{"id":"eed5cabf-1da5-42fc-9499-d3af367969aa","arxiv_id":"2502.10067","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Slow, nonadiabatic chromatin-state transitions generate circular probability currents, hysteresis, and an entropy-production peak that can tune gene-expression fluctuations, explaining the large Nanog heterogeneity in mouse embryonic stem cells.","lead":"This paper uses computer simulations of gene circuits to show that when changes in the DNA packaging around a gene are much slower than changes in the protein it produces, the system develops circular probability flows, hysteresis, and maximum energy dissipation at an intermediate timescale ratio. The model suggests that differences in these timescales between genes can explain why Nanog, but not Oct4 or Sox2, fluctuates strongly in mouse embryonic stem cells.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central entropy-production and hysteresis claims rest on a continuous diffusion approximation of discrete chromatin states (Eq. 4) that is never checked against the original jump master equation; if the approximation fails, the simulated peak in Fig. 9D needs re-evaluation.","rationale":"The reader's weakest-assumption analysis identified exactly the same load-bearing concern: the continuous Langevin treatment of discrete chromatin states is unvalidated against the underlying jump master equation. My independent reading agrees that this is the single most important threat to the paper's central claims, because everything that follows — the landscape pictures, the circular probability currents, the entropy-production maximum in Fig. 9D, and the hysteresis interpretation via the cross-correlation A(t) — is computed from the continuous approximation of Eq. 4. If the approximation does not reproduce the original discrete-state process, then the model's quantitative statements (especially the location and existence of the entropy-production peak) are not established.\n\nThe paper is otherwise internally consistent and does not contain obvious algebraic errors; the phase diagrams and hysteresis trends are plausible qualitative behaviors that could survive a discrete-state check. But the check is absent. The appendices test monomer vs. dimer TF assumptions and two-state vs. three-state chromatin transitions, yet they never test the discrete vs. continuous description of the same three-state process, which is the more foundational assumption. The mES application (Sec. IV) is a fit of omega values to observed heterogeneous fluctuations, so it provides no independent confirmation of the continuous approximation.\n\nMy proposed test is concrete and feasible: simulate the exact jump process with the same rate constants and compare stationary distributions, cross-correlations, and entropy production. This would settle whether the diffusion approximation changes the qualitative predictions. If the test passes, the paper's conclusions survive; if it fails, the central claims need to be re-derived for discrete chromatin states. Either outcome is compatible with the reader's conditional verdict, so I recommend leaving the verdict as CONDITIONAL, i.e., no change from the reader's judgment.","tokens_in":26943,"tokens_out":3649,"duration_ms":37551,"concrete_test":"Simulate the exact continuous-time Markov chain for Circuit A with the same parameters as in Figs. 4 and 9: y in {-1,0,1} with transition rates r^{yy'}_i(x) from Eqs. 2-3 and Table II, and protein concentration p as a birth-death process with synthesis rate g_xy and degradation rate k, using Gillespie or Monte Carlo sampling. For omega in 0.001 to 100, compute the stationary distribution, the cross-correlation A(t) of Eq. 8, and the exact jump-process entropy production rate. Compare with the Langevin results: if the entropy-production peak in Fig. 9D or the eddy-regime peak in A(1/omega) shifts, disappears, or changes substantially, the central claims are artifacts of the continuous approximation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central variables y_i (and later z_i) are genuinely discrete three-state chromatin variables with transition rates r^{yy'}_i(x) given by Eqs. 2-3. Equation 4 replaces each y_i by a continuous variable on [-1,1] using a saddle-point/path-integral Gaussian approximation, effectively a system-size expansion. For a Markov chain that visits only three states, there is no large parameter making the Gaussian noise small; the truncation at second order is uncontrolled. Consequently, the stationary distribution, the circular probability currents of Eq. 7, and the entropy-production rate of Eq. 9 may differ systematically from the exact jump process.