Contribution
Bistable switches in bacteria were known one circuit at a time (the lac and mar operons, sporulation, Caulobacter stalking), and each is thrown by an environmental cue. This paper asks the question for a whole reconstructed regulatory network and for genetic rather than environmental perturbation: can switching one gene off or on for a while, and then restoring it, leave the rest of the network in a lasting new state? Across an ensemble of plausible Boolean rules it finds that this is common in the core of the E. coli network, says which genes can do it (those that reach a positive circuit), and gives a structural predictor of how likely it is. It ends with a concrete experiment nobody has yet run.
Key insight
A transient perturbation is remembered when it flips a positive circuit that then holds itself in the new state after the cause is gone. So the ability of a gene to cause irreversibility is a matter of where it sits: it must reach a strongly connected component that contains a positive circuit, and the more independent routes it has to such components the likelier it is to flip one. Genotype then fixes a set of possible expression states, not one, and history picks among them.
Assumptions
- The network is RegulonDB’s union of potential interactions, with polarity taken as known and presence allowed to vary by condition. Edges of dual or unknown sign (148) are dropped.
- Boolean, synchronous, deterministic dynamics, x_u^{t+1} = B_u(x^t), with three consistency constraints: each rule uses exactly the gene’s regulators, each regulator is essential, and each enters with its sign (monotone in each input). Autorepression is silenced when the gene is off, to exclude artefactual oscillation; this breaks essentiality for some edges. phoB, the only core gene with no input, is given a self-loop.
- Rule ensemble. Rules are sums of products built left to right over the inputs, sorted by their out-degree (ascending: “diffuse control”; descending: “concentrated control”), joining consecutive inputs by ×( with probability r, + with probability (1 − r)s, × with probability (1 − r)(1 − s). M = 20 realizations per non-unique (r, s) setting, on grids along s = 1, r = 0 and r = 1 − s.
- The perturbation protocol. From the first recorded point of each attractor 𝒜, one gene is clamped (1 → 0 for knockout, 0 → 1 for overexpression), the network runs to a new attractor, the clamp is released at the first point reached on it, and the network runs to a final attractor 𝒜′. Irreversible means 𝒜′ ≠ 𝒜. Every attractor counts equally unless stated (basin weighting is a robustness check).
- Only the core matters. Genes outside the 87-gene core cannot affect core genes and change reversibly downstream, so they are trimmed. The largest origon is taken to stand for the whole network: Table S1 shows the other large origons’ cores overlap it almost entirely.
The setting is narrower than the title. “Irreversibility” is a change of attractor in a deterministic model, permanent by construction. Real cells are noisy and divide; the authors expect “temporarily irreversible but long-lived” changes, and how long they last is left to experiment.
Key results
- Network reduction. 1859 genes and 5119 edges; the phoB origon has 1406 genes; its core has 87 genes and 290 edges. The number of consistent rule sets is at least ∏_u 2^{k_u⁺ − 1}, about 10
- Attractor counts. The geometric mean is about 10² attractors under concentrated control and 10³ under diffuse control, largest at intermediate nestedness. Of attractors, 42.2% are fixed points, 24.4% period 2, 1.1% period 3, 27.9% period 4, 4.4% longer.
- Who can be irreversible (Fig. 5). 51 core genes admit an irreversible perturbation for some rules; every gene influencing a positive circuit does. The 36 that never do are 32 leaves and 4 genes (metJ, pdhR, argP, nac) whose only output is an autorepressive leaf. The ensemble estimate is within about 0.1 of its limit (RMSD between independent ensembles of 10 realizations, fig. S2).
- Structure predicts propensity (Fig. 6). With path weight ω(H) = ∏{i=2}^{ℓ} (k⁺{H_i})^{−1} and K_u the sum over paths from u into strongly connected components with a positive circuit, p̂_u = aK_u^b explains 55–62% of the variance in irreversibility probability; b = 0.68 ± 0.09 (diffuse), 0.93 ± 0.14 (concentrated). The weight is the probability of a path transmitting a change if each input is equally likely to flip its target and changes are independent.
- Conditions. Necessary: some gene other than the perturbed one has changed by the time the clamp is released. Sufficient: the released state lies outside 𝒜‘s basin. Both checked in the simulations.
