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  • Perspective
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Statistical mechanics meets single-cell biology

Nature Reviews Genetics volume 22, pages 459–476 (2021)Cite this article

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Abstract

Single-cell omics is transforming our understanding of cell biology and disease, yet the systems-level analysis and interpretation of single-cell data faces many challenges. In this Perspective, we describe the impact that fundamental concepts from statistical mechanics, notably entropy, stochastic processes and critical phenomena, are having on single-cell data analysis. We further advocate the need for more bottom-up modelling of single-cell data and to embrace a statistical mechanics analysis paradigm to help attain a deeper understanding of single-cell systems biology.

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Fig. 1: Statistical mechanical modelling of state manifolds.
Fig. 2: Paradigms for estimating differentiation potency of single cells.
Fig. 3: Single-cell differentiation potency and molecular entropy.
Fig. 4: Critical phenomena in development and disease.

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Acknowledgements

A.E.T. is funded by the NSFC (grant numbers 31571359 and 31771464). A.P.F. is funded by grant DP1 DK119129 from the NIH.

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Authors and Affiliations

  1. CAS Key Laboratory of Computational Biology, Shanghai Institute of Nutrition and Health, Shanghai Institutes for Biological Sciences, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Shanghai, China

    Andrew E. Teschendorff

  2. UCL Cancer Institute, University College London, London, UK

    Andrew E. Teschendorff

  3. Center for Epigenetics, Johns Hopkins University School of Medicine, Baltimore, MD, USA

    Andrew P. Feinberg

  4. Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD, USA

    Andrew P. Feinberg

  5. Department of Medicine, Johns Hopkins University School of Medicine, Baltimore, MD, USA

    Andrew P. Feinberg

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Glossary

Bottom-up modelling

In the context of single-cell data, this refers to an analysis paradigm where one formulates an explicit dynamic (network) model to describe the data within each cell, often without the need to use data from other cells.

Cell-fate probabilities

Probabilities that cells will differentiate and give rise to progeny in a particular cell fate, and which can be estimated from a Markov chain description of single-cell dynamics.

Correlation length

A term used frequently in statistical mechanics to describe the strength of correlations between neighbouring microscopic elements (for example, atoms or molecules) in a system, and which typically decays with distance or time.

Differentiation potency hierarchy

The hierarchical arrangement of cell types according to the number of downstream cell types a given cell could give rise to during development or in a general differentiation process.

Dimensional reduction

A step in the analysis of big data, including single-cell RNA sequencing data, where by a lower-dimensional representation of the data is sought that can capture a high proportion of the variance in the data.

Dropout rate

This refers to the generally low sensitivity to detect gene expression in single-cell RNA sequencing assays, especially for genes expressed at a low number of molecules in a cell, resulting in potentially many zero expression values.

Dynamical systems theory

A branch of classical physics that describes the dynamics of multiple variables (for example, molecular concentrations) in terms of a set of linear or non-linear differential equations.

Entropy

A generic term to quantify the amount of uncertainty associated with a system or the outcome of some measurement.

Feature selection

In the context of single-cell RNA sequencing data analysis, this refers to a filtering step whereby all cells passing quality control are used to select genes according to some statistical criterion (for example, high variability across cells).

Functional cellular states

The states of the complex molecular network within a cell that determine its function, encoded by a multidimensional vector describing properties such as cell type, cell-cycle phase and metabolic state.

Functional pluripotency

A term used to characterize pluripotency at the level of a cell population, and which refers to the ability of this population to give rise to cellular progeny of each of the three main germ layers.

Gene count

The number of expressed genes (that is, the number of genes with non-zero expression values) in a cell, as derived from a single-cell RNA sequencing profile.

Global manifold

The complete state manifold encompassing all developmental stages and cell types within a given organism, to be distinguished from local state manifolds, which refer only to specific subparts.

Hubs

Nodes within a network that possess an abnormally high number of neighbours. They define outliers in the degree distribution, and are a key feature of real biological networks, including protein–protein interaction networks.

Infinite-range spin glasses

Spin glasses where interactions can occur over very long distances encompassing the whole system.

Ising spin model

A special case of a spin glass where interactions are localized to nearest neighbours and where the state space of each particle is only two-dimensional.

Least action principle

A variational principle derived from classical dynamics where the dynamics of the system can be derived from the minimization of an energy function.

Lineage trajectories

One-dimensional trajectories in a high-dimensional phase space describing the dynamics of a cell, but often depicted graphically in a low-dimensional reduced space.

Macrostate

A macroscopic observable of a system. Examples include electrical conductivity of a material and the number of animals within an ecosystem.

Markov chain

A Markov process on a finite discrete state space (often a graph), defined by a probability matrix that describes the probabilities of transition between connecting states (in the graph context, these are the nodes).

Markov process

A memoryless stochastic process where the state of the system (for example, a cell) at any given timepoint is determined only by its immediately previous state.

