We first illustrate our quantitative approach with a simple circuit of two genes x and y that inhibit each other, forming a double-negative feedback loop structure (see Methods). This circuit works as a toggle switch with two stable steady states: one state with high y and low x expression, and the other state with high x and low y expression 40 . Such "bistable" switches formed by mutual antagonism of a pair of key regulatory genes underlie many binary cell fate choices 7 13 . The circuit can be described as a two-variable dynamical system, with the rate of change in expression of each of the two genes given as a function of their expression levels:

If we were able to derive a closed-form potential function V(x,y) for the system in Eq. 1 that satisfied the conditions:

then the local minima on the two-variable potential surface V(x,y) would correspond mathematically to the stable steady states of the system, given that at the local minima on the surface (∂V/∂x = 0; ∂V/∂y = 0), the rates of change in expression of both genes x and y would be zero (per Eq. 2). But such a closed-form potential function can be derived only in the case of a gradient system, defined by the condition 41 :

In general, condition (3) will not be valid for an arbitrary circuit of two genes x and y that regulate each other as per Eq. 1, making it impossible to derive a closed-form potential function.

Therefore, given that a gene circuit is in general a non-gradient system, we define a term V _q that changes incrementally along a trajectory followed by the system in x-y phase space (Figure 2A) as follows:

where Δx and Δy are sufficiently small increments along the trajectory such that and can be assumed to remain unchanged over the interval [(x, x+Δx); (y, y+Δy)]. The quantities Δx and Δy are obtained as the products and , respectively, where Δt is the time increment. We use the term "quasi-potential" to describe V _q , to emphasize its distinction from a closed-form potential function.

The change in the quasi-potential, Δ V _q , can be rewritten from Eq. 4 as:

For positive increments in time Δt, Δ V _q is thus always negative along an evolving trajectory, ensuring that trajectories flow "downhill" along a putative "quasi-potential surface". Stable steady states of the system (dx/dt = 0; dy/dt = 0) would correspond to local minima on this quasi-potential surface, given that at these states Δ V _q = 0 (per Eq. 5). The overall change in the quasi-potential along a trajectory can then be calculated by numerically integrating the quantity Δ V _q in Eq. 4 from a given initial configuration up to a stable steady state, thereby allowing us to map out a temporal trajectory along the putative quasi-potential surface (Figure 2B). The quasi-potential thus defined is a measurable quantity that is minimized along a trajectory from any initial condition to an attractor in the phase space of the two genes, and is in effect a Liapunov function of the dynamical system represented by the two-gene circuit 41 .

The procedure described above was repeated to evaluate the change in the quasi-potential along trajectories originating from different points in x-y phase space. To derive a quasi-potential surface from multiple trajectories, we then make the following assumptions: (i) two trajectories with different initial conditions that converge to the same steady state must also converge to the same final quasi-potential level (Figure 2B); (ii) two trajectories that originate from "adjacent" initial conditions that are sufficiently close in x-y phase space, but converge to different steady states, must start from the same initial quasi-potential level (Figure 2C). Observation (i) allows us to map out a basin of attraction from multiple trajectories converging to a single steady state; while observation (ii) enables the alignment of two adjacent basins of attraction along their shared basin boundary, or separatrix. (Essentially, (i) and (ii) together amount to the assumption that the putative epigenetic landscape is continuous.) The quasi-potential surface can then be obtained by interpolation among the aligned trajectories (Figure 2D, E), yielding the epigenetic landscape with its characteristic ridges and valleys (Figure 3A, B).

The same procedure can be applied to systems with more than two stable steady states - for instance, a "tristable" system produced by a circuit of two genes that induce their own expression, in addition to mutual inhibition (Figure 3C, D). This system has three steady states - two of which represent alternative differentiated cell lineages, while the third state depicts the common progenitor cell of the two lineages 8 13 42 .

To establish that the "elevation" of the computed landscape at a given location in x-y phase space correlates inversely to the probability of occurrence of the corresponding network state, we used stochastic simulations 43 of the underlying gene network. These simulations, which take into account fluctuations in gene expression levels 44 in a population of simulated "cells" (multiple instances of the gene network), showed that the "valleys" of low elevation on the computed epigenetic landscape correspond to stable cellular states, with "deeper" valleys associated with higher probability of occupancy than shallower valleys (Figure 4A-D). On the other hand, the "ridges" separating the valleys represent barriers to stochastic transitions between multiple steady states. Varying the parameters in the network model to increase the height of the ridges relative to the valleys dramatically reduced the probability of transitions between the steady states (Figure 5), even though there was no appreciable change in the relative distance between the steady states on the x-y phase plane (Figure 4A-D, right panels).

The "third dimension" (elevation) of the landscape represented by the quasi-potential, although directly derived from the dynamic rate equations without any additional information, thus yields an interpretation of cellular stability not immediately apparent from two-dimensional phase portrait analysis. The analysis above supports the contention that the length of the "least action trajectory" along the contours of the epigenetic landscape is more important in predicting transitions between alternative cellular states than the simple "aerial distance" in state space 13 . It is also interesting to note that the contours of the quantitatively mapped epigenetic landscape act as a constraint on the extent of stochastic fluctuations in protein levels, with simulated cells "smeared out" on the surface of a shallower basin (Figure 4A, middle panel) compared to a tighter distribution of cells on a deeper valley (Figure 4D, middle panel).