- Basin direction (fig. S4). Weighting each starting attractor by its basin keeps the pattern (R² > 0.91 against unweighted) but roughly halves the rate; initial basins average about one-eighth the size of final ones.
- Rule dependence (Fig. 7). For hns, stpA, crp, rcsB, leuO, bglJ, rhaR, rhaS irreversibility varies monotonically with rule bias, and knockout and overexpression irreversibility are anticorrelated: a gene is more irreversible under the perturbation applicable in more attractors. Hubs upstream of many components (phoB, cra, fis) lose irreversibility at intermediate nestedness, genes inside large components (fnr, fur, fliZ, gadX) gain it.
- Transition types (Supplementary Text). 49.0% of irreversible transitions are between fixed points, 31.7% between partial fixed points differing on their fixed genes, 2.2% fixed point to partial fixed point, 17.1% between partial fixed points with different fixed sets. In every case the attractors differ on a gene fixed in at least one of them. At least 67.9% of transitions keep their initial and final attractors under asynchronous updating.
- Evolved crp-knockout strains (Fig. 8, Tables S2–S3; RNA-seq, GEO GSE152214). The predicted sign of change, minus the polarity of the shortest path from crp, matches the observed log fold change (relative to the mean shift) for 10 of 11 genes with |ln| > 0.5 in batch and 33 of 42 in chemostat; average precision 0.99 and 0.85, each P < 0.01 by shuffling the predicted signs (25,000 shuffles). Among strongly changed genes, irreversible responders matched the predicted sign 8 of 9 times and reversible genes 2 of 2 in batch, 28 of 34 against 5 of 8 in chemostat; P = 0.03 for both conditions together.
- Candidates (figs. S5–S7). Weighting attractors by their similarity to 16 observed instances of crp downregulation in public RNA-seq points to self-activating genes positively regulated by crp. Candidates: zraR, melR, rhaRS. A HillCube model of the motif gives a bistable range that widens with self-activation strength and Hill coefficient.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | In the model, transient single-gene perturbations of the E. coli core network often leave it in a different attractor | strong for the model ensemble | simulations over the rule ensemble, Fig. 5, with convergence check fig. S2 |
| C2 | Only genes that can reach a strongly connected component containing a positive circuit can cause irreversibility | strong (argument plus simulation) | Supplementary Text, with Thomas’s rule that positive circuits are necessary for multiple fixed points (ref. 38) |
| C3 | Propensity rises with the weighted number of paths to such components | moderate | power-law fit, R² 0.55–0.62, Fig. 6 |
| C4 | Irreversible transitions run from small basins to large | moderate | basin-weighted re-analysis, fig. S4 |
| C5 | The conclusions hold under asynchronous updating and in continuous models | weak to moderate | fixed points are preserved (cited); partial fixed points preserved in at least 46.3% of cases by a sufficient condition; one continuous motif |
| C6 | The signed network predicts the direction of expression change after adaptive evolution to a crp knockout | moderate | sign concordance, Fig. 8B, P < 0.01 in each condition |
| C7 | Genes that respond irreversibly to a transient crp knockout are those that change in adaptive evolution to a permanent one | weak | counts of 9 and 2 (batch), 34 and 8 (chemostat), P = 0.03 jointly |
| C8 | Transient crp knockdown will cause lasting expression changes in zraR, melR or rhaRS in cells | untested | a prediction; no experiment reported |
Method
Trim the RegulonDB network to the core of its largest origon by repeatedly deleting genes with no outgoing edges. Sample rule sets with the (r, s) algorithm and two input orderings. Find all attractors of each rule set with a SAT-based method (BoolNet in R). From each attractor, clamp each gene off and on, run to an attractor, release, run to the final attractor, and record whether it changed and which genes differ. Average over attractors weighted by how often each gene is on (eq. 4). Relate the averages to path counts into positive-circuit components, then compare predicted signs and irreversible-responder sets with RNA-seq of evolved strains by permutation tests.
Concepts
- irreversible perturbation: a transient knockout or overexpression after whose removal the network reaches an attractor different from its starting one.
- irreversible response gene: a gene whose state differs between the initial and the final attractor.
- origon: the subnetwork reachable from a root gene with no inputs other than itself.