\n\nThe entropy-production expression in Eq. 9 is the diffusion form integral(J^2/(2DP)). For the original discrete-state process, the correct entropy-production rate is sum over transitions of (forward rate - backward rate) * ln(forward rate / backward rate). These two formulas need not share the same dependence on the adiabaticity parameter omega; the maximum of Sdot/omega shown in Fig. 9D could be an artifact of the continuous drift-diffusion dynamics and the reflecting walls at y=±1. The paper itself notes in Appendix A that the area near the reflecting walls (5% of the total) was excluded from the Eq. 9 integral to remove artifact contributions, which acknowledges that the diffusion treatment produces spurious boundary currents; without the discrete calculation there is no way to know how much of the remaining current is physical.\n\nThe time-ordering/hysteresis result based on the cross-correlation A(t) in Eq. 8 is also computed from continuous sample paths. In the discrete process, the slower y transitions will naturally tend to lead p changes, but the amplitude and omega window of the peak (Fig. 9C) could be qualitatively different.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript extends a stochastic landscape model of eukaryotic gene regulation from a single gene to multi-gene circuits. The authors define an adiabaticity parameter omega as the ratio of the chromatin-state transition rate to the protein-concentration relaxation rate, replace discrete chromatin states by continuous variables through a saddle-point approximation, and integrate coupled Langevin equations for protein concentration and chromatin state (Eq. 4). For a self-activating single-gene circuit and a mutually repressing two-gene circuit, they compute stationary landscapes, probability currents, the cross-correlation A(t), and an entropy-production measure (Eq. 9). They report that decreasing omega increases the number of landscape basins, that the eddy regime 0.1-1 hosts circular probability currents and hysteresis, and that the entropy production per chromatin turnover time peaks in the intensely nonadiabatic regime. They then apply the model to a three-gene Oct4-Sox2-Nanog circuit in mouse embryonic stem cells, proposing that heterogeneous per-gene omega values, extended by a 'deep epigenetic' two-layer chromatin model, can explain the wide Nanog fluctuations and narrow Oct4/Sox2 distributions observed experimentally.","tokens_in":27327,"tokens_out":5177,"duration_ms":55636,"significance":"If its central claims survive validation, the paper offers a physically suggestive control parameter, omega, for nonequilibrium fluctuations in gene circuits, and it makes concrete, testable predictions about circular probability currents and hysteresis in chromatin-protein dynamics. The manuscript is transparent in its parameter choices, tabulates all rates, and includes useful robustness checks (monomer versus dimer transcription factors, two-state versus three-state chromatin, alternative parameterizations in Appendices C-E). The main strength is the breadth of the conceptual framework: it connects a single quantitative parameter to basin structure, current circulation, entropy production, and cell-to-cell heterogeneity in pluripotency genes. The main limitations are that the central quantitative claims rest on an uncontrolled continuous approximation of discrete chromatin variables, and the mES cell application currently selects parameters after the fact rather than validating them against independent data.","major_comments":[{"comment":"The continuous Gaussian description of the discrete chromatin variables is an uncontrolled approximation. The underlying process is a three-state Markov chain for y_i with rates given by Eqs. 2-3. The saddle-point/truncation at second order has no small parameter for y_i, and the noise amplitude D_yi = (G_yi + F_yi)/(2 omega_i) actually grows as omega_i decreases, i.e., in the regime where the paper's central peak in Fig. 9D appears. The reflecting-wall treatment in Appendix A, which excludes 5% of the domain from the Eq. 9 integral, demonstrates that the diffusion dynamics generate spurious boundary currents; without a direct simulation of the original jump master equation (or a Gillespie simulation of the discrete-state process) there is no way to know whether the circular currents, the hysteresis measure A(t), and the entropy-production peak are physical or artifacts. I request an explicit comparison of P(p,y), J, and Sdot between Eq. 4 and the discrete-state