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Quasi-potential landscape in complex multi-stable systems

Joseph Xu Zhou, M. D. S. Aliyu, Erik Aurell, Sui Huang

arXiv:1206.2311v1q-bio.MNcond-mat.other

TL;DR

Complex multi-stable biological systems need a global way to compare attractor stabilities, beyond theories limited to transitions between pairs of attractors. The paper reviews existing approaches and proposes a constrained vector-field decomposition whose quasi-potential captures relative stability between adjacent attractors, while noting that global ordering is not guaranteed.

  • Problem

    Existing transition theories address pairs of attractors but lack a global potential function for comparing relative stabilities across more than two attractors.

  • Method

    The paper reviews existing landscape methods and proposes constrained decompositions of non-gradient vector fields to compute a quasi-potential equivalent to the Freidlin-Wentzell potential.

  • Results

    The normal-decomposition quasi-potential has a non-local meaning: the sign of its difference indicates relative stability and spontaneous transition direction between adjacent attractors.

  • Takeaways & Limitations

    The proposed quasi-potential supports comparisons of adjacent attractor stabilities in multi-attractor systems, including spontaneous transitions without explicit external driving.

  • Takeaways & Limitations

    The decomposition is not guaranteed to exist for all dynamical systems, and transitivity required for global ordering is not guaranteed across systems.

Abstract

from arXiv · show

Developmental dynamics of multicellular organism is a process that takes place in a multi-stable system in which each attractor state represents a cell type and attractor transitions correspond to cell differentiation paths. This new understanding has revived the idea of a quasi-potential landscape, first proposed by Waddington as a metaphor. To describe development one is interested in the "relative stabilities" of N attractors (N>2). Existing theories of state transition between local minima on some potential landscape deal with the exit in the transition between a pair attractor but do not offer the notion of a global potential function that relate more than two attractors to each other. Several ad hoc methods have been used in systems biology to compute a landscape in non-gradient systems, such as gene regulatory networks. Here we present an overview of the currently available methods, discuss their limitations and propose a new decomposition of vector fields that permit the computation of a quasi-potential function that is equivalent to the Freidlin-Wentzell potential but is not limited to two attractors. Several examples of decomposition are given and the significance of such a quasi-potential function is discussed.

1. INTRODUCTION: The need for a quasi-potential function · 2. THE PHYSICS PROBLEM: THE QUASI-POTENTIAL FUNCTION

The paper motivates a global quasi-potential function for ordering metastable attractors and relating transitions in high-dimensional, far-from-equilibrium systems. It reviews existing constructions and proposes a normal decomposition intended to satisfy global metastability and Freidlin-Wentzell transition-rate requirements.

  • 1. INTRODUCTION: The need for a quasi-potential function: Gene regulatory networks exemplify high-dimensional nonlinear systems whose attractors represent cell types and whose transitions represent developmental cell-state changes.The relevant behavior concerns transitions between attractors across large distances in phase space.
  • 1. INTRODUCTION: The need for a quasi-potential function: A global energy-like function would order metastable attractors by relative depth and inform the probability and direction of noisy or perturbed transitions.This framework is intended to illuminate spontaneous cell-fate choices among alternative nearby attractors.
  • 1. INTRODUCTION: The need for a quasi-potential function: Equilibrium potential theory does not directly address far-from-equilibrium, non-conservative, high-dimensional open systems driven by internal interactions.Even gradient systems can have path-dependent transition rates determined by attractor and intervening saddle potentials.
  • 1. INTRODUCTION: The need for a quasi-potential function: A meaningful non-gradient quasi-potential should characterize metastability ordering and connect pairs of states to Freidlin-Wentzell least-action transition behavior.The paper also conjectures a transitivity property for comparisons among multiple attractor states, though only a subset of systems may satisfy it.
  • 2. THE PHYSICS PROBLEM: THE QUASI-POTENTIAL FUNCTION: Because high-dimensional nonequilibrium systems generally are not gradient systems, their vector fields require a decomposition into potential-gradient and remainder components.The decomposition assumes a finite, smooth vector field, with the remainder representing driving forces beyond the potential-gradient component.
  • 2. THE PHYSICS PROBLEM: THE QUASI-POTENTIAL FUNCTION: Existing approaches include measures of global stability, Lyapunov-function analyses, mappings to Hamiltonian systems, and stochastic decompositions for gene networks, but they differ in method and scope.The paper notes that some prior work mischaracterized Ao’s method as a generalized Helmholtz decomposition.
  • 2. THE PHYSICS PROBLEM: THE QUASI-POTENTIAL FUNCTION: The paper evaluates ad hoc landscape-construction methods against global metastability and agreement with Freidlin-Wentzell transition rates.It then presents the normal decomposition as a systematic method, supported by mathematical derivation and examples.