- core network: what remains of an origon after recursively removing genes with no outgoing edges.
- positive circuit: a directed cycle whose product of edge signs is positive (an even number of repressions).
- canalization depth: the expected number of input states needed to fix a rule’s output; the paper’s measure of nestedness.
- rule bias: the probability that a rule outputs 1.
- concentrated / diffuse control: input orderings in which the regulator with the largest / smallest out-degree tends to canalize.
- partial fixed point: a periodic attractor in which some genes are constant.
Connections
The necessity of positive circuits for multiple stable states is René Thomas’s rule, cited (ref. 38) rather than proved anew; the Supplementary Text adapts it to partial fixed points. The rule ensembles relate to random Boolean networks with bias (Kauffman’s line, refs. 29–30) and to threshold networks (ref. 31), and the paper says how each corresponds to an (r, s) setting. The irreversibility argument is the bacterial counterpart of eukaryotic cell-fate commitment, without chromatin marks. Steering between attractors is the network-control programme of the senior author’s group (refs. 49–51). The evolution data are from an earlier adaptive-laboratory- evolution study of crp knockouts (ref. 41).
Bearing on the record
- THEORY-136. That standard says a lasting shift after a temporary disturbance, with slow return excluded, is among the few kinds of evidence that come close to showing alternative stable states, and that positive feedback must be strong to produce them. This paper defines irreversibility as exactly that kind of shift, predicts it across a whole regulatory network, and its continuous model agrees that self-activation must be strong and switch-like. But by that standard it has not yet shown it: the evidence is models, and the one data comparison is evolution under a permanent knockout, which is not a temporary disturbance. Its proposed CRISPRi experiment is the kind THEORY-136 asks for, and the authors’ own caveat (noise turns permanent into long-lived) is THEORY-136’s slow-return confound.
- It produces THEORY-178: that regulation alone can make a bacterium’s expression state depend on its history, with positive circuits necessary, filed Proposed with that experiment as its promotion condition.
- THEORY-058 (disorders as self-maintaining symptom loops) and the record’s ecological alternative-state works (LIT-726, THEORY-132) share the mechanism: a strongly connected positive loop that outlasts its trigger. This is my connection, not the paper’s; it adds a molecular case and changes none of them.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- No transient perturbation was done in cells. C1–C5 are about the model ensemble; C8 is a proposal.
- The data comparison tests something else. C6 tests the signed network’s prediction of the direction of change after a permanent knockout and evolution, which needs no multistability. C7 is the only test touching irreversibility, and it is weak: 68 of the 86 other core genes are irreversible responders to crp knockout, so the reversible comparison groups are tiny (2 genes in batch, 8 in chemostat), and in batch the reversible genes matched at a higher rate (2 of 2 against 8 of 9). The paper’s own statement that evolving after a permanent knockout is “akin to” releasing a transient one is an intuition, not an argument.
- The p-value formula as printed, P = 1 − N_exc/N_samp with N_exc the number of shuffles exceeding the observed statistic, would give a large P for an extreme observation; it reads as a slip for N_exc/N_samp, and the reported values are consistent with the latter.
- The necessary and sufficient conditions are close to definitional. “Some other gene changed while clamped” and “the released state is outside the original basin” restate irreversibility in a deterministic map rather than give a structural test; the structural content is in C2 and C3.
- Synchronous updates produce many period-2 and period-4 attractors; about a third of transitions are not shown to survive asynchronous updating.
- Equal weighting of attractors, most of which may never be visited by a cell; the basin and transcription-weighted analyses address this only partly.
- The rule family is restricted to nested sums of products built in one input order; the true rules are unknown, and RegulonDB’s network is a union over conditions.
Open questions
- Does a transient crp knockdown, released, leave zraR, melR or rhaRS on for many generations in cells, with the environment held fixed and slow return ruled out? That experiment would settle C8 and THEORY-178.
- How long do the predicted states last under noise and division? A stochastic model of transcription and translation mapped from the Boolean transitions would give lifetimes.
- Is the small-to-large basin direction a general property of such networks or of this rule ensemble?
- Would a direct comparison, transient perturbation against permanent perturbation plus evolution in the same strain, support the paper’s “akin to” link between them?