Multivariate Surprisal Analysis of Gene Expression Levels

Density Matrix via Few Dominant Observables for the Ultrafast Non-Radiative Decay in Pyrazine

Unraveling the density matrix of a non-stationary quantum state as an explicit function of a few observables provides a complementary view of quantum dynamics. We have recently developed a practical way to identify the minimal set of the dominant observables that govern the quantal dynamics even in the case of strong non-adiabatic effects and large anharmonicity [Komarova et al., J. Chem. Phys. 155, 204110 (2021)]. Fast convergence in the number of the dominant contributions is achieved when instead of the density matrix we describe the time-evolution of the surprisal, the logarithm of the density operator. In the present work, we illustrate the efficiency of the proposed approach using an example of the early time dynamics in pyrazine in a Hilbert space accounting for up to four vibrational normal modes, {Q10a, Q6a, Q1, and Q9a}, and two coupled electronic states, the optically dark B 1 3 u ( n π * ) and the bright B 1 2 u ( π π * ) states. Dynamics in four-dimensional (4D) configurational space involve 19,600 vibronic eigenstates. Our results reveal that the rate of the ultrafast population decay as well as the shape of the nuclear wave packets in 2D, accounting only for {Q10a,Q6a} normal modes, are accurately captured with only six dominant time-independent observables in the surprisal. Extension of the dynamics to 3D and 4D vibrational subspace requires only five additional constraints. The time-evolution of a quantum state in 4D vibrational space on two electronic states is thus compacted to only 11 time-dependent coefficients of these observables.

Compacting the density matrix in quantum dynamics: Singular value decomposition of the surprisal and the dominant constraints for anharmonic systems

We introduce a practical method for compacting the time evolution of the quantum state of a closed physical system. The density matrix is specified as a function of a few time-independent observables where their coefficients are time-dependent. The key mathematical step is the vectorization of the surprisal, the logarithm of the density matrix, at each time point of interest. The time span used depends on the required spectral resolution. The entire course of the system evolution is represented as a matrix where each column is the vectorized surprisal at the given time point. Using the singular value decomposition (SVD) of this matrix, we generate realistic approximations for the time-independent observables and their respective time-dependent coefficients. This allows for a simplification of the algebraic procedure for determining the dominant constraints (the time-independent observables) in the sense of the maximal entropy approach. A non-stationary coherent initial state of a Morse oscillator is used to introduce the approach. We derive the analytical exact expression for the surprisal as a function of time, and this offers a benchmark for comparison with the accurate but approximate SVD results. We discuss two examples of a Morse potential of different anharmonicities, H₂ and I₂ molecules. We further demonstrate the approach for a two-coupled electronic state problem, the well-studied non-radiative decay of pyrazine from its bright state. Five constraints are found to be enough to capture the ultrafast electronic population exchange and to recover the dynamics of the wave packet in both electronic states.

TMEA: A Thermodynamically Motivated Framework for Functional Characterization of Biological Responses to System Acclimation

The objective of gene set enrichment analysis (GSEA) in modern biological studies is to identify functional profiles in huge sets of biomolecules generated by high-throughput measurements of genes, transcripts, metabolites, and proteins. GSEA is based on a two-stage process using classical statistical analysis to score the input data and subsequent testing for overrepresentation of the enrichment score within a given functional coherent set. However, enrichment scores computed by different methods are merely statistically motivated and often elusive to direct biological interpretation. Here, we propose a novel approach, called Thermodynamically Motivated Enrichment Analysis (TMEA), to account for the energy investment in biological relevant processes. Therefore, TMEA is based on surprisal analysis, which offers a thermodynamic-free energy-based representation of the biological steady state and of the biological change. The contribution of each biomolecule underlying the changes in free energy is used in a Monte Carlo resampling procedure resulting in a functional characterization directly coupled to the thermodynamic characterization of biological responses to system perturbations. To illustrate the utility of our method on real experimental data, we benchmark our approach on plant acclimation to high light and compare the performance of TMEA with the most frequently used method for GSEA.

Translating Cancer Molecular Variability into Personalized Information Using Bulk and Single Cell Approaches

Cancer research is striving toward new frontiers of assigning the correct personalized drug(s) to a given patient. However, extensive tumor heterogeneity poses a major obstacle. Tumors of the same type often respond differently to therapy, due to patient-specific molecular aberrations and/or untargeted tumor subpopulations. It is frequently not possible to determine a priori which patients will respond to a certain therapy or how an efficient patient-specific combined therapy should be designed. Large-scale datasets have been growing at an accelerated pace and various technologies and analytical tools for single cell and bulk level analyses are being developed to extract significant individualized signals from such heterogeneous data. However, personalized therapies that dramatically alter the course of the disease remain scarce, and most tumors still respond poorly to medical care. In this review, the basic concepts of bulk and single cell approaches are discussed, as well as their emerging role in individualized designs of drug therapies, including the advantages and limitations of their applications in personalized medicine.