{"id":"56f66f5d-0cab-4bc7-804c-f7657c2bb91e","arxiv_id":"2608.08626","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"General 3D domains now admit three-term asymptotic expansions for diffusive capture rates, coupling local curvatures, patch shape and reactivity, and global Green's function interactions.","lead":"This paper derives three-term asymptotic formulas for the narrow escape and Berg-Purcell capture problems on arbitrary smooth three-dimensional surfaces with small reactive patches. The formulas show how local curvature and receptor arrangement control capture rates, and they are validated against numerical simulations on spheroids.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"","rationale":"The reader's ACCEPT verdict rests on the Green's function uniformity assumption, which is a genuine formal limitation but is not contradicted within the paper. The more load-bearing issue is an internal inconsistency in the definition of E_i-, a coefficient that is part of the stated central claim for arbitrary patch shapes. Appendix D and the explicit ellipse calculation (E.11) use a squared denominator |y'−y|^2, while Lemma 2.3's Eq. (2.39) uses |y'−y|. These cannot both be correct, and the difference changes the numerical value of the O(ε) curvature-anisotropy correction for every non-circular patch. The paper's validation cases all have E_i- = 0, so they do not exercise this part of the theory. This is not an ad hominem or a consensus dispute; it is a precisely locatable algebraic inconsistency in the supplied definitions. The appropriate disposition is CONDITIONAL: the central matched-asymptotics framework and the circular-patch results have substantial independent support (reduction to sphere, agreement with known single-patch results, numerical validation on spheroids), but Proposition 1 and Proposition 2 as written for arbitrary patch shapes require correction and re-verification of the E_i- contribution before the stated claim can be accepted.","tokens_in":44386,"tokens_out":19214,"duration_ms":187835,"concrete_test":"Independently re-derive Eq. (2.39) from Appendix D: substitute ∂_{η'}w|_{η'=0}=−2q_i into Eq. (D.22) and then into (D.13). This yields E_i- = (1/(4π^2)) ∫∫ q_i(y)q_i(y') [(ξ'_1−ξ_1)^2−(ξ'_2−ξ_2)^2]/|y'−y|^2 dy' dy. Compare this with Lemma 2.3 as printed. A direct numerical evaluation for the perfectly absorbing ellipse of Lemma E.1 with a_1=2, a_2=1 would also settle the issue: Eq. (E.11)/(E.13) gives E_i-/C_i^2 = (2−1)/(4(2+1)) = 1/12, while the printed Eq. (2.39) gives a different value. If the mismatch is confirmed, correct Eq. (2.39) and re-verify the O(ε) term for an elliptical patch on a surface with κ_1 ≠ κ_2.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops matched asymptotic expansions for two diffusive capture problems on smooth closed three-dimensional domains with small, well-separated, partially reactive boundary patches of arbitrary shape: the exterior Berg–Purcell problem and the interior narrow escape problem for the mean first-reaction time. The main results are Proposition 1, a three-term expansion for the inverse capacitance 1/C_T, and Proposition 2, the analogous expansion for the volume-averaged MFRT. Both expansions are expressed in terms of patch capacitances C_i, monopole coefficients E_{i+} and E_{i-}, mean curvature and curvature difference at each patch, and the interior or exterior surface Neumann Green's matrix. The paper also derives a new path-dependent singularity expansion for the surface Neumann Green's function (Appendix B), introduces the new coefficient E_{i-} for non-circular patches (Appendix D), gives explicit formulas for elliptical patches (Appendix E), and validates the asymptotics against a boundary integral solver on prolate spheroids, including a Fibonacci-spiral multi-patch example and an optimization study on nearly spherical domains.","tokens_in":44314,"tokens_out":28255,"duration_ms":276130,"significance":"If correct, the results materially extend the prior theory, which was largely confined to spherical domains, to general smooth geometries, and they identify precisely which local and global geometric data control capture rates at the retained order. The derivation is detailed and internally consistent: the matching through O(1), the switchback logarithms, the new Appendix B singularity analysis, and the E_{i-} lemma all cohere. The paper