master equation at representative omega values, including omega = 0.01, 0.1, and 1.","section":"II.B, Eq. (4), Appendix A"},{"comment":"Equation 9 is the diffusion-form entropy production rate, integral of J^2/(2DP), but for the original jump process the correct entropy production rate is the sum over transitions of (r_forward P_backward_state - r_backward P_forward_state) times ln(r_forward/r_backward). These two expressions need not have the same dependence on omega. Moreover, Fig. 9D plots Sdot/omega, the entropy produced per chromatin turnover time, not the absolute entropy production rate, so the abstract's phrase 'maximum entropy production' is not directly supported by the figure. The authors should either compute the discrete-state entropy production rate and show that it peaks in the same regime, or revise the abstract to say 'maximum entropy production per chromatin turnover time' and clearly state that this is the plotted quantity.","section":"III, Eq. (9), Fig. 9D"},{"comment":"The mES cell result is obtained with adiabaticity parameters that appear to be chosen after the fact: omega1 = omega2 = 10 and omega3 = 0.5 are set so that the desired wide-Nanog/narrow-Oct4 pattern emerges, and Appendix E shows that neighboring parameterizations destroy the match. As presented, this is a proof of principle that heterogeneous omega values can tune fluctuations in a circuit, not a validation of the proposed explanation for the experimental data. To make the explanatory claim load-bearing, the authors should state this status explicitly and provide a falsifiable prediction that can be tested against independent data, such as measured chromatin-state lifetimes at the three loci, single-cell time-series cross-correlations, or the projected current divergence pattern of Fig. 14D.","section":"IV, Figs. 10-11, Appendix E"}],"minor_comments":[{"comment":"In the second line of Eq. 13 the variable is written as v_i, while the text and the rest of the equation use y_i; the notation should be made consistent.","section":"Eq. (13)"},{"comment":"The numerical section does not state the stochastic integration convention. The Euler discretization implicitly uses the Ito convention; this should be stated explicitly, since the noise amplitudes depend on p and y and the convention affects boundary behavior and current statistics.","section":"Appendix A"},{"comment":"The caption lists panel (C) twice and omits a caption for panel (F); the labels should be corrected, and the definition of Sdot/omega should be repeated in the caption for clarity.","section":"Fig. 9"},{"comment":"There are several typographical errors, including 'dacetylated', 'calcutated', 'adiavaticiy', 'landscapt', and 'swiching' in Sections V and Appendix E; these should be corrected in a final proofreading pass.","section":"Throughout"},{"comment":"References [27] and [29] appear to share the same title 'Enhancer dynamics: Unraveling the mechanism of transcriptional bursting' but cite different venues; please verify that both entries are correct and distinct.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is a promising modeling contribution, and the requested revisions are within scope: a discrete-state simulation check, a corrected entropy-production statement, and an explicit falsifiable prediction for the mES circuit. I do not see grounds for rejection, but the central quantitative claims should not be accepted without the validation described in the major comments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real advance over the group's earlier single-gene work. The new content is the multi-gene generalization, the phase diagrams of basin number versus omega and pbar, and the deep epigenetic regulation model with the z variable; those are concrete and worth having. The mES application is also conceptually nice: per-gene differences in omega produce wide Nanog and narrow Oct4/Sox2 distributions without invoking the disputed Nanog self-repression. The predicted projected probability currents in Fig. 14D are genuinely testable, and the paper says so. Credit where due: equations are internally consistent, appendices are unusually thorough (monomer TF, two-state chromatin, alternative parameterizations), and the authors flag several assumptions themselves.