3. COMPARISON OF VECTOR FIELD DECOMPOSITION · 3a. The Helmholtz decomposition · 3b. Decomposition based on the flux of the probability, U~ -LnP

The section compares Helmholtz and probability-flux-based vector-field decompositions, emphasizing that neither generally provides a potential satisfying Lyapunov stability or measuring global metastability. The probability-based construction additionally depends on noise conditions and is not itself a Helmholtz decomposition.

  • 3a. The Helmholtz decomposition: The Hodge theorem decomposes sufficiently smooth, rapidly decaying fields into curl-free potential and divergence-free curl components, with Helmholtz as its three-dimensional special case.The potential is obtained from Poisson equations, and uniqueness requires boundary conditions; without them, harmonic functions create nonuniqueness.
  • 3a. The Helmholtz decomposition: Helmholtz decomposition separates conservative fields connecting attractors or repellors from solenoidal fields uninfluenced by sinks or sources.The curl component can also represent the driving force of gradient-free limit cycles, so its interpretation is not restricted to rotational effects around attractors.
  • 3a. The Helmholtz decomposition: The Helmholtz-derived potential is not necessarily non-increasing, so Lyapunov stability is not guaranteed and it is not a measure of global metastability.This conclusion follows from the two-dimensional time-derivative analysis, where the relevant sign can be positive or negative.
  • 3b. Decomposition based on the flux of the probability, U~ -LnP: The probability-based approach defines a quasi-potential from steady-state probabilities using the Boltzmann-law intuition that more stable states are more probable and have lower potential.Its decomposition uses the probability flux as an additional component because the driving force is generally not purely a gradient.
  • 3b. Decomposition based on the flux of the probability, U~ -LnP: The Fokker–Planck-based decomposition yields a gradient potential term and a flux-like curl term, but only the latter’s specified component is divergence-free.Because the remainder is generally not divergence-free, this construction is not a Helmholtz decomposition as claimed elsewhere.
  • 3b. Decomposition based on the flux of the probability, U~ -LnP: The two fields are not generally perpendicular, and the probability-derived potential is not necessarily non-increasing; therefore Lyapunov stability and global-metastability interpretation are not guaranteed.Perpendicularity requires an additional condition, while the general two-dimensional analysis permits a positive time derivative.
  • 3b. Decomposition based on the flux of the probability, U~ -LnP: In the zero-noise limit, the gradient and remainder become perpendicular and the Lyapunov criterion is satisfied, but attractor transitions cease and the distribution is no longer global.With vanishing noise, only local-attractor probabilities remain; thus noise cannot simply be made arbitrarily small to rescue the construction.

3c. The normal decomposition

The normal decomposition separates a vector field into a conservative potential field and remaining forces, yielding a quasi-potential framework for comparing attractor stability in nonequilibrium multistable systems. Its existence is not guaranteed, and computing it requires solving a generally nonlinear Hamilton-Jacobi equation numerically.