reduces to known sphere results as special cases and numerically confirms the predicted O(epsilon^2 log epsilon) error for several configurations on prolate spheroids. These strengths give confidence in the central claims. The main results are formal asymptotics rather than theorems with rigorous remainder estimates, which is standard for this literature and is mitigated by the numerical verification. The new E_{i-} term is not directly exercised by the numerical tests, since all boundary-integral validations use circular patches, but the analytical derivation in Appendix D is careful and explicit.","major_comments":[],"minor_comments":[{"comment":"The three-term singularity expansion of the surface Neumann Green's function, including the path-dependent term e(x;x_i), is assumed to hold uniformly on the O(epsilon)-scale inner region around each patch. Because this uniformity underlies the matching at O(1) and the U3 monopole condition, I suggest adding one sentence stating this assumption explicitly and, if possible, a numerical check comparing the truncated expansion with a computed Green's function on a non-spherical surface at small distances.","section":"Appendix B, Eqs. (B.26)-(B.29)"},{"comment":"The new coefficient E_{i-}, which is one of the paper's main novelties, is never exercised by the numerical validation because all boundary-integral tests use circular patches, for which E_{i-}=0. A validation with an elliptical patch at a location with nonzero curvature difference Q_i, or an independent check of the closed-form ratios E_{1+/-}/C_1^2 in Lemma E.1, would further strengthen confidence in this term.","section":"Lemma 2.3 and Section 4"},{"comment":"The pointwise expansion (3.15) contains an undetermined O(epsilon^2 log epsilon) constant U2, as the paper notes. Since the paper's central claim concerns the volume-averaged MFRT (3.16), this does not affect Proposition 2, but I recommend emphasizing in the statement of Proposition 2 that only the global quantity is fully determined to the stated order.","section":"Proposition 2 and Eq. (3.15)"},{"comment":"The text describes a boundary integral method, but the caption of Fig. 4.1(b) refers to 'FEM'. Please make the terminology consistent.","section":"Section 4.1 and Fig. 4.1"},{"comment":"There are several typographical errors: 'encloded' in the abstract, 'incling' in the introduction, 'fill fix' in Section 4.3, and 'capacita nce' in the caption of Fig. 4.3. These should be corrected.","section":"Throughout"},{"comment":"Several references are cited as '2026', 'under revision', or 'accepted' (e.g., [14], [16], [29], [42]). If the journal requires final publication data, please update these before publication.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"This is a strong formal asymptotic contribution with careful internal consistency and useful numerical verification. The main caveats are the unproven uniformity of the Green's singularity expansion and the lack of a direct numerical test of E_{i-}; neither undermines the central claims, but both are worth addressing in a revision. The paper fits well within the scope of an applied mathematics journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a genuine step beyond sphere-only results, and the machinery holds up. The two three-term expansions for capacitance and global mean first-reaction time on general smooth closed surfaces are new as far as I can tell, and the curvature-anisotropy term Q_i E_{i-} is a real addition to the theory. The paper earns its length: the boundary-fitted coordinate expansion, the Green's function singularity analysis in Appendix B, and the new E_{i-} calculation all feed directly into the matching, and the numerical validation on prolate spheroids shows the predicted O(ε^2 log ε) error. It also reduces cleanly to the known spherical and single-patch cases, including the single-patch result of Nursultanov–Tzou–Tzou.\n\nWhere I would be careful: the whole edifice rests on a uniform three-term expansion of the surface Neumann Green's function near each patch (B.26)–(B.29), including the path-dependent term e(x;x_i). The derivation is formal and there is no remainder bound. That is standard for matched asymptotics and I do not think it is load-bearing here, but it is the spot a skeptic should inspect first. Two smaller caveats: the patch-level coefficients C_i and E_{i+} are imported from the authors' own earlier papers, and no code is shipped. Neither is a defect—the prior results are referenced and the genuinely new pieces are derived in this paper—but the paper is not fully self-contained at the patch scale.