Quantifying Waddington's epigenetic landscape: a comparison of single-cell potency measures.

MOTIVATION: Estimating differentiation potency of single cells is a task of great biological and clinical significance, as it may allow identification of normal and cancer stem cell phenotypes. However, very few single-cell potency models have been proposed, and their robustness and reliability across independent studies have not yet been fully assessed. RESULTS: Using nine independent single-cell RNA-Seq experiments, we here compare four different single-cell potency models to each other, in their ability to discriminate cells that ought to differ in terms of differentiation potency. Two of the potency models approximate potency via network entropy measures that integrate the single-cell RNA-Seq profile of a cell with a protein interaction network. The comparison between the four models reveals that integration of RNA-Seq data with a protein interaction network dramatically improves the robustness and reliability of single-cell potency estimates. We demonstrate that underlying this robustness is a correlation relationship, according to which high differentiation potency is positively associated with overexpression of network hubs. We further show that overexpressed network hubs are strongly enriched for ribosomal mitochondrial proteins, suggesting that their mRNA levels may provide a universal marker of a cell's potency. Thus, this study provides novel systems-biological insight into cellular potency and may provide a foundation for improved models of differentiation potency with far-reaching implications for the discovery of novel stem cell or progenitor cell phenotypes.

Mean-field analysis of Stuart-Landau oscillator networks with symmetric coupling and dynamical noise.

This paper presents analyses of networks composed of homogeneous Stuart-Landau oscillators with symmetric linear coupling and dynamical Gaussian noise. With a simple mean-field approximation, the original system is transformed into a surrogate system that describes uncorrelated oscillation/fluctuation modes of the original system. The steady-state probability distribution for these modes is described using an exponential family, and the dynamics of the system are mainly determined by the eigenvalue spectrum of the coupling matrix and the noise level. The variances of the modes can be expressed as functions of the eigenvalues and noise level, yielding the relation between the covariance matrix and the coupling matrix of the oscillators. With decreasing noise, the leading mode changes from fluctuation to oscillation, generating apparent synchrony of the coupled oscillators, and the condition for such a transition is derived. Finally, the approximate analyses are examined via numerical simulation of the oscillator networks with weak coupling to verify the utility of the approximation in outlining the basic properties of the considered coupled oscillator networks. These results are potentially useful for the modeling and analysis of indirectly measured data of neurodynamics, e.g., via functional magnetic resonance imaging and electroencephalography, as a counterpart of the frequently used Ising model.

Quantifying pluripotency landscape of cell differentiation from scRNA-seq data by continuous birth-death process.

Modeling cell differentiation from omics data is an essential problem in systems biology research. Although many algorithms have been established to analyze scRNA-seq data, approaches to infer the pseudo-time of cells or quantify their potency have not yet been satisfactorily solved. Here, we propose the Landscape of Differentiation Dynamics (LDD) method, which calculates cell potentials and constructs their differentiation landscape by a continuous birth-death process from scRNA-seq data. From the viewpoint of stochastic dynamics, we exploited the features of the differentiation process and quantified the differentiation landscape based on the source-sink diffusion process. In comparison with other scRNA-seq methods in seven benchmark datasets, we found that LDD could accurately and efficiently build the evolution tree of cells with pseudo-time, in particular quantifying their differentiation landscape in terms of potency. This study provides not only a computational tool to quantify cell potency or the Waddington potential landscape based on scRNA-seq data, but also novel insights to understand the cell differentiation process from a dynamic perspective.

Embedding entropy: a nonlinear measure of dynamical causality.

Research on concepts and computational methods of causality has a long history, and there are various concepts of causality as well as corresponding computing algorithms based on measured data. Here, by considering causes and effects from a dynamical perspective, we present a unified mathematical framework for the so-called dynamical causality (DC), which can detect causal interactions over time; notably, this framework covers Granger causality, transfer entropy, embedding causality and their conditional versions. Based on this framework, we further propose a causality criterion called embedding entropy (EE) to measure the DC between two variables. Moreover, its conditional version, conditional embedding entropy (cEE), is also derived for detecting conditional/direct causality. The significant advantages of EE and cEE are that they can be employed for solving not only nonlinear causal inference but also the non-separability problem, and they can reduce the scale bias in numerical calculation. The performance and robustness of EE and cEE were demonstrated through numerical simulations, and causal inference on various real-world datasets validated their effectiveness.

