The Epigenetic Landscape
Waddington drew a developing cell as a ball rolling down a hillside cut by valleys, each valley a fate it could end in. The question this page asks is whether a single number can redraw that hillside. In the model run here it can, in three ways that do three different things.
A number shared by both genes enters them the same way, so it keeps the two fates evenly matched. Moving it changes how many valleys there are, and when the middle valley closes the cells split evenly between the two that remain. That evenness comes from the symmetry of this circuit: a single number that entered the two genes unequally could favour one fate. A number on one gene, such as how heavily that gene's promoter is methylated, breaks the symmetry, and its effect grows with the dose. At 0.20 on gene X almost no cell reaches the X valley even though that valley still exists: a valley can be there and still be out of reach from where the cells start. And when the marks follow the genes, the state of each gene sets its own marks and the marks reshape what that gene can do next. The cell's history becomes part of the hillside it stands on.
The biology is quoted from the two sources read, listed at the bottom with the date. The panel runs a model built on the circuit the first of them describes. Its equation form and every number in it were chosen for this page, and every result quoted here was measured from the panel's own code before the page was written. It is a teaching model; nothing in it was fitted to a real cell, and every result on it describes this circuit only.
Layer 1 of 2 · the classical picture
Two genes that hold a fate
Blood progenitor cells make a choice between two families of blood cell, and two transcription factors, GATA1 and PU.1, sit at the fork. Huang and colleagues modelled that fork in 2007 as a "gene-circuit containing auto-stimulation and cross-inhibition": each factor switches its own gene on and switches the other's off.
gene X switches itself on · gene Y switches itself on · each switches the other off
They report that this circuit "generates stable attractors corresponding to erythroid and myelomonocytic fates, as well as an uncommitted metastable state characterized by coexpression of both regulators." Three valleys: one for each fate, and one in the middle where both genes are on and the cell has not yet chosen. Commitment, in their account, happens in two stages: "first, the progenitor state is destabilized in an almost symmetrical bifurcation event", then the cell is driven into one of the two fates. The middle valley closes, and the cell has to go somewhere.
What a methyl mark does
DNA can carry a methyl group on the C of a CG pair. Near the start of a gene, these marks shut the gene down. Bird's 2002 review: "A small but significant proportion of all CpG islands become methylated during development, and when this happens the associated promoter is stably silent."
The marks are copied when the cell divides. The enzyme that does it, DNMT1, "prefers to methylate those new CpGs whose partners on the parental strand already carry a methyl group," so a daughter cell inherits the mother's pattern. And they can be removed by drugs: "both X inactivation and retroviral silencing can be relieved by treatment of somatic cells with demethylating agents."
So a gene's methylation acts as a number sitting on that gene, set by the cell's history, kept through division, and able to hold the gene off.
Drawing a hillside from a circuit
Waddington's hillside was a drawing (see the Waddington page on this site). To get a hillside out of a circuit, the panel does the following: it computes the long-run chance of finding a noisy cell at each pair of gene levels, and draws the height as minus the logarithm of that chance. Where cells crowd, the ground is low. Where they almost never go, it is high. Wang and colleagues used this construction in 2011 for circuits of this kind; the page did not read their paper and takes nothing else from it.
A caution that matters later: this hillside describes where cells end up, and a cell does not simply roll straight downhill on it. Part of the motion goes around. The Honest limits section says how much.
The hillside, run here
Four hundred cells, each running the same two-gene circuit with its own small random noise. One number, self-activation, sets how strongly each gene keeps itself on, and it is shared by both genes. Two more numbers are methylation levels, one on each gene's promoter; a gene with methylation m makes only (1 − m) of what it would otherwise make. Fifteen units of model time pass every second.
change in X = (1 − mX) [ a·on(X) + off(Y) ] − X
change in Y = (1 − mY) [ a·on(Y) + off(X) ] − Y
Here on(v) rises steeply from 0 to 1 as v passes 0.5, and off(v) is 1 − on(v). The first term is the gene switching itself on, the second is the other gene failing to switch it off, and the last is the protein breaking down.
How to read the picture. The ground is the hillside for the current settings: low, warm ground is where cells collect, and the flat dark top is ground they practically never reach. Gold rings mark the bottoms of the valleys that exist at these settings, each labelled with the gene levels it holds. The white dots are the cells. Drag to turn it.