\n\nThe soft spots are real but not fatal. The biggest is the saddle-point/Langevin treatment of y, a discrete three-state variable, without any check against the original jump master equation. There is no large parameter that justifies the Gaussian noise; the entropy production integral (Eq. 9) and the circular currents may be artifacts of the diffusion approximation plus reflecting walls. Appendix A's exclusion of the 5% boundary region is an admission that the diffusion treatment produces spurious currents, and the paper never quantifies how much of the remaining current is physical. The stress-test note hits this accurately. I would want a direct discrete-state calculation of Sdot and A(t) before trusting the peak in Fig. 9D.\n\nThe second issue is presentation: the abstract's \"maximum entropy production\" overstates what is plotted, which is entropy production per chromatin turnover time Sdot/omega. The peak may simply reflect the definition; the wording needs to be corrected. Third, Circuit C's adiabaticity parameters are chosen after the fact to match the observed heterogeneity; Appendix E honestly shows the feature disappears for other parameterizations, so the claim is conditional, not a prediction.\n\nThe citation pattern is fine; earlier work is built on explicitly, and the multi-gene extension is not disguised. No code or data is provided, which matters here because the numerical claims are central.\n\nWho is this for: biophysicists working on gene circuit noise, epigenetic landscapes, and nonadiabatic effects. It deserves a serious referee: it is important enough, the model is coherent, and the flaws are fixable with a discrete-state check and softened claims. My recommendation is to send it to review, with a request for the jump-process comparison.","headline":"A useful multi-gene extension of the eddy/landscape model, with a compelling omega-based explanation for Nanog heterogeneity, but the central entropy-production claim rests on an unvalidated continuous approximation of discrete chromatin states.","tokens_in":27898,"tokens_out":1693,"would_cite":true,"duration_ms":16547,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single ratio of chromatin-switch rate to protein turnover rate shapes gene-regulation landscapes and fluctuations.","keywords":["gene regulation","chromatin-state transitions","nonequilibrium fluctuations","stochastic gene expression","adiabaticity","probability current","entropy production","pluripotency"],"falsifier":"Simulate the original discrete-state jump master equation (for example, by the Gillespie algorithm) with the same transition rates and compare the cross-correlation $A(1/\\omega)$ and entropy production $\\dot{S}/\\omega$: if the positive time-ordering and the dissipation peak near $0.01 \\lesssim \\omega \\lesssim 0.1$ disappear or move outside the eddy regime, the continuous saddle-point approximation is the source of the claimed nonequilibrium effects.","tokens_in":26735,"feed_emoji":"🧬","tokens_out":9714,"duration_ms":83037,"temperature":0.7,"pith_summary":"The paper argues that eukaryotic gene regulation cannot be understood from circuit wiring alone: the ratio of the chromatin-transition rate to the protein-turnover rate, encoded in the adiabaticity parameter $\\omega$, reshapes the regulatory landscape and controls fluctuation behavior. Extending a previously developed stochastic model from a single gene to multi-gene circuits, it shows that in the mildly nonadiabatic regime $0.1 \\lesssim \\omega \\lesssim 1$ the probability current develops a global circular flow, entropy production rises, and gene switching becomes hysteretic with chromatin-state change preceding protein-concentration change. In the intensely nonadiabatic regime $0.01 \\lesssim \\omega \\lesssim 0.1$, entropy production per chromatin turnover time peaks. Applied to the three core pluripotency genes of mouse embryonic stem cells, the model reproduces the observed wide fluctuations of Nanog coexisting with narrow single-peak distributions of Oct4 and Sox2 when the three genes have different $\\omega$ values. A deeper two-layer model ties this heterogeneity to domain-wide versus local epigenetic regulation and predicts measurable probability currents.","feed_headline":"Slow chromatin switches drive gene fluctuation patterns","feed_subtitle":"A single timescale ratio explains wide Nanog noise and narrow Oct4/Sox2 in mouse stem cells.","key_machinery":"The load-bearing object is the extended landscape $U(p,y)=-\\log P(p,y)$ built from the stationary distribution of coupled Langevin equations for continuous protein concentration $p_i$ and continuous chromatin state $y_i$ (and later the enhancer-promoter state $z_i$). The equations are derived by writing the discrete-state master equation in path-integral form and applying the saddle-point approximation, which truncates fluctuations at second order and yields Gaussian noise