Landscape
Stability shapes the space of possible cell states.
Single-cell dynamics
Landscape–Flux
Framework
A visual guide to how cell states emerge, move, and change.
Interactive state space stochastic trajectories
Now viewingProgenitor basin
Flux
Non-equilibrium flow drives transitions between states.
Cell states
Attractors correspond to stable cellular phenotypes.
The central idea
One map. Two kinds of force.
A landscape shows where cells accumulate. Flux shows how probability keeps moving. You need both to reconstruct non-equilibrium dynamics.
Observed drift
Single-cell dynamics combine stabilization with directional, driven flow.
Infer the steady distribution
Estimate Pss(x) from a transition matrix, RNA-velocity field, optimal transport model, or an ensemble of trajectories.
Convert density into landscape
U(x) = −ln Pss(x). Deep basins mark frequently occupied, stable phenotypic regions.
Recover the missing direction
The steady current Jss distinguishes genuine circulation from equilibrium relaxation.
Quantitative laboratory
Compare two dynamical landscapes.
Switch between cell-cycle circulation and branching differentiation. Every basin reports its steady probability and relative potential; the controls update flux, transition rate, and entropy production.
U(x) = −ln Pss(x)Jₛₛ = 0.084
0.184 keff (a.u.⁻¹)0.084 |Jss|0.443 Ṡtot (kB/a.u.)
Illustrative numerical model
Cell cycle
A nonzero cyclic current captures the arrow of cell-cycle progression even though the phase distribution is stationary.
StatePssU − Umin
G10.450.000
S0.280.474
G2/M0.270.511
Calculationkeff = exp(−ΔU‡/D) × drive factorValues are pedagogical and dimensionless—not fitted experimental estimates.
Mathematical core
Six equations to keep nearby.
These relations connect observed single-cell motion to probability, force, irreversibility, and transition paths.
Force decomposition
F = −D∇U + Jss / Pss
Gradient stabilizes. Flux circulates.
The first term follows the potential; the second is the non-equilibrium driving force.
Probability current
J = FP − D∇P
Drift transports probability while diffusion spreads it.
Continuity
∂tP = −∇·J
Probability changes wherever currents converge or diverge.
Landscape
U = −ln Pss
Highly occupied states become low-potential basins.
Entropy production
Ṡtot = ∫ JᵀD⁻¹JP dx ≥ 0
A global measure of time-reversal breaking and dissipation.
Path action
S[x] = ¼∫‖ẋ − F‖²D⁻¹dt
Lower-action paths are more probable transition routes.
Important
The landscape depends on coordinates, sampling, and the diffusion model. Treat it as an effective probabilistic potential, and test biological conclusions for robustness.
Lab reference
Shared language for the group.
Open any term for a collaborator-ready definition. The distinctions between density, current, and force are especially important.
Data→Dynamics→Steady state→Landscape + flux
01Attractor+
A stable region of state space toward which nearby cellular trajectories tend to move.
02Detailed balance+
The equilibrium condition in which every microscopic probability flow is balanced by its reverse, so Jₛₛ = 0.
03Landscape+
U = −ln Pₛₛ: a probabilistic potential derived from the steady-state distribution. It is not automatically a thermodynamic energy.
04Probability flux+
The directed current J that transports probability through state space. At steady state it can circulate even when P no longer changes.
05Barrier+
A high-potential region separating attractors; larger barriers generally imply rarer transitions, with a noise-dependent rate.
06Entropy production+
A nonnegative measure of irreversibility and energetic dissipation associated with sustained probability currents.
Selected literature
References
Foundational theory and recent single-cell applications of the landscape–flux framework.
- 01Potential landscape and flux framework of nonequilibrium networks: Robustness, dissipation, and coherence of biochemical oscillations
- 02Quantifying the Waddington landscape and biological paths for development and differentiation
- 03Landscape and flux theory of non-equilibrium dynamical systems with application to biology
- 04Nonequilibrium physics in biology
- 05Quantifying Landscape-Flux via Single-Cell Transcriptomics Uncovers the Underlying Mechanism of Cell Cycle
- 06Uncovering underlying physical principles and driving forces of cell differentiation and reprogramming from single-cell transcriptomics
- 07Quantifying Landscape and Flux from Single-Cell Omics: Unraveling the Physical Mechanisms of Cell Function