These results suggest that calculating the relative heights of the ridges and valleys on the computed epigenetic landscape of a multi-gene system can help predict the probability of trans-differentiation from one cell lineage to another, or de-differentiation of a particular cell type to its progenitor state. Current efforts to reprogram cell fate with potential application in regenerative medicine suffer from a low rate of successful reprogramming 36 and a trial-and-error approach to choice of a reprogramming strategy 13 . Computing the epigenetic landscape for the critical gene interactions regulating the transition between two cellular states may indicate particular genetic manipulations that would lower the barriers separating the two states, thereby increasing the efficiency of the reprogramming process. It can also help characterize the relative ease or difficulty of alternative routes of cell fate transition 7 9 36 . For instance, comparison of the elevation of the barriers separating two terminally-differentiated cell lineages on the epigenetic landscape might suggest that de-differentiation of cells of one lineage to the common progenitor cell of the two lineages followed by redirection to the second lineage would lead to more efficient reprogramming than direct trans-differentiation (Figure S2, Additional File 1).

The computed epigenetic landscape derived above should not be interpreted as a static surface 45 . Alterations in gene interactions in course of development or experimental manipulation will change the shape of the landscape, in turn altering the stability of individual steady states or creating novel steady states. For instance, increasing the basal expression of one gene in the tristable gene network sharply lowers the elevation of the corresponding attractor state relative to the other attractors (Figure 6A, B). As a result, cells located in the shallower attractor are destabilized and "roll into" the valley representing the deeper, more stable attractor state. This may explain the phenomenon of trans-differentiation of cells of one lineage into another by forced expression of a gene regulating the second lineage or by conditional deletion of a gene required for the first lineage 29 30 .

Interestingly, this flexibility of the quasi-potential surface under gene manipulation gives a quantitative interpretation of the revised image of the epigenetic landscape proposed by Waddington (Figure S3, Additional File 1), which showed an array of pegs representing genes, holding up a sheet of fabric (the landscape) through a network of guy ropes (gene interactions) - meant to convey the idea that "the modelling of the epigenetic landscape ... is controlled by the pull of these numerous guy-ropes which are ultimately anchored to the genes 5 ". Similar changes in the shape of the epigenetic landscape may also be brought about by external signals - for example endogenous cytokines or environmental chemicals - which by transiently altering the landscape could have an instructive effect on cell fate choice.

To illustrate the derivation of the epigenetic landscape, we used a simplified mathematical model of a bistable network of two genes, x and y, that suppress each other to form a double-negative feedback loop. The dynamics of the model are described by the two rate equations:

where variables x and y represent the concentrations of the two gene products, and parameters B _X and B _Y denote the basal (constitutive) expression rates of genes x and y, respectively. The parameters fold _YX and fold _XY represent the rate constants, and K _DYX and K _DXY the effective affinity constants, for the suppressive effects of gene y on gene x, and of gene x on gene y, respectively. The mutual suppression of the two genes is quantified by the Hill-coefficient n _H (the interaction is ultrasensitive for values of n _H > 1). Parameters deg _X and deg _Y represent the first-order degradation rate constants for the two gene products x and y, respectively. For this simplified model we used dimensionless parameters with the following values: fold _YX = fold _XY = 2; K _DYX = 0.7; K _DXY = 0.5; B _X = B _Y = 0.2; deg _X = deg _Y = 1; n _H = 4. These values were tuned to ensure bistable switching behavior in the model.

The tristable network model consisted of two genes, x and y, that in addition to mutual suppression, induce their own expression (positive autoregulation). The dynamics of this model are described by:

where the new parameters fold _XX and fold _YY represent the rate constants, and K _DXX and K _XYY the effective affinity constants, for the positive autoregulation of genes x and y, respectively. The default parameter values chosen to ensure three robust stable states in this model were as follows: fold _XX = fold _YY = fold _YX = fold _XY = 10; K _DXX = K _DYY = K _DYX = K _DXY = 4; B _X = B _Y = 0; deg _X = deg _Y = 1; n _H = 4. This system has been modeled previously 42 48 49 in the context of mutual inhibition of the transcription factors PU.1 and GATA1 in common myeloid progenitor (CMP) cells, which gives rise to either bipotential granulocyte/macrophage progenitor (GMP) cells or megakaryocyte/erythroid progenitor (MEP) cells.

To evaluate the change in the quasi-potential along each trajectory in x-y phase space by numerical integration, the initial level of the quasi-potential at time t = 0 at the origin of the trajectory was arbitrarily set to zero (the same initial quasi-potential level was used for all trajectories so that the drop in the quasi-potential along each trajectory could be compared and used as a basis for alignment of multiple trajectories along a basin of attraction).

Thereafter, at each time step:

• The rates and were updated to the current value of x and y according to Eqs. 8 and 9.

• Expression levels x and y were updated as:

where for increments in time Δt (fixed for a simulation to ensure convergence), the changes in x and y are given by:

• The quasi-potential V _q was updated as:

where:

The above steps were repeated until the quasi-potential V _q converged to a minimum (decided by a pre-set tolerance). Multiple trajectories thus obtained were aligned into basins of attraction according to the process described in the main text. The quasi-potential surface was then derived by linear interpolation among the aligned trajectories.

The deterministic models were implemented and simulated on the MATLAB^® (R2009a, The MathWorks, Inc., Natick, MA) platform, while the BioNetS program 50 , based on the Gillespie algorithm 51 52 , was used for stochastic simulations. All graphics were rendered on MATLAB^®.

The stochastically simulated "cells" (i.e. individual realizations of the stochastic network model) were overlaid on the quasi-potential surface at the x and y values predicted for each cell. Since stochastic simulations yield integral values, we added a small random "deviation" term [= (rand*0.5) where rand is a MATLAB^® function that draws pseudorandom values from the standard uniform distribution on the open interval (0,1)] to each simulated x and y value to visualize multiple cells situated at the same point in x-y phase space. The appropriate "elevation" for each cell on the quasi-potential surface was calculated by linear interpolation between the two points on the deterministic trajectories "closest to" the location of the cell in x-y phase space. Source code for the model in MATLAB^® format is appended in Additional File 2.