  • Normal decomposition: The normal decomposition requires the gradient term to be perpendicular to the remainder force.This constraint gives the decomposition a direct geometric interpretation.
  • Normal decomposition: When the normality condition holds, the potential decreases monotonically toward steady states and satisfies Lyapunov’s condition for global metastability.The potential can therefore represent global metastability in the relevant dynamical systems.
  • Biological interpretation: The quasi-potential represents the epigenetic landscape and can describe the driving force behind cell differentiation in multicellular organisms.The decomposition assigns the least-energy exit process to the gradient field while the remainder force does not contribute.
  • Quasi-potential construction: The Hamilton-Jacobi equation provides a general framework for constructing a quasi-potential that compares the relative stability of different attractors.This extends Lyapunov-style global stability analysis from individual systems to multistable nonequilibrium dynamics.
  • Limitations: Existence of the Hamilton-Jacobi solution is not guaranteed, and the proposed developmental systems require further conditions and biological verification.The authors restrict their proposal to a subset of biologically encountered systems.
  • Limitations: Nonlinear Hamilton-Jacobi equations generally lack analytical solutions, so numerical methods require careful convergence and stability analysis.This paper uses the Newton-Raphson method, described in the Supplement.

3d. The relationship between the normal decomposition and the symmetric-antisymmetric decomposition

The symmetric-antisymmetric decomposition is mathematically equivalent to the normal decomposition when the diffusion matrix is the identity. For non-identity diffusion matrices, its quasi-potential may remain a Lyapunov function but cannot compute non-equilibrium transition rates.

  • Relationship between decompositions: The decomposition represents dynamics using a symmetric operator S and an antisymmetric operator T, which can be approximated as symmetric and antisymmetric matrices in finite dimensions.D is the diffusion matrix in the governing equations.
  • Relationship between decompositions: Under an identity diffusion matrix, the symmetric-antisymmetric decomposition is mathematically identical to the normal decomposition because the quasi-potential gradient is perpendicular to the remainder.The equivalence requires setting the diffusion matrix D to the identity matrix.
  • Relationship between decompositions: When the solution exists, the quasi-potential from the symmetric-antisymmetric decomposition decreases monotonically toward steady states and serves as a Lyapunov function representing global metastability.The monotonic-decrease condition is used to assess the approach to steady states.
  • Relationship between decompositions: For a non-identity diffusion matrix, the remainder is not perpendicular to the quasi-potential gradient and contributes to state transitions.This breaks the perpendicularity condition underlying the identity-matrix equivalence.
  • Relationship between decompositions: Consequently, the symmetric-antisymmetric quasi-potential cannot compute transition rates in non-equilibrium dynamical systems when the diffusion matrix is non-identity.This limitation follows from Freidlin-Wentzell large-deviation theory as described in the section.

4b. The relationship between the Freidlin-Wentzell potential and normal decomposition

The Freidlin-Wentzell potential and normal potential are mathematically related but differ in how they treat gradient and remainder components. Within an attractor, the Freidlin-Wentzell potential is twice the normal-decomposition potential, while transitions accumulate only uphill contributions.

  • 4b. The relationship between the Freidlin-Wentzell potential and normal decomposition: The Freidlin-Wentzell potential and normal potential are defined differently but are mathematically related through a dynamical-system reformulation.The supplied passage introduces an explicit rewriting of the Wentzell potential, though the equation itself is not included.
  • 4b. The relationship between the Freidlin-Wentzell potential and normal decomposition: Least-action paths exiting attractors follow the governing dynamical equation and are driven by a landscape that reverses hills and valleys.This reversal describes the exit path’s effective landscape rather than the original landscape itself.
  • 4b. The relationship between the Freidlin-Wentzell potential and normal decomposition: The landscape reversal applies only to the gradient component, because the normal remainder retains its original direction.This distinction limits a complete reversal of the vector field during attractor exit.
  • 4b. The relationship between the Freidlin-Wentzell potential and normal decomposition: Within the same attractor, the Freidlin-Wentzell potential is exactly twice the normal-decomposition potential.The factor-of-two relationship is stated specifically for motion remaining within one attractor.
  • 4b. The relationship between the Freidlin-Wentzell potential and normal decomposition: During transitions, the Freidlin-Wentzell potential counts uphill energy, becomes zero after the saddle, and sums uphill potentials across intermediate attractors.Downhill free-fall paths contribute nothing to the Wentzell potential.