Mass-action principle

A physical law which states that the probability (rate) of a molecular interaction (reaction) is proportional to the product of the concentrations of each molecule (reactant).

Metastable

In the context of attractors, it refers to locally stable states, which, however, are not stable under higher-energy perturbations. Their stability is therefore often only transient.

Microstate

The instantaneous and dynamic state that each microscopic constituent of the system is in, and which is generally unobserved. Examples include the speed and direction at which each air molecule moves and the electrical activity of each neuron in a brain.

Multifurcations

A higher-order generalization of a bifurcation, whereby a cell lineage describing a multipotent state splits or branches out into three or more daughter lineages.

Multilineage priming

In a cell population, this refers to the existence of cells primed or restricted to differentiate into one of many downstream lineages.

Occam’s razor

Also called the ‘law of parsimony’, it is a principle stated by the philosopher William of Ockham (1285–1347) that gives precedence to simplicity: of two competing theories, the simpler explanation of an entity is to be preferred.

Optimal transport

(OT). A branch of mathematics that deals with optimizing the cost of transporting a distribution of mass (for example, cells) between two successive locations, and which is amenable to solution via numerical programming.

Order parameters

Parameters of a statistical mechanical model, whereby varying a parameter can lead to transitions between different low-energy states.

Phase space

An abstract space (usually high-dimensional) in which each point corresponds to a functional cellular state, with the dynamics of a cell’s state describing a one-dimensional trajectory in this space.

Phase transitions

Abrupt, discontinuous changes in the macroscopic properties of a system, often as a result of energy exchange with the environment, and driven by changes in the microscopic interaction patterns.

Potential energy

A term associated with the Waddington landscape, representing the elevation and correlating with developmental potential, in analogy with the physical potential energy associated with the elevation in geophysical landscapes.

Pseudotime

A temporal variable computed for each cell along a lineage trajectory measuring the differentiation time from a given root state. It correlates with experimental differentiation stage and differentiation potency but is also distinct.

Quasi-potential

The name given to the scalar function obtained by solving sets of non-linear differential equations describing the dynamics of transcription factor concentrations, and which approximates developmental potential.

Random walks

Stochastic Markov processes on a finite discrete state space, often represented as nodes on a network or graph, and describing a trajectory along nodes, with transitions between nodes following a specific probability distribution.

Regulatory network motifs

Topological regulatory interaction patterns involving activation and/or inhibition between transcription factors, as specified in a gene regulatory network.

Root states

In the context of inferring lineage trajectories from single-cell RNA sequencing data, a root state refers to the cell (or cluster of cells) where the trajectory starts, and which needs to be assigned before trajectories can be inferred.

Saddle-node bifurcations

Dynamical bifurcations associated with the emergence of unstable ‘saddle-node’ states in phase space, often termed ‘tipping points’, and which have been proposed to describe the transitions to disease states.

Scale-free

When pertaining to networks, scale-free describes a type of network where the probability of finding a node with n neighbours decays according to a power law in n (that is, it decays more slowly than an exponential function). The implication is that such networks contain network hubs.

Spin glass

A statistical mechanical model used to describe disordered physical systems composed of many interacting particles in a high-dimensional state space and characterized by a high number of equivalent low-energy states.

State convergence

The existence of two or more distinct paths or cellular lineage trajectories by which the same end point (that is, terminal cell state) can be reached.

State manifold

A generalization of Waddington’s landscape providing a more realistic geometric representation of functional cellular states and of the single-cell dynamics connecting these states.

Statistical mechanics

A discipline of physics which broadly speaking aims to describe macroscopic observables of a general system in terms of the properties of its microscopic constituents, including their interactions.

Stochastic process

‘Stochastic’ refers to our inability to predict with certainty the value that some variable would take if observed. This randomness can be due to incomplete knowledge or can be inherent/intrinsic to the system. A stochastic process describes multiple observations of a stochastic variable, often in time, or in some more abstract temporal space.

TF regulons

A regulon for a given transcription factor (TF) consists of a set of direct (and possibly also indirect) targets of the TF and whose average gene expression provides a faithful measure of the TF’s regulatory activity.

Top-down modelling

In the context of single-cell data, this refers to an analysis paradigm where one analyses the data from many cells together to infer properties of individual cells.

Velocity field

In the context of single-cell RNA sequencing data, a low-dimensional static graphical representation of cell dynamics in which the future transcriptomic state of each cell is indicated by an arrow pointing away from the cell.

Waddington epigenetic landscapes

Three-dimensional representations of cellular development, with differentiation trajectories and cell states described by bifurcating valleys and local basins, and with the elevation in the landscape describing developmental potential.

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Teschendorff, A.E., Feinberg, A.P. Statistical mechanics meets single-cell biology. Nat Rev Genet 22, 459–476 (2021). https://doi.org/10.1038/s41576-021-00341-z

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