with diffusion constants set by the volume $\\Omega$ and the adiabaticity $\\omega_i$. The circular, divergence-free component of the probability current $J$ signals broken detailed balance; entropy production is computed from $\\dot{S}=\\int\\int dp\\,dy\\,(J_p^2/(P D_p)+J_y^2/(P D_y))$; and time ordering is quantified by the cross-correlation $A(t)$, whose peak at $t\\approx 1/\\omega$ marks the chromatin turnover time. These objects carry the argument: the current drives hysteresis, $A(t)$ detects it, and $\\dot{S}$ quantifies dissipation.","core_discovery":"The central claim is that $\\omega$, the ratio of the chromatin-state transition rate to the protein concentration change rate, is a control parameter for nonequilibrium fluctuations in eukaryotic gene circuits, not just a correction to adiabatic gene-network theory. As $\\omega$ is lowered below unity, landscapes $U=-\\log P$ develop additional basins; in the eddy regime $0.1 \\lesssim \\omega \\lesssim 1$ the stationary probability current circulates, detailed balance is broken, and the cross-correlation $A(t)$ between chromatin state $y$ and protein concentration $p$ becomes positive at positive lag, meaning $y$ changes first and $p$ follows in both activation and inactivation. The entropy production rate $\\dot{S}$ per chromatin turnover time is low in the adiabatic limit, rises through the eddy regime, peaks for $0.01 \\lesssim \\omega \\lesssim 0.1$, and decreases at smaller $\\omega$. In the mutually repressing two-gene circuit the circular fluxes appear even before the basin bifurcation, acting as precursors. For the Oct4\\textendash Sox2\\textendash Nanog circuit, heterogeneous adiabaticity\\textemdash slow chromatin transitions for Nanog ($\\omega_3 \\approx 0.2$\\textendash $0.5$) and fast transitions for Oct4 and Sox2 ($\\omega_1 = \\omega_2 \\approx 10$)\\textemdash produces exactly the experimentally observed heterogeneous fluctuation pattern without invoking Nanog self-activation or mutual repression.","pith_inferences":["The same qualitative phenomenology\\textemdash basin proliferation, positive $A(t)$, and a dissipation peak\\textemdash should survive in the original discrete jump master equation, but the location of the peak may shift; a Gillespie simulation of the discrete model with the same parameters would calibrate the continuous approximation.","The per-gene $\\omega$ mechanism suggests a general design principle for synthetic gene circuits: heterogeneity can be programmed by choosing chromatin modification rates rather than promoter strengths, which might be cheaper or more reversible in engineered cells.","If the entropy-production peak is confirmed, it implies that regulation operating at timescales comparable to protein turnover is thermodynamically costly; this could reflect a trade-off between stability (slow switching) and flexibility (fast response) that natural selection tunes.","The projected divergence of the probability current in the Oct4-Nanog plane is a falsifiable signature: time-resolved single-cell data that show no directional flow from high- to low-Nanog would count against the heterogeneous-$\\omega$ hypothesis."],"forward_implications":["If the central claim holds, eukaryotes with $\\omega \\approx 0.1$ are generically in the nonadiabatic regime, so slow histone modifications should produce extra landscape basins and larger expression fluctuations than adiabatic models predict.","The circular probability flux appears before the emergence of the off-diagonal basins in the two-gene toggle, so circular currents are a precursor or early-warning signal of switching; the paper suggests RNA-velocity-style measurements could detect them.","For mouse embryonic stem cells, the observed wide Nanog fluctuations can be explained by slow chromatin transitions at the Nanog locus alone, without Nanog self-activation or mutual repression among the three core genes.","Because $\\omega$ can be tuned by chromatin-modifying enzyme activity, cells could pass from stable nonadiabatic basins through a fluctuating eddy regime to different stabilized states, providing a physical route for cell-type transitions.","In the deep-epigenetic model, the predicted signature is a diagonal circular current between Nanog's enhancer-promoter state $z_3$ and protein concentration $p_3$, plus a projected current flowing from high- to low-Nanog states; the paper argues this is experimentally testable with combined Hi-C, ChIP-seq, and RNA-seq data."],"supporting_citations":[{"why":"supplies the single-gene Langevin equations and extended-landscape method that this paper