4c. Wentzell potential and spontaneous transition rate

The section relates spontaneous transition probabilities under large-deviation perturbations to relative global stability between states. It also explains that the Wentzell potential provides a least action path, while actual transition paths may deviate from it, increasingly so with larger noise.

  • Under random perturbation, a ball remains longer in the globally more stable state, determining the directionality of spontaneous transition toward the other state.
  • Wentzell potential and least action path of spontaneous transition: The Wentzell potential yields both a minimum value and a least action path between two points, providing information beyond the normal potential.
  • Wentzell potential and least action path of spontaneous transition: A state transition need not follow the least action path exactly, and larger noise permits greater deviation of actual paths from it.

5. EXAMPLES

The examples show that the normal decomposition produces quasi-potentials consistent with Freidlin-Wentzell transition barriers and supports global metastability ordering, whereas Helmholtz and noisy -lnP decompositions have important limitations.

  • 5. EXAMPLES: The examples compare Helmholtz, -lnP, and normal decompositions against Wentzell action functions to assess transition barriers and rates.The Wentzell action functions serve as the reference for correct transition rates among attractors.
  • 5. EXAMPLES: The Helmholtz quasi-potential significantly departs from the Wentzell potential and is unsuitable for calculating transition probabilities.Its gradient and remainder components are not necessarily perpendicular, and the decomposition is non-unique.
  • 5. EXAMPLES: For four attractors, the normal decomposition yields a quasi-potential that orders attractors by relative metastability and forms a transitive set in this example.The attractors are designated A, B, C, and D; transitivity is not guaranteed in general.
  • 5. EXAMPLES: The normal decomposition satisfies the Hamilton-Jacobi equation, with gradient components driving attractors and perpendicular components capturing rotational phase-space motion.Because the rotational forces are perpendicular to the gradient components, they do not contribute to transitions between attractors.
  • 5. EXAMPLES: The normal-decomposition transition barriers agree well with half the Freidlin-Wentzell action functions, while the alternative quasi-potential mostly does not.The Freidlin-Wentzell action functions are approximated numerically through least action paths because they are usually analytically intractable.

6. DISCUSSION

The normal-decomposition quasi-potential Unorm provides a non-local comparison of adjacent attractor stability through transition spontaneity, while extensions address nonadjacent attractors. However, lower Unorm indicates higher steady-state occupancy probability but ΔUnorm alone cannot predict stem-cell fate choice amid alternative transitions.

  • 6. DISCUSSION: The sign of ΔUnorm represents the relative stability of adjacent attractors with respect to spontaneous transitions between them.With noise in state variables and no explicit external driving force, the system can move toward the most probable neighboring state.
  • 6. DISCUSSION: The equivalence between potential differences and transition rates is not exact for Uprob ≈ -lnP, unlike the normal-decomposition interpretation.The supplied passage notes that the alternative potential-transition-rate relationship remains useful for practical purposes despite this limitation.
  • 6. DISCUSSION: The adjacent-attractor approach can be expanded to transitions between nonadjacent attractors separated by intermediate attractors and saddle points.Cases with more than one intervening attractor can likewise be calculated using a formula analogous to Eq. (52).
  • 6. DISCUSSION: The quasi-potential is motivated as a tool for comparing relative metastability across systems with more than two attractors and estimating efforts for cell-type reprogramming.Existing Freidlin-Wentzell and Kramer’s Law theories are described as addressing one-to-one relationships between two points.
  • 6a. ΔU and “fate choice” in multi-potent cells: At steady state, attractors with lower Unorm have higher occupation probability, but ΔUnorm alone cannot predict stem-cell fate choice when multiple differentiation transitions are available.Reaching steady state may take a long time in rugged landscapes with small noise.

6b. A global potential landscape?

This section examines whether a global quasi-potential can order attractor states by relative metastability and long-term fate choice, while noting that transitivity is not guaranteed across systems. Such a global landscape may apply to some systems but fails for biological limit cycles and lacks a simple relation to transition-path potentials.

