use of mathematic modeling that comprises equations of a large number of parameters. Second, the binding mechanism of the transcription factor with DNA is another factor that requires detailed modeling. The first issue was tackled by adopting an optimization algorithm, and the second was addressed by comparing the performance of three alternative modeling approaches, namely the S-system, the Michaelis-Menten model and the Mass-action model. The efficiencies of parameter estimation and modeling performance were calculated based on least square error (O(p)), mean relative error (MRE) and Akaike Information Criterion (AIC).

Results

We compared three models to describe gene regulation of the flowering transition process in Arabidopsis. The Mass-action model is the simplest and has the least parameters. It is therefore less computation-intensive with the smallest AIC value. The disadvantage, however, is that it assumes the system is simply a second order reaction which is not the case in our study. The Michaelis-Menten model also assumes the system is homogeneous and ignores the intracellular protein transport process. The S-system model has the best performance and it does describe the diffusion effects. A disadvantage of the S-system is that it involves the most parameters. The largest AIC value also implies an over-fitting may occur in parameter estimation.

Conclusions

Three dynamic models were adopted to describe the dynamics of the gene regulatory network of the flowering transition process in Arabidopsis. Based on MRE, the least square error and global sensitivity analysis, the S-system has the best performance. However, the fact that it has the highest AIC suggests an over-fitting may occur in parameter estimation. The result of this study may need to be applied carefully when modeling complex gene regulatory networks.