\n\nWho this is for: applied mathematicians and biophysicists doing narrow-capture modeling on realistic cell geometries. It deserves a serious referee and, in my view, acceptance after a routine check. If I were handling it, I would ask the authors to add a short paragraph making precise which steps are formal and which are rigorous, and ideally release the boundary-integral code and data so the numerics are reproducible. I would cite this paper.","headline":"Genuinely new three-term expansions for narrow escape and Berg-Purcell on general smooth 3D domains; the formal machinery holds up, with the Green's function uniformity assumption as the main soft spot.","tokens_in":44866,"tokens_out":3216,"would_cite":true,"duration_ms":33803,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B25","35C20","35J05","35Q92"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives three-term asymptotic expansions for the capacitance and the global mean first-reaction time in the small-patch limit, valid on any smooth closed three-dimensional domain with multiple reactive patches.","keywords":["narrow escape problem","Berg-Purcell problem","matched asymptotic expansions","surface Neumann Green's function","reactive boundary patches","mean first-reaction time","principal curvatures","diffusive capture"],"falsifier":"A high-accuracy boundary-integral computation of $C_T$ for an ellipsoid with one non-circular patch at a point with unequal principal curvatures, repeated for a sequence of $\\varepsilon \\to 0$, would settle the claim: the residual against Proposition 1 must shrink like $\\varepsilon^2 \\log \\varepsilon$, and the $O(\\varepsilon)$ coefficient must contain the $Q E_-$ contribution. If either fails, the expansion is wrong.","tokens_in":44188,"feed_emoji":"🧬","tokens_out":8949,"duration_ms":90087,"temperature":0.7,"pith_summary":"This paper derives three-term asymptotic expansions for two diffusive capture problems on a general smooth closed three-dimensional domain with $N$ small reactive patches: the exterior Berg-Purcell problem, which asks how many diffusing ligands a patchy cell surface captures, and the interior narrow escape problem, which asks how quickly a Brownian particle inside a domain reaches a small reactive window. The central claim is that, to order $\\varepsilon^2 \\log \\varepsilon$, both the capacitance $C_T$ and the global mean first-reaction time are controlled by local patch data—reactive capacitance $C_i$, two monopole coefficients $E_{i\\pm}$, mean curvature $H_i$, and curvature difference $Q_i$ at each patch—together with a global surface Neumann Green's matrix that encodes the placement of patches and the shape of the domain. The analysis resolves the weak, path-dependent singularity of the surface Neumann Green's function on a curved boundary, which is the technical step that lets the asymptotics go beyond spheres and constant-curvature surfaces. The formulas reduce to earlier spherical results in the appropriate limits and are checked against a boundary-integral solution on a prolate spheroid, with the predicted $O(\\varepsilon^2 \\log \\varepsilon)$ error.","feed_headline":"Curvature now enters receptor-capture rates to third order","feed_subtitle":"New asymptotic formulas predict capture rate and escape time from patch shape, curvature, and surface interactions.","key_machinery":"The machinery is matched asymptotic expansions in the small patch diameter $\\varepsilon$, organized around boundary-fitted tangential-normal coordinates $(t_1,t_2,d)$ aligned with the principal directions at each patch center. In these coordinates the Laplacian acquires explicit curvature terms proportional to mean curvature $H_i$ and curvature difference $Q_i$, and the near-patch problem becomes a half-space Robin problem for each flattened patch $\\Gamma_i$. Its solution supplies the reactive capacitance $C_i(b_i)$ and dipole vector $p_i$, while the next-order inner problems define the logarithmic monopole coefficient $E_{i+}$ and the new curvature-anisotropy monopole $E_{i-}$, combined as $E_i=H_i E_{i+}+Q_i E_{i-}$. The outer solution is represented using the surface Neumann Green's function $G_e$ or $G_s$; the load-bearing technical result is the three-term singularity expansion of $G$ near a boundary source, including the path-dependent term $e(x;x_i)$ proportional to $Q$, which makes the matching of inner and outer expansions possible to the retained order.","core_discovery":"On the paper's own terms, the main discovery is quantitative and explicit. For the exterior problem, the capacitance satisfies $1/C_T \\sim |U_0|\\varepsilon^{-1}(1+\\varepsilon U_{10}/U_0 \\log\\varepsilon+\\varepsilon U_{11}/U_0)$, with $|U_0|=2/C$, $U_{10}/U_0=-(2C)^{-1}\\sum_j H_j C_j^2$, and $U_{11}/U_0=2\\pi C^{-1}C^T G_e C+E/C$, where $E=\\sum_j(H_j E_{j+}+Q_j E_{j-})$. For the interior problem, the global mean first-reaction time has the same structural expansion with the interior Green's matrix $G_s$ replacing $G_e$ and $U_0=|\\Omega|/(2\\pi C)$. The new ingredient that distinguishes general surfaces from the sphere is the curvature-anisotropy term $Q_i E_{i-}$: it is nonzero only when a patch is non-circular and sits where the two principal curvatures differ, and it encodes how the orientation of an elliptic patch relative to the principal directions changes capture.","pith_inferences":["Editorial inference: if the expansion remains valid while the number of patches grows, it suggests a homogenized boundary condition for dense receptor arrays whose coefficients should depend on an average of local mean curvature and on a Green's-function interaction energy; the paper states this as a conjecture rather than deriving it.","Editorial inference: because the path-dependent term $e(x;x_i)$ enters the inner problem only through local curvature, surface features smaller than the patch scale that alter principal curvatures should change capture rates at $O(\\varepsilon \\log \\varepsilon)$ even when patch area is fixed, a prediction that could be tested on structured surfaces.","Editorial inference: the same inner/outer decomposition should apply to splitting probabilities among competing receptors and to the statistics of the first few binding events, since those quantities depend on the per-patch fluxes that the matched solution already yields; the paper suggests these directions but does not carry them out."],"forward_implications":["For $N$ identical perfectly absorbing circular patches, the expansions collapse to compact formulas in which only the sum of mean curvatures $H$ and the Green's interaction energy $p_e$ or $p_s$ enter, giving a direct analytic account of how patch arrangement and surface curvature compete.","On a prolate spheroid with a single circular patch, the global mean first-reaction time is minimized at the equator, where both the regular part of the Green's function and the mean curvature attain their minima, as verified numerically.","When the patch is elliptic and aligned with principal directions, the ratios $E_{\\pm}/C^2$ have closed forms, so the mean first-reaction time becomes an explicit function of the semi-axis ratio $(a_1-a_2)/(a_1+a_2)$; non-circular patches on curved surfaces are thus analytically tractable.","The same three-term expansion for the global mean first-reaction time yields a three-term expansion for the principal Neumann-Robin eigenvalue $\\lambda_0$ of the Laplacian, extending the spherical result to general domains.","Optimizing patch positions to maximize capture reduces to minimizing a finite-dimensional discrete energy, whose curvature term has weight depending on patch reactivity through $\\gamma(b)$, so the optimal arrangement can shift with receptor properties."],"supporting_citations":[{"why":"Defines the original Berg-Purcell diffusive-capture problem and its capacitance/flux quantity, which Proposition 1 generalizes.","marker":"[6]"},{"why":"Supplies the half-space inner-problem analysis with partially reactive patches, including reactive capacitance $C_i$, the Steklov representation, and the monopole coefficient $E_{i+}$ that enter the three-term expansions.","marker":"[29]"},{"why":"Provides the spherical-domain three-term result for partially reactive patches of arbitrary shape that the general-domain formula reduces to when $H_i=-1$ and $Q_i=0$.","marker":"[31]"},{"why":"Gives the earlier microlocal resolution of the curved-boundary Green's-function singularity and the single-patch mean-first-arrival-time result that the circular-patch specialization reproduces.","marker":"[47]"},{"why":"Supplies