  • Motivation: A global quasi-potential is motivated by assessing attractors’ relative metastability through their ordering under spontaneous transitions.This concerns global system behavior rather than local peri-attractor dynamics, relevant to complex biological processes such as development.
  • Global ordering: Computing Unorm(S) for any state S could potentially provide global information about long-term fate choice beyond kinetic inhibition.This contrasts with the Freidlin-Wentzell potential difference ΔV, which is defined for pairs of states.
  • Limitations: Transitivity can hold in some systems, but it is not guaranteed universally.The example 2 is described as transitive, whereas the general condition may fail.
  • Limitations: Although Δ scales monotonically with the Wentzell action difference ΔV within a basin, no simple mathematical relation connects them along trajectories between attractors.The comparison is most useful between an attractor and an exit state at a saddle, not across the full transition trajectory.
  • Applicability: A global landscape requires consistency in directed transition relationships, a condition not satisfied by systems exhibiting circadian cycles, cell cycles, and other biological limit cycles.The passage nevertheless indicates that some systems may support such a landscape, as illustrated qualitatively by Waddington’s epigenetic landscape.

Figure Legends

The figures illustrate quasi-potential landscapes, vector-field decompositions, attractor transitions, least-action paths, and potential barriers in multistable dynamical systems. They compare reconstructed quasi-potentials and transition-related quantities across several decomposition methods.

  • Quasi-potential landscape: Figure 1 schematizes a quasi-potential and transitions among stable steady states in a one-dimensional multistable dynamical system.It emphasizes that transition rates depend on quantities other than the depicted potential values alone.
  • Vector-field properties: Figure 2 shows that curl is not necessarily a driving force for a limit cycle, while divergence-free vector fields can have open trajectories.The figure also presents the governing dynamical equations.
  • Decomposition methods: Figure 3 compares vector fields and quasi-potentials reconstructed from normal, Helmholtz, -lnP, and related decompositions.Non-vanishing remainder components can make the vector field non-symmetric even when the underlying quasi-potential is symmetric.
  • Normal decomposition: Figure 4 presents the normal-decomposition quasi-potential, its gradient component, its remainder component, and calculated quantities for attractors and saddle points.The attractors are labeled A, B, C, and D, while saddle points are distinguished separately.
  • Transitions and barriers: Figures 5 and 6 examine least-action paths, quasi-potential values, and potential barriers for transitions among four attractors.Figure 5 contrasts paths between attractor pairs, whereas Figure 6 compares -lnP and normal decompositions at noise level D=20 and derives barriers between attractors.

Supplement information

The supplement describes numerical procedures for solving the quasi-potential, Fokker–Planck equation, and least-action trajectories. It covers Newton–Raphson iteration, finite-difference time stepping, and conjugate-gradient minimization of discretized action.

  • Solving Hamilton-Jacobi equation derived from normal decomposition: The Hamilton–Jacobi equation is solved numerically with Newton–Raphson iteration after specifying boundary conditions.The quasi-potential is discretized, and an iterative scheme proceeds from an initial guess until a convergence criterion is met.
  • Solving U~-lnP with finite difference method: Finite difference methods numerically solve the Fokker–Planck equation by dividing a two-dimensional square region into lattice boxes and mesh points.The scheme can be extended to n-dimensional problems and uses an Euler-type discretization for temporal stepping.
  • Solving U~-lnP with finite difference method: Boundary values are specified through Neumann or Dirichlet conditions, while Runge-Kutta time stepping can improve accuracy.The supplement states that the choice between these boundary conditions has no influence on the result in the described calculation.
  • Solving the least-action trajectory in discrete form by conjugate gradient (CG) method: The least-action trajectory is approximated by discretizing the Wentzell action and minimizing it with a conjugate-gradient method.Two attractors are initially connected by a straight line, and the number of time segments is increased from 16 to 32, 64, 128, ... until the action change falls below a threshold.
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