How to read the graph. The share of cells in each valley over recent time: gold for both on (uncommitted), red for X only, blue for Y only. Vertical ticks mark something you did: gold for the shared number moved, red or blue for a methylation slider or a push, white for marks erased or cells started over.
Four things to try.
- The shared number. Slide self-activation down from 1.20 to 0.60. The middle valley flattens and closes, and the cells pour out, about half each way. Slide it back up: the middle valley returns, and most cells stay where they went.
- A number on one gene. Start the cells over, uncommitted, at 1.20. Raise the methylation on gene X to 0.30. Every cell goes to Y, even though the X valley is still there.
- A push. With cells committed and marks off, push toward Y. The X cells are lifted out of their valley, and most land back in the middle.
- Marks that follow the genes. Tick "marks follow the genes", commit the cells, wait half a minute, and push again. Nothing moves. Set the landscape to a cell in the X valley to see why: for that cell, the valley it would be pushed toward is gone. Erase all marks and push again.
Measured from the panel
| what was done | what happened |
|---|---|
| self-activation varied, valleys counted from the equations | three valleys above 0.7745, two below it; the middle valley is the one that goes |
| cells left uncommitted for 300 time units | at 1.20, 400 of 400 stay; at 1.00, 325 stay; at 0.90, 1 stays. With noise, cells leave the middle valley well before it closes |
| self-activation dropped to 0.50 | 95% committed within 7 time units; the split between X and Y was 196/204, 212/187 and 202/198 over three runs |
| raised back to 1.20 after commitment | 32 to 41 of 400 returned to the middle in 100 time units; raised to 1.40, 158 to 166 returned in 200 |
| methylation on gene X raised, cells uncommitted | 0.10: no change. 0.20: 323 to 330 go to Y, none to X. 0.30 and above: all go to Y. The middle valley closes at 0.281; the X valley itself survives until 0.624, and no cell enters it |
| push toward Y on cells committed to X, marks off | 0 of the 181 to 191 X cells still in X afterwards; the middle rose from 32 to 41 cells to 237 to 245 |
| the same push with marks following the genes, after 400 time units | 0 cells moved in three runs; after 50, 100, 150 and 200 time units, also 0 |
| marks on the silent gene of an X cell, over time | 0.27 at the moment of commitment, 0.54 after 100 time units, 0.69 after 400 |
| all marks erased, then pushed toward X | 0 cells left in Y |
| marks never allowed to follow the genes; the same 0.69 written by hand onto each committed cell's silent gene, then pushed toward Y | 0 cells moved in three runs, the same as when the cells wrote the marks themselves |
| the same, with 0.20 written by hand | 100 to 121 of the 181 to 191 X cells stayed in X; the rest were lifted into the middle |
Three rows are worth reading twice. A valley can exist and be out of reach. With 0.20 on gene X, 323 to 330 cells go to Y and none to X, while the X valley is still fully there: the mark tilts the ground the cells cross when they leave the middle, so the route from the middle to X is lost before X itself is. The marks lock a cell in quickly: by the time they reach 0.28 the middle valley is gone for that cell, so a push lifts it up a slope with nowhere to settle, and it rolls back. What locks the cell is the value the marks reach. Marks of 0.69 written by hand lock the cells exactly as well as marks the cells wrote over 400 time units. History matters in this model only through the marks it leaves.
Layer 2 of 2 · the framework's reading
The wrapped reading
The Shape of the Basin
At the default settings (self-activation 1.20, noise 0.008) the hillside has three basins, and the middle one is the deepest: the lowest pass out of it is 12.4 height units above its floor. The two committed basins have floors 6.5 units up, with 5.9 units to climb to the same passes. That is why, at 1.20, committed cells drift back into the middle (32 to 41 of 400 in 100 time units) and uncommitted cells do not leave it. At 1.00 the order reverses: the committed floors are now the lowest ground and the middle floor sits 0.8 above them. At 0.90 the middle floor is 4.0 up with its rim only 2.6 above that, and at 0.7745 the floor meets the rim and the basin is gone. Mark gene X and the whole hillside leans toward Y, while the X basin keeps a floor of its own until the mark reaches 0.624.
With marks following the genes, a committed cell's marks keep growing (0.27 at commitment, 0.69 after 400 time units), and the middle valley it left closes once they reach 0.28. Recovery is then measured the plain way: push the cells and count how many are back in the valley they left. With marks off, 0 of the X cells return to X after a push toward Y. With marks on, all of them do.