the numerical computation of the surface Neumann Green's function and its regular part used to evaluate and validate the Green's-matrix terms on the spheroid.","marker":"[42]"},{"why":"Provides the spherical narrow-escape expansion for perfectly absorbing patches whose single-patch and sphere limits the mean-first-reaction-time formula recovers.","marker":"[20]"},{"why":"Gives the elliptic-patch charge density from which Appendix E obtains closed-form expressions for $C$, $E_+/C^2$, and $E_-/C^2$.","marker":"[61]"},{"why":"The previous two-term narrow-escape expansion on nonspherical domains that the present three-term result extends.","marker":"[22]"}],"fun_headline_variants":["Third-order formulas tie capture rates to patch curvature","Curved patches reorder capture-rate expansions","Orientation of elliptical patches now matters in capture","Beyond spheres: curvature terms in reaction rates","New expansions for capture on curved surfaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central premise is that the refined local formula for the surface Neumann Green's function near each patch remains accurate all the way down to the tiny inner region, so that no extra global-shape contribution shows up at the orders kept.","fun_headline_variants_meta":{"raw":{"variants":["Third-order formulas tie capture rates to patch curvature","Curved patches reorder capture-rate expansions","Orientation of elliptical patches now matters in capture","Beyond spheres: curvature terms in reaction rates","New expansions for capture on curved surfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000532,"raw_usage":{"total_tokens":2595,"prompt_tokens":1013,"completion_tokens":1582,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":629,"completion_tokens_details":{"reasoning_tokens":1515}},"tokens_in":629,"tokens_out":1582,"duration_ms":13335,"temperature":1.0,"reasoning_tokens":1515,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:29:48.826421+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A high-accuracy boundary-integral computation of $C_T$ for an ellipsoid with one non-circular patch at a point with unequal principal curvatures, repeated for a sequence of $\\varepsilon \\to 0$, would settle the claim: the residual against Proposition 1 must shrink like $\\varepsilon^2 \\log \\varepsilon$, and the $O(\\varepsilon)$ coefficient must contain the $Q E_-$ contribution. If either fails, the expansion is wrong.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the original Berg-Purcell diffusive-capture problem and its capacitance/flux quantity, which Proposition 1 generalizes."},{"cited_title":"Dobramysl and D","cited_arxiv_id":null,"evidence_quote":"Supplies the half-space inner-problem analysis with partially reactive patches, including reactive capacitance $C_i$, the Steklov representation, and the monopole coefficient $E_{i+}$ that enter the three-term expansions."},{"cited_title":"Gonz´ alez","cited_arxiv_id":null,"evidence_quote":"Provides the spherical-domain three-term result for partially reactive patches of arbitrary shape that the general-domain formula reduces to when $H_i=-1$ and $Q_i=0$."},{"cited_title":"In our analysis below, for ease of notation we omit the subscriptefor this Green’s function","cited_arxiv_id":null,"evidence_quote":"Gives the earlier microlocal resolution of the curved-boundary Green's-function singularity and the single-patch mean-first-arrival-time result that the circular-patch specialization reproduces."},{"cited_title":"Holcman and Z","cited_arxiv_id":null,"evidence_quote":"Supplies the numerical computation of the surface Neumann Green's function and its regular part used to evaluate and validate the Green's-matrix terms on the spheroid."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the spherical narrow-escape expansion for perfectly absorbing patches whose single-patch and sphere limits the mean-first-reaction-time formula recovers."},{"cited_title":"Schamberger, A","cited_arxiv_id":null,"evidence_quote":"Gives the elliptic-patch charge density from which Appendix E obtains closed-form expressions for $C$, $E_+/C^2$, and $E_-/C^2$."},{"cited_title":"The narrow escape problem in arbitrary dimension, 2026","cited_arxiv_id":null,"evidence_quote":"The previous two-term narrow-escape expansion on nonspherical domains that the present three-term result extends."}],"review_version":1}