Energy landscape decomposition for cell differentiation with proliferation effect.

Complex interactions between genes determine the development and differentiation of cells. We establish a landscape theory for cell differentiation with proliferation effect, in which the developmental process is modeled as a stochastic dynamical system with a birth-death term. We find that two different energy landscapes, denoted U and V, collectively contribute to the establishment of non-equilibrium steady differentiation. The potential U is known as the energy landscape leading to the steady distribution, whose metastable states stand for cell types, while V indicates the differentiation direction from pluripotent to differentiated cells. This interpretation of cell differentiation is different from the previous landscape theory without the proliferation effect. We propose feasible numerical methods and a mean-field approximation for constructing landscapes U and V. Successful applications to typical biological models demonstrate the energy landscape decomposition's validity and reveal biological insights into the considered processes.

Dynamics-based data science in biology.

With the increasingly accumulated bio-data, dynamics-based data-science has been progressing as an efficient way to reveal mechanisms of dynamical biological processes. We review three applications on detecting the tipping-points of diseases, quantifying cell's potency, and predicting time-series, to show the importance of dynamics-based data-science.

Criticality in the Healthy Brain.

The excellence of the brain is its robustness under various types of noise and its flexibility under various environments. However, how the brain works is still a mystery. The critical brain hypothesis proposes a possible mechanism and states that criticality plays an important role in the healthy brain. Herein, using an electroencephalography dataset obtained from patients with psychotic disorders (PDs), ultra-high risk (UHR) individuals and healthy controls (HCs), and its dynamical network analysis, we show that the brain of HCs remains around a critical state, whereas that of patients with PD falls into more stable states. Meanwhile, the brain of UHR individuals is similar to that of PD in terms of entropy but is analogous to that of HCs in causality patterns. These results not only provide evidence for the criticality of the normal brain but also highlight the practicability of using an analytic biophysical tool to study the dynamical properties of mental diseases.

Quantifying Direct Dependencies in Biological Networks by Multiscale Association Analysis.

Partial correlation (PC) or conditional mutual information (CMI) is widely used in detecting direct dependencies between the observed variables in biological networks by eliminating indirect correlations/associations, but it fails whenever there are some strong correlations in a network. In this paper, we theoretically develop a multiscale association analysis to overcome this flaw. We propose a new measure, partial association (PA), based on the multiscale conditional mutual information. We show that linear PA and nonlinear PA have clear advantages over PC and CMI from both theoretical and computational aspects. Both simulated models and real omics datasets demonstrate that PA is superior to PC and CMI in terms of accuracy, and is a powerful tool to identify the direct associations or reconstruct molecular networks based on the observed data. Survival and functional analyses of the hub genes in the gene networks reconstructed from TCGA data for different cancers also validated the effectiveness of our method.

Towards a critical transition theory under different temporal scales and noise strengths.

The mechanism of critical phenomena or critical transitions has been recently studied from various aspects, in particular considering slow parameter change and small noise. In this article, we systematically classify critical transitions into three types based on temporal scales and noise strengths of dynamical systems. Specifically, the classification is made by comparing three important time scales τ(λ), τ(tran), and τ(ergo), where τ(λ) is the time scale of parameter change (e.g., the change of environment), τ(tran) is the time scale when a particle or state transits from a metastable state into another, and τ(ergo) is the time scale when the system becomes ergodic. According to the time scales, we classify the critical transition behaviors as three types, i.e., state transition, basin transition, and distribution transition. Moreover, for each type of transition, there are two cases, i.e., single-trajectory transition and multitrajectory ensemble transition, which correspond to the transition of individual behavior and population behavior, respectively. We also define the critical point for each type of critical transition, derive several properties, and further propose the indicators for predicting critical transitions with numerical simulations. In addition, we show that the noise-to-signal ratio is effective to make the classification of critical transitions for real systems.