The Morph of the Basin
How the set of basins changes as each number is moved, at noise 0.008:
| setting | basins | where uncommitted cells go |
|---|---|---|
| self-activation 1.20 | 3 | they stay in the middle |
| self-activation 1.00 | 3 | most stay (325 of 400 after 300 time units) |
| self-activation 0.90 | 3 | almost all leave, evenly |
| self-activation below 0.7745 | 2 | all leave, evenly |
| X methylation 0.20 (at 1.20) | 3 | most go to Y, none to X |
| X methylation 0.30 | 2 | all go to Y; the X basin exists and stays empty |
| X methylation above 0.624 | 1 | all go to Y; only the Y basin is left |
What the morph can and cannot say here. The table is drawn inside a model whose genes, rates and noise are all chosen. In this symmetric circuit, the shared self-activation number changes how many basins there are and how stable they are, without favouring X over Y. Methylation on one gene breaks that symmetry, and with growing dose it decides which basin the cells can reach. The table says nothing about how many numbers a real cell uses, or which, and nothing about global numbers in circuits that are not symmetric.
What would show this reading wrong
Against using this model for the biology (tested in cells). The model puts the effect of methylation into its equations, so it cannot fail these tests internally; cells can.
Cells at the GATA1 and PU.1 fork whose decision splits evenly no matter how heavily the promoter of one of the two genes is methylated. The model's methylation term would then not describe that system.
Committed cells whose silent gene carries heavy methylation and which still switch fate readily when pushed. The model would then not be an adequate description of those cells. That result alone would not say why: the methylation may not suppress the gene enough, another route may exist, other regulators may compensate, or the hillside picture may not fit.
Against the framework's reading of the model (tested inside the model).
Committed cells that stay put after a push with every mark erased. The lock would then not come from the marks, and the claim that the cell's own marks close the route back would fail. Measured: with marks erased, the push moves every committed cell.
Hand-written marks that lock the cells differently from marks of the same value the cells wrote themselves. The claim that history acts only through the marks it leaves would then fail. Measured: at 0.69 both lock all cells.
Committed valleys that vanish when self-activation is set to zero, or a middle valley that survives it. The account of which loop holds which valley would then be wrong. Measured: at zero the two committed valleys remain and the middle one is gone.
Honest limits
A hillside only tells part of the story. The flow in this model does not run straight downhill on any surface. The best surface that can be fitted to the flow leaves part of it unexplained: what is left over, the part that circulates, is 18% to 28% of the size of the flow (measured across self-activation 0.60 to 1.20). That best-fit surface puts the committed valleys at 1.73 when the model's resting states are at 2.20, and at 0.60 it keeps a middle valley that the model does not have. The panel therefore draws where cells end up, which puts every valley in the right place, and does not claim that cells roll down it along the steepest line.
Two genes stand in for many. A real methylation mark is on or off at each site in each cell. Real marks are laid and removed by enzymes with their own regulation; here they follow expression at one fixed rate. Histone marks, which also shape which genes can switch on, are left out.
Not read: Wang, Zhang, Xu and Wang (2011), PNAS, on drawing a landscape as minus the logarithm of where cells settle, returned an error. Taylor and Jones (1979), Cell, titled "Multiple new phenotypes induced in 10T1/2 and 3T3 cells treated with 5-azacytidine", a demethylating drug, could not be reached. Nothing on this page rests on either.
Sources
Huang, S., Guo, Y.-P., May, G., & Enver, T. (2007). Bifurcation dynamics in lineage-commitment in bipotent progenitor cells. Developmental Biology, 305(2), 695–713. Abstract read. Source of the circuit of auto-stimulation and cross-inhibition, the two fate states and the uncommitted state, and the two-stage account of commitment. doi.org/10.1016/j.ydbio.2007.02.036
Bird, A. (2002). DNA methylation patterns and epigenetic memory. Genes & Development, 16(1), 6–21. Source of promoter silencing by methylation, the copying of marks through division by DNMT1, and the relief of silencing by demethylating agents. genesdev.cshlp.org/content/16/1/6
Related on this site: Life & Physiology; Waddington; The Shape of a Rule, where one number per cell sets a body's shape; and Homeostasis, where the body moves its own target.