extends to circuits of multiple genes.","marker":"[13]"},{"why":"introduces the eddy regime of comparable chromatin-transition and diffusion rates that the paper identifies with $0.1 \\lesssim \\omega \\lesssim 1$.","marker":"[8]"},{"why":"gives the exactly solvable self-regulating gene model used to frame nonadiabatic switching behavior.","marker":"[10]"},{"why":"states the adiabatic approximation for transcription-factor binding that the paper argues is insufficient for eukaryotic regulation.","marker":"[14]"},{"why":"measures the timescales of histone modification (hours to days) that set eukaryotic adiabaticity values near 0.1.","marker":"[21, 22]"},{"why":"reports two-way feedback between chromatin compaction and histone modification, motivating delayed nonadiabatic regulation.","marker":"[47]"},{"why":"documents wide Nanog fluctuations with narrow Oct4 and Sox2 distributions that Circuit C is designed to reproduce.","marker":"[56]"},{"why":"shows A/B compartment switching for the core-gene chromatin domains, grounding the heterogeneous adiabaticity assumption.","marker":"[65, 66]"},{"why":"supplies a single-cell method for inferring vector fields and currents from transcriptomic data, the proposed experimental test.","marker":"[75]"}],"fun_headline_variants":["Timescale ratio tunes gene noise in stem cells","Slow chromatin state changes drive gene circuit fluctuations","One ratio predicts Nanog vs Oct4/Sox2 noise pattern","Nonadiabatic chromatin transitions shape gene noise landscape"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire calculation depends on treating the discrete chromatin state $y$ (and, in the deeper model, the enhancer-promoter state $z$) as continuous variables with Gaussian noise through the saddle-point approximation; if real chromatin transitions are genuinely discrete and rare, the continuous landscape, the circular probability current, and the entropy-production formula may describe an artifact of that approximation rather than the cell.","fun_headline_variants_meta":{"raw":{"variants":["Timescale ratio tunes gene noise in stem cells","Slow chromatin state changes drive gene circuit fluctuations","One ratio predicts Nanog vs Oct4/Sox2 noise pattern","Nonadiabatic chromatin transitions shape gene noise landscape"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000209,"raw_usage":{"total_tokens":1456,"prompt_tokens":1044,"completion_tokens":412,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":660,"completion_tokens_details":{"reasoning_tokens":349}},"tokens_in":660,"tokens_out":412,"duration_ms":4704,"temperature":1.0,"reasoning_tokens":349,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T19:31:12.079153+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the original discrete-state jump master equation (for example, by the Gillespie algorithm) with the same transition rates and compare the cross-correlation $A(1/\\omega)$ and entropy production $\\dot{S}/\\omega$: if the positive time-ordering and the dissipation peak near $0.01 \\lesssim \\omega \\lesssim 0.1$ disappear or move outside the eddy regime, the continuous saddle-point approximation is the source of the claimed nonequilibrium effects.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the single-gene Langevin equations and extended-landscape method that this paper extends to circuits of multiple genes."},{"cited_title":"Hastings, Timescales, dynamics, and ecological under- standing, Ecology 91, 3471 (2010)","cited_arxiv_id":null,"evidence_quote":"introduces the eddy regime of comparable chromatin-transition and diffusion rates that the paper identifies with $0.1 \\lesssim \\omega \\lesssim 1$."},{"cited_title":"Senkowski and A","cited_arxiv_id":null,"evidence_quote":"gives the exactly solvable self-regulating gene model used to frame nonadiabatic switching behavior."},{"cited_title":"Shi and H","cited_arxiv_id":null,"evidence_quote":"states the adiabatic approximation for transcription-factor binding that the paper argues is insufficient for eukaryotic regulation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"reports two-way feedback between chromatin compaction and histone modification, motivating delayed nonadiabatic regulation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"documents wide Nanog fluctuations with narrow Oct4 and Sox2 distributions that Circuit C is designed to reproduce."},{"cited_title":"Krishnamurthy, T","cited_arxiv_id":null,"evidence_quote":"supplies a single-cell method for inferring vector fields and currents from transcriptomic data, the proposed experimental test."}],"review_version":1}